LMFDB - Embedded newform 784.6.f.c.783.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 $ x^14 - 3x^13 + 691x^12 - 8602x^11 + 416261x^10 - 3521447x^9 + 66162087x^8 - 31847546x^7 + 3819865057x^6 + 14320412589x^5 + 280505627703x^4 + 1315235310354x^3 + 6574080917124x^2 + 11537944725888*x + 17213603549184
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{27}\cdot 3^{6}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.13
Root			$7.14120 - 12.3689i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.6.f.c.783.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+23.1254 q^{3} -87.5115i q^{5} +291.784 q^{9} +405.208i q^{11} +640.275i q^{13} -2023.74i q^{15} -1485.88i q^{17} -1399.95 q^{19} -1200.64i q^{23} -4533.26 q^{25} +1128.15 q^{27} -1968.20 q^{29} -813.914 q^{31} +9370.60i q^{33} -13912.8 q^{37} +14806.6i q^{39} -17041.2i q^{41} -19253.7i q^{43} -25534.5i q^{45} +22650.7 q^{47} -34361.5i q^{51} -16018.7 q^{53} +35460.4 q^{55} -32374.3 q^{57} -30122.1 q^{59} -43658.2i q^{61} +56031.5 q^{65} +10606.4i q^{67} -27765.4i q^{69} +57405.1i q^{71} -65014.3i q^{73} -104833. q^{75} -18465.2i q^{79} -44814.6 q^{81} +17093.5 q^{83} -130031. q^{85} -45515.3 q^{87} -47195.1i q^{89} -18822.1 q^{93} +122511. i q^{95} +52847.4i q^{97} +118233. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{3} + 1076 q^{9} - 4270 q^{19} - 10868 q^{25} - 17910 q^{27} + 684 q^{29} + 6238 q^{31} - 6862 q^{37} + 16818 q^{47} - 3414 q^{53} + 56622 q^{55} + 68450 q^{57} - 98610 q^{59} - 34896 q^{65}+ \cdots - 585746 q^{93}+O(q^{100}) $ 14 * q - 18 * q^3 + 1076 * q^9 - 4270 * q^19 - 10868 * q^25 - 17910 * q^27 + 684 * q^29 + 6238 * q^31 - 6862 * q^37 + 16818 * q^47 - 3414 * q^53 + 56622 * q^55 + 68450 * q^57 - 98610 * q^59 - 34896 * q^65 - 295668 * q^75 + 291134 * q^81 + 305400 * q^83 - 176454 * q^85 - 190068 * q^87 - 585746 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	23.1254	1.48350	0.741748	−	0.670679i	$-0.233997\pi$
$3$	23.1254	1.48350	0.741748	+	0.670679i	$0.233997\pi$
$4$	0	0
$5$	− 87.5115i	− 1.56545i	−0.622366	−	0.782727i	$-0.713828\pi$
$5$	− 87.5115i	− 1.56545i	0.622366	−	0.782727i	$-0.286172\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	291.784	1.20076
$10$	0	0
$11$	405.208i	1.00971i	0.863204	+	0.504855i	$0.168454\pi$
$11$	405.208i	1.00971i	−0.863204	+	0.504855i	$0.831546\pi$
$12$	0	0
$13$	640.275i	1.05077i	0.850864	+	0.525386i	$0.176079\pi$
$13$	640.275i	1.05077i	−0.850864	+	0.525386i	$0.823921\pi$
$14$	0	0
$15$	− 2023.74i	− 2.32234i
$16$	0	0
$17$	− 1485.88i	− 1.24698i	−0.781830	−	0.623492i	$-0.785713\pi$
$17$	− 1485.88i	− 1.24698i	0.781830	−	0.623492i	$-0.214287\pi$
$18$	0	0
$19$	−1399.95	−0.889667	−0.444833	−	0.895613i	$-0.646737\pi$
$19$	−1399.95	−0.889667	−0.444833	+	0.895613i	$0.646737\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1200.64i	− 0.473255i	−0.971601	−	0.236627i	$-0.923958\pi$
$23$	− 1200.64i	− 0.473255i	0.971601	−	0.236627i	$-0.0760420\pi$
$24$	0	0
$25$	−4533.26	−1.45064
$26$	0	0
$27$	1128.15	0.297823
$28$	0	0
$29$	−1968.20	−0.434584	−0.217292	−	0.976107i	$-0.569722\pi$
$29$	−1968.20	−0.434584	−0.217292	+	0.976107i	$0.569722\pi$
$30$	0	0
$31$	−813.914	−0.152116	−0.0760579	−	0.997103i	$-0.524233\pi$
$31$	−813.914	−0.152116	−0.0760579	+	0.997103i	$0.524233\pi$
$32$	0	0
$33$	9370.60i	1.49790i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13912.8	−1.67074	−0.835372	−	0.549686i	$-0.814747\pi$
$37$	−13912.8	−1.67074	−0.835372	+	0.549686i	$0.814747\pi$
$38$	0	0
$39$	14806.6i	1.55882i
$40$	0	0
$41$	− 17041.2i	− 1.58322i	−0.611026	−	0.791611i	$-0.709243\pi$
$41$	− 17041.2i	− 1.58322i	0.611026	−	0.791611i	$-0.290757\pi$
$42$	0	0
$43$	− 19253.7i	− 1.58797i	−0.607937	−	0.793986i	$-0.708003\pi$
$43$	− 19253.7i	− 1.58797i	0.607937	−	0.793986i	$-0.291997\pi$
$44$	0	0
$45$	− 25534.5i	− 1.87973i
$46$	0	0
$47$	22650.7	1.49567	0.747837	−	0.663882i	$-0.231092\pi$
$47$	22650.7	1.49567	0.747837	+	0.663882i	$0.231092\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 34361.5i	− 1.84989i
$52$	0	0
$53$	−16018.7	−0.783318	−0.391659	−	0.920110i	$-0.628099\pi$
$53$	−16018.7	−0.783318	−0.391659	+	0.920110i	$0.628099\pi$
$54$	0	0
$55$	35460.4	1.58065
$56$	0	0
$57$	−32374.3	−1.31982
$58$	0	0
$59$	−30122.1	−1.12656	−0.563281	−	0.826265i	$-0.690461\pi$
$59$	−30122.1	−1.12656	−0.563281	+	0.826265i	$0.690461\pi$
$60$	0	0
$61$	− 43658.2i	− 1.50225i	−0.660161	−	0.751124i	$-0.729512\pi$
$61$	− 43658.2i	− 1.50225i	0.660161	−	0.751124i	$-0.270488\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	56031.5	1.64493
$66$	0	0
$67$	10606.4i	0.288655i	0.989530	+	0.144328i	$0.0461019\pi$
$67$	10606.4i	0.288655i	−0.989530	+	0.144328i	$0.953898\pi$
$68$	0	0
$69$	− 27765.4i	− 0.702071i
$70$	0	0
$71$	57405.1i	1.35146i	0.737148	+	0.675732i	$0.236172\pi$
$71$	57405.1i	1.35146i	−0.737148	+	0.675732i	$0.763828\pi$
$72$	0	0
$73$	− 65014.3i	− 1.42791i	−0.700190	−	0.713957i	$-0.746901\pi$
$73$	− 65014.3i	− 1.42791i	0.700190	−	0.713957i	$-0.253099\pi$
$74$	0	0
$75$	−104833.	−2.15202
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 18465.2i	− 0.332878i	−0.986052	−	0.166439i	$-0.946773\pi$
$79$	− 18465.2i	− 0.332878i	0.986052	−	0.166439i	$-0.0532269\pi$
$80$	0	0
$81$	−44814.6	−0.758939
$82$	0	0
$83$	17093.5	0.272355	0.136178	−	0.990684i	$-0.456518\pi$
$83$	17093.5	0.272355	0.136178	+	0.990684i	$0.456518\pi$
$84$	0	0
$85$	−130031.	−1.95209
$86$	0	0
$87$	−45515.3	−0.644703
$88$	0	0
$89$	− 47195.1i	− 0.631571i	−0.948831	−	0.315785i	$-0.897732\pi$
$89$	− 47195.1i	− 0.631571i	0.948831	−	0.315785i	$-0.102268\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18822.1	−0.225663
$94$	0	0
$95$	122511.i	1.39273i
$96$	0	0
$97$	52847.4i	0.570288i	0.958485	+	0.285144i	$0.0920414\pi$
$97$	52847.4i	0.570288i	−0.958485	+	0.285144i	$0.907959\pi$
$98$	0	0
$99$	118233.i	1.21242i
$100$	0	0
$101$	180107.i	1.75682i	0.477904	+	0.878412i	$0.341397\pi$
$101$	180107.i	1.75682i	−0.477904	+	0.878412i	$0.658603\pi$
$102$	0	0
$103$	−24679.3	−0.229214	−0.114607	−	0.993411i	$-0.536561\pi$
$103$	−24679.3	−0.229214	−0.114607	+	0.993411i	$0.536561\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14238.3i	− 0.120226i	−0.998192	−	0.0601132i	$-0.980854\pi$
$107$	− 14238.3i	− 0.120226i	0.998192	−	0.0601132i	$-0.0191462\pi$
$108$	0	0
$109$	−180797.	−1.45755	−0.728777	−	0.684752i	$-0.759911\pi$
$109$	−180797.	−1.45755	−0.728777	+	0.684752i	$0.759911\pi$
$110$	0	0
$111$	−321739.	−2.47854
$112$	0	0
$113$	−46481.6	−0.342440	−0.171220	−	0.985233i	$-0.554771\pi$
$113$	−46481.6	−0.342440	−0.171220	+	0.985233i	$0.554771\pi$
$114$	0	0
$115$	−105070.	−0.740858
$116$	0	0
$117$	186822.i	1.26172i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3142.56	−0.0195128
$122$	0	0
$123$	− 394086.i	− 2.34870i
$124$	0	0
$125$	123239.i	0.705462i
$126$	0	0
$127$	− 129698.i	− 0.713551i	−0.934190	−	0.356775i	$-0.883876\pi$
$127$	− 129698.i	− 0.713551i	0.934190	−	0.356775i	$-0.116124\pi$
$128$	0	0
$129$	− 445249.i	− 2.35575i
$130$	0	0
$131$	−159087.	−0.809945	−0.404973	−	0.914329i	$-0.632719\pi$
$131$	−159087.	−0.809945	−0.404973	+	0.914329i	$0.632719\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 98726.3i	− 0.466228i
$136$	0	0
$137$	−130226.	−0.592782	−0.296391	−	0.955067i	$-0.595783\pi$
$137$	−130226.	−0.592782	−0.296391	+	0.955067i	$0.595783\pi$
$138$	0	0
$139$	317422.	1.39348	0.696739	−	0.717325i	$-0.254634\pi$
$139$	317422.	1.39348	0.696739	+	0.717325i	$0.254634\pi$
$140$	0	0
$141$	523807.	2.21883
$142$	0	0
$143$	−259445.	−1.06097
$144$	0	0
$145$	172240.i	0.680320i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19846.2	−0.0732339	−0.0366169	−	0.999329i	$-0.511658\pi$
$149$	−19846.2	−0.0732339	−0.0366169	+	0.999329i	$0.511658\pi$
$150$	0	0
$151$	210098.i	0.749858i	0.927053	+	0.374929i	$0.122333\pi$
$151$	210098.i	0.749858i	−0.927053	+	0.374929i	$0.877667\pi$
$152$	0	0
$153$	− 433556.i	− 1.49733i
$154$	0	0
$155$	71226.8i	0.238130i
$156$	0	0
$157$	395680.i	1.28114i	0.767902	+	0.640568i	$0.221301\pi$
$157$	395680.i	1.28114i	−0.767902	+	0.640568i	$0.778699\pi$
$158$	0	0
$159$	−370439.	−1.16205
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 219700.i	− 0.647680i	−0.946112	−	0.323840i	$-0.895026\pi$
$163$	− 219700.i	− 0.647680i	0.946112	−	0.323840i	$-0.104974\pi$
$164$	0	0
$165$	820035.	2.34489
$166$	0	0
$167$	−28110.7	−0.0779974	−0.0389987	−	0.999239i	$-0.512417\pi$
$167$	−28110.7	−0.0779974	−0.0389987	+	0.999239i	$0.512417\pi$
$168$	0	0
$169$	−38659.7	−0.104122
$170$	0	0
$171$	−408482.	−1.06827
$172$	0	0
$173$	430301.i	1.09309i	0.837429	+	0.546546i	$0.184058\pi$
$173$	430301.i	1.09309i	−0.837429	+	0.546546i	$0.815942\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−696586.	−1.67125
$178$	0	0
$179$	41573.3i	0.0969800i	0.998824	+	0.0484900i	$0.0154409\pi$
$179$	41573.3i	0.0969800i	−0.998824	+	0.0484900i	$0.984559\pi$
$180$	0	0
$181$	273268.i	0.620001i	0.950736	+	0.310000i	$0.100329\pi$
$181$	273268.i	0.620001i	−0.950736	+	0.310000i	$0.899671\pi$
$182$	0	0
$183$	− 1.00961e6i	− 2.22858i
$184$	0	0
$185$	1.21753e6i	2.61547i
$186$	0	0
$187$	602090.	1.25909
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 496861.i	− 0.985488i	−0.870174	−	0.492744i	$-0.835994\pi$
$191$	− 496861.i	− 0.985488i	0.870174	−	0.492744i	$-0.164006\pi$
$192$	0	0
$193$	−229972.	−0.444408	−0.222204	−	0.975000i	$-0.571325\pi$
$193$	−229972.	−0.444408	−0.222204	+	0.975000i	$0.571325\pi$
$194$	0	0
$195$	1.29575e6	2.44025
$196$	0	0
$197$	67861.0	0.124582	0.0622909	−	0.998058i	$-0.480159\pi$
$197$	67861.0	0.124582	0.0622909	+	0.998058i	$0.480159\pi$
$198$	0	0
$199$	663781.	1.18821	0.594103	−	0.804389i	$-0.297507\pi$
$199$	663781.	1.18821	0.594103	+	0.804389i	$0.297507\pi$
$200$	0	0
$201$	245276.i	0.428219i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.49130e6	−2.47846
$206$	0	0
$207$	− 350329.i	− 0.568264i
$208$	0	0
$209$	− 567269.i	− 0.898305i
$210$	0	0
$211$	− 129670.i	− 0.200509i	−0.994962	−	0.100254i	$-0.968034\pi$
$211$	− 129670.i	− 0.200509i	0.994962	−	0.100254i	$-0.0319656\pi$
$212$	0	0
$213$	1.32751e6i	2.00489i
$214$	0	0
$215$	−1.68492e6	−2.48589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 1.50348e6i	− 2.11830i
$220$	0	0
$221$	951371.	1.31030
$222$	0	0
$223$	1.16536e6	1.56927	0.784637	−	0.619956i	$-0.212849\pi$
$223$	1.16536e6	1.56927	0.784637	+	0.619956i	$0.212849\pi$
$224$	0	0
$225$	−1.32273e6	−1.74187
$226$	0	0
$227$	1.09066e6	1.40484	0.702420	−	0.711763i	$-0.252103\pi$
$227$	1.09066e6	1.40484	0.702420	+	0.711763i	$0.252103\pi$
$228$	0	0
$229$	− 43063.3i	− 0.0542649i	−0.999632	−	0.0271324i	$-0.991362\pi$
$229$	− 43063.3i	− 0.0542649i	0.999632	−	0.0271324i	$-0.00863758\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	801449.	0.967133	0.483567	−	0.875308i	$-0.339341\pi$
$233$	801449.	0.967133	0.483567	+	0.875308i	$0.339341\pi$
$234$	0	0
$235$	− 1.98220e6i	− 2.34141i
$236$	0	0
$237$	− 427014.i	− 0.493823i
$238$	0	0
$239$	− 841735.i	− 0.953193i	−0.879122	−	0.476596i	$-0.841870\pi$
$239$	− 841735.i	− 0.953193i	0.879122	−	0.476596i	$-0.158130\pi$
$240$	0	0
$241$	− 826079.i	− 0.916176i	−0.888907	−	0.458088i	$-0.848534\pi$
$241$	− 826079.i	− 0.916176i	0.888907	−	0.458088i	$-0.151466\pi$
$242$	0	0
$243$	−1.31050e6	−1.42370
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 896351.i	− 0.934837i
$248$	0	0
$249$	395294.	0.404038
$250$	0	0
$251$	−1.51586e6	−1.51871	−0.759357	−	0.650674i	$-0.774487\pi$
$251$	−1.51586e6	−1.51871	−0.759357	+	0.650674i	$0.774487\pi$
$252$	0	0
$253$	486511.	0.477849
$254$	0	0
$255$	−3.00703e6	−2.89592
$256$	0	0
$257$	− 419956.i	− 0.396617i	−0.980140	−	0.198309i	$-0.936455\pi$
$257$	− 419956.i	− 0.396617i	0.980140	−	0.198309i	$-0.0635448\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−574288.	−0.521830
$262$	0	0
$263$	1.25652e6i	1.12016i	0.828440	+	0.560078i	$0.189229\pi$
$263$	1.25652e6i	1.12016i	−0.828440	+	0.560078i	$0.810771\pi$
$264$	0	0
$265$	1.40182e6i	1.22625i
$266$	0	0
$267$	− 1.09141e6i	− 0.936932i
$268$	0	0
$269$	273397.i	0.230363i	0.993344	+	0.115182i	$0.0367450\pi$
$269$	273397.i	0.230363i	−0.993344	+	0.115182i	$0.963255\pi$
$270$	0	0
$271$	−772690.	−0.639119	−0.319560	−	0.947566i	$-0.603535\pi$
$271$	−772690.	−0.639119	−0.319560	+	0.947566i	$0.603535\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.83691e6i	− 1.46473i
$276$	0	0
$277$	31861.9	0.0249501	0.0124750	−	0.999922i	$-0.496029\pi$
$277$	31861.9	0.0249501	0.0124750	+	0.999922i	$0.496029\pi$
$278$	0	0
$279$	−237487.	−0.182654
$280$	0	0
$281$	510770.	0.385887	0.192943	−	0.981210i	$-0.438197\pi$
$281$	510770.	0.385887	0.192943	+	0.981210i	$0.438197\pi$
$282$	0	0
$283$	−1.94759e6	−1.44554	−0.722771	−	0.691088i	$-0.757132\pi$
$283$	−1.94759e6	−1.44554	−0.722771	+	0.691088i	$0.757132\pi$
$284$	0	0
$285$	2.83312e6i	2.06611i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−787976.	−0.554968
$290$	0	0
$291$	1.22212e6i	0.846020i
$292$	0	0
$293$	− 1.59231e6i	− 1.08357i	−0.840516	−	0.541787i	$-0.817748\pi$
$293$	− 1.59231e6i	− 1.08357i	0.840516	−	0.541787i	$-0.182252\pi$
$294$	0	0
$295$	2.63603e6i	1.76358i
$296$	0	0
$297$	457136.i	0.300715i
$298$	0	0
$299$	768743.	0.497283
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.16506e6i	2.60624i
$304$	0	0
$305$	−3.82060e6	−2.35170
$306$	0	0
$307$	−148691.	−0.0900404	−0.0450202	−	0.998986i	$-0.514335\pi$
$307$	−148691.	−0.0900404	−0.0450202	+	0.998986i	$0.514335\pi$
$308$	0	0
$309$	−570720.	−0.340037
$310$	0	0
$311$	−729282.	−0.427557	−0.213779	−	0.976882i	$-0.568577\pi$
$311$	−729282.	−0.427557	−0.213779	+	0.976882i	$0.568577\pi$
$312$	0	0
$313$	1.82131e6i	1.05081i	0.850853	+	0.525404i	$0.176086\pi$
$313$	1.82131e6i	1.05081i	−0.850853	+	0.525404i	$0.823914\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.58021e6	0.883213	0.441607	−	0.897209i	$-0.354409\pi$
$317$	1.58021e6	0.883213	0.441607	+	0.897209i	$0.354409\pi$
$318$	0	0
$319$	− 797529.i	− 0.438803i
$320$	0	0
$321$	− 329268.i	− 0.178355i
$322$	0	0
$323$	2.08015e6i	1.10940i
$324$	0	0
$325$	− 2.90254e6i	− 1.52430i
$326$	0	0
$327$	−4.18100e6	−2.16227
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 3.13334e6i	− 1.57195i	−0.618260	−	0.785974i	$-0.712162\pi$
$331$	− 3.13334e6i	− 1.57195i	0.618260	−	0.785974i	$-0.287838\pi$
$332$	0	0
$333$	−4.05953e6	−2.00616
$334$	0	0
$335$	928178.	0.451876
$336$	0	0
$337$	3.16854e6	1.51979	0.759897	−	0.650043i	$-0.225249\pi$
$337$	3.16854e6	1.51979	0.759897	+	0.650043i	$0.225249\pi$
$338$	0	0
$339$	−1.07490e6	−0.508008
$340$	0	0
$341$	− 329804.i	− 0.153593i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.42979e6	−1.09906
$346$	0	0
$347$	− 274345.i	− 0.122313i	−0.998128	−	0.0611565i	$-0.980521\pi$
$347$	− 274345.i	− 0.122313i	0.998128	−	0.0611565i	$-0.0194789\pi$
$348$	0	0
$349$	528446.i	0.232240i	0.993235	+	0.116120i	$0.0370457\pi$
$349$	528446.i	0.232240i	−0.993235	+	0.116120i	$0.962954\pi$
$350$	0	0
$351$	722328.i	0.312944i
$352$	0	0
$353$	155996.i	0.0666310i	0.999445	+	0.0333155i	$0.0106066\pi$
$353$	155996.i	0.0666310i	−0.999445	+	0.0333155i	$0.989393\pi$
$354$	0	0
$355$	5.02360e6	2.11565
$356$	0	0
$357$	0	0
$358$	0	0
$359$	298899.i	0.122402i	0.998125	+	0.0612009i	$0.0194930\pi$
$359$	298899.i	0.122402i	−0.998125	+	0.0612009i	$0.980507\pi$
$360$	0	0
$361$	−516250.	−0.208493
$362$	0	0
$363$	−72672.9	−0.0289472
$364$	0	0
$365$	−5.68950e6	−2.23533
$366$	0	0
$367$	−2.72918e6	−1.05771	−0.528855	−	0.848712i	$-0.677379\pi$
$367$	−2.72918e6	−1.05771	−0.528855	+	0.848712i	$0.677379\pi$
$368$	0	0
$369$	− 4.97237e6i	− 1.90107i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.95224e6	0.726541	0.363270	−	0.931684i	$-0.381660\pi$
$373$	1.95224e6	0.726541	0.363270	+	0.931684i	$0.381660\pi$
$374$	0	0
$375$	2.84995e6i	1.04655i
$376$	0	0
$377$	− 1.26019e6i	− 0.456648i
$378$	0	0
$379$	− 223755.i	− 0.0800157i	−0.999199	−	0.0400079i	$-0.987262\pi$
$379$	− 223755.i	− 0.0800157i	0.999199	−	0.0400079i	$-0.0127383\pi$
$380$	0	0
$381$	− 2.99932e6i	− 1.05855i
$382$	0	0
$383$	−4.09699e6	−1.42714	−0.713572	−	0.700582i	$-0.752924\pi$
$383$	−4.09699e6	−1.42714	−0.713572	+	0.700582i	$0.752924\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.61792e6i	− 1.90677i
$388$	0	0
$389$	1.37706e6	0.461400	0.230700	−	0.973025i	$-0.425898\pi$
$389$	1.37706e6	0.461400	0.230700	+	0.973025i	$0.425898\pi$
$390$	0	0
$391$	−1.78401e6	−0.590141
$392$	0	0
$393$	−3.67894e6	−1.20155
$394$	0	0
$395$	−1.61591e6	−0.521105
$396$	0	0
$397$	− 4.47452e6i	− 1.42485i	−0.701747	−	0.712427i	$-0.747596\pi$
$397$	− 4.47452e6i	− 1.42485i	0.701747	−	0.712427i	$-0.252404\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.96522e6	1.85253	0.926266	−	0.376869i	$-0.122999\pi$
$401$	5.96522e6	1.85253	0.926266	+	0.376869i	$0.122999\pi$
$402$	0	0
$403$	− 521129.i	− 0.159839i
$404$	0	0
$405$	3.92179e6i	1.18808i
$406$	0	0
$407$	− 5.63757e6i	− 1.68696i
$408$	0	0
$409$	1.22131e6i	0.361008i	0.983574	+	0.180504i	$0.0577729\pi$
$409$	1.22131e6i	0.361008i	−0.983574	+	0.180504i	$0.942227\pi$
$410$	0	0
$411$	−3.01152e6	−0.879389
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 1.49588e6i	− 0.426359i
$416$	0	0
$417$	7.34051e6	2.06722
$418$	0	0
$419$	3.12252e6	0.868901	0.434450	−	0.900696i	$-0.356943\pi$
$419$	3.12252e6	0.868901	0.434450	+	0.900696i	$0.356943\pi$
$420$	0	0
$421$	−937555.	−0.257805	−0.128903	−	0.991657i	$-0.541145\pi$
$421$	−937555.	−0.257805	−0.128903	+	0.991657i	$0.541145\pi$
$422$	0	0
$423$	6.60912e6	1.79594
$424$	0	0
$425$	6.73587e6i	1.80893i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.99976e6	−1.57395
$430$	0	0
$431$	272216.i	0.0705864i	0.999377	+	0.0352932i	$0.0112365\pi$
$431$	272216.i	0.0705864i	−0.999377	+	0.0352932i	$0.988763\pi$
$432$	0	0
$433$	− 6.33614e6i	− 1.62407i	−0.583608	−	0.812035i	$-0.698360\pi$
$433$	− 6.33614e6i	− 1.62407i	0.583608	−	0.812035i	$-0.301640\pi$
$434$	0	0
$435$	3.98311e6i	1.00925i
$436$	0	0
$437$	1.68084e6i	0.421039i
$438$	0	0
$439$	118231.	0.0292798	0.0146399	−	0.999893i	$-0.495340\pi$
$439$	118231.	0.0292798	0.0146399	+	0.999893i	$0.495340\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.47333e6i	1.08298i	0.840706	+	0.541492i	$0.182140\pi$
$443$	4.47333e6i	1.08298i	−0.840706	+	0.541492i	$0.817860\pi$
$444$	0	0
$445$	−4.13011e6	−0.988694
$446$	0	0
$447$	−458952.	−0.108642
$448$	0	0
$449$	1.27906e6	0.299416	0.149708	−	0.988730i	$-0.452167\pi$
$449$	1.27906e6	0.299416	0.149708	+	0.988730i	$0.452167\pi$
$450$	0	0
$451$	6.90525e6	1.59859
$452$	0	0
$453$	4.85860e6i	1.11241i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.52132e6	−1.01269	−0.506343	−	0.862332i	$-0.669003\pi$
$457$	−4.52132e6	−1.01269	−0.506343	+	0.862332i	$0.669003\pi$
$458$	0	0
$459$	− 1.67630e6i	− 0.371381i
$460$	0	0
$461$	1.58045e6i	0.346360i	0.984890	+	0.173180i	$0.0554043\pi$
$461$	1.58045e6i	0.346360i	−0.984890	+	0.173180i	$0.944596\pi$
$462$	0	0
$463$	− 7.54852e6i	− 1.63648i	−0.574880	−	0.818238i	$-0.694951\pi$
$463$	− 7.54852e6i	− 1.63648i	0.574880	−	0.818238i	$-0.305049\pi$
$464$	0	0
$465$	1.64715e6i	0.353265i
$466$	0	0
$467$	6.55082e6	1.38996	0.694982	−	0.719028i	$-0.255412\pi$
$467$	6.55082e6	1.38996	0.694982	+	0.719028i	$0.255412\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.15026e6i	1.90056i
$472$	0	0
$473$	7.80175e6	1.60339
$474$	0	0
$475$	6.34632e6	1.29059
$476$	0	0
$477$	−4.67401e6	−0.940576
$478$	0	0
$479$	4.19540e6	0.835477	0.417739	−	0.908567i	$-0.362823\pi$
$479$	4.19540e6	0.835477	0.417739	+	0.908567i	$0.362823\pi$
$480$	0	0
$481$	− 8.90801e6i	− 1.75557i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.62476e6	0.892759
$486$	0	0
$487$	− 2.79453e6i	− 0.533932i	−0.963706	−	0.266966i	$-0.913979\pi$
$487$	− 2.79453e6i	− 0.533932i	0.963706	−	0.266966i	$-0.0860212\pi$
$488$	0	0
$489$	− 5.08064e6i	− 0.960830i
$490$	0	0
$491$	− 3.03606e6i	− 0.568338i	−0.958774	−	0.284169i	$-0.908282\pi$
$491$	− 3.03606e6i	− 0.568338i	0.958774	−	0.284169i	$-0.0917177\pi$
$492$	0	0
$493$	2.92450e6i	0.541919i
$494$	0	0
$495$	1.03468e7	1.89798
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 4.51171e6i	− 0.811128i	−0.914067	−	0.405564i	$-0.867075\pi$
$499$	− 4.51171e6i	− 0.811128i	0.914067	−	0.405564i	$-0.132925\pi$
$500$	0	0
$501$	−650070.	−0.115709
$502$	0	0
$503$	−5.19848e6	−0.916129	−0.458065	−	0.888919i	$-0.651457\pi$
$503$	−5.19848e6	−0.916129	−0.458065	+	0.888919i	$0.651457\pi$
$504$	0	0
$505$	1.57615e7	2.75023
$506$	0	0
$507$	−894021.	−0.154464
$508$	0	0
$509$	− 4.14280e6i	− 0.708761i	−0.935101	−	0.354380i	$-0.884692\pi$
$509$	− 4.14280e6i	− 0.708761i	0.935101	−	0.354380i	$-0.115308\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.57935e6	−0.264963
$514$	0	0
$515$	2.15973e6i	0.358823i
$516$	0	0
$517$	9.17825e6i	1.51020i
$518$	0	0
$519$	9.95088e6i	1.62160i
$520$	0	0
$521$	4.90738e6i	0.792055i	0.918239	+	0.396027i	$0.129611\pi$
$521$	4.90738e6i	0.792055i	−0.918239	+	0.396027i	$0.870389\pi$
$522$	0	0
$523$	8.99644e6	1.43819	0.719095	−	0.694911i	$-0.244556\pi$
$523$	8.99644e6	1.43819	0.719095	+	0.694911i	$0.244556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.20938e6i	0.189686i
$528$	0	0
$529$	4.99480e6	0.776030
$530$	0	0
$531$	−8.78915e6	−1.35273
$532$	0	0
$533$	1.09111e7	1.66360
$534$	0	0
$535$	−1.24602e6	−0.188209
$536$	0	0
$537$	961400.i	0.143869i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.11507e6	0.163799	0.0818993	−	0.996641i	$-0.473901\pi$
$541$	1.11507e6	0.163799	0.0818993	+	0.996641i	$0.473901\pi$
$542$	0	0
$543$	6.31943e6i	0.919768i
$544$	0	0
$545$	1.58218e7i	2.28173i
$546$	0	0
$547$	164832.i	0.0235545i	0.999931	+	0.0117772i	$0.00374889\pi$
$547$	164832.i	0.0235545i	−0.999931	+	0.0117772i	$0.996251\pi$
$548$	0	0
$549$	− 1.27388e7i	− 1.80384i
$550$	0	0
$551$	2.75537e6	0.386634
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.81558e7i	3.88004i
$556$	0	0
$557$	−9.13565e6	−1.24768	−0.623838	−	0.781554i	$-0.714427\pi$
$557$	−9.13565e6	−1.24768	−0.623838	+	0.781554i	$0.714427\pi$
$558$	0	0
$559$	1.23277e7	1.66860
$560$	0	0
$561$	1.39236e7	1.86786
$562$	0	0
$563$	7.69699e6	1.02341	0.511705	−	0.859161i	$-0.329014\pi$
$563$	7.69699e6	1.02341	0.511705	+	0.859161i	$0.329014\pi$
$564$	0	0
$565$	4.06767e6i	0.536074i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.92396e6	0.767064	0.383532	−	0.923528i	$-0.374708\pi$
$569$	5.92396e6	0.767064	0.383532	+	0.923528i	$0.374708\pi$
$570$	0	0
$571$	− 8.59815e6i	− 1.10361i	−0.833974	−	0.551804i	$-0.813940\pi$
$571$	− 8.59815e6i	− 1.10361i	0.833974	−	0.551804i	$-0.186060\pi$
$572$	0	0
$573$	− 1.14901e7i	− 1.46197i
$574$	0	0
$575$	5.44283e6i	0.686524i
$576$	0	0
$577$	6.90585e6i	0.863531i	0.901986	+	0.431765i	$0.142109\pi$
$577$	6.90585e6i	0.863531i	−0.901986	+	0.431765i	$0.857891\pi$
$578$	0	0
$579$	−5.31819e6	−0.659277
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 6.49092e6i	− 0.790924i
$584$	0	0
$585$	1.63491e7	1.97517
$586$	0	0
$587$	−9.16568e6	−1.09792	−0.548958	−	0.835850i	$-0.684975\pi$
$587$	−9.16568e6	−1.09792	−0.548958	+	0.835850i	$0.684975\pi$
$588$	0	0
$589$	1.13944e6	0.135332
$590$	0	0
$591$	1.56931e6	0.184817
$592$	0	0
$593$	3.81010e6i	0.444938i	0.974940	+	0.222469i	$0.0714116\pi$
$593$	3.81010e6i	0.444938i	−0.974940	+	0.222469i	$0.928588\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.53502e7	1.76270
$598$	0	0
$599$	− 3.66114e6i	− 0.416917i	−0.978031	−	0.208459i	$-0.933155\pi$
$599$	− 3.66114e6i	− 0.416917i	0.978031	−	0.208459i	$-0.0668446\pi$
$600$	0	0
$601$	− 1.35098e6i	− 0.152568i	−0.997086	−	0.0762840i	$-0.975694\pi$
$601$	− 1.35098e6i	− 0.152568i	0.997086	−	0.0762840i	$-0.0243056\pi$
$602$	0	0
$603$	3.09477e6i	0.346605i
$604$	0	0
$605$	275010.i	0.0305464i
$606$	0	0
$607$	1.21732e7	1.34102	0.670509	−	0.741901i	$-0.266076\pi$
$607$	1.21732e7	1.34102	0.670509	+	0.741901i	$0.266076\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.45027e7i	1.57161i
$612$	0	0
$613$	−2.77702e6	−0.298489	−0.149245	−	0.988800i	$-0.547684\pi$
$613$	−2.77702e6	−0.298489	−0.149245	+	0.988800i	$0.547684\pi$
$614$	0	0
$615$	−3.44870e7	−3.67678
$616$	0	0
$617$	−3.36595e6	−0.355955	−0.177978	−	0.984035i	$-0.556955\pi$
$617$	−3.36595e6	−0.355955	−0.177978	+	0.984035i	$0.556955\pi$
$618$	0	0
$619$	1.67628e7	1.75841	0.879207	−	0.476440i	$-0.158073\pi$
$619$	1.67628e7	1.75841	0.879207	+	0.476440i	$0.158073\pi$
$620$	0	0
$621$	− 1.35451e6i	− 0.140946i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.38161e6	−0.346277
$626$	0	0
$627$	− 1.31183e7i	− 1.33263i
$628$	0	0
$629$	2.06727e7i	2.08339i
$630$	0	0
$631$	7.07244e6i	0.707124i	0.935411	+	0.353562i	$0.115030\pi$
$631$	7.07244e6i	0.707124i	−0.935411	+	0.353562i	$0.884970\pi$
$632$	0	0
$633$	− 2.99867e6i	− 0.297454i
$634$	0	0
$635$	−1.13501e7	−1.11703
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.67499e7i	1.62278i
$640$	0	0
$641$	−1.75827e7	−1.69021	−0.845105	−	0.534600i	$-0.820462\pi$
$641$	−1.75827e7	−1.69021	−0.845105	+	0.534600i	$0.820462\pi$
$642$	0	0
$643$	−1.07973e7	−1.02988	−0.514940	−	0.857226i	$-0.672186\pi$
$643$	−1.07973e7	−1.02988	−0.514940	+	0.857226i	$0.672186\pi$
$644$	0	0
$645$	−3.89644e7	−3.68781
$646$	0	0
$647$	−3.53076e6	−0.331594	−0.165797	−	0.986160i	$-0.553020\pi$
$647$	−3.53076e6	−0.331594	−0.165797	+	0.986160i	$0.553020\pi$
$648$	0	0
$649$	− 1.22057e7i	− 1.13750i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.41325e7	−1.29699	−0.648493	−	0.761221i	$-0.724600\pi$
$653$	−1.41325e7	−1.29699	−0.648493	+	0.761221i	$0.724600\pi$
$654$	0	0
$655$	1.39219e7i	1.26793i
$656$	0	0
$657$	− 1.89702e7i	− 1.71458i
$658$	0	0
$659$	− 8.40793e6i	− 0.754181i	−0.926176	−	0.377090i	$-0.876925\pi$
$659$	− 8.40793e6i	− 0.754181i	0.926176	−	0.377090i	$-0.123075\pi$
$660$	0	0
$661$	− 6.54101e6i	− 0.582293i	−0.956679	−	0.291146i	$-0.905963\pi$
$661$	− 6.54101e6i	− 0.582293i	0.956679	−	0.291146i	$-0.0940367\pi$
$662$	0	0
$663$	2.20008e7	1.94382
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.36310e6i	0.205669i
$668$	0	0
$669$	2.69495e7	2.32801
$670$	0	0
$671$	1.76907e7	1.51683
$672$	0	0
$673$	−1.09027e6	−0.0927890	−0.0463945	−	0.998923i	$-0.514773\pi$
$673$	−1.09027e6	−0.0927890	−0.0463945	+	0.998923i	$0.514773\pi$
$674$	0	0
$675$	−5.11421e6	−0.432035
$676$	0	0
$677$	− 1.62072e6i	− 0.135905i	−0.997689	−	0.0679525i	$-0.978353\pi$
$677$	− 1.62072e6i	− 0.135905i	0.997689	−	0.0679525i	$-0.0216466\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.52221e7	2.08407
$682$	0	0
$683$	2.11839e7i	1.73761i	0.495150	+	0.868807i	$0.335113\pi$
$683$	2.11839e7i	1.73761i	−0.495150	+	0.868807i	$0.664887\pi$
$684$	0	0
$685$	1.13962e7i	0.927973i
$686$	0	0
$687$	− 995856.i	− 0.0805017i
$688$	0	0
$689$	− 1.02564e7i	− 0.823089i
$690$	0	0
$691$	−9.47759e6	−0.755097	−0.377549	−	0.925990i	$-0.623233\pi$
$691$	−9.47759e6	−0.755097	−0.377549	+	0.925990i	$0.623233\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 2.77781e7i	− 2.18142i
$696$	0	0
$697$	−2.53212e7	−1.97425
$698$	0	0
$699$	1.85338e7	1.43474
$700$	0	0
$701$	1.44960e7	1.11417	0.557087	−	0.830454i	$-0.311919\pi$
$701$	1.44960e7	1.11417	0.557087	+	0.830454i	$0.311919\pi$
$702$	0	0
$703$	1.94771e7	1.48640
$704$	0	0
$705$	− 4.58391e7i	− 3.47347i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.32615e7	1.73789	0.868945	−	0.494908i	$-0.164798\pi$
$709$	2.32615e7	1.73789	0.868945	+	0.494908i	$0.164798\pi$
$710$	0	0
$711$	− 5.38784e6i	− 0.399706i
$712$	0	0
$713$	977221.i	0.0719895i
$714$	0	0
$715$	2.27044e7i	1.66091i
$716$	0	0
$717$	− 1.94655e7i	− 1.41406i
$718$	0	0
$719$	−3.79131e6	−0.273506	−0.136753	−	0.990605i	$-0.543667\pi$
$719$	−3.79131e6	−0.273506	−0.136753	+	0.990605i	$0.543667\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 1.91034e7i	− 1.35914i
$724$	0	0
$725$	8.92235e6	0.630426
$726$	0	0
$727$	1.04927e7	0.736294	0.368147	−	0.929768i	$-0.379992\pi$
$727$	1.04927e7	0.736294	0.368147	+	0.929768i	$0.379992\pi$
$728$	0	0
$729$	−1.94158e7	−1.35312
$730$	0	0
$731$	−2.86086e7	−1.98017
$732$	0	0
$733$	9.17803e6i	0.630942i	0.948935	+	0.315471i	$0.102163\pi$
$733$	9.17803e6i	0.630942i	−0.948935	+	0.315471i	$0.897837\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.29778e6	−0.291458
$738$	0	0
$739$	2.28875e7i	1.54165i	0.637044	+	0.770827i	$0.280157\pi$
$739$	2.28875e7i	1.54165i	−0.637044	+	0.770827i	$0.719843\pi$
$740$	0	0
$741$	− 2.07285e7i	− 1.38683i
$742$	0	0
$743$	8.13890e6i	0.540871i	0.962738	+	0.270435i	$0.0871676\pi$
$743$	8.13890e6i	0.540871i	−0.962738	+	0.270435i	$0.912832\pi$
$744$	0	0
$745$	1.73677e6i	0.114644i
$746$	0	0
$747$	4.98761e6	0.327033
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 8.00773e6i	− 0.518095i	−0.965865	−	0.259048i	$-0.916591\pi$
$751$	− 8.00773e6i	− 0.518095i	0.965865	−	0.259048i	$-0.0834086\pi$
$752$	0	0
$753$	−3.50550e7	−2.25301
$754$	0	0
$755$	1.83860e7	1.17387
$756$	0	0
$757$	−1.42534e7	−0.904020	−0.452010	−	0.892013i	$-0.649293\pi$
$757$	−1.42534e7	−0.904020	−0.452010	+	0.892013i	$0.649293\pi$
$758$	0	0
$759$	1.12508e7	0.708887
$760$	0	0
$761$	− 8.02161e6i	− 0.502111i	−0.967973	−	0.251056i	$-0.919222\pi$
$761$	− 8.02161e6i	− 0.502111i	0.967973	−	0.251056i	$-0.0807777\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.79411e7	−2.34399
$766$	0	0
$767$	− 1.92864e7i	− 1.18376i
$768$	0	0
$769$	− 6.02804e6i	− 0.367587i	−0.982965	−	0.183794i	$-0.941162\pi$
$769$	− 6.02804e6i	− 0.367587i	0.982965	−	0.183794i	$-0.0588378\pi$
$770$	0	0
$771$	− 9.71166e6i	− 0.588380i
$772$	0	0
$773$	− 5.98145e6i	− 0.360046i	−0.983662	−	0.180023i	$-0.942383\pi$
$773$	− 5.98145e6i	− 0.360046i	0.983662	−	0.180023i	$-0.0576172\pi$
$774$	0	0
$775$	3.68968e6	0.220666
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.38568e7i	1.40854i
$780$	0	0
$781$	−2.32610e7	−1.36459
$782$	0	0
$783$	−2.22043e6	−0.129429
$784$	0	0
$785$	3.46265e7	2.00556
$786$	0	0
$787$	−1.08999e7	−0.627317	−0.313659	−	0.949536i	$-0.601555\pi$
$787$	−1.08999e7	−0.627317	−0.313659	+	0.949536i	$0.601555\pi$
$788$	0	0
$789$	2.90574e7i	1.66175i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.79533e7	1.57852
$794$	0	0
$795$	3.24177e7i	1.81913i
$796$	0	0
$797$	9.04729e6i	0.504514i	0.967660	+	0.252257i	$0.0811728\pi$
$797$	9.04729e6i	0.504514i	−0.967660	+	0.252257i	$0.918827\pi$
$798$	0	0
$799$	− 3.36562e7i	− 1.86508i
$800$	0	0
$801$	− 1.37708e7i	− 0.758363i
$802$	0	0
$803$	2.63443e7	1.44178
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.32242e6i	0.341743i
$808$	0	0
$809$	2.67854e7	1.43889	0.719445	−	0.694549i	$-0.244396\pi$
$809$	2.67854e7	1.43889	0.719445	+	0.694549i	$0.244396\pi$
$810$	0	0
$811$	−1.53751e7	−0.820853	−0.410427	−	0.911894i	$-0.634620\pi$
$811$	−1.53751e7	−0.820853	−0.410427	+	0.911894i	$0.634620\pi$
$812$	0	0
$813$	−1.78688e7	−0.948130
$814$	0	0
$815$	−1.92262e7	−1.01391
$816$	0	0
$817$	2.69541e7i	1.41276i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.20149e7	−1.13988	−0.569940	−	0.821686i	$-0.693034\pi$
$821$	−2.20149e7	−1.13988	−0.569940	+	0.821686i	$0.693034\pi$
$822$	0	0
$823$	− 4.40589e6i	− 0.226743i	−0.993553	−	0.113371i	$-0.963835\pi$
$823$	− 4.40589e6i	− 0.226743i	0.993553	−	0.113371i	$-0.0361650\pi$
$824$	0	0
$825$	− 4.24794e7i	− 2.17292i
$826$	0	0
$827$	− 1.96164e7i	− 0.997369i	−0.866783	−	0.498685i	$-0.833817\pi$
$827$	− 1.96164e7i	− 0.997369i	0.866783	−	0.498685i	$-0.166183\pi$
$828$	0	0
$829$	− 5.76695e6i	− 0.291447i	−0.989325	−	0.145724i	$-0.953449\pi$
$829$	− 5.76695e6i	− 0.291447i	0.989325	−	0.145724i	$-0.0465511\pi$
$830$	0	0
$831$	736819.	0.0370133
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.46001e6i	0.122101i
$836$	0	0
$837$	−918219.	−0.0453036
$838$	0	0
$839$	−1.04959e6	−0.0514774	−0.0257387	−	0.999669i	$-0.508194\pi$
$839$	−1.04959e6	−0.0514774	−0.0257387	+	0.999669i	$0.508194\pi$
$840$	0	0
$841$	−1.66374e7	−0.811137
$842$	0	0
$843$	1.18118e7	0.572461
$844$	0	0
$845$	3.38317e6i	0.162998i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.50387e7	−2.14445
$850$	0	0
$851$	1.67043e7i	0.790687i
$852$	0	0
$853$	2.19938e7i	1.03497i	0.855692	+	0.517485i	$0.173132\pi$
$853$	2.19938e7i	1.03497i	−0.855692	+	0.517485i	$0.826868\pi$
$854$	0	0
$855$	3.57469e7i	1.67233i
$856$	0	0
$857$	− 1.63145e7i	− 0.758789i	−0.925235	−	0.379394i	$-0.876132\pi$
$857$	− 1.63145e7i	− 0.758789i	0.925235	−	0.379394i	$-0.123868\pi$
$858$	0	0
$859$	−3.14463e7	−1.45408	−0.727038	−	0.686597i	$-0.759104\pi$
$859$	−3.14463e7	−1.45408	−0.727038	+	0.686597i	$0.759104\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.94425e7i	− 0.888640i	−0.895868	−	0.444320i	$-0.853445\pi$
$863$	− 1.94425e7i	− 0.888640i	0.895868	−	0.444320i	$-0.146555\pi$
$864$	0	0
$865$	3.76563e7	1.71119
$866$	0	0
$867$	−1.82222e7	−0.823293
$868$	0	0
$869$	7.48223e6	0.336110
$870$	0	0
$871$	−6.79099e6	−0.303311
$872$	0	0
$873$	1.54200e7i	0.684778i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.15776e7	1.82541	0.912704	−	0.408621i	$-0.133990\pi$
$877$	4.15776e7	1.82541	0.912704	+	0.408621i	$0.133990\pi$
$878$	0	0
$879$	− 3.68228e7i	− 1.60748i
$880$	0	0
$881$	− 7.22354e6i	− 0.313553i	−0.987634	−	0.156776i	$-0.949890\pi$
$881$	− 7.22354e6i	− 0.313553i	0.987634	−	0.156776i	$-0.0501102\pi$
$882$	0	0
$883$	− 2.73652e7i	− 1.18113i	−0.806992	−	0.590563i	$-0.798906\pi$
$883$	− 2.73652e7i	− 1.18113i	0.806992	−	0.590563i	$-0.201094\pi$
$884$	0	0
$885$	6.09592e7i	2.61626i
$886$	0	0
$887$	2.10859e6	0.0899878	0.0449939	−	0.998987i	$-0.485673\pi$
$887$	2.10859e6	0.0899878	0.0449939	+	0.998987i	$0.485673\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1.81592e7i	− 0.766307i
$892$	0	0
$893$	−3.17098e7	−1.33065
$894$	0	0
$895$	3.63814e6	0.151818
$896$	0	0
$897$	1.77775e7	0.737716
$898$	0	0
$899$	1.60194e6	0.0661070
$900$	0	0
$901$	2.38019e7i	0.976785i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.39141e7	0.970582
$906$	0	0
$907$	− 3.70799e7i	− 1.49665i	−0.663333	−	0.748324i	$-0.730859\pi$
$907$	− 3.70799e7i	− 1.49665i	0.663333	−	0.748324i	$-0.269141\pi$
$908$	0	0
$909$	5.25525e7i	2.10952i
$910$	0	0
$911$	− 8.94944e6i	− 0.357273i	−0.983915	−	0.178636i	$-0.942831\pi$
$911$	− 8.94944e6i	− 0.357273i	0.983915	−	0.178636i	$-0.0571686\pi$
$912$	0	0
$913$	6.92642e6i	0.275000i
$914$	0	0
$915$	−8.83528e7	−3.48873
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.75668e7i	1.07671i	0.842719	+	0.538353i	$0.180953\pi$
$919$	2.75668e7i	1.07671i	−0.842719	+	0.538353i	$0.819047\pi$
$920$	0	0
$921$	−3.43853e6	−0.133575
$922$	0	0
$923$	−3.67550e7	−1.42008
$924$	0	0
$925$	6.30703e7	2.42365
$926$	0	0
$927$	−7.20104e6	−0.275230
$928$	0	0
$929$	− 3.45828e7i	− 1.31468i	−0.753593	−	0.657341i	$-0.771681\pi$
$929$	− 3.45828e7i	− 1.31468i	0.753593	−	0.657341i	$-0.228319\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.68649e7	−0.634279
$934$	0	0
$935$	− 5.26898e7i	− 1.97105i
$936$	0	0
$937$	2.64955e7i	0.985877i	0.870064	+	0.492938i	$0.164077\pi$
$937$	2.64955e7i	0.985877i	−0.870064	+	0.492938i	$0.835923\pi$
$938$	0	0
$939$	4.21185e7i	1.55887i
$940$	0	0
$941$	2.41317e7i	0.888410i	0.895925	+	0.444205i	$0.146514\pi$
$941$	2.41317e7i	0.888410i	−0.895925	+	0.444205i	$0.853486\pi$
$942$	0	0
$943$	−2.04605e7	−0.749267
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.25752e7i	− 1.18036i	−0.807273	−	0.590178i	$-0.799058\pi$
$947$	− 3.25752e7i	− 1.18036i	0.807273	−	0.590178i	$-0.200942\pi$
$948$	0	0
$949$	4.16271e7	1.50041
$950$	0	0
$951$	3.65429e7	1.31024
$952$	0	0
$953$	2.20095e7	0.785016	0.392508	−	0.919749i	$-0.371608\pi$
$953$	2.20095e7	0.785016	0.392508	+	0.919749i	$0.371608\pi$
$954$	0	0
$955$	−4.34811e7	−1.54274
$956$	0	0
$957$	− 1.84432e7i	− 0.650962i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.79667e7	−0.976861
$962$	0	0
$963$	− 4.15452e6i	− 0.144363i
$964$	0	0
$965$	2.01252e7i	0.695699i
$966$	0	0
$967$	4.35658e7i	1.49823i	0.662439	+	0.749116i	$0.269522\pi$
$967$	4.35658e7i	1.49823i	−0.662439	+	0.749116i	$0.730478\pi$
$968$	0	0
$969$	4.81043e7i	1.64579i
$970$	0	0
$971$	3.50418e7	1.19272	0.596360	−	0.802717i	$-0.296613\pi$
$971$	3.50418e7	1.19272	0.596360	+	0.802717i	$0.296613\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 6.71223e7i	− 2.26129i
$976$	0	0
$977$	8.28519e6	0.277694	0.138847	−	0.990314i	$-0.455660\pi$
$977$	8.28519e6	0.277694	0.138847	+	0.990314i	$0.455660\pi$
$978$	0	0
$979$	1.91238e7	0.637703
$980$	0	0
$981$	−5.27536e7	−1.75017
$982$	0	0
$983$	7.67863e6	0.253454	0.126727	−	0.991938i	$-0.459553\pi$
$983$	7.67863e6	0.253454	0.126727	+	0.991938i	$0.459553\pi$
$984$	0	0
$985$	− 5.93862e6i	− 0.195027i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.31168e7	−0.751515
$990$	0	0
$991$	4.54415e7i	1.46983i	0.678157	+	0.734917i	$0.262779\pi$
$991$	4.54415e7i	1.46983i	−0.678157	+	0.734917i	$0.737221\pi$
$992$	0	0
$993$	− 7.24598e7i	− 2.33198i
$994$	0	0
$995$	− 5.80884e7i	− 1.86008i
$996$	0	0
$997$	5.29962e7i	1.68852i	0.535932	+	0.844261i	$0.319960\pi$
$997$	5.29962e7i	1.68852i	−0.535932	+	0.844261i	$0.680040\pi$
$998$	0	0
$999$	−1.56957e7	−0.497586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.f.c.783.13		14
4.3	odd	2		784.6.f.d.783.1		14
7.2	even	3		112.6.p.c.31.1	yes	14
7.3	odd	6		112.6.p.b.47.7	yes	14
7.6	odd	2		784.6.f.d.783.2		14
28.3	even	6		112.6.p.c.47.1	yes	14
28.23	odd	6		112.6.p.b.31.7	✓	14
28.27	even	2	inner	784.6.f.c.783.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.b.31.7	✓	14	28.23	odd	6
112.6.p.b.47.7	yes	14	7.3	odd	6
112.6.p.c.31.1	yes	14	7.2	even	3
112.6.p.c.47.1	yes	14	28.3	even	6
784.6.f.c.783.13		14	1.1	even	1	trivial
784.6.f.c.783.14		14	28.27	even	2	inner
784.6.f.d.783.1		14	4.3	odd	2
784.6.f.d.783.2		14	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+23.1254 q^{3} -87.5115i q^{5} +291.784 q^{9} +405.208i q^{11} +640.275i q^{13} -2023.74i q^{15} -1485.88i q^{17} -1399.95 q^{19} -1200.64i q^{23} -4533.26 q^{25} +1128.15 q^{27} -1968.20 q^{29} -813.914 q^{31} +9370.60i q^{33} -13912.8 q^{37} +14806.6i q^{39} -17041.2i q^{41} -19253.7i q^{43} -25534.5i q^{45} +22650.7 q^{47} -34361.5i q^{51} -16018.7 q^{53} +35460.4 q^{55} -32374.3 q^{57} -30122.1 q^{59} -43658.2i q^{61} +56031.5 q^{65} +10606.4i q^{67} -27765.4i q^{69} +57405.1i q^{71} -65014.3i q^{73} -104833. q^{75} -18465.2i q^{79} -44814.6 q^{81} +17093.5 q^{83} -130031. q^{85} -45515.3 q^{87} -47195.1i q^{89} -18822.1 q^{93} +122511. i q^{95} +52847.4i q^{97} +118233. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{3} + 1076 q^{9} - 4270 q^{19} - 10868 q^{25} - 17910 q^{27} + 684 q^{29} + 6238 q^{31} - 6862 q^{37} + 16818 q^{47} - 3414 q^{53} + 56622 q^{55} + 68450 q^{57} - 98610 q^{59} - 34896 q^{65}+ \cdots - 585746 q^{93}+O(q^{100}) \) 14 * q - 18 * q^3 + 1076 * q^9 - 4270 * q^19 - 10868 * q^25 - 17910 * q^27 + 684 * q^29 + 6238 * q^31 - 6862 * q^37 + 16818 * q^47 - 3414 * q^53 + 56622 * q^55 + 68450 * q^57 - 98610 * q^59 - 34896 * q^65 - 295668 * q^75 + 291134 * q^81 + 305400 * q^83 - 176454 * q^85 - 190068 * q^87 - 585746 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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