LMFDB - Embedded newform 7623.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} -3.00000 q^{8} +2.85410 q^{10} -1.23607 q^{13} +1.00000 q^{14} -1.00000 q^{16} -7.85410 q^{17} +2.61803 q^{19} -2.85410 q^{20} +3.09017 q^{23} +3.14590 q^{25} -1.23607 q^{26} -1.00000 q^{28} +2.00000 q^{29} -7.32624 q^{31} +5.00000 q^{32} -7.85410 q^{34} +2.85410 q^{35} -12.0902 q^{37} +2.61803 q^{38} -8.56231 q^{40} +10.8541 q^{41} -1.23607 q^{43} +3.09017 q^{46} -2.00000 q^{47} +1.00000 q^{49} +3.14590 q^{50} +1.23607 q^{52} -12.1803 q^{53} -3.00000 q^{56} +2.00000 q^{58} -6.76393 q^{59} +8.94427 q^{61} -7.32624 q^{62} +7.00000 q^{64} -3.52786 q^{65} +8.00000 q^{67} +7.85410 q^{68} +2.85410 q^{70} -8.94427 q^{71} -5.23607 q^{73} -12.0902 q^{74} -2.61803 q^{76} -14.0000 q^{79} -2.85410 q^{80} +10.8541 q^{82} -2.94427 q^{83} -22.4164 q^{85} -1.23607 q^{86} -1.09017 q^{89} -1.23607 q^{91} -3.09017 q^{92} -2.00000 q^{94} +7.47214 q^{95} -3.52786 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 6 q^{8} - q^{10} + 2 q^{13} + 2 q^{14} - 2 q^{16} - 9 q^{17} + 3 q^{19} + q^{20} - 5 q^{23} + 13 q^{25} + 2 q^{26} - 2 q^{28} + 4 q^{29} + q^{31} + 10 q^{32} - 9 q^{34} - q^{35} - 13 q^{37} + 3 q^{38} + 3 q^{40} + 15 q^{41} + 2 q^{43} - 5 q^{46} - 4 q^{47} + 2 q^{49} + 13 q^{50} - 2 q^{52} - 2 q^{53} - 6 q^{56} + 4 q^{58} - 18 q^{59} + q^{62} + 14 q^{64} - 16 q^{65} + 16 q^{67} + 9 q^{68} - q^{70} - 6 q^{73} - 13 q^{74} - 3 q^{76} - 28 q^{79} + q^{80} + 15 q^{82} + 12 q^{83} - 18 q^{85} + 2 q^{86} + 9 q^{89} + 2 q^{91} + 5 q^{92} - 4 q^{94} + 6 q^{95} - 16 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 - 6 * q^8 - q^10 + 2 * q^13 + 2 * q^14 - 2 * q^16 - 9 * q^17 + 3 * q^19 + q^20 - 5 * q^23 + 13 * q^25 + 2 * q^26 - 2 * q^28 + 4 * q^29 + q^31 + 10 * q^32 - 9 * q^34 - q^35 - 13 * q^37 + 3 * q^38 + 3 * q^40 + 15 * q^41 + 2 * q^43 - 5 * q^46 - 4 * q^47 + 2 * q^49 + 13 * q^50 - 2 * q^52 - 2 * q^53 - 6 * q^56 + 4 * q^58 - 18 * q^59 + q^62 + 14 * q^64 - 16 * q^65 + 16 * q^67 + 9 * q^68 - q^70 - 6 * q^73 - 13 * q^74 - 3 * q^76 - 28 * q^79 + q^80 + 15 * q^82 + 12 * q^83 - 18 * q^85 + 2 * q^86 + 9 * q^89 + 2 * q^91 + 5 * q^92 - 4 * q^94 + 6 * q^95 - 16 * 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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.85410	0.902546
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	0	0
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	−2.85410	−0.638197
$21$	0	0
$22$	0	0
$23$	3.09017	0.644345	0.322172	−	0.946681i	$-0.395587\pi$
$23$	3.09017	0.644345	0.322172	+	0.946681i	$0.395587\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	−1.23607	−0.242413
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−7.32624	−1.31583	−0.657916	−	0.753092i	$-0.728562\pi$
$31$	−7.32624	−1.31583	−0.657916	+	0.753092i	$0.728562\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−7.85410	−1.34697
$35$	2.85410	0.482431
$36$	0	0
$37$	−12.0902	−1.98761	−0.993806	−	0.111130i	$-0.964553\pi$
$37$	−12.0902	−1.98761	−0.993806	+	0.111130i	$0.964553\pi$
$38$	2.61803	0.424701
$39$	0	0
$40$	−8.56231	−1.35382
$41$	10.8541	1.69513	0.847563	−	0.530695i	$-0.178069\pi$
$41$	10.8541	1.69513	0.847563	+	0.530695i	$0.178069\pi$
$42$	0	0
$43$	−1.23607	−0.188499	−0.0942493	−	0.995549i	$-0.530045\pi$
$43$	−1.23607	−0.188499	−0.0942493	+	0.995549i	$0.530045\pi$
$44$	0	0
$45$	0	0
$46$	3.09017	0.455621
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.14590	0.444897
$51$	0	0
$52$	1.23607	0.171412
$53$	−12.1803	−1.67310	−0.836549	−	0.547892i	$-0.815431\pi$
$53$	−12.1803	−1.67310	−0.836549	+	0.547892i	$0.815431\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	2.00000	0.262613
$59$	−6.76393	−0.880589	−0.440294	−	0.897853i	$-0.645126\pi$
$59$	−6.76393	−0.880589	−0.440294	+	0.897853i	$0.645126\pi$
$60$	0	0
$61$	8.94427	1.14520	0.572598	−	0.819836i	$-0.305935\pi$
$61$	8.94427	1.14520	0.572598	+	0.819836i	$0.305935\pi$
$62$	−7.32624	−0.930433
$63$	0	0
$64$	7.00000	0.875000
$65$	−3.52786	−0.437578
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	7.85410	0.952450
$69$	0	0
$70$	2.85410	0.341130
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	−5.23607	−0.612835	−0.306418	−	0.951897i	$-0.599130\pi$
$73$	−5.23607	−0.612835	−0.306418	+	0.951897i	$0.599130\pi$
$74$	−12.0902	−1.40545
$75$	0	0
$76$	−2.61803	−0.300309
$77$	0	0
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−2.85410	−0.319098
$81$	0	0
$82$	10.8541	1.19864
$83$	−2.94427	−0.323176	−0.161588	−	0.986858i	$-0.551662\pi$
$83$	−2.94427	−0.323176	−0.161588	+	0.986858i	$0.551662\pi$
$84$	0	0
$85$	−22.4164	−2.43140
$86$	−1.23607	−0.133289
$87$	0	0
$88$	0	0
$89$	−1.09017	−0.115558	−0.0577789	−	0.998329i	$-0.518402\pi$
$89$	−1.09017	−0.115558	−0.0577789	+	0.998329i	$0.518402\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	−3.09017	−0.322172
$93$	0	0
$94$	−2.00000	−0.206284
$95$	7.47214	0.766625
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−3.14590	−0.314590
$101$	2.09017	0.207980	0.103990	−	0.994578i	$-0.466839\pi$
$101$	2.09017	0.207980	0.103990	+	0.994578i	$0.466839\pi$
$102$	0	0
$103$	−10.3262	−1.01747	−0.508737	−	0.860922i	$-0.669888\pi$
$103$	−10.3262	−1.01747	−0.508737	+	0.860922i	$0.669888\pi$
$104$	3.70820	0.363619
$105$	0	0
$106$	−12.1803	−1.18306
$107$	−4.90983	−0.474651	−0.237326	−	0.971430i	$-0.576271\pi$
$107$	−4.90983	−0.474651	−0.237326	+	0.971430i	$0.576271\pi$
$108$	0	0
$109$	10.5623	1.01169	0.505843	−	0.862626i	$-0.331182\pi$
$109$	10.5623	1.01169	0.505843	+	0.862626i	$0.331182\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−0.763932	−0.0718647	−0.0359323	−	0.999354i	$-0.511440\pi$
$113$	−0.763932	−0.0718647	−0.0359323	+	0.999354i	$0.511440\pi$
$114$	0	0
$115$	8.81966	0.822438
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−6.76393	−0.622670
$119$	−7.85410	−0.719984
$120$	0	0
$121$	0	0
$122$	8.94427	0.809776
$123$	0	0
$124$	7.32624	0.657916
$125$	−5.29180	−0.473313
$126$	0	0
$127$	−3.23607	−0.287155	−0.143577	−	0.989639i	$-0.545861\pi$
$127$	−3.23607	−0.287155	−0.143577	+	0.989639i	$0.545861\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−3.52786	−0.309414
$131$	−2.29180	−0.200235	−0.100118	−	0.994976i	$-0.531922\pi$
$131$	−2.29180	−0.200235	−0.100118	+	0.994976i	$0.531922\pi$
$132$	0	0
$133$	2.61803	0.227012
$134$	8.00000	0.691095
$135$	0	0
$136$	23.5623	2.02045
$137$	−6.76393	−0.577882	−0.288941	−	0.957347i	$-0.593303\pi$
$137$	−6.76393	−0.577882	−0.288941	+	0.957347i	$0.593303\pi$
$138$	0	0
$139$	4.79837	0.406993	0.203496	−	0.979076i	$-0.434769\pi$
$139$	4.79837	0.406993	0.203496	+	0.979076i	$0.434769\pi$
$140$	−2.85410	−0.241216
$141$	0	0
$142$	−8.94427	−0.750587
$143$	0	0
$144$	0	0
$145$	5.70820	0.474041
$146$	−5.23607	−0.433340
$147$	0	0
$148$	12.0902	0.993806
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−7.85410	−0.637052
$153$	0	0
$154$	0	0
$155$	−20.9098	−1.67952
$156$	0	0
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	−14.0000	−1.11378
$159$	0	0
$160$	14.2705	1.12818
$161$	3.09017	0.243540
$162$	0	0
$163$	−5.41641	−0.424246	−0.212123	−	0.977243i	$-0.568038\pi$
$163$	−5.41641	−0.424246	−0.212123	+	0.977243i	$0.568038\pi$
$164$	−10.8541	−0.847563
$165$	0	0
$166$	−2.94427	−0.228520
$167$	−8.18034	−0.633014	−0.316507	−	0.948590i	$-0.602510\pi$
$167$	−8.18034	−0.633014	−0.316507	+	0.948590i	$0.602510\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	−22.4164	−1.71926
$171$	0	0
$172$	1.23607	0.0942493
$173$	−19.8541	−1.50948	−0.754740	−	0.656024i	$-0.772237\pi$
$173$	−19.8541	−1.50948	−0.754740	+	0.656024i	$0.772237\pi$
$174$	0	0
$175$	3.14590	0.237808
$176$	0	0
$177$	0	0
$178$	−1.09017	−0.0817117
$179$	−8.38197	−0.626498	−0.313249	−	0.949671i	$-0.601417\pi$
$179$	−8.38197	−0.626498	−0.313249	+	0.949671i	$0.601417\pi$
$180$	0	0
$181$	−18.9443	−1.40812	−0.704058	−	0.710142i	$-0.748631\pi$
$181$	−18.9443	−1.40812	−0.704058	+	0.710142i	$0.748631\pi$
$182$	−1.23607	−0.0916235
$183$	0	0
$184$	−9.27051	−0.683431
$185$	−34.5066	−2.53697
$186$	0	0
$187$	0	0
$188$	2.00000	0.145865
$189$	0	0
$190$	7.47214	0.542086
$191$	18.2705	1.32201	0.661004	−	0.750383i	$-0.270131\pi$
$191$	18.2705	1.32201	0.661004	+	0.750383i	$0.270131\pi$
$192$	0	0
$193$	18.3820	1.32316	0.661581	−	0.749873i	$-0.269886\pi$
$193$	18.3820	1.32316	0.661581	+	0.749873i	$0.269886\pi$
$194$	−3.52786	−0.253286
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−4.00000	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$198$	0	0
$199$	−2.61803	−0.185588	−0.0927938	−	0.995685i	$-0.529580\pi$
$199$	−2.61803	−0.185588	−0.0927938	+	0.995685i	$0.529580\pi$
$200$	−9.43769	−0.667346
$201$	0	0
$202$	2.09017	0.147064
$203$	2.00000	0.140372
$204$	0	0
$205$	30.9787	2.16365
$206$	−10.3262	−0.719463
$207$	0	0
$208$	1.23607	0.0857059
$209$	0	0
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	12.1803	0.836549
$213$	0	0
$214$	−4.90983	−0.335629
$215$	−3.52786	−0.240598
$216$	0	0
$217$	−7.32624	−0.497337
$218$	10.5623	0.715370
$219$	0	0
$220$	0	0
$221$	9.70820	0.653044
$222$	0	0
$223$	20.7984	1.39276	0.696381	−	0.717672i	$-0.254792\pi$
$223$	20.7984	1.39276	0.696381	+	0.717672i	$0.254792\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	−0.763932	−0.0508160
$227$	−21.4164	−1.42146	−0.710728	−	0.703466i	$-0.751635\pi$
$227$	−21.4164	−1.42146	−0.710728	+	0.703466i	$0.751635\pi$
$228$	0	0
$229$	21.2361	1.40332	0.701659	−	0.712512i	$-0.252443\pi$
$229$	21.2361	1.40332	0.701659	+	0.712512i	$0.252443\pi$
$230$	8.81966	0.581551
$231$	0	0
$232$	−6.00000	−0.393919
$233$	5.70820	0.373957	0.186978	−	0.982364i	$-0.440131\pi$
$233$	5.70820	0.373957	0.186978	+	0.982364i	$0.440131\pi$
$234$	0	0
$235$	−5.70820	−0.372362
$236$	6.76393	0.440294
$237$	0	0
$238$	−7.85410	−0.509106
$239$	1.56231	0.101057	0.0505286	−	0.998723i	$-0.483909\pi$
$239$	1.56231	0.101057	0.0505286	+	0.998723i	$0.483909\pi$
$240$	0	0
$241$	4.94427	0.318489	0.159244	−	0.987239i	$-0.449094\pi$
$241$	4.94427	0.318489	0.159244	+	0.987239i	$0.449094\pi$
$242$	0	0
$243$	0	0
$244$	−8.94427	−0.572598
$245$	2.85410	0.182342
$246$	0	0
$247$	−3.23607	−0.205906
$248$	21.9787	1.39565
$249$	0	0
$250$	−5.29180	−0.334683
$251$	−24.1803	−1.52625	−0.763125	−	0.646251i	$-0.776336\pi$
$251$	−24.1803	−1.52625	−0.763125	+	0.646251i	$0.776336\pi$
$252$	0	0
$253$	0	0
$254$	−3.23607	−0.203049
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−2.61803	−0.163308	−0.0816542	−	0.996661i	$-0.526020\pi$
$257$	−2.61803	−0.163308	−0.0816542	+	0.996661i	$0.526020\pi$
$258$	0	0
$259$	−12.0902	−0.751247
$260$	3.52786	0.218789
$261$	0	0
$262$	−2.29180	−0.141588
$263$	−2.43769	−0.150315	−0.0751573	−	0.997172i	$-0.523946\pi$
$263$	−2.43769	−0.150315	−0.0751573	+	0.997172i	$0.523946\pi$
$264$	0	0
$265$	−34.7639	−2.13553
$266$	2.61803	0.160522
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−27.8885	−1.70039	−0.850197	−	0.526464i	$-0.823517\pi$
$269$	−27.8885	−1.70039	−0.850197	+	0.526464i	$0.823517\pi$
$270$	0	0
$271$	24.0902	1.46337	0.731687	−	0.681641i	$-0.238733\pi$
$271$	24.0902	1.46337	0.731687	+	0.681641i	$0.238733\pi$
$272$	7.85410	0.476225
$273$	0	0
$274$	−6.76393	−0.408624
$275$	0	0
$276$	0	0
$277$	−10.4377	−0.627140	−0.313570	−	0.949565i	$-0.601525\pi$
$277$	−10.4377	−0.627140	−0.313570	+	0.949565i	$0.601525\pi$
$278$	4.79837	0.287787
$279$	0	0
$280$	−8.56231	−0.511696
$281$	−1.05573	−0.0629795	−0.0314897	−	0.999504i	$-0.510025\pi$
$281$	−1.05573	−0.0629795	−0.0314897	+	0.999504i	$0.510025\pi$
$282$	0	0
$283$	13.4377	0.798788	0.399394	−	0.916779i	$-0.369221\pi$
$283$	13.4377	0.798788	0.399394	+	0.916779i	$0.369221\pi$
$284$	8.94427	0.530745
$285$	0	0
$286$	0	0
$287$	10.8541	0.640697
$288$	0	0
$289$	44.6869	2.62864
$290$	5.70820	0.335197
$291$	0	0
$292$	5.23607	0.306418
$293$	−19.7984	−1.15663	−0.578317	−	0.815812i	$-0.696290\pi$
$293$	−19.7984	−1.15663	−0.578317	+	0.815812i	$0.696290\pi$
$294$	0	0
$295$	−19.3050	−1.12398
$296$	36.2705	2.10818
$297$	0	0
$298$	20.0000	1.15857
$299$	−3.81966	−0.220897
$300$	0	0
$301$	−1.23607	−0.0712458
$302$	−2.00000	−0.115087
$303$	0	0
$304$	−2.61803	−0.150155
$305$	25.5279	1.46172
$306$	0	0
$307$	−24.2705	−1.38519	−0.692596	−	0.721326i	$-0.743533\pi$
$307$	−24.2705	−1.38519	−0.692596	+	0.721326i	$0.743533\pi$
$308$	0	0
$309$	0	0
$310$	−20.9098	−1.18760
$311$	0.763932	0.0433186	0.0216593	−	0.999765i	$-0.493105\pi$
$311$	0.763932	0.0433186	0.0216593	+	0.999765i	$0.493105\pi$
$312$	0	0
$313$	−9.81966	−0.555040	−0.277520	−	0.960720i	$-0.589512\pi$
$313$	−9.81966	−0.555040	−0.277520	+	0.960720i	$0.589512\pi$
$314$	9.70820	0.547866
$315$	0	0
$316$	14.0000	0.787562
$317$	27.5967	1.54999	0.774994	−	0.631969i	$-0.217753\pi$
$317$	27.5967	1.54999	0.774994	+	0.631969i	$0.217753\pi$
$318$	0	0
$319$	0	0
$320$	19.9787	1.11684
$321$	0	0
$322$	3.09017	0.172208
$323$	−20.5623	−1.14412
$324$	0	0
$325$	−3.88854	−0.215698
$326$	−5.41641	−0.299987
$327$	0	0
$328$	−32.5623	−1.79795
$329$	−2.00000	−0.110264
$330$	0	0
$331$	30.3607	1.66877	0.834387	−	0.551179i	$-0.185822\pi$
$331$	30.3607	1.66877	0.834387	+	0.551179i	$0.185822\pi$
$332$	2.94427	0.161588
$333$	0	0
$334$	−8.18034	−0.447608
$335$	22.8328	1.24749
$336$	0	0
$337$	−27.1459	−1.47873	−0.739366	−	0.673304i	$-0.764874\pi$
$337$	−27.1459	−1.47873	−0.739366	+	0.673304i	$0.764874\pi$
$338$	−11.4721	−0.624002
$339$	0	0
$340$	22.4164	1.21570
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.70820	0.199933
$345$	0	0
$346$	−19.8541	−1.06736
$347$	12.3820	0.664699	0.332349	−	0.943156i	$-0.392159\pi$
$347$	12.3820	0.664699	0.332349	+	0.943156i	$0.392159\pi$
$348$	0	0
$349$	22.7639	1.21853	0.609263	−	0.792968i	$-0.291466\pi$
$349$	22.7639	1.21853	0.609263	+	0.792968i	$0.291466\pi$
$350$	3.14590	0.168155
$351$	0	0
$352$	0	0
$353$	−12.4721	−0.663825	−0.331912	−	0.943310i	$-0.607694\pi$
$353$	−12.4721	−0.663825	−0.331912	+	0.943310i	$0.607694\pi$
$354$	0	0
$355$	−25.5279	−1.35488
$356$	1.09017	0.0577789
$357$	0	0
$358$	−8.38197	−0.443001
$359$	−7.03444	−0.371264	−0.185632	−	0.982619i	$-0.559433\pi$
$359$	−7.03444	−0.371264	−0.185632	+	0.982619i	$0.559433\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	−18.9443	−0.995689
$363$	0	0
$364$	1.23607	0.0647876
$365$	−14.9443	−0.782219
$366$	0	0
$367$	16.2705	0.849314	0.424657	−	0.905354i	$-0.360395\pi$
$367$	16.2705	0.849314	0.424657	+	0.905354i	$0.360395\pi$
$368$	−3.09017	−0.161086
$369$	0	0
$370$	−34.5066	−1.79391
$371$	−12.1803	−0.632372
$372$	0	0
$373$	−23.3262	−1.20779	−0.603893	−	0.797065i	$-0.706385\pi$
$373$	−23.3262	−1.20779	−0.603893	+	0.797065i	$0.706385\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	−2.47214	−0.127321
$378$	0	0
$379$	2.47214	0.126985	0.0634925	−	0.997982i	$-0.479776\pi$
$379$	2.47214	0.126985	0.0634925	+	0.997982i	$0.479776\pi$
$380$	−7.47214	−0.383312
$381$	0	0
$382$	18.2705	0.934801
$383$	13.0557	0.667117	0.333558	−	0.942729i	$-0.391751\pi$
$383$	13.0557	0.667117	0.333558	+	0.942729i	$0.391751\pi$
$384$	0	0
$385$	0	0
$386$	18.3820	0.935617
$387$	0	0
$388$	3.52786	0.179100
$389$	−22.1803	−1.12459	−0.562294	−	0.826937i	$-0.690081\pi$
$389$	−22.1803	−1.12459	−0.562294	+	0.826937i	$0.690081\pi$
$390$	0	0
$391$	−24.2705	−1.22741
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−4.00000	−0.201517
$395$	−39.9574	−2.01048
$396$	0	0
$397$	14.6525	0.735387	0.367693	−	0.929947i	$-0.380148\pi$
$397$	14.6525	0.735387	0.367693	+	0.929947i	$0.380148\pi$
$398$	−2.61803	−0.131230
$399$	0	0
$400$	−3.14590	−0.157295
$401$	−18.4721	−0.922454	−0.461227	−	0.887282i	$-0.652591\pi$
$401$	−18.4721	−0.922454	−0.461227	+	0.887282i	$0.652591\pi$
$402$	0	0
$403$	9.05573	0.451098
$404$	−2.09017	−0.103990
$405$	0	0
$406$	2.00000	0.0992583
$407$	0	0
$408$	0	0
$409$	28.7639	1.42228	0.711142	−	0.703048i	$-0.248178\pi$
$409$	28.7639	1.42228	0.711142	+	0.703048i	$0.248178\pi$
$410$	30.9787	1.52993
$411$	0	0
$412$	10.3262	0.508737
$413$	−6.76393	−0.332831
$414$	0	0
$415$	−8.40325	−0.412499
$416$	−6.18034	−0.303016
$417$	0	0
$418$	0	0
$419$	−14.7639	−0.721265	−0.360633	−	0.932708i	$-0.617439\pi$
$419$	−14.7639	−0.721265	−0.360633	+	0.932708i	$0.617439\pi$
$420$	0	0
$421$	−4.32624	−0.210848	−0.105424	−	0.994427i	$-0.533620\pi$
$421$	−4.32624	−0.210848	−0.105424	+	0.994427i	$0.533620\pi$
$422$	−2.00000	−0.0973585
$423$	0	0
$424$	36.5410	1.77459
$425$	−24.7082	−1.19852
$426$	0	0
$427$	8.94427	0.432844
$428$	4.90983	0.237326
$429$	0	0
$430$	−3.52786	−0.170129
$431$	2.09017	0.100680	0.0503400	−	0.998732i	$-0.483970\pi$
$431$	2.09017	0.100680	0.0503400	+	0.998732i	$0.483970\pi$
$432$	0	0
$433$	−17.2361	−0.828313	−0.414156	−	0.910206i	$-0.635923\pi$
$433$	−17.2361	−0.828313	−0.414156	+	0.910206i	$0.635923\pi$
$434$	−7.32624	−0.351671
$435$	0	0
$436$	−10.5623	−0.505843
$437$	8.09017	0.387005
$438$	0	0
$439$	−21.7984	−1.04038	−0.520190	−	0.854051i	$-0.674139\pi$
$439$	−21.7984	−1.04038	−0.520190	+	0.854051i	$0.674139\pi$
$440$	0	0
$441$	0	0
$442$	9.70820	0.461772
$443$	−4.90983	−0.233273	−0.116637	−	0.993175i	$-0.537211\pi$
$443$	−4.90983	−0.233273	−0.116637	+	0.993175i	$0.537211\pi$
$444$	0	0
$445$	−3.11146	−0.147497
$446$	20.7984	0.984832
$447$	0	0
$448$	7.00000	0.330719
$449$	−41.8885	−1.97684	−0.988421	−	0.151734i	$-0.951514\pi$
$449$	−41.8885	−1.97684	−0.988421	+	0.151734i	$0.951514\pi$
$450$	0	0
$451$	0	0
$452$	0.763932	0.0359323
$453$	0	0
$454$	−21.4164	−1.00512
$455$	−3.52786	−0.165389
$456$	0	0
$457$	12.4721	0.583422	0.291711	−	0.956507i	$-0.405775\pi$
$457$	12.4721	0.583422	0.291711	+	0.956507i	$0.405775\pi$
$458$	21.2361	0.992296
$459$	0	0
$460$	−8.81966	−0.411219
$461$	6.36068	0.296246	0.148123	−	0.988969i	$-0.452677\pi$
$461$	6.36068	0.296246	0.148123	+	0.988969i	$0.452677\pi$
$462$	0	0
$463$	11.4164	0.530565	0.265283	−	0.964171i	$-0.414535\pi$
$463$	11.4164	0.530565	0.265283	+	0.964171i	$0.414535\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	5.70820	0.264427
$467$	−19.2361	−0.890139	−0.445070	−	0.895496i	$-0.646821\pi$
$467$	−19.2361	−0.890139	−0.445070	+	0.895496i	$0.646821\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	−5.70820	−0.263300
$471$	0	0
$472$	20.2918	0.934006
$473$	0	0
$474$	0	0
$475$	8.23607	0.377897
$476$	7.85410	0.359992
$477$	0	0
$478$	1.56231	0.0714582
$479$	11.0557	0.505149	0.252575	−	0.967577i	$-0.418723\pi$
$479$	11.0557	0.505149	0.252575	+	0.967577i	$0.418723\pi$
$480$	0	0
$481$	14.9443	0.681400
$482$	4.94427	0.225205
$483$	0	0
$484$	0	0
$485$	−10.0689	−0.457204
$486$	0	0
$487$	−39.2361	−1.77796	−0.888978	−	0.457950i	$-0.848584\pi$
$487$	−39.2361	−1.77796	−0.888978	+	0.457950i	$0.848584\pi$
$488$	−26.8328	−1.21466
$489$	0	0
$490$	2.85410	0.128935
$491$	0.326238	0.0147229	0.00736146	−	0.999973i	$-0.497657\pi$
$491$	0.326238	0.0147229	0.00736146	+	0.999973i	$0.497657\pi$
$492$	0	0
$493$	−15.7082	−0.707462
$494$	−3.23607	−0.145598
$495$	0	0
$496$	7.32624	0.328958
$497$	−8.94427	−0.401205
$498$	0	0
$499$	26.1803	1.17199	0.585996	−	0.810314i	$-0.300703\pi$
$499$	26.1803	1.17199	0.585996	+	0.810314i	$0.300703\pi$
$500$	5.29180	0.236656
$501$	0	0
$502$	−24.1803	−1.07922
$503$	0.583592	0.0260211	0.0130105	−	0.999915i	$-0.495858\pi$
$503$	0.583592	0.0260211	0.0130105	+	0.999915i	$0.495858\pi$
$504$	0	0
$505$	5.96556	0.265464
$506$	0	0
$507$	0	0
$508$	3.23607	0.143577
$509$	5.90983	0.261949	0.130974	−	0.991386i	$-0.458189\pi$
$509$	5.90983	0.261949	0.130974	+	0.991386i	$0.458189\pi$
$510$	0	0
$511$	−5.23607	−0.231630
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−2.61803	−0.115477
$515$	−29.4721	−1.29870
$516$	0	0
$517$	0	0
$518$	−12.0902	−0.531212
$519$	0	0
$520$	10.5836	0.464121
$521$	11.6180	0.508995	0.254498	−	0.967073i	$-0.418090\pi$
$521$	11.6180	0.508995	0.254498	+	0.967073i	$0.418090\pi$
$522$	0	0
$523$	2.90983	0.127238	0.0636190	−	0.997974i	$-0.479736\pi$
$523$	2.90983	0.127238	0.0636190	+	0.997974i	$0.479736\pi$
$524$	2.29180	0.100118
$525$	0	0
$526$	−2.43769	−0.106289
$527$	57.5410	2.50653
$528$	0	0
$529$	−13.4508	−0.584820
$530$	−34.7639	−1.51005
$531$	0	0
$532$	−2.61803	−0.113506
$533$	−13.4164	−0.581129
$534$	0	0
$535$	−14.0132	−0.605842
$536$	−24.0000	−1.03664
$537$	0	0
$538$	−27.8885	−1.20236
$539$	0	0
$540$	0	0
$541$	16.8541	0.724614	0.362307	−	0.932059i	$-0.381989\pi$
$541$	16.8541	0.724614	0.362307	+	0.932059i	$0.381989\pi$
$542$	24.0902	1.03476
$543$	0	0
$544$	−39.2705	−1.68371
$545$	30.1459	1.29131
$546$	0	0
$547$	−36.4721	−1.55944	−0.779718	−	0.626131i	$-0.784638\pi$
$547$	−36.4721	−1.55944	−0.779718	+	0.626131i	$0.784638\pi$
$548$	6.76393	0.288941
$549$	0	0
$550$	0	0
$551$	5.23607	0.223064
$552$	0	0
$553$	−14.0000	−0.595341
$554$	−10.4377	−0.443455
$555$	0	0
$556$	−4.79837	−0.203496
$557$	21.4164	0.907442	0.453721	−	0.891144i	$-0.350096\pi$
$557$	21.4164	0.907442	0.453721	+	0.891144i	$0.350096\pi$
$558$	0	0
$559$	1.52786	0.0646218
$560$	−2.85410	−0.120608
$561$	0	0
$562$	−1.05573	−0.0445332
$563$	−40.3607	−1.70100	−0.850500	−	0.525975i	$-0.823700\pi$
$563$	−40.3607	−1.70100	−0.850500	+	0.525975i	$0.823700\pi$
$564$	0	0
$565$	−2.18034	−0.0917276
$566$	13.4377	0.564828
$567$	0	0
$568$	26.8328	1.12588
$569$	18.1803	0.762159	0.381080	−	0.924542i	$-0.375552\pi$
$569$	18.1803	0.762159	0.381080	+	0.924542i	$0.375552\pi$
$570$	0	0
$571$	28.5410	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$571$	28.5410	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$572$	0	0
$573$	0	0
$574$	10.8541	0.453041
$575$	9.72136	0.405409
$576$	0	0
$577$	−14.6525	−0.609991	−0.304995	−	0.952354i	$-0.598655\pi$
$577$	−14.6525	−0.609991	−0.304995	+	0.952354i	$0.598655\pi$
$578$	44.6869	1.85873
$579$	0	0
$580$	−5.70820	−0.237020
$581$	−2.94427	−0.122149
$582$	0	0
$583$	0	0
$584$	15.7082	0.650010
$585$	0	0
$586$	−19.7984	−0.817863
$587$	−3.88854	−0.160497	−0.0802487	−	0.996775i	$-0.525571\pi$
$587$	−3.88854	−0.160497	−0.0802487	+	0.996775i	$0.525571\pi$
$588$	0	0
$589$	−19.1803	−0.790312
$590$	−19.3050	−0.794772
$591$	0	0
$592$	12.0902	0.496903
$593$	−3.43769	−0.141169	−0.0705846	−	0.997506i	$-0.522486\pi$
$593$	−3.43769	−0.141169	−0.0705846	+	0.997506i	$0.522486\pi$
$594$	0	0
$595$	−22.4164	−0.918983
$596$	−20.0000	−0.819232
$597$	0	0
$598$	−3.81966	−0.156198
$599$	1.56231	0.0638341	0.0319170	−	0.999491i	$-0.489839\pi$
$599$	1.56231	0.0638341	0.0319170	+	0.999491i	$0.489839\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	−1.23607	−0.0503784
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	−10.3262	−0.419129	−0.209565	−	0.977795i	$-0.567205\pi$
$607$	−10.3262	−0.419129	−0.209565	+	0.977795i	$0.567205\pi$
$608$	13.0902	0.530876
$609$	0	0
$610$	25.5279	1.03359
$611$	2.47214	0.100012
$612$	0	0
$613$	−5.85410	−0.236445	−0.118222	−	0.992987i	$-0.537720\pi$
$613$	−5.85410	−0.236445	−0.118222	+	0.992987i	$0.537720\pi$
$614$	−24.2705	−0.979478
$615$	0	0
$616$	0	0
$617$	−44.6525	−1.79764	−0.898820	−	0.438317i	$-0.855575\pi$
$617$	−44.6525	−1.79764	−0.898820	+	0.438317i	$0.855575\pi$
$618$	0	0
$619$	−11.5066	−0.462488	−0.231244	−	0.972896i	$-0.574280\pi$
$619$	−11.5066	−0.462488	−0.231244	+	0.972896i	$0.574280\pi$
$620$	20.9098	0.839759
$621$	0	0
$622$	0.763932	0.0306309
$623$	−1.09017	−0.0436767
$624$	0	0
$625$	−30.8328	−1.23331
$626$	−9.81966	−0.392473
$627$	0	0
$628$	−9.70820	−0.387400
$629$	94.9574	3.78620
$630$	0	0
$631$	29.8885	1.18984	0.594922	−	0.803783i	$-0.297183\pi$
$631$	29.8885	1.18984	0.594922	+	0.803783i	$0.297183\pi$
$632$	42.0000	1.67067
$633$	0	0
$634$	27.5967	1.09601
$635$	−9.23607	−0.366522
$636$	0	0
$637$	−1.23607	−0.0489748
$638$	0	0
$639$	0	0
$640$	−8.56231	−0.338455
$641$	−8.65248	−0.341752	−0.170876	−	0.985293i	$-0.554660\pi$
$641$	−8.65248	−0.341752	−0.170876	+	0.985293i	$0.554660\pi$
$642$	0	0
$643$	47.9230	1.88990	0.944949	−	0.327218i	$-0.106111\pi$
$643$	47.9230	1.88990	0.944949	+	0.327218i	$0.106111\pi$
$644$	−3.09017	−0.121770
$645$	0	0
$646$	−20.5623	−0.809013
$647$	30.3607	1.19360	0.596801	−	0.802389i	$-0.296438\pi$
$647$	30.3607	1.19360	0.596801	+	0.802389i	$0.296438\pi$
$648$	0	0
$649$	0	0
$650$	−3.88854	−0.152521
$651$	0	0
$652$	5.41641	0.212123
$653$	25.8885	1.01310	0.506549	−	0.862211i	$-0.330921\pi$
$653$	25.8885	1.01310	0.506549	+	0.862211i	$0.330921\pi$
$654$	0	0
$655$	−6.54102	−0.255579
$656$	−10.8541	−0.423781
$657$	0	0
$658$	−2.00000	−0.0779681
$659$	35.0344	1.36475	0.682374	−	0.731003i	$-0.260948\pi$
$659$	35.0344	1.36475	0.682374	+	0.731003i	$0.260948\pi$
$660$	0	0
$661$	8.29180	0.322513	0.161257	−	0.986912i	$-0.448445\pi$
$661$	8.29180	0.322513	0.161257	+	0.986912i	$0.448445\pi$
$662$	30.3607	1.18000
$663$	0	0
$664$	8.83282	0.342780
$665$	7.47214	0.289757
$666$	0	0
$667$	6.18034	0.239304
$668$	8.18034	0.316507
$669$	0	0
$670$	22.8328	0.882109
$671$	0	0
$672$	0	0
$673$	35.8885	1.38340	0.691701	−	0.722184i	$-0.256862\pi$
$673$	35.8885	1.38340	0.691701	+	0.722184i	$0.256862\pi$
$674$	−27.1459	−1.04562
$675$	0	0
$676$	11.4721	0.441236
$677$	−23.8885	−0.918111	−0.459056	−	0.888408i	$-0.651812\pi$
$677$	−23.8885	−0.918111	−0.459056	+	0.888408i	$0.651812\pi$
$678$	0	0
$679$	−3.52786	−0.135387
$680$	67.2492	2.57889
$681$	0	0
$682$	0	0
$683$	−40.9230	−1.56587	−0.782937	−	0.622101i	$-0.786279\pi$
$683$	−40.9230	−1.56587	−0.782937	+	0.622101i	$0.786279\pi$
$684$	0	0
$685$	−19.3050	−0.737604
$686$	1.00000	0.0381802
$687$	0	0
$688$	1.23607	0.0471246
$689$	15.0557	0.573578
$690$	0	0
$691$	39.9230	1.51874	0.759371	−	0.650658i	$-0.225507\pi$
$691$	39.9230	1.51874	0.759371	+	0.650658i	$0.225507\pi$
$692$	19.8541	0.754740
$693$	0	0
$694$	12.3820	0.470013
$695$	13.6950	0.519483
$696$	0	0
$697$	−85.2492	−3.22904
$698$	22.7639	0.861628
$699$	0	0
$700$	−3.14590	−0.118904
$701$	−24.0689	−0.909069	−0.454535	−	0.890729i	$-0.650194\pi$
$701$	−24.0689	−0.909069	−0.454535	+	0.890729i	$0.650194\pi$
$702$	0	0
$703$	−31.6525	−1.19380
$704$	0	0
$705$	0	0
$706$	−12.4721	−0.469395
$707$	2.09017	0.0786089
$708$	0	0
$709$	−19.6869	−0.739358	−0.369679	−	0.929160i	$-0.620532\pi$
$709$	−19.6869	−0.739358	−0.369679	+	0.929160i	$0.620532\pi$
$710$	−25.5279	−0.958044
$711$	0	0
$712$	3.27051	0.122568
$713$	−22.6393	−0.847849
$714$	0	0
$715$	0	0
$716$	8.38197	0.313249
$717$	0	0
$718$	−7.03444	−0.262523
$719$	−4.87539	−0.181821	−0.0909106	−	0.995859i	$-0.528978\pi$
$719$	−4.87539	−0.181821	−0.0909106	+	0.995859i	$0.528978\pi$
$720$	0	0
$721$	−10.3262	−0.384569
$722$	−12.1459	−0.452024
$723$	0	0
$724$	18.9443	0.704058
$725$	6.29180	0.233671
$726$	0	0
$727$	25.7426	0.954742	0.477371	−	0.878702i	$-0.341590\pi$
$727$	25.7426	0.954742	0.477371	+	0.878702i	$0.341590\pi$
$728$	3.70820	0.137435
$729$	0	0
$730$	−14.9443	−0.553112
$731$	9.70820	0.359071
$732$	0	0
$733$	31.1246	1.14961	0.574807	−	0.818289i	$-0.305077\pi$
$733$	31.1246	1.14961	0.574807	+	0.818289i	$0.305077\pi$
$734$	16.2705	0.600555
$735$	0	0
$736$	15.4508	0.569526
$737$	0	0
$738$	0	0
$739$	49.0132	1.80298	0.901489	−	0.432802i	$-0.142475\pi$
$739$	49.0132	1.80298	0.901489	+	0.432802i	$0.142475\pi$
$740$	34.5066	1.26849
$741$	0	0
$742$	−12.1803	−0.447154
$743$	−17.3262	−0.635638	−0.317819	−	0.948151i	$-0.602950\pi$
$743$	−17.3262	−0.635638	−0.317819	+	0.948151i	$0.602950\pi$
$744$	0	0
$745$	57.0820	2.09132
$746$	−23.3262	−0.854034
$747$	0	0
$748$	0	0
$749$	−4.90983	−0.179401
$750$	0	0
$751$	−10.4721	−0.382134	−0.191067	−	0.981577i	$-0.561195\pi$
$751$	−10.4721	−0.382134	−0.191067	+	0.981577i	$0.561195\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	−2.47214	−0.0900299
$755$	−5.70820	−0.207743
$756$	0	0
$757$	0.326238	0.0118573	0.00592866	−	0.999982i	$-0.498113\pi$
$757$	0.326238	0.0118573	0.00592866	+	0.999982i	$0.498113\pi$
$758$	2.47214	0.0897920
$759$	0	0
$760$	−22.4164	−0.813129
$761$	−45.7771	−1.65942	−0.829709	−	0.558196i	$-0.811494\pi$
$761$	−45.7771	−1.65942	−0.829709	+	0.558196i	$0.811494\pi$
$762$	0	0
$763$	10.5623	0.382381
$764$	−18.2705	−0.661004
$765$	0	0
$766$	13.0557	0.471723
$767$	8.36068	0.301887
$768$	0	0
$769$	14.5836	0.525898	0.262949	−	0.964810i	$-0.415305\pi$
$769$	14.5836	0.525898	0.262949	+	0.964810i	$0.415305\pi$
$770$	0	0
$771$	0	0
$772$	−18.3820	−0.661581
$773$	38.9443	1.40073	0.700364	−	0.713786i	$-0.253021\pi$
$773$	38.9443	1.40073	0.700364	+	0.713786i	$0.253021\pi$
$774$	0	0
$775$	−23.0476	−0.827894
$776$	10.5836	0.379929
$777$	0	0
$778$	−22.1803	−0.795204
$779$	28.4164	1.01812
$780$	0	0
$781$	0	0
$782$	−24.2705	−0.867912
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	27.7082	0.988948
$786$	0	0
$787$	12.2705	0.437396	0.218698	−	0.975793i	$-0.429819\pi$
$787$	12.2705	0.437396	0.218698	+	0.975793i	$0.429819\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	−39.9574	−1.42162
$791$	−0.763932	−0.0271623
$792$	0	0
$793$	−11.0557	−0.392600
$794$	14.6525	0.519997
$795$	0	0
$796$	2.61803	0.0927938
$797$	−10.2705	−0.363800	−0.181900	−	0.983317i	$-0.558225\pi$
$797$	−10.2705	−0.363800	−0.181900	+	0.983317i	$0.558225\pi$
$798$	0	0
$799$	15.7082	0.555716
$800$	15.7295	0.556121
$801$	0	0
$802$	−18.4721	−0.652274
$803$	0	0
$804$	0	0
$805$	8.81966	0.310852
$806$	9.05573	0.318974
$807$	0	0
$808$	−6.27051	−0.220596
$809$	28.6525	1.00737	0.503684	−	0.863888i	$-0.331978\pi$
$809$	28.6525	1.00737	0.503684	+	0.863888i	$0.331978\pi$
$810$	0	0
$811$	−16.3607	−0.574501	−0.287251	−	0.957855i	$-0.592741\pi$
$811$	−16.3607	−0.574501	−0.287251	+	0.957855i	$0.592741\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0	0
$815$	−15.4590	−0.541504
$816$	0	0
$817$	−3.23607	−0.113216
$818$	28.7639	1.00571
$819$	0	0
$820$	−30.9787	−1.08182
$821$	−30.3607	−1.05960	−0.529798	−	0.848124i	$-0.677732\pi$
$821$	−30.3607	−1.05960	−0.529798	+	0.848124i	$0.677732\pi$
$822$	0	0
$823$	24.4721	0.853045	0.426523	−	0.904477i	$-0.359738\pi$
$823$	24.4721	0.853045	0.426523	+	0.904477i	$0.359738\pi$
$824$	30.9787	1.07919
$825$	0	0
$826$	−6.76393	−0.235347
$827$	4.32624	0.150438	0.0752190	−	0.997167i	$-0.476034\pi$
$827$	4.32624	0.150438	0.0752190	+	0.997167i	$0.476034\pi$
$828$	0	0
$829$	28.9443	1.00528	0.502638	−	0.864497i	$-0.332363\pi$
$829$	28.9443	1.00528	0.502638	+	0.864497i	$0.332363\pi$
$830$	−8.40325	−0.291681
$831$	0	0
$832$	−8.65248	−0.299971
$833$	−7.85410	−0.272129
$834$	0	0
$835$	−23.3475	−0.807974
$836$	0	0
$837$	0	0
$838$	−14.7639	−0.510012
$839$	−33.4164	−1.15366	−0.576831	−	0.816863i	$-0.695711\pi$
$839$	−33.4164	−1.15366	−0.576831	+	0.816863i	$0.695711\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−4.32624	−0.149092
$843$	0	0
$844$	2.00000	0.0688428
$845$	−32.7426	−1.12638
$846$	0	0
$847$	0	0
$848$	12.1803	0.418275
$849$	0	0
$850$	−24.7082	−0.847484
$851$	−37.3607	−1.28071
$852$	0	0
$853$	−15.7082	−0.537839	−0.268919	−	0.963163i	$-0.586667\pi$
$853$	−15.7082	−0.537839	−0.268919	+	0.963163i	$0.586667\pi$
$854$	8.94427	0.306067
$855$	0	0
$856$	14.7295	0.503444
$857$	18.9443	0.647124	0.323562	−	0.946207i	$-0.395120\pi$
$857$	18.9443	0.647124	0.323562	+	0.946207i	$0.395120\pi$
$858$	0	0
$859$	−5.88854	−0.200915	−0.100457	−	0.994941i	$-0.532031\pi$
$859$	−5.88854	−0.200915	−0.100457	+	0.994941i	$0.532031\pi$
$860$	3.52786	0.120299
$861$	0	0
$862$	2.09017	0.0711915
$863$	−37.6869	−1.28288	−0.641439	−	0.767174i	$-0.721662\pi$
$863$	−37.6869	−1.28288	−0.641439	+	0.767174i	$0.721662\pi$
$864$	0	0
$865$	−56.6656	−1.92669
$866$	−17.2361	−0.585705
$867$	0	0
$868$	7.32624	0.248669
$869$	0	0
$870$	0	0
$871$	−9.88854	−0.335061
$872$	−31.6869	−1.07305
$873$	0	0
$874$	8.09017	0.273654
$875$	−5.29180	−0.178895
$876$	0	0
$877$	−46.3607	−1.56549	−0.782744	−	0.622343i	$-0.786181\pi$
$877$	−46.3607	−1.56549	−0.782744	+	0.622343i	$0.786181\pi$
$878$	−21.7984	−0.735659
$879$	0	0
$880$	0	0
$881$	33.4508	1.12699	0.563494	−	0.826120i	$-0.309457\pi$
$881$	33.4508	1.12699	0.563494	+	0.826120i	$0.309457\pi$
$882$	0	0
$883$	7.70820	0.259402	0.129701	−	0.991553i	$-0.458598\pi$
$883$	7.70820	0.259402	0.129701	+	0.991553i	$0.458598\pi$
$884$	−9.70820	−0.326522
$885$	0	0
$886$	−4.90983	−0.164949
$887$	46.5410	1.56269	0.781347	−	0.624097i	$-0.214533\pi$
$887$	46.5410	1.56269	0.781347	+	0.624097i	$0.214533\pi$
$888$	0	0
$889$	−3.23607	−0.108534
$890$	−3.11146	−0.104296
$891$	0	0
$892$	−20.7984	−0.696381
$893$	−5.23607	−0.175218
$894$	0	0
$895$	−23.9230	−0.799657
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−41.8885	−1.39784
$899$	−14.6525	−0.488687
$900$	0	0
$901$	95.6656	3.18708
$902$	0	0
$903$	0	0
$904$	2.29180	0.0762240
$905$	−54.0689	−1.79731
$906$	0	0
$907$	−47.2361	−1.56845	−0.784224	−	0.620478i	$-0.786939\pi$
$907$	−47.2361	−1.56845	−0.784224	+	0.620478i	$0.786939\pi$
$908$	21.4164	0.710728
$909$	0	0
$910$	−3.52786	−0.116948
$911$	11.0557	0.366293	0.183146	−	0.983086i	$-0.441372\pi$
$911$	11.0557	0.366293	0.183146	+	0.983086i	$0.441372\pi$
$912$	0	0
$913$	0	0
$914$	12.4721	0.412542
$915$	0	0
$916$	−21.2361	−0.701659
$917$	−2.29180	−0.0756818
$918$	0	0
$919$	7.41641	0.244645	0.122322	−	0.992490i	$-0.460966\pi$
$919$	7.41641	0.244645	0.122322	+	0.992490i	$0.460966\pi$
$920$	−26.4590	−0.872327
$921$	0	0
$922$	6.36068	0.209478
$923$	11.0557	0.363904
$924$	0	0
$925$	−38.0344	−1.25056
$926$	11.4164	0.375166
$927$	0	0
$928$	10.0000	0.328266
$929$	28.7984	0.944844	0.472422	−	0.881372i	$-0.343380\pi$
$929$	28.7984	0.944844	0.472422	+	0.881372i	$0.343380\pi$
$930$	0	0
$931$	2.61803	0.0858026
$932$	−5.70820	−0.186978
$933$	0	0
$934$	−19.2361	−0.629423
$935$	0	0
$936$	0	0
$937$	11.7082	0.382490	0.191245	−	0.981542i	$-0.438747\pi$
$937$	11.7082	0.382490	0.191245	+	0.981542i	$0.438747\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	5.70820	0.186181
$941$	11.8541	0.386433	0.193216	−	0.981156i	$-0.438108\pi$
$941$	11.8541	0.386433	0.193216	+	0.981156i	$0.438108\pi$
$942$	0	0
$943$	33.5410	1.09225
$944$	6.76393	0.220147
$945$	0	0
$946$	0	0
$947$	23.4377	0.761623	0.380811	−	0.924653i	$-0.375645\pi$
$947$	23.4377	0.761623	0.380811	+	0.924653i	$0.375645\pi$
$948$	0	0
$949$	6.47214	0.210094
$950$	8.23607	0.267213
$951$	0	0
$952$	23.5623	0.763659
$953$	−4.83282	−0.156550	−0.0782751	−	0.996932i	$-0.524941\pi$
$953$	−4.83282	−0.156550	−0.0782751	+	0.996932i	$0.524941\pi$
$954$	0	0
$955$	52.1459	1.68740
$956$	−1.56231	−0.0505286
$957$	0	0
$958$	11.0557	0.357194
$959$	−6.76393	−0.218419
$960$	0	0
$961$	22.6738	0.731412
$962$	14.9443	0.481823
$963$	0	0
$964$	−4.94427	−0.159244
$965$	52.4640	1.68888
$966$	0	0
$967$	49.3050	1.58554	0.792770	−	0.609521i	$-0.208638\pi$
$967$	49.3050	1.58554	0.792770	+	0.609521i	$0.208638\pi$
$968$	0	0
$969$	0	0
$970$	−10.0689	−0.323292
$971$	−42.0689	−1.35005	−0.675027	−	0.737793i	$-0.735868\pi$
$971$	−42.0689	−1.35005	−0.675027	+	0.737793i	$0.735868\pi$
$972$	0	0
$973$	4.79837	0.153829
$974$	−39.2361	−1.25720
$975$	0	0
$976$	−8.94427	−0.286299
$977$	26.5836	0.850484	0.425242	−	0.905080i	$-0.360189\pi$
$977$	26.5836	0.850484	0.425242	+	0.905080i	$0.360189\pi$
$978$	0	0
$979$	0	0
$980$	−2.85410	−0.0911709
$981$	0	0
$982$	0.326238	0.0104107
$983$	−48.4721	−1.54602	−0.773011	−	0.634393i	$-0.781250\pi$
$983$	−48.4721	−1.54602	−0.773011	+	0.634393i	$0.781250\pi$
$984$	0	0
$985$	−11.4164	−0.363757
$986$	−15.7082	−0.500251
$987$	0	0
$988$	3.23607	0.102953
$989$	−3.81966	−0.121458
$990$	0	0
$991$	18.6525	0.592515	0.296258	−	0.955108i	$-0.404261\pi$
$991$	18.6525	0.592515	0.296258	+	0.955108i	$0.404261\pi$
$992$	−36.6312	−1.16304
$993$	0	0
$994$	−8.94427	−0.283695
$995$	−7.47214	−0.236883
$996$	0	0
$997$	−52.5410	−1.66399	−0.831995	−	0.554783i	$-0.812801\pi$
$997$	−52.5410	−1.66399	−0.831995	+	0.554783i	$0.812801\pi$
$998$	26.1803	0.828724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bu.1.2		2
3.2	odd	2		2541.2.a.n.1.1		2
11.5	even	5		693.2.m.c.190.1		4
11.9	even	5		693.2.m.c.631.1		4
11.10	odd	2		7623.2.a.w.1.2		2
33.5	odd	10		231.2.j.c.190.1	yes	4
33.20	odd	10		231.2.j.c.169.1	✓	4
33.32	even	2		2541.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.c.169.1	✓	4	33.20	odd	10
231.2.j.c.190.1	yes	4	33.5	odd	10
693.2.m.c.190.1		4	11.5	even	5
693.2.m.c.631.1		4	11.9	even	5
2541.2.a.n.1.1		2	3.2	odd	2
2541.2.a.bd.1.1		2	33.32	even	2
7623.2.a.w.1.2		2	11.10	odd	2
7623.2.a.bu.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} +1.00000 q^{7} -3.00000 q^{8} +2.85410 q^{10} -1.23607 q^{13} +1.00000 q^{14} -1.00000 q^{16} -7.85410 q^{17} +2.61803 q^{19} -2.85410 q^{20} +3.09017 q^{23} +3.14590 q^{25} -1.23607 q^{26} -1.00000 q^{28} +2.00000 q^{29} -7.32624 q^{31} +5.00000 q^{32} -7.85410 q^{34} +2.85410 q^{35} -12.0902 q^{37} +2.61803 q^{38} -8.56231 q^{40} +10.8541 q^{41} -1.23607 q^{43} +3.09017 q^{46} -2.00000 q^{47} +1.00000 q^{49} +3.14590 q^{50} +1.23607 q^{52} -12.1803 q^{53} -3.00000 q^{56} +2.00000 q^{58} -6.76393 q^{59} +8.94427 q^{61} -7.32624 q^{62} +7.00000 q^{64} -3.52786 q^{65} +8.00000 q^{67} +7.85410 q^{68} +2.85410 q^{70} -8.94427 q^{71} -5.23607 q^{73} -12.0902 q^{74} -2.61803 q^{76} -14.0000 q^{79} -2.85410 q^{80} +10.8541 q^{82} -2.94427 q^{83} -22.4164 q^{85} -1.23607 q^{86} -1.09017 q^{89} -1.23607 q^{91} -3.09017 q^{92} -2.00000 q^{94} +7.47214 q^{95} -3.52786 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 6 q^{8} - q^{10} + 2 q^{13} + 2 q^{14} - 2 q^{16} - 9 q^{17} + 3 q^{19} + q^{20} - 5 q^{23} + 13 q^{25} + 2 q^{26} - 2 q^{28} + 4 q^{29} + q^{31} + 10 q^{32} - 9 q^{34} - q^{35} - 13 q^{37} + 3 q^{38} + 3 q^{40} + 15 q^{41} + 2 q^{43} - 5 q^{46} - 4 q^{47} + 2 q^{49} + 13 q^{50} - 2 q^{52} - 2 q^{53} - 6 q^{56} + 4 q^{58} - 18 q^{59} + q^{62} + 14 q^{64} - 16 q^{65} + 16 q^{67} + 9 q^{68} - q^{70} - 6 q^{73} - 13 q^{74} - 3 q^{76} - 28 q^{79} + q^{80} + 15 q^{82} + 12 q^{83} - 18 q^{85} + 2 q^{86} + 9 q^{89} + 2 q^{91} + 5 q^{92} - 4 q^{94} + 6 q^{95} - 16 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - q^5 + 2 * q^7 - 6 * q^8 - q^10 + 2 * q^13 + 2 * q^14 - 2 * q^16 - 9 * q^17 + 3 * q^19 + q^20 - 5 * q^23 + 13 * q^25 + 2 * q^26 - 2 * q^28 + 4 * q^29 + q^31 + 10 * q^32 - 9 * q^34 - q^35 - 13 * q^37 + 3 * q^38 + 3 * q^40 + 15 * q^41 + 2 * q^43 - 5 * q^46 - 4 * q^47 + 2 * q^49 + 13 * q^50 - 2 * q^52 - 2 * q^53 - 6 * q^56 + 4 * q^58 - 18 * q^59 + q^62 + 14 * q^64 - 16 * q^65 + 16 * q^67 + 9 * q^68 - q^70 - 6 * q^73 - 13 * q^74 - 3 * q^76 - 28 * q^79 + q^80 + 15 * q^82 + 12 * q^83 - 18 * q^85 + 2 * q^86 + 9 * q^89 + 2 * q^91 + 5 * q^92 - 4 * q^94 + 6 * q^95 - 16 * q^97 + 2 * q^98

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