LMFDB - Embedded newform 7623.2.a.cz.1.1

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 $ x^12 - 22x^10 + 181x^8 - 692x^6 + 1240x^4 - 936*x^2 + 244
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.79401$ 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)

$f(q)$	$=$	$q-2.79401 q^{2} +5.80650 q^{4} +1.62657 q^{5} -1.00000 q^{7} -10.6354 q^{8} +O(q^{10})$	$q-2.79401 q^{2} +5.80650 q^{4} +1.62657 q^{5} -1.00000 q^{7} -10.6354 q^{8} -4.54465 q^{10} -6.23267 q^{13} +2.79401 q^{14} +18.1024 q^{16} +6.00681 q^{17} +0.305878 q^{19} +9.44466 q^{20} +7.84138 q^{23} -2.35428 q^{25} +17.4141 q^{26} -5.80650 q^{28} +2.94628 q^{29} -5.31322 q^{31} -29.3076 q^{32} -16.7831 q^{34} -1.62657 q^{35} -6.41825 q^{37} -0.854627 q^{38} -17.2992 q^{40} +4.04048 q^{41} -0.640934 q^{43} -21.9089 q^{46} -7.86467 q^{47} +1.00000 q^{49} +6.57787 q^{50} -36.1900 q^{52} -0.251121 q^{53} +10.6354 q^{56} -8.23194 q^{58} -5.46616 q^{59} +2.45477 q^{61} +14.8452 q^{62} +45.6808 q^{64} -10.1379 q^{65} +6.54902 q^{67} +34.8785 q^{68} +4.54465 q^{70} -5.61219 q^{71} -10.4714 q^{73} +17.9327 q^{74} +1.77608 q^{76} +7.16047 q^{79} +29.4448 q^{80} -11.2891 q^{82} +11.7565 q^{83} +9.77049 q^{85} +1.79078 q^{86} -9.59605 q^{89} +6.23267 q^{91} +45.5310 q^{92} +21.9740 q^{94} +0.497532 q^{95} +16.0963 q^{97} -2.79401 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{4} - 12 q^{7}+O(q^{10}) $ 12 * q + 20 * q^4 - 12 * q^7	$ 12 q + 20 q^{4} - 12 q^{7} - 20 q^{10} - 20 q^{13} + 28 q^{16} - 12 q^{19} + 32 q^{25} - 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} - 48 q^{40} - 16 q^{43} - 24 q^{46} + 12 q^{49} - 96 q^{52} + 20 q^{58} - 44 q^{61} + 76 q^{64} + 20 q^{70} - 52 q^{73} + 8 q^{79} - 68 q^{82} - 72 q^{85} + 20 q^{91} - 20 q^{94} - 32 q^{97}+O(q^{100}) $ 12 * q + 20 * q^4 - 12 * q^7 - 20 * q^10 - 20 * q^13 + 28 * q^16 - 12 * q^19 + 32 * q^25 - 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 - 48 * q^40 - 16 * q^43 - 24 * q^46 + 12 * q^49 - 96 * q^52 + 20 * q^58 - 44 * q^61 + 76 * q^64 + 20 * q^70 - 52 * q^73 + 8 * q^79 - 68 * q^82 - 72 * q^85 + 20 * q^91 - 20 * q^94 - 32 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.79401	−1.97566	−0.987832	−	0.155524i	$-0.950293\pi$
$2$	−2.79401	−1.97566	−0.987832	+	0.155524i	$0.950293\pi$
$3$	0	0
$3$	0	0
$4$	5.80650	2.90325
$5$	1.62657	0.727423	0.363712	−	0.931512i	$-0.381509\pi$
$5$	1.62657	0.727423	0.363712	+	0.931512i	$0.381509\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−10.6354	−3.76018
$9$	0	0
$10$	−4.54465	−1.43714
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.23267	−1.72863	−0.864316	−	0.502949i	$-0.832248\pi$
$13$	−6.23267	−1.72863	−0.864316	+	0.502949i	$0.832248\pi$
$14$	2.79401	0.746731
$15$	0	0
$16$	18.1024	4.52560
$17$	6.00681	1.45687	0.728433	−	0.685117i	$-0.240249\pi$
$17$	6.00681	1.45687	0.728433	+	0.685117i	$0.240249\pi$
$18$	0	0
$19$	0.305878	0.0701733	0.0350866	−	0.999384i	$-0.488829\pi$
$19$	0.305878	0.0701733	0.0350866	+	0.999384i	$0.488829\pi$
$20$	9.44466	2.11189
$21$	0	0
$22$	0	0
$23$	7.84138	1.63504	0.817521	−	0.575899i	$-0.195348\pi$
$23$	7.84138	1.63504	0.817521	+	0.575899i	$0.195348\pi$
$24$	0	0
$25$	−2.35428	−0.470855
$26$	17.4141	3.41520
$27$	0	0
$28$	−5.80650	−1.09732
$29$	2.94628	0.547111	0.273555	−	0.961856i	$-0.411800\pi$
$29$	2.94628	0.547111	0.273555	+	0.961856i	$0.411800\pi$
$30$	0	0
$31$	−5.31322	−0.954282	−0.477141	−	0.878827i	$-0.658327\pi$
$31$	−5.31322	−0.954282	−0.477141	+	0.878827i	$0.658327\pi$
$32$	−29.3076	−5.18089
$33$	0	0
$34$	−16.7831	−2.87828
$35$	−1.62657	−0.274940
$36$	0	0
$37$	−6.41825	−1.05515	−0.527577	−	0.849507i	$-0.676899\pi$
$37$	−6.41825	−1.05515	−0.527577	+	0.849507i	$0.676899\pi$
$38$	−0.854627	−0.138639
$39$	0	0
$40$	−17.2992	−2.73524
$41$	4.04048	0.631016	0.315508	−	0.948923i	$-0.397825\pi$
$41$	4.04048	0.631016	0.315508	+	0.948923i	$0.397825\pi$
$42$	0	0
$43$	−0.640934	−0.0977416	−0.0488708	−	0.998805i	$-0.515562\pi$
$43$	−0.640934	−0.0977416	−0.0488708	+	0.998805i	$0.515562\pi$
$44$	0	0
$45$	0	0
$46$	−21.9089	−3.23029
$47$	−7.86467	−1.14718	−0.573590	−	0.819142i	$-0.694450\pi$
$47$	−7.86467	−1.14718	−0.573590	+	0.819142i	$0.694450\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.57787	0.930251
$51$	0	0
$52$	−36.1900	−5.01865
$53$	−0.251121	−0.0344941	−0.0172471	−	0.999851i	$-0.505490\pi$
$53$	−0.251121	−0.0344941	−0.0172471	+	0.999851i	$0.505490\pi$
$54$	0	0
$55$	0	0
$56$	10.6354	1.42121
$57$	0	0
$58$	−8.23194	−1.08091
$59$	−5.46616	−0.711634	−0.355817	−	0.934556i	$-0.615797\pi$
$59$	−5.46616	−0.711634	−0.355817	+	0.934556i	$0.615797\pi$
$60$	0	0
$61$	2.45477	0.314301	0.157151	−	0.987575i	$-0.449769\pi$
$61$	2.45477	0.314301	0.157151	+	0.987575i	$0.449769\pi$
$62$	14.8452	1.88534
$63$	0	0
$64$	45.6808	5.71010
$65$	−10.1379	−1.25745
$66$	0	0
$67$	6.54902	0.800090	0.400045	−	0.916496i	$-0.368995\pi$
$67$	6.54902	0.800090	0.400045	+	0.916496i	$0.368995\pi$
$68$	34.8785	4.22964
$69$	0	0
$70$	4.54465	0.543190
$71$	−5.61219	−0.666045	−0.333022	−	0.942919i	$-0.608068\pi$
$71$	−5.61219	−0.666045	−0.333022	+	0.942919i	$0.608068\pi$
$72$	0	0
$73$	−10.4714	−1.22559	−0.612795	−	0.790242i	$-0.709955\pi$
$73$	−10.4714	−1.22559	−0.612795	+	0.790242i	$0.709955\pi$
$74$	17.9327	2.08463
$75$	0	0
$76$	1.77608	0.203731
$77$	0	0
$78$	0	0
$79$	7.16047	0.805616	0.402808	−	0.915285i	$-0.368034\pi$
$79$	7.16047	0.805616	0.402808	+	0.915285i	$0.368034\pi$
$80$	29.4448	3.29203
$81$	0	0
$82$	−11.2891	−1.24668
$83$	11.7565	1.29045	0.645224	−	0.763993i	$-0.276764\pi$
$83$	11.7565	1.29045	0.645224	+	0.763993i	$0.276764\pi$
$84$	0	0
$85$	9.77049	1.05976
$86$	1.79078	0.193104
$87$	0	0
$88$	0	0
$89$	−9.59605	−1.01718	−0.508590	−	0.861009i	$-0.669833\pi$
$89$	−9.59605	−1.01718	−0.508590	+	0.861009i	$0.669833\pi$
$90$	0	0
$91$	6.23267	0.653361
$92$	45.5310	4.74693
$93$	0	0
$94$	21.9740	2.26644
$95$	0.497532	0.0510457
$96$	0	0
$97$	16.0963	1.63433	0.817167	−	0.576402i	$-0.195544\pi$
$97$	16.0963	1.63433	0.817167	+	0.576402i	$0.195544\pi$
$98$	−2.79401	−0.282238
$99$	0	0
$100$	−13.6701	−1.36701
$101$	−11.2701	−1.12141	−0.560707	−	0.828014i	$-0.689471\pi$
$101$	−11.2701	−1.12141	−0.560707	+	0.828014i	$0.689471\pi$
$102$	0	0
$103$	−12.7879	−1.26003	−0.630014	−	0.776584i	$-0.716951\pi$
$103$	−12.7879	−1.26003	−0.630014	+	0.776584i	$0.716951\pi$
$104$	66.2869	6.49997
$105$	0	0
$106$	0.701635	0.0681488
$107$	−7.68911	−0.743335	−0.371667	−	0.928366i	$-0.621214\pi$
$107$	−7.68911	−0.743335	−0.371667	+	0.928366i	$0.621214\pi$
$108$	0	0
$109$	15.1634	1.45239	0.726196	−	0.687487i	$-0.241286\pi$
$109$	15.1634	1.45239	0.726196	+	0.687487i	$0.241286\pi$
$110$	0	0
$111$	0	0
$112$	−18.1024	−1.71052
$113$	−1.50471	−0.141551	−0.0707756	−	0.997492i	$-0.522547\pi$
$113$	−1.50471	−0.141551	−0.0707756	+	0.997492i	$0.522547\pi$
$114$	0	0
$115$	12.7545	1.18937
$116$	17.1076	1.58840
$117$	0	0
$118$	15.2725	1.40595
$119$	−6.00681	−0.550644
$120$	0	0
$121$	0	0
$122$	−6.85866	−0.620954
$123$	0	0
$124$	−30.8512	−2.77052
$125$	−11.9622	−1.06993
$126$	0	0
$127$	−1.93093	−0.171342	−0.0856710	−	0.996323i	$-0.527303\pi$
$127$	−1.93093	−0.171342	−0.0856710	+	0.996323i	$0.527303\pi$
$128$	−69.0176	−6.10035
$129$	0	0
$130$	28.3253	2.48429
$131$	12.7038	1.10993	0.554966	−	0.831873i	$-0.312731\pi$
$131$	12.7038	1.10993	0.554966	+	0.831873i	$0.312731\pi$
$132$	0	0
$133$	−0.305878	−0.0265230
$134$	−18.2980	−1.58071
$135$	0	0
$136$	−63.8848	−5.47808
$137$	−0.345262	−0.0294977	−0.0147489	−	0.999891i	$-0.504695\pi$
$137$	−0.345262	−0.0294977	−0.0147489	+	0.999891i	$0.504695\pi$
$138$	0	0
$139$	−19.0710	−1.61758	−0.808790	−	0.588097i	$-0.799877\pi$
$139$	−19.0710	−1.61758	−0.808790	+	0.588097i	$0.799877\pi$
$140$	−9.44466	−0.798220
$141$	0	0
$142$	15.6805	1.31588
$143$	0	0
$144$	0	0
$145$	4.79233	0.397981
$146$	29.2573	2.42135
$147$	0	0
$148$	−37.2676	−3.06338
$149$	6.46872	0.529938	0.264969	−	0.964257i	$-0.414638\pi$
$149$	6.46872	0.529938	0.264969	+	0.964257i	$0.414638\pi$
$150$	0	0
$151$	4.36030	0.354837	0.177418	−	0.984136i	$-0.443225\pi$
$151$	4.36030	0.354837	0.177418	+	0.984136i	$0.443225\pi$
$152$	−3.25314	−0.263864
$153$	0	0
$154$	0	0
$155$	−8.64231	−0.694167
$156$	0	0
$157$	−6.03397	−0.481563	−0.240782	−	0.970579i	$-0.577404\pi$
$157$	−6.03397	−0.481563	−0.240782	+	0.970579i	$0.577404\pi$
$158$	−20.0064	−1.59163
$159$	0	0
$160$	−47.6708	−3.76870
$161$	−7.84138	−0.617988
$162$	0	0
$163$	−17.5925	−1.37795	−0.688974	−	0.724786i	$-0.741939\pi$
$163$	−17.5925	−1.37795	−0.688974	+	0.724786i	$0.741939\pi$
$164$	23.4610	1.83200
$165$	0	0
$166$	−32.8479	−2.54949
$167$	−0.870792	−0.0673839	−0.0336920	−	0.999432i	$-0.510727\pi$
$167$	−0.870792	−0.0673839	−0.0336920	+	0.999432i	$0.510727\pi$
$168$	0	0
$169$	25.8462	1.98817
$170$	−27.2989	−2.09373
$171$	0	0
$172$	−3.72158	−0.283768
$173$	−0.432174	−0.0328576	−0.0164288	−	0.999865i	$-0.505230\pi$
$173$	−0.432174	−0.0328576	−0.0164288	+	0.999865i	$0.505230\pi$
$174$	0	0
$175$	2.35428	0.177966
$176$	0	0
$177$	0	0
$178$	26.8115	2.00960
$179$	7.87633	0.588705	0.294352	−	0.955697i	$-0.404896\pi$
$179$	7.87633	0.588705	0.294352	+	0.955697i	$0.404896\pi$
$180$	0	0
$181$	13.7505	1.02207	0.511035	−	0.859560i	$-0.329262\pi$
$181$	13.7505	1.02207	0.511035	+	0.859560i	$0.329262\pi$
$182$	−17.4141	−1.29082
$183$	0	0
$184$	−83.3962	−6.14805
$185$	−10.4397	−0.767544
$186$	0	0
$187$	0	0
$188$	−45.6662	−3.33055
$189$	0	0
$190$	−1.39011	−0.100849
$191$	27.2449	1.97137	0.985685	−	0.168597i	$-0.0539237\pi$
$191$	27.2449	1.97137	0.985685	+	0.168597i	$0.0539237\pi$
$192$	0	0
$193$	−20.4927	−1.47510	−0.737548	−	0.675294i	$-0.764017\pi$
$193$	−20.4927	−1.47510	−0.737548	+	0.675294i	$0.764017\pi$
$194$	−44.9733	−3.22889
$195$	0	0
$196$	5.80650	0.414750
$197$	−12.5543	−0.894455	−0.447227	−	0.894420i	$-0.647589\pi$
$197$	−12.5543	−0.894455	−0.447227	+	0.894420i	$0.647589\pi$
$198$	0	0
$199$	5.65670	0.400993	0.200497	−	0.979694i	$-0.435744\pi$
$199$	5.65670	0.400993	0.200497	+	0.979694i	$0.435744\pi$
$200$	25.0387	1.77050
$201$	0	0
$202$	31.4887	2.21554
$203$	−2.94628	−0.206788
$204$	0	0
$205$	6.57211	0.459016
$206$	35.7295	2.48939
$207$	0	0
$208$	−112.826	−7.82310
$209$	0	0
$210$	0	0
$211$	−9.86223	−0.678944	−0.339472	−	0.940616i	$-0.610248\pi$
$211$	−9.86223	−0.678944	−0.339472	+	0.940616i	$0.610248\pi$
$212$	−1.45813	−0.100145
$213$	0	0
$214$	21.4835	1.46858
$215$	−1.04252	−0.0710995
$216$	0	0
$217$	5.31322	0.360685
$218$	−42.3668	−2.86944
$219$	0	0
$220$	0	0
$221$	−37.4385	−2.51838
$222$	0	0
$223$	−12.4235	−0.831936	−0.415968	−	0.909379i	$-0.636557\pi$
$223$	−12.4235	−0.831936	−0.415968	+	0.909379i	$0.636557\pi$
$224$	29.3076	1.95819
$225$	0	0
$226$	4.20418	0.279658
$227$	−8.47261	−0.562347	−0.281173	−	0.959657i	$-0.590724\pi$
$227$	−8.47261	−0.562347	−0.281173	+	0.959657i	$0.590724\pi$
$228$	0	0
$229$	−11.0760	−0.731920	−0.365960	−	0.930631i	$-0.619259\pi$
$229$	−11.0760	−0.731920	−0.365960	+	0.930631i	$0.619259\pi$
$230$	−35.6363	−2.34979
$231$	0	0
$232$	−31.3349	−2.05723
$233$	7.59103	0.497305	0.248652	−	0.968593i	$-0.420012\pi$
$233$	7.59103	0.497305	0.248652	+	0.968593i	$0.420012\pi$
$234$	0	0
$235$	−12.7924	−0.834486
$236$	−31.7393	−2.06605
$237$	0	0
$238$	16.7831	1.08789
$239$	−27.8021	−1.79837	−0.899184	−	0.437571i	$-0.855839\pi$
$239$	−27.8021	−1.79837	−0.899184	+	0.437571i	$0.855839\pi$
$240$	0	0
$241$	−0.242223	−0.0156030	−0.00780149	−	0.999970i	$-0.502483\pi$
$241$	−0.242223	−0.0156030	−0.00780149	+	0.999970i	$0.502483\pi$
$242$	0	0
$243$	0	0
$244$	14.2536	0.912495
$245$	1.62657	0.103918
$246$	0	0
$247$	−1.90644	−0.121304
$248$	56.5082	3.58827
$249$	0	0
$250$	33.4226	2.11383
$251$	18.0334	1.13826	0.569130	−	0.822247i	$-0.307280\pi$
$251$	18.0334	1.13826	0.569130	+	0.822247i	$0.307280\pi$
$252$	0	0
$253$	0	0
$254$	5.39503	0.338514
$255$	0	0
$256$	101.474	6.34214
$257$	−8.27384	−0.516108	−0.258054	−	0.966131i	$-0.583081\pi$
$257$	−8.27384	−0.516108	−0.258054	+	0.966131i	$0.583081\pi$
$258$	0	0
$259$	6.41825	0.398811
$260$	−58.8655	−3.65068
$261$	0	0
$262$	−35.4944	−2.19285
$263$	3.42958	0.211477	0.105738	−	0.994394i	$-0.466279\pi$
$263$	3.42958	0.211477	0.105738	+	0.994394i	$0.466279\pi$
$264$	0	0
$265$	−0.408466	−0.0250918
$266$	0.854627	0.0524006
$267$	0	0
$268$	38.0268	2.32286
$269$	−10.4629	−0.637934	−0.318967	−	0.947766i	$-0.603336\pi$
$269$	−10.4629	−0.637934	−0.318967	+	0.947766i	$0.603336\pi$
$270$	0	0
$271$	22.4886	1.36608	0.683042	−	0.730380i	$-0.260657\pi$
$271$	22.4886	1.36608	0.683042	+	0.730380i	$0.260657\pi$
$272$	108.738	6.59320
$273$	0	0
$274$	0.964666	0.0582776
$275$	0	0
$276$	0	0
$277$	−16.0639	−0.965188	−0.482594	−	0.875844i	$-0.660305\pi$
$277$	−16.0639	−0.965188	−0.482594	+	0.875844i	$0.660305\pi$
$278$	53.2846	3.19580
$279$	0	0
$280$	17.2992	1.03382
$281$	−9.57248	−0.571046	−0.285523	−	0.958372i	$-0.592167\pi$
$281$	−9.57248	−0.571046	−0.285523	+	0.958372i	$0.592167\pi$
$282$	0	0
$283$	−12.2182	−0.726296	−0.363148	−	0.931731i	$-0.618298\pi$
$283$	−12.2182	−0.726296	−0.363148	+	0.931731i	$0.618298\pi$
$284$	−32.5872	−1.93369
$285$	0	0
$286$	0	0
$287$	−4.04048	−0.238502
$288$	0	0
$289$	19.0818	1.12246
$290$	−13.3898	−0.786277
$291$	0	0
$292$	−60.8024	−3.55819
$293$	30.0285	1.75428	0.877142	−	0.480232i	$-0.159447\pi$
$293$	30.0285	1.75428	0.877142	+	0.480232i	$0.159447\pi$
$294$	0	0
$295$	−8.89109	−0.517659
$296$	68.2607	3.96757
$297$	0	0
$298$	−18.0737	−1.04698
$299$	−48.8728	−2.82638
$300$	0	0
$301$	0.640934	0.0369428
$302$	−12.1827	−0.701038
$303$	0	0
$304$	5.53714	0.317577
$305$	3.99285	0.228630
$306$	0	0
$307$	−17.5503	−1.00165	−0.500825	−	0.865549i	$-0.666970\pi$
$307$	−17.5503	−1.00165	−0.500825	+	0.865549i	$0.666970\pi$
$308$	0	0
$309$	0	0
$310$	24.1467	1.37144
$311$	17.9991	1.02063	0.510317	−	0.859986i	$-0.329528\pi$
$311$	17.9991	1.02063	0.510317	+	0.859986i	$0.329528\pi$
$312$	0	0
$313$	−26.1718	−1.47932	−0.739659	−	0.672982i	$-0.765013\pi$
$313$	−26.1718	−1.47932	−0.739659	+	0.672982i	$0.765013\pi$
$314$	16.8590	0.951408
$315$	0	0
$316$	41.5772	2.33890
$317$	12.2998	0.690824	0.345412	−	0.938451i	$-0.387739\pi$
$317$	12.2998	0.690824	0.345412	+	0.938451i	$0.387739\pi$
$318$	0	0
$319$	0	0
$320$	74.3030	4.15366
$321$	0	0
$322$	21.9089	1.22094
$323$	1.83735	0.102233
$324$	0	0
$325$	14.6734	0.813935
$326$	49.1536	2.72236
$327$	0	0
$328$	−42.9721	−2.37274
$329$	7.86467	0.433593
$330$	0	0
$331$	10.1006	0.555177	0.277589	−	0.960700i	$-0.410465\pi$
$331$	10.1006	0.555177	0.277589	+	0.960700i	$0.410465\pi$
$332$	68.2643	3.74649
$333$	0	0
$334$	2.43300	0.133128
$335$	10.6524	0.582004
$336$	0	0
$337$	−22.1452	−1.20633	−0.603163	−	0.797618i	$-0.706093\pi$
$337$	−22.1452	−1.20633	−0.603163	+	0.797618i	$0.706093\pi$
$338$	−72.2145	−3.92795
$339$	0	0
$340$	56.7323	3.07674
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.81659	0.367526
$345$	0	0
$346$	1.20750	0.0649156
$347$	10.5620	0.567000	0.283500	−	0.958972i	$-0.408504\pi$
$347$	10.5620	0.567000	0.283500	+	0.958972i	$0.408504\pi$
$348$	0	0
$349$	0.188962	0.0101149	0.00505746	−	0.999987i	$-0.498390\pi$
$349$	0.188962	0.0101149	0.00505746	+	0.999987i	$0.498390\pi$
$350$	−6.57787	−0.351602
$351$	0	0
$352$	0	0
$353$	−33.2121	−1.76770	−0.883851	−	0.467768i	$-0.845058\pi$
$353$	−33.2121	−1.76770	−0.883851	+	0.467768i	$0.845058\pi$
$354$	0	0
$355$	−9.12862	−0.484497
$356$	−55.7194	−2.95312
$357$	0	0
$358$	−22.0066	−1.16308
$359$	24.7059	1.30393	0.651965	−	0.758249i	$-0.273945\pi$
$359$	24.7059	1.30393	0.651965	+	0.758249i	$0.273945\pi$
$360$	0	0
$361$	−18.9064	−0.995076
$362$	−38.4192	−2.01927
$363$	0	0
$364$	36.1900	1.89687
$365$	−17.0325	−0.891523
$366$	0	0
$367$	0.563769	0.0294285	0.0147143	−	0.999892i	$-0.495316\pi$
$367$	0.563769	0.0294285	0.0147143	+	0.999892i	$0.495316\pi$
$368$	141.948	7.39955
$369$	0	0
$370$	29.1687	1.51641
$371$	0.251121	0.0130376
$372$	0	0
$373$	18.5228	0.959073	0.479536	−	0.877522i	$-0.340805\pi$
$373$	18.5228	0.959073	0.479536	+	0.877522i	$0.340805\pi$
$374$	0	0
$375$	0	0
$376$	83.6439	4.31361
$377$	−18.3632	−0.945753
$378$	0	0
$379$	−22.2210	−1.14141	−0.570707	−	0.821153i	$-0.693331\pi$
$379$	−22.2210	−1.14141	−0.570707	+	0.821153i	$0.693331\pi$
$380$	2.88892	0.148198
$381$	0	0
$382$	−76.1225	−3.89477
$383$	−27.3023	−1.39508	−0.697540	−	0.716546i	$-0.745722\pi$
$383$	−27.3023	−1.39508	−0.697540	+	0.716546i	$0.745722\pi$
$384$	0	0
$385$	0	0
$386$	57.2568	2.91430
$387$	0	0
$388$	93.4632	4.74488
$389$	27.9317	1.41620	0.708098	−	0.706115i	$-0.249554\pi$
$389$	27.9317	1.41620	0.708098	+	0.706115i	$0.249554\pi$
$390$	0	0
$391$	47.1017	2.38204
$392$	−10.6354	−0.537169
$393$	0	0
$394$	35.0768	1.76714
$395$	11.6470	0.586024
$396$	0	0
$397$	15.0029	0.752975	0.376488	−	0.926422i	$-0.377132\pi$
$397$	15.0029	0.752975	0.376488	+	0.926422i	$0.377132\pi$
$398$	−15.8049	−0.792227
$399$	0	0
$400$	−42.6181	−2.13090
$401$	−28.5554	−1.42599	−0.712996	−	0.701169i	$-0.752662\pi$
$401$	−28.5554	−1.42599	−0.712996	+	0.701169i	$0.752662\pi$
$402$	0	0
$403$	33.1155	1.64960
$404$	−65.4397	−3.25575
$405$	0	0
$406$	8.23194	0.408544
$407$	0	0
$408$	0	0
$409$	−23.6910	−1.17145	−0.585723	−	0.810511i	$-0.699190\pi$
$409$	−23.6910	−1.17145	−0.585723	+	0.810511i	$0.699190\pi$
$410$	−18.3626	−0.906862
$411$	0	0
$412$	−74.2528	−3.65817
$413$	5.46616	0.268972
$414$	0	0
$415$	19.1228	0.938702
$416$	182.664	8.95586
$417$	0	0
$418$	0	0
$419$	−4.47054	−0.218400	−0.109200	−	0.994020i	$-0.534829\pi$
$419$	−4.47054	−0.218400	−0.109200	+	0.994020i	$0.534829\pi$
$420$	0	0
$421$	13.7968	0.672416	0.336208	−	0.941788i	$-0.390856\pi$
$421$	13.7968	0.672416	0.336208	+	0.941788i	$0.390856\pi$
$422$	27.5552	1.34137
$423$	0	0
$424$	2.67077	0.129704
$425$	−14.1417	−0.685973
$426$	0	0
$427$	−2.45477	−0.118795
$428$	−44.6468	−2.15809
$429$	0	0
$430$	2.91282	0.140469
$431$	−20.1019	−0.968275	−0.484137	−	0.874992i	$-0.660866\pi$
$431$	−20.1019	−0.968275	−0.484137	+	0.874992i	$0.660866\pi$
$432$	0	0
$433$	−13.4237	−0.645103	−0.322552	−	0.946552i	$-0.604541\pi$
$433$	−13.4237	−0.645103	−0.322552	+	0.946552i	$0.604541\pi$
$434$	−14.8452	−0.712592
$435$	0	0
$436$	88.0464	4.21666
$437$	2.39851	0.114736
$438$	0	0
$439$	−25.3166	−1.20829	−0.604147	−	0.796873i	$-0.706486\pi$
$439$	−25.3166	−1.20829	−0.604147	+	0.796873i	$0.706486\pi$
$440$	0	0
$441$	0	0
$442$	104.604	4.97548
$443$	−14.9425	−0.709939	−0.354970	−	0.934878i	$-0.615509\pi$
$443$	−14.9425	−0.709939	−0.354970	+	0.934878i	$0.615509\pi$
$444$	0	0
$445$	−15.6086	−0.739920
$446$	34.7113	1.64363
$447$	0	0
$448$	−45.6808	−2.15822
$449$	32.4752	1.53260	0.766300	−	0.642482i	$-0.222095\pi$
$449$	32.4752	1.53260	0.766300	+	0.642482i	$0.222095\pi$
$450$	0	0
$451$	0	0
$452$	−8.73709	−0.410958
$453$	0	0
$454$	23.6726	1.11101
$455$	10.1379	0.475270
$456$	0	0
$457$	−1.97016	−0.0921603	−0.0460802	−	0.998938i	$-0.514673\pi$
$457$	−1.97016	−0.0921603	−0.0460802	+	0.998938i	$0.514673\pi$
$458$	30.9464	1.44603
$459$	0	0
$460$	74.0593	3.45303
$461$	−31.8240	−1.48219	−0.741095	−	0.671400i	$-0.765693\pi$
$461$	−31.8240	−1.48219	−0.741095	+	0.671400i	$0.765693\pi$
$462$	0	0
$463$	11.1084	0.516250	0.258125	−	0.966112i	$-0.416895\pi$
$463$	11.1084	0.516250	0.258125	+	0.966112i	$0.416895\pi$
$464$	53.3348	2.47601
$465$	0	0
$466$	−21.2094	−0.982507
$467$	−27.2112	−1.25918	−0.629592	−	0.776926i	$-0.716778\pi$
$467$	−27.2112	−1.25918	−0.629592	+	0.776926i	$0.716778\pi$
$468$	0	0
$469$	−6.54902	−0.302405
$470$	35.7422	1.64866
$471$	0	0
$472$	58.1348	2.67587
$473$	0	0
$474$	0	0
$475$	−0.720122	−0.0330415
$476$	−34.8785	−1.59866
$477$	0	0
$478$	77.6793	3.55297
$479$	13.3861	0.611626	0.305813	−	0.952092i	$-0.401072\pi$
$479$	13.3861	0.611626	0.305813	+	0.952092i	$0.401072\pi$
$480$	0	0
$481$	40.0029	1.82397
$482$	0.676775	0.0308262
$483$	0	0
$484$	0	0
$485$	26.1818	1.18885
$486$	0	0
$487$	17.2741	0.782762	0.391381	−	0.920229i	$-0.371997\pi$
$487$	17.2741	0.782762	0.391381	+	0.920229i	$0.371997\pi$
$488$	−26.1075	−1.18183
$489$	0	0
$490$	−4.54465	−0.205306
$491$	−2.45742	−0.110902	−0.0554510	−	0.998461i	$-0.517660\pi$
$491$	−2.45742	−0.110902	−0.0554510	+	0.998461i	$0.517660\pi$
$492$	0	0
$493$	17.6978	0.797067
$494$	5.32661	0.239656
$495$	0	0
$496$	−96.1821	−4.31870
$497$	5.61219	0.251741
$498$	0	0
$499$	−16.5539	−0.741055	−0.370527	−	0.928822i	$-0.620823\pi$
$499$	−16.5539	−0.741055	−0.370527	+	0.928822i	$0.620823\pi$
$500$	−69.4587	−3.10629
$501$	0	0
$502$	−50.3856	−2.24882
$503$	−2.75734	−0.122944	−0.0614718	−	0.998109i	$-0.519579\pi$
$503$	−2.75734	−0.122944	−0.0614718	+	0.998109i	$0.519579\pi$
$504$	0	0
$505$	−18.3316	−0.815743
$506$	0	0
$507$	0	0
$508$	−11.2119	−0.497449
$509$	−32.6055	−1.44522	−0.722608	−	0.691258i	$-0.757057\pi$
$509$	−32.6055	−1.44522	−0.722608	+	0.691258i	$0.757057\pi$
$510$	0	0
$511$	10.4714	0.463229
$512$	−145.485	−6.42959
$513$	0	0
$514$	23.1172	1.01966
$515$	−20.8004	−0.916574
$516$	0	0
$517$	0	0
$518$	−17.9327	−0.787916
$519$	0	0
$520$	107.820	4.72823
$521$	−5.53662	−0.242564	−0.121282	−	0.992618i	$-0.538700\pi$
$521$	−5.53662	−0.242564	−0.121282	+	0.992618i	$0.538700\pi$
$522$	0	0
$523$	0.800091	0.0349856	0.0174928	−	0.999847i	$-0.494432\pi$
$523$	0.800091	0.0349856	0.0174928	+	0.999847i	$0.494432\pi$
$524$	73.7644	3.22241
$525$	0	0
$526$	−9.58228	−0.417807
$527$	−31.9155	−1.39026
$528$	0	0
$529$	38.4873	1.67336
$530$	1.14126	0.0495730
$531$	0	0
$532$	−1.77608	−0.0770029
$533$	−25.1830	−1.09079
$534$	0	0
$535$	−12.5069	−0.540719
$536$	−69.6514	−3.00848
$537$	0	0
$538$	29.2335	1.26034
$539$	0	0
$540$	0	0
$541$	−34.0294	−1.46304	−0.731519	−	0.681821i	$-0.761188\pi$
$541$	−34.0294	−1.46304	−0.731519	+	0.681821i	$0.761188\pi$
$542$	−62.8333	−2.69892
$543$	0	0
$544$	−176.045	−7.54787
$545$	24.6643	1.05650
$546$	0	0
$547$	−26.2897	−1.12406	−0.562032	−	0.827115i	$-0.689980\pi$
$547$	−26.2897	−1.12406	−0.562032	+	0.827115i	$0.689980\pi$
$548$	−2.00476	−0.0856392
$549$	0	0
$550$	0	0
$551$	0.901203	0.0383926
$552$	0	0
$553$	−7.16047	−0.304494
$554$	44.8828	1.90689
$555$	0	0
$556$	−110.736	−4.69624
$557$	15.3838	0.651832	0.325916	−	0.945399i	$-0.394327\pi$
$557$	15.3838	0.651832	0.325916	+	0.945399i	$0.394327\pi$
$558$	0	0
$559$	3.99473	0.168959
$560$	−29.4448	−1.24427
$561$	0	0
$562$	26.7456	1.12820
$563$	3.55750	0.149931	0.0749654	−	0.997186i	$-0.476115\pi$
$563$	3.55750	0.149931	0.0749654	+	0.997186i	$0.476115\pi$
$564$	0	0
$565$	−2.44751	−0.102968
$566$	34.1378	1.43492
$567$	0	0
$568$	59.6879	2.50445
$569$	−4.30125	−0.180318	−0.0901590	−	0.995927i	$-0.528738\pi$
$569$	−4.30125	−0.180318	−0.0901590	+	0.995927i	$0.528738\pi$
$570$	0	0
$571$	−20.8296	−0.871692	−0.435846	−	0.900021i	$-0.643551\pi$
$571$	−20.8296	−0.871692	−0.435846	+	0.900021i	$0.643551\pi$
$572$	0	0
$573$	0	0
$574$	11.2891	0.471199
$575$	−18.4608	−0.769868
$576$	0	0
$577$	−28.5025	−1.18657	−0.593287	−	0.804991i	$-0.702170\pi$
$577$	−28.5025	−1.18657	−0.593287	+	0.804991i	$0.702170\pi$
$578$	−53.3147	−2.21760
$579$	0	0
$580$	27.8266	1.15544
$581$	−11.7565	−0.487744
$582$	0	0
$583$	0	0
$584$	111.368	4.60844
$585$	0	0
$586$	−83.9000	−3.46588
$587$	18.1401	0.748721	0.374360	−	0.927283i	$-0.377862\pi$
$587$	18.1401	0.748721	0.374360	+	0.927283i	$0.377862\pi$
$588$	0	0
$589$	−1.62520	−0.0669651
$590$	24.8418	1.02272
$591$	0	0
$592$	−116.186	−4.77521
$593$	28.2014	1.15809	0.579047	−	0.815294i	$-0.303425\pi$
$593$	28.2014	1.15809	0.579047	+	0.815294i	$0.303425\pi$
$594$	0	0
$595$	−9.77049	−0.400551
$596$	37.5606	1.53854
$597$	0	0
$598$	136.551	5.58399
$599$	−3.67415	−0.150122	−0.0750608	−	0.997179i	$-0.523915\pi$
$599$	−3.67415	−0.150122	−0.0750608	+	0.997179i	$0.523915\pi$
$600$	0	0
$601$	13.1861	0.537872	0.268936	−	0.963158i	$-0.413328\pi$
$601$	13.1861	0.537872	0.268936	+	0.963158i	$0.413328\pi$
$602$	−1.79078	−0.0729866
$603$	0	0
$604$	25.3181	1.03018
$605$	0	0
$606$	0	0
$607$	10.7655	0.436958	0.218479	−	0.975842i	$-0.429890\pi$
$607$	10.7655	0.436958	0.218479	+	0.975842i	$0.429890\pi$
$608$	−8.96455	−0.363560
$609$	0	0
$610$	−11.1561	−0.451696
$611$	49.0179	1.98305
$612$	0	0
$613$	−11.9610	−0.483100	−0.241550	−	0.970388i	$-0.577656\pi$
$613$	−11.9610	−0.483100	−0.241550	+	0.970388i	$0.577656\pi$
$614$	49.0358	1.97892
$615$	0	0
$616$	0	0
$617$	−19.0580	−0.767247	−0.383623	−	0.923490i	$-0.625324\pi$
$617$	−19.0580	−0.767247	−0.383623	+	0.923490i	$0.625324\pi$
$618$	0	0
$619$	17.8089	0.715799	0.357900	−	0.933760i	$-0.383493\pi$
$619$	17.8089	0.715799	0.357900	+	0.933760i	$0.383493\pi$
$620$	−50.1816	−2.01534
$621$	0	0
$622$	−50.2896	−2.01643
$623$	9.59605	0.384458
$624$	0	0
$625$	−7.68601	−0.307440
$626$	73.1243	2.92263
$627$	0	0
$628$	−35.0362	−1.39810
$629$	−38.5532	−1.53722
$630$	0	0
$631$	−20.7960	−0.827874	−0.413937	−	0.910306i	$-0.635847\pi$
$631$	−20.7960	−0.827874	−0.413937	+	0.910306i	$0.635847\pi$
$632$	−76.1544	−3.02926
$633$	0	0
$634$	−34.3657	−1.36484
$635$	−3.14078	−0.124638
$636$	0	0
$637$	−6.23267	−0.246947
$638$	0	0
$639$	0	0
$640$	−112.262	−4.43754
$641$	35.3225	1.39515	0.697577	−	0.716510i	$-0.254262\pi$
$641$	35.3225	1.39515	0.697577	+	0.716510i	$0.254262\pi$
$642$	0	0
$643$	33.6899	1.32860	0.664299	−	0.747467i	$-0.268730\pi$
$643$	33.6899	1.32860	0.664299	+	0.747467i	$0.268730\pi$
$644$	−45.5310	−1.79417
$645$	0	0
$646$	−5.13359	−0.201978
$647$	−28.6950	−1.12812	−0.564059	−	0.825735i	$-0.690761\pi$
$647$	−28.6950	−1.12812	−0.564059	+	0.825735i	$0.690761\pi$
$648$	0	0
$649$	0	0
$650$	−40.9977	−1.60806
$651$	0	0
$652$	−102.151	−4.00053
$653$	1.07891	0.0422212	0.0211106	−	0.999777i	$-0.493280\pi$
$653$	1.07891	0.0422212	0.0211106	+	0.999777i	$0.493280\pi$
$654$	0	0
$655$	20.6635	0.807391
$656$	73.1424	2.85573
$657$	0	0
$658$	−21.9740	−0.856635
$659$	4.07523	0.158748	0.0793742	−	0.996845i	$-0.474708\pi$
$659$	4.07523	0.158748	0.0793742	+	0.996845i	$0.474708\pi$
$660$	0	0
$661$	19.3678	0.753321	0.376660	−	0.926351i	$-0.377072\pi$
$661$	19.3678	0.753321	0.376660	+	0.926351i	$0.377072\pi$
$662$	−28.2211	−1.09684
$663$	0	0
$664$	−125.035	−4.85232
$665$	−0.497532	−0.0192935
$666$	0	0
$667$	23.1029	0.894549
$668$	−5.05625	−0.195632
$669$	0	0
$670$	−29.7630	−1.14984
$671$	0	0
$672$	0	0
$673$	−7.39835	−0.285185	−0.142593	−	0.989781i	$-0.545544\pi$
$673$	−7.39835	−0.285185	−0.142593	+	0.989781i	$0.545544\pi$
$674$	61.8740	2.38330
$675$	0	0
$676$	150.076	5.77214
$677$	17.7449	0.681990	0.340995	−	0.940065i	$-0.389236\pi$
$677$	17.7449	0.681990	0.340995	+	0.940065i	$0.389236\pi$
$678$	0	0
$679$	−16.0963	−0.617720
$680$	−103.913	−3.98488
$681$	0	0
$682$	0	0
$683$	34.6595	1.32621	0.663104	−	0.748527i	$-0.269239\pi$
$683$	34.6595	1.32621	0.663104	+	0.748527i	$0.269239\pi$
$684$	0	0
$685$	−0.561592	−0.0214573
$686$	2.79401	0.106676
$687$	0	0
$688$	−11.6025	−0.442340
$689$	1.56515	0.0596276
$690$	0	0
$691$	10.5107	0.399845	0.199923	−	0.979812i	$-0.435931\pi$
$691$	10.5107	0.399845	0.199923	+	0.979812i	$0.435931\pi$
$692$	−2.50942	−0.0953938
$693$	0	0
$694$	−29.5105	−1.12020
$695$	−31.0203	−1.17667
$696$	0	0
$697$	24.2704	0.919306
$698$	−0.527963	−0.0199837
$699$	0	0
$700$	13.6701	0.516681
$701$	−28.1203	−1.06209	−0.531044	−	0.847344i	$-0.678200\pi$
$701$	−28.1203	−1.06209	−0.531044	+	0.847344i	$0.678200\pi$
$702$	0	0
$703$	−1.96320	−0.0740437
$704$	0	0
$705$	0	0
$706$	92.7950	3.49239
$707$	11.2701	0.423855
$708$	0	0
$709$	4.44375	0.166889	0.0834443	−	0.996512i	$-0.473408\pi$
$709$	4.44375	0.166889	0.0834443	+	0.996512i	$0.473408\pi$
$710$	25.5055	0.957203
$711$	0	0
$712$	102.058	3.82478
$713$	−41.6630	−1.56029
$714$	0	0
$715$	0	0
$716$	45.7339	1.70916
$717$	0	0
$718$	−69.0286	−2.57613
$719$	2.11794	0.0789857	0.0394929	−	0.999220i	$-0.487426\pi$
$719$	2.11794	0.0789857	0.0394929	+	0.999220i	$0.487426\pi$
$720$	0	0
$721$	12.7879	0.476246
$722$	52.8248	1.96594
$723$	0	0
$724$	79.8425	2.96732
$725$	−6.93636	−0.257610
$726$	0	0
$727$	24.3015	0.901292	0.450646	−	0.892703i	$-0.351194\pi$
$727$	24.3015	0.901292	0.450646	+	0.892703i	$0.351194\pi$
$728$	−66.2869	−2.45676
$729$	0	0
$730$	47.5890	1.76135
$731$	−3.84997	−0.142396
$732$	0	0
$733$	2.14483	0.0792212	0.0396106	−	0.999215i	$-0.487388\pi$
$733$	2.14483	0.0792212	0.0396106	+	0.999215i	$0.487388\pi$
$734$	−1.57518	−0.0581408
$735$	0	0
$736$	−229.812	−8.47098
$737$	0	0
$738$	0	0
$739$	42.4005	1.55973	0.779864	−	0.625949i	$-0.215288\pi$
$739$	42.4005	1.55973	0.779864	+	0.625949i	$0.215288\pi$
$740$	−60.6183	−2.22837
$741$	0	0
$742$	−0.701635	−0.0257578
$743$	11.5589	0.424054	0.212027	−	0.977264i	$-0.431994\pi$
$743$	11.5589	0.424054	0.212027	+	0.977264i	$0.431994\pi$
$744$	0	0
$745$	10.5218	0.385489
$746$	−51.7528	−1.89481
$747$	0	0
$748$	0	0
$749$	7.68911	0.280954
$750$	0	0
$751$	−26.5439	−0.968602	−0.484301	−	0.874902i	$-0.660926\pi$
$751$	−26.5439	−0.968602	−0.484301	+	0.874902i	$0.660926\pi$
$752$	−142.370	−5.19168
$753$	0	0
$754$	51.3070	1.86849
$755$	7.09233	0.258116
$756$	0	0
$757$	6.32802	0.229996	0.114998	−	0.993366i	$-0.463314\pi$
$757$	6.32802	0.229996	0.114998	+	0.993366i	$0.463314\pi$
$758$	62.0857	2.25505
$759$	0	0
$760$	−5.29145	−0.191941
$761$	−11.2562	−0.408037	−0.204018	−	0.978967i	$-0.565400\pi$
$761$	−11.2562	−0.408037	−0.204018	+	0.978967i	$0.565400\pi$
$762$	0	0
$763$	−15.1634	−0.548953
$764$	158.197	5.72338
$765$	0	0
$766$	76.2828	2.75621
$767$	34.0688	1.23015
$768$	0	0
$769$	−42.7122	−1.54024	−0.770121	−	0.637897i	$-0.779804\pi$
$769$	−42.7122	−1.54024	−0.770121	+	0.637897i	$0.779804\pi$
$770$	0	0
$771$	0	0
$772$	−118.991	−4.28257
$773$	32.6637	1.17483	0.587415	−	0.809286i	$-0.300146\pi$
$773$	32.6637	1.17483	0.587415	+	0.809286i	$0.300146\pi$
$774$	0	0
$775$	12.5088	0.449329
$776$	−171.191	−6.14539
$777$	0	0
$778$	−78.0416	−2.79793
$779$	1.23589	0.0442805
$780$	0	0
$781$	0	0
$782$	−131.603	−4.70610
$783$	0	0
$784$	18.1024	0.646515
$785$	−9.81467	−0.350301
$786$	0	0
$787$	−35.5832	−1.26840	−0.634201	−	0.773168i	$-0.718671\pi$
$787$	−35.5832	−1.26840	−0.634201	+	0.773168i	$0.718671\pi$
$788$	−72.8963	−2.59682
$789$	0	0
$790$	−32.5418	−1.15779
$791$	1.50471	0.0535013
$792$	0	0
$793$	−15.2998	−0.543311
$794$	−41.9183	−1.48763
$795$	0	0
$796$	32.8456	1.16418
$797$	−31.7624	−1.12508	−0.562541	−	0.826769i	$-0.690176\pi$
$797$	−31.7624	−1.12508	−0.562541	+	0.826769i	$0.690176\pi$
$798$	0	0
$799$	−47.2416	−1.67129
$800$	68.9981	2.43945
$801$	0	0
$802$	79.7842	2.81728
$803$	0	0
$804$	0	0
$805$	−12.7545	−0.449539
$806$	−92.5252	−3.25906
$807$	0	0
$808$	119.862	4.21672
$809$	8.48704	0.298388	0.149194	−	0.988808i	$-0.452332\pi$
$809$	8.48704	0.298388	0.149194	+	0.988808i	$0.452332\pi$
$810$	0	0
$811$	7.72728	0.271341	0.135671	−	0.990754i	$-0.456681\pi$
$811$	7.72728	0.271341	0.135671	+	0.990754i	$0.456681\pi$
$812$	−17.1076	−0.600358
$813$	0	0
$814$	0	0
$815$	−28.6154	−1.00235
$816$	0	0
$817$	−0.196048	−0.00685885
$818$	66.1931	2.31439
$819$	0	0
$820$	38.1609	1.33264
$821$	7.37351	0.257337	0.128669	−	0.991688i	$-0.458930\pi$
$821$	7.37351	0.257337	0.128669	+	0.991688i	$0.458930\pi$
$822$	0	0
$823$	−20.5473	−0.716232	−0.358116	−	0.933677i	$-0.616581\pi$
$823$	−20.5473	−0.716232	−0.358116	+	0.933677i	$0.616581\pi$
$824$	136.004	4.73793
$825$	0	0
$826$	−15.2725	−0.531399
$827$	−5.14784	−0.179008	−0.0895040	−	0.995986i	$-0.528528\pi$
$827$	−5.14784	−0.179008	−0.0895040	+	0.995986i	$0.528528\pi$
$828$	0	0
$829$	7.86695	0.273231	0.136615	−	0.990624i	$-0.456378\pi$
$829$	7.86695	0.273231	0.136615	+	0.990624i	$0.456378\pi$
$830$	−53.4294	−1.85456
$831$	0	0
$832$	−284.713	−9.87066
$833$	6.00681	0.208124
$834$	0	0
$835$	−1.41640	−0.0490166
$836$	0	0
$837$	0	0
$838$	12.4907	0.431485
$839$	−46.3300	−1.59949	−0.799744	−	0.600341i	$-0.795031\pi$
$839$	−46.3300	−1.59949	−0.799744	+	0.600341i	$0.795031\pi$
$840$	0	0
$841$	−20.3194	−0.700670
$842$	−38.5484	−1.32847
$843$	0	0
$844$	−57.2650	−1.97114
$845$	42.0406	1.44624
$846$	0	0
$847$	0	0
$848$	−4.54590	−0.156107
$849$	0	0
$850$	39.5120	1.35525
$851$	−50.3280	−1.72522
$852$	0	0
$853$	−45.9539	−1.57343	−0.786715	−	0.617316i	$-0.788220\pi$
$853$	−45.9539	−1.57343	−0.786715	+	0.617316i	$0.788220\pi$
$854$	6.85866	0.234698
$855$	0	0
$856$	81.7768	2.79507
$857$	−10.4781	−0.357925	−0.178963	−	0.983856i	$-0.557274\pi$
$857$	−10.4781	−0.357925	−0.178963	+	0.983856i	$0.557274\pi$
$858$	0	0
$859$	−40.6385	−1.38657	−0.693283	−	0.720665i	$-0.743837\pi$
$859$	−40.6385	−1.38657	−0.693283	+	0.720665i	$0.743837\pi$
$860$	−6.05341	−0.206420
$861$	0	0
$862$	56.1649	1.91299
$863$	−40.6737	−1.38455	−0.692274	−	0.721634i	$-0.743391\pi$
$863$	−40.6737	−1.38455	−0.692274	+	0.721634i	$0.743391\pi$
$864$	0	0
$865$	−0.702961	−0.0239014
$866$	37.5060	1.27451
$867$	0	0
$868$	30.8512	1.04716
$869$	0	0
$870$	0	0
$871$	−40.8179	−1.38306
$872$	−161.269	−5.46126
$873$	0	0
$874$	−6.70146	−0.226680
$875$	11.9622	0.404397
$876$	0	0
$877$	−31.5951	−1.06689	−0.533445	−	0.845835i	$-0.679103\pi$
$877$	−31.5951	−1.06689	−0.533445	+	0.845835i	$0.679103\pi$
$878$	70.7348	2.38718
$879$	0	0
$880$	0	0
$881$	−35.1210	−1.18326	−0.591628	−	0.806211i	$-0.701515\pi$
$881$	−35.1210	−1.18326	−0.591628	+	0.806211i	$0.701515\pi$
$882$	0	0
$883$	−47.5879	−1.60146	−0.800730	−	0.599026i	$-0.795555\pi$
$883$	−47.5879	−1.60146	−0.800730	+	0.599026i	$0.795555\pi$
$884$	−217.386	−7.31150
$885$	0	0
$886$	41.7495	1.40260
$887$	−1.94873	−0.0654318	−0.0327159	−	0.999465i	$-0.510416\pi$
$887$	−1.94873	−0.0654318	−0.0327159	+	0.999465i	$0.510416\pi$
$888$	0	0
$889$	1.93093	0.0647612
$890$	43.6107	1.46183
$891$	0	0
$892$	−72.1368	−2.41532
$893$	−2.40563	−0.0805014
$894$	0	0
$895$	12.8114	0.428238
$896$	69.0176	2.30572
$897$	0	0
$898$	−90.7362	−3.02790
$899$	−15.6542	−0.522098
$900$	0	0
$901$	−1.50844	−0.0502533
$902$	0	0
$903$	0	0
$904$	16.0032	0.532258
$905$	22.3662	0.743478
$906$	0	0
$907$	−19.1044	−0.634351	−0.317175	−	0.948367i	$-0.602734\pi$
$907$	−19.1044	−0.634351	−0.317175	+	0.948367i	$0.602734\pi$
$908$	−49.1962	−1.63263
$909$	0	0
$910$	−28.3253	−0.938975
$911$	−38.3457	−1.27045	−0.635225	−	0.772327i	$-0.719093\pi$
$911$	−38.3457	−1.27045	−0.635225	+	0.772327i	$0.719093\pi$
$912$	0	0
$913$	0	0
$914$	5.50466	0.182078
$915$	0	0
$916$	−64.3126	−2.12495
$917$	−12.7038	−0.419515
$918$	0	0
$919$	14.3891	0.474652	0.237326	−	0.971430i	$-0.423729\pi$
$919$	14.3891	0.474652	0.237326	+	0.971430i	$0.423729\pi$
$920$	−135.650	−4.47224
$921$	0	0
$922$	88.9166	2.92831
$923$	34.9789	1.15135
$924$	0	0
$925$	15.1103	0.496825
$926$	−31.0369	−1.01994
$927$	0	0
$928$	−86.3483	−2.83452
$929$	31.6841	1.03952	0.519761	−	0.854311i	$-0.326021\pi$
$929$	31.6841	1.03952	0.519761	+	0.854311i	$0.326021\pi$
$930$	0	0
$931$	0.305878	0.0100248
$932$	44.0773	1.44380
$933$	0	0
$934$	76.0284	2.48772
$935$	0	0
$936$	0	0
$937$	1.41304	0.0461620	0.0230810	−	0.999734i	$-0.492652\pi$
$937$	1.41304	0.0461620	0.0230810	+	0.999734i	$0.492652\pi$
$938$	18.2980	0.597452
$939$	0	0
$940$	−74.2792	−2.42272
$941$	−21.9730	−0.716298	−0.358149	−	0.933664i	$-0.616592\pi$
$941$	−21.9730	−0.716298	−0.358149	+	0.933664i	$0.616592\pi$
$942$	0	0
$943$	31.6829	1.03174
$944$	−98.9508	−3.22057
$945$	0	0
$946$	0	0
$947$	−53.5540	−1.74027	−0.870136	−	0.492812i	$-0.835969\pi$
$947$	−53.5540	−1.74027	−0.870136	+	0.492812i	$0.835969\pi$
$948$	0	0
$949$	65.2650	2.11859
$950$	2.01203	0.0652788
$951$	0	0
$952$	63.8848	2.07052
$953$	16.0586	0.520189	0.260095	−	0.965583i	$-0.416246\pi$
$953$	16.0586	0.520189	0.260095	+	0.965583i	$0.416246\pi$
$954$	0	0
$955$	44.3157	1.43402
$956$	−161.433	−5.22111
$957$	0	0
$958$	−37.4009	−1.20837
$959$	0.345262	0.0111491
$960$	0	0
$961$	−2.76971	−0.0893456
$962$	−111.768	−3.60356
$963$	0	0
$964$	−1.40647	−0.0452993
$965$	−33.3328	−1.07302
$966$	0	0
$967$	−0.923832	−0.0297084	−0.0148542	−	0.999890i	$-0.504728\pi$
$967$	−0.923832	−0.0297084	−0.0148542	+	0.999890i	$0.504728\pi$
$968$	0	0
$969$	0	0
$970$	−73.1521	−2.34877
$971$	−15.1444	−0.486007	−0.243004	−	0.970025i	$-0.578133\pi$
$971$	−15.1444	−0.486007	−0.243004	+	0.970025i	$0.578133\pi$
$972$	0	0
$973$	19.0710	0.611388
$974$	−48.2639	−1.54647
$975$	0	0
$976$	44.4373	1.42240
$977$	−33.2665	−1.06429	−0.532145	−	0.846653i	$-0.678614\pi$
$977$	−33.2665	−1.06429	−0.532145	+	0.846653i	$0.678614\pi$
$978$	0	0
$979$	0	0
$980$	9.44466	0.301699
$981$	0	0
$982$	6.86607	0.219105
$983$	0.187510	0.00598064	0.00299032	−	0.999996i	$-0.499048\pi$
$983$	0.187510	0.00598064	0.00299032	+	0.999996i	$0.499048\pi$
$984$	0	0
$985$	−20.4204	−0.650647
$986$	−49.4477	−1.57474
$987$	0	0
$988$	−11.0697	−0.352175
$989$	−5.02581	−0.159812
$990$	0	0
$991$	3.61914	0.114966	0.0574829	−	0.998346i	$-0.481693\pi$
$991$	3.61914	0.114966	0.0574829	+	0.998346i	$0.481693\pi$
$992$	155.717	4.94403
$993$	0	0
$994$	−15.6805	−0.497356
$995$	9.20101	0.291692
$996$	0	0
$997$	12.7426	0.403561	0.201780	−	0.979431i	$-0.435327\pi$
$997$	12.7426	0.403561	0.201780	+	0.979431i	$0.435327\pi$
$998$	46.2518	1.46408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cz.1.1	✓	12
3.2	odd	2	inner	7623.2.a.cz.1.12	yes	12
11.10	odd	2		7623.2.a.da.1.12	yes	12
33.32	even	2		7623.2.a.da.1.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cz.1.1	✓	12	1.1	even	1	trivial
7623.2.a.cz.1.12	yes	12	3.2	odd	2	inner
7623.2.a.da.1.1	yes	12	33.32	even	2
7623.2.a.da.1.12	yes	12	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)

\(f(q)\)	\(=\)	\(q-2.79401 q^{2} +5.80650 q^{4} +1.62657 q^{5} -1.00000 q^{7} -10.6354 q^{8} +O(q^{10})\)	\(q-2.79401 q^{2} +5.80650 q^{4} +1.62657 q^{5} -1.00000 q^{7} -10.6354 q^{8} -4.54465 q^{10} -6.23267 q^{13} +2.79401 q^{14} +18.1024 q^{16} +6.00681 q^{17} +0.305878 q^{19} +9.44466 q^{20} +7.84138 q^{23} -2.35428 q^{25} +17.4141 q^{26} -5.80650 q^{28} +2.94628 q^{29} -5.31322 q^{31} -29.3076 q^{32} -16.7831 q^{34} -1.62657 q^{35} -6.41825 q^{37} -0.854627 q^{38} -17.2992 q^{40} +4.04048 q^{41} -0.640934 q^{43} -21.9089 q^{46} -7.86467 q^{47} +1.00000 q^{49} +6.57787 q^{50} -36.1900 q^{52} -0.251121 q^{53} +10.6354 q^{56} -8.23194 q^{58} -5.46616 q^{59} +2.45477 q^{61} +14.8452 q^{62} +45.6808 q^{64} -10.1379 q^{65} +6.54902 q^{67} +34.8785 q^{68} +4.54465 q^{70} -5.61219 q^{71} -10.4714 q^{73} +17.9327 q^{74} +1.77608 q^{76} +7.16047 q^{79} +29.4448 q^{80} -11.2891 q^{82} +11.7565 q^{83} +9.77049 q^{85} +1.79078 q^{86} -9.59605 q^{89} +6.23267 q^{91} +45.5310 q^{92} +21.9740 q^{94} +0.497532 q^{95} +16.0963 q^{97} -2.79401 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} - 12 q^{7}+O(q^{10}) \) 12 * q + 20 * q^4 - 12 * q^7	\( 12 q + 20 q^{4} - 12 q^{7} - 20 q^{10} - 20 q^{13} + 28 q^{16} - 12 q^{19} + 32 q^{25} - 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} - 48 q^{40} - 16 q^{43} - 24 q^{46} + 12 q^{49} - 96 q^{52} + 20 q^{58} - 44 q^{61} + 76 q^{64} + 20 q^{70} - 52 q^{73} + 8 q^{79} - 68 q^{82} - 72 q^{85} + 20 q^{91} - 20 q^{94} - 32 q^{97}+O(q^{100}) \) 12 * q + 20 * q^4 - 12 * q^7 - 20 * q^10 - 20 * q^13 + 28 * q^16 - 12 * q^19 + 32 * q^25 - 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 - 48 * q^40 - 16 * q^43 - 24 * q^46 + 12 * q^49 - 96 * q^52 + 20 * q^58 - 44 * q^61 + 76 * q^64 + 20 * q^70 - 52 * q^73 + 8 * q^79 - 68 * q^82 - 72 * q^85 + 20 * q^91 - 20 * q^94 - 32 * q^97

Introduction

L-functions

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