LMFDB - Embedded newform 5445.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.737640$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.737640 q^{2} -1.45589 q^{4} +1.00000 q^{5} +1.03138 q^{7} -2.54920 q^{8} +O(q^{10})$	$q+0.737640 q^{2} -1.45589 q^{4} +1.00000 q^{5} +1.03138 q^{7} -2.54920 q^{8} +0.737640 q^{10} +3.44899 q^{13} +0.760787 q^{14} +1.03138 q^{16} -2.39822 q^{17} -7.66881 q^{19} -1.45589 q^{20} -2.45589 q^{23} +1.00000 q^{25} +2.54411 q^{26} -1.50157 q^{28} +5.95431 q^{29} -3.68820 q^{31} +5.85919 q^{32} -1.76902 q^{34} +1.03138 q^{35} +5.95858 q^{37} -5.65682 q^{38} -2.54920 q^{40} -3.93626 q^{41} -7.64941 q^{43} -1.81156 q^{46} -5.84294 q^{47} -5.93626 q^{49} +0.737640 q^{50} -5.02134 q^{52} +11.8480 q^{53} -2.62920 q^{56} +4.39214 q^{58} -2.94630 q^{59} +2.48037 q^{61} -2.72057 q^{62} +2.25922 q^{64} +3.44899 q^{65} -6.14702 q^{67} +3.49153 q^{68} +0.760787 q^{70} -2.02315 q^{71} -0.825867 q^{73} +4.39529 q^{74} +11.1649 q^{76} -12.0782 q^{79} +1.03138 q^{80} -2.90354 q^{82} -1.61002 q^{83} -2.39822 q^{85} -5.64252 q^{86} -8.16116 q^{89} +3.55722 q^{91} +3.57549 q^{92} -4.30999 q^{94} -7.66881 q^{95} +2.44278 q^{97} -4.37882 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 + 3 * q^8	$ 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8} + q^{10} - q^{13} - 2 q^{14} - 3 q^{16} - q^{17} - 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} - 13 q^{28} + 12 q^{29} - 5 q^{31} - 8 q^{32} + 2 q^{34} - 3 q^{35} + 7 q^{37} - 20 q^{38} + 3 q^{40} + 11 q^{41} - 19 q^{43} + 4 q^{46} - 5 q^{47} + 3 q^{49} + q^{50} + 11 q^{52} + 11 q^{53} - 11 q^{56} - 14 q^{58} - 9 q^{59} - 12 q^{61} - 35 q^{62} - 3 q^{64} - q^{65} - 19 q^{67} - 3 q^{68} - 2 q^{70} - 5 q^{71} - 11 q^{73} - 34 q^{79} - 3 q^{80} - 6 q^{82} - 11 q^{83} - q^{85} - q^{86} + 8 q^{89} - 8 q^{91} + 12 q^{92} + q^{94} - 20 q^{95} + 32 q^{97} - 8 q^{98}+O(q^{100}) $ 4 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 + 3 * q^8 + q^10 - q^13 - 2 * q^14 - 3 * q^16 - q^17 - 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 - 13 * q^28 + 12 * q^29 - 5 * q^31 - 8 * q^32 + 2 * q^34 - 3 * q^35 + 7 * q^37 - 20 * q^38 + 3 * q^40 + 11 * q^41 - 19 * q^43 + 4 * q^46 - 5 * q^47 + 3 * q^49 + q^50 + 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 - 12 * q^61 - 35 * q^62 - 3 * q^64 - q^65 - 19 * q^67 - 3 * q^68 - 2 * q^70 - 5 * q^71 - 11 * q^73 - 34 * q^79 - 3 * q^80 - 6 * q^82 - 11 * q^83 - q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 + q^94 - 20 * q^95 + 32 * q^97 - 8 * 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 $

$2$	0.737640	0.521590	0.260795	−	0.965394i	$-0.416015\pi$
$2$	0.737640	0.521590	0.260795	+	0.965394i	$0.416015\pi$
$3$	0	0
$3$	0	0
$4$	−1.45589	−0.727943
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.03138	0.389825	0.194912	−	0.980821i	$-0.437558\pi$
$7$	1.03138	0.389825	0.194912	+	0.980821i	$0.437558\pi$
$8$	−2.54920	−0.901279
$9$	0	0
$10$	0.737640	0.233262
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.44899	0.956577	0.478289	−	0.878203i	$-0.341257\pi$
$13$	3.44899	0.956577	0.478289	+	0.878203i	$0.341257\pi$
$14$	0.760787	0.203329
$15$	0	0
$16$	1.03138	0.257845
$17$	−2.39822	−0.581653	−0.290826	−	0.956776i	$-0.593930\pi$
$17$	−2.39822	−0.581653	−0.290826	+	0.956776i	$0.593930\pi$
$18$	0	0
$19$	−7.66881	−1.75935	−0.879673	−	0.475580i	$-0.842238\pi$
$19$	−7.66881	−1.75935	−0.879673	+	0.475580i	$0.842238\pi$
$20$	−1.45589	−0.325546
$21$	0	0
$22$	0	0
$23$	−2.45589	−0.512088	−0.256044	−	0.966665i	$-0.582419\pi$
$23$	−2.45589	−0.512088	−0.256044	+	0.966665i	$0.582419\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.54411	0.498942
$27$	0	0
$28$	−1.50157	−0.283770
$29$	5.95431	1.10569	0.552844	−	0.833285i	$-0.313542\pi$
$29$	5.95431	1.10569	0.552844	+	0.833285i	$0.313542\pi$
$30$	0	0
$31$	−3.68820	−0.662421	−0.331210	−	0.943557i	$-0.607457\pi$
$31$	−3.68820	−0.662421	−0.331210	+	0.943557i	$0.607457\pi$
$32$	5.85919	1.03577
$33$	0	0
$34$	−1.76902	−0.303384
$35$	1.03138	0.174335
$36$	0	0
$37$	5.95858	0.979584	0.489792	−	0.871839i	$-0.337073\pi$
$37$	5.95858	0.979584	0.489792	+	0.871839i	$0.337073\pi$
$38$	−5.65682	−0.917658
$39$	0	0
$40$	−2.54920	−0.403064
$41$	−3.93626	−0.614740	−0.307370	−	0.951590i	$-0.599449\pi$
$41$	−3.93626	−0.614740	−0.307370	+	0.951590i	$0.599449\pi$
$42$	0	0
$43$	−7.64941	−1.16652	−0.583262	−	0.812284i	$-0.698224\pi$
$43$	−7.64941	−1.16652	−0.583262	+	0.812284i	$0.698224\pi$
$44$	0	0
$45$	0	0
$46$	−1.81156	−0.267100
$47$	−5.84294	−0.852281	−0.426140	−	0.904657i	$-0.640127\pi$
$47$	−5.84294	−0.852281	−0.426140	+	0.904657i	$0.640127\pi$
$48$	0	0
$49$	−5.93626	−0.848037
$50$	0.737640	0.104318
$51$	0	0
$52$	−5.02134	−0.696334
$53$	11.8480	1.62745	0.813726	−	0.581249i	$-0.197436\pi$
$53$	11.8480	1.62745	0.813726	+	0.581249i	$0.197436\pi$
$54$	0	0
$55$	0	0
$56$	−2.62920	−0.351341
$57$	0	0
$58$	4.39214	0.576717
$59$	−2.94630	−0.383575	−0.191788	−	0.981436i	$-0.561428\pi$
$59$	−2.94630	−0.383575	−0.191788	+	0.981436i	$0.561428\pi$
$60$	0	0
$61$	2.48037	0.317579	0.158789	−	0.987312i	$-0.449241\pi$
$61$	2.48037	0.317579	0.158789	+	0.987312i	$0.449241\pi$
$62$	−2.72057	−0.345512
$63$	0	0
$64$	2.25922	0.282402
$65$	3.44899	0.427794
$66$	0	0
$67$	−6.14702	−0.750978	−0.375489	−	0.926827i	$-0.622525\pi$
$67$	−6.14702	−0.750978	−0.375489	+	0.926827i	$0.622525\pi$
$68$	3.49153	0.423410
$69$	0	0
$70$	0.760787	0.0909315
$71$	−2.02315	−0.240103	−0.120052	−	0.992768i	$-0.538306\pi$
$71$	−2.02315	−0.240103	−0.120052	+	0.992768i	$0.538306\pi$
$72$	0	0
$73$	−0.825867	−0.0966604	−0.0483302	−	0.998831i	$-0.515390\pi$
$73$	−0.825867	−0.0966604	−0.0483302	+	0.998831i	$0.515390\pi$
$74$	4.39529	0.510942
$75$	0	0
$76$	11.1649	1.28070
$77$	0	0
$78$	0	0
$79$	−12.0782	−1.35890	−0.679451	−	0.733721i	$-0.737782\pi$
$79$	−12.0782	−1.35890	−0.679451	+	0.733721i	$0.737782\pi$
$80$	1.03138	0.115312
$81$	0	0
$82$	−2.90354	−0.320642
$83$	−1.61002	−0.176722	−0.0883612	−	0.996088i	$-0.528163\pi$
$83$	−1.61002	−0.176722	−0.0883612	+	0.996088i	$0.528163\pi$
$84$	0	0
$85$	−2.39822	−0.260123
$86$	−5.64252	−0.608448
$87$	0	0
$88$	0	0
$89$	−8.16116	−0.865081	−0.432541	−	0.901614i	$-0.642383\pi$
$89$	−8.16116	−0.865081	−0.432541	+	0.901614i	$0.642383\pi$
$90$	0	0
$91$	3.55722	0.372898
$92$	3.57549	0.372771
$93$	0	0
$94$	−4.30999	−0.444541
$95$	−7.66881	−0.786803
$96$	0	0
$97$	2.44278	0.248027	0.124013	−	0.992281i	$-0.460423\pi$
$97$	2.44278	0.248027	0.124013	+	0.992281i	$0.460423\pi$
$98$	−4.37882	−0.442328
$99$	0	0
$100$	−1.45589	−0.145589
$101$	7.52373	0.748640	0.374320	−	0.927300i	$-0.377876\pi$
$101$	7.52373	0.748640	0.374320	+	0.927300i	$0.377876\pi$
$102$	0	0
$103$	−9.48231	−0.934320	−0.467160	−	0.884173i	$-0.654723\pi$
$103$	−9.48231	−0.934320	−0.467160	+	0.884173i	$0.654723\pi$
$104$	−8.79217	−0.862143
$105$	0	0
$106$	8.73958	0.848863
$107$	4.64678	0.449221	0.224611	−	0.974449i	$-0.427889\pi$
$107$	4.64678	0.449221	0.224611	+	0.974449i	$0.427889\pi$
$108$	0	0
$109$	−5.32826	−0.510355	−0.255178	−	0.966894i	$-0.582134\pi$
$109$	−5.32826	−0.510355	−0.255178	+	0.966894i	$0.582134\pi$
$110$	0	0
$111$	0	0
$112$	1.06374	0.100514
$113$	−0.304901	−0.0286826	−0.0143413	−	0.999897i	$-0.504565\pi$
$113$	−0.304901	−0.0286826	−0.0143413	+	0.999897i	$0.504565\pi$
$114$	0	0
$115$	−2.45589	−0.229013
$116$	−8.66881	−0.804879
$117$	0	0
$118$	−2.17331	−0.200069
$119$	−2.47347	−0.226743
$120$	0	0
$121$	0	0
$122$	1.82962	0.165646
$123$	0	0
$124$	5.36960	0.482205
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.9100	−1.05684	−0.528419	−	0.848984i	$-0.677215\pi$
$127$	−11.9100	−1.05684	−0.528419	+	0.848984i	$0.677215\pi$
$128$	−10.0519	−0.888470
$129$	0	0
$130$	2.54411	0.223133
$131$	11.1875	0.977452	0.488726	−	0.872437i	$-0.337462\pi$
$131$	11.1875	0.977452	0.488726	+	0.872437i	$0.337462\pi$
$132$	0	0
$133$	−7.90945	−0.685837
$134$	−4.53429	−0.391703
$135$	0	0
$136$	6.11353	0.524231
$137$	4.28124	0.365771	0.182886	−	0.983134i	$-0.441456\pi$
$137$	4.28124	0.365771	0.182886	+	0.983134i	$0.441456\pi$
$138$	0	0
$139$	−5.95320	−0.504943	−0.252472	−	0.967604i	$-0.581243\pi$
$139$	−5.95320	−0.504943	−0.252472	+	0.967604i	$0.581243\pi$
$140$	−1.50157	−0.126906
$141$	0	0
$142$	−1.49235	−0.125236
$143$	0	0
$144$	0	0
$145$	5.95431	0.494479
$146$	−0.609193	−0.0504171
$147$	0	0
$148$	−8.67501	−0.713082
$149$	3.78444	0.310034	0.155017	−	0.987912i	$-0.450457\pi$
$149$	3.78444	0.310034	0.155017	+	0.987912i	$0.450457\pi$
$150$	0	0
$151$	−24.3566	−1.98211	−0.991057	−	0.133437i	$-0.957399\pi$
$151$	−24.3566	−1.98211	−0.991057	+	0.133437i	$0.957399\pi$
$152$	19.5493	1.58566
$153$	0	0
$154$	0	0
$155$	−3.68820	−0.296243
$156$	0	0
$157$	7.23210	0.577184	0.288592	−	0.957452i	$-0.406813\pi$
$157$	7.23210	0.577184	0.288592	+	0.957452i	$0.406813\pi$
$158$	−8.90936	−0.708790
$159$	0	0
$160$	5.85919	0.463210
$161$	−2.53295	−0.199625
$162$	0	0
$163$	18.6892	1.46385	0.731924	−	0.681386i	$-0.238623\pi$
$163$	18.6892	1.46385	0.731924	+	0.681386i	$0.238623\pi$
$164$	5.73074	0.447496
$165$	0	0
$166$	−1.18761	−0.0921767
$167$	−7.85328	−0.607705	−0.303852	−	0.952719i	$-0.598273\pi$
$167$	−7.85328	−0.607705	−0.303852	+	0.952719i	$0.598273\pi$
$168$	0	0
$169$	−1.10448	−0.0849597
$170$	−1.76902	−0.135678
$171$	0	0
$172$	11.1367	0.849164
$173$	−11.2047	−0.851877	−0.425938	−	0.904752i	$-0.640056\pi$
$173$	−11.2047	−0.851877	−0.425938	+	0.904752i	$0.640056\pi$
$174$	0	0
$175$	1.03138	0.0779650
$176$	0	0
$177$	0	0
$178$	−6.02000	−0.451218
$179$	−1.46463	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$179$	−1.46463	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$180$	0	0
$181$	9.28900	0.690446	0.345223	−	0.938521i	$-0.387803\pi$
$181$	9.28900	0.690446	0.345223	+	0.938521i	$0.387803\pi$
$182$	2.62395	0.194500
$183$	0	0
$184$	6.26055	0.461534
$185$	5.95858	0.438083
$186$	0	0
$187$	0	0
$188$	8.50666	0.620412
$189$	0	0
$190$	−5.65682	−0.410389
$191$	4.47296	0.323652	0.161826	−	0.986819i	$-0.448262\pi$
$191$	4.47296	0.323652	0.161826	+	0.986819i	$0.448262\pi$
$192$	0	0
$193$	−22.6660	−1.63154	−0.815768	−	0.578380i	$-0.803685\pi$
$193$	−22.6660	−1.63154	−0.815768	+	0.578380i	$0.803685\pi$
$194$	1.80189	0.129368
$195$	0	0
$196$	8.64252	0.617323
$197$	11.2080	0.798535	0.399267	−	0.916835i	$-0.369265\pi$
$197$	11.2080	0.798535	0.399267	+	0.916835i	$0.369265\pi$
$198$	0	0
$199$	−7.81979	−0.554330	−0.277165	−	0.960822i	$-0.589395\pi$
$199$	−7.81979	−0.554330	−0.277165	+	0.960822i	$0.589395\pi$
$200$	−2.54920	−0.180256
$201$	0	0
$202$	5.54981	0.390483
$203$	6.14116	0.431025
$204$	0	0
$205$	−3.93626	−0.274920
$206$	−6.99454	−0.487332
$207$	0	0
$208$	3.55722	0.246649
$209$	0	0
$210$	0	0
$211$	−22.7670	−1.56734	−0.783672	−	0.621175i	$-0.786656\pi$
$211$	−22.7670	−1.56734	−0.783672	+	0.621175i	$0.786656\pi$
$212$	−17.2494	−1.18469
$213$	0	0
$214$	3.42765	0.234309
$215$	−7.64941	−0.521686
$216$	0	0
$217$	−3.80394	−0.258228
$218$	−3.93034	−0.266196
$219$	0	0
$220$	0	0
$221$	−8.27142	−0.556396
$222$	0	0
$223$	−16.0427	−1.07430	−0.537148	−	0.843488i	$-0.680499\pi$
$223$	−16.0427	−1.07430	−0.537148	+	0.843488i	$0.680499\pi$
$224$	6.04305	0.403768
$225$	0	0
$226$	−0.224907	−0.0149606
$227$	24.1562	1.60330	0.801652	−	0.597791i	$-0.203955\pi$
$227$	24.1562	1.60330	0.801652	+	0.597791i	$0.203955\pi$
$228$	0	0
$229$	−15.0143	−0.992172	−0.496086	−	0.868274i	$-0.665230\pi$
$229$	−15.0143	−0.992172	−0.496086	+	0.868274i	$0.665230\pi$
$230$	−1.81156	−0.119451
$231$	0	0
$232$	−15.1787	−0.996534
$233$	−11.4259	−0.748538	−0.374269	−	0.927320i	$-0.622106\pi$
$233$	−11.4259	−0.748538	−0.374269	+	0.927320i	$0.622106\pi$
$234$	0	0
$235$	−5.84294	−0.381151
$236$	4.28948	0.279221
$237$	0	0
$238$	−1.82453	−0.118267
$239$	−27.4067	−1.77279	−0.886397	−	0.462927i	$-0.846799\pi$
$239$	−27.4067	−1.77279	−0.886397	+	0.462927i	$0.846799\pi$
$240$	0	0
$241$	10.9387	0.704624	0.352312	−	0.935883i	$-0.385396\pi$
$241$	10.9387	0.704624	0.352312	+	0.935883i	$0.385396\pi$
$242$	0	0
$243$	0	0
$244$	−3.61114	−0.231179
$245$	−5.93626	−0.379253
$246$	0	0
$247$	−26.4496	−1.68295
$248$	9.40197	0.597026
$249$	0	0
$250$	0.737640	0.0466525
$251$	−17.2311	−1.08762	−0.543809	−	0.839209i	$-0.683018\pi$
$251$	−17.2311	−1.08762	−0.543809	+	0.839209i	$0.683018\pi$
$252$	0	0
$253$	0	0
$254$	−8.78527	−0.551237
$255$	0	0
$256$	−11.9331	−0.745819
$257$	−18.4954	−1.15371	−0.576856	−	0.816846i	$-0.695721\pi$
$257$	−18.4954	−1.15371	−0.576856	+	0.816846i	$0.695721\pi$
$258$	0	0
$259$	6.14556	0.381866
$260$	−5.02134	−0.311410
$261$	0	0
$262$	8.25232	0.509830
$263$	−3.69135	−0.227618	−0.113809	−	0.993503i	$-0.536305\pi$
$263$	−3.69135	−0.227618	−0.113809	+	0.993503i	$0.536305\pi$
$264$	0	0
$265$	11.8480	0.727819
$266$	−5.83433	−0.357726
$267$	0	0
$268$	8.94936	0.546669
$269$	9.94510	0.606363	0.303182	−	0.952933i	$-0.401951\pi$
$269$	9.94510	0.606363	0.303182	+	0.952933i	$0.401951\pi$
$270$	0	0
$271$	1.25365	0.0761539	0.0380770	−	0.999275i	$-0.487877\pi$
$271$	1.25365	0.0761539	0.0380770	+	0.999275i	$0.487877\pi$
$272$	−2.47347	−0.149976
$273$	0	0
$274$	3.15802	0.190783
$275$	0	0
$276$	0	0
$277$	−8.09331	−0.486280	−0.243140	−	0.969991i	$-0.578177\pi$
$277$	−8.09331	−0.486280	−0.243140	+	0.969991i	$0.578177\pi$
$278$	−4.39132	−0.263374
$279$	0	0
$280$	−2.62920	−0.157124
$281$	−25.3702	−1.51346	−0.756731	−	0.653726i	$-0.773205\pi$
$281$	−25.3702	−1.51346	−0.756731	+	0.653726i	$0.773205\pi$
$282$	0	0
$283$	−20.4424	−1.21517	−0.607587	−	0.794253i	$-0.707863\pi$
$283$	−20.4424	−1.21517	−0.607587	+	0.794253i	$0.707863\pi$
$284$	2.94547	0.174782
$285$	0	0
$286$	0	0
$287$	−4.05977	−0.239641
$288$	0	0
$289$	−11.2486	−0.661680
$290$	4.39214	0.257915
$291$	0	0
$292$	1.20237	0.0703633
$293$	−2.54907	−0.148918	−0.0744591	−	0.997224i	$-0.523723\pi$
$293$	−2.54907	−0.148918	−0.0744591	+	0.997224i	$0.523723\pi$
$294$	0	0
$295$	−2.94630	−0.171540
$296$	−15.1896	−0.882878
$297$	0	0
$298$	2.79156	0.161711
$299$	−8.47033	−0.489852
$300$	0	0
$301$	−7.88945	−0.454740
$302$	−17.9664	−1.03385
$303$	0	0
$304$	−7.90945	−0.453638
$305$	2.48037	0.142026
$306$	0	0
$307$	8.99273	0.513242	0.256621	−	0.966512i	$-0.417391\pi$
$307$	8.99273	0.513242	0.256621	+	0.966512i	$0.417391\pi$
$308$	0	0
$309$	0	0
$310$	−2.72057	−0.154518
$311$	20.1232	1.14108	0.570540	−	0.821270i	$-0.306734\pi$
$311$	20.1232	1.14108	0.570540	+	0.821270i	$0.306734\pi$
$312$	0	0
$313$	7.10483	0.401589	0.200794	−	0.979633i	$-0.435648\pi$
$313$	7.10483	0.401589	0.200794	+	0.979633i	$0.435648\pi$
$314$	5.33469	0.301054
$315$	0	0
$316$	17.5845	0.989204
$317$	−2.29323	−0.128801	−0.0644003	−	0.997924i	$-0.520513\pi$
$317$	−2.29323	−0.128801	−0.0644003	+	0.997924i	$0.520513\pi$
$318$	0	0
$319$	0	0
$320$	2.25922	0.126294
$321$	0	0
$322$	−1.86841	−0.104122
$323$	18.3915	1.02333
$324$	0	0
$325$	3.44899	0.191315
$326$	13.7859	0.763529
$327$	0	0
$328$	10.0343	0.554052
$329$	−6.02629	−0.332240
$330$	0	0
$331$	15.3951	0.846192	0.423096	−	0.906085i	$-0.360943\pi$
$331$	15.3951	0.846192	0.423096	+	0.906085i	$0.360943\pi$
$332$	2.34400	0.128644
$333$	0	0
$334$	−5.79289	−0.316973
$335$	−6.14702	−0.335847
$336$	0	0
$337$	−19.4968	−1.06206	−0.531030	−	0.847353i	$-0.678195\pi$
$337$	−19.4968	−1.06206	−0.531030	+	0.847353i	$0.678195\pi$
$338$	−0.814706	−0.0443141
$339$	0	0
$340$	3.49153	0.189355
$341$	0	0
$342$	0	0
$343$	−13.3422	−0.720411
$344$	19.4999	1.05136
$345$	0	0
$346$	−8.26503	−0.444331
$347$	2.16905	0.116440	0.0582202	−	0.998304i	$-0.481457\pi$
$347$	2.16905	0.116440	0.0582202	+	0.998304i	$0.481457\pi$
$348$	0	0
$349$	−25.0520	−1.34100	−0.670502	−	0.741908i	$-0.733921\pi$
$349$	−25.0520	−1.34100	−0.670502	+	0.741908i	$0.733921\pi$
$350$	0.760787	0.0406658
$351$	0	0
$352$	0	0
$353$	23.2532	1.23764	0.618821	−	0.785532i	$-0.287611\pi$
$353$	23.2532	1.23764	0.618821	+	0.785532i	$0.287611\pi$
$354$	0	0
$355$	−2.02315	−0.107377
$356$	11.8817	0.629730
$357$	0	0
$358$	−1.08037	−0.0570993
$359$	−10.1224	−0.534239	−0.267119	−	0.963663i	$-0.586072\pi$
$359$	−10.1224	−0.534239	−0.267119	+	0.963663i	$0.586072\pi$
$360$	0	0
$361$	39.8106	2.09530
$362$	6.85194	0.360130
$363$	0	0
$364$	−5.17891	−0.271448
$365$	−0.825867	−0.0432278
$366$	0	0
$367$	3.70925	0.193621	0.0968105	−	0.995303i	$-0.469136\pi$
$367$	3.70925	0.193621	0.0968105	+	0.995303i	$0.469136\pi$
$368$	−2.53295	−0.132039
$369$	0	0
$370$	4.39529	0.228500
$371$	12.2198	0.634421
$372$	0	0
$373$	−9.34017	−0.483616	−0.241808	−	0.970324i	$-0.577740\pi$
$373$	−9.34017	−0.483616	−0.241808	+	0.970324i	$0.577740\pi$
$374$	0	0
$375$	0	0
$376$	14.8948	0.768142
$377$	20.5364	1.05768
$378$	0	0
$379$	−9.81169	−0.503993	−0.251996	−	0.967728i	$-0.581087\pi$
$379$	−9.81169	−0.503993	−0.251996	+	0.967728i	$0.581087\pi$
$380$	11.1649	0.572748
$381$	0	0
$382$	3.29944	0.168814
$383$	18.0468	0.922149	0.461074	−	0.887362i	$-0.347464\pi$
$383$	18.0468	0.922149	0.461074	+	0.887362i	$0.347464\pi$
$384$	0	0
$385$	0	0
$386$	−16.7194	−0.850993
$387$	0	0
$388$	−3.55641	−0.180550
$389$	−31.1915	−1.58147	−0.790737	−	0.612156i	$-0.790302\pi$
$389$	−31.1915	−1.58147	−0.790737	+	0.612156i	$0.790302\pi$
$390$	0	0
$391$	5.88974	0.297857
$392$	15.1327	0.764317
$393$	0	0
$394$	8.26745	0.416508
$395$	−12.0782	−0.607719
$396$	0	0
$397$	−10.6212	−0.533062	−0.266531	−	0.963826i	$-0.585877\pi$
$397$	−10.6212	−0.533062	−0.266531	+	0.963826i	$0.585877\pi$
$398$	−5.76820	−0.289133
$399$	0	0
$400$	1.03138	0.0515690
$401$	27.5679	1.37668	0.688338	−	0.725390i	$-0.258341\pi$
$401$	27.5679	1.37668	0.688338	+	0.725390i	$0.258341\pi$
$402$	0	0
$403$	−12.7206	−0.633657
$404$	−10.9537	−0.544967
$405$	0	0
$406$	4.52997	0.224818
$407$	0	0
$408$	0	0
$409$	14.3682	0.710460	0.355230	−	0.934779i	$-0.384402\pi$
$409$	14.3682	0.710460	0.355230	+	0.934779i	$0.384402\pi$
$410$	−2.90354	−0.143396
$411$	0	0
$412$	13.8052	0.680132
$413$	−3.03875	−0.149527
$414$	0	0
$415$	−1.61002	−0.0790327
$416$	20.2083	0.990793
$417$	0	0
$418$	0	0
$419$	31.4707	1.53744	0.768722	−	0.639584i	$-0.220893\pi$
$419$	31.4707	1.53744	0.768722	+	0.639584i	$0.220893\pi$
$420$	0	0
$421$	26.5712	1.29500	0.647500	−	0.762065i	$-0.275815\pi$
$421$	26.5712	1.29500	0.647500	+	0.762065i	$0.275815\pi$
$422$	−16.7939	−0.817512
$423$	0	0
$424$	−30.2030	−1.46679
$425$	−2.39822	−0.116331
$426$	0	0
$427$	2.55820	0.123800
$428$	−6.76518	−0.327008
$429$	0	0
$430$	−5.64252	−0.272106
$431$	3.86870	0.186349	0.0931744	−	0.995650i	$-0.470299\pi$
$431$	3.86870	0.186349	0.0931744	+	0.995650i	$0.470299\pi$
$432$	0	0
$433$	40.1388	1.92895	0.964474	−	0.264179i	$-0.0851010\pi$
$433$	40.1388	1.92895	0.964474	+	0.264179i	$0.0851010\pi$
$434$	−2.80594	−0.134689
$435$	0	0
$436$	7.75735	0.371510
$437$	18.8337	0.900939
$438$	0	0
$439$	1.02336	0.0488425	0.0244212	−	0.999702i	$-0.492226\pi$
$439$	1.02336	0.0488425	0.0244212	+	0.999702i	$0.492226\pi$
$440$	0	0
$441$	0	0
$442$	−6.10133	−0.290211
$443$	16.0728	0.763644	0.381822	−	0.924236i	$-0.375297\pi$
$443$	16.0728	0.763644	0.381822	+	0.924236i	$0.375297\pi$
$444$	0	0
$445$	−8.16116	−0.386876
$446$	−11.8337	−0.560343
$447$	0	0
$448$	2.33011	0.110087
$449$	−35.8421	−1.69149	−0.845746	−	0.533585i	$-0.820844\pi$
$449$	−35.8421	−1.69149	−0.845746	+	0.533585i	$0.820844\pi$
$450$	0	0
$451$	0	0
$452$	0.443901	0.0208793
$453$	0	0
$454$	17.8186	0.836268
$455$	3.55722	0.166765
$456$	0	0
$457$	−25.1525	−1.17658	−0.588291	−	0.808649i	$-0.700199\pi$
$457$	−25.1525	−1.17658	−0.588291	+	0.808649i	$0.700199\pi$
$458$	−11.0751	−0.517507
$459$	0	0
$460$	3.57549	0.166708
$461$	−6.65631	−0.310015	−0.155008	−	0.987913i	$-0.549540\pi$
$461$	−6.65631	−0.310015	−0.155008	+	0.987913i	$0.549540\pi$
$462$	0	0
$463$	38.7730	1.80194	0.900968	−	0.433886i	$-0.142858\pi$
$463$	38.7730	1.80194	0.900968	+	0.433886i	$0.142858\pi$
$464$	6.14116	0.285096
$465$	0	0
$466$	−8.42823	−0.390430
$467$	−22.5494	−1.04346	−0.521731	−	0.853110i	$-0.674714\pi$
$467$	−22.5494	−1.04346	−0.521731	+	0.853110i	$0.674714\pi$
$468$	0	0
$469$	−6.33991	−0.292750
$470$	−4.30999	−0.198805
$471$	0	0
$472$	7.51071	0.345708
$473$	0	0
$474$	0	0
$475$	−7.66881	−0.351869
$476$	3.60109	0.165056
$477$	0	0
$478$	−20.2163	−0.924672
$479$	1.63186	0.0745618	0.0372809	−	0.999305i	$-0.488130\pi$
$479$	1.63186	0.0745618	0.0372809	+	0.999305i	$0.488130\pi$
$480$	0	0
$481$	20.5511	0.937048
$482$	8.06883	0.367525
$483$	0	0
$484$	0	0
$485$	2.44278	0.110921
$486$	0	0
$487$	1.05030	0.0475936	0.0237968	−	0.999717i	$-0.492425\pi$
$487$	1.05030	0.0475936	0.0237968	+	0.999717i	$0.492425\pi$
$488$	−6.32296	−0.286227
$489$	0	0
$490$	−4.37882	−0.197815
$491$	13.9141	0.627934	0.313967	−	0.949434i	$-0.398342\pi$
$491$	13.9141	0.627934	0.313967	+	0.949434i	$0.398342\pi$
$492$	0	0
$493$	−14.2797	−0.643127
$494$	−19.5103	−0.877811
$495$	0	0
$496$	−3.80394	−0.170802
$497$	−2.08663	−0.0935983
$498$	0	0
$499$	17.3673	0.777466	0.388733	−	0.921350i	$-0.372913\pi$
$499$	17.3673	0.777466	0.388733	+	0.921350i	$0.372913\pi$
$500$	−1.45589	−0.0651092
$501$	0	0
$502$	−12.7104	−0.567291
$503$	−35.8536	−1.59863	−0.799316	−	0.600910i	$-0.794805\pi$
$503$	−35.8536	−1.59863	−0.799316	+	0.600910i	$0.794805\pi$
$504$	0	0
$505$	7.52373	0.334802
$506$	0	0
$507$	0	0
$508$	17.3396	0.769319
$509$	−2.13878	−0.0947999	−0.0474000	−	0.998876i	$-0.515094\pi$
$509$	−2.13878	−0.0947999	−0.0474000	+	0.998876i	$0.515094\pi$
$510$	0	0
$511$	−0.851782	−0.0376806
$512$	11.3014	0.499458
$513$	0	0
$514$	−13.6430	−0.601765
$515$	−9.48231	−0.417841
$516$	0	0
$517$	0	0
$518$	4.53321	0.199178
$519$	0	0
$520$	−8.79217	−0.385562
$521$	12.7352	0.557940	0.278970	−	0.960300i	$-0.410007\pi$
$521$	12.7352	0.557940	0.278970	+	0.960300i	$0.410007\pi$
$522$	0	0
$523$	23.9088	1.04546	0.522729	−	0.852499i	$-0.324914\pi$
$523$	23.9088	1.04546	0.522729	+	0.852499i	$0.324914\pi$
$524$	−16.2877	−0.711530
$525$	0	0
$526$	−2.72289	−0.118723
$527$	8.84510	0.385299
$528$	0	0
$529$	−16.9686	−0.737766
$530$	8.73958	0.379623
$531$	0	0
$532$	11.5153	0.499250
$533$	−13.5761	−0.588046
$534$	0	0
$535$	4.64678	0.200898
$536$	15.6700	0.676840
$537$	0	0
$538$	7.33590	0.316273
$539$	0	0
$540$	0	0
$541$	1.31921	0.0567171	0.0283586	−	0.999598i	$-0.490972\pi$
$541$	1.31921	0.0567171	0.0283586	+	0.999598i	$0.490972\pi$
$542$	0.924744	0.0397212
$543$	0	0
$544$	−14.0516	−0.602457
$545$	−5.32826	−0.228238
$546$	0	0
$547$	9.32128	0.398549	0.199275	−	0.979944i	$-0.436141\pi$
$547$	9.32128	0.398549	0.199275	+	0.979944i	$0.436141\pi$
$548$	−6.23301	−0.266261
$549$	0	0
$550$	0	0
$551$	−45.6625	−1.94529
$552$	0	0
$553$	−12.4572	−0.529734
$554$	−5.96996	−0.253639
$555$	0	0
$556$	8.66718	0.367570
$557$	−39.3172	−1.66592	−0.832961	−	0.553331i	$-0.813356\pi$
$557$	−39.3172	−1.66592	−0.832961	+	0.553331i	$0.813356\pi$
$558$	0	0
$559$	−26.3827	−1.11587
$560$	1.06374	0.0449514
$561$	0	0
$562$	−18.7141	−0.789407
$563$	−19.9810	−0.842097	−0.421048	−	0.907038i	$-0.638338\pi$
$563$	−19.9810	−0.842097	−0.421048	+	0.907038i	$0.638338\pi$
$564$	0	0
$565$	−0.304901	−0.0128273
$566$	−15.0791	−0.633824
$567$	0	0
$568$	5.15741	0.216400
$569$	34.5647	1.44903	0.724515	−	0.689260i	$-0.242064\pi$
$569$	34.5647	1.44903	0.724515	+	0.689260i	$0.242064\pi$
$570$	0	0
$571$	3.15090	0.131861	0.0659306	−	0.997824i	$-0.478998\pi$
$571$	3.15090	0.131861	0.0659306	+	0.997824i	$0.478998\pi$
$572$	0	0
$573$	0	0
$574$	−2.99465	−0.124994
$575$	−2.45589	−0.102418
$576$	0	0
$577$	27.3226	1.13745	0.568727	−	0.822526i	$-0.307436\pi$
$577$	27.3226	1.13745	0.568727	+	0.822526i	$0.307436\pi$
$578$	−8.29739	−0.345126
$579$	0	0
$580$	−8.66881	−0.359953
$581$	−1.66054	−0.0688908
$582$	0	0
$583$	0	0
$584$	2.10530	0.0871180
$585$	0	0
$586$	−1.88030	−0.0776743
$587$	46.1679	1.90555	0.952776	−	0.303675i	$-0.0982138\pi$
$587$	46.1679	1.90555	0.952776	+	0.303675i	$0.0982138\pi$
$588$	0	0
$589$	28.2841	1.16543
$590$	−2.17331	−0.0894737
$591$	0	0
$592$	6.14556	0.252581
$593$	−39.4265	−1.61905	−0.809525	−	0.587085i	$-0.800275\pi$
$593$	−39.4265	−1.61905	−0.809525	+	0.587085i	$0.800275\pi$
$594$	0	0
$595$	−2.47347	−0.101402
$596$	−5.50972	−0.225687
$597$	0	0
$598$	−6.24805	−0.255502
$599$	−1.04875	−0.0428507	−0.0214253	−	0.999770i	$-0.506820\pi$
$599$	−1.04875	−0.0428507	−0.0214253	+	0.999770i	$0.506820\pi$
$600$	0	0
$601$	27.2498	1.11154	0.555771	−	0.831336i	$-0.312423\pi$
$601$	27.2498	1.11154	0.555771	+	0.831336i	$0.312423\pi$
$602$	−5.81958	−0.237188
$603$	0	0
$604$	35.4605	1.44287
$605$	0	0
$606$	0	0
$607$	21.9217	0.889774	0.444887	−	0.895587i	$-0.353244\pi$
$607$	21.9217	0.889774	0.444887	+	0.895587i	$0.353244\pi$
$608$	−44.9330	−1.82227
$609$	0	0
$610$	1.82962	0.0740791
$611$	−20.1522	−0.815272
$612$	0	0
$613$	10.5921	0.427809	0.213905	−	0.976855i	$-0.431382\pi$
$613$	10.5921	0.427809	0.213905	+	0.976855i	$0.431382\pi$
$614$	6.63340	0.267702
$615$	0	0
$616$	0	0
$617$	−4.60402	−0.185351	−0.0926755	−	0.995696i	$-0.529542\pi$
$617$	−4.60402	−0.185351	−0.0926755	+	0.995696i	$0.529542\pi$
$618$	0	0
$619$	37.0037	1.48731	0.743653	−	0.668566i	$-0.233092\pi$
$619$	37.0037	1.48731	0.743653	+	0.668566i	$0.233092\pi$
$620$	5.36960	0.215648
$621$	0	0
$622$	14.8437	0.595177
$623$	−8.41726	−0.337230
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.24081	0.209465
$627$	0	0
$628$	−10.5291	−0.420157
$629$	−14.2900	−0.569778
$630$	0	0
$631$	−24.8406	−0.988889	−0.494444	−	0.869209i	$-0.664628\pi$
$631$	−24.8406	−0.988889	−0.494444	+	0.869209i	$0.664628\pi$
$632$	30.7897	1.22475
$633$	0	0
$634$	−1.69158	−0.0671812
$635$	−11.9100	−0.472632
$636$	0	0
$637$	−20.4741	−0.811213
$638$	0	0
$639$	0	0
$640$	−10.0519	−0.397336
$641$	44.4293	1.75485	0.877425	−	0.479714i	$-0.159260\pi$
$641$	44.4293	1.75485	0.877425	+	0.479714i	$0.159260\pi$
$642$	0	0
$643$	25.8610	1.01986	0.509929	−	0.860217i	$-0.329672\pi$
$643$	25.8610	1.01986	0.509929	+	0.860217i	$0.329672\pi$
$644$	3.68769	0.145315
$645$	0	0
$646$	13.5663	0.533758
$647$	−19.4865	−0.766093	−0.383047	−	0.923729i	$-0.625125\pi$
$647$	−19.4865	−0.766093	−0.383047	+	0.923729i	$0.625125\pi$
$648$	0	0
$649$	0	0
$650$	2.54411	0.0997883
$651$	0	0
$652$	−27.2093	−1.06560
$653$	16.7298	0.654687	0.327344	−	0.944905i	$-0.393847\pi$
$653$	16.7298	0.654687	0.327344	+	0.944905i	$0.393847\pi$
$654$	0	0
$655$	11.1875	0.437130
$656$	−4.05977	−0.158508
$657$	0	0
$658$	−4.44524	−0.173293
$659$	1.66127	0.0647137	0.0323569	−	0.999476i	$-0.489699\pi$
$659$	1.66127	0.0647137	0.0323569	+	0.999476i	$0.489699\pi$
$660$	0	0
$661$	−44.0130	−1.71191	−0.855953	−	0.517053i	$-0.827029\pi$
$661$	−44.0130	−1.71191	−0.855953	+	0.517053i	$0.827029\pi$
$662$	11.3561	0.441365
$663$	0	0
$664$	4.10426	0.159276
$665$	−7.90945	−0.306715
$666$	0	0
$667$	−14.6231	−0.566210
$668$	11.4335	0.442375
$669$	0	0
$670$	−4.53429	−0.175175
$671$	0	0
$672$	0	0
$673$	−38.5949	−1.48772	−0.743862	−	0.668333i	$-0.767008\pi$
$673$	−38.5949	−1.48772	−0.743862	+	0.668333i	$0.767008\pi$
$674$	−14.3817	−0.553960
$675$	0	0
$676$	1.60799	0.0618458
$677$	38.5988	1.48347	0.741737	−	0.670691i	$-0.234002\pi$
$677$	38.5988	1.48347	0.741737	+	0.670691i	$0.234002\pi$
$678$	0	0
$679$	2.51944	0.0966871
$680$	6.11353	0.234443
$681$	0	0
$682$	0	0
$683$	0.748158	0.0286275	0.0143137	−	0.999898i	$-0.495444\pi$
$683$	0.748158	0.0286275	0.0143137	+	0.999898i	$0.495444\pi$
$684$	0	0
$685$	4.28124	0.163578
$686$	−9.84174	−0.375759
$687$	0	0
$688$	−7.88945	−0.300783
$689$	40.8637	1.55678
$690$	0	0
$691$	5.22184	0.198648	0.0993242	−	0.995055i	$-0.468332\pi$
$691$	5.22184	0.198648	0.0993242	+	0.995055i	$0.468332\pi$
$692$	16.3128	0.620118
$693$	0	0
$694$	1.59998	0.0607342
$695$	−5.95320	−0.225818
$696$	0	0
$697$	9.43999	0.357565
$698$	−18.4794	−0.699455
$699$	0	0
$700$	−1.50157	−0.0567541
$701$	14.1580	0.534740	0.267370	−	0.963594i	$-0.413845\pi$
$701$	14.1580	0.534740	0.267370	+	0.963594i	$0.413845\pi$
$702$	0	0
$703$	−45.6952	−1.72343
$704$	0	0
$705$	0	0
$706$	17.1525	0.645542
$707$	7.75983	0.291838
$708$	0	0
$709$	−17.2144	−0.646499	−0.323249	−	0.946314i	$-0.604775\pi$
$709$	−17.2144	−0.646499	−0.323249	+	0.946314i	$0.604775\pi$
$710$	−1.49235	−0.0560071
$711$	0	0
$712$	20.8044	0.779680
$713$	9.05781	0.339217
$714$	0	0
$715$	0	0
$716$	2.13233	0.0796891
$717$	0	0
$718$	−7.46667	−0.278654
$719$	26.5559	0.990369	0.495185	−	0.868788i	$-0.335100\pi$
$719$	26.5559	0.990369	0.495185	+	0.868788i	$0.335100\pi$
$720$	0	0
$721$	−9.77987	−0.364221
$722$	29.3659	1.09289
$723$	0	0
$724$	−13.5237	−0.502606
$725$	5.95431	0.221138
$726$	0	0
$727$	44.1917	1.63898	0.819490	−	0.573094i	$-0.194257\pi$
$727$	44.1917	1.63898	0.819490	+	0.573094i	$0.194257\pi$
$728$	−9.06806	−0.336085
$729$	0	0
$730$	−0.609193	−0.0225472
$731$	18.3449	0.678512
$732$	0	0
$733$	24.9604	0.921935	0.460968	−	0.887417i	$-0.347502\pi$
$733$	24.9604	0.921935	0.460968	+	0.887417i	$0.347502\pi$
$734$	2.73609	0.100991
$735$	0	0
$736$	−14.3895	−0.530404
$737$	0	0
$738$	0	0
$739$	21.2342	0.781114	0.390557	−	0.920579i	$-0.372282\pi$
$739$	21.2342	0.781114	0.390557	+	0.920579i	$0.372282\pi$
$740$	−8.67501	−0.318900
$741$	0	0
$742$	9.01383	0.330908
$743$	30.6527	1.12454	0.562270	−	0.826954i	$-0.309928\pi$
$743$	30.6527	1.12454	0.562270	+	0.826954i	$0.309928\pi$
$744$	0	0
$745$	3.78444	0.138651
$746$	−6.88968	−0.252249
$747$	0	0
$748$	0	0
$749$	4.79259	0.175118
$750$	0	0
$751$	−30.3073	−1.10593	−0.552965	−	0.833204i	$-0.686504\pi$
$751$	−30.3073	−1.10593	−0.552965	+	0.833204i	$0.686504\pi$
$752$	−6.02629	−0.219756
$753$	0	0
$754$	15.1485	0.551674
$755$	−24.3566	−0.886429
$756$	0	0
$757$	−34.8694	−1.26735	−0.633674	−	0.773600i	$-0.718454\pi$
$757$	−34.8694	−1.26735	−0.633674	+	0.773600i	$0.718454\pi$
$758$	−7.23750	−0.262878
$759$	0	0
$760$	19.5493	0.709129
$761$	2.68972	0.0975022	0.0487511	−	0.998811i	$-0.484476\pi$
$761$	2.68972	0.0975022	0.0487511	+	0.998811i	$0.484476\pi$
$762$	0	0
$763$	−5.49546	−0.198949
$764$	−6.51212	−0.235600
$765$	0	0
$766$	13.3121	0.480984
$767$	−10.1617	−0.366920
$768$	0	0
$769$	−32.5735	−1.17463	−0.587315	−	0.809359i	$-0.699815\pi$
$769$	−32.5735	−1.17463	−0.587315	+	0.809359i	$0.699815\pi$
$770$	0	0
$771$	0	0
$772$	32.9991	1.18767
$773$	−41.6637	−1.49854	−0.749270	−	0.662265i	$-0.769595\pi$
$773$	−41.6637	−1.49854	−0.749270	+	0.662265i	$0.769595\pi$
$774$	0	0
$775$	−3.68820	−0.132484
$776$	−6.22714	−0.223541
$777$	0	0
$778$	−23.0081	−0.824882
$779$	30.1864	1.08154
$780$	0	0
$781$	0	0
$782$	4.34451	0.155359
$783$	0	0
$784$	−6.12253	−0.218662
$785$	7.23210	0.258125
$786$	0	0
$787$	−35.9207	−1.28044	−0.640218	−	0.768193i	$-0.721156\pi$
$787$	−35.9207	−1.28044	−0.640218	+	0.768193i	$0.721156\pi$
$788$	−16.3175	−0.581288
$789$	0	0
$790$	−8.90936	−0.316981
$791$	−0.314468	−0.0111812
$792$	0	0
$793$	8.55476	0.303789
$794$	−7.83461	−0.278040
$795$	0	0
$796$	11.3847	0.403521
$797$	31.7197	1.12357	0.561785	−	0.827283i	$-0.310115\pi$
$797$	31.7197	1.12357	0.561785	+	0.827283i	$0.310115\pi$
$798$	0	0
$799$	14.0126	0.495731
$800$	5.85919	0.207154
$801$	0	0
$802$	20.3352	0.718061
$803$	0	0
$804$	0	0
$805$	−2.53295	−0.0892748
$806$	−9.38320	−0.330509
$807$	0	0
$808$	−19.1795	−0.674733
$809$	8.51572	0.299397	0.149698	−	0.988732i	$-0.452170\pi$
$809$	8.51572	0.299397	0.149698	+	0.988732i	$0.452170\pi$
$810$	0	0
$811$	9.03030	0.317097	0.158548	−	0.987351i	$-0.449319\pi$
$811$	9.03030	0.317097	0.158548	+	0.987351i	$0.449319\pi$
$812$	−8.94083	−0.313762
$813$	0	0
$814$	0	0
$815$	18.6892	0.654653
$816$	0	0
$817$	58.6619	2.05232
$818$	10.5985	0.370569
$819$	0	0
$820$	5.73074	0.200126
$821$	54.3225	1.89587	0.947935	−	0.318465i	$-0.103167\pi$
$821$	54.3225	1.89587	0.947935	+	0.318465i	$0.103167\pi$
$822$	0	0
$823$	17.8594	0.622541	0.311271	−	0.950321i	$-0.399245\pi$
$823$	17.8594	0.622541	0.311271	+	0.950321i	$0.399245\pi$
$824$	24.1723	0.842083
$825$	0	0
$826$	−2.24151	−0.0779920
$827$	51.9494	1.80646	0.903229	−	0.429159i	$-0.141190\pi$
$827$	51.9494	1.80646	0.903229	+	0.429159i	$0.141190\pi$
$828$	0	0
$829$	−19.7460	−0.685807	−0.342904	−	0.939371i	$-0.611410\pi$
$829$	−19.7460	−0.685807	−0.342904	+	0.939371i	$0.611410\pi$
$830$	−1.18761	−0.0412227
$831$	0	0
$832$	7.79201	0.270139
$833$	14.2364	0.493263
$834$	0	0
$835$	−7.85328	−0.271774
$836$	0	0
$837$	0	0
$838$	23.2140	0.801916
$839$	4.11650	0.142117	0.0710586	−	0.997472i	$-0.477362\pi$
$839$	4.11650	0.142117	0.0710586	+	0.997472i	$0.477362\pi$
$840$	0	0
$841$	6.45386	0.222547
$842$	19.6000	0.675460
$843$	0	0
$844$	33.1462	1.14094
$845$	−1.10448	−0.0379951
$846$	0	0
$847$	0	0
$848$	12.2198	0.419630
$849$	0	0
$850$	−1.76902	−0.0606769
$851$	−14.6336	−0.501633
$852$	0	0
$853$	5.50285	0.188414	0.0942070	−	0.995553i	$-0.469968\pi$
$853$	5.50285	0.188414	0.0942070	+	0.995553i	$0.469968\pi$
$854$	1.88703	0.0645729
$855$	0	0
$856$	−11.8456	−0.404873
$857$	26.9281	0.919847	0.459924	−	0.887959i	$-0.347877\pi$
$857$	26.9281	0.919847	0.459924	+	0.887959i	$0.347877\pi$
$858$	0	0
$859$	19.1519	0.653456	0.326728	−	0.945118i	$-0.394054\pi$
$859$	19.1519	0.653456	0.326728	+	0.945118i	$0.394054\pi$
$860$	11.1367	0.379758
$861$	0	0
$862$	2.85371	0.0971977
$863$	−4.96151	−0.168892	−0.0844458	−	0.996428i	$-0.526912\pi$
$863$	−4.96151	−0.168892	−0.0844458	+	0.996428i	$0.526912\pi$
$864$	0	0
$865$	−11.2047	−0.380971
$866$	29.6080	1.00612
$867$	0	0
$868$	5.53810	0.187975
$869$	0	0
$870$	0	0
$871$	−21.2010	−0.718368
$872$	13.5828	0.459972
$873$	0	0
$874$	13.8925	0.469921
$875$	1.03138	0.0348670
$876$	0	0
$877$	−27.2053	−0.918659	−0.459329	−	0.888266i	$-0.651910\pi$
$877$	−27.2053	−0.918659	−0.459329	+	0.888266i	$0.651910\pi$
$878$	0.754874	0.0254758
$879$	0	0
$880$	0	0
$881$	−10.3570	−0.348935	−0.174467	−	0.984663i	$-0.555820\pi$
$881$	−10.3570	−0.348935	−0.174467	+	0.984663i	$0.555820\pi$
$882$	0	0
$883$	7.39489	0.248858	0.124429	−	0.992229i	$-0.460290\pi$
$883$	7.39489	0.248858	0.124429	+	0.992229i	$0.460290\pi$
$884$	12.0422	0.405025
$885$	0	0
$886$	11.8560	0.398309
$887$	18.0590	0.606363	0.303181	−	0.952933i	$-0.401951\pi$
$887$	18.0590	0.606363	0.303181	+	0.952933i	$0.401951\pi$
$888$	0	0
$889$	−12.2837	−0.411982
$890$	−6.02000	−0.201791
$891$	0	0
$892$	23.3563	0.782027
$893$	44.8084	1.49946
$894$	0	0
$895$	−1.46463	−0.0489572
$896$	−10.3673	−0.346348
$897$	0	0
$898$	−26.4386	−0.882267
$899$	−21.9607	−0.732431
$900$	0	0
$901$	−28.4141	−0.946612
$902$	0	0
$903$	0	0
$904$	0.777253	0.0258511
$905$	9.28900	0.308777
$906$	0	0
$907$	26.2971	0.873179	0.436590	−	0.899661i	$-0.356186\pi$
$907$	26.2971	0.873179	0.436590	+	0.899661i	$0.356186\pi$
$908$	−35.1687	−1.16711
$909$	0	0
$910$	2.62395	0.0869830
$911$	−28.6489	−0.949179	−0.474589	−	0.880207i	$-0.657403\pi$
$911$	−28.6489	−0.949179	−0.474589	+	0.880207i	$0.657403\pi$
$912$	0	0
$913$	0	0
$914$	−18.5535	−0.613694
$915$	0	0
$916$	21.8591	0.722245
$917$	11.5385	0.381035
$918$	0	0
$919$	38.8568	1.28177	0.640884	−	0.767638i	$-0.278568\pi$
$919$	38.8568	1.28177	0.640884	+	0.767638i	$0.278568\pi$
$920$	6.26055	0.206404
$921$	0	0
$922$	−4.90996	−0.161701
$923$	−6.97781	−0.229677
$924$	0	0
$925$	5.95858	0.195917
$926$	28.6006	0.939873
$927$	0	0
$928$	34.8875	1.14524
$929$	−7.42132	−0.243486	−0.121743	−	0.992562i	$-0.538848\pi$
$929$	−7.42132	−0.243486	−0.121743	+	0.992562i	$0.538848\pi$
$930$	0	0
$931$	45.5240	1.49199
$932$	16.6349	0.544893
$933$	0	0
$934$	−16.6334	−0.544260
$935$	0	0
$936$	0	0
$937$	−14.9360	−0.487937	−0.243968	−	0.969783i	$-0.578449\pi$
$937$	−14.9360	−0.487937	−0.243968	+	0.969783i	$0.578449\pi$
$938$	−4.67657	−0.152696
$939$	0	0
$940$	8.50666	0.277457
$941$	52.3409	1.70626	0.853132	−	0.521695i	$-0.174700\pi$
$941$	52.3409	1.70626	0.853132	+	0.521695i	$0.174700\pi$
$942$	0	0
$943$	9.66700	0.314801
$944$	−3.03875	−0.0989030
$945$	0	0
$946$	0	0
$947$	3.69553	0.120088	0.0600442	−	0.998196i	$-0.480876\pi$
$947$	3.69553	0.120088	0.0600442	+	0.998196i	$0.480876\pi$
$948$	0	0
$949$	−2.84841	−0.0924631
$950$	−5.65682	−0.183532
$951$	0	0
$952$	6.30538	0.204358
$953$	−43.1526	−1.39785	−0.698925	−	0.715195i	$-0.746338\pi$
$953$	−43.1526	−1.39785	−0.698925	+	0.715195i	$0.746338\pi$
$954$	0	0
$955$	4.47296	0.144742
$956$	39.9011	1.29049
$957$	0	0
$958$	1.20373	0.0388907
$959$	4.41559	0.142587
$960$	0	0
$961$	−17.3972	−0.561199
$962$	15.1593	0.488755
$963$	0	0
$964$	−15.9255	−0.512927
$965$	−22.6660	−0.729645
$966$	0	0
$967$	−29.2144	−0.939471	−0.469736	−	0.882807i	$-0.655651\pi$
$967$	−29.2144	−0.939471	−0.469736	+	0.882807i	$0.655651\pi$
$968$	0	0
$969$	0	0
$970$	1.80189	0.0578554
$971$	26.4046	0.847365	0.423683	−	0.905811i	$-0.360737\pi$
$971$	26.4046	0.847365	0.423683	+	0.905811i	$0.360737\pi$
$972$	0	0
$973$	−6.14001	−0.196840
$974$	0.774743	0.0248244
$975$	0	0
$976$	2.55820	0.0818861
$977$	15.8434	0.506875	0.253437	−	0.967352i	$-0.418439\pi$
$977$	15.8434	0.506875	0.253437	+	0.967352i	$0.418439\pi$
$978$	0	0
$979$	0	0
$980$	8.64252	0.276075
$981$	0	0
$982$	10.2636	0.327525
$983$	−35.1039	−1.11964	−0.559821	−	0.828614i	$-0.689130\pi$
$983$	−35.1039	−1.11964	−0.559821	+	0.828614i	$0.689130\pi$
$984$	0	0
$985$	11.2080	0.357116
$986$	−10.5333	−0.335449
$987$	0	0
$988$	38.5077	1.22509
$989$	18.7861	0.597363
$990$	0	0
$991$	18.9700	0.602600	0.301300	−	0.953529i	$-0.402579\pi$
$991$	18.9700	0.602600	0.301300	+	0.953529i	$0.402579\pi$
$992$	−21.6099	−0.686114
$993$	0	0
$994$	−1.53918	−0.0488200
$995$	−7.81979	−0.247904
$996$	0	0
$997$	30.1347	0.954375	0.477188	−	0.878801i	$-0.341656\pi$
$997$	30.1347	0.954375	0.477188	+	0.878801i	$0.341656\pi$
$998$	12.8108	0.405519
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bp.1.3		4
3.2	odd	2		605.2.a.j.1.2		4
11.5	even	5		495.2.n.e.91.1		8
11.9	even	5		495.2.n.e.136.1		8
11.10	odd	2		5445.2.a.bi.1.2		4
12.11	even	2		9680.2.a.cn.1.4		4
15.14	odd	2		3025.2.a.bd.1.3		4
33.2	even	10		605.2.g.k.81.1		8
33.5	odd	10		55.2.g.b.36.2	yes	8
33.8	even	10		605.2.g.e.251.2		8
33.14	odd	10		605.2.g.m.251.1		8
33.17	even	10		605.2.g.k.366.1		8
33.20	odd	10		55.2.g.b.26.2	✓	8
33.26	odd	10		605.2.g.m.511.1		8
33.29	even	10		605.2.g.e.511.2		8
33.32	even	2		605.2.a.k.1.3		4
132.71	even	10		880.2.bo.h.641.2		8
132.119	even	10		880.2.bo.h.81.2		8
132.131	odd	2		9680.2.a.cm.1.4		4
165.38	even	20		275.2.z.a.124.2		16
165.53	even	20		275.2.z.a.224.3		16
165.104	odd	10		275.2.h.a.201.1		8
165.119	odd	10		275.2.h.a.26.1		8
165.137	even	20		275.2.z.a.124.3		16
165.152	even	20		275.2.z.a.224.2		16
165.164	even	2		3025.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.26.2	✓	8	33.20	odd	10
55.2.g.b.36.2	yes	8	33.5	odd	10
275.2.h.a.26.1		8	165.119	odd	10
275.2.h.a.201.1		8	165.104	odd	10
275.2.z.a.124.2		16	165.38	even	20
275.2.z.a.124.3		16	165.137	even	20
275.2.z.a.224.2		16	165.152	even	20
275.2.z.a.224.3		16	165.53	even	20
495.2.n.e.91.1		8	11.5	even	5
495.2.n.e.136.1		8	11.9	even	5
605.2.a.j.1.2		4	3.2	odd	2
605.2.a.k.1.3		4	33.32	even	2
605.2.g.e.251.2		8	33.8	even	10
605.2.g.e.511.2		8	33.29	even	10
605.2.g.k.81.1		8	33.2	even	10
605.2.g.k.366.1		8	33.17	even	10
605.2.g.m.251.1		8	33.14	odd	10
605.2.g.m.511.1		8	33.26	odd	10
880.2.bo.h.81.2		8	132.119	even	10
880.2.bo.h.641.2		8	132.71	even	10
3025.2.a.w.1.2		4	165.164	even	2
3025.2.a.bd.1.3		4	15.14	odd	2
5445.2.a.bi.1.2		4	11.10	odd	2
5445.2.a.bp.1.3		4	1.1	even	1	trivial
9680.2.a.cm.1.4		4	132.131	odd	2
9680.2.a.cn.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.737640 q^{2} -1.45589 q^{4} +1.00000 q^{5} +1.03138 q^{7} -2.54920 q^{8} +O(q^{10})\)	\(q+0.737640 q^{2} -1.45589 q^{4} +1.00000 q^{5} +1.03138 q^{7} -2.54920 q^{8} +0.737640 q^{10} +3.44899 q^{13} +0.760787 q^{14} +1.03138 q^{16} -2.39822 q^{17} -7.66881 q^{19} -1.45589 q^{20} -2.45589 q^{23} +1.00000 q^{25} +2.54411 q^{26} -1.50157 q^{28} +5.95431 q^{29} -3.68820 q^{31} +5.85919 q^{32} -1.76902 q^{34} +1.03138 q^{35} +5.95858 q^{37} -5.65682 q^{38} -2.54920 q^{40} -3.93626 q^{41} -7.64941 q^{43} -1.81156 q^{46} -5.84294 q^{47} -5.93626 q^{49} +0.737640 q^{50} -5.02134 q^{52} +11.8480 q^{53} -2.62920 q^{56} +4.39214 q^{58} -2.94630 q^{59} +2.48037 q^{61} -2.72057 q^{62} +2.25922 q^{64} +3.44899 q^{65} -6.14702 q^{67} +3.49153 q^{68} +0.760787 q^{70} -2.02315 q^{71} -0.825867 q^{73} +4.39529 q^{74} +11.1649 q^{76} -12.0782 q^{79} +1.03138 q^{80} -2.90354 q^{82} -1.61002 q^{83} -2.39822 q^{85} -5.64252 q^{86} -8.16116 q^{89} +3.55722 q^{91} +3.57549 q^{92} -4.30999 q^{94} -7.66881 q^{95} +2.44278 q^{97} -4.37882 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 + 3 * q^8	\( 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8} + q^{10} - q^{13} - 2 q^{14} - 3 q^{16} - q^{17} - 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} - 13 q^{28} + 12 q^{29} - 5 q^{31} - 8 q^{32} + 2 q^{34} - 3 q^{35} + 7 q^{37} - 20 q^{38} + 3 q^{40} + 11 q^{41} - 19 q^{43} + 4 q^{46} - 5 q^{47} + 3 q^{49} + q^{50} + 11 q^{52} + 11 q^{53} - 11 q^{56} - 14 q^{58} - 9 q^{59} - 12 q^{61} - 35 q^{62} - 3 q^{64} - q^{65} - 19 q^{67} - 3 q^{68} - 2 q^{70} - 5 q^{71} - 11 q^{73} - 34 q^{79} - 3 q^{80} - 6 q^{82} - 11 q^{83} - q^{85} - q^{86} + 8 q^{89} - 8 q^{91} + 12 q^{92} + q^{94} - 20 q^{95} + 32 q^{97} - 8 q^{98}+O(q^{100}) \) 4 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 + 3 * q^8 + q^10 - q^13 - 2 * q^14 - 3 * q^16 - q^17 - 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 - 13 * q^28 + 12 * q^29 - 5 * q^31 - 8 * q^32 + 2 * q^34 - 3 * q^35 + 7 * q^37 - 20 * q^38 + 3 * q^40 + 11 * q^41 - 19 * q^43 + 4 * q^46 - 5 * q^47 + 3 * q^49 + q^50 + 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 - 12 * q^61 - 35 * q^62 - 3 * q^64 - q^65 - 19 * q^67 - 3 * q^68 - 2 * q^70 - 5 * q^71 - 11 * q^73 - 34 * q^79 - 3 * q^80 - 6 * q^82 - 11 * q^83 - q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 + q^94 - 20 * q^95 + 32 * q^97 - 8 * q^98

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