LMFDB - Embedded newform 117.5.j.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.0942856808$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.1
Root			$-5.39509 + 5.39509i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.39509 + 5.39509i) q^{2} -42.2140i q^{4} +(17.6256 - 17.6256i) q^{5} +(26.0542 + 26.0542i) q^{7} +(141.427 + 141.427i) q^{8} +190.184i q^{10} +(-141.872 - 141.872i) q^{11} +(-154.603 - 68.2564i) q^{13} -281.130 q^{14} -850.600 q^{16} +216.815i q^{17} +(-101.874 + 101.874i) q^{19} +(-744.048 - 744.048i) q^{20} +1530.83 q^{22} -31.5880i q^{23} +3.67533i q^{25} +(1202.35 - 465.848i) q^{26} +(1099.85 - 1099.85i) q^{28} -634.240 q^{29} +(-750.494 + 750.494i) q^{31} +(2326.23 - 2326.23i) q^{32} +(-1169.74 - 1169.74i) q^{34} +918.442 q^{35} +(-1868.84 - 1868.84i) q^{37} -1099.24i q^{38} +4985.48 q^{40} +(399.410 - 399.410i) q^{41} -1574.71i q^{43} +(-5989.00 + 5989.00i) q^{44} +(170.420 + 170.420i) q^{46} +(-661.037 - 661.037i) q^{47} -1043.36i q^{49} +(-19.8287 - 19.8287i) q^{50} +(-2881.38 + 6526.41i) q^{52} +2764.37 q^{53} -5001.17 q^{55} +7369.54i q^{56} +(3421.79 - 3421.79i) q^{58} +(-2131.00 - 2131.00i) q^{59} -1556.43 q^{61} -8097.96i q^{62} +11490.9i q^{64} +(-3928.03 + 1521.91i) q^{65} +(81.1721 - 81.1721i) q^{67} +9152.65 q^{68} +(-4955.08 + 4955.08i) q^{70} +(-6425.97 + 6425.97i) q^{71} +(2565.20 + 2565.20i) q^{73} +20165.1 q^{74} +(4300.51 + 4300.51i) q^{76} -7392.73i q^{77} +1133.16 q^{79} +(-14992.4 + 14992.4i) q^{80} +4309.71i q^{82} +(-4665.02 + 4665.02i) q^{83} +(3821.50 + 3821.50i) q^{85} +(8495.72 + 8495.72i) q^{86} -40129.2i q^{88} +(-9509.38 - 9509.38i) q^{89} +(-2249.69 - 5806.42i) q^{91} -1333.46 q^{92} +7132.72 q^{94} +3591.18i q^{95} +(-2592.99 + 2592.99i) q^{97} +(5629.01 + 5629.01i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−5.39509	+	5.39509i	−1.34877	+	1.34877i	−0.461777	+	0.886996i	$0.652788\pi$
$2$	−5.39509	+	5.39509i	−1.34877	+	1.34877i	−0.886996	+	0.461777i	$0.847212\pi$
$3$		0			0
$3$		0			0
$4$		−	42.2140i		−	2.63838i
$5$	17.6256	−	17.6256i	0.705025	−	0.705025i	−0.260460	−	0.965485i	$-0.583874\pi$
$5$	17.6256	−	17.6256i	0.705025	−	0.705025i	0.965485	+	0.260460i	$0.0838742\pi$
$6$		0			0
$7$	26.0542	+	26.0542i	0.531718	+	0.531718i	0.921083	−	0.389365i	$-0.127306\pi$
$7$	26.0542	+	26.0542i	0.531718	+	0.531718i	−0.389365	+	0.921083i	$0.627306\pi$
$8$	141.427	+	141.427i	2.20980	+	2.20980i
$9$		0			0
$10$			190.184i			1.90184i
$11$	−141.872	−	141.872i	−1.17250	−	1.17250i	−0.981612	−	0.190885i	$-0.938864\pi$
$11$	−141.872	−	141.872i	−1.17250	−	1.17250i	−0.190885	−	0.981612i	$-0.561136\pi$
$12$		0			0
$13$	−154.603	−	68.2564i	−0.914810	−	0.403884i
$13$	−154.603	−	68.2564i	−0.914810	−	0.403884i
$14$	−281.130			−1.43433
$15$		0			0
$16$	−850.600			−3.32266
$17$			216.815i			0.750226i	0.926979	+	0.375113i	$0.122396\pi$
$17$			216.815i			0.750226i	−0.926979	+	0.375113i	$0.877604\pi$
$18$		0			0
$19$	−101.874	+	101.874i	−0.282199	+	0.282199i	−0.833986	−	0.551786i	$-0.813946\pi$
$19$	−101.874	+	101.874i	−0.282199	+	0.282199i	0.551786	+	0.833986i	$0.313946\pi$
$20$	−744.048	−	744.048i	−1.86012	−	1.86012i
$21$		0			0
$22$	1530.83			3.16287
$23$		−	31.5880i		−	0.0597126i	−0.999554	−	0.0298563i	$-0.990495\pi$
$23$		−	31.5880i		−	0.0597126i	0.999554	−	0.0298563i	$-0.00950497\pi$
$24$		0			0
$25$			3.67533i			0.00588053i
$26$	1202.35	−	465.848i	1.77862	−	0.689124i
$27$		0			0
$28$	1099.85	−	1099.85i	1.40287	−	1.40287i
$29$	−634.240			−0.754150			−0.377075	−	0.926183i	$-0.623070\pi$
$29$	−634.240			−0.754150			−0.377075	+	0.926183i	$0.623070\pi$
$30$		0			0
$31$	−750.494	+	750.494i	−0.780951	+	0.780951i	−0.979991	−	0.199041i	$-0.936217\pi$
$31$	−750.494	+	750.494i	−0.780951	+	0.780951i	0.199041	+	0.979991i	$0.436217\pi$
$32$	2326.23	−	2326.23i	2.27171	−	2.27171i
$33$		0			0
$34$	−1169.74	−	1169.74i	−1.01188	−	1.01188i
$35$	918.442			0.749749
$36$		0			0
$37$	−1868.84	−	1868.84i	−1.36511	−	1.36511i	−0.867266	−	0.497845i	$-0.834125\pi$
$37$	−1868.84	−	1868.84i	−1.36511	−	1.36511i	−0.497845	−	0.867266i	$-0.665875\pi$
$38$		−	1099.24i		−	0.761246i
$39$		0			0
$40$	4985.48			3.11593
$41$	399.410	−	399.410i	0.237603	−	0.237603i	−0.578254	−	0.815857i	$-0.696266\pi$
$41$	399.410	−	399.410i	0.237603	−	0.237603i	0.815857	+	0.578254i	$0.196266\pi$
$42$		0			0
$43$		−	1574.71i		−	0.851656i	−0.904804	−	0.425828i	$-0.859983\pi$
$43$		−	1574.71i		−	0.851656i	0.904804	−	0.425828i	$-0.140017\pi$
$44$	−5989.00	+	5989.00i	−3.09349	+	3.09349i
$45$		0			0
$46$	170.420	+	170.420i	0.0805388	+	0.0805388i
$47$	−661.037	−	661.037i	−0.299247	−	0.299247i	0.541472	−	0.840719i	$-0.317867\pi$
$47$	−661.037	−	661.037i	−0.299247	−	0.299247i	−0.840719	+	0.541472i	$0.817867\pi$
$48$		0			0
$49$		−	1043.36i		−	0.434551i
$50$	−19.8287	−	19.8287i	−0.00793150	−	0.00793150i
$51$		0			0
$52$	−2881.38	+	6526.41i	−1.06560	+	2.41361i
$53$	2764.37			0.984111			0.492056	−	0.870564i	$-0.336246\pi$
$53$	2764.37			0.984111			0.492056	+	0.870564i	$0.336246\pi$
$54$		0			0
$55$	−5001.17			−1.65328
$56$			7369.54i			2.34998i
$57$		0			0
$58$	3421.79	−	3421.79i	1.01718	−	1.01718i
$59$	−2131.00	−	2131.00i	−0.612181	−	0.612181i	0.331333	−	0.943514i	$-0.392502\pi$
$59$	−2131.00	−	2131.00i	−0.612181	−	0.612181i	−0.943514	+	0.331333i	$0.892502\pi$
$60$		0			0
$61$	−1556.43			−0.418283			−0.209142	−	0.977885i	$-0.567067\pi$
$61$	−1556.43			−0.418283			−0.209142	+	0.977885i	$0.567067\pi$
$62$		−	8097.96i		−	2.10665i
$63$		0			0
$64$			11490.9i			2.80539i
$65$	−3928.03	+	1521.91i	−0.929712	+	0.360216i
$66$		0			0
$67$	81.1721	−	81.1721i	0.0180824	−	0.0180824i	−0.698008	−	0.716090i	$-0.745930\pi$
$67$	81.1721	−	81.1721i	0.0180824	−	0.0180824i	0.716090	+	0.698008i	$0.245930\pi$
$68$	9152.65			1.97938
$69$		0			0
$70$	−4955.08	+	4955.08i	−1.01124	+	1.01124i
$71$	−6425.97	+	6425.97i	−1.27474	+	1.27474i	−0.331169	+	0.943571i	$0.607443\pi$
$71$	−6425.97	+	6425.97i	−1.27474	+	1.27474i	−0.943571	+	0.331169i	$0.892557\pi$
$72$		0			0
$73$	2565.20	+	2565.20i	0.481366	+	0.481366i	0.905568	−	0.424201i	$-0.139445\pi$
$73$	2565.20	+	2565.20i	0.481366	+	0.481366i	−0.424201	+	0.905568i	$0.639445\pi$
$74$	20165.1			3.68245
$75$		0			0
$76$	4300.51	+	4300.51i	0.744549	+	0.744549i
$77$		−	7392.73i		−	1.24688i
$78$		0			0
$79$	1133.16			0.181568			0.0907839	−	0.995871i	$-0.471063\pi$
$79$	1133.16			0.181568			0.0907839	+	0.995871i	$0.471063\pi$
$80$	−14992.4	+	14992.4i	−2.34256	+	2.34256i
$81$		0			0
$82$			4309.71i			0.640945i
$83$	−4665.02	+	4665.02i	−0.677169	+	0.677169i	−0.959359	−	0.282190i	$-0.908939\pi$
$83$	−4665.02	+	4665.02i	−0.677169	+	0.677169i	0.282190	+	0.959359i	$0.408939\pi$
$84$		0			0
$85$	3821.50	+	3821.50i	0.528928	+	0.528928i
$86$	8495.72	+	8495.72i	1.14869	+	1.14869i
$87$		0			0
$88$		−	40129.2i		−	5.18197i
$89$	−9509.38	−	9509.38i	−1.20053	−	1.20053i	−0.974006	−	0.226521i	$-0.927265\pi$
$89$	−9509.38	−	9509.38i	−1.20053	−	1.20053i	−0.226521	−	0.974006i	$-0.572735\pi$
$90$		0			0
$91$	−2249.69	−	5806.42i	−0.271669	−	0.701174i
$92$	−1333.46			−0.157544
$93$		0			0
$94$	7132.72			0.807234
$95$			3591.18i			0.397915i
$96$		0			0
$97$	−2592.99	+	2592.99i	−0.275586	+	0.275586i	−0.831344	−	0.555758i	$-0.812428\pi$
$97$	−2592.99	+	2592.99i	−0.275586	+	0.275586i	0.555758	+	0.831344i	$0.312428\pi$
$98$	5629.01	+	5629.01i	0.586111	+	0.586111i
$99$		0			0
$100$	155.150			0.0155150
$101$		−	3169.30i		−	0.310685i	−0.987861	−	0.155342i	$-0.950352\pi$
$101$		−	3169.30i		−	0.310685i	0.987861	−	0.155342i	$-0.0496481\pi$
$102$		0			0
$103$			837.656i			0.0789571i	0.999220	+	0.0394785i	$0.0125697\pi$
$103$			837.656i			0.0789571i	−0.999220	+	0.0394785i	$0.987430\pi$
$104$	−12211.8	−	31518.4i	−1.12905	−	2.91405i
$105$		0			0
$106$	−14914.0	+	14914.0i	−1.32734	+	1.32734i
$107$	−2575.64			−0.224966			−0.112483	−	0.993654i	$-0.535880\pi$
$107$	−2575.64			−0.224966			−0.112483	+	0.993654i	$0.535880\pi$
$108$		0			0
$109$	14711.8	−	14711.8i	1.23826	−	1.23826i	0.277555	−	0.960710i	$-0.410476\pi$
$109$	14711.8	−	14711.8i	1.23826	−	1.23826i	0.960710	−	0.277555i	$-0.0895240\pi$
$110$	26981.8	−	26981.8i	2.22990	−	2.22990i
$111$		0			0
$112$	−22161.7	−	22161.7i	−1.76672	−	1.76672i
$113$	−2115.72			−0.165692			−0.0828458	−	0.996562i	$-0.526401\pi$
$113$	−2115.72			−0.165692			−0.0828458	+	0.996562i	$0.526401\pi$
$114$		0			0
$115$	−556.757	−	556.757i	−0.0420989	−	0.0420989i
$116$			26773.9i			1.98973i
$117$		0			0
$118$	22993.9			1.65139
$119$	−5648.95	+	5648.95i	−0.398909	+	0.398909i
$120$		0			0
$121$			25614.4i			1.74950i
$122$	8397.09	−	8397.09i	0.564169	−	0.564169i
$123$		0			0
$124$	31681.4	+	31681.4i	2.06044	+	2.06044i
$125$	11080.8	+	11080.8i	0.709171	+	0.709171i
$126$		0			0
$127$		−	8241.86i		−	0.510996i	−0.966810	−	0.255498i	$-0.917761\pi$
$127$		−	8241.86i		−	0.510996i	0.966810	−	0.255498i	$-0.0822394\pi$
$128$	−24774.6	−	24774.6i	−1.51212	−	1.51212i
$129$		0			0
$130$	12981.2	−	29403.0i	0.768121	−	1.73982i
$131$	14460.4			0.842629			0.421315	−	0.906915i	$-0.361569\pi$
$131$	14460.4			0.842629			0.421315	+	0.906915i	$0.361569\pi$
$132$		0			0
$133$	−5308.49			−0.300101
$134$			875.862i			0.0487782i
$135$		0			0
$136$	−30663.6	+	30663.6i	−1.65785	+	1.65785i
$137$	−6841.36	−	6841.36i	−0.364503	−	0.364503i	0.500965	−	0.865468i	$-0.332979\pi$
$137$	−6841.36	−	6841.36i	−0.364503	−	0.364503i	−0.865468	+	0.500965i	$0.832979\pi$
$138$		0			0
$139$	−20952.3			−1.08443			−0.542216	−	0.840239i	$-0.682415\pi$
$139$	−20952.3			−1.08443			−0.542216	+	0.840239i	$0.682415\pi$
$140$		−	38771.2i		−	1.97812i
$141$		0			0
$142$		−	69337.4i		−	3.43867i
$143$	12250.2	+	31617.5i	0.599060	+	1.54617i
$144$		0			0
$145$	−11178.9	+	11178.9i	−0.531695	+	0.531695i
$146$	−27679.0			−1.29851
$147$		0			0
$148$	−78891.2	+	78891.2i	−3.60168	+	3.60168i
$149$	3142.63	−	3142.63i	0.141553	−	0.141553i	−0.632779	−	0.774332i	$-0.718086\pi$
$149$	3142.63	−	3142.63i	0.141553	−	0.141553i	0.774332	+	0.632779i	$0.218086\pi$
$150$		0			0
$151$	15748.1	+	15748.1i	0.690678	+	0.690678i	0.962381	−	0.271703i	$-0.0875869\pi$
$151$	15748.1	+	15748.1i	0.690678	+	0.690678i	−0.271703	+	0.962381i	$0.587587\pi$
$152$	−28815.5			−1.24721
$153$		0			0
$154$	39884.5	+	39884.5i	1.68175	+	1.68175i
$155$			26455.8i			1.10118i
$156$		0			0
$157$	1199.88			0.0486786			0.0243393	−	0.999704i	$-0.492252\pi$
$157$	1199.88			0.0486786			0.0243393	+	0.999704i	$0.492252\pi$
$158$	−6113.52	+	6113.52i	−0.244894	+	0.244894i
$159$		0			0
$160$		−	82002.6i		−	3.20323i
$161$	822.999	−	822.999i	0.0317503	−	0.0317503i
$162$		0			0
$163$	9254.30	+	9254.30i	0.348312	+	0.348312i	0.859481	−	0.511169i	$-0.170787\pi$
$163$	9254.30	+	9254.30i	0.348312	+	0.348312i	−0.511169	+	0.859481i	$0.670787\pi$
$164$	−16860.7	−	16860.7i	−0.626886	−	0.626886i
$165$		0			0
$166$		−	50336.4i		−	1.82669i
$167$	3520.69	+	3520.69i	0.126239	+	0.126239i	0.767404	−	0.641164i	$-0.221548\pi$
$167$	3520.69	+	3520.69i	0.126239	+	0.126239i	−0.641164	+	0.767404i	$0.721548\pi$
$168$		0			0
$169$	19243.1	+	21105.3i	0.673756	+	0.738954i
$170$	−41234.7			−1.42681
$171$		0			0
$172$	−66475.0			−2.24699
$173$		−	48893.1i		−	1.63364i	−0.576896	−	0.816818i	$-0.695736\pi$
$173$		−	48893.1i		−	1.63364i	0.576896	−	0.816818i	$-0.304264\pi$
$174$		0			0
$175$	−95.7577	+	95.7577i	−0.00312678	+	0.00312678i
$176$	120677.	+	120677.i	3.89581	+	3.89581i
$177$		0			0
$178$	102608.			3.23848
$179$		−	2018.23i		−	0.0629891i	−0.999504	−	0.0314946i	$-0.989973\pi$
$179$		−	2018.23i		−	0.0629891i	0.999504	−	0.0314946i	$-0.0100267\pi$
$180$		0			0
$181$			16380.3i			0.499994i	0.968247	+	0.249997i	$0.0804296\pi$
$181$			16380.3i			0.499994i	−0.968247	+	0.249997i	$0.919570\pi$
$182$	43463.5	+	19188.9i	1.31214	+	0.579304i
$183$		0			0
$184$	4467.40	−	4467.40i	0.131953	−	0.131953i
$185$	−65878.8			−1.92487
$186$		0			0
$187$	30760.1	−	30760.1i	0.879638	−	0.879638i
$188$	−27905.1	+	27905.1i	−0.789528	+	0.789528i
$189$		0			0
$190$	−19374.8	−	19374.8i	−0.536697	−	0.536697i
$191$	57513.6			1.57654			0.788268	−	0.615332i	$-0.210978\pi$
$191$	57513.6			1.57654			0.788268	+	0.615332i	$0.210978\pi$
$192$		0			0
$193$	−16470.6	−	16470.6i	−0.442175	−	0.442175i	0.450567	−	0.892742i	$-0.351222\pi$
$193$	−16470.6	−	16470.6i	−0.442175	−	0.442175i	−0.892742	+	0.450567i	$0.851222\pi$
$194$		−	27978.8i		−	0.743406i
$195$		0			0
$196$	−44044.4			−1.14651
$197$	24506.7	−	24506.7i	0.631469	−	0.631469i	−0.316968	−	0.948436i	$-0.602665\pi$
$197$	24506.7	−	24506.7i	0.631469	−	0.631469i	0.948436	+	0.316968i	$0.102665\pi$
$198$		0			0
$199$		−	27560.8i		−	0.695963i	−0.937501	−	0.347982i	$-0.886867\pi$
$199$		−	27560.8i		−	0.695963i	0.937501	−	0.347982i	$-0.113133\pi$
$200$	−519.791	+	519.791i	−0.0129948	+	0.0129948i
$201$		0			0
$202$	17098.6	+	17098.6i	0.419043	+	0.419043i
$203$	−16524.6	−	16524.6i	−0.400996	−	0.400996i
$204$		0			0
$205$		−	14079.7i		−	0.335032i
$206$	−4519.23	−	4519.23i	−0.106495	−	0.106495i
$207$		0			0
$208$	131505.	+	58058.9i	3.03960	+	1.34197i
$209$	28906.2			0.661756
$210$		0			0
$211$	48729.8			1.09454			0.547268	−	0.836957i	$-0.315668\pi$
$211$	48729.8			1.09454			0.547268	+	0.836957i	$0.315668\pi$
$212$		−	116695.i		−	2.59646i
$213$		0			0
$214$	13895.8	−	13895.8i	0.303429	−	0.303429i
$215$	−27755.3	−	27755.3i	−0.600439	−	0.600439i
$216$		0			0
$217$	−39107.0			−0.830491
$218$			158743.i			3.34028i
$219$		0			0
$220$			211120.i			4.36197i
$221$	14799.0	−	33520.3i	0.303004	−	0.686315i
$222$		0			0
$223$	−58941.0	+	58941.0i	−1.18524	+	1.18524i	−0.206876	+	0.978367i	$0.566330\pi$
$223$	−58941.0	+	58941.0i	−1.18524	+	1.18524i	−0.978367	+	0.206876i	$0.933670\pi$
$224$	121216.			2.41582
$225$		0			0
$226$	11414.5	−	11414.5i	0.223480	−	0.223480i
$227$	−19785.2	+	19785.2i	−0.383962	+	0.383962i	−0.872527	−	0.488565i	$-0.837520\pi$
$227$	−19785.2	+	19785.2i	−0.383962	+	0.383962i	0.488565	+	0.872527i	$0.337520\pi$
$228$		0			0
$229$	−62319.6	−	62319.6i	−1.18837	−	1.18837i	−0.977517	−	0.210858i	$-0.932374\pi$
$229$	−62319.6	−	62319.6i	−1.18837	−	1.18837i	−0.210858	−	0.977517i	$-0.567626\pi$
$230$	6007.52			0.113564
$231$		0			0
$232$	−89698.8	−	89698.8i	−1.66652	−	1.66652i
$233$			11660.7i			0.214789i	0.994216	+	0.107394i	$0.0342508\pi$
$233$			11660.7i			0.214789i	−0.994216	+	0.107394i	$0.965749\pi$
$234$		0			0
$235$	−23302.4			−0.421954
$236$	−89958.2	+	89958.2i	−1.61516	+	1.61516i
$237$		0			0
$238$		−	60953.2i		−	1.07608i
$239$	−35963.3	+	35963.3i	−0.629599	+	0.629599i	−0.947967	−	0.318368i	$-0.896865\pi$
$239$	−35963.3	+	35963.3i	−0.629599	+	0.629599i	0.318368	+	0.947967i	$0.396865\pi$
$240$		0			0
$241$	−5021.39	−	5021.39i	−0.0864550	−	0.0864550i	0.662557	−	0.749012i	$-0.269471\pi$
$241$	−5021.39	−	5021.39i	−0.0864550	−	0.0864550i	−0.749012	+	0.662557i	$0.769471\pi$
$242$	−138192.	−	138192.i	−2.35968	−	2.35968i
$243$		0			0
$244$			65703.2i			1.10359i
$245$	−18389.8	−	18389.8i	−0.306369	−	0.306369i
$246$		0			0
$247$	22703.6	−	8796.47i	0.372135	−	0.144183i
$248$	−212280.			−3.45149
$249$		0			0
$250$	−119564.			−1.91302
$251$			33336.1i			0.529136i	0.964367	+	0.264568i	$0.0852294\pi$
$251$			33336.1i			0.529136i	−0.964367	+	0.264568i	$0.914771\pi$
$252$		0			0
$253$	−4481.45	+	4481.45i	−0.0700129	+	0.0700129i
$254$	44465.6	+	44465.6i	0.689218	+	0.689218i
$255$		0			0
$256$	83468.7			1.27363
$257$			46144.5i			0.698640i	0.937003	+	0.349320i	$0.113587\pi$
$257$			46144.5i			0.698640i	−0.937003	+	0.349320i	$0.886413\pi$
$258$		0			0
$259$		−	97382.1i		−	1.45171i
$260$	64246.0	+	165818.i	0.950385	+	2.45293i
$261$		0			0
$262$	−78015.0	+	78015.0i	−1.13652	+	1.13652i
$263$	86819.8			1.25518			0.627592	−	0.778543i	$-0.284041\pi$
$263$	86819.8			1.25518			0.627592	+	0.778543i	$0.284041\pi$
$264$		0			0
$265$	48723.7	−	48723.7i	0.693823	−	0.693823i
$266$	28639.8	−	28639.8i	0.404768	−	0.404768i
$267$		0			0
$268$	−3426.60	−	3426.60i	−0.0477083	−	0.0477083i
$269$	66198.6			0.914838			0.457419	−	0.889251i	$-0.348774\pi$
$269$	66198.6			0.914838			0.457419	+	0.889251i	$0.348774\pi$
$270$		0			0
$271$	37598.3	+	37598.3i	0.511952	+	0.511952i	0.915124	−	0.403172i	$-0.132092\pi$
$271$	37598.3	+	37598.3i	0.511952	+	0.511952i	−0.403172	+	0.915124i	$0.632092\pi$
$272$		−	184423.i		−	2.49275i
$273$		0			0
$274$	73819.5			0.983264
$275$	521.427	−	521.427i	0.00689490	−	0.00689490i
$276$		0			0
$277$			48047.1i			0.626192i	0.949721	+	0.313096i	$0.101366\pi$
$277$			48047.1i			0.626192i	−0.949721	+	0.313096i	$0.898634\pi$
$278$	113040.	−	113040.i	1.46265	−	1.46265i
$279$		0			0
$280$	129893.	+	129893.i	1.65679	+	1.65679i
$281$	3560.42	+	3560.42i	0.0450909	+	0.0450909i	0.729293	−	0.684202i	$-0.239849\pi$
$281$	3560.42	+	3560.42i	0.0450909	+	0.0450909i	−0.684202	+	0.729293i	$0.739849\pi$
$282$		0			0
$283$			4833.10i			0.0603466i	0.999545	+	0.0301733i	$0.00960592\pi$
$283$			4833.10i			0.0603466i	−0.999545	+	0.0301733i	$0.990394\pi$
$284$	271266.	+	271266.i	3.36325	+	3.36325i
$285$		0			0
$286$	−236670.	−	104489.i	−2.89342	−	1.27743i
$287$	20812.6			0.252676
$288$		0			0
$289$	36512.1			0.437161
$290$		−	120622.i		−	1.43427i
$291$		0			0
$292$	108288.	−	108288.i	1.27003	−	1.27003i
$293$	−78570.2	−	78570.2i	−0.915213	−	0.915213i	0.0814629	−	0.996676i	$-0.474041\pi$
$293$	−78570.2	−	78570.2i	−0.915213	−	0.915213i	−0.996676	+	0.0814629i	$0.974041\pi$
$294$		0			0
$295$	−75120.5			−0.863206
$296$		−	528609.i		−	6.03324i
$297$		0			0
$298$			33909.5i			0.381847i
$299$	−2156.08	+	4883.59i	−0.0241170	+	0.0546257i
$300$		0			0
$301$	41027.9	−	41027.9i	0.452841	−	0.452841i
$302$	−169925.			−1.86314
$303$		0			0
$304$	86654.1	−	86654.1i	0.937652	−	0.937652i
$305$	−27433.1	+	27433.1i	−0.294900	+	0.294900i
$306$		0			0
$307$	−36999.3	−	36999.3i	−0.392570	−	0.392570i	0.483032	−	0.875602i	$-0.339535\pi$
$307$	−36999.3	−	36999.3i	−0.392570	−	0.392570i	−0.875602	+	0.483032i	$0.839535\pi$
$308$	−312077.			−3.28973
$309$		0			0
$310$	−142732.	−	142732.i	−1.48524	−	1.48524i
$311$		−	50001.9i		−	0.516971i	−0.966015	−	0.258485i	$-0.916777\pi$
$311$		−	50001.9i		−	0.516971i	0.966015	−	0.258485i	$-0.0832234\pi$
$312$		0			0
$313$	99016.7			1.01069			0.505347	−	0.862916i	$-0.331364\pi$
$313$	99016.7			1.01069			0.505347	+	0.862916i	$0.331364\pi$
$314$	−6473.46	+	6473.46i	−0.0656564	+	0.0656564i
$315$		0			0
$316$		−	47835.4i		−	0.479044i
$317$	−64312.4	+	64312.4i	−0.639994	+	0.639994i	−0.950554	−	0.310560i	$-0.899484\pi$
$317$	−64312.4	+	64312.4i	−0.639994	+	0.639994i	0.310560	+	0.950554i	$0.399484\pi$
$318$		0			0
$319$	89981.1	+	89981.1i	0.884239	+	0.884239i
$320$	202534.	+	202534.i	1.97787	+	1.97787i
$321$		0			0
$322$			8880.31i			0.0856479i
$323$	−22087.8	−	22087.8i	−0.211713	−	0.211713i
$324$		0			0
$325$	250.865	−	568.217i	0.00237505	−	0.00537957i
$326$	−99855.6			−0.939588
$327$		0			0
$328$	112975.			1.05011
$329$		−	34445.6i		−	0.318231i
$330$		0			0
$331$	−132098.	+	132098.i	−1.20570	+	1.20570i	−0.233297	+	0.972405i	$0.574951\pi$
$331$	−132098.	+	132098.i	−1.20570	+	1.20570i	−0.972405	+	0.233297i	$0.925049\pi$
$332$	196929.	+	196929.i	1.78663	+	1.78663i
$333$		0			0
$334$	−37988.9			−0.340536
$335$		−	2861.42i		−	0.0254971i
$336$		0			0
$337$		−	150689.i		−	1.32685i	−0.748244	−	0.663423i	$-0.769103\pi$
$337$		−	150689.i		−	1.32685i	0.748244	−	0.663423i	$-0.230897\pi$
$338$	−217683.	−	10046.3i	−1.90543	−	0.0879375i
$339$		0			0
$340$	161321.	−	161321.i	1.39551	−	1.39551i
$341$	212948.			1.83133
$342$		0			0
$343$	89740.0	−	89740.0i	0.762777	−	0.762777i
$344$	222707.	−	222707.i	1.88199	−	1.88199i
$345$		0			0
$346$	263783.	+	263783.i	2.20340	+	2.20340i
$347$	−59463.6			−0.493847			−0.246923	−	0.969035i	$-0.579420\pi$
$347$	−59463.6			−0.493847			−0.246923	+	0.969035i	$0.579420\pi$
$348$		0			0
$349$	43463.0	+	43463.0i	0.356836	+	0.356836i	0.862645	−	0.505809i	$-0.168806\pi$
$349$	43463.0	+	43463.0i	0.356836	+	0.356836i	−0.505809	+	0.862645i	$0.668806\pi$
$350$		−	1033.24i		−	0.00843464i
$351$		0			0
$352$	−660056.			−5.32715
$353$	64237.6	−	64237.6i	0.515513	−	0.515513i	−0.400697	−	0.916210i	$-0.631232\pi$
$353$	64237.6	−	64237.6i	0.515513	−	0.515513i	0.916210	+	0.400697i	$0.131232\pi$
$354$		0			0
$355$			226523.i			1.79745i
$356$	−401429.	+	401429.i	−3.16744	+	3.16744i
$357$		0			0
$358$	10888.6	+	10888.6i	0.0849581	+	0.0849581i
$359$	−18911.7	−	18911.7i	−0.146738	−	0.146738i	0.629921	−	0.776659i	$-0.283087\pi$
$359$	−18911.7	−	18911.7i	−0.146738	−	0.146738i	−0.776659	+	0.629921i	$0.783087\pi$
$360$		0			0
$361$			109564.i			0.840727i
$362$	−88373.2	−	88373.2i	−0.674378	−	0.674378i
$363$		0			0
$364$	−245112.	+	94968.5i	−1.84996	+	0.716765i
$365$	90426.5			0.678750
$366$		0			0
$367$	159803.			1.18646			0.593229	−	0.805034i	$-0.297853\pi$
$367$	159803.			1.18646			0.593229	+	0.805034i	$0.297853\pi$
$368$			26868.7i			0.198405i
$369$		0			0
$370$	355422.	−	355422.i	2.59622	−	2.59622i
$371$	72023.4	+	72023.4i	0.523270	+	0.523270i
$372$		0			0
$373$	15973.6			0.114812			0.0574059	−	0.998351i	$-0.481717\pi$
$373$	15973.6			0.114812			0.0574059	+	0.998351i	$0.481717\pi$
$374$			331907.i			2.37286i
$375$		0			0
$376$		−	186977.i		−	1.32255i
$377$	98055.4	+	43290.9i	0.689905	+	0.304589i
$378$		0			0
$379$	−152306.	+	152306.i	−1.06032	+	1.06032i	−0.0622636	+	0.998060i	$0.519832\pi$
$379$	−152306.	+	152306.i	−1.06032	+	1.06032i	−0.998060	+	0.0622636i	$0.980168\pi$
$380$	151598.			1.04985
$381$		0			0
$382$	−310291.	+	310291.i	−2.12639	+	2.12639i
$383$	113449.	−	113449.i	0.773399	−	0.773399i	−0.205300	−	0.978699i	$-0.565817\pi$
$383$	113449.	−	113449.i	0.773399	−	0.773399i	0.978699	+	0.205300i	$0.0658169\pi$
$384$		0			0
$385$	−130301.	−	130301.i	−0.879079	−	0.879079i
$386$	177721.			1.19279
$387$		0			0
$388$	109461.	+	109461.i	0.727100	+	0.727100i
$389$			72934.4i			0.481985i	0.970527	+	0.240992i	$0.0774728\pi$
$389$			72934.4i			0.481985i	−0.970527	+	0.240992i	$0.922527\pi$
$390$		0			0
$391$	6848.76			0.0447980
$392$	147559.	−	147559.i	0.960272	−	0.960272i
$393$		0			0
$394$			264431.i			1.70342i
$395$	19972.7	−	19972.7i	0.128010	−	0.128010i
$396$		0			0
$397$	−157471.	−	157471.i	−0.999124	−	0.999124i	0.000875739	−	1.00000i	$-0.499721\pi$
$397$	−157471.	−	157471.i	−0.999124	−	0.999124i	−1.00000		0.000875739i	$0.999721\pi$
$398$	148693.	+	148693.i	0.938696	+	0.938696i
$399$		0			0
$400$		−	3126.24i		−	0.0195390i
$401$	−215812.	−	215812.i	−1.34210	−	1.34210i	−0.893963	−	0.448140i	$-0.852086\pi$
$401$	−215812.	−	215812.i	−1.34210	−	1.34210i	−0.448140	−	0.893963i	$-0.647914\pi$
$402$		0			0
$403$	167254.	−	64802.6i	1.02984	−	0.399008i
$404$	−133789.			−0.819704
$405$		0			0
$406$	178304.			1.08170
$407$			530272.i			3.20118i
$408$		0			0
$409$	189086.	−	189086.i	1.13035	−	1.13035i	0.140227	−	0.990119i	$-0.455217\pi$
$409$	189086.	−	189086.i	1.13035	−	1.13035i	0.990119	−	0.140227i	$-0.0447833\pi$
$410$	75961.3	+	75961.3i	0.451882	+	0.451882i
$411$		0			0
$412$	35360.8			0.208319
$413$		−	111043.i		−	0.651016i
$414$		0			0
$415$			164448.i			0.954842i
$416$	−518423.	+	200862.i	−2.99569	+	1.16068i
$417$		0			0
$418$	−155951.	+	155951.i	−0.892559	+	0.892559i
$419$	−128365.			−0.731173			−0.365586	−	0.930777i	$-0.619131\pi$
$419$	−128365.			−0.731173			−0.365586	+	0.930777i	$0.619131\pi$
$420$		0			0
$421$	−143608.	+	143608.i	−0.810244	+	0.810244i	−0.984670	−	0.174426i	$-0.944193\pi$
$421$	−143608.	+	143608.i	−0.810244	+	0.810244i	0.174426	+	0.984670i	$0.444193\pi$
$422$	−262902.	+	262902.i	−1.47628	+	1.47628i
$423$		0			0
$424$	390957.	+	390957.i	2.17469	+	2.17469i
$425$	−796.868			−0.00441172
$426$		0			0
$427$	−40551.6	−	40551.6i	−0.222409	−	0.222409i
$428$			108728.i			0.593546i
$429$		0			0
$430$	299485.			1.61971
$431$	−97112.9	+	97112.9i	−0.522784	+	0.522784i	−0.918411	−	0.395627i	$-0.870527\pi$
$431$	−97112.9	+	97112.9i	−0.522784	+	0.522784i	0.395627	+	0.918411i	$0.370527\pi$
$432$		0			0
$433$		−	207477.i		−	1.10661i	−0.832980	−	0.553304i	$-0.813367\pi$
$433$		−	207477.i		−	1.10661i	0.832980	−	0.553304i	$-0.186633\pi$
$434$	210986.	−	210986.i	1.12014	−	1.12014i
$435$		0			0
$436$	−621045.	−	621045.i	−3.26701	−	3.26701i
$437$	3217.99	+	3217.99i	0.0168509	+	0.0168509i
$438$		0			0
$439$			278696.i			1.44611i	0.690791	+	0.723054i	$0.257262\pi$
$439$			278696.i			1.44611i	−0.690791	+	0.723054i	$0.742738\pi$
$440$	−707301.	−	707301.i	−3.65342	−	3.65342i
$441$		0			0
$442$	101003.	+	260687.i	0.516999	+	1.33437i
$443$	343219.			1.74889			0.874447	−	0.485121i	$-0.161224\pi$
$443$	343219.			1.74889			0.874447	+	0.485121i	$0.161224\pi$
$444$		0			0
$445$	−335217.			−1.69280
$446$		−	635984.i		−	3.19725i
$447$		0			0
$448$	−299386.	+	299386.i	−1.49168	+	1.49168i
$449$	−176502.	−	176502.i	−0.875502	−	0.875502i	0.117563	−	0.993065i	$-0.462492\pi$
$449$	−176502.	−	176502.i	−0.875502	−	0.875502i	−0.993065	+	0.117563i	$0.962492\pi$
$450$		0			0
$451$	−113330.			−0.557177
$452$			89312.9i			0.437157i
$453$		0			0
$454$		−	213486.i		−	1.03576i
$455$	−141994.	−	62689.5i	−0.685878	−	0.302811i
$456$		0			0
$457$	284006.	−	284006.i	1.35986	−	1.35986i	0.485784	−	0.874079i	$-0.338534\pi$
$457$	284006.	−	284006.i	1.35986	−	1.35986i	0.874079	−	0.485784i	$-0.161466\pi$
$458$	672440.			3.20570
$459$		0			0
$460$	−23503.0	+	23503.0i	−0.111073	+	0.111073i
$461$	244619.	−	244619.i	1.15103	−	1.15103i	0.164686	−	0.986346i	$-0.447339\pi$
$461$	244619.	−	244619.i	1.15103	−	1.15103i	0.986346	−	0.164686i	$-0.0526612\pi$
$462$		0			0
$463$	89908.4	+	89908.4i	0.419409	+	0.419409i	0.885000	−	0.465591i	$-0.154158\pi$
$463$	89908.4	+	89908.4i	0.419409	+	0.419409i	−0.465591	+	0.885000i	$0.654158\pi$
$464$	539485.			2.50578
$465$		0			0
$466$	−62910.4	−	62910.4i	−0.289702	−	0.289702i
$467$			135947.i			0.623357i	0.950188	+	0.311679i	$0.100891\pi$
$467$			135947.i			0.623357i	−0.950188	+	0.311679i	$0.899109\pi$
$468$		0			0
$469$	4229.75			0.0192295
$470$	125719.	−	125719.i	0.569120	−	0.569120i
$471$		0			0
$472$		−	602763.i		−	2.70559i
$473$	−223408.	+	223408.i	−0.998565	+	0.998565i
$474$		0			0
$475$	−374.420	−	374.420i	−0.00165948	−	0.00165948i
$476$	238465.	+	238465.i	1.05247	+	1.05247i
$477$		0			0
$478$		−	388051.i		−	1.69837i
$479$	245796.	+	245796.i	1.07128	+	1.07128i	0.997256	+	0.0740240i	$0.0235842\pi$
$479$	245796.	+	245796.i	1.07128	+	1.07128i	0.0740240	+	0.997256i	$0.476416\pi$
$480$		0			0
$481$	161368.	+	416488.i	0.697472	+	1.80016i
$482$	54181.7			0.233216
$483$		0			0
$484$	1.08129e6			4.61584
$485$			91406.0i			0.388590i
$486$		0			0
$487$	57471.5	−	57471.5i	0.242323	−	0.242323i	−0.575488	−	0.817810i	$-0.695188\pi$
$487$	57471.5	−	57471.5i	0.242323	−	0.242323i	0.817810	+	0.575488i	$0.195188\pi$
$488$	−220122.	−	220122.i	−0.924322	−	0.924322i
$489$		0			0
$490$	198430.			0.826446
$491$		−	273989.i		−	1.13650i	−0.822855	−	0.568251i	$-0.807620\pi$
$491$		−	273989.i		−	1.13650i	0.822855	−	0.568251i	$-0.192380\pi$
$492$		0			0
$493$		−	137513.i		−	0.565783i
$494$	−75030.0	+	169946.i	−0.307455	+	0.696396i
$495$		0			0
$496$	638370.	−	638370.i	2.59483	−	2.59483i
$497$	−334847.			−1.35561
$498$		0			0
$499$	−3284.85	+	3284.85i	−0.0131921	+	0.0131921i	−0.713672	−	0.700480i	$-0.752969\pi$
$499$	−3284.85	+	3284.85i	−0.0131921	+	0.0131921i	0.700480	+	0.713672i	$0.252969\pi$
$500$	467765.	−	467765.i	1.87106	−	1.87106i
$501$		0			0
$502$	−179851.	−	179851.i	−0.713685	−	0.713685i
$503$	−464818.			−1.83716			−0.918580	−	0.395235i	$-0.870663\pi$
$503$	−464818.			−1.83716			−0.918580	+	0.395235i	$0.870663\pi$
$504$		0			0
$505$	−55860.8	−	55860.8i	−0.219041	−	0.219041i
$506$		−	48355.7i		−	0.188863i
$507$		0			0
$508$	−347922.			−1.34820
$509$	−138065.	+	138065.i	−0.532903	+	0.532903i	−0.921435	−	0.388532i	$-0.872982\pi$
$509$	−138065.	+	138065.i	−0.532903	+	0.532903i	0.388532	+	0.921435i	$0.372982\pi$
$510$		0			0
$511$			133669.i			0.511903i
$512$	−53927.2	+	53927.2i	−0.205716	+	0.205716i
$513$		0			0
$514$	−248954.	−	248954.i	−0.942307	−	0.942307i
$515$	14764.2	+	14764.2i	0.0556667	+	0.0556667i
$516$		0			0
$517$			187566.i			0.701734i
$518$	525385.	+	525385.i	1.95803	+	1.95803i
$519$		0			0
$520$	−770770.	−	340291.i	−2.85048	−	1.25847i
$521$	−7527.60			−0.0277320			−0.0138660	−	0.999904i	$-0.504414\pi$
$521$	−7527.60			−0.0277320			−0.0138660	+	0.999904i	$0.504414\pi$
$522$		0			0
$523$	101541.			0.371226			0.185613	−	0.982623i	$-0.440573\pi$
$523$	101541.			0.371226			0.185613	+	0.982623i	$0.440573\pi$
$524$		−	610430.i		−	2.22317i
$525$		0			0
$526$	−468401.	+	468401.i	−1.69296	+	1.69296i
$527$	−162719.	−	162719.i	−0.585890	−	0.585890i
$528$		0			0
$529$	278843.			0.996434
$530$			525738.i			1.87162i
$531$		0			0
$532$			224093.i			0.791780i
$533$	−89012.3	+	34487.7i	−0.313325	+	0.121398i
$534$		0			0
$535$	−45397.3	+	45397.3i	−0.158607	+	0.158607i
$536$	22959.9			0.0799171
$537$		0			0
$538$	−357148.	+	357148.i	−1.23391	+	1.23391i
$539$	−148023.	+	148023.i	−0.509510	+	0.509510i
$540$		0			0
$541$	−319986.	−	319986.i	−1.09329	−	1.09329i	−0.995175	−	0.0981197i	$-0.968717\pi$
$541$	−319986.	−	319986.i	−1.09329	−	1.09329i	−0.0981197	−	0.995175i	$-0.531283\pi$
$542$	−405693.			−1.38102
$543$		0			0
$544$	504363.	+	504363.i	1.70430	+	1.70430i
$545$		−	518610.i		−	1.74601i
$546$		0			0
$547$	−4392.73			−0.0146811			−0.00734056	−	0.999973i	$-0.502337\pi$
$547$	−4392.73			−0.0146811			−0.00734056	+	0.999973i	$0.502337\pi$
$548$	−288801.	+	288801.i	−0.961697	+	0.961697i
$549$		0			0
$550$			5626.29i			0.0185993i
$551$	64612.6	−	64612.6i	0.212821	−	0.212821i
$552$		0			0
$553$	29523.7	+	29523.7i	0.0965429	+	0.0965429i
$554$	−259219.	−	259219.i	−0.844592	−	0.844592i
$555$		0			0
$556$			884482.i			2.86114i
$557$	−149881.	−	149881.i	−0.483101	−	0.483101i	0.423020	−	0.906120i	$-0.360970\pi$
$557$	−149881.	−	149881.i	−0.483101	−	0.483101i	−0.906120	+	0.423020i	$0.860970\pi$
$558$		0			0
$559$	−107484.	+	243455.i	−0.343970	+	0.779104i
$560$	−781228.			−2.49116
$561$		0			0
$562$	−38417.6			−0.121635
$563$			229570.i			0.724266i	0.932126	+	0.362133i	$0.117951\pi$
$563$			229570.i			0.724266i	−0.932126	+	0.362133i	$0.882049\pi$
$564$		0			0
$565$	−37290.8	+	37290.8i	−0.116817	+	0.116817i
$566$	−26075.0	−	26075.0i	−0.0813939	−	0.0813939i
$567$		0			0
$568$	−1.81761e6			−5.63384
$569$			81898.7i			0.252960i	0.991969	+	0.126480i	$0.0403680\pi$
$569$			81898.7i			0.252960i	−0.991969	+	0.126480i	$0.959632\pi$
$570$		0			0
$571$		−	508689.i		−	1.56020i	−0.625655	−	0.780100i	$-0.715168\pi$
$571$		−	508689.i		−	1.56020i	0.625655	−	0.780100i	$-0.284832\pi$
$572$	1.33470e6	−	517130.i	4.07937	−	1.58055i
$573$		0			0
$574$	−112286.	+	112286.i	−0.340802	+	0.340802i
$575$	116.096			0.000351142
$576$		0			0
$577$	160019.	−	160019.i	0.480639	−	0.480639i	−0.424697	−	0.905336i	$-0.639619\pi$
$577$	160019.	−	160019.i	0.480639	−	0.480639i	0.905336	+	0.424697i	$0.139619\pi$
$578$	−196986.	+	196986.i	−0.589631	+	0.589631i
$579$		0			0
$580$	471906.	+	471906.i	1.40281	+	1.40281i
$581$	−243087.			−0.720126
$582$		0			0
$583$	−392187.	−	392187.i	−1.15387	−	1.15387i
$584$			725578.i			2.12745i
$585$		0			0
$586$	847787.			2.46883
$587$	12874.3	−	12874.3i	0.0373635	−	0.0373635i	−0.688178	−	0.725542i	$-0.741589\pi$
$587$	12874.3	−	12874.3i	0.0373635	−	0.0373635i	0.725542	+	0.688178i	$0.241589\pi$
$588$		0			0
$589$		−	152912.i		−	0.440768i
$590$	405282.	−	405282.i	1.16427	−	1.16427i
$591$		0			0
$592$	1.58963e6	+	1.58963e6i	4.53580	+	4.53580i
$593$	4806.93	+	4806.93i	0.0136697	+	0.0136697i	0.713909	−	0.700239i	$-0.246923\pi$
$593$	4806.93	+	4806.93i	0.0136697	+	0.0136697i	−0.700239	+	0.713909i	$0.746923\pi$
$594$		0			0
$595$			199132.i			0.562481i
$596$	−132663.	−	132663.i	−0.373471	−	0.373471i
$597$		0			0
$598$	−14715.2	−	37979.7i	−0.0411494	−	0.106206i
$599$	−295770.			−0.824330			−0.412165	−	0.911109i	$-0.635227\pi$
$599$	−295770.			−0.824330			−0.412165	+	0.911109i	$0.635227\pi$
$600$		0			0
$601$	−504060.			−1.39551			−0.697756	−	0.716336i	$-0.745818\pi$
$601$	−504060.			−1.39551			−0.697756	+	0.716336i	$0.745818\pi$
$602$			442698.i			1.22156i
$603$		0			0
$604$	664793.	−	664793.i	1.82227	−	1.82227i
$605$	451470.	+	451470.i	1.23344	+	1.23344i
$606$		0			0
$607$	−224718.			−0.609903			−0.304952	−	0.952368i	$-0.598640\pi$
$607$	−224718.			−0.609903			−0.304952	+	0.952368i	$0.598640\pi$
$608$			473965.i			1.28215i
$609$		0			0
$610$		−	296008.i		−	0.795506i
$611$	57078.3	+	147318.i	0.152893	+	0.394616i
$612$		0			0
$613$	−354062.	+	354062.i	−0.942232	+	0.942232i	−0.998420	−	0.0561878i	$-0.982105\pi$
$613$	−354062.	+	354062.i	−0.942232	+	0.942232i	0.0561878	+	0.998420i	$0.482105\pi$
$614$	399230.			1.05898
$615$		0			0
$616$	1.04553e6	−	1.04553e6i	2.75535	−	2.75535i
$617$	159963.	−	159963.i	0.420195	−	0.420195i	−0.465076	−	0.885271i	$-0.653973\pi$
$617$	159963.	−	159963.i	0.420195	−	0.420195i	0.885271	+	0.465076i	$0.153973\pi$
$618$		0			0
$619$	−493946.	−	493946.i	−1.28913	−	1.28913i	−0.935313	−	0.353821i	$-0.884882\pi$
$619$	−493946.	−	493946.i	−1.28913	−	1.28913i	−0.353821	−	0.935313i	$-0.615118\pi$
$620$	1.11681e6			2.90533
$621$		0			0
$622$	269765.	+	269765.i	0.697276	+	0.697276i
$623$		−	495518.i		−	1.27668i
$624$		0			0
$625$	388314.			0.994085
$626$	−534204.	+	534204.i	−1.36320	+	1.36320i
$627$		0			0
$628$		−	50651.7i		−	0.128433i
$629$	405193.	−	405193.i	1.02414	−	1.02414i
$630$		0			0
$631$	−337240.	−	337240.i	−0.846993	−	0.846993i	0.142763	−	0.989757i	$-0.454401\pi$
$631$	−337240.	−	337240.i	−0.846993	−	0.846993i	−0.989757	+	0.142763i	$0.954401\pi$
$632$	160260.	+	160260.i	0.401228	+	0.401228i
$633$		0			0
$634$		−	693942.i		−	1.72641i
$635$	−145268.	−	145268.i	−0.360265	−	0.360265i
$636$		0			0
$637$	−71215.8	+	161306.i	−0.175508	+	0.397532i
$638$	−970913.			−2.38528
$639$		0			0
$640$	−873336.			−2.13217
$641$		−	121982.i		−	0.296880i	−0.988921	−	0.148440i	$-0.952575\pi$
$641$		−	121982.i		−	0.296880i	0.988921	−	0.148440i	$-0.0474252\pi$
$642$		0			0
$643$	496461.	−	496461.i	1.20078	−	1.20078i	0.226850	−	0.973930i	$-0.427157\pi$
$643$	496461.	−	496461.i	1.20078	−	1.20078i	0.973930	−	0.226850i	$-0.0728427\pi$
$644$	−34742.1	−	34742.1i	−0.0837692	−	0.0837692i
$645$		0			0
$646$	238332.			0.571107
$647$			367939.i			0.878956i	0.898253	+	0.439478i	$0.144837\pi$
$647$			367939.i			0.878956i	−0.898253	+	0.439478i	$0.855163\pi$
$648$		0			0
$649$			604660.i			1.43556i
$650$	1712.14	+	4419.02i	0.00405241	+	0.0104592i
$651$		0			0
$652$	390661.	−	390661.i	0.918979	−	0.918979i
$653$	−24379.6			−0.0571743			−0.0285872	−	0.999591i	$-0.509101\pi$
$653$	−24379.6			−0.0571743			−0.0285872	+	0.999591i	$0.509101\pi$
$654$		0			0
$655$	254873.	−	254873.i	0.594074	−	0.594074i
$656$	−339739.	+	339739.i	−0.789473	+	0.789473i
$657$		0			0
$658$	185837.	+	185837.i	0.429221	+	0.429221i
$659$	230704.			0.531231			0.265616	−	0.964079i	$-0.414425\pi$
$659$	230704.			0.531231			0.265616	+	0.964079i	$0.414425\pi$
$660$		0			0
$661$	−94783.2	−	94783.2i	−0.216934	−	0.216934i	0.590271	−	0.807205i	$-0.299021\pi$
$661$	−94783.2	−	94783.2i	−0.216934	−	0.216934i	−0.807205	+	0.590271i	$0.799021\pi$
$662$		−	1.42536e6i		−	3.25244i
$663$		0			0
$664$	−1.31952e6			−2.99282
$665$	−93565.4	+	93565.4i	−0.211579	+	0.211579i
$666$		0			0
$667$			20034.4i			0.0450323i
$668$	148622.	−	148622.i	0.333067	−	0.333067i
$669$		0			0
$670$	15437.6	+	15437.6i	0.0343899	+	0.0343899i
$671$	220814.	+	220814.i	0.490436	+	0.490436i
$672$		0			0
$673$			583502.i			1.28829i	0.764905	+	0.644143i	$0.222786\pi$
$673$			583502.i			1.28829i	−0.764905	+	0.644143i	$0.777214\pi$
$674$	812979.	+	812979.i	1.78961	+	1.78961i
$675$		0			0
$676$	890939.	−	812331.i	1.94964	−	1.77762i
$677$	317649.			0.693060			0.346530	−	0.938039i	$-0.387360\pi$
$677$	317649.			0.693060			0.346530	+	0.938039i	$0.387360\pi$
$678$		0			0
$679$	−135116.			−0.293068
$680$			1.08093e6i			2.33765i
$681$		0			0
$682$	−1.14888e6	+	1.14888e6i	−2.47004	+	2.47004i
$683$	580148.	+	580148.i	1.24365	+	1.24365i	0.958476	+	0.285172i	$0.0920508\pi$
$683$	580148.	+	580148.i	1.24365	+	1.24365i	0.285172	+	0.958476i	$0.407949\pi$
$684$		0			0
$685$	−241166.			−0.513968
$686$			968311.i			2.05763i
$687$		0			0
$688$			1.33945e6i			2.82976i
$689$	−427379.	−	188686.i	−0.900275	−	0.397466i
$690$		0			0
$691$	−254193.	+	254193.i	−0.532363	+	0.532363i	−0.921275	−	0.388912i	$-0.872851\pi$
$691$	−254193.	+	254193.i	−0.532363	+	0.532363i	0.388912	+	0.921275i	$0.372851\pi$
$692$	−2.06397e6			−4.31015
$693$		0			0
$694$	320812.	−	320812.i	0.666088	−	0.666088i
$695$	−369298.	+	369298.i	−0.764552	+	0.764552i
$696$		0			0
$697$	86598.3	+	86598.3i	0.178256	+	0.178256i
$698$	−468974.			−0.962582
$699$		0			0
$700$	4042.32	+	4042.32i	0.00824963	+	0.00824963i
$701$		−	756736.i		−	1.53996i	−0.638070	−	0.769978i	$-0.720267\pi$
$701$		−	756736.i		−	1.53996i	0.638070	−	0.769978i	$-0.279733\pi$
$702$		0			0
$703$	380772.			0.770467
$704$	1.63024e6	−	1.63024e6i	3.28931	−	3.28931i
$705$		0			0
$706$			693135.i			1.39062i
$707$	82573.5	−	82573.5i	0.165197	−	0.165197i
$708$		0			0
$709$	144273.	+	144273.i	0.287006	+	0.287006i	0.835895	−	0.548889i	$-0.184949\pi$
$709$	144273.	+	144273.i	0.287006	+	0.287006i	−0.548889	+	0.835895i	$0.684949\pi$
$710$	−1.22211e6	−	1.22211e6i	−2.42435	−	2.42435i
$711$		0			0
$712$		−	2.68977e6i		−	5.30585i
$713$	23706.6	+	23706.6i	0.0466326	+	0.0466326i
$714$		0			0
$715$	773196.	+	341362.i	1.51244	+	0.667733i
$716$	−85197.9			−0.166189
$717$		0			0
$718$	204061.			0.395832
$719$			73910.3i			0.142971i	0.997442	+	0.0714853i	$0.0227739\pi$
$719$			73910.3i			0.142971i	−0.997442	+	0.0714853i	$0.977226\pi$
$720$		0			0
$721$	−21824.4	+	21824.4i	−0.0419829	+	0.0419829i
$722$	−591110.	−	591110.i	−1.13395	−	1.13395i
$723$		0			0
$724$	691479.			1.31917
$725$		−	2331.04i		−	0.00443480i
$726$		0			0
$727$		−	714716.i		−	1.35227i	−0.736776	−	0.676137i	$-0.763653\pi$
$727$		−	714716.i		−	1.35227i	0.736776	−	0.676137i	$-0.236347\pi$
$728$	503018.	−	1.13935e6i	0.949119	−	2.14979i
$729$		0			0
$730$	−487859.	+	487859.i	−0.915480	+	0.915480i
$731$	341422.			0.638935
$732$		0			0
$733$	−180248.	+	180248.i	−0.335476	+	0.335476i	−0.854662	−	0.519186i	$-0.826235\pi$
$733$	−180248.	+	180248.i	−0.335476	+	0.335476i	0.519186	+	0.854662i	$0.326235\pi$
$734$	−862150.	+	862150.i	−1.60026	+	1.60026i
$735$		0			0
$736$	−73481.0	−	73481.0i	−0.135650	−	0.135650i
$737$	−23032.1			−0.0424032
$738$		0			0
$739$	538336.	+	538336.i	0.985744	+	0.985744i	0.999900	−	0.0141555i	$-0.00450600\pi$
$739$	538336.	+	538336.i	0.985744	+	0.985744i	−0.0141555	+	0.999900i	$0.504506\pi$
$740$			2.78101e6i			5.07854i
$741$		0			0
$742$	−777145.			−1.41154
$743$	53537.5	−	53537.5i	0.0969796	−	0.0969796i	−0.656952	−	0.753932i	$-0.728155\pi$
$743$	53537.5	−	53537.5i	0.0969796	−	0.0969796i	0.753932	+	0.656952i	$0.228155\pi$
$744$		0			0
$745$		−	110781.i		−	0.199597i
$746$	−86179.3	+	86179.3i	−0.154855	+	0.154855i
$747$		0			0
$748$	−1.29851e6	−	1.29851e6i	−2.32082	−	2.32082i
$749$	−67106.2	−	67106.2i	−0.119619	−	0.119619i
$750$		0			0
$751$			334548.i			0.593169i	0.955007	+	0.296585i	$0.0958477\pi$
$751$			334548.i			0.593169i	−0.955007	+	0.296585i	$0.904152\pi$
$752$	562279.	+	562279.i	0.994297	+	0.994297i
$753$		0			0
$754$	−762577.	+	295460.i	−1.34135	+	0.519703i
$755$	555141.			0.973890
$756$		0			0
$757$	−335226.			−0.584986			−0.292493	−	0.956268i	$-0.594485\pi$
$757$	−335226.			−0.584986			−0.292493	+	0.956268i	$0.594485\pi$
$758$		−	1.64341e6i		−	2.86027i
$759$		0			0
$760$	−507891.	+	507891.i	−0.879312	+	0.879312i
$761$	243144.	+	243144.i	0.419851	+	0.419851i	0.885152	−	0.465302i	$-0.154054\pi$
$761$	243144.	+	243144.i	0.419851	+	0.419851i	−0.465302	+	0.885152i	$0.654054\pi$
$762$		0			0
$763$	766609.			1.31682
$764$		−	2.42788e6i		−	4.15950i
$765$		0			0
$766$			1.22414e6i			2.08628i
$767$	184005.	+	474914.i	0.312780	+	0.807280i
$768$		0			0
$769$	134055.	−	134055.i	0.226688	−	0.226688i	−0.584620	−	0.811308i	$-0.698756\pi$
$769$	134055.	−	134055.i	0.226688	−	0.226688i	0.811308	+	0.584620i	$0.198756\pi$
$770$	1.40598e6			2.37136
$771$		0			0
$772$	−695290.	+	695290.i	−1.16662	+	1.16662i
$773$	488147.	−	488147.i	0.816943	−	0.816943i	−0.168721	−	0.985664i	$-0.553964\pi$
$773$	488147.	−	488147.i	0.816943	−	0.816943i	0.985664	+	0.168721i	$0.0539637\pi$
$774$		0			0
$775$	−2758.31	−	2758.31i	−0.00459240	−	0.00459240i
$776$	−733438.			−1.21798
$777$		0			0
$778$	−393488.	−	393488.i	−0.650088	−	0.650088i
$779$			81379.0i			0.134103i
$780$		0			0
$781$	1.82333e6			2.98926
$782$	−36949.7	+	36949.7i	−0.0604223	+	0.0604223i
$783$		0			0
$784$			887481.i			1.44387i
$785$	21148.6	−	21148.6i	0.0343196	−	0.0343196i
$786$		0			0
$787$	644084.	+	644084.i	1.03990	+	1.03990i	0.999170	+	0.0407338i	$0.0129695\pi$
$787$	644084.	+	644084.i	1.03990	+	1.03990i	0.0407338	+	0.999170i	$0.487030\pi$
$788$	−1.03453e6	−	1.03453e6i	−1.66605	−	1.66605i
$789$		0			0
$790$			215509.i			0.345312i
$791$	−55123.3	−	55123.3i	−0.0881013	−	0.0881013i
$792$		0			0
$793$	240629.	+	106236.i	0.382650	+	0.168938i
$794$	1.69914e6			2.69518
$795$		0			0
$796$	−1.16345e6			−1.83621
$797$			519061.i			0.817150i	0.912725	+	0.408575i	$0.133974\pi$
$797$			519061.i			0.817150i	−0.912725	+	0.408575i	$0.866026\pi$
$798$		0			0
$799$	143323.	−	143323.i	0.224503	−	0.224503i
$800$	8549.67	+	8549.67i	0.0133589	+	0.0133589i
$801$		0			0
$802$	2.32865e6			3.62039
$803$		−	727862.i		−	1.12880i
$804$		0			0
$805$		−	29011.7i		−	0.0447695i
$806$	−552738.	+	1.25197e6i	−0.850842	+	1.92719i
$807$		0			0
$808$	448225.	−	448225.i	0.686551	−	0.686551i
$809$	−786528.			−1.20176			−0.600879	−	0.799340i	$-0.705183\pi$
$809$	−786528.			−1.20176			−0.600879	+	0.799340i	$0.705183\pi$
$810$		0			0
$811$	−820989.	+	820989.i	−1.24823	+	1.24823i	−0.291733	+	0.956500i	$0.594232\pi$
$811$	−820989.	+	820989.i	−1.24823	+	1.24823i	−0.956500	+	0.291733i	$0.905768\pi$
$812$	−697571.	+	697571.i	−1.05798	+	1.05798i
$813$		0			0
$814$	−2.86087e6	−	2.86087e6i	−4.31766	−	4.31766i
$815$	326226.			0.491137
$816$		0			0
$817$	160422.	+	160422.i	0.240337	+	0.240337i
$818$			2.04027e6i			3.04916i
$819$		0			0
$820$	−594361.			−0.883940
$821$	−266133.	+	266133.i	−0.394833	+	0.394833i	−0.876406	−	0.481573i	$-0.840066\pi$
$821$	−266133.	+	266133.i	−0.394833	+	0.394833i	0.481573	+	0.876406i	$0.340066\pi$
$822$		0			0
$823$		−	756731.i		−	1.11723i	−0.829428	−	0.558614i	$-0.811333\pi$
$823$		−	756731.i		−	1.11723i	0.829428	−	0.558614i	$-0.188667\pi$
$824$	−118467.	+	118467.i	−0.174479	+	0.174479i
$825$		0			0
$826$	599088.	+	599088.i	0.878072	+	0.878072i
$827$	200899.	+	200899.i	0.293743	+	0.293743i	0.838557	−	0.544814i	$-0.183400\pi$
$827$	200899.	+	200899.i	0.293743	+	0.293743i	−0.544814	+	0.838557i	$0.683400\pi$
$828$		0			0
$829$			58319.1i			0.0848598i	0.999099	+	0.0424299i	$0.0135099\pi$
$829$			58319.1i			0.0848598i	−0.999099	+	0.0424299i	$0.986490\pi$
$830$	−887210.	−	887210.i	−1.28786	−	1.28786i
$831$		0			0
$832$	784326.	−	1.77652e6i	1.13305	−	2.56640i
$833$	226216.			0.326012
$834$		0			0
$835$	124109.			0.178004
$836$		−	1.22025e6i		−	1.74596i
$837$		0			0
$838$	692543.	−	692543.i	0.986186	−	0.986186i
$839$	−860228.	−	860228.i	−1.22205	−	1.22205i	−0.966901	−	0.255151i	$-0.917875\pi$
$839$	−860228.	−	860228.i	−1.22205	−	1.22205i	−0.255151	−	0.966901i	$-0.582125\pi$
$840$		0			0
$841$	−305020.			−0.431257
$842$		−	1.54956e6i		−	2.18567i
$843$		0			0
$844$		−	2.05708e6i		−	2.88780i
$845$	711166.	+	32821.1i	0.995995	+	0.0459663i
$846$		0			0
$847$	−667364.	+	667364.i	−0.930241	+	0.930241i
$848$	−2.35137e6			−3.26986
$849$		0			0
$850$	4299.17	−	4299.17i	0.00595042	−	0.00595042i
$851$	−59032.8	+	59032.8i	−0.0815144	+	0.0815144i
$852$		0			0
$853$	33807.8	+	33807.8i	0.0464642	+	0.0464642i	0.729957	−	0.683493i	$-0.239540\pi$
$853$	33807.8	+	33807.8i	0.0464642	+	0.0464642i	−0.683493	+	0.729957i	$0.739540\pi$
$854$	437559.			0.599958
$855$		0			0
$856$	−364266.	−	364266.i	−0.497131	−	0.497131i
$857$			754201.i			1.02689i	0.858121	+	0.513447i	$0.171632\pi$
$857$			754201.i			1.02689i	−0.858121	+	0.513447i	$0.828368\pi$
$858$		0			0
$859$	−665748.			−0.902243			−0.451121	−	0.892463i	$-0.648976\pi$
$859$	−665748.			−0.902243			−0.451121	+	0.892463i	$0.648976\pi$
$860$	−1.17166e6	+	1.17166e6i	−1.58418	+	1.58418i
$861$		0			0
$862$		−	1.04787e6i		−	1.41023i
$863$	−429331.	+	429331.i	−0.576462	+	0.576462i	−0.933927	−	0.357465i	$-0.883641\pi$
$863$	−429331.	+	429331.i	−0.576462	+	0.576462i	0.357465	+	0.933927i	$0.383641\pi$
$864$		0			0
$865$	−861770.	−	861770.i	−1.15175	−	1.15175i
$866$	1.11936e6	+	1.11936e6i	1.49256	+	1.49256i
$867$		0			0
$868$			1.65086e6i			2.19115i
$869$	−160764.	−	160764.i	−0.212888	−	0.212888i
$870$		0			0
$871$	−18090.0	+	7008.93i	−0.0238452	+	0.00923880i
$872$	4.16130e6			5.47263
$873$		0			0
$874$	−34722.7			−0.0454560
$875$			577402.i			0.754158i
$876$		0			0
$877$	450147.	−	450147.i	0.585268	−	0.585268i	−0.351078	−	0.936346i	$-0.614185\pi$
$877$	450147.	−	450147.i	0.585268	−	0.585268i	0.936346	+	0.351078i	$0.114185\pi$
$878$	−1.50359e6	−	1.50359e6i	−1.95047	−	1.95047i
$879$		0			0
$880$	4.25400e6			5.49328
$881$			1.29733e6i			1.67147i	0.549130	+	0.835737i	$0.314959\pi$
$881$			1.29733e6i			1.67147i	−0.549130	+	0.835737i	$0.685041\pi$
$882$		0			0
$883$		−	1.21426e6i		−	1.55736i	−0.627421	−	0.778680i	$-0.715890\pi$
$883$		−	1.21426e6i		−	1.55736i	0.627421	−	0.778680i	$-0.284110\pi$
$884$	−1.41503e6	−	624727.i	−1.81076	−	0.799439i
$885$		0			0
$886$	−1.85170e6	+	1.85170e6i	−2.35886	+	2.35886i
$887$	613983.			0.780385			0.390192	−	0.920733i	$-0.372408\pi$
$887$	613983.			0.780385			0.390192	+	0.920733i	$0.372408\pi$
$888$		0			0
$889$	214735.	−	214735.i	0.271706	−	0.271706i
$890$	1.80853e6	−	1.80853e6i	2.28321	−	2.28321i
$891$		0			0
$892$	2.48814e6	+	2.48814e6i	3.12712	+	3.12712i
$893$	134685.			0.168895
$894$		0			0
$895$	−35572.6	−	35572.6i	−0.0444089	−	0.0444089i
$896$		−	1.29097e6i		−	1.60805i
$897$		0			0
$898$	1.90449e6			2.36171
$899$	475993.	−	475993.i	0.588954	−	0.588954i
$900$		0			0
$901$			599357.i			0.738306i
$902$	611428.	−	611428.i	0.751506	−	0.751506i
$903$		0			0
$904$	−299220.	−	299220.i	−0.366145	−	0.366145i
$905$	288713.	+	288713.i	0.352508	+	0.352508i
$906$		0			0
$907$		−	715019.i		−	0.869167i	−0.900632	−	0.434583i	$-0.856896\pi$
$907$		−	715019.i		−	0.869167i	0.900632	−	0.434583i	$-0.143104\pi$
$908$	835213.	+	835213.i	1.01304	+	1.01304i
$909$		0			0
$910$	1.10429e6	−	427854.i	1.33352	−	0.516670i
$911$	−665942.			−0.802416			−0.401208	−	0.915987i	$-0.631409\pi$
$911$	−665942.			−0.802416			−0.401208	+	0.915987i	$0.631409\pi$
$912$		0			0
$913$	1.32367e6			1.58796
$914$			3.06448e6i			3.66829i
$915$		0			0
$916$	−2.63076e6	+	2.63076e6i	−3.13538	+	3.13538i
$917$	376753.	+	376753.i	0.448041	+	0.448041i
$918$		0			0
$919$	−740588.			−0.876891			−0.438446	−	0.898758i	$-0.644471\pi$
$919$	−740588.			−0.876891			−0.438446	+	0.898758i	$0.644471\pi$
$920$		−	157481.i		−	0.186060i
$921$		0			0
$922$			2.63948e6i			3.10496i
$923$	1.43209e6	−	554860.i	1.68099	−	0.651299i
$924$		0			0
$925$	6868.59	−	6868.59i	0.00802757	−	0.00802757i
$926$	−970128.			−1.13138
$927$		0			0
$928$	−1.47539e6	+	1.47539e6i	−1.71321	+	1.71321i
$929$	1.04076e6	−	1.04076e6i	1.20592	−	1.20592i	0.233581	−	0.972337i	$-0.424956\pi$
$929$	1.04076e6	−	1.04076e6i	1.20592	−	1.20592i	0.972337	−	0.233581i	$-0.0750444\pi$
$930$		0			0
$931$	106291.	+	106291.i	0.122630	+	0.122630i
$932$	492244.			0.566694
$933$		0			0
$934$	−733449.	−	733449.i	−0.840768	−	0.840768i
$935$		−	1.08433e6i		−	1.24033i
$936$		0			0
$937$	−324332.			−0.369412			−0.184706	−	0.982794i	$-0.559133\pi$
$937$	−324332.			−0.369412			−0.184706	+	0.982794i	$0.559133\pi$
$938$	−22819.9	+	22819.9i	−0.0259363	+	0.0259363i
$939$		0			0
$940$			983688.i			1.11327i
$941$	−710945.	+	710945.i	−0.802891	+	0.802891i	−0.983546	−	0.180656i	$-0.942178\pi$
$941$	−710945.	+	710945.i	−0.802891	+	0.802891i	0.180656	+	0.983546i	$0.442178\pi$
$942$		0			0
$943$	−12616.6	−	12616.6i	−0.0141879	−	0.0141879i
$944$	1.81263e6	+	1.81263e6i	2.03407	+	2.03407i
$945$		0			0
$946$		−	2.41061e6i		−	2.69368i
$947$	−741248.	−	741248.i	−0.826540	−	0.826540i	0.160497	−	0.987036i	$-0.448690\pi$
$947$	−741248.	−	741248.i	−0.826540	−	0.826540i	−0.987036	+	0.160497i	$0.948690\pi$
$948$		0			0
$949$	−221496.	−	571679.i	−0.245943	−	0.634775i
$950$	4040.06			0.00447653
$951$		0			0
$952$	−1.59783e6			−1.76302
$953$		−	1.16535e6i		−	1.28313i	−0.767068	−	0.641566i	$-0.778285\pi$
$953$		−	1.16535e6i		−	1.28313i	0.767068	−	0.641566i	$-0.221715\pi$
$954$		0			0
$955$	1.01371e6	−	1.01371e6i	1.11150	−	1.11150i
$956$	1.51816e6	+	1.51816e6i	1.66112	+	1.66112i
$957$		0			0
$958$	−2.65218e6			−2.88983
$959$		−	356492.i		−	0.387626i
$960$		0			0
$961$		−	202960.i		−	0.219768i
$962$	−3.11758e6	−	1.37640e6i	−3.36874	−	1.48728i
$963$		0			0
$964$	−211973.	+	211973.i	−0.228101	+	0.228101i
$965$	−580608.			−0.623489
$966$		0			0
$967$	−973688.	+	973688.i	−1.04128	+	1.04128i	−0.0421682	+	0.999111i	$0.513427\pi$
$967$	−973688.	+	973688.i	−1.04128	+	1.04128i	−0.999111	+	0.0421682i	$0.986573\pi$
$968$	−3.62258e6	+	3.62258e6i	−3.86605	+	3.86605i
$969$		0			0
$970$	−493144.	−	493144.i	−0.524119	−	0.524119i
$971$	−1.67477e6			−1.77630			−0.888150	−	0.459553i	$-0.848010\pi$
$971$	−1.67477e6			−1.77630			−0.888150	+	0.459553i	$0.848010\pi$
$972$		0			0
$973$	−545896.	−	545896.i	−0.576613	−	0.576613i
$974$			620128.i			0.653677i
$975$		0			0
$976$	1.32390e6			1.38981
$977$	−85903.0	+	85903.0i	−0.0899952	+	0.0899952i	−0.750671	−	0.660676i	$-0.770270\pi$
$977$	−85903.0	+	85903.0i	−0.0899952	+	0.0899952i	0.660676	+	0.750671i	$0.270270\pi$
$978$		0			0
$979$			2.69823e6i			2.81523i
$980$	−776309.	+	776309.i	−0.808318	+	0.808318i
$981$		0			0
$982$	1.47820e6	+	1.47820e6i	1.53288	+	1.53288i
$983$	−1.21292e6	−	1.21292e6i	−1.25524	−	1.25524i	−0.953339	−	0.301900i	$-0.902379\pi$
$983$	−1.21292e6	−	1.21292e6i	−1.25524	−	1.25524i	−0.301900	−	0.953339i	$-0.597621\pi$
$984$		0			0
$985$		−	863890.i		−	0.890402i
$986$	741896.	+	741896.i	0.763113	+	0.763113i
$987$		0			0
$988$	−371335.	−	958409.i	−0.380410	−	0.981832i
$989$	−49742.0			−0.0508546
$990$		0			0
$991$	−504840.			−0.514051			−0.257026	−	0.966405i	$-0.582742\pi$
$991$	−504840.			−0.514051			−0.257026	+	0.966405i	$0.582742\pi$
$992$			3.49165e6i			3.54819i
$993$		0			0
$994$	1.80653e6	−	1.80653e6i	1.82840	−	1.82840i
$995$	−485777.	−	485777.i	−0.490671	−	0.490671i
$996$		0			0
$997$	−1.39880e6			−1.40723			−0.703614	−	0.710583i	$-0.748431\pi$
$997$	−1.39880e6			−1.40723			−0.703614	+	0.710583i	$0.748431\pi$
$998$		−	35444.1i		−	0.0355863i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.109.1		20
3.2	odd	2		39.5.g.a.31.10	✓	20
13.8	odd	4	inner	117.5.j.b.73.1		20
39.8	even	4		39.5.g.a.34.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.10	✓	20	3.2	odd	2
39.5.g.a.34.10	yes	20	39.8	even	4
117.5.j.b.73.1		20	13.8	odd	4	inner
117.5.j.b.109.1		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.39509 + 5.39509i) q^{2} -42.2140i q^{4} +(17.6256 - 17.6256i) q^{5} +(26.0542 + 26.0542i) q^{7} +(141.427 + 141.427i) q^{8} +190.184i q^{10} +(-141.872 - 141.872i) q^{11} +(-154.603 - 68.2564i) q^{13} -281.130 q^{14} -850.600 q^{16} +216.815i q^{17} +(-101.874 + 101.874i) q^{19} +(-744.048 - 744.048i) q^{20} +1530.83 q^{22} -31.5880i q^{23} +3.67533i q^{25} +(1202.35 - 465.848i) q^{26} +(1099.85 - 1099.85i) q^{28} -634.240 q^{29} +(-750.494 + 750.494i) q^{31} +(2326.23 - 2326.23i) q^{32} +(-1169.74 - 1169.74i) q^{34} +918.442 q^{35} +(-1868.84 - 1868.84i) q^{37} -1099.24i q^{38} +4985.48 q^{40} +(399.410 - 399.410i) q^{41} -1574.71i q^{43} +(-5989.00 + 5989.00i) q^{44} +(170.420 + 170.420i) q^{46} +(-661.037 - 661.037i) q^{47} -1043.36i q^{49} +(-19.8287 - 19.8287i) q^{50} +(-2881.38 + 6526.41i) q^{52} +2764.37 q^{53} -5001.17 q^{55} +7369.54i q^{56} +(3421.79 - 3421.79i) q^{58} +(-2131.00 - 2131.00i) q^{59} -1556.43 q^{61} -8097.96i q^{62} +11490.9i q^{64} +(-3928.03 + 1521.91i) q^{65} +(81.1721 - 81.1721i) q^{67} +9152.65 q^{68} +(-4955.08 + 4955.08i) q^{70} +(-6425.97 + 6425.97i) q^{71} +(2565.20 + 2565.20i) q^{73} +20165.1 q^{74} +(4300.51 + 4300.51i) q^{76} -7392.73i q^{77} +1133.16 q^{79} +(-14992.4 + 14992.4i) q^{80} +4309.71i q^{82} +(-4665.02 + 4665.02i) q^{83} +(3821.50 + 3821.50i) q^{85} +(8495.72 + 8495.72i) q^{86} -40129.2i q^{88} +(-9509.38 - 9509.38i) q^{89} +(-2249.69 - 5806.42i) q^{91} -1333.46 q^{92} +7132.72 q^{94} +3591.18i q^{95} +(-2592.99 + 2592.99i) q^{97} +(5629.01 + 5629.01i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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