LMFDB - Embedded newform 2252.3.d.b.1125.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,3,Mod(1125,2252)] 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("2252.1125"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2252 = 2^{2} \cdot 563 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2252.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$61.3625555339$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1125.19
Character	$\chi$	$=$	2252.1125
Dual form			2252.3.d.b.1125.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46129 q^{3} -6.41237i q^{5} -2.35196 q^{7} -2.94204 q^{9} +18.8286 q^{11} -13.2836 q^{13} +15.7827i q^{15} -1.57683 q^{17} +22.8108 q^{19} +5.78886 q^{21} -29.0053 q^{23} -16.1185 q^{25} +29.3929 q^{27} +3.79120i q^{29} +36.5354i q^{31} -46.3427 q^{33} +15.0816i q^{35} +65.6876i q^{37} +32.6949 q^{39} -4.45877i q^{41} -56.6162i q^{43} +18.8655i q^{45} -45.8070 q^{47} -43.4683 q^{49} +3.88105 q^{51} +36.2087i q^{53} -120.736i q^{55} -56.1441 q^{57} +61.7401 q^{59} +47.6548 q^{61} +6.91956 q^{63} +85.1795i q^{65} -52.2962 q^{67} +71.3904 q^{69} -23.2796 q^{71} +117.011i q^{73} +39.6724 q^{75} -44.2841 q^{77} +92.8192i q^{79} -45.8660 q^{81} +42.4434i q^{83} +10.1113i q^{85} -9.33126i q^{87} +71.4541i q^{89} +31.2425 q^{91} -89.9244i q^{93} -146.272i q^{95} -186.081i q^{97} -55.3946 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) $ 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times$.

$n$	$565$	$1127$
$\chi(n)$	$-1$	$1$

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
$3$	−2.46129			−0.820431			−0.410215	−	0.911989i	$-0.634546\pi$
$3$	−2.46129			−0.820431			−0.410215	+	0.911989i	$0.634546\pi$
$4$		0			0
$5$		−	6.41237i		−	1.28247i	−0.767343	−	0.641237i	$-0.778421\pi$
$5$		−	6.41237i		−	1.28247i	0.767343	−	0.641237i	$-0.221579\pi$
$6$		0			0
$7$	−2.35196			−0.335994			−0.167997	−	0.985788i	$-0.553730\pi$
$7$	−2.35196			−0.335994			−0.167997	+	0.985788i	$0.553730\pi$
$8$		0			0
$9$	−2.94204			−0.326893
$10$		0			0
$11$	18.8286			1.71169			0.855846	−	0.517230i	$-0.173037\pi$
$11$	18.8286			1.71169			0.855846	+	0.517230i	$0.173037\pi$
$12$		0			0
$13$	−13.2836			−1.02182			−0.510908	−	0.859635i	$-0.670691\pi$
$13$	−13.2836			−1.02182			−0.510908	+	0.859635i	$0.670691\pi$
$14$		0			0
$15$			15.7827i			1.05218i
$16$		0			0
$17$	−1.57683			−0.0927550			−0.0463775	−	0.998924i	$-0.514768\pi$
$17$	−1.57683			−0.0927550			−0.0463775	+	0.998924i	$0.514768\pi$
$18$		0			0
$19$	22.8108			1.20057			0.600285	−	0.799786i	$-0.295054\pi$
$19$	22.8108			1.20057			0.600285	+	0.799786i	$0.295054\pi$
$20$		0			0
$21$	5.78886			0.275660
$22$		0			0
$23$	−29.0053			−1.26110			−0.630549	−	0.776149i	$-0.717170\pi$
$23$	−29.0053			−1.26110			−0.630549	+	0.776149i	$0.717170\pi$
$24$		0			0
$25$	−16.1185			−0.644741
$26$		0			0
$27$	29.3929			1.08862
$28$		0			0
$29$			3.79120i			0.130731i	0.997861	+	0.0653656i	$0.0208214\pi$
$29$			3.79120i			0.130731i	−0.997861	+	0.0653656i	$0.979179\pi$
$30$		0			0
$31$			36.5354i			1.17856i	0.807928	+	0.589281i	$0.200589\pi$
$31$			36.5354i			1.17856i	−0.807928	+	0.589281i	$0.799411\pi$
$32$		0			0
$33$	−46.3427			−1.40433
$34$		0			0
$35$			15.0816i			0.430904i
$36$		0			0
$37$			65.6876i			1.77534i	0.460479	+	0.887671i	$0.347678\pi$
$37$			65.6876i			1.77534i	−0.460479	+	0.887671i	$0.652322\pi$
$38$		0			0
$39$	32.6949			0.838330
$40$		0			0
$41$		−	4.45877i		−	0.108751i	−0.998521	−	0.0543753i	$-0.982683\pi$
$41$		−	4.45877i		−	0.108751i	0.998521	−	0.0543753i	$-0.0173167\pi$
$42$		0			0
$43$		−	56.6162i		−	1.31665i	−0.752732	−	0.658327i	$-0.771264\pi$
$43$		−	56.6162i		−	1.31665i	0.752732	−	0.658327i	$-0.228736\pi$
$44$		0			0
$45$			18.8655i			0.419233i
$46$		0			0
$47$	−45.8070			−0.974616			−0.487308	−	0.873230i	$-0.662021\pi$
$47$	−45.8070			−0.974616			−0.487308	+	0.873230i	$0.662021\pi$
$48$		0			0
$49$	−43.4683			−0.887108
$50$		0			0
$51$	3.88105			0.0760990
$52$		0			0
$53$			36.2087i			0.683184i	0.939848	+	0.341592i	$0.110966\pi$
$53$			36.2087i			0.683184i	−0.939848	+	0.341592i	$0.889034\pi$
$54$		0			0
$55$		−	120.736i		−	2.19520i
$56$		0			0
$57$	−56.1441			−0.984984
$58$		0			0
$59$	61.7401			1.04644			0.523221	−	0.852197i	$-0.324730\pi$
$59$	61.7401			1.04644			0.523221	+	0.852197i	$0.324730\pi$
$60$		0			0
$61$	47.6548			0.781227			0.390613	−	0.920555i	$-0.372263\pi$
$61$	47.6548			0.781227			0.390613	+	0.920555i	$0.372263\pi$
$62$		0			0
$63$	6.91956			0.109834
$64$		0			0
$65$			85.1795i			1.31045i
$66$		0			0
$67$	−52.2962			−0.780540			−0.390270	−	0.920701i	$-0.627618\pi$
$67$	−52.2962			−0.780540			−0.390270	+	0.920701i	$0.627618\pi$
$68$		0			0
$69$	71.3904			1.03464
$70$		0			0
$71$	−23.2796			−0.327881			−0.163941	−	0.986470i	$-0.552421\pi$
$71$	−23.2796			−0.327881			−0.163941	+	0.986470i	$0.552421\pi$
$72$		0			0
$73$			117.011i			1.60289i	0.598069	+	0.801444i	$0.295935\pi$
$73$			117.011i			1.60289i	−0.598069	+	0.801444i	$0.704065\pi$
$74$		0			0
$75$	39.6724			0.528966
$76$		0			0
$77$	−44.2841			−0.575118
$78$		0			0
$79$			92.8192i			1.17493i	0.809251	+	0.587463i	$0.199873\pi$
$79$			92.8192i			1.17493i	−0.809251	+	0.587463i	$0.800127\pi$
$80$		0			0
$81$	−45.8660			−0.566247
$82$		0			0
$83$			42.4434i			0.511366i	0.966761	+	0.255683i	$0.0823004\pi$
$83$			42.4434i			0.511366i	−0.966761	+	0.255683i	$0.917700\pi$
$84$		0			0
$85$			10.1113i			0.118956i
$86$		0			0
$87$		−	9.33126i		−	0.107256i
$88$		0			0
$89$			71.4541i			0.802855i	0.915891	+	0.401427i	$0.131486\pi$
$89$			71.4541i			0.802855i	−0.915891	+	0.401427i	$0.868514\pi$
$90$		0			0
$91$	31.2425			0.343324
$92$		0			0
$93$		−	89.9244i		−	0.966929i
$94$		0			0
$95$		−	146.272i		−	1.53970i
$96$		0			0
$97$		−	186.081i		−	1.91836i	−0.282798	−	0.959179i	$-0.591263\pi$
$97$		−	186.081i		−	1.91836i	0.282798	−	0.959179i	$-0.408737\pi$
$98$		0			0
$99$	−55.3946			−0.559541
$100$		0			0
$101$	98.5186			0.975431			0.487716	−	0.873003i	$-0.337830\pi$
$101$	98.5186			0.975431			0.487716	+	0.873003i	$0.337830\pi$
$102$		0			0
$103$	−16.0891			−0.156205			−0.0781024	−	0.996945i	$-0.524886\pi$
$103$	−16.0891			−0.156205			−0.0781024	+	0.996945i	$0.524886\pi$
$104$		0			0
$105$		−	37.1203i		−	0.353527i
$106$		0			0
$107$	192.218			1.79643			0.898214	−	0.439559i	$-0.144865\pi$
$107$	192.218			1.79643			0.898214	+	0.439559i	$0.144865\pi$
$108$		0			0
$109$		−	183.480i		−	1.68330i	−0.540024	−	0.841649i	$-0.681585\pi$
$109$		−	183.480i		−	1.68330i	0.540024	−	0.841649i	$-0.318415\pi$
$110$		0			0
$111$		−	161.676i		−	1.45654i
$112$		0			0
$113$	66.7186			0.590430			0.295215	−	0.955431i	$-0.404609\pi$
$113$	66.7186			0.590430			0.295215	+	0.955431i	$0.404609\pi$
$114$		0			0
$115$			185.993i			1.61733i
$116$		0			0
$117$	39.0809			0.334025
$118$		0			0
$119$	3.70865			0.0311651
$120$		0			0
$121$	233.517			1.92989
$122$		0			0
$123$			10.9743i			0.0892223i
$124$		0			0
$125$		−	56.9513i		−	0.455610i
$126$		0			0
$127$	−58.2177			−0.458407			−0.229203	−	0.973379i	$-0.573612\pi$
$127$	−58.2177			−0.458407			−0.229203	+	0.973379i	$0.573612\pi$
$128$		0			0
$129$			139.349i			1.08022i
$130$		0			0
$131$			201.499i			1.53816i	0.639152	+	0.769080i	$0.279285\pi$
$131$			201.499i			1.53816i	−0.639152	+	0.769080i	$0.720715\pi$
$132$		0			0
$133$	−53.6501			−0.403384
$134$		0			0
$135$		−	188.478i		−	1.39613i
$136$		0			0
$137$	−21.4574			−0.156623			−0.0783116	−	0.996929i	$-0.524953\pi$
$137$	−21.4574			−0.156623			−0.0783116	+	0.996929i	$0.524953\pi$
$138$		0			0
$139$		−	171.899i		−	1.23668i	−0.785910	−	0.618341i	$-0.787805\pi$
$139$		−	171.899i		−	1.23668i	0.785910	−	0.618341i	$-0.212195\pi$
$140$		0			0
$141$	112.744			0.799605
$142$		0			0
$143$	−250.112			−1.74904
$144$		0			0
$145$	24.3106			0.167659
$146$		0			0
$147$	106.988			0.727811
$148$		0			0
$149$	62.3906			0.418729			0.209364	−	0.977838i	$-0.432861\pi$
$149$	62.3906			0.418729			0.209364	+	0.977838i	$0.432861\pi$
$150$		0			0
$151$		−	125.217i		−	0.829255i	−0.909991	−	0.414627i	$-0.863912\pi$
$151$		−	125.217i		−	0.829255i	0.909991	−	0.414627i	$-0.136088\pi$
$152$		0			0
$153$	4.63911			0.0303210
$154$		0			0
$155$	234.279			1.51148
$156$		0			0
$157$		−	85.6193i		−	0.545346i	−0.962107	−	0.272673i	$-0.912092\pi$
$157$		−	85.6193i		−	0.545346i	0.962107	−	0.272673i	$-0.0879077\pi$
$158$		0			0
$159$		−	89.1203i		−	0.560505i
$160$		0			0
$161$	68.2192			0.423721
$162$		0			0
$163$			35.1502i			0.215645i	0.994170	+	0.107823i	$0.0343878\pi$
$163$			35.1502i			0.215645i	−0.994170	+	0.107823i	$0.965612\pi$
$164$		0			0
$165$			297.167i			1.80101i
$166$		0			0
$167$			110.540i			0.661918i	0.943645	+	0.330959i	$0.107372\pi$
$167$			110.540i			0.661918i	−0.943645	+	0.330959i	$0.892628\pi$
$168$		0			0
$169$	7.45440			0.0441089
$170$		0			0
$171$	−67.1104			−0.392458
$172$		0			0
$173$			96.0218i			0.555039i	0.960720	+	0.277520i	$0.0895124\pi$
$173$			96.0218i			0.555039i	−0.960720	+	0.277520i	$0.910488\pi$
$174$		0			0
$175$	37.9101			0.216629
$176$		0			0
$177$	−151.960			−0.858533
$178$		0			0
$179$	296.216			1.65484			0.827420	−	0.561583i	$-0.189808\pi$
$179$	296.216			1.65484			0.827420	+	0.561583i	$0.189808\pi$
$180$		0			0
$181$	249.843			1.38035			0.690175	−	0.723643i	$-0.257534\pi$
$181$	249.843			1.38035			0.690175	+	0.723643i	$0.257534\pi$
$182$		0			0
$183$	−117.292			−0.640943
$184$		0			0
$185$	421.214			2.27683
$186$		0			0
$187$	−29.6896			−0.158768
$188$		0			0
$189$	−69.1308			−0.365771
$190$		0			0
$191$	−37.3020			−0.195298			−0.0976491	−	0.995221i	$-0.531132\pi$
$191$	−37.3020			−0.195298			−0.0976491	+	0.995221i	$0.531132\pi$
$192$		0			0
$193$	−221.523			−1.14779			−0.573894	−	0.818930i	$-0.694568\pi$
$193$	−221.523			−1.14779			−0.573894	+	0.818930i	$0.694568\pi$
$194$		0			0
$195$		−	209.652i		−	1.07514i
$196$		0			0
$197$	210.456			1.06831			0.534153	−	0.845388i	$-0.320631\pi$
$197$	210.456			1.06831			0.534153	+	0.845388i	$0.320631\pi$
$198$		0			0
$199$			245.711i			1.23473i	0.786677	+	0.617365i	$0.211800\pi$
$199$			245.711i			1.23473i	−0.786677	+	0.617365i	$0.788200\pi$
$200$		0			0
$201$	128.716			0.640379
$202$		0			0
$203$		−	8.91675i		−	0.0439249i
$204$		0			0
$205$	−28.5913			−0.139470
$206$		0			0
$207$	85.3347			0.412245
$208$		0			0
$209$	429.496			2.05501
$210$		0			0
$211$	249.766			1.18373			0.591863	−	0.806039i	$-0.298393\pi$
$211$	249.766			1.18373			0.591863	+	0.806039i	$0.298393\pi$
$212$		0			0
$213$	57.2978			0.269004
$214$		0			0
$215$	−363.044			−1.68858
$216$		0			0
$217$		−	85.9298i		−	0.395990i
$218$		0			0
$219$		−	287.998i		−	1.31506i
$220$		0			0
$221$	20.9461			0.0947786
$222$		0			0
$223$	−6.64740			−0.0298090			−0.0149045	−	0.999889i	$-0.504744\pi$
$223$	−6.64740			−0.0298090			−0.0149045	+	0.999889i	$0.504744\pi$
$224$		0			0
$225$	47.4214			0.210762
$226$		0			0
$227$		−	334.659i		−	1.47427i	−0.675745	−	0.737135i	$-0.736178\pi$
$227$		−	334.659i		−	1.47427i	0.675745	−	0.737135i	$-0.263822\pi$
$228$		0			0
$229$			212.545i			0.928145i	0.885797	+	0.464073i	$0.153612\pi$
$229$			212.545i			0.928145i	−0.885797	+	0.464073i	$0.846388\pi$
$230$		0			0
$231$	108.996			0.471845
$232$		0			0
$233$			35.5874i			0.152736i	0.997080	+	0.0763678i	$0.0243323\pi$
$233$			35.5874i			0.152736i	−0.997080	+	0.0763678i	$0.975668\pi$
$234$		0			0
$235$			293.731i			1.24992i
$236$		0			0
$237$		−	228.455i		−	0.963946i
$238$		0			0
$239$			228.954i			0.957967i	0.877824	+	0.478984i	$0.158995\pi$
$239$			228.954i			0.957967i	−0.877824	+	0.478984i	$0.841005\pi$
$240$		0			0
$241$	455.068			1.88825			0.944124	−	0.329589i	$-0.106910\pi$
$241$	455.068			1.88825			0.944124	+	0.329589i	$0.106910\pi$
$242$		0			0
$243$	−151.646			−0.624058
$244$		0			0
$245$			278.735i			1.13769i
$246$		0			0
$247$	−303.010			−1.22676
$248$		0			0
$249$		−	104.466i		−	0.419541i
$250$		0			0
$251$	−313.152			−1.24762			−0.623808	−	0.781577i	$-0.714415\pi$
$251$	−313.152			−1.24762			−0.623808	+	0.781577i	$0.714415\pi$
$252$		0			0
$253$	−546.129			−2.15861
$254$		0			0
$255$		−	24.8867i		−	0.0975951i
$256$		0			0
$257$	456.068			1.77458			0.887292	−	0.461209i	$-0.152584\pi$
$257$	456.068			1.77458			0.887292	+	0.461209i	$0.152584\pi$
$258$		0			0
$259$		−	154.495i		−	0.596504i
$260$		0			0
$261$		−	11.1539i		−	0.0427352i
$262$		0			0
$263$		−	141.763i		−	0.539024i	−0.962997	−	0.269512i	$-0.913138\pi$
$263$		−	141.763i		−	0.539024i	0.962997	−	0.269512i	$-0.0868624\pi$
$264$		0			0
$265$	232.184			0.876166
$266$		0			0
$267$		−	175.869i		−	0.658687i
$268$		0			0
$269$	241.408			0.897428			0.448714	−	0.893675i	$-0.351882\pi$
$269$	241.408			0.897428			0.448714	+	0.893675i	$0.351882\pi$
$270$		0			0
$271$	118.692			0.437980			0.218990	−	0.975727i	$-0.429724\pi$
$271$	118.692			0.437980			0.218990	+	0.975727i	$0.429724\pi$
$272$		0			0
$273$	−76.8969			−0.281674
$274$		0			0
$275$	−303.490			−1.10360
$276$		0			0
$277$	391.656			1.41392			0.706960	−	0.707253i	$-0.250066\pi$
$277$	391.656			1.41392			0.706960	+	0.707253i	$0.250066\pi$
$278$		0			0
$279$		−	107.489i		−	0.385264i
$280$		0			0
$281$	138.410			0.492561			0.246281	−	0.969199i	$-0.420792\pi$
$281$	138.410			0.492561			0.246281	+	0.969199i	$0.420792\pi$
$282$		0			0
$283$			502.473i			1.77552i	0.460304	+	0.887761i	$0.347740\pi$
$283$			502.473i			1.77552i	−0.460304	+	0.887761i	$0.652260\pi$
$284$		0			0
$285$			360.017i			1.26322i
$286$		0			0
$287$			10.4868i			0.0365395i
$288$		0			0
$289$	−286.514			−0.991397
$290$		0			0
$291$			457.999i			1.57388i
$292$		0			0
$293$			449.339i			1.53358i	0.641898	+	0.766790i	$0.278147\pi$
$293$			449.339i			1.53358i	−0.641898	+	0.766790i	$0.721853\pi$
$294$		0			0
$295$		−	395.900i		−	1.34204i
$296$		0			0
$297$	553.427			1.86339
$298$		0			0
$299$	385.295			1.28861
$300$		0			0
$301$			133.159i			0.442388i
$302$		0			0
$303$	−242.483			−0.800274
$304$		0			0
$305$		−	305.581i		−	1.00190i
$306$		0			0
$307$			470.310i			1.53195i	0.642868	+	0.765977i	$0.277744\pi$
$307$			470.310i			1.53195i	−0.642868	+	0.765977i	$0.722256\pi$
$308$		0			0
$309$	39.6000			0.128155
$310$		0			0
$311$		−	512.584i		−	1.64818i	−0.566459	−	0.824090i	$-0.691687\pi$
$311$		−	512.584i		−	1.64818i	0.566459	−	0.824090i	$-0.308313\pi$
$312$		0			0
$313$		−	337.796i		−	1.07922i	−0.841915	−	0.539610i	$-0.818572\pi$
$313$		−	337.796i		−	1.07922i	0.841915	−	0.539610i	$-0.181428\pi$
$314$		0			0
$315$		−	44.3708i		−	0.140860i
$316$		0			0
$317$			114.280i			0.360506i	0.983620	+	0.180253i	$0.0576916\pi$
$317$			114.280i			0.360506i	−0.983620	+	0.180253i	$0.942308\pi$
$318$		0			0
$319$			71.3831i			0.223772i
$320$		0			0
$321$	−473.104			−1.47384
$322$		0			0
$323$	−35.9689			−0.111359
$324$		0			0
$325$	214.112			0.658807
$326$		0			0
$327$			451.597i			1.38103i
$328$		0			0
$329$	107.736			0.327465
$330$		0			0
$331$		−	76.8109i		−	0.232057i	−0.993246	−	0.116029i	$-0.962984\pi$
$331$		−	76.8109i		−	0.232057i	0.993246	−	0.116029i	$-0.0370164\pi$
$332$		0			0
$333$		−	193.256i		−	0.580347i
$334$		0			0
$335$			335.342i			1.00102i
$336$		0			0
$337$	280.875			0.833456			0.416728	−	0.909031i	$-0.363177\pi$
$337$	280.875			0.833456			0.416728	+	0.909031i	$0.363177\pi$
$338$		0			0
$339$	−164.214			−0.484407
$340$		0			0
$341$			687.912i			2.01734i
$342$		0			0
$343$	217.482			0.634057
$344$		0			0
$345$		−	457.782i		−	1.32690i
$346$		0			0
$347$	425.988			1.22763			0.613816	−	0.789449i	$-0.289634\pi$
$347$	425.988			1.22763			0.613816	+	0.789449i	$0.289634\pi$
$348$		0			0
$349$	−204.290			−0.585359			−0.292679	−	0.956211i	$-0.594547\pi$
$349$	−204.290			−0.585359			−0.292679	+	0.956211i	$0.594547\pi$
$350$		0			0
$351$	−390.443			−1.11237
$352$		0			0
$353$			403.004i			1.14165i	0.821070	+	0.570827i	$0.193377\pi$
$353$			403.004i			1.14165i	−0.821070	+	0.570827i	$0.806623\pi$
$354$		0			0
$355$			149.277i			0.420499i
$356$		0			0
$357$	−9.12807			−0.0255688
$358$		0			0
$359$			94.6757i			0.263721i	0.991268	+	0.131860i	$0.0420950\pi$
$359$			94.6757i			0.263721i	−0.991268	+	0.131860i	$0.957905\pi$
$360$		0			0
$361$	159.334			0.441368
$362$		0			0
$363$	−574.753			−1.58334
$364$		0			0
$365$	750.317			2.05566
$366$		0			0
$367$		−	598.444i		−	1.63064i	−0.579013	−	0.815318i	$-0.696562\pi$
$367$		−	598.444i		−	1.63064i	0.579013	−	0.815318i	$-0.303438\pi$
$368$		0			0
$369$			13.1179i			0.0355498i
$370$		0			0
$371$		−	85.1614i		−	0.229546i
$372$		0			0
$373$		−	511.665i		−	1.37176i	−0.727716	−	0.685878i	$-0.759418\pi$
$373$		−	511.665i		−	1.37176i	0.727716	−	0.685878i	$-0.240582\pi$
$374$		0			0
$375$			140.174i			0.373797i
$376$		0			0
$377$		−	50.3609i		−	0.133583i
$378$		0			0
$379$	−101.372			−0.267471			−0.133735	−	0.991017i	$-0.542697\pi$
$379$	−101.372			−0.267471			−0.133735	+	0.991017i	$0.542697\pi$
$380$		0			0
$381$	143.291			0.376091
$382$		0			0
$383$	−548.729			−1.43271			−0.716356	−	0.697735i	$-0.754191\pi$
$383$	−548.729			−1.43271			−0.716356	+	0.697735i	$0.754191\pi$
$384$		0			0
$385$			283.966i			0.737575i
$386$		0			0
$387$			166.567i			0.430406i
$388$		0			0
$389$			169.196i			0.434950i	0.976066	+	0.217475i	$0.0697821\pi$
$389$			169.196i			0.434950i	−0.976066	+	0.217475i	$0.930218\pi$
$390$		0			0
$391$	45.7365			0.116973
$392$		0			0
$393$		−	495.948i		−	1.26195i
$394$		0			0
$395$	595.191			1.50681
$396$		0			0
$397$		−	281.507i		−	0.709085i	−0.935040	−	0.354543i	$-0.884637\pi$
$397$		−	281.507i		−	0.709085i	0.935040	−	0.354543i	$-0.115363\pi$
$398$		0			0
$399$	132.049			0.330949
$400$		0			0
$401$	−38.2780			−0.0954564			−0.0477282	−	0.998860i	$-0.515198\pi$
$401$	−38.2780			−0.0954564			−0.0477282	+	0.998860i	$0.515198\pi$
$402$		0			0
$403$		−	485.323i		−	1.20427i
$404$		0			0
$405$			294.110i			0.726198i
$406$		0			0
$407$			1236.81i			3.03884i
$408$		0			0
$409$	206.313			0.504433			0.252216	−	0.967671i	$-0.418841\pi$
$409$	206.313			0.504433			0.252216	+	0.967671i	$0.418841\pi$
$410$		0			0
$411$	52.8129			0.128499
$412$		0			0
$413$	−145.210			−0.351598
$414$		0			0
$415$	272.163			0.655814
$416$		0			0
$417$			423.093i			1.01461i
$418$		0			0
$419$		−	617.752i		−	1.47435i	−0.675703	−	0.737174i	$-0.736160\pi$
$419$		−	617.752i		−	1.47435i	0.675703	−	0.737174i	$-0.263840\pi$
$420$		0			0
$421$	−116.885			−0.277638			−0.138819	−	0.990318i	$-0.544331\pi$
$421$	−116.885			−0.277638			−0.138819	+	0.990318i	$0.544331\pi$
$422$		0			0
$423$	134.766			0.318596
$424$		0			0
$425$	25.4163			0.0598030
$426$		0			0
$427$	−112.082			−0.262488
$428$		0			0
$429$	615.599			1.43496
$430$		0			0
$431$		−	306.125i		−	0.710267i	−0.934816	−	0.355133i	$-0.884435\pi$
$431$		−	306.125i		−	0.710267i	0.934816	−	0.355133i	$-0.115565\pi$
$432$		0			0
$433$			190.174i			0.439200i	0.975590	+	0.219600i	$0.0704752\pi$
$433$			190.174i			0.439200i	−0.975590	+	0.219600i	$0.929525\pi$
$434$		0			0
$435$	−59.8355			−0.137553
$436$		0			0
$437$	−661.634			−1.51404
$438$		0			0
$439$	817.839			1.86296			0.931480	−	0.363792i	$-0.118518\pi$
$439$	817.839			1.86296			0.931480	+	0.363792i	$0.118518\pi$
$440$		0			0
$441$	127.885			0.289990
$442$		0			0
$443$			275.040i			0.620858i	0.950596	+	0.310429i	$0.100473\pi$
$443$			275.040i			0.620858i	−0.950596	+	0.310429i	$0.899527\pi$
$444$		0			0
$445$	458.190			1.02964
$446$		0			0
$447$	−153.561			−0.343538
$448$		0			0
$449$	87.5874			0.195072			0.0975361	−	0.995232i	$-0.468904\pi$
$449$	87.5874			0.195072			0.0975361	+	0.995232i	$0.468904\pi$
$450$		0			0
$451$		−	83.9525i		−	0.186147i
$452$		0			0
$453$			308.197i			0.680346i
$454$		0			0
$455$		−	200.339i		−	0.440305i
$456$		0			0
$457$			441.844i			0.966835i	0.875390	+	0.483418i	$0.160605\pi$
$457$			441.844i			0.966835i	−0.875390	+	0.483418i	$0.839395\pi$
$458$		0			0
$459$	−46.3477			−0.100975
$460$		0			0
$461$	−72.0062			−0.156196			−0.0780979	−	0.996946i	$-0.524885\pi$
$461$	−72.0062			−0.156196			−0.0780979	+	0.996946i	$0.524885\pi$
$462$		0			0
$463$		−	13.9925i		−	0.0302215i	−0.999886	−	0.0151107i	$-0.995190\pi$
$463$		−	13.9925i		−	0.0302215i	0.999886	−	0.0151107i	$-0.00481008\pi$
$464$		0			0
$465$	−576.629			−1.24006
$466$		0			0
$467$	−565.101			−1.21007			−0.605033	−	0.796200i	$-0.706840\pi$
$467$	−565.101			−1.21007			−0.605033	+	0.796200i	$0.706840\pi$
$468$		0			0
$469$	122.998			0.262257
$470$		0			0
$471$			210.734i			0.447419i
$472$		0			0
$473$		−	1066.00i		−	2.25371i
$474$		0			0
$475$	−367.677			−0.774057
$476$		0			0
$477$		−	106.528i		−	0.223328i
$478$		0			0
$479$			590.304i			1.23237i	0.787602	+	0.616184i	$0.211322\pi$
$479$			590.304i			1.23237i	−0.787602	+	0.616184i	$0.788678\pi$
$480$		0			0
$481$		−	872.569i		−	1.81407i
$482$		0			0
$483$	−167.907			−0.347634
$484$		0			0
$485$	−1193.22			−2.46025
$486$		0			0
$487$		−	345.252i		−	0.708937i	−0.935068	−	0.354469i	$-0.884662\pi$
$487$		−	345.252i		−	0.708937i	0.935068	−	0.354469i	$-0.115338\pi$
$488$		0			0
$489$		−	86.5148i		−	0.176922i
$490$		0			0
$491$	99.1684			0.201972			0.100986	−	0.994888i	$-0.467800\pi$
$491$	99.1684			0.201972			0.100986	+	0.994888i	$0.467800\pi$
$492$		0			0
$493$		−	5.97810i		−	0.0121260i
$494$		0			0
$495$			355.211i			0.717597i
$496$		0			0
$497$	54.7526			0.110166
$498$		0			0
$499$		−	866.813i		−	1.73710i	−0.495601	−	0.868550i	$-0.665052\pi$
$499$		−	866.813i		−	1.73710i	0.495601	−	0.868550i	$-0.334948\pi$
$500$		0			0
$501$		−	272.072i		−	0.543058i
$502$		0			0
$503$	−782.722			−1.55611			−0.778053	−	0.628198i	$-0.783793\pi$
$503$	−782.722			−1.55611			−0.778053	+	0.628198i	$0.783793\pi$
$504$		0			0
$505$		−	631.738i		−	1.25097i
$506$		0			0
$507$	−18.3475			−0.0361883
$508$		0			0
$509$	−814.862			−1.60091			−0.800454	−	0.599395i	$-0.795408\pi$
$509$	−814.862			−1.60091			−0.800454	+	0.599395i	$0.795408\pi$
$510$		0			0
$511$		−	275.205i		−	0.538561i
$512$		0			0
$513$	670.475			1.30697
$514$		0			0
$515$			103.169i			0.200329i
$516$		0			0
$517$	−862.482			−1.66824
$518$		0			0
$519$		−	236.338i		−	0.455371i
$520$		0			0
$521$	−732.987			−1.40688			−0.703442	−	0.710753i	$-0.748355\pi$
$521$	−732.987			−1.40688			−0.703442	+	0.710753i	$0.748355\pi$
$522$		0			0
$523$		−	768.191i		−	1.46882i	−0.678708	−	0.734408i	$-0.737460\pi$
$523$		−	768.191i		−	1.46882i	0.678708	−	0.734408i	$-0.262540\pi$
$524$		0			0
$525$	−93.3079			−0.177729
$526$		0			0
$527$		−	57.6103i		−	0.109318i
$528$		0			0
$529$	312.305			0.590368
$530$		0			0
$531$	−181.642			−0.342075
$532$		0			0
$533$			59.2286i			0.111123i
$534$		0			0
$535$		−	1232.57i		−	2.30387i
$536$		0			0
$537$	−729.075			−1.35768
$538$		0			0
$539$	−818.448			−1.51846
$540$		0			0
$541$	−261.921			−0.484143			−0.242071	−	0.970258i	$-0.577827\pi$
$541$	−261.921			−0.484143			−0.242071	+	0.970258i	$0.577827\pi$
$542$		0			0
$543$	−614.937			−1.13248
$544$		0			0
$545$	−1176.54			−2.15879
$546$		0			0
$547$			539.196i			0.985733i	0.870105	+	0.492866i	$0.164051\pi$
$547$			539.196i			0.985733i	−0.870105	+	0.492866i	$0.835949\pi$
$548$		0			0
$549$	−140.203			−0.255378
$550$		0			0
$551$			86.4805i			0.156952i
$552$		0			0
$553$		−	218.307i		−	0.394768i
$554$		0			0
$555$	−1036.73			−1.86798
$556$		0			0
$557$	−530.054			−0.951622			−0.475811	−	0.879548i	$-0.657845\pi$
$557$	−530.054			−0.951622			−0.475811	+	0.879548i	$0.657845\pi$
$558$		0			0
$559$			752.067i			1.34538i
$560$		0			0
$561$	73.0748			0.130258
$562$		0			0
$563$	−561.840	+	36.1145i	−0.997940	+	0.0641466i
$563$	−561.840	+	36.1145i	−0.997940	+	0.0641466i
$564$		0			0
$565$		−	427.824i		−	0.757211i
$566$		0			0
$567$	107.875			0.190256
$568$		0			0
$569$			43.1849i			0.0758961i	0.999280	+	0.0379480i	$0.0120821\pi$
$569$			43.1849i			0.0758961i	−0.999280	+	0.0379480i	$0.987918\pi$
$570$		0			0
$571$			398.314i			0.697572i	0.937202	+	0.348786i	$0.113406\pi$
$571$			398.314i			0.697572i	−0.937202	+	0.348786i	$0.886594\pi$
$572$		0			0
$573$	91.8110			0.160229
$574$		0			0
$575$	467.522			0.813082
$576$		0			0
$577$			364.314i			0.631393i	0.948860	+	0.315696i	$0.102238\pi$
$577$			364.314i			0.631393i	−0.948860	+	0.315696i	$0.897762\pi$
$578$		0			0
$579$	545.233			0.941680
$580$		0			0
$581$		−	99.8251i		−	0.171816i
$582$		0			0
$583$			681.760i			1.16940i
$584$		0			0
$585$		−	250.602i		−	0.428379i
$586$		0			0
$587$		−	43.3734i		−	0.0738899i	−0.999317	−	0.0369449i	$-0.988237\pi$
$587$		−	43.3734i		−	0.0738899i	0.999317	−	0.0369449i	$-0.0117626\pi$
$588$		0			0
$589$			833.403i			1.41495i
$590$		0			0
$591$	−517.995			−0.876471
$592$		0			0
$593$	749.953			1.26468			0.632338	−	0.774692i	$-0.282095\pi$
$593$	749.953			1.26468			0.632338	+	0.774692i	$0.282095\pi$
$594$		0			0
$595$		−	23.7812i		−	0.0399685i
$596$		0			0
$597$		−	604.767i		−	1.01301i
$598$		0			0
$599$	509.824			0.851125			0.425562	−	0.904929i	$-0.360076\pi$
$599$	509.824			0.851125			0.425562	+	0.904929i	$0.360076\pi$
$600$		0			0
$601$			812.631i			1.35213i	0.736841	+	0.676066i	$0.236316\pi$
$601$			812.631i			1.35213i	−0.736841	+	0.676066i	$0.763684\pi$
$602$		0			0
$603$	153.857			0.255153
$604$		0			0
$605$		−	1497.40i		−	2.47504i
$606$		0			0
$607$	199.220			0.328205			0.164103	−	0.986443i	$-0.447527\pi$
$607$	199.220			0.328205			0.164103	+	0.986443i	$0.447527\pi$
$608$		0			0
$609$			21.9467i			0.0360373i
$610$		0			0
$611$	608.482			0.995879
$612$		0			0
$613$			51.0999i			0.0833603i	0.999131	+	0.0416801i	$0.0132710\pi$
$613$			51.0999i			0.0833603i	−0.999131	+	0.0416801i	$0.986729\pi$
$614$		0			0
$615$	70.3716			0.114425
$616$		0			0
$617$			634.902i			1.02901i	0.857486	+	0.514507i	$0.172025\pi$
$617$			634.902i			1.02901i	−0.857486	+	0.514507i	$0.827975\pi$
$618$		0			0
$619$			153.822i			0.248501i	0.992251	+	0.124250i	$0.0396526\pi$
$619$			153.822i			0.248501i	−0.992251	+	0.124250i	$0.960347\pi$
$620$		0			0
$621$	−852.547			−1.37286
$622$		0			0
$623$		−	168.057i		−	0.269754i
$624$		0			0
$625$	−768.156			−1.22905
$626$		0			0
$627$	−1057.12			−1.68599
$628$		0			0
$629$		−	103.579i		−	0.164672i
$630$		0			0
$631$	150.792			0.238973			0.119487	−	0.992836i	$-0.461875\pi$
$631$	150.792			0.238973			0.119487	+	0.992836i	$0.461875\pi$
$632$		0			0
$633$	−614.747			−0.971165
$634$		0			0
$635$			373.313i			0.587895i
$636$		0			0
$637$	577.416			0.906462
$638$		0			0
$639$	68.4895			0.107182
$640$		0			0
$641$		−	269.936i		−	0.421118i	−0.977581	−	0.210559i	$-0.932472\pi$
$641$		−	269.936i		−	0.421118i	0.977581	−	0.210559i	$-0.0675283\pi$
$642$		0			0
$643$		−	844.578i		−	1.31350i	−0.754110	−	0.656748i	$-0.771931\pi$
$643$		−	844.578i		−	1.31350i	0.754110	−	0.656748i	$-0.228069\pi$
$644$		0			0
$645$	893.557			1.38536
$646$		0			0
$647$	−677.202			−1.04668			−0.523340	−	0.852124i	$-0.675314\pi$
$647$	−677.202			−1.04668			−0.523340	+	0.852124i	$0.675314\pi$
$648$		0			0
$649$	1162.48			1.79119
$650$		0			0
$651$			211.498i			0.324882i
$652$		0			0
$653$	1145.40			1.75405			0.877027	−	0.480440i	$-0.159523\pi$
$653$	1145.40			1.75405			0.877027	+	0.480440i	$0.159523\pi$
$654$		0			0
$655$	1292.09			1.97265
$656$		0			0
$657$		−	344.251i		−	0.523974i
$658$		0			0
$659$			555.216i			0.842512i	0.906942	+	0.421256i	$0.138411\pi$
$659$			555.216i			0.842512i	−0.906942	+	0.421256i	$0.861589\pi$
$660$		0			0
$661$			824.671i			1.24761i	0.781580	+	0.623806i	$0.214414\pi$
$661$			824.671i			1.24761i	−0.781580	+	0.623806i	$0.785586\pi$
$662$		0			0
$663$	−51.5544			−0.0777592
$664$		0			0
$665$			344.025i			0.517330i
$666$		0			0
$667$		−	109.965i		−	0.164865i
$668$		0			0
$669$	16.3612			0.0244562
$670$		0			0
$671$	897.275			1.33722
$672$		0			0
$673$	651.956			0.968731			0.484365	−	0.874866i	$-0.339051\pi$
$673$	651.956			0.968731			0.484365	+	0.874866i	$0.339051\pi$
$674$		0			0
$675$	−473.770			−0.701881
$676$		0			0
$677$		−	954.052i		−	1.40923i	−0.709588	−	0.704617i	$-0.751119\pi$
$677$		−	954.052i		−	1.40923i	0.709588	−	0.704617i	$-0.248881\pi$
$678$		0			0
$679$			437.654i			0.644557i
$680$		0			0
$681$			823.695i			1.20954i
$682$		0			0
$683$	−491.572			−0.719725			−0.359862	−	0.933005i	$-0.617176\pi$
$683$	−491.572			−0.719725			−0.359862	+	0.933005i	$0.617176\pi$
$684$		0			0
$685$			137.593i			0.200865i
$686$		0			0
$687$		−	523.136i		−	0.761479i
$688$		0			0
$689$		−	480.983i		−	0.698088i
$690$		0			0
$691$		−	903.372i		−	1.30734i	−0.756780	−	0.653670i	$-0.773228\pi$
$691$		−	903.372i		−	1.30734i	0.756780	−	0.653670i	$-0.226772\pi$
$692$		0			0
$693$	130.286			0.188002
$694$		0			0
$695$	−1102.28			−1.58601
$696$		0			0
$697$			7.03075i			0.0100872i
$698$		0			0
$699$		−	87.5909i		−	0.125309i
$700$		0			0
$701$			864.232i			1.23286i	0.787411	+	0.616428i	$0.211421\pi$
$701$			864.232i			1.23286i	−0.787411	+	0.616428i	$0.788579\pi$
$702$		0			0
$703$			1498.39i			2.13142i
$704$		0			0
$705$		−	722.959i		−	1.02547i
$706$		0			0
$707$	−231.712			−0.327739
$708$		0			0
$709$	661.895			0.933562			0.466781	−	0.884373i	$-0.345414\pi$
$709$	661.895			0.933562			0.466781	+	0.884373i	$0.345414\pi$
$710$		0			0
$711$		−	273.078i		−	0.384076i
$712$		0			0
$713$		−	1059.72i		−	1.48628i
$714$		0			0
$715$			1603.81i			2.24309i
$716$		0			0
$717$		−	563.523i		−	0.785946i
$718$		0			0
$719$	61.6252			0.0857096			0.0428548	−	0.999081i	$-0.486355\pi$
$719$	61.6252			0.0857096			0.0428548	+	0.999081i	$0.486355\pi$
$720$		0			0
$721$	37.8409			0.0524839
$722$		0			0
$723$	−1120.06			−1.54918
$724$		0			0
$725$		−	61.1086i		−	0.0842878i
$726$		0			0
$727$			1383.59i			1.90315i	0.307422	+	0.951573i	$0.400534\pi$
$727$			1383.59i			1.90315i	−0.307422	+	0.951573i	$0.599466\pi$
$728$		0			0
$729$	786.039			1.07824
$730$		0			0
$731$			89.2743i			0.122126i
$732$		0			0
$733$	−247.773			−0.338026			−0.169013	−	0.985614i	$-0.554058\pi$
$733$	−247.773			−0.338026			−0.169013	+	0.985614i	$0.554058\pi$
$734$		0			0
$735$		−	686.048i		−	0.933399i
$736$		0			0
$737$	−984.664			−1.33604
$738$		0			0
$739$	−231.878			−0.313773			−0.156886	−	0.987617i	$-0.550146\pi$
$739$	−231.878			−0.313773			−0.156886	+	0.987617i	$0.550146\pi$
$740$		0			0
$741$	745.797			1.00647
$742$		0			0
$743$		−	135.054i		−	0.181769i	−0.995861	−	0.0908843i	$-0.971031\pi$
$743$		−	135.054i		−	0.181769i	0.995861	−	0.0908843i	$-0.0289693\pi$
$744$		0			0
$745$		−	400.072i		−	0.537009i
$746$		0			0
$747$		−	124.870i		−	0.167162i
$748$		0			0
$749$	−452.088			−0.603589
$750$		0			0
$751$	−657.248			−0.875163			−0.437582	−	0.899179i	$-0.644165\pi$
$751$	−657.248			−0.875163			−0.437582	+	0.899179i	$0.644165\pi$
$752$		0			0
$753$	770.758			1.02358
$754$		0			0
$755$	−802.941			−1.06350
$756$		0			0
$757$	1375.45			1.81697			0.908484	−	0.417919i	$-0.137241\pi$
$757$	1375.45			1.81697			0.908484	+	0.417919i	$0.137241\pi$
$758$		0			0
$759$	1344.18			1.77099
$760$		0			0
$761$			945.145i			1.24198i	0.783819	+	0.620989i	$0.213269\pi$
$761$			945.145i			1.24198i	−0.783819	+	0.620989i	$0.786731\pi$
$762$		0			0
$763$			431.536i			0.565578i
$764$		0			0
$765$		−	29.7477i		−	0.0388859i
$766$		0			0
$767$	−820.131			−1.06927
$768$		0			0
$769$			909.580i			1.18281i	0.806375	+	0.591404i	$0.201426\pi$
$769$			909.580i			1.18281i	−0.806375	+	0.591404i	$0.798574\pi$
$770$		0			0
$771$	−1122.52			−1.45592
$772$		0			0
$773$	−1294.26			−1.67433			−0.837167	−	0.546948i	$-0.815790\pi$
$773$	−1294.26			−1.67433			−0.837167	+	0.546948i	$0.815790\pi$
$774$		0			0
$775$		−	588.898i		−	0.759868i
$776$		0			0
$777$			380.256i			0.489390i
$778$		0			0
$779$		−	101.708i		−	0.130563i
$780$		0			0
$781$	−438.322			−0.561232
$782$		0			0
$783$			111.434i			0.142317i
$784$		0			0
$785$	−549.023			−0.699393
$786$		0			0
$787$			1204.82i			1.53090i	0.643496	+	0.765449i	$0.277483\pi$
$787$			1204.82i			1.53090i	−0.643496	+	0.765449i	$0.722517\pi$
$788$		0			0
$789$			348.921i			0.442232i
$790$		0			0
$791$	−156.919			−0.198381
$792$		0			0
$793$	−633.029			−0.798271
$794$		0			0
$795$	−571.472			−0.718833
$796$		0			0
$797$		−	326.182i		−	0.409262i	−0.978839	−	0.204631i	$-0.934401\pi$
$797$		−	326.182i		−	0.409262i	0.978839	−	0.204631i	$-0.0655995\pi$
$798$		0			0
$799$	72.2300			0.0904005
$800$		0			0
$801$		−	210.221i		−	0.262448i
$802$		0			0
$803$			2203.15i			2.74365i
$804$		0			0
$805$		−	437.447i		−	0.543412i
$806$		0			0
$807$	−594.176			−0.736278
$808$		0			0
$809$	157.001			0.194069			0.0970343	−	0.995281i	$-0.469064\pi$
$809$	157.001			0.194069			0.0970343	+	0.995281i	$0.469064\pi$
$810$		0			0
$811$	−737.734			−0.909659			−0.454830	−	0.890578i	$-0.650300\pi$
$811$	−737.734			−0.909659			−0.454830	+	0.890578i	$0.650300\pi$
$812$		0			0
$813$	−292.137			−0.359332
$814$		0			0
$815$	225.396			0.276559
$816$		0			0
$817$		−	1291.46i		−	1.58074i
$818$		0			0
$819$	−91.9167			−0.112230
$820$		0			0
$821$	−483.724			−0.589189			−0.294594	−	0.955622i	$-0.595185\pi$
$821$	−483.724			−0.589189			−0.294594	+	0.955622i	$0.595185\pi$
$822$		0			0
$823$		−	351.661i		−	0.427291i	−0.976911	−	0.213646i	$-0.931466\pi$
$823$		−	351.661i		−	0.427291i	0.976911	−	0.213646i	$-0.0685338\pi$
$824$		0			0
$825$	746.977			0.905426
$826$		0			0
$827$		−	537.116i		−	0.649475i	−0.945804	−	0.324738i	$-0.894724\pi$
$827$		−	537.116i		−	0.649475i	0.945804	−	0.324738i	$-0.105276\pi$
$828$		0			0
$829$		−	342.852i		−	0.413573i	−0.978386	−	0.206787i	$-0.933699\pi$
$829$		−	342.852i		−	0.413573i	0.978386	−	0.206787i	$-0.0663006\pi$
$830$		0			0
$831$	−963.980			−1.16002
$832$		0			0
$833$	68.5423			0.0822837
$834$		0			0
$835$	708.826			0.848894
$836$		0			0
$837$			1073.88i			1.28301i
$838$		0			0
$839$	347.140			0.413755			0.206877	−	0.978367i	$-0.433670\pi$
$839$	347.140			0.413755			0.206877	+	0.978367i	$0.433670\pi$
$840$		0			0
$841$	826.627			0.982909
$842$		0			0
$843$	−340.667			−0.404112
$844$		0			0
$845$		−	47.8004i		−	0.0565685i
$846$		0			0
$847$	−549.222			−0.648432
$848$		0			0
$849$		−	1236.73i		−	1.45669i
$850$		0			0
$851$		−	1905.29i		−	2.23888i
$852$		0			0
$853$			370.797i			0.434697i	0.976094	+	0.217349i	$0.0697409\pi$
$853$			370.797i			0.434697i	−0.976094	+	0.217349i	$0.930259\pi$
$854$		0			0
$855$			430.337i			0.503318i
$856$		0			0
$857$		−	421.331i		−	0.491635i	−0.969316	−	0.245817i	$-0.920944\pi$
$857$		−	421.331i		−	0.491635i	0.969316	−	0.245817i	$-0.0790564\pi$
$858$		0			0
$859$	386.117			0.449496			0.224748	−	0.974417i	$-0.427844\pi$
$859$	386.117			0.449496			0.224748	+	0.974417i	$0.427844\pi$
$860$		0			0
$861$		−	25.8112i		−	0.0299782i
$862$		0			0
$863$	452.819			0.524704			0.262352	−	0.964972i	$-0.415502\pi$
$863$	452.819			0.524704			0.262352	+	0.964972i	$0.415502\pi$
$864$		0			0
$865$	615.728			0.711824
$866$		0			0
$867$	705.194			0.813372
$868$		0			0
$869$			1747.66i			2.01111i
$870$		0			0
$871$	694.682			0.797568
$872$		0			0
$873$			547.457i			0.627099i
$874$		0			0
$875$			133.947i			0.153082i
$876$		0			0
$877$	1525.02			1.73891			0.869454	−	0.494014i	$-0.164471\pi$
$877$	1525.02			1.73891			0.869454	+	0.494014i	$0.164471\pi$
$878$		0			0
$879$		−	1105.95i		−	1.25820i
$880$		0			0
$881$	−1124.87			−1.27681			−0.638403	−	0.769703i	$-0.720405\pi$
$881$	−1124.87			−1.27681			−0.638403	+	0.769703i	$0.720405\pi$
$882$		0			0
$883$		−	615.507i		−	0.697063i	−0.937297	−	0.348532i	$-0.886680\pi$
$883$		−	615.507i		−	0.697063i	0.937297	−	0.348532i	$-0.113320\pi$
$884$		0			0
$885$			974.427i			1.10105i
$886$		0			0
$887$	1223.33			1.37918			0.689588	−	0.724202i	$-0.257792\pi$
$887$	1223.33			1.37918			0.689588	+	0.724202i	$0.257792\pi$
$888$		0			0
$889$	136.926			0.154022
$890$		0			0
$891$	−863.594			−0.969241
$892$		0			0
$893$	−1044.89			−1.17010
$894$		0			0
$895$		−	1899.45i		−	2.12229i
$896$		0			0
$897$	−948.323			−1.05722
$898$		0			0
$899$	−138.513			−0.154075
$900$		0			0
$901$		−	57.0952i		−	0.0633687i
$902$		0			0
$903$		−	327.743i		−	0.362949i
$904$		0			0
$905$		−	1602.09i		−	1.77026i
$906$		0			0
$907$	388.129			0.427926			0.213963	−	0.976842i	$-0.431363\pi$
$907$	388.129			0.427926			0.213963	+	0.976842i	$0.431363\pi$
$908$		0			0
$909$	−289.846			−0.318862
$910$		0			0
$911$			1325.20i			1.45466i	0.686286	+	0.727332i	$0.259240\pi$
$911$			1325.20i			1.45466i	−0.686286	+	0.727332i	$0.740760\pi$
$912$		0			0
$913$			799.150i			0.875302i
$914$		0			0
$915$			752.123i			0.821993i
$916$		0			0
$917$		−	473.917i		−	0.516813i
$918$		0			0
$919$			740.693i			0.805977i	0.915205	+	0.402988i	$0.132028\pi$
$919$			740.693i			0.805977i	−0.915205	+	0.402988i	$0.867972\pi$
$920$		0			0
$921$		−	1157.57i		−	1.25686i
$922$		0			0
$923$	309.237			0.335035
$924$		0			0
$925$		−	1058.79i		−	1.14464i
$926$		0			0
$927$	47.3348			0.0510624
$928$		0			0
$929$		−	315.228i		−	0.339320i	−0.985503	−	0.169660i	$-0.945733\pi$
$929$		−	315.228i		−	0.339320i	0.985503	−	0.169660i	$-0.0542670\pi$
$930$		0			0
$931$	−991.548			−1.06504
$932$		0			0
$933$			1261.62i			1.35222i
$934$		0			0
$935$			190.381i			0.203616i
$936$		0			0
$937$			1540.72i			1.64431i	0.569262	+	0.822156i	$0.307229\pi$
$937$			1540.72i			1.64431i	−0.569262	+	0.822156i	$0.692771\pi$
$938$		0			0
$939$			831.414i			0.885425i
$940$		0			0
$941$			1519.14i			1.61439i	0.590283	+	0.807196i	$0.299016\pi$
$941$			1519.14i			1.61439i	−0.590283	+	0.807196i	$0.700984\pi$
$942$		0			0
$943$			129.328i			0.137145i
$944$		0			0
$945$			443.292i			0.469092i
$946$		0			0
$947$		−	1279.36i		−	1.35096i	−0.737379	−	0.675479i	$-0.763937\pi$
$947$		−	1279.36i		−	1.35096i	0.737379	−	0.675479i	$-0.236063\pi$
$948$		0			0
$949$		−	1554.33i		−	1.63786i
$950$		0			0
$951$		−	281.277i		−	0.295770i
$952$		0			0
$953$	65.0206			0.0682272			0.0341136	−	0.999418i	$-0.489139\pi$
$953$	65.0206			0.0682272			0.0341136	+	0.999418i	$0.489139\pi$
$954$		0			0
$955$			239.194i			0.250465i
$956$		0			0
$957$		−	175.695i		−	0.183589i
$958$		0			0
$959$	50.4669			0.0526245
$960$		0			0
$961$	−373.838			−0.389009
$962$		0			0
$963$	−565.512			−0.587240
$964$		0			0
$965$			1420.49i			1.47201i
$966$		0			0
$967$	−532.548			−0.550721			−0.275361	−	0.961341i	$-0.588797\pi$
$967$	−532.548			−0.550721			−0.275361	+	0.961341i	$0.588797\pi$
$968$		0			0
$969$	88.5300			0.0913622
$970$		0			0
$971$		−	1421.97i		−	1.46444i	−0.681067	−	0.732221i	$-0.738484\pi$
$971$		−	1421.97i		−	1.46444i	0.681067	−	0.732221i	$-0.261516\pi$
$972$		0			0
$973$			404.299i			0.415518i
$974$		0			0
$975$	−526.993			−0.540506
$976$		0			0
$977$			1341.58i			1.37316i	0.727053	+	0.686581i	$0.240889\pi$
$977$			1341.58i			1.37316i	−0.727053	+	0.686581i	$0.759111\pi$
$978$		0			0
$979$			1345.38i			1.37424i
$980$		0			0
$981$			539.804i			0.550259i
$982$		0			0
$983$		−	265.739i		−	0.270335i	−0.990823	−	0.135167i	$-0.956843\pi$
$983$		−	265.739i		−	0.270335i	0.990823	−	0.135167i	$-0.0431572\pi$
$984$		0			0
$985$		−	1349.52i		−	1.37008i
$986$		0			0
$987$	−265.170			−0.268663
$988$		0			0
$989$			1642.17i			1.66043i
$990$		0			0
$991$	−737.282			−0.743978			−0.371989	−	0.928237i	$-0.621324\pi$
$991$	−737.282			−0.743978			−0.371989	+	0.928237i	$0.621324\pi$
$992$		0			0
$993$			189.054i			0.190387i
$994$		0			0
$995$	1575.59			1.58351
$996$		0			0
$997$	890.548			0.893228			0.446614	−	0.894727i	$-0.352630\pi$
$997$	890.548			0.893228			0.446614	+	0.894727i	$0.352630\pi$
$998$		0			0
$999$			1930.75i			1.93268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2252.3.d.b.1125.19	✓	76
563.562	odd	2	inner	2252.3.d.b.1125.20	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2252.3.d.b.1125.19	✓	76	1.1	even	1	trivial
2252.3.d.b.1125.20	yes	76	563.562	odd	2	inner

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:	\( N \)	\(=\)	\( 2252 = 2^{2} \cdot 563 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.46129 q^{3} -6.41237i q^{5} -2.35196 q^{7} -2.94204 q^{9} +18.8286 q^{11} -13.2836 q^{13} +15.7827i q^{15} -1.57683 q^{17} +22.8108 q^{19} +5.78886 q^{21} -29.0053 q^{23} -16.1185 q^{25} +29.3929 q^{27} +3.79120i q^{29} +36.5354i q^{31} -46.3427 q^{33} +15.0816i q^{35} +65.6876i q^{37} +32.6949 q^{39} -4.45877i q^{41} -56.6162i q^{43} +18.8655i q^{45} -45.8070 q^{47} -43.4683 q^{49} +3.88105 q^{51} +36.2087i q^{53} -120.736i q^{55} -56.1441 q^{57} +61.7401 q^{59} +47.6548 q^{61} +6.91956 q^{63} +85.1795i q^{65} -52.2962 q^{67} +71.3904 q^{69} -23.2796 q^{71} +117.011i q^{73} +39.6724 q^{75} -44.2841 q^{77} +92.8192i q^{79} -45.8660 q^{81} +42.4434i q^{83} +10.1113i q^{85} -9.33126i q^{87} +71.4541i q^{89} +31.2425 q^{91} -89.9244i q^{93} -146.272i q^{95} -186.081i q^{97} -55.3946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) \) 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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