LMFDB - Embedded newform 1323.2.a.bb.1.2

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:	$1$
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			1.2
Root			$-0.874032$ 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)

$f(q)$	$=$	$q-0.874032 q^{2} -1.23607 q^{4} +0.236068 q^{5} +2.82843 q^{8} -0.206331 q^{10} -0.540182 q^{11} -0.874032 q^{13} -5.00000 q^{17} +4.03631 q^{19} -0.291796 q^{20} +0.472136 q^{22} -5.99070 q^{23} -4.94427 q^{25} +0.763932 q^{26} +8.61280 q^{29} +6.53089 q^{31} -5.65685 q^{32} +4.37016 q^{34} +8.70820 q^{37} -3.52786 q^{38} +0.667701 q^{40} -8.70820 q^{41} -2.23607 q^{43} +0.667701 q^{44} +5.23607 q^{46} -7.47214 q^{47} +4.32145 q^{50} +1.08036 q^{52} -3.16228 q^{53} -0.127520 q^{55} -7.52786 q^{58} -13.9443 q^{59} -0.540182 q^{61} -5.70820 q^{62} +4.94427 q^{64} -0.206331 q^{65} -6.76393 q^{67} +6.18034 q^{68} +6.73722 q^{71} -13.3956 q^{73} -7.61125 q^{74} -4.98915 q^{76} -2.52786 q^{79} +7.61125 q^{82} -4.23607 q^{83} -1.18034 q^{85} +1.95440 q^{86} -1.52786 q^{88} -11.7082 q^{89} +7.40492 q^{92} +6.53089 q^{94} +0.952843 q^{95} -5.11667 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.874032	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$2$	−0.874032	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$3$	0	0
$3$	0	0
$4$	−1.23607	−0.618034
$5$	0.236068	0.105573	0.0527864	−	0.998606i	$-0.483190\pi$
$5$	0.236068	0.105573	0.0527864	+	0.998606i	$0.483190\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	0	0
$10$	−0.206331	−0.0652476
$11$	−0.540182	−0.162871	−0.0814354	−	0.996679i	$-0.525950\pi$
$11$	−0.540182	−0.162871	−0.0814354	+	0.996679i	$0.525950\pi$
$12$	0	0
$13$	−0.874032	−0.242413	−0.121206	−	0.992627i	$-0.538676\pi$
$13$	−0.874032	−0.242413	−0.121206	+	0.992627i	$0.538676\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	4.03631	0.925993	0.462996	−	0.886360i	$-0.346774\pi$
$19$	4.03631	0.925993	0.462996	+	0.886360i	$0.346774\pi$
$20$	−0.291796	−0.0652476
$21$	0	0
$22$	0.472136	0.100660
$23$	−5.99070	−1.24915	−0.624574	−	0.780966i	$-0.714727\pi$
$23$	−5.99070	−1.24915	−0.624574	+	0.780966i	$0.714727\pi$
$24$	0	0
$25$	−4.94427	−0.988854
$26$	0.763932	0.149819
$27$	0	0
$28$	0	0
$29$	8.61280	1.59936	0.799678	−	0.600428i	$-0.205003\pi$
$29$	8.61280	1.59936	0.799678	+	0.600428i	$0.205003\pi$
$30$	0	0
$31$	6.53089	1.17298	0.586491	−	0.809956i	$-0.300509\pi$
$31$	6.53089	1.17298	0.586491	+	0.809956i	$0.300509\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	4.37016	0.749476
$35$	0	0
$36$	0	0
$37$	8.70820	1.43162	0.715810	−	0.698295i	$-0.246058\pi$
$37$	8.70820	1.43162	0.715810	+	0.698295i	$0.246058\pi$
$38$	−3.52786	−0.572295
$39$	0	0
$40$	0.667701	0.105573
$41$	−8.70820	−1.35999	−0.679996	−	0.733215i	$-0.738019\pi$
$41$	−8.70820	−1.35999	−0.679996	+	0.733215i	$0.738019\pi$
$42$	0	0
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	0.667701	0.100660
$45$	0	0
$46$	5.23607	0.772016
$47$	−7.47214	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$47$	−7.47214	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$48$	0	0
$49$	0	0
$50$	4.32145	0.611146
$51$	0	0
$52$	1.08036	0.149819
$53$	−3.16228	−0.434372	−0.217186	−	0.976130i	$-0.569688\pi$
$53$	−3.16228	−0.434372	−0.217186	+	0.976130i	$0.569688\pi$
$54$	0	0
$55$	−0.127520	−0.0171947
$56$	0	0
$57$	0	0
$58$	−7.52786	−0.988457
$59$	−13.9443	−1.81539	−0.907695	−	0.419631i	$-0.862159\pi$
$59$	−13.9443	−1.81539	−0.907695	+	0.419631i	$0.862159\pi$
$60$	0	0
$61$	−0.540182	−0.0691632	−0.0345816	−	0.999402i	$-0.511010\pi$
$61$	−0.540182	−0.0691632	−0.0345816	+	0.999402i	$0.511010\pi$
$62$	−5.70820	−0.724943
$63$	0	0
$64$	4.94427	0.618034
$65$	−0.206331	−0.0255922
$66$	0	0
$67$	−6.76393	−0.826346	−0.413173	−	0.910653i	$-0.635579\pi$
$67$	−6.76393	−0.826346	−0.413173	+	0.910653i	$0.635579\pi$
$68$	6.18034	0.749476
$69$	0	0
$70$	0	0
$71$	6.73722	0.799561	0.399780	−	0.916611i	$-0.369086\pi$
$71$	6.73722	0.799561	0.399780	+	0.916611i	$0.369086\pi$
$72$	0	0
$73$	−13.3956	−1.56784	−0.783920	−	0.620862i	$-0.786783\pi$
$73$	−13.3956	−1.56784	−0.783920	+	0.620862i	$0.786783\pi$
$74$	−7.61125	−0.884790
$75$	0	0
$76$	−4.98915	−0.572295
$77$	0	0
$78$	0	0
$79$	−2.52786	−0.284407	−0.142203	−	0.989837i	$-0.545419\pi$
$79$	−2.52786	−0.284407	−0.142203	+	0.989837i	$0.545419\pi$
$80$	0	0
$81$	0	0
$82$	7.61125	0.840522
$83$	−4.23607	−0.464969	−0.232484	−	0.972600i	$-0.574685\pi$
$83$	−4.23607	−0.464969	−0.232484	+	0.972600i	$0.574685\pi$
$84$	0	0
$85$	−1.18034	−0.128026
$86$	1.95440	0.210748
$87$	0	0
$88$	−1.52786	−0.162871
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	0	0
$92$	7.40492	0.772016
$93$	0	0
$94$	6.53089	0.673609
$95$	0.952843	0.0977597
$96$	0	0
$97$	−5.11667	−0.519519	−0.259760	−	0.965673i	$-0.583643\pi$
$97$	−5.11667	−0.519519	−0.259760	+	0.965673i	$0.583643\pi$
$98$	0	0
$99$	0	0
$100$	6.11146	0.611146
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	0	0
$103$	−16.8918	−1.66439	−0.832197	−	0.554480i	$-0.812917\pi$
$103$	−16.8918	−1.66439	−0.832197	+	0.554480i	$0.812917\pi$
$104$	−2.47214	−0.242413
$105$	0	0
$106$	2.76393	0.268457
$107$	−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
$109$	14.2361	1.36357	0.681784	−	0.731554i	$-0.261204\pi$
$109$	14.2361	1.36357	0.681784	+	0.731554i	$0.261204\pi$
$110$	0.111456	0.0106269
$111$	0	0
$112$	0	0
$113$	−8.27895	−0.778818	−0.389409	−	0.921065i	$-0.627321\pi$
$113$	−8.27895	−0.778818	−0.389409	+	0.921065i	$0.627321\pi$
$114$	0	0
$115$	−1.41421	−0.131876
$116$	−10.6460	−0.988457
$117$	0	0
$118$	12.1877	1.12197
$119$	0	0
$120$	0	0
$121$	−10.7082	−0.973473
$122$	0.472136	0.0427452
$123$	0	0
$124$	−8.07262	−0.724943
$125$	−2.34752	−0.209969
$126$	0	0
$127$	−3.47214	−0.308102	−0.154051	−	0.988063i	$-0.549232\pi$
$127$	−3.47214	−0.308102	−0.154051	+	0.988063i	$0.549232\pi$
$128$	6.99226	0.618034
$129$	0	0
$130$	0.180340	0.0158169
$131$	6.94427	0.606724	0.303362	−	0.952875i	$-0.401891\pi$
$131$	6.94427	0.606724	0.303362	+	0.952875i	$0.401891\pi$
$132$	0	0
$133$	0	0
$134$	5.91189	0.510710
$135$	0	0
$136$	−14.1421	−1.21268
$137$	13.9358	1.19062	0.595308	−	0.803498i	$-0.297030\pi$
$137$	13.9358	1.19062	0.595308	+	0.803498i	$0.297030\pi$
$138$	0	0
$139$	−9.28050	−0.787162	−0.393581	−	0.919290i	$-0.628764\pi$
$139$	−9.28050	−0.787162	−0.393581	+	0.919290i	$0.628764\pi$
$140$	0	0
$141$	0	0
$142$	−5.88854	−0.494156
$143$	0.472136	0.0394820
$144$	0	0
$145$	2.03321	0.168849
$146$	11.7082	0.968978
$147$	0	0
$148$	−10.7639	−0.884790
$149$	−6.45207	−0.528575	−0.264287	−	0.964444i	$-0.585137\pi$
$149$	−6.45207	−0.528575	−0.264287	+	0.964444i	$0.585137\pi$
$150$	0	0
$151$	10.4164	0.847675	0.423838	−	0.905738i	$-0.360683\pi$
$151$	10.4164	0.847675	0.423838	+	0.905738i	$0.360683\pi$
$152$	11.4164	0.925993
$153$	0	0
$154$	0	0
$155$	1.54173	0.123835
$156$	0	0
$157$	17.5107	1.39751	0.698755	−	0.715361i	$-0.253738\pi$
$157$	17.5107	1.39751	0.698755	+	0.715361i	$0.253738\pi$
$158$	2.20943	0.175773
$159$	0	0
$160$	−1.33540	−0.105573
$161$	0	0
$162$	0	0
$163$	17.1803	1.34567	0.672834	−	0.739793i	$-0.265077\pi$
$163$	17.1803	1.34567	0.672834	+	0.739793i	$0.265077\pi$
$164$	10.7639	0.840522
$165$	0	0
$166$	3.70246	0.287367
$167$	8.23607	0.637326	0.318663	−	0.947868i	$-0.396766\pi$
$167$	8.23607	0.637326	0.318663	+	0.947868i	$0.396766\pi$
$168$	0	0
$169$	−12.2361	−0.941236
$170$	1.03165	0.0791243
$171$	0	0
$172$	2.76393	0.210748
$173$	−9.52786	−0.724390	−0.362195	−	0.932102i	$-0.617973\pi$
$173$	−9.52786	−0.724390	−0.362195	+	0.932102i	$0.617973\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.2333	0.767022
$179$	17.8446	1.33377	0.666884	−	0.745162i	$-0.267628\pi$
$179$	17.8446	1.33377	0.666884	+	0.745162i	$0.267628\pi$
$180$	0	0
$181$	8.74032	0.649663	0.324831	−	0.945772i	$-0.394692\pi$
$181$	8.74032	0.649663	0.324831	+	0.945772i	$0.394692\pi$
$182$	0	0
$183$	0	0
$184$	−16.9443	−1.24915
$185$	2.05573	0.151140
$186$	0	0
$187$	2.70091	0.197510
$188$	9.23607	0.673609
$189$	0	0
$190$	−0.832816	−0.0604188
$191$	−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
$193$	−5.18034	−0.372889	−0.186445	−	0.982465i	$-0.559696\pi$
$193$	−5.18034	−0.372889	−0.186445	+	0.982465i	$0.559696\pi$
$194$	4.47214	0.321081
$195$	0	0
$196$	0	0
$197$	−21.6746	−1.54425	−0.772125	−	0.635471i	$-0.780806\pi$
$197$	−21.6746	−1.54425	−0.772125	+	0.635471i	$0.780806\pi$
$198$	0	0
$199$	−26.1235	−1.85185	−0.925925	−	0.377709i	$-0.876712\pi$
$199$	−26.1235	−1.85185	−0.925925	+	0.377709i	$0.876712\pi$
$200$	−13.9845	−0.988854
$201$	0	0
$202$	4.16383	0.292966
$203$	0	0
$204$	0	0
$205$	−2.05573	−0.143578
$206$	14.7639	1.02865
$207$	0	0
$208$	0	0
$209$	−2.18034	−0.150817
$210$	0	0
$211$	9.41641	0.648252	0.324126	−	0.946014i	$-0.394930\pi$
$211$	9.41641	0.648252	0.324126	+	0.946014i	$0.394930\pi$
$212$	3.90879	0.268457
$213$	0	0
$214$	8.65248	0.591471
$215$	−0.527864	−0.0360000
$216$	0	0
$217$	0	0
$218$	−12.4428	−0.842731
$219$	0	0
$220$	0.157623	0.0106269
$221$	4.37016	0.293969
$222$	0	0
$223$	−15.2712	−1.02264	−0.511318	−	0.859392i	$-0.670842\pi$
$223$	−15.2712	−1.02264	−0.511318	+	0.859392i	$0.670842\pi$
$224$	0	0
$225$	0	0
$226$	7.23607	0.481336
$227$	10.1803	0.675693	0.337846	−	0.941201i	$-0.390302\pi$
$227$	10.1803	0.675693	0.337846	+	0.941201i	$0.390302\pi$
$228$	0	0
$229$	−23.2951	−1.53938	−0.769692	−	0.638415i	$-0.779590\pi$
$229$	−23.2951	−1.53938	−0.769692	+	0.638415i	$0.779590\pi$
$230$	1.23607	0.0815039
$231$	0	0
$232$	24.3607	1.59936
$233$	15.6839	1.02748	0.513742	−	0.857945i	$-0.328259\pi$
$233$	15.6839	1.02748	0.513742	+	0.857945i	$0.328259\pi$
$234$	0	0
$235$	−1.76393	−0.115066
$236$	17.2361	1.12197
$237$	0	0
$238$	0	0
$239$	−9.77198	−0.632097	−0.316048	−	0.948743i	$-0.602356\pi$
$239$	−9.77198	−0.632097	−0.316048	+	0.948743i	$0.602356\pi$
$240$	0	0
$241$	2.62210	0.168904	0.0844520	−	0.996428i	$-0.473086\pi$
$241$	2.62210	0.168904	0.0844520	+	0.996428i	$0.473086\pi$
$242$	9.35931	0.601639
$243$	0	0
$244$	0.667701	0.0427452
$245$	0	0
$246$	0	0
$247$	−3.52786	−0.224473
$248$	18.4721	1.17298
$249$	0	0
$250$	2.05181	0.129768
$251$	22.7082	1.43333	0.716665	−	0.697418i	$-0.245668\pi$
$251$	22.7082	1.43333	0.716665	+	0.697418i	$0.245668\pi$
$252$	0	0
$253$	3.23607	0.203450
$254$	3.03476	0.190418
$255$	0	0
$256$	−16.0000	−1.00000
$257$	−21.4164	−1.33592	−0.667959	−	0.744198i	$-0.732832\pi$
$257$	−21.4164	−1.33592	−0.667959	+	0.744198i	$0.732832\pi$
$258$	0	0
$259$	0	0
$260$	0.255039	0.0158169
$261$	0	0
$262$	−6.06952	−0.374976
$263$	27.2526	1.68047	0.840234	−	0.542224i	$-0.182417\pi$
$263$	27.2526	1.68047	0.840234	+	0.542224i	$0.182417\pi$
$264$	0	0
$265$	−0.746512	−0.0458579
$266$	0	0
$267$	0	0
$268$	8.36068	0.510710
$269$	−7.00000	−0.426798	−0.213399	−	0.976965i	$-0.568453\pi$
$269$	−7.00000	−0.426798	−0.213399	+	0.976965i	$0.568453\pi$
$270$	0	0
$271$	19.7202	1.19792	0.598958	−	0.800781i	$-0.295582\pi$
$271$	19.7202	1.19792	0.598958	+	0.800781i	$0.295582\pi$
$272$	0	0
$273$	0	0
$274$	−12.1803	−0.735841
$275$	2.67080	0.161056
$276$	0	0
$277$	−18.2361	−1.09570	−0.547850	−	0.836577i	$-0.684553\pi$
$277$	−18.2361	−1.09570	−0.547850	+	0.836577i	$0.684553\pi$
$278$	8.11146	0.486493
$279$	0	0
$280$	0	0
$281$	−28.6181	−1.70721	−0.853607	−	0.520918i	$-0.825590\pi$
$281$	−28.6181	−1.70721	−0.853607	+	0.520918i	$0.825590\pi$
$282$	0	0
$283$	12.1089	0.719801	0.359901	−	0.932991i	$-0.382811\pi$
$283$	12.1089	0.719801	0.359901	+	0.932991i	$0.382811\pi$
$284$	−8.32766	−0.494156
$285$	0	0
$286$	−0.412662	−0.0244012
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	−1.77709	−0.104354
$291$	0	0
$292$	16.5579	0.968978
$293$	1.47214	0.0860031	0.0430016	−	0.999075i	$-0.486308\pi$
$293$	1.47214	0.0860031	0.0430016	+	0.999075i	$0.486308\pi$
$294$	0	0
$295$	−3.29180	−0.191656
$296$	24.6305	1.43162
$297$	0	0
$298$	5.63932	0.326677
$299$	5.23607	0.302810
$300$	0	0
$301$	0	0
$302$	−9.10427	−0.523892
$303$	0	0
$304$	0	0
$305$	−0.127520	−0.00730175
$306$	0	0
$307$	22.5486	1.28692	0.643458	−	0.765481i	$-0.277499\pi$
$307$	22.5486	1.28692	0.643458	+	0.765481i	$0.277499\pi$
$308$	0	0
$309$	0	0
$310$	−1.34752	−0.0765342
$311$	31.0689	1.76175	0.880877	−	0.473345i	$-0.156953\pi$
$311$	31.0689	1.76175	0.880877	+	0.473345i	$0.156953\pi$
$312$	0	0
$313$	16.3029	0.921492	0.460746	−	0.887532i	$-0.347582\pi$
$313$	16.3029	0.921492	0.460746	+	0.887532i	$0.347582\pi$
$314$	−15.3050	−0.863708
$315$	0	0
$316$	3.12461	0.175773
$317$	10.5672	0.593513	0.296756	−	0.954953i	$-0.404095\pi$
$317$	10.5672	0.593513	0.296756	+	0.954953i	$0.404095\pi$
$318$	0	0
$319$	−4.65248	−0.260489
$320$	1.16718	0.0652476
$321$	0	0
$322$	0	0
$323$	−20.1815	−1.12293
$324$	0	0
$325$	4.32145	0.239711
$326$	−15.0162	−0.831669
$327$	0	0
$328$	−24.6305	−1.35999
$329$	0	0
$330$	0	0
$331$	−8.41641	−0.462608	−0.231304	−	0.972882i	$-0.574299\pi$
$331$	−8.41641	−0.462608	−0.231304	+	0.972882i	$0.574299\pi$
$332$	5.23607	0.287367
$333$	0	0
$334$	−7.19859	−0.393889
$335$	−1.59675	−0.0872396
$336$	0	0
$337$	8.23607	0.448647	0.224324	−	0.974515i	$-0.427983\pi$
$337$	8.23607	0.448647	0.224324	+	0.974515i	$0.427983\pi$
$338$	10.6947	0.581716
$339$	0	0
$340$	1.45898	0.0791243
$341$	−3.52786	−0.191045
$342$	0	0
$343$	0	0
$344$	−6.32456	−0.340997
$345$	0	0
$346$	8.32766	0.447698
$347$	26.3299	1.41346	0.706731	−	0.707482i	$-0.250169\pi$
$347$	26.3299	1.41346	0.706731	+	0.707482i	$0.250169\pi$
$348$	0	0
$349$	8.07262	0.432117	0.216059	−	0.976380i	$-0.430680\pi$
$349$	8.07262	0.432117	0.216059	+	0.976380i	$0.430680\pi$
$350$	0	0
$351$	0	0
$352$	3.05573	0.162871
$353$	10.1246	0.538879	0.269439	−	0.963017i	$-0.413162\pi$
$353$	10.1246	0.538879	0.269439	+	0.963017i	$0.413162\pi$
$354$	0	0
$355$	1.59044	0.0844119
$356$	14.4721	0.767022
$357$	0	0
$358$	−15.5967	−0.824314
$359$	−23.6290	−1.24709	−0.623545	−	0.781788i	$-0.714308\pi$
$359$	−23.6290	−1.24709	−0.623545	+	0.781788i	$0.714308\pi$
$360$	0	0
$361$	−2.70820	−0.142537
$362$	−7.63932	−0.401514
$363$	0	0
$364$	0	0
$365$	−3.16228	−0.165521
$366$	0	0
$367$	12.9830	0.677705	0.338853	−	0.940839i	$-0.389961\pi$
$367$	12.9830	0.677705	0.338853	+	0.940839i	$0.389961\pi$
$368$	0	0
$369$	0	0
$370$	−1.79677	−0.0934097
$371$	0	0
$372$	0	0
$373$	17.6525	0.914011	0.457005	−	0.889464i	$-0.348922\pi$
$373$	17.6525	0.914011	0.457005	+	0.889464i	$0.348922\pi$
$374$	−2.36068	−0.122068
$375$	0	0
$376$	−21.1344	−1.08992
$377$	−7.52786	−0.387705
$378$	0	0
$379$	4.88854	0.251108	0.125554	−	0.992087i	$-0.459929\pi$
$379$	4.88854	0.251108	0.125554	+	0.992087i	$0.459929\pi$
$380$	−1.17778	−0.0604188
$381$	0	0
$382$	3.70820	0.189728
$383$	−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
$386$	4.52778	0.230458
$387$	0	0
$388$	6.32456	0.321081
$389$	0.952843	0.0483111	0.0241555	−	0.999708i	$-0.492310\pi$
$389$	0.952843	0.0483111	0.0241555	+	0.999708i	$0.492310\pi$
$390$	0	0
$391$	29.9535	1.51481
$392$	0	0
$393$	0	0
$394$	18.9443	0.954399
$395$	−0.596748	−0.0300256
$396$	0	0
$397$	16.4791	0.827062	0.413531	−	0.910490i	$-0.364295\pi$
$397$	16.4791	0.827062	0.413531	+	0.910490i	$0.364295\pi$
$398$	22.8328	1.14451
$399$	0	0
$400$	0	0
$401$	30.0323	1.49974	0.749872	−	0.661583i	$-0.230115\pi$
$401$	30.0323	1.49974	0.749872	+	0.661583i	$0.230115\pi$
$402$	0	0
$403$	−5.70820	−0.284346
$404$	5.88854	0.292966
$405$	0	0
$406$	0	0
$407$	−4.70401	−0.233169
$408$	0	0
$409$	12.2364	0.605053	0.302527	−	0.953141i	$-0.402170\pi$
$409$	12.2364	0.605053	0.302527	+	0.953141i	$0.402170\pi$
$410$	1.79677	0.0887363
$411$	0	0
$412$	20.8794	1.02865
$413$	0	0
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	4.94427	0.242413
$417$	0	0
$418$	1.90569	0.0932102
$419$	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	$-0.465242\pi$
$421$	4.47214	0.217959	0.108979	+	0.994044i	$0.465242\pi$
$422$	−8.23024	−0.400642
$423$	0	0
$424$	−8.94427	−0.434372
$425$	24.7214	1.19916
$426$	0	0
$427$	0	0
$428$	12.2364	0.591471
$429$	0	0
$430$	0.461370	0.0222492
$431$	−11.5687	−0.557247	−0.278623	−	0.960400i	$-0.589878\pi$
$431$	−11.5687	−0.557247	−0.278623	+	0.960400i	$0.589878\pi$
$432$	0	0
$433$	−37.4860	−1.80146	−0.900730	−	0.434379i	$-0.856968\pi$
$433$	−37.4860	−1.80146	−0.900730	+	0.434379i	$0.856968\pi$
$434$	0	0
$435$	0	0
$436$	−17.5967	−0.842731
$437$	−24.1803	−1.15670
$438$	0	0
$439$	19.3863	0.925259	0.462629	−	0.886552i	$-0.346906\pi$
$439$	19.3863	0.925259	0.462629	+	0.886552i	$0.346906\pi$
$440$	−0.360680	−0.0171947
$441$	0	0
$442$	−3.81966	−0.181683
$443$	−25.8384	−1.22762	−0.613810	−	0.789454i	$-0.710364\pi$
$443$	−25.8384	−1.22762	−0.613810	+	0.789454i	$0.710364\pi$
$444$	0	0
$445$	−2.76393	−0.131023
$446$	13.3475	0.632024
$447$	0	0
$448$	0	0
$449$	−6.27585	−0.296176	−0.148088	−	0.988974i	$-0.547312\pi$
$449$	−6.27585	−0.296176	−0.148088	+	0.988974i	$0.547312\pi$
$450$	0	0
$451$	4.70401	0.221503
$452$	10.2333	0.481336
$453$	0	0
$454$	−8.89794	−0.417601
$455$	0	0
$456$	0	0
$457$	−20.8328	−0.974518	−0.487259	−	0.873257i	$-0.662003\pi$
$457$	−20.8328	−0.974518	−0.487259	+	0.873257i	$0.662003\pi$
$458$	20.3607	0.951392
$459$	0	0
$460$	1.74806	0.0815039
$461$	−31.4721	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$461$	−31.4721	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$462$	0	0
$463$	14.8197	0.688728	0.344364	−	0.938836i	$-0.388095\pi$
$463$	14.8197	0.688728	0.344364	+	0.938836i	$0.388095\pi$
$464$	0	0
$465$	0	0
$466$	−13.7082	−0.635020
$467$	−11.5967	−0.536633	−0.268317	−	0.963331i	$-0.586467\pi$
$467$	−11.5967	−0.536633	−0.268317	+	0.963331i	$0.586467\pi$
$468$	0	0
$469$	0	0
$470$	1.54173	0.0711148
$471$	0	0
$472$	−39.4404	−1.81539
$473$	1.20788	0.0555385
$474$	0	0
$475$	−19.9566	−0.915672
$476$	0	0
$477$	0	0
$478$	8.54102	0.390657
$479$	−27.0689	−1.23681	−0.618404	−	0.785860i	$-0.712221\pi$
$479$	−27.0689	−1.23681	−0.618404	+	0.785860i	$0.712221\pi$
$480$	0	0
$481$	−7.61125	−0.347043
$482$	−2.29180	−0.104388
$483$	0	0
$484$	13.2361	0.601639
$485$	−1.20788	−0.0548471
$486$	0	0
$487$	9.88854	0.448093	0.224046	−	0.974578i	$-0.428073\pi$
$487$	9.88854	0.448093	0.224046	+	0.974578i	$0.428073\pi$
$488$	−1.52786	−0.0691632
$489$	0	0
$490$	0	0
$491$	39.6467	1.78923	0.894615	−	0.446838i	$-0.147450\pi$
$491$	39.6467	1.78923	0.894615	+	0.446838i	$0.147450\pi$
$492$	0	0
$493$	−43.0640	−1.93951
$494$	3.08347	0.138732
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.11146	0.0497556	0.0248778	−	0.999690i	$-0.492080\pi$
$499$	1.11146	0.0497556	0.0248778	+	0.999690i	$0.492080\pi$
$500$	2.90170	0.129768
$501$	0	0
$502$	−19.8477	−0.885846
$503$	17.8328	0.795126	0.397563	−	0.917575i	$-0.369856\pi$
$503$	17.8328	0.795126	0.397563	+	0.917575i	$0.369856\pi$
$504$	0	0
$505$	−1.12461	−0.0500446
$506$	−2.82843	−0.125739
$507$	0	0
$508$	4.29180	0.190418
$509$	−27.3607	−1.21274	−0.606370	−	0.795182i	$-0.707375\pi$
$509$	−27.3607	−1.21274	−0.606370	+	0.795182i	$0.707375\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	18.7186	0.825643
$515$	−3.98760	−0.175715
$516$	0	0
$517$	4.03631	0.177517
$518$	0	0
$519$	0	0
$520$	−0.583592	−0.0255922
$521$	17.4721	0.765468	0.382734	−	0.923859i	$-0.374983\pi$
$521$	17.4721	0.765468	0.382734	+	0.923859i	$0.374983\pi$
$522$	0	0
$523$	−36.8183	−1.60995	−0.804975	−	0.593309i	$-0.797821\pi$
$523$	−36.8183	−1.60995	−0.804975	+	0.593309i	$0.797821\pi$
$524$	−8.58359	−0.374976
$525$	0	0
$526$	−23.8197	−1.03859
$527$	−32.6544	−1.42245
$528$	0	0
$529$	12.8885	0.560371
$530$	0.652476	0.0283417
$531$	0	0
$532$	0	0
$533$	7.61125	0.329680
$534$	0	0
$535$	−2.33695	−0.101035
$536$	−19.1313	−0.826346
$537$	0	0
$538$	6.11822	0.263775
$539$	0	0
$540$	0	0
$541$	−34.3050	−1.47489	−0.737443	−	0.675410i	$-0.763967\pi$
$541$	−34.3050	−1.47489	−0.737443	+	0.675410i	$0.763967\pi$
$542$	−17.2361	−0.740353
$543$	0	0
$544$	28.2843	1.21268
$545$	3.36068	0.143956
$546$	0	0
$547$	−22.1246	−0.945980	−0.472990	−	0.881068i	$-0.656825\pi$
$547$	−22.1246	−0.945980	−0.472990	+	0.881068i	$0.656825\pi$
$548$	−17.2256	−0.735841
$549$	0	0
$550$	−2.33437	−0.0995378
$551$	34.7639	1.48099
$552$	0	0
$553$	0	0
$554$	15.9389	0.677179
$555$	0	0
$556$	11.4713	0.486493
$557$	−6.32456	−0.267980	−0.133990	−	0.990983i	$-0.542779\pi$
$557$	−6.32456	−0.267980	−0.133990	+	0.990983i	$0.542779\pi$
$558$	0	0
$559$	1.95440	0.0826621
$560$	0	0
$561$	0	0
$562$	25.0132	1.05512
$563$	−11.8197	−0.498139	−0.249070	−	0.968486i	$-0.580125\pi$
$563$	−11.8197	−0.498139	−0.249070	+	0.968486i	$0.580125\pi$
$564$	0	0
$565$	−1.95440	−0.0822220
$566$	−10.5836	−0.444862
$567$	0	0
$568$	19.0557	0.799561
$569$	0.0788114	0.00330395	0.00165197	−	0.999999i	$-0.499474\pi$
$569$	0.0788114	0.00330395	0.00165197	+	0.999999i	$0.499474\pi$
$570$	0	0
$571$	39.0689	1.63498	0.817491	−	0.575941i	$-0.195364\pi$
$571$	39.0689	1.63498	0.817491	+	0.575941i	$0.195364\pi$
$572$	−0.583592	−0.0244012
$573$	0	0
$574$	0	0
$575$	29.6197	1.23523
$576$	0	0
$577$	18.5610	0.772705	0.386352	−	0.922351i	$-0.373735\pi$
$577$	18.5610	0.772705	0.386352	+	0.922351i	$0.373735\pi$
$578$	−6.99226	−0.290840
$579$	0	0
$580$	−2.51318	−0.104354
$581$	0	0
$582$	0	0
$583$	1.70820	0.0707466
$584$	−37.8885	−1.56784
$585$	0	0
$586$	−1.28669	−0.0531528
$587$	9.52786	0.393257	0.196629	−	0.980478i	$-0.437001\pi$
$587$	9.52786	0.393257	0.196629	+	0.980478i	$0.437001\pi$
$588$	0	0
$589$	26.3607	1.08617
$590$	2.87714	0.118450
$591$	0	0
$592$	0	0
$593$	−2.52786	−0.103807	−0.0519035	−	0.998652i	$-0.516529\pi$
$593$	−2.52786	−0.103807	−0.0519035	+	0.998652i	$0.516529\pi$
$594$	0	0
$595$	0	0
$596$	7.97520	0.326677
$597$	0	0
$598$	−4.57649	−0.187147
$599$	−2.54328	−0.103916	−0.0519579	−	0.998649i	$-0.516546\pi$
$599$	−2.54328	−0.103916	−0.0519579	+	0.998649i	$0.516546\pi$
$600$	0	0
$601$	41.1397	1.67812	0.839062	−	0.544036i	$-0.183104\pi$
$601$	41.1397	1.67812	0.839062	+	0.544036i	$0.183104\pi$
$602$	0	0
$603$	0	0
$604$	−12.8754	−0.523892
$605$	−2.52786	−0.102772
$606$	0	0
$607$	−17.1769	−0.697189	−0.348594	−	0.937274i	$-0.613341\pi$
$607$	−17.1769	−0.697189	−0.348594	+	0.937274i	$0.613341\pi$
$608$	−22.8328	−0.925993
$609$	0	0
$610$	0.111456	0.00451273
$611$	6.53089	0.264211
$612$	0	0
$613$	3.88854	0.157057	0.0785284	−	0.996912i	$-0.474978\pi$
$613$	3.88854	0.157057	0.0785284	+	0.996912i	$0.474978\pi$
$614$	−19.7082	−0.795358
$615$	0	0
$616$	0	0
$617$	−2.08191	−0.0838147	−0.0419074	−	0.999122i	$-0.513343\pi$
$617$	−2.08191	−0.0838147	−0.0419074	+	0.999122i	$0.513343\pi$
$618$	0	0
$619$	−8.86784	−0.356429	−0.178214	−	0.983992i	$-0.557032\pi$
$619$	−8.86784	−0.356429	−0.178214	+	0.983992i	$0.557032\pi$
$620$	−1.90569	−0.0765342
$621$	0	0
$622$	−27.1552	−1.08882
$623$	0	0
$624$	0	0
$625$	24.1672	0.966687
$626$	−14.2492	−0.569514
$627$	0	0
$628$	−21.6445	−0.863708
$629$	−43.5410	−1.73609
$630$	0	0
$631$	−21.7639	−0.866408	−0.433204	−	0.901296i	$-0.642617\pi$
$631$	−21.7639	−0.866408	−0.433204	+	0.901296i	$0.642617\pi$
$632$	−7.14988	−0.284407
$633$	0	0
$634$	−9.23607	−0.366811
$635$	−0.819660	−0.0325272
$636$	0	0
$637$	0	0
$638$	4.06641	0.160991
$639$	0	0
$640$	1.65065	0.0652476
$641$	−31.6529	−1.25021	−0.625107	−	0.780539i	$-0.714945\pi$
$641$	−31.6529	−1.25021	−0.625107	+	0.780539i	$0.714945\pi$
$642$	0	0
$643$	8.56409	0.337735	0.168867	−	0.985639i	$-0.445989\pi$
$643$	8.56409	0.337735	0.168867	+	0.985639i	$0.445989\pi$
$644$	0	0
$645$	0	0
$646$	17.6393	0.694010
$647$	−40.3607	−1.58674	−0.793371	−	0.608738i	$-0.791676\pi$
$647$	−40.3607	−1.58674	−0.793371	+	0.608738i	$0.791676\pi$
$648$	0	0
$649$	7.53244	0.295674
$650$	−3.77709	−0.148150
$651$	0	0
$652$	−21.2361	−0.831669
$653$	22.6761	0.887385	0.443693	−	0.896179i	$-0.353668\pi$
$653$	22.6761	0.887385	0.443693	+	0.896179i	$0.353668\pi$
$654$	0	0
$655$	1.63932	0.0640535
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.1931	1.25406	0.627032	−	0.778994i	$-0.284270\pi$
$659$	32.1931	1.25406	0.627032	+	0.778994i	$0.284270\pi$
$660$	0	0
$661$	−30.7487	−1.19599	−0.597994	−	0.801501i	$-0.704035\pi$
$661$	−30.7487	−1.19599	−0.597994	+	0.801501i	$0.704035\pi$
$662$	7.35621	0.285907
$663$	0	0
$664$	−11.9814	−0.464969
$665$	0	0
$666$	0	0
$667$	−51.5967	−1.99783
$668$	−10.1803	−0.393889
$669$	0	0
$670$	1.39561	0.0539171
$671$	0.291796	0.0112647
$672$	0	0
$673$	−36.7639	−1.41715	−0.708573	−	0.705638i	$-0.750661\pi$
$673$	−36.7639	−1.41715	−0.708573	+	0.705638i	$0.750661\pi$
$674$	−7.19859	−0.277279
$675$	0	0
$676$	15.1246	0.581716
$677$	−47.8885	−1.84051	−0.920253	−	0.391324i	$-0.872017\pi$
$677$	−47.8885	−1.84051	−0.920253	+	0.391324i	$0.872017\pi$
$678$	0	0
$679$	0	0
$680$	−3.33851	−0.128026
$681$	0	0
$682$	3.08347	0.118072
$683$	−14.8585	−0.568546	−0.284273	−	0.958743i	$-0.591752\pi$
$683$	−14.8585	−0.568546	−0.284273	+	0.958743i	$0.591752\pi$
$684$	0	0
$685$	3.28980	0.125697
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.76393	0.105297
$690$	0	0
$691$	27.6653	1.05244	0.526218	−	0.850349i	$-0.323609\pi$
$691$	27.6653	1.05244	0.526218	+	0.850349i	$0.323609\pi$
$692$	11.7771	0.447698
$693$	0	0
$694$	−23.0132	−0.873567
$695$	−2.19083	−0.0831029
$696$	0	0
$697$	43.5410	1.64923
$698$	−7.05573	−0.267063
$699$	0	0
$700$	0	0
$701$	31.5741	1.19254	0.596268	−	0.802785i	$-0.296650\pi$
$701$	31.5741	1.19254	0.596268	+	0.802785i	$0.296650\pi$
$702$	0	0
$703$	35.1490	1.32567
$704$	−2.67080	−0.100660
$705$	0	0
$706$	−8.84924	−0.333045
$707$	0	0
$708$	0	0
$709$	−12.5279	−0.470494	−0.235247	−	0.971936i	$-0.575590\pi$
$709$	−12.5279	−0.470494	−0.235247	+	0.971936i	$0.575590\pi$
$710$	−1.39010	−0.0521694
$711$	0	0
$712$	−33.1158	−1.24107
$713$	−39.1246	−1.46523
$714$	0	0
$715$	0.111456	0.00416822
$716$	−22.0571	−0.824314
$717$	0	0
$718$	20.6525	0.770744
$719$	26.4164	0.985166	0.492583	−	0.870266i	$-0.336053\pi$
$719$	26.4164	0.985166	0.492583	+	0.870266i	$0.336053\pi$
$720$	0	0
$721$	0	0
$722$	2.36706	0.0880927
$723$	0	0
$724$	−10.8036	−0.401514
$725$	−42.5840	−1.58153
$726$	0	0
$727$	−19.6715	−0.729574	−0.364787	−	0.931091i	$-0.618858\pi$
$727$	−19.6715	−0.729574	−0.364787	+	0.931091i	$0.618858\pi$
$728$	0	0
$729$	0	0
$730$	2.76393	0.102298
$731$	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	$0.462736\pi$
$734$	−11.3475	−0.418845
$735$	0	0
$736$	33.8885	1.24915
$737$	3.65375	0.134588
$738$	0	0
$739$	−48.3607	−1.77898	−0.889488	−	0.456958i	$-0.848939\pi$
$739$	−48.3607	−1.77898	−0.889488	+	0.456958i	$0.848939\pi$
$740$	−2.54102	−0.0934097
$741$	0	0
$742$	0	0
$743$	3.78127	0.138721	0.0693607	−	0.997592i	$-0.477904\pi$
$743$	3.78127	0.138721	0.0693607	+	0.997592i	$0.477904\pi$
$744$	0	0
$745$	−1.52313	−0.0558031
$746$	−15.4288	−0.564890
$747$	0	0
$748$	−3.33851	−0.122068
$749$	0	0
$750$	0	0
$751$	13.3475	0.487058	0.243529	−	0.969894i	$-0.421695\pi$
$751$	13.3475	0.487058	0.243529	+	0.969894i	$0.421695\pi$
$752$	0	0
$753$	0	0
$754$	6.57959	0.239615
$755$	2.45898	0.0894915
$756$	0	0
$757$	9.36068	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$757$	9.36068	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$758$	−4.27274	−0.155193
$759$	0	0
$760$	2.69505	0.0977597
$761$	26.5279	0.961634	0.480817	−	0.876821i	$-0.340340\pi$
$761$	26.5279	0.961634	0.480817	+	0.876821i	$0.340340\pi$
$762$	0	0
$763$	0	0
$764$	5.24419	0.189728
$765$	0	0
$766$	9.77198	0.353076
$767$	12.1877	0.440074
$768$	0	0
$769$	47.6706	1.71905	0.859523	−	0.511097i	$-0.170761\pi$
$769$	47.6706	1.71905	0.859523	+	0.511097i	$0.170761\pi$
$770$	0	0
$771$	0	0
$772$	6.40325	0.230458
$773$	−48.5967	−1.74790	−0.873952	−	0.486013i	$-0.838451\pi$
$773$	−48.5967	−1.74790	−0.873952	+	0.486013i	$0.838451\pi$
$774$	0	0
$775$	−32.2905	−1.15991
$776$	−14.4721	−0.519519
$777$	0	0
$778$	−0.832816	−0.0298579
$779$	−35.1490	−1.25934
$780$	0	0
$781$	−3.63932	−0.130225
$782$	−26.1803	−0.936207
$783$	0	0
$784$	0	0
$785$	4.13373	0.147539
$786$	0	0
$787$	−34.5300	−1.23086	−0.615431	−	0.788191i	$-0.711018\pi$
$787$	−34.5300	−1.23086	−0.615431	+	0.788191i	$0.711018\pi$
$788$	26.7912	0.954399
$789$	0	0
$790$	0.521577	0.0185569
$791$	0	0
$792$	0	0
$793$	0.472136	0.0167660
$794$	−14.4033	−0.511152
$795$	0	0
$796$	32.2905	1.14451
$797$	−42.0689	−1.49016	−0.745078	−	0.666977i	$-0.767588\pi$
$797$	−42.0689	−1.49016	−0.745078	+	0.666977i	$0.767588\pi$
$798$	0	0
$799$	37.3607	1.32173
$800$	27.9690	0.988854
$801$	0	0
$802$	−26.2492	−0.926892
$803$	7.23607	0.255355
$804$	0	0
$805$	0	0
$806$	4.98915	0.175735
$807$	0	0
$808$	−13.4744	−0.474029
$809$	2.08191	0.0731962	0.0365981	−	0.999330i	$-0.488348\pi$
$809$	2.08191	0.0731962	0.0365981	+	0.999330i	$0.488348\pi$
$810$	0	0
$811$	−28.4906	−1.00044	−0.500220	−	0.865898i	$-0.666748\pi$
$811$	−28.4906	−1.00044	−0.500220	+	0.865898i	$0.666748\pi$
$812$	0	0
$813$	0	0
$814$	4.11146	0.144106
$815$	4.05573	0.142066
$816$	0	0
$817$	−9.02546	−0.315761
$818$	−10.6950	−0.373944
$819$	0	0
$820$	2.54102	0.0887363
$821$	32.0354	1.11804	0.559022	−	0.829153i	$-0.311177\pi$
$821$	32.0354	1.11804	0.559022	+	0.829153i	$0.311177\pi$
$822$	0	0
$823$	−36.4164	−1.26940	−0.634698	−	0.772760i	$-0.718876\pi$
$823$	−36.4164	−1.26940	−0.634698	+	0.772760i	$0.718876\pi$
$824$	−47.7771	−1.66439
$825$	0	0
$826$	0	0
$827$	7.45363	0.259188	0.129594	−	0.991567i	$-0.458633\pi$
$827$	7.45363	0.259188	0.129594	+	0.991567i	$0.458633\pi$
$828$	0	0
$829$	32.3206	1.12254	0.561270	−	0.827633i	$-0.310313\pi$
$829$	32.3206	1.12254	0.561270	+	0.827633i	$0.310313\pi$
$830$	0.874032	0.0303381
$831$	0	0
$832$	−4.32145	−0.149819
$833$	0	0
$834$	0	0
$835$	1.94427	0.0672843
$836$	2.69505	0.0932102
$837$	0	0
$838$	−13.6808	−0.472594
$839$	47.1803	1.62885	0.814423	−	0.580271i	$-0.197054\pi$
$839$	47.1803	1.62885	0.814423	+	0.580271i	$0.197054\pi$
$840$	0	0
$841$	45.1803	1.55794
$842$	−3.90879	−0.134706
$843$	0	0
$844$	−11.6393	−0.400642
$845$	−2.88854	−0.0993689
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−21.6073	−0.741123
$851$	−52.1683	−1.78831
$852$	0	0
$853$	−35.5316	−1.21658	−0.608289	−	0.793716i	$-0.708144\pi$
$853$	−35.5316	−1.21658	−0.608289	+	0.793716i	$0.708144\pi$
$854$	0	0
$855$	0	0
$856$	−28.0000	−0.957020
$857$	−19.1803	−0.655188	−0.327594	−	0.944819i	$-0.606238\pi$
$857$	−19.1803	−0.655188	−0.327594	+	0.944819i	$0.606238\pi$
$858$	0	0
$859$	3.31990	0.113274	0.0566368	−	0.998395i	$-0.481962\pi$
$859$	3.31990	0.113274	0.0566368	+	0.998395i	$0.481962\pi$
$860$	0.652476	0.0222492
$861$	0	0
$862$	10.1115	0.344398
$863$	−9.07417	−0.308888	−0.154444	−	0.988002i	$-0.549359\pi$
$863$	−9.07417	−0.308888	−0.154444	+	0.988002i	$0.549359\pi$
$864$	0	0
$865$	−2.24922	−0.0764759
$866$	32.7639	1.11336
$867$	0	0
$868$	0	0
$869$	1.36551	0.0463216
$870$	0	0
$871$	5.91189	0.200317
$872$	40.2657	1.36357
$873$	0	0
$874$	21.1344	0.714881
$875$	0	0
$876$	0	0
$877$	26.0132	0.878402	0.439201	−	0.898389i	$-0.355262\pi$
$877$	26.0132	0.878402	0.439201	+	0.898389i	$0.355262\pi$
$878$	−16.9443	−0.571841
$879$	0	0
$880$	0	0
$881$	36.8328	1.24093	0.620465	−	0.784234i	$-0.286944\pi$
$881$	36.8328	1.24093	0.620465	+	0.784234i	$0.286944\pi$
$882$	0	0
$883$	−18.8885	−0.635650	−0.317825	−	0.948149i	$-0.602952\pi$
$883$	−18.8885	−0.635650	−0.317825	+	0.948149i	$0.602952\pi$
$884$	−5.40182	−0.181683
$885$	0	0
$886$	22.5836	0.758711
$887$	46.1935	1.55103	0.775513	−	0.631332i	$-0.217491\pi$
$887$	46.1935	1.55103	0.775513	+	0.631332i	$0.217491\pi$
$888$	0	0
$889$	0	0
$890$	2.41577	0.0809766
$891$	0	0
$892$	18.8762	0.632024
$893$	−30.1599	−1.00926
$894$	0	0
$895$	4.21254	0.140810
$896$	0	0
$897$	0	0
$898$	5.48529	0.183047
$899$	56.2492	1.87602
$900$	0	0
$901$	15.8114	0.526754
$902$	−4.11146	−0.136897
$903$	0	0
$904$	−23.4164	−0.778818
$905$	2.06331	0.0685867
$906$	0	0
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	−12.5836	−0.417601
$909$	0	0
$910$	0	0
$911$	32.3206	1.07083	0.535414	−	0.844590i	$-0.320155\pi$
$911$	32.3206	1.07083	0.535414	+	0.844590i	$0.320155\pi$
$912$	0	0
$913$	2.28825	0.0757299
$914$	18.2085	0.602285
$915$	0	0
$916$	28.7943	0.951392
$917$	0	0
$918$	0	0
$919$	−15.9443	−0.525953	−0.262976	−	0.964802i	$-0.584704\pi$
$919$	−15.9443	−0.525953	−0.262976	+	0.964802i	$0.584704\pi$
$920$	−4.00000	−0.131876
$921$	0	0
$922$	27.5077	0.905916
$923$	−5.88854	−0.193824
$924$	0	0
$925$	−43.0557	−1.41566
$926$	−12.9529	−0.425657
$927$	0	0
$928$	−48.7214	−1.59936
$929$	−35.8328	−1.17564	−0.587818	−	0.808993i	$-0.700013\pi$
$929$	−35.8328	−1.17564	−0.587818	+	0.808993i	$0.700013\pi$
$930$	0	0
$931$	0	0
$932$	−19.3863	−0.635020
$933$	0	0
$934$	10.1359	0.331658
$935$	0.637598	0.0208517
$936$	0	0
$937$	−14.6823	−0.479650	−0.239825	−	0.970816i	$-0.577090\pi$
$937$	−14.6823	−0.479650	−0.239825	+	0.970816i	$0.577090\pi$
$938$	0	0
$939$	0	0
$940$	2.18034	0.0711148
$941$	−7.29180	−0.237706	−0.118853	−	0.992912i	$-0.537922\pi$
$941$	−7.29180	−0.237706	−0.118853	+	0.992912i	$0.537922\pi$
$942$	0	0
$943$	52.1683	1.69883
$944$	0	0
$945$	0	0
$946$	−1.05573	−0.0343247
$947$	−35.1189	−1.14121	−0.570606	−	0.821224i	$-0.693291\pi$
$947$	−35.1189	−1.14121	−0.570606	+	0.821224i	$0.693291\pi$
$948$	0	0
$949$	11.7082	0.380064
$950$	17.4427	0.565917
$951$	0	0
$952$	0	0
$953$	−27.5378	−0.892036	−0.446018	−	0.895024i	$-0.647158\pi$
$953$	−27.5378	−0.892036	−0.446018	+	0.895024i	$0.647158\pi$
$954$	0	0
$955$	−1.00155	−0.0324094
$956$	12.0788	0.390657
$957$	0	0
$958$	23.6591	0.764390
$959$	0	0
$960$	0	0
$961$	11.6525	0.375886
$962$	6.65248	0.214484
$963$	0	0
$964$	−3.24109	−0.104388
$965$	−1.22291	−0.0393669
$966$	0	0
$967$	14.4721	0.465393	0.232696	−	0.972549i	$-0.425245\pi$
$967$	14.4721	0.465393	0.232696	+	0.972549i	$0.425245\pi$
$968$	−30.2874	−0.973473
$969$	0	0
$970$	1.05573	0.0338974
$971$	29.4721	0.945806	0.472903	−	0.881115i	$-0.343206\pi$
$971$	29.4721	0.945806	0.472903	+	0.881115i	$0.343206\pi$
$972$	0	0
$973$	0	0
$974$	−8.64290	−0.276937
$975$	0	0
$976$	0	0
$977$	33.1158	1.05947	0.529734	−	0.848164i	$-0.322292\pi$
$977$	33.1158	1.05947	0.529734	+	0.848164i	$0.322292\pi$
$978$	0	0
$979$	6.32456	0.202134
$980$	0	0
$981$	0	0
$982$	−34.6525	−1.10580
$983$	43.9443	1.40160	0.700802	−	0.713356i	$-0.252826\pi$
$983$	43.9443	1.40160	0.700802	+	0.713356i	$0.252826\pi$
$984$	0	0
$985$	−5.11667	−0.163031
$986$	37.6393	1.19868
$987$	0	0
$988$	4.36068	0.138732
$989$	13.3956	0.425956
$990$	0	0
$991$	−40.1246	−1.27460	−0.637300	−	0.770616i	$-0.719949\pi$
$991$	−40.1246	−1.27460	−0.637300	+	0.770616i	$0.719949\pi$
$992$	−36.9443	−1.17298
$993$	0	0
$994$	0	0
$995$	−6.16693	−0.195505
$996$	0	0
$997$	17.6082	0.557656	0.278828	−	0.960341i	$-0.410054\pi$
$997$	17.6082	0.557656	0.278828	+	0.960341i	$0.410054\pi$
$998$	−0.971448	−0.0307507
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.bb.1.2	✓	4
3.2	odd	2		1323.2.a.be.1.3	yes	4
7.6	odd	2		1323.2.a.be.1.2	yes	4
21.20	even	2	inner	1323.2.a.bb.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.a.bb.1.2	✓	4	1.1	even	1	trivial
1323.2.a.bb.1.3	yes	4	21.20	even	2	inner
1323.2.a.be.1.2	yes	4	7.6	odd	2
1323.2.a.be.1.3	yes	4	3.2	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-0.874032 q^{2} -1.23607 q^{4} +0.236068 q^{5} +2.82843 q^{8} -0.206331 q^{10} -0.540182 q^{11} -0.874032 q^{13} -5.00000 q^{17} +4.03631 q^{19} -0.291796 q^{20} +0.472136 q^{22} -5.99070 q^{23} -4.94427 q^{25} +0.763932 q^{26} +8.61280 q^{29} +6.53089 q^{31} -5.65685 q^{32} +4.37016 q^{34} +8.70820 q^{37} -3.52786 q^{38} +0.667701 q^{40} -8.70820 q^{41} -2.23607 q^{43} +0.667701 q^{44} +5.23607 q^{46} -7.47214 q^{47} +4.32145 q^{50} +1.08036 q^{52} -3.16228 q^{53} -0.127520 q^{55} -7.52786 q^{58} -13.9443 q^{59} -0.540182 q^{61} -5.70820 q^{62} +4.94427 q^{64} -0.206331 q^{65} -6.76393 q^{67} +6.18034 q^{68} +6.73722 q^{71} -13.3956 q^{73} -7.61125 q^{74} -4.98915 q^{76} -2.52786 q^{79} +7.61125 q^{82} -4.23607 q^{83} -1.18034 q^{85} +1.95440 q^{86} -1.52786 q^{88} -11.7082 q^{89} +7.40492 q^{92} +6.53089 q^{94} +0.952843 q^{95} -5.11667 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 8 q^{5} - 20 q^{17} - 28 q^{20} - 16 q^{22} + 16 q^{25} + 12 q^{26} + 8 q^{37} - 32 q^{38} - 8 q^{41} + 12 q^{46} - 12 q^{47} - 48 q^{58} - 20 q^{59} + 4 q^{62} - 16 q^{64} - 36 q^{67} - 20 q^{68}+ \cdots - 20 q^{89}+O(q^{100}) \) 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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