LMFDB - Embedded newform 784.3.c.g.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.c (of order $2$, degree $1$, 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:	$21.3624527258$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 597x^{4} - 384x^{3} - 4236x^{2} + 688x + 13378 $ x^8 - 4x^7 - 30x^6 + 84x^5 + 597x^4 - 384x^3 - 4236x^2 + 688*x + 13378
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$5.16940 + 1.84776i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.3.c.g.97.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34336i q^{3} -6.62165i q^{5} +7.19538 q^{9} +O(q^{10})$	$q-1.34336i q^{3} -6.62165i q^{5} +7.19538 q^{9} +12.5385 q^{11} +2.63066i q^{13} -8.89526 q^{15} -26.4811i q^{17} +10.2708i q^{19} +43.4955 q^{23} -18.8462 q^{25} -21.7562i q^{27} +10.1226 q^{29} +47.1721i q^{31} -16.8438i q^{33} +8.26470 q^{37} +3.53392 q^{39} -66.9187i q^{41} -80.9734 q^{43} -47.6453i q^{45} +27.9374i q^{47} -35.5737 q^{51} -74.8533 q^{53} -83.0258i q^{55} +13.7974 q^{57} -70.6398i q^{59} -22.2629i q^{61} +17.4193 q^{65} -62.3319 q^{67} -58.4301i q^{69} -17.1596 q^{71} -34.8985i q^{73} +25.3173i q^{75} +109.450 q^{79} +35.5320 q^{81} -10.8247i q^{83} -175.349 q^{85} -13.5982i q^{87} +116.218i q^{89} +63.3691 q^{93} +68.0097 q^{95} +55.5922i q^{97} +90.2196 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{9}+O(q^{10}) $ 8 * q - 64 * q^9	$ 8 q - 64 q^{9} + 24 q^{11} - 40 q^{15} + 136 q^{23} - 80 q^{25} + 64 q^{29} - 64 q^{37} + 376 q^{39} - 136 q^{43} + 408 q^{51} - 104 q^{53} - 240 q^{57} + 224 q^{65} + 16 q^{67} - 352 q^{71} + 112 q^{79} + 584 q^{81} + 200 q^{85} + 784 q^{93} - 120 q^{95} + 880 q^{99}+O(q^{100}) $ 8 * q - 64 * q^9 + 24 * q^11 - 40 * q^15 + 136 * q^23 - 80 * q^25 + 64 * q^29 - 64 * q^37 + 376 * q^39 - 136 * q^43 + 408 * q^51 - 104 * q^53 - 240 * q^57 + 224 * q^65 + 16 * q^67 - 352 * q^71 + 112 * q^79 + 584 * q^81 + 200 * q^85 + 784 * q^93 - 120 * q^95 + 880 * q^99

Display 100 coefficients 10 coefficients

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
$3$	− 1.34336i	− 0.447787i	−0.974614	−	0.223893i	$-0.928123\pi$
$3$	− 1.34336i	− 0.447787i	0.974614	−	0.223893i	$-0.0718767\pi$
$4$	0	0
$5$	− 6.62165i	− 1.32433i	−0.749358	−	0.662165i	$-0.769638\pi$
$5$	− 6.62165i	− 1.32433i	0.749358	−	0.662165i	$-0.230362\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.19538	0.799487
$10$	0	0
$11$	12.5385	1.13987	0.569933	−	0.821691i	$-0.306969\pi$
$11$	12.5385	1.13987	0.569933	+	0.821691i	$0.306969\pi$
$12$	0	0
$13$	2.63066i	0.202359i	0.994868	+	0.101179i	$0.0322616\pi$
$13$	2.63066i	0.202359i	−0.994868	+	0.101179i	$0.967738\pi$
$14$	0	0
$15$	−8.89526	−0.593017
$16$	0	0
$17$	− 26.4811i	− 1.55771i	−0.627202	−	0.778857i	$-0.715800\pi$
$17$	− 26.4811i	− 1.55771i	0.627202	−	0.778857i	$-0.284200\pi$
$18$	0	0
$19$	10.2708i	0.540569i	0.962780	+	0.270285i	$0.0871178\pi$
$19$	10.2708i	0.540569i	−0.962780	+	0.270285i	$0.912882\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	43.4955	1.89111	0.945555	−	0.325463i	$-0.105520\pi$
$23$	43.4955	1.89111	0.945555	+	0.325463i	$0.105520\pi$
$24$	0	0
$25$	−18.8462	−0.753850
$26$	0	0
$27$	− 21.7562i	− 0.805786i
$28$	0	0
$29$	10.1226	0.349054	0.174527	−	0.984652i	$-0.444160\pi$
$29$	10.1226	0.349054	0.174527	+	0.984652i	$0.444160\pi$
$30$	0	0
$31$	47.1721i	1.52168i	0.648939	+	0.760840i	$0.275213\pi$
$31$	47.1721i	1.52168i	−0.648939	+	0.760840i	$0.724787\pi$
$32$	0	0
$33$	− 16.8438i	− 0.510417i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.26470	0.223370	0.111685	−	0.993744i	$-0.464375\pi$
$37$	8.26470	0.223370	0.111685	+	0.993744i	$0.464375\pi$
$38$	0	0
$39$	3.53392	0.0906134
$40$	0	0
$41$	− 66.9187i	− 1.63216i	−0.577937	−	0.816081i	$-0.696142\pi$
$41$	− 66.9187i	− 1.63216i	0.577937	−	0.816081i	$-0.303858\pi$
$42$	0	0
$43$	−80.9734	−1.88310	−0.941551	−	0.336870i	$-0.890632\pi$
$43$	−80.9734	−1.88310	−0.941551	+	0.336870i	$0.890632\pi$
$44$	0	0
$45$	− 47.6453i	− 1.05878i
$46$	0	0
$47$	27.9374i	0.594412i	0.954813	+	0.297206i	$0.0960548\pi$
$47$	27.9374i	0.594412i	−0.954813	+	0.297206i	$0.903945\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−35.5737	−0.697523
$52$	0	0
$53$	−74.8533	−1.41233	−0.706164	−	0.708049i	$-0.749576\pi$
$53$	−74.8533	−1.41233	−0.706164	+	0.708049i	$0.749576\pi$
$54$	0	0
$55$	− 83.0258i	− 1.50956i
$56$	0	0
$57$	13.7974	0.242060
$58$	0	0
$59$	− 70.6398i	− 1.19728i	−0.801017	−	0.598642i	$-0.795707\pi$
$59$	− 70.6398i	− 1.19728i	0.801017	−	0.598642i	$-0.204293\pi$
$60$	0	0
$61$	− 22.2629i	− 0.364965i	−0.983209	−	0.182482i	$-0.941587\pi$
$61$	− 22.2629i	− 0.364965i	0.983209	−	0.182482i	$-0.0584133\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.4193	0.267990
$66$	0	0
$67$	−62.3319	−0.930327	−0.465163	−	0.885225i	$-0.654004\pi$
$67$	−62.3319	−0.930327	−0.465163	+	0.885225i	$0.654004\pi$
$68$	0	0
$69$	− 58.4301i	− 0.846813i
$70$	0	0
$71$	−17.1596	−0.241684	−0.120842	−	0.992672i	$-0.538559\pi$
$71$	−17.1596	−0.241684	−0.120842	+	0.992672i	$0.538559\pi$
$72$	0	0
$73$	− 34.8985i	− 0.478062i	−0.971012	−	0.239031i	$-0.923170\pi$
$73$	− 34.8985i	− 0.478062i	0.971012	−	0.239031i	$-0.0768297\pi$
$74$	0	0
$75$	25.3173i	0.337564i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	109.450	1.38544	0.692719	−	0.721208i	$-0.256413\pi$
$79$	109.450	1.38544	0.692719	+	0.721208i	$0.256413\pi$
$80$	0	0
$81$	35.5320	0.438667
$82$	0	0
$83$	− 10.8247i	− 0.130418i	−0.997872	−	0.0652088i	$-0.979229\pi$
$83$	− 10.8247i	− 0.130418i	0.997872	−	0.0652088i	$-0.0207713\pi$
$84$	0	0
$85$	−175.349	−2.06293
$86$	0	0
$87$	− 13.5982i	− 0.156302i
$88$	0	0
$89$	116.218i	1.30582i	0.757435	+	0.652910i	$0.226452\pi$
$89$	116.218i	1.30582i	−0.757435	+	0.652910i	$0.773548\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	63.3691	0.681388
$94$	0	0
$95$	68.0097	0.715892
$96$	0	0
$97$	55.5922i	0.573116i	0.958063	+	0.286558i	$0.0925110\pi$
$97$	55.5922i	0.573116i	−0.958063	+	0.286558i	$0.907489\pi$
$98$	0	0
$99$	90.2196	0.911309
$100$	0	0
$101$	− 10.0374i	− 0.0993798i	−0.998765	−	0.0496899i	$-0.984177\pi$
$101$	− 10.0374i	− 0.0993798i	0.998765	−	0.0496899i	$-0.0158233\pi$
$102$	0	0
$103$	− 69.9520i	− 0.679145i	−0.940580	−	0.339573i	$-0.889718\pi$
$103$	− 69.9520i	− 0.679145i	0.940580	−	0.339573i	$-0.110282\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0152	0.140329	0.0701645	−	0.997535i	$-0.477648\pi$
$107$	15.0152	0.140329	0.0701645	+	0.997535i	$0.477648\pi$
$108$	0	0
$109$	123.641	1.13432	0.567160	−	0.823608i	$-0.308042\pi$
$109$	123.641	1.13432	0.567160	+	0.823608i	$0.308042\pi$
$110$	0	0
$111$	− 11.1025i	− 0.100022i
$112$	0	0
$113$	−87.6080	−0.775292	−0.387646	−	0.921808i	$-0.626712\pi$
$113$	−87.6080	−0.775292	−0.387646	+	0.921808i	$0.626712\pi$
$114$	0	0
$115$	− 288.012i	− 2.50445i
$116$	0	0
$117$	18.9286i	0.161783i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	36.2148	0.299295
$122$	0	0
$123$	−89.8958	−0.730860
$124$	0	0
$125$	− 40.7480i	− 0.325984i
$126$	0	0
$127$	−151.557	−1.19336	−0.596679	−	0.802480i	$-0.703514\pi$
$127$	−151.557	−1.19336	−0.596679	+	0.802480i	$0.703514\pi$
$128$	0	0
$129$	108.776i	0.843228i
$130$	0	0
$131$	− 124.133i	− 0.947580i	−0.880638	−	0.473790i	$-0.842886\pi$
$131$	− 124.133i	− 0.947580i	0.880638	−	0.473790i	$-0.157114\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−144.062	−1.06713
$136$	0	0
$137$	109.282	0.797679	0.398840	−	0.917021i	$-0.369413\pi$
$137$	109.282	0.797679	0.398840	+	0.917021i	$0.369413\pi$
$138$	0	0
$139$	17.7902i	0.127987i	0.997950	+	0.0639935i	$0.0203837\pi$
$139$	17.7902i	0.127987i	−0.997950	+	0.0639935i	$0.979616\pi$
$140$	0	0
$141$	37.5299	0.266170
$142$	0	0
$143$	32.9846i	0.230662i
$144$	0	0
$145$	− 67.0281i	− 0.462262i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	108.377	0.727359	0.363680	−	0.931524i	$-0.381520\pi$
$149$	108.377	0.727359	0.363680	+	0.931524i	$0.381520\pi$
$150$	0	0
$151$	−55.2634	−0.365983	−0.182991	−	0.983115i	$-0.558578\pi$
$151$	−55.2634	−0.365983	−0.182991	+	0.983115i	$0.558578\pi$
$152$	0	0
$153$	− 190.542i	− 1.24537i
$154$	0	0
$155$	312.357	2.01521
$156$	0	0
$157$	220.451i	1.40415i	0.712104	+	0.702074i	$0.247743\pi$
$157$	220.451i	1.40415i	−0.712104	+	0.702074i	$0.752257\pi$
$158$	0	0
$159$	100.555i	0.632421i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	58.6895	0.360058	0.180029	−	0.983661i	$-0.442381\pi$
$163$	58.6895	0.360058	0.180029	+	0.983661i	$0.442381\pi$
$164$	0	0
$165$	−111.533	−0.675960
$166$	0	0
$167$	99.5134i	0.595888i	0.954583	+	0.297944i	$0.0963009\pi$
$167$	99.5134i	0.595888i	−0.954583	+	0.297944i	$0.903699\pi$
$168$	0	0
$169$	162.080	0.959051
$170$	0	0
$171$	73.9025i	0.432178i
$172$	0	0
$173$	142.523i	0.823830i	0.911222	+	0.411915i	$0.135140\pi$
$173$	142.523i	0.823830i	−0.911222	+	0.411915i	$0.864860\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−94.8946	−0.536128
$178$	0	0
$179$	−273.297	−1.52680	−0.763401	−	0.645925i	$-0.776472\pi$
$179$	−273.297	−1.52680	−0.763401	+	0.645925i	$0.776472\pi$
$180$	0	0
$181$	289.628i	1.60015i	0.599898	+	0.800077i	$0.295208\pi$
$181$	289.628i	1.60015i	−0.599898	+	0.800077i	$0.704792\pi$
$182$	0	0
$183$	−29.9070	−0.163426
$184$	0	0
$185$	− 54.7259i	− 0.295816i
$186$	0	0
$187$	− 332.034i	− 1.77558i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−69.5686	−0.364234	−0.182117	−	0.983277i	$-0.558295\pi$
$191$	−69.5686	−0.364234	−0.182117	+	0.983277i	$0.558295\pi$
$192$	0	0
$193$	−131.805	−0.682929	−0.341464	−	0.939895i	$-0.610923\pi$
$193$	−131.805	−0.682929	−0.341464	+	0.939895i	$0.610923\pi$
$194$	0	0
$195$	− 23.4004i	− 0.120002i
$196$	0	0
$197$	39.1914	0.198941	0.0994706	−	0.995041i	$-0.468285\pi$
$197$	39.1914	0.198941	0.0994706	+	0.995041i	$0.468285\pi$
$198$	0	0
$199$	290.424i	1.45942i	0.683758	+	0.729709i	$0.260344\pi$
$199$	290.424i	1.45942i	−0.683758	+	0.729709i	$0.739656\pi$
$200$	0	0
$201$	83.7342i	0.416588i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−443.112	−2.16152
$206$	0	0
$207$	312.967	1.51192
$208$	0	0
$209$	128.781i	0.616177i
$210$	0	0
$211$	114.121	0.540860	0.270430	−	0.962740i	$-0.412834\pi$
$211$	114.121	0.540860	0.270430	+	0.962740i	$0.412834\pi$
$212$	0	0
$213$	23.0515i	0.108223i
$214$	0	0
$215$	536.178i	2.49385i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−46.8813	−0.214070
$220$	0	0
$221$	69.6629	0.315217
$222$	0	0
$223$	− 28.6810i	− 0.128615i	−0.997930	−	0.0643073i	$-0.979516\pi$
$223$	− 28.6810i	− 0.128615i	0.997930	−	0.0643073i	$-0.0204838\pi$
$224$	0	0
$225$	−135.606	−0.602693
$226$	0	0
$227$	266.857i	1.17558i	0.809013	+	0.587791i	$0.200002\pi$
$227$	266.857i	1.17558i	−0.809013	+	0.587791i	$0.799998\pi$
$228$	0	0
$229$	− 333.709i	− 1.45724i	−0.684917	−	0.728621i	$-0.740161\pi$
$229$	− 333.709i	− 1.45724i	0.684917	−	0.728621i	$-0.259839\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	171.449	0.735834	0.367917	−	0.929859i	$-0.380071\pi$
$233$	171.449	0.735834	0.367917	+	0.929859i	$0.380071\pi$
$234$	0	0
$235$	184.992	0.787198
$236$	0	0
$237$	− 147.030i	− 0.620380i
$238$	0	0
$239$	−143.422	−0.600092	−0.300046	−	0.953925i	$-0.597002\pi$
$239$	−143.422	−0.600092	−0.300046	+	0.953925i	$0.597002\pi$
$240$	0	0
$241$	133.666i	0.554631i	0.960779	+	0.277315i	$0.0894447\pi$
$241$	133.666i	0.554631i	−0.960779	+	0.277315i	$0.910555\pi$
$242$	0	0
$243$	− 243.538i	− 1.00222i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−27.0190	−0.109389
$248$	0	0
$249$	−14.5414	−0.0583993
$250$	0	0
$251$	330.489i	1.31669i	0.752716	+	0.658345i	$0.228743\pi$
$251$	330.489i	1.31669i	−0.752716	+	0.658345i	$0.771257\pi$
$252$	0	0
$253$	545.370	2.15561
$254$	0	0
$255$	235.556i	0.923751i
$256$	0	0
$257$	− 314.754i	− 1.22473i	−0.790577	−	0.612363i	$-0.790219\pi$
$257$	− 314.754i	− 1.22473i	0.790577	−	0.612363i	$-0.209781\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	72.8357	0.279064
$262$	0	0
$263$	367.843	1.39864	0.699320	−	0.714808i	$-0.253486\pi$
$263$	367.843	1.39864	0.699320	+	0.714808i	$0.253486\pi$
$264$	0	0
$265$	495.653i	1.87039i
$266$	0	0
$267$	156.123	0.584729
$268$	0	0
$269$	− 345.780i	− 1.28543i	−0.766107	−	0.642713i	$-0.777809\pi$
$269$	− 345.780i	− 1.28543i	0.766107	−	0.642713i	$-0.222191\pi$
$270$	0	0
$271$	285.011i	1.05170i	0.850577	+	0.525851i	$0.176253\pi$
$271$	285.011i	1.05170i	−0.850577	+	0.525851i	$0.823747\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−236.304	−0.859288
$276$	0	0
$277$	44.2716	0.159825	0.0799126	−	0.996802i	$-0.474536\pi$
$277$	44.2716	0.159825	0.0799126	+	0.996802i	$0.474536\pi$
$278$	0	0
$279$	339.421i	1.21656i
$280$	0	0
$281$	328.085	1.16756	0.583781	−	0.811911i	$-0.301573\pi$
$281$	328.085	1.16756	0.583781	+	0.811911i	$0.301573\pi$
$282$	0	0
$283$	− 492.334i	− 1.73970i	−0.493319	−	0.869849i	$-0.664216\pi$
$283$	− 492.334i	− 1.73970i	0.493319	−	0.869849i	$-0.335784\pi$
$284$	0	0
$285$	− 91.3615i	− 0.320567i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−412.250	−1.42647
$290$	0	0
$291$	74.6803	0.256633
$292$	0	0
$293$	381.762i	1.30294i	0.758674	+	0.651470i	$0.225848\pi$
$293$	381.762i	1.30294i	−0.758674	+	0.651470i	$0.774152\pi$
$294$	0	0
$295$	−467.752	−1.58560
$296$	0	0
$297$	− 272.791i	− 0.918489i
$298$	0	0
$299$	114.422i	0.382682i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−13.4838	−0.0445009
$304$	0	0
$305$	−147.417	−0.483334
$306$	0	0
$307$	− 121.613i	− 0.396134i	−0.980188	−	0.198067i	$-0.936534\pi$
$307$	− 121.613i	− 0.396134i	0.980188	−	0.198067i	$-0.0634664\pi$
$308$	0	0
$309$	−93.9706	−0.304112
$310$	0	0
$311$	534.326i	1.71809i	0.511901	+	0.859044i	$0.328941\pi$
$311$	534.326i	1.71809i	−0.511901	+	0.859044i	$0.671059\pi$
$312$	0	0
$313$	171.823i	0.548956i	0.961593	+	0.274478i	$0.0885051\pi$
$313$	171.823i	0.548956i	−0.961593	+	0.274478i	$0.911495\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	114.739	0.361954	0.180977	−	0.983487i	$-0.442074\pi$
$317$	114.739	0.361954	0.180977	+	0.983487i	$0.442074\pi$
$318$	0	0
$319$	126.922	0.397875
$320$	0	0
$321$	− 20.1708i	− 0.0628374i
$322$	0	0
$323$	271.983	0.842052
$324$	0	0
$325$	− 49.5781i	− 0.152548i
$326$	0	0
$327$	− 166.094i	− 0.507933i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	84.0141	0.253819	0.126910	−	0.991914i	$-0.459494\pi$
$331$	84.0141	0.253819	0.126910	+	0.991914i	$0.459494\pi$
$332$	0	0
$333$	59.4677	0.178582
$334$	0	0
$335$	412.740i	1.23206i
$336$	0	0
$337$	2.76645	0.00820906	0.00410453	−	0.999992i	$-0.498693\pi$
$337$	2.76645	0.00820906	0.00410453	+	0.999992i	$0.498693\pi$
$338$	0	0
$339$	117.689i	0.347166i
$340$	0	0
$341$	591.469i	1.73451i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−386.904	−1.12146
$346$	0	0
$347$	−440.603	−1.26975	−0.634875	−	0.772615i	$-0.718948\pi$
$347$	−440.603	−1.26975	−0.634875	+	0.772615i	$0.718948\pi$
$348$	0	0
$349$	− 400.441i	− 1.14740i	−0.819067	−	0.573698i	$-0.805508\pi$
$349$	− 400.441i	− 1.14740i	0.819067	−	0.573698i	$-0.194492\pi$
$350$	0	0
$351$	57.2333	0.163058
$352$	0	0
$353$	94.2209i	0.266915i	0.991055	+	0.133457i	$0.0426079\pi$
$353$	94.2209i	0.266915i	−0.991055	+	0.133457i	$0.957392\pi$
$354$	0	0
$355$	113.625i	0.320070i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	369.162	1.02831	0.514153	−	0.857699i	$-0.328106\pi$
$359$	369.162	1.02831	0.514153	+	0.857699i	$0.328106\pi$
$360$	0	0
$361$	255.510	0.707785
$362$	0	0
$363$	− 48.6494i	− 0.134020i
$364$	0	0
$365$	−231.086	−0.633112
$366$	0	0
$367$	267.748i	0.729557i	0.931094	+	0.364779i	$0.118855\pi$
$367$	267.748i	0.729557i	−0.931094	+	0.364779i	$0.881145\pi$
$368$	0	0
$369$	− 481.506i	− 1.30489i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−105.916	−0.283957	−0.141979	−	0.989870i	$-0.545346\pi$
$373$	−105.916	−0.283957	−0.141979	+	0.989870i	$0.545346\pi$
$374$	0	0
$375$	−54.7392	−0.145971
$376$	0	0
$377$	26.6290i	0.0706340i
$378$	0	0
$379$	−8.04037	−0.0212147	−0.0106074	−	0.999944i	$-0.503376\pi$
$379$	−8.04037	−0.0212147	−0.0106074	+	0.999944i	$0.503376\pi$
$380$	0	0
$381$	203.595i	0.534370i
$382$	0	0
$383$	− 317.869i	− 0.829946i	−0.909834	−	0.414973i	$-0.863791\pi$
$383$	− 317.869i	− 0.829946i	0.909834	−	0.414973i	$-0.136209\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−582.635	−1.50552
$388$	0	0
$389$	−393.345	−1.01117	−0.505584	−	0.862777i	$-0.668723\pi$
$389$	−393.345	−1.01117	−0.505584	+	0.862777i	$0.668723\pi$
$390$	0	0
$391$	− 1151.81i	− 2.94581i
$392$	0	0
$393$	−166.755	−0.424313
$394$	0	0
$395$	− 724.737i	− 1.83478i
$396$	0	0
$397$	− 72.2070i	− 0.181882i	−0.995856	−	0.0909408i	$-0.971013\pi$
$397$	− 72.2070i	− 0.181882i	0.995856	−	0.0909408i	$-0.0289874\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	480.257	1.19765	0.598824	−	0.800881i	$-0.295635\pi$
$401$	480.257	1.19765	0.598824	+	0.800881i	$0.295635\pi$
$402$	0	0
$403$	−124.094	−0.307925
$404$	0	0
$405$	− 235.281i	− 0.580940i
$406$	0	0
$407$	103.627	0.254612
$408$	0	0
$409$	719.727i	1.75972i	0.475230	+	0.879862i	$0.342365\pi$
$409$	719.727i	1.75972i	−0.475230	+	0.879862i	$0.657635\pi$
$410$	0	0
$411$	− 146.805i	− 0.357190i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−71.6771	−0.172716
$416$	0	0
$417$	23.8986	0.0573108
$418$	0	0
$419$	473.040i	1.12897i	0.825442	+	0.564487i	$0.190926\pi$
$419$	473.040i	1.12897i	−0.825442	+	0.564487i	$0.809074\pi$
$420$	0	0
$421$	179.357	0.426027	0.213014	−	0.977049i	$-0.431672\pi$
$421$	179.357	0.426027	0.213014	+	0.977049i	$0.431672\pi$
$422$	0	0
$423$	201.020i	0.475225i
$424$	0	0
$425$	499.070i	1.17428i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	44.3102	0.103287
$430$	0	0
$431$	−58.2137	−0.135067	−0.0675333	−	0.997717i	$-0.521513\pi$
$431$	−58.2137	−0.135067	−0.0675333	+	0.997717i	$0.521513\pi$
$432$	0	0
$433$	− 241.364i	− 0.557422i	−0.960375	−	0.278711i	$-0.910093\pi$
$433$	− 241.364i	− 0.557422i	0.960375	−	0.278711i	$-0.0899071\pi$
$434$	0	0
$435$	−90.0428	−0.206995
$436$	0	0
$437$	446.734i	1.02228i
$438$	0	0
$439$	215.956i	0.491927i	0.969279	+	0.245964i	$0.0791044\pi$
$439$	215.956i	0.491927i	−0.969279	+	0.245964i	$0.920896\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−213.818	−0.482658	−0.241329	−	0.970443i	$-0.577583\pi$
$443$	−213.818	−0.482658	−0.241329	+	0.970443i	$0.577583\pi$
$444$	0	0
$445$	769.555	1.72934
$446$	0	0
$447$	− 145.589i	− 0.325702i
$448$	0	0
$449$	−625.904	−1.39400	−0.696998	−	0.717073i	$-0.745481\pi$
$449$	−625.904	−1.39400	−0.696998	+	0.717073i	$0.745481\pi$
$450$	0	0
$451$	− 839.062i	− 1.86045i
$452$	0	0
$453$	74.2386i	0.163882i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	606.081	1.32622	0.663108	−	0.748523i	$-0.269237\pi$
$457$	606.081	1.32622	0.663108	+	0.748523i	$0.269237\pi$
$458$	0	0
$459$	−576.129	−1.25518
$460$	0	0
$461$	− 268.244i	− 0.581874i	−0.956742	−	0.290937i	$-0.906033\pi$
$461$	− 268.244i	− 0.581874i	0.956742	−	0.290937i	$-0.0939670\pi$
$462$	0	0
$463$	417.587	0.901916	0.450958	−	0.892545i	$-0.351082\pi$
$463$	417.587	0.901916	0.450958	+	0.892545i	$0.351082\pi$
$464$	0	0
$465$	− 419.608i	− 0.902382i
$466$	0	0
$467$	236.015i	0.505386i	0.967547	+	0.252693i	$0.0813162\pi$
$467$	236.015i	0.505386i	−0.967547	+	0.252693i	$0.918684\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	296.145	0.628759
$472$	0	0
$473$	−1015.29	−2.14649
$474$	0	0
$475$	− 193.566i	− 0.407508i
$476$	0	0
$477$	−538.599	−1.12914
$478$	0	0
$479$	− 570.507i	− 1.19104i	−0.803342	−	0.595519i	$-0.796947\pi$
$479$	− 570.507i	− 1.19104i	0.803342	−	0.595519i	$-0.203053\pi$
$480$	0	0
$481$	21.7416i	0.0452009i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	368.112	0.758994
$486$	0	0
$487$	−56.0345	−0.115061	−0.0575303	−	0.998344i	$-0.518323\pi$
$487$	−56.0345	−0.115061	−0.0575303	+	0.998344i	$0.518323\pi$
$488$	0	0
$489$	− 78.8411i	− 0.161229i
$490$	0	0
$491$	70.3733	0.143326	0.0716632	−	0.997429i	$-0.477169\pi$
$491$	70.3733	0.143326	0.0716632	+	0.997429i	$0.477169\pi$
$492$	0	0
$493$	− 268.057i	− 0.543726i
$494$	0	0
$495$	− 597.402i	− 1.20687i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	716.646	1.43617	0.718083	−	0.695958i	$-0.245020\pi$
$499$	716.646	1.43617	0.718083	+	0.695958i	$0.245020\pi$
$500$	0	0
$501$	133.682	0.266831
$502$	0	0
$503$	46.1333i	0.0917164i	0.998948	+	0.0458582i	$0.0146022\pi$
$503$	46.1333i	0.0917164i	−0.998948	+	0.0458582i	$0.985398\pi$
$504$	0	0
$505$	−66.4639	−0.131612
$506$	0	0
$507$	− 217.731i	− 0.429450i
$508$	0	0
$509$	− 336.883i	− 0.661852i	−0.943657	−	0.330926i	$-0.892639\pi$
$509$	− 336.883i	− 0.661852i	0.943657	−	0.330926i	$-0.107361\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	223.454	0.435583
$514$	0	0
$515$	−463.197	−0.899412
$516$	0	0
$517$	350.294i	0.677551i
$518$	0	0
$519$	191.459	0.368900
$520$	0	0
$521$	− 237.877i	− 0.456578i	−0.973593	−	0.228289i	$-0.926687\pi$
$521$	− 237.877i	− 0.456578i	0.973593	−	0.228289i	$-0.0733131\pi$
$522$	0	0
$523$	− 847.694i	− 1.62083i	−0.585856	−	0.810415i	$-0.699242\pi$
$523$	− 847.694i	− 1.62083i	0.585856	−	0.810415i	$-0.300758\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1249.17	2.37034
$528$	0	0
$529$	1362.86	2.57629
$530$	0	0
$531$	− 508.280i	− 0.957213i
$532$	0	0
$533$	176.040	0.330282
$534$	0	0
$535$	− 99.4254i	− 0.185842i
$536$	0	0
$537$	367.137i	0.683681i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−44.8830	−0.0829630	−0.0414815	−	0.999139i	$-0.513208\pi$
$541$	−44.8830	−0.0829630	−0.0414815	+	0.999139i	$0.513208\pi$
$542$	0	0
$543$	389.074	0.716527
$544$	0	0
$545$	− 818.706i	− 1.50221i
$546$	0	0
$547$	−400.295	−0.731800	−0.365900	−	0.930654i	$-0.619239\pi$
$547$	−400.295	−0.731800	−0.365900	+	0.930654i	$0.619239\pi$
$548$	0	0
$549$	− 160.190i	− 0.291785i
$550$	0	0
$551$	103.967i	0.188688i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−73.5166	−0.132462
$556$	0	0
$557$	−185.832	−0.333631	−0.166815	−	0.985988i	$-0.553348\pi$
$557$	−185.832	−0.333631	−0.166815	+	0.985988i	$0.553348\pi$
$558$	0	0
$559$	− 213.014i	− 0.381062i
$560$	0	0
$561$	−446.042	−0.795083
$562$	0	0
$563$	73.7683i	0.131027i	0.997852	+	0.0655136i	$0.0208686\pi$
$563$	73.7683i	0.131027i	−0.997852	+	0.0655136i	$0.979131\pi$
$564$	0	0
$565$	580.110i	1.02674i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	128.128	0.225180	0.112590	−	0.993642i	$-0.464085\pi$
$569$	128.128	0.225180	0.112590	+	0.993642i	$0.464085\pi$
$570$	0	0
$571$	1139.98	1.99646	0.998232	−	0.0594309i	$-0.0189286\pi$
$571$	1139.98	1.99646	0.998232	+	0.0594309i	$0.0189286\pi$
$572$	0	0
$573$	93.4557i	0.163099i
$574$	0	0
$575$	−819.727	−1.42561
$576$	0	0
$577$	− 287.859i	− 0.498890i	−0.968389	−	0.249445i	$-0.919752\pi$
$577$	− 287.859i	− 0.498890i	0.968389	−	0.249445i	$-0.0802482\pi$
$578$	0	0
$579$	177.062i	0.305806i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−938.551	−1.60986
$584$	0	0
$585$	125.339	0.214254
$586$	0	0
$587$	− 11.5536i	− 0.0196825i	−0.999952	−	0.00984123i	$-0.996867\pi$
$587$	− 11.5536i	− 0.0196825i	0.999952	−	0.00984123i	$-0.00313261\pi$
$588$	0	0
$589$	−484.496	−0.822573
$590$	0	0
$591$	− 52.6482i	− 0.0890832i
$592$	0	0
$593$	128.281i	0.216325i	0.994133	+	0.108163i	$0.0344967\pi$
$593$	128.281i	0.216325i	−0.994133	+	0.108163i	$0.965503\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	390.144	0.653507
$598$	0	0
$599$	−633.531	−1.05765	−0.528824	−	0.848731i	$-0.677367\pi$
$599$	−633.531	−1.05765	−0.528824	+	0.848731i	$0.677367\pi$
$600$	0	0
$601$	− 323.899i	− 0.538933i	−0.963010	−	0.269467i	$-0.913153\pi$
$601$	− 323.899i	− 0.538933i	0.963010	−	0.269467i	$-0.0868474\pi$
$602$	0	0
$603$	−448.502	−0.743784
$604$	0	0
$605$	− 239.801i	− 0.396366i
$606$	0	0
$607$	638.673i	1.05218i	0.850429	+	0.526090i	$0.176342\pi$
$607$	638.673i	1.05218i	−0.850429	+	0.526090i	$0.823658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−73.4938	−0.120284
$612$	0	0
$613$	−981.273	−1.60077	−0.800386	−	0.599485i	$-0.795372\pi$
$613$	−981.273	−1.60077	−0.800386	+	0.599485i	$0.795372\pi$
$614$	0	0
$615$	595.259i	0.967900i
$616$	0	0
$617$	208.446	0.337837	0.168919	−	0.985630i	$-0.445972\pi$
$617$	208.446	0.337837	0.168919	+	0.985630i	$0.445972\pi$
$618$	0	0
$619$	425.590i	0.687545i	0.939053	+	0.343772i	$0.111705\pi$
$619$	425.590i	0.687545i	−0.939053	+	0.343772i	$0.888295\pi$
$620$	0	0
$621$	− 946.298i	− 1.52383i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−740.975	−1.18556
$626$	0	0
$627$	172.999	0.275916
$628$	0	0
$629$	− 218.858i	− 0.347947i
$630$	0	0
$631$	−575.230	−0.911616	−0.455808	−	0.890078i	$-0.650650\pi$
$631$	−575.230	−0.911616	−0.455808	+	0.890078i	$0.650650\pi$
$632$	0	0
$633$	− 153.306i	− 0.242190i
$634$	0	0
$635$	1003.55i	1.58040i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−123.470	−0.193224
$640$	0	0
$641$	−199.166	−0.310711	−0.155356	−	0.987859i	$-0.549652\pi$
$641$	−199.166	−0.310711	−0.155356	+	0.987859i	$0.549652\pi$
$642$	0	0
$643$	− 898.859i	− 1.39791i	−0.715164	−	0.698957i	$-0.753648\pi$
$643$	− 898.859i	− 1.39791i	0.715164	−	0.698957i	$-0.246352\pi$
$644$	0	0
$645$	720.279	1.11671
$646$	0	0
$647$	− 386.160i	− 0.596848i	−0.954433	−	0.298424i	$-0.903539\pi$
$647$	− 386.160i	− 0.596848i	0.954433	−	0.298424i	$-0.0964609\pi$
$648$	0	0
$649$	− 885.719i	− 1.36474i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	143.549	0.219830	0.109915	−	0.993941i	$-0.464942\pi$
$653$	143.549	0.219830	0.109915	+	0.993941i	$0.464942\pi$
$654$	0	0
$655$	−821.965	−1.25491
$656$	0	0
$657$	− 251.108i	− 0.382204i
$658$	0	0
$659$	251.484	0.381614	0.190807	−	0.981628i	$-0.438890\pi$
$659$	251.484	0.381614	0.190807	+	0.981628i	$0.438890\pi$
$660$	0	0
$661$	− 366.721i	− 0.554797i	−0.960755	−	0.277399i	$-0.910528\pi$
$661$	− 366.721i	− 0.554797i	0.960755	−	0.277399i	$-0.0894723\pi$
$662$	0	0
$663$	− 93.5823i	− 0.141150i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	440.286	0.660099
$668$	0	0
$669$	−38.5290	−0.0575919
$670$	0	0
$671$	− 279.144i	− 0.416011i
$672$	0	0
$673$	1082.98	1.60919	0.804593	−	0.593827i	$-0.202383\pi$
$673$	1082.98	1.60919	0.804593	+	0.593827i	$0.202383\pi$
$674$	0	0
$675$	410.023i	0.607442i
$676$	0	0
$677$	953.601i	1.40857i	0.709918	+	0.704284i	$0.248732\pi$
$677$	953.601i	1.40857i	−0.709918	+	0.704284i	$0.751268\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	358.485	0.526410
$682$	0	0
$683$	173.809	0.254479	0.127240	−	0.991872i	$-0.459388\pi$
$683$	173.809	0.254479	0.127240	+	0.991872i	$0.459388\pi$
$684$	0	0
$685$	− 723.627i	− 1.05639i
$686$	0	0
$687$	−448.291	−0.652534
$688$	0	0
$689$	− 196.914i	− 0.285796i
$690$	0	0
$691$	− 211.539i	− 0.306135i	−0.988216	−	0.153067i	$-0.951085\pi$
$691$	− 211.539i	− 0.306135i	0.988216	−	0.153067i	$-0.0489151\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	117.800	0.169497
$696$	0	0
$697$	−1772.08	−2.54244
$698$	0	0
$699$	− 230.318i	− 0.329496i
$700$	0	0
$701$	−757.766	−1.08098	−0.540489	−	0.841351i	$-0.681761\pi$
$701$	−757.766	−1.08098	−0.540489	+	0.841351i	$0.681761\pi$
$702$	0	0
$703$	84.8852i	0.120747i
$704$	0	0
$705$	− 248.510i	− 0.352497i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−462.742	−0.652669	−0.326334	−	0.945254i	$-0.605814\pi$
$709$	−462.742	−0.652669	−0.326334	+	0.945254i	$0.605814\pi$
$710$	0	0
$711$	787.532	1.10764
$712$	0	0
$713$	2051.77i	2.87766i
$714$	0	0
$715$	218.413	0.305472
$716$	0	0
$717$	192.667i	0.268713i
$718$	0	0
$719$	1255.46i	1.74611i	0.487618	+	0.873057i	$0.337866\pi$
$719$	1255.46i	1.74611i	−0.487618	+	0.873057i	$0.662134\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	179.562	0.248356
$724$	0	0
$725$	−190.772	−0.263134
$726$	0	0
$727$	190.560i	0.262118i	0.991375	+	0.131059i	$0.0418377\pi$
$727$	190.560i	0.262118i	−0.991375	+	0.131059i	$0.958162\pi$
$728$	0	0
$729$	−7.37128	−0.0101115
$730$	0	0
$731$	2144.27i	2.93333i
$732$	0	0
$733$	− 892.327i	− 1.21736i	−0.793415	−	0.608681i	$-0.791699\pi$
$733$	− 892.327i	− 1.21736i	0.793415	−	0.608681i	$-0.208301\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−781.550	−1.06045
$738$	0	0
$739$	−25.4208	−0.0343989	−0.0171995	−	0.999852i	$-0.505475\pi$
$739$	−25.4208	−0.0343989	−0.0171995	+	0.999852i	$0.505475\pi$
$740$	0	0
$741$	36.2963i	0.0489828i
$742$	0	0
$743$	662.948	0.892258	0.446129	−	0.894969i	$-0.352802\pi$
$743$	662.948	0.892258	0.446129	+	0.894969i	$0.352802\pi$
$744$	0	0
$745$	− 717.631i	− 0.963264i
$746$	0	0
$747$	− 77.8876i	− 0.104267i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1163.23	−1.54891	−0.774457	−	0.632627i	$-0.781977\pi$
$751$	−1163.23	−1.54891	−0.774457	+	0.632627i	$0.781977\pi$
$752$	0	0
$753$	443.966	0.589596
$754$	0	0
$755$	365.935i	0.484682i
$756$	0	0
$757$	−670.569	−0.885825	−0.442912	−	0.896565i	$-0.646055\pi$
$757$	−670.569	−0.885825	−0.442912	+	0.896565i	$0.646055\pi$
$758$	0	0
$759$	− 732.628i	− 0.965254i
$760$	0	0
$761$	1358.97i	1.78577i	0.450288	+	0.892883i	$0.351321\pi$
$761$	1358.97i	1.78577i	−0.450288	+	0.892883i	$0.648679\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1261.70	−1.64928
$766$	0	0
$767$	185.829	0.242281
$768$	0	0
$769$	− 1174.06i	− 1.52674i	−0.645961	−	0.763370i	$-0.723543\pi$
$769$	− 1174.06i	− 1.52674i	0.645961	−	0.763370i	$-0.276457\pi$
$770$	0	0
$771$	−422.828	−0.548416
$772$	0	0
$773$	− 1147.55i	− 1.48455i	−0.670098	−	0.742273i	$-0.733748\pi$
$773$	− 1147.55i	− 1.48455i	0.670098	−	0.742273i	$-0.266252\pi$
$774$	0	0
$775$	− 889.017i	− 1.14712i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	687.309	0.882297
$780$	0	0
$781$	−215.156	−0.275488
$782$	0	0
$783$	− 220.229i	− 0.281263i
$784$	0	0
$785$	1459.75	1.85956
$786$	0	0
$787$	910.396i	1.15679i	0.815756	+	0.578397i	$0.196321\pi$
$787$	910.396i	1.15679i	−0.815756	+	0.578397i	$0.803679\pi$
$788$	0	0
$789$	− 494.145i	− 0.626292i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	58.5660	0.0738538
$794$	0	0
$795$	665.840	0.837534
$796$	0	0
$797$	1079.02i	1.35386i	0.736050	+	0.676928i	$0.236689\pi$
$797$	1079.02i	1.35386i	−0.736050	+	0.676928i	$0.763311\pi$
$798$	0	0
$799$	739.813	0.925924
$800$	0	0
$801$	836.233i	1.04399i
$802$	0	0
$803$	− 437.576i	− 0.544927i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−464.507	−0.575597
$808$	0	0
$809$	1027.32	1.26986	0.634930	−	0.772569i	$-0.281029\pi$
$809$	1027.32	1.26986	0.634930	+	0.772569i	$0.281029\pi$
$810$	0	0
$811$	984.329i	1.21372i	0.794808	+	0.606861i	$0.207572\pi$
$811$	984.329i	1.21372i	−0.794808	+	0.606861i	$0.792428\pi$
$812$	0	0
$813$	382.872	0.470938
$814$	0	0
$815$	− 388.621i	− 0.476836i
$816$	0	0
$817$	− 831.663i	− 1.01795i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1118.94	1.36290	0.681449	−	0.731866i	$-0.261350\pi$
$821$	1118.94	1.36290	0.681449	+	0.731866i	$0.261350\pi$
$822$	0	0
$823$	−110.690	−0.134496	−0.0672481	−	0.997736i	$-0.521422\pi$
$823$	−110.690	−0.134496	−0.0672481	+	0.997736i	$0.521422\pi$
$824$	0	0
$825$	317.442i	0.384778i
$826$	0	0
$827$	−600.694	−0.726353	−0.363177	−	0.931720i	$-0.618308\pi$
$827$	−600.694	−0.726353	−0.363177	+	0.931720i	$0.618308\pi$
$828$	0	0
$829$	1051.77i	1.26872i	0.773037	+	0.634361i	$0.218737\pi$
$829$	1051.77i	1.26872i	−0.773037	+	0.634361i	$0.781263\pi$
$830$	0	0
$831$	− 59.4727i	− 0.0715676i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	658.943	0.789153
$836$	0	0
$837$	1026.29	1.22615
$838$	0	0
$839$	147.774i	0.176131i	0.996115	+	0.0880656i	$0.0280685\pi$
$839$	147.774i	0.176131i	−0.996115	+	0.0880656i	$0.971931\pi$
$840$	0	0
$841$	−738.534	−0.878161
$842$	0	0
$843$	− 440.736i	− 0.522818i
$844$	0	0
$845$	− 1073.23i	− 1.27010i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−661.382	−0.779013
$850$	0	0
$851$	359.477	0.422417
$852$	0	0
$853$	773.138i	0.906376i	0.891415	+	0.453188i	$0.149713\pi$
$853$	773.138i	0.906376i	−0.891415	+	0.453188i	$0.850287\pi$
$854$	0	0
$855$	489.356	0.572346
$856$	0	0
$857$	941.605i	1.09872i	0.835585	+	0.549361i	$0.185129\pi$
$857$	941.605i	1.09872i	−0.835585	+	0.549361i	$0.814871\pi$
$858$	0	0
$859$	733.892i	0.854356i	0.904167	+	0.427178i	$0.140492\pi$
$859$	733.892i	0.854356i	−0.904167	+	0.427178i	$0.859508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−739.946	−0.857412	−0.428706	−	0.903444i	$-0.641030\pi$
$863$	−739.946	−0.857412	−0.428706	+	0.903444i	$0.641030\pi$
$864$	0	0
$865$	943.734	1.09102
$866$	0	0
$867$	553.800i	0.638754i
$868$	0	0
$869$	1372.34	1.57921
$870$	0	0
$871$	− 163.974i	− 0.188260i
$872$	0	0
$873$	400.007i	0.458199i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	506.992	0.578098	0.289049	−	0.957314i	$-0.406661\pi$
$877$	506.992	0.578098	0.289049	+	0.957314i	$0.406661\pi$
$878$	0	0
$879$	512.843	0.583439
$880$	0	0
$881$	112.813i	0.128052i	0.997948	+	0.0640258i	$0.0203940\pi$
$881$	112.813i	0.128052i	−0.997948	+	0.0640258i	$0.979606\pi$
$882$	0	0
$883$	−133.113	−0.150751	−0.0753754	−	0.997155i	$-0.524016\pi$
$883$	−133.113	−0.150751	−0.0753754	+	0.997155i	$0.524016\pi$
$884$	0	0
$885$	628.359i	0.710010i
$886$	0	0
$887$	− 489.556i	− 0.551923i	−0.961169	−	0.275962i	$-0.911004\pi$
$887$	− 489.556i	− 0.551923i	0.961169	−	0.275962i	$-0.0889963\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	445.519	0.500022
$892$	0	0
$893$	−286.940	−0.321321
$894$	0	0
$895$	1809.68i	2.02199i
$896$	0	0
$897$	153.710	0.171360
$898$	0	0
$899$	477.502i	0.531148i
$900$	0	0
$901$	1982.20i	2.20000i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1917.81	2.11913
$906$	0	0
$907$	259.207	0.285785	0.142892	−	0.989738i	$-0.454360\pi$
$907$	259.207	0.285785	0.142892	+	0.989738i	$0.454360\pi$
$908$	0	0
$909$	− 72.2227i	− 0.0794529i
$910$	0	0
$911$	752.359	0.825860	0.412930	−	0.910763i	$-0.364505\pi$
$911$	752.359	0.825860	0.412930	+	0.910763i	$0.364505\pi$
$912$	0	0
$913$	− 135.725i	− 0.148659i
$914$	0	0
$915$	198.034i	0.216430i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−330.959	−0.360130	−0.180065	−	0.983655i	$-0.557631\pi$
$919$	−330.959	−0.360130	−0.180065	+	0.983655i	$0.557631\pi$
$920$	0	0
$921$	−163.370	−0.177384
$922$	0	0
$923$	− 45.1411i	− 0.0489069i
$924$	0	0
$925$	−155.759	−0.168388
$926$	0	0
$927$	− 503.331i	− 0.542968i
$928$	0	0
$929$	− 229.551i	− 0.247094i	−0.992339	−	0.123547i	$-0.960573\pi$
$929$	− 229.551i	− 0.247094i	0.992339	−	0.123547i	$-0.0394270\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	717.792	0.769337
$934$	0	0
$935$	−2198.62	−2.35146
$936$	0	0
$937$	503.860i	0.537738i	0.963177	+	0.268869i	$0.0866499\pi$
$937$	503.860i	0.537738i	−0.963177	+	0.268869i	$0.913350\pi$
$938$	0	0
$939$	230.820	0.245815
$940$	0	0
$941$	1272.07i	1.35183i	0.736979	+	0.675916i	$0.236252\pi$
$941$	1272.07i	1.35183i	−0.736979	+	0.675916i	$0.763748\pi$
$942$	0	0
$943$	− 2910.66i	− 3.08660i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−972.545	−1.02698	−0.513488	−	0.858097i	$-0.671647\pi$
$947$	−972.545	−1.02698	−0.513488	+	0.858097i	$0.671647\pi$
$948$	0	0
$949$	91.8062	0.0967399
$950$	0	0
$951$	− 154.136i	− 0.162078i
$952$	0	0
$953$	−345.615	−0.362660	−0.181330	−	0.983422i	$-0.558040\pi$
$953$	−345.615	−0.362660	−0.181330	+	0.983422i	$0.558040\pi$
$954$	0	0
$955$	460.659i	0.482366i
$956$	0	0
$957$	− 170.502i	− 0.178163i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1264.21	−1.31551
$962$	0	0
$963$	108.040	0.112191
$964$	0	0
$965$	872.768i	0.904423i
$966$	0	0
$967$	1385.34	1.43261	0.716306	−	0.697786i	$-0.245831\pi$
$967$	1385.34	1.43261	0.716306	+	0.697786i	$0.245831\pi$
$968$	0	0
$969$	− 365.371i	− 0.377059i
$970$	0	0
$971$	1121.88i	1.15538i	0.816256	+	0.577691i	$0.196046\pi$
$971$	1121.88i	1.15538i	−0.816256	+	0.577691i	$0.803954\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−66.6012	−0.0683089
$976$	0	0
$977$	648.848	0.664123	0.332061	−	0.943258i	$-0.392256\pi$
$977$	648.848	0.664123	0.332061	+	0.943258i	$0.392256\pi$
$978$	0	0
$979$	1457.20i	1.48846i
$980$	0	0
$981$	889.643	0.906874
$982$	0	0
$983$	− 1913.46i	− 1.94655i	−0.229637	−	0.973276i	$-0.573754\pi$
$983$	− 1913.46i	− 1.94655i	0.229637	−	0.973276i	$-0.426246\pi$
$984$	0	0
$985$	− 259.512i	− 0.263464i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3521.98	−3.56115
$990$	0	0
$991$	−243.968	−0.246184	−0.123092	−	0.992395i	$-0.539281\pi$
$991$	−243.968	−0.246184	−0.123092	+	0.992395i	$0.539281\pi$
$992$	0	0
$993$	− 112.861i	− 0.113657i
$994$	0	0
$995$	1923.09	1.93275
$996$	0	0
$997$	384.602i	0.385759i	0.981222	+	0.192880i	$0.0617827\pi$
$997$	384.602i	0.385759i	−0.981222	+	0.192880i	$0.938217\pi$
$998$	0	0
$999$	− 179.809i	− 0.179989i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.c.g.97.4		8
4.3	odd	2		392.3.c.b.97.5	yes	8
7.2	even	3		784.3.s.k.129.5		16
7.3	odd	6		784.3.s.k.705.5		16
7.4	even	3		784.3.s.k.705.4		16
7.5	odd	6		784.3.s.k.129.4		16
7.6	odd	2	inner	784.3.c.g.97.5		8
12.11	even	2		3528.3.f.e.2449.7		8
28.3	even	6		392.3.o.d.313.4		16
28.11	odd	6		392.3.o.d.313.5		16
28.19	even	6		392.3.o.d.129.5		16
28.23	odd	6		392.3.o.d.129.4		16
28.27	even	2		392.3.c.b.97.4	✓	8
84.83	odd	2		3528.3.f.e.2449.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.3.c.b.97.4	✓	8	28.27	even	2
392.3.c.b.97.5	yes	8	4.3	odd	2
392.3.o.d.129.4		16	28.23	odd	6
392.3.o.d.129.5		16	28.19	even	6
392.3.o.d.313.4		16	28.3	even	6
392.3.o.d.313.5		16	28.11	odd	6
784.3.c.g.97.4		8	1.1	even	1	trivial
784.3.c.g.97.5		8	7.6	odd	2	inner
784.3.s.k.129.4		16	7.5	odd	6
784.3.s.k.129.5		16	7.2	even	3
784.3.s.k.705.4		16	7.4	even	3
784.3.s.k.705.5		16	7.3	odd	6
3528.3.f.e.2449.2		8	84.83	odd	2
3528.3.f.e.2449.7		8	12.11	even	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.34336i q^{3} -6.62165i q^{5} +7.19538 q^{9} +O(q^{10})\)	\(q-1.34336i q^{3} -6.62165i q^{5} +7.19538 q^{9} +12.5385 q^{11} +2.63066i q^{13} -8.89526 q^{15} -26.4811i q^{17} +10.2708i q^{19} +43.4955 q^{23} -18.8462 q^{25} -21.7562i q^{27} +10.1226 q^{29} +47.1721i q^{31} -16.8438i q^{33} +8.26470 q^{37} +3.53392 q^{39} -66.9187i q^{41} -80.9734 q^{43} -47.6453i q^{45} +27.9374i q^{47} -35.5737 q^{51} -74.8533 q^{53} -83.0258i q^{55} +13.7974 q^{57} -70.6398i q^{59} -22.2629i q^{61} +17.4193 q^{65} -62.3319 q^{67} -58.4301i q^{69} -17.1596 q^{71} -34.8985i q^{73} +25.3173i q^{75} +109.450 q^{79} +35.5320 q^{81} -10.8247i q^{83} -175.349 q^{85} -13.5982i q^{87} +116.218i q^{89} +63.3691 q^{93} +68.0097 q^{95} +55.5922i q^{97} +90.2196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{9}+O(q^{10}) \) 8 * q - 64 * q^9	\( 8 q - 64 q^{9} + 24 q^{11} - 40 q^{15} + 136 q^{23} - 80 q^{25} + 64 q^{29} - 64 q^{37} + 376 q^{39} - 136 q^{43} + 408 q^{51} - 104 q^{53} - 240 q^{57} + 224 q^{65} + 16 q^{67} - 352 q^{71} + 112 q^{79} + 584 q^{81} + 200 q^{85} + 784 q^{93} - 120 q^{95} + 880 q^{99}+O(q^{100}) \) 8 * q - 64 * q^9 + 24 * q^11 - 40 * q^15 + 136 * q^23 - 80 * q^25 + 64 * q^29 - 64 * q^37 + 376 * q^39 - 136 * q^43 + 408 * q^51 - 104 * q^53 - 240 * q^57 + 224 * q^65 + 16 * q^67 - 352 * q^71 + 112 * q^79 + 584 * q^81 + 200 * q^85 + 784 * q^93 - 120 * q^95 + 880 * q^99

Introduction

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