LMFDB - Embedded newform 5445.2.a.bx.1.6

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.27433728.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 15x^{2} - 3 $ x^6 - 9x^4 + 15x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.62383$ 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)

$f(q)$	$=$	$q+2.62383 q^{2} +4.88448 q^{4} -1.00000 q^{5} +2.62383 q^{7} +7.56839 q^{8} +O(q^{10})$	$q+2.62383 q^{2} +4.88448 q^{4} -1.00000 q^{5} +2.62383 q^{7} +7.56839 q^{8} -2.62383 q^{10} +6.72812 q^{13} +6.88448 q^{14} +10.0892 q^{16} +1.78356 q^{17} +1.48046 q^{19} -4.88448 q^{20} -5.20473 q^{23} +1.00000 q^{25} +17.6535 q^{26} +12.8161 q^{28} +1.17737 q^{29} -4.56424 q^{31} +11.3356 q^{32} +4.67975 q^{34} -2.62383 q^{35} -0.679754 q^{37} +3.88448 q^{38} -7.56839 q^{40} +5.49925 q^{41} -3.80120 q^{43} -13.6563 q^{46} -6.66012 q^{47} -0.115516 q^{49} +2.62383 q^{50} +32.8634 q^{52} -14.2939 q^{53} +19.8582 q^{56} +3.08921 q^{58} +3.88448 q^{59} -3.21251 q^{61} -11.9758 q^{62} +9.56424 q^{64} -6.72812 q^{65} +4.66012 q^{67} +8.71176 q^{68} -6.88448 q^{70} -3.88448 q^{71} +3.56712 q^{73} -1.78356 q^{74} +7.23130 q^{76} +15.1368 q^{79} -10.0892 q^{80} +14.4291 q^{82} +7.33431 q^{83} -1.78356 q^{85} -9.97370 q^{86} -3.44872 q^{89} +17.6535 q^{91} -25.4224 q^{92} -17.4750 q^{94} -1.48046 q^{95} +1.47502 q^{97} -0.303095 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 6 * q^4 - 6 * q^5	$ 6 q + 6 q^{4} - 6 q^{5} + 18 q^{14} + 18 q^{16} - 6 q^{20} - 12 q^{23} + 6 q^{25} + 36 q^{26} + 24 q^{34} - 42 q^{47} - 24 q^{49} - 24 q^{53} + 30 q^{56} - 24 q^{58} + 30 q^{64} + 30 q^{67} - 18 q^{70} - 18 q^{80} + 42 q^{82} + 6 q^{86} + 30 q^{89} + 36 q^{91} - 36 q^{92} + 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 - 6 * q^5 + 18 * q^14 + 18 * q^16 - 6 * q^20 - 12 * q^23 + 6 * q^25 + 36 * q^26 + 24 * q^34 - 42 * q^47 - 24 * q^49 - 24 * q^53 + 30 * q^56 - 24 * q^58 + 30 * q^64 + 30 * q^67 - 18 * q^70 - 18 * q^80 + 42 * q^82 + 6 * q^86 + 30 * q^89 + 36 * q^91 - 36 * q^92 + 24 * 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 $

$2$	2.62383	1.85533	0.927664	−	0.373416i	$-0.121814\pi$
$2$	2.62383	1.85533	0.927664	+	0.373416i	$0.121814\pi$
$3$	0	0
$3$	0	0
$4$	4.88448	2.44224
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.62383	0.991715	0.495857	−	0.868404i	$-0.334854\pi$
$7$	2.62383	0.991715	0.495857	+	0.868404i	$0.334854\pi$
$8$	7.56839	2.67583
$9$	0	0
$10$	−2.62383	−0.829728
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.72812	1.86605	0.933023	−	0.359817i	$-0.117161\pi$
$13$	6.72812	1.86605	0.933023	+	0.359817i	$0.117161\pi$
$14$	6.88448	1.83996
$15$	0	0
$16$	10.0892	2.52230
$17$	1.78356	0.432576	0.216288	−	0.976330i	$-0.430605\pi$
$17$	1.78356	0.432576	0.216288	+	0.976330i	$0.430605\pi$
$18$	0	0
$19$	1.48046	0.339642	0.169821	−	0.985475i	$-0.445681\pi$
$19$	1.48046	0.339642	0.169821	+	0.985475i	$0.445681\pi$
$20$	−4.88448	−1.09220
$21$	0	0
$22$	0	0
$23$	−5.20473	−1.08526	−0.542631	−	0.839971i	$-0.682572\pi$
$23$	−5.20473	−1.08526	−0.542631	+	0.839971i	$0.682572\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	17.6535	3.46213
$27$	0	0
$28$	12.8161	2.42201
$29$	1.17737	0.218632	0.109316	−	0.994007i	$-0.465134\pi$
$29$	1.17737	0.218632	0.109316	+	0.994007i	$0.465134\pi$
$30$	0	0
$31$	−4.56424	−0.819761	−0.409881	−	0.912139i	$-0.634430\pi$
$31$	−4.56424	−0.819761	−0.409881	+	0.912139i	$0.634430\pi$
$32$	11.3356	2.00387
$33$	0	0
$34$	4.67975	0.802571
$35$	−2.62383	−0.443508
$36$	0	0
$37$	−0.679754	−0.111751	−0.0558754	−	0.998438i	$-0.517795\pi$
$37$	−0.679754	−0.111751	−0.0558754	+	0.998438i	$0.517795\pi$
$38$	3.88448	0.630146
$39$	0	0
$40$	−7.56839	−1.19667
$41$	5.49925	0.858838	0.429419	−	0.903105i	$-0.358718\pi$
$41$	5.49925	0.858838	0.429419	+	0.903105i	$0.358718\pi$
$42$	0	0
$43$	−3.80120	−0.579677	−0.289839	−	0.957076i	$-0.593602\pi$
$43$	−3.80120	−0.579677	−0.289839	+	0.957076i	$0.593602\pi$
$44$	0	0
$45$	0	0
$46$	−13.6563	−2.01352
$47$	−6.66012	−0.971479	−0.485739	−	0.874104i	$-0.661450\pi$
$47$	−6.66012	−0.971479	−0.485739	+	0.874104i	$0.661450\pi$
$48$	0	0
$49$	−0.115516	−0.0165023
$50$	2.62383	0.371066
$51$	0	0
$52$	32.8634	4.55733
$53$	−14.2939	−1.96342	−0.981712	−	0.190372i	$-0.939031\pi$
$53$	−14.2939	−1.96342	−0.981712	+	0.190372i	$0.939031\pi$
$54$	0	0
$55$	0	0
$56$	19.8582	2.65366
$57$	0	0
$58$	3.08921	0.405634
$59$	3.88448	0.505717	0.252858	−	0.967503i	$-0.418629\pi$
$59$	3.88448	0.505717	0.252858	+	0.967503i	$0.418629\pi$
$60$	0	0
$61$	−3.21251	−0.411320	−0.205660	−	0.978623i	$-0.565934\pi$
$61$	−3.21251	−0.411320	−0.205660	+	0.978623i	$0.565934\pi$
$62$	−11.9758	−1.52093
$63$	0	0
$64$	9.56424	1.19553
$65$	−6.72812	−0.834521
$66$	0	0
$67$	4.66012	0.569325	0.284662	−	0.958628i	$-0.408119\pi$
$67$	4.66012	0.569325	0.284662	+	0.958628i	$0.408119\pi$
$68$	8.71176	1.05646
$69$	0	0
$70$	−6.88448	−0.822853
$71$	−3.88448	−0.461003	−0.230502	−	0.973072i	$-0.574037\pi$
$71$	−3.88448	−0.461003	−0.230502	+	0.973072i	$0.574037\pi$
$72$	0	0
$73$	3.56712	0.417499	0.208750	−	0.977969i	$-0.433061\pi$
$73$	3.56712	0.417499	0.208750	+	0.977969i	$0.433061\pi$
$74$	−1.78356	−0.207334
$75$	0	0
$76$	7.23130	0.829487
$77$	0	0
$78$	0	0
$79$	15.1368	1.70302	0.851511	−	0.524337i	$-0.175687\pi$
$79$	15.1368	1.70302	0.851511	+	0.524337i	$0.175687\pi$
$80$	−10.0892	−1.12801
$81$	0	0
$82$	14.4291	1.59343
$83$	7.33431	0.805045	0.402523	−	0.915410i	$-0.368133\pi$
$83$	7.33431	0.805045	0.402523	+	0.915410i	$0.368133\pi$
$84$	0	0
$85$	−1.78356	−0.193454
$86$	−9.97370	−1.07549
$87$	0	0
$88$	0	0
$89$	−3.44872	−0.365564	−0.182782	−	0.983153i	$-0.558510\pi$
$89$	−3.44872	−0.365564	−0.182782	+	0.983153i	$0.558510\pi$
$90$	0	0
$91$	17.6535	1.85058
$92$	−25.4224	−2.65047
$93$	0	0
$94$	−17.4750	−1.80241
$95$	−1.48046	−0.151892
$96$	0	0
$97$	1.47502	0.149766	0.0748830	−	0.997192i	$-0.476142\pi$
$97$	1.47502	0.149766	0.0748830	+	0.997192i	$0.476142\pi$
$98$	−0.303095	−0.0306172
$99$	0	0
$100$	4.88448	0.488448
$101$	−3.41259	−0.339566	−0.169783	−	0.985481i	$-0.554307\pi$
$101$	−3.41259	−0.339566	−0.169783	+	0.985481i	$0.554307\pi$
$102$	0	0
$103$	−4.97370	−0.490073	−0.245036	−	0.969514i	$-0.578800\pi$
$103$	−4.97370	−0.490073	−0.245036	+	0.969514i	$0.578800\pi$
$104$	50.9211	4.99322
$105$	0	0
$106$	−37.5049	−3.64280
$107$	7.06522	0.683021	0.341510	−	0.939878i	$-0.389062\pi$
$107$	7.06522	0.683021	0.341510	+	0.939878i	$0.389062\pi$
$108$	0	0
$109$	10.6439	1.01950	0.509750	−	0.860323i	$-0.329738\pi$
$109$	10.6439	1.01950	0.509750	+	0.860323i	$0.329738\pi$
$110$	0	0
$111$	0	0
$112$	26.4724	2.50140
$113$	13.7690	1.29528	0.647638	−	0.761948i	$-0.275757\pi$
$113$	13.7690	1.29528	0.647638	+	0.761948i	$0.275757\pi$
$114$	0	0
$115$	5.20473	0.485344
$116$	5.75084	0.533952
$117$	0	0
$118$	10.1922	0.938270
$119$	4.67975	0.428992
$120$	0	0
$121$	0	0
$122$	−8.42909	−0.763134
$123$	0	0
$124$	−22.2939	−2.00206
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.3129	1.09259	0.546296	−	0.837592i	$-0.316037\pi$
$127$	12.3129	1.09259	0.546296	+	0.837592i	$0.316037\pi$
$128$	2.42375	0.214231
$129$	0	0
$130$	−17.6535	−1.54831
$131$	19.5102	1.70461	0.852306	−	0.523043i	$-0.175203\pi$
$131$	19.5102	1.70461	0.852306	+	0.523043i	$0.175203\pi$
$132$	0	0
$133$	3.88448	0.336827
$134$	12.2274	1.05628
$135$	0	0
$136$	13.4987	1.15750
$137$	8.56424	0.731692	0.365846	−	0.930675i	$-0.380780\pi$
$137$	8.56424	0.731692	0.365846	+	0.930675i	$0.380780\pi$
$138$	0	0
$139$	−1.98364	−0.168250	−0.0841250	−	0.996455i	$-0.526810\pi$
$139$	−1.98364	−0.168250	−0.0841250	+	0.996455i	$0.526810\pi$
$140$	−12.8161	−1.08315
$141$	0	0
$142$	−10.1922	−0.855313
$143$	0	0
$144$	0	0
$145$	−1.17737	−0.0977751
$146$	9.35951	0.774598
$147$	0	0
$148$	−3.32025	−0.272923
$149$	−18.3493	−1.50323	−0.751617	−	0.659600i	$-0.770726\pi$
$149$	−18.3493	−1.50323	−0.751617	+	0.659600i	$0.770726\pi$
$150$	0	0
$151$	−17.2234	−1.40162	−0.700812	−	0.713346i	$-0.747179\pi$
$151$	−17.2234	−1.40162	−0.700812	+	0.713346i	$0.747179\pi$
$152$	11.2047	0.908824
$153$	0	0
$154$	0	0
$155$	4.56424	0.366608
$156$	0	0
$157$	−20.8974	−1.66780	−0.833899	−	0.551917i	$-0.813896\pi$
$157$	−20.8974	−1.66780	−0.833899	+	0.551917i	$0.813896\pi$
$158$	39.7164	3.15966
$159$	0	0
$160$	−11.3356	−0.896157
$161$	−13.6563	−1.07627
$162$	0	0
$163$	9.45539	0.740604	0.370302	−	0.928912i	$-0.379254\pi$
$163$	9.45539	0.740604	0.370302	+	0.928912i	$0.379254\pi$
$164$	26.8610	2.09749
$165$	0	0
$166$	19.2440	1.49362
$167$	−9.44902	−0.731187	−0.365593	−	0.930775i	$-0.619134\pi$
$167$	−9.44902	−0.731187	−0.365593	+	0.930775i	$0.619134\pi$
$168$	0	0
$169$	32.2676	2.48213
$170$	−4.67975	−0.358921
$171$	0	0
$172$	−18.5669	−1.41571
$173$	−9.08286	−0.690557	−0.345279	−	0.938500i	$-0.612216\pi$
$173$	−9.08286	−0.690557	−0.345279	+	0.938500i	$0.612216\pi$
$174$	0	0
$175$	2.62383	0.198343
$176$	0	0
$177$	0	0
$178$	−9.04886	−0.678241
$179$	13.7690	1.02914	0.514570	−	0.857448i	$-0.327951\pi$
$179$	13.7690	1.02914	0.514570	+	0.857448i	$0.327951\pi$
$180$	0	0
$181$	−23.8582	−1.77336	−0.886682	−	0.462379i	$-0.846996\pi$
$181$	−23.8582	−1.77336	−0.886682	+	0.462379i	$0.846996\pi$
$182$	46.3197	3.43344
$183$	0	0
$184$	−39.3915	−2.90398
$185$	0.679754	0.0499765
$186$	0	0
$187$	0	0
$188$	−32.5313	−2.37259
$189$	0	0
$190$	−3.88448	−0.281810
$191$	−16.0629	−1.16227	−0.581136	−	0.813807i	$-0.697391\pi$
$191$	−16.0629	−1.16227	−0.581136	+	0.813807i	$0.697391\pi$
$192$	0	0
$193$	−21.4588	−1.54464	−0.772319	−	0.635235i	$-0.780903\pi$
$193$	−21.4588	−1.54464	−0.772319	+	0.635235i	$0.780903\pi$
$194$	3.87021	0.277865
$195$	0	0
$196$	−0.564237	−0.0403027
$197$	9.21494	0.656537	0.328269	−	0.944584i	$-0.393535\pi$
$197$	9.21494	0.656537	0.328269	+	0.944584i	$0.393535\pi$
$198$	0	0
$199$	−16.4880	−1.16880	−0.584401	−	0.811465i	$-0.698670\pi$
$199$	−16.4880	−1.16880	−0.584401	+	0.811465i	$0.698670\pi$
$200$	7.56839	0.535166
$201$	0	0
$202$	−8.95407	−0.630006
$203$	3.08921	0.216820
$204$	0	0
$205$	−5.49925	−0.384084
$206$	−13.0501	−0.909246
$207$	0	0
$208$	67.8815	4.70673
$209$	0	0
$210$	0	0
$211$	−14.8337	−1.02119	−0.510597	−	0.859820i	$-0.670576\pi$
$211$	−14.8337	−1.02119	−0.510597	+	0.859820i	$0.670576\pi$
$212$	−69.8185	−4.79516
$213$	0	0
$214$	18.5379	1.26723
$215$	3.80120	0.259240
$216$	0	0
$217$	−11.9758	−0.812969
$218$	27.9278	1.89151
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	23.4028	1.56717	0.783583	−	0.621287i	$-0.213390\pi$
$223$	23.4028	1.56717	0.783583	+	0.621287i	$0.213390\pi$
$224$	29.7427	1.98727
$225$	0	0
$226$	36.1274	2.40316
$227$	−2.62383	−0.174150	−0.0870749	−	0.996202i	$-0.527752\pi$
$227$	−2.62383	−0.174150	−0.0870749	+	0.996202i	$0.527752\pi$
$228$	0	0
$229$	−22.0366	−1.45622	−0.728110	−	0.685460i	$-0.759601\pi$
$229$	−22.0366	−1.45622	−0.728110	+	0.685460i	$0.759601\pi$
$230$	13.6563	0.900471
$231$	0	0
$232$	8.91079	0.585022
$233$	5.34473	0.350145	0.175072	−	0.984556i	$-0.443984\pi$
$233$	5.34473	0.350145	0.175072	+	0.984556i	$0.443984\pi$
$234$	0	0
$235$	6.66012	0.434459
$236$	18.9737	1.23508
$237$	0	0
$238$	12.2789	0.795921
$239$	−12.5820	−0.813860	−0.406930	−	0.913459i	$-0.633401\pi$
$239$	−12.5820	−0.813860	−0.406930	+	0.913459i	$0.633401\pi$
$240$	0	0
$241$	4.28687	0.276141	0.138071	−	0.990422i	$-0.455910\pi$
$241$	4.28687	0.276141	0.138071	+	0.990422i	$0.455910\pi$
$242$	0	0
$243$	0	0
$244$	−15.6915	−1.00454
$245$	0.115516	0.00738007
$246$	0	0
$247$	9.96074	0.633787
$248$	−34.5440	−2.19354
$249$	0	0
$250$	−2.62383	−0.165946
$251$	−10.0629	−0.635165	−0.317583	−	0.948231i	$-0.602871\pi$
$251$	−10.0629	−0.635165	−0.317583	+	0.948231i	$0.602871\pi$
$252$	0	0
$253$	0	0
$254$	32.3069	2.02712
$255$	0	0
$256$	−12.7690	−0.798060
$257$	3.16547	0.197457	0.0987283	−	0.995114i	$-0.468523\pi$
$257$	3.16547	0.197457	0.0987283	+	0.995114i	$0.468523\pi$
$258$	0	0
$259$	−1.78356	−0.110825
$260$	−32.8634	−2.03810
$261$	0	0
$262$	51.1914	3.16261
$263$	22.3681	1.37928	0.689638	−	0.724155i	$-0.257770\pi$
$263$	22.3681	1.37928	0.689638	+	0.724155i	$0.257770\pi$
$264$	0	0
$265$	14.2939	0.878070
$266$	10.1922	0.624925
$267$	0	0
$268$	22.7623	1.39043
$269$	−15.9737	−0.973934	−0.486967	−	0.873421i	$-0.661897\pi$
$269$	−15.9737	−0.973934	−0.486967	+	0.873421i	$0.661897\pi$
$270$	0	0
$271$	15.6400	0.950060	0.475030	−	0.879970i	$-0.342437\pi$
$271$	15.6400	0.950060	0.475030	+	0.879970i	$0.342437\pi$
$272$	17.9947	1.09109
$273$	0	0
$274$	22.4711	1.35753
$275$	0	0
$276$	0	0
$277$	−23.8836	−1.43502	−0.717512	−	0.696546i	$-0.754719\pi$
$277$	−23.8836	−1.43502	−0.717512	+	0.696546i	$0.754719\pi$
$278$	−5.20473	−0.312159
$279$	0	0
$280$	−19.8582	−1.18675
$281$	12.2439	0.730408	0.365204	−	0.930928i	$-0.380999\pi$
$281$	12.2439	0.730408	0.365204	+	0.930928i	$0.380999\pi$
$282$	0	0
$283$	20.8536	1.23962	0.619810	−	0.784752i	$-0.287210\pi$
$283$	20.8536	1.23962	0.619810	+	0.784752i	$0.287210\pi$
$284$	−18.9737	−1.12588
$285$	0	0
$286$	0	0
$287$	14.4291	0.851722
$288$	0	0
$289$	−13.8189	−0.812878
$290$	−3.08921	−0.181405
$291$	0	0
$292$	17.4235	1.01963
$293$	14.3305	0.837198	0.418599	−	0.908171i	$-0.362521\pi$
$293$	14.3305	0.837198	0.418599	+	0.908171i	$0.362521\pi$
$294$	0	0
$295$	−3.88448	−0.226163
$296$	−5.14464	−0.299026
$297$	0	0
$298$	−48.1455	−2.78899
$299$	−35.0181	−2.02515
$300$	0	0
$301$	−9.97370	−0.574874
$302$	−45.1914	−2.60047
$303$	0	0
$304$	14.9367	0.856679
$305$	3.21251	0.183948
$306$	0	0
$307$	−10.2952	−0.587580	−0.293790	−	0.955870i	$-0.594917\pi$
$307$	−10.2952	−0.587580	−0.293790	+	0.955870i	$0.594917\pi$
$308$	0	0
$309$	0	0
$310$	11.9758	0.680179
$311$	19.1521	1.08602	0.543009	−	0.839727i	$-0.317285\pi$
$311$	19.1521	1.08602	0.543009	+	0.839727i	$0.317285\pi$
$312$	0	0
$313$	−14.9737	−0.846363	−0.423182	−	0.906045i	$-0.639087\pi$
$313$	−14.9737	−0.846363	−0.423182	+	0.906045i	$0.639087\pi$
$314$	−54.8313	−3.09431
$315$	0	0
$316$	73.9354	4.15919
$317$	−12.1784	−0.684009	−0.342004	−	0.939698i	$-0.611106\pi$
$317$	−12.1784	−0.684009	−0.342004	+	0.939698i	$0.611106\pi$
$318$	0	0
$319$	0	0
$320$	−9.56424	−0.534657
$321$	0	0
$322$	−35.8319	−1.99683
$323$	2.64049	0.146921
$324$	0	0
$325$	6.72812	0.373209
$326$	24.8093	1.37406
$327$	0	0
$328$	41.6205	2.29811
$329$	−17.4750	−0.963430
$330$	0	0
$331$	−15.8319	−0.870199	−0.435099	−	0.900382i	$-0.643287\pi$
$331$	−15.8319	−0.870199	−0.435099	+	0.900382i	$0.643287\pi$
$332$	35.8243	1.96612
$333$	0	0
$334$	−24.7926	−1.35659
$335$	−4.66012	−0.254610
$336$	0	0
$337$	−6.72812	−0.366504	−0.183252	−	0.983066i	$-0.558662\pi$
$337$	−6.72812	−0.366504	−0.183252	+	0.983066i	$0.558662\pi$
$338$	84.6648	4.60516
$339$	0	0
$340$	−8.71176	−0.472462
$341$	0	0
$342$	0	0
$343$	−18.6699	−1.00808
$344$	−28.7690	−1.55112
$345$	0	0
$346$	−23.8319	−1.28121
$347$	33.5036	1.79857	0.899284	−	0.437366i	$-0.144088\pi$
$347$	33.5036	1.79857	0.899284	+	0.437366i	$0.144088\pi$
$348$	0	0
$349$	−22.7742	−1.21907	−0.609537	−	0.792757i	$-0.708645\pi$
$349$	−22.7742	−1.21907	−0.609537	+	0.792757i	$0.708645\pi$
$350$	6.88448	0.367991
$351$	0	0
$352$	0	0
$353$	15.4617	0.822942	0.411471	−	0.911423i	$-0.365015\pi$
$353$	15.4617	0.822942	0.411471	+	0.911423i	$0.365015\pi$
$354$	0	0
$355$	3.88448	0.206167
$356$	−16.8452	−0.892795
$357$	0	0
$358$	36.1274	1.90939
$359$	9.58603	0.505932	0.252966	−	0.967475i	$-0.418594\pi$
$359$	9.58603	0.505932	0.252966	+	0.967475i	$0.418594\pi$
$360$	0	0
$361$	−16.8082	−0.884644
$362$	−62.5998	−3.29017
$363$	0	0
$364$	86.2280	4.51957
$365$	−3.56712	−0.186711
$366$	0	0
$367$	−10.8386	−0.565768	−0.282884	−	0.959154i	$-0.591291\pi$
$367$	−10.8386	−0.565768	−0.282884	+	0.959154i	$0.591291\pi$
$368$	−52.5116	−2.73736
$369$	0	0
$370$	1.78356	0.0927228
$371$	−37.5049	−1.94716
$372$	0	0
$373$	10.0212	0.518878	0.259439	−	0.965759i	$-0.416462\pi$
$373$	10.0212	0.518878	0.259439	+	0.965759i	$0.416462\pi$
$374$	0	0
$375$	0	0
$376$	−50.4064	−2.59951
$377$	7.92148	0.407977
$378$	0	0
$379$	4.19177	0.215317	0.107658	−	0.994188i	$-0.465665\pi$
$379$	4.19177	0.215317	0.107658	+	0.994188i	$0.465665\pi$
$380$	−7.23130	−0.370958
$381$	0	0
$382$	−42.1463	−2.15639
$383$	−19.2440	−0.983322	−0.491661	−	0.870787i	$-0.663610\pi$
$383$	−19.2440	−0.983322	−0.491661	+	0.870787i	$0.663610\pi$
$384$	0	0
$385$	0	0
$386$	−56.3042	−2.86581
$387$	0	0
$388$	7.20473	0.365765
$389$	28.4224	1.44107	0.720537	−	0.693417i	$-0.243895\pi$
$389$	28.4224	1.44107	0.720537	+	0.693417i	$0.243895\pi$
$390$	0	0
$391$	−9.28294	−0.469458
$392$	−0.874273	−0.0441575
$393$	0	0
$394$	24.1784	1.21809
$395$	−15.1368	−0.761615
$396$	0	0
$397$	−22.1784	−1.11310	−0.556552	−	0.830813i	$-0.687876\pi$
$397$	−22.1784	−1.11310	−0.556552	+	0.830813i	$0.687876\pi$
$398$	−43.2617	−2.16851
$399$	0	0
$400$	10.0892	0.504461
$401$	−22.0759	−1.10242	−0.551208	−	0.834368i	$-0.685833\pi$
$401$	−22.0759	−1.10242	−0.551208	+	0.834368i	$0.685833\pi$
$402$	0	0
$403$	−30.7088	−1.52971
$404$	−16.6688	−0.829302
$405$	0	0
$406$	8.10557	0.402273
$407$	0	0
$408$	0	0
$409$	−24.6713	−1.21992	−0.609959	−	0.792433i	$-0.708814\pi$
$409$	−24.6713	−1.21992	−0.609959	+	0.792433i	$0.708814\pi$
$410$	−14.4291	−0.712602
$411$	0	0
$412$	−24.2939	−1.19688
$413$	10.1922	0.501527
$414$	0	0
$415$	−7.33431	−0.360027
$416$	76.2673	3.73931
$417$	0	0
$418$	0	0
$419$	−17.6535	−0.862428	−0.431214	−	0.902250i	$-0.641915\pi$
$419$	−17.6535	−0.862428	−0.431214	+	0.902250i	$0.641915\pi$
$420$	0	0
$421$	1.52498	0.0743228	0.0371614	−	0.999309i	$-0.488168\pi$
$421$	1.52498	0.0743228	0.0371614	+	0.999309i	$0.488168\pi$
$422$	−38.9211	−1.89465
$423$	0	0
$424$	−108.182	−5.25379
$425$	1.78356	0.0865153
$426$	0	0
$427$	−8.42909	−0.407912
$428$	34.5099	1.66810
$429$	0	0
$430$	9.97370	0.480974
$431$	−16.7493	−0.806787	−0.403393	−	0.915027i	$-0.632169\pi$
$431$	−16.7493	−0.806787	−0.403393	+	0.915027i	$0.632169\pi$
$432$	0	0
$433$	7.47502	0.359227	0.179613	−	0.983737i	$-0.442515\pi$
$433$	7.47502	0.359227	0.179613	+	0.983737i	$0.442515\pi$
$434$	−31.4224	−1.50832
$435$	0	0
$436$	51.9899	2.48987
$437$	−7.70541	−0.368600
$438$	0	0
$439$	−17.4585	−0.833250	−0.416625	−	0.909078i	$-0.636787\pi$
$439$	−17.4585	−0.833250	−0.416625	+	0.909078i	$0.636787\pi$
$440$	0	0
$441$	0	0
$442$	31.4860	1.49763
$443$	14.6838	0.697647	0.348824	−	0.937188i	$-0.386581\pi$
$443$	14.6838	0.697647	0.348824	+	0.937188i	$0.386581\pi$
$444$	0	0
$445$	3.44872	0.163485
$446$	61.4049	2.90761
$447$	0	0
$448$	25.0949	1.18562
$449$	1.30729	0.0616947	0.0308474	−	0.999524i	$-0.490179\pi$
$449$	1.30729	0.0616947	0.0308474	+	0.999524i	$0.490179\pi$
$450$	0	0
$451$	0	0
$452$	67.2543	3.16338
$453$	0	0
$454$	−6.88448	−0.323105
$455$	−17.6535	−0.827607
$456$	0	0
$457$	−10.1922	−0.476772	−0.238386	−	0.971170i	$-0.576618\pi$
$457$	−10.1922	−0.476772	−0.238386	+	0.971170i	$0.576618\pi$
$458$	−57.8203	−2.70177
$459$	0	0
$460$	25.4224	1.18533
$461$	−4.12180	−0.191971	−0.0959857	−	0.995383i	$-0.530600\pi$
$461$	−4.12180	−0.191971	−0.0959857	+	0.995383i	$0.530600\pi$
$462$	0	0
$463$	32.7623	1.52259	0.761296	−	0.648404i	$-0.224563\pi$
$463$	32.7623	1.52259	0.761296	+	0.648404i	$0.224563\pi$
$464$	11.8787	0.551456
$465$	0	0
$466$	14.0236	0.649633
$467$	−37.9144	−1.75447	−0.877235	−	0.480061i	$-0.840614\pi$
$467$	−37.9144	−1.75447	−0.877235	+	0.480061i	$0.840614\pi$
$468$	0	0
$469$	12.2274	0.564608
$470$	17.4750	0.806063
$471$	0	0
$472$	29.3993	1.35321
$473$	0	0
$474$	0	0
$475$	1.48046	0.0679283
$476$	22.8582	1.04770
$477$	0	0
$478$	−33.0130	−1.50998
$479$	−6.46004	−0.295167	−0.147583	−	0.989050i	$-0.547149\pi$
$479$	−6.46004	−0.295167	−0.147583	+	0.989050i	$0.547149\pi$
$480$	0	0
$481$	−4.57347	−0.208532
$482$	11.2480	0.512333
$483$	0	0
$484$	0	0
$485$	−1.47502	−0.0669774
$486$	0	0
$487$	43.4617	1.96944	0.984718	−	0.174154i	$-0.0557192\pi$
$487$	43.4617	1.96944	0.984718	+	0.174154i	$0.0557192\pi$
$488$	−24.3136	−1.10062
$489$	0	0
$490$	0.303095	0.0136924
$491$	−5.45369	−0.246122	−0.123061	−	0.992399i	$-0.539271\pi$
$491$	−5.45369	−0.246122	−0.123061	+	0.992399i	$0.539271\pi$
$492$	0	0
$493$	2.09990	0.0945749
$494$	26.1353	1.17588
$495$	0	0
$496$	−46.0496	−2.06769
$497$	−10.1922	−0.457184
$498$	0	0
$499$	12.4880	0.559039	0.279519	−	0.960140i	$-0.409825\pi$
$499$	12.4880	0.559039	0.279519	+	0.960140i	$0.409825\pi$
$500$	−4.88448	−0.218441
$501$	0	0
$502$	−26.4034	−1.17844
$503$	−20.8536	−0.929817	−0.464909	−	0.885359i	$-0.653913\pi$
$503$	−20.8536	−0.929817	−0.464909	+	0.885359i	$0.653913\pi$
$504$	0	0
$505$	3.41259	0.151858
$506$	0	0
$507$	0	0
$508$	60.1421	2.66837
$509$	26.5772	1.17801	0.589007	−	0.808128i	$-0.299519\pi$
$509$	26.5772	1.17801	0.589007	+	0.808128i	$0.299519\pi$
$510$	0	0
$511$	9.35951	0.414040
$512$	−38.3511	−1.69490
$513$	0	0
$514$	8.30565	0.366347
$515$	4.97370	0.219167
$516$	0	0
$517$	0	0
$518$	−4.67975	−0.205617
$519$	0	0
$520$	−50.9211	−2.23304
$521$	23.0393	1.00937	0.504684	−	0.863304i	$-0.331609\pi$
$521$	23.0393	1.00937	0.504684	+	0.863304i	$0.331609\pi$
$522$	0	0
$523$	1.18332	0.0517429	0.0258714	−	0.999665i	$-0.491764\pi$
$523$	1.18332	0.0517429	0.0258714	+	0.999665i	$0.491764\pi$
$524$	95.2971	4.16307
$525$	0	0
$526$	58.6901	2.55901
$527$	−8.14058	−0.354609
$528$	0	0
$529$	4.08921	0.177792
$530$	37.5049	1.62911
$531$	0	0
$532$	18.9737	0.822614
$533$	36.9996	1.60263
$534$	0	0
$535$	−7.06522	−0.305456
$536$	35.2697	1.52342
$537$	0	0
$538$	−41.9123	−1.80697
$539$	0	0
$540$	0	0
$541$	31.6055	1.35883	0.679413	−	0.733756i	$-0.262235\pi$
$541$	31.6055	1.35883	0.679413	+	0.733756i	$0.262235\pi$
$542$	41.0366	1.76267
$543$	0	0
$544$	20.2177	0.866826
$545$	−10.6439	−0.455934
$546$	0	0
$547$	4.97958	0.212911	0.106456	−	0.994317i	$-0.466050\pi$
$547$	4.97958	0.212911	0.106456	+	0.994317i	$0.466050\pi$
$548$	41.8319	1.78697
$549$	0	0
$550$	0	0
$551$	1.74305	0.0742564
$552$	0	0
$553$	39.7164	1.68891
$554$	−62.6664	−2.66244
$555$	0	0
$556$	−9.68905	−0.410907
$557$	−25.1289	−1.06475	−0.532374	−	0.846510i	$-0.678700\pi$
$557$	−25.1289	−1.06475	−0.532374	+	0.846510i	$0.678700\pi$
$558$	0	0
$559$	−25.5749	−1.08170
$560$	−26.4724	−1.11866
$561$	0	0
$562$	32.1258	1.35515
$563$	−11.3356	−0.477738	−0.238869	−	0.971052i	$-0.576777\pi$
$563$	−11.3356	−0.477738	−0.238869	+	0.971052i	$0.576777\pi$
$564$	0	0
$565$	−13.7690	−0.579265
$566$	54.7164	2.29990
$567$	0	0
$568$	−29.3993	−1.23357
$569$	−11.9593	−0.501359	−0.250680	−	0.968070i	$-0.580654\pi$
$569$	−11.9593	−0.501359	−0.250680	+	0.968070i	$0.580654\pi$
$570$	0	0
$571$	29.8054	1.24732	0.623659	−	0.781697i	$-0.285645\pi$
$571$	29.8054	1.24732	0.623659	+	0.781697i	$0.285645\pi$
$572$	0	0
$573$	0	0
$574$	37.8595	1.58022
$575$	−5.20473	−0.217052
$576$	0	0
$577$	−18.4857	−0.769570	−0.384785	−	0.923006i	$-0.625724\pi$
$577$	−18.4857	−0.769570	−0.384785	+	0.923006i	$0.625724\pi$
$578$	−36.2585	−1.50815
$579$	0	0
$580$	−5.75084	−0.238790
$581$	19.2440	0.798375
$582$	0	0
$583$	0	0
$584$	26.9973	1.11716
$585$	0	0
$586$	37.6008	1.55328
$587$	2.68377	0.110771	0.0553856	−	0.998465i	$-0.482361\pi$
$587$	2.68377	0.110771	0.0553856	+	0.998465i	$0.482361\pi$
$588$	0	0
$589$	−6.75719	−0.278425
$590$	−10.1922	−0.419607
$591$	0	0
$592$	−6.85818	−0.281870
$593$	−28.7251	−1.17960	−0.589800	−	0.807550i	$-0.700793\pi$
$593$	−28.7251	−1.17960	−0.589800	+	0.807550i	$0.700793\pi$
$594$	0	0
$595$	−4.67975	−0.191851
$596$	−89.6269	−3.67126
$597$	0	0
$598$	−91.8814	−3.75731
$599$	34.1807	1.39659	0.698293	−	0.715812i	$-0.253943\pi$
$599$	34.1807	1.39659	0.698293	+	0.715812i	$0.253943\pi$
$600$	0	0
$601$	32.4573	1.32396	0.661980	−	0.749521i	$-0.269716\pi$
$601$	32.4573	1.32396	0.661980	+	0.749521i	$0.269716\pi$
$602$	−26.1693	−1.06658
$603$	0	0
$604$	−84.1276	−3.42310
$605$	0	0
$606$	0	0
$607$	−5.34473	−0.216936	−0.108468	−	0.994100i	$-0.534594\pi$
$607$	−5.34473	−0.216936	−0.108468	+	0.994100i	$0.534594\pi$
$608$	16.7819	0.680597
$609$	0	0
$610$	8.42909	0.341284
$611$	−44.8101	−1.81282
$612$	0	0
$613$	30.2056	1.21999	0.609996	−	0.792405i	$-0.291171\pi$
$613$	30.2056	1.21999	0.609996	+	0.792405i	$0.291171\pi$
$614$	−27.0130	−1.09015
$615$	0	0
$616$	0	0
$617$	−0.973697	−0.0391996	−0.0195998	−	0.999808i	$-0.506239\pi$
$617$	−0.973697	−0.0391996	−0.0195998	+	0.999808i	$0.506239\pi$
$618$	0	0
$619$	1.82157	0.0732152	0.0366076	−	0.999330i	$-0.488345\pi$
$619$	1.82157	0.0732152	0.0366076	+	0.999330i	$0.488345\pi$
$620$	22.2939	0.895346
$621$	0	0
$622$	50.2519	2.01492
$623$	−9.04886	−0.362535
$624$	0	0
$625$	1.00000	0.0400000
$626$	−39.2884	−1.57028
$627$	0	0
$628$	−102.073	−4.07316
$629$	−1.21238	−0.0483408
$630$	0	0
$631$	−32.3569	−1.28811	−0.644053	−	0.764981i	$-0.722748\pi$
$631$	−32.3569	−1.28811	−0.644053	+	0.764981i	$0.722748\pi$
$632$	114.561	4.55700
$633$	0	0
$634$	−31.9541	−1.26906
$635$	−12.3129	−0.488622
$636$	0	0
$637$	−0.777208	−0.0307941
$638$	0	0
$639$	0	0
$640$	−2.42375	−0.0958071
$641$	13.7690	0.543842	0.271921	−	0.962320i	$-0.412341\pi$
$641$	13.7690	0.543842	0.271921	+	0.962320i	$0.412341\pi$
$642$	0	0
$643$	32.1611	1.26831	0.634154	−	0.773207i	$-0.281348\pi$
$643$	32.1611	1.26831	0.634154	+	0.773207i	$0.281348\pi$
$644$	−66.7041	−2.62851
$645$	0	0
$646$	6.92820	0.272587
$647$	−19.4813	−0.765889	−0.382945	−	0.923771i	$-0.625090\pi$
$647$	−19.4813	−0.765889	−0.382945	+	0.923771i	$0.625090\pi$
$648$	0	0
$649$	0	0
$650$	17.6535	0.692425
$651$	0	0
$652$	46.1847	1.80873
$653$	−0.779658	−0.0305104	−0.0152552	−	0.999884i	$-0.504856\pi$
$653$	−0.779658	−0.0305104	−0.0152552	+	0.999884i	$0.504856\pi$
$654$	0	0
$655$	−19.5102	−0.762326
$656$	55.4831	2.16625
$657$	0	0
$658$	−45.8515	−1.78748
$659$	−28.1929	−1.09824	−0.549119	−	0.835744i	$-0.685037\pi$
$659$	−28.1929	−1.09824	−0.549119	+	0.835744i	$0.685037\pi$
$660$	0	0
$661$	1.70340	0.0662547	0.0331274	−	0.999451i	$-0.489453\pi$
$661$	1.70340	0.0662547	0.0331274	+	0.999451i	$0.489453\pi$
$662$	−41.5402	−1.61450
$663$	0	0
$664$	55.5090	2.15417
$665$	−3.88448	−0.150634
$666$	0	0
$667$	−6.12788	−0.237273
$668$	−46.1536	−1.78574
$669$	0	0
$670$	−12.2274	−0.472385
$671$	0	0
$672$	0	0
$673$	−11.0044	−0.424190	−0.212095	−	0.977249i	$-0.568029\pi$
$673$	−11.0044	−0.424190	−0.212095	+	0.977249i	$0.568029\pi$
$674$	−17.6535	−0.679986
$675$	0	0
$676$	157.611	6.06195
$677$	−21.3908	−0.822115	−0.411058	−	0.911609i	$-0.634840\pi$
$677$	−21.3908	−0.822115	−0.411058	+	0.911609i	$0.634840\pi$
$678$	0	0
$679$	3.87021	0.148525
$680$	−13.4987	−0.517650
$681$	0	0
$682$	0	0
$683$	−34.0959	−1.30464	−0.652321	−	0.757942i	$-0.726205\pi$
$683$	−34.0959	−1.30464	−0.652321	+	0.757942i	$0.726205\pi$
$684$	0	0
$685$	−8.56424	−0.327223
$686$	−48.9867	−1.87032
$687$	0	0
$688$	−38.3511	−1.46212
$689$	−96.1714	−3.66384
$690$	0	0
$691$	46.3199	1.76209	0.881045	−	0.473032i	$-0.156840\pi$
$691$	46.3199	1.76209	0.881045	+	0.473032i	$0.156840\pi$
$692$	−44.3651	−1.68651
$693$	0	0
$694$	87.9077	3.33693
$695$	1.98364	0.0752437
$696$	0	0
$697$	9.80823	0.371513
$698$	−59.7556	−2.26178
$699$	0	0
$700$	12.8161	0.484401
$701$	43.3647	1.63786	0.818931	−	0.573893i	$-0.194567\pi$
$701$	43.3647	1.63786	0.818931	+	0.573893i	$0.194567\pi$
$702$	0	0
$703$	−1.00635	−0.0379552
$704$	0	0
$705$	0	0
$706$	40.5688	1.52683
$707$	−8.95407	−0.336752
$708$	0	0
$709$	19.1784	0.720261	0.360130	−	0.932902i	$-0.382732\pi$
$709$	19.1784	0.720261	0.360130	+	0.932902i	$0.382732\pi$
$710$	10.1922	0.382507
$711$	0	0
$712$	−26.1013	−0.978187
$713$	23.7556	0.889655
$714$	0	0
$715$	0	0
$716$	67.2543	2.51341
$717$	0	0
$718$	25.1521	0.938669
$719$	32.2939	1.20436	0.602180	−	0.798360i	$-0.294299\pi$
$719$	32.2939	1.20436	0.602180	+	0.798360i	$0.294299\pi$
$720$	0	0
$721$	−13.0501	−0.486012
$722$	−44.1019	−1.64130
$723$	0	0
$724$	−116.535	−4.33099
$725$	1.17737	0.0437264
$726$	0	0
$727$	−40.8622	−1.51550	−0.757748	−	0.652548i	$-0.773700\pi$
$727$	−40.8622	−1.51550	−0.757748	+	0.652548i	$0.773700\pi$
$728$	133.608	4.95185
$729$	0	0
$730$	−9.35951	−0.346411
$731$	−6.77966	−0.250755
$732$	0	0
$733$	9.92414	0.366557	0.183278	−	0.983061i	$-0.441329\pi$
$733$	9.92414	0.366557	0.183278	+	0.983061i	$0.441329\pi$
$734$	−28.4385	−1.04968
$735$	0	0
$736$	−58.9987	−2.17472
$737$	0	0
$738$	0	0
$739$	−33.3725	−1.22763	−0.613814	−	0.789450i	$-0.710366\pi$
$739$	−33.3725	−1.22763	−0.613814	+	0.789450i	$0.710366\pi$
$740$	3.32025	0.122055
$741$	0	0
$742$	−98.4064	−3.61261
$743$	3.80715	0.139671	0.0698354	−	0.997559i	$-0.477753\pi$
$743$	3.80715	0.139671	0.0698354	+	0.997559i	$0.477753\pi$
$744$	0	0
$745$	18.3493	0.672266
$746$	26.2939	0.962690
$747$	0	0
$748$	0	0
$749$	18.5379	0.677361
$750$	0	0
$751$	22.5094	0.821378	0.410689	−	0.911775i	$-0.365288\pi$
$751$	22.5094	0.821378	0.410689	+	0.911775i	$0.365288\pi$
$752$	−67.1954	−2.45036
$753$	0	0
$754$	20.7846	0.756931
$755$	17.2234	0.626825
$756$	0	0
$757$	−33.6927	−1.22458	−0.612291	−	0.790632i	$-0.709752\pi$
$757$	−33.6927	−1.22458	−0.612291	+	0.790632i	$0.709752\pi$
$758$	10.9985	0.399483
$759$	0	0
$760$	−11.2047	−0.406438
$761$	18.7719	0.680481	0.340241	−	0.940338i	$-0.389491\pi$
$761$	18.7719	0.680481	0.340241	+	0.940338i	$0.389491\pi$
$762$	0	0
$763$	27.9278	1.01105
$764$	−78.4590	−2.83855
$765$	0	0
$766$	−50.4930	−1.82438
$767$	26.1353	0.943690
$768$	0	0
$769$	21.5618	0.777539	0.388770	−	0.921335i	$-0.372900\pi$
$769$	21.5618	0.777539	0.388770	+	0.921335i	$0.372900\pi$
$770$	0	0
$771$	0	0
$772$	−104.815	−3.77238
$773$	23.3832	0.841034	0.420517	−	0.907285i	$-0.361849\pi$
$773$	23.3832	0.841034	0.420517	+	0.907285i	$0.361849\pi$
$774$	0	0
$775$	−4.56424	−0.163952
$776$	11.1636	0.400749
$777$	0	0
$778$	74.5756	2.67366
$779$	8.14143	0.291697
$780$	0	0
$781$	0	0
$782$	−24.3569	−0.870999
$783$	0	0
$784$	−1.16547	−0.0416239
$785$	20.8974	0.745862
$786$	0	0
$787$	−25.0949	−0.894538	−0.447269	−	0.894400i	$-0.647603\pi$
$787$	−25.0949	−0.894538	−0.447269	+	0.894400i	$0.647603\pi$
$788$	45.0102	1.60342
$789$	0	0
$790$	−39.7164	−1.41304
$791$	36.1274	1.28454
$792$	0	0
$793$	−21.6142	−0.767542
$794$	−58.1924	−2.06517
$795$	0	0
$796$	−80.5353	−2.85450
$797$	12.0763	0.427763	0.213881	−	0.976860i	$-0.431389\pi$
$797$	12.0763	0.427763	0.213881	+	0.976860i	$0.431389\pi$
$798$	0	0
$799$	−11.8787	−0.420239
$800$	11.3356	0.400774
$801$	0	0
$802$	−57.9233	−2.04534
$803$	0	0
$804$	0	0
$805$	13.6563	0.481322
$806$	−80.5745	−2.83812
$807$	0	0
$808$	−25.8279	−0.908621
$809$	9.11192	0.320358	0.160179	−	0.987088i	$-0.448793\pi$
$809$	9.11192	0.320358	0.160179	+	0.987088i	$0.448793\pi$
$810$	0	0
$811$	−2.98999	−0.104993	−0.0524964	−	0.998621i	$-0.516718\pi$
$811$	−2.98999	−0.104993	−0.0524964	+	0.998621i	$0.516718\pi$
$812$	15.0892	0.529528
$813$	0	0
$814$	0	0
$815$	−9.45539	−0.331208
$816$	0	0
$817$	−5.62753	−0.196882
$818$	−64.7333	−2.26335
$819$	0	0
$820$	−26.8610	−0.938026
$821$	35.2697	1.23092	0.615460	−	0.788168i	$-0.288970\pi$
$821$	35.2697	1.23092	0.615460	+	0.788168i	$0.288970\pi$
$822$	0	0
$823$	15.5576	0.542303	0.271151	−	0.962537i	$-0.412596\pi$
$823$	15.5576	0.542303	0.271151	+	0.962537i	$0.412596\pi$
$824$	−37.6429	−1.31135
$825$	0	0
$826$	26.7427	0.930496
$827$	−10.5293	−0.366140	−0.183070	−	0.983100i	$-0.558604\pi$
$827$	−10.5293	−0.366140	−0.183070	+	0.983100i	$0.558604\pi$
$828$	0	0
$829$	25.6249	0.889989	0.444994	−	0.895533i	$-0.353206\pi$
$829$	25.6249	0.889989	0.444994	+	0.895533i	$0.353206\pi$
$830$	−19.2440	−0.667969
$831$	0	0
$832$	64.3494	2.23091
$833$	−0.206030	−0.00713852
$834$	0	0
$835$	9.44902	0.326997
$836$	0	0
$837$	0	0
$838$	−46.3197	−1.60009
$839$	−26.1258	−0.901964	−0.450982	−	0.892533i	$-0.648926\pi$
$839$	−26.1258	−0.901964	−0.450982	+	0.892533i	$0.648926\pi$
$840$	0	0
$841$	−27.6138	−0.952200
$842$	4.00128	0.137893
$843$	0	0
$844$	−72.4549	−2.49400
$845$	−32.2676	−1.11004
$846$	0	0
$847$	0	0
$848$	−144.215	−4.95235
$849$	0	0
$850$	4.67975	0.160514
$851$	3.53793	0.121279
$852$	0	0
$853$	−2.75490	−0.0943259	−0.0471629	−	0.998887i	$-0.515018\pi$
$853$	−2.75490	−0.0943259	−0.0471629	+	0.998887i	$0.515018\pi$
$854$	−22.1165	−0.756811
$855$	0	0
$856$	53.4724	1.82765
$857$	20.3164	0.693997	0.346998	−	0.937866i	$-0.387201\pi$
$857$	20.3164	0.693997	0.346998	+	0.937866i	$0.387201\pi$
$858$	0	0
$859$	22.6249	0.771951	0.385975	−	0.922509i	$-0.373865\pi$
$859$	22.6249	0.771951	0.385975	+	0.922509i	$0.373865\pi$
$860$	18.5669	0.633126
$861$	0	0
$862$	−43.9474	−1.49685
$863$	−43.1088	−1.46744	−0.733721	−	0.679451i	$-0.762218\pi$
$863$	−43.1088	−1.46744	−0.733721	+	0.679451i	$0.762218\pi$
$864$	0	0
$865$	9.08286	0.308826
$866$	19.6132	0.666483
$867$	0	0
$868$	−58.4955	−1.98547
$869$	0	0
$870$	0	0
$871$	31.3539	1.06239
$872$	80.5572	2.72801
$873$	0	0
$874$	−20.2177	−0.683874
$875$	−2.62383	−0.0887016
$876$	0	0
$877$	16.6853	0.563421	0.281711	−	0.959499i	$-0.409098\pi$
$877$	16.6853	0.563421	0.281711	+	0.959499i	$0.409098\pi$
$878$	−45.8082	−1.54595
$879$	0	0
$880$	0	0
$881$	43.5116	1.46594	0.732972	−	0.680259i	$-0.238133\pi$
$881$	43.5116	1.46594	0.732972	+	0.680259i	$0.238133\pi$
$882$	0	0
$883$	−26.9996	−0.908609	−0.454305	−	0.890846i	$-0.650112\pi$
$883$	−26.9996	−0.908609	−0.454305	+	0.890846i	$0.650112\pi$
$884$	58.6138	1.97140
$885$	0	0
$886$	38.5277	1.29436
$887$	−22.9403	−0.770259	−0.385130	−	0.922863i	$-0.625843\pi$
$887$	−22.9403	−0.770259	−0.385130	+	0.922863i	$0.625843\pi$
$888$	0	0
$889$	32.3069	1.08354
$890$	9.04886	0.303318
$891$	0	0
$892$	114.311	3.82740
$893$	−9.86007	−0.329955
$894$	0	0
$895$	−13.7690	−0.460246
$896$	6.35951	0.212456
$897$	0	0
$898$	3.43010	0.114464
$899$	−5.37379	−0.179226
$900$	0	0
$901$	−25.4941	−0.849331
$902$	0	0
$903$	0	0
$904$	104.209	3.46594
$905$	23.8582	0.793073
$906$	0	0
$907$	41.7337	1.38575	0.692873	−	0.721060i	$-0.256345\pi$
$907$	41.7337	1.38575	0.692873	+	0.721060i	$0.256345\pi$
$908$	−12.8161	−0.425316
$909$	0	0
$910$	−46.3197	−1.53548
$911$	38.7271	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$911$	38.7271	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$912$	0	0
$913$	0	0
$914$	−26.7427	−0.884569
$915$	0	0
$916$	−107.637	−3.55644
$917$	51.1914	1.69049
$918$	0	0
$919$	2.85196	0.0940775	0.0470388	−	0.998893i	$-0.485022\pi$
$919$	2.85196	0.0940775	0.0470388	+	0.998893i	$0.485022\pi$
$920$	39.3915	1.29870
$921$	0	0
$922$	−10.8149	−0.356170
$923$	−26.1353	−0.860253
$924$	0	0
$925$	−0.679754	−0.0223502
$926$	85.9627	2.82491
$927$	0	0
$928$	13.3462	0.438109
$929$	13.2284	0.434009	0.217005	−	0.976171i	$-0.430371\pi$
$929$	13.2284	0.434009	0.217005	+	0.976171i	$0.430371\pi$
$930$	0	0
$931$	−0.171018	−0.00560488
$932$	26.1062	0.855138
$933$	0	0
$934$	−99.4810	−3.25512
$935$	0	0
$936$	0	0
$937$	−1.68054	−0.0549010	−0.0274505	−	0.999623i	$-0.508739\pi$
$937$	−1.68054	−0.0549010	−0.0274505	+	0.999623i	$0.508739\pi$
$938$	32.0825	1.04753
$939$	0	0
$940$	32.5313	1.06105
$941$	−14.6851	−0.478721	−0.239361	−	0.970931i	$-0.576938\pi$
$941$	−14.6851	−0.478721	−0.239361	+	0.970931i	$0.576938\pi$
$942$	0	0
$943$	−28.6221	−0.932064
$944$	39.1914	1.27557
$945$	0	0
$946$	0	0
$947$	−17.6535	−0.573660	−0.286830	−	0.957981i	$-0.592601\pi$
$947$	−17.6535	−0.573660	−0.286830	+	0.957981i	$0.592601\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	3.88448	0.126029
$951$	0	0
$952$	35.4182	1.14791
$953$	21.7678	0.705130	0.352565	−	0.935787i	$-0.385310\pi$
$953$	21.7678	0.705130	0.352565	+	0.935787i	$0.385310\pi$
$954$	0	0
$955$	16.0629	0.519784
$956$	−61.4564	−1.98764
$957$	0	0
$958$	−16.9500	−0.547631
$959$	22.4711	0.725630
$960$	0	0
$961$	−10.1677	−0.327991
$962$	−12.0000	−0.386896
$963$	0	0
$964$	20.9391	0.674404
$965$	21.4588	0.690783
$966$	0	0
$967$	14.0684	0.452409	0.226204	−	0.974080i	$-0.427368\pi$
$967$	14.0684	0.452409	0.226204	+	0.974080i	$0.427368\pi$
$968$	0	0
$969$	0	0
$970$	−3.87021	−0.124265
$971$	14.7324	0.472784	0.236392	−	0.971658i	$-0.424035\pi$
$971$	14.7324	0.472784	0.236392	+	0.971658i	$0.424035\pi$
$972$	0	0
$973$	−5.20473	−0.166856
$974$	114.036	3.65395
$975$	0	0
$976$	−32.4117	−1.03747
$977$	−13.1418	−0.420444	−0.210222	−	0.977654i	$-0.567419\pi$
$977$	−13.1418	−0.420444	−0.210222	+	0.977654i	$0.567419\pi$
$978$	0	0
$979$	0	0
$980$	0.564237	0.0180239
$981$	0	0
$982$	−14.3096	−0.456636
$983$	−18.4157	−0.587371	−0.293686	−	0.955902i	$-0.594882\pi$
$983$	−18.4157	−0.587371	−0.293686	+	0.955902i	$0.594882\pi$
$984$	0	0
$985$	−9.21494	−0.293612
$986$	5.50979	0.175468
$987$	0	0
$988$	48.6531	1.54786
$989$	19.7842	0.629101
$990$	0	0
$991$	44.3591	1.40911	0.704557	−	0.709647i	$-0.251146\pi$
$991$	44.3591	1.40911	0.704557	+	0.709647i	$0.251146\pi$
$992$	−51.7383	−1.64269
$993$	0	0
$994$	−26.7427	−0.848226
$995$	16.4880	0.522704
$996$	0	0
$997$	−50.4930	−1.59913	−0.799564	−	0.600581i	$-0.794936\pi$
$997$	−50.4930	−1.59913	−0.799564	+	0.600581i	$0.794936\pi$
$998$	32.7663	1.03720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bx.1.6		6
3.2	odd	2		605.2.a.m.1.1	✓	6
11.10	odd	2	inner	5445.2.a.bx.1.1		6
12.11	even	2		9680.2.a.cw.1.3		6
15.14	odd	2		3025.2.a.bg.1.6		6
33.2	even	10		605.2.g.q.81.1		24
33.5	odd	10		605.2.g.q.366.6		24
33.8	even	10		605.2.g.q.251.6		24
33.14	odd	10		605.2.g.q.251.1		24
33.17	even	10		605.2.g.q.366.1		24
33.20	odd	10		605.2.g.q.81.6		24
33.26	odd	10		605.2.g.q.511.1		24
33.29	even	10		605.2.g.q.511.6		24
33.32	even	2		605.2.a.m.1.6	yes	6
132.131	odd	2		9680.2.a.cw.1.4		6
165.164	even	2		3025.2.a.bg.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.m.1.1	✓	6	3.2	odd	2
605.2.a.m.1.6	yes	6	33.32	even	2
605.2.g.q.81.1		24	33.2	even	10
605.2.g.q.81.6		24	33.20	odd	10
605.2.g.q.251.1		24	33.14	odd	10
605.2.g.q.251.6		24	33.8	even	10
605.2.g.q.366.1		24	33.17	even	10
605.2.g.q.366.6		24	33.5	odd	10
605.2.g.q.511.1		24	33.26	odd	10
605.2.g.q.511.6		24	33.29	even	10
3025.2.a.bg.1.1		6	165.164	even	2
3025.2.a.bg.1.6		6	15.14	odd	2
5445.2.a.bx.1.1		6	11.10	odd	2	inner
5445.2.a.bx.1.6		6	1.1	even	1	trivial
9680.2.a.cw.1.3		6	12.11	even	2
9680.2.a.cw.1.4		6	132.131	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)

\(f(q)\)	\(=\)	\(q+2.62383 q^{2} +4.88448 q^{4} -1.00000 q^{5} +2.62383 q^{7} +7.56839 q^{8} +O(q^{10})\)	\(q+2.62383 q^{2} +4.88448 q^{4} -1.00000 q^{5} +2.62383 q^{7} +7.56839 q^{8} -2.62383 q^{10} +6.72812 q^{13} +6.88448 q^{14} +10.0892 q^{16} +1.78356 q^{17} +1.48046 q^{19} -4.88448 q^{20} -5.20473 q^{23} +1.00000 q^{25} +17.6535 q^{26} +12.8161 q^{28} +1.17737 q^{29} -4.56424 q^{31} +11.3356 q^{32} +4.67975 q^{34} -2.62383 q^{35} -0.679754 q^{37} +3.88448 q^{38} -7.56839 q^{40} +5.49925 q^{41} -3.80120 q^{43} -13.6563 q^{46} -6.66012 q^{47} -0.115516 q^{49} +2.62383 q^{50} +32.8634 q^{52} -14.2939 q^{53} +19.8582 q^{56} +3.08921 q^{58} +3.88448 q^{59} -3.21251 q^{61} -11.9758 q^{62} +9.56424 q^{64} -6.72812 q^{65} +4.66012 q^{67} +8.71176 q^{68} -6.88448 q^{70} -3.88448 q^{71} +3.56712 q^{73} -1.78356 q^{74} +7.23130 q^{76} +15.1368 q^{79} -10.0892 q^{80} +14.4291 q^{82} +7.33431 q^{83} -1.78356 q^{85} -9.97370 q^{86} -3.44872 q^{89} +17.6535 q^{91} -25.4224 q^{92} -17.4750 q^{94} -1.48046 q^{95} +1.47502 q^{97} -0.303095 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 6 * q^4 - 6 * q^5	\( 6 q + 6 q^{4} - 6 q^{5} + 18 q^{14} + 18 q^{16} - 6 q^{20} - 12 q^{23} + 6 q^{25} + 36 q^{26} + 24 q^{34} - 42 q^{47} - 24 q^{49} - 24 q^{53} + 30 q^{56} - 24 q^{58} + 30 q^{64} + 30 q^{67} - 18 q^{70} - 18 q^{80} + 42 q^{82} + 6 q^{86} + 30 q^{89} + 36 q^{91} - 36 q^{92} + 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 - 6 * q^5 + 18 * q^14 + 18 * q^16 - 6 * q^20 - 12 * q^23 + 6 * q^25 + 36 * q^26 + 24 * q^34 - 42 * q^47 - 24 * q^49 - 24 * q^53 + 30 * q^56 - 24 * q^58 + 30 * q^64 + 30 * q^67 - 18 * q^70 - 18 * q^80 + 42 * q^82 + 6 * q^86 + 30 * q^89 + 36 * q^91 - 36 * q^92 + 24 * q^97

Introduction

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