LMFDB - Embedded newform 3100.3.d.g.1301.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.5
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.g.1301.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.53436i q^{3} +7.77177 q^{7} -3.49167 q^{9} +O(q^{10})$	$q-3.53436i q^{3} +7.77177 q^{7} -3.49167 q^{9} -10.7776i q^{11} -10.7132i q^{13} -21.5623i q^{17} +18.1421 q^{19} -27.4682i q^{21} +13.0683i q^{23} -19.4684i q^{27} -38.9378i q^{29} +(26.0829 - 16.7536i) q^{31} -38.0920 q^{33} +48.3699i q^{37} -37.8643 q^{39} -63.7021 q^{41} +2.27071i q^{43} +72.8538 q^{47} +11.4005 q^{49} -76.2089 q^{51} +100.964i q^{53} -64.1206i q^{57} +117.059 q^{59} +20.9013i q^{61} -27.1365 q^{63} -15.3857 q^{67} +46.1879 q^{69} +61.1605 q^{71} -40.9351i q^{73} -83.7613i q^{77} -95.6498i q^{79} -100.233 q^{81} -28.0391i q^{83} -137.620 q^{87} +16.8598i q^{89} -83.2606i q^{91} +(-59.2131 - 92.1863i) q^{93} -171.203 q^{97} +37.6319i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) $ 22 * q + 4 * q^7 - 72 * q^9	$ 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) $ 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.53436i		−	1.17812i	−0.808090	−	0.589059i	$-0.799498\pi$
$3$		−	3.53436i		−	1.17812i	0.808090	−	0.589059i	$-0.200502\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.77177			1.11025			0.555127	−	0.831766i	$-0.312670\pi$
$7$	7.77177			1.11025			0.555127	+	0.831766i	$0.312670\pi$
$8$		0			0
$9$	−3.49167			−0.387963
$10$		0			0
$11$		−	10.7776i		−	0.979784i	−0.871783	−	0.489892i	$-0.837036\pi$
$11$		−	10.7776i		−	0.979784i	0.871783	−	0.489892i	$-0.162964\pi$
$12$		0			0
$13$		−	10.7132i		−	0.824093i	−0.911163	−	0.412047i	$-0.864814\pi$
$13$		−	10.7132i		−	0.824093i	0.911163	−	0.412047i	$-0.135186\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	21.5623i		−	1.26837i	−0.773181	−	0.634186i	$-0.781336\pi$
$17$		−	21.5623i		−	1.26837i	0.773181	−	0.634186i	$-0.218664\pi$
$18$		0			0
$19$	18.1421			0.954846			0.477423	−	0.878674i	$-0.341571\pi$
$19$	18.1421			0.954846			0.477423	+	0.878674i	$0.341571\pi$
$20$		0			0
$21$		−	27.4682i		−	1.30801i
$22$		0			0
$23$			13.0683i			0.568185i	0.958797	+	0.284093i	$0.0916924\pi$
$23$			13.0683i			0.568185i	−0.958797	+	0.284093i	$0.908308\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	19.4684i		−	0.721052i
$28$		0			0
$29$		−	38.9378i		−	1.34268i	−0.741148	−	0.671342i	$-0.765718\pi$
$29$		−	38.9378i		−	1.34268i	0.741148	−	0.671342i	$-0.234282\pi$
$30$		0			0
$31$	26.0829	−	16.7536i	0.841384	−	0.540438i
$31$	26.0829	−	16.7536i	0.841384	−	0.540438i
$32$		0			0
$33$	−38.0920			−1.15430
$34$		0			0
$35$		0			0
$36$		0			0
$37$			48.3699i			1.30729i	0.756799	+	0.653647i	$0.226762\pi$
$37$			48.3699i			1.30729i	−0.756799	+	0.653647i	$0.773238\pi$
$38$		0			0
$39$	−37.8643			−0.970879
$40$		0			0
$41$	−63.7021			−1.55371			−0.776855	−	0.629680i	$-0.783186\pi$
$41$	−63.7021			−1.55371			−0.776855	+	0.629680i	$0.783186\pi$
$42$		0			0
$43$			2.27071i			0.0528073i	0.999651	+	0.0264036i	$0.00840552\pi$
$43$			2.27071i			0.0528073i	−0.999651	+	0.0264036i	$0.991594\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	72.8538			1.55008			0.775040	−	0.631912i	$-0.217730\pi$
$47$	72.8538			1.55008			0.775040	+	0.631912i	$0.217730\pi$
$48$		0			0
$49$	11.4005			0.232662
$50$		0			0
$51$	−76.2089			−1.49429
$52$		0			0
$53$			100.964i			1.90499i	0.304557	+	0.952494i	$0.401492\pi$
$53$			100.964i			1.90499i	−0.304557	+	0.952494i	$0.598508\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	64.1206i		−	1.12492i
$58$		0			0
$59$	117.059			1.98406			0.992028	−	0.126020i	$-0.0402203\pi$
$59$	117.059			1.98406			0.992028	+	0.126020i	$0.0402203\pi$
$60$		0			0
$61$			20.9013i			0.342645i	0.985215	+	0.171322i	$0.0548039\pi$
$61$			20.9013i			0.342645i	−0.985215	+	0.171322i	$0.945196\pi$
$62$		0			0
$63$	−27.1365			−0.430738
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−15.3857			−0.229638			−0.114819	−	0.993386i	$-0.536629\pi$
$67$	−15.3857			−0.229638			−0.114819	+	0.993386i	$0.536629\pi$
$68$		0			0
$69$	46.1879			0.669390
$70$		0			0
$71$	61.1605			0.861416			0.430708	−	0.902491i	$-0.358264\pi$
$71$	61.1605			0.861416			0.430708	+	0.902491i	$0.358264\pi$
$72$		0			0
$73$		−	40.9351i		−	0.560754i	−0.959890	−	0.280377i	$-0.909540\pi$
$73$		−	40.9351i		−	0.560754i	0.959890	−	0.280377i	$-0.0904595\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	83.7613i		−	1.08781i
$78$		0			0
$79$		−	95.6498i		−	1.21076i	−0.795938	−	0.605379i	$-0.793022\pi$
$79$		−	95.6498i		−	1.21076i	0.795938	−	0.605379i	$-0.206978\pi$
$80$		0			0
$81$	−100.233			−1.23745
$82$		0			0
$83$		−	28.0391i		−	0.337821i	−0.985631	−	0.168910i	$-0.945975\pi$
$83$		−	28.0391i		−	0.337821i	0.985631	−	0.168910i	$-0.0540248\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−137.620			−1.58184
$88$		0			0
$89$			16.8598i			0.189436i	0.995504	+	0.0947179i	$0.0301949\pi$
$89$			16.8598i			0.189436i	−0.995504	+	0.0947179i	$0.969805\pi$
$90$		0			0
$91$		−	83.2606i		−	0.914952i
$92$		0			0
$93$	−59.2131	−	92.1863i	−0.636700	−	0.991250i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−171.203			−1.76498			−0.882490	−	0.470331i	$-0.844135\pi$
$97$	−171.203			−1.76498			−0.882490	+	0.470331i	$0.844135\pi$
$98$		0			0
$99$			37.6319i			0.380120i
$100$		0			0
$101$	97.9009			0.969315			0.484658	−	0.874704i	$-0.338944\pi$
$101$	97.9009			0.969315			0.484658	+	0.874704i	$0.338944\pi$
$102$		0			0
$103$	−140.117			−1.36036			−0.680181	−	0.733044i	$-0.738099\pi$
$103$	−140.117			−1.36036			−0.680181	+	0.733044i	$0.738099\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	96.7388			0.904101			0.452051	−	0.891992i	$-0.350693\pi$
$107$	96.7388			0.904101			0.452051	+	0.891992i	$0.350693\pi$
$108$		0			0
$109$	−3.89207			−0.0357071			−0.0178535	−	0.999841i	$-0.505683\pi$
$109$	−3.89207			−0.0357071			−0.0178535	+	0.999841i	$0.505683\pi$
$110$		0			0
$111$	170.956			1.54015
$112$		0			0
$113$	76.3471			0.675638			0.337819	−	0.941211i	$-0.390311\pi$
$113$	76.3471			0.675638			0.337819	+	0.941211i	$0.390311\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			37.4070i			0.319718i
$118$		0			0
$119$		−	167.577i		−	1.40821i
$120$		0			0
$121$	4.84278			0.0400230
$122$		0			0
$123$			225.146i			1.83045i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	93.9787i		−	0.739990i	−0.929034	−	0.369995i	$-0.879359\pi$
$127$		−	93.9787i		−	0.739990i	0.929034	−	0.369995i	$-0.120641\pi$
$128$		0			0
$129$	8.02551			0.0622132
$130$		0			0
$131$	−103.496			−0.790049			−0.395024	−	0.918671i	$-0.629264\pi$
$131$	−103.496			−0.790049			−0.395024	+	0.918671i	$0.629264\pi$
$132$		0			0
$133$	140.996			1.06012
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	133.135i		−	0.971787i	−0.874018	−	0.485894i	$-0.838494\pi$
$137$		−	133.135i		−	0.971787i	0.874018	−	0.485894i	$-0.161506\pi$
$138$		0			0
$139$		−	12.4978i		−	0.0899120i	−0.998989	−	0.0449560i	$-0.985685\pi$
$139$		−	12.4978i		−	0.0899120i	0.998989	−	0.0449560i	$-0.0143148\pi$
$140$		0			0
$141$		−	257.491i		−	1.82618i
$142$		0			0
$143$	−115.463			−0.807433
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	40.2933i		−	0.274104i
$148$		0			0
$149$	219.246			1.47145			0.735725	−	0.677280i	$-0.236841\pi$
$149$	219.246			1.47145			0.735725	+	0.677280i	$0.236841\pi$
$150$		0			0
$151$			202.050i			1.33808i	0.743227	+	0.669040i	$0.233294\pi$
$151$			202.050i			1.33808i	−0.743227	+	0.669040i	$0.766706\pi$
$152$		0			0
$153$			75.2885i			0.492082i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−284.722			−1.81352			−0.906760	−	0.421648i	$-0.861452\pi$
$157$	−284.722			−1.81352			−0.906760	+	0.421648i	$0.861452\pi$
$158$		0			0
$159$	356.844			2.24430
$160$		0			0
$161$			101.564i			0.630830i
$162$		0			0
$163$	−317.599			−1.94846			−0.974230	−	0.225558i	$-0.927580\pi$
$163$	−317.599			−1.94846			−0.974230	+	0.225558i	$0.927580\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	153.261i		−	0.917729i	−0.888506	−	0.458864i	$-0.848256\pi$
$167$		−	153.261i		−	0.917729i	0.888506	−	0.458864i	$-0.151744\pi$
$168$		0			0
$169$	54.2271			0.320871
$170$		0			0
$171$	−63.3462			−0.370445
$172$		0			0
$173$	−26.1139			−0.150947			−0.0754736	−	0.997148i	$-0.524047\pi$
$173$	−26.1139			−0.150947			−0.0754736	+	0.997148i	$0.524047\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	413.729i		−	2.33745i
$178$		0			0
$179$			258.026i			1.44149i	0.693203	+	0.720743i	$0.256199\pi$
$179$			258.026i			1.44149i	−0.693203	+	0.720743i	$0.743801\pi$
$180$		0			0
$181$		−	246.887i		−	1.36402i	−0.731345	−	0.682008i	$-0.761107\pi$
$181$		−	246.887i		−	1.36402i	0.731345	−	0.682008i	$-0.238893\pi$
$182$		0			0
$183$	73.8727			0.403676
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−232.390			−1.24273
$188$		0			0
$189$		−	151.304i		−	0.800550i
$190$		0			0
$191$	188.761			0.988280			0.494140	−	0.869382i	$-0.335483\pi$
$191$	188.761			0.988280			0.494140	+	0.869382i	$0.335483\pi$
$192$		0			0
$193$	371.772			1.92628			0.963140	−	0.269002i	$-0.0866938\pi$
$193$	371.772			1.92628			0.963140	+	0.269002i	$0.0866938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			343.630i			1.74432i	0.489224	+	0.872158i	$0.337280\pi$
$197$			343.630i			1.74432i	−0.489224	+	0.872158i	$0.662720\pi$
$198$		0			0
$199$			329.717i			1.65687i	0.560087	+	0.828434i	$0.310768\pi$
$199$			329.717i			1.65687i	−0.560087	+	0.828434i	$0.689232\pi$
$200$		0			0
$201$			54.3787i			0.270541i
$202$		0			0
$203$		−	302.616i		−	1.49072i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	45.6301i		−	0.220435i
$208$		0			0
$209$		−	195.529i		−	0.935543i
$210$		0			0
$211$	−357.657			−1.69505			−0.847527	−	0.530752i	$-0.821910\pi$
$211$	−357.657			−1.69505			−0.847527	+	0.530752i	$0.821910\pi$
$212$		0			0
$213$		−	216.163i		−	1.01485i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	202.710	−	130.205i	0.934150	−	0.600023i
$218$		0			0
$219$	−144.679			−0.660635
$220$		0			0
$221$	−231.002			−1.04526
$222$		0			0
$223$			130.880i			0.586907i	0.955973	+	0.293454i	$0.0948046\pi$
$223$			130.880i			0.586907i	−0.955973	+	0.293454i	$0.905195\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−97.0525			−0.427544			−0.213772	−	0.976884i	$-0.568575\pi$
$227$	−97.0525			−0.427544			−0.213772	+	0.976884i	$0.568575\pi$
$228$		0			0
$229$			3.62387i			0.0158248i	0.999969	+	0.00791238i	$0.00251861\pi$
$229$			3.62387i			0.0158248i	−0.999969	+	0.00791238i	$0.997481\pi$
$230$		0			0
$231$	−296.042			−1.28157
$232$		0			0
$233$	−226.722			−0.973056			−0.486528	−	0.873665i	$-0.661737\pi$
$233$	−226.722			−0.973056			−0.486528	+	0.873665i	$0.661737\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−338.061			−1.42642
$238$		0			0
$239$			408.954i			1.71110i	0.517718	+	0.855552i	$0.326782\pi$
$239$			408.954i			1.71110i	−0.517718	+	0.855552i	$0.673218\pi$
$240$		0			0
$241$		−	412.154i		−	1.71018i	−0.518479	−	0.855091i	$-0.673501\pi$
$241$		−	412.154i		−	1.71018i	0.518479	−	0.855091i	$-0.326499\pi$
$242$		0			0
$243$			179.044i			0.736809i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	194.360i		−	0.786882i
$248$		0			0
$249$	−99.1002			−0.397993
$250$		0			0
$251$			62.8482i			0.250391i	0.992132	+	0.125196i	$0.0399559\pi$
$251$			62.8482i			0.250391i	−0.992132	+	0.125196i	$0.960044\pi$
$252$		0			0
$253$	140.845			0.556699
$254$		0			0
$255$		0			0
$256$		0			0
$257$	120.299			0.468090			0.234045	−	0.972226i	$-0.424804\pi$
$257$	120.299			0.468090			0.234045	+	0.972226i	$0.424804\pi$
$258$		0			0
$259$			375.920i			1.45143i
$260$		0			0
$261$			135.958i			0.520912i
$262$		0			0
$263$		−	160.324i		−	0.609598i	−0.952417	−	0.304799i	$-0.901411\pi$
$263$		−	160.324i		−	0.609598i	0.952417	−	0.304799i	$-0.0985893\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	59.5885			0.223178
$268$		0			0
$269$			80.9441i			0.300907i	0.988617	+	0.150454i	$0.0480734\pi$
$269$			80.9441i			0.300907i	−0.988617	+	0.150454i	$0.951927\pi$
$270$		0			0
$271$		−	215.268i		−	0.794348i	−0.917743	−	0.397174i	$-0.869991\pi$
$271$		−	215.268i		−	0.794348i	0.917743	−	0.397174i	$-0.130009\pi$
$272$		0			0
$273$	−294.273			−1.07792
$274$		0			0
$275$		0			0
$276$		0			0
$277$			275.893i			0.996003i	0.867176	+	0.498001i	$0.165933\pi$
$277$			275.893i			0.996003i	−0.867176	+	0.498001i	$0.834067\pi$
$278$		0			0
$279$	−91.0729	+	58.4979i	−0.326426	+	0.209670i
$280$		0			0
$281$	−41.9733			−0.149371			−0.0746857	−	0.997207i	$-0.523795\pi$
$281$	−41.9733			−0.149371			−0.0746857	+	0.997207i	$0.523795\pi$
$282$		0			0
$283$	56.6413			0.200146			0.100073	−	0.994980i	$-0.468092\pi$
$283$	56.6413			0.200146			0.100073	+	0.994980i	$0.468092\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−495.078			−1.72501
$288$		0			0
$289$	−175.933			−0.608765
$290$		0			0
$291$			605.093i			2.07936i
$292$		0			0
$293$	−166.989			−0.569929			−0.284964	−	0.958538i	$-0.591982\pi$
$293$	−166.989			−0.569929			−0.284964	+	0.958538i	$0.591982\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−209.823			−0.706475
$298$		0			0
$299$	140.003			0.468238
$300$		0			0
$301$			17.6475i			0.0586294i
$302$		0			0
$303$		−	346.016i		−	1.14197i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−239.329			−0.779575			−0.389787	−	0.920905i	$-0.627452\pi$
$307$	−239.329			−0.779575			−0.389787	+	0.920905i	$0.627452\pi$
$308$		0			0
$309$			495.224i			1.60267i
$310$		0			0
$311$	−598.556			−1.92462			−0.962308	−	0.271961i	$-0.912328\pi$
$311$	−598.556			−1.92462			−0.962308	+	0.271961i	$0.912328\pi$
$312$		0			0
$313$		−	302.161i		−	0.965371i	−0.875794	−	0.482686i	$-0.839661\pi$
$313$		−	302.161i		−	0.965371i	0.875794	−	0.482686i	$-0.160339\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−49.1362			−0.155004			−0.0775019	−	0.996992i	$-0.524694\pi$
$317$	−49.1362			−0.155004			−0.0775019	+	0.996992i	$0.524694\pi$
$318$		0			0
$319$	−419.657			−1.31554
$320$		0			0
$321$		−	341.909i		−	1.06514i
$322$		0			0
$323$		−	391.185i		−	1.21110i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			13.7560i			0.0420672i
$328$		0			0
$329$	566.203			1.72098
$330$		0			0
$331$		−	453.959i		−	1.37148i	−0.727848	−	0.685739i	$-0.759479\pi$
$331$		−	453.959i		−	1.37148i	0.727848	−	0.685739i	$-0.240521\pi$
$332$		0			0
$333$		−	168.892i		−	0.507182i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			641.793i			1.90443i	0.305426	+	0.952216i	$0.401201\pi$
$337$			641.793i			1.90443i	−0.305426	+	0.952216i	$0.598799\pi$
$338$		0			0
$339$		−	269.838i		−	0.795982i
$340$		0			0
$341$	−180.564	−	281.112i	−0.529512	−	0.824375i
$342$		0			0
$343$	−292.215			−0.851939
$344$		0			0
$345$		0			0
$346$		0			0
$347$			146.717i			0.422816i	0.977398	+	0.211408i	$0.0678049\pi$
$347$			146.717i			0.422816i	−0.977398	+	0.211408i	$0.932195\pi$
$348$		0			0
$349$	125.183			0.358691			0.179345	−	0.983786i	$-0.442602\pi$
$349$	125.183			0.358691			0.179345	+	0.983786i	$0.442602\pi$
$350$		0			0
$351$	−208.569			−0.594214
$352$		0			0
$353$		−	157.219i		−	0.445380i	−0.974889	−	0.222690i	$-0.928516\pi$
$353$		−	157.219i		−	0.445380i	0.974889	−	0.222690i	$-0.0714838\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−592.278			−1.65904
$358$		0			0
$359$	118.549			0.330221			0.165110	−	0.986275i	$-0.447202\pi$
$359$	118.549			0.330221			0.165110	+	0.986275i	$0.447202\pi$
$360$		0			0
$361$	−31.8649			−0.0882686
$362$		0			0
$363$		−	17.1161i		−	0.0471518i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	416.786i		−	1.13566i	−0.823147	−	0.567829i	$-0.807784\pi$
$367$		−	416.786i		−	1.13566i	0.823147	−	0.567829i	$-0.192216\pi$
$368$		0			0
$369$	222.427			0.602783
$370$		0			0
$371$			784.672i			2.11502i
$372$		0			0
$373$	−10.3665			−0.0277922			−0.0138961	−	0.999903i	$-0.504423\pi$
$373$	−10.3665			−0.0277922			−0.0138961	+	0.999903i	$0.504423\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−417.149			−1.10650
$378$		0			0
$379$	−123.950			−0.327045			−0.163523	−	0.986540i	$-0.552286\pi$
$379$	−123.950			−0.327045			−0.163523	+	0.986540i	$0.552286\pi$
$380$		0			0
$381$	−332.154			−0.871796
$382$		0			0
$383$			66.0355i			0.172416i	0.996277	+	0.0862082i	$0.0274750\pi$
$383$			66.0355i			0.172416i	−0.996277	+	0.0862082i	$0.972525\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	7.92858i		−	0.0204873i
$388$		0			0
$389$			474.133i			1.21885i	0.792843	+	0.609426i	$0.208600\pi$
$389$			474.133i			1.21885i	−0.792843	+	0.609426i	$0.791400\pi$
$390$		0			0
$391$	281.782			0.720670
$392$		0			0
$393$			365.793i			0.930771i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−260.721			−0.656728			−0.328364	−	0.944551i	$-0.606497\pi$
$397$	−260.721			−0.656728			−0.328364	+	0.944551i	$0.606497\pi$
$398$		0			0
$399$		−	498.330i		−	1.24895i
$400$		0			0
$401$		−	515.595i		−	1.28577i	−0.765961	−	0.642887i	$-0.777737\pi$
$401$		−	515.595i		−	1.28577i	0.765961	−	0.642887i	$-0.222263\pi$
$402$		0			0
$403$	−179.484	−	279.432i	−0.445371	−	0.693379i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	521.313			1.28087
$408$		0			0
$409$		−	523.025i		−	1.27879i	−0.768878	−	0.639395i	$-0.779185\pi$
$409$		−	523.025i		−	1.27879i	0.768878	−	0.639395i	$-0.220815\pi$
$410$		0			0
$411$	−470.546			−1.14488
$412$		0			0
$413$	909.758			2.20280
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−44.1716			−0.105927
$418$		0			0
$419$	−168.296			−0.401661			−0.200831	−	0.979626i	$-0.564364\pi$
$419$	−168.296			−0.401661			−0.200831	+	0.979626i	$0.564364\pi$
$420$		0			0
$421$	112.886			0.268138			0.134069	−	0.990972i	$-0.457196\pi$
$421$	112.886			0.268138			0.134069	+	0.990972i	$0.457196\pi$
$422$		0			0
$423$	−254.381			−0.601374
$424$		0			0
$425$		0			0
$426$		0			0
$427$			162.440i			0.380422i
$428$		0			0
$429$			408.087i			0.951252i
$430$		0			0
$431$	438.923			1.01838			0.509191	−	0.860653i	$-0.329945\pi$
$431$	438.923			1.01838			0.509191	+	0.860653i	$0.329945\pi$
$432$		0			0
$433$			441.896i			1.02054i	0.860013	+	0.510272i	$0.170455\pi$
$433$			441.896i			1.02054i	−0.860013	+	0.510272i	$0.829545\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			237.085i			0.542530i
$438$		0			0
$439$	200.444			0.456591			0.228296	−	0.973592i	$-0.426685\pi$
$439$	200.444			0.456591			0.228296	+	0.973592i	$0.426685\pi$
$440$		0			0
$441$	−39.8067			−0.0902645
$442$		0			0
$443$	−638.542			−1.44140			−0.720702	−	0.693245i	$-0.756180\pi$
$443$	−638.542			−1.44140			−0.720702	+	0.693245i	$0.756180\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	774.894i		−	1.73354i
$448$		0			0
$449$			803.959i			1.79056i	0.445509	+	0.895278i	$0.353023\pi$
$449$			803.959i			1.79056i	−0.445509	+	0.895278i	$0.646977\pi$
$450$		0			0
$451$			686.558i			1.52230i
$452$		0			0
$453$	714.116			1.57642
$454$		0			0
$455$		0			0
$456$		0			0
$457$			254.439i			0.556759i	0.960471	+	0.278380i	$0.0897973\pi$
$457$			254.439i			0.556759i	−0.960471	+	0.278380i	$0.910203\pi$
$458$		0			0
$459$	−419.784			−0.914561
$460$		0			0
$461$		−	342.783i		−	0.743565i	−0.928320	−	0.371782i	$-0.878747\pi$
$461$		−	342.783i		−	0.743565i	0.928320	−	0.371782i	$-0.121253\pi$
$462$		0			0
$463$			306.900i			0.662850i	0.943482	+	0.331425i	$0.107529\pi$
$463$			306.900i			0.662850i	−0.943482	+	0.331425i	$0.892471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−418.500			−0.896145			−0.448072	−	0.893997i	$-0.647889\pi$
$467$	−418.500			−0.896145			−0.448072	+	0.893997i	$0.647889\pi$
$468$		0			0
$469$	−119.574			−0.254956
$470$		0			0
$471$			1006.31i			2.13654i
$472$		0			0
$473$	24.4729			0.0517397
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	352.534i		−	0.739066i
$478$		0			0
$479$	171.694			0.358443			0.179222	−	0.983809i	$-0.442642\pi$
$479$	171.694			0.358443			0.179222	+	0.983809i	$0.442642\pi$
$480$		0			0
$481$	518.197			1.07733
$482$		0			0
$483$	358.962			0.743192
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	600.828i		−	1.23373i	−0.787068	−	0.616867i	$-0.788402\pi$
$487$		−	600.828i		−	1.23373i	0.787068	−	0.616867i	$-0.211598\pi$
$488$		0			0
$489$			1122.51i			2.29552i
$490$		0			0
$491$		−	606.073i		−	1.23437i	−0.786820	−	0.617183i	$-0.788274\pi$
$491$		−	606.073i		−	1.23437i	0.786820	−	0.617183i	$-0.211726\pi$
$492$		0			0
$493$	−839.589			−1.70302
$494$		0			0
$495$		0			0
$496$		0			0
$497$	475.326			0.956390
$498$		0			0
$499$			213.965i			0.428788i	0.976747	+	0.214394i	$0.0687776\pi$
$499$			213.965i			0.428788i	−0.976747	+	0.214394i	$0.931222\pi$
$500$		0			0
$501$	−541.678			−1.08119
$502$		0			0
$503$	222.431			0.442209			0.221105	−	0.975250i	$-0.429034\pi$
$503$	222.431			0.442209			0.221105	+	0.975250i	$0.429034\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	191.658i		−	0.378024i
$508$		0			0
$509$		−	246.927i		−	0.485121i	−0.970136	−	0.242561i	$-0.922013\pi$
$509$		−	246.927i		−	0.485121i	0.970136	−	0.242561i	$-0.0779874\pi$
$510$		0			0
$511$		−	318.138i		−	0.622579i
$512$		0			0
$513$		−	353.197i		−	0.688494i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	785.191i		−	1.51874i
$518$		0			0
$519$			92.2957i			0.177834i
$520$		0			0
$521$	746.086			1.43203			0.716013	−	0.698086i	$-0.245965\pi$
$521$	746.086			1.43203			0.716013	+	0.698086i	$0.245965\pi$
$522$		0			0
$523$			14.1002i			0.0269602i	0.999909	+	0.0134801i	$0.00429098\pi$
$523$			14.1002i			0.0269602i	−0.999909	+	0.0134801i	$0.995709\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−361.246	−	562.408i	−0.685475	−	1.06719i
$528$		0			0
$529$	358.220			0.677165
$530$		0			0
$531$	−408.732			−0.769741
$532$		0			0
$533$			682.454i			1.28040i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	911.955			1.69824
$538$		0			0
$539$		−	122.870i		−	0.227959i
$540$		0			0
$541$	498.837			0.922065			0.461032	−	0.887383i	$-0.347479\pi$
$541$	498.837			0.922065			0.461032	+	0.887383i	$0.347479\pi$
$542$		0			0
$543$	−872.586			−1.60697
$544$		0			0
$545$		0			0
$546$		0			0
$547$	90.2575			0.165005			0.0825023	−	0.996591i	$-0.473709\pi$
$547$	90.2575			0.165005			0.0825023	+	0.996591i	$0.473709\pi$
$548$		0			0
$549$		−	72.9805i		−	0.132934i
$550$		0			0
$551$		−	706.413i		−	1.28206i
$552$		0			0
$553$		−	743.369i		−	1.34425i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			377.939i			0.678526i	0.940692	+	0.339263i	$0.110178\pi$
$557$			377.939i			0.678526i	−0.940692	+	0.339263i	$0.889822\pi$
$558$		0			0
$559$	24.3266			0.0435181
$560$		0			0
$561$			821.351i			1.46408i
$562$		0			0
$563$	44.9407			0.0798237			0.0399119	−	0.999203i	$-0.487292\pi$
$563$	44.9407			0.0798237			0.0399119	+	0.999203i	$0.487292\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−778.990			−1.37388
$568$		0			0
$569$		−	355.490i		−	0.624762i	−0.949957	−	0.312381i	$-0.898873\pi$
$569$		−	355.490i		−	0.624762i	0.949957	−	0.312381i	$-0.101127\pi$
$570$		0			0
$571$			271.176i			0.474915i	0.971398	+	0.237458i	$0.0763141\pi$
$571$			271.176i			0.474915i	−0.971398	+	0.237458i	$0.923686\pi$
$572$		0			0
$573$		−	667.150i		−	1.16431i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−970.595			−1.68214			−0.841070	−	0.540926i	$-0.818074\pi$
$577$	−970.595			−1.68214			−0.841070	+	0.540926i	$0.818074\pi$
$578$		0			0
$579$		−	1313.97i		−	2.26939i
$580$		0			0
$581$		−	217.914i		−	0.375066i
$582$		0			0
$583$	1088.16			1.86648
$584$		0			0
$585$		0			0
$586$		0			0
$587$			789.105i			1.34430i	0.740414	+	0.672151i	$0.234629\pi$
$587$			789.105i			1.34430i	−0.740414	+	0.672151i	$0.765371\pi$
$588$		0			0
$589$	473.198	−	303.945i	0.803393	−	0.516035i
$590$		0			0
$591$	1214.51			2.05501
$592$		0			0
$593$	245.273			0.413613			0.206807	−	0.978382i	$-0.433693\pi$
$593$	245.273			0.413613			0.206807	+	0.978382i	$0.433693\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1165.34			1.95199
$598$		0			0
$599$	757.884			1.26525			0.632625	−	0.774459i	$-0.281978\pi$
$599$	757.884			1.26525			0.632625	+	0.774459i	$0.281978\pi$
$600$		0			0
$601$		−	638.595i		−	1.06255i	−0.847198	−	0.531277i	$-0.821712\pi$
$601$		−	638.595i		−	1.06255i	0.847198	−	0.531277i	$-0.178288\pi$
$602$		0			0
$603$	53.7219			0.0890911
$604$		0			0
$605$		0			0
$606$		0			0
$607$	322.153			0.530729			0.265365	−	0.964148i	$-0.414508\pi$
$607$	322.153			0.530729			0.265365	+	0.964148i	$0.414508\pi$
$608$		0			0
$609$	−1069.55			−1.75624
$610$		0			0
$611$		−	780.498i		−	1.27741i
$612$		0			0
$613$		−	371.537i		−	0.606096i	−0.952975	−	0.303048i	$-0.901996\pi$
$613$		−	371.537i		−	0.606096i	0.952975	−	0.303048i	$-0.0980043\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−119.323			−0.193392			−0.0966959	−	0.995314i	$-0.530827\pi$
$617$	−119.323			−0.193392			−0.0966959	+	0.995314i	$0.530827\pi$
$618$		0			0
$619$			67.3230i			0.108761i	0.998520	+	0.0543805i	$0.0173184\pi$
$619$			67.3230i			0.108761i	−0.998520	+	0.0543805i	$0.982682\pi$
$620$		0			0
$621$	254.418			0.409691
$622$		0			0
$623$			131.031i			0.210322i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−691.067			−1.10218
$628$		0			0
$629$	1042.97			1.65813
$630$		0			0
$631$			988.201i			1.56609i	0.621967	+	0.783043i	$0.286334\pi$
$631$			988.201i			1.56609i	−0.621967	+	0.783043i	$0.713666\pi$
$632$		0			0
$633$			1264.09i			1.99698i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	122.136i		−	0.191736i
$638$		0			0
$639$	−213.552			−0.334198
$640$		0			0
$641$			533.349i			0.832058i	0.909351	+	0.416029i	$0.136579\pi$
$641$			533.349i			0.832058i	−0.909351	+	0.416029i	$0.863421\pi$
$642$		0			0
$643$			182.986i			0.284582i	0.989825	+	0.142291i	$0.0454469\pi$
$643$			182.986i			0.284582i	−0.989825	+	0.142291i	$0.954553\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			565.516i			0.874059i	0.899447	+	0.437029i	$0.143969\pi$
$647$			565.516i			0.874059i	−0.899447	+	0.437029i	$0.856031\pi$
$648$		0			0
$649$		−	1261.62i		−	1.94395i
$650$		0			0
$651$	−460.190	−	716.451i	−0.706898	−	1.10054i
$652$		0			0
$653$	1092.30			1.67275			0.836374	−	0.548159i	$-0.184671\pi$
$653$	1092.30			1.67275			0.836374	+	0.548159i	$0.184671\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			142.932i			0.217552i
$658$		0			0
$659$	−463.466			−0.703287			−0.351644	−	0.936134i	$-0.614377\pi$
$659$	−463.466			−0.703287			−0.351644	+	0.936134i	$0.614377\pi$
$660$		0			0
$661$	−844.548			−1.27768			−0.638841	−	0.769339i	$-0.720586\pi$
$661$	−844.548			−1.27768			−0.638841	+	0.769339i	$0.720586\pi$
$662$		0			0
$663$			816.442i			1.23144i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	508.850			0.762893
$668$		0			0
$669$	462.578			0.691446
$670$		0			0
$671$	225.267			0.335718
$672$		0			0
$673$			148.153i			0.220138i	0.993924	+	0.110069i	$0.0351071\pi$
$673$			148.153i			0.220138i	−0.993924	+	0.110069i	$0.964893\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			373.795i			0.552134i	0.961138	+	0.276067i	$0.0890312\pi$
$677$			373.795i			0.552134i	−0.961138	+	0.276067i	$0.910969\pi$
$678$		0			0
$679$	−1330.55			−1.95958
$680$		0			0
$681$			343.018i			0.503698i
$682$		0			0
$683$	−541.974			−0.793520			−0.396760	−	0.917922i	$-0.629866\pi$
$683$	−541.974			−0.793520			−0.396760	+	0.917922i	$0.629866\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	12.8080			0.0186434
$688$		0			0
$689$	1081.65			1.56989
$690$		0			0
$691$	262.755			0.380253			0.190127	−	0.981760i	$-0.439110\pi$
$691$	262.755			0.380253			0.190127	+	0.981760i	$0.439110\pi$
$692$		0			0
$693$			292.467i			0.422030i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			1373.56i			1.97068i
$698$		0			0
$699$			801.316i			1.14638i
$700$		0			0
$701$	268.602			0.383170			0.191585	−	0.981476i	$-0.438637\pi$
$701$	268.602			0.383170			0.191585	+	0.981476i	$0.438637\pi$
$702$		0			0
$703$			877.531i			1.24827i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	760.863			1.07619
$708$		0			0
$709$		−	730.233i		−	1.02995i	−0.857206	−	0.514974i	$-0.827802\pi$
$709$		−	730.233i		−	1.02995i	0.857206	−	0.514974i	$-0.172198\pi$
$710$		0			0
$711$			333.978i			0.469730i
$712$		0			0
$713$	218.940	+	340.858i	0.307069	+	0.478062i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1445.39			2.01588
$718$		0			0
$719$		−	158.978i		−	0.221111i	−0.993870	−	0.110555i	$-0.964737\pi$
$719$		−	158.978i		−	0.221111i	0.993870	−	0.110555i	$-0.0352629\pi$
$720$		0			0
$721$	−1088.96			−1.51035
$722$		0			0
$723$	−1456.70			−2.01480
$724$		0			0
$725$		0			0
$726$		0			0
$727$	97.9014			0.134665			0.0673325	−	0.997731i	$-0.478551\pi$
$727$	97.9014			0.134665			0.0673325	+	0.997731i	$0.478551\pi$
$728$		0			0
$729$	−269.293			−0.369400
$730$		0			0
$731$	48.9618			0.0669792
$732$		0			0
$733$	−809.987			−1.10503			−0.552515	−	0.833503i	$-0.686332\pi$
$733$	−809.987			−1.10503			−0.552515	+	0.833503i	$0.686332\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			165.822i			0.224996i
$738$		0			0
$739$		−	836.523i		−	1.13197i	−0.824417	−	0.565983i	$-0.808497\pi$
$739$		−	836.523i		−	1.13197i	0.824417	−	0.565983i	$-0.191503\pi$
$740$		0			0
$741$	−686.937			−0.927040
$742$		0			0
$743$		−	638.071i		−	0.858776i	−0.903120	−	0.429388i	$-0.858729\pi$
$743$		−	638.071i		−	0.858776i	0.903120	−	0.429388i	$-0.141271\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			97.9033i			0.131062i
$748$		0			0
$749$	751.832			1.00378
$750$		0			0
$751$	501.974			0.668407			0.334204	−	0.942501i	$-0.391533\pi$
$751$	501.974			0.668407			0.334204	+	0.942501i	$0.391533\pi$
$752$		0			0
$753$	222.128			0.294991
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	345.125i		−	0.455911i	−0.973672	−	0.227956i	$-0.926796\pi$
$757$		−	345.125i		−	0.455911i	0.973672	−	0.227956i	$-0.0732041\pi$
$758$		0			0
$759$		−	497.796i		−	0.655858i
$760$		0			0
$761$		−	117.829i		−	0.154834i	−0.996999	−	0.0774172i	$-0.975333\pi$
$761$		−	117.829i		−	0.154834i	0.996999	−	0.0774172i	$-0.0246673\pi$
$762$		0			0
$763$	−30.2483			−0.0396439
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1254.08i		−	1.63505i
$768$		0			0
$769$	1192.54			1.55077			0.775385	−	0.631489i	$-0.217556\pi$
$769$	1192.54			1.55077			0.775385	+	0.631489i	$0.217556\pi$
$770$		0			0
$771$		−	425.180i		−	0.551465i
$772$		0			0
$773$			146.194i			0.189125i	0.995519	+	0.0945624i	$0.0301452\pi$
$773$			146.194i			0.189125i	−0.995519	+	0.0945624i	$0.969855\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1328.63			1.70995
$778$		0			0
$779$	−1155.69			−1.48355
$780$		0			0
$781$		−	659.165i		−	0.844002i
$782$		0			0
$783$	−758.057			−0.968144
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	523.598i		−	0.665309i	−0.943049	−	0.332655i	$-0.892056\pi$
$787$		−	523.598i		−	0.665309i	0.943049	−	0.332655i	$-0.107944\pi$
$788$		0			0
$789$	−566.643			−0.718178
$790$		0			0
$791$	593.352			0.750129
$792$		0			0
$793$	223.920			0.282371
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	788.937i		−	0.989883i	−0.868926	−	0.494941i	$-0.835190\pi$
$797$		−	788.937i		−	0.989883i	0.868926	−	0.494941i	$-0.164810\pi$
$798$		0			0
$799$		−	1570.90i		−	1.96608i
$800$		0			0
$801$		−	58.8688i		−	0.0734942i
$802$		0			0
$803$	−441.183			−0.549418
$804$		0			0
$805$		0			0
$806$		0			0
$807$	286.085			0.354505
$808$		0			0
$809$			223.393i			0.276135i	0.990423	+	0.138067i	$0.0440891\pi$
$809$			223.393i			0.276135i	−0.990423	+	0.138067i	$0.955911\pi$
$810$		0			0
$811$	710.310			0.875845			0.437922	−	0.899013i	$-0.355715\pi$
$811$	710.310			0.875845			0.437922	+	0.899013i	$0.355715\pi$
$812$		0			0
$813$	−760.835			−0.935837
$814$		0			0
$815$		0			0
$816$		0			0
$817$			41.1955i			0.0504228i
$818$		0			0
$819$			290.719i			0.354968i
$820$		0			0
$821$		−	544.898i		−	0.663700i	−0.943332	−	0.331850i	$-0.892327\pi$
$821$		−	544.898i		−	0.663700i	0.943332	−	0.331850i	$-0.107673\pi$
$822$		0			0
$823$			158.981i			0.193172i	0.995325	+	0.0965860i	$0.0307923\pi$
$823$			158.981i			0.193172i	−0.995325	+	0.0965860i	$0.969208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1101.09i			1.33143i	0.746205	+	0.665716i	$0.231874\pi$
$827$			1101.09i			1.33143i	−0.746205	+	0.665716i	$0.768126\pi$
$828$		0			0
$829$		−	460.325i		−	0.555278i	−0.960686	−	0.277639i	$-0.910448\pi$
$829$		−	460.325i		−	0.555278i	0.960686	−	0.277639i	$-0.0895519\pi$
$830$		0			0
$831$	975.103			1.17341
$832$		0			0
$833$		−	245.820i		−	0.295102i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−326.165	−	507.792i	−0.389683	−	0.606681i
$838$		0			0
$839$	−832.764			−0.992567			−0.496284	−	0.868160i	$-0.665302\pi$
$839$	−832.764			−0.992567			−0.496284	+	0.868160i	$0.665302\pi$
$840$		0			0
$841$	−675.154			−0.802799
$842$		0			0
$843$			148.349i			0.175977i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	37.6370			0.0444356
$848$		0			0
$849$		−	200.190i		−	0.235796i
$850$		0			0
$851$	−632.111			−0.742786
$852$		0			0
$853$	1340.27			1.57124			0.785620	−	0.618709i	$-0.212344\pi$
$853$	1340.27			1.57124			0.785620	+	0.618709i	$0.212344\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1514.40			1.76710			0.883548	−	0.468341i	$-0.155148\pi$
$857$	1514.40			1.76710			0.883548	+	0.468341i	$0.155148\pi$
$858$		0			0
$859$			945.224i			1.10038i	0.835040	+	0.550189i	$0.185444\pi$
$859$			945.224i			1.10038i	−0.835040	+	0.550189i	$0.814556\pi$
$860$		0			0
$861$			1749.78i			2.03227i
$862$		0			0
$863$			665.986i			0.771711i	0.922559	+	0.385855i	$0.126094\pi$
$863$			665.986i			0.771711i	−0.922559	+	0.385855i	$0.873906\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			621.810i			0.717197i
$868$		0			0
$869$	−1030.88			−1.18628
$870$		0			0
$871$			164.831i			0.189243i
$872$		0			0
$873$	597.785			0.684748
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1550.64			1.76812			0.884060	−	0.467374i	$-0.154800\pi$
$877$	1550.64			1.76812			0.884060	+	0.467374i	$0.154800\pi$
$878$		0			0
$879$			590.199i			0.671443i
$880$		0			0
$881$		−	584.156i		−	0.663060i	−0.943445	−	0.331530i	$-0.892435\pi$
$881$		−	584.156i		−	0.663060i	0.943445	−	0.331530i	$-0.107565\pi$
$882$		0			0
$883$		−	1400.97i		−	1.58660i	−0.608829	−	0.793301i	$-0.708361\pi$
$883$		−	1400.97i		−	1.58660i	0.608829	−	0.793301i	$-0.291639\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	845.078			0.952737			0.476369	−	0.879246i	$-0.341953\pi$
$887$	845.078			0.952737			0.476369	+	0.879246i	$0.341953\pi$
$888$		0			0
$889$		−	730.381i		−	0.821576i
$890$		0			0
$891$			1080.28i			1.21243i
$892$		0			0
$893$	1321.72			1.48009
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	494.821i		−	0.551639i
$898$		0			0
$899$	−652.347	−	1015.61i	−0.725637	−	1.12971i
$900$		0			0
$901$	2177.03			2.41623
$902$		0			0
$903$	62.3724			0.0690724
$904$		0			0
$905$		0			0
$906$		0			0
$907$	558.711			0.615999			0.308000	−	0.951387i	$-0.400340\pi$
$907$	558.711			0.615999			0.308000	+	0.951387i	$0.400340\pi$
$908$		0			0
$909$	−341.838			−0.376059
$910$		0			0
$911$		−	220.594i		−	0.242145i	−0.992644	−	0.121072i	$-0.961367\pi$
$911$		−	220.594i		−	0.242145i	0.992644	−	0.121072i	$-0.0386333\pi$
$912$		0			0
$913$	−302.195			−0.330991
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−804.350			−0.877154
$918$		0			0
$919$	1113.31			1.21144			0.605720	−	0.795678i	$-0.292885\pi$
$919$	1113.31			1.21144			0.605720	+	0.795678i	$0.292885\pi$
$920$		0			0
$921$			845.875i			0.918432i
$922$		0			0
$923$		−	655.225i		−	0.709887i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	489.243			0.527770
$928$		0			0
$929$		−	1167.23i		−	1.25644i	−0.778037	−	0.628219i	$-0.783784\pi$
$929$		−	1167.23i		−	1.25644i	0.778037	−	0.628219i	$-0.216216\pi$
$930$		0			0
$931$	206.828			0.222157
$932$		0			0
$933$			2115.51i			2.26743i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−72.3598			−0.0772250			−0.0386125	−	0.999254i	$-0.512294\pi$
$937$	−72.3598			−0.0772250			−0.0386125	+	0.999254i	$0.512294\pi$
$938$		0			0
$939$	−1067.95			−1.13732
$940$		0			0
$941$		−	647.944i		−	0.688569i	−0.938865	−	0.344285i	$-0.888121\pi$
$941$		−	647.944i		−	0.688569i	0.938865	−	0.344285i	$-0.111879\pi$
$942$		0			0
$943$		−	832.476i		−	0.882795i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			417.811i			0.441195i	0.975365	+	0.220597i	$0.0708007\pi$
$947$			417.811i			0.441195i	−0.975365	+	0.220597i	$0.929199\pi$
$948$		0			0
$949$	−438.546			−0.462114
$950$		0			0
$951$			173.665i			0.182613i
$952$		0			0
$953$			1387.13i			1.45554i	0.685820	+	0.727771i	$0.259444\pi$
$953$			1387.13i			1.45554i	−0.685820	+	0.727771i	$0.740556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			1483.22i			1.54986i
$958$		0			0
$959$		−	1034.69i		−	1.07893i
$960$		0			0
$961$	399.636	−	873.963i	0.415854	−	0.909431i
$962$		0			0
$963$	−337.780			−0.350758
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1531.45i			1.58371i	0.610708	+	0.791856i	$0.290885\pi$
$967$			1531.45i			1.58371i	−0.610708	+	0.791856i	$0.709115\pi$
$968$		0			0
$969$	−1382.59			−1.42682
$970$		0			0
$971$	−860.652			−0.886356			−0.443178	−	0.896434i	$-0.646149\pi$
$971$	−860.652			−0.886356			−0.443178	+	0.896434i	$0.646149\pi$
$972$		0			0
$973$		−	97.1299i		−	0.0998251i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−594.614			−0.608612			−0.304306	−	0.952574i	$-0.598425\pi$
$977$	−594.614			−0.608612			−0.304306	+	0.952574i	$0.598425\pi$
$978$		0			0
$979$	181.709			0.185606
$980$		0			0
$981$	13.5898			0.0138530
$982$		0			0
$983$		−	655.927i		−	0.667270i	−0.942702	−	0.333635i	$-0.891725\pi$
$983$		−	655.927i		−	0.667270i	0.942702	−	0.333635i	$-0.108275\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	2001.16i		−	2.02752i
$988$		0			0
$989$	−29.6743			−0.0300043
$990$		0			0
$991$		−	1002.24i		−	1.01135i	−0.862725	−	0.505673i	$-0.831244\pi$
$991$		−	1002.24i		−	1.01135i	0.862725	−	0.505673i	$-0.168756\pi$
$992$		0			0
$993$	−1604.45			−1.61576
$994$		0			0
$995$		0			0
$996$		0			0
$997$	179.983			0.180525			0.0902624	−	0.995918i	$-0.471229\pi$
$997$	179.983			0.180525			0.0902624	+	0.995918i	$0.471229\pi$
$998$		0			0
$999$	941.684			0.942627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.g.1301.5	yes	22
5.2	odd	4		3100.3.f.d.1549.9		44
5.3	odd	4		3100.3.f.d.1549.36		44
5.4	even	2		3100.3.d.f.1301.18	yes	22
31.30	odd	2	inner	3100.3.d.g.1301.18	yes	22
155.92	even	4		3100.3.f.d.1549.35		44
155.123	even	4		3100.3.f.d.1549.10		44
155.154	odd	2		3100.3.d.f.1301.5	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.5	✓	22	155.154	odd	2
3100.3.d.f.1301.18	yes	22	5.4	even	2
3100.3.d.g.1301.5	yes	22	1.1	even	1	trivial
3100.3.d.g.1301.18	yes	22	31.30	odd	2	inner
3100.3.f.d.1549.9		44	5.2	odd	4
3100.3.f.d.1549.10		44	155.123	even	4
3100.3.f.d.1549.35		44	155.92	even	4
3100.3.f.d.1549.36		44	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.53436i q^{3} +7.77177 q^{7} -3.49167 q^{9} +O(q^{10})\)	\(q-3.53436i q^{3} +7.77177 q^{7} -3.49167 q^{9} -10.7776i q^{11} -10.7132i q^{13} -21.5623i q^{17} +18.1421 q^{19} -27.4682i q^{21} +13.0683i q^{23} -19.4684i q^{27} -38.9378i q^{29} +(26.0829 - 16.7536i) q^{31} -38.0920 q^{33} +48.3699i q^{37} -37.8643 q^{39} -63.7021 q^{41} +2.27071i q^{43} +72.8538 q^{47} +11.4005 q^{49} -76.2089 q^{51} +100.964i q^{53} -64.1206i q^{57} +117.059 q^{59} +20.9013i q^{61} -27.1365 q^{63} -15.3857 q^{67} +46.1879 q^{69} +61.1605 q^{71} -40.9351i q^{73} -83.7613i q^{77} -95.6498i q^{79} -100.233 q^{81} -28.0391i q^{83} -137.620 q^{87} +16.8598i q^{89} -83.2606i q^{91} +(-59.2131 - 92.1863i) q^{93} -171.203 q^{97} +37.6319i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) \) 22 * q + 4 * q^7 - 72 * q^9	\( 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) \) 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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