LMFDB - Embedded newform 9702.2.a.ds.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.a (trivial)

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:	yes
Analytic conductor:	$77.4708600410$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1386)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{8} +2.41421 q^{10} +1.00000 q^{11} -6.82843 q^{13} +1.00000 q^{16} -7.82843 q^{17} -4.82843 q^{19} +2.41421 q^{20} +1.00000 q^{22} +5.24264 q^{23} +0.828427 q^{25} -6.82843 q^{26} -2.82843 q^{29} +10.4853 q^{31} +1.00000 q^{32} -7.82843 q^{34} -1.65685 q^{37} -4.82843 q^{38} +2.41421 q^{40} -6.65685 q^{41} +2.82843 q^{43} +1.00000 q^{44} +5.24264 q^{46} -0.757359 q^{47} +0.828427 q^{50} -6.82843 q^{52} -10.8284 q^{53} +2.41421 q^{55} -2.82843 q^{58} +7.65685 q^{59} -4.41421 q^{61} +10.4853 q^{62} +1.00000 q^{64} -16.4853 q^{65} +3.48528 q^{67} -7.82843 q^{68} -9.31371 q^{71} +0.828427 q^{73} -1.65685 q^{74} -4.82843 q^{76} +13.2426 q^{79} +2.41421 q^{80} -6.65685 q^{82} -4.17157 q^{83} -18.8995 q^{85} +2.82843 q^{86} +1.00000 q^{88} -4.48528 q^{89} +5.24264 q^{92} -0.757359 q^{94} -11.6569 q^{95} -15.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} - 10 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} - 8 q^{26} + 4 q^{31} + 2 q^{32} - 10 q^{34} + 8 q^{37} - 4 q^{38} + 2 q^{40} - 2 q^{41} + 2 q^{44} + 2 q^{46} - 10 q^{47} - 4 q^{50} - 8 q^{52} - 16 q^{53} + 2 q^{55} + 4 q^{59} - 6 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 10 q^{67} - 10 q^{68} + 4 q^{71} - 4 q^{73} + 8 q^{74} - 4 q^{76} + 18 q^{79} + 2 q^{80} - 2 q^{82} - 14 q^{83} - 18 q^{85} + 2 q^{88} + 8 q^{89} + 2 q^{92} - 10 q^{94} - 12 q^{95} - 26 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 - 8 * q^13 + 2 * q^16 - 10 * q^17 - 4 * q^19 + 2 * q^20 + 2 * q^22 + 2 * q^23 - 4 * q^25 - 8 * q^26 + 4 * q^31 + 2 * q^32 - 10 * q^34 + 8 * q^37 - 4 * q^38 + 2 * q^40 - 2 * q^41 + 2 * q^44 + 2 * q^46 - 10 * q^47 - 4 * q^50 - 8 * q^52 - 16 * q^53 + 2 * q^55 + 4 * q^59 - 6 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 - 10 * q^67 - 10 * q^68 + 4 * q^71 - 4 * q^73 + 8 * q^74 - 4 * q^76 + 18 * q^79 + 2 * q^80 - 2 * q^82 - 14 * q^83 - 18 * q^85 + 2 * q^88 + 8 * q^89 + 2 * q^92 - 10 * q^94 - 12 * q^95 - 26 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.41421	1.07967	0.539835	−	0.841771i	$-0.318487\pi$
$5$	2.41421	1.07967	0.539835	+	0.841771i	$0.318487\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.41421	0.763441
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.82843	−1.89386	−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843	−1.89386	−0.946932	+	0.321433i	$0.895836\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.82843	−1.89867	−0.949336	−	0.314262i	$-0.898243\pi$
$17$	−7.82843	−1.89867	−0.949336	+	0.314262i	$0.898243\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	2.41421	0.539835
$21$	0	0
$22$	1.00000	0.213201
$23$	5.24264	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$23$	5.24264	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$24$	0	0
$25$	0.828427	0.165685
$26$	−6.82843	−1.33916
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.82843	−1.34256
$35$	0	0
$36$	0	0
$37$	−1.65685	−0.272385	−0.136193	−	0.990682i	$-0.543487\pi$
$37$	−1.65685	−0.272385	−0.136193	+	0.990682i	$0.543487\pi$
$38$	−4.82843	−0.783274
$39$	0	0
$40$	2.41421	0.381721
$41$	−6.65685	−1.03963	−0.519813	−	0.854280i	$-0.673998\pi$
$41$	−6.65685	−1.03963	−0.519813	+	0.854280i	$0.673998\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	5.24264	0.772985
$47$	−0.757359	−0.110472	−0.0552361	−	0.998473i	$-0.517591\pi$
$47$	−0.757359	−0.110472	−0.0552361	+	0.998473i	$0.517591\pi$
$48$	0	0
$49$	0	0
$50$	0.828427	0.117157
$51$	0	0
$52$	−6.82843	−0.946932
$53$	−10.8284	−1.48740	−0.743699	−	0.668514i	$-0.766931\pi$
$53$	−10.8284	−1.48740	−0.743699	+	0.668514i	$0.766931\pi$
$54$	0	0
$55$	2.41421	0.325532
$56$	0	0
$57$	0	0
$58$	−2.82843	−0.371391
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	−4.41421	−0.565182	−0.282591	−	0.959240i	$-0.591194\pi$
$61$	−4.41421	−0.565182	−0.282591	+	0.959240i	$0.591194\pi$
$62$	10.4853	1.33163
$63$	0	0
$64$	1.00000	0.125000
$65$	−16.4853	−2.04475
$66$	0	0
$67$	3.48528	0.425795	0.212897	−	0.977075i	$-0.431710\pi$
$67$	3.48528	0.425795	0.212897	+	0.977075i	$0.431710\pi$
$68$	−7.82843	−0.949336
$69$	0	0
$70$	0	0
$71$	−9.31371	−1.10533	−0.552667	−	0.833402i	$-0.686390\pi$
$71$	−9.31371	−1.10533	−0.552667	+	0.833402i	$0.686390\pi$
$72$	0	0
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	−1.65685	−0.192605
$75$	0	0
$76$	−4.82843	−0.553859
$77$	0	0
$78$	0	0
$79$	13.2426	1.48991	0.744957	−	0.667113i	$-0.232470\pi$
$79$	13.2426	1.48991	0.744957	+	0.667113i	$0.232470\pi$
$80$	2.41421	0.269917
$81$	0	0
$82$	−6.65685	−0.735127
$83$	−4.17157	−0.457890	−0.228945	−	0.973439i	$-0.573528\pi$
$83$	−4.17157	−0.457890	−0.228945	+	0.973439i	$0.573528\pi$
$84$	0	0
$85$	−18.8995	−2.04994
$86$	2.82843	0.304997
$87$	0	0
$88$	1.00000	0.106600
$89$	−4.48528	−0.475439	−0.237719	−	0.971334i	$-0.576400\pi$
$89$	−4.48528	−0.475439	−0.237719	+	0.971334i	$0.576400\pi$
$90$	0	0
$91$	0	0
$92$	5.24264	0.546583
$93$	0	0
$94$	−0.757359	−0.0781156
$95$	−11.6569	−1.19597
$96$	0	0
$97$	−15.8284	−1.60713	−0.803567	−	0.595215i	$-0.797067\pi$
$97$	−15.8284	−1.60713	−0.803567	+	0.595215i	$0.797067\pi$
$98$	0	0
$99$	0	0
$100$	0.828427	0.0828427
$101$	14.1421	1.40720	0.703598	−	0.710599i	$-0.251576\pi$
$101$	14.1421	1.40720	0.703598	+	0.710599i	$0.251576\pi$
$102$	0	0
$103$	−14.8284	−1.46109	−0.730544	−	0.682865i	$-0.760734\pi$
$103$	−14.8284	−1.46109	−0.730544	+	0.682865i	$0.760734\pi$
$104$	−6.82843	−0.669582
$105$	0	0
$106$	−10.8284	−1.05175
$107$	−5.34315	−0.516541	−0.258271	−	0.966073i	$-0.583153\pi$
$107$	−5.34315	−0.516541	−0.258271	+	0.966073i	$0.583153\pi$
$108$	0	0
$109$	−5.24264	−0.502154	−0.251077	−	0.967967i	$-0.580785\pi$
$109$	−5.24264	−0.502154	−0.251077	+	0.967967i	$0.580785\pi$
$110$	2.41421	0.230186
$111$	0	0
$112$	0	0
$113$	−7.65685	−0.720296	−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685	−0.720296	−0.360148	+	0.932895i	$0.617274\pi$
$114$	0	0
$115$	12.6569	1.18026
$116$	−2.82843	−0.262613
$117$	0	0
$118$	7.65685	0.704871
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.41421	−0.399644
$123$	0	0
$124$	10.4853	0.941606
$125$	−10.0711	−0.900784
$126$	0	0
$127$	−7.24264	−0.642680	−0.321340	−	0.946964i	$-0.604133\pi$
$127$	−7.24264	−0.642680	−0.321340	+	0.946964i	$0.604133\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−16.4853	−1.44585
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	0	0
$134$	3.48528	0.301082
$135$	0	0
$136$	−7.82843	−0.671282
$137$	0.828427	0.0707773	0.0353887	−	0.999374i	$-0.488733\pi$
$137$	0.828427	0.0707773	0.0353887	+	0.999374i	$0.488733\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	−9.31371	−0.781589
$143$	−6.82843	−0.571022
$144$	0	0
$145$	−6.82843	−0.567070
$146$	0.828427	0.0685611
$147$	0	0
$148$	−1.65685	−0.136193
$149$	−11.3137	−0.926855	−0.463428	−	0.886135i	$-0.653381\pi$
$149$	−11.3137	−0.926855	−0.463428	+	0.886135i	$0.653381\pi$
$150$	0	0
$151$	−7.24264	−0.589398	−0.294699	−	0.955590i	$-0.595219\pi$
$151$	−7.24264	−0.589398	−0.294699	+	0.955590i	$0.595219\pi$
$152$	−4.82843	−0.391637
$153$	0	0
$154$	0	0
$155$	25.3137	2.03325
$156$	0	0
$157$	−14.8284	−1.18344	−0.591719	−	0.806145i	$-0.701550\pi$
$157$	−14.8284	−1.18344	−0.591719	+	0.806145i	$0.701550\pi$
$158$	13.2426	1.05353
$159$	0	0
$160$	2.41421	0.190860
$161$	0	0
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	−6.65685	−0.519813
$165$	0	0
$166$	−4.17157	−0.323777
$167$	0.485281	0.0375522	0.0187761	−	0.999824i	$-0.494023\pi$
$167$	0.485281	0.0375522	0.0187761	+	0.999824i	$0.494023\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	−18.8995	−1.44953
$171$	0	0
$172$	2.82843	0.215666
$173$	16.1421	1.22726	0.613632	−	0.789592i	$-0.289708\pi$
$173$	16.1421	1.22726	0.613632	+	0.789592i	$0.289708\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−4.48528	−0.336186
$179$	−8.82843	−0.659868	−0.329934	−	0.944004i	$-0.607026\pi$
$179$	−8.82843	−0.659868	−0.329934	+	0.944004i	$0.607026\pi$
$180$	0	0
$181$	−13.6569	−1.01511	−0.507553	−	0.861621i	$-0.669450\pi$
$181$	−13.6569	−1.01511	−0.507553	+	0.861621i	$0.669450\pi$
$182$	0	0
$183$	0	0
$184$	5.24264	0.386493
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−7.82843	−0.572471
$188$	−0.757359	−0.0552361
$189$	0	0
$190$	−11.6569	−0.845677
$191$	13.3137	0.963346	0.481673	−	0.876351i	$-0.340029\pi$
$191$	13.3137	0.963346	0.481673	+	0.876351i	$0.340029\pi$
$192$	0	0
$193$	−14.8284	−1.06737	−0.533687	−	0.845682i	$-0.679194\pi$
$193$	−14.8284	−1.06737	−0.533687	+	0.845682i	$0.679194\pi$
$194$	−15.8284	−1.13641
$195$	0	0
$196$	0	0
$197$	−20.4853	−1.45952	−0.729758	−	0.683706i	$-0.760367\pi$
$197$	−20.4853	−1.45952	−0.729758	+	0.683706i	$0.760367\pi$
$198$	0	0
$199$	−19.6569	−1.39344	−0.696719	−	0.717344i	$-0.745357\pi$
$199$	−19.6569	−1.39344	−0.696719	+	0.717344i	$0.745357\pi$
$200$	0.828427	0.0585786
$201$	0	0
$202$	14.1421	0.995037
$203$	0	0
$204$	0	0
$205$	−16.0711	−1.12245
$206$	−14.8284	−1.03315
$207$	0	0
$208$	−6.82843	−0.473466
$209$	−4.82843	−0.333989
$210$	0	0
$211$	9.17157	0.631397	0.315699	−	0.948860i	$-0.397761\pi$
$211$	9.17157	0.631397	0.315699	+	0.948860i	$0.397761\pi$
$212$	−10.8284	−0.743699
$213$	0	0
$214$	−5.34315	−0.365250
$215$	6.82843	0.465695
$216$	0	0
$217$	0	0
$218$	−5.24264	−0.355076
$219$	0	0
$220$	2.41421	0.162766
$221$	53.4558	3.59583
$222$	0	0
$223$	9.31371	0.623692	0.311846	−	0.950133i	$-0.399053\pi$
$223$	9.31371	0.623692	0.311846	+	0.950133i	$0.399053\pi$
$224$	0	0
$225$	0	0
$226$	−7.65685	−0.509326
$227$	−10.3137	−0.684545	−0.342272	−	0.939601i	$-0.611197\pi$
$227$	−10.3137	−0.684545	−0.342272	+	0.939601i	$0.611197\pi$
$228$	0	0
$229$	11.3137	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$230$	12.6569	0.834568
$231$	0	0
$232$	−2.82843	−0.185695
$233$	−14.3137	−0.937722	−0.468861	−	0.883272i	$-0.655336\pi$
$233$	−14.3137	−0.937722	−0.468861	+	0.883272i	$0.655336\pi$
$234$	0	0
$235$	−1.82843	−0.119273
$236$	7.65685	0.498419
$237$	0	0
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−8.97056	−0.577845	−0.288922	−	0.957353i	$-0.593297\pi$
$241$	−8.97056	−0.577845	−0.288922	+	0.957353i	$0.593297\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−4.41421	−0.282591
$245$	0	0
$246$	0	0
$247$	32.9706	2.09787
$248$	10.4853	0.665816
$249$	0	0
$250$	−10.0711	−0.636950
$251$	−2.14214	−0.135210	−0.0676052	−	0.997712i	$-0.521536\pi$
$251$	−2.14214	−0.135210	−0.0676052	+	0.997712i	$0.521536\pi$
$252$	0	0
$253$	5.24264	0.329602
$254$	−7.24264	−0.454444
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.8284	1.92302	0.961512	−	0.274762i	$-0.0885991\pi$
$257$	30.8284	1.92302	0.961512	+	0.274762i	$0.0885991\pi$
$258$	0	0
$259$	0	0
$260$	−16.4853	−1.02237
$261$	0	0
$262$	−7.31371	−0.451842
$263$	−13.3137	−0.820958	−0.410479	−	0.911870i	$-0.634639\pi$
$263$	−13.3137	−0.820958	−0.410479	+	0.911870i	$0.634639\pi$
$264$	0	0
$265$	−26.1421	−1.60590
$266$	0	0
$267$	0	0
$268$	3.48528	0.212897
$269$	6.07107	0.370160	0.185080	−	0.982724i	$-0.440746\pi$
$269$	6.07107	0.370160	0.185080	+	0.982724i	$0.440746\pi$
$270$	0	0
$271$	9.31371	0.565767	0.282884	−	0.959154i	$-0.408709\pi$
$271$	9.31371	0.565767	0.282884	+	0.959154i	$0.408709\pi$
$272$	−7.82843	−0.474668
$273$	0	0
$274$	0.828427	0.0500471
$275$	0.828427	0.0499560
$276$	0	0
$277$	−1.17157	−0.0703930	−0.0351965	−	0.999380i	$-0.511206\pi$
$277$	−1.17157	−0.0703930	−0.0351965	+	0.999380i	$0.511206\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	0	0
$281$	19.4853	1.16239	0.581197	−	0.813763i	$-0.302584\pi$
$281$	19.4853	1.16239	0.581197	+	0.813763i	$0.302584\pi$
$282$	0	0
$283$	−30.1421	−1.79176	−0.895882	−	0.444292i	$-0.853455\pi$
$283$	−30.1421	−1.79176	−0.895882	+	0.444292i	$0.853455\pi$
$284$	−9.31371	−0.552667
$285$	0	0
$286$	−6.82843	−0.403773
$287$	0	0
$288$	0	0
$289$	44.2843	2.60496
$290$	−6.82843	−0.400979
$291$	0	0
$292$	0.828427	0.0484800
$293$	7.17157	0.418968	0.209484	−	0.977812i	$-0.432822\pi$
$293$	7.17157	0.418968	0.209484	+	0.977812i	$0.432822\pi$
$294$	0	0
$295$	18.4853	1.07625
$296$	−1.65685	−0.0963027
$297$	0	0
$298$	−11.3137	−0.655386
$299$	−35.7990	−2.07031
$300$	0	0
$301$	0	0
$302$	−7.24264	−0.416767
$303$	0	0
$304$	−4.82843	−0.276929
$305$	−10.6569	−0.610210
$306$	0	0
$307$	10.6274	0.606539	0.303269	−	0.952905i	$-0.401922\pi$
$307$	10.6274	0.606539	0.303269	+	0.952905i	$0.401922\pi$
$308$	0	0
$309$	0	0
$310$	25.3137	1.43772
$311$	−32.2132	−1.82664	−0.913322	−	0.407239i	$-0.866492\pi$
$311$	−32.2132	−1.82664	−0.913322	+	0.407239i	$0.866492\pi$
$312$	0	0
$313$	−6.68629	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$313$	−6.68629	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$314$	−14.8284	−0.836817
$315$	0	0
$316$	13.2426	0.744957
$317$	19.8701	1.11601	0.558007	−	0.829836i	$-0.311566\pi$
$317$	19.8701	1.11601	0.558007	+	0.829836i	$0.311566\pi$
$318$	0	0
$319$	−2.82843	−0.158362
$320$	2.41421	0.134959
$321$	0	0
$322$	0	0
$323$	37.7990	2.10319
$324$	0	0
$325$	−5.65685	−0.313786
$326$	3.00000	0.166155
$327$	0	0
$328$	−6.65685	−0.367563
$329$	0	0
$330$	0	0
$331$	−16.3137	−0.896683	−0.448341	−	0.893862i	$-0.647985\pi$
$331$	−16.3137	−0.896683	−0.448341	+	0.893862i	$0.647985\pi$
$332$	−4.17157	−0.228945
$333$	0	0
$334$	0.485281	0.0265534
$335$	8.41421	0.459718
$336$	0	0
$337$	26.4853	1.44275	0.721373	−	0.692547i	$-0.243512\pi$
$337$	26.4853	1.44275	0.721373	+	0.692547i	$0.243512\pi$
$338$	33.6274	1.82909
$339$	0	0
$340$	−18.8995	−1.02497
$341$	10.4853	0.567810
$342$	0	0
$343$	0	0
$344$	2.82843	0.152499
$345$	0	0
$346$	16.1421	0.867807
$347$	33.1421	1.77916	0.889582	−	0.456776i	$-0.150996\pi$
$347$	33.1421	1.77916	0.889582	+	0.456776i	$0.150996\pi$
$348$	0	0
$349$	9.72792	0.520724	0.260362	−	0.965511i	$-0.416158\pi$
$349$	9.72792	0.520724	0.260362	+	0.965511i	$0.416158\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−11.1716	−0.594603	−0.297301	−	0.954784i	$-0.596087\pi$
$353$	−11.1716	−0.594603	−0.297301	+	0.954784i	$0.596087\pi$
$354$	0	0
$355$	−22.4853	−1.19339
$356$	−4.48528	−0.237719
$357$	0	0
$358$	−8.82843	−0.466597
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	−13.6569	−0.717788
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	22.4853	1.17372	0.586861	−	0.809688i	$-0.300363\pi$
$367$	22.4853	1.17372	0.586861	+	0.809688i	$0.300363\pi$
$368$	5.24264	0.273292
$369$	0	0
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	9.72792	0.503693	0.251846	−	0.967767i	$-0.418962\pi$
$373$	9.72792	0.503693	0.251846	+	0.967767i	$0.418962\pi$
$374$	−7.82843	−0.404798
$375$	0	0
$376$	−0.757359	−0.0390578
$377$	19.3137	0.994707
$378$	0	0
$379$	30.6569	1.57474	0.787368	−	0.616483i	$-0.211443\pi$
$379$	30.6569	1.57474	0.787368	+	0.616483i	$0.211443\pi$
$380$	−11.6569	−0.597984
$381$	0	0
$382$	13.3137	0.681189
$383$	22.2843	1.13867	0.569337	−	0.822105i	$-0.307200\pi$
$383$	22.2843	1.13867	0.569337	+	0.822105i	$0.307200\pi$
$384$	0	0
$385$	0	0
$386$	−14.8284	−0.754747
$387$	0	0
$388$	−15.8284	−0.803567
$389$	23.7279	1.20305	0.601527	−	0.798853i	$-0.294559\pi$
$389$	23.7279	1.20305	0.601527	+	0.798853i	$0.294559\pi$
$390$	0	0
$391$	−41.0416	−2.07556
$392$	0	0
$393$	0	0
$394$	−20.4853	−1.03203
$395$	31.9706	1.60861
$396$	0	0
$397$	−4.48528	−0.225110	−0.112555	−	0.993646i	$-0.535903\pi$
$397$	−4.48528	−0.225110	−0.112555	+	0.993646i	$0.535903\pi$
$398$	−19.6569	−0.985309
$399$	0	0
$400$	0.828427	0.0414214
$401$	23.7990	1.18846	0.594232	−	0.804293i	$-0.297456\pi$
$401$	23.7990	1.18846	0.594232	+	0.804293i	$0.297456\pi$
$402$	0	0
$403$	−71.5980	−3.56655
$404$	14.1421	0.703598
$405$	0	0
$406$	0	0
$407$	−1.65685	−0.0821272
$408$	0	0
$409$	39.4558	1.95097	0.975483	−	0.220075i	$-0.0706302\pi$
$409$	39.4558	1.95097	0.975483	+	0.220075i	$0.0706302\pi$
$410$	−16.0711	−0.793693
$411$	0	0
$412$	−14.8284	−0.730544
$413$	0	0
$414$	0	0
$415$	−10.0711	−0.494369
$416$	−6.82843	−0.334791
$417$	0	0
$418$	−4.82843	−0.236166
$419$	13.7990	0.674125	0.337062	−	0.941482i	$-0.390567\pi$
$419$	13.7990	0.674125	0.337062	+	0.941482i	$0.390567\pi$
$420$	0	0
$421$	9.17157	0.446995	0.223498	−	0.974704i	$-0.428253\pi$
$421$	9.17157	0.446995	0.223498	+	0.974704i	$0.428253\pi$
$422$	9.17157	0.446465
$423$	0	0
$424$	−10.8284	−0.525875
$425$	−6.48528	−0.314582
$426$	0	0
$427$	0	0
$428$	−5.34315	−0.258271
$429$	0	0
$430$	6.82843	0.329296
$431$	−24.8284	−1.19594	−0.597972	−	0.801517i	$-0.704026\pi$
$431$	−24.8284	−1.19594	−0.597972	+	0.801517i	$0.704026\pi$
$432$	0	0
$433$	−15.3431	−0.737345	−0.368672	−	0.929559i	$-0.620188\pi$
$433$	−15.3431	−0.737345	−0.368672	+	0.929559i	$0.620188\pi$
$434$	0	0
$435$	0	0
$436$	−5.24264	−0.251077
$437$	−25.3137	−1.21092
$438$	0	0
$439$	−33.3848	−1.59337	−0.796684	−	0.604396i	$-0.793415\pi$
$439$	−33.3848	−1.59337	−0.796684	+	0.604396i	$0.793415\pi$
$440$	2.41421	0.115093
$441$	0	0
$442$	53.4558	2.54264
$443$	25.6569	1.21899	0.609497	−	0.792788i	$-0.291371\pi$
$443$	25.6569	1.21899	0.609497	+	0.792788i	$0.291371\pi$
$444$	0	0
$445$	−10.8284	−0.513317
$446$	9.31371	0.441017
$447$	0	0
$448$	0	0
$449$	−39.1127	−1.84584	−0.922921	−	0.384989i	$-0.874205\pi$
$449$	−39.1127	−1.84584	−0.922921	+	0.384989i	$0.874205\pi$
$450$	0	0
$451$	−6.65685	−0.313459
$452$	−7.65685	−0.360148
$453$	0	0
$454$	−10.3137	−0.484046
$455$	0	0
$456$	0	0
$457$	−10.6274	−0.497130	−0.248565	−	0.968615i	$-0.579959\pi$
$457$	−10.6274	−0.497130	−0.248565	+	0.968615i	$0.579959\pi$
$458$	11.3137	0.528655
$459$	0	0
$460$	12.6569	0.590129
$461$	−12.3431	−0.574878	−0.287439	−	0.957799i	$-0.592804\pi$
$461$	−12.3431	−0.574878	−0.287439	+	0.957799i	$0.592804\pi$
$462$	0	0
$463$	−39.1127	−1.81772	−0.908861	−	0.417100i	$-0.863046\pi$
$463$	−39.1127	−1.81772	−0.908861	+	0.417100i	$0.863046\pi$
$464$	−2.82843	−0.131306
$465$	0	0
$466$	−14.3137	−0.663070
$467$	30.9706	1.43315	0.716573	−	0.697512i	$-0.245709\pi$
$467$	30.9706	1.43315	0.716573	+	0.697512i	$0.245709\pi$
$468$	0	0
$469$	0	0
$470$	−1.82843	−0.0843391
$471$	0	0
$472$	7.65685	0.352435
$473$	2.82843	0.130051
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	20.0000	0.914779
$479$	10.8284	0.494763	0.247382	−	0.968918i	$-0.420430\pi$
$479$	10.8284	0.494763	0.247382	+	0.968918i	$0.420430\pi$
$480$	0	0
$481$	11.3137	0.515861
$482$	−8.97056	−0.408598
$483$	0	0
$484$	1.00000	0.0454545
$485$	−38.2132	−1.73517
$486$	0	0
$487$	1.31371	0.0595298	0.0297649	−	0.999557i	$-0.490524\pi$
$487$	1.31371	0.0595298	0.0297649	+	0.999557i	$0.490524\pi$
$488$	−4.41421	−0.199822
$489$	0	0
$490$	0	0
$491$	29.6274	1.33707	0.668533	−	0.743682i	$-0.266922\pi$
$491$	29.6274	1.33707	0.668533	+	0.743682i	$0.266922\pi$
$492$	0	0
$493$	22.1421	0.997232
$494$	32.9706	1.48342
$495$	0	0
$496$	10.4853	0.470803
$497$	0	0
$498$	0	0
$499$	25.6569	1.14856	0.574279	−	0.818659i	$-0.305282\pi$
$499$	25.6569	1.14856	0.574279	+	0.818659i	$0.305282\pi$
$500$	−10.0711	−0.450392
$501$	0	0
$502$	−2.14214	−0.0956082
$503$	−24.9706	−1.11338	−0.556691	−	0.830720i	$-0.687929\pi$
$503$	−24.9706	−1.11338	−0.556691	+	0.830720i	$0.687929\pi$
$504$	0	0
$505$	34.1421	1.51931
$506$	5.24264	0.233064
$507$	0	0
$508$	−7.24264	−0.321340
$509$	30.8284	1.36645	0.683223	−	0.730210i	$-0.260578\pi$
$509$	30.8284	1.36645	0.683223	+	0.730210i	$0.260578\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.8284	1.35978
$515$	−35.7990	−1.57749
$516$	0	0
$517$	−0.757359	−0.0333086
$518$	0	0
$519$	0	0
$520$	−16.4853	−0.722927
$521$	6.68629	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$521$	6.68629	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$522$	0	0
$523$	19.4558	0.850745	0.425372	−	0.905018i	$-0.360143\pi$
$523$	19.4558	0.850745	0.425372	+	0.905018i	$0.360143\pi$
$524$	−7.31371	−0.319501
$525$	0	0
$526$	−13.3137	−0.580505
$527$	−82.0833	−3.57560
$528$	0	0
$529$	4.48528	0.195012
$530$	−26.1421	−1.13554
$531$	0	0
$532$	0	0
$533$	45.4558	1.96891
$534$	0	0
$535$	−12.8995	−0.557694
$536$	3.48528	0.150541
$537$	0	0
$538$	6.07107	0.261742
$539$	0	0
$540$	0	0
$541$	−36.5563	−1.57168	−0.785840	−	0.618430i	$-0.787769\pi$
$541$	−36.5563	−1.57168	−0.785840	+	0.618430i	$0.787769\pi$
$542$	9.31371	0.400058
$543$	0	0
$544$	−7.82843	−0.335641
$545$	−12.6569	−0.542160
$546$	0	0
$547$	−7.85786	−0.335978	−0.167989	−	0.985789i	$-0.553727\pi$
$547$	−7.85786	−0.335978	−0.167989	+	0.985789i	$0.553727\pi$
$548$	0.828427	0.0353887
$549$	0	0
$550$	0.828427	0.0353243
$551$	13.6569	0.581802
$552$	0	0
$553$	0	0
$554$	−1.17157	−0.0497754
$555$	0	0
$556$	−6.00000	−0.254457
$557$	12.4853	0.529018	0.264509	−	0.964383i	$-0.414790\pi$
$557$	12.4853	0.529018	0.264509	+	0.964383i	$0.414790\pi$
$558$	0	0
$559$	−19.3137	−0.816883
$560$	0	0
$561$	0	0
$562$	19.4853	0.821937
$563$	14.3431	0.604492	0.302246	−	0.953230i	$-0.402264\pi$
$563$	14.3431	0.604492	0.302246	+	0.953230i	$0.402264\pi$
$564$	0	0
$565$	−18.4853	−0.777682
$566$	−30.1421	−1.26697
$567$	0	0
$568$	−9.31371	−0.390795
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−10.3431	−0.432847	−0.216424	−	0.976300i	$-0.569439\pi$
$571$	−10.3431	−0.432847	−0.216424	+	0.976300i	$0.569439\pi$
$572$	−6.82843	−0.285511
$573$	0	0
$574$	0	0
$575$	4.34315	0.181122
$576$	0	0
$577$	−23.1421	−0.963420	−0.481710	−	0.876331i	$-0.659984\pi$
$577$	−23.1421	−0.963420	−0.481710	+	0.876331i	$0.659984\pi$
$578$	44.2843	1.84198
$579$	0	0
$580$	−6.82843	−0.283535
$581$	0	0
$582$	0	0
$583$	−10.8284	−0.448468
$584$	0.828427	0.0342806
$585$	0	0
$586$	7.17157	0.296255
$587$	0.142136	0.00586657	0.00293328	−	0.999996i	$-0.499066\pi$
$587$	0.142136	0.00586657	0.00293328	+	0.999996i	$0.499066\pi$
$588$	0	0
$589$	−50.6274	−2.08607
$590$	18.4853	0.761027
$591$	0	0
$592$	−1.65685	−0.0680963
$593$	−27.9411	−1.14740	−0.573702	−	0.819064i	$-0.694493\pi$
$593$	−27.9411	−1.14740	−0.573702	+	0.819064i	$0.694493\pi$
$594$	0	0
$595$	0	0
$596$	−11.3137	−0.463428
$597$	0	0
$598$	−35.7990	−1.46393
$599$	−19.0416	−0.778020	−0.389010	−	0.921234i	$-0.627183\pi$
$599$	−19.0416	−0.778020	−0.389010	+	0.921234i	$0.627183\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	0	0
$604$	−7.24264	−0.294699
$605$	2.41421	0.0981517
$606$	0	0
$607$	3.78680	0.153701	0.0768507	−	0.997043i	$-0.475514\pi$
$607$	3.78680	0.153701	0.0768507	+	0.997043i	$0.475514\pi$
$608$	−4.82843	−0.195819
$609$	0	0
$610$	−10.6569	−0.431483
$611$	5.17157	0.209219
$612$	0	0
$613$	−11.1005	−0.448345	−0.224173	−	0.974549i	$-0.571968\pi$
$613$	−11.1005	−0.448345	−0.224173	+	0.974549i	$0.571968\pi$
$614$	10.6274	0.428888
$615$	0	0
$616$	0	0
$617$	−7.51472	−0.302531	−0.151266	−	0.988493i	$-0.548335\pi$
$617$	−7.51472	−0.302531	−0.151266	+	0.988493i	$0.548335\pi$
$618$	0	0
$619$	17.4853	0.702793	0.351396	−	0.936227i	$-0.385707\pi$
$619$	17.4853	0.702793	0.351396	+	0.936227i	$0.385707\pi$
$620$	25.3137	1.01662
$621$	0	0
$622$	−32.2132	−1.29163
$623$	0	0
$624$	0	0
$625$	−28.4558	−1.13823
$626$	−6.68629	−0.267238
$627$	0	0
$628$	−14.8284	−0.591719
$629$	12.9706	0.517170
$630$	0	0
$631$	−34.9706	−1.39216	−0.696078	−	0.717966i	$-0.745073\pi$
$631$	−34.9706	−1.39216	−0.696078	+	0.717966i	$0.745073\pi$
$632$	13.2426	0.526764
$633$	0	0
$634$	19.8701	0.789141
$635$	−17.4853	−0.693882
$636$	0	0
$637$	0	0
$638$	−2.82843	−0.111979
$639$	0	0
$640$	2.41421	0.0954302
$641$	−4.68629	−0.185097	−0.0925487	−	0.995708i	$-0.529501\pi$
$641$	−4.68629	−0.185097	−0.0925487	+	0.995708i	$0.529501\pi$
$642$	0	0
$643$	6.34315	0.250149	0.125075	−	0.992147i	$-0.460083\pi$
$643$	6.34315	0.250149	0.125075	+	0.992147i	$0.460083\pi$
$644$	0	0
$645$	0	0
$646$	37.7990	1.48718
$647$	−16.6985	−0.656485	−0.328243	−	0.944593i	$-0.606456\pi$
$647$	−16.6985	−0.656485	−0.328243	+	0.944593i	$0.606456\pi$
$648$	0	0
$649$	7.65685	0.300558
$650$	−5.65685	−0.221880
$651$	0	0
$652$	3.00000	0.117489
$653$	−26.3553	−1.03136	−0.515682	−	0.856780i	$-0.672461\pi$
$653$	−26.3553	−1.03136	−0.515682	+	0.856780i	$0.672461\pi$
$654$	0	0
$655$	−17.6569	−0.689910
$656$	−6.65685	−0.259906
$657$	0	0
$658$	0	0
$659$	30.9411	1.20530	0.602648	−	0.798007i	$-0.294112\pi$
$659$	30.9411	1.20530	0.602648	+	0.798007i	$0.294112\pi$
$660$	0	0
$661$	−5.02944	−0.195622	−0.0978112	−	0.995205i	$-0.531184\pi$
$661$	−5.02944	−0.195622	−0.0978112	+	0.995205i	$0.531184\pi$
$662$	−16.3137	−0.634050
$663$	0	0
$664$	−4.17157	−0.161888
$665$	0	0
$666$	0	0
$667$	−14.8284	−0.574159
$668$	0.485281	0.0187761
$669$	0	0
$670$	8.41421	0.325069
$671$	−4.41421	−0.170409
$672$	0	0
$673$	−20.6274	−0.795128	−0.397564	−	0.917574i	$-0.630144\pi$
$673$	−20.6274	−0.795128	−0.397564	+	0.917574i	$0.630144\pi$
$674$	26.4853	1.02017
$675$	0	0
$676$	33.6274	1.29336
$677$	−30.7696	−1.18257	−0.591285	−	0.806463i	$-0.701379\pi$
$677$	−30.7696	−1.18257	−0.591285	+	0.806463i	$0.701379\pi$
$678$	0	0
$679$	0	0
$680$	−18.8995	−0.724763
$681$	0	0
$682$	10.4853	0.401502
$683$	6.34315	0.242714	0.121357	−	0.992609i	$-0.461275\pi$
$683$	6.34315	0.242714	0.121357	+	0.992609i	$0.461275\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	2.82843	0.107833
$689$	73.9411	2.81693
$690$	0	0
$691$	0.857864	0.0326347	0.0163173	−	0.999867i	$-0.494806\pi$
$691$	0.857864	0.0326347	0.0163173	+	0.999867i	$0.494806\pi$
$692$	16.1421	0.613632
$693$	0	0
$694$	33.1421	1.25806
$695$	−14.4853	−0.549458
$696$	0	0
$697$	52.1127	1.97391
$698$	9.72792	0.368207
$699$	0	0
$700$	0	0
$701$	−0.142136	−0.00536839	−0.00268419	−	0.999996i	$-0.500854\pi$
$701$	−0.142136	−0.00536839	−0.00268419	+	0.999996i	$0.500854\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	1.00000	0.0376889
$705$	0	0
$706$	−11.1716	−0.420448
$707$	0	0
$708$	0	0
$709$	32.1421	1.20712	0.603562	−	0.797316i	$-0.293748\pi$
$709$	32.1421	1.20712	0.603562	+	0.797316i	$0.293748\pi$
$710$	−22.4853	−0.843858
$711$	0	0
$712$	−4.48528	−0.168093
$713$	54.9706	2.05866
$714$	0	0
$715$	−16.4853	−0.616515
$716$	−8.82843	−0.329934
$717$	0	0
$718$	8.48528	0.316668
$719$	26.2132	0.977588	0.488794	−	0.872399i	$-0.337437\pi$
$719$	26.2132	0.977588	0.488794	+	0.872399i	$0.337437\pi$
$720$	0	0
$721$	0	0
$722$	4.31371	0.160540
$723$	0	0
$724$	−13.6569	−0.507553
$725$	−2.34315	−0.0870222
$726$	0	0
$727$	−15.5147	−0.575409	−0.287705	−	0.957719i	$-0.592892\pi$
$727$	−15.5147	−0.575409	−0.287705	+	0.957719i	$0.592892\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	−22.1421	−0.818956
$732$	0	0
$733$	−23.2426	−0.858487	−0.429243	−	0.903189i	$-0.641220\pi$
$733$	−23.2426	−0.858487	−0.429243	+	0.903189i	$0.641220\pi$
$734$	22.4853	0.829947
$735$	0	0
$736$	5.24264	0.193246
$737$	3.48528	0.128382
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	−2.34315	−0.0859617	−0.0429808	−	0.999076i	$-0.513685\pi$
$743$	−2.34315	−0.0859617	−0.0429808	+	0.999076i	$0.513685\pi$
$744$	0	0
$745$	−27.3137	−1.00070
$746$	9.72792	0.356165
$747$	0	0
$748$	−7.82843	−0.286236
$749$	0	0
$750$	0	0
$751$	−9.79899	−0.357570	−0.178785	−	0.983888i	$-0.557217\pi$
$751$	−9.79899	−0.357570	−0.178785	+	0.983888i	$0.557217\pi$
$752$	−0.757359	−0.0276181
$753$	0	0
$754$	19.3137	0.703364
$755$	−17.4853	−0.636355
$756$	0	0
$757$	8.34315	0.303237	0.151618	−	0.988439i	$-0.451552\pi$
$757$	8.34315	0.303237	0.151618	+	0.988439i	$0.451552\pi$
$758$	30.6569	1.11351
$759$	0	0
$760$	−11.6569	−0.422839
$761$	−26.4558	−0.959024	−0.479512	−	0.877535i	$-0.659186\pi$
$761$	−26.4558	−0.959024	−0.479512	+	0.877535i	$0.659186\pi$
$762$	0	0
$763$	0	0
$764$	13.3137	0.481673
$765$	0	0
$766$	22.2843	0.805163
$767$	−52.2843	−1.88788
$768$	0	0
$769$	9.51472	0.343110	0.171555	−	0.985175i	$-0.445121\pi$
$769$	9.51472	0.343110	0.171555	+	0.985175i	$0.445121\pi$
$770$	0	0
$771$	0	0
$772$	−14.8284	−0.533687
$773$	22.7574	0.818525	0.409263	−	0.912417i	$-0.365786\pi$
$773$	22.7574	0.818525	0.409263	+	0.912417i	$0.365786\pi$
$774$	0	0
$775$	8.68629	0.312021
$776$	−15.8284	−0.568207
$777$	0	0
$778$	23.7279	0.850687
$779$	32.1421	1.15161
$780$	0	0
$781$	−9.31371	−0.333271
$782$	−41.0416	−1.46765
$783$	0	0
$784$	0	0
$785$	−35.7990	−1.27772
$786$	0	0
$787$	34.7696	1.23940	0.619700	−	0.784839i	$-0.287254\pi$
$787$	34.7696	1.23940	0.619700	+	0.784839i	$0.287254\pi$
$788$	−20.4853	−0.729758
$789$	0	0
$790$	31.9706	1.13746
$791$	0	0
$792$	0	0
$793$	30.1421	1.07038
$794$	−4.48528	−0.159177
$795$	0	0
$796$	−19.6569	−0.696719
$797$	33.1005	1.17248	0.586240	−	0.810137i	$-0.300608\pi$
$797$	33.1005	1.17248	0.586240	+	0.810137i	$0.300608\pi$
$798$	0	0
$799$	5.92893	0.209751
$800$	0.828427	0.0292893
$801$	0	0
$802$	23.7990	0.840372
$803$	0.828427	0.0292346
$804$	0	0
$805$	0	0
$806$	−71.5980	−2.52193
$807$	0	0
$808$	14.1421	0.497519
$809$	−20.6569	−0.726256	−0.363128	−	0.931739i	$-0.618291\pi$
$809$	−20.6569	−0.726256	−0.363128	+	0.931739i	$0.618291\pi$
$810$	0	0
$811$	3.65685	0.128410	0.0642048	−	0.997937i	$-0.479549\pi$
$811$	3.65685	0.128410	0.0642048	+	0.997937i	$0.479549\pi$
$812$	0	0
$813$	0	0
$814$	−1.65685	−0.0580727
$815$	7.24264	0.253699
$816$	0	0
$817$	−13.6569	−0.477793
$818$	39.4558	1.37954
$819$	0	0
$820$	−16.0711	−0.561226
$821$	38.4853	1.34315	0.671573	−	0.740939i	$-0.265619\pi$
$821$	38.4853	1.34315	0.671573	+	0.740939i	$0.265619\pi$
$822$	0	0
$823$	−38.4853	−1.34151	−0.670756	−	0.741678i	$-0.734030\pi$
$823$	−38.4853	−1.34151	−0.670756	+	0.741678i	$0.734030\pi$
$824$	−14.8284	−0.516573
$825$	0	0
$826$	0	0
$827$	7.82843	0.272221	0.136111	−	0.990694i	$-0.456540\pi$
$827$	7.82843	0.272221	0.136111	+	0.990694i	$0.456540\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	−10.0711	−0.349572
$831$	0	0
$832$	−6.82843	−0.236733
$833$	0	0
$834$	0	0
$835$	1.17157	0.0405440
$836$	−4.82843	−0.166995
$837$	0	0
$838$	13.7990	0.476678
$839$	−34.2132	−1.18117	−0.590585	−	0.806975i	$-0.701103\pi$
$839$	−34.2132	−1.18117	−0.590585	+	0.806975i	$0.701103\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	9.17157	0.316073
$843$	0	0
$844$	9.17157	0.315699
$845$	81.1838	2.79281
$846$	0	0
$847$	0	0
$848$	−10.8284	−0.371850
$849$	0	0
$850$	−6.48528	−0.222443
$851$	−8.68629	−0.297762
$852$	0	0
$853$	−14.7574	−0.505282	−0.252641	−	0.967560i	$-0.581299\pi$
$853$	−14.7574	−0.505282	−0.252641	+	0.967560i	$0.581299\pi$
$854$	0	0
$855$	0	0
$856$	−5.34315	−0.182625
$857$	44.3137	1.51373	0.756864	−	0.653573i	$-0.226731\pi$
$857$	44.3137	1.51373	0.756864	+	0.653573i	$0.226731\pi$
$858$	0	0
$859$	16.9411	0.578024	0.289012	−	0.957326i	$-0.406673\pi$
$859$	16.9411	0.578024	0.289012	+	0.957326i	$0.406673\pi$
$860$	6.82843	0.232847
$861$	0	0
$862$	−24.8284	−0.845660
$863$	37.1838	1.26575	0.632875	−	0.774254i	$-0.281875\pi$
$863$	37.1838	1.26575	0.632875	+	0.774254i	$0.281875\pi$
$864$	0	0
$865$	38.9706	1.32504
$866$	−15.3431	−0.521381
$867$	0	0
$868$	0	0
$869$	13.2426	0.449226
$870$	0	0
$871$	−23.7990	−0.806398
$872$	−5.24264	−0.177538
$873$	0	0
$874$	−25.3137	−0.856249
$875$	0	0
$876$	0	0
$877$	−6.55635	−0.221392	−0.110696	−	0.993854i	$-0.535308\pi$
$877$	−6.55635	−0.221392	−0.110696	+	0.993854i	$0.535308\pi$
$878$	−33.3848	−1.12668
$879$	0	0
$880$	2.41421	0.0813831
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	17.4853	0.588427	0.294213	−	0.955740i	$-0.404942\pi$
$883$	17.4853	0.588427	0.294213	+	0.955740i	$0.404942\pi$
$884$	53.4558	1.79791
$885$	0	0
$886$	25.6569	0.861959
$887$	19.6569	0.660013	0.330006	−	0.943979i	$-0.392949\pi$
$887$	19.6569	0.660013	0.330006	+	0.943979i	$0.392949\pi$
$888$	0	0
$889$	0	0
$890$	−10.8284	−0.362970
$891$	0	0
$892$	9.31371	0.311846
$893$	3.65685	0.122372
$894$	0	0
$895$	−21.3137	−0.712439
$896$	0	0
$897$	0	0
$898$	−39.1127	−1.30521
$899$	−29.6569	−0.989111
$900$	0	0
$901$	84.7696	2.82408
$902$	−6.65685	−0.221649
$903$	0	0
$904$	−7.65685	−0.254663
$905$	−32.9706	−1.09598
$906$	0	0
$907$	39.1421	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$907$	39.1421	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$908$	−10.3137	−0.342272
$909$	0	0
$910$	0	0
$911$	7.78680	0.257988	0.128994	−	0.991645i	$-0.458825\pi$
$911$	7.78680	0.257988	0.128994	+	0.991645i	$0.458825\pi$
$912$	0	0
$913$	−4.17157	−0.138059
$914$	−10.6274	−0.351524
$915$	0	0
$916$	11.3137	0.373815
$917$	0	0
$918$	0	0
$919$	−3.10051	−0.102276	−0.0511381	−	0.998692i	$-0.516285\pi$
$919$	−3.10051	−0.102276	−0.0511381	+	0.998692i	$0.516285\pi$
$920$	12.6569	0.417284
$921$	0	0
$922$	−12.3431	−0.406500
$923$	63.5980	2.09335
$924$	0	0
$925$	−1.37258	−0.0451303
$926$	−39.1127	−1.28532
$927$	0	0
$928$	−2.82843	−0.0928477
$929$	−43.4558	−1.42574	−0.712870	−	0.701296i	$-0.752605\pi$
$929$	−43.4558	−1.42574	−0.712870	+	0.701296i	$0.752605\pi$
$930$	0	0
$931$	0	0
$932$	−14.3137	−0.468861
$933$	0	0
$934$	30.9706	1.01339
$935$	−18.8995	−0.618080
$936$	0	0
$937$	0.343146	0.0112101	0.00560504	−	0.999984i	$-0.498216\pi$
$937$	0.343146	0.0112101	0.00560504	+	0.999984i	$0.498216\pi$
$938$	0	0
$939$	0	0
$940$	−1.82843	−0.0596367
$941$	56.6274	1.84600	0.923001	−	0.384799i	$-0.125729\pi$
$941$	56.6274	1.84600	0.923001	+	0.384799i	$0.125729\pi$
$942$	0	0
$943$	−34.8995	−1.13648
$944$	7.65685	0.249209
$945$	0	0
$946$	2.82843	0.0919601
$947$	−32.6274	−1.06025	−0.530124	−	0.847920i	$-0.677855\pi$
$947$	−32.6274	−1.06025	−0.530124	+	0.847920i	$0.677855\pi$
$948$	0	0
$949$	−5.65685	−0.183629
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	16.3137	0.528453	0.264226	−	0.964461i	$-0.414883\pi$
$953$	16.3137	0.528453	0.264226	+	0.964461i	$0.414883\pi$
$954$	0	0
$955$	32.1421	1.04010
$956$	20.0000	0.646846
$957$	0	0
$958$	10.8284	0.349851
$959$	0	0
$960$	0	0
$961$	78.9411	2.54649
$962$	11.3137	0.364769
$963$	0	0
$964$	−8.97056	−0.288922
$965$	−35.7990	−1.15241
$966$	0	0
$967$	22.8995	0.736398	0.368199	−	0.929747i	$-0.379974\pi$
$967$	22.8995	0.736398	0.368199	+	0.929747i	$0.379974\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−38.2132	−1.22695
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	0	0
$973$	0	0
$974$	1.31371	0.0420939
$975$	0	0
$976$	−4.41421	−0.141296
$977$	−35.9411	−1.14986	−0.574929	−	0.818203i	$-0.694970\pi$
$977$	−35.9411	−1.14986	−0.574929	+	0.818203i	$0.694970\pi$
$978$	0	0
$979$	−4.48528	−0.143350
$980$	0	0
$981$	0	0
$982$	29.6274	0.945449
$983$	45.2426	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$983$	45.2426	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$984$	0	0
$985$	−49.4558	−1.57579
$986$	22.1421	0.705149
$987$	0	0
$988$	32.9706	1.04893
$989$	14.8284	0.471517
$990$	0	0
$991$	33.6569	1.06915	0.534573	−	0.845123i	$-0.320473\pi$
$991$	33.6569	1.06915	0.534573	+	0.845123i	$0.320473\pi$
$992$	10.4853	0.332908
$993$	0	0
$994$	0	0
$995$	−47.4558	−1.50445
$996$	0	0
$997$	8.20101	0.259729	0.129864	−	0.991532i	$-0.458546\pi$
$997$	8.20101	0.259729	0.129864	+	0.991532i	$0.458546\pi$
$998$	25.6569	0.812154
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.ds.1.2		2
3.2	odd	2		9702.2.a.cj.1.1		2
7.2	even	3		1386.2.k.q.991.1	yes	4
7.4	even	3		1386.2.k.q.793.1	✓	4
7.6	odd	2		9702.2.a.da.1.1		2
21.2	odd	6		1386.2.k.u.991.2	yes	4
21.11	odd	6		1386.2.k.u.793.2	yes	4
21.20	even	2		9702.2.a.cv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.k.q.793.1	✓	4	7.4	even	3
1386.2.k.q.991.1	yes	4	7.2	even	3
1386.2.k.u.793.2	yes	4	21.11	odd	6
1386.2.k.u.991.2	yes	4	21.2	odd	6
9702.2.a.cj.1.1		2	3.2	odd	2
9702.2.a.cv.1.2		2	21.20	even	2
9702.2.a.da.1.1		2	7.6	odd	2
9702.2.a.ds.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{8} +2.41421 q^{10} +1.00000 q^{11} -6.82843 q^{13} +1.00000 q^{16} -7.82843 q^{17} -4.82843 q^{19} +2.41421 q^{20} +1.00000 q^{22} +5.24264 q^{23} +0.828427 q^{25} -6.82843 q^{26} -2.82843 q^{29} +10.4853 q^{31} +1.00000 q^{32} -7.82843 q^{34} -1.65685 q^{37} -4.82843 q^{38} +2.41421 q^{40} -6.65685 q^{41} +2.82843 q^{43} +1.00000 q^{44} +5.24264 q^{46} -0.757359 q^{47} +0.828427 q^{50} -6.82843 q^{52} -10.8284 q^{53} +2.41421 q^{55} -2.82843 q^{58} +7.65685 q^{59} -4.41421 q^{61} +10.4853 q^{62} +1.00000 q^{64} -16.4853 q^{65} +3.48528 q^{67} -7.82843 q^{68} -9.31371 q^{71} +0.828427 q^{73} -1.65685 q^{74} -4.82843 q^{76} +13.2426 q^{79} +2.41421 q^{80} -6.65685 q^{82} -4.17157 q^{83} -18.8995 q^{85} +2.82843 q^{86} +1.00000 q^{88} -4.48528 q^{89} +5.24264 q^{92} -0.757359 q^{94} -11.6569 q^{95} -15.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} - 10 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} - 8 q^{26} + 4 q^{31} + 2 q^{32} - 10 q^{34} + 8 q^{37} - 4 q^{38} + 2 q^{40} - 2 q^{41} + 2 q^{44} + 2 q^{46} - 10 q^{47} - 4 q^{50} - 8 q^{52} - 16 q^{53} + 2 q^{55} + 4 q^{59} - 6 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 10 q^{67} - 10 q^{68} + 4 q^{71} - 4 q^{73} + 8 q^{74} - 4 q^{76} + 18 q^{79} + 2 q^{80} - 2 q^{82} - 14 q^{83} - 18 q^{85} + 2 q^{88} + 8 q^{89} + 2 q^{92} - 10 q^{94} - 12 q^{95} - 26 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 - 8 * q^13 + 2 * q^16 - 10 * q^17 - 4 * q^19 + 2 * q^20 + 2 * q^22 + 2 * q^23 - 4 * q^25 - 8 * q^26 + 4 * q^31 + 2 * q^32 - 10 * q^34 + 8 * q^37 - 4 * q^38 + 2 * q^40 - 2 * q^41 + 2 * q^44 + 2 * q^46 - 10 * q^47 - 4 * q^50 - 8 * q^52 - 16 * q^53 + 2 * q^55 + 4 * q^59 - 6 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 - 10 * q^67 - 10 * q^68 + 4 * q^71 - 4 * q^73 + 8 * q^74 - 4 * q^76 + 18 * q^79 + 2 * q^80 - 2 * q^82 - 14 * q^83 - 18 * q^85 + 2 * q^88 + 8 * q^89 + 2 * q^92 - 10 * q^94 - 12 * q^95 - 26 * q^97

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