LMFDB - Embedded newform 1856.2.e.f.1217.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1856,2,Mod(1217,1856)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1856.1217"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,6,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.2
Root			$2.23607i$ of defining polynomial
Character	$\chi$	$=$	1856.1217
Dual form			1856.2.e.f.1217.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{3} +3.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} -2.23607i q^{11} +1.00000 q^{13} +6.70820i q^{15} +4.47214i q^{17} -4.47214i q^{21} -6.00000 q^{23} +4.00000 q^{25} +2.23607i q^{27} +(3.00000 + 4.47214i) q^{29} +6.70820i q^{31} +5.00000 q^{33} -6.00000 q^{35} +2.23607i q^{39} +4.47214i q^{41} +6.70820i q^{43} -6.00000 q^{45} +2.23607i q^{47} -3.00000 q^{49} -10.0000 q^{51} +9.00000 q^{53} -6.70820i q^{55} +6.00000 q^{59} +13.4164i q^{61} +4.00000 q^{63} +3.00000 q^{65} +8.00000 q^{67} -13.4164i q^{69} +8.94427i q^{75} +4.47214i q^{77} -6.70820i q^{79} -11.0000 q^{81} -6.00000 q^{83} +13.4164i q^{85} +(-10.0000 + 6.70820i) q^{87} +4.47214i q^{89} -2.00000 q^{91} -15.0000 q^{93} -13.4164i q^{97} +4.47214i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} - 4 q^{7} - 4 q^{9} + 2 q^{13} - 12 q^{23} + 8 q^{25} + 6 q^{29} + 10 q^{33} - 12 q^{35} - 12 q^{45} - 6 q^{49} - 20 q^{51} + 18 q^{53} + 12 q^{59} + 8 q^{63} + 6 q^{65} + 16 q^{67} - 22 q^{81}+ \cdots - 30 q^{93}+O(q^{100}) $ 2 * q + 6 * q^5 - 4 * q^7 - 4 * q^9 + 2 * q^13 - 12 * q^23 + 8 * q^25 + 6 * q^29 + 10 * q^33 - 12 * q^35 - 12 * q^45 - 6 * q^49 - 20 * q^51 + 18 * q^53 + 12 * q^59 + 8 * q^63 + 6 * q^65 + 16 * q^67 - 22 * q^81 - 12 * q^83 - 20 * q^87 - 4 * q^91 - 30 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$639$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			2.23607i			1.29099i	0.763763	+	0.645497i	$0.223350\pi$
$3$			2.23607i			1.29099i	−0.763763	+	0.645497i	$0.776650\pi$
$4$		0			0
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$			6.70820i			1.73205i
$16$		0			0
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		−	4.47214i		−	0.975900i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$			2.23607i			0.430331i
$28$		0			0
$29$	3.00000	+	4.47214i	0.557086	+	0.830455i
$29$	3.00000	+	4.47214i	0.557086	+	0.830455i
$30$		0			0
$31$			6.70820i			1.20483i	0.798183	+	0.602414i	$0.205795\pi$
$31$			6.70820i			1.20483i	−0.798183	+	0.602414i	$0.794205\pi$
$32$		0			0
$33$	5.00000			0.870388
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$			2.23607i			0.358057i
$40$		0			0
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$			6.70820i			1.02299i	0.859286	+	0.511496i	$0.170908\pi$
$43$			6.70820i			1.02299i	−0.859286	+	0.511496i	$0.829092\pi$
$44$		0			0
$45$	−6.00000			−0.894427
$46$		0			0
$47$			2.23607i			0.326164i	0.986613	+	0.163082i	$0.0521435\pi$
$47$			2.23607i			0.326164i	−0.986613	+	0.163082i	$0.947856\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$	−10.0000			−1.40028
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$		−	6.70820i		−	0.904534i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	4.00000			0.503953
$64$		0			0
$65$	3.00000			0.372104
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$		−	13.4164i		−	1.61515i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			8.94427i			1.03280i
$76$		0			0
$77$			4.47214i			0.509647i
$78$		0			0
$79$		−	6.70820i		−	0.754732i	−0.926064	−	0.377366i	$-0.876830\pi$
$79$		−	6.70820i		−	0.754732i	0.926064	−	0.377366i	$-0.123170\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$			13.4164i			1.45521i
$86$		0			0
$87$	−10.0000	+	6.70820i	−1.07211	+	0.719195i
$88$		0			0
$89$			4.47214i			0.474045i	0.971504	+	0.237023i	$0.0761716\pi$
$89$			4.47214i			0.474045i	−0.971504	+	0.237023i	$0.923828\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$	−15.0000			−1.55543
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	13.4164i		−	1.36223i	−0.732177	−	0.681115i	$-0.761495\pi$
$97$		−	13.4164i		−	1.36223i	0.732177	−	0.681115i	$-0.238505\pi$
$98$		0			0
$99$			4.47214i			0.449467i
$100$		0			0
$101$		−	17.8885i		−	1.77998i	−0.455983	−	0.889988i	$-0.650712\pi$
$101$		−	17.8885i		−	1.77998i	0.455983	−	0.889988i	$-0.349288\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		−	13.4164i		−	1.30931i
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.94427i		−	0.841406i	−0.907198	−	0.420703i	$-0.861783\pi$
$113$		−	8.94427i		−	0.841406i	0.907198	−	0.420703i	$-0.138217\pi$
$114$		0			0
$115$	−18.0000			−1.67851
$116$		0			0
$117$	−2.00000			−0.184900
$118$		0			0
$119$		−	8.94427i		−	0.819920i
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$	−10.0000			−0.901670
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−15.0000			−1.32068
$130$		0			0
$131$		−	8.94427i		−	0.781465i	−0.920504	−	0.390732i	$-0.872222\pi$
$131$		−	8.94427i		−	0.781465i	0.920504	−	0.390732i	$-0.127778\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			6.70820i			0.577350i
$136$		0			0
$137$		−	8.94427i		−	0.764161i	−0.924129	−	0.382080i	$-0.875208\pi$
$137$		−	8.94427i		−	0.764161i	0.924129	−	0.382080i	$-0.124792\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−5.00000			−0.421076
$142$		0			0
$143$		−	2.23607i		−	0.186989i
$144$		0			0
$145$	9.00000	+	13.4164i	0.747409	+	1.11417i
$146$		0			0
$147$		−	6.70820i		−	0.553283i
$148$		0			0
$149$	−15.0000			−1.22885			−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000			−1.22885			−0.614424	+	0.788976i	$0.710612\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		−	8.94427i		−	0.723102i
$154$		0			0
$155$			20.1246i			1.61645i
$156$		0			0
$157$		−	13.4164i		−	1.07075i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$		−	13.4164i		−	1.07075i	0.844616	−	0.535373i	$-0.179829\pi$
$158$		0			0
$159$			20.1246i			1.59599i
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$			20.1246i			1.57628i	0.615495	+	0.788141i	$0.288956\pi$
$163$			20.1246i			1.57628i	−0.615495	+	0.788141i	$0.711044\pi$
$164$		0			0
$165$	15.0000			1.16775
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	−8.00000			−0.604743
$176$		0			0
$177$			13.4164i			1.00844i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$	−30.0000			−2.21766
$184$		0			0
$185$		0			0
$186$		0			0
$187$	10.0000			0.731272
$188$		0			0
$189$		−	4.47214i		−	0.325300i
$190$		0			0
$191$			8.94427i			0.647185i	0.946197	+	0.323592i	$0.104891\pi$
$191$			8.94427i			0.647185i	−0.946197	+	0.323592i	$0.895109\pi$
$192$		0			0
$193$			13.4164i			0.965734i	0.875694	+	0.482867i	$0.160405\pi$
$193$			13.4164i			0.965734i	−0.875694	+	0.482867i	$0.839595\pi$
$194$		0			0
$195$			6.70820i			0.480384i
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$			17.8885i			1.26176i
$202$		0			0
$203$	−6.00000	−	8.94427i	−0.421117	−	0.627765i
$204$		0			0
$205$			13.4164i			0.937043i
$206$		0			0
$207$	12.0000			0.834058
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	6.70820i		−	0.461812i	−0.972976	−	0.230906i	$-0.925831\pi$
$211$		−	6.70820i		−	0.461812i	0.972976	−	0.230906i	$-0.0741690\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			20.1246i			1.37249i
$216$		0			0
$217$		−	13.4164i		−	0.910765i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.47214i			0.300828i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$		−	13.4164i		−	0.886581i	−0.896378	−	0.443291i	$-0.853811\pi$
$229$		−	13.4164i		−	0.886581i	0.896378	−	0.443291i	$-0.146189\pi$
$230$		0			0
$231$	−10.0000			−0.657952
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$			6.70820i			0.437595i
$236$		0			0
$237$	15.0000			0.974355
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	−25.0000			−1.61039			−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000			−1.61039			−0.805196	+	0.593009i	$0.797940\pi$
$242$		0			0
$243$		−	17.8885i		−	1.14755i
$244$		0			0
$245$	−9.00000			−0.574989
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	13.4164i		−	0.850230i
$250$		0			0
$251$		−	15.6525i		−	0.987976i	−0.869469	−	0.493988i	$-0.835539\pi$
$251$		−	15.6525i		−	0.987976i	0.869469	−	0.493988i	$-0.164461\pi$
$252$		0			0
$253$			13.4164i			0.843482i
$254$		0			0
$255$	−30.0000			−1.87867
$256$		0			0
$257$	−3.00000			−0.187135			−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000			−0.187135			−0.0935674	+	0.995613i	$0.529827\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000	−	8.94427i	−0.371391	−	0.553637i
$262$		0			0
$263$		−	11.1803i		−	0.689409i	−0.938711	−	0.344705i	$-0.887979\pi$
$263$		−	11.1803i		−	0.689409i	0.938711	−	0.344705i	$-0.112021\pi$
$264$		0			0
$265$	27.0000			1.65860
$266$		0			0
$267$	−10.0000			−0.611990
$268$		0			0
$269$		−	4.47214i		−	0.272671i	−0.990663	−	0.136335i	$-0.956467\pi$
$269$		−	4.47214i		−	0.272671i	0.990663	−	0.136335i	$-0.0435325\pi$
$270$		0			0
$271$		−	20.1246i		−	1.22248i	−0.791444	−	0.611242i	$-0.790670\pi$
$271$		−	20.1246i		−	1.22248i	0.791444	−	0.611242i	$-0.209330\pi$
$272$		0			0
$273$		−	4.47214i		−	0.270666i
$274$		0			0
$275$		−	8.94427i		−	0.539360i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$		−	13.4164i		−	0.803219i
$280$		0			0
$281$	3.00000			0.178965			0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000			0.178965			0.0894825	+	0.995988i	$0.471479\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.94427i		−	0.527964i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	30.0000			1.75863
$292$		0			0
$293$			8.94427i			0.522530i	0.965267	+	0.261265i	$0.0841396\pi$
$293$			8.94427i			0.522530i	−0.965267	+	0.261265i	$0.915860\pi$
$294$		0			0
$295$	18.0000			1.04800
$296$		0			0
$297$	5.00000			0.290129
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$		−	13.4164i		−	0.773309i
$302$		0			0
$303$	40.0000			2.29794
$304$		0			0
$305$			40.2492i			2.30466i
$306$		0			0
$307$		−	20.1246i		−	1.14857i	−0.818655	−	0.574286i	$-0.805280\pi$
$307$		−	20.1246i		−	1.14857i	0.818655	−	0.574286i	$-0.194720\pi$
$308$		0			0
$309$			8.94427i			0.508822i
$310$		0			0
$311$			8.94427i			0.507183i	0.967311	+	0.253592i	$0.0816119\pi$
$311$			8.94427i			0.507183i	−0.967311	+	0.253592i	$0.918388\pi$
$312$		0			0
$313$	29.0000			1.63918			0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000			1.63918			0.819588	+	0.572953i	$0.194202\pi$
$314$		0			0
$315$	12.0000			0.676123
$316$		0			0
$317$			22.3607i			1.25590i	0.778253	+	0.627950i	$0.216106\pi$
$317$			22.3607i			1.25590i	−0.778253	+	0.627950i	$0.783894\pi$
$318$		0			0
$319$	10.0000	−	6.70820i	0.559893	−	0.375587i
$320$		0			0
$321$			40.2492i			2.24649i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		−	11.1803i		−	0.618274i
$328$		0			0
$329$		−	4.47214i		−	0.246557i
$330$		0			0
$331$		−	33.5410i		−	1.84358i	−0.387688	−	0.921791i	$-0.626726\pi$
$331$		−	33.5410i		−	1.84358i	0.387688	−	0.921791i	$-0.373274\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000			1.31126
$336$		0			0
$337$			13.4164i			0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$337$			13.4164i			0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$338$		0			0
$339$	20.0000			1.08625
$340$		0			0
$341$	15.0000			0.812296
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		−	40.2492i		−	2.16695i
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$	1.00000			0.0535288			0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000			0.0535288			0.0267644	+	0.999642i	$0.491480\pi$
$350$		0			0
$351$			2.23607i			0.119352i
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	20.0000			1.05851
$358$		0			0
$359$			2.23607i			0.118015i	0.998258	+	0.0590076i	$0.0187936\pi$
$359$			2.23607i			0.118015i	−0.998258	+	0.0590076i	$0.981206\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$			13.4164i			0.704179i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	26.8328i		−	1.40066i	−0.713818	−	0.700331i	$-0.753036\pi$
$367$		−	26.8328i		−	1.40066i	0.713818	−	0.700331i	$-0.246964\pi$
$368$		0			0
$369$		−	8.94427i		−	0.465620i
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$	31.0000			1.60512			0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000			1.60512			0.802560	+	0.596572i	$0.203471\pi$
$374$		0			0
$375$		−	6.70820i		−	0.346410i
$376$		0			0
$377$	3.00000	+	4.47214i	0.154508	+	0.230327i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000			−0.306586			−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000			−0.306586			−0.153293	+	0.988181i	$0.548988\pi$
$384$		0			0
$385$			13.4164i			0.683763i
$386$		0			0
$387$		−	13.4164i		−	0.681994i
$388$		0			0
$389$		−	17.8885i		−	0.906985i	−0.891260	−	0.453493i	$-0.850178\pi$
$389$		−	17.8885i		−	0.906985i	0.891260	−	0.453493i	$-0.149822\pi$
$390$		0			0
$391$		−	26.8328i		−	1.35699i
$392$		0			0
$393$	20.0000			1.00887
$394$		0			0
$395$		−	20.1246i		−	1.01258i
$396$		0			0
$397$	7.00000			0.351320			0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000			0.351320			0.175660	+	0.984451i	$0.443794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000			0.749064			0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000			0.749064			0.374532	+	0.927214i	$0.377803\pi$
$402$		0			0
$403$			6.70820i			0.334159i
$404$		0			0
$405$	−33.0000			−1.63978
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	26.8328i		−	1.32680i	−0.748266	−	0.663399i	$-0.769113\pi$
$409$		−	26.8328i		−	1.32680i	0.748266	−	0.663399i	$-0.230887\pi$
$410$		0			0
$411$	20.0000			0.986527
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	−18.0000			−0.883585
$416$		0			0
$417$		−	22.3607i		−	1.09501i
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$			13.4164i			0.653876i	0.945046	+	0.326938i	$0.106017\pi$
$421$			13.4164i			0.653876i	−0.945046	+	0.326938i	$0.893983\pi$
$422$		0			0
$423$		−	4.47214i		−	0.217443i
$424$		0			0
$425$			17.8885i			0.867722i
$426$		0			0
$427$		−	26.8328i		−	1.29853i
$428$		0			0
$429$	5.00000			0.241402
$430$		0			0
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−30.0000	+	20.1246i	−1.43839	+	0.964901i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	34.0000			1.62273			0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000			1.62273			0.811366	+	0.584539i	$0.198725\pi$
$440$		0			0
$441$	6.00000			0.285714
$442$		0			0
$443$			17.8885i			0.849910i	0.905214	+	0.424955i	$0.139710\pi$
$443$			17.8885i			0.849910i	−0.905214	+	0.424955i	$0.860290\pi$
$444$		0			0
$445$			13.4164i			0.635999i
$446$		0			0
$447$		−	33.5410i		−	1.58644i
$448$		0			0
$449$		−	35.7771i		−	1.68843i	−0.536009	−	0.844213i	$-0.680069\pi$
$449$		−	35.7771i		−	1.68843i	0.536009	−	0.844213i	$-0.319931\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		0			0
$453$			22.3607i			1.05060i
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$	2.00000			0.0935561			0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000			0.0935561			0.0467780	+	0.998905i	$0.485105\pi$
$458$		0			0
$459$	−10.0000			−0.466760
$460$		0			0
$461$			8.94427i			0.416576i	0.978068	+	0.208288i	$0.0667892\pi$
$461$			8.94427i			0.416576i	−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	−26.0000			−1.20832			−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000			−1.20832			−0.604161	+	0.796862i	$0.706492\pi$
$464$		0			0
$465$	−45.0000			−2.08683
$466$		0			0
$467$		−	2.23607i		−	0.103473i	−0.998661	−	0.0517364i	$-0.983524\pi$
$467$		−	2.23607i		−	0.103473i	0.998661	−	0.0517364i	$-0.0164756\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	30.0000			1.38233
$472$		0			0
$473$	15.0000			0.689701
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−18.0000			−0.824163
$478$		0			0
$479$		−	11.1803i		−	0.510843i	−0.966830	−	0.255421i	$-0.917786\pi$
$479$		−	11.1803i		−	0.510843i	0.966830	−	0.255421i	$-0.0822142\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			26.8328i			1.22094i
$484$		0			0
$485$		−	40.2492i		−	1.82762i
$486$		0			0
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$	−45.0000			−2.03497
$490$		0			0
$491$			24.5967i			1.11004i	0.831838	+	0.555018i	$0.187289\pi$
$491$			24.5967i			1.11004i	−0.831838	+	0.555018i	$0.812711\pi$
$492$		0			0
$493$	−20.0000	+	13.4164i	−0.900755	+	0.604245i
$494$		0			0
$495$			13.4164i			0.603023i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$			26.8328i			1.19880i
$502$		0			0
$503$			15.6525i			0.697909i	0.937140	+	0.348955i	$0.113463\pi$
$503$			15.6525i			0.697909i	−0.937140	+	0.348955i	$0.886537\pi$
$504$		0			0
$505$		−	53.6656i		−	2.38809i
$506$		0			0
$507$		−	26.8328i		−	1.19169i
$508$		0			0
$509$	−15.0000			−0.664863			−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000			−0.664863			−0.332432	+	0.943127i	$0.607869\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000			0.528783
$516$		0			0
$517$	5.00000			0.219900
$518$		0			0
$519$			13.4164i			0.588915i
$520$		0			0
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$		−	17.8885i		−	0.780720i
$526$		0			0
$527$	−30.0000			−1.30682
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$			4.47214i			0.193710i
$534$		0			0
$535$	54.0000			2.33462
$536$		0			0
$537$		0			0
$538$		0			0
$539$			6.70820i			0.288943i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	11.1803i		−	0.479794i
$544$		0			0
$545$	−15.0000			−0.642529
$546$		0			0
$547$	38.0000			1.62476			0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000			1.62476			0.812381	+	0.583127i	$0.198171\pi$
$548$		0			0
$549$		−	26.8328i		−	1.14520i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			13.4164i			0.570524i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$			6.70820i			0.283727i
$560$		0			0
$561$			22.3607i			0.944069i
$562$		0			0
$563$			11.1803i			0.471195i	0.971851	+	0.235598i	$0.0757047\pi$
$563$			11.1803i			0.471195i	−0.971851	+	0.235598i	$0.924295\pi$
$564$		0			0
$565$		−	26.8328i		−	1.12887i
$566$		0			0
$567$	22.0000			0.923913
$568$		0			0
$569$			17.8885i			0.749927i	0.927040	+	0.374963i	$0.122345\pi$
$569$			17.8885i			0.749927i	−0.927040	+	0.374963i	$0.877655\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	−20.0000			−0.835512
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$			40.2492i			1.67560i	0.545979	+	0.837799i	$0.316158\pi$
$577$			40.2492i			1.67560i	−0.545979	+	0.837799i	$0.683842\pi$
$578$		0			0
$579$	−30.0000			−1.24676
$580$		0			0
$581$	12.0000			0.497844
$582$		0			0
$583$		−	20.1246i		−	0.833476i
$584$		0			0
$585$	−6.00000			−0.248069
$586$		0			0
$587$	−42.0000			−1.73353			−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000			−1.73353			−0.866763	+	0.498721i	$0.833803\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			40.2492i			1.65563i
$592$		0			0
$593$	9.00000			0.369586			0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000			0.369586			0.184793	+	0.982777i	$0.440839\pi$
$594$		0			0
$595$		−	26.8328i		−	1.10004i
$596$		0			0
$597$		−	31.3050i		−	1.28123i
$598$		0			0
$599$			29.0689i			1.18772i	0.804568	+	0.593861i	$0.202397\pi$
$599$			29.0689i			1.18772i	−0.804568	+	0.593861i	$0.797603\pi$
$600$		0			0
$601$		−	13.4164i		−	0.547267i	−0.961834	−	0.273633i	$-0.911775\pi$
$601$		−	13.4164i		−	0.547267i	0.961834	−	0.273633i	$-0.0882255\pi$
$602$		0			0
$603$	−16.0000			−0.651570
$604$		0			0
$605$	18.0000			0.731804
$606$		0			0
$607$			6.70820i			0.272278i	0.990690	+	0.136139i	$0.0434693\pi$
$607$			6.70820i			0.272278i	−0.990690	+	0.136139i	$0.956531\pi$
$608$		0			0
$609$	20.0000	−	13.4164i	0.810441	−	0.543660i
$610$		0			0
$611$			2.23607i			0.0904616i
$612$		0			0
$613$	31.0000			1.25208			0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000			1.25208			0.626039	+	0.779792i	$0.284675\pi$
$614$		0			0
$615$	−30.0000			−1.20972
$616$		0			0
$617$			44.7214i			1.80041i	0.435462	+	0.900207i	$0.356585\pi$
$617$			44.7214i			1.80041i	−0.435462	+	0.900207i	$0.643415\pi$
$618$		0			0
$619$		−	33.5410i		−	1.34813i	−0.738673	−	0.674064i	$-0.764547\pi$
$619$		−	33.5410i		−	1.34813i	0.738673	−	0.674064i	$-0.235453\pi$
$620$		0			0
$621$		−	13.4164i		−	0.538382i
$622$		0			0
$623$		−	8.94427i		−	0.358345i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$	15.0000			0.596196
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.00000			−0.118864
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	22.3607i		−	0.883194i	−0.897214	−	0.441597i	$-0.854412\pi$
$641$		−	22.3607i		−	0.883194i	0.897214	−	0.441597i	$-0.145588\pi$
$642$		0			0
$643$	26.0000			1.02534			0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000			1.02534			0.512670	+	0.858586i	$0.328656\pi$
$644$		0			0
$645$	−45.0000			−1.77187
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$		0			0
$649$		−	13.4164i		−	0.526640i
$650$		0			0
$651$	30.0000			1.17579
$652$		0			0
$653$			49.1935i			1.92509i	0.271122	+	0.962545i	$0.412605\pi$
$653$			49.1935i			1.92509i	−0.271122	+	0.962545i	$0.587395\pi$
$654$		0			0
$655$		−	26.8328i		−	1.04844i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			38.0132i			1.48078i	0.672176	+	0.740391i	$0.265360\pi$
$659$			38.0132i			1.48078i	−0.672176	+	0.740391i	$0.734640\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$	−10.0000			−0.388368
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.0000	−	26.8328i	−0.696963	−	1.03897i
$668$		0			0
$669$			35.7771i			1.38322i
$670$		0			0
$671$	30.0000			1.15814
$672$		0			0
$673$	11.0000			0.424019			0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000			0.424019			0.212009	+	0.977268i	$0.431999\pi$
$674$		0			0
$675$			8.94427i			0.344265i
$676$		0			0
$677$		−	17.8885i		−	0.687513i	−0.939059	−	0.343756i	$-0.888301\pi$
$677$		−	17.8885i		−	0.687513i	0.939059	−	0.343756i	$-0.111699\pi$
$678$		0			0
$679$			26.8328i			1.02975i
$680$		0			0
$681$			26.8328i			1.02824i
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$		−	26.8328i		−	1.02523i
$686$		0			0
$687$	30.0000			1.14457
$688$		0			0
$689$	9.00000			0.342873
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$		−	8.94427i		−	0.339765i
$694$		0			0
$695$	−30.0000			−1.13796
$696$		0			0
$697$	−20.0000			−0.757554
$698$		0			0
$699$		−	20.1246i		−	0.761183i
$700$		0			0
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−15.0000			−0.564933
$706$		0			0
$707$			35.7771i			1.34554i
$708$		0			0
$709$	−35.0000			−1.31445			−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000			−1.31445			−0.657226	+	0.753693i	$0.728270\pi$
$710$		0			0
$711$			13.4164i			0.503155i
$712$		0			0
$713$		−	40.2492i		−	1.50735i
$714$		0			0
$715$		−	6.70820i		−	0.250873i
$716$		0			0
$717$		−	13.4164i		−	0.501045i
$718$		0			0
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		−	55.9017i		−	2.07901i
$724$		0			0
$725$	12.0000	+	17.8885i	0.445669	+	0.664364i
$726$		0			0
$727$		−	26.8328i		−	0.995174i	−0.867414	−	0.497587i	$-0.834220\pi$
$727$		−	26.8328i		−	0.995174i	0.867414	−	0.497587i	$-0.165780\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$	−30.0000			−1.10959
$732$		0			0
$733$			26.8328i			0.991093i	0.868582	+	0.495546i	$0.165032\pi$
$733$			26.8328i			0.991093i	−0.868582	+	0.495546i	$0.834968\pi$
$734$		0			0
$735$		−	20.1246i		−	0.742307i
$736$		0			0
$737$		−	17.8885i		−	0.658933i
$738$		0			0
$739$			20.1246i			0.740296i	0.928973	+	0.370148i	$0.120693\pi$
$739$			20.1246i			0.740296i	−0.928973	+	0.370148i	$0.879307\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.94427i			0.328134i	0.986449	+	0.164067i	$0.0524612\pi$
$743$			8.94427i			0.328134i	−0.986449	+	0.164067i	$0.947539\pi$
$744$		0			0
$745$	−45.0000			−1.64867
$746$		0			0
$747$	12.0000			0.439057
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$			26.8328i			0.979143i	0.871963	+	0.489572i	$0.162847\pi$
$751$			26.8328i			0.979143i	−0.871963	+	0.489572i	$0.837153\pi$
$752$		0			0
$753$	35.0000			1.27547
$754$		0			0
$755$	30.0000			1.09181
$756$		0			0
$757$		−	40.2492i		−	1.46288i	−0.681904	−	0.731441i	$-0.738848\pi$
$757$		−	40.2492i		−	1.46288i	0.681904	−	0.731441i	$-0.261152\pi$
$758$		0			0
$759$	−30.0000			−1.08893
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$	10.0000			0.362024
$764$		0			0
$765$		−	26.8328i		−	0.970143i
$766$		0			0
$767$	6.00000			0.216647
$768$		0			0
$769$			13.4164i			0.483808i	0.970300	+	0.241904i	$0.0777719\pi$
$769$			13.4164i			0.483808i	−0.970300	+	0.241904i	$0.922228\pi$
$770$		0			0
$771$		−	6.70820i		−	0.241590i
$772$		0			0
$773$			22.3607i			0.804258i	0.915583	+	0.402129i	$0.131730\pi$
$773$			22.3607i			0.804258i	−0.915583	+	0.402129i	$0.868270\pi$
$774$		0			0
$775$			26.8328i			0.963863i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−10.0000	+	6.70820i	−0.357371	+	0.239732i
$784$		0			0
$785$		−	40.2492i		−	1.43656i
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$		0			0
$789$	25.0000			0.890024
$790$		0			0
$791$			17.8885i			0.636043i
$792$		0			0
$793$			13.4164i			0.476431i
$794$		0			0
$795$			60.3738i			2.14124i
$796$		0			0
$797$		−	4.47214i		−	0.158411i	−0.996858	−	0.0792056i	$-0.974762\pi$
$797$		−	4.47214i		−	0.158411i	0.996858	−	0.0792056i	$-0.0252384\pi$
$798$		0			0
$799$	−10.0000			−0.353775
$800$		0			0
$801$		−	8.94427i		−	0.316030i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	36.0000			1.26883
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$			44.7214i			1.57232i	0.618023	+	0.786160i	$0.287934\pi$
$809$			44.7214i			1.57232i	−0.618023	+	0.786160i	$0.712066\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$	45.0000			1.57822
$814$		0			0
$815$			60.3738i			2.11480i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	4.00000			0.139771
$820$		0			0
$821$	33.0000			1.15171			0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000			1.15171			0.575854	+	0.817553i	$0.304670\pi$
$822$		0			0
$823$		−	53.6656i		−	1.87067i	−0.353768	−	0.935333i	$-0.615100\pi$
$823$		−	53.6656i		−	1.87067i	0.353768	−	0.935333i	$-0.384900\pi$
$824$		0			0
$825$	20.0000			0.696311
$826$		0			0
$827$			38.0132i			1.32185i	0.750453	+	0.660923i	$0.229835\pi$
$827$			38.0132i			1.32185i	−0.750453	+	0.660923i	$0.770165\pi$
$828$		0			0
$829$		−	40.2492i		−	1.39791i	−0.715164	−	0.698957i	$-0.753648\pi$
$829$		−	40.2492i		−	1.39791i	0.715164	−	0.698957i	$-0.246352\pi$
$830$		0			0
$831$		−	4.47214i		−	0.155137i
$832$		0			0
$833$		−	13.4164i		−	0.464851i
$834$		0			0
$835$	36.0000			1.24583
$836$		0			0
$837$	−15.0000			−0.518476
$838$		0			0
$839$			2.23607i			0.0771976i	0.999255	+	0.0385988i	$0.0122894\pi$
$839$			2.23607i			0.0771976i	−0.999255	+	0.0385988i	$0.987711\pi$
$840$		0			0
$841$	−11.0000	+	26.8328i	−0.379310	+	0.925270i
$842$		0			0
$843$			6.70820i			0.231043i
$844$		0			0
$845$	−36.0000			−1.23844
$846$		0			0
$847$	−12.0000			−0.412325
$848$		0			0
$849$			31.3050i			1.07438i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	26.8328i		−	0.918738i	−0.888246	−	0.459369i	$-0.848076\pi$
$853$		−	26.8328i		−	0.918738i	0.888246	−	0.459369i	$-0.151924\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	57.0000			1.94708			0.973541	−	0.228510i	$-0.0733855\pi$
$857$	57.0000			1.94708			0.973541	+	0.228510i	$0.0733855\pi$
$858$		0			0
$859$		−	46.9574i		−	1.60217i	−0.598553	−	0.801083i	$-0.704257\pi$
$859$		−	46.9574i		−	1.60217i	0.598553	−	0.801083i	$-0.295743\pi$
$860$		0			0
$861$	20.0000			0.681598
$862$		0			0
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$	18.0000			0.612018
$866$		0			0
$867$		−	6.70820i		−	0.227823i
$868$		0			0
$869$	−15.0000			−0.508840
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$			26.8328i			0.908153i
$874$		0			0
$875$	6.00000			0.202837
$876$		0			0
$877$	13.0000			0.438979			0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000			0.438979			0.219489	+	0.975615i	$0.429561\pi$
$878$		0			0
$879$	−20.0000			−0.674583
$880$		0			0
$881$		−	35.7771i		−	1.20536i	−0.797983	−	0.602680i	$-0.794099\pi$
$881$		−	35.7771i		−	1.20536i	0.797983	−	0.602680i	$-0.205901\pi$
$882$		0			0
$883$	−34.0000			−1.14419			−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000			−1.14419			−0.572096	+	0.820187i	$0.693869\pi$
$884$		0			0
$885$			40.2492i			1.35296i
$886$		0			0
$887$		−	11.1803i		−	0.375399i	−0.982226	−	0.187700i	$-0.939897\pi$
$887$		−	11.1803i		−	0.375399i	0.982226	−	0.187700i	$-0.0601031\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			24.5967i			0.824022i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	13.4164i		−	0.447961i
$898$		0			0
$899$	−30.0000	+	20.1246i	−1.00056	+	0.671193i
$900$		0			0
$901$			40.2492i			1.34090i
$902$		0			0
$903$	30.0000			0.998337
$904$		0			0
$905$	−15.0000			−0.498617
$906$		0			0
$907$		−	53.6656i		−	1.78194i	−0.454064	−	0.890969i	$-0.650026\pi$
$907$		−	53.6656i		−	1.78194i	0.454064	−	0.890969i	$-0.349974\pi$
$908$		0			0
$909$			35.7771i			1.18665i
$910$		0			0
$911$		−	38.0132i		−	1.25943i	−0.776825	−	0.629716i	$-0.783171\pi$
$911$		−	38.0132i		−	1.25943i	0.776825	−	0.629716i	$-0.216829\pi$
$912$		0			0
$913$			13.4164i			0.444018i
$914$		0			0
$915$	−90.0000			−2.97531
$916$		0			0
$917$			17.8885i			0.590732i
$918$		0			0
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$		0			0
$921$	45.0000			1.48280
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−8.00000			−0.262754
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−20.0000			−0.654771
$934$		0			0
$935$	30.0000			0.981105
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$			64.8460i			2.11617i
$940$		0			0
$941$	3.00000			0.0977972			0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000			0.0977972			0.0488986	+	0.998804i	$0.484429\pi$
$942$		0			0
$943$		−	26.8328i		−	0.873797i
$944$		0			0
$945$		−	13.4164i		−	0.436436i
$946$		0			0
$947$		−	2.23607i		−	0.0726624i	−0.999340	−	0.0363312i	$-0.988433\pi$
$947$		−	2.23607i		−	0.0726624i	0.999340	−	0.0363312i	$-0.0115671\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−50.0000			−1.62136
$952$		0			0
$953$	−9.00000			−0.291539			−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000			−0.291539			−0.145769	+	0.989319i	$0.546566\pi$
$954$		0			0
$955$			26.8328i			0.868290i
$956$		0			0
$957$	15.0000	+	22.3607i	0.484881	+	0.722818i
$958$		0			0
$959$			17.8885i			0.577651i
$960$		0			0
$961$	−14.0000			−0.451613
$962$		0			0
$963$	−36.0000			−1.16008
$964$		0			0
$965$			40.2492i			1.29567i
$966$		0			0
$967$			46.9574i			1.51005i	0.655697	+	0.755025i	$0.272375\pi$
$967$			46.9574i			1.51005i	−0.655697	+	0.755025i	$0.727625\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			17.8885i			0.574071i	0.957920	+	0.287035i	$0.0926697\pi$
$971$			17.8885i			0.574071i	−0.957920	+	0.287035i	$0.907330\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$			8.94427i			0.286446i
$976$		0			0
$977$	33.0000			1.05576			0.527882	−	0.849318i	$-0.322986\pi$
$977$	33.0000			1.05576			0.527882	+	0.849318i	$0.322986\pi$
$978$		0			0
$979$	10.0000			0.319601
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$			29.0689i			0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$			29.0689i			0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		0			0
$985$	54.0000			1.72058
$986$		0			0
$987$	10.0000			0.318304
$988$		0			0
$989$		−	40.2492i		−	1.27985i
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$		0			0
$993$	75.0000			2.38005
$994$		0			0
$995$	−42.0000			−1.33149
$996$		0			0
$997$		−	26.8328i		−	0.849804i	−0.905239	−	0.424902i	$-0.860309\pi$
$997$		−	26.8328i		−	0.849804i	0.905239	−	0.424902i	$-0.139691\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.e.f.1217.2		2
4.3	odd	2		1856.2.e.g.1217.1		2
8.3	odd	2		29.2.b.a.28.1	✓	2
8.5	even	2		464.2.e.a.289.1		2
24.5	odd	2		4176.2.o.k.289.1		2
24.11	even	2		261.2.c.a.28.2		2
29.28	even	2	inner	1856.2.e.f.1217.1		2
40.3	even	4		725.2.d.a.724.1		4
40.19	odd	2		725.2.c.c.376.2		2
40.27	even	4		725.2.d.a.724.4		4
56.27	even	2		1421.2.b.b.1275.1		2
116.115	odd	2		1856.2.e.g.1217.2		2
232.3	even	28		841.2.d.h.571.2		12
232.11	even	28		841.2.d.h.778.1		12
232.19	even	28		841.2.d.h.190.2		12
232.27	even	28		841.2.d.h.605.1		12
232.35	odd	14		841.2.e.g.196.2		12
232.43	even	28		841.2.d.h.645.2		12
232.51	odd	14		841.2.e.g.270.1		12
232.67	odd	14		841.2.e.g.267.2		12
232.75	even	4		841.2.a.b.1.1		2
232.83	odd	14		841.2.e.g.651.1		12
232.91	odd	14		841.2.e.g.651.2		12
232.99	even	4		841.2.a.b.1.2		2
232.107	odd	14		841.2.e.g.267.1		12
232.115	odd	2		29.2.b.a.28.2	yes	2
232.123	odd	14		841.2.e.g.270.2		12
232.131	even	28		841.2.d.h.645.1		12
232.139	odd	14		841.2.e.g.196.1		12
232.147	even	28		841.2.d.h.605.2		12
232.155	even	28		841.2.d.h.190.1		12
232.163	even	28		841.2.d.h.778.2		12
232.171	even	28		841.2.d.h.571.1		12
232.173	even	2		464.2.e.a.289.2		2
232.179	odd	14		841.2.e.g.236.1		12
232.187	odd	14		841.2.e.g.63.1		12
232.195	even	28		841.2.d.h.574.1		12
232.211	even	28		841.2.d.h.574.2		12
232.219	odd	14		841.2.e.g.63.2		12
232.227	odd	14		841.2.e.g.236.2		12
696.173	odd	2		4176.2.o.k.289.2		2
696.347	even	2		261.2.c.a.28.1		2
696.539	odd	4		7569.2.a.i.1.2		2
696.563	odd	4		7569.2.a.i.1.1		2
1160.347	even	4		725.2.d.a.724.2		4
1160.579	odd	2		725.2.c.c.376.1		2
1160.1043	even	4		725.2.d.a.724.3		4
1624.811	even	2		1421.2.b.b.1275.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.b.a.28.1	✓	2	8.3	odd	2
29.2.b.a.28.2	yes	2	232.115	odd	2
261.2.c.a.28.1		2	696.347	even	2
261.2.c.a.28.2		2	24.11	even	2
464.2.e.a.289.1		2	8.5	even	2
464.2.e.a.289.2		2	232.173	even	2
725.2.c.c.376.1		2	1160.579	odd	2
725.2.c.c.376.2		2	40.19	odd	2
725.2.d.a.724.1		4	40.3	even	4
725.2.d.a.724.2		4	1160.347	even	4
725.2.d.a.724.3		4	1160.1043	even	4
725.2.d.a.724.4		4	40.27	even	4
841.2.a.b.1.1		2	232.75	even	4
841.2.a.b.1.2		2	232.99	even	4
841.2.d.h.190.1		12	232.155	even	28
841.2.d.h.190.2		12	232.19	even	28
841.2.d.h.571.1		12	232.171	even	28
841.2.d.h.571.2		12	232.3	even	28
841.2.d.h.574.1		12	232.195	even	28
841.2.d.h.574.2		12	232.211	even	28
841.2.d.h.605.1		12	232.27	even	28
841.2.d.h.605.2		12	232.147	even	28
841.2.d.h.645.1		12	232.131	even	28
841.2.d.h.645.2		12	232.43	even	28
841.2.d.h.778.1		12	232.11	even	28
841.2.d.h.778.2		12	232.163	even	28
841.2.e.g.63.1		12	232.187	odd	14
841.2.e.g.63.2		12	232.219	odd	14
841.2.e.g.196.1		12	232.139	odd	14
841.2.e.g.196.2		12	232.35	odd	14
841.2.e.g.236.1		12	232.179	odd	14
841.2.e.g.236.2		12	232.227	odd	14
841.2.e.g.267.1		12	232.107	odd	14
841.2.e.g.267.2		12	232.67	odd	14
841.2.e.g.270.1		12	232.51	odd	14
841.2.e.g.270.2		12	232.123	odd	14
841.2.e.g.651.1		12	232.83	odd	14
841.2.e.g.651.2		12	232.91	odd	14
1421.2.b.b.1275.1		2	56.27	even	2
1421.2.b.b.1275.2		2	1624.811	even	2
1856.2.e.f.1217.1		2	29.28	even	2	inner
1856.2.e.f.1217.2		2	1.1	even	1	trivial
1856.2.e.g.1217.1		2	4.3	odd	2
1856.2.e.g.1217.2		2	116.115	odd	2
4176.2.o.k.289.1		2	24.5	odd	2
4176.2.o.k.289.2		2	696.173	odd	2
7569.2.a.i.1.1		2	696.563	odd	4
7569.2.a.i.1.2		2	696.539	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{3} +3.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} -2.23607i q^{11} +1.00000 q^{13} +6.70820i q^{15} +4.47214i q^{17} -4.47214i q^{21} -6.00000 q^{23} +4.00000 q^{25} +2.23607i q^{27} +(3.00000 + 4.47214i) q^{29} +6.70820i q^{31} +5.00000 q^{33} -6.00000 q^{35} +2.23607i q^{39} +4.47214i q^{41} +6.70820i q^{43} -6.00000 q^{45} +2.23607i q^{47} -3.00000 q^{49} -10.0000 q^{51} +9.00000 q^{53} -6.70820i q^{55} +6.00000 q^{59} +13.4164i q^{61} +4.00000 q^{63} +3.00000 q^{65} +8.00000 q^{67} -13.4164i q^{69} +8.94427i q^{75} +4.47214i q^{77} -6.70820i q^{79} -11.0000 q^{81} -6.00000 q^{83} +13.4164i q^{85} +(-10.0000 + 6.70820i) q^{87} +4.47214i q^{89} -2.00000 q^{91} -15.0000 q^{93} -13.4164i q^{97} +4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 4 q^{7} - 4 q^{9} + 2 q^{13} - 12 q^{23} + 8 q^{25} + 6 q^{29} + 10 q^{33} - 12 q^{35} - 12 q^{45} - 6 q^{49} - 20 q^{51} + 18 q^{53} + 12 q^{59} + 8 q^{63} + 6 q^{65} + 16 q^{67} - 22 q^{81}+ \cdots - 30 q^{93}+O(q^{100}) \) 2 * q + 6 * q^5 - 4 * q^7 - 4 * q^9 + 2 * q^13 - 12 * q^23 + 8 * q^25 + 6 * q^29 + 10 * q^33 - 12 * q^35 - 12 * q^45 - 6 * q^49 - 20 * q^51 + 18 * q^53 + 12 * q^59 + 8 * q^63 + 6 * q^65 + 16 * q^67 - 22 * q^81 - 12 * q^83 - 20 * q^87 - 4 * q^91 - 30 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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