LMFDB - Embedded newform 5445.2.a.bi.1.2

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:	$4$
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			1.2
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	$-0.583985\pi$
$2$	−0.737640	−0.521590	−0.260795	+	0.965394i	$0.583985\pi$
$3$	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	$-0.562442\pi$
$7$	−1.03138	−0.389825	−0.194912	+	0.980821i	$0.562442\pi$
$8$	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	$-0.658743\pi$
$13$	−3.44899	−0.956577	−0.478289	+	0.878203i	$0.658743\pi$
$14$	0.760787	0.203329
$15$	0	0
$16$	1.03138	0.257845
$17$	2.39822	0.581653	0.290826	−	0.956776i	$-0.406070\pi$
$17$	2.39822	0.581653	0.290826	+	0.956776i	$0.406070\pi$
$18$	0	0
$19$	7.66881	1.75935	0.879673	−	0.475580i	$-0.157762\pi$
$19$	7.66881	1.75935	0.879673	+	0.475580i	$0.157762\pi$
$20$	−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	$-0.686458\pi$
$29$	−5.95431	−1.10569	−0.552844	+	0.833285i	$0.686458\pi$
$30$	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	$-0.400551\pi$
$41$	3.93626	0.614740	0.307370	+	0.951590i	$0.400551\pi$
$42$	0	0
$43$	7.64941	1.16652	0.583262	−	0.812284i	$-0.301776\pi$
$43$	7.64941	1.16652	0.583262	+	0.812284i	$0.301776\pi$
$44$	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	$-0.550759\pi$
$61$	−2.48037	−0.317579	−0.158789	+	0.987312i	$0.550759\pi$
$62$	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	$-0.484610\pi$
$73$	0.825867	0.0966604	0.0483302	+	0.998831i	$0.484610\pi$
$74$	−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	$-0.262218\pi$
$79$	12.0782	1.35890	0.679451	+	0.733721i	$0.262218\pi$
$80$	1.03138	0.115312
$81$	0	0
$82$	−2.90354	−0.320642
$83$	1.61002	0.176722	0.0883612	−	0.996088i	$-0.471837\pi$
$83$	1.61002	0.176722	0.0883612	+	0.996088i	$0.471837\pi$
$84$	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	$-0.622124\pi$
$101$	−7.52373	−0.748640	−0.374320	+	0.927300i	$0.622124\pi$
$102$	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	$-0.572111\pi$
$107$	−4.64678	−0.449221	−0.224611	+	0.974449i	$0.572111\pi$
$108$	0	0
$109$	5.32826	0.510355	0.255178	−	0.966894i	$-0.417866\pi$
$109$	5.32826	0.510355	0.255178	+	0.966894i	$0.417866\pi$
$110$	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	$-0.322785\pi$
$127$	11.9100	1.05684	0.528419	+	0.848984i	$0.322785\pi$
$128$	10.0519	0.888470
$129$	0	0
$130$	2.54411	0.223133
$131$	−11.1875	−0.977452	−0.488726	−	0.872437i	$-0.662538\pi$
$131$	−11.1875	−0.977452	−0.488726	+	0.872437i	$0.662538\pi$
$132$	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	$-0.418757\pi$
$139$	5.95320	0.504943	0.252472	+	0.967604i	$0.418757\pi$
$140$	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	$-0.549543\pi$
$149$	−3.78444	−0.310034	−0.155017	+	0.987912i	$0.549543\pi$
$150$	0	0
$151$	24.3566	1.98211	0.991057	−	0.133437i	$-0.0426013\pi$
$151$	24.3566	1.98211	0.991057	+	0.133437i	$0.0426013\pi$
$152$	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	$-0.401727\pi$
$167$	7.85328	0.607705	0.303852	+	0.952719i	$0.401727\pi$
$168$	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	$-0.359944\pi$
$173$	11.2047	0.851877	0.425938	+	0.904752i	$0.359944\pi$
$174$	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	$-0.196315\pi$
$193$	22.6660	1.63154	0.815768	+	0.578380i	$0.196315\pi$
$194$	−1.80189	−0.129368
$195$	0	0
$196$	8.64252	0.617323
$197$	−11.2080	−0.798535	−0.399267	−	0.916835i	$-0.630735\pi$
$197$	−11.2080	−0.798535	−0.399267	+	0.916835i	$0.630735\pi$
$198$	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	$-0.213344\pi$
$211$	22.7670	1.56734	0.783672	+	0.621175i	$0.213344\pi$
$212$	−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	$-0.796045\pi$
$227$	−24.1562	−1.60330	−0.801652	+	0.597791i	$0.796045\pi$
$228$	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	$-0.377894\pi$
$233$	11.4259	0.748538	0.374269	+	0.927320i	$0.377894\pi$
$234$	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	$-0.153201\pi$
$239$	27.4067	1.77279	0.886397	+	0.462927i	$0.153201\pi$
$240$	0	0
$241$	−10.9387	−0.704624	−0.352312	−	0.935883i	$-0.614604\pi$
$241$	−10.9387	−0.704624	−0.352312	+	0.935883i	$0.614604\pi$
$242$	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	$-0.463695\pi$
$263$	3.69135	0.227618	0.113809	+	0.993503i	$0.463695\pi$
$264$	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	$-0.512123\pi$
$271$	−1.25365	−0.0761539	−0.0380770	+	0.999275i	$0.512123\pi$
$272$	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	$-0.421823\pi$
$277$	8.09331	0.486280	0.243140	+	0.969991i	$0.421823\pi$
$278$	−4.39132	−0.263374
$279$	0	0
$280$	−2.62920	−0.157124
$281$	25.3702	1.51346	0.756731	−	0.653726i	$-0.226795\pi$
$281$	25.3702	1.51346	0.756731	+	0.653726i	$0.226795\pi$
$282$	0	0
$283$	20.4424	1.21517	0.607587	−	0.794253i	$-0.292137\pi$
$283$	20.4424	1.21517	0.607587	+	0.794253i	$0.292137\pi$
$284$	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	$-0.476277\pi$
$293$	2.54907	0.148918	0.0744591	+	0.997224i	$0.476277\pi$
$294$	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	$-0.582609\pi$
$307$	−8.99273	−0.513242	−0.256621	+	0.966512i	$0.582609\pi$
$308$	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	$-0.321805\pi$
$337$	19.4968	1.06206	0.531030	+	0.847353i	$0.321805\pi$
$338$	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	$-0.518543\pi$
$347$	−2.16905	−0.116440	−0.0582202	+	0.998304i	$0.518543\pi$
$348$	0	0
$349$	25.0520	1.34100	0.670502	−	0.741908i	$-0.266079\pi$
$349$	25.0520	1.34100	0.670502	+	0.741908i	$0.266079\pi$
$350$	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	$-0.413928\pi$
$359$	10.1224	0.534239	0.267119	+	0.963663i	$0.413928\pi$
$360$	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	$-0.422260\pi$
$373$	9.34017	0.483616	0.241808	+	0.970324i	$0.422260\pi$
$374$	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	$-0.615598\pi$
$409$	−14.3682	−0.710460	−0.355230	+	0.934779i	$0.615598\pi$
$410$	−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	$-0.529701\pi$
$431$	−3.86870	−0.186349	−0.0931744	+	0.995650i	$0.529701\pi$
$432$	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	$-0.507774\pi$
$439$	−1.02336	−0.0488425	−0.0244212	+	0.999702i	$0.507774\pi$
$440$	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	$-0.299801\pi$
$457$	25.1525	1.17658	0.588291	+	0.808649i	$0.299801\pi$
$458$	11.0751	0.517507
$459$	0	0
$460$	3.57549	0.166708
$461$	6.65631	0.310015	0.155008	−	0.987913i	$-0.450460\pi$
$461$	6.65631	0.310015	0.155008	+	0.987913i	$0.450460\pi$
$462$	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	$-0.511870\pi$
$479$	−1.63186	−0.0745618	−0.0372809	+	0.999305i	$0.511870\pi$
$480$	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	$-0.601658\pi$
$491$	−13.9141	−0.627934	−0.313967	+	0.949434i	$0.601658\pi$
$492$	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	$-0.205195\pi$
$503$	35.8536	1.59863	0.799316	+	0.600910i	$0.205195\pi$
$504$	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	$-0.675086\pi$
$523$	−23.9088	−1.04546	−0.522729	+	0.852499i	$0.675086\pi$
$524$	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	$-0.509028\pi$
$541$	−1.31921	−0.0567171	−0.0283586	+	0.999598i	$0.509028\pi$
$542$	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	$-0.563859\pi$
$547$	−9.32128	−0.398549	−0.199275	+	0.979944i	$0.563859\pi$
$548$	−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	$-0.186644\pi$
$557$	39.3172	1.66592	0.832961	+	0.553331i	$0.186644\pi$
$558$	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	$-0.361662\pi$
$563$	19.9810	0.842097	0.421048	+	0.907038i	$0.361662\pi$
$564$	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	$-0.757936\pi$
$569$	−34.5647	−1.44903	−0.724515	+	0.689260i	$0.757936\pi$
$570$	0	0
$571$	−3.15090	−0.131861	−0.0659306	−	0.997824i	$-0.521002\pi$
$571$	−3.15090	−0.131861	−0.0659306	+	0.997824i	$0.521002\pi$
$572$	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	$-0.199725\pi$
$593$	39.4265	1.61905	0.809525	+	0.587085i	$0.199725\pi$
$594$	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	$-0.687577\pi$
$601$	−27.2498	−1.11154	−0.555771	+	0.831336i	$0.687577\pi$
$602$	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	$-0.646756\pi$
$607$	−21.9217	−0.889774	−0.444887	+	0.895587i	$0.646756\pi$
$608$	−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	$-0.568618\pi$
$613$	−10.5921	−0.427809	−0.213905	+	0.976855i	$0.568618\pi$
$614$	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	$-0.510301\pi$
$659$	−1.66127	−0.0647137	−0.0323569	+	0.999476i	$0.510301\pi$
$660$	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	$-0.232992\pi$
$673$	38.5949	1.48772	0.743862	+	0.668333i	$0.232992\pi$
$674$	−14.3817	−0.553960
$675$	0	0
$676$	1.60799	0.0618458
$677$	−38.5988	−1.48347	−0.741737	−	0.670691i	$-0.765998\pi$
$677$	−38.5988	−1.48347	−0.741737	+	0.670691i	$0.765998\pi$
$678$	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	$-0.586155\pi$
$701$	−14.1580	−0.534740	−0.267370	+	0.963594i	$0.586155\pi$
$702$	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	$-0.652498\pi$
$733$	−24.9604	−0.921935	−0.460968	+	0.887417i	$0.652498\pi$
$734$	−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	$-0.627718\pi$
$739$	−21.2342	−0.781114	−0.390557	+	0.920579i	$0.627718\pi$
$740$	−8.67501	−0.318900
$741$	0	0
$742$	9.01383	0.330908
$743$	−30.6527	−1.12454	−0.562270	−	0.826954i	$-0.690072\pi$
$743$	−30.6527	−1.12454	−0.562270	+	0.826954i	$0.690072\pi$
$744$	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	$-0.515524\pi$
$761$	−2.68972	−0.0975022	−0.0487511	+	0.998811i	$0.515524\pi$
$762$	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	$-0.300185\pi$
$769$	32.5735	1.17463	0.587315	+	0.809359i	$0.300185\pi$
$770$	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	$-0.278844\pi$
$787$	35.9207	1.28044	0.640218	+	0.768193i	$0.278844\pi$
$788$	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	$-0.547830\pi$
$809$	−8.51572	−0.299397	−0.149698	+	0.988732i	$0.547830\pi$
$810$	0	0
$811$	−9.03030	−0.317097	−0.158548	−	0.987351i	$-0.550681\pi$
$811$	−9.03030	−0.317097	−0.158548	+	0.987351i	$0.550681\pi$
$812$	−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	$-0.896833\pi$
$821$	−54.3225	−1.89587	−0.947935	+	0.318465i	$0.896833\pi$
$822$	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	$-0.858810\pi$
$827$	−51.9494	−1.80646	−0.903229	+	0.429159i	$0.858810\pi$
$828$	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	$-0.530032\pi$
$853$	−5.50285	−0.188414	−0.0942070	+	0.995553i	$0.530032\pi$
$854$	−1.88703	−0.0645729
$855$	0	0
$856$	−11.8456	−0.404873
$857$	−26.9281	−0.919847	−0.459924	−	0.887959i	$-0.652123\pi$
$857$	−26.9281	−0.919847	−0.459924	+	0.887959i	$0.652123\pi$
$858$	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	$-0.348090\pi$
$877$	27.2053	0.918659	0.459329	+	0.888266i	$0.348090\pi$
$878$	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	$-0.598049\pi$
$887$	−18.0590	−0.606363	−0.303181	+	0.952933i	$0.598049\pi$
$888$	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	$-0.721432\pi$
$919$	−38.8568	−1.28177	−0.640884	+	0.767638i	$0.721432\pi$
$920$	−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	$-0.421551\pi$
$937$	14.9360	0.487937	0.243968	+	0.969783i	$0.421551\pi$
$938$	−4.67657	−0.152696
$939$	0	0
$940$	8.50666	0.277457
$941$	−52.3409	−1.70626	−0.853132	−	0.521695i	$-0.825300\pi$
$941$	−52.3409	−1.70626	−0.853132	+	0.521695i	$0.825300\pi$
$942$	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	$-0.253662\pi$
$953$	43.1526	1.39785	0.698925	+	0.715195i	$0.253662\pi$
$954$	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	$-0.344349\pi$
$967$	29.2144	0.939471	0.469736	+	0.882807i	$0.344349\pi$
$968$	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	$-0.658344\pi$
$997$	−30.1347	−0.954375	−0.477188	+	0.878801i	$0.658344\pi$
$998$	−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.bi.1.2		4
3.2	odd	2		605.2.a.k.1.3		4
11.2	odd	10		495.2.n.e.136.1		8
11.6	odd	10		495.2.n.e.91.1		8
11.10	odd	2		5445.2.a.bp.1.3		4
12.11	even	2		9680.2.a.cm.1.4		4
15.14	odd	2		3025.2.a.w.1.2		4
33.2	even	10		55.2.g.b.26.2	✓	8
33.5	odd	10		605.2.g.k.366.1		8
33.8	even	10		605.2.g.m.251.1		8
33.14	odd	10		605.2.g.e.251.2		8
33.17	even	10		55.2.g.b.36.2	yes	8
33.20	odd	10		605.2.g.k.81.1		8
33.26	odd	10		605.2.g.e.511.2		8
33.29	even	10		605.2.g.m.511.1		8
33.32	even	2		605.2.a.j.1.2		4
132.35	odd	10		880.2.bo.h.81.2		8
132.83	odd	10		880.2.bo.h.641.2		8
132.131	odd	2		9680.2.a.cn.1.4		4
165.2	odd	20		275.2.z.a.224.2		16
165.17	odd	20		275.2.z.a.124.3		16
165.68	odd	20		275.2.z.a.224.3		16
165.83	odd	20		275.2.z.a.124.2		16
165.134	even	10		275.2.h.a.26.1		8
165.149	even	10		275.2.h.a.201.1		8
165.164	even	2		3025.2.a.bd.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.26.2	✓	8	33.2	even	10
55.2.g.b.36.2	yes	8	33.17	even	10
275.2.h.a.26.1		8	165.134	even	10
275.2.h.a.201.1		8	165.149	even	10
275.2.z.a.124.2		16	165.83	odd	20
275.2.z.a.124.3		16	165.17	odd	20
275.2.z.a.224.2		16	165.2	odd	20
275.2.z.a.224.3		16	165.68	odd	20
495.2.n.e.91.1		8	11.6	odd	10
495.2.n.e.136.1		8	11.2	odd	10
605.2.a.j.1.2		4	33.32	even	2
605.2.a.k.1.3		4	3.2	odd	2
605.2.g.e.251.2		8	33.14	odd	10
605.2.g.e.511.2		8	33.26	odd	10
605.2.g.k.81.1		8	33.20	odd	10
605.2.g.k.366.1		8	33.5	odd	10
605.2.g.m.251.1		8	33.8	even	10
605.2.g.m.511.1		8	33.29	even	10
880.2.bo.h.81.2		8	132.35	odd	10
880.2.bo.h.641.2		8	132.83	odd	10
3025.2.a.w.1.2		4	15.14	odd	2
3025.2.a.bd.1.3		4	165.164	even	2
5445.2.a.bi.1.2		4	1.1	even	1	trivial
5445.2.a.bp.1.3		4	11.10	odd	2
9680.2.a.cm.1.4		4	12.11	even	2
9680.2.a.cn.1.4		4	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)

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