LMFDB - Embedded newform 7623.2.a.z.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:	$0$
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:	$\Z[a_1, a_2]$
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			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{5} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{10} -4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} -3.47214 q^{17} -6.47214 q^{19} -2.61803 q^{20} -5.70820 q^{23} -2.38197 q^{25} -2.61803 q^{26} -1.61803 q^{28} -0.236068 q^{29} +3.00000 q^{31} +5.61803 q^{32} -2.14590 q^{34} +1.61803 q^{35} +6.00000 q^{37} -4.00000 q^{38} -3.61803 q^{40} -9.85410 q^{41} +11.4721 q^{43} -3.52786 q^{46} +5.14590 q^{47} +1.00000 q^{49} -1.47214 q^{50} +6.85410 q^{52} +12.5623 q^{53} -2.23607 q^{56} -0.145898 q^{58} +4.61803 q^{59} +7.94427 q^{61} +1.85410 q^{62} -0.236068 q^{64} -6.85410 q^{65} +13.2361 q^{67} +5.61803 q^{68} +1.00000 q^{70} +4.52786 q^{71} -7.38197 q^{73} +3.70820 q^{74} +10.4721 q^{76} +6.09017 q^{79} +3.00000 q^{80} -6.09017 q^{82} +8.47214 q^{83} -5.61803 q^{85} +7.09017 q^{86} -1.32624 q^{89} -4.23607 q^{91} +9.23607 q^{92} +3.18034 q^{94} -10.4721 q^{95} -5.29180 q^{97} +0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^7	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 2 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{20} + 2 q^{23} - 7 q^{25} - 3 q^{26} - q^{28} + 4 q^{29} + 6 q^{31} + 9 q^{32} - 11 q^{34} + q^{35} + 12 q^{37} - 8 q^{38} - 5 q^{40} - 13 q^{41} + 14 q^{43} - 16 q^{46} + 17 q^{47} + 2 q^{49} + 6 q^{50} + 7 q^{52} + 5 q^{53} - 7 q^{58} + 7 q^{59} - 2 q^{61} - 3 q^{62} + 4 q^{64} - 7 q^{65} + 22 q^{67} + 9 q^{68} + 2 q^{70} + 18 q^{71} - 17 q^{73} - 6 q^{74} + 12 q^{76} + q^{79} + 6 q^{80} - q^{82} + 8 q^{83} - 9 q^{85} + 3 q^{86} + 13 q^{89} - 4 q^{91} + 14 q^{92} - 16 q^{94} - 12 q^{95} - 24 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^7 + 2 * q^10 - 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 - 3 * q^20 + 2 * q^23 - 7 * q^25 - 3 * q^26 - q^28 + 4 * q^29 + 6 * q^31 + 9 * q^32 - 11 * q^34 + q^35 + 12 * q^37 - 8 * q^38 - 5 * q^40 - 13 * q^41 + 14 * q^43 - 16 * q^46 + 17 * q^47 + 2 * q^49 + 6 * q^50 + 7 * q^52 + 5 * q^53 - 7 * q^58 + 7 * q^59 - 2 * q^61 - 3 * q^62 + 4 * q^64 - 7 * q^65 + 22 * q^67 + 9 * q^68 + 2 * q^70 + 18 * q^71 - 17 * q^73 - 6 * q^74 + 12 * q^76 + q^79 + 6 * q^80 - q^82 + 8 * q^83 - 9 * q^85 + 3 * q^86 + 13 * q^89 - 4 * q^91 + 14 * q^92 - 16 * q^94 - 12 * q^95 - 24 * q^97 - 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−3.47214	−0.842117	−0.421058	−	0.907034i	$-0.638341\pi$
$17$	−3.47214	−0.842117	−0.421058	+	0.907034i	$0.638341\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	−2.61803	−0.585410
$21$	0	0
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	−2.61803	−0.513439
$27$	0	0
$28$	−1.61803	−0.305780
$29$	−0.236068	−0.0438367	−0.0219184	−	0.999760i	$-0.506977\pi$
$29$	−0.236068	−0.0438367	−0.0219184	+	0.999760i	$0.506977\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−2.14590	−0.368018
$35$	1.61803	0.273498
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−3.61803	−0.572061
$41$	−9.85410	−1.53895	−0.769476	−	0.638676i	$-0.779483\pi$
$41$	−9.85410	−1.53895	−0.769476	+	0.638676i	$0.779483\pi$
$42$	0	0
$43$	11.4721	1.74948	0.874742	−	0.484589i	$-0.161031\pi$
$43$	11.4721	1.74948	0.874742	+	0.484589i	$0.161031\pi$
$44$	0	0
$45$	0	0
$46$	−3.52786	−0.520155
$47$	5.14590	0.750606	0.375303	−	0.926902i	$-0.377539\pi$
$47$	5.14590	0.750606	0.375303	+	0.926902i	$0.377539\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.47214	−0.208191
$51$	0	0
$52$	6.85410	0.950493
$53$	12.5623	1.72557	0.862783	−	0.505575i	$-0.168720\pi$
$53$	12.5623	1.72557	0.862783	+	0.505575i	$0.168720\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−0.145898	−0.0191574
$59$	4.61803	0.601217	0.300608	−	0.953748i	$-0.402810\pi$
$59$	4.61803	0.601217	0.300608	+	0.953748i	$0.402810\pi$
$60$	0	0
$61$	7.94427	1.01716	0.508580	−	0.861015i	$-0.330171\pi$
$61$	7.94427	1.01716	0.508580	+	0.861015i	$0.330171\pi$
$62$	1.85410	0.235471
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−6.85410	−0.850147
$66$	0	0
$67$	13.2361	1.61704	0.808522	−	0.588467i	$-0.200268\pi$
$67$	13.2361	1.61704	0.808522	+	0.588467i	$0.200268\pi$
$68$	5.61803	0.681287
$69$	0	0
$70$	1.00000	0.119523
$71$	4.52786	0.537359	0.268679	−	0.963230i	$-0.413413\pi$
$71$	4.52786	0.537359	0.268679	+	0.963230i	$0.413413\pi$
$72$	0	0
$73$	−7.38197	−0.863994	−0.431997	−	0.901875i	$-0.642191\pi$
$73$	−7.38197	−0.863994	−0.431997	+	0.901875i	$0.642191\pi$
$74$	3.70820	0.431070
$75$	0	0
$76$	10.4721	1.20124
$77$	0	0
$78$	0	0
$79$	6.09017	0.685198	0.342599	−	0.939482i	$-0.388693\pi$
$79$	6.09017	0.685198	0.342599	+	0.939482i	$0.388693\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−6.09017	−0.672547
$83$	8.47214	0.929938	0.464969	−	0.885327i	$-0.346066\pi$
$83$	8.47214	0.929938	0.464969	+	0.885327i	$0.346066\pi$
$84$	0	0
$85$	−5.61803	−0.609361
$86$	7.09017	0.764553
$87$	0	0
$88$	0	0
$89$	−1.32624	−0.140581	−0.0702905	−	0.997527i	$-0.522393\pi$
$89$	−1.32624	−0.140581	−0.0702905	+	0.997527i	$0.522393\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	9.23607	0.962927
$93$	0	0
$94$	3.18034	0.328027
$95$	−10.4721	−1.07442
$96$	0	0
$97$	−5.29180	−0.537300	−0.268650	−	0.963238i	$-0.586578\pi$
$97$	−5.29180	−0.537300	−0.268650	+	0.963238i	$0.586578\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	3.85410	0.385410
$101$	11.2361	1.11803	0.559015	−	0.829157i	$-0.311179\pi$
$101$	11.2361	1.11803	0.559015	+	0.829157i	$0.311179\pi$
$102$	0	0
$103$	−15.4721	−1.52451	−0.762257	−	0.647274i	$-0.775909\pi$
$103$	−15.4721	−1.52451	−0.762257	+	0.647274i	$0.775909\pi$
$104$	9.47214	0.928819
$105$	0	0
$106$	7.76393	0.754100
$107$	5.76393	0.557220	0.278610	−	0.960404i	$-0.410126\pi$
$107$	5.76393	0.557220	0.278610	+	0.960404i	$0.410126\pi$
$108$	0	0
$109$	4.14590	0.397105	0.198553	−	0.980090i	$-0.436376\pi$
$109$	4.14590	0.397105	0.198553	+	0.980090i	$0.436376\pi$
$110$	0	0
$111$	0	0
$112$	1.85410	0.175196
$113$	11.1803	1.05176	0.525879	−	0.850559i	$-0.323736\pi$
$113$	11.1803	1.05176	0.525879	+	0.850559i	$0.323736\pi$
$114$	0	0
$115$	−9.23607	−0.861268
$116$	0.381966	0.0354647
$117$	0	0
$118$	2.85410	0.262741
$119$	−3.47214	−0.318290
$120$	0	0
$121$	0	0
$122$	4.90983	0.444515
$123$	0	0
$124$	−4.85410	−0.435911
$125$	−11.9443	−1.06833
$126$	0	0
$127$	7.32624	0.650098	0.325049	−	0.945697i	$-0.394619\pi$
$127$	7.32624	0.650098	0.325049	+	0.945697i	$0.394619\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−4.23607	−0.371528
$131$	6.85410	0.598846	0.299423	−	0.954121i	$-0.403206\pi$
$131$	6.85410	0.598846	0.299423	+	0.954121i	$0.403206\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	8.18034	0.706674
$135$	0	0
$136$	7.76393	0.665752
$137$	10.4721	0.894695	0.447347	−	0.894360i	$-0.352369\pi$
$137$	10.4721	0.894695	0.447347	+	0.894360i	$0.352369\pi$
$138$	0	0
$139$	−2.61803	−0.222059	−0.111029	−	0.993817i	$-0.535415\pi$
$139$	−2.61803	−0.222059	−0.111029	+	0.993817i	$0.535415\pi$
$140$	−2.61803	−0.221264
$141$	0	0
$142$	2.79837	0.234834
$143$	0	0
$144$	0	0
$145$	−0.381966	−0.0317206
$146$	−4.56231	−0.377579
$147$	0	0
$148$	−9.70820	−0.798009
$149$	−1.38197	−0.113215	−0.0566075	−	0.998397i	$-0.518028\pi$
$149$	−1.38197	−0.113215	−0.0566075	+	0.998397i	$0.518028\pi$
$150$	0	0
$151$	−6.32624	−0.514822	−0.257411	−	0.966302i	$-0.582869\pi$
$151$	−6.32624	−0.514822	−0.257411	+	0.966302i	$0.582869\pi$
$152$	14.4721	1.17385
$153$	0	0
$154$	0	0
$155$	4.85410	0.389891
$156$	0	0
$157$	−6.79837	−0.542569	−0.271285	−	0.962499i	$-0.587448\pi$
$157$	−6.79837	−0.542569	−0.271285	+	0.962499i	$0.587448\pi$
$158$	3.76393	0.299442
$159$	0	0
$160$	9.09017	0.718641
$161$	−5.70820	−0.449869
$162$	0	0
$163$	−18.3262	−1.43542	−0.717711	−	0.696341i	$-0.754810\pi$
$163$	−18.3262	−1.43542	−0.717711	+	0.696341i	$0.754810\pi$
$164$	15.9443	1.24504
$165$	0	0
$166$	5.23607	0.406398
$167$	−2.14590	−0.166055	−0.0830273	−	0.996547i	$-0.526459\pi$
$167$	−2.14590	−0.166055	−0.0830273	+	0.996547i	$0.526459\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	−3.47214	−0.266301
$171$	0	0
$172$	−18.5623	−1.41536
$173$	10.7984	0.820985	0.410493	−	0.911864i	$-0.365357\pi$
$173$	10.7984	0.820985	0.410493	+	0.911864i	$0.365357\pi$
$174$	0	0
$175$	−2.38197	−0.180060
$176$	0	0
$177$	0	0
$178$	−0.819660	−0.0614361
$179$	18.7082	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$179$	18.7082	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$180$	0	0
$181$	−0.527864	−0.0392358	−0.0196179	−	0.999808i	$-0.506245\pi$
$181$	−0.527864	−0.0392358	−0.0196179	+	0.999808i	$0.506245\pi$
$182$	−2.61803	−0.194062
$183$	0	0
$184$	12.7639	0.940970
$185$	9.70820	0.713761
$186$	0	0
$187$	0	0
$188$	−8.32624	−0.607253
$189$	0	0
$190$	−6.47214	−0.469538
$191$	16.2361	1.17480	0.587400	−	0.809297i	$-0.300151\pi$
$191$	16.2361	1.17480	0.587400	+	0.809297i	$0.300151\pi$
$192$	0	0
$193$	−17.4164	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$193$	−17.4164	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$194$	−3.27051	−0.234809
$195$	0	0
$196$	−1.61803	−0.115574
$197$	4.65248	0.331475	0.165738	−	0.986170i	$-0.447000\pi$
$197$	4.65248	0.331475	0.165738	+	0.986170i	$0.447000\pi$
$198$	0	0
$199$	−26.2148	−1.85832	−0.929158	−	0.369682i	$-0.879467\pi$
$199$	−26.2148	−1.85832	−0.929158	+	0.369682i	$0.879467\pi$
$200$	5.32624	0.376622
$201$	0	0
$202$	6.94427	0.488597
$203$	−0.236068	−0.0165687
$204$	0	0
$205$	−15.9443	−1.11360
$206$	−9.56231	−0.666237
$207$	0	0
$208$	−7.85410	−0.544584
$209$	0	0
$210$	0	0
$211$	18.7082	1.28793	0.643963	−	0.765057i	$-0.277289\pi$
$211$	18.7082	1.28793	0.643963	+	0.765057i	$0.277289\pi$
$212$	−20.3262	−1.39601
$213$	0	0
$214$	3.56231	0.243514
$215$	18.5623	1.26594
$216$	0	0
$217$	3.00000	0.203653
$218$	2.56231	0.173541
$219$	0	0
$220$	0	0
$221$	14.7082	0.989381
$222$	0	0
$223$	19.9443	1.33557	0.667784	−	0.744355i	$-0.267243\pi$
$223$	19.9443	1.33557	0.667784	+	0.744355i	$0.267243\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	6.90983	0.459635
$227$	−0.909830	−0.0603875	−0.0301938	−	0.999544i	$-0.509612\pi$
$227$	−0.909830	−0.0603875	−0.0301938	+	0.999544i	$0.509612\pi$
$228$	0	0
$229$	15.1246	0.999462	0.499731	−	0.866181i	$-0.333432\pi$
$229$	15.1246	0.999462	0.499731	+	0.866181i	$0.333432\pi$
$230$	−5.70820	−0.376388
$231$	0	0
$232$	0.527864	0.0346560
$233$	−15.7639	−1.03273	−0.516365	−	0.856369i	$-0.672715\pi$
$233$	−15.7639	−1.03273	−0.516365	+	0.856369i	$0.672715\pi$
$234$	0	0
$235$	8.32624	0.543144
$236$	−7.47214	−0.486395
$237$	0	0
$238$	−2.14590	−0.139098
$239$	−21.6180	−1.39835	−0.699177	−	0.714948i	$-0.746450\pi$
$239$	−21.6180	−1.39835	−0.699177	+	0.714948i	$0.746450\pi$
$240$	0	0
$241$	−10.4164	−0.670980	−0.335490	−	0.942044i	$-0.608902\pi$
$241$	−10.4164	−0.670980	−0.335490	+	0.942044i	$0.608902\pi$
$242$	0	0
$243$	0	0
$244$	−12.8541	−0.822900
$245$	1.61803	0.103372
$246$	0	0
$247$	27.4164	1.74446
$248$	−6.70820	−0.425971
$249$	0	0
$250$	−7.38197	−0.466877
$251$	13.9443	0.880155	0.440077	−	0.897960i	$-0.354951\pi$
$251$	13.9443	0.880155	0.440077	+	0.897960i	$0.354951\pi$
$252$	0	0
$253$	0	0
$254$	4.52786	0.284103
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−23.8885	−1.49013	−0.745063	−	0.666994i	$-0.767581\pi$
$257$	−23.8885	−1.49013	−0.745063	+	0.666994i	$0.767581\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	11.0902	0.687783
$261$	0	0
$262$	4.23607	0.261705
$263$	−9.29180	−0.572957	−0.286478	−	0.958087i	$-0.592485\pi$
$263$	−9.29180	−0.572957	−0.286478	+	0.958087i	$0.592485\pi$
$264$	0	0
$265$	20.3262	1.24863
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−21.4164	−1.30822
$269$	29.1803	1.77916	0.889578	−	0.456783i	$-0.150998\pi$
$269$	29.1803	1.77916	0.889578	+	0.456783i	$0.150998\pi$
$270$	0	0
$271$	22.1246	1.34397	0.671987	−	0.740563i	$-0.265441\pi$
$271$	22.1246	1.34397	0.671987	+	0.740563i	$0.265441\pi$
$272$	−6.43769	−0.390343
$273$	0	0
$274$	6.47214	0.390996
$275$	0	0
$276$	0	0
$277$	−8.47214	−0.509041	−0.254521	−	0.967067i	$-0.581918\pi$
$277$	−8.47214	−0.509041	−0.254521	+	0.967067i	$0.581918\pi$
$278$	−1.61803	−0.0970432
$279$	0	0
$280$	−3.61803	−0.216219
$281$	10.8541	0.647501	0.323751	−	0.946142i	$-0.395056\pi$
$281$	10.8541	0.647501	0.323751	+	0.946142i	$0.395056\pi$
$282$	0	0
$283$	−12.8885	−0.766144	−0.383072	−	0.923718i	$-0.625134\pi$
$283$	−12.8885	−0.766144	−0.383072	+	0.923718i	$0.625134\pi$
$284$	−7.32624	−0.434732
$285$	0	0
$286$	0	0
$287$	−9.85410	−0.581669
$288$	0	0
$289$	−4.94427	−0.290840
$290$	−0.236068	−0.0138624
$291$	0	0
$292$	11.9443	0.698986
$293$	−7.70820	−0.450318	−0.225159	−	0.974322i	$-0.572290\pi$
$293$	−7.70820	−0.450318	−0.225159	+	0.974322i	$0.572290\pi$
$294$	0	0
$295$	7.47214	0.435045
$296$	−13.4164	−0.779813
$297$	0	0
$298$	−0.854102	−0.0494768
$299$	24.1803	1.39839
$300$	0	0
$301$	11.4721	0.661243
$302$	−3.90983	−0.224985
$303$	0	0
$304$	−12.0000	−0.688247
$305$	12.8541	0.736024
$306$	0	0
$307$	−8.05573	−0.459765	−0.229882	−	0.973218i	$-0.573834\pi$
$307$	−8.05573	−0.459765	−0.229882	+	0.973218i	$0.573834\pi$
$308$	0	0
$309$	0	0
$310$	3.00000	0.170389
$311$	−6.79837	−0.385500	−0.192750	−	0.981248i	$-0.561741\pi$
$311$	−6.79837	−0.385500	−0.192750	+	0.981248i	$0.561741\pi$
$312$	0	0
$313$	32.0902	1.81384	0.906922	−	0.421299i	$-0.138426\pi$
$313$	32.0902	1.81384	0.906922	+	0.421299i	$0.138426\pi$
$314$	−4.20163	−0.237111
$315$	0	0
$316$	−9.85410	−0.554337
$317$	34.8885	1.95954	0.979768	−	0.200137i	$-0.0641387\pi$
$317$	34.8885	1.95954	0.979768	+	0.200137i	$0.0641387\pi$
$318$	0	0
$319$	0	0
$320$	−0.381966	−0.0213525
$321$	0	0
$322$	−3.52786	−0.196600
$323$	22.4721	1.25038
$324$	0	0
$325$	10.0902	0.559702
$326$	−11.3262	−0.627302
$327$	0	0
$328$	22.0344	1.21665
$329$	5.14590	0.283703
$330$	0	0
$331$	−19.2918	−1.06037	−0.530187	−	0.847881i	$-0.677878\pi$
$331$	−19.2918	−1.06037	−0.530187	+	0.847881i	$0.677878\pi$
$332$	−13.7082	−0.752335
$333$	0	0
$334$	−1.32624	−0.0725685
$335$	21.4164	1.17010
$336$	0	0
$337$	4.23607	0.230753	0.115377	−	0.993322i	$-0.463192\pi$
$337$	4.23607	0.230753	0.115377	+	0.993322i	$0.463192\pi$
$338$	3.05573	0.166210
$339$	0	0
$340$	9.09017	0.492984
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−25.6525	−1.38309
$345$	0	0
$346$	6.67376	0.358784
$347$	−0.472136	−0.0253456	−0.0126728	−	0.999920i	$-0.504034\pi$
$347$	−0.472136	−0.0253456	−0.0126728	+	0.999920i	$0.504034\pi$
$348$	0	0
$349$	17.3262	0.927452	0.463726	−	0.885979i	$-0.346512\pi$
$349$	17.3262	0.927452	0.463726	+	0.885979i	$0.346512\pi$
$350$	−1.47214	−0.0786890
$351$	0	0
$352$	0	0
$353$	4.12461	0.219531	0.109765	−	0.993958i	$-0.464990\pi$
$353$	4.12461	0.219531	0.109765	+	0.993958i	$0.464990\pi$
$354$	0	0
$355$	7.32624	0.388836
$356$	2.14590	0.113732
$357$	0	0
$358$	11.5623	0.611087
$359$	10.3820	0.547939	0.273970	−	0.961738i	$-0.411663\pi$
$359$	10.3820	0.547939	0.273970	+	0.961738i	$0.411663\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−0.326238	−0.0171467
$363$	0	0
$364$	6.85410	0.359253
$365$	−11.9443	−0.625192
$366$	0	0
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	−10.5836	−0.551708
$369$	0	0
$370$	6.00000	0.311925
$371$	12.5623	0.652202
$372$	0	0
$373$	20.2918	1.05067	0.525335	−	0.850896i	$-0.323940\pi$
$373$	20.2918	1.05067	0.525335	+	0.850896i	$0.323940\pi$
$374$	0	0
$375$	0	0
$376$	−11.5066	−0.593406
$377$	1.00000	0.0515026
$378$	0	0
$379$	−1.90983	−0.0981014	−0.0490507	−	0.998796i	$-0.515620\pi$
$379$	−1.90983	−0.0981014	−0.0490507	+	0.998796i	$0.515620\pi$
$380$	16.9443	0.869223
$381$	0	0
$382$	10.0344	0.513407
$383$	2.67376	0.136623	0.0683114	−	0.997664i	$-0.478239\pi$
$383$	2.67376	0.136623	0.0683114	+	0.997664i	$0.478239\pi$
$384$	0	0
$385$	0	0
$386$	−10.7639	−0.547870
$387$	0	0
$388$	8.56231	0.434685
$389$	17.2361	0.873903	0.436952	−	0.899485i	$-0.356058\pi$
$389$	17.2361	0.873903	0.436952	+	0.899485i	$0.356058\pi$
$390$	0	0
$391$	19.8197	1.00232
$392$	−2.23607	−0.112938
$393$	0	0
$394$	2.87539	0.144860
$395$	9.85410	0.495814
$396$	0	0
$397$	−10.5623	−0.530107	−0.265053	−	0.964234i	$-0.585390\pi$
$397$	−10.5623	−0.530107	−0.265053	+	0.964234i	$0.585390\pi$
$398$	−16.2016	−0.812114
$399$	0	0
$400$	−4.41641	−0.220820
$401$	−27.7426	−1.38540	−0.692701	−	0.721225i	$-0.743579\pi$
$401$	−27.7426	−1.38540	−0.692701	+	0.721225i	$0.743579\pi$
$402$	0	0
$403$	−12.7082	−0.633041
$404$	−18.1803	−0.904506
$405$	0	0
$406$	−0.145898	−0.00724080
$407$	0	0
$408$	0	0
$409$	−8.20163	−0.405544	−0.202772	−	0.979226i	$-0.564995\pi$
$409$	−8.20163	−0.405544	−0.202772	+	0.979226i	$0.564995\pi$
$410$	−9.85410	−0.486659
$411$	0	0
$412$	25.0344	1.23336
$413$	4.61803	0.227239
$414$	0	0
$415$	13.7082	0.672909
$416$	−23.7984	−1.16681
$417$	0	0
$418$	0	0
$419$	35.6180	1.74005	0.870027	−	0.493003i	$-0.164101\pi$
$419$	35.6180	1.74005	0.870027	+	0.493003i	$0.164101\pi$
$420$	0	0
$421$	18.1459	0.884377	0.442188	−	0.896922i	$-0.354202\pi$
$421$	18.1459	0.884377	0.442188	+	0.896922i	$0.354202\pi$
$422$	11.5623	0.562844
$423$	0	0
$424$	−28.0902	−1.36418
$425$	8.27051	0.401179
$426$	0	0
$427$	7.94427	0.384450
$428$	−9.32624	−0.450801
$429$	0	0
$430$	11.4721	0.553236
$431$	−7.14590	−0.344206	−0.172103	−	0.985079i	$-0.555056\pi$
$431$	−7.14590	−0.344206	−0.172103	+	0.985079i	$0.555056\pi$
$432$	0	0
$433$	−6.94427	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$433$	−6.94427	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$434$	1.85410	0.0889997
$435$	0	0
$436$	−6.70820	−0.321265
$437$	36.9443	1.76728
$438$	0	0
$439$	−35.6525	−1.70160	−0.850800	−	0.525490i	$-0.823882\pi$
$439$	−35.6525	−1.70160	−0.850800	+	0.525490i	$0.823882\pi$
$440$	0	0
$441$	0	0
$442$	9.09017	0.432375
$443$	−24.6525	−1.17127	−0.585637	−	0.810573i	$-0.699156\pi$
$443$	−24.6525	−1.17127	−0.585637	+	0.810573i	$0.699156\pi$
$444$	0	0
$445$	−2.14590	−0.101725
$446$	12.3262	0.583664
$447$	0	0
$448$	−0.236068	−0.0111532
$449$	20.5967	0.972021	0.486010	−	0.873953i	$-0.338452\pi$
$449$	20.5967	0.972021	0.486010	+	0.873953i	$0.338452\pi$
$450$	0	0
$451$	0	0
$452$	−18.0902	−0.850890
$453$	0	0
$454$	−0.562306	−0.0263903
$455$	−6.85410	−0.321325
$456$	0	0
$457$	−18.7426	−0.876744	−0.438372	−	0.898794i	$-0.644445\pi$
$457$	−18.7426	−0.876744	−0.438372	+	0.898794i	$0.644445\pi$
$458$	9.34752	0.436781
$459$	0	0
$460$	14.9443	0.696780
$461$	28.3050	1.31829	0.659147	−	0.752015i	$-0.270918\pi$
$461$	28.3050	1.31829	0.659147	+	0.752015i	$0.270918\pi$
$462$	0	0
$463$	−14.9787	−0.696120	−0.348060	−	0.937472i	$-0.613159\pi$
$463$	−14.9787	−0.696120	−0.348060	+	0.937472i	$0.613159\pi$
$464$	−0.437694	−0.0203194
$465$	0	0
$466$	−9.74265	−0.451319
$467$	26.7082	1.23591	0.617954	−	0.786214i	$-0.287962\pi$
$467$	26.7082	1.23591	0.617954	+	0.786214i	$0.287962\pi$
$468$	0	0
$469$	13.2361	0.611185
$470$	5.14590	0.237363
$471$	0	0
$472$	−10.3262	−0.475304
$473$	0	0
$474$	0	0
$475$	15.4164	0.707353
$476$	5.61803	0.257502
$477$	0	0
$478$	−13.3607	−0.611103
$479$	30.3820	1.38819	0.694094	−	0.719885i	$-0.255805\pi$
$479$	30.3820	1.38819	0.694094	+	0.719885i	$0.255805\pi$
$480$	0	0
$481$	−25.4164	−1.15889
$482$	−6.43769	−0.293229
$483$	0	0
$484$	0	0
$485$	−8.56231	−0.388794
$486$	0	0
$487$	−15.3820	−0.697023	−0.348512	−	0.937304i	$-0.613313\pi$
$487$	−15.3820	−0.697023	−0.348512	+	0.937304i	$0.613313\pi$
$488$	−17.7639	−0.804135
$489$	0	0
$490$	1.00000	0.0451754
$491$	−8.29180	−0.374204	−0.187102	−	0.982341i	$-0.559909\pi$
$491$	−8.29180	−0.374204	−0.187102	+	0.982341i	$0.559909\pi$
$492$	0	0
$493$	0.819660	0.0369156
$494$	16.9443	0.762359
$495$	0	0
$496$	5.56231	0.249755
$497$	4.52786	0.203102
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	19.3262	0.864296
$501$	0	0
$502$	8.61803	0.384642
$503$	−42.0132	−1.87327	−0.936637	−	0.350301i	$-0.886079\pi$
$503$	−42.0132	−1.87327	−0.936637	+	0.350301i	$0.886079\pi$
$504$	0	0
$505$	18.1803	0.809015
$506$	0	0
$507$	0	0
$508$	−11.8541	−0.525941
$509$	−12.1246	−0.537414	−0.268707	−	0.963222i	$-0.586596\pi$
$509$	−12.1246	−0.537414	−0.268707	+	0.963222i	$0.586596\pi$
$510$	0	0
$511$	−7.38197	−0.326559
$512$	18.7082	0.826794
$513$	0	0
$514$	−14.7639	−0.651209
$515$	−25.0344	−1.10315
$516$	0	0
$517$	0	0
$518$	3.70820	0.162929
$519$	0	0
$520$	15.3262	0.672100
$521$	−8.41641	−0.368730	−0.184365	−	0.982858i	$-0.559023\pi$
$521$	−8.41641	−0.368730	−0.184365	+	0.982858i	$0.559023\pi$
$522$	0	0
$523$	−29.5410	−1.29174	−0.645869	−	0.763448i	$-0.723505\pi$
$523$	−29.5410	−1.29174	−0.645869	+	0.763448i	$0.723505\pi$
$524$	−11.0902	−0.484476
$525$	0	0
$526$	−5.74265	−0.250391
$527$	−10.4164	−0.453746
$528$	0	0
$529$	9.58359	0.416678
$530$	12.5623	0.545672
$531$	0	0
$532$	10.4721	0.454025
$533$	41.7426	1.80807
$534$	0	0
$535$	9.32624	0.403208
$536$	−29.5967	−1.27838
$537$	0	0
$538$	18.0344	0.777520
$539$	0	0
$540$	0	0
$541$	10.9656	0.471446	0.235723	−	0.971820i	$-0.424254\pi$
$541$	10.9656	0.471446	0.235723	+	0.971820i	$0.424254\pi$
$542$	13.6738	0.587338
$543$	0	0
$544$	−19.5066	−0.836338
$545$	6.70820	0.287348
$546$	0	0
$547$	19.0689	0.815327	0.407663	−	0.913132i	$-0.366344\pi$
$547$	19.0689	0.815327	0.407663	+	0.913132i	$0.366344\pi$
$548$	−16.9443	−0.723823
$549$	0	0
$550$	0	0
$551$	1.52786	0.0650892
$552$	0	0
$553$	6.09017	0.258980
$554$	−5.23607	−0.222459
$555$	0	0
$556$	4.23607	0.179649
$557$	−20.5410	−0.870351	−0.435175	−	0.900346i	$-0.643314\pi$
$557$	−20.5410	−0.870351	−0.435175	+	0.900346i	$0.643314\pi$
$558$	0	0
$559$	−48.5967	−2.05542
$560$	3.00000	0.126773
$561$	0	0
$562$	6.70820	0.282969
$563$	−27.9443	−1.17771	−0.588855	−	0.808238i	$-0.700421\pi$
$563$	−27.9443	−1.17771	−0.588855	+	0.808238i	$0.700421\pi$
$564$	0	0
$565$	18.0902	0.761059
$566$	−7.96556	−0.334817
$567$	0	0
$568$	−10.1246	−0.424819
$569$	−38.7082	−1.62273	−0.811366	−	0.584538i	$-0.801276\pi$
$569$	−38.7082	−1.62273	−0.811366	+	0.584538i	$0.801276\pi$
$570$	0	0
$571$	−35.9443	−1.50422	−0.752110	−	0.659037i	$-0.770964\pi$
$571$	−35.9443	−1.50422	−0.752110	+	0.659037i	$0.770964\pi$
$572$	0	0
$573$	0	0
$574$	−6.09017	−0.254199
$575$	13.5967	0.567024
$576$	0	0
$577$	37.1459	1.54640	0.773202	−	0.634160i	$-0.218654\pi$
$577$	37.1459	1.54640	0.773202	+	0.634160i	$0.218654\pi$
$578$	−3.05573	−0.127102
$579$	0	0
$580$	0.618034	0.0256625
$581$	8.47214	0.351483
$582$	0	0
$583$	0	0
$584$	16.5066	0.683047
$585$	0	0
$586$	−4.76393	−0.196796
$587$	32.2492	1.33107	0.665534	−	0.746368i	$-0.268204\pi$
$587$	32.2492	1.33107	0.665534	+	0.746368i	$0.268204\pi$
$588$	0	0
$589$	−19.4164	−0.800039
$590$	4.61803	0.190121
$591$	0	0
$592$	11.1246	0.457219
$593$	10.0000	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$594$	0	0
$595$	−5.61803	−0.230317
$596$	2.23607	0.0915929
$597$	0	0
$598$	14.9443	0.611117
$599$	14.5623	0.595000	0.297500	−	0.954722i	$-0.403847\pi$
$599$	14.5623	0.595000	0.297500	+	0.954722i	$0.403847\pi$
$600$	0	0
$601$	4.58359	0.186969	0.0934843	−	0.995621i	$-0.470200\pi$
$601$	4.58359	0.186969	0.0934843	+	0.995621i	$0.470200\pi$
$602$	7.09017	0.288974
$603$	0	0
$604$	10.2361	0.416500
$605$	0	0
$606$	0	0
$607$	23.1591	0.939997	0.469998	−	0.882667i	$-0.344254\pi$
$607$	23.1591	0.939997	0.469998	+	0.882667i	$0.344254\pi$
$608$	−36.3607	−1.47462
$609$	0	0
$610$	7.94427	0.321654
$611$	−21.7984	−0.881868
$612$	0	0
$613$	−24.5623	−0.992062	−0.496031	−	0.868305i	$-0.665210\pi$
$613$	−24.5623	−0.992062	−0.496031	+	0.868305i	$0.665210\pi$
$614$	−4.97871	−0.200925
$615$	0	0
$616$	0	0
$617$	1.32624	0.0533923	0.0266962	−	0.999644i	$-0.491501\pi$
$617$	1.32624	0.0533923	0.0266962	+	0.999644i	$0.491501\pi$
$618$	0	0
$619$	−22.1803	−0.891503	−0.445752	−	0.895157i	$-0.647064\pi$
$619$	−22.1803	−0.891503	−0.445752	+	0.895157i	$0.647064\pi$
$620$	−7.85410	−0.315428
$621$	0	0
$622$	−4.20163	−0.168470
$623$	−1.32624	−0.0531346
$624$	0	0
$625$	−7.41641	−0.296656
$626$	19.8328	0.792679
$627$	0	0
$628$	11.0000	0.438948
$629$	−20.8328	−0.830659
$630$	0	0
$631$	0.708204	0.0281932	0.0140966	−	0.999901i	$-0.495513\pi$
$631$	0.708204	0.0281932	0.0140966	+	0.999901i	$0.495513\pi$
$632$	−13.6180	−0.541696
$633$	0	0
$634$	21.5623	0.856349
$635$	11.8541	0.470416
$636$	0	0
$637$	−4.23607	−0.167839
$638$	0	0
$639$	0	0
$640$	−18.4164	−0.727972
$641$	−9.32624	−0.368364	−0.184182	−	0.982892i	$-0.558964\pi$
$641$	−9.32624	−0.368364	−0.184182	+	0.982892i	$0.558964\pi$
$642$	0	0
$643$	5.88854	0.232221	0.116111	−	0.993236i	$-0.462957\pi$
$643$	5.88854	0.232221	0.116111	+	0.993236i	$0.462957\pi$
$644$	9.23607	0.363952
$645$	0	0
$646$	13.8885	0.546437
$647$	−14.8885	−0.585329	−0.292665	−	0.956215i	$-0.594542\pi$
$647$	−14.8885	−0.585329	−0.292665	+	0.956215i	$0.594542\pi$
$648$	0	0
$649$	0	0
$650$	6.23607	0.244599
$651$	0	0
$652$	29.6525	1.16128
$653$	31.1803	1.22018	0.610090	−	0.792332i	$-0.291133\pi$
$653$	31.1803	1.22018	0.610090	+	0.792332i	$0.291133\pi$
$654$	0	0
$655$	11.0902	0.433329
$656$	−18.2705	−0.713344
$657$	0	0
$658$	3.18034	0.123983
$659$	−20.8885	−0.813702	−0.406851	−	0.913495i	$-0.633373\pi$
$659$	−20.8885	−0.813702	−0.406851	+	0.913495i	$0.633373\pi$
$660$	0	0
$661$	18.5967	0.723330	0.361665	−	0.932308i	$-0.382208\pi$
$661$	18.5967	0.723330	0.361665	+	0.932308i	$0.382208\pi$
$662$	−11.9230	−0.463400
$663$	0	0
$664$	−18.9443	−0.735180
$665$	−10.4721	−0.406092
$666$	0	0
$667$	1.34752	0.0521763
$668$	3.47214	0.134341
$669$	0	0
$670$	13.2361	0.511354
$671$	0	0
$672$	0	0
$673$	18.0557	0.695997	0.347999	−	0.937495i	$-0.386861\pi$
$673$	18.0557	0.695997	0.347999	+	0.937495i	$0.386861\pi$
$674$	2.61803	0.100843
$675$	0	0
$676$	−8.00000	−0.307692
$677$	13.6738	0.525525	0.262763	−	0.964860i	$-0.415366\pi$
$677$	13.6738	0.525525	0.262763	+	0.964860i	$0.415366\pi$
$678$	0	0
$679$	−5.29180	−0.203080
$680$	12.5623	0.481742
$681$	0	0
$682$	0	0
$683$	15.4721	0.592025	0.296012	−	0.955184i	$-0.404343\pi$
$683$	15.4721	0.592025	0.296012	+	0.955184i	$0.404343\pi$
$684$	0	0
$685$	16.9443	0.647407
$686$	0.618034	0.0235966
$687$	0	0
$688$	21.2705	0.810931
$689$	−53.2148	−2.02732
$690$	0	0
$691$	−0.965558	−0.0367316	−0.0183658	−	0.999831i	$-0.505846\pi$
$691$	−0.965558	−0.0367316	−0.0183658	+	0.999831i	$0.505846\pi$
$692$	−17.4721	−0.664191
$693$	0	0
$694$	−0.291796	−0.0110764
$695$	−4.23607	−0.160683
$696$	0	0
$697$	34.2148	1.29598
$698$	10.7082	0.405311
$699$	0	0
$700$	3.85410	0.145671
$701$	33.9443	1.28206	0.641029	−	0.767517i	$-0.278508\pi$
$701$	33.9443	1.28206	0.641029	+	0.767517i	$0.278508\pi$
$702$	0	0
$703$	−38.8328	−1.46461
$704$	0	0
$705$	0	0
$706$	2.54915	0.0959385
$707$	11.2361	0.422576
$708$	0	0
$709$	−5.70820	−0.214376	−0.107188	−	0.994239i	$-0.534185\pi$
$709$	−5.70820	−0.214376	−0.107188	+	0.994239i	$0.534185\pi$
$710$	4.52786	0.169928
$711$	0	0
$712$	2.96556	0.111139
$713$	−17.1246	−0.641322
$714$	0	0
$715$	0	0
$716$	−30.2705	−1.13126
$717$	0	0
$718$	6.41641	0.239458
$719$	33.4508	1.24751	0.623753	−	0.781621i	$-0.285607\pi$
$719$	33.4508	1.24751	0.623753	+	0.781621i	$0.285607\pi$
$720$	0	0
$721$	−15.4721	−0.576212
$722$	14.1459	0.526456
$723$	0	0
$724$	0.854102	0.0317424
$725$	0.562306	0.0208835
$726$	0	0
$727$	−2.03444	−0.0754533	−0.0377266	−	0.999288i	$-0.512012\pi$
$727$	−2.03444	−0.0754533	−0.0377266	+	0.999288i	$0.512012\pi$
$728$	9.47214	0.351061
$729$	0	0
$730$	−7.38197	−0.273219
$731$	−39.8328	−1.47327
$732$	0	0
$733$	−17.5836	−0.649465	−0.324732	−	0.945806i	$-0.605274\pi$
$733$	−17.5836	−0.649465	−0.324732	+	0.945806i	$0.605274\pi$
$734$	−3.09017	−0.114060
$735$	0	0
$736$	−32.0689	−1.18207
$737$	0	0
$738$	0	0
$739$	40.1459	1.47679	0.738395	−	0.674368i	$-0.235584\pi$
$739$	40.1459	1.47679	0.738395	+	0.674368i	$0.235584\pi$
$740$	−15.7082	−0.577445
$741$	0	0
$742$	7.76393	0.285023
$743$	38.0902	1.39739	0.698696	−	0.715418i	$-0.253764\pi$
$743$	38.0902	1.39739	0.698696	+	0.715418i	$0.253764\pi$
$744$	0	0
$745$	−2.23607	−0.0819232
$746$	12.5410	0.459159
$747$	0	0
$748$	0	0
$749$	5.76393	0.210609
$750$	0	0
$751$	50.5623	1.84504	0.922522	−	0.385944i	$-0.126124\pi$
$751$	50.5623	1.84504	0.922522	+	0.385944i	$0.126124\pi$
$752$	9.54102	0.347925
$753$	0	0
$754$	0.618034	0.0225075
$755$	−10.2361	−0.372529
$756$	0	0
$757$	51.1246	1.85816	0.929078	−	0.369884i	$-0.120603\pi$
$757$	51.1246	1.85816	0.929078	+	0.369884i	$0.120603\pi$
$758$	−1.18034	−0.0428719
$759$	0	0
$760$	23.4164	0.849402
$761$	30.8541	1.11846	0.559230	−	0.829012i	$-0.311097\pi$
$761$	30.8541	1.11846	0.559230	+	0.829012i	$0.311097\pi$
$762$	0	0
$763$	4.14590	0.150092
$764$	−26.2705	−0.950434
$765$	0	0
$766$	1.65248	0.0597064
$767$	−19.5623	−0.706354
$768$	0	0
$769$	53.7771	1.93925	0.969626	−	0.244594i	$-0.0786545\pi$
$769$	53.7771	1.93925	0.969626	+	0.244594i	$0.0786545\pi$
$770$	0	0
$771$	0	0
$772$	28.1803	1.01423
$773$	30.9443	1.11299	0.556494	−	0.830852i	$-0.312146\pi$
$773$	30.9443	1.11299	0.556494	+	0.830852i	$0.312146\pi$
$774$	0	0
$775$	−7.14590	−0.256688
$776$	11.8328	0.424773
$777$	0	0
$778$	10.6525	0.381910
$779$	63.7771	2.28505
$780$	0	0
$781$	0	0
$782$	12.2492	0.438031
$783$	0	0
$784$	1.85410	0.0662179
$785$	−11.0000	−0.392607
$786$	0	0
$787$	−0.562306	−0.0200440	−0.0100220	−	0.999950i	$-0.503190\pi$
$787$	−0.562306	−0.0200440	−0.0100220	+	0.999950i	$0.503190\pi$
$788$	−7.52786	−0.268169
$789$	0	0
$790$	6.09017	0.216679
$791$	11.1803	0.397527
$792$	0	0
$793$	−33.6525	−1.19503
$794$	−6.52786	−0.231665
$795$	0	0
$796$	42.4164	1.50341
$797$	−37.1459	−1.31578	−0.657888	−	0.753116i	$-0.728550\pi$
$797$	−37.1459	−1.31578	−0.657888	+	0.753116i	$0.728550\pi$
$798$	0	0
$799$	−17.8673	−0.632098
$800$	−13.3820	−0.473124
$801$	0	0
$802$	−17.1459	−0.605443
$803$	0	0
$804$	0	0
$805$	−9.23607	−0.325529
$806$	−7.85410	−0.276649
$807$	0	0
$808$	−25.1246	−0.883881
$809$	−54.3951	−1.91243	−0.956215	−	0.292664i	$-0.905458\pi$
$809$	−54.3951	−1.91243	−0.956215	+	0.292664i	$0.905458\pi$
$810$	0	0
$811$	−47.7082	−1.67526	−0.837631	−	0.546237i	$-0.816060\pi$
$811$	−47.7082	−1.67526	−0.837631	+	0.546237i	$0.816060\pi$
$812$	0.381966	0.0134044
$813$	0	0
$814$	0	0
$815$	−29.6525	−1.03868
$816$	0	0
$817$	−74.2492	−2.59765
$818$	−5.06888	−0.177229
$819$	0	0
$820$	25.7984	0.900918
$821$	−13.8197	−0.482309	−0.241155	−	0.970487i	$-0.577526\pi$
$821$	−13.8197	−0.482309	−0.241155	+	0.970487i	$0.577526\pi$
$822$	0	0
$823$	22.4721	0.783329	0.391665	−	0.920108i	$-0.371899\pi$
$823$	22.4721	0.783329	0.391665	+	0.920108i	$0.371899\pi$
$824$	34.5967	1.20523
$825$	0	0
$826$	2.85410	0.0993069
$827$	41.9443	1.45855	0.729273	−	0.684223i	$-0.239859\pi$
$827$	41.9443	1.45855	0.729273	+	0.684223i	$0.239859\pi$
$828$	0	0
$829$	−0.673762	−0.0234007	−0.0117004	−	0.999932i	$-0.503724\pi$
$829$	−0.673762	−0.0234007	−0.0117004	+	0.999932i	$0.503724\pi$
$830$	8.47214	0.294072
$831$	0	0
$832$	1.00000	0.0346688
$833$	−3.47214	−0.120302
$834$	0	0
$835$	−3.47214	−0.120158
$836$	0	0
$837$	0	0
$838$	22.0132	0.760432
$839$	14.5279	0.501558	0.250779	−	0.968044i	$-0.419313\pi$
$839$	14.5279	0.501558	0.250779	+	0.968044i	$0.419313\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	11.2148	0.386487
$843$	0	0
$844$	−30.2705	−1.04195
$845$	8.00000	0.275208
$846$	0	0
$847$	0	0
$848$	23.2918	0.799844
$849$	0	0
$850$	5.11146	0.175322
$851$	−34.2492	−1.17405
$852$	0	0
$853$	−24.7984	−0.849080	−0.424540	−	0.905409i	$-0.639564\pi$
$853$	−24.7984	−0.849080	−0.424540	+	0.905409i	$0.639564\pi$
$854$	4.90983	0.168011
$855$	0	0
$856$	−12.8885	−0.440521
$857$	−31.9443	−1.09120	−0.545598	−	0.838047i	$-0.683697\pi$
$857$	−31.9443	−1.09120	−0.545598	+	0.838047i	$0.683697\pi$
$858$	0	0
$859$	1.65248	0.0563817	0.0281909	−	0.999603i	$-0.491025\pi$
$859$	1.65248	0.0563817	0.0281909	+	0.999603i	$0.491025\pi$
$860$	−30.0344	−1.02417
$861$	0	0
$862$	−4.41641	−0.150423
$863$	−31.2492	−1.06374	−0.531868	−	0.846827i	$-0.678510\pi$
$863$	−31.2492	−1.06374	−0.531868	+	0.846827i	$0.678510\pi$
$864$	0	0
$865$	17.4721	0.594070
$866$	−4.29180	−0.145841
$867$	0	0
$868$	−4.85410	−0.164759
$869$	0	0
$870$	0	0
$871$	−56.0689	−1.89982
$872$	−9.27051	−0.313939
$873$	0	0
$874$	22.8328	0.772332
$875$	−11.9443	−0.403790
$876$	0	0
$877$	12.1459	0.410138	0.205069	−	0.978748i	$-0.434258\pi$
$877$	12.1459	0.410138	0.205069	+	0.978748i	$0.434258\pi$
$878$	−22.0344	−0.743626
$879$	0	0
$880$	0	0
$881$	5.29180	0.178285	0.0891426	−	0.996019i	$-0.471587\pi$
$881$	5.29180	0.178285	0.0891426	+	0.996019i	$0.471587\pi$
$882$	0	0
$883$	−44.0902	−1.48375	−0.741876	−	0.670537i	$-0.766064\pi$
$883$	−44.0902	−1.48375	−0.741876	+	0.670537i	$0.766064\pi$
$884$	−23.7984	−0.800426
$885$	0	0
$886$	−15.2361	−0.511866
$887$	38.7082	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$887$	38.7082	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$888$	0	0
$889$	7.32624	0.245714
$890$	−1.32624	−0.0444556
$891$	0	0
$892$	−32.2705	−1.08050
$893$	−33.3050	−1.11451
$894$	0	0
$895$	30.2705	1.01183
$896$	−11.3820	−0.380245
$897$	0	0
$898$	12.7295	0.424789
$899$	−0.708204	−0.0236199
$900$	0	0
$901$	−43.6180	−1.45313
$902$	0	0
$903$	0	0
$904$	−25.0000	−0.831488
$905$	−0.854102	−0.0283913
$906$	0	0
$907$	−11.4934	−0.381633	−0.190816	−	0.981626i	$-0.561114\pi$
$907$	−11.4934	−0.381633	−0.190816	+	0.981626i	$0.561114\pi$
$908$	1.47214	0.0488545
$909$	0	0
$910$	−4.23607	−0.140424
$911$	31.9098	1.05722	0.528610	−	0.848865i	$-0.322713\pi$
$911$	31.9098	1.05722	0.528610	+	0.848865i	$0.322713\pi$
$912$	0	0
$913$	0	0
$914$	−11.5836	−0.383151
$915$	0	0
$916$	−24.4721	−0.808582
$917$	6.85410	0.226342
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	20.6525	0.680892
$921$	0	0
$922$	17.4934	0.576115
$923$	−19.1803	−0.631329
$924$	0	0
$925$	−14.2918	−0.469911
$926$	−9.25735	−0.304216
$927$	0	0
$928$	−1.32624	−0.0435359
$929$	9.88854	0.324433	0.162216	−	0.986755i	$-0.448136\pi$
$929$	9.88854	0.324433	0.162216	+	0.986755i	$0.448136\pi$
$930$	0	0
$931$	−6.47214	−0.212116
$932$	25.5066	0.835496
$933$	0	0
$934$	16.5066	0.540112
$935$	0	0
$936$	0	0
$937$	22.1246	0.722780	0.361390	−	0.932415i	$-0.382302\pi$
$937$	22.1246	0.722780	0.361390	+	0.932415i	$0.382302\pi$
$938$	8.18034	0.267098
$939$	0	0
$940$	−13.4721	−0.439413
$941$	−2.40325	−0.0783438	−0.0391719	−	0.999232i	$-0.512472\pi$
$941$	−2.40325	−0.0783438	−0.0391719	+	0.999232i	$0.512472\pi$
$942$	0	0
$943$	56.2492	1.83173
$944$	8.56231	0.278679
$945$	0	0
$946$	0	0
$947$	−4.97871	−0.161786	−0.0808932	−	0.996723i	$-0.525777\pi$
$947$	−4.97871	−0.161786	−0.0808932	+	0.996723i	$0.525777\pi$
$948$	0	0
$949$	31.2705	1.01508
$950$	9.52786	0.309125
$951$	0	0
$952$	7.76393	0.251630
$953$	−10.0344	−0.325047	−0.162524	−	0.986705i	$-0.551963\pi$
$953$	−10.0344	−0.325047	−0.162524	+	0.986705i	$0.551963\pi$
$954$	0	0
$955$	26.2705	0.850094
$956$	34.9787	1.13129
$957$	0	0
$958$	18.7771	0.606660
$959$	10.4721	0.338163
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−15.7082	−0.506453
$963$	0	0
$964$	16.8541	0.542834
$965$	−28.1803	−0.907157
$966$	0	0
$967$	48.6525	1.56456	0.782279	−	0.622928i	$-0.214057\pi$
$967$	48.6525	1.56456	0.782279	+	0.622928i	$0.214057\pi$
$968$	0	0
$969$	0	0
$970$	−5.29180	−0.169909
$971$	10.4164	0.334278	0.167139	−	0.985933i	$-0.446547\pi$
$971$	10.4164	0.334278	0.167139	+	0.985933i	$0.446547\pi$
$972$	0	0
$973$	−2.61803	−0.0839303
$974$	−9.50658	−0.304610
$975$	0	0
$976$	14.7295	0.471479
$977$	−42.7771	−1.36856	−0.684280	−	0.729219i	$-0.739883\pi$
$977$	−42.7771	−1.36856	−0.684280	+	0.729219i	$0.739883\pi$
$978$	0	0
$979$	0	0
$980$	−2.61803	−0.0836300
$981$	0	0
$982$	−5.12461	−0.163533
$983$	−50.5967	−1.61379	−0.806893	−	0.590698i	$-0.798853\pi$
$983$	−50.5967	−1.61379	−0.806893	+	0.590698i	$0.798853\pi$
$984$	0	0
$985$	7.52786	0.239858
$986$	0.506578	0.0161327
$987$	0	0
$988$	−44.3607	−1.41130
$989$	−65.4853	−2.08231
$990$	0	0
$991$	−28.6180	−0.909082	−0.454541	−	0.890726i	$-0.650197\pi$
$991$	−28.6180	−0.909082	−0.454541	+	0.890726i	$0.650197\pi$
$992$	16.8541	0.535118
$993$	0	0
$994$	2.79837	0.0887590
$995$	−42.4164	−1.34469
$996$	0	0
$997$	41.1935	1.30461	0.652306	−	0.757956i	$-0.273802\pi$
$997$	41.1935	1.30461	0.652306	+	0.757956i	$0.273802\pi$
$998$	4.32624	0.136945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.z.1.2		2
3.2	odd	2		2541.2.a.x.1.1		2
11.2	odd	10		693.2.m.d.631.1		4
11.6	odd	10		693.2.m.d.190.1		4
11.10	odd	2		7623.2.a.bo.1.1		2
33.2	even	10		231.2.j.b.169.1	✓	4
33.17	even	10		231.2.j.b.190.1	yes	4
33.32	even	2		2541.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.b.169.1	✓	4	33.2	even	10
231.2.j.b.190.1	yes	4	33.17	even	10
693.2.m.d.190.1		4	11.6	odd	10
693.2.m.d.631.1		4	11.2	odd	10
2541.2.a.p.1.2		2	33.32	even	2
2541.2.a.x.1.1		2	3.2	odd	2
7623.2.a.z.1.2		2	1.1	even	1	trivial
7623.2.a.bo.1.1		2	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{5} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{10} -4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} -3.47214 q^{17} -6.47214 q^{19} -2.61803 q^{20} -5.70820 q^{23} -2.38197 q^{25} -2.61803 q^{26} -1.61803 q^{28} -0.236068 q^{29} +3.00000 q^{31} +5.61803 q^{32} -2.14590 q^{34} +1.61803 q^{35} +6.00000 q^{37} -4.00000 q^{38} -3.61803 q^{40} -9.85410 q^{41} +11.4721 q^{43} -3.52786 q^{46} +5.14590 q^{47} +1.00000 q^{49} -1.47214 q^{50} +6.85410 q^{52} +12.5623 q^{53} -2.23607 q^{56} -0.145898 q^{58} +4.61803 q^{59} +7.94427 q^{61} +1.85410 q^{62} -0.236068 q^{64} -6.85410 q^{65} +13.2361 q^{67} +5.61803 q^{68} +1.00000 q^{70} +4.52786 q^{71} -7.38197 q^{73} +3.70820 q^{74} +10.4721 q^{76} +6.09017 q^{79} +3.00000 q^{80} -6.09017 q^{82} +8.47214 q^{83} -5.61803 q^{85} +7.09017 q^{86} -1.32624 q^{89} -4.23607 q^{91} +9.23607 q^{92} +3.18034 q^{94} -10.4721 q^{95} -5.29180 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^7	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 2 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{20} + 2 q^{23} - 7 q^{25} - 3 q^{26} - q^{28} + 4 q^{29} + 6 q^{31} + 9 q^{32} - 11 q^{34} + q^{35} + 12 q^{37} - 8 q^{38} - 5 q^{40} - 13 q^{41} + 14 q^{43} - 16 q^{46} + 17 q^{47} + 2 q^{49} + 6 q^{50} + 7 q^{52} + 5 q^{53} - 7 q^{58} + 7 q^{59} - 2 q^{61} - 3 q^{62} + 4 q^{64} - 7 q^{65} + 22 q^{67} + 9 q^{68} + 2 q^{70} + 18 q^{71} - 17 q^{73} - 6 q^{74} + 12 q^{76} + q^{79} + 6 q^{80} - q^{82} + 8 q^{83} - 9 q^{85} + 3 q^{86} + 13 q^{89} - 4 q^{91} + 14 q^{92} - 16 q^{94} - 12 q^{95} - 24 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^7 + 2 * q^10 - 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 - 3 * q^20 + 2 * q^23 - 7 * q^25 - 3 * q^26 - q^28 + 4 * q^29 + 6 * q^31 + 9 * q^32 - 11 * q^34 + q^35 + 12 * q^37 - 8 * q^38 - 5 * q^40 - 13 * q^41 + 14 * q^43 - 16 * q^46 + 17 * q^47 + 2 * q^49 + 6 * q^50 + 7 * q^52 + 5 * q^53 - 7 * q^58 + 7 * q^59 - 2 * q^61 - 3 * q^62 + 4 * q^64 - 7 * q^65 + 22 * q^67 + 9 * q^68 + 2 * q^70 + 18 * q^71 - 17 * q^73 - 6 * q^74 + 12 * q^76 + q^79 + 6 * q^80 - q^82 + 8 * q^83 - 9 * q^85 + 3 * q^86 + 13 * q^89 - 4 * q^91 + 14 * q^92 - 16 * q^94 - 12 * q^95 - 24 * q^97 - q^98

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