LMFDB - Embedded newform 3744.2.c.m.3457.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3744,2,Mod(3457,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3744.3457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-12,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 1248)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3457.5
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	3744.3457
Dual form			3744.2.c.m.3457.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49396i q^{5} +1.10992i q^{7} +3.60388i q^{11} +(3.49396 - 0.890084i) q^{13} -2.00000 q^{17} -1.10992i q^{19} +7.20775 q^{23} -1.21983 q^{25} +5.20775 q^{29} +6.09783i q^{31} -2.76809 q^{35} -11.2078i q^{37} +7.48188i q^{41} +9.42758 q^{43} +7.60388i q^{47} +5.76809 q^{49} -4.76809 q^{53} -8.98792 q^{55} -1.38404i q^{59} -10.1957 q^{61} +(2.21983 + 8.71379i) q^{65} +11.8780i q^{67} -1.82371i q^{71} -4.98792i q^{73} -4.00000 q^{77} +0.439665 q^{79} +0.591794i q^{83} -4.98792i q^{85} +1.06638i q^{89} +(0.987918 + 3.87800i) q^{91} +2.76809 q^{95} +3.01208i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{13} - 12 q^{17} + 8 q^{23} - 10 q^{25} - 4 q^{29} + 24 q^{35} + 24 q^{43} - 6 q^{49} + 12 q^{53} - 16 q^{55} + 12 q^{61} + 16 q^{65} - 24 q^{77} + 8 q^{79} - 32 q^{91} - 24 q^{95}+O(q^{100}) $ 6 * q + 2 * q^13 - 12 * q^17 + 8 * q^23 - 10 * q^25 - 4 * q^29 + 24 * q^35 + 24 * q^43 - 6 * q^49 + 12 * q^53 - 16 * q^55 + 12 * q^61 + 16 * q^65 - 24 * q^77 + 8 * q^79 - 32 * q^91 - 24 * q^95

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
$5$			2.49396i			1.11533i	0.830065	+	0.557666i	$0.188303\pi$
$5$			2.49396i			1.11533i	−0.830065	+	0.557666i	$0.811697\pi$
$6$		0			0
$7$			1.10992i			0.419509i	0.977754	+	0.209754i	$0.0672665\pi$
$7$			1.10992i			0.419509i	−0.977754	+	0.209754i	$0.932734\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.60388i			1.08661i	0.839536	+	0.543305i	$0.182827\pi$
$11$			3.60388i			1.08661i	−0.839536	+	0.543305i	$0.817173\pi$
$12$		0			0
$13$	3.49396	−	0.890084i	0.969050	−	0.246865i
$13$	3.49396	−	0.890084i	0.969050	−	0.246865i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		−	1.10992i		−	0.254632i	−0.991862	−	0.127316i	$-0.959364\pi$
$19$		−	1.10992i		−	0.254632i	0.991862	−	0.127316i	$-0.0406363\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.20775			1.50292			0.751460	−	0.659779i	$-0.229350\pi$
$23$	7.20775			1.50292			0.751460	+	0.659779i	$0.229350\pi$
$24$		0			0
$25$	−1.21983			−0.243967
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.20775			0.967055			0.483528	−	0.875329i	$-0.339355\pi$
$29$	5.20775			0.967055			0.483528	+	0.875329i	$0.339355\pi$
$30$		0			0
$31$			6.09783i			1.09520i	0.836739	+	0.547602i	$0.184459\pi$
$31$			6.09783i			1.09520i	−0.836739	+	0.547602i	$0.815541\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.76809			−0.467892
$36$		0			0
$37$		−	11.2078i		−	1.84254i	−0.388920	−	0.921271i	$-0.627152\pi$
$37$		−	11.2078i		−	1.84254i	0.388920	−	0.921271i	$-0.372848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.48188i			1.16847i	0.811583	+	0.584236i	$0.198606\pi$
$41$			7.48188i			1.16847i	−0.811583	+	0.584236i	$0.801394\pi$
$42$		0			0
$43$	9.42758			1.43769			0.718847	−	0.695169i	$-0.244670\pi$
$43$	9.42758			1.43769			0.718847	+	0.695169i	$0.244670\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.60388i			1.10914i	0.832137	+	0.554570i	$0.187117\pi$
$47$			7.60388i			1.10914i	−0.832137	+	0.554570i	$0.812883\pi$
$48$		0			0
$49$	5.76809			0.824012
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.76809			−0.654947			−0.327473	−	0.944860i	$-0.606197\pi$
$53$	−4.76809			−0.654947			−0.327473	+	0.944860i	$0.606197\pi$
$54$		0			0
$55$	−8.98792			−1.21193
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.38404i		−	0.180187i	−0.995933	−	0.0900935i	$-0.971283\pi$
$59$		−	1.38404i		−	0.180187i	0.995933	−	0.0900935i	$-0.0287166\pi$
$60$		0			0
$61$	−10.1957			−1.30542			−0.652711	−	0.757607i	$-0.726368\pi$
$61$	−10.1957			−1.30542			−0.652711	+	0.757607i	$0.726368\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.21983	+	8.71379i	0.275336	+	1.08081i
$66$		0			0
$67$			11.8780i			1.45113i	0.688154	+	0.725564i	$0.258421\pi$
$67$			11.8780i			1.45113i	−0.688154	+	0.725564i	$0.741579\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.82371i		−	0.216434i	−0.994127	−	0.108217i	$-0.965486\pi$
$71$		−	1.82371i		−	0.216434i	0.994127	−	0.108217i	$-0.0345142\pi$
$72$		0			0
$73$		−	4.98792i		−	0.583792i	−0.956450	−	0.291896i	$-0.905714\pi$
$73$		−	4.98792i		−	0.583792i	0.956450	−	0.291896i	$-0.0942861\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$	0.439665			0.0494662			0.0247331	−	0.999694i	$-0.492126\pi$
$79$	0.439665			0.0494662			0.0247331	+	0.999694i	$0.492126\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.591794i			0.0649578i	0.999472	+	0.0324789i	$0.0103402\pi$
$83$			0.591794i			0.0649578i	−0.999472	+	0.0324789i	$0.989660\pi$
$84$		0			0
$85$		−	4.98792i		−	0.541016i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.06638i			0.113036i	0.998402	+	0.0565178i	$0.0179998\pi$
$89$			1.06638i			0.113036i	−0.998402	+	0.0565178i	$0.982000\pi$
$90$		0			0
$91$	0.987918	+	3.87800i	0.103562	+	0.406525i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.76809			0.284000
$96$		0			0
$97$			3.01208i			0.305831i	0.988239	+	0.152915i	$0.0488662\pi$
$97$			3.01208i			0.305831i	−0.988239	+	0.152915i	$0.951134\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−19.1836			−1.90884			−0.954419	−	0.298470i	$-0.903524\pi$
$101$	−19.1836			−1.90884			−0.954419	+	0.298470i	$0.903524\pi$
$102$		0			0
$103$	−8.54825			−0.842284			−0.421142	−	0.906995i	$-0.638371\pi$
$103$	−8.54825			−0.842284			−0.421142	+	0.906995i	$0.638371\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.20775			0.310105			0.155052	−	0.987906i	$-0.450445\pi$
$107$	3.20775			0.310105			0.155052	+	0.987906i	$0.450445\pi$
$108$		0			0
$109$		−	1.78017i		−	0.170509i	−0.996359	−	0.0852546i	$-0.972830\pi$
$109$		−	1.78017i		−	0.170509i	0.996359	−	0.0852546i	$-0.0271704\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.97584			0.750304			0.375152	−	0.926963i	$-0.377591\pi$
$113$	7.97584			0.750304			0.375152	+	0.926963i	$0.377591\pi$
$114$		0			0
$115$			17.9758i			1.67626i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.21983i		−	0.203492i
$120$		0			0
$121$	−1.98792			−0.180720
$122$		0			0
$123$		0			0
$124$		0			0
$125$			9.42758i			0.843229i
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−21.1836			−1.85082			−0.925409	−	0.378969i	$-0.876279\pi$
$131$	−21.1836			−1.85082			−0.925409	+	0.378969i	$0.876279\pi$
$132$		0			0
$133$	1.23191			0.106821
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11.9215i		−	1.01853i	−0.860611	−	0.509263i	$-0.829918\pi$
$137$		−	11.9215i		−	1.01853i	0.860611	−	0.509263i	$-0.170082\pi$
$138$		0			0
$139$	−12.9879			−1.10162			−0.550810	−	0.834630i	$-0.685681\pi$
$139$	−12.9879			−1.10162			−0.550810	+	0.834630i	$0.685681\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.20775	+	12.5918i	0.268246	+	1.05298i
$144$		0			0
$145$			12.9879i			1.07859i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	18.4940i		−	1.51508i	−0.652786	−	0.757542i	$-0.726400\pi$
$149$		−	18.4940i		−	1.51508i	0.652786	−	0.757542i	$-0.273600\pi$
$150$		0			0
$151$			12.5133i			1.01832i	0.860672	+	0.509160i	$0.170044\pi$
$151$			12.5133i			1.01832i	−0.860672	+	0.509160i	$0.829956\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.2078			−1.22152
$156$		0			0
$157$	9.64742			0.769948			0.384974	−	0.922927i	$-0.374210\pi$
$157$	9.64742			0.769948			0.384974	+	0.922927i	$0.374210\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			8.00000i			0.630488i
$162$		0			0
$163$			16.8659i			1.32104i	0.750808	+	0.660520i	$0.229664\pi$
$163$			16.8659i			1.32104i	−0.750808	+	0.660520i	$0.770336\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.16421i			0.244854i	0.992478	+	0.122427i	$0.0390677\pi$
$167$			3.16421i			0.244854i	−0.992478	+	0.122427i	$0.960932\pi$
$168$		0			0
$169$	11.4155	−	6.21983i	0.878116	−	0.478449i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.20775			−0.0918236			−0.0459118	−	0.998945i	$-0.514619\pi$
$173$	−1.20775			−0.0918236			−0.0459118	+	0.998945i	$0.514619\pi$
$174$		0			0
$175$		−	1.35391i		−	0.102346i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.76809			−0.505870			−0.252935	−	0.967483i	$-0.581396\pi$
$179$	−6.76809			−0.505870			−0.252935	+	0.967483i	$0.581396\pi$
$180$		0			0
$181$	0.768086			0.0570914			0.0285457	−	0.999592i	$-0.490912\pi$
$181$	0.768086			0.0570914			0.0285457	+	0.999592i	$0.490912\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	27.9517			2.05505
$186$		0			0
$187$		−	7.20775i		−	0.527083i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.7681			−0.779151			−0.389576	−	0.920994i	$-0.627378\pi$
$191$	−10.7681			−0.779151			−0.389576	+	0.920994i	$0.627378\pi$
$192$		0			0
$193$			6.41550i			0.461798i	0.972978	+	0.230899i	$0.0741667\pi$
$193$			6.41550i			0.461798i	−0.972978	+	0.230899i	$0.925833\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	19.0422i		−	1.35670i	−0.734738	−	0.678351i	$-0.762695\pi$
$197$		−	19.0422i		−	1.35670i	0.734738	−	0.678351i	$-0.237305\pi$
$198$		0			0
$199$	−12.0000			−0.850657			−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000			−0.850657			−0.425329	+	0.905039i	$0.639842\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			5.78017i			0.405688i
$204$		0			0
$205$	−18.6595			−1.30324
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	20.3913			1.40380			0.701899	−	0.712277i	$-0.252336\pi$
$211$	20.3913			1.40380			0.701899	+	0.712277i	$0.252336\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			23.5120i			1.60351i
$216$		0			0
$217$	−6.76809			−0.459448
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.98792	+	1.78017i	−0.470058	+	0.119747i
$222$		0			0
$223$			12.1220i			0.811749i	0.913929	+	0.405875i	$0.133033\pi$
$223$			12.1220i			0.811749i	−0.913929	+	0.405875i	$0.866967\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			18.0194i			1.19599i	0.801501	+	0.597994i	$0.204035\pi$
$227$			18.0194i			1.19599i	−0.801501	+	0.597994i	$0.795965\pi$
$228$		0			0
$229$			18.1715i			1.20081i	0.799697	+	0.600403i	$0.204993\pi$
$229$			18.1715i			1.20081i	−0.799697	+	0.600403i	$0.795007\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	23.9758			1.57071			0.785355	−	0.619046i	$-0.212481\pi$
$233$	23.9758			1.57071			0.785355	+	0.619046i	$0.212481\pi$
$234$		0			0
$235$	−18.9638			−1.23706
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	23.3599i		−	1.51103i	−0.655134	−	0.755513i	$-0.727388\pi$
$239$		−	23.3599i		−	1.51103i	0.655134	−	0.755513i	$-0.272612\pi$
$240$		0			0
$241$			8.54825i			0.550641i	0.961352	+	0.275321i	$0.0887840\pi$
$241$			8.54825i			0.550641i	−0.961352	+	0.275321i	$0.911216\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			14.3854i			0.919048i
$246$		0			0
$247$	−0.987918	−	3.87800i	−0.0628597	−	0.246751i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.3913			0.782134			0.391067	−	0.920362i	$-0.372106\pi$
$251$	12.3913			0.782134			0.391067	+	0.920362i	$0.372106\pi$
$252$		0			0
$253$			25.9758i			1.63309i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.8552			−0.801883			−0.400942	−	0.916104i	$-0.631317\pi$
$257$	−12.8552			−0.801883			−0.400942	+	0.916104i	$0.631317\pi$
$258$		0			0
$259$	12.4397			0.772963
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.97584			0.615136			0.307568	−	0.951526i	$-0.400485\pi$
$263$	9.97584			0.615136			0.307568	+	0.951526i	$0.400485\pi$
$264$		0			0
$265$		−	11.8914i		−	0.730484i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.64742			−0.344329			−0.172165	−	0.985068i	$-0.555076\pi$
$269$	−5.64742			−0.344329			−0.172165	+	0.985068i	$0.555076\pi$
$270$		0			0
$271$			4.67025i			0.283698i	0.989888	+	0.141849i	$0.0453047\pi$
$271$			4.67025i			0.283698i	−0.989888	+	0.141849i	$0.954695\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.39612i		−	0.265096i
$276$		0			0
$277$	−15.5362			−0.933478			−0.466739	−	0.884395i	$-0.654571\pi$
$277$	−15.5362			−0.933478			−0.466739	+	0.884395i	$0.654571\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.36121i		−	0.498788i	−0.968402	−	0.249394i	$-0.919769\pi$
$281$		−	8.36121i		−	0.498788i	0.968402	−	0.249394i	$-0.0802314\pi$
$282$		0			0
$283$	−7.56033			−0.449415			−0.224708	−	0.974426i	$-0.572143\pi$
$283$	−7.56033			−0.449415			−0.224708	+	0.974426i	$0.572143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.30426			−0.490185
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.06638i		−	0.529663i	−0.964295	−	0.264832i	$-0.914684\pi$
$293$		−	9.06638i		−	0.529663i	0.964295	−	0.264832i	$-0.0853164\pi$
$294$		0			0
$295$	3.45175			0.200968
$296$		0			0
$297$		0			0
$298$		0			0
$299$	25.1836	−	6.41550i	1.45640	−	0.371018i
$300$		0			0
$301$			10.4638i			0.603125i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	25.4276i		−	1.45598i
$306$		0			0
$307$		−	32.9530i		−	1.88073i	−0.340171	−	0.940364i	$-0.610485\pi$
$307$		−	32.9530i		−	1.88073i	0.340171	−	0.940364i	$-0.389515\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.2319			0.750313			0.375157	−	0.926961i	$-0.377589\pi$
$311$	13.2319			0.750313			0.375157	+	0.926961i	$0.377589\pi$
$312$		0			0
$313$	21.8431			1.23464			0.617322	−	0.786711i	$-0.288218\pi$
$313$	21.8431			1.23464			0.617322	+	0.786711i	$0.288218\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.45771i		−	0.531198i	−0.964084	−	0.265599i	$-0.914430\pi$
$317$		−	9.45771i		−	0.531198i	0.964084	−	0.265599i	$-0.0855697\pi$
$318$		0			0
$319$			18.7681i			1.05081i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.21983i			0.123515i
$324$		0			0
$325$	−4.26205	+	1.08575i	−0.236416	+	0.0602267i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.43967			−0.465294
$330$		0			0
$331$			1.19700i			0.0657929i	0.999459	+	0.0328965i	$0.0104732\pi$
$331$			1.19700i			0.0657929i	−0.999459	+	0.0328965i	$0.989527\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−29.6233			−1.61849
$336$		0			0
$337$	2.98792			0.162762			0.0813811	−	0.996683i	$-0.474067\pi$
$337$	2.98792			0.162762			0.0813811	+	0.996683i	$0.474067\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.9758			−1.19006
$342$		0			0
$343$			14.1715i			0.765189i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.41550			−0.129671			−0.0648355	−	0.997896i	$-0.520652\pi$
$347$	−2.41550			−0.129671			−0.0648355	+	0.997896i	$0.520652\pi$
$348$		0			0
$349$		−	12.7922i		−	0.684753i	−0.939563	−	0.342377i	$-0.888768\pi$
$349$		−	12.7922i		−	0.684753i	0.939563	−	0.342377i	$-0.111232\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	32.9095i		−	1.75159i	−0.482680	−	0.875797i	$-0.660336\pi$
$353$		−	32.9095i		−	1.75159i	0.482680	−	0.875797i	$-0.339664\pi$
$354$		0			0
$355$	4.54825			0.241396
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.0435i			0.635634i	0.948152	+	0.317817i	$0.102950\pi$
$359$			12.0435i			0.635634i	−0.948152	+	0.317817i	$0.897050\pi$
$360$		0			0
$361$	17.7681			0.935162
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.4397			0.651122
$366$		0			0
$367$	27.4034			1.43045			0.715223	−	0.698896i	$-0.246325\pi$
$367$	27.4034			1.43045			0.715223	+	0.698896i	$0.246325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	5.29218i		−	0.274756i
$372$		0			0
$373$	3.42758			0.177474			0.0887368	−	0.996055i	$-0.471717\pi$
$373$	3.42758			0.177474			0.0887368	+	0.996055i	$0.471717\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.1957	−	4.63533i	0.937125	−	0.238732i
$378$		0			0
$379$			31.2814i			1.60682i	0.595427	+	0.803409i	$0.296983\pi$
$379$			31.2814i			1.60682i	−0.595427	+	0.803409i	$0.703017\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	18.3720i		−	0.938763i	−0.882995	−	0.469382i	$-0.844477\pi$
$383$		−	18.3720i		−	0.938763i	0.882995	−	0.469382i	$-0.155523\pi$
$384$		0			0
$385$		−	9.97584i		−	0.508416i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−21.6474			−1.09757			−0.548784	−	0.835964i	$-0.684909\pi$
$389$	−21.6474			−1.09757			−0.548784	+	0.835964i	$0.684909\pi$
$390$		0			0
$391$	−14.4155			−0.729023
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.09651i			0.0551712i
$396$		0			0
$397$			33.6233i			1.68750i	0.536734	+	0.843751i	$0.319658\pi$
$397$			33.6233i			1.68750i	−0.536734	+	0.843751i	$0.680342\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			31.4819i			1.57213i	0.618144	+	0.786065i	$0.287885\pi$
$401$			31.4819i			1.57213i	−0.618144	+	0.786065i	$0.712115\pi$
$402$		0			0
$403$	5.42758	+	21.3056i	0.270367	+	1.06131i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	40.3913			2.00212
$408$		0			0
$409$			24.5483i			1.21383i	0.794766	+	0.606917i	$0.207594\pi$
$409$			24.5483i			1.21383i	−0.794766	+	0.606917i	$0.792406\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.53617			0.0755901
$414$		0			0
$415$	−1.47591			−0.0724496
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.7439			1.20882			0.604410	−	0.796674i	$-0.293409\pi$
$419$	24.7439			1.20882			0.604410	+	0.796674i	$0.293409\pi$
$420$		0			0
$421$		−	33.2922i		−	1.62256i	−0.584657	−	0.811281i	$-0.698771\pi$
$421$		−	33.2922i		−	1.62256i	0.584657	−	0.811281i	$-0.301229\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.43967			0.118341
$426$		0			0
$427$		−	11.3163i		−	0.547636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			32.8745i			1.58351i	0.610838	+	0.791756i	$0.290833\pi$
$431$			32.8745i			1.58351i	−0.610838	+	0.791756i	$0.709167\pi$
$432$		0			0
$433$	8.21983			0.395020			0.197510	−	0.980301i	$-0.436715\pi$
$433$	8.21983			0.395020			0.197510	+	0.980301i	$0.436715\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	25.4276			1.21359			0.606796	−	0.794857i	$-0.292454\pi$
$439$	25.4276			1.21359			0.606796	+	0.794857i	$0.292454\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.67158			0.269465			0.134732	−	0.990882i	$-0.456983\pi$
$443$	5.67158			0.269465			0.134732	+	0.990882i	$0.456983\pi$
$444$		0			0
$445$	−2.65950			−0.126072
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.3370i			0.865379i	0.901543	+	0.432689i	$0.142435\pi$
$449$			18.3370i			0.865379i	−0.901543	+	0.432689i	$0.857565\pi$
$450$		0			0
$451$	−26.9638			−1.26967
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.67158	+	2.46383i	−0.453411	+	0.115506i
$456$		0			0
$457$			34.5241i			1.61497i	0.589889	+	0.807484i	$0.299172\pi$
$457$			34.5241i			1.61497i	−0.589889	+	0.807484i	$0.700828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			15.3250i			0.713755i	0.934151	+	0.356877i	$0.116159\pi$
$461$			15.3250i			0.713755i	−0.934151	+	0.356877i	$0.883841\pi$
$462$		0			0
$463$		−	3.32975i		−	0.154747i	−0.997002	−	0.0773733i	$-0.975347\pi$
$463$		−	3.32975i		−	0.154747i	0.997002	−	0.0773733i	$-0.0246533\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0965			−0.976230			−0.488115	−	0.872779i	$-0.662315\pi$
$467$	−21.0965			−0.976230			−0.488115	+	0.872779i	$0.662315\pi$
$468$		0			0
$469$	−13.1836			−0.608762
$470$		0			0
$471$		0			0
$472$		0			0
$473$			33.9758i			1.56221i
$474$		0			0
$475$			1.35391i			0.0621217i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	23.3599i		−	1.06734i	−0.845693	−	0.533670i	$-0.820812\pi$
$479$		−	23.3599i		−	1.06734i	0.845693	−	0.533670i	$-0.179188\pi$
$480$		0			0
$481$	−9.97584	−	39.1594i	−0.454859	−	1.78552i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.51201			−0.341103
$486$		0			0
$487$		−	34.2935i		−	1.55399i	−0.629509	−	0.776994i	$-0.716744\pi$
$487$		−	34.2935i		−	1.55399i	0.629509	−	0.776994i	$-0.283256\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	43.5120			1.96367			0.981835	−	0.189738i	$-0.0607639\pi$
$491$	43.5120			1.96367			0.981835	+	0.189738i	$0.0607639\pi$
$492$		0			0
$493$	−10.4155			−0.469091
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.02416			0.0907961
$498$		0			0
$499$		−	32.0737i		−	1.43581i	−0.696139	−	0.717907i	$-0.745100\pi$
$499$		−	32.0737i		−	1.43581i	0.696139	−	0.717907i	$-0.254900\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.9517			−1.60301			−0.801503	−	0.597991i	$-0.795966\pi$
$503$	−35.9517			−1.60301			−0.801503	+	0.597991i	$0.795966\pi$
$504$		0			0
$505$		−	47.8431i		−	2.12899i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	27.9215i		−	1.23760i	−0.785548	−	0.618800i	$-0.787619\pi$
$509$		−	27.9215i		−	1.23760i	0.785548	−	0.618800i	$-0.212381\pi$
$510$		0			0
$511$	5.53617			0.244906
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	21.3190i		−	0.939427i
$516$		0			0
$517$	−27.4034			−1.20520
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.39134			0.280010			0.140005	−	0.990151i	$-0.455288\pi$
$521$	6.39134			0.280010			0.140005	+	0.990151i	$0.455288\pi$
$522$		0			0
$523$	−39.3551			−1.72088			−0.860439	−	0.509553i	$-0.829811\pi$
$523$	−39.3551			−1.72088			−0.860439	+	0.509553i	$0.829811\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.1957i		−	0.531252i
$528$		0			0
$529$	28.9517			1.25877
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.65950	+	26.1414i	0.288455	+	1.13231i
$534$		0			0
$535$			8.00000i			0.345870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			20.7875i			0.895379i
$540$		0			0
$541$		−	35.0508i		−	1.50695i	−0.657475	−	0.753477i	$-0.728375\pi$
$541$		−	35.0508i		−	1.50695i	0.657475	−	0.753477i	$-0.271625\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.43967			0.190174
$546$		0			0
$547$	6.57242			0.281016			0.140508	−	0.990080i	$-0.455126\pi$
$547$	6.57242			0.281016			0.140508	+	0.990080i	$0.455126\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	5.78017i		−	0.246243i
$552$		0			0
$553$			0.487991i			0.0207515i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			22.0543i			0.934470i	0.884133	+	0.467235i	$0.154750\pi$
$557$			22.0543i			0.934470i	−0.884133	+	0.467235i	$0.845250\pi$
$558$		0			0
$559$	32.9396	−	8.39134i	1.39320	−	0.354916i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.67158			−0.239029			−0.119514	−	0.992832i	$-0.538134\pi$
$563$	−5.67158			−0.239029			−0.119514	+	0.992832i	$0.538134\pi$
$564$		0			0
$565$			19.8914i			0.836838i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	44.8068			1.87840			0.939200	−	0.343370i	$-0.111568\pi$
$569$	44.8068			1.87840			0.939200	+	0.343370i	$0.111568\pi$
$570$		0			0
$571$	−13.2707			−0.555360			−0.277680	−	0.960674i	$-0.589566\pi$
$571$	−13.2707			−0.555360			−0.277680	+	0.960674i	$0.589566\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.79225			−0.366662
$576$		0			0
$577$			27.2465i			1.13429i	0.823619	+	0.567143i	$0.191951\pi$
$577$			27.2465i			1.13429i	−0.823619	+	0.567143i	$0.808049\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.656842			−0.0272504
$582$		0			0
$583$		−	17.1836i		−	0.711671i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	23.0944i		−	0.953207i	−0.879118	−	0.476603i	$-0.841868\pi$
$587$		−	23.0944i		−	0.953207i	0.879118	−	0.476603i	$-0.158132\pi$
$588$		0			0
$589$	6.76809			0.278874
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.94571i			0.408421i	0.978927	+	0.204211i	$0.0654627\pi$
$593$			9.94571i			0.408421i	−0.978927	+	0.204211i	$0.934537\pi$
$594$		0			0
$595$	5.53617			0.226961
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−25.2707			−1.03253			−0.516266	−	0.856428i	$-0.672678\pi$
$599$	−25.2707			−1.03253			−0.516266	+	0.856428i	$0.672678\pi$
$600$		0			0
$601$	20.0844			0.819261			0.409630	−	0.912252i	$-0.365658\pi$
$601$	20.0844			0.819261			0.409630	+	0.912252i	$0.365658\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	4.95779i		−	0.201563i
$606$		0			0
$607$	36.9879			1.50129			0.750647	−	0.660704i	$-0.229742\pi$
$607$	36.9879			1.50129			0.750647	+	0.660704i	$0.229742\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.76809	+	26.5676i	0.273807	+	1.07481i
$612$		0			0
$613$		−	34.0146i		−	1.37384i	−0.726735	−	0.686918i	$-0.758963\pi$
$613$		−	34.0146i		−	1.37384i	0.726735	−	0.686918i	$-0.241037\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.4940i		−	0.744539i	−0.928125	−	0.372269i	$-0.878580\pi$
$617$		−	18.4940i		−	0.744539i	0.928125	−	0.372269i	$-0.121420\pi$
$618$		0			0
$619$		−	15.6823i		−	0.630326i	−0.949038	−	0.315163i	$-0.897941\pi$
$619$		−	15.6823i		−	0.630326i	0.949038	−	0.315163i	$-0.102059\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.18359			−0.0474194
$624$		0			0
$625$	−29.6112			−1.18445
$626$		0			0
$627$		0			0
$628$		0			0
$629$			22.4155i			0.893765i
$630$		0			0
$631$		−	12.2693i		−	0.488435i	−0.969721	−	0.244217i	$-0.921469\pi$
$631$		−	12.2693i		−	0.488435i	0.969721	−	0.244217i	$-0.0785310\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	9.97584i		−	0.395879i
$636$		0			0
$637$	20.1535	−	5.13408i	0.798509	−	0.203420i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.14483			−0.282204			−0.141102	−	0.989995i	$-0.545065\pi$
$641$	−7.14483			−0.282204			−0.141102	+	0.989995i	$0.545065\pi$
$642$		0			0
$643$		−	30.0978i		−	1.18694i	−0.804855	−	0.593471i	$-0.797757\pi$
$643$		−	30.0978i		−	1.18694i	0.804855	−	0.593471i	$-0.202243\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.391339			0.0153851			0.00769256	−	0.999970i	$-0.497551\pi$
$647$	0.391339			0.0153851			0.00769256	+	0.999970i	$0.497551\pi$
$648$		0			0
$649$	4.98792			0.195793
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.6233			−1.23751			−0.618757	−	0.785583i	$-0.712363\pi$
$653$	−31.6233			−1.23751			−0.618757	+	0.785583i	$0.712363\pi$
$654$		0			0
$655$		−	52.8310i		−	2.06428i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.831004			−0.0323713			−0.0161857	−	0.999869i	$-0.505152\pi$
$659$	−0.831004			−0.0323713			−0.0161857	+	0.999869i	$0.505152\pi$
$660$		0			0
$661$		−	26.3284i		−	1.02406i	−0.858968	−	0.512028i	$-0.828894\pi$
$661$		−	26.3284i		−	1.02406i	0.858968	−	0.512028i	$-0.171106\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			3.07234i			0.119140i
$666$		0			0
$667$	37.5362			1.45341
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	36.7439i		−	1.41848i
$672$		0			0
$673$	−36.0146			−1.38826			−0.694130	−	0.719849i	$-0.744211\pi$
$673$	−36.0146			−1.38826			−0.694130	+	0.719849i	$0.744211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0871			−0.695143			−0.347571	−	0.937653i	$-0.612994\pi$
$677$	−18.0871			−0.695143			−0.347571	+	0.937653i	$0.612994\pi$
$678$		0			0
$679$	−3.34316			−0.128299
$680$		0			0
$681$		0			0
$682$		0			0
$683$			43.1159i			1.64978i	0.565291	+	0.824892i	$0.308764\pi$
$683$			43.1159i			1.64978i	−0.565291	+	0.824892i	$0.691236\pi$
$684$		0			0
$685$	29.7318			1.13600
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.6595	+	4.24400i	−0.634676	+	0.161683i
$690$		0			0
$691$		−	46.8805i		−	1.78342i	−0.452608	−	0.891709i	$-0.649506\pi$
$691$		−	46.8805i		−	1.78342i	0.452608	−	0.891709i	$-0.350494\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	32.3913i		−	1.22867i
$696$		0			0
$697$		−	14.9638i		−	0.566793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.8888			−0.751188			−0.375594	−	0.926784i	$-0.622561\pi$
$701$	−19.8888			−0.751188			−0.375594	+	0.926784i	$0.622561\pi$
$702$		0			0
$703$	−12.4397			−0.469171
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	21.2922i		−	0.800775i
$708$		0			0
$709$			3.75600i			0.141060i	0.997510	+	0.0705299i	$0.0224690\pi$
$709$			3.75600i			0.141060i	−0.997510	+	0.0705299i	$0.977531\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			43.9517i			1.64600i
$714$		0			0
$715$	−31.4034	+	8.00000i	−1.17442	+	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.67158			0.360689			0.180345	−	0.983604i	$-0.442279\pi$
$719$	9.67158			0.360689			0.180345	+	0.983604i	$0.442279\pi$
$720$		0			0
$721$		−	9.48785i		−	0.353346i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.35258			−0.235929
$726$		0			0
$727$	−45.2707			−1.67900			−0.839498	−	0.543363i	$-0.817151\pi$
$727$	−45.2707			−1.67900			−0.839498	+	0.543363i	$0.817151\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.8552			−0.697384
$732$		0			0
$733$		−	47.4905i		−	1.75410i	−0.480398	−	0.877051i	$-0.659508\pi$
$733$		−	47.4905i		−	1.75410i	0.480398	−	0.877051i	$-0.340492\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.8068			−1.57681
$738$		0			0
$739$			14.7332i			0.541968i	0.962584	+	0.270984i	$0.0873491\pi$
$739$			14.7332i			0.541968i	−0.962584	+	0.270984i	$0.912651\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	31.7512i		−	1.16484i	−0.812888	−	0.582420i	$-0.802106\pi$
$743$		−	31.7512i		−	1.16484i	0.812888	−	0.582420i	$-0.197894\pi$
$744$		0			0
$745$	46.1232			1.68982
$746$		0			0
$747$		0			0
$748$		0			0
$749$			3.56033i			0.130092i
$750$		0			0
$751$	0.156915			0.00572590			0.00286295	−	0.999996i	$-0.499089\pi$
$751$	0.156915			0.00572590			0.00286295	+	0.999996i	$0.499089\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.2078			−1.13577
$756$		0			0
$757$	27.7318			1.00793			0.503965	−	0.863724i	$-0.331874\pi$
$757$	27.7318			1.00793			0.503965	+	0.863724i	$0.331874\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.05429i			0.219468i	0.993961	+	0.109734i	$0.0349999\pi$
$761$			6.05429i			0.219468i	−0.993961	+	0.109734i	$0.965000\pi$
$762$		0			0
$763$	1.97584			0.0715301
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.23191	−	4.83579i	−0.0444818	−	0.174610i
$768$		0			0
$769$			32.3913i			1.16806i	0.811732	+	0.584031i	$0.198525\pi$
$769$			32.3913i			1.16806i	−0.811732	+	0.584031i	$0.801475\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	48.9095i		−	1.75915i	−0.475759	−	0.879576i	$-0.657827\pi$
$773$		−	48.9095i		−	1.75915i	0.475759	−	0.879576i	$-0.342173\pi$
$774$		0			0
$775$		−	7.43834i		−	0.267193i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.30426			0.297531
$780$		0			0
$781$	6.57242			0.235180
$782$		0			0
$783$		0			0
$784$		0			0
$785$			24.0603i			0.858748i
$786$		0			0
$787$			25.9022i			0.923312i	0.887059	+	0.461656i	$0.152745\pi$
$787$			25.9022i			0.923312i	−0.887059	+	0.461656i	$0.847255\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			8.85251i			0.314759i
$792$		0			0
$793$	−35.6233	+	9.07500i	−1.26502	+	0.322263i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	45.5991			1.61520			0.807601	−	0.589729i	$-0.200765\pi$
$797$	45.5991			1.61520			0.807601	+	0.589729i	$0.200765\pi$
$798$		0			0
$799$		−	15.2078i		−	0.538012i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.9758			0.634354
$804$		0			0
$805$	−19.9517			−0.703204
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.3672			−0.997337			−0.498668	−	0.866793i	$-0.666177\pi$
$809$	−28.3672			−0.997337			−0.498668	+	0.866793i	$0.666177\pi$
$810$		0			0
$811$		−	54.6461i		−	1.91888i	−0.281907	−	0.959442i	$-0.590967\pi$
$811$		−	54.6461i		−	1.91888i	0.281907	−	0.959442i	$-0.409033\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−42.0629			−1.47340
$816$		0			0
$817$		−	10.4638i		−	0.366083i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.33704i			0.0815634i	0.999168	+	0.0407817i	$0.0129848\pi$
$821$			2.33704i			0.0815634i	−0.999168	+	0.0407817i	$0.987015\pi$
$822$		0			0
$823$	17.8189			0.621129			0.310564	−	0.950552i	$-0.399482\pi$
$823$	17.8189			0.621129			0.310564	+	0.950552i	$0.399482\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.51679i		−	0.122291i	−0.998129	−	0.0611455i	$-0.980525\pi$
$827$		−	3.51679i		−	0.122291i	0.998129	−	0.0611455i	$-0.0194754\pi$
$828$		0			0
$829$	21.4905			0.746396			0.373198	−	0.927752i	$-0.378261\pi$
$829$	21.4905			0.746396			0.373198	+	0.927752i	$0.378261\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.5362			−0.399705
$834$		0			0
$835$	−7.89141			−0.273094
$836$		0			0
$837$		0			0
$838$		0			0
$839$			10.6160i			0.366504i	0.983066	+	0.183252i	$0.0586624\pi$
$839$			10.6160i			0.366504i	−0.983066	+	0.183252i	$0.941338\pi$
$840$		0			0
$841$	−1.87933			−0.0648045
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.5120	+	28.4698i	0.533629	+	0.979391i
$846$		0			0
$847$		−	2.20642i		−	0.0758136i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	80.7827i		−	2.76919i
$852$		0			0
$853$			38.4543i			1.31665i	0.752734	+	0.658324i	$0.228734\pi$
$853$			38.4543i			1.31665i	−0.752734	+	0.658324i	$0.771266\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.2413			0.418156			0.209078	−	0.977899i	$-0.432954\pi$
$857$	12.2413			0.418156			0.209078	+	0.977899i	$0.432954\pi$
$858$		0			0
$859$	30.8552			1.05276			0.526382	−	0.850248i	$-0.323548\pi$
$859$	30.8552			1.05276			0.526382	+	0.850248i	$0.323548\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.9952i			0.816807i	0.912802	+	0.408403i	$0.133914\pi$
$863$			23.9952i			0.816807i	−0.912802	+	0.408403i	$0.866086\pi$
$864$		0			0
$865$		−	3.01208i		−	0.102414i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.58450i			0.0537504i
$870$		0			0
$871$	10.5724	+	41.5013i	0.358233	+	1.40622i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.4638			−0.353742
$876$		0			0
$877$			41.8404i			1.41285i	0.707787	+	0.706425i	$0.249693\pi$
$877$			41.8404i			1.41285i	−0.707787	+	0.706425i	$0.750307\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.3913			1.02391			0.511955	−	0.859012i	$-0.328921\pi$
$881$	30.3913			1.02391			0.511955	+	0.859012i	$0.328921\pi$
$882$		0			0
$883$	42.4155			1.42740			0.713698	−	0.700454i	$-0.247019\pi$
$883$	42.4155			1.42740			0.713698	+	0.700454i	$0.247019\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	22.3284			0.749715			0.374857	−	0.927082i	$-0.377692\pi$
$887$	22.3284			0.749715			0.374857	+	0.927082i	$0.377692\pi$
$888$		0			0
$889$		−	4.43967i		−	0.148902i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.43967			0.282423
$894$		0			0
$895$		−	16.8793i		−	0.564214i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			31.7560i			1.05912i
$900$		0			0
$901$	9.53617			0.317696
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.91557i			0.0636759i
$906$		0			0
$907$	−38.8552			−1.29016			−0.645082	−	0.764113i	$-0.723177\pi$
$907$	−38.8552			−1.29016			−0.645082	+	0.764113i	$0.723177\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	10.3767			0.343797			0.171898	−	0.985115i	$-0.445010\pi$
$911$	10.3767			0.343797			0.171898	+	0.985115i	$0.445010\pi$
$912$		0			0
$913$	−2.13275			−0.0705838
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	23.5120i		−	0.776435i
$918$		0			0
$919$	−18.8068			−0.620380			−0.310190	−	0.950675i	$-0.600393\pi$
$919$	−18.8068			−0.620380			−0.310190	+	0.950675i	$0.600393\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.62325	−	6.37196i	−0.0534300	−	0.209736i
$924$		0			0
$925$			13.6716i			0.449519i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	48.0301i		−	1.57582i	−0.615792	−	0.787909i	$-0.711164\pi$
$929$		−	48.0301i		−	1.57582i	0.615792	−	0.787909i	$-0.288836\pi$
$930$		0			0
$931$		−	6.40209i		−	0.209820i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	17.9758			0.587873
$936$		0			0
$937$	−44.6112			−1.45738			−0.728692	−	0.684842i	$-0.759871\pi$
$937$	−44.6112			−1.45738			−0.728692	+	0.684842i	$0.759871\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			3.53020i			0.115081i	0.998343	+	0.0575407i	$0.0183259\pi$
$941$			3.53020i			0.115081i	−0.998343	+	0.0575407i	$0.981674\pi$
$942$		0			0
$943$			53.9275i			1.75612i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			18.3505i			0.596310i	0.954518	+	0.298155i	$0.0963712\pi$
$947$			18.3505i			0.596310i	−0.954518	+	0.298155i	$0.903629\pi$
$948$		0			0
$949$	−4.43967	−	17.4276i	−0.144118	−	0.565723i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.8310			0.869141			0.434571	−	0.900638i	$-0.356900\pi$
$953$	26.8310			0.869141			0.434571	+	0.900638i	$0.356900\pi$
$954$		0			0
$955$		−	26.8552i		−	0.869013i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.2319			0.427281
$960$		0			0
$961$	−6.18359			−0.199471
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.0000			−0.515058
$966$		0			0
$967$		−	30.5590i		−	0.982711i	−0.870959	−	0.491356i	$-0.836502\pi$
$967$		−	30.5590i		−	0.982711i	0.870959	−	0.491356i	$-0.163498\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	51.6862			1.65869			0.829344	−	0.558738i	$-0.188714\pi$
$971$	51.6862			1.65869			0.829344	+	0.558738i	$0.188714\pi$
$972$		0			0
$973$		−	14.4155i		−	0.462140i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.78879i			0.0572285i	0.999591	+	0.0286143i	$0.00910944\pi$
$977$			1.78879i			0.0572285i	−0.999591	+	0.0286143i	$0.990891\pi$
$978$		0			0
$979$	−3.84309			−0.122826
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	8.94438i		−	0.285281i	−0.989775	−	0.142641i	$-0.954441\pi$
$983$		−	8.94438i		−	0.285281i	0.989775	−	0.142641i	$-0.0455593\pi$
$984$		0			0
$985$	47.4905			1.51317
$986$		0			0
$987$		0			0
$988$		0			0
$989$	67.9517			2.16074
$990$		0			0
$991$	23.4517			0.744969			0.372485	−	0.928038i	$-0.378506\pi$
$991$	23.4517			0.744969			0.372485	+	0.928038i	$0.378506\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	29.9275i		−	0.948766i
$996$		0			0
$997$	−25.1207			−0.795580			−0.397790	−	0.917477i	$-0.630223\pi$
$997$	−25.1207			−0.795580			−0.397790	+	0.917477i	$0.630223\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.c.m.3457.5		6
3.2	odd	2		1248.2.c.b.961.2	yes	6
4.3	odd	2		3744.2.c.l.3457.5		6
12.11	even	2		1248.2.c.a.961.2	✓	6
13.12	even	2	inner	3744.2.c.m.3457.2		6
24.5	odd	2		2496.2.c.o.961.5		6
24.11	even	2		2496.2.c.p.961.5		6
39.38	odd	2		1248.2.c.b.961.5	yes	6
52.51	odd	2		3744.2.c.l.3457.2		6
156.155	even	2		1248.2.c.a.961.5	yes	6
312.77	odd	2		2496.2.c.o.961.2		6
312.155	even	2		2496.2.c.p.961.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.c.a.961.2	✓	6	12.11	even	2
1248.2.c.a.961.5	yes	6	156.155	even	2
1248.2.c.b.961.2	yes	6	3.2	odd	2
1248.2.c.b.961.5	yes	6	39.38	odd	2
2496.2.c.o.961.2		6	312.77	odd	2
2496.2.c.o.961.5		6	24.5	odd	2
2496.2.c.p.961.2		6	312.155	even	2
2496.2.c.p.961.5		6	24.11	even	2
3744.2.c.l.3457.2		6	52.51	odd	2
3744.2.c.l.3457.5		6	4.3	odd	2
3744.2.c.m.3457.2		6	13.12	even	2	inner
3744.2.c.m.3457.5		6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.49396i q^{5} +1.10992i q^{7} +3.60388i q^{11} +(3.49396 - 0.890084i) q^{13} -2.00000 q^{17} -1.10992i q^{19} +7.20775 q^{23} -1.21983 q^{25} +5.20775 q^{29} +6.09783i q^{31} -2.76809 q^{35} -11.2078i q^{37} +7.48188i q^{41} +9.42758 q^{43} +7.60388i q^{47} +5.76809 q^{49} -4.76809 q^{53} -8.98792 q^{55} -1.38404i q^{59} -10.1957 q^{61} +(2.21983 + 8.71379i) q^{65} +11.8780i q^{67} -1.82371i q^{71} -4.98792i q^{73} -4.00000 q^{77} +0.439665 q^{79} +0.591794i q^{83} -4.98792i q^{85} +1.06638i q^{89} +(0.987918 + 3.87800i) q^{91} +2.76809 q^{95} +3.01208i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{13} - 12 q^{17} + 8 q^{23} - 10 q^{25} - 4 q^{29} + 24 q^{35} + 24 q^{43} - 6 q^{49} + 12 q^{53} - 16 q^{55} + 12 q^{61} + 16 q^{65} - 24 q^{77} + 8 q^{79} - 32 q^{91} - 24 q^{95}+O(q^{100}) \) 6 * q + 2 * q^13 - 12 * q^17 + 8 * q^23 - 10 * q^25 - 4 * q^29 + 24 * q^35 + 24 * q^43 - 6 * q^49 + 12 * q^53 - 16 * q^55 + 12 * q^61 + 16 * q^65 - 24 * q^77 + 8 * q^79 - 32 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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