LMFDB - Embedded newform 5445.2.a.ca.1.1

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 $ x^8 - 4x^7 - 3x^6 + 22x^5 - 3x^4 - 32x^3 + 9x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75229$ 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)

$f(q)$	$=$	$q-2.75229 q^{2} +5.57509 q^{4} -1.00000 q^{5} +2.73187 q^{7} -9.83966 q^{8} +O(q^{10})$	$q-2.75229 q^{2} +5.57509 q^{4} -1.00000 q^{5} +2.73187 q^{7} -9.83966 q^{8} +2.75229 q^{10} -2.77482 q^{13} -7.51889 q^{14} +15.9314 q^{16} +3.78118 q^{17} -4.25186 q^{19} -5.57509 q^{20} +1.37658 q^{23} +1.00000 q^{25} +7.63712 q^{26} +15.2304 q^{28} -8.86735 q^{29} +5.46722 q^{31} -24.1685 q^{32} -10.4069 q^{34} -2.73187 q^{35} -2.30142 q^{37} +11.7023 q^{38} +9.83966 q^{40} -8.61928 q^{41} +12.3981 q^{43} -3.78875 q^{46} +5.59315 q^{47} +0.463108 q^{49} -2.75229 q^{50} -15.4699 q^{52} +0.543807 q^{53} -26.8807 q^{56} +24.4055 q^{58} -0.389279 q^{59} -14.3414 q^{61} -15.0474 q^{62} +34.6558 q^{64} +2.77482 q^{65} +3.65454 q^{67} +21.0804 q^{68} +7.51889 q^{70} +9.16556 q^{71} +8.16025 q^{73} +6.33417 q^{74} -23.7045 q^{76} -3.44708 q^{79} -15.9314 q^{80} +23.7227 q^{82} +4.65472 q^{83} -3.78118 q^{85} -34.1231 q^{86} -6.62318 q^{89} -7.58046 q^{91} +7.67457 q^{92} -15.3940 q^{94} +4.25186 q^{95} -16.5938 q^{97} -1.27461 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 + 6 * q^13 - 14 * q^14 + 14 * q^16 - 8 * q^17 - 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 + 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 + 20 * q^38 + 12 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^46 - 10 * q^47 - 4 * q^50 - 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 - 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 + 2 * q^71 + 16 * q^73 - 24 * q^74 - 22 * q^76 - 16 * q^79 - 14 * q^80 + 10 * q^82 - 46 * q^83 + 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 - 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.75229	−1.94616	−0.973081	−	0.230465i	$-0.925975\pi$
$2$	−2.75229	−1.94616	−0.973081	+	0.230465i	$0.925975\pi$
$3$	0	0
$3$	0	0
$4$	5.57509	2.78754
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.73187	1.03255	0.516275	−	0.856423i	$-0.327318\pi$
$7$	2.73187	1.03255	0.516275	+	0.856423i	$0.327318\pi$
$8$	−9.83966	−3.47885
$9$	0	0
$10$	2.75229	0.870350
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.77482	−0.769598	−0.384799	−	0.923000i	$-0.625729\pi$
$13$	−2.77482	−0.769598	−0.384799	+	0.923000i	$0.625729\pi$
$14$	−7.51889	−2.00951
$15$	0	0
$16$	15.9314	3.98285
$17$	3.78118	0.917071	0.458536	−	0.888676i	$-0.348374\pi$
$17$	3.78118	0.917071	0.458536	+	0.888676i	$0.348374\pi$
$18$	0	0
$19$	−4.25186	−0.975444	−0.487722	−	0.872999i	$-0.662172\pi$
$19$	−4.25186	−0.975444	−0.487722	+	0.872999i	$0.662172\pi$
$20$	−5.57509	−1.24663
$21$	0	0
$22$	0	0
$23$	1.37658	0.287038	0.143519	−	0.989648i	$-0.454158\pi$
$23$	1.37658	0.287038	0.143519	+	0.989648i	$0.454158\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	7.63712	1.49776
$27$	0	0
$28$	15.2304	2.87828
$29$	−8.86735	−1.64663	−0.823313	−	0.567588i	$-0.807877\pi$
$29$	−8.86735	−1.64663	−0.823313	+	0.567588i	$0.807877\pi$
$30$	0	0
$31$	5.46722	0.981942	0.490971	−	0.871176i	$-0.336642\pi$
$31$	5.46722	0.981942	0.490971	+	0.871176i	$0.336642\pi$
$32$	−24.1685	−4.27243
$33$	0	0
$34$	−10.4069	−1.78477
$35$	−2.73187	−0.461770
$36$	0	0
$37$	−2.30142	−0.378351	−0.189176	−	0.981943i	$-0.560582\pi$
$37$	−2.30142	−0.378351	−0.189176	+	0.981943i	$0.560582\pi$
$38$	11.7023	1.89837
$39$	0	0
$40$	9.83966	1.55579
$41$	−8.61928	−1.34611	−0.673053	−	0.739594i	$-0.735017\pi$
$41$	−8.61928	−1.34611	−0.673053	+	0.739594i	$0.735017\pi$
$42$	0	0
$43$	12.3981	1.89069	0.945346	−	0.326070i	$-0.105724\pi$
$43$	12.3981	1.89069	0.945346	+	0.326070i	$0.105724\pi$
$44$	0	0
$45$	0	0
$46$	−3.78875	−0.558621
$47$	5.59315	0.815845	0.407923	−	0.913017i	$-0.366253\pi$
$47$	5.59315	0.815845	0.407923	+	0.913017i	$0.366253\pi$
$48$	0	0
$49$	0.463108	0.0661582
$50$	−2.75229	−0.389232
$51$	0	0
$52$	−15.4699	−2.14529
$53$	0.543807	0.0746976	0.0373488	−	0.999302i	$-0.488109\pi$
$53$	0.543807	0.0746976	0.0373488	+	0.999302i	$0.488109\pi$
$54$	0	0
$55$	0	0
$56$	−26.8807	−3.59208
$57$	0	0
$58$	24.4055	3.20460
$59$	−0.389279	−0.0506797	−0.0253399	−	0.999679i	$-0.508067\pi$
$59$	−0.389279	−0.0506797	−0.0253399	+	0.999679i	$0.508067\pi$
$60$	0	0
$61$	−14.3414	−1.83622	−0.918112	−	0.396320i	$-0.870287\pi$
$61$	−14.3414	−1.83622	−0.918112	+	0.396320i	$0.870287\pi$
$62$	−15.0474	−1.91102
$63$	0	0
$64$	34.6558	4.33198
$65$	2.77482	0.344175
$66$	0	0
$67$	3.65454	0.446473	0.223237	−	0.974764i	$-0.428338\pi$
$67$	3.65454	0.446473	0.223237	+	0.974764i	$0.428338\pi$
$68$	21.0804	2.55638
$69$	0	0
$70$	7.51889	0.898679
$71$	9.16556	1.08775	0.543876	−	0.839166i	$-0.316956\pi$
$71$	9.16556	1.08775	0.543876	+	0.839166i	$0.316956\pi$
$72$	0	0
$73$	8.16025	0.955085	0.477542	−	0.878609i	$-0.341528\pi$
$73$	8.16025	0.955085	0.477542	+	0.878609i	$0.341528\pi$
$74$	6.33417	0.736333
$75$	0	0
$76$	−23.7045	−2.71909
$77$	0	0
$78$	0	0
$79$	−3.44708	−0.387827	−0.193913	−	0.981019i	$-0.562118\pi$
$79$	−3.44708	−0.387827	−0.193913	+	0.981019i	$0.562118\pi$
$80$	−15.9314	−1.78119
$81$	0	0
$82$	23.7227	2.61974
$83$	4.65472	0.510922	0.255461	−	0.966819i	$-0.417773\pi$
$83$	4.65472	0.510922	0.255461	+	0.966819i	$0.417773\pi$
$84$	0	0
$85$	−3.78118	−0.410127
$86$	−34.1231	−3.67959
$87$	0	0
$88$	0	0
$89$	−6.62318	−0.702056	−0.351028	−	0.936365i	$-0.614168\pi$
$89$	−6.62318	−0.702056	−0.351028	+	0.936365i	$0.614168\pi$
$90$	0	0
$91$	−7.58046	−0.794648
$92$	7.67457	0.800129
$93$	0	0
$94$	−15.3940	−1.58777
$95$	4.25186	0.436232
$96$	0	0
$97$	−16.5938	−1.68484	−0.842421	−	0.538820i	$-0.818870\pi$
$97$	−16.5938	−1.68484	−0.842421	+	0.538820i	$0.818870\pi$
$98$	−1.27461	−0.128755
$99$	0	0
$100$	5.57509	0.557509
$101$	−0.539052	−0.0536376	−0.0268188	−	0.999640i	$-0.508538\pi$
$101$	−0.539052	−0.0536376	−0.0268188	+	0.999640i	$0.508538\pi$
$102$	0	0
$103$	−6.11063	−0.602098	−0.301049	−	0.953609i	$-0.597337\pi$
$103$	−6.11063	−0.602098	−0.301049	+	0.953609i	$0.597337\pi$
$104$	27.3033	2.67731
$105$	0	0
$106$	−1.49671	−0.145374
$107$	−9.87705	−0.954851	−0.477425	−	0.878672i	$-0.658430\pi$
$107$	−9.87705	−0.954851	−0.477425	+	0.878672i	$0.658430\pi$
$108$	0	0
$109$	−5.44683	−0.521712	−0.260856	−	0.965378i	$-0.584005\pi$
$109$	−5.44683	−0.521712	−0.260856	+	0.965378i	$0.584005\pi$
$110$	0	0
$111$	0	0
$112$	43.5225	4.11249
$113$	−0.777605	−0.0731510	−0.0365755	−	0.999331i	$-0.511645\pi$
$113$	−0.777605	−0.0731510	−0.0365755	+	0.999331i	$0.511645\pi$
$114$	0	0
$115$	−1.37658	−0.128367
$116$	−49.4362	−4.59004
$117$	0	0
$118$	1.07141	0.0986310
$119$	10.3297	0.946921
$120$	0	0
$121$	0	0
$122$	39.4716	3.57359
$123$	0	0
$124$	30.4802	2.73721
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.33061	0.384279	0.192140	−	0.981368i	$-0.438457\pi$
$127$	4.33061	0.384279	0.192140	+	0.981368i	$0.438457\pi$
$128$	−47.0458	−4.15830
$129$	0	0
$130$	−7.63712	−0.669819
$131$	11.7312	1.02496	0.512478	−	0.858700i	$-0.328728\pi$
$131$	11.7312	1.02496	0.512478	+	0.858700i	$0.328728\pi$
$132$	0	0
$133$	−11.6155	−1.00719
$134$	−10.0583	−0.868909
$135$	0	0
$136$	−37.2056	−3.19035
$137$	−13.5841	−1.16057	−0.580285	−	0.814413i	$-0.697059\pi$
$137$	−13.5841	−1.16057	−0.580285	+	0.814413i	$0.697059\pi$
$138$	0	0
$139$	−12.0322	−1.02056	−0.510278	−	0.860009i	$-0.670458\pi$
$139$	−12.0322	−1.02056	−0.510278	+	0.860009i	$0.670458\pi$
$140$	−15.2304	−1.28720
$141$	0	0
$142$	−25.2263	−2.11694
$143$	0	0
$144$	0	0
$145$	8.86735	0.736393
$146$	−22.4594	−1.85875
$147$	0	0
$148$	−12.8306	−1.05467
$149$	13.7934	1.13000	0.564999	−	0.825092i	$-0.308877\pi$
$149$	13.7934	1.13000	0.564999	+	0.825092i	$0.308877\pi$
$150$	0	0
$151$	−17.0063	−1.38395	−0.691976	−	0.721921i	$-0.743259\pi$
$151$	−17.0063	−1.38395	−0.691976	+	0.721921i	$0.743259\pi$
$152$	41.8369	3.39342
$153$	0	0
$154$	0	0
$155$	−5.46722	−0.439138
$156$	0	0
$157$	−3.61081	−0.288174	−0.144087	−	0.989565i	$-0.546025\pi$
$157$	−3.61081	−0.288174	−0.144087	+	0.989565i	$0.546025\pi$
$158$	9.48735	0.754773
$159$	0	0
$160$	24.1685	1.91069
$161$	3.76065	0.296380
$162$	0	0
$163$	−3.62347	−0.283812	−0.141906	−	0.989880i	$-0.545323\pi$
$163$	−3.62347	−0.283812	−0.141906	+	0.989880i	$0.545323\pi$
$164$	−48.0532	−3.75233
$165$	0	0
$166$	−12.8111	−0.994337
$167$	−2.27168	−0.175788	−0.0878939	−	0.996130i	$-0.528014\pi$
$167$	−2.27168	−0.175788	−0.0878939	+	0.996130i	$0.528014\pi$
$168$	0	0
$169$	−5.30035	−0.407719
$170$	10.4069	0.798173
$171$	0	0
$172$	69.1204	5.27038
$173$	19.2705	1.46511	0.732554	−	0.680709i	$-0.238328\pi$
$173$	19.2705	1.46511	0.732554	+	0.680709i	$0.238328\pi$
$174$	0	0
$175$	2.73187	0.206510
$176$	0	0
$177$	0	0
$178$	18.2289	1.36631
$179$	7.26883	0.543298	0.271649	−	0.962396i	$-0.412431\pi$
$179$	7.26883	0.543298	0.271649	+	0.962396i	$0.412431\pi$
$180$	0	0
$181$	−8.11886	−0.603470	−0.301735	−	0.953392i	$-0.597566\pi$
$181$	−8.11886	−0.603470	−0.301735	+	0.953392i	$0.597566\pi$
$182$	20.8636	1.54651
$183$	0	0
$184$	−13.5451	−0.998560
$185$	2.30142	0.169204
$186$	0	0
$187$	0	0
$188$	31.1823	2.27420
$189$	0	0
$190$	−11.7023	−0.848977
$191$	20.0211	1.44868	0.724339	−	0.689444i	$-0.242145\pi$
$191$	20.0211	1.44868	0.724339	+	0.689444i	$0.242145\pi$
$192$	0	0
$193$	−19.5724	−1.40885	−0.704425	−	0.709779i	$-0.748795\pi$
$193$	−19.5724	−1.40885	−0.704425	+	0.709779i	$0.748795\pi$
$194$	45.6708	3.27897
$195$	0	0
$196$	2.58186	0.184419
$197$	−22.9044	−1.63187	−0.815937	−	0.578141i	$-0.803778\pi$
$197$	−22.9044	−1.63187	−0.815937	+	0.578141i	$0.803778\pi$
$198$	0	0
$199$	−17.3671	−1.23112	−0.615560	−	0.788090i	$-0.711070\pi$
$199$	−17.3671	−1.23112	−0.615560	+	0.788090i	$0.711070\pi$
$200$	−9.83966	−0.695769
$201$	0	0
$202$	1.48363	0.104388
$203$	−24.2244	−1.70022
$204$	0	0
$205$	8.61928	0.601997
$206$	16.8182	1.17178
$207$	0	0
$208$	−44.2069	−3.06520
$209$	0	0
$210$	0	0
$211$	16.6106	1.14352	0.571760	−	0.820421i	$-0.306261\pi$
$211$	16.6106	1.14352	0.571760	+	0.820421i	$0.306261\pi$
$212$	3.03177	0.208223
$213$	0	0
$214$	27.1845	1.85829
$215$	−12.3981	−0.845543
$216$	0	0
$217$	14.9357	1.01390
$218$	14.9912	1.01533
$219$	0	0
$220$	0	0
$221$	−10.4921	−0.705776
$222$	0	0
$223$	−8.96047	−0.600037	−0.300019	−	0.953933i	$-0.596993\pi$
$223$	−8.96047	−0.600037	−0.300019	+	0.953933i	$0.596993\pi$
$224$	−66.0252	−4.41149
$225$	0	0
$226$	2.14019	0.142364
$227$	−1.06154	−0.0704571	−0.0352286	−	0.999379i	$-0.511216\pi$
$227$	−1.06154	−0.0704571	−0.0352286	+	0.999379i	$0.511216\pi$
$228$	0	0
$229$	−5.85354	−0.386813	−0.193406	−	0.981119i	$-0.561954\pi$
$229$	−5.85354	−0.386813	−0.193406	+	0.981119i	$0.561954\pi$
$230$	3.78875	0.249823
$231$	0	0
$232$	87.2518	5.72836
$233$	−1.87722	−0.122981	−0.0614904	−	0.998108i	$-0.519585\pi$
$233$	−1.87722	−0.122981	−0.0614904	+	0.998108i	$0.519585\pi$
$234$	0	0
$235$	−5.59315	−0.364857
$236$	−2.17026	−0.141272
$237$	0	0
$238$	−28.4303	−1.84286
$239$	−2.04058	−0.131994	−0.0659970	−	0.997820i	$-0.521023\pi$
$239$	−2.04058	−0.131994	−0.0659970	+	0.997820i	$0.521023\pi$
$240$	0	0
$241$	15.3922	0.991500	0.495750	−	0.868465i	$-0.334893\pi$
$241$	15.3922	0.991500	0.495750	+	0.868465i	$0.334893\pi$
$242$	0	0
$243$	0	0
$244$	−79.9544	−5.11856
$245$	−0.463108	−0.0295869
$246$	0	0
$247$	11.7982	0.750700
$248$	−53.7956	−3.41603
$249$	0	0
$250$	2.75229	0.174070
$251$	−13.4172	−0.846889	−0.423444	−	0.905922i	$-0.639179\pi$
$251$	−13.4172	−0.846889	−0.423444	+	0.905922i	$0.639179\pi$
$252$	0	0
$253$	0	0
$254$	−11.9191	−0.747869
$255$	0	0
$256$	60.1719	3.76074
$257$	−9.61357	−0.599678	−0.299839	−	0.953990i	$-0.596933\pi$
$257$	−9.61357	−0.599678	−0.299839	+	0.953990i	$0.596933\pi$
$258$	0	0
$259$	−6.28718	−0.390666
$260$	15.4699	0.959402
$261$	0	0
$262$	−32.2875	−1.99473
$263$	23.4178	1.44400	0.722002	−	0.691891i	$-0.243222\pi$
$263$	23.4178	1.44400	0.722002	+	0.691891i	$0.243222\pi$
$264$	0	0
$265$	−0.543807	−0.0334058
$266$	31.9693	1.96016
$267$	0	0
$268$	20.3744	1.24456
$269$	−9.46210	−0.576914	−0.288457	−	0.957493i	$-0.593142\pi$
$269$	−9.46210	−0.576914	−0.288457	+	0.957493i	$0.593142\pi$
$270$	0	0
$271$	25.5459	1.55180	0.775902	−	0.630853i	$-0.217295\pi$
$271$	25.5459	1.55180	0.775902	+	0.630853i	$0.217295\pi$
$272$	60.2396	3.65256
$273$	0	0
$274$	37.3874	2.25866
$275$	0	0
$276$	0	0
$277$	−4.65032	−0.279411	−0.139705	−	0.990193i	$-0.544616\pi$
$277$	−4.65032	−0.279411	−0.139705	+	0.990193i	$0.544616\pi$
$278$	33.1160	1.98617
$279$	0	0
$280$	26.8807	1.60643
$281$	−10.8043	−0.644533	−0.322266	−	0.946649i	$-0.604445\pi$
$281$	−10.8043	−0.644533	−0.322266	+	0.946649i	$0.604445\pi$
$282$	0	0
$283$	2.00667	0.119284	0.0596420	−	0.998220i	$-0.481004\pi$
$283$	2.00667	0.119284	0.0596420	+	0.998220i	$0.481004\pi$
$284$	51.0988	3.03215
$285$	0	0
$286$	0	0
$287$	−23.5467	−1.38992
$288$	0	0
$289$	−2.70266	−0.158980
$290$	−24.4055	−1.43314
$291$	0	0
$292$	45.4941	2.66234
$293$	0.602557	0.0352018	0.0176009	−	0.999845i	$-0.494397\pi$
$293$	0.602557	0.0352018	0.0176009	+	0.999845i	$0.494397\pi$
$294$	0	0
$295$	0.389279	0.0226647
$296$	22.6452	1.31623
$297$	0	0
$298$	−37.9633	−2.19916
$299$	−3.81978	−0.220903
$300$	0	0
$301$	33.8700	1.95223
$302$	46.8062	2.69339
$303$	0	0
$304$	−67.7382	−3.88505
$305$	14.3414	0.821185
$306$	0	0
$307$	30.7715	1.75622	0.878112	−	0.478454i	$-0.158803\pi$
$307$	30.7715	1.75622	0.878112	+	0.478454i	$0.158803\pi$
$308$	0	0
$309$	0	0
$310$	15.0474	0.854633
$311$	1.44287	0.0818175	0.0409087	−	0.999163i	$-0.486975\pi$
$311$	1.44287	0.0818175	0.0409087	+	0.999163i	$0.486975\pi$
$312$	0	0
$313$	23.4992	1.32826	0.664128	−	0.747619i	$-0.268803\pi$
$313$	23.4992	1.32826	0.664128	+	0.747619i	$0.268803\pi$
$314$	9.93800	0.560834
$315$	0	0
$316$	−19.2178	−1.08108
$317$	14.3181	0.804184	0.402092	−	0.915599i	$-0.368283\pi$
$317$	14.3181	0.804184	0.402092	+	0.915599i	$0.368283\pi$
$318$	0	0
$319$	0	0
$320$	−34.6558	−1.93732
$321$	0	0
$322$	−10.3504	−0.576804
$323$	−16.0771	−0.894552
$324$	0	0
$325$	−2.77482	−0.153920
$326$	9.97282	0.552344
$327$	0	0
$328$	84.8108	4.68289
$329$	15.2798	0.842400
$330$	0	0
$331$	−4.93282	−0.271132	−0.135566	−	0.990768i	$-0.543285\pi$
$331$	−4.93282	−0.271132	−0.135566	+	0.990768i	$0.543285\pi$
$332$	25.9505	1.42422
$333$	0	0
$334$	6.25231	0.342111
$335$	−3.65454	−0.199669
$336$	0	0
$337$	7.69258	0.419042	0.209521	−	0.977804i	$-0.432810\pi$
$337$	7.69258	0.419042	0.209521	+	0.977804i	$0.432810\pi$
$338$	14.5881	0.793487
$339$	0	0
$340$	−21.0804	−1.14325
$341$	0	0
$342$	0	0
$343$	−17.8579	−0.964238
$344$	−121.993	−6.57743
$345$	0	0
$346$	−53.0379	−2.85134
$347$	−17.1661	−0.921522	−0.460761	−	0.887524i	$-0.652424\pi$
$347$	−17.1661	−0.921522	−0.460761	+	0.887524i	$0.652424\pi$
$348$	0	0
$349$	24.2805	1.29971	0.649853	−	0.760060i	$-0.274830\pi$
$349$	24.2805	1.29971	0.649853	+	0.760060i	$0.274830\pi$
$350$	−7.51889	−0.401901
$351$	0	0
$352$	0	0
$353$	5.84602	0.311152	0.155576	−	0.987824i	$-0.450277\pi$
$353$	5.84602	0.311152	0.155576	+	0.987824i	$0.450277\pi$
$354$	0	0
$355$	−9.16556	−0.486457
$356$	−36.9248	−1.95701
$357$	0	0
$358$	−20.0059	−1.05735
$359$	−8.14440	−0.429845	−0.214922	−	0.976631i	$-0.568950\pi$
$359$	−8.14440	−0.429845	−0.214922	+	0.976631i	$0.568950\pi$
$360$	0	0
$361$	−0.921675	−0.0485092
$362$	22.3454	1.17445
$363$	0	0
$364$	−42.2617	−2.21512
$365$	−8.16025	−0.427127
$366$	0	0
$367$	22.6747	1.18361	0.591806	−	0.806081i	$-0.298415\pi$
$367$	22.6747	1.18361	0.591806	+	0.806081i	$0.298415\pi$
$368$	21.9309	1.14323
$369$	0	0
$370$	−6.33417	−0.329298
$371$	1.48561	0.0771290
$372$	0	0
$373$	−14.0566	−0.727825	−0.363913	−	0.931433i	$-0.618559\pi$
$373$	−14.0566	−0.727825	−0.363913	+	0.931433i	$0.618559\pi$
$374$	0	0
$375$	0	0
$376$	−55.0347	−2.83820
$377$	24.6053	1.26724
$378$	0	0
$379$	−25.2171	−1.29532	−0.647659	−	0.761931i	$-0.724252\pi$
$379$	−25.2171	−1.29532	−0.647659	+	0.761931i	$0.724252\pi$
$380$	23.7045	1.21601
$381$	0	0
$382$	−55.1039	−2.81936
$383$	24.5315	1.25350	0.626750	−	0.779220i	$-0.284385\pi$
$383$	24.5315	1.25350	0.626750	+	0.779220i	$0.284385\pi$
$384$	0	0
$385$	0	0
$386$	53.8688	2.74185
$387$	0	0
$388$	−92.5116	−4.69657
$389$	−24.1607	−1.22500	−0.612499	−	0.790472i	$-0.709836\pi$
$389$	−24.1607	−1.22500	−0.612499	+	0.790472i	$0.709836\pi$
$390$	0	0
$391$	5.20511	0.263234
$392$	−4.55682	−0.230154
$393$	0	0
$394$	63.0396	3.17589
$395$	3.44708	0.173441
$396$	0	0
$397$	−29.4680	−1.47896	−0.739479	−	0.673179i	$-0.764928\pi$
$397$	−29.4680	−1.47896	−0.739479	+	0.673179i	$0.764928\pi$
$398$	47.7992	2.39596
$399$	0	0
$400$	15.9314	0.796571
$401$	4.57792	0.228610	0.114305	−	0.993446i	$-0.463536\pi$
$401$	4.57792	0.228610	0.114305	+	0.993446i	$0.463536\pi$
$402$	0	0
$403$	−15.1706	−0.755701
$404$	−3.00526	−0.149517
$405$	0	0
$406$	66.6726	3.30891
$407$	0	0
$408$	0	0
$409$	−19.0113	−0.940050	−0.470025	−	0.882653i	$-0.655755\pi$
$409$	−19.0113	−0.940050	−0.470025	+	0.882653i	$0.655755\pi$
$410$	−23.7227	−1.17158
$411$	0	0
$412$	−34.0673	−1.67838
$413$	−1.06346	−0.0523293
$414$	0	0
$415$	−4.65472	−0.228491
$416$	67.0633	3.28805
$417$	0	0
$418$	0	0
$419$	2.54104	0.124138	0.0620690	−	0.998072i	$-0.480230\pi$
$419$	2.54104	0.124138	0.0620690	+	0.998072i	$0.480230\pi$
$420$	0	0
$421$	2.05374	0.100093	0.0500466	−	0.998747i	$-0.484063\pi$
$421$	2.05374	0.100093	0.0500466	+	0.998747i	$0.484063\pi$
$422$	−45.7171	−2.22547
$423$	0	0
$424$	−5.35088	−0.259862
$425$	3.78118	0.183414
$426$	0	0
$427$	−39.1788	−1.89599
$428$	−55.0654	−2.66169
$429$	0	0
$430$	34.1231	1.64556
$431$	−37.8911	−1.82515	−0.912576	−	0.408907i	$-0.865910\pi$
$431$	−37.8911	−1.82515	−0.912576	+	0.408907i	$0.865910\pi$
$432$	0	0
$433$	9.10654	0.437633	0.218816	−	0.975766i	$-0.429780\pi$
$433$	9.10654	0.437633	0.218816	+	0.975766i	$0.429780\pi$
$434$	−41.1074	−1.97322
$435$	0	0
$436$	−30.3665	−1.45429
$437$	−5.85304	−0.279989
$438$	0	0
$439$	31.0675	1.48277	0.741385	−	0.671079i	$-0.234169\pi$
$439$	31.0675	1.48277	0.741385	+	0.671079i	$0.234169\pi$
$440$	0	0
$441$	0	0
$442$	28.8773	1.37355
$443$	14.9113	0.708458	0.354229	−	0.935159i	$-0.384743\pi$
$443$	14.9113	0.708458	0.354229	+	0.935159i	$0.384743\pi$
$444$	0	0
$445$	6.62318	0.313969
$446$	24.6618	1.16777
$447$	0	0
$448$	94.6752	4.47298
$449$	−25.6099	−1.20861	−0.604303	−	0.796755i	$-0.706548\pi$
$449$	−25.6099	−1.20861	−0.604303	+	0.796755i	$0.706548\pi$
$450$	0	0
$451$	0	0
$452$	−4.33522	−0.203911
$453$	0	0
$454$	2.92167	0.137121
$455$	7.58046	0.355377
$456$	0	0
$457$	8.02615	0.375447	0.187724	−	0.982222i	$-0.439889\pi$
$457$	8.02615	0.375447	0.187724	+	0.982222i	$0.439889\pi$
$458$	16.1106	0.752800
$459$	0	0
$460$	−7.67457	−0.357829
$461$	−16.8858	−0.786449	−0.393225	−	0.919442i	$-0.628640\pi$
$461$	−16.8858	−0.786449	−0.393225	+	0.919442i	$0.628640\pi$
$462$	0	0
$463$	−4.70959	−0.218873	−0.109437	−	0.993994i	$-0.534905\pi$
$463$	−4.70959	−0.218873	−0.109437	+	0.993994i	$0.534905\pi$
$464$	−141.269	−6.55827
$465$	0	0
$466$	5.16665	0.239340
$467$	−15.4149	−0.713314	−0.356657	−	0.934235i	$-0.616084\pi$
$467$	−15.4149	−0.713314	−0.356657	+	0.934235i	$0.616084\pi$
$468$	0	0
$469$	9.98373	0.461006
$470$	15.3940	0.710071
$471$	0	0
$472$	3.83037	0.176307
$473$	0	0
$474$	0	0
$475$	−4.25186	−0.195089
$476$	57.5889	2.63958
$477$	0	0
$478$	5.61626	0.256882
$479$	9.84576	0.449864	0.224932	−	0.974374i	$-0.427784\pi$
$479$	9.84576	0.449864	0.224932	+	0.974374i	$0.427784\pi$
$480$	0	0
$481$	6.38604	0.291178
$482$	−42.3638	−1.92962
$483$	0	0
$484$	0	0
$485$	16.5938	0.753484
$486$	0	0
$487$	−8.17219	−0.370317	−0.185159	−	0.982709i	$-0.559280\pi$
$487$	−8.17219	−0.370317	−0.185159	+	0.982709i	$0.559280\pi$
$488$	141.114	6.38794
$489$	0	0
$490$	1.27461	0.0575808
$491$	−38.5774	−1.74097	−0.870487	−	0.492192i	$-0.836196\pi$
$491$	−38.5774	−1.74097	−0.870487	+	0.492192i	$0.836196\pi$
$492$	0	0
$493$	−33.5291	−1.51007
$494$	−32.4720	−1.46098
$495$	0	0
$496$	87.1006	3.91093
$497$	25.0391	1.12316
$498$	0	0
$499$	−2.34961	−0.105183	−0.0525916	−	0.998616i	$-0.516748\pi$
$499$	−2.34961	−0.105183	−0.0525916	+	0.998616i	$0.516748\pi$
$500$	−5.57509	−0.249325
$501$	0	0
$502$	36.9281	1.64818
$503$	−6.35233	−0.283237	−0.141618	−	0.989921i	$-0.545231\pi$
$503$	−6.35233	−0.283237	−0.141618	+	0.989921i	$0.545231\pi$
$504$	0	0
$505$	0.539052	0.0239875
$506$	0	0
$507$	0	0
$508$	24.1435	1.07119
$509$	−30.0048	−1.32994	−0.664970	−	0.746870i	$-0.731556\pi$
$509$	−30.0048	−1.32994	−0.664970	+	0.746870i	$0.731556\pi$
$510$	0	0
$511$	22.2927	0.986172
$512$	−71.5187	−3.16071
$513$	0	0
$514$	26.4593	1.16707
$515$	6.11063	0.269267
$516$	0	0
$517$	0	0
$518$	17.3041	0.760300
$519$	0	0
$520$	−27.3033	−1.19733
$521$	23.9434	1.04898	0.524489	−	0.851417i	$-0.324256\pi$
$521$	23.9434	1.04898	0.524489	+	0.851417i	$0.324256\pi$
$522$	0	0
$523$	−16.9932	−0.743060	−0.371530	−	0.928421i	$-0.621167\pi$
$523$	−16.9932	−0.743060	−0.371530	+	0.928421i	$0.621167\pi$
$524$	65.4022	2.85711
$525$	0	0
$526$	−64.4526	−2.81026
$527$	20.6726	0.900511
$528$	0	0
$529$	−21.1050	−0.917609
$530$	1.49671	0.0650131
$531$	0	0
$532$	−64.7576	−2.80760
$533$	23.9170	1.03596
$534$	0	0
$535$	9.87705	0.427022
$536$	−35.9594	−1.55321
$537$	0	0
$538$	26.0424	1.12277
$539$	0	0
$540$	0	0
$541$	1.24537	0.0535427	0.0267714	−	0.999642i	$-0.491477\pi$
$541$	1.24537	0.0535427	0.0267714	+	0.999642i	$0.491477\pi$
$542$	−70.3097	−3.02006
$543$	0	0
$544$	−91.3855	−3.91812
$545$	5.44683	0.233317
$546$	0	0
$547$	7.71616	0.329919	0.164960	−	0.986300i	$-0.447251\pi$
$547$	7.71616	0.329919	0.164960	+	0.986300i	$0.447251\pi$
$548$	−75.7327	−3.23514
$549$	0	0
$550$	0	0
$551$	37.7027	1.60619
$552$	0	0
$553$	−9.41696	−0.400450
$554$	12.7990	0.543779
$555$	0	0
$556$	−67.0804	−2.84484
$557$	−10.9219	−0.462775	−0.231388	−	0.972862i	$-0.574327\pi$
$557$	−10.9219	−0.462775	−0.231388	+	0.972862i	$0.574327\pi$
$558$	0	0
$559$	−34.4025	−1.45507
$560$	−43.5225	−1.83916
$561$	0	0
$562$	29.7366	1.25436
$563$	−12.2033	−0.514309	−0.257154	−	0.966370i	$-0.582785\pi$
$563$	−12.2033	−0.514309	−0.257154	+	0.966370i	$0.582785\pi$
$564$	0	0
$565$	0.777605	0.0327141
$566$	−5.52293	−0.232146
$567$	0	0
$568$	−90.1860	−3.78412
$569$	−6.66849	−0.279558	−0.139779	−	0.990183i	$-0.544639\pi$
$569$	−6.66849	−0.279558	−0.139779	+	0.990183i	$0.544639\pi$
$570$	0	0
$571$	−9.35392	−0.391450	−0.195725	−	0.980659i	$-0.562706\pi$
$571$	−9.35392	−0.391450	−0.195725	+	0.980659i	$0.562706\pi$
$572$	0	0
$573$	0	0
$574$	64.8074	2.70501
$575$	1.37658	0.0574075
$576$	0	0
$577$	−10.9547	−0.456048	−0.228024	−	0.973656i	$-0.573226\pi$
$577$	−10.9547	−0.456048	−0.228024	+	0.973656i	$0.573226\pi$
$578$	7.43851	0.309401
$579$	0	0
$580$	49.4362	2.05273
$581$	12.7161	0.527553
$582$	0	0
$583$	0	0
$584$	−80.2941	−3.32259
$585$	0	0
$586$	−1.65841	−0.0685083
$587$	23.9718	0.989420	0.494710	−	0.869058i	$-0.335274\pi$
$587$	23.9718	0.989420	0.494710	+	0.869058i	$0.335274\pi$
$588$	0	0
$589$	−23.2459	−0.957830
$590$	−1.07141	−0.0441091
$591$	0	0
$592$	−36.6649	−1.50692
$593$	2.17705	0.0894006	0.0447003	−	0.999000i	$-0.485767\pi$
$593$	2.17705	0.0894006	0.0447003	+	0.999000i	$0.485767\pi$
$594$	0	0
$595$	−10.3297	−0.423476
$596$	76.8993	3.14992
$597$	0	0
$598$	10.5131	0.429914
$599$	28.7881	1.17625	0.588124	−	0.808770i	$-0.299867\pi$
$599$	28.7881	1.17625	0.588124	+	0.808770i	$0.299867\pi$
$600$	0	0
$601$	−10.4372	−0.425744	−0.212872	−	0.977080i	$-0.568282\pi$
$601$	−10.4372	−0.425744	−0.212872	+	0.977080i	$0.568282\pi$
$602$	−93.2199	−3.79936
$603$	0	0
$604$	−94.8115	−3.85782
$605$	0	0
$606$	0	0
$607$	29.3274	1.19036	0.595180	−	0.803592i	$-0.297081\pi$
$607$	29.3274	1.19036	0.595180	+	0.803592i	$0.297081\pi$
$608$	102.761	4.16751
$609$	0	0
$610$	−39.4716	−1.59816
$611$	−15.5200	−0.627873
$612$	0	0
$613$	−27.5911	−1.11439	−0.557197	−	0.830381i	$-0.688123\pi$
$613$	−27.5911	−1.11439	−0.557197	+	0.830381i	$0.688123\pi$
$614$	−84.6921	−3.41790
$615$	0	0
$616$	0	0
$617$	−46.9079	−1.88844	−0.944220	−	0.329315i	$-0.893182\pi$
$617$	−46.9079	−1.88844	−0.944220	+	0.329315i	$0.893182\pi$
$618$	0	0
$619$	−19.1067	−0.767964	−0.383982	−	0.923341i	$-0.625448\pi$
$619$	−19.1067	−0.767964	−0.383982	+	0.923341i	$0.625448\pi$
$620$	−30.4802	−1.22412
$621$	0	0
$622$	−3.97118	−0.159230
$623$	−18.0937	−0.724907
$624$	0	0
$625$	1.00000	0.0400000
$626$	−64.6767	−2.58500
$627$	0	0
$628$	−20.1306	−0.803298
$629$	−8.70209	−0.346975
$630$	0	0
$631$	−30.8127	−1.22663	−0.613317	−	0.789837i	$-0.710165\pi$
$631$	−30.8127	−1.22663	−0.613317	+	0.789837i	$0.710165\pi$
$632$	33.9181	1.34919
$633$	0	0
$634$	−39.4075	−1.56507
$635$	−4.33061	−0.171855
$636$	0	0
$637$	−1.28504	−0.0509152
$638$	0	0
$639$	0	0
$640$	47.0458	1.85965
$641$	5.27921	0.208516	0.104258	−	0.994550i	$-0.466753\pi$
$641$	5.27921	0.208516	0.104258	+	0.994550i	$0.466753\pi$
$642$	0	0
$643$	−43.7670	−1.72600	−0.863001	−	0.505203i	$-0.831418\pi$
$643$	−43.7670	−1.72600	−0.863001	+	0.505203i	$0.831418\pi$
$644$	20.9659	0.826173
$645$	0	0
$646$	44.2487	1.74094
$647$	−31.4016	−1.23453	−0.617263	−	0.786757i	$-0.711758\pi$
$647$	−31.4016	−1.23453	−0.617263	+	0.786757i	$0.711758\pi$
$648$	0	0
$649$	0	0
$650$	7.63712	0.299552
$651$	0	0
$652$	−20.2011	−0.791138
$653$	−32.2029	−1.26020	−0.630099	−	0.776514i	$-0.716986\pi$
$653$	−32.2029	−1.26020	−0.630099	+	0.776514i	$0.716986\pi$
$654$	0	0
$655$	−11.7312	−0.458375
$656$	−137.317	−5.36134
$657$	0	0
$658$	−42.0543	−1.63945
$659$	−20.7286	−0.807472	−0.403736	−	0.914875i	$-0.632289\pi$
$659$	−20.7286	−0.807472	−0.403736	+	0.914875i	$0.632289\pi$
$660$	0	0
$661$	−43.6387	−1.69735	−0.848673	−	0.528917i	$-0.822598\pi$
$661$	−43.6387	−1.69735	−0.848673	+	0.528917i	$0.822598\pi$
$662$	13.5765	0.527667
$663$	0	0
$664$	−45.8009	−1.77742
$665$	11.6155	0.450431
$666$	0	0
$667$	−12.2067	−0.472643
$668$	−12.6648	−0.490016
$669$	0	0
$670$	10.0583	0.388588
$671$	0	0
$672$	0	0
$673$	−28.7584	−1.10855	−0.554277	−	0.832333i	$-0.687005\pi$
$673$	−28.7584	−1.10855	−0.554277	+	0.832333i	$0.687005\pi$
$674$	−21.1722	−0.815523
$675$	0	0
$676$	−29.5499	−1.13653
$677$	13.9609	0.536559	0.268280	−	0.963341i	$-0.413545\pi$
$677$	13.9609	0.536559	0.268280	+	0.963341i	$0.413545\pi$
$678$	0	0
$679$	−45.3320	−1.73968
$680$	37.2056	1.42677
$681$	0	0
$682$	0	0
$683$	−12.1334	−0.464272	−0.232136	−	0.972683i	$-0.574571\pi$
$683$	−12.1334	−0.464272	−0.232136	+	0.972683i	$0.574571\pi$
$684$	0	0
$685$	13.5841	0.519023
$686$	49.1502	1.87656
$687$	0	0
$688$	197.519	7.53035
$689$	−1.50897	−0.0574871
$690$	0	0
$691$	−23.9834	−0.912373	−0.456187	−	0.889884i	$-0.650785\pi$
$691$	−23.9834	−0.912373	−0.456187	+	0.889884i	$0.650785\pi$
$692$	107.435	4.08405
$693$	0	0
$694$	47.2459	1.79343
$695$	12.0322	0.456407
$696$	0	0
$697$	−32.5911	−1.23447
$698$	−66.8270	−2.52944
$699$	0	0
$700$	15.2304	0.575655
$701$	42.7155	1.61334	0.806671	−	0.591001i	$-0.201267\pi$
$701$	42.7155	1.61334	0.806671	+	0.591001i	$0.201267\pi$
$702$	0	0
$703$	9.78532	0.369060
$704$	0	0
$705$	0	0
$706$	−16.0899	−0.605552
$707$	−1.47262	−0.0553835
$708$	0	0
$709$	−4.14512	−0.155673	−0.0778366	−	0.996966i	$-0.524801\pi$
$709$	−4.14512	−0.155673	−0.0778366	+	0.996966i	$0.524801\pi$
$710$	25.2263	0.946724
$711$	0	0
$712$	65.1699	2.44234
$713$	7.52609	0.281854
$714$	0	0
$715$	0	0
$716$	40.5244	1.51447
$717$	0	0
$718$	22.4157	0.836547
$719$	−26.3332	−0.982062	−0.491031	−	0.871142i	$-0.663380\pi$
$719$	−26.3332	−0.982062	−0.491031	+	0.871142i	$0.663380\pi$
$720$	0	0
$721$	−16.6934	−0.621696
$722$	2.53671	0.0944067
$723$	0	0
$724$	−45.2634	−1.68220
$725$	−8.86735	−0.329325
$726$	0	0
$727$	14.3772	0.533220	0.266610	−	0.963804i	$-0.414096\pi$
$727$	14.3772	0.533220	0.266610	+	0.963804i	$0.414096\pi$
$728$	74.5892	2.76446
$729$	0	0
$730$	22.4594	0.831258
$731$	46.8794	1.73390
$732$	0	0
$733$	34.4717	1.27324	0.636621	−	0.771177i	$-0.280332\pi$
$733$	34.4717	1.27324	0.636621	+	0.771177i	$0.280332\pi$
$734$	−62.4074	−2.30350
$735$	0	0
$736$	−33.2700	−1.22635
$737$	0	0
$738$	0	0
$739$	−5.01642	−0.184532	−0.0922659	−	0.995734i	$-0.529411\pi$
$739$	−5.01642	−0.184532	−0.0922659	+	0.995734i	$0.529411\pi$
$740$	12.8306	0.471663
$741$	0	0
$742$	−4.08882	−0.150105
$743$	−38.6622	−1.41838	−0.709190	−	0.705018i	$-0.750939\pi$
$743$	−38.6622	−1.41838	−0.709190	+	0.705018i	$0.750939\pi$
$744$	0	0
$745$	−13.7934	−0.505350
$746$	38.6879	1.41647
$747$	0	0
$748$	0	0
$749$	−26.9828	−0.985930
$750$	0	0
$751$	−5.20080	−0.189780	−0.0948899	−	0.995488i	$-0.530250\pi$
$751$	−5.20080	−0.189780	−0.0948899	+	0.995488i	$0.530250\pi$
$752$	89.1068	3.24939
$753$	0	0
$754$	−67.7210	−2.46625
$755$	17.0063	0.618922
$756$	0	0
$757$	6.19362	0.225111	0.112556	−	0.993645i	$-0.464096\pi$
$757$	6.19362	0.225111	0.112556	+	0.993645i	$0.464096\pi$
$758$	69.4048	2.52090
$759$	0	0
$760$	−41.8369	−1.51758
$761$	11.8213	0.428520	0.214260	−	0.976777i	$-0.431266\pi$
$761$	11.8213	0.428520	0.214260	+	0.976777i	$0.431266\pi$
$762$	0	0
$763$	−14.8800	−0.538693
$764$	111.620	4.03825
$765$	0	0
$766$	−67.5177	−2.43951
$767$	1.08018	0.0390030
$768$	0	0
$769$	2.73424	0.0985992	0.0492996	−	0.998784i	$-0.484301\pi$
$769$	2.73424	0.0985992	0.0492996	+	0.998784i	$0.484301\pi$
$770$	0	0
$771$	0	0
$772$	−109.118	−3.92723
$773$	1.54037	0.0554032	0.0277016	−	0.999616i	$-0.491181\pi$
$773$	1.54037	0.0554032	0.0277016	+	0.999616i	$0.491181\pi$
$774$	0	0
$775$	5.46722	0.196388
$776$	163.277	5.86130
$777$	0	0
$778$	66.4973	2.38404
$779$	36.6480	1.31305
$780$	0	0
$781$	0	0
$782$	−14.3260	−0.512296
$783$	0	0
$784$	7.37796	0.263498
$785$	3.61081	0.128875
$786$	0	0
$787$	−47.4293	−1.69067	−0.845336	−	0.534235i	$-0.820600\pi$
$787$	−47.4293	−1.69067	−0.845336	+	0.534235i	$0.820600\pi$
$788$	−127.694	−4.54892
$789$	0	0
$790$	−9.48735	−0.337545
$791$	−2.12432	−0.0755320
$792$	0	0
$793$	39.7948	1.41315
$794$	81.1045	2.87829
$795$	0	0
$796$	−96.8230	−3.43180
$797$	−51.5701	−1.82671	−0.913353	−	0.407170i	$-0.866516\pi$
$797$	−51.5701	−1.82671	−0.913353	+	0.407170i	$0.866516\pi$
$798$	0	0
$799$	21.1487	0.748188
$800$	−24.1685	−0.854485
$801$	0	0
$802$	−12.5997	−0.444912
$803$	0	0
$804$	0	0
$805$	−3.76065	−0.132545
$806$	41.7538	1.47072
$807$	0	0
$808$	5.30409	0.186597
$809$	−27.5009	−0.966881	−0.483440	−	0.875377i	$-0.660613\pi$
$809$	−27.5009	−0.966881	−0.483440	+	0.875377i	$0.660613\pi$
$810$	0	0
$811$	17.5146	0.615020	0.307510	−	0.951545i	$-0.400504\pi$
$811$	17.5146	0.615020	0.307510	+	0.951545i	$0.400504\pi$
$812$	−135.053	−4.73944
$813$	0	0
$814$	0	0
$815$	3.62347	0.126925
$816$	0	0
$817$	−52.7150	−1.84426
$818$	52.3247	1.82949
$819$	0	0
$820$	48.0532	1.67809
$821$	−12.7216	−0.443986	−0.221993	−	0.975048i	$-0.571256\pi$
$821$	−12.7216	−0.443986	−0.221993	+	0.975048i	$0.571256\pi$
$822$	0	0
$823$	12.5648	0.437981	0.218991	−	0.975727i	$-0.429724\pi$
$823$	12.5648	0.437981	0.218991	+	0.975727i	$0.429724\pi$
$824$	60.1266	2.09461
$825$	0	0
$826$	2.92694	0.101841
$827$	−27.6527	−0.961579	−0.480790	−	0.876836i	$-0.659650\pi$
$827$	−27.6527	−0.961579	−0.480790	+	0.876836i	$0.659650\pi$
$828$	0	0
$829$	31.8545	1.10635	0.553176	−	0.833064i	$-0.313416\pi$
$829$	31.8545	1.10635	0.553176	+	0.833064i	$0.313416\pi$
$830$	12.8111	0.444681
$831$	0	0
$832$	−96.1638	−3.33388
$833$	1.75109	0.0606718
$834$	0	0
$835$	2.27168	0.0786147
$836$	0	0
$837$	0	0
$838$	−6.99367	−0.241592
$839$	23.1323	0.798617	0.399308	−	0.916817i	$-0.369250\pi$
$839$	23.1323	0.798617	0.399308	+	0.916817i	$0.369250\pi$
$840$	0	0
$841$	49.6299	1.71138
$842$	−5.65248	−0.194797
$843$	0	0
$844$	92.6054	3.18761
$845$	5.30035	0.182337
$846$	0	0
$847$	0	0
$848$	8.66361	0.297510
$849$	0	0
$850$	−10.4069	−0.356954
$851$	−3.16810	−0.108601
$852$	0	0
$853$	−14.0442	−0.480863	−0.240432	−	0.970666i	$-0.577289\pi$
$853$	−14.0442	−0.480863	−0.240432	+	0.970666i	$0.577289\pi$
$854$	107.831	3.68991
$855$	0	0
$856$	97.1869	3.32178
$857$	−26.1013	−0.891603	−0.445801	−	0.895132i	$-0.647081\pi$
$857$	−26.1013	−0.891603	−0.445801	+	0.895132i	$0.647081\pi$
$858$	0	0
$859$	1.98117	0.0675967	0.0337983	−	0.999429i	$-0.489240\pi$
$859$	1.98117	0.0675967	0.0337983	+	0.999429i	$0.489240\pi$
$860$	−69.1204	−2.35699
$861$	0	0
$862$	104.287	3.55204
$863$	−19.4162	−0.660935	−0.330467	−	0.943817i	$-0.607206\pi$
$863$	−19.4162	−0.660935	−0.330467	+	0.943817i	$0.607206\pi$
$864$	0	0
$865$	−19.2705	−0.655216
$866$	−25.0638	−0.851704
$867$	0	0
$868$	83.2680	2.82630
$869$	0	0
$870$	0	0
$871$	−10.1407	−0.343605
$872$	53.5950	1.81495
$873$	0	0
$874$	16.1093	0.544904
$875$	−2.73187	−0.0923540
$876$	0	0
$877$	−10.2208	−0.345132	−0.172566	−	0.984998i	$-0.555206\pi$
$877$	−10.2208	−0.345132	−0.172566	+	0.984998i	$0.555206\pi$
$878$	−85.5067	−2.88571
$879$	0	0
$880$	0	0
$881$	26.4278	0.890377	0.445188	−	0.895437i	$-0.353137\pi$
$881$	26.4278	0.890377	0.445188	+	0.895437i	$0.353137\pi$
$882$	0	0
$883$	44.1883	1.48705	0.743527	−	0.668706i	$-0.233152\pi$
$883$	44.1883	1.48705	0.743527	+	0.668706i	$0.233152\pi$
$884$	−58.4945	−1.96738
$885$	0	0
$886$	−41.0402	−1.37877
$887$	−8.80191	−0.295539	−0.147770	−	0.989022i	$-0.547209\pi$
$887$	−8.80191	−0.295539	−0.147770	+	0.989022i	$0.547209\pi$
$888$	0	0
$889$	11.8306	0.396787
$890$	−18.2289	−0.611034
$891$	0	0
$892$	−49.9554	−1.67263
$893$	−23.7813	−0.795811
$894$	0	0
$895$	−7.26883	−0.242970
$896$	−128.523	−4.29365
$897$	0	0
$898$	70.4858	2.35214
$899$	−48.4798	−1.61689
$900$	0	0
$901$	2.05623	0.0685031
$902$	0	0
$903$	0	0
$904$	7.65138	0.254481
$905$	8.11886	0.269880
$906$	0	0
$907$	18.8345	0.625389	0.312695	−	0.949854i	$-0.398768\pi$
$907$	18.8345	0.625389	0.312695	+	0.949854i	$0.398768\pi$
$908$	−5.91820	−0.196402
$909$	0	0
$910$	−20.8636	−0.691622
$911$	22.0961	0.732077	0.366038	−	0.930600i	$-0.380714\pi$
$911$	22.0961	0.732077	0.366038	+	0.930600i	$0.380714\pi$
$912$	0	0
$913$	0	0
$914$	−22.0903	−0.730681
$915$	0	0
$916$	−32.6340	−1.07826
$917$	32.0480	1.05832
$918$	0	0
$919$	4.04875	0.133556	0.0667779	−	0.997768i	$-0.478728\pi$
$919$	4.04875	0.133556	0.0667779	+	0.997768i	$0.478728\pi$
$920$	13.5451	0.446569
$921$	0	0
$922$	46.4745	1.53056
$923$	−25.4328	−0.837132
$924$	0	0
$925$	−2.30142	−0.0756703
$926$	12.9621	0.425962
$927$	0	0
$928$	214.311	7.03509
$929$	−40.2897	−1.32186	−0.660931	−	0.750447i	$-0.729838\pi$
$929$	−40.2897	−1.32186	−0.660931	+	0.750447i	$0.729838\pi$
$930$	0	0
$931$	−1.96907	−0.0645336
$932$	−10.4657	−0.342814
$933$	0	0
$934$	42.4261	1.38822
$935$	0	0
$936$	0	0
$937$	36.9767	1.20798	0.603988	−	0.796994i	$-0.293578\pi$
$937$	36.9767	1.20798	0.603988	+	0.796994i	$0.293578\pi$
$938$	−27.4781	−0.897191
$939$	0	0
$940$	−31.1823	−1.01705
$941$	−43.1362	−1.40620	−0.703100	−	0.711091i	$-0.748201\pi$
$941$	−43.1362	−1.40620	−0.703100	+	0.711091i	$0.748201\pi$
$942$	0	0
$943$	−11.8652	−0.386383
$944$	−6.20176	−0.201850
$945$	0	0
$946$	0	0
$947$	−42.5933	−1.38410	−0.692048	−	0.721851i	$-0.743291\pi$
$947$	−42.5933	−1.38410	−0.692048	+	0.721851i	$0.743291\pi$
$948$	0	0
$949$	−22.6433	−0.735031
$950$	11.7023	0.379674
$951$	0	0
$952$	−101.641	−3.29419
$953$	−27.3349	−0.885466	−0.442733	−	0.896654i	$-0.645991\pi$
$953$	−27.3349	−0.885466	−0.442733	+	0.896654i	$0.645991\pi$
$954$	0	0
$955$	−20.0211	−0.647869
$956$	−11.3764	−0.367939
$957$	0	0
$958$	−27.0984	−0.875509
$959$	−37.1101	−1.19835
$960$	0	0
$961$	−1.10947	−0.0357893
$962$	−17.5762	−0.566680
$963$	0	0
$964$	85.8130	2.76385
$965$	19.5724	0.630057
$966$	0	0
$967$	20.7000	0.665668	0.332834	−	0.942985i	$-0.391995\pi$
$967$	20.7000	0.665668	0.332834	+	0.942985i	$0.391995\pi$
$968$	0	0
$969$	0	0
$970$	−45.6708	−1.46640
$971$	27.5250	0.883318	0.441659	−	0.897183i	$-0.354390\pi$
$971$	27.5250	0.883318	0.441659	+	0.897183i	$0.354390\pi$
$972$	0	0
$973$	−32.8703	−1.05377
$974$	22.4922	0.720697
$975$	0	0
$976$	−228.478	−7.31341
$977$	36.4948	1.16757	0.583786	−	0.811908i	$-0.301571\pi$
$977$	36.4948	1.16757	0.583786	+	0.811908i	$0.301571\pi$
$978$	0	0
$979$	0	0
$980$	−2.58186	−0.0824746
$981$	0	0
$982$	106.176	3.38822
$983$	49.1047	1.56620	0.783098	−	0.621899i	$-0.213638\pi$
$983$	49.1047	1.56620	0.783098	+	0.621899i	$0.213638\pi$
$984$	0	0
$985$	22.9044	0.729796
$986$	92.2816	2.93885
$987$	0	0
$988$	65.7758	2.09261
$989$	17.0670	0.542699
$990$	0	0
$991$	−5.08586	−0.161558	−0.0807788	−	0.996732i	$-0.525741\pi$
$991$	−5.08586	−0.161558	−0.0807788	+	0.996732i	$0.525741\pi$
$992$	−132.135	−4.19528
$993$	0	0
$994$	−68.9148	−2.18585
$995$	17.3671	0.550573
$996$	0	0
$997$	−18.3350	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$997$	−18.3350	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$998$	6.46681	0.204703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.ca.1.1		8
3.2	odd	2		5445.2.a.cd.1.8		8
11.3	even	5		495.2.n.h.361.1	yes	16
11.4	even	5		495.2.n.h.181.1	yes	16
11.10	odd	2		5445.2.a.cc.1.8		8
33.14	odd	10		495.2.n.g.361.4	yes	16
33.26	odd	10		495.2.n.g.181.4	✓	16
33.32	even	2		5445.2.a.cb.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.181.4	✓	16	33.26	odd	10
495.2.n.g.361.4	yes	16	33.14	odd	10
495.2.n.h.181.1	yes	16	11.4	even	5
495.2.n.h.361.1	yes	16	11.3	even	5
5445.2.a.ca.1.1		8	1.1	even	1	trivial
5445.2.a.cb.1.1		8	33.32	even	2
5445.2.a.cc.1.8		8	11.10	odd	2
5445.2.a.cd.1.8		8	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75229 q^{2} +5.57509 q^{4} -1.00000 q^{5} +2.73187 q^{7} -9.83966 q^{8} +O(q^{10})\)	\(q-2.75229 q^{2} +5.57509 q^{4} -1.00000 q^{5} +2.73187 q^{7} -9.83966 q^{8} +2.75229 q^{10} -2.77482 q^{13} -7.51889 q^{14} +15.9314 q^{16} +3.78118 q^{17} -4.25186 q^{19} -5.57509 q^{20} +1.37658 q^{23} +1.00000 q^{25} +7.63712 q^{26} +15.2304 q^{28} -8.86735 q^{29} +5.46722 q^{31} -24.1685 q^{32} -10.4069 q^{34} -2.73187 q^{35} -2.30142 q^{37} +11.7023 q^{38} +9.83966 q^{40} -8.61928 q^{41} +12.3981 q^{43} -3.78875 q^{46} +5.59315 q^{47} +0.463108 q^{49} -2.75229 q^{50} -15.4699 q^{52} +0.543807 q^{53} -26.8807 q^{56} +24.4055 q^{58} -0.389279 q^{59} -14.3414 q^{61} -15.0474 q^{62} +34.6558 q^{64} +2.77482 q^{65} +3.65454 q^{67} +21.0804 q^{68} +7.51889 q^{70} +9.16556 q^{71} +8.16025 q^{73} +6.33417 q^{74} -23.7045 q^{76} -3.44708 q^{79} -15.9314 q^{80} +23.7227 q^{82} +4.65472 q^{83} -3.78118 q^{85} -34.1231 q^{86} -6.62318 q^{89} -7.58046 q^{91} +7.67457 q^{92} -15.3940 q^{94} +4.25186 q^{95} -16.5938 q^{97} -1.27461 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 + 6 * q^13 - 14 * q^14 + 14 * q^16 - 8 * q^17 - 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 + 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 + 20 * q^38 + 12 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^46 - 10 * q^47 - 4 * q^50 - 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 - 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 + 2 * q^71 + 16 * q^73 - 24 * q^74 - 22 * q^76 - 16 * q^79 - 14 * q^80 + 10 * q^82 - 46 * q^83 + 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 - 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * q^98

Introduction

L-functions

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