LMFDB - Embedded newform 9702.2.a.ec.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.14336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 14 $ x^4 - 8*x^2 + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.32685$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +3.29066 q^{10} +1.00000 q^{11} +6.06791 q^{13} +1.00000 q^{16} +6.11908 q^{17} +0.0511786 q^{19} +3.29066 q^{20} +1.00000 q^{22} +6.75605 q^{23} +5.82843 q^{25} +6.06791 q^{26} -2.82843 q^{29} +5.87644 q^{31} +1.00000 q^{32} +6.11908 q^{34} -8.31055 q^{37} +0.0511786 q^{38} +3.29066 q^{40} -6.11908 q^{41} +2.90080 q^{43} +1.00000 q^{44} +6.75605 q^{46} -1.22275 q^{47} +5.82843 q^{50} +6.06791 q^{52} -3.00316 q^{53} +3.29066 q^{55} -2.82843 q^{58} -10.8284 q^{59} +7.23948 q^{61} +5.87644 q^{62} +1.00000 q^{64} +19.9674 q^{65} -1.68051 q^{67} +6.11908 q^{68} +6.47896 q^{71} -11.6736 q^{73} -8.31055 q^{74} +0.0511786 q^{76} -9.13897 q^{79} +3.29066 q^{80} -6.11908 q^{82} -0.951983 q^{83} +20.1358 q^{85} +2.90080 q^{86} +1.00000 q^{88} -16.5469 q^{89} +6.75605 q^{92} -1.22275 q^{94} +0.168411 q^{95} -14.7216 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} + 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} + 12 q^{50} - 8 q^{53} - 32 q^{59} + 16 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} - 16 q^{89} + 8 q^{92} - 16 q^{94} + 16 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^11 + 4 * q^16 + 4 * q^22 + 8 * q^23 + 12 * q^25 + 16 * q^31 + 4 * q^32 + 8 * q^37 + 8 * q^43 + 4 * q^44 + 8 * q^46 - 16 * q^47 + 12 * q^50 - 8 * q^53 - 32 * q^59 + 16 * q^61 + 16 * q^62 + 4 * q^64 + 16 * q^65 + 16 * q^67 + 8 * q^74 + 16 * q^79 + 32 * q^85 + 8 * q^86 + 4 * q^88 - 16 * q^89 + 8 * q^92 - 16 * q^94 + 16 * q^95 - 16 * 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
$5$	3.29066	1.47163	0.735813	−	0.677184i	$-0.236800\pi$
$5$	3.29066	1.47163	0.735813	+	0.677184i	$0.236800\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.29066	1.04060
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.06791	1.68293	0.841467	−	0.540308i	$-0.181692\pi$
$13$	6.06791	1.68293	0.841467	+	0.540308i	$0.181692\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.11908	1.48410	0.742048	−	0.670347i	$-0.233855\pi$
$17$	6.11908	1.48410	0.742048	+	0.670347i	$0.233855\pi$
$18$	0	0
$19$	0.0511786	0.0117412	0.00587059	−	0.999983i	$-0.498131\pi$
$19$	0.0511786	0.0117412	0.00587059	+	0.999983i	$0.498131\pi$
$20$	3.29066	0.735813
$21$	0	0
$22$	1.00000	0.213201
$23$	6.75605	1.40873	0.704367	−	0.709836i	$-0.251231\pi$
$23$	6.75605	1.40873	0.704367	+	0.709836i	$0.251231\pi$
$24$	0	0
$25$	5.82843	1.16569
$26$	6.06791	1.19001
$27$	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
$31$	5.87644	1.05544	0.527720	−	0.849418i	$-0.323047\pi$
$31$	5.87644	1.05544	0.527720	+	0.849418i	$0.323047\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.11908	1.04941
$35$	0	0
$36$	0	0
$37$	−8.31055	−1.36625	−0.683123	−	0.730304i	$-0.739379\pi$
$37$	−8.31055	−1.36625	−0.683123	+	0.730304i	$0.739379\pi$
$38$	0.0511786	0.00830226
$39$	0	0
$40$	3.29066	0.520299
$41$	−6.11908	−0.955640	−0.477820	−	0.878458i	$-0.658573\pi$
$41$	−6.11908	−0.955640	−0.477820	+	0.878458i	$0.658573\pi$
$42$	0	0
$43$	2.90080	0.442369	0.221184	−	0.975232i	$-0.429008\pi$
$43$	2.90080	0.442369	0.221184	+	0.975232i	$0.429008\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.75605	0.996125
$47$	−1.22275	−0.178357	−0.0891783	−	0.996016i	$-0.528424\pi$
$47$	−1.22275	−0.178357	−0.0891783	+	0.996016i	$0.528424\pi$
$48$	0	0
$49$	0	0
$50$	5.82843	0.824264
$51$	0	0
$52$	6.06791	0.841467
$53$	−3.00316	−0.412516	−0.206258	−	0.978498i	$-0.566129\pi$
$53$	−3.00316	−0.412516	−0.206258	+	0.978498i	$0.566129\pi$
$54$	0	0
$55$	3.29066	0.443712
$56$	0	0
$57$	0	0
$58$	−2.82843	−0.371391
$59$	−10.8284	−1.40974	−0.704871	−	0.709336i	$-0.748995\pi$
$59$	−10.8284	−1.40974	−0.704871	+	0.709336i	$0.748995\pi$
$60$	0	0
$61$	7.23948	0.926920	0.463460	−	0.886118i	$-0.346608\pi$
$61$	7.23948	0.926920	0.463460	+	0.886118i	$0.346608\pi$
$62$	5.87644	0.746309
$63$	0	0
$64$	1.00000	0.125000
$65$	19.9674	2.47665
$66$	0	0
$67$	−1.68051	−0.205307	−0.102654	−	0.994717i	$-0.532733\pi$
$67$	−1.68051	−0.205307	−0.102654	+	0.994717i	$0.532733\pi$
$68$	6.11908	0.742048
$69$	0	0
$70$	0	0
$71$	6.47896	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$71$	6.47896	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$72$	0	0
$73$	−11.6736	−1.36629	−0.683145	−	0.730283i	$-0.739388\pi$
$73$	−11.6736	−1.36629	−0.683145	+	0.730283i	$0.739388\pi$
$74$	−8.31055	−0.966081
$75$	0	0
$76$	0.0511786	0.00587059
$77$	0	0
$78$	0	0
$79$	−9.13897	−1.02821	−0.514107	−	0.857726i	$-0.671877\pi$
$79$	−9.13897	−1.02821	−0.514107	+	0.857726i	$0.671877\pi$
$80$	3.29066	0.367907
$81$	0	0
$82$	−6.11908	−0.675740
$83$	−0.951983	−0.104494	−0.0522469	−	0.998634i	$-0.516638\pi$
$83$	−0.951983	−0.104494	−0.0522469	+	0.998634i	$0.516638\pi$
$84$	0	0
$85$	20.1358	2.18404
$86$	2.90080	0.312802
$87$	0	0
$88$	1.00000	0.106600
$89$	−16.5469	−1.75396	−0.876982	−	0.480523i	$-0.840447\pi$
$89$	−16.5469	−1.75396	−0.876982	+	0.480523i	$0.840447\pi$
$90$	0	0
$91$	0	0
$92$	6.75605	0.704367
$93$	0	0
$94$	−1.22275	−0.126117
$95$	0.168411	0.0172786
$96$	0	0
$97$	−14.7216	−1.49475	−0.747376	−	0.664401i	$-0.768687\pi$
$97$	−14.7216	−1.49475	−0.747376	+	0.664401i	$0.768687\pi$
$98$	0	0
$99$	0	0
$100$	5.82843	0.582843
$101$	−5.51657	−0.548919	−0.274460	−	0.961599i	$-0.588499\pi$
$101$	−5.51657	−0.548919	−0.274460	+	0.961599i	$0.588499\pi$
$102$	0	0
$103$	11.6781	1.15067	0.575336	−	0.817917i	$-0.304871\pi$
$103$	11.6781	1.15067	0.575336	+	0.817917i	$0.304871\pi$
$104$	6.06791	0.595007
$105$	0	0
$106$	−3.00316	−0.291693
$107$	−15.9611	−1.54302	−0.771508	−	0.636220i	$-0.780497\pi$
$107$	−15.9611	−1.54302	−0.771508	+	0.636220i	$0.780497\pi$
$108$	0	0
$109$	−5.65053	−0.541223	−0.270611	−	0.962689i	$-0.587226\pi$
$109$	−5.65053	−0.541223	−0.270611	+	0.962689i	$0.587226\pi$
$110$	3.29066	0.313752
$111$	0	0
$112$	0	0
$113$	0.247112	0.0232463	0.0116232	−	0.999932i	$-0.496300\pi$
$113$	0.247112	0.0232463	0.0116232	+	0.999932i	$0.496300\pi$
$114$	0	0
$115$	22.2318	2.07313
$116$	−2.82843	−0.262613
$117$	0	0
$118$	−10.8284	−0.996838
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.23948	0.655432
$123$	0	0
$124$	5.87644	0.527720
$125$	2.72607	0.243827
$126$	0	0
$127$	13.3310	1.18294	0.591469	−	0.806328i	$-0.298548\pi$
$127$	13.3310	1.18294	0.591469	+	0.806328i	$0.298548\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	19.9674	1.75126
$131$	−3.04802	−0.266306	−0.133153	−	0.991095i	$-0.542510\pi$
$131$	−3.04802	−0.266306	−0.133153	+	0.991095i	$0.542510\pi$
$132$	0	0
$133$	0	0
$134$	−1.68051	−0.145174
$135$	0	0
$136$	6.11908	0.524707
$137$	−8.47896	−0.724406	−0.362203	−	0.932099i	$-0.617975\pi$
$137$	−8.47896	−0.724406	−0.362203	+	0.932099i	$0.617975\pi$
$138$	0	0
$139$	−3.80407	−0.322657	−0.161328	−	0.986901i	$-0.551578\pi$
$139$	−3.80407	−0.322657	−0.161328	+	0.986901i	$0.551578\pi$
$140$	0	0
$141$	0	0
$142$	6.47896	0.543702
$143$	6.06791	0.507424
$144$	0	0
$145$	−9.30739	−0.772936
$146$	−11.6736	−0.966113
$147$	0	0
$148$	−8.31055	−0.683123
$149$	−24.1269	−1.97655	−0.988275	−	0.152684i	$-0.951208\pi$
$149$	−24.1269	−1.97655	−0.988275	+	0.152684i	$0.951208\pi$
$150$	0	0
$151$	7.85525	0.639251	0.319625	−	0.947544i	$-0.396443\pi$
$151$	7.85525	0.639251	0.319625	+	0.947544i	$0.396443\pi$
$152$	0.0511786	0.00415113
$153$	0	0
$154$	0	0
$155$	19.3374	1.55321
$156$	0	0
$157$	−18.2549	−1.45690	−0.728450	−	0.685099i	$-0.759759\pi$
$157$	−18.2549	−1.45690	−0.728450	+	0.685099i	$0.759759\pi$
$158$	−9.13897	−0.727058
$159$	0	0
$160$	3.29066	0.260149
$161$	0	0
$162$	0	0
$163$	2.34315	0.183529	0.0917647	−	0.995781i	$-0.470749\pi$
$163$	2.34315	0.183529	0.0917647	+	0.995781i	$0.470749\pi$
$164$	−6.11908	−0.477820
$165$	0	0
$166$	−0.951983	−0.0738882
$167$	−19.7863	−1.53111	−0.765557	−	0.643369i	$-0.777536\pi$
$167$	−19.7863	−1.53111	−0.765557	+	0.643369i	$0.777536\pi$
$168$	0	0
$169$	23.8195	1.83227
$170$	20.1358	1.54435
$171$	0	0
$172$	2.90080	0.221184
$173$	−10.5768	−0.804143	−0.402071	−	0.915608i	$-0.631710\pi$
$173$	−10.5768	−0.804143	−0.402071	+	0.915608i	$0.631710\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−16.5469	−1.24024
$179$	−18.3016	−1.36793	−0.683963	−	0.729517i	$-0.739745\pi$
$179$	−18.3016	−1.36793	−0.683963	+	0.729517i	$0.739745\pi$
$180$	0	0
$181$	25.0499	1.86194	0.930971	−	0.365093i	$-0.118963\pi$
$181$	25.0499	1.86194	0.930971	+	0.365093i	$0.118963\pi$
$182$	0	0
$183$	0	0
$184$	6.75605	0.498063
$185$	−27.3472	−2.01060
$186$	0	0
$187$	6.11908	0.447472
$188$	−1.22275	−0.0891783
$189$	0	0
$190$	0.168411	0.0122178
$191$	21.8650	1.58210	0.791050	−	0.611752i	$-0.209535\pi$
$191$	21.8650	1.58210	0.791050	+	0.611752i	$0.209535\pi$
$192$	0	0
$193$	20.8195	1.49862	0.749310	−	0.662220i	$-0.230385\pi$
$193$	20.8195	1.49862	0.749310	+	0.662220i	$0.230385\pi$
$194$	−14.7216	−1.05695
$195$	0	0
$196$	0	0
$197$	20.4790	1.45907	0.729533	−	0.683946i	$-0.239738\pi$
$197$	20.4790	1.45907	0.729533	+	0.683946i	$0.239738\pi$
$198$	0	0
$199$	15.5333	1.10113	0.550563	−	0.834794i	$-0.314413\pi$
$199$	15.5333	1.10113	0.550563	+	0.834794i	$0.314413\pi$
$200$	5.82843	0.412132
$201$	0	0
$202$	−5.51657	−0.388145
$203$	0	0
$204$	0	0
$205$	−20.1358	−1.40635
$206$	11.6781	0.813649
$207$	0	0
$208$	6.06791	0.420734
$209$	0.0511786	0.00354010
$210$	0	0
$211$	−6.40658	−0.441047	−0.220524	−	0.975382i	$-0.570777\pi$
$211$	−6.40658	−0.441047	−0.220524	+	0.975382i	$0.570777\pi$
$212$	−3.00316	−0.206258
$213$	0	0
$214$	−15.9611	−1.09108
$215$	9.54555	0.651001
$216$	0	0
$217$	0	0
$218$	−5.65053	−0.382702
$219$	0	0
$220$	3.29066	0.221856
$221$	37.1300	2.49764
$222$	0	0
$223$	28.4912	1.90791	0.953956	−	0.299945i	$-0.0969684\pi$
$223$	28.4912	1.90791	0.953956	+	0.299945i	$0.0969684\pi$
$224$	0	0
$225$	0	0
$226$	0.247112	0.0164376
$227$	2.90326	0.192696	0.0963481	−	0.995348i	$-0.469284\pi$
$227$	2.90326	0.192696	0.0963481	+	0.995348i	$0.469284\pi$
$228$	0	0
$229$	6.11908	0.404360	0.202180	−	0.979348i	$-0.435197\pi$
$229$	6.11908	0.404360	0.202180	+	0.979348i	$0.435197\pi$
$230$	22.2318	1.46592
$231$	0	0
$232$	−2.82843	−0.185695
$233$	11.6569	0.763666	0.381833	−	0.924231i	$-0.375293\pi$
$233$	11.6569	0.763666	0.381833	+	0.924231i	$0.375293\pi$
$234$	0	0
$235$	−4.02366	−0.262474
$236$	−10.8284	−0.704871
$237$	0	0
$238$	0	0
$239$	21.3074	1.37826	0.689130	−	0.724638i	$-0.257993\pi$
$239$	21.3074	1.37826	0.689130	+	0.724638i	$0.257993\pi$
$240$	0	0
$241$	−11.8783	−0.765148	−0.382574	−	0.923925i	$-0.624962\pi$
$241$	−11.8783	−0.765148	−0.382574	+	0.923925i	$0.624962\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	7.23948	0.463460
$245$	0	0
$246$	0	0
$247$	0.310547	0.0197596
$248$	5.87644	0.373155
$249$	0	0
$250$	2.72607	0.172412
$251$	7.21135	0.455176	0.227588	−	0.973757i	$-0.426916\pi$
$251$	7.21135	0.455176	0.227588	+	0.973757i	$0.426916\pi$
$252$	0	0
$253$	6.75605	0.424749
$254$	13.3310	0.836464
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.08840	−0.566919	−0.283459	−	0.958984i	$-0.591482\pi$
$257$	−9.08840	−0.566919	−0.283459	+	0.958984i	$0.591482\pi$
$258$	0	0
$259$	0	0
$260$	19.9674	1.23833
$261$	0	0
$262$	−3.04802	−0.188307
$263$	−5.28373	−0.325809	−0.162904	−	0.986642i	$-0.552086\pi$
$263$	−5.28373	−0.325809	−0.162904	+	0.986642i	$0.552086\pi$
$264$	0	0
$265$	−9.88238	−0.607070
$266$	0	0
$267$	0	0
$268$	−1.68051	−0.102654
$269$	−20.2151	−1.23254	−0.616269	−	0.787536i	$-0.711356\pi$
$269$	−20.2151	−1.23254	−0.616269	+	0.787536i	$0.711356\pi$
$270$	0	0
$271$	11.1779	0.679009	0.339504	−	0.940604i	$-0.389741\pi$
$271$	11.1779	0.679009	0.339504	+	0.940604i	$0.389741\pi$
$272$	6.11908	0.371024
$273$	0	0
$274$	−8.47896	−0.512233
$275$	5.82843	0.351467
$276$	0	0
$277$	25.7376	1.54642	0.773212	−	0.634148i	$-0.218649\pi$
$277$	25.7376	1.54642	0.773212	+	0.634148i	$0.218649\pi$
$278$	−3.80407	−0.228153
$279$	0	0
$280$	0	0
$281$	12.0063	0.716237	0.358119	−	0.933676i	$-0.383418\pi$
$281$	12.0063	0.716237	0.358119	+	0.933676i	$0.383418\pi$
$282$	0	0
$283$	−28.7746	−1.71047	−0.855237	−	0.518237i	$-0.826589\pi$
$283$	−28.7746	−1.71047	−0.855237	+	0.518237i	$0.826589\pi$
$284$	6.47896	0.384455
$285$	0	0
$286$	6.06791	0.358803
$287$	0	0
$288$	0	0
$289$	20.4432	1.20254
$290$	−9.30739	−0.546548
$291$	0	0
$292$	−11.6736	−0.683145
$293$	−20.2337	−1.18207	−0.591033	−	0.806648i	$-0.701280\pi$
$293$	−20.2337	−1.18207	−0.591033	+	0.806648i	$0.701280\pi$
$294$	0	0
$295$	−35.6326	−2.07461
$296$	−8.31055	−0.483041
$297$	0	0
$298$	−24.1269	−1.39763
$299$	40.9951	2.37081
$300$	0	0
$301$	0	0
$302$	7.85525	0.452019
$303$	0	0
$304$	0.0511786	0.00293529
$305$	23.8226	1.36408
$306$	0	0
$307$	−17.5033	−0.998967	−0.499484	−	0.866323i	$-0.666477\pi$
$307$	−17.5033	−0.998967	−0.499484	+	0.866323i	$0.666477\pi$
$308$	0	0
$309$	0	0
$310$	19.3374	1.09829
$311$	−30.0486	−1.70390	−0.851949	−	0.523625i	$-0.824579\pi$
$311$	−30.0486	−1.70390	−0.851949	+	0.523625i	$0.824579\pi$
$312$	0	0
$313$	24.6256	1.39192	0.695960	−	0.718081i	$-0.254979\pi$
$313$	24.6256	1.39192	0.695960	+	0.718081i	$0.254979\pi$
$314$	−18.2549	−1.03018
$315$	0	0
$316$	−9.13897	−0.514107
$317$	−11.1479	−0.626129	−0.313065	−	0.949732i	$-0.601356\pi$
$317$	−11.1479	−0.626129	−0.313065	+	0.949732i	$0.601356\pi$
$318$	0	0
$319$	−2.82843	−0.158362
$320$	3.29066	0.183953
$321$	0	0
$322$	0	0
$323$	0.313166	0.0174250
$324$	0	0
$325$	35.3663	1.96177
$326$	2.34315	0.129775
$327$	0	0
$328$	−6.11908	−0.337870
$329$	0	0
$330$	0	0
$331$	−4.16841	−0.229117	−0.114558	−	0.993417i	$-0.536545\pi$
$331$	−4.16841	−0.229117	−0.114558	+	0.993417i	$0.536545\pi$
$332$	−0.951983	−0.0522469
$333$	0	0
$334$	−19.7863	−1.08266
$335$	−5.52998	−0.302135
$336$	0	0
$337$	−15.8163	−0.861570	−0.430785	−	0.902455i	$-0.641763\pi$
$337$	−15.8163	−0.861570	−0.430785	+	0.902455i	$0.641763\pi$
$338$	23.8195	1.29561
$339$	0	0
$340$	20.1358	1.09202
$341$	5.87644	0.318227
$342$	0	0
$343$	0	0
$344$	2.90080	0.156401
$345$	0	0
$346$	−10.5768	−0.568615
$347$	16.1058	0.864606	0.432303	−	0.901728i	$-0.357701\pi$
$347$	16.1058	0.864606	0.432303	+	0.901728i	$0.357701\pi$
$348$	0	0
$349$	−9.19076	−0.491970	−0.245985	−	0.969274i	$-0.579111\pi$
$349$	−9.19076	−0.491970	−0.245985	+	0.969274i	$0.579111\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−24.8874	−1.32462	−0.662311	−	0.749229i	$-0.730424\pi$
$353$	−24.8874	−1.32462	−0.662311	+	0.749229i	$0.730424\pi$
$354$	0	0
$355$	21.3200	1.13155
$356$	−16.5469	−0.876982
$357$	0	0
$358$	−18.3016	−0.967270
$359$	6.37945	0.336694	0.168347	−	0.985728i	$-0.446157\pi$
$359$	6.37945	0.336694	0.168347	+	0.985728i	$0.446157\pi$
$360$	0	0
$361$	−18.9974	−0.999862
$362$	25.0499	1.31659
$363$	0	0
$364$	0	0
$365$	−38.4138	−2.01067
$366$	0	0
$367$	6.26831	0.327203	0.163602	−	0.986526i	$-0.447689\pi$
$367$	6.26831	0.327203	0.163602	+	0.986526i	$0.447689\pi$
$368$	6.75605	0.352183
$369$	0	0
$370$	−27.3472	−1.42171
$371$	0	0
$372$	0	0
$373$	−10.8348	−0.561002	−0.280501	−	0.959854i	$-0.590501\pi$
$373$	−10.8348	−0.561002	−0.280501	+	0.959854i	$0.590501\pi$
$374$	6.11908	0.316410
$375$	0	0
$376$	−1.22275	−0.0630586
$377$	−17.1626	−0.883920
$378$	0	0
$379$	12.7658	0.655738	0.327869	−	0.944723i	$-0.393670\pi$
$379$	12.7658	0.655738	0.327869	+	0.944723i	$0.393670\pi$
$380$	0.168411	0.00863931
$381$	0	0
$382$	21.8650	1.11871
$383$	32.6633	1.66902	0.834509	−	0.550994i	$-0.185751\pi$
$383$	32.6633	1.66902	0.834509	+	0.550994i	$0.185751\pi$
$384$	0	0
$385$	0	0
$386$	20.8195	1.05968
$387$	0	0
$388$	−14.7216	−0.747376
$389$	−31.0179	−1.57267	−0.786334	−	0.617801i	$-0.788024\pi$
$389$	−31.0179	−1.57267	−0.786334	+	0.617801i	$0.788024\pi$
$390$	0	0
$391$	41.3408	2.09070
$392$	0	0
$393$	0	0
$394$	20.4790	1.03171
$395$	−30.0732	−1.51315
$396$	0	0
$397$	−15.0986	−0.757777	−0.378888	−	0.925442i	$-0.623694\pi$
$397$	−15.0986	−0.757777	−0.378888	+	0.925442i	$0.623694\pi$
$398$	15.5333	0.778614
$399$	0	0
$400$	5.82843	0.291421
$401$	9.16525	0.457691	0.228845	−	0.973463i	$-0.426505\pi$
$401$	9.16525	0.457691	0.228845	+	0.973463i	$0.426505\pi$
$402$	0	0
$403$	35.6577	1.77624
$404$	−5.51657	−0.274460
$405$	0	0
$406$	0	0
$407$	−8.31055	−0.411939
$408$	0	0
$409$	−9.32149	−0.460918	−0.230459	−	0.973082i	$-0.574023\pi$
$409$	−9.32149	−0.460918	−0.230459	+	0.973082i	$0.574023\pi$
$410$	−20.1358	−0.994437
$411$	0	0
$412$	11.6781	0.575336
$413$	0	0
$414$	0	0
$415$	−3.13265	−0.153776
$416$	6.06791	0.297504
$417$	0	0
$418$	0.0511786	0.00250323
$419$	13.5456	0.661744	0.330872	−	0.943676i	$-0.392657\pi$
$419$	13.5456	0.661744	0.330872	+	0.943676i	$0.392657\pi$
$420$	0	0
$421$	−9.10899	−0.443945	−0.221973	−	0.975053i	$-0.571250\pi$
$421$	−9.10899	−0.443945	−0.221973	+	0.975053i	$0.571250\pi$
$422$	−6.40658	−0.311867
$423$	0	0
$424$	−3.00316	−0.145846
$425$	35.6646	1.72999
$426$	0	0
$427$	0	0
$428$	−15.9611	−0.771508
$429$	0	0
$430$	9.54555	0.460328
$431$	−4.02366	−0.193813	−0.0969064	−	0.995294i	$-0.530895\pi$
$431$	−4.02366	−0.193813	−0.0969064	+	0.995294i	$0.530895\pi$
$432$	0	0
$433$	−8.09789	−0.389160	−0.194580	−	0.980887i	$-0.562334\pi$
$433$	−8.09789	−0.389160	−0.194580	+	0.980887i	$0.562334\pi$
$434$	0	0
$435$	0	0
$436$	−5.65053	−0.270611
$437$	0.345765	0.0165402
$438$	0	0
$439$	−6.82843	−0.325903	−0.162952	−	0.986634i	$-0.552101\pi$
$439$	−6.82843	−0.325903	−0.162952	+	0.986634i	$0.552101\pi$
$440$	3.29066	0.156876
$441$	0	0
$442$	37.1300	1.76610
$443$	1.62055	0.0769947	0.0384974	−	0.999259i	$-0.487743\pi$
$443$	1.62055	0.0769947	0.0384974	+	0.999259i	$0.487743\pi$
$444$	0	0
$445$	−54.4501	−2.58118
$446$	28.4912	1.34910
$447$	0	0
$448$	0	0
$449$	−30.1756	−1.42407	−0.712037	−	0.702142i	$-0.752227\pi$
$449$	−30.1756	−1.42407	−0.712037	+	0.702142i	$0.752227\pi$
$450$	0	0
$451$	−6.11908	−0.288136
$452$	0.247112	0.0116232
$453$	0	0
$454$	2.90326	0.136257
$455$	0	0
$456$	0	0
$457$	11.3200	0.529529	0.264764	−	0.964313i	$-0.414706\pi$
$457$	11.3200	0.529529	0.264764	+	0.964313i	$0.414706\pi$
$458$	6.11908	0.285926
$459$	0	0
$460$	22.2318	1.03657
$461$	7.51025	0.349787	0.174894	−	0.984587i	$-0.444042\pi$
$461$	7.51025	0.349787	0.174894	+	0.984587i	$0.444042\pi$
$462$	0	0
$463$	−3.10899	−0.144487	−0.0722436	−	0.997387i	$-0.523016\pi$
$463$	−3.10899	−0.144487	−0.0722436	+	0.997387i	$0.523016\pi$
$464$	−2.82843	−0.131306
$465$	0	0
$466$	11.6569	0.539993
$467$	−36.3189	−1.68064	−0.840320	−	0.542091i	$-0.817633\pi$
$467$	−36.3189	−1.68064	−0.840320	+	0.542091i	$0.817633\pi$
$468$	0	0
$469$	0	0
$470$	−4.02366	−0.185597
$471$	0	0
$472$	−10.8284	−0.498419
$473$	2.90080	0.133379
$474$	0	0
$475$	0.298291	0.0136865
$476$	0	0
$477$	0	0
$478$	21.3074	0.974577
$479$	30.5035	1.39374	0.696870	−	0.717198i	$-0.254576\pi$
$479$	30.5035	1.39374	0.696870	+	0.717198i	$0.254576\pi$
$480$	0	0
$481$	−50.4276	−2.29930
$482$	−11.8783	−0.541042
$483$	0	0
$484$	1.00000	0.0454545
$485$	−48.4437	−2.19972
$486$	0	0
$487$	−12.6001	−0.570963	−0.285482	−	0.958384i	$-0.592154\pi$
$487$	−12.6001	−0.570963	−0.285482	+	0.958384i	$0.592154\pi$
$488$	7.23948	0.327716
$489$	0	0
$490$	0	0
$491$	31.3137	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$491$	31.3137	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$492$	0	0
$493$	−17.3074	−0.779485
$494$	0.310547	0.0139722
$495$	0	0
$496$	5.87644	0.263860
$497$	0	0
$498$	0	0
$499$	−3.49477	−0.156447	−0.0782236	−	0.996936i	$-0.524925\pi$
$499$	−3.49477	−0.156447	−0.0782236	+	0.996936i	$0.524925\pi$
$500$	2.72607	0.121914
$501$	0	0
$502$	7.21135	0.321858
$503$	−20.2627	−0.903468	−0.451734	−	0.892153i	$-0.649194\pi$
$503$	−20.2627	−0.903468	−0.451734	+	0.892153i	$0.649194\pi$
$504$	0	0
$505$	−18.1531	−0.807804
$506$	6.75605	0.300343
$507$	0	0
$508$	13.3310	0.591469
$509$	18.4123	0.816111	0.408055	−	0.912957i	$-0.366207\pi$
$509$	18.4123	0.816111	0.408055	+	0.912957i	$0.366207\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.08840	−0.400872
$515$	38.4285	1.69336
$516$	0	0
$517$	−1.22275	−0.0537765
$518$	0	0
$519$	0	0
$520$	19.9674	0.875628
$521$	24.0053	1.05169	0.525846	−	0.850580i	$-0.323749\pi$
$521$	24.0053	1.05169	0.525846	+	0.850580i	$0.323749\pi$
$522$	0	0
$523$	39.3894	1.72238	0.861189	−	0.508285i	$-0.169720\pi$
$523$	39.3894	1.72238	0.861189	+	0.508285i	$0.169720\pi$
$524$	−3.04802	−0.133153
$525$	0	0
$526$	−5.28373	−0.230382
$527$	35.9585	1.56638
$528$	0	0
$529$	22.6442	0.984531
$530$	−9.88238	−0.429263
$531$	0	0
$532$	0	0
$533$	−37.1300	−1.60828
$534$	0	0
$535$	−52.5224	−2.27074
$536$	−1.68051	−0.0725870
$537$	0	0
$538$	−20.2151	−0.871536
$539$	0	0
$540$	0	0
$541$	35.1064	1.50934	0.754670	−	0.656104i	$-0.227797\pi$
$541$	35.1064	1.50934	0.754670	+	0.656104i	$0.227797\pi$
$542$	11.1779	0.480132
$543$	0	0
$544$	6.11908	0.262354
$545$	−18.5940	−0.796478
$546$	0	0
$547$	30.3342	1.29700	0.648498	−	0.761216i	$-0.275397\pi$
$547$	30.3342	1.29700	0.648498	+	0.761216i	$0.275397\pi$
$548$	−8.47896	−0.362203
$549$	0	0
$550$	5.82843	0.248525
$551$	−0.144755	−0.00616676
$552$	0	0
$553$	0	0
$554$	25.7376	1.09349
$555$	0	0
$556$	−3.80407	−0.161328
$557$	36.0643	1.52809	0.764047	−	0.645161i	$-0.223210\pi$
$557$	36.0643	1.52809	0.764047	+	0.645161i	$0.223210\pi$
$558$	0	0
$559$	17.6018	0.744477
$560$	0	0
$561$	0	0
$562$	12.0063	0.506456
$563$	−38.7908	−1.63484	−0.817418	−	0.576046i	$-0.804595\pi$
$563$	−38.7908	−1.63484	−0.817418	+	0.576046i	$0.804595\pi$
$564$	0	0
$565$	0.813161	0.0342099
$566$	−28.7746	−1.20949
$567$	0	0
$568$	6.47896	0.271851
$569$	12.4101	0.520257	0.260128	−	0.965574i	$-0.416235\pi$
$569$	12.4101	0.520257	0.260128	+	0.965574i	$0.416235\pi$
$570$	0	0
$571$	−35.7863	−1.49761	−0.748806	−	0.662789i	$-0.769372\pi$
$571$	−35.7863	−1.49761	−0.748806	+	0.662789i	$0.769372\pi$
$572$	6.06791	0.253712
$573$	0	0
$574$	0	0
$575$	39.3771	1.64214
$576$	0	0
$577$	−24.1313	−1.00460	−0.502300	−	0.864693i	$-0.667513\pi$
$577$	−24.1313	−1.00460	−0.502300	+	0.864693i	$0.667513\pi$
$578$	20.4432	0.850325
$579$	0	0
$580$	−9.30739	−0.386468
$581$	0	0
$582$	0	0
$583$	−3.00316	−0.124378
$584$	−11.6736	−0.483056
$585$	0	0
$586$	−20.2337	−0.835846
$587$	31.0539	1.28173	0.640867	−	0.767652i	$-0.278575\pi$
$587$	31.0539	1.28173	0.640867	+	0.767652i	$0.278575\pi$
$588$	0	0
$589$	0.300748	0.0123921
$590$	−35.6326	−1.46697
$591$	0	0
$592$	−8.31055	−0.341561
$593$	23.5378	0.966580	0.483290	−	0.875460i	$-0.339442\pi$
$593$	23.5378	0.966580	0.483290	+	0.875460i	$0.339442\pi$
$594$	0	0
$595$	0	0
$596$	−24.1269	−0.988275
$597$	0	0
$598$	40.9951	1.67641
$599$	20.0037	0.817329	0.408665	−	0.912685i	$-0.365995\pi$
$599$	20.0037	0.817329	0.408665	+	0.912685i	$0.365995\pi$
$600$	0	0
$601$	−0.799053	−0.0325940	−0.0162970	−	0.999867i	$-0.505188\pi$
$601$	−0.799053	−0.0325940	−0.0162970	+	0.999867i	$0.505188\pi$
$602$	0	0
$603$	0	0
$604$	7.85525	0.319625
$605$	3.29066	0.133784
$606$	0	0
$607$	21.3200	0.865353	0.432677	−	0.901549i	$-0.357569\pi$
$607$	21.3200	0.865353	0.432677	+	0.901549i	$0.357569\pi$
$608$	0.0511786	0.00207557
$609$	0	0
$610$	23.8226	0.964551
$611$	−7.41954	−0.300162
$612$	0	0
$613$	−8.37398	−0.338222	−0.169111	−	0.985597i	$-0.554090\pi$
$613$	−8.37398	−0.338222	−0.169111	+	0.985597i	$0.554090\pi$
$614$	−17.5033	−0.706376
$615$	0	0
$616$	0	0
$617$	−3.74395	−0.150726	−0.0753628	−	0.997156i	$-0.524011\pi$
$617$	−3.74395	−0.150726	−0.0753628	+	0.997156i	$0.524011\pi$
$618$	0	0
$619$	0.924461	0.0371572	0.0185786	−	0.999827i	$-0.494086\pi$
$619$	0.924461	0.0371572	0.0185786	+	0.999827i	$0.494086\pi$
$620$	19.3374	0.776607
$621$	0	0
$622$	−30.0486	−1.20484
$623$	0	0
$624$	0	0
$625$	−20.1716	−0.806863
$626$	24.6256	0.984236
$627$	0	0
$628$	−18.2549	−0.728450
$629$	−50.8529	−2.02764
$630$	0	0
$631$	40.4374	1.60979	0.804894	−	0.593419i	$-0.202222\pi$
$631$	40.4374	1.60979	0.804894	+	0.593419i	$0.202222\pi$
$632$	−9.13897	−0.363529
$633$	0	0
$634$	−11.1479	−0.442740
$635$	43.8679	1.74084
$636$	0	0
$637$	0	0
$638$	−2.82843	−0.111979
$639$	0	0
$640$	3.29066	0.130075
$641$	−13.6777	−0.540235	−0.270118	−	0.962827i	$-0.587063\pi$
$641$	−13.6777	−0.540235	−0.270118	+	0.962827i	$0.587063\pi$
$642$	0	0
$643$	−40.2627	−1.58781	−0.793903	−	0.608045i	$-0.791954\pi$
$643$	−40.2627	−1.58781	−0.793903	+	0.608045i	$0.791954\pi$
$644$	0	0
$645$	0	0
$646$	0.313166	0.0123214
$647$	6.61985	0.260253	0.130127	−	0.991497i	$-0.458462\pi$
$647$	6.61985	0.260253	0.130127	+	0.991497i	$0.458462\pi$
$648$	0	0
$649$	−10.8284	−0.425053
$650$	35.3663	1.38718
$651$	0	0
$652$	2.34315	0.0917647
$653$	15.3200	0.599519	0.299760	−	0.954015i	$-0.403094\pi$
$653$	15.3200	0.599519	0.299760	+	0.954015i	$0.403094\pi$
$654$	0	0
$655$	−10.0300	−0.391904
$656$	−6.11908	−0.238910
$657$	0	0
$658$	0	0
$659$	−31.7690	−1.23754	−0.618772	−	0.785570i	$-0.712370\pi$
$659$	−31.7690	−1.23754	−0.618772	+	0.785570i	$0.712370\pi$
$660$	0	0
$661$	−29.8296	−1.16024	−0.580118	−	0.814532i	$-0.696994\pi$
$661$	−29.8296	−1.16024	−0.580118	+	0.814532i	$0.696994\pi$
$662$	−4.16841	−0.162010
$663$	0	0
$664$	−0.951983	−0.0369441
$665$	0	0
$666$	0	0
$667$	−19.1090	−0.739903
$668$	−19.7863	−0.765557
$669$	0	0
$670$	−5.52998	−0.213642
$671$	7.23948	0.279477
$672$	0	0
$673$	23.7690	0.916228	0.458114	−	0.888893i	$-0.348525\pi$
$673$	23.7690	0.916228	0.458114	+	0.888893i	$0.348525\pi$
$674$	−15.8163	−0.609222
$675$	0	0
$676$	23.8195	0.916134
$677$	22.6368	0.870003	0.435002	−	0.900430i	$-0.356748\pi$
$677$	22.6368	0.870003	0.435002	+	0.900430i	$0.356748\pi$
$678$	0	0
$679$	0	0
$680$	20.1358	0.772173
$681$	0	0
$682$	5.87644	0.225021
$683$	−6.79583	−0.260035	−0.130018	−	0.991512i	$-0.541503\pi$
$683$	−6.79583	−0.260035	−0.130018	+	0.991512i	$0.541503\pi$
$684$	0	0
$685$	−27.9013	−1.06606
$686$	0	0
$687$	0	0
$688$	2.90080	0.110592
$689$	−18.2229	−0.694237
$690$	0	0
$691$	16.2292	0.617389	0.308694	−	0.951161i	$-0.400108\pi$
$691$	16.2292	0.617389	0.308694	+	0.951161i	$0.400108\pi$
$692$	−10.5768	−0.402071
$693$	0	0
$694$	16.1058	0.611369
$695$	−12.5179	−0.474830
$696$	0	0
$697$	−37.4432	−1.41826
$698$	−9.19076	−0.347875
$699$	0	0
$700$	0	0
$701$	32.3253	1.22091	0.610454	−	0.792052i	$-0.290987\pi$
$701$	32.3253	1.22091	0.610454	+	0.792052i	$0.290987\pi$
$702$	0	0
$703$	−0.425322	−0.0160413
$704$	1.00000	0.0376889
$705$	0	0
$706$	−24.8874	−0.936649
$707$	0	0
$708$	0	0
$709$	−10.8458	−0.407321	−0.203661	−	0.979042i	$-0.565284\pi$
$709$	−10.8458	−0.407321	−0.203661	+	0.979042i	$0.565284\pi$
$710$	21.3200	0.800127
$711$	0	0
$712$	−16.5469	−0.620120
$713$	39.7015	1.48683
$714$	0	0
$715$	19.9674	0.746738
$716$	−18.3016	−0.683963
$717$	0	0
$718$	6.37945	0.238079
$719$	−3.69277	−0.137717	−0.0688585	−	0.997626i	$-0.521936\pi$
$719$	−3.69277	−0.137717	−0.0688585	+	0.997626i	$0.521936\pi$
$720$	0	0
$721$	0	0
$722$	−18.9974	−0.707009
$723$	0	0
$724$	25.0499	0.930971
$725$	−16.4853	−0.612248
$726$	0	0
$727$	34.8470	1.29240	0.646202	−	0.763166i	$-0.276356\pi$
$727$	34.8470	1.29240	0.646202	+	0.763166i	$0.276356\pi$
$728$	0	0
$729$	0	0
$730$	−38.4138	−1.42176
$731$	17.7503	0.656517
$732$	0	0
$733$	−39.9168	−1.47436	−0.737181	−	0.675696i	$-0.763843\pi$
$733$	−39.9168	−1.47436	−0.737181	+	0.675696i	$0.763843\pi$
$734$	6.26831	0.231368
$735$	0	0
$736$	6.75605	0.249031
$737$	−1.68051	−0.0619024
$738$	0	0
$739$	4.77479	0.175644	0.0878218	−	0.996136i	$-0.472009\pi$
$739$	4.77479	0.175644	0.0878218	+	0.996136i	$0.472009\pi$
$740$	−27.3472	−1.00530
$741$	0	0
$742$	0	0
$743$	25.1153	0.921392	0.460696	−	0.887558i	$-0.347600\pi$
$743$	25.1153	0.921392	0.460696	+	0.887558i	$0.347600\pi$
$744$	0	0
$745$	−79.3933	−2.90874
$746$	−10.8348	−0.396688
$747$	0	0
$748$	6.11908	0.223736
$749$	0	0
$750$	0	0
$751$	54.7154	1.99659	0.998296	−	0.0583532i	$-0.0185850\pi$
$751$	54.7154	1.99659	0.998296	+	0.0583532i	$0.0185850\pi$
$752$	−1.22275	−0.0445892
$753$	0	0
$754$	−17.1626	−0.625026
$755$	25.8489	0.940739
$756$	0	0
$757$	16.9316	0.615391	0.307695	−	0.951485i	$-0.400442\pi$
$757$	16.9316	0.615391	0.307695	+	0.951485i	$0.400442\pi$
$758$	12.7658	0.463676
$759$	0	0
$760$	0.168411	0.00610892
$761$	4.16781	0.151083	0.0755414	−	0.997143i	$-0.475931\pi$
$761$	4.16781	0.151083	0.0755414	+	0.997143i	$0.475931\pi$
$762$	0	0
$763$	0	0
$764$	21.8650	0.791050
$765$	0	0
$766$	32.6633	1.18017
$767$	−65.7059	−2.37250
$768$	0	0
$769$	16.8273	0.606807	0.303403	−	0.952862i	$-0.401877\pi$
$769$	16.8273	0.606807	0.303403	+	0.952862i	$0.401877\pi$
$770$	0	0
$771$	0	0
$772$	20.8195	0.749310
$773$	1.60929	0.0578822	0.0289411	−	0.999581i	$-0.490786\pi$
$773$	1.60929	0.0578822	0.0289411	+	0.999581i	$0.490786\pi$
$774$	0	0
$775$	34.2504	1.23031
$776$	−14.7216	−0.528475
$777$	0	0
$778$	−31.0179	−1.11204
$779$	−0.313166	−0.0112203
$780$	0	0
$781$	6.47896	0.231835
$782$	41.3408	1.47835
$783$	0	0
$784$	0	0
$785$	−60.0706	−2.14401
$786$	0	0
$787$	−32.6123	−1.16250	−0.581252	−	0.813724i	$-0.697437\pi$
$787$	−32.6123	−1.16250	−0.581252	+	0.813724i	$0.697437\pi$
$788$	20.4790	0.729533
$789$	0	0
$790$	−30.0732	−1.06996
$791$	0	0
$792$	0	0
$793$	43.9285	1.55995
$794$	−15.0986	−0.535829
$795$	0	0
$796$	15.5333	0.550563
$797$	−35.4599	−1.25605	−0.628027	−	0.778191i	$-0.716137\pi$
$797$	−35.4599	−1.25605	−0.628027	+	0.778191i	$0.716137\pi$
$798$	0	0
$799$	−7.48212	−0.264698
$800$	5.82843	0.206066
$801$	0	0
$802$	9.16525	0.323636
$803$	−11.6736	−0.411952
$804$	0	0
$805$	0	0
$806$	35.6577	1.25599
$807$	0	0
$808$	−5.51657	−0.194072
$809$	53.3953	1.87728	0.938640	−	0.344899i	$-0.112087\pi$
$809$	53.3953	1.87728	0.938640	+	0.344899i	$0.112087\pi$
$810$	0	0
$811$	0.178048	0.00625211	0.00312606	−	0.999995i	$-0.499005\pi$
$811$	0.178048	0.00625211	0.00312606	+	0.999995i	$0.499005\pi$
$812$	0	0
$813$	0	0
$814$	−8.31055	−0.291285
$815$	7.71049	0.270087
$816$	0	0
$817$	0.148459	0.00519392
$818$	−9.32149	−0.325918
$819$	0	0
$820$	−20.1358	−0.703173
$821$	−4.15740	−0.145094	−0.0725472	−	0.997365i	$-0.523113\pi$
$821$	−4.15740	−0.145094	−0.0725472	+	0.997365i	$0.523113\pi$
$822$	0	0
$823$	0.0210370	0.000733305 0	0.000366653	−	1.00000i	$-0.499883\pi$
$823$	0.0210370	0.000733305 0	0.000366653		1.00000i	$0.499883\pi$
$824$	11.6781	0.406824
$825$	0	0
$826$	0	0
$827$	43.7774	1.52229	0.761145	−	0.648582i	$-0.224638\pi$
$827$	43.7774	1.52229	0.761145	+	0.648582i	$0.224638\pi$
$828$	0	0
$829$	32.0602	1.11350	0.556749	−	0.830681i	$-0.312049\pi$
$829$	32.0602	1.11350	0.556749	+	0.830681i	$0.312049\pi$
$830$	−3.13265	−0.108736
$831$	0	0
$832$	6.06791	0.210367
$833$	0	0
$834$	0	0
$835$	−65.1101	−2.25323
$836$	0.0511786	0.00177005
$837$	0	0
$838$	13.5456	0.467923
$839$	−6.92200	−0.238974	−0.119487	−	0.992836i	$-0.538125\pi$
$839$	−6.92200	−0.238974	−0.119487	+	0.992836i	$0.538125\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	−9.10899	−0.313917
$843$	0	0
$844$	−6.40658	−0.220524
$845$	78.3818	2.69641
$846$	0	0
$847$	0	0
$848$	−3.00316	−0.103129
$849$	0	0
$850$	35.6646	1.22329
$851$	−56.1465	−1.92468
$852$	0	0
$853$	−27.4727	−0.940648	−0.470324	−	0.882494i	$-0.655863\pi$
$853$	−27.4727	−0.940648	−0.470324	+	0.882494i	$0.655863\pi$
$854$	0	0
$855$	0	0
$856$	−15.9611	−0.545538
$857$	30.1949	1.03144	0.515720	−	0.856757i	$-0.327525\pi$
$857$	30.1949	1.03144	0.515720	+	0.856757i	$0.327525\pi$
$858$	0	0
$859$	42.2869	1.44281	0.721405	−	0.692513i	$-0.243497\pi$
$859$	42.2869	1.44281	0.721405	+	0.692513i	$0.243497\pi$
$860$	9.54555	0.325501
$861$	0	0
$862$	−4.02366	−0.137046
$863$	−45.0340	−1.53298	−0.766488	−	0.642259i	$-0.777997\pi$
$863$	−45.0340	−1.53298	−0.766488	+	0.642259i	$0.777997\pi$
$864$	0	0
$865$	−34.8048	−1.18340
$866$	−8.09789	−0.275177
$867$	0	0
$868$	0	0
$869$	−9.13897	−0.310018
$870$	0	0
$871$	−10.1972	−0.345518
$872$	−5.65053	−0.191351
$873$	0	0
$874$	0.345765	0.0116957
$875$	0	0
$876$	0	0
$877$	−42.6684	−1.44081	−0.720405	−	0.693554i	$-0.756044\pi$
$877$	−42.6684	−1.44081	−0.720405	+	0.693554i	$0.756044\pi$
$878$	−6.82843	−0.230448
$879$	0	0
$880$	3.29066	0.110928
$881$	−11.1844	−0.376813	−0.188407	−	0.982091i	$-0.560332\pi$
$881$	−11.1844	−0.376813	−0.188407	+	0.982091i	$0.560332\pi$
$882$	0	0
$883$	−24.2843	−0.817231	−0.408615	−	0.912707i	$-0.633988\pi$
$883$	−24.2843	−0.817231	−0.408615	+	0.912707i	$0.633988\pi$
$884$	37.1300	1.24882
$885$	0	0
$886$	1.62055	0.0544435
$887$	26.2779	0.882327	0.441164	−	0.897427i	$-0.354566\pi$
$887$	26.2779	0.882327	0.441164	+	0.897427i	$0.354566\pi$
$888$	0	0
$889$	0	0
$890$	−54.4501	−1.82517
$891$	0	0
$892$	28.4912	0.953956
$893$	−0.0625787	−0.00209412
$894$	0	0
$895$	−60.2243	−2.01308
$896$	0	0
$897$	0	0
$898$	−30.1756	−1.00697
$899$	−16.6211	−0.554345
$900$	0	0
$901$	−18.3766	−0.612213
$902$	−6.11908	−0.203743
$903$	0	0
$904$	0.247112	0.00821882
$905$	82.4305	2.74008
$906$	0	0
$907$	−48.9164	−1.62424	−0.812121	−	0.583489i	$-0.801687\pi$
$907$	−48.9164	−1.62424	−0.812121	+	0.583489i	$0.801687\pi$
$908$	2.90326	0.0963481
$909$	0	0
$910$	0	0
$911$	−8.95444	−0.296674	−0.148337	−	0.988937i	$-0.547392\pi$
$911$	−8.95444	−0.296674	−0.148337	+	0.988937i	$0.547392\pi$
$912$	0	0
$913$	−0.951983	−0.0315060
$914$	11.3200	0.374433
$915$	0	0
$916$	6.11908	0.202180
$917$	0	0
$918$	0	0
$919$	−11.9222	−0.393276	−0.196638	−	0.980476i	$-0.563002\pi$
$919$	−11.9222	−0.393276	−0.196638	+	0.980476i	$0.563002\pi$
$920$	22.2318	0.732962
$921$	0	0
$922$	7.51025	0.247337
$923$	39.3137	1.29403
$924$	0	0
$925$	−48.4374	−1.59261
$926$	−3.10899	−0.102168
$927$	0	0
$928$	−2.82843	−0.0928477
$929$	13.9105	0.456389	0.228194	−	0.973616i	$-0.426718\pi$
$929$	13.9105	0.456389	0.228194	+	0.973616i	$0.426718\pi$
$930$	0	0
$931$	0	0
$932$	11.6569	0.381833
$933$	0	0
$934$	−36.3189	−1.18839
$935$	20.1358	0.658511
$936$	0	0
$937$	20.0104	0.653711	0.326856	−	0.945074i	$-0.394011\pi$
$937$	20.0104	0.653711	0.326856	+	0.945074i	$0.394011\pi$
$938$	0	0
$939$	0	0
$940$	−4.02366	−0.131237
$941$	54.9084	1.78996	0.894982	−	0.446103i	$-0.147188\pi$
$941$	54.9084	1.78996	0.894982	+	0.446103i	$0.147188\pi$
$942$	0	0
$943$	−41.3408	−1.34624
$944$	−10.8284	−0.352435
$945$	0	0
$946$	2.90080	0.0943133
$947$	7.39850	0.240419	0.120210	−	0.992749i	$-0.461643\pi$
$947$	7.39850	0.240419	0.120210	+	0.992749i	$0.461643\pi$
$948$	0	0
$949$	−70.8342	−2.29938
$950$	0.298291	0.00967782
$951$	0	0
$952$	0	0
$953$	−21.1416	−0.684843	−0.342422	−	0.939546i	$-0.611247\pi$
$953$	−21.1416	−0.684843	−0.342422	+	0.939546i	$0.611247\pi$
$954$	0	0
$955$	71.9504	2.32826
$956$	21.3074	0.689130
$957$	0	0
$958$	30.5035	0.985522
$959$	0	0
$960$	0	0
$961$	3.53259	0.113955
$962$	−50.4276	−1.62585
$963$	0	0
$964$	−11.8783	−0.382574
$965$	68.5098	2.20541
$966$	0	0
$967$	11.7174	0.376807	0.188404	−	0.982092i	$-0.439669\pi$
$967$	11.7174	0.376807	0.188404	+	0.982092i	$0.439669\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−48.4437	−1.55543
$971$	−35.9487	−1.15365	−0.576824	−	0.816869i	$-0.695708\pi$
$971$	−35.9487	−1.15365	−0.576824	+	0.816869i	$0.695708\pi$
$972$	0	0
$973$	0	0
$974$	−12.6001	−0.403732
$975$	0	0
$976$	7.23948	0.231730
$977$	4.33561	0.138709	0.0693543	−	0.997592i	$-0.477906\pi$
$977$	4.33561	0.138709	0.0693543	+	0.997592i	$0.477906\pi$
$978$	0	0
$979$	−16.5469	−0.528840
$980$	0	0
$981$	0	0
$982$	31.3137	0.999261
$983$	16.3444	0.521305	0.260653	−	0.965433i	$-0.416062\pi$
$983$	16.3444	0.521305	0.260653	+	0.965433i	$0.416062\pi$
$984$	0	0
$985$	67.3892	2.14720
$986$	−17.3074	−0.551179
$987$	0	0
$988$	0.310547	0.00987981
$989$	19.5980	0.623180
$990$	0	0
$991$	11.5657	0.367398	0.183699	−	0.982983i	$-0.441193\pi$
$991$	11.5657	0.367398	0.183699	+	0.982983i	$0.441193\pi$
$992$	5.87644	0.186577
$993$	0	0
$994$	0	0
$995$	51.1148	1.62045
$996$	0	0
$997$	41.6956	1.32051	0.660257	−	0.751040i	$-0.270447\pi$
$997$	41.6956	1.32051	0.660257	+	0.751040i	$0.270447\pi$
$998$	−3.49477	−0.110625
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.ec.1.4		4
3.2	odd	2		3234.2.a.bk.1.1	yes	4
7.6	odd	2		9702.2.a.eb.1.1		4
21.20	even	2		3234.2.a.bj.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bj.1.4	✓	4	21.20	even	2
3234.2.a.bk.1.1	yes	4	3.2	odd	2
9702.2.a.eb.1.1		4	7.6	odd	2
9702.2.a.ec.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +3.29066 q^{10} +1.00000 q^{11} +6.06791 q^{13} +1.00000 q^{16} +6.11908 q^{17} +0.0511786 q^{19} +3.29066 q^{20} +1.00000 q^{22} +6.75605 q^{23} +5.82843 q^{25} +6.06791 q^{26} -2.82843 q^{29} +5.87644 q^{31} +1.00000 q^{32} +6.11908 q^{34} -8.31055 q^{37} +0.0511786 q^{38} +3.29066 q^{40} -6.11908 q^{41} +2.90080 q^{43} +1.00000 q^{44} +6.75605 q^{46} -1.22275 q^{47} +5.82843 q^{50} +6.06791 q^{52} -3.00316 q^{53} +3.29066 q^{55} -2.82843 q^{58} -10.8284 q^{59} +7.23948 q^{61} +5.87644 q^{62} +1.00000 q^{64} +19.9674 q^{65} -1.68051 q^{67} +6.11908 q^{68} +6.47896 q^{71} -11.6736 q^{73} -8.31055 q^{74} +0.0511786 q^{76} -9.13897 q^{79} +3.29066 q^{80} -6.11908 q^{82} -0.951983 q^{83} +20.1358 q^{85} +2.90080 q^{86} +1.00000 q^{88} -16.5469 q^{89} +6.75605 q^{92} -1.22275 q^{94} +0.168411 q^{95} -14.7216 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} + 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} + 12 q^{50} - 8 q^{53} - 32 q^{59} + 16 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} - 16 q^{89} + 8 q^{92} - 16 q^{94} + 16 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^11 + 4 * q^16 + 4 * q^22 + 8 * q^23 + 12 * q^25 + 16 * q^31 + 4 * q^32 + 8 * q^37 + 8 * q^43 + 4 * q^44 + 8 * q^46 - 16 * q^47 + 12 * q^50 - 8 * q^53 - 32 * q^59 + 16 * q^61 + 16 * q^62 + 4 * q^64 + 16 * q^65 + 16 * q^67 + 8 * q^74 + 16 * q^79 + 32 * q^85 + 8 * q^86 + 4 * q^88 - 16 * q^89 + 8 * q^92 - 16 * q^94 + 16 * q^95 - 16 * q^97

Introduction

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