LMFDB - Embedded newform 2736.2.f.g.1025.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2736,2,Mod(1025,2736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 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("2736.1025"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,-8,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.131310567424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 $ x^8 + 18x^6 + 37x^4 + 22*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.1
Root			$1.18847i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.g.1025.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.42865i q^{5} +0.394100 q^{7} +2.33407i q^{11} +3.36566i q^{13} -0.494357i q^{17} +(0.300874 + 4.34850i) q^{19} +3.79116i q^{23} -6.75561 q^{25} +2.14971 q^{29} +7.04509i q^{31} -1.35123i q^{35} +7.30877i q^{37} +3.81355 q^{41} +8.45474 q^{43} +2.43993i q^{47} -6.84469 q^{49} -10.9095 q^{53} +8.00270 q^{55} +6.33616 q^{59} -10.5154 q^{61} +11.5397 q^{65} -5.01758i q^{67} +0.759763 q^{71} +3.06479 q^{73} +0.919857i q^{77} +6.40581i q^{79} +1.54018i q^{83} -1.69497 q^{85} +13.2456 q^{89} +1.32641i q^{91} +(14.9095 - 1.03159i) q^{95} -1.71374i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) $ 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$		−	3.42865i		−	1.53334i	−0.642043	−	0.766668i	$-0.721913\pi$
$5$		−	3.42865i		−	1.53334i	0.642043	−	0.766668i	$-0.278087\pi$
$6$		0			0
$7$	0.394100			0.148956			0.0744779	−	0.997223i	$-0.476271\pi$
$7$	0.394100			0.148956			0.0744779	+	0.997223i	$0.476271\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.33407i			0.703749i	0.936047	+	0.351874i	$0.114456\pi$
$11$			2.33407i			0.703749i	−0.936047	+	0.351874i	$0.885544\pi$
$12$		0			0
$13$			3.36566i			0.933466i	0.884398	+	0.466733i	$0.154569\pi$
$13$			3.36566i			0.933466i	−0.884398	+	0.466733i	$0.845431\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.494357i		−	0.119899i	−0.998201	−	0.0599496i	$-0.980906\pi$
$17$		−	0.494357i		−	0.119899i	0.998201	−	0.0599496i	$-0.0190940\pi$
$18$		0			0
$19$	0.300874	+	4.34850i	0.0690252	+	0.997615i
$19$	0.300874	+	4.34850i	0.0690252	+	0.997615i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.79116i			0.790512i	0.918571	+	0.395256i	$0.129344\pi$
$23$			3.79116i			0.790512i	−0.918571	+	0.395256i	$0.870656\pi$
$24$		0			0
$25$	−6.75561			−1.35112
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.14971			0.399191			0.199596	−	0.979878i	$-0.436037\pi$
$29$	2.14971			0.399191			0.199596	+	0.979878i	$0.436037\pi$
$30$		0			0
$31$			7.04509i			1.26534i	0.774423	+	0.632668i	$0.218040\pi$
$31$			7.04509i			1.26534i	−0.774423	+	0.632668i	$0.781960\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.35123i		−	0.228399i
$36$		0			0
$37$			7.30877i			1.20155i	0.799416	+	0.600777i	$0.205142\pi$
$37$			7.30877i			1.20155i	−0.799416	+	0.600777i	$0.794858\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.81355			0.595576			0.297788	−	0.954632i	$-0.403751\pi$
$41$	3.81355			0.595576			0.297788	+	0.954632i	$0.403751\pi$
$42$		0			0
$43$	8.45474			1.28934			0.644668	−	0.764463i	$-0.276996\pi$
$43$	8.45474			1.28934			0.644668	+	0.764463i	$0.276996\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.43993i			0.355901i	0.984039	+	0.177950i	$0.0569466\pi$
$47$			2.43993i			0.355901i	−0.984039	+	0.177950i	$0.943053\pi$
$48$		0			0
$49$	−6.84469			−0.977812
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.9095			−1.49853			−0.749266	−	0.662269i	$-0.769593\pi$
$53$	−10.9095			−1.49853			−0.749266	+	0.662269i	$0.769593\pi$
$54$		0			0
$55$	8.00270			1.07908
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.33616			0.824898			0.412449	−	0.910981i	$-0.364673\pi$
$59$	6.33616			0.824898			0.412449	+	0.910981i	$0.364673\pi$
$60$		0			0
$61$	−10.5154			−1.34636			−0.673178	−	0.739481i	$-0.735071\pi$
$61$	−10.5154			−1.34636			−0.673178	+	0.739481i	$0.735071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	11.5397			1.43132
$66$		0			0
$67$		−	5.01758i		−	0.612995i	−0.951872	−	0.306497i	$-0.900843\pi$
$67$		−	5.01758i		−	0.612995i	0.951872	−	0.306497i	$-0.0991571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.759763			0.0901673			0.0450837	−	0.998983i	$-0.485645\pi$
$71$	0.759763			0.0901673			0.0450837	+	0.998983i	$0.485645\pi$
$72$		0			0
$73$	3.06479			0.358706			0.179353	−	0.983785i	$-0.442599\pi$
$73$	3.06479			0.358706			0.179353	+	0.983785i	$0.442599\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.919857i			0.104827i
$78$		0			0
$79$			6.40581i			0.720710i	0.932815	+	0.360355i	$0.117344\pi$
$79$			6.40581i			0.720710i	−0.932815	+	0.360355i	$0.882656\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.54018i			0.169057i	0.996421	+	0.0845285i	$0.0269384\pi$
$83$			1.54018i			0.169057i	−0.996421	+	0.0845285i	$0.973062\pi$
$84$		0			0
$85$	−1.69497			−0.183846
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.2456			1.40403			0.702017	−	0.712160i	$-0.252283\pi$
$89$	13.2456			1.40403			0.702017	+	0.712160i	$0.252283\pi$
$90$		0			0
$91$			1.32641i			0.139045i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.9095	−	1.03159i	1.52968	−	0.105839i
$96$		0			0
$97$		−	1.71374i		−	0.174004i	−0.996208	−	0.0870022i	$-0.972271\pi$
$97$		−	1.71374i		−	0.174004i	0.996208	−	0.0870022i	$-0.0277287\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.4769i		−	1.34100i	−0.741910	−	0.670500i	$-0.766080\pi$
$101$		−	13.4769i		−	1.34100i	0.741910	−	0.670500i	$-0.233920\pi$
$102$		0			0
$103$			0.577449i			0.0568977i	0.999595	+	0.0284489i	$0.00905678\pi$
$103$			0.577449i			0.0568977i	−0.999595	+	0.0284489i	$0.990943\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.53966			0.728886			0.364443	−	0.931226i	$-0.381259\pi$
$107$	7.53966			0.728886			0.364443	+	0.931226i	$0.381259\pi$
$108$		0			0
$109$		−	2.29119i		−	0.219457i	−0.993962	−	0.109728i	$-0.965002\pi$
$109$		−	2.29119i		−	0.219457i	0.993962	−	0.109728i	$-0.0349981\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	19.2089			1.80702			0.903511	−	0.428566i	$-0.140981\pi$
$113$	19.2089			1.80702			0.903511	+	0.428566i	$0.140981\pi$
$114$		0			0
$115$	12.9985			1.21212
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.194826i		−	0.0178597i
$120$		0			0
$121$	5.55212			0.504738
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.01937i			0.538389i
$126$		0			0
$127$			2.29119i			0.203311i	0.994820	+	0.101655i	$0.0324139\pi$
$127$			2.29119i			0.203311i	−0.994820	+	0.101655i	$0.967586\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.3766i			1.25609i	0.778176	+	0.628046i	$0.216145\pi$
$131$			14.3766i			1.25609i	−0.778176	+	0.628046i	$0.783855\pi$
$132$		0			0
$133$	0.118574	+	1.71374i	0.0102817	+	0.148600i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	7.05799i		−	0.603005i	−0.953465	−	0.301502i	$-0.902512\pi$
$137$		−	7.05799i		−	0.603005i	0.953465	−	0.301502i	$-0.0974881\pi$
$138$		0			0
$139$	−10.3574			−0.878500			−0.439250	−	0.898365i	$-0.644756\pi$
$139$	−10.3574			−0.878500			−0.439250	+	0.898365i	$0.644756\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.85569			−0.656926
$144$		0			0
$145$		−	7.37060i		−	0.612095i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	3.51058i		−	0.287598i	−0.989607	−	0.143799i	$-0.954068\pi$
$149$		−	3.51058i		−	0.287598i	0.989607	−	0.143799i	$-0.0459319\pi$
$150$		0			0
$151$		−	19.9489i		−	1.62342i	−0.584063	−	0.811709i	$-0.698538\pi$
$151$		−	19.9489i		−	1.62342i	0.584063	−	0.811709i	$-0.301462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	24.1551			1.94019
$156$		0			0
$157$	7.24024			0.577834			0.288917	−	0.957354i	$-0.406705\pi$
$157$	7.24024			0.577834			0.288917	+	0.957354i	$0.406705\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.49410i			0.117751i
$162$		0			0
$163$	−23.9971			−1.87960			−0.939799	−	0.341728i	$-0.888988\pi$
$163$	−23.9971			−1.87960			−0.939799	+	0.341728i	$0.888988\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	18.6356			1.44206			0.721032	−	0.692901i	$-0.243668\pi$
$167$	18.6356			1.44206			0.721032	+	0.692901i	$0.243668\pi$
$168$		0			0
$169$	1.67233			0.128640
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.48587			−0.645169			−0.322585	−	0.946541i	$-0.604552\pi$
$173$	−8.48587			−0.645169			−0.322585	+	0.946541i	$0.604552\pi$
$174$		0			0
$175$	−2.66239			−0.201257
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.0959			−1.42730			−0.713648	−	0.700504i	$-0.752958\pi$
$179$	−19.0959			−1.42730			−0.713648	+	0.700504i	$0.752958\pi$
$180$		0			0
$181$			16.0058i			1.18970i	0.803837	+	0.594850i	$0.202789\pi$
$181$			16.0058i			1.18970i	−0.803837	+	0.594850i	$0.797211\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	25.0592			1.84239
$186$		0			0
$187$	1.15386			0.0843788
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.6335i			1.34827i	0.738607	+	0.674137i	$0.235484\pi$
$191$			18.6335i			1.34827i	−0.738607	+	0.674137i	$0.764516\pi$
$192$		0			0
$193$			9.33628i			0.672040i	0.941855	+	0.336020i	$0.109081\pi$
$193$			9.33628i			0.672040i	−0.941855	+	0.336020i	$0.890919\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.7062i			1.11902i	0.828822	+	0.559512i	$0.189011\pi$
$197$			15.7062i			1.11902i	−0.828822	+	0.559512i	$0.810989\pi$
$198$		0			0
$199$	4.45474			0.315788			0.157894	−	0.987456i	$-0.449530\pi$
$199$	4.45474			0.315788			0.157894	+	0.987456i	$0.449530\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.847201			0.0594619
$204$		0			0
$205$		−	13.0753i		−	0.913219i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.1497	+	0.702261i	−0.702070	+	0.0485764i
$210$		0			0
$211$			25.8391i			1.77884i	0.457095	+	0.889418i	$0.348890\pi$
$211$			25.8391i			1.77884i	−0.457095	+	0.889418i	$0.651110\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	28.9883i		−	1.97699i
$216$		0			0
$217$			2.77647i			0.188479i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.66384			0.111922
$222$		0			0
$223$			25.3538i			1.69782i	0.528541	+	0.848908i	$0.322739\pi$
$223$			25.3538i			1.69782i	−0.528541	+	0.848908i	$0.677261\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.81895			−0.651707			−0.325853	−	0.945420i	$-0.605652\pi$
$227$	−9.81895			−0.651707			−0.325853	+	0.945420i	$0.605652\pi$
$228$		0			0
$229$	16.3912			1.08316			0.541580	−	0.840649i	$-0.317826\pi$
$229$	16.3912			1.08316			0.541580	+	0.840649i	$0.317826\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3.27083i			0.214279i	0.994244	+	0.107139i	$0.0341691\pi$
$233$			3.27083i			0.214279i	−0.994244	+	0.107139i	$0.965831\pi$
$234$		0			0
$235$	8.36566			0.545716
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	9.17125i		−	0.593239i	−0.954996	−	0.296620i	$-0.904141\pi$
$239$		−	9.17125i		−	0.593239i	0.954996	−	0.296620i	$-0.0958593\pi$
$240$		0			0
$241$			9.77147i			0.629436i	0.949185	+	0.314718i	$0.101910\pi$
$241$			9.77147i			0.629436i	−0.949185	+	0.314718i	$0.898090\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			23.4680i			1.49932i
$246$		0			0
$247$	−14.6356	+	1.01264i	−0.931240	+	0.0644327i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	25.0093i		−	1.57858i	−0.614023	−	0.789288i	$-0.710450\pi$
$251$		−	25.0093i		−	1.57858i	0.614023	−	0.789288i	$-0.289550\pi$
$252$		0			0
$253$	−8.84884			−0.556322
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.8220			0.675060			0.337530	−	0.941315i	$-0.390409\pi$
$257$	10.8220			0.675060			0.337530	+	0.941315i	$0.390409\pi$
$258$		0			0
$259$			2.88039i			0.178978i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	11.0929i		−	0.684018i	−0.939697	−	0.342009i	$-0.888893\pi$
$263$		−	11.0929i		−	0.684018i	0.939697	−	0.342009i	$-0.111107\pi$
$264$		0			0
$265$			37.4047i			2.29775i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−28.4124			−1.73233			−0.866167	−	0.499754i	$-0.833424\pi$
$269$	−28.4124			−1.73233			−0.866167	+	0.499754i	$0.833424\pi$
$270$		0			0
$271$	−9.81895			−0.596459			−0.298229	−	0.954494i	$-0.596396\pi$
$271$	−9.81895			−0.596459			−0.298229	+	0.954494i	$0.596396\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	15.7681i		−	0.950851i
$276$		0			0
$277$	−10.8883			−0.654213			−0.327107	−	0.944987i	$-0.606074\pi$
$277$	−10.8883			−0.654213			−0.327107	+	0.944987i	$0.606074\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.7060			−0.817630			−0.408815	−	0.912617i	$-0.634058\pi$
$281$	−13.7060			−0.817630			−0.408815	+	0.912617i	$0.634058\pi$
$282$		0			0
$283$	−3.78241			−0.224841			−0.112421	−	0.993661i	$-0.535860\pi$
$283$	−3.78241			−0.224841			−0.112421	+	0.993661i	$0.535860\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.50292			0.0887145
$288$		0			0
$289$	16.7556			0.985624
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.5366			−0.615553			−0.307777	−	0.951459i	$-0.599585\pi$
$293$	−10.5366			−0.615553			−0.307777	+	0.951459i	$0.599585\pi$
$294$		0			0
$295$		−	21.7245i		−	1.26485i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.7598			−0.737916
$300$		0			0
$301$	3.33201			0.192054
$302$		0			0
$303$		0			0
$304$		0			0
$305$			36.0535i			2.06442i
$306$		0			0
$307$			8.70875i			0.497035i	0.968627	+	0.248517i	$0.0799433\pi$
$307$			8.70875i			0.497035i	−0.968627	+	0.248517i	$0.920057\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.7888i			1.34894i	0.738303	+	0.674469i	$0.235628\pi$
$311$			23.7888i			1.34894i	−0.738303	+	0.674469i	$0.764372\pi$
$312$		0			0
$313$	20.2994			1.14739			0.573696	−	0.819069i	$-0.305509\pi$
$313$	20.2994			1.14739			0.573696	+	0.819069i	$0.305509\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.9095			0.837400			0.418700	−	0.908125i	$-0.362486\pi$
$317$	14.9095			0.837400			0.418700	+	0.908125i	$0.362486\pi$
$318$		0			0
$319$			5.01758i			0.280930i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.14971	−	0.148739i	0.119613	−	0.00827607i
$324$		0			0
$325$		−	22.7371i		−	1.26123i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.961577i			0.0530135i
$330$		0			0
$331$		−	2.40087i		−	0.131964i	−0.997821	−	0.0659820i	$-0.978982\pi$
$331$		−	2.40087i		−	0.131964i	0.997821	−	0.0659820i	$-0.0210180\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.2035			−0.939927
$336$		0			0
$337$			2.13719i			0.116420i	0.998304	+	0.0582101i	$0.0185393\pi$
$337$			2.13719i			0.116420i	−0.998304	+	0.0582101i	$0.981461\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.4437			−0.890478
$342$		0			0
$343$	−5.45619			−0.294607
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	15.2796i		−	0.820251i	−0.912029	−	0.410126i	$-0.865485\pi$
$347$		−	15.2796i		−	0.820251i	0.912029	−	0.410126i	$-0.134515\pi$
$348$		0			0
$349$	21.7712			1.16539			0.582693	−	0.812692i	$-0.301999\pi$
$349$	21.7712			1.16539			0.582693	+	0.812692i	$0.301999\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	27.2131i		−	1.44841i	−0.689587	−	0.724203i	$-0.742208\pi$
$353$		−	27.2131i		−	1.44841i	0.689587	−	0.724203i	$-0.257792\pi$
$354$		0			0
$355$		−	2.60496i		−	0.138257i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.19435i			0.274147i	0.990561	+	0.137074i	$0.0437697\pi$
$359$			5.19435i			0.274147i	−0.990561	+	0.137074i	$0.956230\pi$
$360$		0			0
$361$	−18.8189	+	2.61670i	−0.990471	+	0.137721i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	10.5081i		−	0.550018i
$366$		0			0
$367$	−30.4913			−1.59163			−0.795816	−	0.605539i	$-0.792958\pi$
$367$	−30.4913			−1.59163			−0.795816	+	0.605539i	$0.792958\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.29942			−0.223215
$372$		0			0
$373$		−	25.3538i		−	1.31277i	−0.754426	−	0.656385i	$-0.772085\pi$
$373$		−	25.3538i		−	1.31277i	0.754426	−	0.656385i	$-0.227915\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7.23520i			0.372632i
$378$		0			0
$379$		−	23.7019i		−	1.21748i	−0.793368	−	0.608742i	$-0.791674\pi$
$379$		−	23.7019i		−	1.21748i	0.793368	−	0.608742i	$-0.208326\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.816636			0.0417282			0.0208641	−	0.999782i	$-0.493358\pi$
$383$	0.816636			0.0417282			0.0208641	+	0.999782i	$0.493358\pi$
$384$		0			0
$385$	3.15386			0.160736
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	22.3547i		−	1.13343i	−0.823915	−	0.566713i	$-0.808215\pi$
$389$		−	22.3547i		−	1.13343i	0.823915	−	0.566713i	$-0.191785\pi$
$390$		0			0
$391$	1.87419			0.0947817
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.9633			1.10509
$396$		0			0
$397$	32.5731			1.63480			0.817399	−	0.576072i	$-0.195415\pi$
$397$	32.5731			1.63480			0.817399	+	0.576072i	$0.195415\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.81355			0.390190			0.195095	−	0.980784i	$-0.437498\pi$
$401$	7.81355			0.390190			0.195095	+	0.980784i	$0.437498\pi$
$402$		0			0
$403$	−23.7114			−1.18115
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−17.0592			−0.845592
$408$		0			0
$409$			23.3145i			1.15283i	0.817157	+	0.576415i	$0.195549\pi$
$409$			23.3145i			1.15283i	−0.817157	+	0.576415i	$0.804451\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.49708			0.122873
$414$		0			0
$415$	5.28074			0.259221
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6.56950i			0.320941i	0.987041	+	0.160471i	$0.0513012\pi$
$419$			6.56950i			0.320941i	−0.987041	+	0.160471i	$0.948699\pi$
$420$		0			0
$421$		−	36.1763i		−	1.76312i	−0.472069	−	0.881561i	$-0.656493\pi$
$421$		−	36.1763i		−	1.76312i	0.472069	−	0.881561i	$-0.343507\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.33968i			0.161998i
$426$		0			0
$427$	−4.14411			−0.200547
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.2426			−0.975049			−0.487525	−	0.873109i	$-0.662100\pi$
$431$	−20.2426			−0.975049			−0.487525	+	0.873109i	$0.662100\pi$
$432$		0			0
$433$			26.3068i			1.26423i	0.774876	+	0.632113i	$0.217812\pi$
$433$			26.3068i			1.26423i	−0.774876	+	0.632113i	$0.782188\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.4859	+	1.14066i	−0.788626	+	0.0545653i
$438$		0			0
$439$		−	24.8743i		−	1.18719i	−0.804766	−	0.593593i	$-0.797709\pi$
$439$		−	24.8743i		−	1.18719i	0.804766	−	0.593593i	$-0.202291\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			36.1011i			1.71521i	0.514305	+	0.857607i	$0.328050\pi$
$443$			36.1011i			1.71521i	−0.514305	+	0.857607i	$0.671950\pi$
$444$		0			0
$445$		−	45.4146i		−	2.15286i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.42669			0.350487			0.175244	−	0.984525i	$-0.443929\pi$
$449$	7.42669			0.350487			0.175244	+	0.984525i	$0.443929\pi$
$450$		0			0
$451$			8.90109i			0.419136i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.54778			0.213203
$456$		0			0
$457$	22.7542			1.06439			0.532197	−	0.846620i	$-0.321366\pi$
$457$	22.7542			1.06439			0.532197	+	0.846620i	$0.321366\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	17.4771i		−	0.813990i	−0.913430	−	0.406995i	$-0.866577\pi$
$461$		−	17.4771i		−	0.813990i	0.913430	−	0.406995i	$-0.133423\pi$
$462$		0			0
$463$	31.1832			1.44920			0.724602	−	0.689167i	$-0.242024\pi$
$463$	31.1832			1.44920			0.724602	+	0.689167i	$0.242024\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	16.5579i		−	0.766206i	−0.923706	−	0.383103i	$-0.874855\pi$
$467$		−	16.5579i		−	0.766206i	0.923706	−	0.383103i	$-0.125145\pi$
$468$		0			0
$469$		−	1.97743i		−	0.0913091i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			19.7340i			0.907368i
$474$		0			0
$475$	−2.03259	−	29.3768i	−0.0932615	−	1.34790i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	8.37355i		−	0.382597i	−0.981532	−	0.191299i	$-0.938730\pi$
$479$		−	8.37355i		−	0.382597i	0.981532	−	0.191299i	$-0.0612699\pi$
$480$		0			0
$481$	−24.5988			−1.12161
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.87582			−0.266807
$486$		0			0
$487$			21.4107i			0.970211i	0.874456	+	0.485106i	$0.161219\pi$
$487$			21.4107i			0.970211i	−0.874456	+	0.485106i	$0.838781\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			31.9693i			1.44275i	0.692543	+	0.721377i	$0.256490\pi$
$491$			31.9693i			1.44275i	−0.692543	+	0.721377i	$0.743510\pi$
$492$		0			0
$493$		−	1.06272i		−	0.0478627i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.299423			0.0134309
$498$		0			0
$499$	12.7220			0.569513			0.284757	−	0.958600i	$-0.408087\pi$
$499$	12.7220			0.569513			0.284757	+	0.958600i	$0.408087\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.3089i		−	0.816353i	−0.912903	−	0.408177i	$-0.866165\pi$
$503$		−	18.3089i		−	0.816353i	0.912903	−	0.408177i	$-0.133835\pi$
$504$		0			0
$505$	−46.2074			−2.05620
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−36.0255			−1.59680			−0.798402	−	0.602125i	$-0.794321\pi$
$509$	−36.0255			−1.59680			−0.798402	+	0.602125i	$0.794321\pi$
$510$		0			0
$511$	1.20783			0.0534314
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.97987			0.0872434
$516$		0			0
$517$	−5.69497			−0.250465
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.0928			0.967905			0.483952	−	0.875094i	$-0.339201\pi$
$521$	22.0928			0.967905			0.483952	+	0.875094i	$0.339201\pi$
$522$		0			0
$523$		−	18.9362i		−	0.828024i	−0.910271	−	0.414012i	$-0.864127\pi$
$523$		−	18.9362i		−	0.828024i	0.910271	−	0.414012i	$-0.135873\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.48279			0.151713
$528$		0			0
$529$	8.62710			0.375091
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.8351i			0.555950i
$534$		0			0
$535$		−	25.8508i		−	1.11763i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	15.9760i		−	0.688134i
$540$		0			0
$541$	−10.7542			−0.462357			−0.231179	−	0.972911i	$-0.574258\pi$
$541$	−10.7542			−0.462357			−0.231179	+	0.972911i	$0.574258\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.85569			−0.336501
$546$		0			0
$547$		−	35.2675i		−	1.50793i	−0.656914	−	0.753965i	$-0.728139\pi$
$547$		−	35.2675i		−	1.50793i	0.656914	−	0.753965i	$-0.271861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.646792	+	9.34803i	0.0275543	+	0.398239i
$552$		0			0
$553$			2.52453i			0.107354i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			17.9202i			0.759304i	0.925129	+	0.379652i	$0.123956\pi$
$557$			17.9202i			0.759304i	−0.925129	+	0.379652i	$0.876044\pi$
$558$		0			0
$559$			28.4558i			1.20355i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−46.2627			−1.94974			−0.974870	−	0.222775i	$-0.928488\pi$
$563$	−46.2627			−1.94974			−0.974870	+	0.222775i	$0.928488\pi$
$564$		0			0
$565$		−	65.8605i		−	2.77077i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.6692			−1.15995			−0.579977	−	0.814633i	$-0.696939\pi$
$569$	−27.6692			−1.15995			−0.579977	+	0.814633i	$0.696939\pi$
$570$		0			0
$571$	34.4913			1.44341			0.721707	−	0.692198i	$-0.243358\pi$
$571$	34.4913			1.44341			0.721707	+	0.692198i	$0.243358\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	25.6116i		−	1.06808i
$576$		0			0
$577$	5.63974			0.234785			0.117393	−	0.993086i	$-0.462546\pi$
$577$	5.63974			0.234785			0.117393	+	0.993086i	$0.462546\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			0.606986i			0.0251820i
$582$		0			0
$583$		−	25.4635i		−	1.05459i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.24286i			0.340219i	0.985425	+	0.170110i	$0.0544122\pi$
$587$			8.24286i			0.340219i	−0.985425	+	0.170110i	$0.945588\pi$
$588$		0			0
$589$	−30.6356	+	2.11968i	−1.26232	+	0.0873401i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	0.815152i		−	0.0334743i	−0.999860	−	0.0167371i	$-0.994672\pi$
$593$		−	0.815152i		−	0.0334743i	0.999860	−	0.0167371i	$-0.00532785\pi$
$594$		0			0
$595$	−0.667989			−0.0273849
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.74316			−0.357236			−0.178618	−	0.983919i	$-0.557163\pi$
$599$	−8.74316			−0.357236			−0.178618	+	0.983919i	$0.557163\pi$
$600$		0			0
$601$		−	17.2225i		−	0.702520i	−0.936278	−	0.351260i	$-0.885753\pi$
$601$		−	17.2225i		−	0.702520i	0.936278	−	0.351260i	$-0.114247\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	19.0362i		−	0.773933i
$606$		0			0
$607$			7.09294i			0.287894i	0.989585	+	0.143947i	$0.0459794\pi$
$607$			7.09294i			0.287894i	−0.989585	+	0.143947i	$0.954021\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.21199			−0.332221
$612$		0			0
$613$	7.47632			0.301966			0.150983	−	0.988536i	$-0.451756\pi$
$613$	7.47632			0.301966			0.150983	+	0.988536i	$0.451756\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	25.3550i		−	1.02075i	−0.859951	−	0.510376i	$-0.829506\pi$
$617$		−	25.3550i		−	1.02075i	0.859951	−	0.510376i	$-0.170494\pi$
$618$		0			0
$619$	−43.4659			−1.74704			−0.873522	−	0.486785i	$-0.838169\pi$
$619$	−43.4659			−1.74704			−0.873522	+	0.486785i	$0.838169\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.22010			0.209139
$624$		0			0
$625$	−13.1398			−0.525591
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.61314			0.144065
$630$		0			0
$631$	44.0889			1.75515			0.877575	−	0.479439i	$-0.159160\pi$
$631$	44.0889			1.75515			0.877575	+	0.479439i	$0.159160\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.85569			0.311744
$636$		0			0
$637$		−	23.0369i		−	0.912755i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.46574			−0.255381			−0.127691	−	0.991814i	$-0.540756\pi$
$641$	−6.46574			−0.255381			−0.127691	+	0.991814i	$0.540756\pi$
$642$		0			0
$643$	41.0536			1.61900			0.809498	−	0.587123i	$-0.199740\pi$
$643$	41.0536			1.61900			0.809498	+	0.587123i	$0.199740\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			34.4192i			1.35316i	0.736370	+	0.676579i	$0.236538\pi$
$647$			34.4192i			1.35316i	−0.736370	+	0.676579i	$0.763462\pi$
$648$		0			0
$649$			14.7891i			0.580521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	9.50663i		−	0.372023i	−0.982548	−	0.186012i	$-0.940444\pi$
$653$		−	9.50663i		−	0.372023i	0.982548	−	0.186012i	$-0.0595562\pi$
$654$		0			0
$655$	49.2924			1.92601
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.8306			−0.499809			−0.249904	−	0.968270i	$-0.580399\pi$
$659$	−12.8306			−0.499809			−0.249904	+	0.968270i	$0.580399\pi$
$660$		0			0
$661$			33.6870i			1.31027i	0.755512	+	0.655135i	$0.227388\pi$
$661$			33.6870i			1.31027i	−0.755512	+	0.655135i	$0.772612\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.87582	−	0.406550i	0.227855	−	0.0157653i
$666$		0			0
$667$			8.14990i			0.315565i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	24.5436i		−	0.947496i
$672$		0			0
$673$			2.92824i			0.112875i	0.998406	+	0.0564377i	$0.0179742\pi$
$673$			2.92824i			0.112875i	−0.998406	+	0.0564377i	$0.982026\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.0905			0.656842			0.328421	−	0.944531i	$-0.393483\pi$
$677$	17.0905			0.656842			0.328421	+	0.944531i	$0.393483\pi$
$678$		0			0
$679$		−	0.675386i		−	0.0259190i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.0793			1.34227			0.671136	−	0.741334i	$-0.265806\pi$
$683$	35.0793			1.34227			0.671136	+	0.741334i	$0.265806\pi$
$684$		0			0
$685$	−24.1993			−0.924609
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	36.7176i		−	1.39883i
$690$		0			0
$691$	19.1612			0.728925			0.364462	−	0.931218i	$-0.381253\pi$
$691$	19.1612			0.728925			0.364462	+	0.931218i	$0.381253\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			35.5117i			1.34704i
$696$		0			0
$697$		−	1.88525i		−	0.0714091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			15.2828i			0.577223i	0.957446	+	0.288612i	$0.0931936\pi$
$701$			15.2828i			0.577223i	−0.957446	+	0.288612i	$0.906806\pi$
$702$		0			0
$703$	−31.7822	+	2.19902i	−1.19869	+	0.0829376i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5.31124i		−	0.199750i
$708$		0			0
$709$	−16.5648			−0.622102			−0.311051	−	0.950393i	$-0.600681\pi$
$709$	−16.5648			−0.622102			−0.311051	+	0.950393i	$0.600681\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−26.7091			−1.00026
$714$		0			0
$715$			26.9344i			1.00729i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	22.4545i		−	0.837410i	−0.908122	−	0.418705i	$-0.862484\pi$
$719$		−	22.4545i		−	0.837410i	0.908122	−	0.418705i	$-0.137516\pi$
$720$		0			0
$721$			0.227573i			0.00847524i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.5226			−0.539356
$726$		0			0
$727$	−26.0365			−0.965642			−0.482821	−	0.875719i	$-0.660388\pi$
$727$	−26.0365			−0.965642			−0.482821	+	0.875719i	$0.660388\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4.17966i		−	0.154590i
$732$		0			0
$733$	−3.89243			−0.143770			−0.0718851	−	0.997413i	$-0.522901\pi$
$733$	−3.89243			−0.143770			−0.0718851	+	0.997413i	$0.522901\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.7114			0.431394
$738$		0			0
$739$	32.5916			1.19890			0.599451	−	0.800412i	$-0.295386\pi$
$739$	32.5916			1.19890			0.599451	+	0.800412i	$0.295386\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−50.0449			−1.83597			−0.917985	−	0.396616i	$-0.870184\pi$
$743$	−50.0449			−1.83597			−0.917985	+	0.396616i	$0.870184\pi$
$744$		0			0
$745$	−12.0365			−0.440985
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.97138			0.108572
$750$		0			0
$751$		−	5.60677i		−	0.204594i	−0.994754	−	0.102297i	$-0.967381\pi$
$751$		−	5.60677i		−	0.204594i	0.994754	−	0.102297i	$-0.0326192\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−68.3977			−2.48925
$756$		0			0
$757$	4.28113			0.155600			0.0778001	−	0.996969i	$-0.475210\pi$
$757$	4.28113			0.155600			0.0778001	+	0.996969i	$0.475210\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	29.5451i		−	1.07101i	−0.844532	−	0.535504i	$-0.820122\pi$
$761$		−	29.5451i		−	1.07101i	0.844532	−	0.535504i	$-0.179878\pi$
$762$		0			0
$763$		−	0.902959i		−	0.0326893i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			21.3254i			0.770015i
$768$		0			0
$769$	15.0326			0.542089			0.271044	−	0.962567i	$-0.412631\pi$
$769$	15.0326			0.542089			0.271044	+	0.962567i	$0.412631\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−49.8445			−1.79278			−0.896391	−	0.443265i	$-0.853820\pi$
$773$	−49.8445			−1.79278			−0.896391	+	0.443265i	$0.853820\pi$
$774$		0			0
$775$		−	47.5939i		−	1.70962i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.14740	+	16.5832i	0.0411098	+	0.594156i
$780$		0			0
$781$			1.77334i			0.0634551i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	24.8242i		−	0.886014i
$786$		0			0
$787$			38.0557i			1.35654i	0.734812	+	0.678270i	$0.237270\pi$
$787$			38.0557i			1.35654i	−0.734812	+	0.678270i	$0.762730\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.57022			0.269166
$792$		0			0
$793$		−	35.3912i		−	1.25678i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−43.9118			−1.55544			−0.777718	−	0.628614i	$-0.783623\pi$
$797$	−43.9118			−1.55544			−0.777718	+	0.628614i	$0.783623\pi$
$798$		0			0
$799$	1.20620			0.0426722
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7.15343i			0.252439i
$804$		0			0
$805$	5.12273			0.180552
$806$		0			0
$807$		0			0
$808$		0			0
$809$			41.5045i			1.45922i	0.683863	+	0.729610i	$0.260298\pi$
$809$			41.5045i			1.45922i	−0.683863	+	0.729610i	$0.739702\pi$
$810$		0			0
$811$			30.1578i			1.05898i	0.848315	+	0.529491i	$0.177617\pi$
$811$			30.1578i			1.05898i	−0.848315	+	0.529491i	$0.822383\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			82.2775i			2.88206i
$816$		0			0
$817$	2.54381	+	36.7654i	0.0889967	+	1.28626i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	49.1525i		−	1.71543i	−0.514123	−	0.857716i	$-0.671883\pi$
$821$		−	49.1525i		−	1.71543i	0.514123	−	0.857716i	$-0.328117\pi$
$822$		0			0
$823$	−7.37415			−0.257047			−0.128523	−	0.991706i	$-0.541024\pi$
$823$	−7.37415			−0.257047			−0.128523	+	0.991706i	$0.541024\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	32.9149			1.14456			0.572281	−	0.820057i	$-0.306059\pi$
$827$	32.9149			1.14456			0.572281	+	0.820057i	$0.306059\pi$
$828$		0			0
$829$		−	12.7824i		−	0.443950i	−0.975052	−	0.221975i	$-0.928750\pi$
$829$		−	12.7824i		−	0.443950i	0.975052	−	0.221975i	$-0.0712504\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.38372i			0.117239i
$834$		0			0
$835$		−	63.8948i		−	2.21117i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−33.5872			−1.15956			−0.579779	−	0.814773i	$-0.696861\pi$
$839$	−33.5872			−1.15956			−0.579779	+	0.814773i	$0.696861\pi$
$840$		0			0
$841$	−24.3787			−0.840646
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	5.73381i		−	0.197249i
$846$		0			0
$847$	2.18809			0.0751836
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.7087			−0.949843
$852$		0			0
$853$	−21.1125			−0.722879			−0.361440	−	0.932395i	$-0.617715\pi$
$853$	−21.1125			−0.722879			−0.361440	+	0.932395i	$0.617715\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.3984			0.696797			0.348399	−	0.937346i	$-0.386726\pi$
$857$	20.3984			0.696797			0.348399	+	0.937346i	$0.386726\pi$
$858$		0			0
$859$	−37.3208			−1.27337			−0.636685	−	0.771124i	$-0.719695\pi$
$859$	−37.3208			−1.27337			−0.636685	+	0.771124i	$0.719695\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.6948			0.534256			0.267128	−	0.963661i	$-0.413925\pi$
$863$	15.6948			0.534256			0.267128	+	0.963661i	$0.413925\pi$
$864$		0			0
$865$			29.0951i			0.989262i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14.9516			−0.507199
$870$		0			0
$871$	16.8875			0.572210
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.37223i			0.0801961i
$876$		0			0
$877$			49.4556i			1.67000i	0.550251	+	0.834999i	$0.314532\pi$
$877$			49.4556i			1.67000i	−0.550251	+	0.834999i	$0.685468\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	13.1709i		−	0.443738i	−0.975077	−	0.221869i	$-0.928784\pi$
$881$		−	13.1709i		−	0.443738i	0.975077	−	0.221869i	$-0.0712157\pi$
$882$		0			0
$883$	−14.9728			−0.503875			−0.251938	−	0.967743i	$-0.581068\pi$
$883$	−14.9728			−0.503875			−0.251938	+	0.967743i	$0.581068\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.77637			−0.294682			−0.147341	−	0.989086i	$-0.547071\pi$
$887$	−8.77637			−0.294682			−0.147341	+	0.989086i	$0.547071\pi$
$888$		0			0
$889$			0.902959i			0.0302843i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.6101	+	0.734112i	−0.355052	+	0.0245661i
$894$		0			0
$895$			65.4732i			2.18853i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			15.1449i			0.505111i
$900$		0			0
$901$			5.39317i			0.179673i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	54.8781			1.82421
$906$		0			0
$907$			42.5145i			1.41167i	0.708376	+	0.705835i	$0.249428\pi$
$907$			42.5145i			1.41167i	−0.708376	+	0.705835i	$0.750572\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	54.3470			1.80060			0.900298	−	0.435275i	$-0.143349\pi$
$911$	54.3470			1.80060			0.900298	+	0.435275i	$0.143349\pi$
$912$		0			0
$913$	−3.59490			−0.118974
$914$		0			0
$915$		0			0
$916$		0			0
$917$			5.66583i			0.187102i
$918$		0			0
$919$	−26.5081			−0.874423			−0.437211	−	0.899359i	$-0.644034\pi$
$919$	−26.5081			−0.874423			−0.437211	+	0.899359i	$0.644034\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.55711i			0.0841682i
$924$		0			0
$925$		−	49.3752i		−	1.62345i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			16.2074i			0.531747i	0.964008	+	0.265873i	$0.0856603\pi$
$929$			16.2074i			0.531747i	−0.964008	+	0.265873i	$0.914340\pi$
$930$		0			0
$931$	−2.05939	−	29.7641i	−0.0674937	−	0.975480i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.95619i		−	0.129381i
$936$		0			0
$937$	25.2969			0.826414			0.413207	−	0.910637i	$-0.364409\pi$
$937$	25.2969			0.826414			0.413207	+	0.910637i	$0.364409\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−54.3923			−1.77314			−0.886569	−	0.462597i	$-0.846918\pi$
$941$	−54.3923			−1.77314			−0.886569	+	0.462597i	$0.846918\pi$
$942$		0			0
$943$			14.4578i			0.470810i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	31.9734i		−	1.03900i	−0.854472	−	0.519498i	$-0.826119\pi$
$947$		−	31.9734i		−	1.03900i	0.854472	−	0.519498i	$-0.173881\pi$
$948$		0			0
$949$			10.3150i			0.334840i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.6387			0.765732			0.382866	−	0.923804i	$-0.374937\pi$
$953$	23.6387			0.765732			0.382866	+	0.923804i	$0.374937\pi$
$954$		0			0
$955$	63.8877			2.06736
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2.78155i		−	0.0898210i
$960$		0			0
$961$	−18.6333			−0.601073
$962$		0			0
$963$		0			0
$964$		0			0
$965$	32.0108			1.03046
$966$		0			0
$967$	−9.41838			−0.302875			−0.151437	−	0.988467i	$-0.548390\pi$
$967$	−9.41838			−0.302875			−0.151437	+	0.988467i	$0.548390\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.7292			−0.793598			−0.396799	−	0.917906i	$-0.629879\pi$
$971$	−24.7292			−0.793598			−0.396799	+	0.917906i	$0.629879\pi$
$972$		0			0
$973$	−4.08183			−0.130858
$974$		0			0
$975$		0			0
$976$		0			0
$977$	17.3899			0.556354			0.278177	−	0.960530i	$-0.410270\pi$
$977$	17.3899			0.556354			0.278177	+	0.960530i	$0.410270\pi$
$978$		0			0
$979$			30.9163i			0.988088i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.8306			−0.919553			−0.459777	−	0.888035i	$-0.652070\pi$
$983$	−28.8306			−0.919553			−0.459777	+	0.888035i	$0.652070\pi$
$984$		0			0
$985$	53.8512			1.71584
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.0533i			1.01923i
$990$		0			0
$991$		−	29.4684i		−	0.936095i	−0.883703	−	0.468048i	$-0.844958\pi$
$991$		−	29.4684i		−	0.936095i	0.883703	−	0.468048i	$-0.155042\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	15.2737i		−	0.484209i
$996$		0			0
$997$	26.9307			0.852903			0.426451	−	0.904510i	$-0.359763\pi$
$997$	26.9307			0.852903			0.426451	+	0.904510i	$0.359763\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.f.g.1025.1		8
3.2	odd	2		2736.2.f.h.1025.8		8
4.3	odd	2		1368.2.f.c.1025.1	✓	8
12.11	even	2		1368.2.f.d.1025.8	yes	8
19.18	odd	2		2736.2.f.h.1025.1		8
57.56	even	2	inner	2736.2.f.g.1025.8		8
76.75	even	2		1368.2.f.d.1025.1	yes	8
228.227	odd	2		1368.2.f.c.1025.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.f.c.1025.1	✓	8	4.3	odd	2
1368.2.f.c.1025.8	yes	8	228.227	odd	2
1368.2.f.d.1025.1	yes	8	76.75	even	2
1368.2.f.d.1025.8	yes	8	12.11	even	2
2736.2.f.g.1025.1		8	1.1	even	1	trivial
2736.2.f.g.1025.8		8	57.56	even	2	inner
2736.2.f.h.1025.1		8	19.18	odd	2
2736.2.f.h.1025.8		8	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.42865i q^{5} +0.394100 q^{7} +2.33407i q^{11} +3.36566i q^{13} -0.494357i q^{17} +(0.300874 + 4.34850i) q^{19} +3.79116i q^{23} -6.75561 q^{25} +2.14971 q^{29} +7.04509i q^{31} -1.35123i q^{35} +7.30877i q^{37} +3.81355 q^{41} +8.45474 q^{43} +2.43993i q^{47} -6.84469 q^{49} -10.9095 q^{53} +8.00270 q^{55} +6.33616 q^{59} -10.5154 q^{61} +11.5397 q^{65} -5.01758i q^{67} +0.759763 q^{71} +3.06479 q^{73} +0.919857i q^{77} +6.40581i q^{79} +1.54018i q^{83} -1.69497 q^{85} +13.2456 q^{89} +1.32641i q^{91} +(14.9095 - 1.03159i) q^{95} -1.71374i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) \) 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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