LMFDB - Embedded newform 3744.2.g.e.1873.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1873,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3744.1873");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1873.12
Root			$-0.654116 - 1.25385i$ of defining polynomial
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.e.1873.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87654i q^{5} -0.584696 q^{7} +O(q^{10})$	$q+1.87654i q^{5} -0.584696 q^{7} +5.86890i q^{11} +1.00000i q^{13} +5.53339 q^{17} +4.96445i q^{19} +8.72906 q^{23} +1.47859 q^{25} -5.17991i q^{29} -7.04716 q^{31} -1.09721i q^{35} +0.398921i q^{37} +5.30337 q^{41} -7.13447i q^{43} -0.461238 q^{47} -6.65813 q^{49} -0.0988330i q^{53} -11.0132 q^{55} +9.74767i q^{59} -3.50613i q^{61} -1.87654 q^{65} +5.49801i q^{67} -15.8809 q^{71} +5.96090 q^{73} -3.43152i q^{77} -11.4116 q^{79} +9.19030i q^{83} +10.3836i q^{85} +11.5088 q^{89} -0.584696i q^{91} -9.31600 q^{95} +3.22953 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.87654i	0.839215i	0.907706	+	0.419608i	$0.137832\pi$
$5$	1.87654i	0.839215i	−0.907706	+	0.419608i	$0.862168\pi$
$6$	0	0
$7$	−0.584696	−0.220994	−0.110497	−	0.993876i	$-0.535244\pi$
$7$	−0.584696	−0.220994	−0.110497	+	0.993876i	$0.535244\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.86890i	1.76954i	0.466028	+	0.884770i	$0.345685\pi$
$11$	5.86890i	1.76954i	−0.466028	+	0.884770i	$0.654315\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.53339	1.34204	0.671022	−	0.741437i	$-0.265856\pi$
$17$	5.53339	1.34204	0.671022	+	0.741437i	$0.265856\pi$
$18$	0	0
$19$	4.96445i	1.13892i	0.822018	+	0.569462i	$0.192848\pi$
$19$	4.96445i	1.13892i	−0.822018	+	0.569462i	$0.807152\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.72906	1.82013	0.910067	−	0.414460i	$-0.136030\pi$
$23$	8.72906	1.82013	0.910067	+	0.414460i	$0.136030\pi$
$24$	0	0
$25$	1.47859	0.295718
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.17991i	− 0.961885i	−0.876752	−	0.480943i	$-0.840295\pi$
$29$	− 5.17991i	− 0.961885i	0.876752	−	0.480943i	$-0.159705\pi$
$30$	0	0
$31$	−7.04716	−1.26571	−0.632853	−	0.774272i	$-0.718116\pi$
$31$	−7.04716	−1.26571	−0.632853	+	0.774272i	$0.718116\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.09721i	− 0.185462i
$36$	0	0
$37$	0.398921i	0.0655823i	0.999462	+	0.0327911i	$0.0104396\pi$
$37$	0.398921i	0.0655823i	−0.999462	+	0.0327911i	$0.989560\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.30337	0.828247	0.414123	−	0.910221i	$-0.364088\pi$
$41$	5.30337	0.828247	0.414123	+	0.910221i	$0.364088\pi$
$42$	0	0
$43$	− 7.13447i	− 1.08800i	−0.839086	−	0.543998i	$-0.816910\pi$
$43$	− 7.13447i	− 1.08800i	0.839086	−	0.543998i	$-0.183090\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.461238	−0.0672785	−0.0336392	−	0.999434i	$-0.510710\pi$
$47$	−0.461238	−0.0672785	−0.0336392	+	0.999434i	$0.510710\pi$
$48$	0	0
$49$	−6.65813	−0.951162
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.0988330i	− 0.0135758i	−0.999977	−	0.00678788i	$-0.997839\pi$
$53$	− 0.0988330i	− 0.0135758i	0.999977	−	0.00678788i	$-0.00216067\pi$
$54$	0	0
$55$	−11.0132	−1.48502
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.74767i	1.26904i	0.772907	+	0.634519i	$0.218802\pi$
$59$	9.74767i	1.26904i	−0.772907	+	0.634519i	$0.781198\pi$
$60$	0	0
$61$	− 3.50613i	− 0.448913i	−0.974484	−	0.224457i	$-0.927939\pi$
$61$	− 3.50613i	− 0.448913i	0.974484	−	0.224457i	$-0.0720607\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.87654	−0.232756
$66$	0	0
$67$	5.49801i	0.671689i	0.941918	+	0.335844i	$0.109022\pi$
$67$	5.49801i	0.671689i	−0.941918	+	0.335844i	$0.890978\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.8809	−1.88471	−0.942357	−	0.334610i	$-0.891396\pi$
$71$	−15.8809	−1.88471	−0.942357	+	0.334610i	$0.891396\pi$
$72$	0	0
$73$	5.96090	0.697670	0.348835	−	0.937184i	$-0.386577\pi$
$73$	5.96090	0.697670	0.348835	+	0.937184i	$0.386577\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.43152i	− 0.391058i
$78$	0	0
$79$	−11.4116	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$79$	−11.4116	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.19030i	1.00877i	0.863480	+	0.504383i	$0.168280\pi$
$83$	9.19030i	1.00877i	−0.863480	+	0.504383i	$0.831720\pi$
$84$	0	0
$85$	10.3836i	1.12626i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.5088	1.21993	0.609963	−	0.792430i	$-0.291184\pi$
$89$	11.5088	1.21993	0.609963	+	0.792430i	$0.291184\pi$
$90$	0	0
$91$	− 0.584696i	− 0.0612928i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−9.31600	−0.955802
$96$	0	0
$97$	3.22953	0.327909	0.163955	−	0.986468i	$-0.447575\pi$
$97$	3.22953	0.327909	0.163955	+	0.986468i	$0.447575\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.7768i	1.66936i	0.550737	+	0.834679i	$0.314347\pi$
$101$	16.7768i	1.66936i	−0.550737	+	0.834679i	$0.685653\pi$
$102$	0	0
$103$	2.20393	0.217160	0.108580	−	0.994088i	$-0.465370\pi$
$103$	2.20393	0.217160	0.108580	+	0.994088i	$0.465370\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.4384i	− 1.10580i	−0.833249	−	0.552898i	$-0.813522\pi$
$107$	− 11.4384i	− 1.10580i	0.833249	−	0.552898i	$-0.186478\pi$
$108$	0	0
$109$	7.17954i	0.687675i	0.939029	+	0.343838i	$0.111727\pi$
$109$	7.17954i	0.687675i	−0.939029	+	0.343838i	$0.888273\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.98246	−0.468711	−0.234355	−	0.972151i	$-0.575298\pi$
$113$	−4.98246	−0.468711	−0.234355	+	0.972151i	$0.575298\pi$
$114$	0	0
$115$	16.3804i	1.52748i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.23535	−0.296584
$120$	0	0
$121$	−23.4440	−2.13127
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1573i	1.08739i
$126$	0	0
$127$	−1.23246	−0.109363	−0.0546814	−	0.998504i	$-0.517414\pi$
$127$	−1.23246	−0.109363	−0.0546814	+	0.998504i	$0.517414\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.4384i	1.34886i	0.738338	+	0.674431i	$0.235611\pi$
$131$	15.4384i	1.34886i	−0.738338	+	0.674431i	$0.764389\pi$
$132$	0	0
$133$	− 2.90270i	− 0.251696i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.61382	0.821364	0.410682	−	0.911779i	$-0.365291\pi$
$137$	9.61382	0.821364	0.410682	+	0.911779i	$0.365291\pi$
$138$	0	0
$139$	− 3.20231i	− 0.271616i	−0.990735	−	0.135808i	$-0.956637\pi$
$139$	− 3.20231i	− 0.271616i	0.990735	−	0.135808i	$-0.0433631\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.86890	−0.490782
$144$	0	0
$145$	9.72032	0.807229
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.78148i	0.145945i	0.997334	+	0.0729724i	$0.0232485\pi$
$149$	1.78148i	0.145945i	−0.997334	+	0.0729724i	$0.976752\pi$
$150$	0	0
$151$	10.2667	0.835491	0.417746	−	0.908564i	$-0.362820\pi$
$151$	10.2667	0.835491	0.417746	+	0.908564i	$0.362820\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 13.2243i	− 1.06220i
$156$	0	0
$157$	− 5.30920i	− 0.423720i	−0.977300	−	0.211860i	$-0.932048\pi$
$157$	− 5.30920i	− 0.423720i	0.977300	−	0.211860i	$-0.0679521\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.10384	−0.402239
$162$	0	0
$163$	3.68911i	0.288954i	0.989508	+	0.144477i	$0.0461499\pi$
$163$	3.68911i	0.288954i	−0.989508	+	0.144477i	$0.953850\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.96280	−0.538798	−0.269399	−	0.963029i	$-0.586825\pi$
$167$	−6.96280	−0.538798	−0.269399	+	0.963029i	$0.586825\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.66724i	− 0.506901i	−0.967348	−	0.253450i	$-0.918435\pi$
$173$	− 6.66724i	− 0.506901i	0.967348	−	0.253450i	$-0.0815655\pi$
$174$	0	0
$175$	−0.864526	−0.0653520
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.09721i	0.0820091i	0.999159	+	0.0410045i	$0.0130558\pi$
$179$	1.09721i	0.0820091i	−0.999159	+	0.0410045i	$0.986944\pi$
$180$	0	0
$181$	− 14.7148i	− 1.09374i	−0.837218	−	0.546870i	$-0.815819\pi$
$181$	− 14.7148i	− 1.09374i	0.837218	−	0.546870i	$-0.184181\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.748593	−0.0550376
$186$	0	0
$187$	32.4749i	2.37480i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.1138	1.02124	0.510621	−	0.859806i	$-0.329416\pi$
$191$	14.1138	1.02124	0.510621	+	0.859806i	$0.329416\pi$
$192$	0	0
$193$	−19.5750	−1.40904	−0.704521	−	0.709683i	$-0.748838\pi$
$193$	−19.5750	−1.40904	−0.704521	+	0.709683i	$0.748838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.1402i	− 1.22119i	−0.791942	−	0.610596i	$-0.790930\pi$
$197$	− 17.1402i	− 1.22119i	0.791942	−	0.610596i	$-0.209070\pi$
$198$	0	0
$199$	17.2879	1.22551	0.612755	−	0.790273i	$-0.290061\pi$
$199$	17.2879	1.22551	0.612755	+	0.790273i	$0.290061\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.02867i	0.212571i
$204$	0	0
$205$	9.95199i	0.695077i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.1359	−2.01537
$210$	0	0
$211$	8.99477i	0.619225i	0.950863	+	0.309613i	$0.100199\pi$
$211$	8.99477i	0.619225i	−0.950863	+	0.309613i	$0.899801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.3881	0.913063
$216$	0	0
$217$	4.12044	0.279714
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.53339i	0.372216i
$222$	0	0
$223$	11.8450	0.793200	0.396600	−	0.917992i	$-0.370190\pi$
$223$	11.8450	0.793200	0.396600	+	0.917992i	$0.370190\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.1088i	− 1.40104i	−0.713633	−	0.700520i	$-0.752952\pi$
$227$	− 21.1088i	− 1.40104i	0.713633	−	0.700520i	$-0.247048\pi$
$228$	0	0
$229$	3.09954i	0.204824i	0.994742	+	0.102412i	$0.0326559\pi$
$229$	3.09954i	0.204824i	−0.994742	+	0.102412i	$0.967344\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.58768	0.104012	0.0520061	−	0.998647i	$-0.483438\pi$
$233$	1.58768	0.104012	0.0520061	+	0.998647i	$0.483438\pi$
$234$	0	0
$235$	− 0.865532i	− 0.0564611i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.8275	−1.02380	−0.511898	−	0.859046i	$-0.671057\pi$
$239$	−15.8275	−1.02380	−0.511898	+	0.859046i	$0.671057\pi$
$240$	0	0
$241$	−16.8046	−1.08248	−0.541239	−	0.840869i	$-0.682045\pi$
$241$	−16.8046	−1.08248	−0.541239	+	0.840869i	$0.682045\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.4943i	− 0.798229i
$246$	0	0
$247$	−4.96445	−0.315881
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.72555i	− 0.298274i	−0.988817	−	0.149137i	$-0.952350\pi$
$251$	− 4.72555i	− 0.298274i	0.988817	−	0.149137i	$-0.0476495\pi$
$252$	0	0
$253$	51.2300i	3.22080i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.3259	−1.51741	−0.758705	−	0.651434i	$-0.774168\pi$
$257$	−24.3259	−1.51741	−0.758705	+	0.651434i	$0.774168\pi$
$258$	0	0
$259$	− 0.233248i	− 0.0144933i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0067	1.11034	0.555171	−	0.831736i	$-0.312653\pi$
$263$	18.0067	1.11034	0.555171	+	0.831736i	$0.312653\pi$
$264$	0	0
$265$	0.185464	0.0113930
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.4804i	1.18774i	0.804561	+	0.593869i	$0.202400\pi$
$269$	19.4804i	1.18774i	−0.804561	+	0.593869i	$0.797600\pi$
$270$	0	0
$271$	−5.66237	−0.343964	−0.171982	−	0.985100i	$-0.555017\pi$
$271$	−5.66237	−0.343964	−0.171982	+	0.985100i	$0.555017\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.67770i	0.523285i
$276$	0	0
$277$	16.6603i	1.00102i	0.865731	+	0.500509i	$0.166854\pi$
$277$	16.6603i	1.00102i	−0.865731	+	0.500509i	$0.833146\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.2503	−1.44665	−0.723326	−	0.690507i	$-0.757388\pi$
$281$	−24.2503	−1.44665	−0.723326	+	0.690507i	$0.757388\pi$
$282$	0	0
$283$	− 7.93216i	− 0.471518i	−0.971812	−	0.235759i	$-0.924242\pi$
$283$	− 7.93216i	− 0.471518i	0.971812	−	0.235759i	$-0.0757576\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.10086	−0.183038
$288$	0	0
$289$	13.6184	0.801082
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.82729i	− 0.340434i	−0.985407	−	0.170217i	$-0.945553\pi$
$293$	− 5.82729i	− 0.340434i	0.985407	−	0.170217i	$-0.0544469\pi$
$294$	0	0
$295$	−18.2919	−1.06500
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.72906i	0.504815i
$300$	0	0
$301$	4.17149i	0.240441i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.57939	0.376735
$306$	0	0
$307$	0.731684i	0.0417594i	0.999782	+	0.0208797i	$0.00664670\pi$
$307$	0.731684i	0.0417594i	−0.999782	+	0.0208797i	$0.993353\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.476654	0.0270285	0.0135143	−	0.999909i	$-0.495698\pi$
$311$	0.476654	0.0270285	0.0135143	+	0.999909i	$0.495698\pi$
$312$	0	0
$313$	11.0406	0.624052	0.312026	−	0.950074i	$-0.398992\pi$
$313$	11.0406	0.624052	0.312026	+	0.950074i	$0.398992\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.80970i	− 0.101643i	−0.998708	−	0.0508213i	$-0.983816\pi$
$317$	− 1.80970i	− 0.101643i	0.998708	−	0.0508213i	$-0.0161839\pi$
$318$	0	0
$319$	30.4004	1.70209
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.4703i	1.52849i
$324$	0	0
$325$	1.47859i	0.0820174i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.269684	0.0148682
$330$	0	0
$331$	− 22.3503i	− 1.22848i	−0.789118	−	0.614241i	$-0.789462\pi$
$331$	− 22.3503i	− 1.22848i	0.789118	−	0.614241i	$-0.210538\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.3172	−0.563691
$336$	0	0
$337$	27.9817	1.52426	0.762129	−	0.647425i	$-0.224154\pi$
$337$	27.9817	1.52426	0.762129	+	0.647425i	$0.224154\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 41.3591i	− 2.23972i
$342$	0	0
$343$	7.98585	0.431196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.72800i	0.253812i	0.991915	+	0.126906i	$0.0405047\pi$
$347$	4.72800i	0.253812i	−0.991915	+	0.126906i	$0.959495\pi$
$348$	0	0
$349$	− 17.9893i	− 0.962943i	−0.876462	−	0.481471i	$-0.840103\pi$
$349$	− 17.9893i	− 0.962943i	0.876462	−	0.481471i	$-0.159897\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.00709	−0.372950	−0.186475	−	0.982460i	$-0.559706\pi$
$353$	−7.00709	−0.372950	−0.186475	+	0.982460i	$0.559706\pi$
$354$	0	0
$355$	− 29.8011i	− 1.58168i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.1881	−1.27660	−0.638301	−	0.769787i	$-0.720362\pi$
$359$	−24.1881	−1.27660	−0.638301	+	0.769787i	$0.720362\pi$
$360$	0	0
$361$	−5.64580	−0.297147
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.1859i	0.585496i
$366$	0	0
$367$	−27.7596	−1.44904	−0.724520	−	0.689254i	$-0.757939\pi$
$367$	−27.7596	−1.44904	−0.724520	+	0.689254i	$0.757939\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.0577873i	0.00300016i
$372$	0	0
$373$	3.80559i	0.197046i	0.995135	+	0.0985229i	$0.0314118\pi$
$373$	3.80559i	0.197046i	−0.995135	+	0.0985229i	$0.968588\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.17991	0.266779
$378$	0	0
$379$	0.736223i	0.0378172i	0.999821	+	0.0189086i	$0.00601916\pi$
$379$	0.736223i	0.0378172i	−0.999821	+	0.0189086i	$0.993981\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.8656	1.01508	0.507542	−	0.861627i	$-0.330554\pi$
$383$	19.8656	1.01508	0.507542	+	0.861627i	$0.330554\pi$
$384$	0	0
$385$	6.43939	0.328182
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.4072i	1.74452i	0.489046	+	0.872258i	$0.337345\pi$
$389$	34.4072i	1.74452i	−0.489046	+	0.872258i	$0.662655\pi$
$390$	0	0
$391$	48.3013	2.44270
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 21.4143i	− 1.07747i
$396$	0	0
$397$	− 23.0143i	− 1.15505i	−0.816372	−	0.577527i	$-0.804018\pi$
$397$	− 23.0143i	− 1.15505i	0.816372	−	0.577527i	$-0.195982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.4191	−1.31931	−0.659655	−	0.751569i	$-0.729297\pi$
$401$	−26.4191	−1.31931	−0.659655	+	0.751569i	$0.729297\pi$
$402$	0	0
$403$	− 7.04716i	− 0.351044i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.34123	−0.116050
$408$	0	0
$409$	−26.6526	−1.31788	−0.658942	−	0.752193i	$-0.728996\pi$
$409$	−26.6526	−1.31788	−0.658942	+	0.752193i	$0.728996\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.69942i	− 0.280450i
$414$	0	0
$415$	−17.2460	−0.846572
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 34.8483i	− 1.70245i	−0.524801	−	0.851225i	$-0.675860\pi$
$419$	− 34.8483i	− 1.70245i	0.524801	−	0.851225i	$-0.324140\pi$
$420$	0	0
$421$	13.5677i	0.661250i	0.943762	+	0.330625i	$0.107260\pi$
$421$	13.5677i	0.661250i	−0.943762	+	0.330625i	$0.892740\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.18162	0.396867
$426$	0	0
$427$	2.05002i	0.0992073i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.7537	0.758828	0.379414	−	0.925227i	$-0.376126\pi$
$431$	15.7537	0.758828	0.379414	+	0.925227i	$0.376126\pi$
$432$	0	0
$433$	39.6015	1.90313	0.951563	−	0.307455i	$-0.0994773\pi$
$433$	39.6015	1.90313	0.951563	+	0.307455i	$0.0994773\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	43.3350i	2.07299i
$438$	0	0
$439$	−30.9712	−1.47818	−0.739088	−	0.673609i	$-0.764743\pi$
$439$	−30.9712	−1.47818	−0.739088	+	0.673609i	$0.764743\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.18372i	0.198775i	0.995049	+	0.0993873i	$0.0316883\pi$
$443$	4.18372i	0.198775i	−0.995049	+	0.0993873i	$0.968312\pi$
$444$	0	0
$445$	21.5967i	1.02378i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.5223	0.732543	0.366272	−	0.930508i	$-0.380634\pi$
$449$	15.5223	0.732543	0.366272	+	0.930508i	$0.380634\pi$
$450$	0	0
$451$	31.1249i	1.46562i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.09721	0.0514378
$456$	0	0
$457$	−9.78021	−0.457499	−0.228749	−	0.973485i	$-0.573464\pi$
$457$	−9.78021	−0.457499	−0.228749	+	0.973485i	$0.573464\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0994i	0.842974i	0.906834	+	0.421487i	$0.138492\pi$
$461$	18.0994i	0.842974i	−0.906834	+	0.421487i	$0.861508\pi$
$462$	0	0
$463$	−27.8634	−1.29492	−0.647460	−	0.762099i	$-0.724169\pi$
$463$	−27.8634	−1.29492	−0.647460	+	0.762099i	$0.724169\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.16806i	0.100326i	0.998741	+	0.0501630i	$0.0159741\pi$
$467$	2.16806i	0.100326i	−0.998741	+	0.0501630i	$0.984026\pi$
$468$	0	0
$469$	− 3.21466i	− 0.148439i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.8715	1.92525
$474$	0	0
$475$	7.34039i	0.336800i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.2863	0.607067	0.303533	−	0.952821i	$-0.401834\pi$
$479$	13.2863	0.607067	0.303533	+	0.952821i	$0.401834\pi$
$480$	0	0
$481$	−0.398921	−0.0181893
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.06035i	0.275186i
$486$	0	0
$487$	15.5621	0.705188	0.352594	−	0.935776i	$-0.385300\pi$
$487$	15.5621	0.705188	0.352594	+	0.935776i	$0.385300\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 7.07752i	− 0.319404i	−0.987165	−	0.159702i	$-0.948947\pi$
$491$	− 7.07752i	− 0.319404i	0.987165	−	0.159702i	$-0.0510534\pi$
$492$	0	0
$493$	− 28.6625i	− 1.29089i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.28548	0.416511
$498$	0	0
$499$	7.32097i	0.327732i	0.986483	+	0.163866i	$0.0523964\pi$
$499$	7.32097i	0.327732i	−0.986483	+	0.163866i	$0.947604\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	43.0093	1.91769	0.958844	−	0.283933i	$-0.0916393\pi$
$503$	43.0093	1.91769	0.958844	+	0.283933i	$0.0916393\pi$
$504$	0	0
$505$	−31.4824	−1.40095
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.43845i	0.108082i	0.998539	+	0.0540412i	$0.0172102\pi$
$509$	2.43845i	0.108082i	−0.998539	+	0.0540412i	$0.982790\pi$
$510$	0	0
$511$	−3.48531	−0.154181
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.13578i	0.182244i
$516$	0	0
$517$	− 2.70696i	− 0.119052i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.4276	1.20162	0.600812	−	0.799390i	$-0.294844\pi$
$521$	27.4276	1.20162	0.600812	+	0.799390i	$0.294844\pi$
$522$	0	0
$523$	3.30931i	0.144706i	0.997379	+	0.0723530i	$0.0230508\pi$
$523$	3.30931i	0.144706i	−0.997379	+	0.0723530i	$0.976949\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.9947	−1.69863
$528$	0	0
$529$	53.1965	2.31289
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.30337i	0.229714i
$534$	0	0
$535$	21.4647	0.928000
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 39.0759i	− 1.68312i
$540$	0	0
$541$	− 32.5121i	− 1.39781i	−0.715217	−	0.698903i	$-0.753672\pi$
$541$	− 32.5121i	− 1.39781i	0.715217	−	0.698903i	$-0.246328\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−13.4727	−0.577107
$546$	0	0
$547$	− 7.04589i	− 0.301261i	−0.988590	−	0.150630i	$-0.951870\pi$
$547$	− 7.04589i	− 0.301261i	0.988590	−	0.150630i	$-0.0481303\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	25.7154	1.09551
$552$	0	0
$553$	6.67231	0.283735
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.4027i	− 1.16109i	−0.814229	−	0.580544i	$-0.802840\pi$
$557$	− 27.4027i	− 1.16109i	0.814229	−	0.580544i	$-0.197160\pi$
$558$	0	0
$559$	7.13447	0.301756
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.5578i	1.37215i	0.727531	+	0.686075i	$0.240668\pi$
$563$	32.5578i	1.37215i	−0.727531	+	0.686075i	$0.759332\pi$
$564$	0	0
$565$	− 9.34980i	− 0.393349i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.8527	−0.454971	−0.227485	−	0.973781i	$-0.573050\pi$
$569$	−10.8527	−0.454971	−0.227485	+	0.973781i	$0.573050\pi$
$570$	0	0
$571$	10.6415i	0.445334i	0.974895	+	0.222667i	$0.0714763\pi$
$571$	10.6415i	0.445334i	−0.974895	+	0.222667i	$0.928524\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.9067	0.538247
$576$	0	0
$577$	−2.15012	−0.0895108	−0.0447554	−	0.998998i	$-0.514251\pi$
$577$	−2.15012	−0.0895108	−0.0447554	+	0.998998i	$0.514251\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.37353i	− 0.222932i
$582$	0	0
$583$	0.580041	0.0240228
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.70656i	0.400633i	0.979731	+	0.200316i	$0.0641970\pi$
$587$	9.70656i	0.400633i	−0.979731	+	0.200316i	$0.935803\pi$
$588$	0	0
$589$	− 34.9853i	− 1.44154i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.61187	0.189387	0.0946934	−	0.995506i	$-0.469813\pi$
$593$	4.61187	0.189387	0.0946934	+	0.995506i	$0.469813\pi$
$594$	0	0
$595$	− 6.07127i	− 0.248898i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.24722	−0.214396	−0.107198	−	0.994238i	$-0.534188\pi$
$599$	−5.24722	−0.214396	−0.107198	+	0.994238i	$0.534188\pi$
$600$	0	0
$601$	−41.6834	−1.70030	−0.850151	−	0.526539i	$-0.823489\pi$
$601$	−41.6834	−1.70030	−0.850151	+	0.526539i	$0.823489\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 43.9936i	− 1.78860i
$606$	0	0
$607$	41.5694	1.68725	0.843625	−	0.536933i	$-0.180417\pi$
$607$	41.5694	1.68725	0.843625	+	0.536933i	$0.180417\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 0.461238i	− 0.0186597i
$612$	0	0
$613$	19.3281i	0.780656i	0.920676	+	0.390328i	$0.127638\pi$
$613$	19.3281i	0.780656i	−0.920676	+	0.390328i	$0.872362\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.1055	0.608125	0.304063	−	0.952652i	$-0.401657\pi$
$617$	15.1055	0.608125	0.304063	+	0.952652i	$0.401657\pi$
$618$	0	0
$619$	− 20.3146i	− 0.816511i	−0.912868	−	0.408256i	$-0.866137\pi$
$619$	− 20.3146i	− 0.816511i	0.912868	−	0.408256i	$-0.133863\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.72913	−0.269597
$624$	0	0
$625$	−15.4208	−0.616833
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.20739i	0.0880143i
$630$	0	0
$631$	−27.3681	−1.08951	−0.544754	−	0.838596i	$-0.683377\pi$
$631$	−27.3681	−1.08951	−0.544754	+	0.838596i	$0.683377\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2.31276i	− 0.0917789i
$636$	0	0
$637$	− 6.65813i	− 0.263805i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.8191	1.29628	0.648138	−	0.761523i	$-0.275548\pi$
$641$	32.8191	1.29628	0.648138	+	0.761523i	$0.275548\pi$
$642$	0	0
$643$	− 14.8115i	− 0.584107i	−0.956402	−	0.292053i	$-0.905662\pi$
$643$	− 14.8115i	− 0.584107i	0.956402	−	0.292053i	$-0.0943385\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.89396	−0.153087	−0.0765436	−	0.997066i	$-0.524388\pi$
$647$	−3.89396	−0.153087	−0.0765436	+	0.997066i	$0.524388\pi$
$648$	0	0
$649$	−57.2081	−2.24561
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.4473i	1.46543i	0.680538	+	0.732713i	$0.261746\pi$
$653$	37.4473i	1.46543i	−0.680538	+	0.732713i	$0.738254\pi$
$654$	0	0
$655$	−28.9709	−1.13199
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.0504i	1.17060i	0.810818	+	0.585298i	$0.199023\pi$
$659$	30.0504i	1.17060i	−0.810818	+	0.585298i	$0.800977\pi$
$660$	0	0
$661$	− 7.22786i	− 0.281131i	−0.990071	−	0.140566i	$-0.955108\pi$
$661$	− 7.22786i	− 0.281131i	0.990071	−	0.140566i	$-0.0448921\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.44703	0.211227
$666$	0	0
$667$	− 45.2157i	− 1.75076i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.5771	0.794370
$672$	0	0
$673$	−26.5306	−1.02268	−0.511339	−	0.859379i	$-0.670850\pi$
$673$	−26.5306	−1.02268	−0.511339	+	0.859379i	$0.670850\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 21.8871i	− 0.841191i	−0.907248	−	0.420595i	$-0.861821\pi$
$677$	− 21.8871i	− 0.841191i	0.907248	−	0.420595i	$-0.138179\pi$
$678$	0	0
$679$	−1.88829	−0.0724660
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 45.6444i	− 1.74654i	−0.487241	−	0.873268i	$-0.661997\pi$
$683$	− 45.6444i	− 1.74654i	0.487241	−	0.873268i	$-0.338003\pi$
$684$	0	0
$685$	18.0407i	0.689301i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.0988330	0.00376524
$690$	0	0
$691$	− 23.1683i	− 0.881365i	−0.897663	−	0.440683i	$-0.854736\pi$
$691$	− 23.1683i	− 0.881365i	0.897663	−	0.440683i	$-0.145264\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.00927	0.227944
$696$	0	0
$697$	29.3456	1.11154
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 28.8081i	− 1.08807i	−0.839064	−	0.544033i	$-0.816896\pi$
$701$	− 28.8081i	− 1.08807i	0.839064	−	0.544033i	$-0.183104\pi$
$702$	0	0
$703$	−1.98043	−0.0746932
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.80935i	− 0.368919i
$708$	0	0
$709$	− 36.5492i	− 1.37263i	−0.727303	−	0.686317i	$-0.759226\pi$
$709$	− 36.5492i	− 1.37263i	0.727303	−	0.686317i	$-0.240774\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−61.5150	−2.30376
$714$	0	0
$715$	− 11.0132i	− 0.411872i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.3903	−0.536667	−0.268333	−	0.963326i	$-0.586473\pi$
$719$	−14.3903	−0.536667	−0.268333	+	0.963326i	$0.586473\pi$
$720$	0	0
$721$	−1.28863	−0.0479911
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.65896i	− 0.284447i
$726$	0	0
$727$	−22.3360	−0.828396	−0.414198	−	0.910187i	$-0.635938\pi$
$727$	−22.3360	−0.828396	−0.414198	+	0.910187i	$0.635938\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 39.4778i	− 1.46014i
$732$	0	0
$733$	50.3142i	1.85840i	0.369582	+	0.929198i	$0.379501\pi$
$733$	50.3142i	1.85840i	−0.369582	+	0.929198i	$0.620499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.2673	−1.18858
$738$	0	0
$739$	44.9967i	1.65523i	0.561295	+	0.827616i	$0.310303\pi$
$739$	44.9967i	1.65523i	−0.561295	+	0.827616i	$0.689697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.6745	−1.16202	−0.581012	−	0.813895i	$-0.697343\pi$
$743$	−31.6745	−1.16202	−0.581012	+	0.813895i	$0.697343\pi$
$744$	0	0
$745$	−3.34303	−0.122479
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.68801i	0.244374i
$750$	0	0
$751$	22.2640	0.812426	0.406213	−	0.913778i	$-0.366849\pi$
$751$	22.2640	0.812426	0.406213	+	0.913778i	$0.366849\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19.2659i	0.701157i
$756$	0	0
$757$	− 2.92018i	− 0.106136i	−0.998591	−	0.0530678i	$-0.983100\pi$
$757$	− 2.92018i	− 0.106136i	0.998591	−	0.0530678i	$-0.0168999\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.5631	0.419164	0.209582	−	0.977791i	$-0.432790\pi$
$761$	11.5631	0.419164	0.209582	+	0.977791i	$0.432790\pi$
$762$	0	0
$763$	− 4.19785i	− 0.151972i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.74767	−0.351968
$768$	0	0
$769$	38.2703	1.38006	0.690031	−	0.723779i	$-0.257597\pi$
$769$	38.2703	1.38006	0.690031	+	0.723779i	$0.257597\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 26.6341i	− 0.957961i	−0.877826	−	0.478980i	$-0.841007\pi$
$773$	− 26.6341i	− 0.957961i	0.877826	−	0.478980i	$-0.158993\pi$
$774$	0	0
$775$	−10.4199	−0.374292
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.3283i	0.943310i
$780$	0	0
$781$	− 93.2033i	− 3.33508i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.96293	0.355592
$786$	0	0
$787$	− 7.83596i	− 0.279322i	−0.990199	−	0.139661i	$-0.955399\pi$
$787$	− 7.83596i	− 0.279322i	0.990199	−	0.139661i	$-0.0446012\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.91323	0.103582
$792$	0	0
$793$	3.50613	0.124506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.0074i	1.09834i	0.835711	+	0.549169i	$0.185056\pi$
$797$	31.0074i	1.09834i	−0.835711	+	0.549169i	$0.814944\pi$
$798$	0	0
$799$	−2.55221	−0.0902906
$800$	0	0
$801$	0	0
$802$	0	0
$803$	34.9839i	1.23456i
$804$	0	0
$805$	− 9.57758i	− 0.337565i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.3701	0.821650	0.410825	−	0.911714i	$-0.365241\pi$
$809$	23.3701	0.821650	0.410825	+	0.911714i	$0.365241\pi$
$810$	0	0
$811$	− 33.9659i	− 1.19270i	−0.802723	−	0.596352i	$-0.796616\pi$
$811$	− 33.9659i	− 1.19270i	0.802723	−	0.596352i	$-0.203384\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.92278	−0.242494
$816$	0	0
$817$	35.4187	1.23914
$818$	0	0
$819$	0	0
$820$	0	0
$821$	54.3650i	1.89735i	0.316250	+	0.948676i	$0.397576\pi$
$821$	54.3650i	1.89735i	−0.316250	+	0.948676i	$0.602424\pi$
$822$	0	0
$823$	−27.1926	−0.947876	−0.473938	−	0.880558i	$-0.657168\pi$
$823$	−27.1926	−0.947876	−0.473938	+	0.880558i	$0.657168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 35.5065i	− 1.23468i	−0.786695	−	0.617342i	$-0.788210\pi$
$827$	− 35.5065i	− 1.23468i	0.786695	−	0.617342i	$-0.211790\pi$
$828$	0	0
$829$	− 36.7086i	− 1.27494i	−0.770474	−	0.637471i	$-0.779981\pi$
$829$	− 36.7086i	− 1.27494i	0.770474	−	0.637471i	$-0.220019\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−36.8420	−1.27650
$834$	0	0
$835$	− 13.0660i	− 0.452167i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.2953	0.873291	0.436645	−	0.899634i	$-0.356166\pi$
$839$	25.2953	0.873291	0.436645	+	0.899634i	$0.356166\pi$
$840$	0	0
$841$	2.16853	0.0747770
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.87654i	− 0.0645550i
$846$	0	0
$847$	13.7076	0.470999
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.48221i	0.119369i
$852$	0	0
$853$	− 8.28730i	− 0.283752i	−0.989884	−	0.141876i	$-0.954687\pi$
$853$	− 8.28730i	− 0.283752i	0.989884	−	0.141876i	$-0.0453134\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.49993	−0.324511	−0.162256	−	0.986749i	$-0.551877\pi$
$857$	−9.49993	−0.324511	−0.162256	+	0.986749i	$0.551877\pi$
$858$	0	0
$859$	− 52.7519i	− 1.79987i	−0.436024	−	0.899935i	$-0.643614\pi$
$859$	− 52.7519i	− 1.79987i	0.436024	−	0.899935i	$-0.356386\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.1777	1.06130	0.530651	−	0.847590i	$-0.321948\pi$
$863$	31.1777	1.06130	0.530651	+	0.847590i	$0.321948\pi$
$864$	0	0
$865$	12.5114	0.425399
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 66.9734i	− 2.27192i
$870$	0	0
$871$	−5.49801	−0.186293
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 7.10835i	− 0.240306i
$876$	0	0
$877$	42.6919i	1.44160i	0.693141	+	0.720802i	$0.256226\pi$
$877$	42.6919i	1.44160i	−0.693141	+	0.720802i	$0.743774\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.7217	1.23718	0.618592	−	0.785712i	$-0.287703\pi$
$881$	36.7217	1.23718	0.618592	+	0.785712i	$0.287703\pi$
$882$	0	0
$883$	− 42.5481i	− 1.43186i	−0.698173	−	0.715929i	$-0.746003\pi$
$883$	− 42.5481i	− 1.43186i	0.698173	−	0.715929i	$-0.253997\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.46068	−0.317658	−0.158829	−	0.987306i	$-0.550772\pi$
$887$	−9.46068	−0.317658	−0.158829	+	0.987306i	$0.550772\pi$
$888$	0	0
$889$	0.720612	0.0241685
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.28979i	− 0.0766250i
$894$	0	0
$895$	−2.05895	−0.0688232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	36.5036i	1.21746i
$900$	0	0
$901$	− 0.546882i	− 0.0182193i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.6129	0.917883
$906$	0	0
$907$	24.1280i	0.801157i	0.916262	+	0.400578i	$0.131191\pi$
$907$	24.1280i	0.801157i	−0.916262	+	0.400578i	$0.868809\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.4233	0.842312	0.421156	−	0.906988i	$-0.361625\pi$
$911$	25.4233	0.842312	0.421156	+	0.906988i	$0.361625\pi$
$912$	0	0
$913$	−53.9369	−1.78505
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.02679i	− 0.298091i
$918$	0	0
$919$	54.7761	1.80690	0.903449	−	0.428696i	$-0.141027\pi$
$919$	54.7761	1.80690	0.903449	+	0.428696i	$0.141027\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 15.8809i	− 0.522726i
$924$	0	0
$925$	0.589841i	0.0193939i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.1052	1.11896	0.559478	−	0.828845i	$-0.311002\pi$
$929$	34.1052	1.11896	0.559478	+	0.828845i	$0.311002\pi$
$930$	0	0
$931$	− 33.0540i	− 1.08330i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−60.9405	−1.99297
$936$	0	0
$937$	7.82032	0.255479	0.127739	−	0.991808i	$-0.459228\pi$
$937$	7.82032	0.255479	0.127739	+	0.991808i	$0.459228\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 33.1389i	− 1.08030i	−0.841569	−	0.540149i	$-0.818368\pi$
$941$	− 33.1389i	− 1.08030i	0.841569	−	0.540149i	$-0.181632\pi$
$942$	0	0
$943$	46.2934	1.50752
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.9958i	0.617280i	0.951179	+	0.308640i	$0.0998739\pi$
$947$	18.9958i	0.617280i	−0.951179	+	0.308640i	$0.900126\pi$
$948$	0	0
$949$	5.96090i	0.193499i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.3936	−1.21130	−0.605648	−	0.795732i	$-0.707086\pi$
$953$	−37.3936	−1.21130	−0.605648	+	0.795732i	$0.707086\pi$
$954$	0	0
$955$	26.4852i	0.857042i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.62116	−0.181517
$960$	0	0
$961$	18.6624	0.602014
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 36.7334i	− 1.18249i
$966$	0	0
$967$	46.6552	1.50033	0.750165	−	0.661251i	$-0.229974\pi$
$967$	46.6552	1.50033	0.750165	+	0.661251i	$0.229974\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 33.1363i	− 1.06339i	−0.846935	−	0.531697i	$-0.821554\pi$
$971$	− 33.1363i	− 1.06339i	0.846935	−	0.531697i	$-0.178446\pi$
$972$	0	0
$973$	1.87238i	0.0600256i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	60.0164	1.92009	0.960047	−	0.279839i	$-0.0902812\pi$
$977$	60.0164	1.92009	0.960047	+	0.279839i	$0.0902812\pi$
$978$	0	0
$979$	67.5438i	2.15871i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.52990	0.0487962	0.0243981	−	0.999702i	$-0.492233\pi$
$983$	1.52990	0.0487962	0.0243981	+	0.999702i	$0.492233\pi$
$984$	0	0
$985$	32.1644	1.02484
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 62.2772i	− 1.98030i
$990$	0	0
$991$	−21.3336	−0.677685	−0.338843	−	0.940843i	$-0.610035\pi$
$991$	−21.3336	−0.677685	−0.338843	+	0.940843i	$0.610035\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32.4415i	1.02847i
$996$	0	0
$997$	2.05485i	0.0650778i	0.999470	+	0.0325389i	$0.0103593\pi$
$997$	2.05485i	0.0650778i	−0.999470	+	0.0325389i	$0.989641\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.g.e.1873.12		16
3.2	odd	2		1248.2.g.b.625.11		16
4.3	odd	2		936.2.g.e.469.2		16
8.3	odd	2		936.2.g.e.469.1		16
8.5	even	2	inner	3744.2.g.e.1873.5		16
12.11	even	2		312.2.g.b.157.15	✓	16
24.5	odd	2		1248.2.g.b.625.6		16
24.11	even	2		312.2.g.b.157.16	yes	16
48.5	odd	4		9984.2.a.bv.1.6		8
48.11	even	4		9984.2.a.bt.1.6		8
48.29	odd	4		9984.2.a.bs.1.3		8
48.35	even	4		9984.2.a.bu.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.15	✓	16	12.11	even	2
312.2.g.b.157.16	yes	16	24.11	even	2
936.2.g.e.469.1		16	8.3	odd	2
936.2.g.e.469.2		16	4.3	odd	2
1248.2.g.b.625.6		16	24.5	odd	2
1248.2.g.b.625.11		16	3.2	odd	2
3744.2.g.e.1873.5		16	8.5	even	2	inner
3744.2.g.e.1873.12		16	1.1	even	1	trivial
9984.2.a.bs.1.3		8	48.29	odd	4
9984.2.a.bt.1.6		8	48.11	even	4
9984.2.a.bu.1.3		8	48.35	even	4
9984.2.a.bv.1.6		8	48.5	odd	4

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.87654i q^{5} -0.584696 q^{7} +O(q^{10})\)	\(q+1.87654i q^{5} -0.584696 q^{7} +5.86890i q^{11} +1.00000i q^{13} +5.53339 q^{17} +4.96445i q^{19} +8.72906 q^{23} +1.47859 q^{25} -5.17991i q^{29} -7.04716 q^{31} -1.09721i q^{35} +0.398921i q^{37} +5.30337 q^{41} -7.13447i q^{43} -0.461238 q^{47} -6.65813 q^{49} -0.0988330i q^{53} -11.0132 q^{55} +9.74767i q^{59} -3.50613i q^{61} -1.87654 q^{65} +5.49801i q^{67} -15.8809 q^{71} +5.96090 q^{73} -3.43152i q^{77} -11.4116 q^{79} +9.19030i q^{83} +10.3836i q^{85} +11.5088 q^{89} -0.584696i q^{91} -9.31600 q^{95} +3.22953 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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