LMFDB - Embedded newform 387.5.b.c.343.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,5,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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("387.343");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	387.b (of order $2$, degree $1$, 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:	$40.0041757134$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			343.2
Root			$-6.72223i$ of defining polynomial
Character	$\chi$	$=$	387.343
Dual form			387.5.b.c.343.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.72223i q^{2} -29.1884 q^{4} +1.48242i q^{5} +13.7963i q^{7} +88.6554i q^{8} +O(q^{10})$	\(q-6.72223i q^{2} -29.1884 q^{4} +1.48242i q^{5} +13.7963i q^{7} +88.6554i q^{8} +9.96518 q^{10} +10.2025 q^{11} +98.4183 q^{13} +92.7416 q^{14} +128.948 q^{16} +286.515 q^{17} +367.004i q^{19} -43.2695i q^{20} -68.5838i q^{22} +242.039 q^{23} +622.802 q^{25} -661.591i q^{26} -402.691i q^{28} -1147.40i q^{29} +895.225 q^{31} +551.669i q^{32} -1926.02i q^{34} -20.4519 q^{35} -2295.69i q^{37} +2467.09 q^{38} -131.425 q^{40} -1692.26 q^{41} +(-1546.34 + 1013.73i) q^{43} -297.796 q^{44} -1627.04i q^{46} -743.419 q^{47} +2210.66 q^{49} -4186.62i q^{50} -2872.67 q^{52} -99.3399 q^{53} +15.1245i q^{55} -1223.11 q^{56} -7713.09 q^{58} -3286.35 q^{59} -3223.22i q^{61} -6017.91i q^{62} +5771.61 q^{64} +145.898i q^{65} +5556.36 q^{67} -8362.90 q^{68} +137.482i q^{70} -2953.33i q^{71} -3618.34i q^{73} -15432.2 q^{74} -10712.3i q^{76} +140.757i q^{77} +2380.74 q^{79} +191.155i q^{80} +11375.7i q^{82} +6427.91 q^{83} +424.736i q^{85} +(6814.51 + 10394.8i) q^{86} +904.510i q^{88} -3293.36i q^{89} +1357.80i q^{91} -7064.72 q^{92} +4997.44i q^{94} -544.055 q^{95} -9329.95 q^{97} -14860.6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 92 q^{4}+O(q^{10}) $ 12 * q - 92 * q^4	$ 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) $ 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	6.72223i		−	1.68056i	−0.542154	−	0.840279i	$-0.682391\pi$
$2$		−	6.72223i		−	1.68056i	0.542154	−	0.840279i	$-0.317609\pi$
$3$		0			0
$3$		0			0
$4$	−29.1884			−1.82427
$5$			1.48242i			0.0592969i	0.999560	+	0.0296484i	$0.00943877\pi$
$5$			1.48242i			0.0592969i	−0.999560	+	0.0296484i	$0.990561\pi$
$6$		0			0
$7$			13.7963i			0.281556i	0.990041	+	0.140778i	$0.0449604\pi$
$7$			13.7963i			0.281556i	−0.990041	+	0.140778i	$0.955040\pi$
$8$			88.6554i			1.38524i
$9$		0			0
$10$	9.96518			0.0996518
$11$	10.2025			0.0843185			0.0421592	−	0.999111i	$-0.486576\pi$
$11$	10.2025			0.0843185			0.0421592	+	0.999111i	$0.486576\pi$
$12$		0			0
$13$	98.4183			0.582357			0.291179	−	0.956669i	$-0.405953\pi$
$13$	98.4183			0.582357			0.291179	+	0.956669i	$0.405953\pi$
$14$	92.7416			0.473172
$15$		0			0
$16$	128.948			0.503703
$17$	286.515			0.991400			0.495700	−	0.868494i	$-0.334911\pi$
$17$	286.515			0.991400			0.495700	+	0.868494i	$0.334911\pi$
$18$		0			0
$19$			367.004i			1.01663i	0.861171	+	0.508316i	$0.169732\pi$
$19$			367.004i			1.01663i	−0.861171	+	0.508316i	$0.830268\pi$
$20$		−	43.2695i		−	0.108174i
$21$		0			0
$22$		−	68.5838i		−	0.141702i
$23$	242.039			0.457540			0.228770	−	0.973480i	$-0.426530\pi$
$23$	242.039			0.457540			0.228770	+	0.973480i	$0.426530\pi$
$24$		0			0
$25$	622.802			0.996484
$26$		−	661.591i		−	0.978685i
$27$		0			0
$28$		−	402.691i		−	0.513636i
$29$		−	1147.40i		−	1.36433i	−0.731199	−	0.682164i	$-0.761039\pi$
$29$		−	1147.40i		−	1.36433i	0.731199	−	0.682164i	$-0.238961\pi$
$30$		0			0
$31$	895.225			0.931556			0.465778	−	0.884902i	$-0.345775\pi$
$31$	895.225			0.931556			0.465778	+	0.884902i	$0.345775\pi$
$32$			551.669i			0.538739i
$33$		0			0
$34$		−	1926.02i		−	1.66611i
$35$	−20.4519			−0.0166954
$36$		0			0
$37$		−	2295.69i		−	1.67691i	−0.544969	−	0.838456i	$-0.683459\pi$
$37$		−	2295.69i		−	1.67691i	0.544969	−	0.838456i	$-0.316541\pi$
$38$	2467.09			1.70851
$39$		0			0
$40$	−131.425			−0.0821405
$41$	−1692.26			−1.00670			−0.503348	−	0.864084i	$-0.667899\pi$
$41$	−1692.26			−1.00670			−0.503348	+	0.864084i	$0.667899\pi$
$42$		0			0
$43$	−1546.34	+	1013.73i	−0.836310	+	0.548257i
$43$	−1546.34	+	1013.73i	−0.836310	+	0.548257i
$44$	−297.796			−0.153820
$45$		0			0
$46$		−	1627.04i		−	0.768923i
$47$	−743.419			−0.336541			−0.168271	−	0.985741i	$-0.553818\pi$
$47$	−743.419			−0.336541			−0.168271	+	0.985741i	$0.553818\pi$
$48$		0			0
$49$	2210.66			0.920726
$50$		−	4186.62i		−	1.67465i
$51$		0			0
$52$	−2872.67			−1.06238
$53$	−99.3399			−0.0353649			−0.0176824	−	0.999844i	$-0.505629\pi$
$53$	−99.3399			−0.0353649			−0.0176824	+	0.999844i	$0.505629\pi$
$54$		0			0
$55$			15.1245i			0.00499982i
$56$	−1223.11			−0.390023
$57$		0			0
$58$	−7713.09			−2.29283
$59$	−3286.35			−0.944082			−0.472041	−	0.881577i	$-0.656483\pi$
$59$	−3286.35			−0.944082			−0.472041	+	0.881577i	$0.656483\pi$
$60$		0			0
$61$		−	3223.22i		−	0.866225i	−0.901340	−	0.433112i	$-0.857415\pi$
$61$		−	3223.22i		−	0.866225i	0.901340	−	0.433112i	$-0.142585\pi$
$62$		−	6017.91i		−	1.56553i
$63$		0			0
$64$	5771.61			1.40908
$65$			145.898i			0.0345320i
$66$		0			0
$67$	5556.36			1.23777			0.618886	−	0.785481i	$-0.287584\pi$
$67$	5556.36			1.23777			0.618886	+	0.785481i	$0.287584\pi$
$68$	−8362.90			−1.80859
$69$		0			0
$70$			137.482i			0.0280576i
$71$		−	2953.33i		−	0.585861i	−0.956134	−	0.292930i	$-0.905370\pi$
$71$		−	2953.33i		−	0.585861i	0.956134	−	0.292930i	$-0.0946305\pi$
$72$		0			0
$73$		−	3618.34i		−	0.678991i	−0.940608	−	0.339496i	$-0.889744\pi$
$73$		−	3618.34i		−	0.678991i	0.940608	−	0.339496i	$-0.110256\pi$
$74$	−15432.2			−2.81815
$75$		0			0
$76$		−	10712.3i		−	1.85461i
$77$			140.757i			0.0237404i
$78$		0			0
$79$	2380.74			0.381468			0.190734	−	0.981642i	$-0.438913\pi$
$79$	2380.74			0.381468			0.190734	+	0.981642i	$0.438913\pi$
$80$			191.155i			0.0298680i
$81$		0			0
$82$			11375.7i			1.69181i
$83$	6427.91			0.933069			0.466535	−	0.884503i	$-0.345502\pi$
$83$	6427.91			0.933069			0.466535	+	0.884503i	$0.345502\pi$
$84$		0			0
$85$			424.736i			0.0587869i
$86$	6814.51	+	10394.8i	0.921377	+	1.40547i
$87$		0			0
$88$			904.510i			0.116801i
$89$		−	3293.36i		−	0.415776i	−0.978153	−	0.207888i	$-0.933341\pi$
$89$		−	3293.36i		−	0.415776i	0.978153	−	0.207888i	$-0.0666590\pi$
$90$		0			0
$91$			1357.80i			0.163966i
$92$	−7064.72			−0.834679
$93$		0			0
$94$			4997.44i			0.565577i
$95$	−544.055			−0.0602831
$96$		0			0
$97$	−9329.95			−0.991599			−0.495799	−	0.868437i	$-0.665125\pi$
$97$	−9329.95			−0.991599			−0.495799	+	0.868437i	$0.665125\pi$
$98$		−	14860.6i		−	1.54733i
$99$		0			0
$100$	−18178.6			−1.81786
$101$	13572.8			1.33054			0.665270	−	0.746603i	$-0.268317\pi$
$101$	13572.8			1.33054			0.665270	+	0.746603i	$0.268317\pi$
$102$		0			0
$103$	−5527.90			−0.521057			−0.260529	−	0.965466i	$-0.583897\pi$
$103$	−5527.90			−0.521057			−0.260529	+	0.965466i	$0.583897\pi$
$104$			8725.32i			0.806705i
$105$		0			0
$106$			667.786i			0.0594327i
$107$	2435.69			0.212742			0.106371	−	0.994326i	$-0.466077\pi$
$107$	2435.69			0.212742			0.106371	+	0.994326i	$0.466077\pi$
$108$		0			0
$109$	3235.93			0.272362			0.136181	−	0.990684i	$-0.456517\pi$
$109$	3235.93			0.272362			0.136181	+	0.990684i	$0.456517\pi$
$110$	101.670			0.00840249
$111$		0			0
$112$			1779.00i			0.141821i
$113$			14680.9i			1.14973i	0.818247	+	0.574867i	$0.194946\pi$
$113$			14680.9i			1.14973i	−0.818247	+	0.574867i	$0.805054\pi$
$114$		0			0
$115$			358.803i			0.0271307i
$116$			33490.7i			2.48891i
$117$		0			0
$118$			22091.6i			1.58658i
$119$			3952.83i			0.279135i
$120$		0			0
$121$	−14536.9			−0.992890
$122$	−21667.2			−1.45574
$123$		0			0
$124$	−26130.2			−1.69941
$125$			1849.77i			0.118385i
$126$		0			0
$127$	27000.0			1.67400			0.837001	−	0.547201i	$-0.184307\pi$
$127$	27000.0			1.67400			0.837001	+	0.547201i	$0.184307\pi$
$128$		−	29971.4i		−	1.82931i
$129$		0			0
$130$	980.757			0.0580329
$131$			17222.3i			1.00357i	0.864991	+	0.501787i	$0.167324\pi$
$131$			17222.3i			1.00357i	−0.864991	+	0.501787i	$0.832676\pi$
$132$		0			0
$133$	−5063.28			−0.286239
$134$		−	37351.1i		−	2.08015i
$135$		0			0
$136$			25401.1i			1.37333i
$137$		−	1880.48i		−	0.100191i	−0.998744	−	0.0500955i	$-0.984047\pi$
$137$		−	1880.48i		−	0.100191i	0.998744	−	0.0500955i	$-0.0159526\pi$
$138$		0			0
$139$	20668.9			1.06977			0.534883	−	0.844926i	$-0.320356\pi$
$139$	20668.9			1.06977			0.534883	+	0.844926i	$0.320356\pi$
$140$	596.957			0.0304570
$141$		0			0
$142$	−19852.9			−0.984573
$143$	1004.12			0.0491035
$144$		0			0
$145$	1700.93			0.0809004
$146$	−24323.3			−1.14108
$147$		0			0
$148$			67007.6i			3.05915i
$149$		−	13651.5i		−	0.614904i	−0.951564	−	0.307452i	$-0.900524\pi$
$149$		−	13651.5i		−	0.614904i	0.951564	−	0.307452i	$-0.0994763\pi$
$150$		0			0
$151$		−	41870.6i		−	1.83635i	−0.396175	−	0.918175i	$-0.629662\pi$
$151$		−	41870.6i		−	1.83635i	0.396175	−	0.918175i	$-0.370338\pi$
$152$	−32536.9			−1.40828
$153$		0			0
$154$	946.200			0.0398971
$155$			1327.10i			0.0552383i
$156$		0			0
$157$		−	41348.7i		−	1.67750i	−0.544517	−	0.838750i	$-0.683287\pi$
$157$		−	41348.7i		−	1.67750i	0.544517	−	0.838750i	$-0.316713\pi$
$158$		−	16003.9i		−	0.641079i
$159$		0			0
$160$	−817.806			−0.0319455
$161$			3339.23i			0.128823i
$162$		0			0
$163$			6111.31i			0.230017i	0.993365	+	0.115008i	$0.0366895\pi$
$163$			6111.31i			0.230017i	−0.993365	+	0.115008i	$0.963311\pi$
$164$	49394.2			1.83649
$165$		0			0
$166$		−	43209.9i		−	1.56808i
$167$	−34991.0			−1.25465			−0.627326	−	0.778757i	$-0.715851\pi$
$167$	−34991.0			−1.25465			−0.627326	+	0.778757i	$0.715851\pi$
$168$		0			0
$169$	−18874.8			−0.660860
$170$	2855.17			0.0987949
$171$		0			0
$172$	45135.1	−	29589.1i	1.52566	−	1.00017i
$173$	4597.15			0.153602			0.0768009	−	0.997046i	$-0.475529\pi$
$173$	4597.15			0.153602			0.0768009	+	0.997046i	$0.475529\pi$
$174$		0			0
$175$			8592.34i			0.280566i
$176$	1315.60			0.0424715
$177$		0			0
$178$	−22138.7			−0.698736
$179$		−	26315.3i		−	0.821301i	−0.911793	−	0.410650i	$-0.865302\pi$
$179$		−	26315.3i		−	0.821301i	0.911793	−	0.410650i	$-0.134698\pi$
$180$		0			0
$181$	48133.4			1.46923			0.734614	−	0.678485i	$-0.237363\pi$
$181$	48133.4			1.46923			0.734614	+	0.678485i	$0.237363\pi$
$182$	9127.48			0.275555
$183$		0			0
$184$			21458.0i			0.633803i
$185$	3403.19			0.0994357
$186$		0			0
$187$	2923.18			0.0835934
$188$	21699.2			0.613943
$189$		0			0
$190$			3657.26i			0.101309i
$191$		−	46954.1i		−	1.28708i	−0.765411	−	0.643542i	$-0.777464\pi$
$191$		−	46954.1i		−	1.28708i	0.765411	−	0.643542i	$-0.222536\pi$
$192$		0			0
$193$	18967.8			0.509215			0.254608	−	0.967044i	$-0.418054\pi$
$193$	18967.8			0.509215			0.254608	+	0.967044i	$0.418054\pi$
$194$			62718.1i			1.66644i
$195$		0			0
$196$	−64525.7			−1.67966
$197$	77177.1			1.98864			0.994320	−	0.106435i	$-0.0339435\pi$
$197$	77177.1			1.98864			0.994320	+	0.106435i	$0.0339435\pi$
$198$		0			0
$199$		−	2250.86i		−	0.0568385i	−0.999596	−	0.0284192i	$-0.990953\pi$
$199$		−	2250.86i		−	0.0568385i	0.999596	−	0.0284192i	$-0.00904734\pi$
$200$			55214.8i			1.38037i
$201$		0			0
$202$		−	91239.7i		−	2.23605i
$203$	15829.8			0.384135
$204$		0			0
$205$		−	2508.64i		−	0.0596939i
$206$			37159.8i			0.875667i
$207$		0			0
$208$	12690.8			0.293335
$209$			3744.37i			0.0857208i
$210$		0			0
$211$			64198.9i			1.44199i	0.692939	+	0.720996i	$0.256315\pi$
$211$			64198.9i			1.44199i	−0.692939	+	0.720996i	$0.743685\pi$
$212$	2899.57			0.0645152
$213$		0			0
$214$		−	16373.2i		−	0.357526i
$215$	−1502.77	−	2292.32i	−0.0325099	−	0.0495906i
$216$		0			0
$217$			12350.8i			0.262285i
$218$		−	21752.7i		−	0.457720i
$219$		0			0
$220$		−	441.459i		−	0.00912105i
$221$	28198.3			0.577349
$222$		0			0
$223$		−	53407.6i		−	1.07397i	−0.843591	−	0.536987i	$-0.819563\pi$
$223$		−	53407.6i		−	1.07397i	0.843591	−	0.536987i	$-0.180437\pi$
$224$	−7610.96			−0.151685
$225$		0			0
$226$	98688.7			1.93219
$227$		−	43494.8i		−	0.844084i	−0.906576	−	0.422042i	$-0.861313\pi$
$227$		−	43494.8i		−	0.844084i	0.906576	−	0.422042i	$-0.138687\pi$
$228$		0			0
$229$	67145.4			1.28040			0.640199	−	0.768209i	$-0.278852\pi$
$229$	67145.4			1.28040			0.640199	+	0.768209i	$0.278852\pi$
$230$	2411.96			0.0455947
$231$		0			0
$232$	101723.			1.88992
$233$			76734.0i			1.41344i	0.707496	+	0.706718i	$0.249825\pi$
$233$			76734.0i			1.41344i	−0.707496	+	0.706718i	$0.750175\pi$
$234$		0			0
$235$		−	1102.06i		−	0.0199558i
$236$	95923.2			1.72226
$237$		0			0
$238$	26571.8			0.469103
$239$	26767.6			0.468613			0.234307	−	0.972163i	$-0.424718\pi$
$239$	26767.6			0.468613			0.234307	+	0.972163i	$0.424718\pi$
$240$		0			0
$241$		−	100445.i		−	1.72939i	−0.502297	−	0.864695i	$-0.667512\pi$
$241$		−	100445.i		−	1.72939i	0.502297	−	0.864695i	$-0.332488\pi$
$242$			97720.5i			1.66861i
$243$		0			0
$244$			94080.7i			1.58023i
$245$			3277.14i			0.0545962i
$246$		0			0
$247$			36119.9i			0.592043i
$248$			79366.5i			1.29043i
$249$		0			0
$250$	12434.6			0.198953
$251$	71737.6			1.13867			0.569337	−	0.822104i	$-0.307200\pi$
$251$	71737.6			1.13867			0.569337	+	0.822104i	$0.307200\pi$
$252$		0			0
$253$	2469.41			0.0385791
$254$		−	181500.i		−	2.81326i
$255$		0			0
$256$	−109129.			−1.66518
$257$		−	7267.88i		−	0.110038i	−0.998485	−	0.0550189i	$-0.982478\pi$
$257$		−	7267.88i		−	0.110038i	0.998485	−	0.0550189i	$-0.0175219\pi$
$258$		0			0
$259$	31672.0			0.472145
$260$		−	4258.51i		−	0.0629958i
$261$		0			0
$262$	115773.			1.68656
$263$			73838.4i			1.06751i	0.845640	+	0.533754i	$0.179219\pi$
$263$			73838.4i			1.06751i	−0.845640	+	0.533754i	$0.820781\pi$
$264$		0			0
$265$		−	147.264i		−	0.00209703i
$266$			34036.6i			0.481041i
$267$		0			0
$268$	−162181.			−2.25804
$269$	−127266.			−1.75877			−0.879384	−	0.476113i	$-0.842045\pi$
$269$	−127266.			−1.75877			−0.879384	+	0.476113i	$0.842045\pi$
$270$		0			0
$271$	44182.9			0.601610			0.300805	−	0.953686i	$-0.402745\pi$
$271$	44182.9			0.601610			0.300805	+	0.953686i	$0.402745\pi$
$272$	36945.5			0.499371
$273$		0			0
$274$	−12641.0			−0.168377
$275$	6354.17			0.0840220
$276$		0			0
$277$			31197.7i			0.406596i	0.979117	+	0.203298i	$0.0651660\pi$
$277$			31197.7i			0.406596i	−0.979117	+	0.203298i	$0.934834\pi$
$278$		−	138941.i		−	1.79780i
$279$		0			0
$280$		−	1813.17i		−	0.0231272i
$281$	140212.			1.77572			0.887859	−	0.460116i	$-0.152192\pi$
$281$	140212.			1.77572			0.887859	+	0.460116i	$0.152192\pi$
$282$		0			0
$283$	2024.17			0.0252740			0.0126370	−	0.999920i	$-0.495977\pi$
$283$	2024.17			0.0252740			0.0126370	+	0.999920i	$0.495977\pi$
$284$			86202.8i			1.06877i
$285$		0			0
$286$		−	6749.91i		−	0.0825212i
$287$		−	23346.8i		−	0.283442i
$288$		0			0
$289$	−1430.31			−0.0171252
$290$		−	11434.0i		−	0.135958i
$291$		0			0
$292$			105614.i			1.23867i
$293$	−75732.9			−0.882164			−0.441082	−	0.897467i	$-0.645405\pi$
$293$	−75732.9			−0.882164			−0.441082	+	0.897467i	$0.645405\pi$
$294$		0			0
$295$		−	4871.76i		−	0.0559811i
$296$	203526.			2.32293
$297$		0			0
$298$	−91768.4			−1.03338
$299$	23821.0			0.266452
$300$		0			0
$301$	−13985.6	−	21333.7i	−0.154365	−	0.235468i
$302$	−281464.			−3.08609
$303$		0			0
$304$			47324.4i			0.512080i
$305$	4778.18			0.0513644
$306$		0			0
$307$	−158838.			−1.68530			−0.842652	−	0.538458i	$-0.819007\pi$
$307$	−158838.			−1.68530			−0.842652	+	0.538458i	$0.819007\pi$
$308$		−	4108.47i		−	0.0433090i
$309$		0			0
$310$	8921.08			0.0928312
$311$	49344.1			0.510169			0.255085	−	0.966919i	$-0.417897\pi$
$311$	49344.1			0.510169			0.255085	+	0.966919i	$0.417897\pi$
$312$		0			0
$313$			66350.6i			0.677261i	0.940919	+	0.338631i	$0.109964\pi$
$313$			66350.6i			0.677261i	−0.940919	+	0.338631i	$0.890036\pi$
$314$	−277955.			−2.81914
$315$		0			0
$316$	−69490.0			−0.695903
$317$	−15717.4			−0.156409			−0.0782046	−	0.996937i	$-0.524919\pi$
$317$	−15717.4			−0.156409			−0.0782046	+	0.996937i	$0.524919\pi$
$318$		0			0
$319$		−	11706.4i		−	0.115038i
$320$			8555.96i			0.0835543i
$321$		0			0
$322$	22447.1			0.216495
$323$			105152.i			1.00789i
$324$		0			0
$325$	61295.2			0.580309
$326$	41081.6			0.386556
$327$		0			0
$328$		−	150028.i		−	1.39452i
$329$		−	10256.4i		−	0.0947553i
$330$		0			0
$331$			85452.6i			0.779954i	0.920825	+	0.389977i	$0.127517\pi$
$331$			85452.6i			0.779954i	−0.920825	+	0.389977i	$0.872483\pi$
$332$	−187620.			−1.70217
$333$		0			0
$334$			235218.i			2.10852i
$335$			8236.87i			0.0733960i
$336$		0			0
$337$	−146925.			−1.29371			−0.646855	−	0.762613i	$-0.723916\pi$
$337$	−146925.			−1.29371			−0.646855	+	0.762613i	$0.723916\pi$
$338$			126881.i			1.11061i
$339$		0			0
$340$		−	12397.4i		−	0.107244i
$341$	9133.57			0.0785474
$342$		0			0
$343$			63623.7i			0.540793i
$344$	−89872.4	−	137091.i	−0.759468	−	1.15849i
$345$		0			0
$346$		−	30903.1i		−	0.258137i
$347$		−	78623.4i		−	0.652969i	−0.945203	−	0.326485i	$-0.894136\pi$
$347$		−	78623.4i		−	0.652969i	0.945203	−	0.326485i	$-0.105864\pi$
$348$		0			0
$349$			165981.i			1.36273i	0.731945	+	0.681363i	$0.238613\pi$
$349$			165981.i			1.36273i	−0.731945	+	0.681363i	$0.761387\pi$
$350$	57759.7			0.471508
$351$		0			0
$352$			5628.42i			0.0454257i
$353$	−53106.9			−0.426188			−0.213094	−	0.977032i	$-0.568354\pi$
$353$	−53106.9			−0.426188			−0.213094	+	0.977032i	$0.568354\pi$
$354$		0			0
$355$	4378.07			0.0347397
$356$			96127.9i			0.758490i
$357$		0			0
$358$	−176897.			−1.38024
$359$	120101.			0.931879			0.465939	−	0.884817i	$-0.345716\pi$
$359$	120101.			0.931879			0.465939	+	0.884817i	$0.345716\pi$
$360$		0			0
$361$	−4370.92			−0.0335396
$362$		−	323564.i		−	2.46912i
$363$		0			0
$364$		−	39632.1i		−	0.299120i
$365$	5363.91			0.0402620
$366$		0			0
$367$	−37702.0			−0.279919			−0.139959	−	0.990157i	$-0.544697\pi$
$367$	−37702.0			−0.279919			−0.139959	+	0.990157i	$0.544697\pi$
$368$	31210.4			0.230464
$369$		0			0
$370$		−	22877.0i		−	0.167107i
$371$		−	1370.52i		−	0.00995720i
$372$		0			0
$373$			139388.i			1.00186i	0.865488	+	0.500930i	$0.167009\pi$
$373$			139388.i			1.00186i	−0.865488	+	0.500930i	$0.832991\pi$
$374$		−	19650.3i		−	0.140484i
$375$		0			0
$376$		−	65908.1i		−	0.466190i
$377$		−	112925.i		−	0.794526i
$378$		0			0
$379$	−88126.5			−0.613519			−0.306759	−	0.951787i	$-0.599245\pi$
$379$	−88126.5			−0.613519			−0.306759	+	0.951787i	$0.599245\pi$
$380$	15880.1			0.109973
$381$		0			0
$382$	−315636.			−2.16302
$383$			250818.i			1.70986i	0.518743	+	0.854930i	$0.326400\pi$
$383$			250818.i			1.70986i	−0.518743	+	0.854930i	$0.673600\pi$
$384$		0			0
$385$	−208.661			−0.00140773
$386$		−	127506.i		−	0.855766i
$387$		0			0
$388$	272326.			1.80895
$389$			34015.7i			0.224792i	0.993664	+	0.112396i	$0.0358524\pi$
$389$			34015.7i			0.224792i	−0.993664	+	0.112396i	$0.964148\pi$
$390$		0			0
$391$	69347.7			0.453605
$392$			195987.i			1.27543i
$393$		0			0
$394$		−	518802.i		−	3.34202i
$395$			3529.26i			0.0226199i
$396$		0			0
$397$	206559.			1.31058			0.655291	−	0.755377i	$-0.272546\pi$
$397$	206559.			1.31058			0.655291	+	0.755377i	$0.272546\pi$
$398$	−15130.8			−0.0955203
$399$		0			0
$400$	80309.1			0.501932
$401$	−85162.0			−0.529611			−0.264806	−	0.964302i	$-0.585308\pi$
$401$	−85162.0			−0.529611			−0.264806	+	0.964302i	$0.585308\pi$
$402$		0			0
$403$	88106.5			0.542498
$404$	−396169.			−2.42727
$405$		0			0
$406$		−	106412.i		−	0.645561i
$407$		−	23421.9i		−	0.141395i
$408$		0			0
$409$		−	216421.i		−	1.29375i	−0.762594	−	0.646877i	$-0.776075\pi$
$409$		−	216421.i		−	1.29375i	0.762594	−	0.646877i	$-0.223925\pi$
$410$	−16863.6			−0.100319
$411$		0			0
$412$	161350.			0.950552
$413$		−	45339.3i		−	0.265812i
$414$		0			0
$415$			9528.88i			0.0553281i
$416$			54294.3i			0.313738i
$417$		0			0
$418$	25170.5			0.144059
$419$			195727.i			1.11487i	0.830222	+	0.557433i	$0.188214\pi$
$419$			195727.i			1.11487i	−0.830222	+	0.557433i	$0.811786\pi$
$420$		0			0
$421$			216454.i			1.22124i	0.791924	+	0.610620i	$0.209080\pi$
$421$			216454.i			1.22124i	−0.791924	+	0.610620i	$0.790920\pi$
$422$	431560.			2.42335
$423$		0			0
$424$		−	8807.02i		−	0.0489889i
$425$	178442.			0.987915
$426$		0			0
$427$	44468.4			0.243891
$428$	−71093.8			−0.388100
$429$		0			0
$430$	−15409.5	+	10102.0i	−0.0833398	+	0.0546348i
$431$	219971.			1.18416			0.592082	−	0.805878i	$-0.298306\pi$
$431$	219971.			1.18416			0.592082	+	0.805878i	$0.298306\pi$
$432$		0			0
$433$		−	141123.i		−	0.752699i	−0.926478	−	0.376349i	$-0.877179\pi$
$433$		−	141123.i		−	0.752699i	0.926478	−	0.376349i	$-0.122821\pi$
$434$	83024.6			0.440786
$435$		0			0
$436$	−94451.7			−0.496863
$437$			88829.2i			0.465150i
$438$		0			0
$439$	−18660.8			−0.0968279			−0.0484139	−	0.998827i	$-0.515417\pi$
$439$	−18660.8			−0.0968279			−0.0484139	+	0.998827i	$0.515417\pi$
$440$	−1340.87			−0.00692596
$441$		0			0
$442$		−	189556.i		−	0.970268i
$443$	−21366.2			−0.108873			−0.0544366	−	0.998517i	$-0.517336\pi$
$443$	−21366.2			−0.108873			−0.0544366	+	0.998517i	$0.517336\pi$
$444$		0			0
$445$	4882.15			0.0246542
$446$	−359018.			−1.80487
$447$		0			0
$448$			79626.7i			0.396737i
$449$			101517.i			0.503553i	0.967785	+	0.251777i	$0.0810148\pi$
$449$			101517.i			0.503553i	−0.967785	+	0.251777i	$0.918985\pi$
$450$		0			0
$451$	−17265.3			−0.0848831
$452$		−	428513.i		−	2.09743i
$453$		0			0
$454$	−292382.			−1.41853
$455$	−2012.84			−0.00972269
$456$		0			0
$457$			308003.i			1.47477i	0.675475	+	0.737383i	$0.263939\pi$
$457$			308003.i			1.47477i	−0.675475	+	0.737383i	$0.736061\pi$
$458$		−	451367.i		−	2.15178i
$459$		0			0
$460$		−	10472.9i		−	0.0494938i
$461$	−156343.			−0.735660			−0.367830	−	0.929893i	$-0.619899\pi$
$461$	−156343.			−0.735660			−0.367830	+	0.929893i	$0.619899\pi$
$462$		0			0
$463$			203393.i			0.948800i	0.880309	+	0.474400i	$0.157335\pi$
$463$			203393.i			0.948800i	−0.880309	+	0.474400i	$0.842665\pi$
$464$		−	147955.i		−	0.687216i
$465$		0			0
$466$	515824.			2.37536
$467$		−	208975.i		−	0.958209i	−0.877758	−	0.479104i	$-0.840962\pi$
$467$		−	208975.i		−	0.958209i	0.877758	−	0.479104i	$-0.159038\pi$
$468$		0			0
$469$			76657.0i			0.348503i
$470$	−7408.31			−0.0335369
$471$		0			0
$472$		−	291353.i		−	1.30778i
$473$	−15776.6	+	10342.6i	−0.0705164	+	0.0462282i
$474$		0			0
$475$			228571.i			1.01306i
$476$		−	115377.i		−	0.509219i
$477$		0			0
$478$		−	179938.i		−	0.787531i
$479$	−98547.6			−0.429512			−0.214756	−	0.976668i	$-0.568896\pi$
$479$	−98547.6			−0.429512			−0.214756	+	0.976668i	$0.568896\pi$
$480$		0			0
$481$		−	225938.i		−	0.976562i
$482$	−675212.			−2.90634
$483$		0			0
$484$	424309.			1.81130
$485$		−	13830.9i		−	0.0587987i
$486$		0			0
$487$	−319902.			−1.34884			−0.674418	−	0.738350i	$-0.735605\pi$
$487$	−319902.			−1.34884			−0.674418	+	0.738350i	$0.735605\pi$
$488$	285756.			1.19993
$489$		0			0
$490$	22029.7			0.0917520
$491$			454263.i			1.88428i	0.335226	+	0.942138i	$0.391187\pi$
$491$			454263.i			1.88428i	−0.335226	+	0.942138i	$0.608813\pi$
$492$		0			0
$493$		−	328747.i		−	1.35260i
$494$	242806.			0.994962
$495$		0			0
$496$	115437.			0.469227
$497$	40744.8			0.164953
$498$		0			0
$499$		−	336510.i		−	1.35144i	−0.737158	−	0.675720i	$-0.763833\pi$
$499$		−	336510.i		−	1.35144i	0.737158	−	0.675720i	$-0.236167\pi$
$500$		−	53991.8i		−	0.215967i
$501$		0			0
$502$		−	482237.i		−	1.91361i
$503$		−	313054.i		−	1.23732i	−0.785658	−	0.618661i	$-0.787675\pi$
$503$		−	313054.i		−	1.23732i	0.785658	−	0.618661i	$-0.212325\pi$
$504$		0			0
$505$			20120.7i			0.0788968i
$506$		−	16599.9i		−	0.0648344i
$507$		0			0
$508$	−788086.			−3.05384
$509$	−406043.			−1.56724			−0.783621	−	0.621239i	$-0.786630\pi$
$509$	−406043.			−1.56724			−0.783621	+	0.621239i	$0.786630\pi$
$510$		0			0
$511$	49919.6			0.191174
$512$			254048.i			0.969114i
$513$		0			0
$514$	−48856.4			−0.184925
$515$		−	8194.68i		−	0.0308971i
$516$		0			0
$517$	−7584.76			−0.0283766
$518$		−	212906.i		−	0.793467i
$519$		0			0
$520$	−12934.6			−0.0478351
$521$			221529.i			0.816122i	0.912955	+	0.408061i	$0.133795\pi$
$521$			221529.i			0.816122i	−0.912955	+	0.408061i	$0.866205\pi$
$522$		0			0
$523$		−	146666.i		−	0.536199i	−0.963391	−	0.268099i	$-0.913604\pi$
$523$		−	146666.i		−	0.536199i	0.963391	−	0.268099i	$-0.0863955\pi$
$524$		−	502692.i		−	1.83080i
$525$		0			0
$526$	496359.			1.79401
$527$	256495.			0.923545
$528$		0			0
$529$	−221258.			−0.790657
$530$	−989.940			−0.00352417
$531$		0			0
$532$	147789.			0.522179
$533$	−166549.			−0.586257
$534$		0			0
$535$			3610.71i			0.0126150i
$536$			492601.i			1.71461i
$537$		0			0
$538$			855513.i			2.95571i
$539$	22554.4			0.0776342
$540$		0			0
$541$	38164.1			0.130395			0.0651974	−	0.997872i	$-0.479232\pi$
$541$	38164.1			0.130395			0.0651974	+	0.997872i	$0.479232\pi$
$542$		−	297007.i		−	1.01104i
$543$		0			0
$544$			158061.i			0.534106i
$545$			4797.02i			0.0161502i
$546$		0			0
$547$	105758.			0.353459			0.176730	−	0.984259i	$-0.443448\pi$
$547$	105758.			0.353459			0.176730	+	0.984259i	$0.443448\pi$
$548$			54888.3i			0.182776i
$549$		0			0
$550$		−	42714.2i		−	0.141204i
$551$	421100.			1.38702
$552$		0			0
$553$			32845.3i			0.107405i
$554$	209718.			0.683308
$555$		0			0
$556$	−603293.			−1.95155
$557$	−280135.			−0.902936			−0.451468	−	0.892287i	$-0.649100\pi$
$557$	−280135.			−0.902936			−0.451468	+	0.892287i	$0.649100\pi$
$558$		0			0
$559$	−152188.	+	99769.3i	−0.487031	+	0.319281i
$560$	−2637.23			−0.00840953
$561$		0			0
$562$		−	942541.i		−	2.98420i
$563$	403332.			1.27247			0.636233	−	0.771497i	$-0.280492\pi$
$563$	403332.			1.27247			0.636233	+	0.771497i	$0.280492\pi$
$564$		0			0
$565$	−21763.3			−0.0681756
$566$		−	13606.9i		−	0.0424744i
$567$		0			0
$568$	261828.			0.811559
$569$	72865.2			0.225059			0.112529	−	0.993648i	$-0.464105\pi$
$569$	72865.2			0.225059			0.112529	+	0.993648i	$0.464105\pi$
$570$		0			0
$571$			280074.i			0.859015i	0.903063	+	0.429507i	$0.141313\pi$
$571$			280074.i			0.859015i	−0.903063	+	0.429507i	$0.858687\pi$
$572$	−29308.6			−0.0895782
$573$		0			0
$574$	−156943.			−0.476340
$575$	150742.			0.455931
$576$		0			0
$577$		−	205999.i		−	0.618748i	−0.950940	−	0.309374i	$-0.899881\pi$
$577$		−	205999.i		−	0.618748i	0.950940	−	0.309374i	$-0.100119\pi$
$578$			9614.89i			0.0287799i
$579$		0			0
$580$	−49647.4			−0.147584
$581$			88681.2i			0.262712i
$582$		0			0
$583$	−1013.52			−0.00298191
$584$	320786.			0.940566
$585$		0			0
$586$			509094.i			1.48253i
$587$		−	466185.i		−	1.35295i	−0.736464	−	0.676476i	$-0.763506\pi$
$587$		−	466185.i		−	1.35295i	0.736464	−	0.676476i	$-0.236494\pi$
$588$		0			0
$589$			328551.i			0.947049i
$590$	−32749.1			−0.0940795
$591$		0			0
$592$		−	296025.i		−	0.844666i
$593$			284311.i			0.808507i	0.914647	+	0.404253i	$0.132469\pi$
$593$			284311.i			0.808507i	−0.914647	+	0.404253i	$0.867531\pi$
$594$		0			0
$595$	−5859.76			−0.0165518
$596$			398465.i			1.12175i
$597$		0			0
$598$		−	160131.i		−	0.447787i
$599$	−438880.			−1.22318			−0.611592	−	0.791173i	$-0.709471\pi$
$599$	−438880.			−1.22318			−0.611592	+	0.791173i	$0.709471\pi$
$600$		0			0
$601$		−	52157.2i		−	0.144399i	−0.997390	−	0.0721997i	$-0.976998\pi$
$601$		−	52157.2i		−	0.144399i	0.997390	−	0.0721997i	$-0.0230019\pi$
$602$	−143410.	+	94014.7i	−0.395718	+	0.259420i
$603$		0			0
$604$			1.22214e6i			3.35001i
$605$		−	21549.8i		−	0.0588753i
$606$		0			0
$607$			328500.i			0.891575i	0.895139	+	0.445788i	$0.147076\pi$
$607$			328500.i			0.891575i	−0.895139	+	0.445788i	$0.852924\pi$
$608$	−202465.			−0.547699
$609$		0			0
$610$		−	32120.0i		−	0.0863209i
$611$	−73166.1			−0.195987
$612$		0			0
$613$	−194249.			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$613$	−194249.			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$614$			1.06775e6i			2.83225i
$615$		0			0
$616$	−12478.9			−0.0328862
$617$	203384.			0.534253			0.267126	−	0.963662i	$-0.413926\pi$
$617$	203384.			0.534253			0.267126	+	0.963662i	$0.413926\pi$
$618$		0			0
$619$	504248.			1.31602			0.658011	−	0.753008i	$-0.271398\pi$
$619$	504248.			1.31602			0.658011	+	0.753008i	$0.271398\pi$
$620$		−	38735.9i		−	0.100770i
$621$		0			0
$622$		−	331702.i		−	0.857369i
$623$	45436.1			0.117064
$624$		0			0
$625$	386509.			0.989464
$626$	446024.			1.13818
$627$		0			0
$628$			1.20690e6i			3.06022i
$629$		−	657750.i		−	1.66249i
$630$		0			0
$631$			612948.i			1.53945i	0.638376	+	0.769724i	$0.279606\pi$
$631$			612948.i			1.53945i	−0.638376	+	0.769724i	$0.720394\pi$
$632$			211066.i			0.528425i
$633$		0			0
$634$			105656.i			0.262855i
$635$			40025.4i			0.0992631i
$636$		0			0
$637$	217570.			0.536191
$638$	−78693.0			−0.193328
$639$		0			0
$640$	44430.3			0.108472
$641$			571747.i			1.39151i	0.718277	+	0.695757i	$0.244931\pi$
$641$			571747.i			1.39151i	−0.718277	+	0.695757i	$0.755069\pi$
$642$		0			0
$643$	−530650.			−1.28347			−0.641736	−	0.766926i	$-0.721785\pi$
$643$	−530650.			−1.28347			−0.641736	+	0.766926i	$0.721785\pi$
$644$		−	97466.7i		−	0.235009i
$645$		0			0
$646$	706856.			1.69382
$647$			481379.i			1.14995i	0.818171	+	0.574975i	$0.194988\pi$
$647$			481379.i			1.14995i	−0.818171	+	0.574975i	$0.805012\pi$
$648$		0			0
$649$	−33529.1			−0.0796036
$650$		−	412040.i		−	0.975244i
$651$		0			0
$652$		−	178379.i		−	0.419613i
$653$			293363.i			0.687984i	0.938973	+	0.343992i	$0.111779\pi$
$653$			293363.i			0.687984i	−0.938973	+	0.343992i	$0.888221\pi$
$654$		0			0
$655$	−25530.8			−0.0595088
$656$	−218213.			−0.507076
$657$		0			0
$658$	−68945.9			−0.159242
$659$	−657906.			−1.51493			−0.757466	−	0.652875i	$-0.773563\pi$
$659$	−657906.			−1.51493			−0.757466	+	0.652875i	$0.773563\pi$
$660$		0			0
$661$	−267844.			−0.613026			−0.306513	−	0.951866i	$-0.599162\pi$
$661$	−267844.			−0.613026			−0.306513	+	0.951866i	$0.599162\pi$
$662$	574432.			1.31076
$663$		0			0
$664$			569869.i			1.29253i
$665$		−	7505.92i		−	0.0169731i
$666$		0			0
$667$		−	277715.i		−	0.624235i
$668$	1.02133e6			2.28883
$669$		0			0
$670$	55370.1			0.123346
$671$		−	32885.0i		−	0.0730388i
$672$		0			0
$673$			48089.2i			0.106174i	0.998590	+	0.0530869i	$0.0169060\pi$
$673$			48089.2i			0.106174i	−0.998590	+	0.0530869i	$0.983094\pi$
$674$			987667.i			2.17416i
$675$		0			0
$676$	550926.			1.20559
$677$			84879.3i			0.185193i	0.995704	+	0.0925964i	$0.0295166\pi$
$677$			84879.3i			0.185193i	−0.995704	+	0.0925964i	$0.970483\pi$
$678$		0			0
$679$		−	128718.i		−	0.279191i
$680$	−37655.1			−0.0814341
$681$		0			0
$682$		−	61397.9i		−	0.132003i
$683$	707010.			1.51560			0.757799	−	0.652488i	$-0.226275\pi$
$683$	707010.			1.51560			0.757799	+	0.652488i	$0.226275\pi$
$684$		0			0
$685$	2787.67			0.00594101
$686$	427693.			0.908833
$687$		0			0
$688$	−199397.	+	130718.i	−0.421252	+	0.276159i
$689$	−9776.87			−0.0205950
$690$		0			0
$691$		−	316021.i		−	0.661851i	−0.943657	−	0.330925i	$-0.892639\pi$
$691$		−	316021.i		−	0.661851i	0.943657	−	0.330925i	$-0.107361\pi$
$692$	−134183.			−0.280212
$693$		0			0
$694$	−528525.			−1.09735
$695$			30640.1i			0.0634338i
$696$		0			0
$697$	−484856.			−0.998039
$698$	1.11577e6			2.29014
$699$		0			0
$700$		−	250797.i		−	0.511830i
$701$	−790342.			−1.60835			−0.804173	−	0.594396i	$-0.797391\pi$
$701$	−790342.			−1.60835			−0.804173	+	0.594396i	$0.797391\pi$
$702$		0			0
$703$	842529.			1.70480
$704$	58885.1			0.118812
$705$		0			0
$706$			356997.i			0.716234i
$707$			187254.i			0.374622i
$708$		0			0
$709$	218246.			0.434165			0.217082	−	0.976153i	$-0.430346\pi$
$709$	218246.			0.434165			0.217082	+	0.976153i	$0.430346\pi$
$710$		−	29430.4i		−	0.0583821i
$711$		0			0
$712$	291974.			0.575950
$713$	216679.			0.426224
$714$		0			0
$715$			1488.52i			0.00291168i
$716$			768101.i			1.49828i
$717$		0			0
$718$		−	807350.i		−	1.56608i
$719$	−733807.			−1.41946			−0.709732	−	0.704472i	$-0.751184\pi$
$719$	−733807.			−1.41946			−0.709732	+	0.704472i	$0.751184\pi$
$720$		0			0
$721$		−	76264.3i		−	0.146707i
$722$			29382.3i			0.0563653i
$723$		0			0
$724$	−1.40494e6			−2.68027
$725$		−	714603.i		−	1.35953i
$726$		0			0
$727$		−	250259.i		−	0.473501i	−0.971570	−	0.236750i	$-0.923918\pi$
$727$		−	250259.i		−	0.473501i	0.971570	−	0.236750i	$-0.0760824\pi$
$728$	−120377.			−0.227133
$729$		0			0
$730$		−	36057.5i		−	0.0676627i
$731$	−443048.	+	290448.i	−0.829118	+	0.543542i
$732$		0			0
$733$		−	610120.i		−	1.13555i	−0.823183	−	0.567776i	$-0.807804\pi$
$733$		−	610120.i		−	1.13555i	0.823183	−	0.567776i	$-0.192196\pi$
$734$			253441.i			0.470419i
$735$		0			0
$736$			133525.i			0.246495i
$737$	56689.0			0.104367
$738$		0			0
$739$		−	67964.4i		−	0.124449i	−0.998062	−	0.0622246i	$-0.980180\pi$
$739$		−	67964.4i		−	0.124449i	0.998062	−	0.0622246i	$-0.0198195\pi$
$740$	−99333.5			−0.181398
$741$		0			0
$742$	−9212.95			−0.0167337
$743$		−	707585.i		−	1.28174i	−0.767648	−	0.640871i	$-0.778573\pi$
$743$		−	707585.i		−	1.28174i	0.767648	−	0.640871i	$-0.221427\pi$
$744$		0			0
$745$	20237.3			0.0364619
$746$	936998.			1.68368
$747$		0			0
$748$	−85322.8			−0.152497
$749$			33603.4i			0.0598989i
$750$		0			0
$751$		−	613506.i		−	1.08778i	−0.839158	−	0.543888i	$-0.816952\pi$
$751$		−	613506.i		−	1.08778i	0.839158	−	0.543888i	$-0.183048\pi$
$752$	−95862.4			−0.169517
$753$		0			0
$754$	−759109.			−1.33525
$755$	62069.9			0.108890
$756$		0			0
$757$		−	132879.i		−	0.231880i	−0.993256	−	0.115940i	$-0.963012\pi$
$757$		−	132879.i		−	0.231880i	0.993256	−	0.115940i	$-0.0369881\pi$
$758$			592406.i			1.03105i
$759$		0			0
$760$		−	48233.4i		−	0.0835066i
$761$		−	790123.i		−	1.36435i	−0.731190	−	0.682174i	$-0.761035\pi$
$761$		−	790123.i		−	1.36435i	0.731190	−	0.682174i	$-0.238965\pi$
$762$		0			0
$763$			44643.8i			0.0766853i
$764$			1.37051e6i			2.34799i
$765$		0			0
$766$	1.68605e6			2.87352
$767$	−323437.			−0.549793
$768$		0			0
$769$	−204245.			−0.345382			−0.172691	−	0.984976i	$-0.555246\pi$
$769$	−204245.			−0.345382			−0.172691	+	0.984976i	$0.555246\pi$
$770$			1402.67i			0.00236577i
$771$		0			0
$772$	−553638.			−0.928948
$773$		−	824077.i		−	1.37914i	−0.724218	−	0.689571i	$-0.757799\pi$
$773$		−	824077.i		−	1.37914i	0.724218	−	0.689571i	$-0.242201\pi$
$774$		0			0
$775$	557548.			0.928280
$776$		−	827151.i		−	1.37360i
$777$		0			0
$778$	228661.			0.377775
$779$		−	621065.i		−	1.02344i
$780$		0			0
$781$		−	30131.4i		−	0.0493989i
$782$		−	466171.i		−	0.762310i
$783$		0			0
$784$	285061.			0.463772
$785$	61296.2			0.0994705
$786$		0			0
$787$	347326.			0.560774			0.280387	−	0.959887i	$-0.409537\pi$
$787$	347326.			0.560774			0.280387	+	0.959887i	$0.409537\pi$
$788$	−2.25268e6			−3.62782
$789$		0			0
$790$	23724.5			0.0380140
$791$	−202542.			−0.323715
$792$		0			0
$793$		−	317224.i		−	0.504452i
$794$		−	1.38854e6i		−	2.20251i
$795$		0			0
$796$			65699.0i			0.103689i
$797$	550267.			0.866278			0.433139	−	0.901327i	$-0.357406\pi$
$797$	550267.			0.866278			0.433139	+	0.901327i	$0.357406\pi$
$798$		0			0
$799$	−213001.			−0.333647
$800$			343581.i			0.536845i
$801$		0			0
$802$			572479.i			0.890042i
$803$		−	36916.3i		−	0.0572515i
$804$		0			0
$805$	−4950.15			−0.00763882
$806$		−	592273.i		−	0.911699i
$807$		0			0
$808$			1.20331e6i			1.84312i
$809$	237489.			0.362866			0.181433	−	0.983403i	$-0.441926\pi$
$809$	237489.			0.362866			0.181433	+	0.983403i	$0.441926\pi$
$810$		0			0
$811$			1.00555e6i			1.52884i	0.644717	+	0.764422i	$0.276975\pi$
$811$			1.00555e6i			1.52884i	−0.644717	+	0.764422i	$0.723025\pi$
$812$	−462047.			−0.700768
$813$		0			0
$814$	−157447.			−0.237622
$815$	−9059.54			−0.0136393
$816$		0			0
$817$	−372042.	−	567512.i	−0.557375	−	0.850219i
$818$	−1.45483e6			−2.17423
$819$		0			0
$820$			73223.1i			0.108898i
$821$	259037.			0.384305			0.192152	−	0.981365i	$-0.438453\pi$
$821$	259037.			0.384305			0.192152	+	0.981365i	$0.438453\pi$
$822$		0			0
$823$	−70048.9			−0.103419			−0.0517097	−	0.998662i	$-0.516467\pi$
$823$	−70048.9			−0.103419			−0.0517097	+	0.998662i	$0.516467\pi$
$824$		−	490078.i		−	0.721790i
$825$		0			0
$826$	−304781.			−0.446713
$827$	771588.			1.12817			0.564085	−	0.825717i	$-0.309229\pi$
$827$	771588.			1.12817			0.564085	+	0.825717i	$0.309229\pi$
$828$		0			0
$829$			131193.i			0.190898i	0.995434	+	0.0954489i	$0.0304286\pi$
$829$			131193.i			0.190898i	−0.995434	+	0.0954489i	$0.969571\pi$
$830$	64055.3			0.0929821
$831$		0			0
$832$	568033.			0.820591
$833$	633388.			0.912808
$834$		0			0
$835$		−	51871.4i		−	0.0743970i
$836$		−	109292.i		−	0.156378i
$837$		0			0
$838$	1.31572e6			1.87360
$839$			473423.i			0.672552i	0.941764	+	0.336276i	$0.109167\pi$
$839$			473423.i			0.672552i	−0.941764	+	0.336276i	$0.890833\pi$
$840$		0			0
$841$	−609245.			−0.861390
$842$	1.45505e6			2.05236
$843$		0			0
$844$		−	1.87386e6i		−	2.63059i
$845$		−	27980.5i		−	0.0391869i
$846$		0			0
$847$		−	200555.i		−	0.279555i
$848$	−12809.7			−0.0178134
$849$		0			0
$850$		−	1.19953e6i		−	1.66025i
$851$		−	555647.i		−	0.767255i
$852$		0			0
$853$	−535509.			−0.735985			−0.367992	−	0.929829i	$-0.619955\pi$
$853$	−535509.			−0.735985			−0.367992	+	0.929829i	$0.619955\pi$
$854$		−	298927.i		−	0.409873i
$855$		0			0
$856$			215937.i			0.294699i
$857$	−1.28191e6			−1.74540			−0.872699	−	0.488258i	$-0.837632\pi$
$857$	−1.28191e6			−1.74540			−0.872699	+	0.488258i	$0.837632\pi$
$858$		0			0
$859$		−	683886.i		−	0.926825i	−0.886143	−	0.463412i	$-0.846625\pi$
$859$		−	683886.i		−	0.926825i	0.886143	−	0.463412i	$-0.153375\pi$
$860$	43863.5	+	66909.3i	0.0593070	+	0.0904668i
$861$		0			0
$862$		−	1.47870e6i		−	1.99005i
$863$			423128.i			0.568134i	0.958804	+	0.284067i	$0.0916838\pi$
$863$			423128.i			0.568134i	−0.958804	+	0.284067i	$0.908316\pi$
$864$		0			0
$865$			6814.91i			0.00910810i
$866$	−948660.			−1.26495
$867$		0			0
$868$		−	360499.i		−	0.478480i
$869$	24289.6			0.0321648
$870$		0			0
$871$	546848.			0.720825
$872$			286883.i			0.377287i
$873$		0			0
$874$	597130.			0.781711
$875$	−25519.9			−0.0333321
$876$		0			0
$877$	543060.			0.706071			0.353036	−	0.935610i	$-0.385149\pi$
$877$	543060.			0.706071			0.353036	+	0.935610i	$0.385149\pi$
$878$			125442.i			0.162725i
$879$		0			0
$880$			1950.27i			0.00251843i
$881$	747183.			0.962665			0.481333	−	0.876538i	$-0.340153\pi$
$881$	747183.			0.962665			0.481333	+	0.876538i	$0.340153\pi$
$882$		0			0
$883$	908500.			1.16521			0.582604	−	0.812756i	$-0.302034\pi$
$883$	908500.			1.16521			0.582604	+	0.812756i	$0.302034\pi$
$884$	−823063.			−1.05324
$885$		0			0
$886$			143629.i			0.182968i
$887$		−	283934.i		−	0.360886i	−0.983585	−	0.180443i	$-0.942247\pi$
$887$		−	283934.i		−	0.360886i	0.983585	−	0.180443i	$-0.0577531\pi$
$888$		0			0
$889$			372499.i			0.471326i
$890$		−	32819.0i		−	0.0414328i
$891$		0			0
$892$			1.55888e6i			1.95922i
$893$		−	272838.i		−	0.342138i
$894$		0			0
$895$	39010.4			0.0487006
$896$	413493.			0.515054
$897$		0			0
$898$	682420.			0.846250
$899$		−	1.02718e6i		−	1.27095i
$900$		0			0
$901$	−28462.3			−0.0350607
$902$			116061.i			0.142651i
$903$		0			0
$904$	−1.30155e6			−1.59266
$905$			71354.0i			0.0871206i
$906$		0			0
$907$	−329783.			−0.400880			−0.200440	−	0.979706i	$-0.564237\pi$
$907$	−329783.			−0.400880			−0.200440	+	0.979706i	$0.564237\pi$
$908$			1.26954e6i			1.53984i
$909$		0			0
$910$			13530.8i			0.0163395i
$911$			1.20039e6i			1.44639i	0.690642	+	0.723197i	$0.257328\pi$
$911$			1.20039e6i			1.44639i	−0.690642	+	0.723197i	$0.742672\pi$
$912$		0			0
$913$	65581.0			0.0786750
$914$	2.07047e6			2.47843
$915$		0			0
$916$	−1.95986e6			−2.33580
$917$	−237604.			−0.282563
$918$		0			0
$919$	−775338.			−0.918037			−0.459018	−	0.888427i	$-0.651799\pi$
$919$	−775338.			−0.918037			−0.459018	+	0.888427i	$0.651799\pi$
$920$	−31809.9			−0.0375826
$921$		0			0
$922$			1.05098e6i			1.23632i
$923$		−	290661.i		−	0.341180i
$924$		0			0
$925$		−	1.42976e6i		−	1.67102i
$926$	1.36726e6			1.59451
$927$		0			0
$928$	632985.			0.735017
$929$			763445.i			0.884599i	0.896867	+	0.442300i	$0.145837\pi$
$929$			763445.i			0.884599i	−0.896867	+	0.442300i	$0.854163\pi$
$930$		0			0
$931$			811322.i			0.936039i
$932$		−	2.23974e6i		−	2.57849i
$933$		0			0
$934$	−1.40478e6			−1.61033
$935$			4333.38i			0.00495683i
$936$		0			0
$937$		−	1.72023e6i		−	1.95932i	−0.200654	−	0.979662i	$-0.564307\pi$
$937$		−	1.72023e6i		−	1.95932i	0.200654	−	0.979662i	$-0.435693\pi$
$938$	515306.			0.585679
$939$		0			0
$940$			32167.4i			0.0364049i
$941$	−657189.			−0.742183			−0.371092	−	0.928596i	$-0.621016\pi$
$941$	−657189.			−0.742183			−0.371092	+	0.928596i	$0.621016\pi$
$942$		0			0
$943$	−409591.			−0.460604
$944$	−423768.			−0.475537
$945$		0			0
$946$	69525.3	+	106054.i	0.0776891	+	0.118507i
$947$	829200.			0.924612			0.462306	−	0.886721i	$-0.347022\pi$
$947$	829200.			0.924612			0.462306	+	0.886721i	$0.347022\pi$
$948$		0			0
$949$		−	356111.i		−	0.395415i
$950$	1.53651e6			1.70250
$951$		0			0
$952$	−350440.			−0.386669
$953$		−	1.33377e6i		−	1.46858i	−0.678838	−	0.734288i	$-0.737516\pi$
$953$		−	1.33377e6i		−	1.46858i	0.678838	−	0.734288i	$-0.262484\pi$
$954$		0			0
$955$	69605.8			0.0763200
$956$	−781305.			−0.854879
$957$		0			0
$958$			662460.i			0.721819i
$959$	25943.6			0.0282094
$960$		0			0
$961$	−122093.			−0.132204
$962$	−1.51881e6			−1.64117
$963$		0			0
$964$			2.93182e6i			3.15488i
$965$			28118.2i			0.0301949i
$966$		0			0
$967$	788889.			0.843652			0.421826	−	0.906677i	$-0.361389\pi$
$967$	788889.			0.843652			0.421826	+	0.906677i	$0.361389\pi$
$968$		−	1.28878e6i		−	1.37539i
$969$		0			0
$970$	−92974.7			−0.0988146
$971$	1.20251e6			1.27541			0.637704	−	0.770282i	$-0.279885\pi$
$971$	1.20251e6			1.27541			0.637704	+	0.770282i	$0.279885\pi$
$972$		0			0
$973$			285154.i			0.301199i
$974$			2.15046e6i			2.26680i
$975$		0			0
$976$		−	415628.i		−	0.436320i
$977$	−634645.			−0.664878			−0.332439	−	0.943125i	$-0.607872\pi$
$977$	−634645.			−0.664878			−0.332439	+	0.943125i	$0.607872\pi$
$978$		0			0
$979$		−	33600.7i		−	0.0350576i
$980$		−	95654.3i		−	0.0995984i
$981$		0			0
$982$	3.05366e6			3.16663
$983$			1.63591e6i			1.69298i	0.532403	+	0.846491i	$0.321289\pi$
$983$			1.63591e6i			1.69298i	−0.532403	+	0.846491i	$0.678711\pi$
$984$		0			0
$985$			114409.i			0.117920i
$986$	−2.20991e6			−2.27311
$987$		0			0
$988$		−	1.05428e6i		−	1.08005i
$989$	−374273.	+	245361.i	−0.382645	+	0.250849i
$990$		0			0
$991$		−	1.20041e6i		−	1.22231i	−0.791510	−	0.611156i	$-0.790705\pi$
$991$		−	1.20041e6i		−	1.22231i	0.791510	−	0.611156i	$-0.209295\pi$
$992$			493868.i			0.501865i
$993$		0			0
$994$		−	273896.i		−	0.277213i
$995$	3336.72			0.00337034
$996$		0			0
$997$		−	1.25293e6i		−	1.26048i	−0.776401	−	0.630239i	$-0.782957\pi$
$997$		−	1.25293e6i		−	1.26048i	0.776401	−	0.630239i	$-0.217043\pi$
$998$	−2.26210e6			−2.27117
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.5.b.c.343.2		12
3.2	odd	2		43.5.b.b.42.11	yes	12
12.11	even	2		688.5.b.d.257.4		12
43.42	odd	2	inner	387.5.b.c.343.11		12
129.128	even	2		43.5.b.b.42.2	✓	12
516.515	odd	2		688.5.b.d.257.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.2	✓	12	129.128	even	2
43.5.b.b.42.11	yes	12	3.2	odd	2
387.5.b.c.343.2		12	1.1	even	1	trivial
387.5.b.c.343.11		12	43.42	odd	2	inner
688.5.b.d.257.4		12	12.11	even	2
688.5.b.d.257.9		12	516.515	odd	2

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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.72223i q^{2} -29.1884 q^{4} +1.48242i q^{5} +13.7963i q^{7} +88.6554i q^{8} +O(q^{10})\)	\(q-6.72223i q^{2} -29.1884 q^{4} +1.48242i q^{5} +13.7963i q^{7} +88.6554i q^{8} +9.96518 q^{10} +10.2025 q^{11} +98.4183 q^{13} +92.7416 q^{14} +128.948 q^{16} +286.515 q^{17} +367.004i q^{19} -43.2695i q^{20} -68.5838i q^{22} +242.039 q^{23} +622.802 q^{25} -661.591i q^{26} -402.691i q^{28} -1147.40i q^{29} +895.225 q^{31} +551.669i q^{32} -1926.02i q^{34} -20.4519 q^{35} -2295.69i q^{37} +2467.09 q^{38} -131.425 q^{40} -1692.26 q^{41} +(-1546.34 + 1013.73i) q^{43} -297.796 q^{44} -1627.04i q^{46} -743.419 q^{47} +2210.66 q^{49} -4186.62i q^{50} -2872.67 q^{52} -99.3399 q^{53} +15.1245i q^{55} -1223.11 q^{56} -7713.09 q^{58} -3286.35 q^{59} -3223.22i q^{61} -6017.91i q^{62} +5771.61 q^{64} +145.898i q^{65} +5556.36 q^{67} -8362.90 q^{68} +137.482i q^{70} -2953.33i q^{71} -3618.34i q^{73} -15432.2 q^{74} -10712.3i q^{76} +140.757i q^{77} +2380.74 q^{79} +191.155i q^{80} +11375.7i q^{82} +6427.91 q^{83} +424.736i q^{85} +(6814.51 + 10394.8i) q^{86} +904.510i q^{88} -3293.36i q^{89} +1357.80i q^{91} -7064.72 q^{92} +4997.44i q^{94} -544.055 q^{95} -9329.95 q^{97} -14860.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 92 q^{4}+O(q^{10}) \) 12 * q - 92 * q^4	\( 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) \) 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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