LMFDB - Embedded newform 5445.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.27648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 3 $ x^4 - 6*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.741964$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} -3.30136 q^{7} -2.55940 q^{8} +O(q^{10})$	$q+0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} -3.30136 q^{7} -2.55940 q^{8} -0.741964 q^{10} +1.81743 q^{13} -2.44949 q^{14} +1.00000 q^{16} +3.30136 q^{17} -1.48393 q^{19} +1.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +1.34847 q^{26} +4.78529 q^{28} +8.08665 q^{29} -2.00000 q^{31} +5.86076 q^{32} +2.44949 q^{34} +3.30136 q^{35} +2.89898 q^{37} -1.10102 q^{38} +2.55940 q^{40} -5.11879 q^{41} +3.30136 q^{43} +3.63487 q^{46} -9.79796 q^{47} +3.89898 q^{49} +0.741964 q^{50} -2.63435 q^{52} -1.10102 q^{53} +8.44949 q^{56} +6.00000 q^{58} -1.10102 q^{59} +9.57058 q^{61} -1.48393 q^{62} +2.34847 q^{64} -1.81743 q^{65} -0.898979 q^{67} -4.78529 q^{68} +2.44949 q^{70} -10.8990 q^{71} -11.3880 q^{73} +2.15094 q^{74} +2.15094 q^{76} -17.6572 q^{79} -1.00000 q^{80} -3.79796 q^{82} -4.78529 q^{83} -3.30136 q^{85} +2.44949 q^{86} +15.7980 q^{89} -6.00000 q^{91} -7.10102 q^{92} -7.26973 q^{94} +1.48393 q^{95} -12.6969 q^{97} +2.89290 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^4 - 4 * q^5	$ 4 q + 4 q^{4} - 4 q^{5} + 4 q^{16} - 4 q^{20} + 4 q^{25} - 24 q^{26} - 8 q^{31} - 8 q^{37} - 24 q^{38} - 4 q^{49} - 24 q^{53} + 24 q^{56} + 24 q^{58} - 24 q^{59} - 20 q^{64} + 16 q^{67} - 24 q^{71} - 4 q^{80} + 24 q^{82} + 24 q^{89} - 24 q^{91} - 48 q^{92} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^5 + 4 * q^16 - 4 * q^20 + 4 * q^25 - 24 * q^26 - 8 * q^31 - 8 * q^37 - 24 * q^38 - 4 * q^49 - 24 * q^53 + 24 * q^56 + 24 * q^58 - 24 * q^59 - 20 * q^64 + 16 * q^67 - 24 * q^71 - 4 * q^80 + 24 * q^82 + 24 * q^89 - 24 * q^91 - 48 * q^92 + 8 * 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 $

$2$	0.741964	0.524648	0.262324	−	0.964980i	$-0.415511\pi$
$2$	0.741964	0.524648	0.262324	+	0.964980i	$0.415511\pi$
$3$	0	0
$3$	0	0
$4$	−1.44949	−0.724745
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.30136	−1.24780	−0.623898	−	0.781505i	$-0.714452\pi$
$7$	−3.30136	−1.24780	−0.623898	+	0.781505i	$0.714452\pi$
$8$	−2.55940	−0.904883
$9$	0	0
$10$	−0.741964	−0.234630
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.81743	0.504065	0.252033	−	0.967719i	$-0.418901\pi$
$13$	1.81743	0.504065	0.252033	+	0.967719i	$0.418901\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	3.30136	0.800697	0.400349	−	0.916363i	$-0.368889\pi$
$17$	3.30136	0.800697	0.400349	+	0.916363i	$0.368889\pi$
$18$	0	0
$19$	−1.48393	−0.340436	−0.170218	−	0.985406i	$-0.554447\pi$
$19$	−1.48393	−0.340436	−0.170218	+	0.985406i	$0.554447\pi$
$20$	1.44949	0.324116
$21$	0	0
$22$	0	0
$23$	4.89898	1.02151	0.510754	−	0.859727i	$-0.329366\pi$
$23$	4.89898	1.02151	0.510754	+	0.859727i	$0.329366\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.34847	0.264457
$27$	0	0
$28$	4.78529	0.904334
$29$	8.08665	1.50165	0.750826	−	0.660500i	$-0.229655\pi$
$29$	8.08665	1.50165	0.750826	+	0.660500i	$0.229655\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.86076	1.03605
$33$	0	0
$34$	2.44949	0.420084
$35$	3.30136	0.558032
$36$	0	0
$37$	2.89898	0.476589	0.238295	−	0.971193i	$-0.423412\pi$
$37$	2.89898	0.476589	0.238295	+	0.971193i	$0.423412\pi$
$38$	−1.10102	−0.178609
$39$	0	0
$40$	2.55940	0.404676
$41$	−5.11879	−0.799421	−0.399711	−	0.916641i	$-0.630889\pi$
$41$	−5.11879	−0.799421	−0.399711	+	0.916641i	$0.630889\pi$
$42$	0	0
$43$	3.30136	0.503453	0.251726	−	0.967798i	$-0.419002\pi$
$43$	3.30136	0.503453	0.251726	+	0.967798i	$0.419002\pi$
$44$	0	0
$45$	0	0
$46$	3.63487	0.535932
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	3.89898	0.556997
$50$	0.741964	0.104930
$51$	0	0
$52$	−2.63435	−0.365319
$53$	−1.10102	−0.151237	−0.0756184	−	0.997137i	$-0.524093\pi$
$53$	−1.10102	−0.151237	−0.0756184	+	0.997137i	$0.524093\pi$
$54$	0	0
$55$	0	0
$56$	8.44949	1.12911
$57$	0	0
$58$	6.00000	0.787839
$59$	−1.10102	−0.143341	−0.0716703	−	0.997428i	$-0.522833\pi$
$59$	−1.10102	−0.143341	−0.0716703	+	0.997428i	$0.522833\pi$
$60$	0	0
$61$	9.57058	1.22539	0.612693	−	0.790321i	$-0.290086\pi$
$61$	9.57058	1.22539	0.612693	+	0.790321i	$0.290086\pi$
$62$	−1.48393	−0.188459
$63$	0	0
$64$	2.34847	0.293559
$65$	−1.81743	−0.225425
$66$	0	0
$67$	−0.898979	−0.109828	−0.0549139	−	0.998491i	$-0.517488\pi$
$67$	−0.898979	−0.109828	−0.0549139	+	0.998491i	$0.517488\pi$
$68$	−4.78529	−0.580301
$69$	0	0
$70$	2.44949	0.292770
$71$	−10.8990	−1.29347	−0.646735	−	0.762714i	$-0.723866\pi$
$71$	−10.8990	−1.29347	−0.646735	+	0.762714i	$0.723866\pi$
$72$	0	0
$73$	−11.3880	−1.33287	−0.666433	−	0.745565i	$-0.732180\pi$
$73$	−11.3880	−1.33287	−0.666433	+	0.745565i	$0.732180\pi$
$74$	2.15094	0.250041
$75$	0	0
$76$	2.15094	0.246729
$77$	0	0
$78$	0	0
$79$	−17.6572	−1.98659	−0.993296	−	0.115595i	$-0.963123\pi$
$79$	−17.6572	−1.98659	−0.993296	+	0.115595i	$0.963123\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−3.79796	−0.419414
$83$	−4.78529	−0.525254	−0.262627	−	0.964897i	$-0.584589\pi$
$83$	−4.78529	−0.525254	−0.262627	+	0.964897i	$0.584589\pi$
$84$	0	0
$85$	−3.30136	−0.358083
$86$	2.44949	0.264135
$87$	0	0
$88$	0	0
$89$	15.7980	1.67458	0.837290	−	0.546759i	$-0.184139\pi$
$89$	15.7980	1.67458	0.837290	+	0.546759i	$0.184139\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−7.10102	−0.740333
$93$	0	0
$94$	−7.26973	−0.749815
$95$	1.48393	0.152248
$96$	0	0
$97$	−12.6969	−1.28918	−0.644589	−	0.764529i	$-0.722972\pi$
$97$	−12.6969	−1.28918	−0.644589	+	0.764529i	$0.722972\pi$
$98$	2.89290	0.292227
$99$	0	0
$100$	−1.44949	−0.144949
$101$	8.75366	0.871022	0.435511	−	0.900184i	$-0.356568\pi$
$101$	8.75366	0.871022	0.435511	+	0.900184i	$0.356568\pi$
$102$	0	0
$103$	3.10102	0.305553	0.152776	−	0.988261i	$-0.451179\pi$
$103$	3.10102	0.305553	0.152776	+	0.988261i	$0.451179\pi$
$104$	−4.65153	−0.456120
$105$	0	0
$106$	−0.816917	−0.0793460
$107$	8.42015	0.814007	0.407003	−	0.913427i	$-0.366574\pi$
$107$	8.42015	0.814007	0.407003	+	0.913427i	$0.366574\pi$
$108$	0	0
$109$	−10.2376	−0.980583	−0.490291	−	0.871559i	$-0.663110\pi$
$109$	−10.2376	−0.980583	−0.490291	+	0.871559i	$0.663110\pi$
$110$	0	0
$111$	0	0
$112$	−3.30136	−0.311949
$113$	−10.8990	−1.02529	−0.512645	−	0.858601i	$-0.671334\pi$
$113$	−10.8990	−1.02529	−0.512645	+	0.858601i	$0.671334\pi$
$114$	0	0
$115$	−4.89898	−0.456832
$116$	−11.7215	−1.08832
$117$	0	0
$118$	−0.816917	−0.0752033
$119$	−10.8990	−0.999108
$120$	0	0
$121$	0	0
$122$	7.10102	0.642896
$123$	0	0
$124$	2.89898	0.260336
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.63435	0.233761	0.116880	−	0.993146i	$-0.462711\pi$
$127$	2.63435	0.233761	0.116880	+	0.993146i	$0.462711\pi$
$128$	−9.97903	−0.882030
$129$	0	0
$130$	−1.34847	−0.118269
$131$	−16.8403	−1.47134	−0.735672	−	0.677338i	$-0.763134\pi$
$131$	−16.8403	−1.47134	−0.735672	+	0.677338i	$0.763134\pi$
$132$	0	0
$133$	4.89898	0.424795
$134$	−0.667010	−0.0576209
$135$	0	0
$136$	−8.44949	−0.724538
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	18.3242	1.55424	0.777121	−	0.629352i	$-0.216679\pi$
$139$	18.3242	1.55424	0.777121	+	0.629352i	$0.216679\pi$
$140$	−4.78529	−0.404431
$141$	0	0
$142$	−8.08665	−0.678616
$143$	0	0
$144$	0	0
$145$	−8.08665	−0.671560
$146$	−8.44949	−0.699285
$147$	0	0
$148$	−4.20204	−0.345406
$149$	14.6894	1.20340	0.601700	−	0.798722i	$-0.294490\pi$
$149$	14.6894	1.20340	0.601700	+	0.798722i	$0.294490\pi$
$150$	0	0
$151$	−14.6894	−1.19540	−0.597702	−	0.801718i	$-0.703919\pi$
$151$	−14.6894	−1.19540	−0.597702	+	0.801718i	$0.703919\pi$
$152$	3.79796	0.308055
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−11.7980	−0.941580	−0.470790	−	0.882245i	$-0.656031\pi$
$157$	−11.7980	−0.941580	−0.470790	+	0.882245i	$0.656031\pi$
$158$	−13.1010	−1.04226
$159$	0	0
$160$	−5.86076	−0.463334
$161$	−16.1733	−1.27463
$162$	0	0
$163$	22.6969	1.77776	0.888881	−	0.458139i	$-0.151484\pi$
$163$	22.6969	1.77776	0.888881	+	0.458139i	$0.151484\pi$
$164$	7.41964	0.579376
$165$	0	0
$166$	−3.55051	−0.275573
$167$	−21.6256	−1.67344	−0.836719	−	0.547632i	$-0.815529\pi$
$167$	−21.6256	−1.67344	−0.836719	+	0.547632i	$0.815529\pi$
$168$	0	0
$169$	−9.69694	−0.745918
$170$	−2.44949	−0.187867
$171$	0	0
$172$	−4.78529	−0.364875
$173$	22.4425	1.70627	0.853136	−	0.521688i	$-0.174698\pi$
$173$	22.4425	1.70627	0.853136	+	0.521688i	$0.174698\pi$
$174$	0	0
$175$	−3.30136	−0.249559
$176$	0	0
$177$	0	0
$178$	11.7215	0.878565
$179$	−2.20204	−0.164588	−0.0822941	−	0.996608i	$-0.526225\pi$
$179$	−2.20204	−0.164588	−0.0822941	+	0.996608i	$0.526225\pi$
$180$	0	0
$181$	12.8990	0.958774	0.479387	−	0.877604i	$-0.340859\pi$
$181$	12.8990	0.958774	0.479387	+	0.877604i	$0.340859\pi$
$182$	−4.45178	−0.329988
$183$	0	0
$184$	−12.5384	−0.924345
$185$	−2.89898	−0.213137
$186$	0	0
$187$	0	0
$188$	14.2020	1.03579
$189$	0	0
$190$	1.10102	0.0798764
$191$	−19.5959	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$192$	0	0
$193$	17.3237	1.24699	0.623494	−	0.781828i	$-0.285712\pi$
$193$	17.3237	1.24699	0.623494	+	0.781828i	$0.285712\pi$
$194$	−9.42067	−0.676365
$195$	0	0
$196$	−5.65153	−0.403681
$197$	−2.63435	−0.187690	−0.0938448	−	0.995587i	$-0.529916\pi$
$197$	−2.63435	−0.187690	−0.0938448	+	0.995587i	$0.529916\pi$
$198$	0	0
$199$	1.79796	0.127454	0.0637270	−	0.997967i	$-0.479701\pi$
$199$	1.79796	0.127454	0.0637270	+	0.997967i	$0.479701\pi$
$200$	−2.55940	−0.180977
$201$	0	0
$202$	6.49490	0.456979
$203$	−26.6969	−1.87376
$204$	0	0
$205$	5.11879	0.357512
$206$	2.30084	0.160307
$207$	0	0
$208$	1.81743	0.126016
$209$	0	0
$210$	0	0
$211$	−4.45178	−0.306473	−0.153237	−	0.988190i	$-0.548970\pi$
$211$	−4.45178	−0.306473	−0.153237	+	0.988190i	$0.548970\pi$
$212$	1.59592	0.109608
$213$	0	0
$214$	6.24745	0.427067
$215$	−3.30136	−0.225151
$216$	0	0
$217$	6.60272	0.448222
$218$	−7.59592	−0.514460
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	17.7980	1.19184	0.595920	−	0.803044i	$-0.296788\pi$
$223$	17.7980	1.19184	0.595920	+	0.803044i	$0.296788\pi$
$224$	−19.3485	−1.29277
$225$	0	0
$226$	−8.08665	−0.537916
$227$	11.3880	0.755849	0.377924	−	0.925836i	$-0.376638\pi$
$227$	11.3880	0.755849	0.377924	+	0.925836i	$0.376638\pi$
$228$	0	0
$229$	17.5959	1.16277	0.581385	−	0.813628i	$-0.302511\pi$
$229$	17.5959	1.16277	0.581385	+	0.813628i	$0.302511\pi$
$230$	−3.63487	−0.239676
$231$	0	0
$232$	−20.6969	−1.35882
$233$	0.333505	0.0218486	0.0109243	−	0.999940i	$-0.496523\pi$
$233$	0.333505	0.0218486	0.0109243	+	0.999940i	$0.496523\pi$
$234$	0	0
$235$	9.79796	0.639148
$236$	1.59592	0.103885
$237$	0	0
$238$	−8.08665	−0.524180
$239$	−15.5063	−1.00302	−0.501509	−	0.865152i	$-0.667222\pi$
$239$	−15.5063	−1.00302	−0.501509	+	0.865152i	$0.667222\pi$
$240$	0	0
$241$	12.5384	0.807671	0.403836	−	0.914832i	$-0.367677\pi$
$241$	12.5384	0.807671	0.403836	+	0.914832i	$0.367677\pi$
$242$	0	0
$243$	0	0
$244$	−13.8725	−0.888093
$245$	−3.89898	−0.249097
$246$	0	0
$247$	−2.69694	−0.171602
$248$	5.11879	0.325044
$249$	0	0
$250$	−0.741964	−0.0469259
$251$	−22.8990	−1.44537	−0.722685	−	0.691177i	$-0.757092\pi$
$251$	−22.8990	−1.44537	−0.722685	+	0.691177i	$0.757092\pi$
$252$	0	0
$253$	0	0
$254$	1.95459	0.122642
$255$	0	0
$256$	−12.1010	−0.756314
$257$	−13.5959	−0.848090	−0.424045	−	0.905641i	$-0.639390\pi$
$257$	−13.5959	−0.848090	−0.424045	+	0.905641i	$0.639390\pi$
$258$	0	0
$259$	−9.57058	−0.594687
$260$	2.63435	0.163375
$261$	0	0
$262$	−12.4949	−0.771937
$263$	−17.3237	−1.06823	−0.534113	−	0.845413i	$-0.679354\pi$
$263$	−17.3237	−1.06823	−0.534113	+	0.845413i	$0.679354\pi$
$264$	0	0
$265$	1.10102	0.0676352
$266$	3.63487	0.222868
$267$	0	0
$268$	1.30306	0.0795972
$269$	−31.5959	−1.92644	−0.963219	−	0.268719i	$-0.913400\pi$
$269$	−31.5959	−1.92644	−0.963219	+	0.268719i	$0.913400\pi$
$270$	0	0
$271$	−21.9591	−1.33392	−0.666960	−	0.745093i	$-0.732405\pi$
$271$	−21.9591	−1.33392	−0.666960	+	0.745093i	$0.732405\pi$
$272$	3.30136	0.200174
$273$	0	0
$274$	−13.3553	−0.806826
$275$	0	0
$276$	0	0
$277$	2.48444	0.149276	0.0746379	−	0.997211i	$-0.476220\pi$
$277$	2.48444	0.149276	0.0746379	+	0.997211i	$0.476220\pi$
$278$	13.5959	0.815429
$279$	0	0
$280$	−8.44949	−0.504954
$281$	5.11879	0.305362	0.152681	−	0.988276i	$-0.451209\pi$
$281$	5.11879	0.305362	0.152681	+	0.988276i	$0.451209\pi$
$282$	0	0
$283$	5.60221	0.333017	0.166508	−	0.986040i	$-0.446751\pi$
$283$	5.60221	0.333017	0.166508	+	0.986040i	$0.446751\pi$
$284$	15.7980	0.937436
$285$	0	0
$286$	0	0
$287$	16.8990	0.997515
$288$	0	0
$289$	−6.10102	−0.358884
$290$	−6.00000	−0.352332
$291$	0	0
$292$	16.5068	0.965987
$293$	−5.60221	−0.327284	−0.163642	−	0.986520i	$-0.552324\pi$
$293$	−5.60221	−0.327284	−0.163642	+	0.986520i	$0.552324\pi$
$294$	0	0
$295$	1.10102	0.0641039
$296$	−7.41964	−0.431258
$297$	0	0
$298$	10.8990	0.631361
$299$	8.90357	0.514906
$300$	0	0
$301$	−10.8990	−0.628207
$302$	−10.8990	−0.627166
$303$	0	0
$304$	−1.48393	−0.0851091
$305$	−9.57058	−0.548010
$306$	0	0
$307$	15.8398	0.904025	0.452012	−	0.892012i	$-0.350706\pi$
$307$	15.8398	0.904025	0.452012	+	0.892012i	$0.350706\pi$
$308$	0	0
$309$	0	0
$310$	1.48393	0.0842814
$311$	−13.1010	−0.742891	−0.371445	−	0.928455i	$-0.621138\pi$
$311$	−13.1010	−0.742891	−0.371445	+	0.928455i	$0.621138\pi$
$312$	0	0
$313$	−30.8990	−1.74651	−0.873257	−	0.487260i	$-0.837996\pi$
$313$	−30.8990	−1.74651	−0.873257	+	0.487260i	$0.837996\pi$
$314$	−8.75366	−0.493998
$315$	0	0
$316$	25.5940	1.43977
$317$	−20.6969	−1.16246	−0.581228	−	0.813741i	$-0.697428\pi$
$317$	−20.6969	−1.16246	−0.581228	+	0.813741i	$0.697428\pi$
$318$	0	0
$319$	0	0
$320$	−2.34847	−0.131283
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−4.89898	−0.272587
$324$	0	0
$325$	1.81743	0.100813
$326$	16.8403	0.932698
$327$	0	0
$328$	13.1010	0.723383
$329$	32.3466	1.78333
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	6.93623	0.380675
$333$	0	0
$334$	−16.0454	−0.877966
$335$	0.898979	0.0491165
$336$	0	0
$337$	14.3559	0.782014	0.391007	−	0.920388i	$-0.372127\pi$
$337$	14.3559	0.782014	0.391007	+	0.920388i	$0.372127\pi$
$338$	−7.19478	−0.391344
$339$	0	0
$340$	4.78529	0.259519
$341$	0	0
$342$	0	0
$343$	10.2376	0.552778
$344$	−8.44949	−0.455566
$345$	0	0
$346$	16.6515	0.895192
$347$	7.75314	0.416211	0.208105	−	0.978106i	$-0.433270\pi$
$347$	7.75314	0.416211	0.208105	+	0.978106i	$0.433270\pi$
$348$	0	0
$349$	9.57058	0.512301	0.256151	−	0.966637i	$-0.417546\pi$
$349$	9.57058	0.512301	0.256151	+	0.966637i	$0.417546\pi$
$350$	−2.44949	−0.130931
$351$	0	0
$352$	0	0
$353$	1.10102	0.0586014	0.0293007	−	0.999571i	$-0.490672\pi$
$353$	1.10102	0.0586014	0.0293007	+	0.999571i	$0.490672\pi$
$354$	0	0
$355$	10.8990	0.578458
$356$	−22.8990	−1.21364
$357$	0	0
$358$	−1.63383	−0.0863508
$359$	31.6796	1.67198	0.835992	−	0.548741i	$-0.184893\pi$
$359$	31.6796	1.67198	0.835992	+	0.548741i	$0.184893\pi$
$360$	0	0
$361$	−16.7980	−0.884103
$362$	9.57058	0.503018
$363$	0	0
$364$	8.69694	0.455843
$365$	11.3880	0.596076
$366$	0	0
$367$	15.5959	0.814100	0.407050	−	0.913406i	$-0.366557\pi$
$367$	15.5959	0.814100	0.407050	+	0.913406i	$0.366557\pi$
$368$	4.89898	0.255377
$369$	0	0
$370$	−2.15094	−0.111822
$371$	3.63487	0.188713
$372$	0	0
$373$	15.0229	0.777855	0.388927	−	0.921268i	$-0.372846\pi$
$373$	15.0229	0.777855	0.388927	+	0.921268i	$0.372846\pi$
$374$	0	0
$375$	0	0
$376$	25.0769	1.29324
$377$	14.6969	0.756931
$378$	0	0
$379$	15.5959	0.801108	0.400554	−	0.916273i	$-0.368818\pi$
$379$	15.5959	0.801108	0.400554	+	0.916273i	$0.368818\pi$
$380$	−2.15094	−0.110341
$381$	0	0
$382$	−14.5395	−0.743904
$383$	−7.10102	−0.362845	−0.181423	−	0.983405i	$-0.558070\pi$
$383$	−7.10102	−0.362845	−0.181423	+	0.983405i	$0.558070\pi$
$384$	0	0
$385$	0	0
$386$	12.8536	0.654230
$387$	0	0
$388$	18.4041	0.934326
$389$	−7.59592	−0.385128	−0.192564	−	0.981284i	$-0.561680\pi$
$389$	−7.59592	−0.385128	−0.192564	+	0.981284i	$0.561680\pi$
$390$	0	0
$391$	16.1733	0.817919
$392$	−9.97903	−0.504017
$393$	0	0
$394$	−1.95459	−0.0984709
$395$	17.6572	0.888431
$396$	0	0
$397$	−0.696938	−0.0349783	−0.0174892	−	0.999847i	$-0.505567\pi$
$397$	−0.696938	−0.0349783	−0.0174892	+	0.999847i	$0.505567\pi$
$398$	1.33402	0.0668684
$399$	0	0
$400$	1.00000	0.0500000
$401$	−27.7980	−1.38816	−0.694082	−	0.719896i	$-0.744189\pi$
$401$	−27.7980	−1.38816	−0.694082	+	0.719896i	$0.744189\pi$
$402$	0	0
$403$	−3.63487	−0.181066
$404$	−12.6883	−0.631268
$405$	0	0
$406$	−19.8082	−0.983063
$407$	0	0
$408$	0	0
$409$	23.4430	1.15918	0.579592	−	0.814907i	$-0.303212\pi$
$409$	23.4430	1.15918	0.579592	+	0.814907i	$0.303212\pi$
$410$	3.79796	0.187568
$411$	0	0
$412$	−4.49490	−0.221448
$413$	3.63487	0.178860
$414$	0	0
$415$	4.78529	0.234901
$416$	10.6515	0.522234
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−34.6969	−1.69103	−0.845513	−	0.533955i	$-0.820705\pi$
$421$	−34.6969	−1.69103	−0.845513	+	0.533955i	$0.820705\pi$
$422$	−3.30306	−0.160791
$423$	0	0
$424$	2.81795	0.136852
$425$	3.30136	0.160139
$426$	0	0
$427$	−31.5959	−1.52903
$428$	−12.2049	−0.589947
$429$	0	0
$430$	−2.44949	−0.118125
$431$	−13.2054	−0.636084	−0.318042	−	0.948077i	$-0.603025\pi$
$431$	−13.2054	−0.636084	−0.318042	+	0.948077i	$0.603025\pi$
$432$	0	0
$433$	16.6969	0.802404	0.401202	−	0.915990i	$-0.368593\pi$
$433$	16.6969	0.802404	0.401202	+	0.915990i	$0.368593\pi$
$434$	4.89898	0.235159
$435$	0	0
$436$	14.8393	0.710672
$437$	−7.26973	−0.347758
$438$	0	0
$439$	19.9581	0.952547	0.476273	−	0.879297i	$-0.341987\pi$
$439$	19.9581	0.952547	0.476273	+	0.879297i	$0.341987\pi$
$440$	0	0
$441$	0	0
$442$	4.45178	0.211750
$443$	−26.6969	−1.26841	−0.634205	−	0.773165i	$-0.718672\pi$
$443$	−26.6969	−1.26841	−0.634205	+	0.773165i	$0.718672\pi$
$444$	0	0
$445$	−15.7980	−0.748895
$446$	13.2054	0.625296
$447$	0	0
$448$	−7.75314	−0.366302
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	15.7980	0.743073
$453$	0	0
$454$	8.44949	0.396554
$455$	6.00000	0.281284
$456$	0	0
$457$	−26.8943	−1.25806	−0.629031	−	0.777380i	$-0.716548\pi$
$457$	−26.8943	−1.25806	−0.629031	+	0.777380i	$0.716548\pi$
$458$	13.0555	0.610045
$459$	0	0
$460$	7.10102	0.331087
$461$	24.9270	1.16096	0.580482	−	0.814273i	$-0.302864\pi$
$461$	24.9270	1.16096	0.580482	+	0.814273i	$0.302864\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	8.08665	0.375413
$465$	0	0
$466$	0.247449	0.0114628
$467$	−4.89898	−0.226698	−0.113349	−	0.993555i	$-0.536158\pi$
$467$	−4.89898	−0.226698	−0.113349	+	0.993555i	$0.536158\pi$
$468$	0	0
$469$	2.96786	0.137043
$470$	7.26973	0.335328
$471$	0	0
$472$	2.81795	0.129707
$473$	0	0
$474$	0	0
$475$	−1.48393	−0.0680873
$476$	15.7980	0.724098
$477$	0	0
$478$	−11.5051	−0.526231
$479$	−27.0779	−1.23722	−0.618610	−	0.785698i	$-0.712304\pi$
$479$	−27.0779	−1.23722	−0.618610	+	0.785698i	$0.712304\pi$
$480$	0	0
$481$	5.26870	0.240232
$482$	9.30306	0.423743
$483$	0	0
$484$	0	0
$485$	12.6969	0.576538
$486$	0	0
$487$	−0.898979	−0.0407366	−0.0203683	−	0.999793i	$-0.506484\pi$
$487$	−0.898979	−0.0407366	−0.0203683	+	0.999793i	$0.506484\pi$
$488$	−24.4949	−1.10883
$489$	0	0
$490$	−2.89290	−0.130688
$491$	−11.8714	−0.535750	−0.267875	−	0.963454i	$-0.586321\pi$
$491$	−11.8714	−0.535750	−0.267875	+	0.963454i	$0.586321\pi$
$492$	0	0
$493$	26.6969	1.20237
$494$	−2.00103	−0.0900306
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	35.9815	1.61399
$498$	0	0
$499$	−5.79796	−0.259552	−0.129776	−	0.991543i	$-0.541426\pi$
$499$	−5.79796	−0.259552	−0.129776	+	0.991543i	$0.541426\pi$
$500$	1.44949	0.0648232
$501$	0	0
$502$	−16.9902	−0.758310
$503$	4.11828	0.183625	0.0918125	−	0.995776i	$-0.470734\pi$
$503$	4.11828	0.183625	0.0918125	+	0.995776i	$0.470734\pi$
$504$	0	0
$505$	−8.75366	−0.389533
$506$	0	0
$507$	0	0
$508$	−3.81846	−0.169417
$509$	−9.79796	−0.434287	−0.217143	−	0.976140i	$-0.569674\pi$
$509$	−9.79796	−0.434287	−0.217143	+	0.976140i	$0.569674\pi$
$510$	0	0
$511$	37.5959	1.66315
$512$	10.9795	0.485232
$513$	0	0
$514$	−10.0877	−0.444948
$515$	−3.10102	−0.136647
$516$	0	0
$517$	0	0
$518$	−7.10102	−0.312001
$519$	0	0
$520$	4.65153	0.203983
$521$	35.3939	1.55063	0.775317	−	0.631572i	$-0.217590\pi$
$521$	35.3939	1.55063	0.775317	+	0.631572i	$0.217590\pi$
$522$	0	0
$523$	−36.3150	−1.58794	−0.793971	−	0.607955i	$-0.791990\pi$
$523$	−36.3150	−1.58794	−0.793971	+	0.607955i	$0.791990\pi$
$524$	24.4099	1.06635
$525$	0	0
$526$	−12.8536	−0.560442
$527$	−6.60272	−0.287619
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.816917	0.0354846
$531$	0	0
$532$	−7.10102	−0.307868
$533$	−9.30306	−0.402960
$534$	0	0
$535$	−8.42015	−0.364035
$536$	2.30084	0.0993814
$537$	0	0
$538$	−23.4430	−1.01070
$539$	0	0
$540$	0	0
$541$	−33.6806	−1.44804	−0.724021	−	0.689778i	$-0.757708\pi$
$541$	−33.6806	−1.44804	−0.724021	+	0.689778i	$0.757708\pi$
$542$	−16.2929	−0.699838
$543$	0	0
$544$	19.3485	0.829559
$545$	10.2376	0.438530
$546$	0	0
$547$	28.3782	1.21337	0.606683	−	0.794944i	$-0.292500\pi$
$547$	28.3782	1.21337	0.606683	+	0.794944i	$0.292500\pi$
$548$	26.0908	1.11454
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	58.2929	2.47886
$554$	1.84337	0.0783171
$555$	0	0
$556$	−26.5608	−1.12643
$557$	−16.5068	−0.699416	−0.349708	−	0.936859i	$-0.613719\pi$
$557$	−16.5068	−0.699416	−0.349708	+	0.936859i	$0.613719\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	3.30136	0.139508
$561$	0	0
$562$	3.79796	0.160207
$563$	−37.7989	−1.59303	−0.796517	−	0.604617i	$-0.793326\pi$
$563$	−37.7989	−1.59303	−0.796517	+	0.604617i	$0.793326\pi$
$564$	0	0
$565$	10.8990	0.458524
$566$	4.15663	0.174716
$567$	0	0
$568$	27.8948	1.17044
$569$	−30.8627	−1.29383	−0.646915	−	0.762562i	$-0.723941\pi$
$569$	−30.8627	−1.29383	−0.646915	+	0.762562i	$0.723941\pi$
$570$	0	0
$571$	−18.3242	−0.766845	−0.383423	−	0.923573i	$-0.625255\pi$
$571$	−18.3242	−0.766845	−0.383423	+	0.923573i	$0.625255\pi$
$572$	0	0
$573$	0	0
$574$	12.5384	0.523344
$575$	4.89898	0.204302
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−4.52674	−0.188287
$579$	0	0
$580$	11.7215	0.486709
$581$	15.7980	0.655410
$582$	0	0
$583$	0	0
$584$	29.1464	1.20609
$585$	0	0
$586$	−4.15663	−0.171709
$587$	14.6969	0.606608	0.303304	−	0.952894i	$-0.401910\pi$
$587$	14.6969	0.606608	0.303304	+	0.952894i	$0.401910\pi$
$588$	0	0
$589$	2.96786	0.122288
$590$	0.816917	0.0336320
$591$	0	0
$592$	2.89898	0.119147
$593$	−9.23707	−0.379321	−0.189661	−	0.981850i	$-0.560739\pi$
$593$	−9.23707	−0.379321	−0.189661	+	0.981850i	$0.560739\pi$
$594$	0	0
$595$	10.8990	0.446815
$596$	−21.2921	−0.872158
$597$	0	0
$598$	6.60612	0.270144
$599$	7.59592	0.310361	0.155180	−	0.987886i	$-0.450404\pi$
$599$	7.59592	0.310361	0.155180	+	0.987886i	$0.450404\pi$
$600$	0	0
$601$	22.7760	0.929053	0.464527	−	0.885559i	$-0.346225\pi$
$601$	22.7760	0.929053	0.464527	+	0.885559i	$0.346225\pi$
$602$	−8.08665	−0.329587
$603$	0	0
$604$	21.2921	0.866363
$605$	0	0
$606$	0	0
$607$	15.8398	0.642917	0.321459	−	0.946924i	$-0.395827\pi$
$607$	15.8398	0.642917	0.321459	+	0.946924i	$0.395827\pi$
$608$	−8.69694	−0.352707
$609$	0	0
$610$	−7.10102	−0.287512
$611$	−17.8071	−0.720399
$612$	0	0
$613$	20.2916	0.819569	0.409784	−	0.912182i	$-0.365604\pi$
$613$	20.2916	0.819569	0.409784	+	0.912182i	$0.365604\pi$
$614$	11.7526	0.474294
$615$	0	0
$616$	0	0
$617$	−22.8990	−0.921878	−0.460939	−	0.887432i	$-0.652487\pi$
$617$	−22.8990	−0.921878	−0.460939	+	0.887432i	$0.652487\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−2.89898	−0.116426
$621$	0	0
$622$	−9.72048	−0.389756
$623$	−52.1548	−2.08954
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.9259	−0.916304
$627$	0	0
$628$	17.1010	0.682405
$629$	9.57058	0.381604
$630$	0	0
$631$	−37.3939	−1.48863	−0.744313	−	0.667831i	$-0.767223\pi$
$631$	−37.3939	−1.48863	−0.744313	+	0.667831i	$0.767223\pi$
$632$	45.1918	1.79763
$633$	0	0
$634$	−15.3564	−0.609880
$635$	−2.63435	−0.104541
$636$	0	0
$637$	7.08613	0.280763
$638$	0	0
$639$	0	0
$640$	9.97903	0.394456
$641$	−1.59592	−0.0630350	−0.0315175	−	0.999503i	$-0.510034\pi$
$641$	−1.59592	−0.0630350	−0.0315175	+	0.999503i	$0.510034\pi$
$642$	0	0
$643$	−40.4949	−1.59696	−0.798481	−	0.602019i	$-0.794363\pi$
$643$	−40.4949	−1.59696	−0.798481	+	0.602019i	$0.794363\pi$
$644$	23.4430	0.923785
$645$	0	0
$646$	−3.63487	−0.143012
$647$	−31.1010	−1.22271	−0.611354	−	0.791358i	$-0.709375\pi$
$647$	−31.1010	−1.22271	−0.611354	+	0.791358i	$0.709375\pi$
$648$	0	0
$649$	0	0
$650$	1.34847	0.0528913
$651$	0	0
$652$	−32.8990	−1.28842
$653$	37.5959	1.47124	0.735621	−	0.677393i	$-0.236890\pi$
$653$	37.5959	1.47124	0.735621	+	0.677393i	$0.236890\pi$
$654$	0	0
$655$	16.8403	0.658005
$656$	−5.11879	−0.199855
$657$	0	0
$658$	24.0000	0.935617
$659$	−17.5073	−0.681988	−0.340994	−	0.940065i	$-0.610764\pi$
$659$	−17.5073	−0.681988	−0.340994	+	0.940065i	$0.610764\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−7.41964	−0.288372
$663$	0	0
$664$	12.2474	0.475293
$665$	−4.89898	−0.189974
$666$	0	0
$667$	39.6163	1.53395
$668$	31.3461	1.21282
$669$	0	0
$670$	0.667010	0.0257689
$671$	0	0
$672$	0	0
$673$	−40.7667	−1.57144	−0.785721	−	0.618581i	$-0.787708\pi$
$673$	−40.7667	−1.57144	−0.785721	+	0.618581i	$0.787708\pi$
$674$	10.6515	0.410282
$675$	0	0
$676$	14.0556	0.540600
$677$	−30.3793	−1.16757	−0.583785	−	0.811908i	$-0.698429\pi$
$677$	−30.3793	−1.16757	−0.583785	+	0.811908i	$0.698429\pi$
$678$	0	0
$679$	41.9172	1.60863
$680$	8.44949	0.324023
$681$	0	0
$682$	0	0
$683$	7.59592	0.290650	0.145325	−	0.989384i	$-0.453577\pi$
$683$	7.59592	0.290650	0.145325	+	0.989384i	$0.453577\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	7.59592	0.290013
$687$	0	0
$688$	3.30136	0.125863
$689$	−2.00103	−0.0762332
$690$	0	0
$691$	−37.7980	−1.43790	−0.718951	−	0.695061i	$-0.755377\pi$
$691$	−37.7980	−1.43790	−0.718951	+	0.695061i	$0.755377\pi$
$692$	−32.5302	−1.23661
$693$	0	0
$694$	5.75255	0.218364
$695$	−18.3242	−0.695078
$696$	0	0
$697$	−16.8990	−0.640094
$698$	7.10102	0.268778
$699$	0	0
$700$	4.78529	0.180867
$701$	47.7030	1.80172	0.900858	−	0.434114i	$-0.142938\pi$
$701$	47.7030	1.80172	0.900858	+	0.434114i	$0.142938\pi$
$702$	0	0
$703$	−4.30188	−0.162248
$704$	0	0
$705$	0	0
$706$	0.816917	0.0307451
$707$	−28.8990	−1.08686
$708$	0	0
$709$	−22.6969	−0.852401	−0.426201	−	0.904629i	$-0.640148\pi$
$709$	−22.6969	−0.852401	−0.426201	+	0.904629i	$0.640148\pi$
$710$	8.08665	0.303486
$711$	0	0
$712$	−40.4332	−1.51530
$713$	−9.79796	−0.366936
$714$	0	0
$715$	0	0
$716$	3.19184	0.119285
$717$	0	0
$718$	23.5051	0.877203
$719$	33.7980	1.26045	0.630226	−	0.776412i	$-0.282962\pi$
$719$	33.7980	1.26045	0.630226	+	0.776412i	$0.282962\pi$
$720$	0	0
$721$	−10.2376	−0.381268
$722$	−12.4635	−0.463843
$723$	0	0
$724$	−18.6969	−0.694866
$725$	8.08665	0.300331
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	15.3564	0.569145
$729$	0	0
$730$	8.44949	0.312730
$731$	10.8990	0.403113
$732$	0	0
$733$	43.0676	1.59074	0.795369	−	0.606126i	$-0.207277\pi$
$733$	43.0676	1.59074	0.795369	+	0.606126i	$0.207277\pi$
$734$	11.5716	0.427116
$735$	0	0
$736$	28.7117	1.05833
$737$	0	0
$738$	0	0
$739$	24.2599	0.892416	0.446208	−	0.894929i	$-0.352774\pi$
$739$	24.2599	0.892416	0.446208	+	0.894929i	$0.352774\pi$
$740$	4.20204	0.154470
$741$	0	0
$742$	2.69694	0.0990077
$743$	−28.8953	−1.06007	−0.530033	−	0.847977i	$-0.677821\pi$
$743$	−28.8953	−1.06007	−0.530033	+	0.847977i	$0.677821\pi$
$744$	0	0
$745$	−14.6894	−0.538177
$746$	11.1464	0.408100
$747$	0	0
$748$	0	0
$749$	−27.7980	−1.01572
$750$	0	0
$751$	−47.1918	−1.72205	−0.861027	−	0.508559i	$-0.830178\pi$
$751$	−47.1918	−1.72205	−0.861027	+	0.508559i	$0.830178\pi$
$752$	−9.79796	−0.357295
$753$	0	0
$754$	10.9046	0.397122
$755$	14.6894	0.534601
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	11.5716	0.420300
$759$	0	0
$760$	−3.79796	−0.137766
$761$	−21.9591	−0.796017	−0.398008	−	0.917382i	$-0.630299\pi$
$761$	−21.9591	−0.796017	−0.398008	+	0.917382i	$0.630299\pi$
$762$	0	0
$763$	33.7980	1.22357
$764$	28.4041	1.02762
$765$	0	0
$766$	−5.26870	−0.190366
$767$	−2.00103	−0.0722530
$768$	0	0
$769$	3.63487	0.131077	0.0655383	−	0.997850i	$-0.479124\pi$
$769$	3.63487	0.131077	0.0655383	+	0.997850i	$0.479124\pi$
$770$	0	0
$771$	0	0
$772$	−25.1106	−0.903749
$773$	39.7980	1.43143	0.715717	−	0.698391i	$-0.246100\pi$
$773$	39.7980	1.43143	0.715717	+	0.698391i	$0.246100\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	32.4965	1.16656
$777$	0	0
$778$	−5.63590	−0.202057
$779$	7.59592	0.272152
$780$	0	0
$781$	0	0
$782$	12.0000	0.429119
$783$	0	0
$784$	3.89898	0.139249
$785$	11.7980	0.421087
$786$	0	0
$787$	−11.2381	−0.400595	−0.200298	−	0.979735i	$-0.564191\pi$
$787$	−11.2381	−0.400595	−0.200298	+	0.979735i	$0.564191\pi$
$788$	3.81846	0.136027
$789$	0	0
$790$	13.1010	0.466113
$791$	35.9815	1.27935
$792$	0	0
$793$	17.3939	0.617675
$794$	−0.517103	−0.0183513
$795$	0	0
$796$	−2.60612	−0.0923716
$797$	−10.8990	−0.386062	−0.193031	−	0.981193i	$-0.561832\pi$
$797$	−10.8990	−0.386062	−0.193031	+	0.981193i	$0.561832\pi$
$798$	0	0
$799$	−32.3466	−1.14434
$800$	5.86076	0.207209
$801$	0	0
$802$	−20.6251	−0.728297
$803$	0	0
$804$	0	0
$805$	16.1733	0.570034
$806$	−2.69694	−0.0949956
$807$	0	0
$808$	−22.4041	−0.788173
$809$	46.3690	1.63025	0.815123	−	0.579288i	$-0.196669\pi$
$809$	46.3690	1.63025	0.815123	+	0.579288i	$0.196669\pi$
$810$	0	0
$811$	−2.15094	−0.0755296	−0.0377648	−	0.999287i	$-0.512024\pi$
$811$	−2.15094	−0.0755296	−0.0377648	+	0.999287i	$0.512024\pi$
$812$	38.6969	1.35800
$813$	0	0
$814$	0	0
$815$	−22.6969	−0.795039
$816$	0	0
$817$	−4.89898	−0.171394
$818$	17.3939	0.608163
$819$	0	0
$820$	−7.41964	−0.259105
$821$	−19.9581	−0.696541	−0.348271	−	0.937394i	$-0.613231\pi$
$821$	−19.9581	−0.696541	−0.348271	+	0.937394i	$0.613231\pi$
$822$	0	0
$823$	48.8990	1.70451	0.852256	−	0.523126i	$-0.175234\pi$
$823$	48.8990	1.70451	0.852256	+	0.523126i	$0.175234\pi$
$824$	−7.93674	−0.276489
$825$	0	0
$826$	2.69694	0.0938385
$827$	−10.0540	−0.349611	−0.174806	−	0.984603i	$-0.555930\pi$
$827$	−10.0540	−0.349611	−0.174806	+	0.984603i	$0.555930\pi$
$828$	0	0
$829$	−13.3031	−0.462034	−0.231017	−	0.972950i	$-0.574205\pi$
$829$	−13.3031	−0.462034	−0.231017	+	0.972950i	$0.574205\pi$
$830$	3.55051	0.123240
$831$	0	0
$832$	4.26818	0.147973
$833$	12.8719	0.445986
$834$	0	0
$835$	21.6256	0.748385
$836$	0	0
$837$	0	0
$838$	−8.90357	−0.307569
$839$	−18.4949	−0.638515	−0.319257	−	0.947668i	$-0.603433\pi$
$839$	−18.4949	−0.638515	−0.319257	+	0.947668i	$0.603433\pi$
$840$	0	0
$841$	36.3939	1.25496
$842$	−25.7439	−0.887192
$843$	0	0
$844$	6.45281	0.222115
$845$	9.69694	0.333585
$846$	0	0
$847$	0	0
$848$	−1.10102	−0.0378092
$849$	0	0
$850$	2.44949	0.0840168
$851$	14.2020	0.486840
$852$	0	0
$853$	−30.5292	−1.04530	−0.522649	−	0.852548i	$-0.675056\pi$
$853$	−30.5292	−1.04530	−0.522649	+	0.852548i	$0.675056\pi$
$854$	−23.4430	−0.802204
$855$	0	0
$856$	−21.5505	−0.736581
$857$	−38.6158	−1.31909	−0.659545	−	0.751665i	$-0.729251\pi$
$857$	−38.6158	−1.31909	−0.659545	+	0.751665i	$0.729251\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	4.78529	0.163177
$861$	0	0
$862$	−9.79796	−0.333720
$863$	50.6969	1.72574	0.862872	−	0.505423i	$-0.168663\pi$
$863$	50.6969	1.72574	0.862872	+	0.505423i	$0.168663\pi$
$864$	0	0
$865$	−22.4425	−0.763068
$866$	12.3885	0.420979
$867$	0	0
$868$	−9.57058	−0.324847
$869$	0	0
$870$	0	0
$871$	−1.63383	−0.0553604
$872$	26.2020	0.887313
$873$	0	0
$874$	−5.39388	−0.182451
$875$	3.30136	0.111606
$876$	0	0
$877$	34.8310	1.17616	0.588080	−	0.808803i	$-0.299884\pi$
$877$	34.8310	1.17616	0.588080	+	0.808803i	$0.299884\pi$
$878$	14.8082	0.499751
$879$	0	0
$880$	0	0
$881$	−2.20204	−0.0741886	−0.0370943	−	0.999312i	$-0.511810\pi$
$881$	−2.20204	−0.0741886	−0.0370943	+	0.999312i	$0.511810\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−8.69694	−0.292510
$885$	0	0
$886$	−19.8082	−0.665468
$887$	20.2916	0.681324	0.340662	−	0.940186i	$-0.389349\pi$
$887$	20.2916	0.681324	0.340662	+	0.940186i	$0.389349\pi$
$888$	0	0
$889$	−8.69694	−0.291686
$890$	−11.7215	−0.392906
$891$	0	0
$892$	−25.7980	−0.863780
$893$	14.5395	0.486545
$894$	0	0
$895$	2.20204	0.0736061
$896$	32.9444	1.10059
$897$	0	0
$898$	−13.3553	−0.445674
$899$	−16.1733	−0.539410
$900$	0	0
$901$	−3.63487	−0.121095
$902$	0	0
$903$	0	0
$904$	27.8948	0.927768
$905$	−12.8990	−0.428777
$906$	0	0
$907$	49.3939	1.64010	0.820048	−	0.572294i	$-0.193946\pi$
$907$	49.3939	1.64010	0.820048	+	0.572294i	$0.193946\pi$
$908$	−16.5068	−0.547797
$909$	0	0
$910$	4.45178	0.147575
$911$	−33.7980	−1.11978	−0.559888	−	0.828568i	$-0.689156\pi$
$911$	−33.7980	−1.11978	−0.559888	+	0.828568i	$0.689156\pi$
$912$	0	0
$913$	0	0
$914$	−19.9546	−0.660039
$915$	0	0
$916$	−25.5051	−0.842712
$917$	55.5959	1.83594
$918$	0	0
$919$	44.7351	1.47568	0.737838	−	0.674978i	$-0.235847\pi$
$919$	44.7351	1.47568	0.737838	+	0.674978i	$0.235847\pi$
$920$	12.5384	0.413380
$921$	0	0
$922$	18.4949	0.609097
$923$	−19.8082	−0.651994
$924$	0	0
$925$	2.89898	0.0953179
$926$	5.93571	0.195060
$927$	0	0
$928$	47.3939	1.55578
$929$	5.39388	0.176967	0.0884837	−	0.996078i	$-0.471798\pi$
$929$	5.39388	0.176967	0.0884837	+	0.996078i	$0.471798\pi$
$930$	0	0
$931$	−5.78580	−0.189622
$932$	−0.483412	−0.0158347
$933$	0	0
$934$	−3.63487	−0.118936
$935$	0	0
$936$	0	0
$937$	22.2926	0.728268	0.364134	−	0.931347i	$-0.381365\pi$
$937$	22.2926	0.728268	0.364134	+	0.931347i	$0.381365\pi$
$938$	2.20204	0.0718992
$939$	0	0
$940$	−14.2020	−0.463220
$941$	−42.7341	−1.39309	−0.696546	−	0.717512i	$-0.745281\pi$
$941$	−42.7341	−1.39309	−0.696546	+	0.717512i	$0.745281\pi$
$942$	0	0
$943$	−25.0769	−0.816615
$944$	−1.10102	−0.0358352
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	−20.6969	−0.671851
$950$	−1.10102	−0.0357218
$951$	0	0
$952$	27.8948	0.904076
$953$	−32.6801	−1.05861	−0.529306	−	0.848431i	$-0.677548\pi$
$953$	−32.6801	−1.05861	−0.529306	+	0.848431i	$0.677548\pi$
$954$	0	0
$955$	19.5959	0.634109
$956$	22.4762	0.726932
$957$	0	0
$958$	−20.0908	−0.649105
$959$	59.4245	1.91892
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.90918	0.126037
$963$	0	0
$964$	−18.1743	−0.585356
$965$	−17.3237	−0.557670
$966$	0	0
$967$	−28.3782	−0.912582	−0.456291	−	0.889831i	$-0.650822\pi$
$967$	−28.3782	−0.912582	−0.456291	+	0.889831i	$0.650822\pi$
$968$	0	0
$969$	0	0
$970$	9.42067	0.302479
$971$	29.3939	0.943294	0.471647	−	0.881787i	$-0.343660\pi$
$971$	29.3939	0.943294	0.471647	+	0.881787i	$0.343660\pi$
$972$	0	0
$973$	−60.4949	−1.93938
$974$	−0.667010	−0.0213724
$975$	0	0
$976$	9.57058	0.306347
$977$	45.1918	1.44581	0.722907	−	0.690945i	$-0.242805\pi$
$977$	45.1918	1.44581	0.722907	+	0.690945i	$0.242805\pi$
$978$	0	0
$979$	0	0
$980$	5.65153	0.180532
$981$	0	0
$982$	−8.80816	−0.281080
$983$	26.6969	0.851500	0.425750	−	0.904841i	$-0.360010\pi$
$983$	26.6969	0.851500	0.425750	+	0.904841i	$0.360010\pi$
$984$	0	0
$985$	2.63435	0.0839374
$986$	19.8082	0.630820
$987$	0	0
$988$	3.90918	0.124368
$989$	16.1733	0.514281
$990$	0	0
$991$	−53.1918	−1.68969	−0.844847	−	0.535008i	$-0.820309\pi$
$991$	−53.1918	−1.68969	−0.844847	+	0.535008i	$0.820309\pi$
$992$	−11.7215	−0.372158
$993$	0	0
$994$	26.6969	0.846775
$995$	−1.79796	−0.0569991
$996$	0	0
$997$	−57.6071	−1.82443	−0.912217	−	0.409708i	$-0.865631\pi$
$997$	−57.6071	−1.82443	−0.912217	+	0.409708i	$0.865631\pi$
$998$	−4.30188	−0.136173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bm.1.3	yes	4
3.2	odd	2		5445.2.a.bn.1.2	yes	4
11.10	odd	2	inner	5445.2.a.bm.1.2	✓	4
33.32	even	2		5445.2.a.bn.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.bm.1.2	✓	4	11.10	odd	2	inner
5445.2.a.bm.1.3	yes	4	1.1	even	1	trivial
5445.2.a.bn.1.2	yes	4	3.2	odd	2
5445.2.a.bn.1.3	yes	4	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} -3.30136 q^{7} -2.55940 q^{8} +O(q^{10})\)	\(q+0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} -3.30136 q^{7} -2.55940 q^{8} -0.741964 q^{10} +1.81743 q^{13} -2.44949 q^{14} +1.00000 q^{16} +3.30136 q^{17} -1.48393 q^{19} +1.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +1.34847 q^{26} +4.78529 q^{28} +8.08665 q^{29} -2.00000 q^{31} +5.86076 q^{32} +2.44949 q^{34} +3.30136 q^{35} +2.89898 q^{37} -1.10102 q^{38} +2.55940 q^{40} -5.11879 q^{41} +3.30136 q^{43} +3.63487 q^{46} -9.79796 q^{47} +3.89898 q^{49} +0.741964 q^{50} -2.63435 q^{52} -1.10102 q^{53} +8.44949 q^{56} +6.00000 q^{58} -1.10102 q^{59} +9.57058 q^{61} -1.48393 q^{62} +2.34847 q^{64} -1.81743 q^{65} -0.898979 q^{67} -4.78529 q^{68} +2.44949 q^{70} -10.8990 q^{71} -11.3880 q^{73} +2.15094 q^{74} +2.15094 q^{76} -17.6572 q^{79} -1.00000 q^{80} -3.79796 q^{82} -4.78529 q^{83} -3.30136 q^{85} +2.44949 q^{86} +15.7980 q^{89} -6.00000 q^{91} -7.10102 q^{92} -7.26973 q^{94} +1.48393 q^{95} -12.6969 q^{97} +2.89290 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^4 - 4 * q^5	\( 4 q + 4 q^{4} - 4 q^{5} + 4 q^{16} - 4 q^{20} + 4 q^{25} - 24 q^{26} - 8 q^{31} - 8 q^{37} - 24 q^{38} - 4 q^{49} - 24 q^{53} + 24 q^{56} + 24 q^{58} - 24 q^{59} - 20 q^{64} + 16 q^{67} - 24 q^{71} - 4 q^{80} + 24 q^{82} + 24 q^{89} - 24 q^{91} - 48 q^{92} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^5 + 4 * q^16 - 4 * q^20 + 4 * q^25 - 24 * q^26 - 8 * q^31 - 8 * q^37 - 24 * q^38 - 4 * q^49 - 24 * q^53 + 24 * q^56 + 24 * q^58 - 24 * q^59 - 20 * q^64 + 16 * q^67 - 24 * q^71 - 4 * q^80 + 24 * q^82 + 24 * q^89 - 24 * q^91 - 48 * q^92 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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