LMFDB - Embedded newform 392.6.i.j.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not 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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-115})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 28x^{2} - 29x + 841 $ x^4 - x^3 - 28x^2 - 29x + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$-4.39354 + 3.11396i$ of defining polynomial
Character	$\chi$	$=$	392.361
Dual form			392.6.i.j.177.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.7871 - 18.6838i) q^{3} +(7.36126 + 12.7501i) q^{5} +(-111.223 - 192.643i) q^{9} +O(q^{10})$	\(q+(10.7871 - 18.6838i) q^{3} +(7.36126 + 12.7501i) q^{5} +(-111.223 - 192.643i) q^{9} +(-29.2775 + 50.7101i) q^{11} +1179.39 q^{13} +317.626 q^{15} +(-748.277 + 1296.05i) q^{17} +(-249.419 - 432.006i) q^{19} +(944.670 + 1636.22i) q^{23} +(1454.12 - 2518.62i) q^{25} +443.456 q^{27} +1914.54 q^{29} +(-397.288 + 688.124i) q^{31} +(631.637 + 1094.03i) q^{33} +(-1493.96 - 2587.62i) q^{37} +(12722.2 - 22035.5i) q^{39} +11941.3 q^{41} +9820.19 q^{43} +(1637.48 - 2836.19i) q^{45} +(9817.98 + 17005.2i) q^{47} +(16143.5 + 27961.3i) q^{51} +(9937.48 - 17212.2i) q^{53} -862.077 q^{55} -10762.0 q^{57} +(-17919.3 + 31037.2i) q^{59} +(-24988.0 - 43280.4i) q^{61} +(8681.82 + 15037.4i) q^{65} +(-24088.1 + 41721.8i) q^{67} +40761.0 q^{69} +77179.1 q^{71} +(29833.6 - 51673.4i) q^{73} +(-31371.5 - 54337.1i) q^{75} +(-30371.5 - 52605.1i) q^{79} +(31810.7 - 55097.7i) q^{81} -46134.2 q^{83} -22033.1 q^{85} +(20652.4 - 35770.9i) q^{87} +(-39334.4 - 68129.1i) q^{89} +(8571.17 + 14845.7i) q^{93} +(3672.08 - 6360.22i) q^{95} -43573.3 q^{97} +13025.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} - 82 q^{5} - 222 q^{9}+O(q^{10}) $ 4 * q + 6 * q^3 - 82 * q^5 - 222 * q^9	$ 4 q + 6 q^{3} - 82 q^{5} - 222 q^{9} - 340 q^{11} + 1820 q^{13} + 3648 q^{15} - 3216 q^{17} + 674 q^{19} + 1104 q^{23} - 3322 q^{25} - 6696 q^{27} + 16128 q^{29} + 6212 q^{31} - 3120 q^{33} + 8512 q^{37} + 29640 q^{39} - 2608 q^{41} - 20008 q^{43} + 3318 q^{45} + 12748 q^{47} + 5508 q^{51} + 11220 q^{53} + 52720 q^{55} - 58056 q^{57} + 12018 q^{59} - 102738 q^{61} + 43420 q^{65} - 24136 q^{67} + 105984 q^{69} + 179440 q^{71} + 55588 q^{73} - 159774 q^{75} - 48824 q^{79} + 122562 q^{81} + 71564 q^{83} + 288552 q^{85} - 54468 q^{87} + 18300 q^{89} + 126264 q^{93} + 120784 q^{95} - 139968 q^{97} + 25800 q^{99}+O(q^{100}) $ 4 * q + 6 * q^3 - 82 * q^5 - 222 * q^9 - 340 * q^11 + 1820 * q^13 + 3648 * q^15 - 3216 * q^17 + 674 * q^19 + 1104 * q^23 - 3322 * q^25 - 6696 * q^27 + 16128 * q^29 + 6212 * q^31 - 3120 * q^33 + 8512 * q^37 + 29640 * q^39 - 2608 * q^41 - 20008 * q^43 + 3318 * q^45 + 12748 * q^47 + 5508 * q^51 + 11220 * q^53 + 52720 * q^55 - 58056 * q^57 + 12018 * q^59 - 102738 * q^61 + 43420 * q^65 - 24136 * q^67 + 105984 * q^69 + 179440 * q^71 + 55588 * q^73 - 159774 * q^75 - 48824 * q^79 + 122562 * q^81 + 71564 * q^83 + 288552 * q^85 - 54468 * q^87 + 18300 * q^89 + 126264 * q^93 + 120784 * q^95 - 139968 * q^97 + 25800 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	10.7871	−	18.6838i	0.691992	−	1.19857i	−0.279192	−	0.960235i	$-0.590067\pi$
$3$	10.7871	−	18.6838i	0.691992	−	1.19857i	0.971184	−	0.238330i	$-0.0766000\pi$
$4$		0			0
$5$	7.36126	+	12.7501i	0.131682	+	0.228080i	0.924325	−	0.381606i	$-0.124629\pi$
$5$	7.36126	+	12.7501i	0.131682	+	0.228080i	−0.792643	+	0.609686i	$0.791295\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−111.223	−	192.643i	−0.457706	−	0.792770i
$10$		0			0
$11$	−29.2775	+	50.7101i	−0.0729545	+	0.126361i	−0.900195	−	0.435487i	$-0.856576\pi$
$11$	−29.2775	+	50.7101i	−0.0729545	+	0.126361i	0.827240	+	0.561848i	$0.189909\pi$
$12$		0			0
$13$	1179.39			1.93553			0.967765	−	0.251853i	$-0.0810400\pi$
$13$	1179.39			1.93553			0.967765	+	0.251853i	$0.0810400\pi$
$14$		0			0
$15$	317.626			0.364492
$16$		0			0
$17$	−748.277	+	1296.05i	−0.627972	+	1.08768i	0.359986	+	0.932958i	$0.382781\pi$
$17$	−748.277	+	1296.05i	−0.627972	+	1.08768i	−0.987958	+	0.154722i	$0.950552\pi$
$18$		0			0
$19$	−249.419	−	432.006i	−0.158506	−	0.274540i	0.775824	−	0.630949i	$-0.217334\pi$
$19$	−249.419	−	432.006i	−0.158506	−	0.274540i	−0.934330	+	0.356409i	$0.884001\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	944.670	+	1636.22i	0.372358	+	0.644943i	0.989928	−	0.141573i	$-0.0452160\pi$
$23$	944.670	+	1636.22i	0.372358	+	0.644943i	−0.617570	+	0.786516i	$0.711883\pi$
$24$		0			0
$25$	1454.12	−	2518.62i	0.465320	−	0.805957i
$26$		0			0
$27$	443.456			0.117069
$28$		0			0
$29$	1914.54			0.422737			0.211369	−	0.977406i	$-0.432208\pi$
$29$	1914.54			0.422737			0.211369	+	0.977406i	$0.432208\pi$
$30$		0			0
$31$	−397.288	+	688.124i	−0.0742509	+	0.128606i	−0.900760	−	0.434317i	$-0.856990\pi$
$31$	−397.288	+	688.124i	−0.0742509	+	0.128606i	0.826509	+	0.562923i	$0.190323\pi$
$32$		0			0
$33$	631.637	+	1094.03i	0.100968	+	0.174881i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1493.96	−	2587.62i	−0.179406	−	0.310740i	0.762272	−	0.647257i	$-0.224084\pi$
$37$	−1493.96	−	2587.62i	−0.179406	−	0.310740i	−0.941677	+	0.336518i	$0.890751\pi$
$38$		0			0
$39$	12722.2	−	22035.5i	1.33937	−	2.31986i
$40$		0			0
$41$	11941.3			1.10941			0.554704	−	0.832047i	$-0.312831\pi$
$41$	11941.3			1.10941			0.554704	+	0.832047i	$0.312831\pi$
$42$		0			0
$43$	9820.19			0.809933			0.404966	−	0.914332i	$-0.367283\pi$
$43$	9820.19			0.809933			0.404966	+	0.914332i	$0.367283\pi$
$44$		0			0
$45$	1637.48	−	2836.19i	0.120544	−	0.208787i
$46$		0			0
$47$	9817.98	+	17005.2i	0.648302	+	1.12289i	0.983528	+	0.180755i	$0.0578540\pi$
$47$	9817.98	+	17005.2i	0.648302	+	1.12289i	−0.335226	+	0.942138i	$0.608813\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	16143.5	+	27961.3i	0.869103	+	1.50533i
$52$		0			0
$53$	9937.48	−	17212.2i	0.485945	−	0.841681i	−0.513925	−	0.857835i	$-0.671809\pi$
$53$	9937.48	−	17212.2i	0.485945	−	0.841681i	0.999870	+	0.0161545i	$0.00514235\pi$
$54$		0			0
$55$	−862.077			−0.0384272
$56$		0			0
$57$	−10762.0			−0.438739
$58$		0			0
$59$	−17919.3	+	31037.2i	−0.670180	+	1.16079i	0.307673	+	0.951492i	$0.400450\pi$
$59$	−17919.3	+	31037.2i	−0.670180	+	1.16079i	−0.977853	+	0.209293i	$0.932884\pi$
$60$		0			0
$61$	−24988.0	−	43280.4i	−0.859818	−	1.48925i	−0.872102	−	0.489323i	$-0.837244\pi$
$61$	−24988.0	−	43280.4i	−0.859818	−	1.48925i	0.0122848	−	0.999925i	$-0.496090\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8681.82	+	15037.4i	0.254875	+	0.441457i
$66$		0			0
$67$	−24088.1	+	41721.8i	−0.655565	+	1.13547i	0.326187	+	0.945305i	$0.394236\pi$
$67$	−24088.1	+	41721.8i	−0.655565	+	1.13547i	−0.981752	+	0.190166i	$0.939097\pi$
$68$		0			0
$69$	40761.0			1.03068
$70$		0			0
$71$	77179.1			1.81699			0.908497	−	0.417891i	$-0.137230\pi$
$71$	77179.1			1.81699			0.908497	+	0.417891i	$0.137230\pi$
$72$		0			0
$73$	29833.6	−	51673.4i	0.655238	−	1.13491i	−0.326596	−	0.945164i	$-0.605902\pi$
$73$	29833.6	−	51673.4i	0.655238	−	1.13491i	0.981834	−	0.189742i	$-0.0607650\pi$
$74$		0			0
$75$	−31371.5	−	54337.1i	−0.643995	−	1.11543i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−30371.5	−	52605.1i	−0.547519	−	0.948331i	−0.998444	−	0.0557687i	$-0.982239\pi$
$79$	−30371.5	−	52605.1i	−0.547519	−	0.948331i	0.450925	−	0.892562i	$-0.351094\pi$
$80$		0			0
$81$	31810.7	−	55097.7i	0.538717	−	0.933084i
$82$		0			0
$83$	−46134.2			−0.735068			−0.367534	−	0.930010i	$-0.619798\pi$
$83$	−46134.2			−0.735068			−0.367534	+	0.930010i	$0.619798\pi$
$84$		0			0
$85$	−22033.1			−0.330771
$86$		0			0
$87$	20652.4	−	35770.9i	0.292531	−	0.506678i
$88$		0			0
$89$	−39334.4	−	68129.1i	−0.526377	−	0.911712i	−0.999528	−	0.0307303i	$-0.990217\pi$
$89$	−39334.4	−	68129.1i	−0.526377	−	0.911712i	0.473151	−	0.880982i	$-0.343117\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	8571.17	+	14845.7i	0.102762	+	0.177989i
$94$		0			0
$95$	3672.08	−	6360.22i	0.0417448	−	0.0723042i
$96$		0			0
$97$	−43573.3			−0.470209			−0.235104	−	0.971970i	$-0.575543\pi$
$97$	−43573.3			−0.470209			−0.235104	+	0.971970i	$0.575543\pi$
$98$		0			0
$99$	13025.3			0.133567
$100$		0			0
$101$	82094.2	−	142191.i	0.800772	−	1.38698i	−0.118337	−	0.992974i	$-0.537756\pi$
$101$	82094.2	−	142191.i	0.800772	−	1.38698i	0.919109	−	0.394004i	$-0.128910\pi$
$102$		0			0
$103$	−82273.3	−	142501.i	−0.764127	−	1.32351i	−0.940707	−	0.339221i	$-0.889837\pi$
$103$	−82273.3	−	142501.i	−0.764127	−	1.32351i	0.176579	−	0.984286i	$-0.443497\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	92712.9	+	160583.i	0.782854	+	1.35594i	0.930273	+	0.366868i	$0.119570\pi$
$107$	92712.9	+	160583.i	0.782854	+	1.35594i	−0.147419	+	0.989074i	$0.547097\pi$
$108$		0			0
$109$	−95031.4	+	164599.i	−0.766127	+	1.32697i	0.173522	+	0.984830i	$0.444485\pi$
$109$	−95031.4	+	164599.i	−0.766127	+	1.32697i	−0.939649	+	0.342141i	$0.888848\pi$
$110$		0			0
$111$	−64462.1			−0.496589
$112$		0			0
$113$	−116023.			−0.854769			−0.427384	−	0.904070i	$-0.640565\pi$
$113$	−116023.			−0.854769			−0.427384	+	0.904070i	$0.640565\pi$
$114$		0			0
$115$	−13907.9	+	24089.2i	−0.0980659	+	0.169855i
$116$		0			0
$117$	−131175.	−	227202.i	−0.885904	−	1.53443i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	78811.2	+	136505.i	0.489355	+	0.847588i
$122$		0			0
$123$	128812.	−	223109.i	0.767702	−	1.32970i
$124$		0			0
$125$	88824.6			0.508462
$126$		0			0
$127$	71785.5			0.394936			0.197468	−	0.980309i	$-0.436728\pi$
$127$	71785.5			0.394936			0.197468	+	0.980309i	$0.436728\pi$
$128$		0			0
$129$	105931.	−	183478.i	0.560467	−	0.970757i
$130$		0			0
$131$	1008.11	+	1746.09i	0.00513250	+	0.00888975i	0.868580	−	0.495549i	$-0.165033\pi$
$131$	1008.11	+	1746.09i	0.00513250	+	0.00888975i	−0.863448	+	0.504438i	$0.831700\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3264.40	+	5654.10i	0.0154159	+	0.0267011i
$136$		0			0
$137$	121952.	−	211227.i	0.555120	−	0.961496i	−0.442774	−	0.896633i	$-0.646006\pi$
$137$	121952.	−	211227.i	0.555120	−	0.961496i	0.997894	−	0.0648626i	$-0.0206609\pi$
$138$		0			0
$139$	209413.			0.919319			0.459660	−	0.888095i	$-0.347971\pi$
$139$	209413.			0.919319			0.459660	+	0.888095i	$0.347971\pi$
$140$		0			0
$141$	423630.			1.79448
$142$		0			0
$143$	−34529.6	+	59807.1i	−0.141206	+	0.244575i
$144$		0			0
$145$	14093.5	+	24410.6i	0.0556670	+	0.0964180i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11477.1	−	19879.0i	−0.0423514	−	0.0733547i	0.844073	−	0.536229i	$-0.180152\pi$
$149$	−11477.1	−	19879.0i	−0.0423514	−	0.0733547i	−0.886424	+	0.462874i	$0.846818\pi$
$150$		0			0
$151$	138369.	−	239661.i	0.493850	−	0.855373i	−0.506125	−	0.862460i	$-0.668923\pi$
$151$	138369.	−	239661.i	0.493850	−	0.855373i	0.999975	+	0.00708694i	$0.00225586\pi$
$152$		0			0
$153$	332901.			1.14971
$154$		0			0
$155$	−11698.2			−0.0391101
$156$		0			0
$157$	−138346.	+	239622.i	−0.447937	+	0.775850i	−0.998252	−	0.0591076i	$-0.981174\pi$
$157$	−138346.	+	239622.i	−0.447937	+	0.775850i	0.550315	+	0.834957i	$0.314508\pi$
$158$		0			0
$159$	−214393.	−	371340.i	−0.672540	−	1.16487i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−137129.	−	237514.i	−0.404259	−	0.700197i	0.589976	−	0.807421i	$-0.299137\pi$
$163$	−137129.	−	237514.i	−0.404259	−	0.700197i	−0.994235	+	0.107224i	$0.965804\pi$
$164$		0			0
$165$	−9299.30	+	16106.9i	−0.0265913	+	0.0460576i
$166$		0			0
$167$	−215716.			−0.598537			−0.299268	−	0.954169i	$-0.596743\pi$
$167$	−215716.			−0.598537			−0.299268	+	0.954169i	$0.596743\pi$
$168$		0			0
$169$	1.01967e6			2.74628
$170$		0			0
$171$	−55482.0	+	96097.7i	−0.145098	+	0.251317i
$172$		0			0
$173$	−43126.4	−	74697.1i	−0.109554	−	0.189753i	0.806036	−	0.591867i	$-0.201609\pi$
$173$	−43126.4	−	74697.1i	−0.109554	−	0.189753i	−0.915590	+	0.402114i	$0.868276\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	386594.	+	669601.i	0.927518	+	1.60651i
$178$		0			0
$179$	217911.	−	377433.i	0.508331	−	0.880456i	−0.491622	−	0.870809i	$-0.663596\pi$
$179$	217911.	−	377433.i	0.508331	−	0.880456i	0.999953	−	0.00964720i	$-0.00307085\pi$
$180$		0			0
$181$	56972.2			0.129261			0.0646303	−	0.997909i	$-0.479413\pi$
$181$	56972.2			0.129261			0.0646303	+	0.997909i	$0.479413\pi$
$182$		0			0
$183$	−1.07819e6			−2.37995
$184$		0			0
$185$	21994.9	−	38096.3i	0.0472491	−	0.0818378i
$186$		0			0
$187$	−43815.3	−	75890.4i	−0.0916267	−	0.158702i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	175039.	+	303176.i	0.347177	+	0.601328i	0.985747	−	0.168236i	$-0.0538071\pi$
$191$	175039.	+	303176.i	0.347177	+	0.601328i	−0.638570	+	0.769564i	$0.720474\pi$
$192$		0			0
$193$	311613.	−	539730.i	0.602175	−	1.04300i	−0.390316	−	0.920681i	$-0.627634\pi$
$193$	311613.	−	539730.i	0.602175	−	1.04300i	0.992491	−	0.122317i	$-0.0390324\pi$
$194$		0			0
$195$	374606.			0.705486
$196$		0			0
$197$	223265.			0.409879			0.204939	−	0.978775i	$-0.434300\pi$
$197$	223265.			0.409879			0.204939	+	0.978775i	$0.434300\pi$
$198$		0			0
$199$	−378187.	+	655040.i	−0.676978	+	1.17256i	0.298909	+	0.954282i	$0.403377\pi$
$199$	−378187.	+	655040.i	−0.676978	+	1.17256i	−0.975887	+	0.218278i	$0.929956\pi$
$200$		0			0
$201$	519681.	+	900114.i	0.907291	+	1.57147i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	87903.0	+	152252.i	0.146089	+	0.253034i
$206$		0			0
$207$	210137.	−	363968.i	0.340861	−	0.590388i
$208$		0			0
$209$	29209.4			0.0462549
$210$		0			0
$211$	−30644.3			−0.0473853			−0.0236927	−	0.999719i	$-0.507542\pi$
$211$	−30644.3			−0.0473853			−0.0236927	+	0.999719i	$0.507542\pi$
$212$		0			0
$213$	832537.	−	1.44200e6i	1.25735	−	2.17779i
$214$		0			0
$215$	72289.0	+	125208.i	0.106654	+	0.184730i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−643636.	−	1.11481e6i	−0.906839	−	1.57069i
$220$		0			0
$221$	−882513.	+	1.52856e6i	−1.21546	+	2.10524i
$222$		0			0
$223$	461375.			0.621286			0.310643	−	0.950527i	$-0.399456\pi$
$223$	461375.			0.621286			0.310643	+	0.950527i	$0.399456\pi$
$224$		0			0
$225$	−646925.			−0.851918
$226$		0			0
$227$	−173220.	+	300025.i	−0.223117	+	0.386450i	−0.955753	−	0.294171i	$-0.904957\pi$
$227$	−173220.	+	300025.i	−0.223117	+	0.386450i	0.732636	+	0.680621i	$0.238290\pi$
$228$		0			0
$229$	260825.	+	451762.i	0.328671	+	0.569274i	0.982248	−	0.187585i	$-0.0600661\pi$
$229$	260825.	+	451762.i	0.328671	+	0.569274i	−0.653578	+	0.756859i	$0.726733\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−77060.1	−	133472.i	−0.0929907	−	0.161065i	0.815778	−	0.578366i	$-0.196309\pi$
$233$	−77060.1	−	133472.i	−0.0929907	−	0.161065i	−0.908768	+	0.417301i	$0.862976\pi$
$234$		0			0
$235$	−144545.	+	250360.i	−0.170740	+	0.295730i
$236$		0			0
$237$	−1.31048e6			−1.51551
$238$		0			0
$239$	−478940.			−0.542358			−0.271179	−	0.962529i	$-0.587414\pi$
$239$	−478940.			−0.542358			−0.271179	+	0.962529i	$0.587414\pi$
$240$		0			0
$241$	−43275.9	+	74956.1i	−0.0479958	+	0.0831312i	−0.889025	−	0.457858i	$-0.848617\pi$
$241$	−43275.9	+	74956.1i	−0.0479958	+	0.0831312i	0.841029	+	0.540989i	$0.181950\pi$
$242$		0			0
$243$	−632409.	−	1.09536e6i	−0.687041	−	1.18999i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−294163.	−	509505.i	−0.306793	−	0.531381i
$248$		0			0
$249$	−497653.	+	861961.i	−0.508661	+	0.881027i
$250$		0			0
$251$	−1.90322e6			−1.90679			−0.953397	−	0.301719i	$-0.902439\pi$
$251$	−1.90322e6			−1.90679			−0.953397	+	0.301719i	$0.902439\pi$
$252$		0			0
$253$	−110630.			−0.108661
$254$		0			0
$255$	−237673.	+	411661.i	−0.228891	+	0.396451i
$256$		0			0
$257$	284961.	+	493567.i	0.269124	+	0.466137i	0.968636	−	0.248484i	$-0.0799325\pi$
$257$	284961.	+	493567.i	0.269124	+	0.466137i	−0.699512	+	0.714621i	$0.746599\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−212940.	−	368824.i	−0.193489	−	0.335133i
$262$		0			0
$263$	−649266.	+	1.12456e6i	−0.578806	+	1.00252i	0.416810	+	0.908993i	$0.363148\pi$
$263$	−649266.	+	1.12456e6i	−0.578806	+	1.00252i	−0.995617	+	0.0935284i	$0.970185\pi$
$264$		0			0
$265$	292610.			0.255961
$266$		0			0
$267$	−1.69721e6			−1.45699
$268$		0			0
$269$	26860.8	−	46524.2i	0.0226328	−	0.0392011i	−0.854487	−	0.519472i	$-0.826128\pi$
$269$	26860.8	−	46524.2i	0.0226328	−	0.0392011i	0.877120	+	0.480271i	$0.159462\pi$
$270$		0			0
$271$	379027.	+	656493.i	0.313506	+	0.543009i	0.979119	−	0.203289i	$-0.0651629\pi$
$271$	379027.	+	656493.i	0.313506	+	0.543009i	−0.665612	+	0.746298i	$0.731830\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	85146.1	+	147477.i	0.0678943	+	0.117596i
$276$		0			0
$277$	−1.03799e6	+	1.79785e6i	−0.812819	+	1.40784i	0.0980643	+	0.995180i	$0.468735\pi$
$277$	−1.03799e6	+	1.79785e6i	−0.812819	+	1.40784i	−0.910883	+	0.412664i	$0.864598\pi$
$278$		0			0
$279$	176750.			0.135940
$280$		0			0
$281$	2.61195e6			1.97333			0.986664	−	0.162770i	$-0.0520427\pi$
$281$	2.61195e6			1.97333			0.986664	+	0.162770i	$0.0520427\pi$
$282$		0			0
$283$	−496367.	+	859732.i	−0.368414	+	0.638112i	−0.989318	−	0.145775i	$-0.953433\pi$
$283$	−496367.	+	859732.i	−0.368414	+	0.638112i	0.620903	+	0.783887i	$0.286766\pi$
$284$		0			0
$285$	−79222.0	−	137217.i	−0.0577742	−	0.100068i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−409910.	−	709985.i	−0.288698	−	0.500040i
$290$		0			0
$291$	−470029.	+	814114.i	−0.325381	+	0.563576i
$292$		0			0
$293$	371728.			0.252962			0.126481	−	0.991969i	$-0.459632\pi$
$293$	371728.			0.252962			0.126481	+	0.991969i	$0.459632\pi$
$294$		0			0
$295$	−527635.			−0.353003
$296$		0			0
$297$	−12983.3	+	22487.7i	−0.00854069	+	0.0147929i
$298$		0			0
$299$	1.11414e6	+	1.92974e6i	0.720710	+	1.24831i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1.77111e6	−	3.06766e6i	−1.10826	−	1.91955i
$304$		0			0
$305$	367886.	−	637197.i	0.226446	−	0.392215i
$306$		0			0
$307$	−2.83906e6			−1.71921			−0.859606	−	0.510958i	$-0.829291\pi$
$307$	−2.83906e6			−1.71921			−0.859606	+	0.510958i	$0.829291\pi$
$308$		0			0
$309$	−3.54996e6			−2.11508
$310$		0			0
$311$	−379681.	+	657627.i	−0.222596	+	0.385548i	−0.955596	−	0.294681i	$-0.904787\pi$
$311$	−379681.	+	657627.i	−0.222596	+	0.385548i	0.732999	+	0.680229i	$0.238120\pi$
$312$		0			0
$313$	900954.	+	1.56050e6i	0.519806	+	0.900331i	0.999735	+	0.0230234i	$0.00732923\pi$
$313$	900954.	+	1.56050e6i	0.519806	+	0.900331i	−0.479929	+	0.877308i	$0.659337\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	220950.	+	382696.i	0.123494	+	0.213897i	0.921143	−	0.389224i	$-0.127257\pi$
$317$	220950.	+	382696.i	0.123494	+	0.213897i	−0.797649	+	0.603121i	$0.793923\pi$
$318$		0			0
$319$	−56053.0	+	97086.7i	−0.0308406	+	0.0534174i
$320$		0			0
$321$	4.00041e6			2.16691
$322$		0			0
$323$	746538.			0.398149
$324$		0			0
$325$	1.71498e6	−	2.97044e6i	0.900640	−	1.55995i
$326$		0			0
$327$	2.05022e6	+	3.55109e6i	1.06031	+	1.83651i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.57890e6	+	2.73473e6i	0.792107	+	1.37197i	0.924660	+	0.380793i	$0.124349\pi$
$331$	1.57890e6	+	2.73473e6i	0.792107	+	1.37197i	−0.132553	+	0.991176i	$0.542318\pi$
$332$		0			0
$333$	−332325.	+	575604.i	−0.164230	+	0.284455i
$334$		0			0
$335$	−709275.			−0.345305
$336$		0			0
$337$	−1.66612e6			−0.799158			−0.399579	−	0.916699i	$-0.630844\pi$
$337$	−1.66612e6			−0.799158			−0.399579	+	0.916699i	$0.630844\pi$
$338$		0			0
$339$	−1.25155e6	+	2.16775e6i	−0.591493	+	1.02450i
$340$		0			0
$341$	−23263.2	−	40293.1i	−0.0108339	−	0.0187648i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	300052.	+	519706.i	0.135722	+	0.235077i
$346$		0			0
$347$	1.70977e6	−	2.96140e6i	0.762277	−	1.32030i	−0.179397	−	0.983777i	$-0.557415\pi$
$347$	1.70977e6	−	2.96140e6i	0.762277	−	1.32030i	0.941674	−	0.336526i	$-0.109252\pi$
$348$		0			0
$349$	−3.18396e6			−1.39928			−0.699638	−	0.714497i	$-0.746655\pi$
$349$	−3.18396e6			−1.39928			−0.699638	+	0.714497i	$0.746655\pi$
$350$		0			0
$351$	523009.			0.226590
$352$		0			0
$353$	−1.51253e6	+	2.61978e6i	−0.646051	+	1.11899i	0.338006	+	0.941144i	$0.390247\pi$
$353$	−1.51253e6	+	2.61978e6i	−0.646051	+	1.11899i	−0.984058	+	0.177850i	$0.943086\pi$
$354$		0			0
$355$	568135.	+	984039.i	0.239266	+	0.414421i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	561767.	+	973010.i	0.230049	+	0.398457i	0.957822	−	0.287361i	$-0.0927780\pi$
$359$	561767.	+	973010.i	0.230049	+	0.398457i	−0.727773	+	0.685818i	$0.759445\pi$
$360$		0			0
$361$	1.11363e6	−	1.92886e6i	0.449752	−	0.778993i
$362$		0			0
$363$	3.40057e6			1.35452
$364$		0			0
$365$	878453.			0.345133
$366$		0			0
$367$	−1.69558e6	+	2.93682e6i	−0.657131	+	1.13818i	0.324224	+	0.945980i	$0.394897\pi$
$367$	−1.69558e6	+	2.93682e6i	−0.657131	+	1.13818i	−0.981355	+	0.192204i	$0.938437\pi$
$368$		0			0
$369$	−1.32814e6	−	2.30041e6i	−0.507783	−	0.879506i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−669114.	−	1.15894e6i	−0.249017	−	0.431309i	0.714237	−	0.699904i	$-0.246774\pi$
$373$	−669114.	−	1.15894e6i	−0.249017	−	0.431309i	−0.963253	+	0.268595i	$0.913441\pi$
$374$		0			0
$375$	958159.	−	1.65958e6i	0.351852	−	0.609425i
$376$		0			0
$377$	2.25800e6			0.818221
$378$		0			0
$379$	3.11403e6			1.11359			0.556795	−	0.830650i	$-0.312031\pi$
$379$	3.11403e6			1.11359			0.556795	+	0.830650i	$0.312031\pi$
$380$		0			0
$381$	774356.	−	1.34122e6i	0.273293	−	0.473357i
$382$		0			0
$383$	167267.	+	289714.i	0.0582656	+	0.100919i	0.893687	−	0.448691i	$-0.148110\pi$
$383$	167267.	+	289714.i	0.0582656	+	0.100919i	−0.835421	+	0.549610i	$0.814776\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.09223e6	−	1.89179e6i	−0.370711	−	0.642090i
$388$		0			0
$389$	−799354.	+	1.38452e6i	−0.267834	+	0.463901i	−0.968302	−	0.249782i	$-0.919641\pi$
$389$	−799354.	+	1.38452e6i	−0.267834	+	0.463901i	0.700469	+	0.713683i	$0.252974\pi$
$390$		0			0
$391$	−2.82750e6			−0.935322
$392$		0			0
$393$	43498.2			0.0142066
$394$		0			0
$395$	447146.	−	774479.i	0.144197	−	0.249757i
$396$		0			0
$397$	1.10394e6	+	1.91209e6i	0.351537	+	0.608880i	0.986519	−	0.163647i	$-0.0523258\pi$
$397$	1.10394e6	+	1.91209e6i	0.351537	+	0.608880i	−0.634982	+	0.772527i	$0.718992\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−946358.	−	1.63914e6i	−0.293897	−	0.509044i	0.680831	−	0.732441i	$-0.261619\pi$
$401$	−946358.	−	1.63914e6i	−0.293897	−	0.509044i	−0.974728	+	0.223397i	$0.928285\pi$
$402$		0			0
$403$	−468559.	+	811568.i	−0.143715	+	0.248922i
$404$		0			0
$405$	936667.			0.283758
$406$		0			0
$407$	174958.			0.0523537
$408$		0			0
$409$	−572090.	+	990889.i	−0.169105	+	0.292898i	−0.938105	−	0.346350i	$-0.887421\pi$
$409$	−572090.	+	990889.i	−0.169105	+	0.292898i	0.769001	+	0.639248i	$0.220754\pi$
$410$		0			0
$411$	−2.63101e6	−	4.55704e6i	−0.768277	−	1.33069i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−339606.	−	588215.i	−0.0967955	−	0.167655i
$416$		0			0
$417$	2.25896e6	−	3.91262e6i	0.636162	−	1.10186i
$418$		0			0
$419$	5.17298e6			1.43948			0.719740	−	0.694244i	$-0.244261\pi$
$419$	5.17298e6			1.43948			0.719740	+	0.694244i	$0.244261\pi$
$420$		0			0
$421$	−1.40960e6			−0.387606			−0.193803	−	0.981040i	$-0.562082\pi$
$421$	−1.40960e6			−0.387606			−0.193803	+	0.981040i	$0.562082\pi$
$422$		0			0
$423$	2.18396e6	−	3.78273e6i	0.593463	−	1.02791i
$424$		0			0
$425$	2.17618e6	+	3.76925e6i	0.584415	+	1.01224i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	744949.	+	1.29029e6i	0.195426	+	0.338488i
$430$		0			0
$431$	−3.08538e6	+	5.34403e6i	−0.800047	+	1.38572i	0.119538	+	0.992830i	$0.461859\pi$
$431$	−3.08538e6	+	5.34403e6i	−0.800047	+	1.38572i	−0.919585	+	0.392892i	$0.871475\pi$
$432$		0			0
$433$	387046.			0.0992071			0.0496036	−	0.998769i	$-0.484204\pi$
$433$	387046.			0.0992071			0.0496036	+	0.998769i	$0.484204\pi$
$434$		0			0
$435$	608110.			0.154084
$436$		0			0
$437$	471237.	−	816207.i	0.118042	−	0.204455i
$438$		0			0
$439$	−1.66487e6	−	2.88364e6i	−0.412305	−	0.714134i	0.582836	−	0.812590i	$-0.301943\pi$
$439$	−1.66487e6	−	2.88364e6i	−0.412305	−	0.714134i	−0.995141	+	0.0984559i	$0.968610\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	720013.	+	1.24710e6i	0.174314	+	0.301920i	0.939924	−	0.341385i	$-0.110896\pi$
$443$	720013.	+	1.24710e6i	0.174314	+	0.301920i	−0.765610	+	0.643305i	$0.777563\pi$
$444$		0			0
$445$	579101.	−	1.00303e6i	0.138629	−	0.240113i
$446$		0			0
$447$	−495219.			−0.117227
$448$		0			0
$449$	4.28977e6			1.00420			0.502098	−	0.864811i	$-0.332562\pi$
$449$	4.28977e6			1.00420			0.502098	+	0.864811i	$0.332562\pi$
$450$		0			0
$451$	−349611.	+	605544.i	−0.0809363	+	0.140186i
$452$		0			0
$453$	−2.98519e6	−	5.17050e6i	−0.683480	−	1.18382i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.63944e6	+	4.57165e6i	0.591183	+	1.02396i	0.994073	+	0.108711i	$0.0346723\pi$
$457$	2.63944e6	+	4.57165e6i	0.591183	+	1.02396i	−0.402890	+	0.915248i	$0.631994\pi$
$458$		0			0
$459$	−331828.	+	574743.i	−0.0735159	+	0.127333i
$460$		0			0
$461$	−3.72322e6			−0.815955			−0.407977	−	0.912992i	$-0.633766\pi$
$461$	−3.72322e6			−0.815955			−0.407977	+	0.912992i	$0.633766\pi$
$462$		0			0
$463$	1.04809e6			0.227220			0.113610	−	0.993525i	$-0.463759\pi$
$463$	1.04809e6			0.227220			0.113610	+	0.993525i	$0.463759\pi$
$464$		0			0
$465$	−126189.	+	218566.i	−0.0270639	+	0.0468760i
$466$		0			0
$467$	−3.02122e6	−	5.23291e6i	−0.641048	−	1.11033i	−0.985199	−	0.171413i	$-0.945167\pi$
$467$	−3.02122e6	−	5.23291e6i	−0.641048	−	1.11033i	0.344151	−	0.938914i	$-0.388167\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.98470e6	+	5.16965e6i	0.619938	+	1.07376i
$472$		0			0
$473$	−287510.	+	497983.i	−0.0590882	+	0.102344i
$474$		0			0
$475$	−1.45074e6			−0.295024
$476$		0			0
$477$	−4.42109e6			−0.889679
$478$		0			0
$479$	1.03434e6	−	1.79153e6i	0.205980	−	0.356767i	−0.744465	−	0.667662i	$-0.767295\pi$
$479$	1.03434e6	−	1.79153e6i	0.205980	−	0.356767i	0.950444	+	0.310895i	$0.100629\pi$
$480$		0			0
$481$	−1.76197e6	−	3.05182e6i	−0.347245	−	0.601446i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−320754.	−	555563.i	−0.0619182	−	0.107245i
$486$		0			0
$487$	1.41483e6	−	2.45056e6i	0.270323	−	0.468213i	−0.698622	−	0.715491i	$-0.746203\pi$
$487$	1.41483e6	−	2.45056e6i	0.270323	−	0.468213i	0.968944	+	0.247279i	$0.0795363\pi$
$488$		0			0
$489$	−5.91688e6			−1.11898
$490$		0			0
$491$	−4.44554e6			−0.832186			−0.416093	−	0.909322i	$-0.636601\pi$
$491$	−4.44554e6			−0.832186			−0.416093	+	0.909322i	$0.636601\pi$
$492$		0			0
$493$	−1.43261e6	+	2.48135e6i	−0.265467	+	0.459803i
$494$		0			0
$495$	95882.4	+	166073.i	0.0175884	+	0.0304640i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	673115.	+	1.16587e6i	0.121015	+	0.209603i	0.920168	−	0.391524i	$-0.128052\pi$
$499$	673115.	+	1.16587e6i	0.121015	+	0.209603i	−0.799153	+	0.601127i	$0.794719\pi$
$500$		0			0
$501$	−2.32695e6	+	4.03039e6i	−0.414183	+	0.717385i
$502$		0			0
$503$	−4.88909e6			−0.861604			−0.430802	−	0.902446i	$-0.641769\pi$
$503$	−4.88909e6			−0.861604			−0.430802	+	0.902446i	$0.641769\pi$
$504$		0			0
$505$	2.41727e6			0.421790
$506$		0			0
$507$	1.09993e7	−	1.90514e7i	1.90040	−	3.29160i
$508$		0			0
$509$	−3.27759e6	−	5.67695e6i	−0.560738	−	0.971228i	−0.997432	−	0.0716173i	$-0.977184\pi$
$509$	−3.27759e6	−	5.67695e6i	−0.560738	−	0.971228i	0.436694	−	0.899610i	$-0.356149\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−110606.	−	191576.i	−0.0185561	−	0.0321401i
$514$		0			0
$515$	1.21127e6	−	2.09798e6i	0.201244	−	0.348565i
$516$		0			0
$517$	−1.14978e6			−0.189186
$518$		0			0
$519$	−1.86083e6			−0.303242
$520$		0			0
$521$	2.38645e6	−	4.13345e6i	0.385175	−	0.667142i	−0.606619	−	0.794993i	$-0.707475\pi$
$521$	2.38645e6	−	4.13345e6i	0.385175	−	0.667142i	0.991793	+	0.127851i	$0.0408079\pi$
$522$		0			0
$523$	414067.	+	717186.i	0.0661937	+	0.114651i	0.897223	−	0.441578i	$-0.145581\pi$
$523$	414067.	+	717186.i	0.0661937	+	0.114651i	−0.831029	+	0.556229i	$0.812248\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−594564.	−	1.02982e6i	−0.0932550	−	0.161522i
$528$		0			0
$529$	1.43337e6	−	2.48267e6i	0.222699	−	0.385726i
$530$		0			0
$531$	7.97212e6			1.22698
$532$		0			0
$533$	1.40835e7			2.14730
$534$		0			0
$535$	−1.36497e6	+	2.36419e6i	−0.206176	+	0.357107i
$536$		0			0
$537$	−4.70125e6	−	8.14281e6i	−0.703523	−	1.21854i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.46678e6	−	6.00464e6i	−0.509253	−	0.882051i	−0.999943	−	0.0107172i	$-0.996589\pi$
$541$	−3.46678e6	−	6.00464e6i	−0.509253	−	0.882051i	0.490690	−	0.871334i	$-0.336745\pi$
$542$		0			0
$543$	614564.	−	1.06446e6i	0.0894473	−	0.154927i
$544$		0			0
$545$	−2.79820e6			−0.403541
$546$		0			0
$547$	559077.			0.0798920			0.0399460	−	0.999202i	$-0.487281\pi$
$547$	559077.			0.0798920			0.0399460	+	0.999202i	$0.487281\pi$
$548$		0			0
$549$	−5.55845e6	+	9.62752e6i	−0.787087	+	1.36327i
$550$		0			0
$551$	−477524.	−	827095.i	−0.0670063	−	0.116058i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−474522.	−	821897.i	−0.0653920	−	0.113262i
$556$		0			0
$557$	1.95222e6	−	3.38134e6i	0.266619	−	0.461797i	−0.701368	−	0.712800i	$-0.747427\pi$
$557$	1.95222e6	−	3.38134e6i	0.266619	−	0.461797i	0.967986	+	0.251002i	$0.0807602\pi$
$558$		0			0
$559$	1.15819e7			1.56765
$560$		0			0
$561$	−1.89056e6			−0.253620
$562$		0			0
$563$	2.01154e6	−	3.48408e6i	0.267459	−	0.463252i	−0.700746	−	0.713411i	$-0.747149\pi$
$563$	2.01154e6	−	3.48408e6i	0.267459	−	0.463252i	0.968205	+	0.250158i	$0.0804827\pi$
$564$		0			0
$565$	−854077.	−	1.47930e6i	−0.112558	−	0.194956i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.17900e6	+	2.04210e6i	0.152663	+	0.264421i	0.932206	−	0.361929i	$-0.117882\pi$
$569$	1.17900e6	+	2.04210e6i	0.152663	+	0.264421i	−0.779542	+	0.626349i	$0.784548\pi$
$570$		0			0
$571$	3.88367e6	−	6.72671e6i	0.498484	−	0.863400i	−0.501514	−	0.865149i	$-0.667223\pi$
$571$	3.88367e6	−	6.72671e6i	0.498484	−	0.863400i	0.999998	+	0.00174924i	$0.000556802\pi$
$572$		0			0
$573$	7.55263e6			0.960974
$574$		0			0
$575$	5.49467e6			0.693062
$576$		0			0
$577$	−6.91813e6	+	1.19826e7i	−0.865066	+	1.49834i	0.00191465	+	0.999998i	$0.499391\pi$
$577$	−6.91813e6	+	1.19826e7i	−0.865066	+	1.49834i	−0.866981	+	0.498341i	$0.833943\pi$
$578$		0			0
$579$	−6.72280e6	−	1.16442e7i	−0.833401	−	1.44349i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	581889.	+	1.00786e6i	0.0709037	+	0.122809i
$584$		0			0
$585$	1.93123e6	−	3.34499e6i	0.233316	−	0.404115i
$586$		0			0
$587$	2.13747e6			0.256038			0.128019	−	0.991772i	$-0.459138\pi$
$587$	2.13747e6			0.256038			0.128019	+	0.991772i	$0.459138\pi$
$588$		0			0
$589$	396365.			0.0470768
$590$		0			0
$591$	2.40838e6	−	4.17144e6i	0.283633	−	0.491266i
$592$		0			0
$593$	5.56904e6	+	9.64586e6i	0.650345	+	1.12643i	0.983039	+	0.183395i	$0.0587089\pi$
$593$	5.56904e6	+	9.64586e6i	0.650345	+	1.12643i	−0.332695	+	0.943035i	$0.607958\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	8.15908e6	+	1.41319e7i	0.936926	+	1.62280i
$598$		0			0
$599$	−5.50222e6	+	9.53013e6i	−0.626572	+	1.08525i	0.361663	+	0.932309i	$0.382209\pi$
$599$	−5.50222e6	+	9.53013e6i	−0.626572	+	1.08525i	−0.988235	+	0.152946i	$0.951124\pi$
$600$		0			0
$601$	1.28046e6			0.144604			0.0723019	−	0.997383i	$-0.476965\pi$
$601$	1.28046e6			0.144604			0.0723019	+	0.997383i	$0.476965\pi$
$602$		0			0
$603$	1.07166e7			1.20022
$604$		0			0
$605$	−1.16030e6	+	2.00970e6i	−0.128879	+	0.223225i
$606$		0			0
$607$	1.04164e6	+	1.80417e6i	0.114748	+	0.198749i	0.917679	−	0.397323i	$-0.130061\pi$
$607$	1.04164e6	+	1.80417e6i	0.114748	+	0.198749i	−0.802931	+	0.596072i	$0.796727\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.15793e7	+	2.00559e7i	1.25481	+	2.17339i
$612$		0			0
$613$	6.85581e6	−	1.18746e7i	0.736899	−	1.27635i	−0.216986	−	0.976175i	$-0.569623\pi$
$613$	6.85581e6	−	1.18746e7i	0.736899	−	1.27635i	0.953885	−	0.300172i	$-0.0970440\pi$
$614$		0			0
$615$	3.79287e6			0.404371
$616$		0			0
$617$	−1.65172e7			−1.74672			−0.873361	−	0.487073i	$-0.838064\pi$
$617$	−1.65172e7			−1.74672			−0.873361	+	0.487073i	$0.838064\pi$
$618$		0			0
$619$	−1.75661e6	+	3.04253e6i	−0.184267	+	0.319160i	−0.943329	−	0.331858i	$-0.892324\pi$
$619$	−1.75661e6	+	3.04253e6i	−0.184267	+	0.319160i	0.759062	+	0.651018i	$0.225658\pi$
$620$		0			0
$621$	418920.	+	725590.i	0.0435915	+	0.0755027i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.89027e6	−	6.73815e6i	−0.398364	−	0.689987i
$626$		0			0
$627$	315085.	−	545743.i	0.0320080	−	0.0554395i
$628$		0			0
$629$	4.47160e6			0.450647
$630$		0			0
$631$	−8.54135e6			−0.853991			−0.426995	−	0.904254i	$-0.640428\pi$
$631$	−8.54135e6			−0.853991			−0.426995	+	0.904254i	$0.640428\pi$
$632$		0			0
$633$	−330563.	+	572552.i	−0.0327903	+	0.0567944i
$634$		0			0
$635$	528432.	+	915271.i	0.0520061	+	0.0900773i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−8.58405e6	−	1.48680e7i	−0.831649	−	1.44046i
$640$		0			0
$641$	4.97759e6	−	8.62145e6i	0.478492	−	0.828772i	−0.521204	−	0.853432i	$-0.674517\pi$
$641$	4.97759e6	−	8.62145e6i	0.478492	−	0.828772i	0.999696	+	0.0246601i	$0.00785036\pi$
$642$		0			0
$643$	−3.98313e6			−0.379924			−0.189962	−	0.981791i	$-0.560837\pi$
$643$	−3.98313e6			−0.379924			−0.189962	+	0.981791i	$0.560837\pi$
$644$		0			0
$645$	3.11915e6			0.295214
$646$		0			0
$647$	1.92060e6	−	3.32658e6i	0.180375	−	0.312419i	−0.761633	−	0.648008i	$-0.775602\pi$
$647$	1.92060e6	−	3.32658e6i	0.180375	−	0.312419i	0.942008	+	0.335589i	$0.108935\pi$
$648$		0			0
$649$	−1.04926e6	−	1.81738e6i	−0.0977852	−	0.169369i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.21430e6	−	1.59596e7i	−0.845628	−	1.46467i	−0.885074	−	0.465450i	$-0.845893\pi$
$653$	−9.21430e6	−	1.59596e7i	−0.845628	−	1.46467i	0.0394461	−	0.999222i	$-0.487441\pi$
$654$		0			0
$655$	−14841.9	+	25706.9i	−0.00135172	+	0.00234124i
$656$		0			0
$657$	−1.32727e7			−1.19963
$658$		0			0
$659$	2.15854e7			1.93619			0.968093	−	0.250590i	$-0.0806245\pi$
$659$	2.15854e7			1.93619			0.968093	+	0.250590i	$0.0806245\pi$
$660$		0			0
$661$	−7.20361e6	+	1.24770e7i	−0.641278	+	1.11073i	0.343869	+	0.939017i	$0.388262\pi$
$661$	−7.20361e6	+	1.24770e7i	−0.641278	+	1.11073i	−0.985148	+	0.171709i	$0.945071\pi$
$662$		0			0
$663$	1.90395e7	+	3.29774e7i	1.68218	+	2.91362i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.80861e6	+	3.13261e6i	0.157410	+	0.272641i
$668$		0			0
$669$	4.97689e6	−	8.62023e6i	0.429925	−	0.744652i
$670$		0			0
$671$	2.92634e6			0.250910
$672$		0			0
$673$	−5.55230e6			−0.472536			−0.236268	−	0.971688i	$-0.575924\pi$
$673$	−5.55230e6			−0.472536			−0.236268	+	0.971688i	$0.575924\pi$
$674$		0			0
$675$	644840.	−	1.11690e6i	0.0544744	−	0.0943524i
$676$		0			0
$677$	4.96005e6	+	8.59105e6i	0.415924	+	0.720402i	0.995525	−	0.0944989i	$-0.0301249\pi$
$677$	4.96005e6	+	8.59105e6i	0.415924	+	0.720402i	−0.579601	+	0.814900i	$0.696792\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.73707e6	+	6.47280e6i	0.308790	+	0.534841i
$682$		0			0
$683$	5.06457e6	−	8.77209e6i	0.415423	−	0.719534i	−0.580050	−	0.814581i	$-0.696967\pi$
$683$	5.06457e6	−	8.77209e6i	0.415423	−	0.719534i	0.995473	+	0.0950472i	$0.0303002\pi$
$684$		0			0
$685$	3.59088e6			0.292398
$686$		0			0
$687$	1.12542e7			0.909750
$688$		0			0
$689$	1.17202e7	−	2.03000e7i	0.940561	−	1.62910i
$690$		0			0
$691$	−9.93735e6	−	1.72120e7i	−0.791727	−	1.37131i	−0.924897	−	0.380218i	$-0.875849\pi$
$691$	−9.93735e6	−	1.72120e7i	−0.791727	−	1.37131i	0.133170	−	0.991093i	$-0.457485\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.54154e6	+	2.67003e6i	0.121058	+	0.209679i
$696$		0			0
$697$	−8.93540e6	+	1.54766e7i	−0.696678	+	1.20668i
$698$		0			0
$699$	−3.32502e6			−0.257395
$700$		0			0
$701$	−2.13147e7			−1.63826			−0.819132	−	0.573605i	$-0.805545\pi$
$701$	−2.13147e7			−1.63826			−0.819132	+	0.573605i	$0.805545\pi$
$702$		0			0
$703$	−745246.	+	1.29080e6i	−0.0568737	+	0.0985081i
$704$		0			0
$705$	3.11845e6	+	5.40131e6i	0.236301	+	0.409286i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4.91600e6	−	8.51477e6i	−0.367279	−	0.636146i	0.621860	−	0.783129i	$-0.286377\pi$
$709$	−4.91600e6	−	8.51477e6i	−0.367279	−	0.636146i	−0.989139	+	0.146982i	$0.953044\pi$
$710$		0			0
$711$	−6.75600e6	+	1.17017e7i	−0.501205	+	0.868113i
$712$		0			0
$713$	−1.50123e6			−0.110592
$714$		0			0
$715$	−1.01673e6			−0.0743771
$716$		0			0
$717$	−5.16637e6	+	8.94841e6i	−0.375308	+	0.650052i
$718$		0			0
$719$	9.03368e6	+	1.56468e7i	0.651692	+	1.12876i	0.982712	+	0.185140i	$0.0592738\pi$
$719$	9.03368e6	+	1.56468e7i	0.651692	+	1.12876i	−0.331020	+	0.943624i	$0.607393\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	933642.	+	1.61712e6i	0.0664255	+	0.115052i
$724$		0			0
$725$	2.78398e6	−	4.82200e6i	0.196708	−	0.340708i
$726$		0			0
$727$	−1.82003e7			−1.27715			−0.638576	−	0.769559i	$-0.720476\pi$
$727$	−1.82003e7			−1.27715			−0.638576	+	0.769559i	$0.720476\pi$
$728$		0			0
$729$	−1.18274e7			−0.824274
$730$		0			0
$731$	−7.34823e6	+	1.27275e7i	−0.508615	+	0.880947i
$732$		0			0
$733$	−3.35315e6	−	5.80782e6i	−0.230512	−	0.399258i	0.727447	−	0.686164i	$-0.240707\pi$
$733$	−3.35315e6	−	5.80782e6i	−0.230512	−	0.399258i	−0.957959	+	0.286906i	$0.907373\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.41048e6	−	2.44302e6i	−0.0956527	−	0.165675i
$738$		0			0
$739$	−1.08251e7	+	1.87496e7i	−0.729156	+	1.26294i	0.228084	+	0.973641i	$0.426754\pi$
$739$	−1.08251e7	+	1.87496e7i	−0.729156	+	1.26294i	−0.957240	+	0.289294i	$0.906579\pi$
$740$		0			0
$741$	−1.26926e7			−0.849193
$742$		0			0
$743$	−1.40437e7			−0.933276			−0.466638	−	0.884448i	$-0.654535\pi$
$743$	−1.40437e7			−0.933276			−0.466638	+	0.884448i	$0.654535\pi$
$744$		0			0
$745$	168972.	−	292669.i	0.0111539	−	0.0193190i
$746$		0			0
$747$	5.13116e6	+	8.88743e6i	0.336445	+	0.582740i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4.45691e6	−	7.71959e6i	−0.288359	−	0.499453i	0.685059	−	0.728487i	$-0.259776\pi$
$751$	−4.45691e6	−	7.71959e6i	−0.288359	−	0.499453i	−0.973418	+	0.229035i	$0.926443\pi$
$752$		0			0
$753$	−2.05302e7	+	3.55593e7i	−1.31949	+	2.28542i
$754$		0			0
$755$	4.07427e6			0.260125
$756$		0			0
$757$	−1.04419e7			−0.662276			−0.331138	−	0.943582i	$-0.607433\pi$
$757$	−1.04419e7			−0.662276			−0.331138	+	0.943582i	$0.607433\pi$
$758$		0			0
$759$	−1.19338e6	+	2.06699e6i	−0.0751923	+	0.130237i
$760$		0			0
$761$	−1.48107e7	−	2.56529e7i	−0.927072	−	1.60574i	−0.788194	−	0.615427i	$-0.788984\pi$
$761$	−1.48107e7	−	2.56529e7i	−0.927072	−	1.60574i	−0.138878	−	0.990309i	$-0.544350\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.45057e6	+	4.24452e6i	0.151396	+	0.262225i
$766$		0			0
$767$	−2.11339e7	+	3.66050e7i	−1.29715	+	2.24674i
$768$		0			0
$769$	−4.24042e6			−0.258579			−0.129289	−	0.991607i	$-0.541270\pi$
$769$	−4.24042e6			−0.258579			−0.129289	+	0.991607i	$0.541270\pi$
$770$		0			0
$771$	1.22956e7			0.744927
$772$		0			0
$773$	6.85359e6	−	1.18708e7i	0.412543	−	0.714546i	−0.582624	−	0.812742i	$-0.697974\pi$
$773$	6.85359e6	−	1.18708e7i	0.412543	−	0.714546i	0.995167	+	0.0981963i	$0.0313073\pi$
$774$		0			0
$775$	1.15541e6	+	2.00123e6i	0.0691008	+	0.119686i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.97838e6	−	5.15871e6i	−0.175848	−	0.304577i
$780$		0			0
$781$	−2.25961e6	+	3.91376e6i	−0.132558	+	0.229597i
$782$		0			0
$783$	849016.			0.0494893
$784$		0			0
$785$	−4.07360e6			−0.235942

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(10.7871 - 18.6838i) q^{3} +(7.36126 + 12.7501i) q^{5} +(-111.223 - 192.643i) q^{9} +O(q^{10})\)	\(q+(10.7871 - 18.6838i) q^{3} +(7.36126 + 12.7501i) q^{5} +(-111.223 - 192.643i) q^{9} +(-29.2775 + 50.7101i) q^{11} +1179.39 q^{13} +317.626 q^{15} +(-748.277 + 1296.05i) q^{17} +(-249.419 - 432.006i) q^{19} +(944.670 + 1636.22i) q^{23} +(1454.12 - 2518.62i) q^{25} +443.456 q^{27} +1914.54 q^{29} +(-397.288 + 688.124i) q^{31} +(631.637 + 1094.03i) q^{33} +(-1493.96 - 2587.62i) q^{37} +(12722.2 - 22035.5i) q^{39} +11941.3 q^{41} +9820.19 q^{43} +(1637.48 - 2836.19i) q^{45} +(9817.98 + 17005.2i) q^{47} +(16143.5 + 27961.3i) q^{51} +(9937.48 - 17212.2i) q^{53} -862.077 q^{55} -10762.0 q^{57} +(-17919.3 + 31037.2i) q^{59} +(-24988.0 - 43280.4i) q^{61} +(8681.82 + 15037.4i) q^{65} +(-24088.1 + 41721.8i) q^{67} +40761.0 q^{69} +77179.1 q^{71} +(29833.6 - 51673.4i) q^{73} +(-31371.5 - 54337.1i) q^{75} +(-30371.5 - 52605.1i) q^{79} +(31810.7 - 55097.7i) q^{81} -46134.2 q^{83} -22033.1 q^{85} +(20652.4 - 35770.9i) q^{87} +(-39334.4 - 68129.1i) q^{89} +(8571.17 + 14845.7i) q^{93} +(3672.08 - 6360.22i) q^{95} -43573.3 q^{97} +13025.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} - 82 q^{5} - 222 q^{9}+O(q^{10}) \) 4 * q + 6 * q^3 - 82 * q^5 - 222 * q^9	\( 4 q + 6 q^{3} - 82 q^{5} - 222 q^{9} - 340 q^{11} + 1820 q^{13} + 3648 q^{15} - 3216 q^{17} + 674 q^{19} + 1104 q^{23} - 3322 q^{25} - 6696 q^{27} + 16128 q^{29} + 6212 q^{31} - 3120 q^{33} + 8512 q^{37} + 29640 q^{39} - 2608 q^{41} - 20008 q^{43} + 3318 q^{45} + 12748 q^{47} + 5508 q^{51} + 11220 q^{53} + 52720 q^{55} - 58056 q^{57} + 12018 q^{59} - 102738 q^{61} + 43420 q^{65} - 24136 q^{67} + 105984 q^{69} + 179440 q^{71} + 55588 q^{73} - 159774 q^{75} - 48824 q^{79} + 122562 q^{81} + 71564 q^{83} + 288552 q^{85} - 54468 q^{87} + 18300 q^{89} + 126264 q^{93} + 120784 q^{95} - 139968 q^{97} + 25800 q^{99}+O(q^{100}) \) 4 * q + 6 * q^3 - 82 * q^5 - 222 * q^9 - 340 * q^11 + 1820 * q^13 + 3648 * q^15 - 3216 * q^17 + 674 * q^19 + 1104 * q^23 - 3322 * q^25 - 6696 * q^27 + 16128 * q^29 + 6212 * q^31 - 3120 * q^33 + 8512 * q^37 + 29640 * q^39 - 2608 * q^41 - 20008 * q^43 + 3318 * q^45 + 12748 * q^47 + 5508 * q^51 + 11220 * q^53 + 52720 * q^55 - 58056 * q^57 + 12018 * q^59 - 102738 * q^61 + 43420 * q^65 - 24136 * q^67 + 105984 * q^69 + 179440 * q^71 + 55588 * q^73 - 159774 * q^75 - 48824 * q^79 + 122562 * q^81 + 71564 * q^83 + 288552 * q^85 - 54468 * q^87 + 18300 * q^89 + 126264 * q^93 + 120784 * q^95 - 139968 * q^97 + 25800 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more