LMFDB - Embedded newform 1323.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28825 q^{2} +3.23607 q^{4} +4.23607 q^{5} +2.82843 q^{8} +9.69316 q^{10} -3.70246 q^{11} -2.28825 q^{13} +5.00000 q^{17} +5.45052 q^{19} +13.7082 q^{20} -8.47214 q^{22} +0.333851 q^{23} +12.9443 q^{25} -5.23607 q^{26} -7.19859 q^{29} -3.36861 q^{31} -5.65685 q^{32} +11.4412 q^{34} -4.70820 q^{37} +12.4721 q^{38} +11.9814 q^{40} -4.70820 q^{41} +2.23607 q^{43} -11.9814 q^{44} +0.763932 q^{46} -1.47214 q^{47} +29.6197 q^{50} -7.40492 q^{52} +3.16228 q^{53} -15.6839 q^{55} -16.4721 q^{58} -3.94427 q^{59} +3.70246 q^{61} -7.70820 q^{62} -12.9443 q^{64} -9.69316 q^{65} -11.2361 q^{67} +16.1803 q^{68} +13.0618 q^{71} +0.746512 q^{73} -10.7735 q^{74} +17.6383 q^{76} -11.4721 q^{79} -10.7735 q^{82} -0.236068 q^{83} +21.1803 q^{85} +5.11667 q^{86} -10.4721 q^{88} -1.70820 q^{89} +1.08036 q^{92} -3.36861 q^{94} +23.0888 q^{95} +1.95440 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{5} + 20 q^{17} + 28 q^{20} - 16 q^{22} + 16 q^{25} - 12 q^{26} + 8 q^{37} + 32 q^{38} + 8 q^{41} + 12 q^{46} + 12 q^{47} - 48 q^{58} + 20 q^{59} - 4 q^{62} - 16 q^{64} - 36 q^{67} + 20 q^{68}+ \cdots + 20 q^{89}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^5 + 20 * q^17 + 28 * q^20 - 16 * q^22 + 16 * q^25 - 12 * q^26 + 8 * q^37 + 32 * q^38 + 8 * q^41 + 12 * q^46 + 12 * q^47 - 48 * q^58 + 20 * q^59 - 4 * q^62 - 16 * q^64 - 36 * q^67 + 20 * q^68 - 28 * q^79 + 8 * q^83 + 40 * q^85 - 24 * q^88 + 20 * q^89

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 $

$2$	2.28825	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$2$	2.28825	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$3$	0	0
$3$	0	0
$4$	3.23607	1.61803
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	0	0
$10$	9.69316	3.06525
$11$	−3.70246	−1.11633	−0.558167	−	0.829729i	$-0.688495\pi$
$11$	−3.70246	−1.11633	−0.558167	+	0.829729i	$0.688495\pi$
$12$	0	0
$13$	−2.28825	−0.634645	−0.317323	−	0.948318i	$-0.602784\pi$
$13$	−2.28825	−0.634645	−0.317323	+	0.948318i	$0.602784\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	5.45052	1.25044	0.625218	−	0.780450i	$-0.285010\pi$
$19$	5.45052	1.25044	0.625218	+	0.780450i	$0.285010\pi$
$20$	13.7082	3.06525
$21$	0	0
$22$	−8.47214	−1.80627
$23$	0.333851	0.0696126	0.0348063	−	0.999394i	$-0.488919\pi$
$23$	0.333851	0.0696126	0.0348063	+	0.999394i	$0.488919\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	−5.23607	−1.02688
$27$	0	0
$28$	0	0
$29$	−7.19859	−1.33674	−0.668372	−	0.743827i	$-0.733009\pi$
$29$	−7.19859	−1.33674	−0.668372	+	0.743827i	$0.733009\pi$
$30$	0	0
$31$	−3.36861	−0.605020	−0.302510	−	0.953146i	$-0.597825\pi$
$31$	−3.36861	−0.605020	−0.302510	+	0.953146i	$0.597825\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	11.4412	1.96215
$35$	0	0
$36$	0	0
$37$	−4.70820	−0.774024	−0.387012	−	0.922075i	$-0.626493\pi$
$37$	−4.70820	−0.774024	−0.387012	+	0.922075i	$0.626493\pi$
$38$	12.4721	2.02325
$39$	0	0
$40$	11.9814	1.89443
$41$	−4.70820	−0.735298	−0.367649	−	0.929965i	$-0.619837\pi$
$41$	−4.70820	−0.735298	−0.367649	+	0.929965i	$0.619837\pi$
$42$	0	0
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	−11.9814	−1.80627
$45$	0	0
$46$	0.763932	0.112636
$47$	−1.47214	−0.214733	−0.107367	−	0.994220i	$-0.534242\pi$
$47$	−1.47214	−0.214733	−0.107367	+	0.994220i	$0.534242\pi$
$48$	0	0
$49$	0	0
$50$	29.6197	4.18885
$51$	0	0
$52$	−7.40492	−1.02688
$53$	3.16228	0.434372	0.217186	−	0.976130i	$-0.430312\pi$
$53$	3.16228	0.434372	0.217186	+	0.976130i	$0.430312\pi$
$54$	0	0
$55$	−15.6839	−2.11481
$56$	0	0
$57$	0	0
$58$	−16.4721	−2.16290
$59$	−3.94427	−0.513500	−0.256750	−	0.966478i	$-0.582652\pi$
$59$	−3.94427	−0.513500	−0.256750	+	0.966478i	$0.582652\pi$
$60$	0	0
$61$	3.70246	0.474051	0.237026	−	0.971503i	$-0.423827\pi$
$61$	3.70246	0.474051	0.237026	+	0.971503i	$0.423827\pi$
$62$	−7.70820	−0.978943
$63$	0	0
$64$	−12.9443	−1.61803
$65$	−9.69316	−1.20229
$66$	0	0
$67$	−11.2361	−1.37270	−0.686352	−	0.727269i	$-0.740789\pi$
$67$	−11.2361	−1.37270	−0.686352	+	0.727269i	$0.740789\pi$
$68$	16.1803	1.96215
$69$	0	0
$70$	0	0
$71$	13.0618	1.55015	0.775074	−	0.631871i	$-0.217713\pi$
$71$	13.0618	1.55015	0.775074	+	0.631871i	$0.217713\pi$
$72$	0	0
$73$	0.746512	0.0873727	0.0436863	−	0.999045i	$-0.486090\pi$
$73$	0.746512	0.0873727	0.0436863	+	0.999045i	$0.486090\pi$
$74$	−10.7735	−1.25240
$75$	0	0
$76$	17.6383	2.02325
$77$	0	0
$78$	0	0
$79$	−11.4721	−1.29072	−0.645358	−	0.763880i	$-0.723292\pi$
$79$	−11.4721	−1.29072	−0.645358	+	0.763880i	$0.723292\pi$
$80$	0	0
$81$	0	0
$82$	−10.7735	−1.18974
$83$	−0.236068	−0.0259118	−0.0129559	−	0.999916i	$-0.504124\pi$
$83$	−0.236068	−0.0259118	−0.0129559	+	0.999916i	$0.504124\pi$
$84$	0	0
$85$	21.1803	2.29733
$86$	5.11667	0.551745
$87$	0	0
$88$	−10.4721	−1.11633
$89$	−1.70820	−0.181069	−0.0905346	−	0.995893i	$-0.528858\pi$
$89$	−1.70820	−0.181069	−0.0905346	+	0.995893i	$0.528858\pi$
$90$	0	0
$91$	0	0
$92$	1.08036	0.112636
$93$	0	0
$94$	−3.36861	−0.347445
$95$	23.0888	2.36886
$96$	0	0
$97$	1.95440	0.198439	0.0992194	−	0.995066i	$-0.468365\pi$
$97$	1.95440	0.198439	0.0992194	+	0.995066i	$0.468365\pi$
$98$	0	0
$99$	0	0
$100$	41.8885	4.18885
$101$	9.23607	0.919023	0.459512	−	0.888172i	$-0.348024\pi$
$101$	9.23607	0.919023	0.459512	+	0.888172i	$0.348024\pi$
$102$	0	0
$103$	−8.40647	−0.828314	−0.414157	−	0.910205i	$-0.635924\pi$
$103$	−8.40647	−0.828314	−0.414157	+	0.910205i	$0.635924\pi$
$104$	−6.47214	−0.634645
$105$	0	0
$106$	7.23607	0.702829
$107$	−9.89949	−0.957020	−0.478510	−	0.878082i	$-0.658823\pi$
$107$	−9.89949	−0.957020	−0.478510	+	0.878082i	$0.658823\pi$
$108$	0	0
$109$	9.76393	0.935215	0.467608	−	0.883936i	$-0.345116\pi$
$109$	9.76393	0.935215	0.467608	+	0.883936i	$0.345116\pi$
$110$	−35.8885	−3.42184
$111$	0	0
$112$	0	0
$113$	1.20788	0.113628	0.0568140	−	0.998385i	$-0.481906\pi$
$113$	1.20788	0.113628	0.0568140	+	0.998385i	$0.481906\pi$
$114$	0	0
$115$	1.41421	0.131876
$116$	−23.2951	−2.16290
$117$	0	0
$118$	−9.02546	−0.830861
$119$	0	0
$120$	0	0
$121$	2.70820	0.246200
$122$	8.47214	0.767031
$123$	0	0
$124$	−10.9010	−0.978943
$125$	33.6525	3.00997
$126$	0	0
$127$	5.47214	0.485574	0.242787	−	0.970080i	$-0.421938\pi$
$127$	5.47214	0.485574	0.242787	+	0.970080i	$0.421938\pi$
$128$	−18.3060	−1.61803
$129$	0	0
$130$	−22.1803	−1.94534
$131$	10.9443	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$131$	10.9443	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$132$	0	0
$133$	0	0
$134$	−25.7109	−2.22108
$135$	0	0
$136$	14.1421	1.21268
$137$	4.44897	0.380101	0.190051	−	0.981774i	$-0.439135\pi$
$137$	4.44897	0.380101	0.190051	+	0.981774i	$0.439135\pi$
$138$	0	0
$139$	−19.1800	−1.62683	−0.813413	−	0.581687i	$-0.802393\pi$
$139$	−19.1800	−1.62683	−0.813413	+	0.581687i	$0.802393\pi$
$140$	0	0
$141$	0	0
$142$	29.8885	2.50819
$143$	8.47214	0.708476
$144$	0	0
$145$	−30.4937	−2.53236
$146$	1.70820	0.141372
$147$	0	0
$148$	−15.2361	−1.25240
$149$	22.0084	1.80300	0.901500	−	0.432779i	$-0.142467\pi$
$149$	22.0084	1.80300	0.901500	+	0.432779i	$0.142467\pi$
$150$	0	0
$151$	−16.4164	−1.33595	−0.667974	−	0.744184i	$-0.732838\pi$
$151$	−16.4164	−1.33595	−0.667974	+	0.744184i	$0.732838\pi$
$152$	15.4164	1.25044
$153$	0	0
$154$	0	0
$155$	−14.2697	−1.14617
$156$	0	0
$157$	−20.6730	−1.64989	−0.824943	−	0.565215i	$-0.808793\pi$
$157$	−20.6730	−1.64989	−0.824943	+	0.565215i	$0.808793\pi$
$158$	−26.2511	−2.08842
$159$	0	0
$160$	−23.9628	−1.89443
$161$	0	0
$162$	0	0
$163$	−5.18034	−0.405756	−0.202878	−	0.979204i	$-0.565029\pi$
$163$	−5.18034	−0.405756	−0.202878	+	0.979204i	$0.565029\pi$
$164$	−15.2361	−1.18974
$165$	0	0
$166$	−0.540182	−0.0419262
$167$	−3.76393	−0.291262	−0.145631	−	0.989339i	$-0.546521\pi$
$167$	−3.76393	−0.291262	−0.145631	+	0.989339i	$0.546521\pi$
$168$	0	0
$169$	−7.76393	−0.597226
$170$	48.4658	3.71716
$171$	0	0
$172$	7.23607	0.551745
$173$	18.4721	1.40441	0.702205	−	0.711975i	$-0.252199\pi$
$173$	18.4721	1.40441	0.702205	+	0.711975i	$0.252199\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−3.90879	−0.292976
$179$	14.6823	1.09741	0.548704	−	0.836017i	$-0.315121\pi$
$179$	14.6823	1.09741	0.548704	+	0.836017i	$0.315121\pi$
$180$	0	0
$181$	22.8825	1.70084	0.850420	−	0.526105i	$-0.176348\pi$
$181$	22.8825	1.70084	0.850420	+	0.526105i	$0.176348\pi$
$182$	0	0
$183$	0	0
$184$	0.944272	0.0696126
$185$	−19.9443	−1.46633
$186$	0	0
$187$	−18.5123	−1.35375
$188$	−4.76393	−0.347445
$189$	0	0
$190$	52.8328	3.83290
$191$	−4.24264	−0.306987	−0.153493	−	0.988150i	$-0.549052\pi$
$191$	−4.24264	−0.306987	−0.153493	+	0.988150i	$0.549052\pi$
$192$	0	0
$193$	17.1803	1.23667	0.618334	−	0.785915i	$-0.287808\pi$
$193$	17.1803	1.23667	0.618334	+	0.785915i	$0.287808\pi$
$194$	4.47214	0.321081
$195$	0	0
$196$	0	0
$197$	0.461370	0.0328713	0.0164356	−	0.999865i	$-0.494768\pi$
$197$	0.461370	0.0328713	0.0164356	+	0.999865i	$0.494768\pi$
$198$	0	0
$199$	13.4744	0.955177	0.477589	−	0.878584i	$-0.341511\pi$
$199$	13.4744	0.955177	0.477589	+	0.878584i	$0.341511\pi$
$200$	36.6119	2.58885
$201$	0	0
$202$	21.1344	1.48701
$203$	0	0
$204$	0	0
$205$	−19.9443	−1.39297
$206$	−19.2361	−1.34024
$207$	0	0
$208$	0	0
$209$	−20.1803	−1.39590
$210$	0	0
$211$	−17.4164	−1.19899	−0.599497	−	0.800377i	$-0.704633\pi$
$211$	−17.4164	−1.19899	−0.599497	+	0.800377i	$0.704633\pi$
$212$	10.2333	0.702829
$213$	0	0
$214$	−22.6525	−1.54849
$215$	9.47214	0.645994
$216$	0	0
$217$	0	0
$218$	22.3423	1.51321
$219$	0	0
$220$	−50.7541	−3.42184
$221$	−11.4412	−0.769620
$222$	0	0
$223$	−19.5138	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$223$	−19.5138	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$224$	0	0
$225$	0	0
$226$	2.76393	0.183854
$227$	12.1803	0.808438	0.404219	−	0.914662i	$-0.367543\pi$
$227$	12.1803	0.808438	0.404219	+	0.914662i	$0.367543\pi$
$228$	0	0
$229$	10.6460	0.703508	0.351754	−	0.936092i	$-0.385585\pi$
$229$	10.6460	0.703508	0.351754	+	0.936092i	$0.385585\pi$
$230$	3.23607	0.213380
$231$	0	0
$232$	−20.3607	−1.33674
$233$	−0.127520	−0.00835408	−0.00417704	−	0.999991i	$-0.501330\pi$
$233$	−0.127520	−0.00835408	−0.00417704	+	0.999991i	$0.501330\pi$
$234$	0	0
$235$	−6.23607	−0.406796
$236$	−12.7639	−0.830861
$237$	0	0
$238$	0	0
$239$	−25.5834	−1.65485	−0.827425	−	0.561576i	$-0.810195\pi$
$239$	−25.5834	−1.65485	−0.827425	+	0.561576i	$0.810195\pi$
$240$	0	0
$241$	6.86474	0.442197	0.221098	−	0.975252i	$-0.429036\pi$
$241$	6.86474	0.442197	0.221098	+	0.975252i	$0.429036\pi$
$242$	6.19704	0.398361
$243$	0	0
$244$	11.9814	0.767031
$245$	0	0
$246$	0	0
$247$	−12.4721	−0.793583
$248$	−9.52786	−0.605020
$249$	0	0
$250$	77.0051	4.87023
$251$	−9.29180	−0.586493	−0.293246	−	0.956037i	$-0.594736\pi$
$251$	−9.29180	−0.586493	−0.293246	+	0.956037i	$0.594736\pi$
$252$	0	0
$253$	−1.23607	−0.0777109
$254$	12.5216	0.785675
$255$	0	0
$256$	−16.0000	−1.00000
$257$	−5.41641	−0.337866	−0.168933	−	0.985628i	$-0.554032\pi$
$257$	−5.41641	−0.337866	−0.168933	+	0.985628i	$0.554032\pi$
$258$	0	0
$259$	0	0
$260$	−31.3677	−1.94534
$261$	0	0
$262$	25.0432	1.54717
$263$	−20.1815	−1.24445	−0.622224	−	0.782839i	$-0.713771\pi$
$263$	−20.1815	−1.24445	−0.622224	+	0.782839i	$0.713771\pi$
$264$	0	0
$265$	13.3956	0.822887
$266$	0	0
$267$	0	0
$268$	−36.3607	−2.22108
$269$	7.00000	0.426798	0.213399	−	0.976965i	$-0.431547\pi$
$269$	7.00000	0.426798	0.213399	+	0.976965i	$0.431547\pi$
$270$	0	0
$271$	5.57804	0.338842	0.169421	−	0.985544i	$-0.445810\pi$
$271$	5.57804	0.338842	0.169421	+	0.985544i	$0.445810\pi$
$272$	0	0
$273$	0	0
$274$	10.1803	0.615017
$275$	−47.9256	−2.89002
$276$	0	0
$277$	−13.7639	−0.826995	−0.413497	−	0.910505i	$-0.635693\pi$
$277$	−13.7639	−0.826995	−0.413497	+	0.910505i	$0.635693\pi$
$278$	−43.8885	−2.63226
$279$	0	0
$280$	0	0
$281$	−22.2936	−1.32992	−0.664961	−	0.746878i	$-0.731552\pi$
$281$	−22.2936	−1.32992	−0.664961	+	0.746878i	$0.731552\pi$
$282$	0	0
$283$	16.3516	0.972000	0.486000	−	0.873959i	$-0.338456\pi$
$283$	16.3516	0.972000	0.486000	+	0.873959i	$0.338456\pi$
$284$	42.2688	2.50819
$285$	0	0
$286$	19.3863	1.14634
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	−69.7771	−4.09745
$291$	0	0
$292$	2.41577	0.141372
$293$	7.47214	0.436527	0.218263	−	0.975890i	$-0.429961\pi$
$293$	7.47214	0.436527	0.218263	+	0.975890i	$0.429961\pi$
$294$	0	0
$295$	−16.7082	−0.972789
$296$	−13.3168	−0.774024
$297$	0	0
$298$	50.3607	2.91732
$299$	−0.763932	−0.0441793
$300$	0	0
$301$	0	0
$302$	−37.5648	−2.16161
$303$	0	0
$304$	0	0
$305$	15.6839	0.898056
$306$	0	0
$307$	2.74962	0.156929	0.0784644	−	0.996917i	$-0.474998\pi$
$307$	2.74962	0.156929	0.0784644	+	0.996917i	$0.474998\pi$
$308$	0	0
$309$	0	0
$310$	−32.6525	−1.85454
$311$	27.0689	1.53494	0.767468	−	0.641088i	$-0.221516\pi$
$311$	27.0689	1.53494	0.767468	+	0.641088i	$0.221516\pi$
$312$	0	0
$313$	−28.9520	−1.63646	−0.818231	−	0.574889i	$-0.805045\pi$
$313$	−28.9520	−1.63646	−0.818231	+	0.574889i	$0.805045\pi$
$314$	−47.3050	−2.66957
$315$	0	0
$316$	−37.1246	−2.08842
$317$	−2.08191	−0.116932	−0.0584660	−	0.998289i	$-0.518621\pi$
$317$	−2.08191	−0.116932	−0.0584660	+	0.998289i	$0.518621\pi$
$318$	0	0
$319$	26.6525	1.49225
$320$	−54.8328	−3.06525
$321$	0	0
$322$	0	0
$323$	27.2526	1.51638
$324$	0	0
$325$	−29.6197	−1.64300
$326$	−11.8539	−0.656526
$327$	0	0
$328$	−13.3168	−0.735298
$329$	0	0
$330$	0	0
$331$	18.4164	1.01226	0.506129	−	0.862458i	$-0.331076\pi$
$331$	18.4164	1.01226	0.506129	+	0.862458i	$0.331076\pi$
$332$	−0.763932	−0.0419262
$333$	0	0
$334$	−8.61280	−0.471271
$335$	−47.5967	−2.60049
$336$	0	0
$337$	3.76393	0.205034	0.102517	−	0.994731i	$-0.467310\pi$
$337$	3.76393	0.205034	0.102517	+	0.994731i	$0.467310\pi$
$338$	−17.7658	−0.966331
$339$	0	0
$340$	68.5410	3.71716
$341$	12.4721	0.675404
$342$	0	0
$343$	0	0
$344$	6.32456	0.340997
$345$	0	0
$346$	42.2688	2.27238
$347$	23.1676	1.24370	0.621851	−	0.783136i	$-0.286381\pi$
$347$	23.1676	1.24370	0.621851	+	0.783136i	$0.286381\pi$
$348$	0	0
$349$	10.9010	0.583520	0.291760	−	0.956492i	$-0.405759\pi$
$349$	10.9010	0.583520	0.291760	+	0.956492i	$0.405759\pi$
$350$	0	0
$351$	0	0
$352$	20.9443	1.11633
$353$	30.1246	1.60337	0.801686	−	0.597746i	$-0.203937\pi$
$353$	30.1246	1.60337	0.801686	+	0.597746i	$0.203937\pi$
$354$	0	0
$355$	55.3306	2.93664
$356$	−5.52786	−0.292976
$357$	0	0
$358$	33.5967	1.77564
$359$	−4.65530	−0.245697	−0.122849	−	0.992425i	$-0.539203\pi$
$359$	−4.65530	−0.245697	−0.122849	+	0.992425i	$0.539203\pi$
$360$	0	0
$361$	10.7082	0.563590
$362$	52.3607	2.75202
$363$	0	0
$364$	0	0
$365$	3.16228	0.165521
$366$	0	0
$367$	18.6398	0.972990	0.486495	−	0.873683i	$-0.338275\pi$
$367$	18.6398	0.972990	0.486495	+	0.873683i	$0.338275\pi$
$368$	0	0
$369$	0	0
$370$	−45.6374	−2.37258
$371$	0	0
$372$	0	0
$373$	−13.6525	−0.706898	−0.353449	−	0.935454i	$-0.614991\pi$
$373$	−13.6525	−0.706898	−0.353449	+	0.935454i	$0.614991\pi$
$374$	−42.3607	−2.19042
$375$	0	0
$376$	−4.16383	−0.214733
$377$	16.4721	0.848358
$378$	0	0
$379$	−30.8885	−1.58664	−0.793319	−	0.608806i	$-0.791649\pi$
$379$	−30.8885	−1.58664	−0.793319	+	0.608806i	$0.791649\pi$
$380$	74.7169	3.83290
$381$	0	0
$382$	−9.70820	−0.496715
$383$	−11.1803	−0.571289	−0.285644	−	0.958336i	$-0.592208\pi$
$383$	−11.1803	−0.571289	−0.285644	+	0.958336i	$0.592208\pi$
$384$	0	0
$385$	0	0
$386$	39.3128	2.00097
$387$	0	0
$388$	6.32456	0.321081
$389$	23.0888	1.17065	0.585324	−	0.810800i	$-0.300967\pi$
$389$	23.0888	1.17065	0.585324	+	0.810800i	$0.300967\pi$
$390$	0	0
$391$	1.66925	0.0844177
$392$	0	0
$393$	0	0
$394$	1.05573	0.0531868
$395$	−48.5967	−2.44517
$396$	0	0
$397$	27.7928	1.39488	0.697440	−	0.716643i	$-0.254322\pi$
$397$	27.7928	1.39488	0.697440	+	0.716643i	$0.254322\pi$
$398$	30.8328	1.54551
$399$	0	0
$400$	0	0
$401$	23.7078	1.18391	0.591955	−	0.805971i	$-0.298356\pi$
$401$	23.7078	1.18391	0.591955	+	0.805971i	$0.298356\pi$
$402$	0	0
$403$	7.70820	0.383973
$404$	29.8885	1.48701
$405$	0	0
$406$	0	0
$407$	17.4319	0.864069
$408$	0	0
$409$	32.0354	1.58405	0.792025	−	0.610488i	$-0.209027\pi$
$409$	32.0354	1.58405	0.792025	+	0.610488i	$0.209027\pi$
$410$	−45.6374	−2.25387
$411$	0	0
$412$	−27.2039	−1.34024
$413$	0	0
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	12.9443	0.634645
$417$	0	0
$418$	−46.1776	−2.25862
$419$	15.6525	0.764673	0.382337	−	0.924023i	$-0.375119\pi$
$419$	15.6525	0.764673	0.382337	+	0.924023i	$0.375119\pi$
$420$	0	0
$421$	−4.47214	−0.217959	−0.108979	−	0.994044i	$-0.534758\pi$
$421$	−4.47214	−0.217959	−0.108979	+	0.994044i	$0.534758\pi$
$422$	−39.8530	−1.94001
$423$	0	0
$424$	8.94427	0.434372
$425$	64.7214	3.13945
$426$	0	0
$427$	0	0
$428$	−32.0354	−1.54849
$429$	0	0
$430$	21.6746	1.04524
$431$	20.0540	0.965969	0.482984	−	0.875629i	$-0.339553\pi$
$431$	20.0540	0.965969	0.482984	+	0.875629i	$0.339553\pi$
$432$	0	0
$433$	−16.2728	−0.782019	−0.391009	−	0.920387i	$-0.627874\pi$
$433$	−16.2728	−0.782019	−0.391009	+	0.920387i	$0.627874\pi$
$434$	0	0
$435$	0	0
$436$	31.5967	1.51321
$437$	1.81966	0.0870461
$438$	0	0
$439$	−0.412662	−0.0196953	−0.00984764	−	0.999952i	$-0.503135\pi$
$439$	−0.412662	−0.0196953	−0.00984764	+	0.999952i	$0.503135\pi$
$440$	−44.3607	−2.11481
$441$	0	0
$442$	−26.1803	−1.24527
$443$	21.5958	1.02605	0.513023	−	0.858375i	$-0.328526\pi$
$443$	21.5958	1.02605	0.513023	+	0.858375i	$0.328526\pi$
$444$	0	0
$445$	−7.23607	−0.343023
$446$	−44.6525	−2.11436
$447$	0	0
$448$	0	0
$449$	−34.7363	−1.63931	−0.819655	−	0.572858i	$-0.805835\pi$
$449$	−34.7363	−1.63931	−0.819655	+	0.572858i	$0.805835\pi$
$450$	0	0
$451$	17.4319	0.820838
$452$	3.90879	0.183854
$453$	0	0
$454$	27.8716	1.30808
$455$	0	0
$456$	0	0
$457$	32.8328	1.53585	0.767927	−	0.640537i	$-0.221288\pi$
$457$	32.8328	1.53585	0.767927	+	0.640537i	$0.221288\pi$
$458$	24.3607	1.13830
$459$	0	0
$460$	4.57649	0.213380
$461$	22.5279	1.04923	0.524614	−	0.851340i	$-0.324210\pi$
$461$	22.5279	1.04923	0.524614	+	0.851340i	$0.324210\pi$
$462$	0	0
$463$	37.1803	1.72792	0.863958	−	0.503563i	$-0.167978\pi$
$463$	37.1803	1.72792	0.863958	+	0.503563i	$0.167978\pi$
$464$	0	0
$465$	0	0
$466$	−0.291796	−0.0135172
$467$	−37.5967	−1.73977	−0.869885	−	0.493255i	$-0.835807\pi$
$467$	−37.5967	−1.73977	−0.869885	+	0.493255i	$0.835807\pi$
$468$	0	0
$469$	0	0
$470$	−14.2697	−0.658210
$471$	0	0
$472$	−11.1561	−0.513500
$473$	−8.27895	−0.380667
$474$	0	0
$475$	70.5531	3.23720
$476$	0	0
$477$	0	0
$478$	−58.5410	−2.67760
$479$	−31.0689	−1.41957	−0.709787	−	0.704417i	$-0.751209\pi$
$479$	−31.0689	−1.41957	−0.709787	+	0.704417i	$0.751209\pi$
$480$	0	0
$481$	10.7735	0.491231
$482$	15.7082	0.715489
$483$	0	0
$484$	8.76393	0.398361
$485$	8.27895	0.375928
$486$	0	0
$487$	−25.8885	−1.17312	−0.586561	−	0.809905i	$-0.699519\pi$
$487$	−25.8885	−1.17312	−0.586561	+	0.809905i	$0.699519\pi$
$488$	10.4721	0.474051
$489$	0	0
$490$	0	0
$491$	−1.46292	−0.0660207	−0.0330104	−	0.999455i	$-0.510509\pi$
$491$	−1.46292	−0.0660207	−0.0330104	+	0.999455i	$0.510509\pi$
$492$	0	0
$493$	−35.9929	−1.62104
$494$	−28.5393	−1.28404
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.8885	1.65136	0.825679	−	0.564140i	$-0.190792\pi$
$499$	36.8885	1.65136	0.825679	+	0.564140i	$0.190792\pi$
$500$	108.902	4.87023
$501$	0	0
$502$	−21.2619	−0.948966
$503$	35.8328	1.59771	0.798853	−	0.601526i	$-0.205440\pi$
$503$	35.8328	1.59771	0.798853	+	0.601526i	$0.205440\pi$
$504$	0	0
$505$	39.1246	1.74102
$506$	−2.82843	−0.125739
$507$	0	0
$508$	17.7082	0.785675
$509$	−17.3607	−0.769499	−0.384749	−	0.923021i	$-0.625712\pi$
$509$	−17.3607	−0.769499	−0.384749	+	0.923021i	$0.625712\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	−12.3941	−0.546679
$515$	−35.6104	−1.56918
$516$	0	0
$517$	5.45052	0.239714
$518$	0	0
$519$	0	0
$520$	−27.4164	−1.20229
$521$	−8.52786	−0.373613	−0.186806	−	0.982397i	$-0.559814\pi$
$521$	−8.52786	−0.373613	−0.186806	+	0.982397i	$0.559814\pi$
$522$	0	0
$523$	−4.29135	−0.187648	−0.0938238	−	0.995589i	$-0.529909\pi$
$523$	−4.29135	−0.187648	−0.0938238	+	0.995589i	$0.529909\pi$
$524$	35.4164	1.54717
$525$	0	0
$526$	−46.1803	−2.01356
$527$	−16.8430	−0.733694
$528$	0	0
$529$	−22.8885	−0.995154
$530$	30.6525	1.33146
$531$	0	0
$532$	0	0
$533$	10.7735	0.466653
$534$	0	0
$535$	−41.9349	−1.81301
$536$	−31.7804	−1.37270
$537$	0	0
$538$	16.0177	0.690573
$539$	0	0
$540$	0	0
$541$	28.3050	1.21692	0.608462	−	0.793583i	$-0.291787\pi$
$541$	28.3050	1.21692	0.608462	+	0.793583i	$0.291787\pi$
$542$	12.7639	0.548258
$543$	0	0
$544$	−28.2843	−1.21268
$545$	41.3607	1.77170
$546$	0	0
$547$	18.1246	0.774952	0.387476	−	0.921880i	$-0.373347\pi$
$547$	18.1246	0.774952	0.387476	+	0.921880i	$0.373347\pi$
$548$	14.3972	0.615017
$549$	0	0
$550$	−109.666	−4.67616
$551$	−39.2361	−1.67151
$552$	0	0
$553$	0	0
$554$	−31.4953	−1.33811
$555$	0	0
$556$	−62.0678	−2.63226
$557$	6.32456	0.267980	0.133990	−	0.990983i	$-0.457221\pi$
$557$	6.32456	0.267980	0.133990	+	0.990983i	$0.457221\pi$
$558$	0	0
$559$	−5.11667	−0.216412
$560$	0	0
$561$	0	0
$562$	−51.0132	−2.15186
$563$	34.1803	1.44053	0.720265	−	0.693699i	$-0.244020\pi$
$563$	34.1803	1.44053	0.720265	+	0.693699i	$0.244020\pi$
$564$	0	0
$565$	5.11667	0.215260
$566$	37.4164	1.57273
$567$	0	0
$568$	36.9443	1.55015
$569$	25.3770	1.06386	0.531930	−	0.846788i	$-0.321467\pi$
$569$	25.3770	1.06386	0.531930	+	0.846788i	$0.321467\pi$
$570$	0	0
$571$	−19.0689	−0.798008	−0.399004	−	0.916949i	$-0.630644\pi$
$571$	−19.0689	−0.798008	−0.399004	+	0.916949i	$0.630644\pi$
$572$	27.4164	1.14634
$573$	0	0
$574$	0	0
$575$	4.32145	0.180217
$576$	0	0
$577$	38.3600	1.59695	0.798474	−	0.602030i	$-0.205641\pi$
$577$	38.3600	1.59695	0.798474	+	0.602030i	$0.205641\pi$
$578$	18.3060	0.761428
$579$	0	0
$580$	−98.6797	−4.09745
$581$	0	0
$582$	0	0
$583$	−11.7082	−0.484904
$584$	2.11146	0.0873727
$585$	0	0
$586$	17.0981	0.706315
$587$	−18.4721	−0.762427	−0.381213	−	0.924487i	$-0.624494\pi$
$587$	−18.4721	−0.762427	−0.381213	+	0.924487i	$0.624494\pi$
$588$	0	0
$589$	−18.3607	−0.756539
$590$	−38.2325	−1.57401
$591$	0	0
$592$	0	0
$593$	11.4721	0.471104	0.235552	−	0.971862i	$-0.424310\pi$
$593$	11.4721	0.471104	0.235552	+	0.971862i	$0.424310\pi$
$594$	0	0
$595$	0	0
$596$	71.2208	2.91732
$597$	0	0
$598$	−1.74806	−0.0714837
$599$	32.2418	1.31736	0.658681	−	0.752422i	$-0.271114\pi$
$599$	32.2418	1.31736	0.658681	+	0.752422i	$0.271114\pi$
$600$	0	0
$601$	−25.3283	−1.03316	−0.516582	−	0.856238i	$-0.672796\pi$
$601$	−25.3283	−1.03316	−0.516582	+	0.856238i	$0.672796\pi$
$602$	0	0
$603$	0	0
$604$	−53.1246	−2.16161
$605$	11.4721	0.466409
$606$	0	0
$607$	26.6637	1.08225	0.541124	−	0.840943i	$-0.317999\pi$
$607$	26.6637	1.08225	0.541124	+	0.840943i	$0.317999\pi$
$608$	−30.8328	−1.25044
$609$	0	0
$610$	35.8885	1.45308
$611$	3.36861	0.136279
$612$	0	0
$613$	−31.8885	−1.28797	−0.643983	−	0.765040i	$-0.722719\pi$
$613$	−31.8885	−1.28797	−0.643983	+	0.765040i	$0.722719\pi$
$614$	6.29180	0.253916
$615$	0	0
$616$	0	0
$617$	10.5672	0.425419	0.212710	−	0.977115i	$-0.431771\pi$
$617$	10.5672	0.425419	0.212710	+	0.977115i	$0.431771\pi$
$618$	0	0
$619$	−38.5663	−1.55011	−0.775056	−	0.631893i	$-0.782278\pi$
$619$	−38.5663	−1.55011	−0.775056	+	0.631893i	$0.782278\pi$
$620$	−46.1776	−1.85454
$621$	0	0
$622$	61.9403	2.48358
$623$	0	0
$624$	0	0
$625$	77.8328	3.11331
$626$	−66.2492	−2.64785
$627$	0	0
$628$	−66.8993	−2.66957
$629$	−23.5410	−0.938642
$630$	0	0
$631$	−26.2361	−1.04444	−0.522221	−	0.852810i	$-0.674896\pi$
$631$	−26.2361	−1.04444	−0.522221	+	0.852810i	$0.674896\pi$
$632$	−32.4481	−1.29072
$633$	0	0
$634$	−4.76393	−0.189200
$635$	23.1803	0.919884
$636$	0	0
$637$	0	0
$638$	60.9874	2.41451
$639$	0	0
$640$	−77.5453	−3.06525
$641$	−34.8152	−1.37512	−0.687558	−	0.726129i	$-0.741317\pi$
$641$	−34.8152	−1.37512	−0.687558	+	0.726129i	$0.741317\pi$
$642$	0	0
$643$	−33.8623	−1.33540	−0.667700	−	0.744431i	$-0.732721\pi$
$643$	−33.8623	−1.33540	−0.667700	+	0.744431i	$0.732721\pi$
$644$	0	0
$645$	0	0
$646$	62.3607	2.45355
$647$	−4.36068	−0.171436	−0.0857180	−	0.996319i	$-0.527318\pi$
$647$	−4.36068	−0.171436	−0.0857180	+	0.996319i	$0.527318\pi$
$648$	0	0
$649$	14.6035	0.573238
$650$	−67.7771	−2.65844
$651$	0	0
$652$	−16.7639	−0.656526
$653$	−18.4335	−0.721358	−0.360679	−	0.932690i	$-0.617455\pi$
$653$	−18.4335	−0.721358	−0.360679	+	0.932690i	$0.617455\pi$
$654$	0	0
$655$	46.3607	1.81146
$656$	0	0
$657$	0	0
$658$	0	0
$659$	38.5176	1.50043	0.750217	−	0.661192i	$-0.229949\pi$
$659$	38.5176	1.50043	0.750217	+	0.661192i	$0.229949\pi$
$660$	0	0
$661$	−29.3345	−1.14098	−0.570491	−	0.821304i	$-0.693247\pi$
$661$	−29.3345	−1.14098	−0.570491	+	0.821304i	$0.693247\pi$
$662$	42.1413	1.63787
$663$	0	0
$664$	−0.667701	−0.0259118
$665$	0	0
$666$	0	0
$667$	−2.40325	−0.0930543
$668$	−12.1803	−0.471271
$669$	0	0
$670$	−108.913	−4.20768
$671$	−13.7082	−0.529199
$672$	0	0
$673$	−41.2361	−1.58953	−0.794767	−	0.606915i	$-0.792407\pi$
$673$	−41.2361	−1.58953	−0.794767	+	0.606915i	$0.792407\pi$
$674$	8.61280	0.331753
$675$	0	0
$676$	−25.1246	−0.966331
$677$	12.1115	0.465481	0.232741	−	0.972539i	$-0.425231\pi$
$677$	12.1115	0.465481	0.232741	+	0.972539i	$0.425231\pi$
$678$	0	0
$679$	0	0
$680$	59.9070	2.29733
$681$	0	0
$682$	28.5393	1.09283
$683$	38.9002	1.48847	0.744237	−	0.667916i	$-0.232813\pi$
$683$	38.9002	1.48847	0.744237	+	0.667916i	$0.232813\pi$
$684$	0	0
$685$	18.8461	0.720074
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.23607	−0.275672
$690$	0	0
$691$	0.795221	0.0302516	0.0151258	−	0.999886i	$-0.495185\pi$
$691$	0.795221	0.0302516	0.0151258	+	0.999886i	$0.495185\pi$
$692$	59.7771	2.27238
$693$	0	0
$694$	53.0132	2.01235
$695$	−81.2478	−3.08190
$696$	0	0
$697$	−23.5410	−0.891680
$698$	24.9443	0.944155
$699$	0	0
$700$	0	0
$701$	9.43812	0.356473	0.178237	−	0.983988i	$-0.442961\pi$
$701$	9.43812	0.356473	0.178237	+	0.983988i	$0.442961\pi$
$702$	0	0
$703$	−25.6622	−0.967867
$704$	47.9256	1.80627
$705$	0	0
$706$	68.9325	2.59431
$707$	0	0
$708$	0	0
$709$	−21.4721	−0.806403	−0.403201	−	0.915111i	$-0.632103\pi$
$709$	−21.4721	−0.806403	−0.403201	+	0.915111i	$0.632103\pi$
$710$	126.610	4.75159
$711$	0	0
$712$	−4.83153	−0.181069
$713$	−1.12461	−0.0421170
$714$	0	0
$715$	35.8885	1.34216
$716$	47.5130	1.77564
$717$	0	0
$718$	−10.6525	−0.397547
$719$	0.416408	0.0155294	0.00776470	−	0.999970i	$-0.497528\pi$
$719$	0.416408	0.0155294	0.00776470	+	0.999970i	$0.497528\pi$
$720$	0	0
$721$	0	0
$722$	24.5030	0.911907
$723$	0	0
$724$	74.0492	2.75202
$725$	−93.1805	−3.46064
$726$	0	0
$727$	35.4829	1.31599	0.657993	−	0.753024i	$-0.271406\pi$
$727$	35.4829	1.31599	0.657993	+	0.753024i	$0.271406\pi$
$728$	0	0
$729$	0	0
$730$	7.23607	0.267819
$731$	11.1803	0.413520
$732$	0	0
$733$	6.32456	0.233603	0.116801	−	0.993155i	$-0.462736\pi$
$733$	6.32456	0.233603	0.116801	+	0.993155i	$0.462736\pi$
$734$	42.6525	1.57433
$735$	0	0
$736$	−1.88854	−0.0696126
$737$	41.6011	1.53240
$738$	0	0
$739$	−3.63932	−0.133875	−0.0669373	−	0.997757i	$-0.521323\pi$
$739$	−3.63932	−0.133875	−0.0669373	+	0.997757i	$0.521323\pi$
$740$	−64.5410	−2.37258
$741$	0	0
$742$	0	0
$743$	25.9172	0.950810	0.475405	−	0.879767i	$-0.342301\pi$
$743$	25.9172	0.950810	0.475405	+	0.879767i	$0.342301\pi$
$744$	0	0
$745$	93.2292	3.41565
$746$	−31.2402	−1.14379
$747$	0	0
$748$	−59.9070	−2.19042
$749$	0	0
$750$	0	0
$751$	44.6525	1.62939	0.814696	−	0.579888i	$-0.196904\pi$
$751$	44.6525	1.62939	0.814696	+	0.579888i	$0.196904\pi$
$752$	0	0
$753$	0	0
$754$	37.6923	1.37267
$755$	−69.5410	−2.53086
$756$	0	0
$757$	−35.3607	−1.28521	−0.642603	−	0.766199i	$-0.722145\pi$
$757$	−35.3607	−1.28521	−0.642603	+	0.766199i	$0.722145\pi$
$758$	−70.6806	−2.56723
$759$	0	0
$760$	65.3050	2.36886
$761$	−35.4721	−1.28586	−0.642932	−	0.765923i	$-0.722282\pi$
$761$	−35.4721	−1.28586	−0.642932	+	0.765923i	$0.722282\pi$
$762$	0	0
$763$	0	0
$764$	−13.7295	−0.496715
$765$	0	0
$766$	−25.5834	−0.924365
$767$	9.02546	0.325891
$768$	0	0
$769$	−28.6969	−1.03484	−0.517419	−	0.855732i	$-0.673107\pi$
$769$	−28.6969	−1.03484	−0.517419	+	0.855732i	$0.673107\pi$
$770$	0	0
$771$	0	0
$772$	55.5967	2.00097
$773$	−0.596748	−0.0214635	−0.0107318	−	0.999942i	$-0.503416\pi$
$773$	−0.596748	−0.0214635	−0.0107318	+	0.999942i	$0.503416\pi$
$774$	0	0
$775$	−43.6042	−1.56631
$776$	5.52786	0.198439
$777$	0	0
$778$	52.8328	1.89415
$779$	−25.6622	−0.919443
$780$	0	0
$781$	−48.3607	−1.73048
$782$	3.81966	0.136591
$783$	0	0
$784$	0	0
$785$	−87.5723	−3.12559
$786$	0	0
$787$	−3.41732	−0.121814	−0.0609071	−	0.998143i	$-0.519399\pi$
$787$	−3.41732	−0.121814	−0.0609071	+	0.998143i	$0.519399\pi$
$788$	1.49302	0.0531868
$789$	0	0
$790$	−111.201	−3.95636
$791$	0	0
$792$	0	0
$793$	−8.47214	−0.300854
$794$	63.5967	2.25696
$795$	0	0
$796$	43.6042	1.54551
$797$	−16.0689	−0.569189	−0.284595	−	0.958648i	$-0.591859\pi$
$797$	−16.0689	−0.569189	−0.284595	+	0.958648i	$0.591859\pi$
$798$	0	0
$799$	−7.36068	−0.260402
$800$	−73.2239	−2.58885
$801$	0	0
$802$	54.2492	1.91561
$803$	−2.76393	−0.0975370
$804$	0	0
$805$	0	0
$806$	17.6383	0.621281
$807$	0	0
$808$	26.1235	0.919023
$809$	−10.5672	−0.371523	−0.185761	−	0.982595i	$-0.559475\pi$
$809$	−10.5672	−0.371523	−0.185761	+	0.982595i	$0.559475\pi$
$810$	0	0
$811$	37.9774	1.33357	0.666784	−	0.745251i	$-0.267670\pi$
$811$	37.9774	1.33357	0.666784	+	0.745251i	$0.267670\pi$
$812$	0	0
$813$	0	0
$814$	39.8885	1.39809
$815$	−21.9443	−0.768674
$816$	0	0
$817$	12.1877	0.426395
$818$	73.3050	2.56305
$819$	0	0
$820$	−64.5410	−2.25387
$821$	−12.2364	−0.427055	−0.213528	−	0.976937i	$-0.568495\pi$
$821$	−12.2364	−0.427055	−0.213528	+	0.976937i	$0.568495\pi$
$822$	0	0
$823$	−9.58359	−0.334063	−0.167032	−	0.985952i	$-0.553418\pi$
$823$	−9.58359	−0.334063	−0.167032	+	0.985952i	$0.553418\pi$
$824$	−23.7771	−0.828314
$825$	0	0
$826$	0	0
$827$	−39.9805	−1.39026	−0.695130	−	0.718884i	$-0.744653\pi$
$827$	−39.9805	−1.39026	−0.695130	+	0.718884i	$0.744653\pi$
$828$	0	0
$829$	−22.8337	−0.793049	−0.396524	−	0.918024i	$-0.629784\pi$
$829$	−22.8337	−0.793049	−0.396524	+	0.918024i	$0.629784\pi$
$830$	−2.28825	−0.0794262
$831$	0	0
$832$	29.6197	1.02688
$833$	0	0
$834$	0	0
$835$	−15.9443	−0.551774
$836$	−65.3050	−2.25862
$837$	0	0
$838$	35.8167	1.23727
$839$	−24.8197	−0.856870	−0.428435	−	0.903573i	$-0.640935\pi$
$839$	−24.8197	−0.856870	−0.428435	+	0.903573i	$0.640935\pi$
$840$	0	0
$841$	22.8197	0.786885
$842$	−10.2333	−0.352664
$843$	0	0
$844$	−56.3607	−1.94001
$845$	−32.8885	−1.13140
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	148.098	5.07973
$851$	−1.57184	−0.0538819
$852$	0	0
$853$	−21.3894	−0.732360	−0.366180	−	0.930544i	$-0.619335\pi$
$853$	−21.3894	−0.732360	−0.366180	+	0.930544i	$0.619335\pi$
$854$	0	0
$855$	0	0
$856$	−28.0000	−0.957020
$857$	−3.18034	−0.108638	−0.0543192	−	0.998524i	$-0.517299\pi$
$857$	−3.18034	−0.108638	−0.0543192	+	0.998524i	$0.517299\pi$
$858$	0	0
$859$	−47.5918	−1.62381	−0.811905	−	0.583789i	$-0.801570\pi$
$859$	−47.5918	−1.62381	−0.811905	+	0.583789i	$0.801570\pi$
$860$	30.6525	1.04524
$861$	0	0
$862$	45.8885	1.56297
$863$	28.8732	0.982854	0.491427	−	0.870919i	$-0.336475\pi$
$863$	28.8732	0.982854	0.491427	+	0.870919i	$0.336475\pi$
$864$	0	0
$865$	78.2492	2.66055
$866$	−37.2361	−1.26533
$867$	0	0
$868$	0	0
$869$	42.4751	1.44087
$870$	0	0
$871$	25.7109	0.871180
$872$	27.6166	0.935215
$873$	0	0
$874$	4.16383	0.140844
$875$	0	0
$876$	0	0
$877$	−50.0132	−1.68882	−0.844412	−	0.535694i	$-0.820050\pi$
$877$	−50.0132	−1.68882	−0.844412	+	0.535694i	$0.820050\pi$
$878$	−0.944272	−0.0318676
$879$	0	0
$880$	0	0
$881$	16.8328	0.567112	0.283556	−	0.958956i	$-0.408486\pi$
$881$	16.8328	0.567112	0.283556	+	0.958956i	$0.408486\pi$
$882$	0	0
$883$	16.8885	0.568345	0.284172	−	0.958773i	$-0.408281\pi$
$883$	16.8885	0.568345	0.284172	+	0.958773i	$0.408281\pi$
$884$	−37.0246	−1.24527
$885$	0	0
$886$	49.4164	1.66018
$887$	52.1935	1.75249	0.876243	−	0.481869i	$-0.160042\pi$
$887$	52.1935	1.75249	0.876243	+	0.481869i	$0.160042\pi$
$888$	0	0
$889$	0	0
$890$	−16.5579	−0.555022
$891$	0	0
$892$	−63.1481	−2.11436
$893$	−8.02391	−0.268510
$894$	0	0
$895$	62.1953	2.07896
$896$	0	0
$897$	0	0
$898$	−79.4853	−2.65246
$899$	24.2492	0.808757
$900$	0	0
$901$	15.8114	0.526754
$902$	39.8885	1.32814
$903$	0	0
$904$	3.41641	0.113628
$905$	96.9316	3.22212
$906$	0	0
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	39.4164	1.30808
$909$	0	0
$910$	0	0
$911$	22.8337	0.756516	0.378258	−	0.925700i	$-0.376523\pi$
$911$	22.8337	0.756516	0.378258	+	0.925700i	$0.376523\pi$
$912$	0	0
$913$	0.874032	0.0289262
$914$	75.1295	2.48506
$915$	0	0
$916$	34.4512	1.13830
$917$	0	0
$918$	0	0
$919$	1.94427	0.0641356	0.0320678	−	0.999486i	$-0.489791\pi$
$919$	1.94427	0.0641356	0.0320678	+	0.999486i	$0.489791\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	51.5493	1.69769
$923$	−29.8885	−0.983793
$924$	0	0
$925$	−60.9443	−2.00384
$926$	85.0777	2.79583
$927$	0	0
$928$	40.7214	1.33674
$929$	−17.8328	−0.585076	−0.292538	−	0.956254i	$-0.594500\pi$
$929$	−17.8328	−0.585076	−0.292538	+	0.956254i	$0.594500\pi$
$930$	0	0
$931$	0	0
$932$	−0.412662	−0.0135172
$933$	0	0
$934$	−86.0306	−2.81501
$935$	−78.4193	−2.56459
$936$	0	0
$937$	17.8446	0.582958	0.291479	−	0.956577i	$-0.405853\pi$
$937$	17.8446	0.582958	0.291479	+	0.956577i	$0.405853\pi$
$938$	0	0
$939$	0	0
$940$	−20.1803	−0.658210
$941$	20.7082	0.675068	0.337534	−	0.941313i	$-0.390407\pi$
$941$	20.7082	0.675068	0.337534	+	0.941313i	$0.390407\pi$
$942$	0	0
$943$	−1.57184	−0.0511860
$944$	0	0
$945$	0	0
$946$	−18.9443	−0.615931
$947$	40.7758	1.32503	0.662517	−	0.749047i	$-0.269488\pi$
$947$	40.7758	1.32503	0.662517	+	0.749047i	$0.269488\pi$
$948$	0	0
$949$	−1.70820	−0.0554506
$950$	161.443	5.23789
$951$	0	0
$952$	0	0
$953$	−14.8886	−0.482291	−0.241145	−	0.970489i	$-0.577523\pi$
$953$	−14.8886	−0.482291	−0.241145	+	0.970489i	$0.577523\pi$
$954$	0	0
$955$	−17.9721	−0.581564
$956$	−82.7895	−2.67760
$957$	0	0
$958$	−71.0932	−2.29692
$959$	0	0
$960$	0	0
$961$	−19.6525	−0.633951
$962$	24.6525	0.794828
$963$	0	0
$964$	22.2148	0.715489
$965$	72.7771	2.34278
$966$	0	0
$967$	5.52786	0.177764	0.0888821	−	0.996042i	$-0.471671\pi$
$967$	5.52786	0.177764	0.0888821	+	0.996042i	$0.471671\pi$
$968$	7.65996	0.246200
$969$	0	0
$970$	18.9443	0.608264
$971$	−20.5279	−0.658771	−0.329385	−	0.944196i	$-0.606841\pi$
$971$	−20.5279	−0.658771	−0.329385	+	0.944196i	$0.606841\pi$
$972$	0	0
$973$	0	0
$974$	−59.2393	−1.89815
$975$	0	0
$976$	0	0
$977$	−4.83153	−0.154574	−0.0772872	−	0.997009i	$-0.524626\pi$
$977$	−4.83153	−0.154574	−0.0772872	+	0.997009i	$0.524626\pi$
$978$	0	0
$979$	6.32456	0.202134
$980$	0	0
$981$	0	0
$982$	−3.34752	−0.106824
$983$	−26.0557	−0.831049	−0.415524	−	0.909582i	$-0.636402\pi$
$983$	−26.0557	−0.831049	−0.415524	+	0.909582i	$0.636402\pi$
$984$	0	0
$985$	1.95440	0.0622722
$986$	−82.3607	−2.62290
$987$	0	0
$988$	−40.3607	−1.28404
$989$	0.746512	0.0237377
$990$	0	0
$991$	0.124612	0.00395842	0.00197921	−	0.999998i	$-0.499370\pi$
$991$	0.124612	0.00395842	0.00197921	+	0.999998i	$0.499370\pi$
$992$	19.0557	0.605020
$993$	0	0
$994$	0	0
$995$	57.0786	1.80951
$996$	0	0
$997$	61.4488	1.94610	0.973051	−	0.230589i	$-0.0740653\pi$
$997$	61.4488	1.94610	0.973051	+	0.230589i	$0.0740653\pi$
$998$	84.4100	2.67195
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.be.1.4	yes	4
3.2	odd	2		1323.2.a.bb.1.1	✓	4
7.6	odd	2		1323.2.a.bb.1.4	yes	4
21.20	even	2	inner	1323.2.a.be.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.a.bb.1.1	✓	4	3.2	odd	2
1323.2.a.bb.1.4	yes	4	7.6	odd	2
1323.2.a.be.1.1	yes	4	21.20	even	2	inner
1323.2.a.be.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28825 q^{2} +3.23607 q^{4} +4.23607 q^{5} +2.82843 q^{8} +9.69316 q^{10} -3.70246 q^{11} -2.28825 q^{13} +5.00000 q^{17} +5.45052 q^{19} +13.7082 q^{20} -8.47214 q^{22} +0.333851 q^{23} +12.9443 q^{25} -5.23607 q^{26} -7.19859 q^{29} -3.36861 q^{31} -5.65685 q^{32} +11.4412 q^{34} -4.70820 q^{37} +12.4721 q^{38} +11.9814 q^{40} -4.70820 q^{41} +2.23607 q^{43} -11.9814 q^{44} +0.763932 q^{46} -1.47214 q^{47} +29.6197 q^{50} -7.40492 q^{52} +3.16228 q^{53} -15.6839 q^{55} -16.4721 q^{58} -3.94427 q^{59} +3.70246 q^{61} -7.70820 q^{62} -12.9443 q^{64} -9.69316 q^{65} -11.2361 q^{67} +16.1803 q^{68} +13.0618 q^{71} +0.746512 q^{73} -10.7735 q^{74} +17.6383 q^{76} -11.4721 q^{79} -10.7735 q^{82} -0.236068 q^{83} +21.1803 q^{85} +5.11667 q^{86} -10.4721 q^{88} -1.70820 q^{89} +1.08036 q^{92} -3.36861 q^{94} +23.0888 q^{95} +1.95440 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{5} + 20 q^{17} + 28 q^{20} - 16 q^{22} + 16 q^{25} - 12 q^{26} + 8 q^{37} + 32 q^{38} + 8 q^{41} + 12 q^{46} + 12 q^{47} - 48 q^{58} + 20 q^{59} - 4 q^{62} - 16 q^{64} - 36 q^{67} + 20 q^{68}+ \cdots + 20 q^{89}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^5 + 20 * q^17 + 28 * q^20 - 16 * q^22 + 16 * q^25 - 12 * q^26 + 8 * q^37 + 32 * q^38 + 8 * q^41 + 12 * q^46 + 12 * q^47 - 48 * q^58 + 20 * q^59 - 4 * q^62 - 16 * q^64 - 36 * q^67 + 20 * q^68 - 28 * q^79 + 8 * q^83 + 40 * q^85 - 24 * q^88 + 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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