LMFDB - Embedded newform 9450.2.a.ec.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -5.29150 q^{11} -0.645751 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +4.29150 q^{19} +5.29150 q^{22} +3.64575 q^{23} +0.645751 q^{26} -1.00000 q^{28} -5.35425 q^{29} +3.29150 q^{31} -1.00000 q^{32} -1.00000 q^{34} +4.35425 q^{37} -4.29150 q^{38} -3.29150 q^{41} -5.64575 q^{43} -5.29150 q^{44} -3.64575 q^{46} +4.64575 q^{47} +1.00000 q^{49} -0.645751 q^{52} +2.64575 q^{53} +1.00000 q^{56} +5.35425 q^{58} -2.70850 q^{59} -2.64575 q^{61} -3.29150 q^{62} +1.00000 q^{64} +10.9373 q^{67} +1.00000 q^{68} -0.708497 q^{71} -5.29150 q^{73} -4.35425 q^{74} +4.29150 q^{76} +5.29150 q^{77} -0.0627461 q^{79} +3.29150 q^{82} +1.64575 q^{83} +5.64575 q^{86} +5.29150 q^{88} -1.00000 q^{89} +0.645751 q^{91} +3.64575 q^{92} -4.64575 q^{94} -2.93725 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{23} - 4 q^{26} - 2 q^{28} - 16 q^{29} - 4 q^{31} - 2 q^{32} - 2 q^{34} + 14 q^{37} + 2 q^{38} + 4 q^{41} - 6 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{52} + 2 q^{56} + 16 q^{58} - 16 q^{59} + 4 q^{62} + 2 q^{64} + 6 q^{67} + 2 q^{68} - 12 q^{71} - 14 q^{74} - 2 q^{76} - 16 q^{79} - 4 q^{82} - 2 q^{83} + 6 q^{86} - 2 q^{89} - 4 q^{91} + 2 q^{92} - 4 q^{94} + 10 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^23 - 4 * q^26 - 2 * q^28 - 16 * q^29 - 4 * q^31 - 2 * q^32 - 2 * q^34 + 14 * q^37 + 2 * q^38 + 4 * q^41 - 6 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^49 + 4 * q^52 + 2 * q^56 + 16 * q^58 - 16 * q^59 + 4 * q^62 + 2 * q^64 + 6 * q^67 + 2 * q^68 - 12 * q^71 - 14 * q^74 - 2 * q^76 - 16 * q^79 - 4 * q^82 - 2 * q^83 + 6 * q^86 - 2 * q^89 - 4 * q^91 + 2 * q^92 - 4 * q^94 + 10 * q^97 - 2 * q^98

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−5.29150	−1.59545	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	−5.29150	−1.59545	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	−0.645751	−0.179099	−0.0895496	−	0.995982i	$-0.528543\pi$
$13$	−0.645751	−0.179099	−0.0895496	+	0.995982i	$0.528543\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.29150	0.984538	0.492269	−	0.870443i	$-0.336168\pi$
$19$	4.29150	0.984538	0.492269	+	0.870443i	$0.336168\pi$
$20$	0	0
$21$	0	0
$22$	5.29150	1.12815
$23$	3.64575	0.760192	0.380096	−	0.924947i	$-0.375891\pi$
$23$	3.64575	0.760192	0.380096	+	0.924947i	$0.375891\pi$
$24$	0	0
$25$	0	0
$26$	0.645751	0.126642
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−5.35425	−0.994259	−0.497130	−	0.867676i	$-0.665613\pi$
$29$	−5.35425	−0.994259	−0.497130	+	0.867676i	$0.665613\pi$
$30$	0	0
$31$	3.29150	0.591171	0.295586	−	0.955316i	$-0.404485\pi$
$31$	3.29150	0.591171	0.295586	+	0.955316i	$0.404485\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	4.35425	0.715834	0.357917	−	0.933753i	$-0.383487\pi$
$37$	4.35425	0.715834	0.357917	+	0.933753i	$0.383487\pi$
$38$	−4.29150	−0.696174
$39$	0	0
$40$	0	0
$41$	−3.29150	−0.514046	−0.257023	−	0.966405i	$-0.582742\pi$
$41$	−3.29150	−0.514046	−0.257023	+	0.966405i	$0.582742\pi$
$42$	0	0
$43$	−5.64575	−0.860969	−0.430485	−	0.902598i	$-0.641657\pi$
$43$	−5.64575	−0.860969	−0.430485	+	0.902598i	$0.641657\pi$
$44$	−5.29150	−0.797724
$45$	0	0
$46$	−3.64575	−0.537537
$47$	4.64575	0.677652	0.338826	−	0.940849i	$-0.389970\pi$
$47$	4.64575	0.677652	0.338826	+	0.940849i	$0.389970\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−0.645751	−0.0895496
$53$	2.64575	0.363422	0.181711	−	0.983352i	$-0.441836\pi$
$53$	2.64575	0.363422	0.181711	+	0.983352i	$0.441836\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	5.35425	0.703047
$59$	−2.70850	−0.352616	−0.176308	−	0.984335i	$-0.556416\pi$
$59$	−2.70850	−0.352616	−0.176308	+	0.984335i	$0.556416\pi$
$60$	0	0
$61$	−2.64575	−0.338754	−0.169377	−	0.985551i	$-0.554176\pi$
$61$	−2.64575	−0.338754	−0.169377	+	0.985551i	$0.554176\pi$
$62$	−3.29150	−0.418021
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.9373	1.33620	0.668099	−	0.744072i	$-0.267108\pi$
$67$	10.9373	1.33620	0.668099	+	0.744072i	$0.267108\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−0.708497	−0.0840832	−0.0420416	−	0.999116i	$-0.513386\pi$
$71$	−0.708497	−0.0840832	−0.0420416	+	0.999116i	$0.513386\pi$
$72$	0	0
$73$	−5.29150	−0.619324	−0.309662	−	0.950847i	$-0.600216\pi$
$73$	−5.29150	−0.619324	−0.309662	+	0.950847i	$0.600216\pi$
$74$	−4.35425	−0.506171
$75$	0	0
$76$	4.29150	0.492269
$77$	5.29150	0.603023
$78$	0	0
$79$	−0.0627461	−0.00705948	−0.00352974	−	0.999994i	$-0.501124\pi$
$79$	−0.0627461	−0.00705948	−0.00352974	+	0.999994i	$0.501124\pi$
$80$	0	0
$81$	0	0
$82$	3.29150	0.363486
$83$	1.64575	0.180645	0.0903223	−	0.995913i	$-0.471210\pi$
$83$	1.64575	0.180645	0.0903223	+	0.995913i	$0.471210\pi$
$84$	0	0
$85$	0	0
$86$	5.64575	0.608797
$87$	0	0
$88$	5.29150	0.564076
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0.645751	0.0676931
$92$	3.64575	0.380096
$93$	0	0
$94$	−4.64575	−0.479173
$95$	0	0
$96$	0	0
$97$	−2.93725	−0.298233	−0.149116	−	0.988820i	$-0.547643\pi$
$97$	−2.93725	−0.298233	−0.149116	+	0.988820i	$0.547643\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	−1.06275	−0.105747	−0.0528736	−	0.998601i	$-0.516838\pi$
$101$	−1.06275	−0.105747	−0.0528736	+	0.998601i	$0.516838\pi$
$102$	0	0
$103$	13.8745	1.36710	0.683548	−	0.729906i	$-0.260436\pi$
$103$	13.8745	1.36710	0.683548	+	0.729906i	$0.260436\pi$
$104$	0.645751	0.0633211
$105$	0	0
$106$	−2.64575	−0.256978
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	3.64575	0.349200	0.174600	−	0.984639i	$-0.444137\pi$
$109$	3.64575	0.349200	0.174600	+	0.984639i	$0.444137\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	12.5830	1.18371	0.591855	−	0.806045i	$-0.298396\pi$
$113$	12.5830	1.18371	0.591855	+	0.806045i	$0.298396\pi$
$114$	0	0
$115$	0	0
$116$	−5.35425	−0.497130
$117$	0	0
$118$	2.70850	0.249337
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	17.0000	1.54545
$122$	2.64575	0.239535
$123$	0	0
$124$	3.29150	0.295586
$125$	0	0
$126$	0	0
$127$	9.35425	0.830055	0.415028	−	0.909809i	$-0.363772\pi$
$127$	9.35425	0.830055	0.415028	+	0.909809i	$0.363772\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−3.06275	−0.267593	−0.133797	−	0.991009i	$-0.542717\pi$
$131$	−3.06275	−0.267593	−0.133797	+	0.991009i	$0.542717\pi$
$132$	0	0
$133$	−4.29150	−0.372120
$134$	−10.9373	−0.944835
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−2.35425	−0.201137	−0.100569	−	0.994930i	$-0.532066\pi$
$137$	−2.35425	−0.201137	−0.100569	+	0.994930i	$0.532066\pi$
$138$	0	0
$139$	−16.5830	−1.40655	−0.703276	−	0.710917i	$-0.748280\pi$
$139$	−16.5830	−1.40655	−0.703276	+	0.710917i	$0.748280\pi$
$140$	0	0
$141$	0	0
$142$	0.708497	0.0594558
$143$	3.41699	0.285743
$144$	0	0
$145$	0	0
$146$	5.29150	0.437928
$147$	0	0
$148$	4.35425	0.357917
$149$	−22.5203	−1.84493	−0.922466	−	0.386079i	$-0.873829\pi$
$149$	−22.5203	−1.84493	−0.922466	+	0.386079i	$0.873829\pi$
$150$	0	0
$151$	−13.2288	−1.07654	−0.538270	−	0.842772i	$-0.680922\pi$
$151$	−13.2288	−1.07654	−0.538270	+	0.842772i	$0.680922\pi$
$152$	−4.29150	−0.348087
$153$	0	0
$154$	−5.29150	−0.426401
$155$	0	0
$156$	0	0
$157$	−1.35425	−0.108081	−0.0540404	−	0.998539i	$-0.517210\pi$
$157$	−1.35425	−0.108081	−0.0540404	+	0.998539i	$0.517210\pi$
$158$	0.0627461	0.00499181
$159$	0	0
$160$	0	0
$161$	−3.64575	−0.287325
$162$	0	0
$163$	11.8745	0.930083	0.465042	−	0.885289i	$-0.346039\pi$
$163$	11.8745	0.930083	0.465042	+	0.885289i	$0.346039\pi$
$164$	−3.29150	−0.257023
$165$	0	0
$166$	−1.64575	−0.127735
$167$	−17.1660	−1.32835	−0.664173	−	0.747579i	$-0.731216\pi$
$167$	−17.1660	−1.32835	−0.664173	+	0.747579i	$0.731216\pi$
$168$	0	0
$169$	−12.5830	−0.967923
$170$	0	0
$171$	0	0
$172$	−5.64575	−0.430485
$173$	1.64575	0.125124	0.0625621	−	0.998041i	$-0.480073\pi$
$173$	1.64575	0.125124	0.0625621	+	0.998041i	$0.480073\pi$
$174$	0	0
$175$	0	0
$176$	−5.29150	−0.398862
$177$	0	0
$178$	1.00000	0.0749532
$179$	17.5830	1.31422	0.657108	−	0.753797i	$-0.271780\pi$
$179$	17.5830	1.31422	0.657108	+	0.753797i	$0.271780\pi$
$180$	0	0
$181$	22.6458	1.68325	0.841623	−	0.540066i	$-0.181601\pi$
$181$	22.6458	1.68325	0.841623	+	0.540066i	$0.181601\pi$
$182$	−0.645751	−0.0478663
$183$	0	0
$184$	−3.64575	−0.268768
$185$	0	0
$186$	0	0
$187$	−5.29150	−0.386953
$188$	4.64575	0.338826
$189$	0	0
$190$	0	0
$191$	−14.5830	−1.05519	−0.527595	−	0.849496i	$-0.676906\pi$
$191$	−14.5830	−1.05519	−0.527595	+	0.849496i	$0.676906\pi$
$192$	0	0
$193$	14.8745	1.07069	0.535345	−	0.844633i	$-0.320182\pi$
$193$	14.8745	1.07069	0.535345	+	0.844633i	$0.320182\pi$
$194$	2.93725	0.210883
$195$	0	0
$196$	1.00000	0.0714286
$197$	−13.3542	−0.951451	−0.475725	−	0.879594i	$-0.657814\pi$
$197$	−13.3542	−0.951451	−0.475725	+	0.879594i	$0.657814\pi$
$198$	0	0
$199$	−17.5203	−1.24198	−0.620989	−	0.783819i	$-0.713269\pi$
$199$	−17.5203	−1.24198	−0.620989	+	0.783819i	$0.713269\pi$
$200$	0	0
$201$	0	0
$202$	1.06275	0.0747746
$203$	5.35425	0.375795
$204$	0	0
$205$	0	0
$206$	−13.8745	−0.966683
$207$	0	0
$208$	−0.645751	−0.0447748
$209$	−22.7085	−1.57078
$210$	0	0
$211$	14.7085	1.01257	0.506287	−	0.862365i	$-0.331018\pi$
$211$	14.7085	1.01257	0.506287	+	0.862365i	$0.331018\pi$
$212$	2.64575	0.181711
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	−3.29150	−0.223442
$218$	−3.64575	−0.246921
$219$	0	0
$220$	0	0
$221$	−0.645751	−0.0434379
$222$	0	0
$223$	−15.5203	−1.03931	−0.519657	−	0.854375i	$-0.673940\pi$
$223$	−15.5203	−1.03931	−0.519657	+	0.854375i	$0.673940\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−12.5830	−0.837009
$227$	−11.6458	−0.772956	−0.386478	−	0.922299i	$-0.626308\pi$
$227$	−11.6458	−0.772956	−0.386478	+	0.922299i	$0.626308\pi$
$228$	0	0
$229$	−24.5203	−1.62034	−0.810172	−	0.586192i	$-0.800626\pi$
$229$	−24.5203	−1.62034	−0.810172	+	0.586192i	$0.800626\pi$
$230$	0	0
$231$	0	0
$232$	5.35425	0.351524
$233$	−3.64575	−0.238841	−0.119421	−	0.992844i	$-0.538104\pi$
$233$	−3.64575	−0.238841	−0.119421	+	0.992844i	$0.538104\pi$
$234$	0	0
$235$	0	0
$236$	−2.70850	−0.176308
$237$	0	0
$238$	1.00000	0.0648204
$239$	−19.6458	−1.27078	−0.635389	−	0.772192i	$-0.719160\pi$
$239$	−19.6458	−1.27078	−0.635389	+	0.772192i	$0.719160\pi$
$240$	0	0
$241$	2.93725	0.189205	0.0946026	−	0.995515i	$-0.469842\pi$
$241$	2.93725	0.189205	0.0946026	+	0.995515i	$0.469842\pi$
$242$	−17.0000	−1.09280
$243$	0	0
$244$	−2.64575	−0.169377
$245$	0	0
$246$	0	0
$247$	−2.77124	−0.176330
$248$	−3.29150	−0.209011
$249$	0	0
$250$	0	0
$251$	−27.8745	−1.75942	−0.879712	−	0.475507i	$-0.842264\pi$
$251$	−27.8745	−1.75942	−0.879712	+	0.475507i	$0.842264\pi$
$252$	0	0
$253$	−19.2915	−1.21285
$254$	−9.35425	−0.586938
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.29150	0.517210	0.258605	−	0.965983i	$-0.416737\pi$
$257$	8.29150	0.517210	0.258605	+	0.965983i	$0.416737\pi$
$258$	0	0
$259$	−4.35425	−0.270560
$260$	0	0
$261$	0	0
$262$	3.06275	0.189217
$263$	−11.8745	−0.732214	−0.366107	−	0.930573i	$-0.619310\pi$
$263$	−11.8745	−0.732214	−0.366107	+	0.930573i	$0.619310\pi$
$264$	0	0
$265$	0	0
$266$	4.29150	0.263129
$267$	0	0
$268$	10.9373	0.668099
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−4.70850	−0.286021	−0.143010	−	0.989721i	$-0.545678\pi$
$271$	−4.70850	−0.286021	−0.143010	+	0.989721i	$0.545678\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	2.35425	0.142225
$275$	0	0
$276$	0	0
$277$	25.2915	1.51962	0.759810	−	0.650146i	$-0.225292\pi$
$277$	25.2915	1.51962	0.759810	+	0.650146i	$0.225292\pi$
$278$	16.5830	0.994583
$279$	0	0
$280$	0	0
$281$	23.5203	1.40310	0.701551	−	0.712620i	$-0.252491\pi$
$281$	23.5203	1.40310	0.701551	+	0.712620i	$0.252491\pi$
$282$	0	0
$283$	−30.1660	−1.79318	−0.896592	−	0.442858i	$-0.853964\pi$
$283$	−30.1660	−1.79318	−0.896592	+	0.442858i	$0.853964\pi$
$284$	−0.708497	−0.0420416
$285$	0	0
$286$	−3.41699	−0.202051
$287$	3.29150	0.194291
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−5.29150	−0.309662
$293$	−5.52026	−0.322497	−0.161248	−	0.986914i	$-0.551552\pi$
$293$	−5.52026	−0.322497	−0.161248	+	0.986914i	$0.551552\pi$
$294$	0	0
$295$	0	0
$296$	−4.35425	−0.253086
$297$	0	0
$298$	22.5203	1.30456
$299$	−2.35425	−0.136150
$300$	0	0
$301$	5.64575	0.325416
$302$	13.2288	0.761229
$303$	0	0
$304$	4.29150	0.246135
$305$	0	0
$306$	0	0
$307$	−6.70850	−0.382874	−0.191437	−	0.981505i	$-0.561315\pi$
$307$	−6.70850	−0.382874	−0.191437	+	0.981505i	$0.561315\pi$
$308$	5.29150	0.301511
$309$	0	0
$310$	0	0
$311$	10.7085	0.607223	0.303612	−	0.952796i	$-0.401807\pi$
$311$	10.7085	0.607223	0.303612	+	0.952796i	$0.401807\pi$
$312$	0	0
$313$	14.2288	0.804257	0.402128	−	0.915583i	$-0.368271\pi$
$313$	14.2288	0.804257	0.402128	+	0.915583i	$0.368271\pi$
$314$	1.35425	0.0764247
$315$	0	0
$316$	−0.0627461	−0.00352974
$317$	−14.0627	−0.789843	−0.394921	−	0.918715i	$-0.629228\pi$
$317$	−14.0627	−0.789843	−0.394921	+	0.918715i	$0.629228\pi$
$318$	0	0
$319$	28.3320	1.58629
$320$	0	0
$321$	0	0
$322$	3.64575	0.203170
$323$	4.29150	0.238786
$324$	0	0
$325$	0	0
$326$	−11.8745	−0.657668
$327$	0	0
$328$	3.29150	0.181743
$329$	−4.64575	−0.256129
$330$	0	0
$331$	−4.35425	−0.239331	−0.119666	−	0.992814i	$-0.538182\pi$
$331$	−4.35425	−0.239331	−0.119666	+	0.992814i	$0.538182\pi$
$332$	1.64575	0.0903223
$333$	0	0
$334$	17.1660	0.939282
$335$	0	0
$336$	0	0
$337$	−16.1660	−0.880619	−0.440309	−	0.897846i	$-0.645131\pi$
$337$	−16.1660	−0.880619	−0.440309	+	0.897846i	$0.645131\pi$
$338$	12.5830	0.684425
$339$	0	0
$340$	0	0
$341$	−17.4170	−0.943183
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.64575	0.304399
$345$	0	0
$346$	−1.64575	−0.0884761
$347$	−20.2915	−1.08930	−0.544652	−	0.838662i	$-0.683338\pi$
$347$	−20.2915	−1.08930	−0.544652	+	0.838662i	$0.683338\pi$
$348$	0	0
$349$	−14.6458	−0.783969	−0.391985	−	0.919972i	$-0.628211\pi$
$349$	−14.6458	−0.783969	−0.391985	+	0.919972i	$0.628211\pi$
$350$	0	0
$351$	0	0
$352$	5.29150	0.282038
$353$	36.4575	1.94044	0.970219	−	0.242230i	$-0.0778789\pi$
$353$	36.4575	1.94044	0.970219	+	0.242230i	$0.0778789\pi$
$354$	0	0
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−17.5830	−0.929291
$359$	−11.4170	−0.602566	−0.301283	−	0.953535i	$-0.597415\pi$
$359$	−11.4170	−0.602566	−0.301283	+	0.953535i	$0.597415\pi$
$360$	0	0
$361$	−0.583005	−0.0306845
$362$	−22.6458	−1.19023
$363$	0	0
$364$	0.645751	0.0338466
$365$	0	0
$366$	0	0
$367$	14.9373	0.779718	0.389859	−	0.920874i	$-0.372524\pi$
$367$	14.9373	0.779718	0.389859	+	0.920874i	$0.372524\pi$
$368$	3.64575	0.190048
$369$	0	0
$370$	0	0
$371$	−2.64575	−0.137361
$372$	0	0
$373$	−7.87451	−0.407727	−0.203863	−	0.978999i	$-0.565350\pi$
$373$	−7.87451	−0.407727	−0.203863	+	0.978999i	$0.565350\pi$
$374$	5.29150	0.273617
$375$	0	0
$376$	−4.64575	−0.239586
$377$	3.45751	0.178071
$378$	0	0
$379$	24.2288	1.24455	0.622274	−	0.782800i	$-0.286209\pi$
$379$	24.2288	1.24455	0.622274	+	0.782800i	$0.286209\pi$
$380$	0	0
$381$	0	0
$382$	14.5830	0.746131
$383$	16.7085	0.853764	0.426882	−	0.904307i	$-0.359612\pi$
$383$	16.7085	0.853764	0.426882	+	0.904307i	$0.359612\pi$
$384$	0	0
$385$	0	0
$386$	−14.8745	−0.757093
$387$	0	0
$388$	−2.93725	−0.149116
$389$	−34.3948	−1.74388	−0.871942	−	0.489609i	$-0.837139\pi$
$389$	−34.3948	−1.74388	−0.871942	+	0.489609i	$0.837139\pi$
$390$	0	0
$391$	3.64575	0.184374
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	13.3542	0.672777
$395$	0	0
$396$	0	0
$397$	32.5203	1.63215	0.816073	−	0.577950i	$-0.196147\pi$
$397$	32.5203	1.63215	0.816073	+	0.577950i	$0.196147\pi$
$398$	17.5203	0.878211
$399$	0	0
$400$	0	0
$401$	9.06275	0.452572	0.226286	−	0.974061i	$-0.427342\pi$
$401$	9.06275	0.452572	0.226286	+	0.974061i	$0.427342\pi$
$402$	0	0
$403$	−2.12549	−0.105878
$404$	−1.06275	−0.0528736
$405$	0	0
$406$	−5.35425	−0.265727
$407$	−23.0405	−1.14208
$408$	0	0
$409$	18.2288	0.901354	0.450677	−	0.892687i	$-0.351183\pi$
$409$	18.2288	0.901354	0.450677	+	0.892687i	$0.351183\pi$
$410$	0	0
$411$	0	0
$412$	13.8745	0.683548
$413$	2.70850	0.133276
$414$	0	0
$415$	0	0
$416$	0.645751	0.0316606
$417$	0	0
$418$	22.7085	1.11071
$419$	−27.6458	−1.35058	−0.675292	−	0.737551i	$-0.735982\pi$
$419$	−27.6458	−1.35058	−0.675292	+	0.737551i	$0.735982\pi$
$420$	0	0
$421$	16.5830	0.808206	0.404103	−	0.914713i	$-0.367584\pi$
$421$	16.5830	0.808206	0.404103	+	0.914713i	$0.367584\pi$
$422$	−14.7085	−0.715998
$423$	0	0
$424$	−2.64575	−0.128489
$425$	0	0
$426$	0	0
$427$	2.64575	0.128037
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	−2.12549	−0.102381	−0.0511907	−	0.998689i	$-0.516302\pi$
$431$	−2.12549	−0.102381	−0.0511907	+	0.998689i	$0.516302\pi$
$432$	0	0
$433$	−11.2915	−0.542635	−0.271317	−	0.962490i	$-0.587459\pi$
$433$	−11.2915	−0.542635	−0.271317	+	0.962490i	$0.587459\pi$
$434$	3.29150	0.157997
$435$	0	0
$436$	3.64575	0.174600
$437$	15.6458	0.748438
$438$	0	0
$439$	−30.9373	−1.47655	−0.738277	−	0.674497i	$-0.764360\pi$
$439$	−30.9373	−1.47655	−0.738277	+	0.674497i	$0.764360\pi$
$440$	0	0
$441$	0	0
$442$	0.645751	0.0307153
$443$	−33.5830	−1.59558	−0.797788	−	0.602938i	$-0.793997\pi$
$443$	−33.5830	−1.59558	−0.797788	+	0.602938i	$0.793997\pi$
$444$	0	0
$445$	0	0
$446$	15.5203	0.734906
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	18.2288	0.860268	0.430134	−	0.902765i	$-0.358466\pi$
$449$	18.2288	0.860268	0.430134	+	0.902765i	$0.358466\pi$
$450$	0	0
$451$	17.4170	0.820134
$452$	12.5830	0.591855
$453$	0	0
$454$	11.6458	0.546562
$455$	0	0
$456$	0	0
$457$	−7.12549	−0.333316	−0.166658	−	0.986015i	$-0.553298\pi$
$457$	−7.12549	−0.333316	−0.166658	+	0.986015i	$0.553298\pi$
$458$	24.5203	1.14576
$459$	0	0
$460$	0	0
$461$	33.5203	1.56119	0.780597	−	0.625035i	$-0.214915\pi$
$461$	33.5203	1.56119	0.780597	+	0.625035i	$0.214915\pi$
$462$	0	0
$463$	1.35425	0.0629373	0.0314686	−	0.999505i	$-0.489982\pi$
$463$	1.35425	0.0629373	0.0314686	+	0.999505i	$0.489982\pi$
$464$	−5.35425	−0.248565
$465$	0	0
$466$	3.64575	0.168886
$467$	−39.2915	−1.81819	−0.909097	−	0.416585i	$-0.863227\pi$
$467$	−39.2915	−1.81819	−0.909097	+	0.416585i	$0.863227\pi$
$468$	0	0
$469$	−10.9373	−0.505035
$470$	0	0
$471$	0	0
$472$	2.70850	0.124669
$473$	29.8745	1.37363
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	19.6458	0.898576
$479$	−12.6458	−0.577799	−0.288900	−	0.957359i	$-0.593289\pi$
$479$	−12.6458	−0.577799	−0.288900	+	0.957359i	$0.593289\pi$
$480$	0	0
$481$	−2.81176	−0.128205
$482$	−2.93725	−0.133788
$483$	0	0
$484$	17.0000	0.772727
$485$	0	0
$486$	0	0
$487$	−11.2915	−0.511667	−0.255833	−	0.966721i	$-0.582350\pi$
$487$	−11.2915	−0.511667	−0.255833	+	0.966721i	$0.582350\pi$
$488$	2.64575	0.119768
$489$	0	0
$490$	0	0
$491$	18.8745	0.851795	0.425897	−	0.904772i	$-0.359958\pi$
$491$	18.8745	0.851795	0.425897	+	0.904772i	$0.359958\pi$
$492$	0	0
$493$	−5.35425	−0.241143
$494$	2.77124	0.124684
$495$	0	0
$496$	3.29150	0.147793
$497$	0.708497	0.0317805
$498$	0	0
$499$	28.9373	1.29541	0.647705	−	0.761891i	$-0.275729\pi$
$499$	28.9373	1.29541	0.647705	+	0.761891i	$0.275729\pi$
$500$	0	0
$501$	0	0
$502$	27.8745	1.24410
$503$	−13.2915	−0.592639	−0.296319	−	0.955089i	$-0.595759\pi$
$503$	−13.2915	−0.592639	−0.296319	+	0.955089i	$0.595759\pi$
$504$	0	0
$505$	0	0
$506$	19.2915	0.857612
$507$	0	0
$508$	9.35425	0.415028
$509$	−29.5203	−1.30846	−0.654231	−	0.756295i	$-0.727008\pi$
$509$	−29.5203	−1.30846	−0.654231	+	0.756295i	$0.727008\pi$
$510$	0	0
$511$	5.29150	0.234082
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.29150	−0.365723
$515$	0	0
$516$	0	0
$517$	−24.5830	−1.08116
$518$	4.35425	0.191315
$519$	0	0
$520$	0	0
$521$	−8.41699	−0.368755	−0.184378	−	0.982855i	$-0.559027\pi$
$521$	−8.41699	−0.368755	−0.184378	+	0.982855i	$0.559027\pi$
$522$	0	0
$523$	−36.7490	−1.60692	−0.803461	−	0.595357i	$-0.797011\pi$
$523$	−36.7490	−1.60692	−0.803461	+	0.595357i	$0.797011\pi$
$524$	−3.06275	−0.133797
$525$	0	0
$526$	11.8745	0.517753
$527$	3.29150	0.143380
$528$	0	0
$529$	−9.70850	−0.422109
$530$	0	0
$531$	0	0
$532$	−4.29150	−0.186060
$533$	2.12549	0.0920653
$534$	0	0
$535$	0	0
$536$	−10.9373	−0.472417
$537$	0	0
$538$	6.00000	0.258678
$539$	−5.29150	−0.227921
$540$	0	0
$541$	−8.35425	−0.359177	−0.179589	−	0.983742i	$-0.557477\pi$
$541$	−8.35425	−0.359177	−0.179589	+	0.983742i	$0.557477\pi$
$542$	4.70850	0.202247
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−2.35425	−0.100569
$549$	0	0
$550$	0	0
$551$	−22.9778	−0.978886
$552$	0	0
$553$	0.0627461	0.00266823
$554$	−25.2915	−1.07453
$555$	0	0
$556$	−16.5830	−0.703276
$557$	−32.5203	−1.37793	−0.688964	−	0.724796i	$-0.741934\pi$
$557$	−32.5203	−1.37793	−0.688964	+	0.724796i	$0.741934\pi$
$558$	0	0
$559$	3.64575	0.154199
$560$	0	0
$561$	0	0
$562$	−23.5203	−0.992142
$563$	38.9373	1.64101	0.820505	−	0.571640i	$-0.193692\pi$
$563$	38.9373	1.64101	0.820505	+	0.571640i	$0.193692\pi$
$564$	0	0
$565$	0	0
$566$	30.1660	1.26797
$567$	0	0
$568$	0.708497	0.0297279
$569$	5.16601	0.216570	0.108285	−	0.994120i	$-0.465464\pi$
$569$	5.16601	0.216570	0.108285	+	0.994120i	$0.465464\pi$
$570$	0	0
$571$	−8.58301	−0.359188	−0.179594	−	0.983741i	$-0.557478\pi$
$571$	−8.58301	−0.359188	−0.179594	+	0.983741i	$0.557478\pi$
$572$	3.41699	0.142872
$573$	0	0
$574$	−3.29150	−0.137385
$575$	0	0
$576$	0	0
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	−1.64575	−0.0682773
$582$	0	0
$583$	−14.0000	−0.579821
$584$	5.29150	0.218964
$585$	0	0
$586$	5.52026	0.228040
$587$	5.52026	0.227845	0.113923	−	0.993490i	$-0.463658\pi$
$587$	5.52026	0.227845	0.113923	+	0.993490i	$0.463658\pi$
$588$	0	0
$589$	14.1255	0.582031
$590$	0	0
$591$	0	0
$592$	4.35425	0.178959
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	0	0
$595$	0	0
$596$	−22.5203	−0.922466
$597$	0	0
$598$	2.35425	0.0962724
$599$	8.81176	0.360039	0.180019	−	0.983663i	$-0.442384\pi$
$599$	8.81176	0.360039	0.180019	+	0.983663i	$0.442384\pi$
$600$	0	0
$601$	−4.70850	−0.192064	−0.0960318	−	0.995378i	$-0.530615\pi$
$601$	−4.70850	−0.192064	−0.0960318	+	0.995378i	$0.530615\pi$
$602$	−5.64575	−0.230104
$603$	0	0
$604$	−13.2288	−0.538270
$605$	0	0
$606$	0	0
$607$	−46.8118	−1.90003	−0.950015	−	0.312203i	$-0.898933\pi$
$607$	−46.8118	−1.90003	−0.950015	+	0.312203i	$0.898933\pi$
$608$	−4.29150	−0.174043
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	−7.06275	−0.285262	−0.142631	−	0.989776i	$-0.545556\pi$
$613$	−7.06275	−0.285262	−0.142631	+	0.989776i	$0.545556\pi$
$614$	6.70850	0.270733
$615$	0	0
$616$	−5.29150	−0.213201
$617$	20.9373	0.842902	0.421451	−	0.906851i	$-0.361521\pi$
$617$	20.9373	0.842902	0.421451	+	0.906851i	$0.361521\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	−10.7085	−0.429372
$623$	1.00000	0.0400642
$624$	0	0
$625$	0	0
$626$	−14.2288	−0.568695
$627$	0	0
$628$	−1.35425	−0.0540404
$629$	4.35425	0.173615
$630$	0	0
$631$	19.8118	0.788694	0.394347	−	0.918962i	$-0.370971\pi$
$631$	19.8118	0.788694	0.394347	+	0.918962i	$0.370971\pi$
$632$	0.0627461	0.00249590
$633$	0	0
$634$	14.0627	0.558503
$635$	0	0
$636$	0	0
$637$	−0.645751	−0.0255856
$638$	−28.3320	−1.12168
$639$	0	0
$640$	0	0
$641$	−24.7085	−0.975927	−0.487963	−	0.872864i	$-0.662260\pi$
$641$	−24.7085	−0.975927	−0.487963	+	0.872864i	$0.662260\pi$
$642$	0	0
$643$	8.29150	0.326985	0.163492	−	0.986545i	$-0.447724\pi$
$643$	8.29150	0.326985	0.163492	+	0.986545i	$0.447724\pi$
$644$	−3.64575	−0.143663
$645$	0	0
$646$	−4.29150	−0.168847
$647$	−16.6458	−0.654412	−0.327206	−	0.944953i	$-0.606107\pi$
$647$	−16.6458	−0.654412	−0.327206	+	0.944953i	$0.606107\pi$
$648$	0	0
$649$	14.3320	0.562581
$650$	0	0
$651$	0	0
$652$	11.8745	0.465042
$653$	−19.8118	−0.775294	−0.387647	−	0.921808i	$-0.626712\pi$
$653$	−19.8118	−0.775294	−0.387647	+	0.921808i	$0.626712\pi$
$654$	0	0
$655$	0	0
$656$	−3.29150	−0.128512
$657$	0	0
$658$	4.64575	0.181110
$659$	−15.5830	−0.607028	−0.303514	−	0.952827i	$-0.598160\pi$
$659$	−15.5830	−0.607028	−0.303514	+	0.952827i	$0.598160\pi$
$660$	0	0
$661$	39.1660	1.52338	0.761691	−	0.647941i	$-0.224370\pi$
$661$	39.1660	1.52338	0.761691	+	0.647941i	$0.224370\pi$
$662$	4.35425	0.169233
$663$	0	0
$664$	−1.64575	−0.0638675
$665$	0	0
$666$	0	0
$667$	−19.5203	−0.755827
$668$	−17.1660	−0.664173
$669$	0	0
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	−5.41699	−0.208810	−0.104405	−	0.994535i	$-0.533294\pi$
$673$	−5.41699	−0.208810	−0.104405	+	0.994535i	$0.533294\pi$
$674$	16.1660	0.622691
$675$	0	0
$676$	−12.5830	−0.483962
$677$	−1.29150	−0.0496365	−0.0248182	−	0.999692i	$-0.507901\pi$
$677$	−1.29150	−0.0496365	−0.0248182	+	0.999692i	$0.507901\pi$
$678$	0	0
$679$	2.93725	0.112721
$680$	0	0
$681$	0	0
$682$	17.4170	0.666931
$683$	18.5830	0.711059	0.355529	−	0.934665i	$-0.384301\pi$
$683$	18.5830	0.711059	0.355529	+	0.934665i	$0.384301\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	−5.64575	−0.215242
$689$	−1.70850	−0.0650886
$690$	0	0
$691$	−33.2915	−1.26647	−0.633234	−	0.773960i	$-0.718273\pi$
$691$	−33.2915	−1.26647	−0.633234	+	0.773960i	$0.718273\pi$
$692$	1.64575	0.0625621
$693$	0	0
$694$	20.2915	0.770255
$695$	0	0
$696$	0	0
$697$	−3.29150	−0.124675
$698$	14.6458	0.554350
$699$	0	0
$700$	0	0
$701$	−2.70850	−0.102299	−0.0511493	−	0.998691i	$-0.516288\pi$
$701$	−2.70850	−0.102299	−0.0511493	+	0.998691i	$0.516288\pi$
$702$	0	0
$703$	18.6863	0.704766
$704$	−5.29150	−0.199431
$705$	0	0
$706$	−36.4575	−1.37210
$707$	1.06275	0.0399687
$708$	0	0
$709$	−29.8745	−1.12196	−0.560980	−	0.827829i	$-0.689576\pi$
$709$	−29.8745	−1.12196	−0.560980	+	0.827829i	$0.689576\pi$
$710$	0	0
$711$	0	0
$712$	1.00000	0.0374766
$713$	12.0000	0.449404
$714$	0	0
$715$	0	0
$716$	17.5830	0.657108
$717$	0	0
$718$	11.4170	0.426078
$719$	23.3542	0.870967	0.435483	−	0.900197i	$-0.356578\pi$
$719$	23.3542	0.870967	0.435483	+	0.900197i	$0.356578\pi$
$720$	0	0
$721$	−13.8745	−0.516714
$722$	0.583005	0.0216972
$723$	0	0
$724$	22.6458	0.841623
$725$	0	0
$726$	0	0
$727$	−20.3542	−0.754897	−0.377449	−	0.926031i	$-0.623199\pi$
$727$	−20.3542	−0.754897	−0.377449	+	0.926031i	$0.623199\pi$
$728$	−0.645751	−0.0239331
$729$	0	0
$730$	0	0
$731$	−5.64575	−0.208816
$732$	0	0
$733$	9.81176	0.362406	0.181203	−	0.983446i	$-0.442001\pi$
$733$	9.81176	0.362406	0.181203	+	0.983446i	$0.442001\pi$
$734$	−14.9373	−0.551344
$735$	0	0
$736$	−3.64575	−0.134384
$737$	−57.8745	−2.13183
$738$	0	0
$739$	−36.3320	−1.33649	−0.668247	−	0.743939i	$-0.732955\pi$
$739$	−36.3320	−1.33649	−0.668247	+	0.743939i	$0.732955\pi$
$740$	0	0
$741$	0	0
$742$	2.64575	0.0971286
$743$	13.7490	0.504402	0.252201	−	0.967675i	$-0.418846\pi$
$743$	13.7490	0.504402	0.252201	+	0.967675i	$0.418846\pi$
$744$	0	0
$745$	0	0
$746$	7.87451	0.288306
$747$	0	0
$748$	−5.29150	−0.193476
$749$	6.00000	0.219235
$750$	0	0
$751$	−1.41699	−0.0517069	−0.0258534	−	0.999666i	$-0.508230\pi$
$751$	−1.41699	−0.0517069	−0.0258534	+	0.999666i	$0.508230\pi$
$752$	4.64575	0.169413
$753$	0	0
$754$	−3.45751	−0.125915
$755$	0	0
$756$	0	0
$757$	18.8118	0.683725	0.341863	−	0.939750i	$-0.388942\pi$
$757$	18.8118	0.683725	0.341863	+	0.939750i	$0.388942\pi$
$758$	−24.2288	−0.880028
$759$	0	0
$760$	0	0
$761$	−44.1660	−1.60102	−0.800508	−	0.599322i	$-0.795437\pi$
$761$	−44.1660	−1.60102	−0.800508	+	0.599322i	$0.795437\pi$
$762$	0	0
$763$	−3.64575	−0.131985
$764$	−14.5830	−0.527595
$765$	0	0
$766$	−16.7085	−0.603703
$767$	1.74902	0.0631533
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	14.8745	0.535345
$773$	−17.1660	−0.617418	−0.308709	−	0.951156i	$-0.599897\pi$
$773$	−17.1660	−0.617418	−0.308709	+	0.951156i	$0.599897\pi$
$774$	0	0
$775$	0	0
$776$	2.93725	0.105441
$777$	0	0
$778$	34.3948	1.23311
$779$	−14.1255	−0.506098
$780$	0	0
$781$	3.74902	0.134150
$782$	−3.64575	−0.130372
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−40.5830	−1.44663	−0.723314	−	0.690519i	$-0.757382\pi$
$787$	−40.5830	−1.44663	−0.723314	+	0.690519i	$0.757382\pi$
$788$	−13.3542	−0.475725
$789$	0	0
$790$	0	0
$791$	−12.5830	−0.447400
$792$	0	0
$793$	1.70850	0.0606705
$794$	−32.5203	−1.15410
$795$	0	0
$796$	−17.5203	−0.620989
$797$	50.8118	1.79984	0.899922	−	0.436050i	$-0.143623\pi$
$797$	50.8118	1.79984	0.899922	+	0.436050i	$0.143623\pi$
$798$	0	0
$799$	4.64575	0.164355
$800$	0	0
$801$	0	0
$802$	−9.06275	−0.320017
$803$	28.0000	0.988099
$804$	0	0
$805$	0	0
$806$	2.12549	0.0748673
$807$	0	0
$808$	1.06275	0.0373873
$809$	3.41699	0.120135	0.0600676	−	0.998194i	$-0.480868\pi$
$809$	3.41699	0.120135	0.0600676	+	0.998194i	$0.480868\pi$
$810$	0	0
$811$	−27.1660	−0.953928	−0.476964	−	0.878923i	$-0.658263\pi$
$811$	−27.1660	−0.953928	−0.476964	+	0.878923i	$0.658263\pi$
$812$	5.35425	0.187897
$813$	0	0
$814$	23.0405	0.807570
$815$	0	0
$816$	0	0
$817$	−24.2288	−0.847657
$818$	−18.2288	−0.637354
$819$	0	0
$820$	0	0
$821$	20.7712	0.724921	0.362461	−	0.931999i	$-0.381937\pi$
$821$	20.7712	0.724921	0.362461	+	0.931999i	$0.381937\pi$
$822$	0	0
$823$	−53.0405	−1.84888	−0.924438	−	0.381332i	$-0.875465\pi$
$823$	−53.0405	−1.84888	−0.924438	+	0.381332i	$0.875465\pi$
$824$	−13.8745	−0.483341
$825$	0	0
$826$	−2.70850	−0.0942407
$827$	−43.5830	−1.51553	−0.757765	−	0.652528i	$-0.773709\pi$
$827$	−43.5830	−1.51553	−0.757765	+	0.652528i	$0.773709\pi$
$828$	0	0
$829$	7.35425	0.255424	0.127712	−	0.991811i	$-0.459237\pi$
$829$	7.35425	0.255424	0.127712	+	0.991811i	$0.459237\pi$
$830$	0	0
$831$	0	0
$832$	−0.645751	−0.0223874
$833$	1.00000	0.0346479
$834$	0	0
$835$	0	0
$836$	−22.7085	−0.785390
$837$	0	0
$838$	27.6458	0.955007
$839$	−49.8118	−1.71969	−0.859846	−	0.510553i	$-0.829441\pi$
$839$	−49.8118	−1.71969	−0.859846	+	0.510553i	$0.829441\pi$
$840$	0	0
$841$	−0.332021	−0.0114490
$842$	−16.5830	−0.571488
$843$	0	0
$844$	14.7085	0.506287
$845$	0	0
$846$	0	0
$847$	−17.0000	−0.584127
$848$	2.64575	0.0908555
$849$	0	0
$850$	0	0
$851$	15.8745	0.544171
$852$	0	0
$853$	54.3320	1.86029	0.930146	−	0.367189i	$-0.119680\pi$
$853$	54.3320	1.86029	0.930146	+	0.367189i	$0.119680\pi$
$854$	−2.64575	−0.0905357
$855$	0	0
$856$	6.00000	0.205076
$857$	34.2915	1.17138	0.585688	−	0.810537i	$-0.300825\pi$
$857$	34.2915	1.17138	0.585688	+	0.810537i	$0.300825\pi$
$858$	0	0
$859$	−34.3320	−1.17139	−0.585697	−	0.810530i	$-0.699179\pi$
$859$	−34.3320	−1.17139	−0.585697	+	0.810530i	$0.699179\pi$
$860$	0	0
$861$	0	0
$862$	2.12549	0.0723945
$863$	−30.3542	−1.03327	−0.516635	−	0.856206i	$-0.672816\pi$
$863$	−30.3542	−1.03327	−0.516635	+	0.856206i	$0.672816\pi$
$864$	0	0
$865$	0	0
$866$	11.2915	0.383701
$867$	0	0
$868$	−3.29150	−0.111721
$869$	0.332021	0.0112630
$870$	0	0
$871$	−7.06275	−0.239312
$872$	−3.64575	−0.123461
$873$	0	0
$874$	−15.6458	−0.529225
$875$	0	0
$876$	0	0
$877$	5.87451	0.198368	0.0991840	−	0.995069i	$-0.468377\pi$
$877$	5.87451	0.198368	0.0991840	+	0.995069i	$0.468377\pi$
$878$	30.9373	1.04408
$879$	0	0
$880$	0	0
$881$	15.1255	0.509591	0.254795	−	0.966995i	$-0.417992\pi$
$881$	15.1255	0.509591	0.254795	+	0.966995i	$0.417992\pi$
$882$	0	0
$883$	−31.2915	−1.05304	−0.526521	−	0.850162i	$-0.676504\pi$
$883$	−31.2915	−1.05304	−0.526521	+	0.850162i	$0.676504\pi$
$884$	−0.645751	−0.0217190
$885$	0	0
$886$	33.5830	1.12824
$887$	49.8745	1.67462	0.837311	−	0.546727i	$-0.184126\pi$
$887$	49.8745	1.67462	0.837311	+	0.546727i	$0.184126\pi$
$888$	0	0
$889$	−9.35425	−0.313731
$890$	0	0
$891$	0	0
$892$	−15.5203	−0.519657
$893$	19.9373	0.667175
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−18.2288	−0.608301
$899$	−17.6235	−0.587777
$900$	0	0
$901$	2.64575	0.0881428
$902$	−17.4170	−0.579922
$903$	0	0
$904$	−12.5830	−0.418505
$905$	0	0
$906$	0	0
$907$	−23.2915	−0.773382	−0.386691	−	0.922209i	$-0.626382\pi$
$907$	−23.2915	−0.773382	−0.386691	+	0.922209i	$0.626382\pi$
$908$	−11.6458	−0.386478
$909$	0	0
$910$	0	0
$911$	29.1660	0.966313	0.483157	−	0.875534i	$-0.339490\pi$
$911$	29.1660	0.966313	0.483157	+	0.875534i	$0.339490\pi$
$912$	0	0
$913$	−8.70850	−0.288209
$914$	7.12549	0.235690
$915$	0	0
$916$	−24.5203	−0.810172
$917$	3.06275	0.101141
$918$	0	0
$919$	−25.0405	−0.826010	−0.413005	−	0.910729i	$-0.635521\pi$
$919$	−25.0405	−0.826010	−0.413005	+	0.910729i	$0.635521\pi$
$920$	0	0
$921$	0	0
$922$	−33.5203	−1.10393
$923$	0.457513	0.0150592
$924$	0	0
$925$	0	0
$926$	−1.35425	−0.0445034
$927$	0	0
$928$	5.35425	0.175762
$929$	20.8745	0.684870	0.342435	−	0.939541i	$-0.388748\pi$
$929$	20.8745	0.684870	0.342435	+	0.939541i	$0.388748\pi$
$930$	0	0
$931$	4.29150	0.140648
$932$	−3.64575	−0.119421
$933$	0	0
$934$	39.2915	1.28566
$935$	0	0
$936$	0	0
$937$	−10.2288	−0.334159	−0.167079	−	0.985943i	$-0.553434\pi$
$937$	−10.2288	−0.334159	−0.167079	+	0.985943i	$0.553434\pi$
$938$	10.9373	0.357114
$939$	0	0
$940$	0	0
$941$	−10.3542	−0.337539	−0.168769	−	0.985656i	$-0.553979\pi$
$941$	−10.3542	−0.337539	−0.168769	+	0.985656i	$0.553979\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−2.70850	−0.0881541
$945$	0	0
$946$	−29.8745	−0.971304
$947$	29.2915	0.951846	0.475923	−	0.879487i	$-0.342114\pi$
$947$	29.2915	0.951846	0.475923	+	0.879487i	$0.342114\pi$
$948$	0	0
$949$	3.41699	0.110920
$950$	0	0
$951$	0	0
$952$	1.00000	0.0324102
$953$	40.1033	1.29907	0.649536	−	0.760331i	$-0.274963\pi$
$953$	40.1033	1.29907	0.649536	+	0.760331i	$0.274963\pi$
$954$	0	0
$955$	0	0
$956$	−19.6458	−0.635389
$957$	0	0
$958$	12.6458	0.408566
$959$	2.35425	0.0760227
$960$	0	0
$961$	−20.1660	−0.650516
$962$	2.81176	0.0906548
$963$	0	0
$964$	2.93725	0.0946026
$965$	0	0
$966$	0	0
$967$	−1.81176	−0.0582623	−0.0291312	−	0.999576i	$-0.509274\pi$
$967$	−1.81176	−0.0582623	−0.0291312	+	0.999576i	$0.509274\pi$
$968$	−17.0000	−0.546401
$969$	0	0
$970$	0	0
$971$	−30.2288	−0.970087	−0.485043	−	0.874490i	$-0.661196\pi$
$971$	−30.2288	−0.970087	−0.485043	+	0.874490i	$0.661196\pi$
$972$	0	0
$973$	16.5830	0.531627
$974$	11.2915	0.361803
$975$	0	0
$976$	−2.64575	−0.0846884
$977$	7.06275	0.225957	0.112979	−	0.993597i	$-0.463961\pi$
$977$	7.06275	0.225957	0.112979	+	0.993597i	$0.463961\pi$
$978$	0	0
$979$	5.29150	0.169117
$980$	0	0
$981$	0	0
$982$	−18.8745	−0.602310
$983$	50.5203	1.61135	0.805673	−	0.592361i	$-0.201804\pi$
$983$	50.5203	1.61135	0.805673	+	0.592361i	$0.201804\pi$
$984$	0	0
$985$	0	0
$986$	5.35425	0.170514
$987$	0	0
$988$	−2.77124	−0.0881650
$989$	−20.5830	−0.654501
$990$	0	0
$991$	−18.0627	−0.573782	−0.286891	−	0.957963i	$-0.592622\pi$
$991$	−18.0627	−0.573782	−0.286891	+	0.957963i	$0.592622\pi$
$992$	−3.29150	−0.104505
$993$	0	0
$994$	−0.708497	−0.0224722
$995$	0	0
$996$	0	0
$997$	2.45751	0.0778302	0.0389151	−	0.999243i	$-0.487610\pi$
$997$	2.45751	0.0778302	0.0389151	+	0.999243i	$0.487610\pi$
$998$	−28.9373	−0.915993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.ec.1.1	✓	2
3.2	odd	2		9450.2.a.eq.1.2	yes	2
5.4	even	2		9450.2.a.et.1.1	yes	2
15.14	odd	2		9450.2.a.ej.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9450.2.a.ec.1.1	✓	2	1.1	even	1	trivial
9450.2.a.ej.1.2	yes	2	15.14	odd	2
9450.2.a.eq.1.2	yes	2	3.2	odd	2
9450.2.a.et.1.1	yes	2	5.4	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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -5.29150 q^{11} -0.645751 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +4.29150 q^{19} +5.29150 q^{22} +3.64575 q^{23} +0.645751 q^{26} -1.00000 q^{28} -5.35425 q^{29} +3.29150 q^{31} -1.00000 q^{32} -1.00000 q^{34} +4.35425 q^{37} -4.29150 q^{38} -3.29150 q^{41} -5.64575 q^{43} -5.29150 q^{44} -3.64575 q^{46} +4.64575 q^{47} +1.00000 q^{49} -0.645751 q^{52} +2.64575 q^{53} +1.00000 q^{56} +5.35425 q^{58} -2.70850 q^{59} -2.64575 q^{61} -3.29150 q^{62} +1.00000 q^{64} +10.9373 q^{67} +1.00000 q^{68} -0.708497 q^{71} -5.29150 q^{73} -4.35425 q^{74} +4.29150 q^{76} +5.29150 q^{77} -0.0627461 q^{79} +3.29150 q^{82} +1.64575 q^{83} +5.64575 q^{86} +5.29150 q^{88} -1.00000 q^{89} +0.645751 q^{91} +3.64575 q^{92} -4.64575 q^{94} -2.93725 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{23} - 4 q^{26} - 2 q^{28} - 16 q^{29} - 4 q^{31} - 2 q^{32} - 2 q^{34} + 14 q^{37} + 2 q^{38} + 4 q^{41} - 6 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{52} + 2 q^{56} + 16 q^{58} - 16 q^{59} + 4 q^{62} + 2 q^{64} + 6 q^{67} + 2 q^{68} - 12 q^{71} - 14 q^{74} - 2 q^{76} - 16 q^{79} - 4 q^{82} - 2 q^{83} + 6 q^{86} - 2 q^{89} - 4 q^{91} + 2 q^{92} - 4 q^{94} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^23 - 4 * q^26 - 2 * q^28 - 16 * q^29 - 4 * q^31 - 2 * q^32 - 2 * q^34 + 14 * q^37 + 2 * q^38 + 4 * q^41 - 6 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^49 + 4 * q^52 + 2 * q^56 + 16 * q^58 - 16 * q^59 + 4 * q^62 + 2 * q^64 + 6 * q^67 + 2 * q^68 - 12 * q^71 - 14 * q^74 - 2 * q^76 - 16 * q^79 - 4 * q^82 - 2 * q^83 + 6 * q^86 - 2 * q^89 - 4 * q^91 + 2 * q^92 - 4 * q^94 + 10 * q^97 - 2 * q^98

Introduction

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