LMFDB - Embedded newform 5445.2.a.cb.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	$-0.672682\pi$
$7$	−2.73187	−1.03255	−0.516275	+	0.856423i	$0.672682\pi$
$8$	−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	$-0.374271\pi$
$13$	2.77482	0.769598	0.384799	+	0.923000i	$0.374271\pi$
$14$	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	$-0.337828\pi$
$19$	4.25186	0.975444	0.487722	+	0.872999i	$0.337828\pi$
$20$	5.57509	1.24663
$21$	0	0
$22$	0	0
$23$	−1.37658	−0.287038	−0.143519	−	0.989648i	$-0.545842\pi$
$23$	−1.37658	−0.287038	−0.143519	+	0.989648i	$0.545842\pi$
$24$	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	$-0.894276\pi$
$43$	−12.3981	−1.89069	−0.945346	+	0.326070i	$0.894276\pi$
$44$	0	0
$45$	0	0
$46$	3.78875	0.558621
$47$	−5.59315	−0.815845	−0.407923	−	0.913017i	$-0.633747\pi$
$47$	−5.59315	−0.815845	−0.407923	+	0.913017i	$0.633747\pi$
$48$	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	$-0.511891\pi$
$53$	−0.543807	−0.0746976	−0.0373488	+	0.999302i	$0.511891\pi$
$54$	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	$-0.491933\pi$
$59$	0.389279	0.0506797	0.0253399	+	0.999679i	$0.491933\pi$
$60$	0	0
$61$	14.3414	1.83622	0.918112	−	0.396320i	$-0.129713\pi$
$61$	14.3414	1.83622	0.918112	+	0.396320i	$0.129713\pi$
$62$	−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	$-0.683044\pi$
$71$	−9.16556	−1.08775	−0.543876	+	0.839166i	$0.683044\pi$
$72$	0	0
$73$	−8.16025	−0.955085	−0.477542	−	0.878609i	$-0.658472\pi$
$73$	−8.16025	−0.955085	−0.477542	+	0.878609i	$0.658472\pi$
$74$	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	$-0.437882\pi$
$79$	3.44708	0.387827	0.193913	+	0.981019i	$0.437882\pi$
$80$	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	$-0.385832\pi$
$89$	6.62318	0.702056	0.351028	+	0.936365i	$0.385832\pi$
$90$	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	$-0.415995\pi$
$109$	5.44683	0.521712	0.260856	+	0.965378i	$0.415995\pi$
$110$	0	0
$111$	0	0
$112$	−43.5225	−4.11249
$113$	0.777605	0.0731510	0.0365755	−	0.999331i	$-0.488355\pi$
$113$	0.777605	0.0731510	0.0365755	+	0.999331i	$0.488355\pi$
$114$	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	$-0.561543\pi$
$127$	−4.33061	−0.384279	−0.192140	+	0.981368i	$0.561543\pi$
$128$	−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	$-0.302941\pi$
$137$	13.5841	1.16057	0.580285	+	0.814413i	$0.302941\pi$
$138$	0	0
$139$	12.0322	1.02056	0.510278	−	0.860009i	$-0.329542\pi$
$139$	12.0322	1.02056	0.510278	+	0.860009i	$0.329542\pi$
$140$	−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	$-0.256741\pi$
$151$	17.0063	1.38395	0.691976	+	0.721921i	$0.256741\pi$
$152$	−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	$-0.587569\pi$
$179$	−7.26883	−0.543298	−0.271649	+	0.962396i	$0.587569\pi$
$180$	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	$-0.757855\pi$
$191$	−20.0211	−1.44868	−0.724339	+	0.689444i	$0.757855\pi$
$192$	0	0
$193$	19.5724	1.40885	0.704425	−	0.709779i	$-0.251205\pi$
$193$	19.5724	1.40885	0.704425	+	0.709779i	$0.251205\pi$
$194$	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	$-0.693739\pi$
$211$	−16.6106	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$212$	−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	$-0.665107\pi$
$241$	−15.3922	−0.991500	−0.495750	+	0.868465i	$0.665107\pi$
$242$	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	$-0.360821\pi$
$251$	13.4172	0.846889	0.423444	+	0.905922i	$0.360821\pi$
$252$	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	$-0.403067\pi$
$257$	9.61357	0.599678	0.299839	+	0.953990i	$0.403067\pi$
$258$	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	$-0.406858\pi$
$269$	9.46210	0.576914	0.288457	+	0.957493i	$0.406858\pi$
$270$	0	0
$271$	−25.5459	−1.55180	−0.775902	−	0.630853i	$-0.782705\pi$
$271$	−25.5459	−1.55180	−0.775902	+	0.630853i	$0.782705\pi$
$272$	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	$-0.455384\pi$
$277$	4.65032	0.279411	0.139705	+	0.990193i	$0.455384\pi$
$278$	−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	$-0.518996\pi$
$283$	−2.00667	−0.119284	−0.0596420	+	0.998220i	$0.518996\pi$
$284$	−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	$-0.841197\pi$
$307$	−30.7715	−1.75622	−0.878112	+	0.478454i	$0.841197\pi$
$308$	0	0
$309$	0	0
$310$	−15.0474	−0.854633
$311$	−1.44287	−0.0818175	−0.0409087	−	0.999163i	$-0.513025\pi$
$311$	−1.44287	−0.0818175	−0.0409087	+	0.999163i	$0.513025\pi$
$312$	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	$-0.631717\pi$
$317$	−14.3181	−0.804184	−0.402092	+	0.915599i	$0.631717\pi$
$318$	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	$-0.567190\pi$
$337$	−7.69258	−0.419042	−0.209521	+	0.977804i	$0.567190\pi$
$338$	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	$-0.725170\pi$
$349$	−24.2805	−1.29971	−0.649853	+	0.760060i	$0.725170\pi$
$350$	7.51889	0.401901
$351$	0	0
$352$	0	0
$353$	−5.84602	−0.311152	−0.155576	−	0.987824i	$-0.549723\pi$
$353$	−5.84602	−0.311152	−0.155576	+	0.987824i	$0.549723\pi$
$354$	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	$-0.381441\pi$
$373$	14.0566	0.727825	0.363913	+	0.931433i	$0.381441\pi$
$374$	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	$-0.715615\pi$
$383$	−24.5315	−1.25350	−0.626750	+	0.779220i	$0.715615\pi$
$384$	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	$-0.290164\pi$
$389$	24.1607	1.22500	0.612499	+	0.790472i	$0.290164\pi$
$390$	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	$-0.536464\pi$
$401$	−4.57792	−0.228610	−0.114305	+	0.993446i	$0.536464\pi$
$402$	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	$-0.344245\pi$
$409$	19.0113	0.940050	0.470025	+	0.882653i	$0.344245\pi$
$410$	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	$-0.519770\pi$
$419$	−2.54104	−0.124138	−0.0620690	+	0.998072i	$0.519770\pi$
$420$	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	$-0.765831\pi$
$439$	−31.0675	−1.48277	−0.741385	+	0.671079i	$0.765831\pi$
$440$	0	0
$441$	0	0
$442$	−28.8773	−1.37355
$443$	−14.9113	−0.708458	−0.354229	−	0.935159i	$-0.615257\pi$
$443$	−14.9113	−0.708458	−0.354229	+	0.935159i	$0.615257\pi$
$444$	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	$-0.293452\pi$
$449$	25.6099	1.20861	0.604303	+	0.796755i	$0.293452\pi$
$450$	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	$-0.560111\pi$
$457$	−8.02615	−0.375447	−0.187724	+	0.982222i	$0.560111\pi$
$458$	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	$-0.383916\pi$
$467$	15.4149	0.713314	0.356657	+	0.934235i	$0.383916\pi$
$468$	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	$-0.268444\pi$
$509$	30.0048	1.32994	0.664970	+	0.746870i	$0.268444\pi$
$510$	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	$-0.675744\pi$
$521$	−23.9434	−1.04898	−0.524489	+	0.851417i	$0.675744\pi$
$522$	0	0
$523$	16.9932	0.743060	0.371530	−	0.928421i	$-0.378833\pi$
$523$	16.9932	0.743060	0.371530	+	0.928421i	$0.378833\pi$
$524$	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	$-0.508523\pi$
$541$	−1.24537	−0.0535427	−0.0267714	+	0.999642i	$0.508523\pi$
$542$	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	$-0.552749\pi$
$547$	−7.71616	−0.329919	−0.164960	+	0.986300i	$0.552749\pi$
$548$	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	$-0.437294\pi$
$571$	9.35392	0.391450	0.195725	+	0.980659i	$0.437294\pi$
$572$	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	$-0.664726\pi$
$587$	−23.9718	−0.989420	−0.494710	+	0.869058i	$0.664726\pi$
$588$	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	$-0.700133\pi$
$599$	−28.7881	−1.17625	−0.588124	+	0.808770i	$0.700133\pi$
$600$	0	0
$601$	10.4372	0.425744	0.212872	−	0.977080i	$-0.431718\pi$
$601$	10.4372	0.425744	0.212872	+	0.977080i	$0.431718\pi$
$602$	−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	$-0.702919\pi$
$607$	−29.3274	−1.19036	−0.595180	+	0.803592i	$0.702919\pi$
$608$	−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	$-0.311877\pi$
$613$	27.5911	1.11439	0.557197	+	0.830381i	$0.311877\pi$
$614$	84.6921	3.41790
$615$	0	0
$616$	0	0
$617$	46.9079	1.88844	0.944220	−	0.329315i	$-0.106818\pi$
$617$	46.9079	1.88844	0.944220	+	0.329315i	$0.106818\pi$
$618$	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	$-0.533247\pi$
$641$	−5.27921	−0.208516	−0.104258	+	0.994550i	$0.533247\pi$
$642$	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	$-0.288242\pi$
$647$	31.4016	1.23453	0.617263	+	0.786757i	$0.288242\pi$
$648$	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	$-0.283014\pi$
$653$	32.2029	1.26020	0.630099	+	0.776514i	$0.283014\pi$
$654$	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	$-0.312995\pi$
$673$	28.7584	1.10855	0.554277	+	0.832333i	$0.312995\pi$
$674$	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	$-0.425429\pi$
$683$	12.1334	0.464272	0.232136	+	0.972683i	$0.425429\pi$
$684$	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	$-0.336620\pi$
$719$	26.3332	0.982062	0.491031	+	0.871142i	$0.336620\pi$
$720$	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	$-0.719668\pi$
$733$	−34.4717	−1.27324	−0.636621	+	0.771177i	$0.719668\pi$
$734$	−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	$-0.470589\pi$
$739$	5.01642	0.184532	0.0922659	+	0.995734i	$0.470589\pi$
$740$	−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	$-0.515699\pi$
$769$	−2.73424	−0.0985992	−0.0492996	+	0.998784i	$0.515699\pi$
$770$	0	0
$771$	0	0
$772$	109.118	3.92723
$773$	−1.54037	−0.0554032	−0.0277016	−	0.999616i	$-0.508819\pi$
$773$	−1.54037	−0.0554032	−0.0277016	+	0.999616i	$0.508819\pi$
$774$	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	$-0.179400\pi$
$787$	47.4293	1.69067	0.845336	+	0.534235i	$0.179400\pi$
$788$	−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	$-0.133484\pi$
$797$	51.5701	1.82671	0.913353	+	0.407170i	$0.133484\pi$
$798$	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	$-0.599496\pi$
$811$	−17.5146	−0.615020	−0.307510	+	0.951545i	$0.599496\pi$
$812$	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	$-0.630750\pi$
$839$	−23.1323	−0.798617	−0.399308	+	0.916817i	$0.630750\pi$
$840$	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	$-0.422711\pi$
$853$	14.0442	0.480863	0.240432	+	0.970666i	$0.422711\pi$
$854$	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	$-0.392794\pi$
$863$	19.4162	0.660935	0.330467	+	0.943817i	$0.392794\pi$
$864$	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	$-0.444794\pi$
$877$	10.2208	0.345132	0.172566	+	0.984998i	$0.444794\pi$
$878$	85.5067	2.88571
$879$	0	0
$880$	0	0
$881$	−26.4278	−0.890377	−0.445188	−	0.895437i	$-0.646863\pi$
$881$	−26.4278	−0.890377	−0.445188	+	0.895437i	$0.646863\pi$
$882$	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	$-0.619286\pi$
$911$	−22.0961	−0.732077	−0.366038	+	0.930600i	$0.619286\pi$
$912$	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	$-0.521272\pi$
$919$	−4.04875	−0.133556	−0.0667779	+	0.997768i	$0.521272\pi$
$920$	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	$-0.270162\pi$
$929$	40.2897	1.32186	0.660931	+	0.750447i	$0.270162\pi$
$930$	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	$-0.706422\pi$
$937$	−36.9767	−1.20798	−0.603988	+	0.796994i	$0.706422\pi$
$938$	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	$-0.256709\pi$
$947$	42.5933	1.38410	0.692048	+	0.721851i	$0.256709\pi$
$948$	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	$-0.608005\pi$
$967$	−20.7000	−0.665668	−0.332834	+	0.942985i	$0.608005\pi$
$968$	0	0
$969$	0	0
$970$	45.6708	1.46640
$971$	−27.5250	−0.883318	−0.441659	−	0.897183i	$-0.645610\pi$
$971$	−27.5250	−0.883318	−0.441659	+	0.897183i	$0.645610\pi$
$972$	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	$-0.698429\pi$
$977$	−36.4948	−1.16757	−0.583786	+	0.811908i	$0.698429\pi$
$978$	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	$-0.786362\pi$
$983$	−49.1047	−1.56620	−0.783098	+	0.621899i	$0.786362\pi$
$984$	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	$-0.406232\pi$
$997$	18.3350	0.580677	0.290338	+	0.956924i	$0.406232\pi$
$998$	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.cb.1.1		8
3.2	odd	2		5445.2.a.cc.1.8		8
11.7	odd	10		495.2.n.g.181.4	✓	16
11.8	odd	10		495.2.n.g.361.4	yes	16
11.10	odd	2		5445.2.a.cd.1.8		8
33.8	even	10		495.2.n.h.361.1	yes	16
33.29	even	10		495.2.n.h.181.1	yes	16
33.32	even	2		5445.2.a.ca.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.181.4	✓	16	11.7	odd	10
495.2.n.g.361.4	yes	16	11.8	odd	10
495.2.n.h.181.1	yes	16	33.29	even	10
495.2.n.h.361.1	yes	16	33.8	even	10
5445.2.a.ca.1.1		8	33.32	even	2
5445.2.a.cb.1.1		8	1.1	even	1	trivial
5445.2.a.cc.1.8		8	3.2	odd	2
5445.2.a.cd.1.8		8	11.10	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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