LMFDB - Embedded newform 2268.3.bg.c.2213.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,3,Mod(701,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.701"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2268.bg (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$61.7985239569$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2213.1
Root			$-2.23256 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	2268.2213
Dual form			2268.3.bg.c.701.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.46512 + 2.57794i) q^{5} +(1.32288 - 2.29129i) q^{7} +(5.25600 + 3.03455i) q^{11} +(6.29150 + 10.8972i) q^{13} -17.2941i q^{17} -25.8745 q^{19} +(2.09247 - 1.20809i) q^{23} +(0.791503 - 1.37092i) q^{25} +(48.3254 + 27.9007i) q^{29} +(-7.64575 - 13.2428i) q^{31} +13.6412i q^{35} -23.7490 q^{37} +(-13.3953 + 7.73381i) q^{41} +(-13.8745 + 24.0314i) q^{43} +(24.6486 + 14.2309i) q^{47} +(-3.50000 - 6.06218i) q^{49} +46.6690i q^{53} -31.2915 q^{55} +(69.8601 - 40.3337i) q^{59} +(31.3542 - 54.3072i) q^{61} +(-56.1846 - 32.4382i) q^{65} +(-58.0405 - 100.529i) q^{67} +49.7896i q^{71} +30.7085 q^{73} +(13.9061 - 8.02867i) q^{77} +(-71.3320 + 123.551i) q^{79} +(-24.7478 - 14.2882i) q^{83} +(44.5830 + 77.2200i) q^{85} -168.203i q^{89} +33.2915 q^{91} +(115.533 - 66.7028i) q^{95} +(-56.9778 + 98.6884i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} - 80 q^{19} - 36 q^{25} - 40 q^{31} + 64 q^{37} + 16 q^{43} - 28 q^{49} - 208 q^{55} + 272 q^{61} - 168 q^{67} + 288 q^{73} - 232 q^{79} + 272 q^{85} + 224 q^{91} - 96 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 80 * q^19 - 36 * q^25 - 40 * q^31 + 64 * q^37 + 16 * q^43 - 28 * q^49 - 208 * q^55 + 272 * q^61 - 168 * q^67 + 288 * q^73 - 232 * q^79 + 272 * q^85 + 224 * q^91 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−4.46512	+	2.57794i	−0.893023	+	0.515587i	−0.874930	−	0.484249i	$-0.839093\pi$
$5$	−4.46512	+	2.57794i	−0.893023	+	0.515587i	−0.0180929	+	0.999836i	$0.505759\pi$
$6$		0			0
$7$	1.32288	−	2.29129i	0.188982	−	0.327327i
$7$	1.32288	−	2.29129i	0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.25600	+	3.03455i	0.477818	+	0.275868i	0.719507	−	0.694486i	$-0.244368\pi$
$11$	5.25600	+	3.03455i	0.477818	+	0.275868i	−0.241689	+	0.970354i	$0.577701\pi$
$12$		0			0
$13$	6.29150	+	10.8972i	0.483962	+	0.838246i	0.999830	−	0.0184214i	$-0.00586405\pi$
$13$	6.29150	+	10.8972i	0.483962	+	0.838246i	−0.515869	+	0.856668i	$0.672531\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	17.2941i		−	1.01730i	−0.860974	−	0.508649i	$-0.830145\pi$
$17$		−	17.2941i		−	1.01730i	0.860974	−	0.508649i	$-0.169855\pi$
$18$		0			0
$19$	−25.8745			−1.36182			−0.680908	−	0.732369i	$-0.738415\pi$
$19$	−25.8745			−1.36182			−0.680908	+	0.732369i	$0.738415\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.09247	−	1.20809i	0.0909771	−	0.0525257i	−0.453821	−	0.891093i	$-0.649940\pi$
$23$	2.09247	−	1.20809i	0.0909771	−	0.0525257i	0.544798	+	0.838567i	$0.316606\pi$
$24$		0			0
$25$	0.791503	−	1.37092i	0.0316601	−	0.0548369i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	48.3254	+	27.9007i	1.66639	+	0.962092i	0.969558	+	0.244860i	$0.0787420\pi$
$29$	48.3254	+	27.9007i	1.66639	+	0.962092i	0.696834	+	0.717232i	$0.254591\pi$
$30$		0			0
$31$	−7.64575	−	13.2428i	−0.246637	−	0.427188i	0.715953	−	0.698148i	$-0.245992\pi$
$31$	−7.64575	−	13.2428i	−0.246637	−	0.427188i	−0.962591	+	0.270960i	$0.912659\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			13.6412i			0.389747i
$36$		0			0
$37$	−23.7490			−0.641865			−0.320933	−	0.947102i	$-0.603996\pi$
$37$	−23.7490			−0.641865			−0.320933	+	0.947102i	$0.603996\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−13.3953	+	7.73381i	−0.326716	+	0.188629i	−0.654382	−	0.756164i	$-0.727071\pi$
$41$	−13.3953	+	7.73381i	−0.326716	+	0.188629i	0.327666	+	0.944794i	$0.393738\pi$
$42$		0			0
$43$	−13.8745	+	24.0314i	−0.322663	+	0.558869i	−0.981037	−	0.193823i	$-0.937911\pi$
$43$	−13.8745	+	24.0314i	−0.322663	+	0.558869i	0.658374	+	0.752691i	$0.271245\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	24.6486	+	14.2309i	0.524438	+	0.302785i	0.738749	−	0.673981i	$-0.235417\pi$
$47$	24.6486	+	14.2309i	0.524438	+	0.302785i	−0.214310	+	0.976766i	$0.568750\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			46.6690i			0.880548i	0.897863	+	0.440274i	$0.145119\pi$
$53$			46.6690i			0.880548i	−0.897863	+	0.440274i	$0.854881\pi$
$54$		0			0
$55$	−31.2915			−0.568936
$56$		0			0
$57$		0			0
$58$		0			0
$59$	69.8601	−	40.3337i	1.18407	−	0.683623i	0.227117	−	0.973868i	$-0.427070\pi$
$59$	69.8601	−	40.3337i	1.18407	−	0.683623i	0.956952	+	0.290245i	$0.0937368\pi$
$60$		0			0
$61$	31.3542	−	54.3072i	0.514004	−	0.890281i	−0.485864	−	0.874034i	$-0.661495\pi$
$61$	31.3542	−	54.3072i	0.514004	−	0.890281i	0.999868	−	0.0162467i	$-0.00517172\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−56.1846	−	32.4382i	−0.864378	−	0.499049i
$66$		0			0
$67$	−58.0405	−	100.529i	−0.866276	−	1.50043i	−0.865774	−	0.500435i	$-0.833173\pi$
$67$	−58.0405	−	100.529i	−0.866276	−	1.50043i	−0.000502198	−	1.00000i	$-0.500160\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			49.7896i			0.701261i	0.936514	+	0.350631i	$0.114033\pi$
$71$			49.7896i			0.701261i	−0.936514	+	0.350631i	$0.885967\pi$
$72$		0			0
$73$	30.7085			0.420664			0.210332	−	0.977630i	$-0.432545\pi$
$73$	30.7085			0.420664			0.210332	+	0.977630i	$0.432545\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.9061	−	8.02867i	0.180598	−	0.104268i
$78$		0			0
$79$	−71.3320	+	123.551i	−0.902937	+	1.56393i	−0.0792742	+	0.996853i	$0.525260\pi$
$79$	−71.3320	+	123.551i	−0.902937	+	1.56393i	−0.823663	+	0.567080i	$0.808073\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−24.7478	−	14.2882i	−0.298167	−	0.172147i	0.343452	−	0.939170i	$-0.388404\pi$
$83$	−24.7478	−	14.2882i	−0.298167	−	0.172147i	−0.641619	+	0.767023i	$0.721737\pi$
$84$		0			0
$85$	44.5830	+	77.2200i	0.524506	+	0.908471i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	168.203i		−	1.88992i	−0.327189	−	0.944959i	$-0.606101\pi$
$89$		−	168.203i		−	1.88992i	0.327189	−	0.944959i	$-0.393899\pi$
$90$		0			0
$91$	33.2915			0.365841
$92$		0			0
$93$		0			0
$94$		0			0
$95$	115.533	−	66.7028i	1.21613	−	0.702135i
$96$		0			0
$97$	−56.9778	+	98.6884i	−0.587400	+	1.01741i	0.407172	+	0.913352i	$0.366515\pi$
$97$	−56.9778	+	98.6884i	−0.587400	+	1.01741i	−0.994572	+	0.104055i	$0.966818\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−34.8804	−	20.1382i	−0.345351	−	0.199388i	0.317285	−	0.948330i	$-0.397229\pi$
$101$	−34.8804	−	20.1382i	−0.345351	−	0.199388i	−0.662636	+	0.748942i	$0.730562\pi$
$102$		0			0
$103$	−14.5608	−	25.2200i	−0.141367	−	0.244854i	0.786645	−	0.617406i	$-0.211816\pi$
$103$	−14.5608	−	25.2200i	−0.141367	−	0.244854i	−0.928012	+	0.372551i	$0.878483\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			181.901i			1.70001i	0.526774	+	0.850005i	$0.323401\pi$
$107$			181.901i			1.70001i	−0.526774	+	0.850005i	$0.676599\pi$
$108$		0			0
$109$	−88.1621			−0.808826			−0.404413	−	0.914576i	$-0.632524\pi$
$109$	−88.1621			−0.808826			−0.404413	+	0.914576i	$0.632524\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−56.1349	+	32.4095i	−0.496769	+	0.286810i	−0.727378	−	0.686237i	$-0.759261\pi$
$113$	−56.1349	+	32.4095i	−0.496769	+	0.286810i	0.230609	+	0.973046i	$0.425928\pi$
$114$		0			0
$115$	−6.22876	+	10.7885i	−0.0541631	+	0.0938132i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−39.6257	−	22.8779i	−0.332989	−	0.192251i
$120$		0			0
$121$	−42.0830	−	72.8899i	−0.347793	−	0.602396i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	120.735i		−	0.965880i
$126$		0			0
$127$	−223.579			−1.76047			−0.880233	−	0.474543i	$-0.842613\pi$
$127$	−223.579			−1.76047			−0.880233	+	0.474543i	$0.842613\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−179.527	+	103.650i	−1.37043	+	0.791220i	−0.990982	−	0.133992i	$-0.957220\pi$
$131$	−179.527	+	103.650i	−1.37043	+	0.791220i	−0.379451	+	0.925212i	$0.623887\pi$
$132$		0			0
$133$	−34.2288	+	59.2859i	−0.257359	+	0.445759i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−152.225	−	87.8874i	−1.11113	−	0.641514i	−0.172011	−	0.985095i	$-0.555027\pi$
$137$	−152.225	−	87.8874i	−1.11113	−	0.641514i	−0.939123	+	0.343581i	$0.888360\pi$
$138$		0			0
$139$	58.5830	+	101.469i	0.421460	+	0.729991i	0.996083	−	0.0884279i	$-0.0281843\pi$
$139$	58.5830	+	101.469i	0.421460	+	0.729991i	−0.574622	+	0.818419i	$0.694851\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			76.3675i			0.534039i
$144$		0			0
$145$	−287.705			−1.98417
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−161.354	+	93.1578i	−1.08291	+	0.625220i	−0.931681	−	0.363278i	$-0.881658\pi$
$149$	−161.354	+	93.1578i	−1.08291	+	0.625220i	−0.151233	+	0.988498i	$0.548324\pi$
$150$		0			0
$151$	−91.0405	+	157.687i	−0.602917	+	1.04428i	0.389460	+	0.921044i	$0.372662\pi$
$151$	−91.0405	+	157.687i	−0.602917	+	1.04428i	−0.992377	+	0.123240i	$0.960672\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	68.2783	+	39.4205i	0.440505	+	0.254326i
$156$		0			0
$157$	−34.1882	−	59.2158i	−0.217759	−	0.377170i	0.736363	−	0.676586i	$-0.236542\pi$
$157$	−34.1882	−	59.2158i	−0.217759	−	0.377170i	−0.954123	+	0.299416i	$0.903208\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.39261i		−	0.0397057i
$162$		0			0
$163$	−36.7451			−0.225430			−0.112715	−	0.993627i	$-0.535955\pi$
$163$	−36.7451			−0.225430			−0.112715	+	0.993627i	$0.535955\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−208.460	+	120.354i	−1.24826	+	0.720684i	−0.970762	−	0.240042i	$-0.922839\pi$
$167$	−208.460	+	120.354i	−1.24826	+	0.720684i	−0.277498	+	0.960726i	$0.589505\pi$
$168$		0			0
$169$	5.33399	−	9.23874i	0.0315621	−	0.0546671i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	295.340	+	170.514i	1.70717	+	0.985632i	0.938035	+	0.346540i	$0.112644\pi$
$173$	295.340	+	170.514i	1.70717	+	0.985632i	0.769130	+	0.639092i	$0.220690\pi$
$174$		0			0
$175$	−2.09412	−	3.62712i	−0.0119664	−	0.0207264i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	138.410i		−	0.773240i	−0.922239	−	0.386620i	$-0.873643\pi$
$179$		−	138.410i		−	0.773240i	0.922239	−	0.386620i	$-0.126357\pi$
$180$		0			0
$181$	−135.328			−0.747669			−0.373834	−	0.927495i	$-0.621957\pi$
$181$	−135.328			−0.747669			−0.373834	+	0.927495i	$0.621957\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	106.042	−	61.2234i	0.573200	−	0.330937i
$186$		0			0
$187$	52.4797	−	90.8976i	0.280640	−	0.486083i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−145.898	−	84.2344i	−0.763866	−	0.441018i	0.0668163	−	0.997765i	$-0.478716\pi$
$191$	−145.898	−	84.2344i	−0.763866	−	0.441018i	−0.830682	+	0.556747i	$0.812049\pi$
$192$		0			0
$193$	−71.7490	−	124.273i	−0.371757	−	0.643901i	0.618079	−	0.786116i	$-0.287911\pi$
$193$	−71.7490	−	124.273i	−0.371757	−	0.643901i	−0.989836	+	0.142215i	$0.954578\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	296.510i		−	1.50513i	−0.658521	−	0.752563i	$-0.728817\pi$
$197$		−	296.510i		−	1.50513i	0.658521	−	0.752563i	$-0.271183\pi$
$198$		0			0
$199$	−142.243			−0.714789			−0.357395	−	0.933953i	$-0.616335\pi$
$199$	−142.243			−0.714789			−0.357395	+	0.933953i	$0.616335\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	127.857	−	73.8182i	0.629837	−	0.363637i
$204$		0			0
$205$	39.8745	−	69.0647i	0.194510	−	0.336901i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−135.996	−	78.5175i	−0.650700	−	0.375682i
$210$		0			0
$211$	−55.4980	−	96.1254i	−0.263024	−	0.455571i	0.704020	−	0.710180i	$-0.251386\pi$
$211$	−55.4980	−	96.1254i	−0.263024	−	0.455571i	−0.967044	+	0.254609i	$0.918053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	143.070i		−	0.665443i
$216$		0			0
$217$	−40.4575			−0.186440
$218$		0			0
$219$		0			0
$220$		0			0
$221$	188.457	−	108.806i	0.852747	−	0.492334i
$222$		0			0
$223$	52.1255	−	90.2840i	0.233747	−	0.404861i	−0.725161	−	0.688579i	$-0.758235\pi$
$223$	52.1255	−	90.2840i	0.233747	−	0.404861i	0.958908	+	0.283718i	$0.0915681\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	141.302	+	81.5807i	0.622475	+	0.359386i	0.777832	−	0.628472i	$-0.216319\pi$
$227$	141.302	+	81.5807i	0.622475	+	0.359386i	−0.155357	+	0.987858i	$0.549653\pi$
$228$		0			0
$229$	139.207	+	241.113i	0.607889	+	1.05289i	0.991588	+	0.129436i	$0.0413167\pi$
$229$	139.207	+	241.113i	0.607889	+	1.05289i	−0.383699	+	0.923458i	$0.625350\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			404.042i			1.73408i	0.498235	+	0.867042i	$0.333982\pi$
$233$			404.042i			1.73408i	−0.498235	+	0.867042i	$0.666018\pi$
$234$		0			0
$235$	−146.745			−0.624447
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−222.745	+	128.602i	−0.931988	+	0.538083i	−0.887440	−	0.460924i	$-0.847518\pi$
$239$	−222.745	+	128.602i	−0.931988	+	0.538083i	−0.0445480	+	0.999007i	$0.514185\pi$
$240$		0			0
$241$	−124.808	+	216.173i	−0.517875	+	0.896985i	0.481910	+	0.876221i	$0.339943\pi$
$241$	−124.808	+	216.173i	−0.517875	+	0.896985i	−0.999784	+	0.0207645i	$0.993390\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	31.2558	+	18.0455i	0.127575	+	0.0736553i
$246$		0			0
$247$	−162.790	−	281.960i	−0.659067	−	1.14154i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			69.2909i			0.276059i	0.990428	+	0.138030i	$0.0440769\pi$
$251$			69.2909i			0.276059i	−0.990428	+	0.138030i	$0.955923\pi$
$252$		0			0
$253$	14.6640			0.0579606
$254$		0			0
$255$		0			0
$256$		0			0
$257$	221.592	−	127.936i	0.862227	−	0.497807i	−0.00253044	−	0.999997i	$-0.500805\pi$
$257$	221.592	−	127.936i	0.862227	−	0.497807i	0.864757	+	0.502190i	$0.167472\pi$
$258$		0			0
$259$	−31.4170	+	54.4158i	−0.121301	+	0.210100i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.7037	−	7.33449i	−0.0483031	−	0.0278878i	0.475654	−	0.879632i	$-0.342211\pi$
$263$	−12.7037	−	7.33449i	−0.0483031	−	0.0278878i	−0.523957	+	0.851745i	$0.675545\pi$
$264$		0			0
$265$	−120.310	−	208.383i	−0.453999	−	0.786350i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	74.9792i		−	0.278733i	−0.990241	−	0.139367i	$-0.955493\pi$
$269$		−	74.9792i		−	0.278733i	0.990241	−	0.139367i	$-0.0445066\pi$
$270$		0			0
$271$	−211.203			−0.779345			−0.389673	−	0.920953i	$-0.627412\pi$
$271$	−211.203			−0.779345			−0.389673	+	0.920953i	$0.627412\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.32027	−	4.80371i	0.0302555	−	0.0174680i
$276$		0			0
$277$	−204.247	+	353.766i	−0.737354	+	1.27713i	0.216329	+	0.976321i	$0.430592\pi$
$277$	−204.247	+	353.766i	−0.737354	+	1.27713i	−0.953683	+	0.300814i	$0.902742\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−173.547	−	100.197i	−0.617605	−	0.356575i	0.158331	−	0.987386i	$-0.449389\pi$
$281$	−173.547	−	100.197i	−0.617605	−	0.356575i	−0.775936	+	0.630812i	$0.782722\pi$
$282$		0			0
$283$	−179.225	−	310.426i	−0.633303	−	1.09691i	−0.986872	−	0.161505i	$-0.948365\pi$
$283$	−179.225	−	310.426i	−0.633303	−	1.09691i	0.353569	−	0.935409i	$-0.384968\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			40.9235i			0.142590i
$288$		0			0
$289$	−10.0850			−0.0348961
$290$		0			0
$291$		0			0
$292$		0			0
$293$	281.763	−	162.676i	0.961650	−	0.555209i	0.0649693	−	0.997887i	$-0.479305\pi$
$293$	281.763	−	162.676i	0.961650	−	0.555209i	0.896680	+	0.442679i	$0.145972\pi$
$294$		0			0
$295$	−207.956	+	360.190i	−0.704934	+	1.22098i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.3296	+	15.2014i	0.0880589	+	0.0508408i
$300$		0			0
$301$	36.7085	+	63.5810i	0.121955	+	0.211233i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			323.317i			1.06006i
$306$		0			0
$307$	−338.037			−1.10110			−0.550548	−	0.834803i	$-0.685581\pi$
$307$	−338.037			−1.10110			−0.550548	+	0.834803i	$0.685581\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	269.489	−	155.589i	0.866523	−	0.500287i	0.000331809	−	1.00000i	$-0.499894\pi$
$311$	269.489	−	155.589i	0.866523	−	0.500287i	0.866191	+	0.499713i	$0.166561\pi$
$312$		0			0
$313$	−200.535	+	347.336i	−0.640686	+	1.10970i	0.344594	+	0.938752i	$0.388016\pi$
$313$	−200.535	+	347.336i	−0.640686	+	1.10970i	−0.985280	+	0.170948i	$0.945317\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	72.8747	+	42.0742i	0.229889	+	0.132726i	0.610521	−	0.792000i	$-0.290960\pi$
$317$	72.8747	+	42.0742i	0.229889	+	0.132726i	−0.380632	+	0.924727i	$0.624294\pi$
$318$		0			0
$319$	169.332	+	293.292i	0.530821	+	0.919410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			447.476i			1.38537i
$324$		0			0
$325$	19.9190			0.0612891
$326$		0			0
$327$		0			0
$328$		0			0
$329$	65.2140	−	37.6513i	0.198219	−	0.114442i
$330$		0			0
$331$	111.749	−	193.555i	0.337610	−	0.584758i	−0.646372	−	0.763022i	$-0.723715\pi$
$331$	111.749	−	193.555i	0.337610	−	0.584758i	0.983983	+	0.178264i	$0.0570481\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	518.315	+	299.249i	1.54721	+	0.893282i
$336$		0			0
$337$	117.292	+	203.155i	0.348046	+	0.602833i	0.985902	−	0.167322i	$-0.0535120\pi$
$337$	117.292	+	203.155i	0.348046	+	0.602833i	−0.637856	+	0.770155i	$0.720179\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	92.8057i		−	0.272157i
$342$		0			0
$343$	−18.5203			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−72.4136	+	41.8080i	−0.208685	+	0.120484i	−0.600700	−	0.799474i	$-0.705111\pi$
$347$	−72.4136	+	41.8080i	−0.208685	+	0.120484i	0.392015	+	0.919959i	$0.371778\pi$
$348$		0			0
$349$	−149.480	+	258.907i	−0.428309	+	0.741852i	−0.996723	−	0.0808900i	$-0.974224\pi$
$349$	−149.480	+	258.907i	−0.428309	+	0.741852i	0.568414	+	0.822742i	$0.307557\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−370.704	−	214.026i	−1.05015	−	0.606306i	−0.127460	−	0.991844i	$-0.540683\pi$
$353$	−370.704	−	214.026i	−1.05015	−	0.606306i	−0.922692	+	0.385538i	$0.874016\pi$
$354$		0			0
$355$	−128.354	−	222.316i	−0.361561	−	0.626242i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	144.536i		−	0.402608i	−0.979529	−	0.201304i	$-0.935482\pi$
$359$		−	144.536i		−	0.402608i	0.979529	−	0.201304i	$-0.0645179\pi$
$360$		0			0
$361$	308.490			0.854543
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−137.117	+	79.1645i	−0.375663	+	0.216889i
$366$		0			0
$367$	305.122	−	528.486i	0.831394	−	1.44002i	−0.0655391	−	0.997850i	$-0.520877\pi$
$367$	305.122	−	528.486i	0.831394	−	1.44002i	0.896933	−	0.442166i	$-0.145790\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	106.932	+	61.7373i	0.288227	+	0.166408i
$372$		0			0
$373$	−0.166010	−	0.287539i	−0.000445068	−	0.000770881i	0.865803	−	0.500385i	$-0.166808\pi$
$373$	−0.166010	−	0.287539i	−0.000445068	−	0.000770881i	−0.866248	+	0.499615i	$0.833475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			702.149i			1.86246i
$378$		0			0
$379$	−393.652			−1.03866			−0.519330	−	0.854574i	$-0.673818\pi$
$379$	−393.652			−1.03866			−0.519330	+	0.854574i	$0.673818\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	468.259	−	270.350i	1.22261	−	0.705874i	0.257135	−	0.966375i	$-0.417221\pi$
$383$	468.259	−	270.350i	1.22261	−	0.705874i	0.965473	+	0.260502i	$0.0838880\pi$
$384$		0			0
$385$	−41.3948	+	71.6978i	−0.107519	+	0.186228i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	410.840	+	237.199i	1.05614	+	0.609765i	0.924363	−	0.381513i	$-0.124597\pi$
$389$	410.840	+	237.199i	1.05614	+	0.609765i	0.131781	+	0.991279i	$0.457930\pi$
$390$		0			0
$391$	−20.8928	−	36.1874i	−0.0534343	−	0.0925509i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	735.557i		−	1.86217i
$396$		0			0
$397$	400.288			1.00828			0.504141	−	0.863622i	$-0.331809\pi$
$397$	400.288			1.00828			0.504141	+	0.863622i	$0.331809\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−563.839	+	325.533i	−1.40608	+	0.811802i	−0.995008	−	0.0997999i	$-0.968180\pi$
$401$	−563.839	+	325.533i	−1.40608	+	0.811802i	−0.411075	+	0.911602i	$0.634846\pi$
$402$		0			0
$403$	96.2065	−	166.635i	0.238726	−	0.413485i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−124.825	−	72.0676i	−0.306695	−	0.177070i
$408$		0			0
$409$	94.5568	+	163.777i	0.231190	+	0.400433i	0.958159	−	0.286238i	$-0.0924047\pi$
$409$	94.5568	+	163.777i	0.231190	+	0.400433i	−0.726968	+	0.686671i	$0.759071\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	213.426i		−	0.516770i
$414$		0			0
$415$	147.336			0.355026
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−718.603	+	414.886i	−1.71504	+	0.990181i	−0.787626	+	0.616153i	$0.788690\pi$
$419$	−718.603	+	414.886i	−1.71504	+	0.990181i	−0.927417	+	0.374028i	$0.877976\pi$
$420$		0			0
$421$	113.162	−	196.002i	0.268794	−	0.465564i	−0.699757	−	0.714381i	$-0.746708\pi$
$421$	113.162	−	196.002i	0.268794	−	0.465564i	0.968551	+	0.248817i	$0.0800417\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−23.7088	−	13.6883i	−0.0557855	−	0.0322078i
$426$		0			0
$427$	−82.9555	−	143.683i	−0.194275	−	0.336495i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	436.460i		−	1.01267i	−0.862338	−	0.506334i	$-0.831000\pi$
$431$		−	436.460i		−	1.01267i	0.862338	−	0.506334i	$-0.169000\pi$
$432$		0			0
$433$	404.243			0.933587			0.466793	−	0.884366i	$-0.345409\pi$
$433$	404.243			0.933587			0.466793	+	0.884366i	$0.345409\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−54.1417	+	31.2587i	−0.123894	+	0.0715303i
$438$		0			0
$439$	91.4091	−	158.325i	0.208221	−	0.360650i	−0.742933	−	0.669366i	$-0.766566\pi$
$439$	91.4091	−	158.325i	0.208221	−	0.360650i	0.951154	+	0.308716i	$0.0998993\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−286.838	−	165.606i	−0.647491	−	0.373829i	0.140003	−	0.990151i	$-0.455289\pi$
$443$	−286.838	−	165.606i	−0.647491	−	0.373829i	−0.787494	+	0.616322i	$0.788622\pi$
$444$		0			0
$445$	433.616	+	751.044i	0.974417	+	1.68774i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.8425i		−	0.0286025i	−0.999898	−	0.0143013i	$-0.995448\pi$
$449$		−	12.8425i		−	0.0286025i	0.999898	−	0.0143013i	$-0.00455239\pi$
$450$		0			0
$451$	−93.8745			−0.208147
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−148.650	+	85.8233i	−0.326704	+	0.188623i
$456$		0			0
$457$	155.336	−	269.050i	0.339904	−	0.588730i	−0.644511	−	0.764595i	$-0.722939\pi$
$457$	155.336	−	269.050i	0.339904	−	0.588730i	0.984414	+	0.175865i	$0.0562722\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−337.551	−	194.885i	−0.732215	−	0.422744i	0.0870172	−	0.996207i	$-0.472266\pi$
$461$	−337.551	−	194.885i	−0.732215	−	0.422744i	−0.819232	+	0.573463i	$0.805600\pi$
$462$		0			0
$463$	−76.0405	−	131.706i	−0.164234	−	0.284462i	0.772149	−	0.635442i	$-0.219182\pi$
$463$	−76.0405	−	131.706i	−0.164234	−	0.284462i	−0.936383	+	0.350980i	$0.885849\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			86.8679i			0.186013i	0.995666	+	0.0930063i	$0.0296477\pi$
$467$			86.8679i			0.186013i	−0.995666	+	0.0930063i	$0.970352\pi$
$468$		0			0
$469$	−307.122			−0.654843
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−145.849	+	84.2058i	−0.308348	+	0.178025i
$474$		0			0
$475$	−20.4797	+	35.4720i	−0.0431152	+	0.0746778i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	476.005	+	274.821i	0.993747	+	0.573740i	0.906392	−	0.422437i	$-0.138825\pi$
$479$	476.005	+	274.821i	0.993747	+	0.573740i	0.0873545	+	0.996177i	$0.472159\pi$
$480$		0			0
$481$	−149.417	−	258.798i	−0.310638	−	0.538041i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	587.540i		−	1.21142i
$486$		0			0
$487$	−690.745			−1.41837			−0.709184	−	0.705024i	$-0.750936\pi$
$487$	−690.745			−1.41837			−0.709184	+	0.705024i	$0.750936\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	634.259	−	366.190i	1.29177	−	0.745804i	0.312803	−	0.949818i	$-0.398732\pi$
$491$	634.259	−	366.190i	1.29177	−	0.745804i	0.978968	+	0.204014i	$0.0653988\pi$
$492$		0			0
$493$	482.516	−	835.743i	0.978735	−	1.69522i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	114.082	+	65.8654i	0.229542	+	0.132526i
$498$		0			0
$499$	227.203	+	393.526i	0.455316	+	0.788630i	0.998706	−	0.0508499i	$-0.0161930\pi$
$499$	227.203	+	393.526i	0.455316	+	0.788630i	−0.543390	+	0.839480i	$0.682860\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	304.594i		−	0.605554i	−0.953061	−	0.302777i	$-0.902086\pi$
$503$		−	304.594i		−	0.605554i	0.953061	−	0.302777i	$-0.0979138\pi$
$504$		0			0
$505$	207.660			0.411208
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−64.2102	+	37.0718i	−0.126150	+	0.0728326i	−0.561747	−	0.827309i	$-0.689871\pi$
$509$	−64.2102	+	37.0718i	−0.126150	+	0.0728326i	0.435597	+	0.900142i	$0.356537\pi$
$510$		0			0
$511$	40.6235	−	70.3620i	0.0794981	−	0.137695i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	130.031	+	75.0735i	0.252488	+	0.145774i
$516$		0			0
$517$	86.3686	+	149.595i	0.167057	+	0.289352i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			214.511i			0.411730i	0.978580	+	0.205865i	$0.0660007\pi$
$521$			214.511i			0.411730i	−0.978580	+	0.205865i	$0.933999\pi$
$522$		0			0
$523$	932.486			1.78296			0.891478	−	0.453064i	$-0.149669\pi$
$523$	932.486			1.78296			0.891478	+	0.453064i	$0.149669\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−229.022	+	132.226i	−0.434578	+	0.250904i
$528$		0			0
$529$	−261.581	+	453.072i	−0.494482	+	0.856468i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−168.554	−	97.3145i	−0.316236	−	0.182579i
$534$		0			0
$535$	−468.929	−	812.210i	−0.876504	−	1.51815i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	42.4837i		−	0.0788195i
$540$		0			0
$541$	−186.340			−0.344436			−0.172218	−	0.985059i	$-0.555093\pi$
$541$	−186.340			−0.344436			−0.172218	+	0.985059i	$0.555093\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	393.654	−	227.276i	0.722301	−	0.417020i
$546$		0			0
$547$	−396.085	+	686.039i	−0.724104	+	1.25419i	0.235238	+	0.971938i	$0.424413\pi$
$547$	−396.085	+	686.039i	−0.724104	+	1.25419i	−0.959342	+	0.282247i	$0.908920\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1250.40	−	721.916i	−2.26932	−	1.31019i
$552$		0			0
$553$	188.727	+	326.884i	0.341278	+	0.591111i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	399.627i		−	0.717463i	−0.933441	−	0.358732i	$-0.883209\pi$
$557$		−	399.627i		−	0.717463i	0.933441	−	0.358732i	$-0.116791\pi$
$558$		0			0
$559$	−349.166			−0.624626
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−302.571	+	174.690i	−0.537427	+	0.310283i	−0.744035	−	0.668140i	$-0.767091\pi$
$563$	−302.571	+	174.690i	−0.537427	+	0.310283i	0.206609	+	0.978424i	$0.433757\pi$
$564$		0			0
$565$	167.099	−	289.425i	0.295751	−	0.512256i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	654.163	+	377.681i	1.14967	+	0.663763i	0.948807	−	0.315856i	$-0.102292\pi$
$569$	654.163	+	377.681i	1.14967	+	0.663763i	0.200864	+	0.979619i	$0.435625\pi$
$570$		0			0
$571$	−47.5751	−	82.4025i	−0.0833190	−	0.144313i	0.821355	−	0.570418i	$-0.193219\pi$
$571$	−47.5751	−	82.4025i	−0.0833190	−	0.144313i	−0.904674	+	0.426105i	$0.859885\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.82483i		−	0.00665187i
$576$		0			0
$577$	756.065			1.31034			0.655169	−	0.755482i	$-0.272597\pi$
$577$	756.065			1.31034			0.655169	+	0.755482i	$0.272597\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−65.4766	+	37.8030i	−0.112696	+	0.0650653i
$582$		0			0
$583$	−141.620	+	245.292i	−0.242915	+	0.420742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−596.382	−	344.321i	−1.01598	−	0.586578i	−0.103045	−	0.994677i	$-0.532859\pi$
$587$	−596.382	−	344.321i	−1.01598	−	0.586578i	−0.912938	+	0.408099i	$0.866192\pi$
$588$		0			0
$589$	197.830	+	342.652i	0.335874	+	0.581752i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1016.01i			1.71334i	0.515866	+	0.856669i	$0.327470\pi$
$593$			1016.01i			1.71334i	−0.515866	+	0.856669i	$0.672530\pi$
$594$		0			0
$595$	235.911			0.396489
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−511.180	+	295.130i	−0.853388	+	0.492704i	−0.861793	−	0.507261i	$-0.830658\pi$
$599$	−511.180	+	295.130i	−0.853388	+	0.492704i	0.00840423	+	0.999965i	$0.497325\pi$
$600$		0			0
$601$	−150.380	+	260.467i	−0.250217	+	0.433389i	−0.963585	−	0.267401i	$-0.913835\pi$
$601$	−150.380	+	260.467i	−0.250217	+	0.433389i	0.713368	+	0.700789i	$0.247169\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	375.811	+	216.975i	0.621175	+	0.358636i
$606$		0			0
$607$	−348.207	−	603.111i	−0.573652	−	0.993594i	−0.996187	−	0.0872472i	$-0.972193\pi$
$607$	−348.207	−	603.111i	−0.573652	−	0.993594i	0.422535	−	0.906347i	$-0.361140\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			358.134i			0.586144i
$612$		0			0
$613$	557.393			0.909288			0.454644	−	0.890673i	$-0.349767\pi$
$613$	557.393			0.909288			0.454644	+	0.890673i	$0.349767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−79.6629	+	45.9934i	−0.129113	+	0.0745436i	−0.563165	−	0.826344i	$-0.690417\pi$
$617$	−79.6629	+	45.9934i	−0.129113	+	0.0745436i	0.434052	+	0.900888i	$0.357083\pi$
$618$		0			0
$619$	558.708	−	967.712i	0.902599	−	1.56335i	0.0784898	−	0.996915i	$-0.474990\pi$
$619$	558.708	−	967.712i	0.902599	−	1.56335i	0.824109	−	0.566432i	$-0.191676\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−385.401	−	222.511i	−0.618621	−	0.357161i
$624$		0			0
$625$	331.035	+	573.369i	0.529655	+	0.917390i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			410.717i			0.652969i
$630$		0			0
$631$	962.219			1.52491			0.762456	−	0.647040i	$-0.223993\pi$
$631$	962.219			1.52491			0.762456	+	0.647040i	$0.223993\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	998.306	−	576.372i	1.57214	−	0.907673i
$636$		0			0
$637$	44.0405	−	76.2804i	0.0691374	−	0.119749i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−194.373	−	112.221i	−0.303233	−	0.175072i	0.340661	−	0.940186i	$-0.389349\pi$
$641$	−194.373	−	112.221i	−0.303233	−	0.175072i	−0.643895	+	0.765114i	$0.722682\pi$
$642$		0			0
$643$	166.192	+	287.853i	0.258464	+	0.447672i	0.965831	−	0.259174i	$-0.0834504\pi$
$643$	166.192	+	287.853i	0.258464	+	0.447672i	−0.707367	+	0.706847i	$0.750117\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			910.554i			1.40735i	0.710523	+	0.703674i	$0.248458\pi$
$647$			910.554i			1.40735i	−0.710523	+	0.703674i	$0.751542\pi$
$648$		0			0
$649$	489.579			0.754359
$650$		0			0
$651$		0			0
$652$		0			0
$653$	128.435	−	74.1519i	0.196684	−	0.113556i	−0.398424	−	0.917202i	$-0.630443\pi$
$653$	128.435	−	74.1519i	0.196684	−	0.113556i	0.595108	+	0.803646i	$0.297109\pi$
$654$		0			0
$655$	534.405	−	925.617i	0.815886	−	1.41316i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.4592	−	22.7818i	−0.0598775	−	0.0345703i	0.469762	−	0.882793i	$-0.344340\pi$
$659$	−39.4592	−	22.7818i	−0.0598775	−	0.0345703i	−0.529640	+	0.848223i	$0.677673\pi$
$660$		0			0
$661$	−202.107	−	350.060i	−0.305760	−	0.529591i	0.671670	−	0.740850i	$-0.265577\pi$
$661$	−202.107	−	350.060i	−0.305760	−	0.529591i	−0.977430	+	0.211259i	$0.932244\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	352.958i		−	0.530764i
$666$		0			0
$667$	134.826			0.202138
$668$		0			0
$669$		0			0
$670$		0			0
$671$	329.596	−	190.292i	0.491201	−	0.283595i
$672$		0			0
$673$	−508.243	+	880.303i	−0.755190	+	1.30803i	0.190089	+	0.981767i	$0.439122\pi$
$673$	−508.243	+	880.303i	−0.755190	+	1.30803i	−0.945280	+	0.326261i	$0.894211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−201.391	−	116.273i	−0.297476	−	0.171748i	0.343832	−	0.939031i	$-0.388275\pi$
$677$	−201.391	−	116.273i	−0.297476	−	0.171748i	−0.641308	+	0.767283i	$0.721608\pi$
$678$		0			0
$679$	150.749	+	261.105i	0.222016	+	0.384543i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			118.245i			0.173126i	0.996246	+	0.0865628i	$0.0275883\pi$
$683$			118.245i			0.173126i	−0.996246	+	0.0865628i	$0.972412\pi$
$684$		0			0
$685$	906.272			1.32302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−508.562	+	293.618i	−0.738116	+	0.426152i
$690$		0			0
$691$	−24.1255	+	41.7866i	−0.0349139	+	0.0604726i	−0.882954	−	0.469459i	$-0.844449\pi$
$691$	−24.1255	+	41.7866i	−0.0349139	+	0.0604726i	0.848040	+	0.529931i	$0.177782\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−523.160	−	302.046i	−0.752748	−	0.434599i
$696$		0			0
$697$	133.749	+	231.660i	0.191892	+	0.332367i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	236.466i		−	0.337326i	−0.985674	−	0.168663i	$-0.946055\pi$
$701$		−	236.466i		−	0.337326i	0.985674	−	0.168663i	$-0.0539450\pi$
$702$		0			0
$703$	614.494			0.874103
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−92.2849	+	53.2807i	−0.130530	+	0.0753617i
$708$		0			0
$709$	670.988	−	1162.19i	0.946387	−	1.63919i	0.193436	−	0.981113i	$-0.438037\pi$
$709$	670.988	−	1162.19i	0.946387	−	1.63919i	0.752951	−	0.658077i	$-0.228630\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31.9971	−	18.4735i	−0.0448767	−	0.0259096i
$714$		0			0
$715$	−196.871	−	340.990i	−0.275343	−	0.476909i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			284.617i			0.395852i	0.980217	+	0.197926i	$0.0634205\pi$
$719$			284.617i			0.395852i	−0.980217	+	0.197926i	$0.936579\pi$
$720$		0			0
$721$	−77.0484			−0.106863
$722$		0			0
$723$		0			0
$724$		0			0
$725$	76.4993	−	44.1669i	0.105516	−	0.0609199i
$726$		0			0
$727$	−215.225	+	372.780i	−0.296045	+	0.512765i	−0.975227	−	0.221204i	$-0.929001\pi$
$727$	−215.225	+	372.780i	−0.296045	+	0.512765i	0.679182	+	0.733970i	$0.262335\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	415.600	+	239.947i	0.568536	+	0.328245i
$732$		0			0
$733$	193.907	+	335.857i	0.264539	+	0.458195i	0.967443	−	0.253090i	$-0.0814468\pi$
$733$	193.907	+	335.857i	0.264539	+	0.458195i	−0.702904	+	0.711285i	$0.748114\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	704.508i		−	0.955913i
$738$		0			0
$739$	352.907			0.477547			0.238773	−	0.971075i	$-0.423255\pi$
$739$	352.907			0.477547			0.238773	+	0.971075i	$0.423255\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	925.432	−	534.298i	1.24553	−	0.719109i	0.275318	−	0.961353i	$-0.411217\pi$
$743$	925.432	−	534.298i	1.24553	−	0.719109i	0.970215	+	0.242244i	$0.0778834\pi$
$744$		0			0
$745$	480.310	−	831.921i	0.644711	−	1.11667i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	416.788	+	240.633i	0.556459	+	0.321272i
$750$		0			0
$751$	−612.073	−	1060.14i	−0.815011	−	1.41164i	−0.909320	−	0.416097i	$-0.863398\pi$
$751$	−612.073	−	1060.14i	−0.815011	−	1.41164i	0.0943092	−	0.995543i	$-0.469936\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	938.786i		−	1.24343i
$756$		0			0
$757$	−96.3399			−0.127265			−0.0636327	−	0.997973i	$-0.520269\pi$
$757$	−96.3399			−0.127265			−0.0636327	+	0.997973i	$0.520269\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−821.762	+	474.445i	−1.07985	+	0.623449i	−0.930855	−	0.365390i	$-0.880936\pi$
$761$	−821.762	+	474.445i	−1.07985	+	0.623449i	−0.148990	+	0.988839i	$0.547602\pi$
$762$		0			0
$763$	−116.627	+	202.005i	−0.152854	+	0.264751i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	879.050	+	507.520i	1.14609	+	0.661694i
$768$		0			0
$769$	499.316	+	864.841i	0.649306	+	1.12463i	0.983289	+	0.182052i	$0.0582739\pi$
$769$	499.316	+	864.841i	0.649306	+	1.12463i	−0.333983	+	0.942579i	$0.608393\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	86.4704i		−	0.111863i	−0.998435	−	0.0559317i	$-0.982187\pi$
$773$		−	86.4704i		−	0.111863i	0.998435	−	0.0559317i	$-0.0178129\pi$
$774$		0			0
$775$	−24.2065			−0.0312342
$776$		0			0
$777$		0			0
$778$		0			0
$779$	346.598	−	200.108i	0.444927	−	0.256879i
$780$		0			0
$781$	−151.089	+	261.694i	−0.193456	+	0.335075i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	305.309	+	176.270i	0.388928	+	0.224548i
$786$		0			0
$787$	244.907	+	424.192i	0.311191	+	0.538998i	0.978620	−	0.205675i	$-0.0659388\pi$
$787$	244.907	+	424.192i	0.311191	+	0.538998i	−0.667430	+	0.744673i	$0.732606\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			171.495i			0.216808i
$792$		0			0
$793$	789.061			0.995033
$794$		0			0
$795$		0			0
$796$		0			0
$797$	433.180	−	250.097i	0.543514	−	0.313798i	−0.202988	−	0.979181i	$-0.565065\pi$
$797$	433.180	−	250.097i	0.543514	−	0.313798i	0.746502	+	0.665384i	$0.231732\pi$
$798$		0			0
$799$	246.110	−	426.275i	0.308022	−	0.533510i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	161.404	+	93.1865i	0.201001	+	0.116048i
$804$		0			0
$805$	16.4797	+	28.5437i	0.0204717	+	0.0354581i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	310.515i		−	0.383826i	−0.981412	−	0.191913i	$-0.938531\pi$
$809$		−	310.515i		−	0.383826i	0.981412	−	0.191913i	$-0.0614691\pi$
$810$		0			0
$811$	−45.9190			−0.0566202			−0.0283101	−	0.999599i	$-0.509013\pi$
$811$	−45.9190			−0.0566202			−0.0283101	+	0.999599i	$0.509013\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	164.071	−	94.7264i	0.201314	−	0.116229i
$816$		0			0
$817$	358.996	−	621.799i	0.439408	−	0.761076i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1046.73	+	604.330i	1.27495	+	0.736091i	0.975915	−	0.218153i	$-0.0700030\pi$
$821$	1046.73	+	604.330i	1.27495	+	0.736091i	0.299032	+	0.954243i	$0.403336\pi$
$822$		0			0
$823$	59.5909	+	103.214i	0.0724069	+	0.125412i	0.899956	−	0.435981i	$-0.143599\pi$
$823$	59.5909	+	103.214i	0.0724069	+	0.125412i	−0.827549	+	0.561394i	$0.810265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			872.026i			1.05445i	0.849727	+	0.527223i	$0.176767\pi$
$827$			872.026i			1.05445i	−0.849727	+	0.527223i	$0.823233\pi$
$828$		0			0
$829$	737.822			0.890015			0.445007	−	0.895527i	$-0.353201\pi$
$829$	737.822			0.890015			0.445007	+	0.895527i	$0.353201\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−104.840	+	60.5293i	−0.125858	+	0.0726642i
$834$		0			0
$835$	620.531	−	1074.79i	0.743151	−	1.28717i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	776.072	+	448.065i	0.924996	+	0.534047i	0.885226	−	0.465162i	$-0.154004\pi$
$839$	776.072	+	448.065i	0.924996	+	0.534047i	0.0397708	+	0.999209i	$0.487337\pi$
$840$		0			0
$841$	1136.40	+	1968.29i	1.35124	+	2.34042i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			55.0027i			0.0650920i
$846$		0			0
$847$	−222.682			−0.262907
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−49.6942	+	28.6910i	−0.0583950	+	0.0337144i
$852$		0			0
$853$	−155.937	+	270.091i	−0.182810	+	0.316637i	−0.942836	−	0.333256i	$-0.891853\pi$
$853$	−155.937	+	270.091i	−0.182810	+	0.316637i	0.760026	+	0.649893i	$0.225186\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−312.540	−	180.445i	−0.364691	−	0.210555i	0.306445	−	0.951888i	$-0.400860\pi$
$857$	−312.540	−	180.445i	−0.364691	−	0.210555i	−0.671137	+	0.741334i	$0.734194\pi$
$858$		0			0
$859$	−605.549	−	1048.84i	−0.704946	−	1.22100i	−0.966711	−	0.255871i	$-0.917638\pi$
$859$	−605.549	−	1048.84i	−0.704946	−	1.22100i	0.261764	−	0.965132i	$-0.415696\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	511.500i		−	0.592699i	−0.955079	−	0.296350i	$-0.904231\pi$
$863$		−	511.500i		−	0.592699i	0.955079	−	0.296350i	$-0.0957694\pi$
$864$		0			0
$865$	−1758.30			−2.03272
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−749.842	+	432.921i	−0.862879	+	0.498183i
$870$		0			0
$871$	730.324	−	1264.96i	0.838489	−	1.45231i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−276.639	−	159.717i	−0.316158	−	0.182534i
$876$		0			0
$877$	−67.2954	−	116.559i	−0.0767337	−	0.132907i	0.825105	−	0.564979i	$-0.191116\pi$
$877$	−67.2954	−	116.559i	−0.0767337	−	0.132907i	−0.901839	+	0.432072i	$0.857782\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			405.447i			0.460212i	0.973166	+	0.230106i	$0.0739073\pi$
$881$			405.447i			0.460212i	−0.973166	+	0.230106i	$0.926093\pi$
$882$		0			0
$883$	−552.235			−0.625408			−0.312704	−	0.949851i	$-0.601235\pi$
$883$	−552.235			−0.625408			−0.312704	+	0.949851i	$0.601235\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1042.69	+	602.000i	−1.17553	+	0.678692i	−0.954976	−	0.296682i	$-0.904120\pi$
$887$	−1042.69	+	602.000i	−1.17553	+	0.678692i	−0.220554	+	0.975375i	$0.570786\pi$
$888$		0			0
$889$	−295.767	+	512.284i	−0.332697	+	0.576247i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−637.770	−	368.217i	−0.714188	−	0.412337i
$894$		0			0
$895$	356.812	+	618.016i	0.398672	+	0.690521i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	853.286i		−	0.949151i
$900$		0			0
$901$	807.098			0.895780
$902$		0			0
$903$		0			0
$904$		0			0
$905$	604.255	−	348.867i	0.667686	−	0.385488i
$906$		0			0
$907$	−660.207	+	1143.51i	−0.727901	+	1.26076i	0.229867	+	0.973222i	$0.426171\pi$
$907$	−660.207	+	1143.51i	−0.727901	+	1.26076i	−0.957769	+	0.287540i	$0.907163\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	835.732	+	482.510i	0.917379	+	0.529649i	0.882798	−	0.469753i	$-0.155657\pi$
$911$	835.732	+	482.510i	0.917379	+	0.529649i	0.0345810	+	0.999402i	$0.488990\pi$
$912$		0			0
$913$	−86.7164	−	150.197i	−0.0949796	−	0.164509i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			548.463i			0.598106i
$918$		0			0
$919$	720.089			0.783557			0.391779	−	0.920060i	$-0.371860\pi$
$919$	720.089			0.783557			0.391779	+	0.920060i	$0.371860\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−542.567	+	313.251i	−0.587830	+	0.339384i
$924$		0			0
$925$	−18.7974	+	32.5581i	−0.0203215	+	0.0351979i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	806.867	+	465.845i	0.868532	+	0.501447i	0.866860	−	0.498551i	$-0.166134\pi$
$929$	806.867	+	465.845i	0.868532	+	0.501447i	0.00167220	+	0.999999i	$0.499468\pi$
$930$		0			0
$931$	90.5608	+	156.856i	0.0972726	+	0.168481i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			541.158i			0.578778i
$936$		0			0
$937$	1370.97			1.46315			0.731575	−	0.681760i	$-0.238785\pi$
$937$	1370.97			1.46315			0.731575	+	0.681760i	$0.238785\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	197.008	−	113.743i	0.209360	−	0.120874i	−0.391654	−	0.920113i	$-0.628097\pi$
$941$	197.008	−	113.743i	0.209360	−	0.120874i	0.601014	+	0.799239i	$0.294764\pi$
$942$		0			0
$943$	−18.6863	+	32.3656i	−0.0198158	+	0.0343219i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	570.662	+	329.472i	0.602600	+	0.347911i	0.770064	−	0.637967i	$-0.220224\pi$
$947$	570.662	+	329.472i	0.602600	+	0.347911i	−0.167464	+	0.985878i	$0.553558\pi$
$948$		0			0
$949$	193.203	+	334.637i	0.203585	+	0.352620i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	217.669i		−	0.228404i	−0.993458	−	0.114202i	$-0.963569\pi$
$953$		−	217.669i		−	0.228404i	0.993458	−	0.114202i	$-0.0364311\pi$
$954$		0			0
$955$	868.604			0.909533
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−402.750	+	232.528i	−0.419969	+	0.242469i
$960$		0			0
$961$	363.585	−	629.748i	0.378340	−	0.655305i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	640.735	+	369.929i	0.663974	+	0.383346i
$966$		0			0
$967$	509.376	+	882.266i	0.526760	+	0.912374i	0.999514	+	0.0311800i	$0.00992650\pi$
$967$	509.376	+	882.266i	0.526760	+	0.912374i	−0.472754	+	0.881194i	$0.656740\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	408.547i		−	0.420749i	−0.977621	−	0.210374i	$-0.932532\pi$
$971$		−	408.547i		−	0.420749i	0.977621	−	0.210374i	$-0.0674683\pi$
$972$		0			0
$973$	309.992			0.318594
$974$		0			0
$975$		0			0
$976$		0			0
$977$	391.760	−	226.183i	0.400982	−	0.231507i	−0.285925	−	0.958252i	$-0.592301\pi$
$977$	391.760	−	226.183i	0.400982	−	0.231507i	0.686908	+	0.726745i	$0.258968\pi$
$978$		0			0
$979$	510.420	−	884.073i	0.521368	−	0.903036i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1500.44	−	866.281i	−1.52639	−	0.881262i	−0.999509	−	0.0313233i	$-0.990028\pi$
$983$	−1500.44	−	866.281i	−1.52639	−	0.881262i	−0.526881	−	0.849939i	$-0.676639\pi$
$984$		0			0
$985$	764.383	+	1323.95i	0.776023	+	1.34411i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			67.0466i			0.0677923i
$990$		0			0
$991$	394.745			0.398330			0.199165	−	0.979966i	$-0.436177\pi$
$991$	394.745			0.398330			0.199165	+	0.979966i	$0.436177\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	635.132	−	366.694i	0.638323	−	0.368536i
$996$		0			0
$997$	−82.6745	+	143.196i	−0.0829232	+	0.143627i	−0.904504	−	0.426464i	$-0.859759\pi$
$997$	−82.6745	+	143.196i	−0.0829232	+	0.143627i	0.821581	+	0.570092i	$0.193092\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.3.bg.c.2213.1		8
3.2	odd	2	inner	2268.3.bg.c.2213.4		8
9.2	odd	6		252.3.c.a.197.4	yes	4
9.4	even	3	inner	2268.3.bg.c.701.4		8
9.5	odd	6	inner	2268.3.bg.c.701.1		8
9.7	even	3		252.3.c.a.197.1	✓	4
36.7	odd	6		1008.3.d.c.449.1		4
36.11	even	6		1008.3.d.c.449.4		4
63.2	odd	6		1764.3.bk.d.557.4		8
63.11	odd	6		1764.3.bk.d.1745.1		8
63.16	even	3		1764.3.bk.d.557.1		8
63.20	even	6		1764.3.c.f.197.1		4
63.25	even	3		1764.3.bk.d.1745.4		8
63.34	odd	6		1764.3.c.f.197.4		4
63.38	even	6		1764.3.bk.e.1745.4		8
63.47	even	6		1764.3.bk.e.557.1		8
63.52	odd	6		1764.3.bk.e.1745.1		8
63.61	odd	6		1764.3.bk.e.557.4		8
72.11	even	6		4032.3.d.e.449.1		4
72.29	odd	6		4032.3.d.h.449.1		4
72.43	odd	6		4032.3.d.e.449.4		4
72.61	even	6		4032.3.d.h.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.c.a.197.1	✓	4	9.7	even	3
252.3.c.a.197.4	yes	4	9.2	odd	6
1008.3.d.c.449.1		4	36.7	odd	6
1008.3.d.c.449.4		4	36.11	even	6
1764.3.c.f.197.1		4	63.20	even	6
1764.3.c.f.197.4		4	63.34	odd	6
1764.3.bk.d.557.1		8	63.16	even	3
1764.3.bk.d.557.4		8	63.2	odd	6
1764.3.bk.d.1745.1		8	63.11	odd	6
1764.3.bk.d.1745.4		8	63.25	even	3
1764.3.bk.e.557.1		8	63.47	even	6
1764.3.bk.e.557.4		8	63.61	odd	6
1764.3.bk.e.1745.1		8	63.52	odd	6
1764.3.bk.e.1745.4		8	63.38	even	6
2268.3.bg.c.701.1		8	9.5	odd	6	inner
2268.3.bg.c.701.4		8	9.4	even	3	inner
2268.3.bg.c.2213.1		8	1.1	even	1	trivial
2268.3.bg.c.2213.4		8	3.2	odd	2	inner
4032.3.d.e.449.1		4	72.11	even	6
4032.3.d.e.449.4		4	72.43	odd	6
4032.3.d.h.449.1		4	72.29	odd	6
4032.3.d.h.449.4		4	72.61	even	6

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2268.bg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.46512 + 2.57794i) q^{5} +(1.32288 - 2.29129i) q^{7} +(5.25600 + 3.03455i) q^{11} +(6.29150 + 10.8972i) q^{13} -17.2941i q^{17} -25.8745 q^{19} +(2.09247 - 1.20809i) q^{23} +(0.791503 - 1.37092i) q^{25} +(48.3254 + 27.9007i) q^{29} +(-7.64575 - 13.2428i) q^{31} +13.6412i q^{35} -23.7490 q^{37} +(-13.3953 + 7.73381i) q^{41} +(-13.8745 + 24.0314i) q^{43} +(24.6486 + 14.2309i) q^{47} +(-3.50000 - 6.06218i) q^{49} +46.6690i q^{53} -31.2915 q^{55} +(69.8601 - 40.3337i) q^{59} +(31.3542 - 54.3072i) q^{61} +(-56.1846 - 32.4382i) q^{65} +(-58.0405 - 100.529i) q^{67} +49.7896i q^{71} +30.7085 q^{73} +(13.9061 - 8.02867i) q^{77} +(-71.3320 + 123.551i) q^{79} +(-24.7478 - 14.2882i) q^{83} +(44.5830 + 77.2200i) q^{85} -168.203i q^{89} +33.2915 q^{91} +(115.533 - 66.7028i) q^{95} +(-56.9778 + 98.6884i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} - 80 q^{19} - 36 q^{25} - 40 q^{31} + 64 q^{37} + 16 q^{43} - 28 q^{49} - 208 q^{55} + 272 q^{61} - 168 q^{67} + 288 q^{73} - 232 q^{79} + 272 q^{85} + 224 q^{91} - 96 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 80 * q^19 - 36 * q^25 - 40 * q^31 + 64 * q^37 + 16 * q^43 - 28 * q^49 - 208 * q^55 + 272 * q^61 - 168 * q^67 + 288 * q^73 - 232 * q^79 + 272 * q^85 + 224 * q^91 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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