LMFDB - Embedded newform 9450.2.a.ej.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:	$0$
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -5.29150 q^{11} -4.64575 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.29150 q^{19} +5.29150 q^{22} -1.64575 q^{23} +4.64575 q^{26} +1.00000 q^{28} +10.6458 q^{29} -7.29150 q^{31} -1.00000 q^{32} -1.00000 q^{34} -9.64575 q^{37} +6.29150 q^{38} -7.29150 q^{41} +0.354249 q^{43} -5.29150 q^{44} +1.64575 q^{46} -0.645751 q^{47} +1.00000 q^{49} -4.64575 q^{52} -2.64575 q^{53} -1.00000 q^{56} -10.6458 q^{58} +13.2915 q^{59} +2.64575 q^{61} +7.29150 q^{62} +1.00000 q^{64} +4.93725 q^{67} +1.00000 q^{68} +11.2915 q^{71} -5.29150 q^{73} +9.64575 q^{74} -6.29150 q^{76} -5.29150 q^{77} -15.9373 q^{79} +7.29150 q^{82} -3.64575 q^{83} -0.354249 q^{86} +5.29150 q^{88} +1.00000 q^{89} -4.64575 q^{91} -1.64575 q^{92} +0.645751 q^{94} -12.9373 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−4.64575	−1.28850	−0.644250	−	0.764815i	$-0.722830\pi$
$13$	−4.64575	−1.28850	−0.644250	+	0.764815i	$0.722830\pi$
$14$	−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
$19$	−6.29150	−1.44337	−0.721685	−	0.692222i	$-0.756632\pi$
$19$	−6.29150	−1.44337	−0.721685	+	0.692222i	$0.756632\pi$
$20$	0	0
$21$	0	0
$22$	5.29150	1.12815
$23$	−1.64575	−0.343163	−0.171581	−	0.985170i	$-0.554888\pi$
$23$	−1.64575	−0.343163	−0.171581	+	0.985170i	$0.554888\pi$
$24$	0	0
$25$	0	0
$26$	4.64575	0.911107
$27$	0	0
$28$	1.00000	0.188982
$29$	10.6458	1.97687	0.988433	−	0.151657i	$-0.0484609\pi$
$29$	10.6458	1.97687	0.988433	+	0.151657i	$0.0484609\pi$
$30$	0	0
$31$	−7.29150	−1.30959	−0.654796	−	0.755805i	$-0.727246\pi$
$31$	−7.29150	−1.30959	−0.654796	+	0.755805i	$0.727246\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	−9.64575	−1.58575	−0.792876	−	0.609383i	$-0.791417\pi$
$37$	−9.64575	−1.58575	−0.792876	+	0.609383i	$0.791417\pi$
$38$	6.29150	1.02062
$39$	0	0
$40$	0	0
$41$	−7.29150	−1.13874	−0.569371	−	0.822081i	$-0.692813\pi$
$41$	−7.29150	−1.13874	−0.569371	+	0.822081i	$0.692813\pi$
$42$	0	0
$43$	0.354249	0.0540224	0.0270112	−	0.999635i	$-0.491401\pi$
$43$	0.354249	0.0540224	0.0270112	+	0.999635i	$0.491401\pi$
$44$	−5.29150	−0.797724
$45$	0	0
$46$	1.64575	0.242653
$47$	−0.645751	−0.0941925	−0.0470963	−	0.998890i	$-0.514997\pi$
$47$	−0.645751	−0.0941925	−0.0470963	+	0.998890i	$0.514997\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−4.64575	−0.644250
$53$	−2.64575	−0.363422	−0.181711	−	0.983352i	$-0.558164\pi$
$53$	−2.64575	−0.363422	−0.181711	+	0.983352i	$0.558164\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−10.6458	−1.39786
$59$	13.2915	1.73041	0.865203	−	0.501422i	$-0.167189\pi$
$59$	13.2915	1.73041	0.865203	+	0.501422i	$0.167189\pi$
$60$	0	0
$61$	2.64575	0.338754	0.169377	−	0.985551i	$-0.445824\pi$
$61$	2.64575	0.338754	0.169377	+	0.985551i	$0.445824\pi$
$62$	7.29150	0.926022
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.93725	0.603182	0.301591	−	0.953437i	$-0.402482\pi$
$67$	4.93725	0.603182	0.301591	+	0.953437i	$0.402482\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	11.2915	1.34005	0.670027	−	0.742336i	$-0.266282\pi$
$71$	11.2915	1.34005	0.670027	+	0.742336i	$0.266282\pi$
$72$	0	0
$73$	−5.29150	−0.619324	−0.309662	−	0.950847i	$-0.600216\pi$
$73$	−5.29150	−0.619324	−0.309662	+	0.950847i	$0.600216\pi$
$74$	9.64575	1.12130
$75$	0	0
$76$	−6.29150	−0.721685
$77$	−5.29150	−0.603023
$78$	0	0
$79$	−15.9373	−1.79308	−0.896541	−	0.442962i	$-0.853928\pi$
$79$	−15.9373	−1.79308	−0.896541	+	0.442962i	$0.853928\pi$
$80$	0	0
$81$	0	0
$82$	7.29150	0.805212
$83$	−3.64575	−0.400173	−0.200087	−	0.979778i	$-0.564122\pi$
$83$	−3.64575	−0.400173	−0.200087	+	0.979778i	$0.564122\pi$
$84$	0	0
$85$	0	0
$86$	−0.354249	−0.0381996
$87$	0	0
$88$	5.29150	0.564076
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	−4.64575	−0.487007
$92$	−1.64575	−0.171581
$93$	0	0
$94$	0.645751	0.0666042
$95$	0	0
$96$	0	0
$97$	−12.9373	−1.31358	−0.656790	−	0.754074i	$-0.728086\pi$
$97$	−12.9373	−1.31358	−0.656790	+	0.754074i	$0.728086\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	16.9373	1.68532	0.842660	−	0.538446i	$-0.180988\pi$
$101$	16.9373	1.68532	0.842660	+	0.538446i	$0.180988\pi$
$102$	0	0
$103$	17.8745	1.76123	0.880614	−	0.473835i	$-0.157131\pi$
$103$	17.8745	1.76123	0.880614	+	0.473835i	$0.157131\pi$
$104$	4.64575	0.455553
$105$	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
$109$	−1.64575	−0.157634	−0.0788172	−	0.996889i	$-0.525114\pi$
$109$	−1.64575	−0.157634	−0.0788172	+	0.996889i	$0.525114\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−8.58301	−0.807421	−0.403711	−	0.914887i	$-0.632280\pi$
$113$	−8.58301	−0.807421	−0.403711	+	0.914887i	$0.632280\pi$
$114$	0	0
$115$	0	0
$116$	10.6458	0.988433
$117$	0	0
$118$	−13.2915	−1.22358
$119$	1.00000	0.0916698
$120$	0	0
$121$	17.0000	1.54545
$122$	−2.64575	−0.239535
$123$	0	0
$124$	−7.29150	−0.654796
$125$	0	0
$126$	0	0
$127$	−14.6458	−1.29960	−0.649800	−	0.760105i	$-0.725147\pi$
$127$	−14.6458	−1.29960	−0.649800	+	0.760105i	$0.725147\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	18.9373	1.65456	0.827278	−	0.561793i	$-0.189888\pi$
$131$	18.9373	1.65456	0.827278	+	0.561793i	$0.189888\pi$
$132$	0	0
$133$	−6.29150	−0.545542
$134$	−4.93725	−0.426514
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−7.64575	−0.653221	−0.326610	−	0.945159i	$-0.605906\pi$
$137$	−7.64575	−0.653221	−0.326610	+	0.945159i	$0.605906\pi$
$138$	0	0
$139$	4.58301	0.388725	0.194363	−	0.980930i	$-0.437736\pi$
$139$	4.58301	0.388725	0.194363	+	0.980930i	$0.437736\pi$
$140$	0	0
$141$	0	0
$142$	−11.2915	−0.947562
$143$	24.5830	2.05573
$144$	0	0
$145$	0	0
$146$	5.29150	0.437928
$147$	0	0
$148$	−9.64575	−0.792876
$149$	−14.5203	−1.18955	−0.594773	−	0.803894i	$-0.702758\pi$
$149$	−14.5203	−1.18955	−0.594773	+	0.803894i	$0.702758\pi$
$150$	0	0
$151$	13.2288	1.07654	0.538270	−	0.842772i	$-0.319078\pi$
$151$	13.2288	1.07654	0.538270	+	0.842772i	$0.319078\pi$
$152$	6.29150	0.510308
$153$	0	0
$154$	5.29150	0.426401
$155$	0	0
$156$	0	0
$157$	6.64575	0.530389	0.265194	−	0.964195i	$-0.414564\pi$
$157$	6.64575	0.530389	0.265194	+	0.964195i	$0.414564\pi$
$158$	15.9373	1.26790
$159$	0	0
$160$	0	0
$161$	−1.64575	−0.129703
$162$	0	0
$163$	19.8745	1.55669	0.778346	−	0.627836i	$-0.216059\pi$
$163$	19.8745	1.55669	0.778346	+	0.627836i	$0.216059\pi$
$164$	−7.29150	−0.569371
$165$	0	0
$166$	3.64575	0.282965
$167$	25.1660	1.94740	0.973702	−	0.227825i	$-0.0731613\pi$
$167$	25.1660	1.94740	0.973702	+	0.227825i	$0.0731613\pi$
$168$	0	0
$169$	8.58301	0.660231
$170$	0	0
$171$	0	0
$172$	0.354249	0.0270112
$173$	−3.64575	−0.277181	−0.138591	−	0.990350i	$-0.544257\pi$
$173$	−3.64575	−0.277181	−0.138591	+	0.990350i	$0.544257\pi$
$174$	0	0
$175$	0	0
$176$	−5.29150	−0.398862
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	3.58301	0.267806	0.133903	−	0.990994i	$-0.457249\pi$
$179$	3.58301	0.267806	0.133903	+	0.990994i	$0.457249\pi$
$180$	0	0
$181$	17.3542	1.28993	0.644966	−	0.764212i	$-0.276872\pi$
$181$	17.3542	1.28993	0.644966	+	0.764212i	$0.276872\pi$
$182$	4.64575	0.344366
$183$	0	0
$184$	1.64575	0.121326
$185$	0	0
$186$	0	0
$187$	−5.29150	−0.386953
$188$	−0.645751	−0.0470963
$189$	0	0
$190$	0	0
$191$	−6.58301	−0.476330	−0.238165	−	0.971225i	$-0.576546\pi$
$191$	−6.58301	−0.476330	−0.238165	+	0.971225i	$0.576546\pi$
$192$	0	0
$193$	16.8745	1.21465	0.607327	−	0.794452i	$-0.292242\pi$
$193$	16.8745	1.21465	0.607327	+	0.794452i	$0.292242\pi$
$194$	12.9373	0.928841
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.6458	−1.32845	−0.664227	−	0.747531i	$-0.731239\pi$
$197$	−18.6458	−1.32845	−0.664227	+	0.747531i	$0.731239\pi$
$198$	0	0
$199$	19.5203	1.38375	0.691877	−	0.722015i	$-0.256784\pi$
$199$	19.5203	1.38375	0.691877	+	0.722015i	$0.256784\pi$
$200$	0	0
$201$	0	0
$202$	−16.9373	−1.19170
$203$	10.6458	0.747185
$204$	0	0
$205$	0	0
$206$	−17.8745	−1.24538
$207$	0	0
$208$	−4.64575	−0.322125
$209$	33.2915	2.30282
$210$	0	0
$211$	25.2915	1.74114	0.870569	−	0.492046i	$-0.163751\pi$
$211$	25.2915	1.74114	0.870569	+	0.492046i	$0.163751\pi$
$212$	−2.64575	−0.181711
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	−7.29150	−0.494979
$218$	1.64575	0.111464
$219$	0	0
$220$	0	0
$221$	−4.64575	−0.312507
$222$	0	0
$223$	−21.5203	−1.44110	−0.720552	−	0.693401i	$-0.756111\pi$
$223$	−21.5203	−1.44110	−0.720552	+	0.693401i	$0.756111\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	8.58301	0.570933
$227$	−6.35425	−0.421746	−0.210873	−	0.977513i	$-0.567631\pi$
$227$	−6.35425	−0.421746	−0.210873	+	0.977513i	$0.567631\pi$
$228$	0	0
$229$	12.5203	0.827362	0.413681	−	0.910422i	$-0.364243\pi$
$229$	12.5203	0.827362	0.413681	+	0.910422i	$0.364243\pi$
$230$	0	0
$231$	0	0
$232$	−10.6458	−0.698928
$233$	1.64575	0.107817	0.0539084	−	0.998546i	$-0.482832\pi$
$233$	1.64575	0.107817	0.0539084	+	0.998546i	$0.482832\pi$
$234$	0	0
$235$	0	0
$236$	13.2915	0.865203
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	14.3542	0.928499	0.464250	−	0.885704i	$-0.346324\pi$
$239$	14.3542	0.928499	0.464250	+	0.885704i	$0.346324\pi$
$240$	0	0
$241$	−12.9373	−0.833362	−0.416681	−	0.909053i	$-0.636807\pi$
$241$	−12.9373	−0.833362	−0.416681	+	0.909053i	$0.636807\pi$
$242$	−17.0000	−1.09280
$243$	0	0
$244$	2.64575	0.169377
$245$	0	0
$246$	0	0
$247$	29.2288	1.85978
$248$	7.29150	0.463011
$249$	0	0
$250$	0	0
$251$	−3.87451	−0.244557	−0.122278	−	0.992496i	$-0.539020\pi$
$251$	−3.87451	−0.244557	−0.122278	+	0.992496i	$0.539020\pi$
$252$	0	0
$253$	8.70850	0.547499
$254$	14.6458	0.918956
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.29150	−0.142940	−0.0714700	−	0.997443i	$-0.522769\pi$
$257$	−2.29150	−0.142940	−0.0714700	+	0.997443i	$0.522769\pi$
$258$	0	0
$259$	−9.64575	−0.599358
$260$	0	0
$261$	0	0
$262$	−18.9373	−1.16995
$263$	19.8745	1.22551	0.612757	−	0.790271i	$-0.290060\pi$
$263$	19.8745	1.22551	0.612757	+	0.790271i	$0.290060\pi$
$264$	0	0
$265$	0	0
$266$	6.29150	0.385757
$267$	0	0
$268$	4.93725	0.301591
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−15.2915	−0.928893	−0.464446	−	0.885601i	$-0.653747\pi$
$271$	−15.2915	−0.928893	−0.464446	+	0.885601i	$0.653747\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	7.64575	0.461897
$275$	0	0
$276$	0	0
$277$	−14.7085	−0.883748	−0.441874	−	0.897077i	$-0.645686\pi$
$277$	−14.7085	−0.883748	−0.441874	+	0.897077i	$0.645686\pi$
$278$	−4.58301	−0.274870
$279$	0	0
$280$	0	0
$281$	13.5203	0.806551	0.403276	−	0.915079i	$-0.367872\pi$
$281$	13.5203	0.806551	0.403276	+	0.915079i	$0.367872\pi$
$282$	0	0
$283$	−12.1660	−0.723194	−0.361597	−	0.932334i	$-0.617768\pi$
$283$	−12.1660	−0.723194	−0.361597	+	0.932334i	$0.617768\pi$
$284$	11.2915	0.670027
$285$	0	0
$286$	−24.5830	−1.45362
$287$	−7.29150	−0.430404
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−5.29150	−0.309662
$293$	31.5203	1.84143	0.920717	−	0.390232i	$-0.127605\pi$
$293$	31.5203	1.84143	0.920717	+	0.390232i	$0.127605\pi$
$294$	0	0
$295$	0	0
$296$	9.64575	0.560648
$297$	0	0
$298$	14.5203	0.841136
$299$	7.64575	0.442165
$300$	0	0
$301$	0.354249	0.0204186
$302$	−13.2288	−0.761229
$303$	0	0
$304$	−6.29150	−0.360842
$305$	0	0
$306$	0	0
$307$	17.2915	0.986878	0.493439	−	0.869780i	$-0.335740\pi$
$307$	17.2915	0.986878	0.493439	+	0.869780i	$0.335740\pi$
$308$	−5.29150	−0.301511
$309$	0	0
$310$	0	0
$311$	−21.2915	−1.20733	−0.603665	−	0.797238i	$-0.706294\pi$
$311$	−21.2915	−1.20733	−0.603665	+	0.797238i	$0.706294\pi$
$312$	0	0
$313$	12.2288	0.691210	0.345605	−	0.938380i	$-0.387674\pi$
$313$	12.2288	0.691210	0.345605	+	0.938380i	$0.387674\pi$
$314$	−6.64575	−0.375041
$315$	0	0
$316$	−15.9373	−0.896541
$317$	−29.9373	−1.68144	−0.840722	−	0.541467i	$-0.817869\pi$
$317$	−29.9373	−1.68144	−0.840722	+	0.541467i	$0.817869\pi$
$318$	0	0
$319$	−56.3320	−3.15399
$320$	0	0
$321$	0	0
$322$	1.64575	0.0917141
$323$	−6.29150	−0.350069
$324$	0	0
$325$	0	0
$326$	−19.8745	−1.10075
$327$	0	0
$328$	7.29150	0.402606
$329$	−0.645751	−0.0356014
$330$	0	0
$331$	−9.64575	−0.530178	−0.265089	−	0.964224i	$-0.585401\pi$
$331$	−9.64575	−0.530178	−0.265089	+	0.964224i	$0.585401\pi$
$332$	−3.64575	−0.200087
$333$	0	0
$334$	−25.1660	−1.37702
$335$	0	0
$336$	0	0
$337$	−26.1660	−1.42535	−0.712677	−	0.701493i	$-0.752517\pi$
$337$	−26.1660	−1.42535	−0.712677	+	0.701493i	$0.752517\pi$
$338$	−8.58301	−0.466854
$339$	0	0
$340$	0	0
$341$	38.5830	2.08939
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.354249	−0.0190998
$345$	0	0
$346$	3.64575	0.195997
$347$	−9.70850	−0.521179	−0.260590	−	0.965450i	$-0.583917\pi$
$347$	−9.70850	−0.521179	−0.260590	+	0.965450i	$0.583917\pi$
$348$	0	0
$349$	−9.35425	−0.500721	−0.250361	−	0.968153i	$-0.580549\pi$
$349$	−9.35425	−0.500721	−0.250361	+	0.968153i	$0.580549\pi$
$350$	0	0
$351$	0	0
$352$	5.29150	0.282038
$353$	−16.4575	−0.875945	−0.437973	−	0.898988i	$-0.644303\pi$
$353$	−16.4575	−0.875945	−0.437973	+	0.898988i	$0.644303\pi$
$354$	0	0
$355$	0	0
$356$	1.00000	0.0529999
$357$	0	0
$358$	−3.58301	−0.189368
$359$	32.5830	1.71967	0.859833	−	0.510576i	$-0.170568\pi$
$359$	32.5830	1.71967	0.859833	+	0.510576i	$0.170568\pi$
$360$	0	0
$361$	20.5830	1.08332
$362$	−17.3542	−0.912119
$363$	0	0
$364$	−4.64575	−0.243504
$365$	0	0
$366$	0	0
$367$	0.937254	0.0489243	0.0244621	−	0.999701i	$-0.492213\pi$
$367$	0.937254	0.0489243	0.0244621	+	0.999701i	$0.492213\pi$
$368$	−1.64575	−0.0857907
$369$	0	0
$370$	0	0
$371$	−2.64575	−0.137361
$372$	0	0
$373$	−23.8745	−1.23618	−0.618088	−	0.786109i	$-0.712092\pi$
$373$	−23.8745	−1.23618	−0.618088	+	0.786109i	$0.712092\pi$
$374$	5.29150	0.273617
$375$	0	0
$376$	0.645751	0.0333021
$377$	−49.4575	−2.54719
$378$	0	0
$379$	−2.22876	−0.114484	−0.0572418	−	0.998360i	$-0.518231\pi$
$379$	−2.22876	−0.114484	−0.0572418	+	0.998360i	$0.518231\pi$
$380$	0	0
$381$	0	0
$382$	6.58301	0.336816
$383$	27.2915	1.39453	0.697265	−	0.716813i	$-0.254400\pi$
$383$	27.2915	1.39453	0.697265	+	0.716813i	$0.254400\pi$
$384$	0	0
$385$	0	0
$386$	−16.8745	−0.858890
$387$	0	0
$388$	−12.9373	−0.656790
$389$	−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
$391$	−1.64575	−0.0832292
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	18.6458	0.939359
$395$	0	0
$396$	0	0
$397$	4.52026	0.226865	0.113433	−	0.993546i	$-0.463815\pi$
$397$	4.52026	0.226865	0.113433	+	0.993546i	$0.463815\pi$
$398$	−19.5203	−0.978462
$399$	0	0
$400$	0	0
$401$	−24.9373	−1.24531	−0.622654	−	0.782498i	$-0.713945\pi$
$401$	−24.9373	−1.24531	−0.622654	+	0.782498i	$0.713945\pi$
$402$	0	0
$403$	33.8745	1.68741
$404$	16.9373	0.842660
$405$	0	0
$406$	−10.6458	−0.528340
$407$	51.0405	2.52998
$408$	0	0
$409$	−8.22876	−0.406886	−0.203443	−	0.979087i	$-0.565213\pi$
$409$	−8.22876	−0.406886	−0.203443	+	0.979087i	$0.565213\pi$
$410$	0	0
$411$	0	0
$412$	17.8745	0.880614
$413$	13.2915	0.654032
$414$	0	0
$415$	0	0
$416$	4.64575	0.227777
$417$	0	0
$418$	−33.2915	−1.62834
$419$	22.3542	1.09208	0.546038	−	0.837760i	$-0.316135\pi$
$419$	22.3542	1.09208	0.546038	+	0.837760i	$0.316135\pi$
$420$	0	0
$421$	−4.58301	−0.223362	−0.111681	−	0.993744i	$-0.535623\pi$
$421$	−4.58301	−0.223362	−0.111681	+	0.993744i	$0.535623\pi$
$422$	−25.2915	−1.23117
$423$	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
$431$	33.8745	1.63168	0.815839	−	0.578279i	$-0.196276\pi$
$431$	33.8745	1.63168	0.815839	+	0.578279i	$0.196276\pi$
$432$	0	0
$433$	0.708497	0.0340482	0.0170241	−	0.999855i	$-0.494581\pi$
$433$	0.708497	0.0340482	0.0170241	+	0.999855i	$0.494581\pi$
$434$	7.29150	0.350003
$435$	0	0
$436$	−1.64575	−0.0788172
$437$	10.3542	0.495311
$438$	0	0
$439$	−15.0627	−0.718906	−0.359453	−	0.933163i	$-0.617037\pi$
$439$	−15.0627	−0.718906	−0.359453	+	0.933163i	$0.617037\pi$
$440$	0	0
$441$	0	0
$442$	4.64575	0.220976
$443$	−12.4170	−0.589949	−0.294975	−	0.955505i	$-0.595311\pi$
$443$	−12.4170	−0.589949	−0.294975	+	0.955505i	$0.595311\pi$
$444$	0	0
$445$	0	0
$446$	21.5203	1.01901
$447$	0	0
$448$	1.00000	0.0472456
$449$	8.22876	0.388339	0.194170	−	0.980968i	$-0.437799\pi$
$449$	8.22876	0.388339	0.194170	+	0.980968i	$0.437799\pi$
$450$	0	0
$451$	38.5830	1.81680
$452$	−8.58301	−0.403711
$453$	0	0
$454$	6.35425	0.298220
$455$	0	0
$456$	0	0
$457$	38.8745	1.81847	0.909236	−	0.416280i	$-0.136666\pi$
$457$	38.8745	1.81847	0.909236	+	0.416280i	$0.136666\pi$
$458$	−12.5203	−0.585033
$459$	0	0
$460$	0	0
$461$	3.52026	0.163955	0.0819774	−	0.996634i	$-0.473876\pi$
$461$	3.52026	0.163955	0.0819774	+	0.996634i	$0.473876\pi$
$462$	0	0
$463$	−6.64575	−0.308854	−0.154427	−	0.988004i	$-0.549353\pi$
$463$	−6.64575	−0.308854	−0.154427	+	0.988004i	$0.549353\pi$
$464$	10.6458	0.494217
$465$	0	0
$466$	−1.64575	−0.0762380
$467$	−28.7085	−1.32847	−0.664235	−	0.747523i	$-0.731243\pi$
$467$	−28.7085	−1.32847	−0.664235	+	0.747523i	$0.731243\pi$
$468$	0	0
$469$	4.93725	0.227981
$470$	0	0
$471$	0	0
$472$	−13.2915	−0.611791
$473$	−1.87451	−0.0861900
$474$	0	0
$475$	0	0
$476$	1.00000	0.0458349
$477$	0	0
$478$	−14.3542	−0.656548
$479$	7.35425	0.336024	0.168012	−	0.985785i	$-0.446265\pi$
$479$	7.35425	0.336024	0.168012	+	0.985785i	$0.446265\pi$
$480$	0	0
$481$	44.8118	2.04324
$482$	12.9373	0.589276
$483$	0	0
$484$	17.0000	0.772727
$485$	0	0
$486$	0	0
$487$	0.708497	0.0321051	0.0160525	−	0.999871i	$-0.494890\pi$
$487$	0.708497	0.0321051	0.0160525	+	0.999871i	$0.494890\pi$
$488$	−2.64575	−0.119768
$489$	0	0
$490$	0	0
$491$	12.8745	0.581018	0.290509	−	0.956872i	$-0.406175\pi$
$491$	12.8745	0.581018	0.290509	+	0.956872i	$0.406175\pi$
$492$	0	0
$493$	10.6458	0.479461
$494$	−29.2288	−1.31506
$495$	0	0
$496$	−7.29150	−0.327398
$497$	11.2915	0.506493
$498$	0	0
$499$	13.0627	0.584769	0.292384	−	0.956301i	$-0.405551\pi$
$499$	13.0627	0.584769	0.292384	+	0.956301i	$0.405551\pi$
$500$	0	0
$501$	0	0
$502$	3.87451	0.172928
$503$	−2.70850	−0.120766	−0.0603830	−	0.998175i	$-0.519232\pi$
$503$	−2.70850	−0.120766	−0.0603830	+	0.998175i	$0.519232\pi$
$504$	0	0
$505$	0	0
$506$	−8.70850	−0.387140
$507$	0	0
$508$	−14.6458	−0.649800
$509$	−7.52026	−0.333330	−0.166665	−	0.986014i	$-0.553300\pi$
$509$	−7.52026	−0.333330	−0.166665	+	0.986014i	$0.553300\pi$
$510$	0	0
$511$	−5.29150	−0.234082
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.29150	0.101074
$515$	0	0
$516$	0	0
$517$	3.41699	0.150279
$518$	9.64575	0.423810
$519$	0	0
$520$	0	0
$521$	29.5830	1.29605	0.648027	−	0.761617i	$-0.275594\pi$
$521$	29.5830	1.29605	0.648027	+	0.761617i	$0.275594\pi$
$522$	0	0
$523$	−26.7490	−1.16965	−0.584826	−	0.811158i	$-0.698837\pi$
$523$	−26.7490	−1.16965	−0.584826	+	0.811158i	$0.698837\pi$
$524$	18.9373	0.827278
$525$	0	0
$526$	−19.8745	−0.866570
$527$	−7.29150	−0.317623
$528$	0	0
$529$	−20.2915	−0.882239
$530$	0	0
$531$	0	0
$532$	−6.29150	−0.272771
$533$	33.8745	1.46727
$534$	0	0
$535$	0	0
$536$	−4.93725	−0.213257
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−5.29150	−0.227921
$540$	0	0
$541$	−13.6458	−0.586677	−0.293338	−	0.956009i	$-0.594766\pi$
$541$	−13.6458	−0.586677	−0.293338	+	0.956009i	$0.594766\pi$
$542$	15.2915	0.656826
$543$	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	$0.5\pi$
$548$	−7.64575	−0.326610
$549$	0	0
$550$	0	0
$551$	−66.9778	−2.85335
$552$	0	0
$553$	−15.9373	−0.677721
$554$	14.7085	0.624904
$555$	0	0
$556$	4.58301	0.194363
$557$	4.52026	0.191530	0.0957648	−	0.995404i	$-0.469470\pi$
$557$	4.52026	0.191530	0.0957648	+	0.995404i	$0.469470\pi$
$558$	0	0
$559$	−1.64575	−0.0696079
$560$	0	0
$561$	0	0
$562$	−13.5203	−0.570318
$563$	23.0627	0.971979	0.485989	−	0.873965i	$-0.338459\pi$
$563$	23.0627	0.971979	0.485989	+	0.873965i	$0.338459\pi$
$564$	0	0
$565$	0	0
$566$	12.1660	0.511376
$567$	0	0
$568$	−11.2915	−0.473781
$569$	37.1660	1.55808	0.779040	−	0.626974i	$-0.215707\pi$
$569$	37.1660	1.55808	0.779040	+	0.626974i	$0.215707\pi$
$570$	0	0
$571$	12.5830	0.526582	0.263291	−	0.964716i	$-0.415192\pi$
$571$	12.5830	0.526582	0.263291	+	0.964716i	$0.415192\pi$
$572$	24.5830	1.02787
$573$	0	0
$574$	7.29150	0.304341
$575$	0	0
$576$	0	0
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	−3.64575	−0.151251
$582$	0	0
$583$	14.0000	0.579821
$584$	5.29150	0.218964
$585$	0	0
$586$	−31.5203	−1.30209
$587$	−31.5203	−1.30098	−0.650490	−	0.759515i	$-0.725436\pi$
$587$	−31.5203	−1.30098	−0.650490	+	0.759515i	$0.725436\pi$
$588$	0	0
$589$	45.8745	1.89023
$590$	0	0
$591$	0	0
$592$	−9.64575	−0.396438
$593$	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
$596$	−14.5203	−0.594773
$597$	0	0
$598$	−7.64575	−0.312658
$599$	38.8118	1.58581	0.792903	−	0.609348i	$-0.208569\pi$
$599$	38.8118	1.58581	0.792903	+	0.609348i	$0.208569\pi$
$600$	0	0
$601$	−15.2915	−0.623753	−0.311877	−	0.950123i	$-0.600958\pi$
$601$	−15.2915	−0.623753	−0.311877	+	0.950123i	$0.600958\pi$
$602$	−0.354249	−0.0144381
$603$	0	0
$604$	13.2288	0.538270
$605$	0	0
$606$	0	0
$607$	−0.811762	−0.0329484	−0.0164742	−	0.999864i	$-0.505244\pi$
$607$	−0.811762	−0.0329484	−0.0164742	+	0.999864i	$0.505244\pi$
$608$	6.29150	0.255154
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	22.9373	0.926427	0.463213	−	0.886247i	$-0.346696\pi$
$613$	22.9373	0.926427	0.463213	+	0.886247i	$0.346696\pi$
$614$	−17.2915	−0.697828
$615$	0	0
$616$	5.29150	0.213201
$617$	5.06275	0.203818	0.101909	−	0.994794i	$-0.467505\pi$
$617$	5.06275	0.203818	0.101909	+	0.994794i	$0.467505\pi$
$618$	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
$622$	21.2915	0.853711
$623$	1.00000	0.0400642
$624$	0	0
$625$	0	0
$626$	−12.2288	−0.488759
$627$	0	0
$628$	6.64575	0.265194
$629$	−9.64575	−0.384601
$630$	0	0
$631$	−27.8118	−1.10717	−0.553584	−	0.832793i	$-0.686740\pi$
$631$	−27.8118	−1.10717	−0.553584	+	0.832793i	$0.686740\pi$
$632$	15.9373	0.633950
$633$	0	0
$634$	29.9373	1.18896
$635$	0	0
$636$	0	0
$637$	−4.64575	−0.184071
$638$	56.3320	2.23021
$639$	0	0
$640$	0	0
$641$	35.2915	1.39393	0.696965	−	0.717105i	$-0.254533\pi$
$641$	35.2915	1.39393	0.696965	+	0.717105i	$0.254533\pi$
$642$	0	0
$643$	2.29150	0.0903680	0.0451840	−	0.998979i	$-0.485613\pi$
$643$	2.29150	0.0903680	0.0451840	+	0.998979i	$0.485613\pi$
$644$	−1.64575	−0.0648517
$645$	0	0
$646$	6.29150	0.247536
$647$	−11.3542	−0.446382	−0.223191	−	0.974775i	$-0.571647\pi$
$647$	−11.3542	−0.446382	−0.223191	+	0.974775i	$0.571647\pi$
$648$	0	0
$649$	−70.3320	−2.76077
$650$	0	0
$651$	0	0
$652$	19.8745	0.778346
$653$	27.8118	1.08836	0.544179	−	0.838969i	$-0.316841\pi$
$653$	27.8118	1.08836	0.544179	+	0.838969i	$0.316841\pi$
$654$	0	0
$655$	0	0
$656$	−7.29150	−0.284685
$657$	0	0
$658$	0.645751	0.0251740
$659$	−5.58301	−0.217483	−0.108742	−	0.994070i	$-0.534682\pi$
$659$	−5.58301	−0.217483	−0.108742	+	0.994070i	$0.534682\pi$
$660$	0	0
$661$	−3.16601	−0.123144	−0.0615718	−	0.998103i	$-0.519611\pi$
$661$	−3.16601	−0.123144	−0.0615718	+	0.998103i	$0.519611\pi$
$662$	9.64575	0.374893
$663$	0	0
$664$	3.64575	0.141483
$665$	0	0
$666$	0	0
$667$	−17.5203	−0.678387
$668$	25.1660	0.973702
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	26.5830	1.02470	0.512350	−	0.858777i	$-0.328775\pi$
$673$	26.5830	1.02470	0.512350	+	0.858777i	$0.328775\pi$
$674$	26.1660	1.00788
$675$	0	0
$676$	8.58301	0.330116
$677$	9.29150	0.357101	0.178551	−	0.983931i	$-0.442859\pi$
$677$	9.29150	0.357101	0.178551	+	0.983931i	$0.442859\pi$
$678$	0	0
$679$	−12.9373	−0.496486
$680$	0	0
$681$	0	0
$682$	−38.5830	−1.47742
$683$	−2.58301	−0.0988359	−0.0494180	−	0.998778i	$-0.515737\pi$
$683$	−2.58301	−0.0988359	−0.0494180	+	0.998778i	$0.515737\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0.354249	0.0135056
$689$	12.2915	0.468269
$690$	0	0
$691$	−22.7085	−0.863872	−0.431936	−	0.901904i	$-0.642169\pi$
$691$	−22.7085	−0.863872	−0.431936	+	0.901904i	$0.642169\pi$
$692$	−3.64575	−0.138591
$693$	0	0
$694$	9.70850	0.368530
$695$	0	0
$696$	0	0
$697$	−7.29150	−0.276185
$698$	9.35425	0.354064
$699$	0	0
$700$	0	0
$701$	13.2915	0.502013	0.251007	−	0.967985i	$-0.419238\pi$
$701$	13.2915	0.502013	0.251007	+	0.967985i	$0.419238\pi$
$702$	0	0
$703$	60.6863	2.28883
$704$	−5.29150	−0.199431
$705$	0	0
$706$	16.4575	0.619387
$707$	16.9373	0.636991
$708$	0	0
$709$	1.87451	0.0703986	0.0351993	−	0.999380i	$-0.488793\pi$
$709$	1.87451	0.0703986	0.0351993	+	0.999380i	$0.488793\pi$
$710$	0	0
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	12.0000	0.449404
$714$	0	0
$715$	0	0
$716$	3.58301	0.133903
$717$	0	0
$718$	−32.5830	−1.21599
$719$	−28.6458	−1.06831	−0.534153	−	0.845388i	$-0.679369\pi$
$719$	−28.6458	−1.06831	−0.534153	+	0.845388i	$0.679369\pi$
$720$	0	0
$721$	17.8745	0.665681
$722$	−20.5830	−0.766020
$723$	0	0
$724$	17.3542	0.644966
$725$	0	0
$726$	0	0
$727$	25.6458	0.951148	0.475574	−	0.879676i	$-0.342240\pi$
$727$	25.6458	0.951148	0.475574	+	0.879676i	$0.342240\pi$
$728$	4.64575	0.172183
$729$	0	0
$730$	0	0
$731$	0.354249	0.0131024
$732$	0	0
$733$	37.8118	1.39661	0.698305	−	0.715801i	$-0.253938\pi$
$733$	37.8118	1.39661	0.698305	+	0.715801i	$0.253938\pi$
$734$	−0.937254	−0.0345947
$735$	0	0
$736$	1.64575	0.0606632
$737$	−26.1255	−0.962345
$738$	0	0
$739$	48.3320	1.77792	0.888961	−	0.457983i	$-0.151428\pi$
$739$	48.3320	1.77792	0.888961	+	0.457983i	$0.151428\pi$
$740$	0	0
$741$	0	0
$742$	2.64575	0.0971286
$743$	−49.7490	−1.82511	−0.912557	−	0.408949i	$-0.865895\pi$
$743$	−49.7490	−1.82511	−0.912557	+	0.408949i	$0.865895\pi$
$744$	0	0
$745$	0	0
$746$	23.8745	0.874108
$747$	0	0
$748$	−5.29150	−0.193476
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−22.5830	−0.824066	−0.412033	−	0.911169i	$-0.635181\pi$
$751$	−22.5830	−0.824066	−0.412033	+	0.911169i	$0.635181\pi$
$752$	−0.645751	−0.0235481
$753$	0	0
$754$	49.4575	1.80114
$755$	0	0
$756$	0	0
$757$	28.8118	1.04718	0.523591	−	0.851970i	$-0.324592\pi$
$757$	28.8118	1.04718	0.523591	+	0.851970i	$0.324592\pi$
$758$	2.22876	0.0809521
$759$	0	0
$760$	0	0
$761$	1.83399	0.0664821	0.0332410	−	0.999447i	$-0.489417\pi$
$761$	1.83399	0.0664821	0.0332410	+	0.999447i	$0.489417\pi$
$762$	0	0
$763$	−1.64575	−0.0595802
$764$	−6.58301	−0.238165
$765$	0	0
$766$	−27.2915	−0.986082
$767$	−61.7490	−2.22963
$768$	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
$772$	16.8745	0.607327
$773$	25.1660	0.905158	0.452579	−	0.891724i	$-0.350504\pi$
$773$	25.1660	0.905158	0.452579	+	0.891724i	$0.350504\pi$
$774$	0	0
$775$	0	0
$776$	12.9373	0.464420
$777$	0	0
$778$	34.3948	1.23311
$779$	45.8745	1.64362
$780$	0	0
$781$	−59.7490	−2.13799
$782$	1.64575	0.0588519
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	19.4170	0.692141	0.346071	−	0.938208i	$-0.387516\pi$
$787$	19.4170	0.692141	0.346071	+	0.938208i	$0.387516\pi$
$788$	−18.6458	−0.664227
$789$	0	0
$790$	0	0
$791$	−8.58301	−0.305177
$792$	0	0
$793$	−12.2915	−0.436484
$794$	−4.52026	−0.160418
$795$	0	0
$796$	19.5203	0.691877
$797$	3.18824	0.112933	0.0564666	−	0.998404i	$-0.482017\pi$
$797$	3.18824	0.112933	0.0564666	+	0.998404i	$0.482017\pi$
$798$	0	0
$799$	−0.645751	−0.0228450
$800$	0	0
$801$	0	0
$802$	24.9373	0.880565
$803$	28.0000	0.988099
$804$	0	0
$805$	0	0
$806$	−33.8745	−1.19318
$807$	0	0
$808$	−16.9373	−0.595851
$809$	−24.5830	−0.864292	−0.432146	−	0.901804i	$-0.642244\pi$
$809$	−24.5830	−0.864292	−0.432146	+	0.901804i	$0.642244\pi$
$810$	0	0
$811$	15.1660	0.532551	0.266275	−	0.963897i	$-0.414207\pi$
$811$	15.1660	0.532551	0.266275	+	0.963897i	$0.414207\pi$
$812$	10.6458	0.373593
$813$	0	0
$814$	−51.0405	−1.78897
$815$	0	0
$816$	0	0
$817$	−2.22876	−0.0779743
$818$	8.22876	0.287712
$819$	0	0
$820$	0	0
$821$	−47.2288	−1.64829	−0.824147	−	0.566375i	$-0.808345\pi$
$821$	−47.2288	−1.64829	−0.824147	+	0.566375i	$0.808345\pi$
$822$	0	0
$823$	−21.0405	−0.733426	−0.366713	−	0.930334i	$-0.619517\pi$
$823$	−21.0405	−0.733426	−0.366713	+	0.930334i	$0.619517\pi$
$824$	−17.8745	−0.622688
$825$	0	0
$826$	−13.2915	−0.462471
$827$	−22.4170	−0.779515	−0.389758	−	0.920917i	$-0.627441\pi$
$827$	−22.4170	−0.779515	−0.389758	+	0.920917i	$0.627441\pi$
$828$	0	0
$829$	12.6458	0.439205	0.219603	−	0.975589i	$-0.429524\pi$
$829$	12.6458	0.439205	0.219603	+	0.975589i	$0.429524\pi$
$830$	0	0
$831$	0	0
$832$	−4.64575	−0.161062
$833$	1.00000	0.0346479
$834$	0	0
$835$	0	0
$836$	33.2915	1.15141
$837$	0	0
$838$	−22.3542	−0.772215
$839$	2.18824	0.0755464	0.0377732	−	0.999286i	$-0.487974\pi$
$839$	2.18824	0.0755464	0.0377732	+	0.999286i	$0.487974\pi$
$840$	0	0
$841$	84.3320	2.90800
$842$	4.58301	0.157941
$843$	0	0
$844$	25.2915	0.870569
$845$	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
$853$	30.3320	1.03855	0.519274	−	0.854608i	$-0.326202\pi$
$853$	30.3320	1.03855	0.519274	+	0.854608i	$0.326202\pi$
$854$	−2.64575	−0.0905357
$855$	0	0
$856$	6.00000	0.205076
$857$	23.7085	0.809867	0.404933	−	0.914346i	$-0.367295\pi$
$857$	23.7085	0.809867	0.404933	+	0.914346i	$0.367295\pi$
$858$	0	0
$859$	50.3320	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$859$	50.3320	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$860$	0	0
$861$	0	0
$862$	−33.8745	−1.15377
$863$	−35.6458	−1.21340	−0.606698	−	0.794933i	$-0.707506\pi$
$863$	−35.6458	−1.21340	−0.606698	+	0.794933i	$0.707506\pi$
$864$	0	0
$865$	0	0
$866$	−0.708497	−0.0240757
$867$	0	0
$868$	−7.29150	−0.247490
$869$	84.3320	2.86077
$870$	0	0
$871$	−22.9373	−0.777199
$872$	1.64575	0.0557322
$873$	0	0
$874$	−10.3542	−0.350238
$875$	0	0
$876$	0	0
$877$	25.8745	0.873720	0.436860	−	0.899529i	$-0.356091\pi$
$877$	25.8745	0.873720	0.436860	+	0.899529i	$0.356091\pi$
$878$	15.0627	0.508343
$879$	0	0
$880$	0	0
$881$	−46.8745	−1.57924	−0.789621	−	0.613595i	$-0.789723\pi$
$881$	−46.8745	−1.57924	−0.789621	+	0.613595i	$0.789723\pi$
$882$	0	0
$883$	20.7085	0.696896	0.348448	−	0.937328i	$-0.386709\pi$
$883$	20.7085	0.696896	0.348448	+	0.937328i	$0.386709\pi$
$884$	−4.64575	−0.156254
$885$	0	0
$886$	12.4170	0.417157
$887$	18.1255	0.608594	0.304297	−	0.952577i	$-0.401578\pi$
$887$	18.1255	0.608594	0.304297	+	0.952577i	$0.401578\pi$
$888$	0	0
$889$	−14.6458	−0.491203
$890$	0	0
$891$	0	0
$892$	−21.5203	−0.720552
$893$	4.06275	0.135955
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−8.22876	−0.274597
$899$	−77.6235	−2.58889
$900$	0	0
$901$	−2.64575	−0.0881428
$902$	−38.5830	−1.28467
$903$	0	0
$904$	8.58301	0.285467
$905$	0	0
$906$	0	0
$907$	12.7085	0.421979	0.210989	−	0.977488i	$-0.432331\pi$
$907$	12.7085	0.421979	0.210989	+	0.977488i	$0.432331\pi$
$908$	−6.35425	−0.210873
$909$	0	0
$910$	0	0
$911$	13.1660	0.436209	0.218105	−	0.975925i	$-0.430013\pi$
$911$	13.1660	0.436209	0.218105	+	0.975925i	$0.430013\pi$
$912$	0	0
$913$	19.2915	0.638456
$914$	−38.8745	−1.28585
$915$	0	0
$916$	12.5203	0.413681
$917$	18.9373	0.625363
$918$	0	0
$919$	49.0405	1.61770	0.808849	−	0.588017i	$-0.200091\pi$
$919$	49.0405	1.61770	0.808849	+	0.588017i	$0.200091\pi$
$920$	0	0
$921$	0	0
$922$	−3.52026	−0.115934
$923$	−52.4575	−1.72666
$924$	0	0
$925$	0	0
$926$	6.64575	0.218393
$927$	0	0
$928$	−10.6458	−0.349464
$929$	10.8745	0.356781	0.178391	−	0.983960i	$-0.442911\pi$
$929$	10.8745	0.356781	0.178391	+	0.983960i	$0.442911\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	1.64575	0.0539084
$933$	0	0
$934$	28.7085	0.939371
$935$	0	0
$936$	0	0
$937$	−16.2288	−0.530170	−0.265085	−	0.964225i	$-0.585400\pi$
$937$	−16.2288	−0.530170	−0.265085	+	0.964225i	$0.585400\pi$
$938$	−4.93725	−0.161207
$939$	0	0
$940$	0	0
$941$	15.6458	0.510037	0.255018	−	0.966936i	$-0.417918\pi$
$941$	15.6458	0.510037	0.255018	+	0.966936i	$0.417918\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	13.2915	0.432602
$945$	0	0
$946$	1.87451	0.0609455
$947$	18.7085	0.607944	0.303972	−	0.952681i	$-0.401687\pi$
$947$	18.7085	0.607944	0.303972	+	0.952681i	$0.401687\pi$
$948$	0	0
$949$	24.5830	0.797998
$950$	0	0
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	−18.1033	−0.586422	−0.293211	−	0.956048i	$-0.594724\pi$
$953$	−18.1033	−0.586422	−0.293211	+	0.956048i	$0.594724\pi$
$954$	0	0
$955$	0	0
$956$	14.3542	0.464250
$957$	0	0
$958$	−7.35425	−0.237605
$959$	−7.64575	−0.246894
$960$	0	0
$961$	22.1660	0.715033
$962$	−44.8118	−1.44479
$963$	0	0
$964$	−12.9373	−0.416681
$965$	0	0
$966$	0	0
$967$	−45.8118	−1.47321	−0.736603	−	0.676325i	$-0.763572\pi$
$967$	−45.8118	−1.47321	−0.736603	+	0.676325i	$0.763572\pi$
$968$	−17.0000	−0.546401
$969$	0	0
$970$	0	0
$971$	3.77124	0.121025	0.0605125	−	0.998167i	$-0.480727\pi$
$971$	3.77124	0.121025	0.0605125	+	0.998167i	$0.480727\pi$
$972$	0	0
$973$	4.58301	0.146924
$974$	−0.708497	−0.0227017
$975$	0	0
$976$	2.64575	0.0846884
$977$	22.9373	0.733828	0.366914	−	0.930255i	$-0.380414\pi$
$977$	22.9373	0.733828	0.366914	+	0.930255i	$0.380414\pi$
$978$	0	0
$979$	−5.29150	−0.169117
$980$	0	0
$981$	0	0
$982$	−12.8745	−0.410842
$983$	13.4797	0.429937	0.214968	−	0.976621i	$-0.431035\pi$
$983$	13.4797	0.429937	0.214968	+	0.976621i	$0.431035\pi$
$984$	0	0
$985$	0	0
$986$	−10.6458	−0.339030
$987$	0	0
$988$	29.2288	0.929891
$989$	−0.583005	−0.0185385
$990$	0	0
$991$	−33.9373	−1.07805	−0.539026	−	0.842289i	$-0.681208\pi$
$991$	−33.9373	−1.07805	−0.539026	+	0.842289i	$0.681208\pi$
$992$	7.29150	0.231505
$993$	0	0
$994$	−11.2915	−0.358145
$995$	0	0
$996$	0	0
$997$	50.4575	1.59801	0.799003	−	0.601327i	$-0.205361\pi$
$997$	50.4575	1.59801	0.799003	+	0.601327i	$0.205361\pi$
$998$	−13.0627	−0.413494
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -5.29150 q^{11} -4.64575 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.29150 q^{19} +5.29150 q^{22} -1.64575 q^{23} +4.64575 q^{26} +1.00000 q^{28} +10.6458 q^{29} -7.29150 q^{31} -1.00000 q^{32} -1.00000 q^{34} -9.64575 q^{37} +6.29150 q^{38} -7.29150 q^{41} +0.354249 q^{43} -5.29150 q^{44} +1.64575 q^{46} -0.645751 q^{47} +1.00000 q^{49} -4.64575 q^{52} -2.64575 q^{53} -1.00000 q^{56} -10.6458 q^{58} +13.2915 q^{59} +2.64575 q^{61} +7.29150 q^{62} +1.00000 q^{64} +4.93725 q^{67} +1.00000 q^{68} +11.2915 q^{71} -5.29150 q^{73} +9.64575 q^{74} -6.29150 q^{76} -5.29150 q^{77} -15.9373 q^{79} +7.29150 q^{82} -3.64575 q^{83} -0.354249 q^{86} +5.29150 q^{88} +1.00000 q^{89} -4.64575 q^{91} -1.64575 q^{92} +0.645751 q^{94} -12.9373 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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

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