LMFDB - Embedded newform 7623.2.a.cz.1.9

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			1.9
Root			$1.60257$ 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)

$f(q)$	$=$	$q+1.60257 q^{2} +0.568238 q^{4} +1.76461 q^{5} -1.00000 q^{7} -2.29450 q^{8} +O(q^{10})$	$q+1.60257 q^{2} +0.568238 q^{4} +1.76461 q^{5} -1.00000 q^{7} -2.29450 q^{8} +2.82792 q^{10} +0.237832 q^{13} -1.60257 q^{14} -4.81358 q^{16} -6.14292 q^{17} +1.53812 q^{19} +1.00272 q^{20} +3.89707 q^{23} -1.88615 q^{25} +0.381142 q^{26} -0.568238 q^{28} +4.45543 q^{29} +8.52985 q^{31} -3.12511 q^{32} -9.84447 q^{34} -1.76461 q^{35} -3.64448 q^{37} +2.46495 q^{38} -4.04890 q^{40} +1.90659 q^{41} -11.7102 q^{43} +6.24534 q^{46} -8.55604 q^{47} +1.00000 q^{49} -3.02269 q^{50} +0.135145 q^{52} +3.14359 q^{53} +2.29450 q^{56} +7.14015 q^{58} -2.56187 q^{59} -6.78951 q^{61} +13.6697 q^{62} +4.61895 q^{64} +0.419680 q^{65} -1.62185 q^{67} -3.49064 q^{68} -2.82792 q^{70} -7.37151 q^{71} +4.60387 q^{73} -5.84054 q^{74} +0.874018 q^{76} -17.2504 q^{79} -8.49410 q^{80} +3.05546 q^{82} -9.16622 q^{83} -10.8399 q^{85} -18.7665 q^{86} -1.14303 q^{89} -0.237832 q^{91} +2.21446 q^{92} -13.7117 q^{94} +2.71418 q^{95} -4.68538 q^{97} +1.60257 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.60257	1.13319	0.566595	−	0.823997i	$-0.308260\pi$
$2$	1.60257	1.13319	0.566595	+	0.823997i	$0.308260\pi$
$3$	0	0
$3$	0	0
$4$	0.568238	0.284119
$5$	1.76461	0.789158	0.394579	−	0.918862i	$-0.370890\pi$
$5$	1.76461	0.789158	0.394579	+	0.918862i	$0.370890\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.29450	−0.811229
$9$	0	0
$10$	2.82792	0.894265
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.237832	0.0659626	0.0329813	−	0.999456i	$-0.489500\pi$
$13$	0.237832	0.0659626	0.0329813	+	0.999456i	$0.489500\pi$
$14$	−1.60257	−0.428305
$15$	0	0
$16$	−4.81358	−1.20340
$17$	−6.14292	−1.48988	−0.744938	−	0.667133i	$-0.767521\pi$
$17$	−6.14292	−1.48988	−0.744938	+	0.667133i	$0.767521\pi$
$18$	0	0
$19$	1.53812	0.352869	0.176434	−	0.984312i	$-0.443544\pi$
$19$	1.53812	0.352869	0.176434	+	0.984312i	$0.443544\pi$
$20$	1.00272	0.224215
$21$	0	0
$22$	0	0
$23$	3.89707	0.812596	0.406298	−	0.913741i	$-0.366819\pi$
$23$	3.89707	0.812596	0.406298	+	0.913741i	$0.366819\pi$
$24$	0	0
$25$	−1.88615	−0.377230
$26$	0.381142	0.0747482
$27$	0	0
$28$	−0.568238	−0.107387
$29$	4.45543	0.827352	0.413676	−	0.910424i	$-0.364245\pi$
$29$	4.45543	0.827352	0.413676	+	0.910424i	$0.364245\pi$
$30$	0	0
$31$	8.52985	1.53201	0.766003	−	0.642837i	$-0.222243\pi$
$31$	8.52985	1.53201	0.766003	+	0.642837i	$0.222243\pi$
$32$	−3.12511	−0.552446
$33$	0	0
$34$	−9.84447	−1.68831
$35$	−1.76461	−0.298274
$36$	0	0
$37$	−3.64448	−0.599149	−0.299574	−	0.954073i	$-0.596845\pi$
$37$	−3.64448	−0.599149	−0.299574	+	0.954073i	$0.596845\pi$
$38$	2.46495	0.399867
$39$	0	0
$40$	−4.04890	−0.640188
$41$	1.90659	0.297760	0.148880	−	0.988855i	$-0.452433\pi$
$41$	1.90659	0.297760	0.148880	+	0.988855i	$0.452433\pi$
$42$	0	0
$43$	−11.7102	−1.78579	−0.892896	−	0.450262i	$-0.851330\pi$
$43$	−11.7102	−1.78579	−0.892896	+	0.450262i	$0.851330\pi$
$44$	0	0
$45$	0	0
$46$	6.24534	0.920826
$47$	−8.55604	−1.24803	−0.624013	−	0.781414i	$-0.714499\pi$
$47$	−8.55604	−1.24803	−0.624013	+	0.781414i	$0.714499\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.02269	−0.427473
$51$	0	0
$52$	0.135145	0.0187412
$53$	3.14359	0.431805	0.215903	−	0.976415i	$-0.430731\pi$
$53$	3.14359	0.431805	0.215903	+	0.976415i	$0.430731\pi$
$54$	0	0
$55$	0	0
$56$	2.29450	0.306616
$57$	0	0
$58$	7.14015	0.937547
$59$	−2.56187	−0.333527	−0.166764	−	0.985997i	$-0.553332\pi$
$59$	−2.56187	−0.333527	−0.166764	+	0.985997i	$0.553332\pi$
$60$	0	0
$61$	−6.78951	−0.869307	−0.434654	−	0.900598i	$-0.643129\pi$
$61$	−6.78951	−0.869307	−0.434654	+	0.900598i	$0.643129\pi$
$62$	13.6697	1.73605
$63$	0	0
$64$	4.61895	0.577369
$65$	0.419680	0.0520549
$66$	0	0
$67$	−1.62185	−0.198141	−0.0990704	−	0.995080i	$-0.531587\pi$
$67$	−1.62185	−0.198141	−0.0990704	+	0.995080i	$0.531587\pi$
$68$	−3.49064	−0.423302
$69$	0	0
$70$	−2.82792	−0.338001
$71$	−7.37151	−0.874837	−0.437418	−	0.899258i	$-0.644107\pi$
$71$	−7.37151	−0.874837	−0.437418	+	0.899258i	$0.644107\pi$
$72$	0	0
$73$	4.60387	0.538842	0.269421	−	0.963023i	$-0.413168\pi$
$73$	4.60387	0.538842	0.269421	+	0.963023i	$0.413168\pi$
$74$	−5.84054	−0.678949
$75$	0	0
$76$	0.874018	0.100257
$77$	0	0
$78$	0	0
$79$	−17.2504	−1.94082	−0.970409	−	0.241468i	$-0.922371\pi$
$79$	−17.2504	−1.94082	−0.970409	+	0.241468i	$0.922371\pi$
$80$	−8.49410	−0.949669
$81$	0	0
$82$	3.05546	0.337419
$83$	−9.16622	−1.00612	−0.503062	−	0.864250i	$-0.667793\pi$
$83$	−9.16622	−1.00612	−0.503062	+	0.864250i	$0.667793\pi$
$84$	0	0
$85$	−10.8399	−1.17575
$86$	−18.7665	−2.02364
$87$	0	0
$88$	0	0
$89$	−1.14303	−0.121161	−0.0605807	−	0.998163i	$-0.519295\pi$
$89$	−1.14303	−0.121161	−0.0605807	+	0.998163i	$0.519295\pi$
$90$	0	0
$91$	−0.237832	−0.0249315
$92$	2.21446	0.230874
$93$	0	0
$94$	−13.7117	−1.41425
$95$	2.71418	0.278469
$96$	0	0
$97$	−4.68538	−0.475728	−0.237864	−	0.971298i	$-0.576447\pi$
$97$	−4.68538	−0.475728	−0.237864	+	0.971298i	$0.576447\pi$
$98$	1.60257	0.161884
$99$	0	0
$100$	−1.07178	−0.107178
$101$	−18.1433	−1.80532	−0.902661	−	0.430352i	$-0.858389\pi$
$101$	−18.1433	−1.80532	−0.902661	+	0.430352i	$0.858389\pi$
$102$	0	0
$103$	4.75190	0.468219	0.234110	−	0.972210i	$-0.424783\pi$
$103$	4.75190	0.468219	0.234110	+	0.972210i	$0.424783\pi$
$104$	−0.545705	−0.0535108
$105$	0	0
$106$	5.03783	0.489317
$107$	2.16093	0.208905	0.104452	−	0.994530i	$-0.466691\pi$
$107$	2.16093	0.208905	0.104452	+	0.994530i	$0.466691\pi$
$108$	0	0
$109$	4.21878	0.404086	0.202043	−	0.979377i	$-0.435242\pi$
$109$	4.21878	0.404086	0.202043	+	0.979377i	$0.435242\pi$
$110$	0	0
$111$	0	0
$112$	4.81358	0.454841
$113$	−7.53163	−0.708516	−0.354258	−	0.935148i	$-0.615267\pi$
$113$	−7.53163	−0.708516	−0.354258	+	0.935148i	$0.615267\pi$
$114$	0	0
$115$	6.87682	0.641267
$116$	2.53174	0.235066
$117$	0	0
$118$	−4.10559	−0.377950
$119$	6.14292	0.563120
$120$	0	0
$121$	0	0
$122$	−10.8807	−0.985090
$123$	0	0
$124$	4.84698	0.435272
$125$	−12.1514	−1.08685
$126$	0	0
$127$	3.86671	0.343115	0.171557	−	0.985174i	$-0.445120\pi$
$127$	3.86671	0.343115	0.171557	+	0.985174i	$0.445120\pi$
$128$	13.6524	1.20671
$129$	0	0
$130$	0.672568	0.0589881
$131$	4.55591	0.398052	0.199026	−	0.979994i	$-0.436222\pi$
$131$	4.55591	0.398052	0.199026	+	0.979994i	$0.436222\pi$
$132$	0	0
$133$	−1.53812	−0.133372
$134$	−2.59914	−0.224531
$135$	0	0
$136$	14.0949	1.20863
$137$	3.34382	0.285682	0.142841	−	0.989746i	$-0.454376\pi$
$137$	3.34382	0.285682	0.142841	+	0.989746i	$0.454376\pi$
$138$	0	0
$139$	−10.7288	−0.910003	−0.455002	−	0.890491i	$-0.650361\pi$
$139$	−10.7288	−0.910003	−0.455002	+	0.890491i	$0.650361\pi$
$140$	−1.00272	−0.0847451
$141$	0	0
$142$	−11.8134	−0.991356
$143$	0	0
$144$	0	0
$145$	7.86210	0.652912
$146$	7.37803	0.610610
$147$	0	0
$148$	−2.07093	−0.170229
$149$	−20.4997	−1.67940	−0.839699	−	0.543052i	$-0.817269\pi$
$149$	−20.4997	−1.67940	−0.839699	+	0.543052i	$0.817269\pi$
$150$	0	0
$151$	−19.6500	−1.59909	−0.799547	−	0.600604i	$-0.794927\pi$
$151$	−19.6500	−1.59909	−0.799547	+	0.600604i	$0.794927\pi$
$152$	−3.52922	−0.286258
$153$	0	0
$154$	0	0
$155$	15.0519	1.20899
$156$	0	0
$157$	13.0140	1.03863	0.519313	−	0.854584i	$-0.326188\pi$
$157$	13.0140	1.03863	0.519313	+	0.854584i	$0.326188\pi$
$158$	−27.6450	−2.19931
$159$	0	0
$160$	−5.51460	−0.435967
$161$	−3.89707	−0.307133
$162$	0	0
$163$	8.09731	0.634230	0.317115	−	0.948387i	$-0.397286\pi$
$163$	8.09731	0.634230	0.317115	+	0.948387i	$0.397286\pi$
$164$	1.08340	0.0845992
$165$	0	0
$166$	−14.6895	−1.14013
$167$	4.97203	0.384747	0.192374	−	0.981322i	$-0.438381\pi$
$167$	4.97203	0.384747	0.192374	+	0.981322i	$0.438381\pi$
$168$	0	0
$169$	−12.9434	−0.995649
$170$	−17.3717	−1.33235
$171$	0	0
$172$	−6.65419	−0.507377
$173$	−14.0437	−1.06772	−0.533861	−	0.845572i	$-0.679259\pi$
$173$	−14.0437	−1.06772	−0.533861	+	0.845572i	$0.679259\pi$
$174$	0	0
$175$	1.88615	0.142580
$176$	0	0
$177$	0	0
$178$	−1.83180	−0.137299
$179$	−19.2574	−1.43937	−0.719684	−	0.694301i	$-0.755714\pi$
$179$	−19.2574	−1.43937	−0.719684	+	0.694301i	$0.755714\pi$
$180$	0	0
$181$	−5.45622	−0.405558	−0.202779	−	0.979225i	$-0.564997\pi$
$181$	−5.45622	−0.405558	−0.202779	+	0.979225i	$0.564997\pi$
$182$	−0.381142	−0.0282522
$183$	0	0
$184$	−8.94185	−0.659202
$185$	−6.43108	−0.472823
$186$	0	0
$187$	0	0
$188$	−4.86187	−0.354588
$189$	0	0
$190$	4.34967	0.315559
$191$	−0.336721	−0.0243643	−0.0121822	−	0.999926i	$-0.503878\pi$
$191$	−0.336721	−0.0243643	−0.0121822	+	0.999926i	$0.503878\pi$
$192$	0	0
$193$	−12.4807	−0.898378	−0.449189	−	0.893437i	$-0.648287\pi$
$193$	−12.4807	−0.898378	−0.449189	+	0.893437i	$0.648287\pi$
$194$	−7.50866	−0.539090
$195$	0	0
$196$	0.568238	0.0405884
$197$	20.9906	1.49552	0.747760	−	0.663969i	$-0.231129\pi$
$197$	20.9906	1.49552	0.747760	+	0.663969i	$0.231129\pi$
$198$	0	0
$199$	−18.6504	−1.32209	−0.661045	−	0.750346i	$-0.729887\pi$
$199$	−18.6504	−1.32209	−0.661045	+	0.750346i	$0.729887\pi$
$200$	4.32778	0.306020
$201$	0	0
$202$	−29.0759	−2.04577
$203$	−4.45543	−0.312710
$204$	0	0
$205$	3.36440	0.234980
$206$	7.61527	0.530581
$207$	0	0
$208$	−1.14482	−0.0793791
$209$	0	0
$210$	0	0
$211$	12.1517	0.836558	0.418279	−	0.908319i	$-0.362634\pi$
$211$	12.1517	0.836558	0.418279	+	0.908319i	$0.362634\pi$
$212$	1.78631	0.122684
$213$	0	0
$214$	3.46304	0.236729
$215$	−20.6640	−1.40927
$216$	0	0
$217$	−8.52985	−0.579044
$218$	6.76090	0.457906
$219$	0	0
$220$	0	0
$221$	−1.46098	−0.0982762
$222$	0	0
$223$	22.0300	1.47524	0.737618	−	0.675219i	$-0.235951\pi$
$223$	22.0300	1.47524	0.737618	+	0.675219i	$0.235951\pi$
$224$	3.12511	0.208805
$225$	0	0
$226$	−12.0700	−0.802883
$227$	0.991072	0.0657798	0.0328899	−	0.999459i	$-0.489529\pi$
$227$	0.991072	0.0657798	0.0328899	+	0.999459i	$0.489529\pi$
$228$	0	0
$229$	26.2425	1.73415	0.867076	−	0.498176i	$-0.165997\pi$
$229$	26.2425	1.73415	0.867076	+	0.498176i	$0.165997\pi$
$230$	11.0206	0.726677
$231$	0	0
$232$	−10.2230	−0.671172
$233$	10.3858	0.680394	0.340197	−	0.940354i	$-0.389506\pi$
$233$	10.3858	0.680394	0.340197	+	0.940354i	$0.389506\pi$
$234$	0	0
$235$	−15.0981	−0.984890
$236$	−1.45575	−0.0947614
$237$	0	0
$238$	9.84447	0.638122
$239$	7.75162	0.501411	0.250705	−	0.968063i	$-0.419337\pi$
$239$	7.75162	0.501411	0.250705	+	0.968063i	$0.419337\pi$
$240$	0	0
$241$	5.25863	0.338738	0.169369	−	0.985553i	$-0.445827\pi$
$241$	5.25863	0.338738	0.169369	+	0.985553i	$0.445827\pi$
$242$	0	0
$243$	0	0
$244$	−3.85805	−0.246987
$245$	1.76461	0.112737
$246$	0	0
$247$	0.365814	0.0232762
$248$	−19.5718	−1.24281
$249$	0	0
$250$	−19.4735	−1.23161
$251$	22.0821	1.39381	0.696905	−	0.717164i	$-0.254560\pi$
$251$	22.0821	1.39381	0.696905	+	0.717164i	$0.254560\pi$
$252$	0	0
$253$	0	0
$254$	6.19668	0.388814
$255$	0	0
$256$	12.6411	0.790068
$257$	−19.1651	−1.19549	−0.597745	−	0.801687i	$-0.703936\pi$
$257$	−19.1651	−1.19549	−0.597745	+	0.801687i	$0.703936\pi$
$258$	0	0
$259$	3.64448	0.226457
$260$	0.238478	0.0147898
$261$	0	0
$262$	7.30118	0.451068
$263$	20.1639	1.24336	0.621679	−	0.783272i	$-0.286451\pi$
$263$	20.1639	1.24336	0.621679	+	0.783272i	$0.286451\pi$
$264$	0	0
$265$	5.54721	0.340763
$266$	−2.46495	−0.151136
$267$	0	0
$268$	−0.921598	−0.0562955
$269$	26.6866	1.62711	0.813556	−	0.581487i	$-0.197529\pi$
$269$	26.6866	1.62711	0.813556	+	0.581487i	$0.197529\pi$
$270$	0	0
$271$	12.5714	0.763659	0.381830	−	0.924233i	$-0.375294\pi$
$271$	12.5714	0.763659	0.381830	+	0.924233i	$0.375294\pi$
$272$	29.5694	1.79291
$273$	0	0
$274$	5.35871	0.323732
$275$	0	0
$276$	0	0
$277$	−3.65785	−0.219779	−0.109890	−	0.993944i	$-0.535050\pi$
$277$	−3.65785	−0.219779	−0.109890	+	0.993944i	$0.535050\pi$
$278$	−17.1936	−1.03121
$279$	0	0
$280$	4.04890	0.241968
$281$	4.74032	0.282784	0.141392	−	0.989954i	$-0.454842\pi$
$281$	4.74032	0.282784	0.141392	+	0.989954i	$0.454842\pi$
$282$	0	0
$283$	4.64873	0.276338	0.138169	−	0.990409i	$-0.455878\pi$
$283$	4.64873	0.276338	0.138169	+	0.990409i	$0.455878\pi$
$284$	−4.18877	−0.248558
$285$	0	0
$286$	0	0
$287$	−1.90659	−0.112543
$288$	0	0
$289$	20.7355	1.21973
$290$	12.5996	0.739873
$291$	0	0
$292$	2.61609	0.153095
$293$	8.26249	0.482700	0.241350	−	0.970438i	$-0.422410\pi$
$293$	8.26249	0.482700	0.241350	+	0.970438i	$0.422410\pi$
$294$	0	0
$295$	−4.52071	−0.263206
$296$	8.36226	0.486047
$297$	0	0
$298$	−32.8522	−1.90308
$299$	0.926848	0.0536010
$300$	0	0
$301$	11.7102	0.674966
$302$	−31.4905	−1.81208
$303$	0	0
$304$	−7.40387	−0.424641
$305$	−11.9808	−0.686020
$306$	0	0
$307$	−17.8463	−1.01854	−0.509272	−	0.860606i	$-0.670085\pi$
$307$	−17.8463	−1.01854	−0.509272	+	0.860606i	$0.670085\pi$
$308$	0	0
$309$	0	0
$310$	24.1217	1.37002
$311$	−12.6005	−0.714511	−0.357256	−	0.934007i	$-0.616288\pi$
$311$	−12.6005	−0.714511	−0.357256	+	0.934007i	$0.616288\pi$
$312$	0	0
$313$	−27.7579	−1.56897	−0.784484	−	0.620149i	$-0.787072\pi$
$313$	−27.7579	−1.56897	−0.784484	+	0.620149i	$0.787072\pi$
$314$	20.8558	1.17696
$315$	0	0
$316$	−9.80231	−0.551423
$317$	−3.05179	−0.171406	−0.0857028	−	0.996321i	$-0.527314\pi$
$317$	−3.05179	−0.171406	−0.0857028	+	0.996321i	$0.527314\pi$
$318$	0	0
$319$	0	0
$320$	8.15065	0.455635
$321$	0	0
$322$	−6.24534	−0.348039
$323$	−9.44855	−0.525731
$324$	0	0
$325$	−0.448586	−0.0248831
$326$	12.9765	0.718703
$327$	0	0
$328$	−4.37469	−0.241552
$329$	8.55604	0.471710
$330$	0	0
$331$	22.7323	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$331$	22.7323	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$332$	−5.20859	−0.285859
$333$	0	0
$334$	7.96804	0.435992
$335$	−2.86194	−0.156364
$336$	0	0
$337$	−11.2682	−0.613815	−0.306908	−	0.951739i	$-0.599294\pi$
$337$	−11.2682	−0.613815	−0.306908	+	0.951739i	$0.599294\pi$
$338$	−20.7428	−1.12826
$339$	0	0
$340$	−6.15962	−0.334052
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	26.8691	1.44869
$345$	0	0
$346$	−22.5060	−1.20993
$347$	27.8666	1.49596	0.747980	−	0.663722i	$-0.231024\pi$
$347$	27.8666	1.49596	0.747980	+	0.663722i	$0.231024\pi$
$348$	0	0
$349$	7.54901	0.404089	0.202045	−	0.979376i	$-0.435241\pi$
$349$	7.54901	0.404089	0.202045	+	0.979376i	$0.435241\pi$
$350$	3.02269	0.161570
$351$	0	0
$352$	0	0
$353$	11.2041	0.596334	0.298167	−	0.954514i	$-0.403625\pi$
$353$	11.2041	0.596334	0.298167	+	0.954514i	$0.403625\pi$
$354$	0	0
$355$	−13.0078	−0.690384
$356$	−0.649515	−0.0344242
$357$	0	0
$358$	−30.8614	−1.63108
$359$	−14.2242	−0.750725	−0.375362	−	0.926878i	$-0.622482\pi$
$359$	−14.2242	−0.750725	−0.375362	+	0.926878i	$0.622482\pi$
$360$	0	0
$361$	−16.6342	−0.875483
$362$	−8.74398	−0.459574
$363$	0	0
$364$	−0.135145	−0.00708352
$365$	8.12403	0.425231
$366$	0	0
$367$	−16.3499	−0.853455	−0.426728	−	0.904380i	$-0.640334\pi$
$367$	−16.3499	−0.853455	−0.426728	+	0.904380i	$0.640334\pi$
$368$	−18.7589	−0.977875
$369$	0	0
$370$	−10.3063	−0.535798
$371$	−3.14359	−0.163207
$372$	0	0
$373$	−22.6303	−1.17175	−0.585877	−	0.810400i	$-0.699250\pi$
$373$	−22.6303	−1.17175	−0.585877	+	0.810400i	$0.699250\pi$
$374$	0	0
$375$	0	0
$376$	19.6319	1.01244
$377$	1.05964	0.0545743
$378$	0	0
$379$	1.82350	0.0936671	0.0468335	−	0.998903i	$-0.485087\pi$
$379$	1.82350	0.0936671	0.0468335	+	0.998903i	$0.485087\pi$
$380$	1.54230	0.0791184
$381$	0	0
$382$	−0.539620	−0.0276094
$383$	−11.1434	−0.569404	−0.284702	−	0.958616i	$-0.591895\pi$
$383$	−11.1434	−0.569404	−0.284702	+	0.958616i	$0.591895\pi$
$384$	0	0
$385$	0	0
$386$	−20.0012	−1.01803
$387$	0	0
$388$	−2.66241	−0.135163
$389$	−28.6576	−1.45300	−0.726499	−	0.687168i	$-0.758854\pi$
$389$	−28.6576	−1.45300	−0.726499	+	0.687168i	$0.758854\pi$
$390$	0	0
$391$	−23.9394	−1.21067
$392$	−2.29450	−0.115890
$393$	0	0
$394$	33.6390	1.69471
$395$	−30.4402	−1.53161
$396$	0	0
$397$	−9.37451	−0.470493	−0.235247	−	0.971936i	$-0.575590\pi$
$397$	−9.37451	−0.470493	−0.235247	+	0.971936i	$0.575590\pi$
$398$	−29.8886	−1.49818
$399$	0	0
$400$	9.07914	0.453957
$401$	17.1824	0.858048	0.429024	−	0.903293i	$-0.358858\pi$
$401$	17.1824	0.858048	0.429024	+	0.903293i	$0.358858\pi$
$402$	0	0
$403$	2.02867	0.101055
$404$	−10.3097	−0.512926
$405$	0	0
$406$	−7.14015	−0.354360
$407$	0	0
$408$	0	0
$409$	0.548440	0.0271186	0.0135593	−	0.999908i	$-0.495684\pi$
$409$	0.548440	0.0271186	0.0135593	+	0.999908i	$0.495684\pi$
$410$	5.39169	0.266276
$411$	0	0
$412$	2.70021	0.133030
$413$	2.56187	0.126061
$414$	0	0
$415$	−16.1748	−0.793990
$416$	−0.743249	−0.0364408
$417$	0	0
$418$	0	0
$419$	5.31528	0.259668	0.129834	−	0.991536i	$-0.458556\pi$
$419$	5.31528	0.259668	0.129834	+	0.991536i	$0.458556\pi$
$420$	0	0
$421$	9.52820	0.464376	0.232188	−	0.972671i	$-0.425412\pi$
$421$	9.52820	0.464376	0.232188	+	0.972671i	$0.425412\pi$
$422$	19.4740	0.947978
$423$	0	0
$424$	−7.21298	−0.350293
$425$	11.5865	0.562026
$426$	0	0
$427$	6.78951	0.328567
$428$	1.22792	0.0593537
$429$	0	0
$430$	−33.1155	−1.59697
$431$	35.6144	1.71548	0.857742	−	0.514081i	$-0.171867\pi$
$431$	35.6144	1.71548	0.857742	+	0.514081i	$0.171867\pi$
$432$	0	0
$433$	1.15006	0.0552682	0.0276341	−	0.999618i	$-0.491203\pi$
$433$	1.15006	0.0552682	0.0276341	+	0.999618i	$0.491203\pi$
$434$	−13.6697	−0.656167
$435$	0	0
$436$	2.39727	0.114808
$437$	5.99417	0.286740
$438$	0	0
$439$	−33.3954	−1.59388	−0.796938	−	0.604061i	$-0.793548\pi$
$439$	−33.3954	−1.59388	−0.796938	+	0.604061i	$0.793548\pi$
$440$	0	0
$441$	0	0
$442$	−2.34133	−0.111366
$443$	−20.8554	−0.990871	−0.495435	−	0.868645i	$-0.664991\pi$
$443$	−20.8554	−0.990871	−0.495435	+	0.868645i	$0.664991\pi$
$444$	0	0
$445$	−2.01701	−0.0956155
$446$	35.3046	1.67172
$447$	0	0
$448$	−4.61895	−0.218225
$449$	32.8374	1.54969	0.774846	−	0.632150i	$-0.217827\pi$
$449$	32.8374	1.54969	0.774846	+	0.632150i	$0.217827\pi$
$450$	0	0
$451$	0	0
$452$	−4.27975	−0.201303
$453$	0	0
$454$	1.58826	0.0745410
$455$	−0.419680	−0.0196749
$456$	0	0
$457$	−22.9232	−1.07230	−0.536151	−	0.844122i	$-0.680122\pi$
$457$	−22.9232	−1.07230	−0.536151	+	0.844122i	$0.680122\pi$
$458$	42.0555	1.96512
$459$	0	0
$460$	3.90767	0.182196
$461$	19.5054	0.908455	0.454228	−	0.890886i	$-0.349915\pi$
$461$	19.5054	0.908455	0.454228	+	0.890886i	$0.349915\pi$
$462$	0	0
$463$	−6.22668	−0.289378	−0.144689	−	0.989477i	$-0.546218\pi$
$463$	−6.22668	−0.289378	−0.144689	+	0.989477i	$0.546218\pi$
$464$	−21.4466	−0.995632
$465$	0	0
$466$	16.6439	0.771016
$467$	−5.20609	−0.240909	−0.120455	−	0.992719i	$-0.538435\pi$
$467$	−5.20609	−0.240909	−0.120455	+	0.992719i	$0.538435\pi$
$468$	0	0
$469$	1.62185	0.0748902
$470$	−24.1958	−1.11607
$471$	0	0
$472$	5.87822	0.270567
$473$	0	0
$474$	0	0
$475$	−2.90113	−0.133113
$476$	3.49064	0.159993
$477$	0	0
$478$	12.4225	0.568193
$479$	−25.9938	−1.18769	−0.593844	−	0.804580i	$-0.702391\pi$
$479$	−25.9938	−1.18769	−0.593844	+	0.804580i	$0.702391\pi$
$480$	0	0
$481$	−0.866772	−0.0395214
$482$	8.42733	0.383855
$483$	0	0
$484$	0	0
$485$	−8.26787	−0.375424
$486$	0	0
$487$	37.1070	1.68148	0.840739	−	0.541441i	$-0.182121\pi$
$487$	37.1070	1.68148	0.840739	+	0.541441i	$0.182121\pi$
$488$	15.5785	0.705207
$489$	0	0
$490$	2.82792	0.127752
$491$	−27.7241	−1.25117	−0.625585	−	0.780156i	$-0.715140\pi$
$491$	−27.7241	−1.25117	−0.625585	+	0.780156i	$0.715140\pi$
$492$	0	0
$493$	−27.3693	−1.23265
$494$	0.586243	0.0263763
$495$	0	0
$496$	−41.0591	−1.84361
$497$	7.37151	0.330657
$498$	0	0
$499$	−18.9850	−0.849885	−0.424942	−	0.905220i	$-0.639706\pi$
$499$	−18.9850	−0.849885	−0.424942	+	0.905220i	$0.639706\pi$
$500$	−6.90487	−0.308795
$501$	0	0
$502$	35.3881	1.57945
$503$	23.2430	1.03635	0.518177	−	0.855273i	$-0.326611\pi$
$503$	23.2430	1.03635	0.518177	+	0.855273i	$0.326611\pi$
$504$	0	0
$505$	−32.0158	−1.42468
$506$	0	0
$507$	0	0
$508$	2.19721	0.0974853
$509$	28.8227	1.27754	0.638771	−	0.769397i	$-0.279443\pi$
$509$	28.8227	1.27754	0.638771	+	0.769397i	$0.279443\pi$
$510$	0	0
$511$	−4.60387	−0.203663
$512$	−7.04659	−0.311418
$513$	0	0
$514$	−30.7135	−1.35472
$515$	8.38526	0.369499
$516$	0	0
$517$	0	0
$518$	5.84054	0.256619
$519$	0	0
$520$	−0.962957	−0.0422285
$521$	41.2984	1.80931	0.904657	−	0.426140i	$-0.140127\pi$
$521$	41.2984	1.80931	0.904657	+	0.426140i	$0.140127\pi$
$522$	0	0
$523$	−21.0083	−0.918629	−0.459315	−	0.888274i	$-0.651905\pi$
$523$	−21.0083	−0.918629	−0.459315	+	0.888274i	$0.651905\pi$
$524$	2.58884	0.113094
$525$	0	0
$526$	32.3141	1.40896
$527$	−52.3982	−2.28250
$528$	0	0
$529$	−7.81281	−0.339687
$530$	8.88981	0.386149
$531$	0	0
$532$	−0.874018	−0.0378935
$533$	0.453449	0.0196410
$534$	0	0
$535$	3.81319	0.164859
$536$	3.72134	0.160738
$537$	0	0
$538$	42.7672	1.84383
$539$	0	0
$540$	0	0
$541$	−7.00282	−0.301075	−0.150537	−	0.988604i	$-0.548100\pi$
$541$	−7.00282	−0.301075	−0.150537	+	0.988604i	$0.548100\pi$
$542$	20.1466	0.865371
$543$	0	0
$544$	19.1973	0.823076
$545$	7.44450	0.318887
$546$	0	0
$547$	15.6735	0.670149	0.335075	−	0.942192i	$-0.391238\pi$
$547$	15.6735	0.670149	0.335075	+	0.942192i	$0.391238\pi$
$548$	1.90008	0.0811676
$549$	0	0
$550$	0	0
$551$	6.85299	0.291947
$552$	0	0
$553$	17.2504	0.733560
$554$	−5.86197	−0.249051
$555$	0	0
$556$	−6.09650	−0.258549
$557$	−43.2989	−1.83463	−0.917317	−	0.398157i	$-0.869650\pi$
$557$	−43.2989	−1.83463	−0.917317	+	0.398157i	$0.869650\pi$
$558$	0	0
$559$	−2.78506	−0.117796
$560$	8.49410	0.358941
$561$	0	0
$562$	7.59671	0.320448
$563$	−10.3329	−0.435482	−0.217741	−	0.976007i	$-0.569869\pi$
$563$	−10.3329	−0.435482	−0.217741	+	0.976007i	$0.569869\pi$
$564$	0	0
$565$	−13.2904	−0.559131
$566$	7.44992	0.313144
$567$	0	0
$568$	16.9139	0.709693
$569$	−21.1095	−0.884957	−0.442479	−	0.896779i	$-0.645901\pi$
$569$	−21.1095	−0.884957	−0.442479	+	0.896779i	$0.645901\pi$
$570$	0	0
$571$	34.8603	1.45886	0.729430	−	0.684056i	$-0.239786\pi$
$571$	34.8603	1.45886	0.729430	+	0.684056i	$0.239786\pi$
$572$	0	0
$573$	0	0
$574$	−3.05546	−0.127532
$575$	−7.35047	−0.306536
$576$	0	0
$577$	13.5157	0.562668	0.281334	−	0.959610i	$-0.409223\pi$
$577$	13.5157	0.562668	0.281334	+	0.959610i	$0.409223\pi$
$578$	33.2301	1.38219
$579$	0	0
$580$	4.46754	0.185504
$581$	9.16622	0.380279
$582$	0	0
$583$	0	0
$584$	−10.5636	−0.437124
$585$	0	0
$586$	13.2412	0.546990
$587$	−2.25845	−0.0932161	−0.0466081	−	0.998913i	$-0.514841\pi$
$587$	−2.25845	−0.0932161	−0.0466081	+	0.998913i	$0.514841\pi$
$588$	0	0
$589$	13.1199	0.540597
$590$	−7.24476	−0.298262
$591$	0	0
$592$	17.5430	0.721013
$593$	−30.2674	−1.24293	−0.621467	−	0.783440i	$-0.713463\pi$
$593$	−30.2674	−1.24293	−0.621467	+	0.783440i	$0.713463\pi$
$594$	0	0
$595$	10.8399	0.444391
$596$	−11.6487	−0.477149
$597$	0	0
$598$	1.48534	0.0607401
$599$	21.3459	0.872169	0.436084	−	0.899906i	$-0.356365\pi$
$599$	21.3459	0.872169	0.436084	+	0.899906i	$0.356365\pi$
$600$	0	0
$601$	−43.9786	−1.79392	−0.896962	−	0.442108i	$-0.854231\pi$
$601$	−43.9786	−1.79392	−0.896962	+	0.442108i	$0.854231\pi$
$602$	18.7665	0.764865
$603$	0	0
$604$	−11.1659	−0.454333
$605$	0	0
$606$	0	0
$607$	−18.7123	−0.759507	−0.379753	−	0.925088i	$-0.623991\pi$
$607$	−18.7123	−0.759507	−0.379753	+	0.925088i	$0.623991\pi$
$608$	−4.80679	−0.194941
$609$	0	0
$610$	−19.2001	−0.777391
$611$	−2.03490	−0.0823231
$612$	0	0
$613$	26.9777	1.08962	0.544810	−	0.838560i	$-0.316602\pi$
$613$	26.9777	1.08962	0.544810	+	0.838560i	$0.316602\pi$
$614$	−28.6000	−1.15420
$615$	0	0
$616$	0	0
$617$	29.1127	1.17203	0.586016	−	0.810300i	$-0.300696\pi$
$617$	29.1127	1.17203	0.586016	+	0.810300i	$0.300696\pi$
$618$	0	0
$619$	9.41996	0.378620	0.189310	−	0.981917i	$-0.439375\pi$
$619$	9.41996	0.378620	0.189310	+	0.981917i	$0.439375\pi$
$620$	8.55303	0.343498
$621$	0	0
$622$	−20.1933	−0.809677
$623$	1.14303	0.0457947
$624$	0	0
$625$	−12.0117	−0.480467
$626$	−44.4840	−1.77794
$627$	0	0
$628$	7.39502	0.295093
$629$	22.3877	0.892657
$630$	0	0
$631$	−1.99811	−0.0795436	−0.0397718	−	0.999209i	$-0.512663\pi$
$631$	−1.99811	−0.0795436	−0.0397718	+	0.999209i	$0.512663\pi$
$632$	39.5810	1.57445
$633$	0	0
$634$	−4.89071	−0.194235
$635$	6.82323	0.270772
$636$	0	0
$637$	0.237832	0.00942323
$638$	0	0
$639$	0	0
$640$	24.0912	0.952288
$641$	45.5981	1.80102	0.900508	−	0.434840i	$-0.143195\pi$
$641$	45.5981	1.80102	0.900508	+	0.434840i	$0.143195\pi$
$642$	0	0
$643$	−18.0073	−0.710140	−0.355070	−	0.934840i	$-0.615543\pi$
$643$	−18.0073	−0.710140	−0.355070	+	0.934840i	$0.615543\pi$
$644$	−2.21446	−0.0872621
$645$	0	0
$646$	−15.1420	−0.595753
$647$	−47.3237	−1.86049	−0.930243	−	0.366944i	$-0.880404\pi$
$647$	−47.3237	−1.86049	−0.930243	+	0.366944i	$0.880404\pi$
$648$	0	0
$649$	0	0
$650$	−0.718892	−0.0281973
$651$	0	0
$652$	4.60119	0.180197
$653$	5.87593	0.229943	0.114972	−	0.993369i	$-0.463322\pi$
$653$	5.87593	0.229943	0.114972	+	0.993369i	$0.463322\pi$
$654$	0	0
$655$	8.03941	0.314126
$656$	−9.17755	−0.358323
$657$	0	0
$658$	13.7117	0.534537
$659$	40.3862	1.57322	0.786612	−	0.617448i	$-0.211833\pi$
$659$	40.3862	1.57322	0.786612	+	0.617448i	$0.211833\pi$
$660$	0	0
$661$	22.8932	0.890442	0.445221	−	0.895421i	$-0.353125\pi$
$661$	22.8932	0.890442	0.445221	+	0.895421i	$0.353125\pi$
$662$	36.4301	1.41590
$663$	0	0
$664$	21.0319	0.816197
$665$	−2.71418	−0.105251
$666$	0	0
$667$	17.3631	0.672303
$668$	2.82530	0.109314
$669$	0	0
$670$	−4.58646	−0.177190
$671$	0	0
$672$	0	0
$673$	−16.1286	−0.621712	−0.310856	−	0.950457i	$-0.600616\pi$
$673$	−16.1286	−0.621712	−0.310856	+	0.950457i	$0.600616\pi$
$674$	−18.0580	−0.695569
$675$	0	0
$676$	−7.35495	−0.282883
$677$	−27.0862	−1.04101	−0.520504	−	0.853859i	$-0.674256\pi$
$677$	−27.0862	−1.04101	−0.520504	+	0.853859i	$0.674256\pi$
$678$	0	0
$679$	4.68538	0.179808
$680$	24.8721	0.953801
$681$	0	0
$682$	0	0
$683$	20.7408	0.793623	0.396811	−	0.917900i	$-0.370117\pi$
$683$	20.7408	0.793623	0.396811	+	0.917900i	$0.370117\pi$
$684$	0	0
$685$	5.90054	0.225448
$686$	−1.60257	−0.0611865
$687$	0	0
$688$	56.3681	2.14901
$689$	0.747645	0.0284830
$690$	0	0
$691$	−16.8328	−0.640348	−0.320174	−	0.947359i	$-0.603741\pi$
$691$	−16.8328	−0.640348	−0.320174	+	0.947359i	$0.603741\pi$
$692$	−7.98015	−0.303360
$693$	0	0
$694$	44.6583	1.69521
$695$	−18.9321	−0.718136
$696$	0	0
$697$	−11.7121	−0.443626
$698$	12.0978	0.457910
$699$	0	0
$700$	1.07178	0.0405095
$701$	39.9602	1.50928	0.754638	−	0.656141i	$-0.227812\pi$
$701$	39.9602	1.50928	0.754638	+	0.656141i	$0.227812\pi$
$702$	0	0
$703$	−5.60565	−0.211421
$704$	0	0
$705$	0	0
$706$	17.9554	0.675760
$707$	18.1433	0.682348
$708$	0	0
$709$	−2.60837	−0.0979595	−0.0489797	−	0.998800i	$-0.515597\pi$
$709$	−2.60837	−0.0979595	−0.0489797	+	0.998800i	$0.515597\pi$
$710$	−20.8460	−0.782336
$711$	0	0
$712$	2.62270	0.0982897
$713$	33.2415	1.24490
$714$	0	0
$715$	0	0
$716$	−10.9428	−0.408952
$717$	0	0
$718$	−22.7953	−0.850714
$719$	38.7626	1.44560	0.722800	−	0.691057i	$-0.242855\pi$
$719$	38.7626	1.44560	0.722800	+	0.691057i	$0.242855\pi$
$720$	0	0
$721$	−4.75190	−0.176970
$722$	−26.6575	−0.992089
$723$	0	0
$724$	−3.10043	−0.115227
$725$	−8.40361	−0.312102
$726$	0	0
$727$	−51.3510	−1.90450	−0.952251	−	0.305317i	$-0.901238\pi$
$727$	−51.3510	−1.90450	−0.952251	+	0.305317i	$0.901238\pi$
$728$	0.545705	0.0202252
$729$	0	0
$730$	13.0193	0.481868
$731$	71.9350	2.66061
$732$	0	0
$733$	−48.8445	−1.80411	−0.902057	−	0.431617i	$-0.857943\pi$
$733$	−48.8445	−1.80411	−0.902057	+	0.431617i	$0.857943\pi$
$734$	−26.2018	−0.967127
$735$	0	0
$736$	−12.1788	−0.448916
$737$	0	0
$738$	0	0
$739$	−15.0365	−0.553127	−0.276563	−	0.960996i	$-0.589196\pi$
$739$	−15.0365	−0.553127	−0.276563	+	0.960996i	$0.589196\pi$
$740$	−3.65438	−0.134338
$741$	0	0
$742$	−5.03783	−0.184945
$743$	45.1304	1.65567	0.827836	−	0.560970i	$-0.189572\pi$
$743$	45.1304	1.65567	0.827836	+	0.560970i	$0.189572\pi$
$744$	0	0
$745$	−36.1739	−1.32531
$746$	−36.2668	−1.32782
$747$	0	0
$748$	0	0
$749$	−2.16093	−0.0789585
$750$	0	0
$751$	6.78179	0.247471	0.123736	−	0.992315i	$-0.460513\pi$
$751$	6.78179	0.247471	0.123736	+	0.992315i	$0.460513\pi$
$752$	41.1852	1.50187
$753$	0	0
$754$	1.69815	0.0618431
$755$	−34.6746	−1.26194
$756$	0	0
$757$	49.9448	1.81528	0.907638	−	0.419754i	$-0.137884\pi$
$757$	49.9448	1.81528	0.907638	+	0.419754i	$0.137884\pi$
$758$	2.92230	0.106143
$759$	0	0
$760$	−6.22770	−0.225902
$761$	28.2853	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$761$	28.2853	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$762$	0	0
$763$	−4.21878	−0.152730
$764$	−0.191338	−0.00692236
$765$	0	0
$766$	−17.8582	−0.645242
$767$	−0.609294	−0.0220003
$768$	0	0
$769$	51.9422	1.87308	0.936542	−	0.350555i	$-0.114007\pi$
$769$	51.9422	1.87308	0.936542	+	0.350555i	$0.114007\pi$
$770$	0	0
$771$	0	0
$772$	−7.09198	−0.255246
$773$	−4.11535	−0.148019	−0.0740095	−	0.997258i	$-0.523580\pi$
$773$	−4.11535	−0.148019	−0.0740095	+	0.997258i	$0.523580\pi$
$774$	0	0
$775$	−16.0886	−0.577919
$776$	10.7506	0.385924
$777$	0	0
$778$	−45.9258	−1.64652
$779$	2.93257	0.105070
$780$	0	0
$781$	0	0
$782$	−38.3646	−1.37192
$783$	0	0
$784$	−4.81358	−0.171914
$785$	22.9646	0.819640
$786$	0	0
$787$	−0.444368	−0.0158400	−0.00792001	−	0.999969i	$-0.502521\pi$
$787$	−0.444368	−0.0158400	−0.00792001	+	0.999969i	$0.502521\pi$
$788$	11.9277	0.424906
$789$	0	0
$790$	−48.7826	−1.73561
$791$	7.53163	0.267794
$792$	0	0
$793$	−1.61476	−0.0573418
$794$	−15.0233	−0.533158
$795$	0	0
$796$	−10.5978	−0.375631
$797$	−16.0300	−0.567810	−0.283905	−	0.958852i	$-0.591630\pi$
$797$	−16.0300	−0.567810	−0.283905	+	0.958852i	$0.591630\pi$
$798$	0	0
$799$	52.5591	1.85941
$800$	5.89442	0.208399
$801$	0	0
$802$	27.5360	0.972331
$803$	0	0
$804$	0	0
$805$	−6.87682	−0.242376
$806$	3.25109	0.114515
$807$	0	0
$808$	41.6298	1.46453
$809$	24.1194	0.847993	0.423997	−	0.905664i	$-0.360627\pi$
$809$	24.1194	0.847993	0.423997	+	0.905664i	$0.360627\pi$
$810$	0	0
$811$	29.6041	1.03954	0.519769	−	0.854307i	$-0.326018\pi$
$811$	29.6041	1.03954	0.519769	+	0.854307i	$0.326018\pi$
$812$	−2.53174	−0.0888468
$813$	0	0
$814$	0	0
$815$	14.2886	0.500507
$816$	0	0
$817$	−18.0117	−0.630151
$818$	0.878914	0.0307305
$819$	0	0
$820$	1.91178	0.0667621
$821$	−44.3006	−1.54610	−0.773051	−	0.634344i	$-0.781270\pi$
$821$	−44.3006	−1.54610	−0.773051	+	0.634344i	$0.781270\pi$
$822$	0	0
$823$	24.1312	0.841162	0.420581	−	0.907255i	$-0.361826\pi$
$823$	24.1312	0.841162	0.420581	+	0.907255i	$0.361826\pi$
$824$	−10.9033	−0.379833
$825$	0	0
$826$	4.10559	0.142852
$827$	30.3326	1.05477	0.527383	−	0.849627i	$-0.323173\pi$
$827$	30.3326	1.05477	0.527383	+	0.849627i	$0.323173\pi$
$828$	0	0
$829$	−46.1560	−1.60306	−0.801532	−	0.597951i	$-0.795982\pi$
$829$	−46.1560	−1.60306	−0.801532	+	0.597951i	$0.795982\pi$
$830$	−25.9213	−0.899741
$831$	0	0
$832$	1.09853	0.0380848
$833$	−6.14292	−0.212840
$834$	0	0
$835$	8.77370	0.303626
$836$	0	0
$837$	0	0
$838$	8.51812	0.294253
$839$	39.3482	1.35845	0.679226	−	0.733930i	$-0.262316\pi$
$839$	39.3482	1.35845	0.679226	+	0.733930i	$0.262316\pi$
$840$	0	0
$841$	−9.14915	−0.315488
$842$	15.2696	0.526226
$843$	0	0
$844$	6.90505	0.237682
$845$	−22.8401	−0.785724
$846$	0	0
$847$	0	0
$848$	−15.1319	−0.519633
$849$	0	0
$850$	18.5682	0.636882
$851$	−14.2028	−0.486866
$852$	0	0
$853$	−12.9953	−0.444950	−0.222475	−	0.974938i	$-0.571414\pi$
$853$	−12.9953	−0.444950	−0.222475	+	0.974938i	$0.571414\pi$
$854$	10.8807	0.372329
$855$	0	0
$856$	−4.95825	−0.169470
$857$	−29.0953	−0.993877	−0.496939	−	0.867786i	$-0.665543\pi$
$857$	−29.0953	−0.993877	−0.496939	+	0.867786i	$0.665543\pi$
$858$	0	0
$859$	23.4404	0.799775	0.399888	−	0.916564i	$-0.369049\pi$
$859$	23.4404	0.799775	0.399888	+	0.916564i	$0.369049\pi$
$860$	−11.7421	−0.400401
$861$	0	0
$862$	57.0746	1.94397
$863$	30.9751	1.05441	0.527203	−	0.849740i	$-0.323241\pi$
$863$	30.9751	1.05441	0.527203	+	0.849740i	$0.323241\pi$
$864$	0	0
$865$	−24.7816	−0.842601
$866$	1.84305	0.0626294
$867$	0	0
$868$	−4.84698	−0.164517
$869$	0	0
$870$	0	0
$871$	−0.385728	−0.0130699
$872$	−9.68000	−0.327806
$873$	0	0
$874$	9.60609	0.324931
$875$	12.1514	0.410791
$876$	0	0
$877$	−45.4618	−1.53514	−0.767568	−	0.640967i	$-0.778533\pi$
$877$	−45.4618	−1.53514	−0.767568	+	0.640967i	$0.778533\pi$
$878$	−53.5186	−1.80616
$879$	0	0
$880$	0	0
$881$	−48.3625	−1.62937	−0.814687	−	0.579901i	$-0.803091\pi$
$881$	−48.3625	−1.62937	−0.814687	+	0.579901i	$0.803091\pi$
$882$	0	0
$883$	−13.9195	−0.468428	−0.234214	−	0.972185i	$-0.575252\pi$
$883$	−13.9195	−0.468428	−0.234214	+	0.972185i	$0.575252\pi$
$884$	−0.830184	−0.0279221
$885$	0	0
$886$	−33.4223	−1.12284
$887$	13.9089	0.467015	0.233508	−	0.972355i	$-0.424980\pi$
$887$	13.9089	0.467015	0.233508	+	0.972355i	$0.424980\pi$
$888$	0	0
$889$	−3.86671	−0.129685
$890$	−3.23241	−0.108350
$891$	0	0
$892$	12.5183	0.419142
$893$	−13.1602	−0.440390
$894$	0	0
$895$	−33.9819	−1.13589
$896$	−13.6524	−0.456095
$897$	0	0
$898$	52.6243	1.75609
$899$	38.0041	1.26751
$900$	0	0
$901$	−19.3108	−0.643337
$902$	0	0
$903$	0	0
$904$	17.2813	0.574769
$905$	−9.62810	−0.320049
$906$	0	0
$907$	46.8698	1.55629	0.778144	−	0.628086i	$-0.216162\pi$
$907$	46.8698	1.55629	0.778144	+	0.628086i	$0.216162\pi$
$908$	0.563165	0.0186893
$909$	0	0
$910$	−0.672568	−0.0222954
$911$	−17.2331	−0.570957	−0.285478	−	0.958385i	$-0.592152\pi$
$911$	−17.2331	−0.570957	−0.285478	+	0.958385i	$0.592152\pi$
$912$	0	0
$913$	0	0
$914$	−36.7361	−1.21512
$915$	0	0
$916$	14.9120	0.492705
$917$	−4.55591	−0.150450
$918$	0	0
$919$	20.0621	0.661786	0.330893	−	0.943668i	$-0.392650\pi$
$919$	20.0621	0.661786	0.330893	+	0.943668i	$0.392650\pi$
$920$	−15.7789	−0.520214
$921$	0	0
$922$	31.2587	1.02945
$923$	−1.75318	−0.0577066
$924$	0	0
$925$	6.87403	0.226017
$926$	−9.97870	−0.327921
$927$	0	0
$928$	−13.9237	−0.457068
$929$	49.7071	1.63084	0.815419	−	0.578871i	$-0.196507\pi$
$929$	49.7071	1.63084	0.815419	+	0.578871i	$0.196507\pi$
$930$	0	0
$931$	1.53812	0.0504099
$932$	5.90158	0.193313
$933$	0	0
$934$	−8.34314	−0.272996
$935$	0	0
$936$	0	0
$937$	−40.9723	−1.33851	−0.669254	−	0.743034i	$-0.733386\pi$
$937$	−40.9723	−1.33851	−0.669254	+	0.743034i	$0.733386\pi$
$938$	2.59914	0.0848648
$939$	0	0
$940$	−8.57930	−0.279826
$941$	−57.7002	−1.88097	−0.940486	−	0.339832i	$-0.889630\pi$
$941$	−57.7002	−1.88097	−0.940486	+	0.339832i	$0.889630\pi$
$942$	0	0
$943$	7.43014	0.241959
$944$	12.3318	0.401365
$945$	0	0
$946$	0	0
$947$	−27.9921	−0.909621	−0.454811	−	0.890588i	$-0.650293\pi$
$947$	−27.9921	−0.909621	−0.454811	+	0.890588i	$0.650293\pi$
$948$	0	0
$949$	1.09495	0.0355434
$950$	−4.64926	−0.150842
$951$	0	0
$952$	−14.0949	−0.456820
$953$	14.9272	0.483538	0.241769	−	0.970334i	$-0.422272\pi$
$953$	14.9272	0.483538	0.241769	+	0.970334i	$0.422272\pi$
$954$	0	0
$955$	−0.594182	−0.0192273
$956$	4.40476	0.142460
$957$	0	0
$958$	−41.6570	−1.34588
$959$	−3.34382	−0.107978
$960$	0	0
$961$	41.7583	1.34704
$962$	−1.38907	−0.0447853
$963$	0	0
$964$	2.98815	0.0962419
$965$	−22.0235	−0.708962
$966$	0	0
$967$	−40.7409	−1.31014	−0.655070	−	0.755568i	$-0.727361\pi$
$967$	−40.7409	−1.31014	−0.655070	+	0.755568i	$0.727361\pi$
$968$	0	0
$969$	0	0
$970$	−13.2499	−0.425427
$971$	32.4367	1.04094	0.520471	−	0.853879i	$-0.325756\pi$
$971$	32.4367	1.04094	0.520471	+	0.853879i	$0.325756\pi$
$972$	0	0
$973$	10.7288	0.343949
$974$	59.4666	1.90543
$975$	0	0
$976$	32.6818	1.04612
$977$	43.6622	1.39688	0.698438	−	0.715670i	$-0.253879\pi$
$977$	43.6622	1.39688	0.698438	+	0.715670i	$0.253879\pi$
$978$	0	0
$979$	0	0
$980$	1.00272	0.0320307
$981$	0	0
$982$	−44.4298	−1.41781
$983$	46.6583	1.48817	0.744085	−	0.668085i	$-0.232886\pi$
$983$	46.6583	1.48817	0.744085	+	0.668085i	$0.232886\pi$
$984$	0	0
$985$	37.0403	1.18020
$986$	−43.8613	−1.39683
$987$	0	0
$988$	0.207869	0.00661320
$989$	−45.6356	−1.45113
$990$	0	0
$991$	23.0760	0.733033	0.366516	−	0.930412i	$-0.380550\pi$
$991$	23.0760	0.733033	0.366516	+	0.930412i	$0.380550\pi$
$992$	−26.6567	−0.846351
$993$	0	0
$994$	11.8134	0.374697
$995$	−32.9106	−1.04334
$996$	0	0
$997$	−29.4136	−0.931538	−0.465769	−	0.884906i	$-0.654222\pi$
$997$	−29.4136	−0.931538	−0.465769	+	0.884906i	$0.654222\pi$
$998$	−30.4248	−0.963081
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cz.1.9	yes	12
3.2	odd	2	inner	7623.2.a.cz.1.4	✓	12
11.10	odd	2		7623.2.a.da.1.4	yes	12
33.32	even	2		7623.2.a.da.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cz.1.4	✓	12	3.2	odd	2	inner
7623.2.a.cz.1.9	yes	12	1.1	even	1	trivial
7623.2.a.da.1.4	yes	12	11.10	odd	2
7623.2.a.da.1.9	yes	12	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.60257 q^{2} +0.568238 q^{4} +1.76461 q^{5} -1.00000 q^{7} -2.29450 q^{8} +O(q^{10})\)	\(q+1.60257 q^{2} +0.568238 q^{4} +1.76461 q^{5} -1.00000 q^{7} -2.29450 q^{8} +2.82792 q^{10} +0.237832 q^{13} -1.60257 q^{14} -4.81358 q^{16} -6.14292 q^{17} +1.53812 q^{19} +1.00272 q^{20} +3.89707 q^{23} -1.88615 q^{25} +0.381142 q^{26} -0.568238 q^{28} +4.45543 q^{29} +8.52985 q^{31} -3.12511 q^{32} -9.84447 q^{34} -1.76461 q^{35} -3.64448 q^{37} +2.46495 q^{38} -4.04890 q^{40} +1.90659 q^{41} -11.7102 q^{43} +6.24534 q^{46} -8.55604 q^{47} +1.00000 q^{49} -3.02269 q^{50} +0.135145 q^{52} +3.14359 q^{53} +2.29450 q^{56} +7.14015 q^{58} -2.56187 q^{59} -6.78951 q^{61} +13.6697 q^{62} +4.61895 q^{64} +0.419680 q^{65} -1.62185 q^{67} -3.49064 q^{68} -2.82792 q^{70} -7.37151 q^{71} +4.60387 q^{73} -5.84054 q^{74} +0.874018 q^{76} -17.2504 q^{79} -8.49410 q^{80} +3.05546 q^{82} -9.16622 q^{83} -10.8399 q^{85} -18.7665 q^{86} -1.14303 q^{89} -0.237832 q^{91} +2.21446 q^{92} -13.7117 q^{94} +2.71418 q^{95} -4.68538 q^{97} +1.60257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} - 12 q^{7}+O(q^{10}) \) 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

Modular forms

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Groups

Database

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