LMFDB - Embedded newform 4950.2.d.n.4751.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(4751,4950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4950.4751"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,8,0,8,0,0,0,8,0,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$39.5259490005$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.231260962816.24
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + x^{6} - 4x^{5} + 41x^{4} - 122x^{3} + 293x^{2} - 244x + 236 $ x^8 - 2x^7 + x^6 - 4x^5 + 41x^4 - 122x^3 + 293x^2 - 244x + 236
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 990)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4751.6
Root			$-2.15569 - 1.81129i$ of defining polynomial
Character	$\chi$	$=$	4950.4751
Dual form			4950.2.d.n.4751.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.41421i q^{7} +1.00000 q^{8} +(-0.772087 - 3.22550i) q^{11} +3.62258i q^{13} +1.41421i q^{14} +1.00000 q^{16} +2.00000 q^{17} +6.07263i q^{19} +(-0.772087 - 3.22550i) q^{22} -7.77769i q^{23} +3.62258i q^{26} +1.41421i q^{28} -5.49966 q^{29} +6.24621 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.54417 q^{37} +6.07263i q^{38} +12.5435 q^{41} +9.45353i q^{43} +(-0.772087 - 3.22550i) q^{44} -7.77769i q^{46} +9.96148i q^{47} +5.00000 q^{49} +3.62258i q^{52} +6.07263i q^{53} +1.41421i q^{56} -5.49966 q^{58} -0.794156i q^{59} -9.96148i q^{61} +6.24621 q^{62} +1.00000 q^{64} -3.08835 q^{67} +2.00000 q^{68} +4.06854i q^{73} -1.54417 q^{74} +6.07263i q^{76} +(4.56155 - 1.09190i) q^{77} +6.07263i q^{79} +12.5435 q^{82} +17.3693 q^{83} +9.45353i q^{86} +(-0.772087 - 3.22550i) q^{88} +11.4878i q^{89} -5.12311 q^{91} -7.77769i q^{92} +9.96148i q^{94} -10.9993 q^{97} +5.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{16} + 16 q^{17} - 16 q^{31} + 8 q^{32} + 16 q^{34} + 40 q^{49} - 16 q^{62} + 8 q^{64} + 16 q^{68} + 20 q^{77} + 40 q^{83} - 8 q^{91} + 40 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 + 8 * q^16 + 16 * q^17 - 16 * q^31 + 8 * q^32 + 16 * q^34 + 40 * q^49 - 16 * q^62 + 8 * q^64 + 16 * q^68 + 20 * q^77 + 40 * q^83 - 8 * q^91 + 40 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$.

$n$	$551$	$2377$	$4501$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$	−0.772087	−	3.22550i	−0.232793	−	0.972526i
$11$	−0.772087	−	3.22550i	−0.232793	−	0.972526i
$12$		0			0
$13$			3.62258i			1.00472i	0.864657	+	0.502362i	$0.167535\pi$
$13$			3.62258i			1.00472i	−0.864657	+	0.502362i	$0.832465\pi$
$14$			1.41421i			0.377964i
$15$		0			0
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$			6.07263i			1.39316i	0.717480	+	0.696579i	$0.245295\pi$
$19$			6.07263i			1.39316i	−0.717480	+	0.696579i	$0.754705\pi$
$20$		0			0
$21$		0			0
$22$	−0.772087	−	3.22550i	−0.164609	−	0.687680i
$23$		−	7.77769i		−	1.62176i	−0.585212	−	0.810880i	$-0.698989\pi$
$23$		−	7.77769i		−	1.62176i	0.585212	−	0.810880i	$-0.301011\pi$
$24$		0			0
$25$		0			0
$26$			3.62258i			0.710447i
$27$		0			0
$28$			1.41421i			0.267261i
$29$	−5.49966			−1.02126			−0.510630	−	0.859800i	$-0.670588\pi$
$29$	−5.49966			−1.02126			−0.510630	+	0.859800i	$0.670588\pi$
$30$		0			0
$31$	6.24621			1.12185			0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621			1.12185			0.560926	+	0.827866i	$0.310445\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	2.00000			0.342997
$35$		0			0
$36$		0			0
$37$	−1.54417			−0.253861			−0.126930	−	0.991912i	$-0.540512\pi$
$37$	−1.54417			−0.253861			−0.126930	+	0.991912i	$0.540512\pi$
$38$			6.07263i			0.985111i
$39$		0			0
$40$		0			0
$41$	12.5435			1.95896			0.979482	−	0.201534i	$-0.0645925\pi$
$41$	12.5435			1.95896			0.979482	+	0.201534i	$0.0645925\pi$
$42$		0			0
$43$			9.45353i			1.44165i	0.693117	+	0.720825i	$0.256237\pi$
$43$			9.45353i			1.44165i	−0.693117	+	0.720825i	$0.743763\pi$
$44$	−0.772087	−	3.22550i	−0.116396	−	0.486263i
$45$		0			0
$46$		−	7.77769i		−	1.14676i
$47$			9.96148i			1.45303i	0.687150	+	0.726515i	$0.258861\pi$
$47$			9.96148i			1.45303i	−0.687150	+	0.726515i	$0.741139\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$			3.62258i			0.502362i
$53$			6.07263i			0.834141i	0.908874	+	0.417070i	$0.136943\pi$
$53$			6.07263i			0.834141i	−0.908874	+	0.417070i	$0.863057\pi$
$54$		0			0
$55$		0			0
$56$			1.41421i			0.188982i
$57$		0			0
$58$	−5.49966			−0.722140
$59$		−	0.794156i		−	0.103390i	−0.998663	−	0.0516951i	$-0.983538\pi$
$59$		−	0.794156i		−	0.103390i	0.998663	−	0.0516951i	$-0.0164624\pi$
$60$		0			0
$61$		−	9.96148i		−	1.27544i	−0.770270	−	0.637718i	$-0.779878\pi$
$61$		−	9.96148i		−	1.27544i	0.770270	−	0.637718i	$-0.220122\pi$
$62$	6.24621			0.793270
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−3.08835			−0.377302			−0.188651	−	0.982044i	$-0.560411\pi$
$67$	−3.08835			−0.377302			−0.188651	+	0.982044i	$0.560411\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			4.06854i			0.476187i	0.971242	+	0.238093i	$0.0765225\pi$
$73$			4.06854i			0.476187i	−0.971242	+	0.238093i	$0.923478\pi$
$74$	−1.54417			−0.179507
$75$		0			0
$76$			6.07263i			0.696579i
$77$	4.56155	−	1.09190i	0.519837	−	0.124433i
$78$		0			0
$79$			6.07263i			0.683225i	0.939841	+	0.341612i	$0.110973\pi$
$79$			6.07263i			0.683225i	−0.939841	+	0.341612i	$0.889027\pi$
$80$		0			0
$81$		0			0
$82$	12.5435			1.38520
$83$	17.3693			1.90653			0.953265	−	0.302135i	$-0.0976994\pi$
$83$	17.3693			1.90653			0.953265	+	0.302135i	$0.0976994\pi$
$84$		0			0
$85$		0			0
$86$			9.45353i			1.01940i
$87$		0			0
$88$	−0.772087	−	3.22550i	−0.0823047	−	0.343840i
$89$			11.4878i			1.21771i	0.793283	+	0.608853i	$0.208370\pi$
$89$			11.4878i			1.21771i	−0.793283	+	0.608853i	$0.791630\pi$
$90$		0			0
$91$	−5.12311			−0.537047
$92$		−	7.77769i		−	0.810880i
$93$		0			0
$94$			9.96148i			1.02745i
$95$		0			0
$96$		0			0
$97$	−10.9993			−1.11681			−0.558405	−	0.829568i	$-0.688587\pi$
$97$	−10.9993			−1.11681			−0.558405	+	0.829568i	$0.688587\pi$
$98$	5.00000			0.505076
$99$		0			0
$100$		0			0
$101$	−16.4990			−1.64171			−0.820854	−	0.571138i	$-0.806502\pi$
$101$	−16.4990			−1.64171			−0.820854	+	0.571138i	$0.806502\pi$
$102$		0			0
$103$	3.95548			0.389745			0.194873	−	0.980829i	$-0.437571\pi$
$103$	3.95548			0.389745			0.194873	+	0.980829i	$0.437571\pi$
$104$			3.62258i			0.355223i
$105$		0			0
$106$			6.07263i			0.589826i
$107$	11.1231			1.07531			0.537656	−	0.843165i	$-0.319310\pi$
$107$	11.1231			1.07531			0.537656	+	0.843165i	$0.319310\pi$
$108$		0			0
$109$		−	2.18379i		−	0.209169i	−0.994516	−	0.104585i	$-0.966649\pi$
$109$		−	2.18379i		−	0.209169i	0.994516	−	0.104585i	$-0.0333513\pi$
$110$		0			0
$111$		0			0
$112$			1.41421i			0.133631i
$113$			8.25643i			0.776699i	0.921512	+	0.388350i	$0.126955\pi$
$113$			8.25643i			0.776699i	−0.921512	+	0.388350i	$0.873045\pi$
$114$		0			0
$115$		0			0
$116$	−5.49966			−0.510630
$117$		0			0
$118$		−	0.794156i		−	0.0731079i
$119$			2.82843i			0.259281i
$120$		0			0
$121$	−9.80776	+	4.98074i	−0.891615	+	0.452794i
$122$		−	9.96148i		−	0.901870i
$123$		0			0
$124$	6.24621			0.560926
$125$		0			0
$126$		0			0
$127$			9.89949i			0.878438i	0.898380	+	0.439219i	$0.144745\pi$
$127$			9.89949i			0.878438i	−0.898380	+	0.439219i	$0.855255\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		0			0
$131$	−1.54417			−0.134915			−0.0674575	−	0.997722i	$-0.521489\pi$
$131$	−1.54417			−0.134915			−0.0674575	+	0.997722i	$0.521489\pi$
$132$		0			0
$133$	−8.58800			−0.744674
$134$	−3.08835			−0.266793
$135$		0			0
$136$	2.00000			0.171499
$137$			0.478739i			0.0409014i	0.999791	+	0.0204507i	$0.00651011\pi$
$137$			0.478739i			0.0409014i	−0.999791	+	0.0204507i	$0.993490\pi$
$138$		0			0
$139$		−	19.4442i		−	1.64924i	−0.565689	−	0.824619i	$-0.691390\pi$
$139$		−	19.4442i		−	1.64924i	0.565689	−	0.824619i	$-0.308610\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.6847	−	2.79695i	0.977120	−	0.233893i
$144$		0			0
$145$		0			0
$146$			4.06854i			0.336715i
$147$		0			0
$148$	−1.54417			−0.126930
$149$	2.41131			0.197542			0.0987710	−	0.995110i	$-0.468509\pi$
$149$	2.41131			0.197542			0.0987710	+	0.995110i	$0.468509\pi$
$150$		0			0
$151$			9.48274i			0.771694i	0.922563	+	0.385847i	$0.126091\pi$
$151$			9.48274i			0.771694i	−0.922563	+	0.385847i	$0.873909\pi$
$152$			6.07263i			0.492556i
$153$		0			0
$154$	4.56155	−	1.09190i	0.367580	−	0.0879875i
$155$		0			0
$156$		0			0
$157$	21.1315			1.68648			0.843238	−	0.537540i	$-0.180646\pi$
$157$	21.1315			1.68648			0.843238	+	0.537540i	$0.180646\pi$
$158$			6.07263i			0.483113i
$159$		0			0
$160$		0			0
$161$	10.9993			0.866867
$162$		0			0
$163$	7.91096			0.619635			0.309817	−	0.950796i	$-0.399732\pi$
$163$	7.91096			0.619635			0.309817	+	0.950796i	$0.399732\pi$
$164$	12.5435			0.979482
$165$		0			0
$166$	17.3693			1.34812
$167$	−7.36932			−0.570255			−0.285127	−	0.958490i	$-0.592036\pi$
$167$	−7.36932			−0.570255			−0.285127	+	0.958490i	$0.592036\pi$
$168$		0			0
$169$	−0.123106			−0.00946966
$170$		0			0
$171$		0			0
$172$			9.45353i			0.720825i
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$		0			0
$176$	−0.772087	−	3.22550i	−0.0581982	−	0.243132i
$177$		0			0
$178$			11.4878i			0.861047i
$179$			10.8677i			0.812294i	0.913808	+	0.406147i	$0.133128\pi$
$179$			10.8677i			0.812294i	−0.913808	+	0.406147i	$0.866872\pi$
$180$		0			0
$181$	−9.12311			−0.678115			−0.339058	−	0.940766i	$-0.610108\pi$
$181$	−9.12311			−0.678115			−0.339058	+	0.940766i	$0.610108\pi$
$182$	−5.12311			−0.379750
$183$		0			0
$184$		−	7.77769i		−	0.573379i
$185$		0			0
$186$		0			0
$187$	−1.54417	−	6.45101i	−0.112921	−	0.471745i
$188$			9.96148i			0.726515i
$189$		0			0
$190$		0			0
$191$		−	20.1472i		−	1.45780i	−0.684621	−	0.728900i	$-0.740032\pi$
$191$		−	20.1472i		−	1.45780i	0.684621	−	0.728900i	$-0.259968\pi$
$192$		0			0
$193$			17.3188i			1.24663i	0.781970	+	0.623316i	$0.214215\pi$
$193$			17.3188i			1.24663i	−0.781970	+	0.623316i	$0.785785\pi$
$194$	−10.9993			−0.789705
$195$		0			0
$196$	5.00000			0.357143
$197$	−0.876894			−0.0624761			−0.0312381	−	0.999512i	$-0.509945\pi$
$197$	−0.876894			−0.0624761			−0.0312381	+	0.999512i	$0.509945\pi$
$198$		0			0
$199$	16.4924			1.16912			0.584558	−	0.811352i	$-0.301268\pi$
$199$	16.4924			1.16912			0.584558	+	0.811352i	$0.301268\pi$
$200$		0			0
$201$		0			0
$202$	−16.4990			−1.16086
$203$		−	7.77769i		−	0.545887i
$204$		0			0
$205$		0			0
$206$	3.95548			0.275591
$207$		0			0
$208$			3.62258i			0.251181i
$209$	19.5873	−	4.68860i	1.35488	−	0.324317i
$210$		0			0
$211$		−	3.88884i		−	0.267719i	−0.991000	−	0.133860i	$-0.957263\pi$
$211$		−	3.88884i		−	0.267719i	0.991000	−	0.133860i	$-0.0427371\pi$
$212$			6.07263i			0.417070i
$213$		0			0
$214$	11.1231			0.760360
$215$		0			0
$216$		0			0
$217$			8.83348i			0.599655i
$218$		−	2.18379i		−	0.147905i
$219$		0			0
$220$		0			0
$221$			7.24517i			0.487363i
$222$		0			0
$223$	3.95548			0.264879			0.132439	−	0.991191i	$-0.457719\pi$
$223$	3.95548			0.264879			0.132439	+	0.991191i	$0.457719\pi$
$224$			1.41421i			0.0944911i
$225$		0			0
$226$			8.25643i			0.549209i
$227$	5.36932			0.356374			0.178187	−	0.983997i	$-0.442977\pi$
$227$	5.36932			0.356374			0.178187	+	0.983997i	$0.442977\pi$
$228$		0			0
$229$	−20.2462			−1.33791			−0.668954	−	0.743304i	$-0.733258\pi$
$229$	−20.2462			−1.33791			−0.668954	+	0.743304i	$0.733258\pi$
$230$		0			0
$231$		0			0
$232$	−5.49966			−0.361070
$233$	14.4924			0.949430			0.474715	−	0.880140i	$-0.342551\pi$
$233$	14.4924			0.949430			0.474715	+	0.880140i	$0.342551\pi$
$234$		0			0
$235$		0			0
$236$		−	0.794156i		−	0.0516951i
$237$		0			0
$238$			2.82843i			0.183340i
$239$	−16.4990			−1.06723			−0.533615	−	0.845728i	$-0.679167\pi$
$239$	−16.4990			−1.06723			−0.533615	+	0.845728i	$0.679167\pi$
$240$		0			0
$241$		−	2.18379i		−	0.140670i	−0.997523	−	0.0703352i	$-0.977593\pi$
$241$		−	2.18379i		−	0.140670i	0.997523	−	0.0703352i	$-0.0224069\pi$
$242$	−9.80776	+	4.98074i	−0.630467	+	0.320174i
$243$		0			0
$244$		−	9.96148i		−	0.637718i
$245$		0			0
$246$		0			0
$247$	−21.9986			−1.39974
$248$	6.24621			0.396635
$249$		0			0
$250$		0			0
$251$		−	3.62258i		−	0.228655i	−0.993443	−	0.114328i	$-0.963529\pi$
$251$		−	3.62258i		−	0.228655i	0.993443	−	0.114328i	$-0.0364714\pi$
$252$		0			0
$253$	−25.0870	+	6.00505i	−1.57720	+	0.377534i
$254$			9.89949i			0.621150i
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.88884i			0.242579i	0.992617	+	0.121290i	$0.0387030\pi$
$257$			3.88884i			0.242579i	−0.992617	+	0.121290i	$0.961297\pi$
$258$		0			0
$259$		−	2.18379i		−	0.135694i
$260$		0			0
$261$		0			0
$262$	−1.54417			−0.0953994
$263$	15.3693			0.947713			0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693			0.947713			0.473856	+	0.880602i	$0.342862\pi$
$264$		0			0
$265$		0			0
$266$	−8.58800			−0.526564
$267$		0			0
$268$	−3.08835			−0.188651
$269$		−	13.8703i		−	0.845685i	−0.906203	−	0.422843i	$-0.861032\pi$
$269$		−	13.8703i		−	0.845685i	0.906203	−	0.422843i	$-0.138968\pi$
$270$		0			0
$271$			17.2604i			1.04850i	0.851566	+	0.524248i	$0.175654\pi$
$271$			17.2604i			1.04850i	−0.851566	+	0.524248i	$0.824346\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$			0.478739i			0.0289217i
$275$		0			0
$276$		0			0
$277$		−	10.8677i		−	0.652980i	−0.945201	−	0.326490i	$-0.894134\pi$
$277$		−	10.8677i		−	0.652980i	0.945201	−	0.326490i	$-0.105866\pi$
$278$		−	19.4442i		−	1.16619i
$279$		0			0
$280$		0			0
$281$	−20.4544			−1.22021			−0.610105	−	0.792321i	$-0.708873\pi$
$281$	−20.4544			−1.22021			−0.610105	+	0.792321i	$0.708873\pi$
$282$		0			0
$283$		−	13.5221i		−	0.803804i	−0.915683	−	0.401902i	$-0.868349\pi$
$283$		−	13.5221i		−	0.803804i	0.915683	−	0.401902i	$-0.131651\pi$
$284$		0			0
$285$		0			0
$286$	11.6847	−	2.79695i	0.690928	−	0.165387i
$287$			17.7392i			1.04711i
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$			4.06854i			0.238093i
$293$	−11.6155			−0.678586			−0.339293	−	0.940681i	$-0.610188\pi$
$293$	−11.6155			−0.678586			−0.339293	+	0.940681i	$0.610188\pi$
$294$		0			0
$295$		0			0
$296$	−1.54417			−0.0897533
$297$		0			0
$298$	2.41131			0.139683
$299$	28.1753			1.62942
$300$		0			0
$301$	−13.3693			−0.770595
$302$			9.48274i			0.545670i
$303$		0			0
$304$			6.07263i			0.348289i
$305$		0			0
$306$		0			0
$307$			10.6937i			0.610319i	0.952301	+	0.305159i	$0.0987098\pi$
$307$			10.6937i			0.610319i	−0.952301	+	0.305159i	$0.901290\pi$
$308$	4.56155	−	1.09190i	0.259919	−	0.0622165i
$309$		0			0
$310$		0			0
$311$			4.41674i			0.250450i	0.992128	+	0.125225i	$0.0399653\pi$
$311$			4.41674i			0.250450i	−0.992128	+	0.125225i	$0.960035\pi$
$312$		0			0
$313$	16.4990			0.932577			0.466288	−	0.884633i	$-0.345591\pi$
$313$	16.4990			0.932577			0.466288	+	0.884633i	$0.345591\pi$
$314$	21.1315			1.19252
$315$		0			0
$316$			6.07263i			0.341612i
$317$		−	27.2219i		−	1.52893i	−0.644663	−	0.764467i	$-0.723002\pi$
$317$		−	27.2219i		−	1.52893i	0.644663	−	0.764467i	$-0.276998\pi$
$318$		0			0
$319$	4.24621	+	17.7392i	0.237742	+	0.993203i
$320$		0			0
$321$		0			0
$322$	10.9993			0.612968
$323$			12.1453i			0.675781i
$324$		0			0
$325$		0			0
$326$	7.91096			0.438148
$327$		0			0
$328$	12.5435			0.692598
$329$	−14.0877			−0.776678
$330$		0			0
$331$	−7.36932			−0.405054			−0.202527	−	0.979277i	$-0.564915\pi$
$331$	−7.36932			−0.405054			−0.202527	+	0.979277i	$0.564915\pi$
$332$	17.3693			0.953265
$333$		0			0
$334$	−7.36932			−0.403231
$335$		0			0
$336$		0			0
$337$			16.0786i			0.875859i	0.899009	+	0.437930i	$0.144288\pi$
$337$			16.0786i			0.875859i	−0.899009	+	0.437930i	$0.855712\pi$
$338$	−0.123106			−0.00669606
$339$		0			0
$340$		0			0
$341$	−4.82262	−	20.1472i	−0.261159	−	1.09103i
$342$		0			0
$343$			16.9706i			0.916324i
$344$			9.45353i			0.509700i
$345$		0			0
$346$	18.0000			0.967686
$347$	−23.6155			−1.26775			−0.633874	−	0.773436i	$-0.718536\pi$
$347$	−23.6155			−1.26775			−0.633874	+	0.773436i	$0.718536\pi$
$348$		0			0
$349$		−	29.8844i		−	1.59968i	−0.600215	−	0.799839i	$-0.704918\pi$
$349$		−	29.8844i		−	1.59968i	0.600215	−	0.799839i	$-0.295082\pi$
$350$		0			0
$351$		0			0
$352$	−0.772087	−	3.22550i	−0.0411524	−	0.171920i
$353$			16.0341i			0.853410i	0.904391	+	0.426705i	$0.140326\pi$
$353$			16.0341i			0.853410i	−0.904391	+	0.426705i	$0.859674\pi$
$354$		0			0
$355$		0			0
$356$			11.4878i			0.608853i
$357$		0			0
$358$			10.8677i			0.574378i
$359$	−21.9986			−1.16104			−0.580521	−	0.814245i	$-0.697151\pi$
$359$	−21.9986			−1.16104			−0.580521	+	0.814245i	$0.697151\pi$
$360$		0			0
$361$	−17.8769			−0.940889
$362$	−9.12311			−0.479500
$363$		0			0
$364$	−5.12311			−0.268524
$365$		0			0
$366$		0			0
$367$	7.04383			0.367685			0.183842	−	0.982956i	$-0.441146\pi$
$367$	7.04383			0.367685			0.183842	+	0.982956i	$0.441146\pi$
$368$		−	7.77769i		−	0.405440i
$369$		0			0
$370$		0			0
$371$	−8.58800			−0.445867
$372$		0			0
$373$		−	24.1180i		−	1.24878i	−0.781112	−	0.624390i	$-0.785347\pi$
$373$		−	24.1180i		−	1.24878i	0.781112	−	0.624390i	$-0.214653\pi$
$374$	−1.54417	−	6.45101i	−0.0798473	−	0.333574i
$375$		0			0
$376$			9.96148i			0.513724i
$377$		−	19.9230i		−	1.02608i
$378$		0			0
$379$	−0.492423			−0.0252940			−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423			−0.0252940			−0.0126470	+	0.999920i	$0.504026\pi$
$380$		0			0
$381$		0			0
$382$		−	20.1472i		−	1.03082i
$383$			22.1067i			1.12960i	0.825227	+	0.564801i	$0.191047\pi$
$383$			22.1067i			1.12960i	−0.825227	+	0.564801i	$0.808953\pi$
$384$		0			0
$385$		0			0
$386$			17.3188i			0.881502i
$387$		0			0
$388$	−10.9993			−0.558405
$389$		−	36.1495i		−	1.83285i	−0.400204	−	0.916426i	$-0.631061\pi$
$389$		−	36.1495i		−	1.83285i	0.400204	−	0.916426i	$-0.368939\pi$
$390$		0			0
$391$		−	15.5554i		−	0.786669i
$392$	5.00000			0.252538
$393$		0			0
$394$	−0.876894			−0.0441773
$395$		0			0
$396$		0			0
$397$	23.5428			1.18158			0.590790	−	0.806826i	$-0.298816\pi$
$397$	23.5428			1.18158			0.590790	+	0.806826i	$0.298816\pi$
$398$	16.4924			0.826690
$399$		0			0
$400$		0			0
$401$			9.89949i			0.494357i	0.968970	+	0.247179i	$0.0795034\pi$
$401$			9.89949i			0.494357i	−0.968970	+	0.247179i	$0.920497\pi$
$402$		0			0
$403$			22.6274i			1.12715i
$404$	−16.4990			−0.820854
$405$		0			0
$406$		−	7.77769i		−	0.386000i
$407$	1.19224	+	4.98074i	0.0590969	+	0.246886i
$408$		0			0
$409$			18.9655i			0.937783i	0.883256	+	0.468891i	$0.155346\pi$
$409$			18.9655i			0.937783i	−0.883256	+	0.468891i	$0.844654\pi$
$410$		0			0
$411$		0			0
$412$	3.95548			0.194873
$413$	1.12311			0.0552644
$414$		0			0
$415$		0			0
$416$			3.62258i			0.177612i
$417$		0			0
$418$	19.5873	−	4.68860i	0.958047	−	0.229327i
$419$			13.3480i			0.652091i	0.945354	+	0.326046i	$0.105716\pi$
$419$			13.3480i			0.652091i	−0.945354	+	0.326046i	$0.894284\pi$
$420$		0			0
$421$	−14.8769			−0.725055			−0.362528	−	0.931973i	$-0.618086\pi$
$421$	−14.8769			−0.725055			−0.362528	+	0.931973i	$0.618086\pi$
$422$		−	3.88884i		−	0.189306i
$423$		0			0
$424$			6.07263i			0.294913i
$425$		0			0
$426$		0			0
$427$	14.0877			0.681750
$428$	11.1231			0.537656
$429$		0			0
$430$		0			0
$431$	36.7633			1.77083			0.885413	−	0.464804i	$-0.153875\pi$
$431$	36.7633			1.77083			0.885413	+	0.464804i	$0.153875\pi$
$432$		0			0
$433$	−5.49966			−0.264297			−0.132148	−	0.991230i	$-0.542188\pi$
$433$	−5.49966			−0.264297			−0.132148	+	0.991230i	$0.542188\pi$
$434$			8.83348i			0.424020i
$435$		0			0
$436$		−	2.18379i		−	0.104585i
$437$	47.2311			2.25937
$438$		0			0
$439$			6.07263i			0.289831i	0.989444	+	0.144916i	$0.0462910\pi$
$439$			6.07263i			0.289831i	−0.989444	+	0.144916i	$0.953709\pi$
$440$		0			0
$441$		0			0
$442$			7.24517i			0.344617i
$443$		−	12.1453i		−	0.577039i	−0.957474	−	0.288520i	$-0.906837\pi$
$443$		−	12.1453i		−	0.577039i	0.957474	−	0.288520i	$-0.0931630\pi$
$444$		0			0
$445$		0			0
$446$	3.95548			0.187298
$447$		0			0
$448$			1.41421i			0.0668153i
$449$		−	21.9096i		−	1.03398i	−0.855992	−	0.516989i	$-0.827053\pi$
$449$		−	21.9096i		−	1.03398i	0.855992	−	0.516989i	$-0.172947\pi$
$450$		0			0
$451$	−9.68466	−	40.4591i	−0.456033	−	1.90514i
$452$			8.25643i			0.388350i
$453$		0			0
$454$	5.36932			0.251995
$455$		0			0
$456$		0			0
$457$		−	12.0101i		−	0.561809i	−0.959736	−	0.280904i	$-0.909366\pi$
$457$		−	12.0101i		−	0.561809i	0.959736	−	0.280904i	$-0.0906344\pi$
$458$	−20.2462			−0.946043
$459$		0			0
$460$		0			0
$461$	14.7647			0.687660			0.343830	−	0.939032i	$-0.388276\pi$
$461$	14.7647			0.687660			0.343830	+	0.939032i	$0.388276\pi$
$462$		0			0
$463$	−22.8658			−1.06266			−0.531331	−	0.847164i	$-0.678308\pi$
$463$	−22.8658			−1.06266			−0.531331	+	0.847164i	$0.678308\pi$
$464$	−5.49966			−0.255315
$465$		0			0
$466$	14.4924			0.671349
$467$		−	12.1453i		−	0.562016i	−0.959705	−	0.281008i	$-0.909331\pi$
$467$		−	12.1453i		−	0.562016i	0.959705	−	0.281008i	$-0.0906688\pi$
$468$		0			0
$469$		−	4.36758i		−	0.201676i
$470$		0			0
$471$		0			0
$472$		−	0.794156i		−	0.0365540i
$473$	30.4924	−	7.29895i	1.40204	−	0.335606i
$474$		0			0
$475$		0			0
$476$			2.82843i			0.129641i
$477$		0			0
$478$	−16.4990			−0.754645
$479$	27.4983			1.25643			0.628214	−	0.778040i	$-0.283786\pi$
$479$	27.4983			1.25643			0.628214	+	0.778040i	$0.283786\pi$
$480$		0			0
$481$		−	5.59390i		−	0.255060i
$482$		−	2.18379i		−	0.0994690i
$483$		0			0
$484$	−9.80776	+	4.98074i	−0.445807	+	0.226397i
$485$		0			0
$486$		0			0
$487$	−41.3958			−1.87582			−0.937912	−	0.346873i	$-0.887244\pi$
$487$	−41.3958			−1.87582			−0.937912	+	0.346873i	$0.887244\pi$
$488$		−	9.96148i		−	0.450935i
$489$		0			0
$490$		0			0
$491$	43.1301			1.94643			0.973217	−	0.229887i	$-0.0738357\pi$
$491$	43.1301			1.94643			0.973217	+	0.229887i	$0.0738357\pi$
$492$		0			0
$493$	−10.9993			−0.495384
$494$	−21.9986			−0.989765
$495$		0			0
$496$	6.24621			0.280463
$497$		0			0
$498$		0			0
$499$	14.8769			0.665981			0.332991	−	0.942930i	$-0.391942\pi$
$499$	14.8769			0.665981			0.332991	+	0.942930i	$0.391942\pi$
$500$		0			0
$501$		0			0
$502$		−	3.62258i		−	0.161684i
$503$	3.50758			0.156395			0.0781976	−	0.996938i	$-0.475084\pi$
$503$	3.50758			0.156395			0.0781976	+	0.996938i	$0.475084\pi$
$504$		0			0
$505$		0			0
$506$	−25.0870	+	6.00505i	−1.11525	+	0.266957i
$507$		0			0
$508$			9.89949i			0.439219i
$509$		−	38.4342i		−	1.70357i	−0.523895	−	0.851783i	$-0.675522\pi$
$509$		−	38.4342i		−	1.70357i	0.523895	−	0.851783i	$-0.324478\pi$
$510$		0			0
$511$	−5.75379			−0.254533
$512$	1.00000			0.0441942
$513$		0			0
$514$			3.88884i			0.171530i
$515$		0			0
$516$		0			0
$517$	32.1308	−	7.69113i	1.41311	−	0.338255i
$518$		−	2.18379i		−	0.0959503i
$519$		0			0
$520$		0			0
$521$		−	28.1102i		−	1.23153i	−0.787930	−	0.615765i	$-0.788847\pi$
$521$		−	28.1102i		−	1.23153i	0.787930	−	0.615765i	$-0.211153\pi$
$522$		0			0
$523$			22.3556i			0.977540i	0.872413	+	0.488770i	$0.162554\pi$
$523$			22.3556i			0.977540i	−0.872413	+	0.488770i	$0.837446\pi$
$524$	−1.54417			−0.0674575
$525$		0			0
$526$	15.3693			0.670134
$527$	12.4924			0.544178
$528$		0			0
$529$	−37.4924			−1.63011
$530$		0			0
$531$		0			0
$532$	−8.58800			−0.372337
$533$			45.4398i			1.96822i
$534$		0			0
$535$		0			0
$536$	−3.08835			−0.133396
$537$		0			0
$538$		−	13.8703i		−	0.597990i
$539$	−3.86043	−	16.1275i	−0.166281	−	0.694662i
$540$		0			0
$541$		−	6.55137i		−	0.281666i	−0.990033	−	0.140833i	$-0.955022\pi$
$541$		−	6.55137i		−	0.281666i	0.990033	−	0.140833i	$-0.0449780\pi$
$542$			17.2604i			0.741399i
$543$		0			0
$544$	2.00000			0.0857493
$545$		0			0
$546$		0			0
$547$		−	8.75714i		−	0.374428i	−0.982319	−	0.187214i	$-0.940054\pi$
$547$		−	8.75714i		−	0.374428i	0.982319	−	0.187214i	$-0.0599459\pi$
$548$			0.478739i			0.0204507i
$549$		0			0
$550$		0			0
$551$		−	33.3974i		−	1.42278i
$552$		0			0
$553$	−8.58800			−0.365199
$554$		−	10.8677i		−	0.461726i
$555$		0			0
$556$		−	19.4442i		−	0.824619i
$557$	6.49242			0.275093			0.137546	−	0.990495i	$-0.456078\pi$
$557$	6.49242			0.275093			0.137546	+	0.990495i	$0.456078\pi$
$558$		0			0
$559$	−34.2462			−1.44846
$560$		0			0
$561$		0			0
$562$	−20.4544			−0.862819
$563$	−38.2462			−1.61189			−0.805943	−	0.591993i	$-0.798341\pi$
$563$	−38.2462			−1.61189			−0.805943	+	0.591993i	$0.798341\pi$
$564$		0			0
$565$		0			0
$566$		−	13.5221i		−	0.568375i
$567$		0			0
$568$		0			0
$569$	10.8092			0.453146			0.226573	−	0.973994i	$-0.427248\pi$
$569$	10.8092			0.453146			0.226573	+	0.973994i	$0.427248\pi$
$570$		0			0
$571$		−	2.66253i		−	0.111423i	−0.998447	−	0.0557117i	$-0.982257\pi$
$571$		−	2.66253i		−	0.111423i	0.998447	−	0.0557117i	$-0.0177428\pi$
$572$	11.6847	−	2.79695i	0.488560	−	0.116946i
$573$		0			0
$574$			17.7392i			0.740418i
$575$		0			0
$576$		0			0
$577$	−6.17669			−0.257139			−0.128570	−	0.991700i	$-0.541039\pi$
$577$	−6.17669			−0.257139			−0.128570	+	0.991700i	$0.541039\pi$
$578$	−13.0000			−0.540729
$579$		0			0
$580$		0			0
$581$			24.5639i			1.01908i
$582$		0			0
$583$	19.5873	−	4.68860i	0.811224	−	0.194182i
$584$			4.06854i			0.168358i
$585$		0			0
$586$	−11.6155			−0.479833
$587$			12.1453i			0.501289i	0.968079	+	0.250644i	$0.0806425\pi$
$587$			12.1453i			0.501289i	−0.968079	+	0.250644i	$0.919357\pi$
$588$		0			0
$589$			37.9310i			1.56292i
$590$		0			0
$591$		0			0
$592$	−1.54417			−0.0634651
$593$	−28.7386			−1.18015			−0.590077	−	0.807347i	$-0.700903\pi$
$593$	−28.7386			−1.18015			−0.590077	+	0.807347i	$0.700903\pi$
$594$		0			0
$595$		0			0
$596$	2.41131			0.0987710
$597$		0			0
$598$	28.1753			1.15217
$599$			17.3188i			0.707625i	0.935316	+	0.353813i	$0.115115\pi$
$599$			17.3188i			0.707625i	−0.935316	+	0.353813i	$0.884885\pi$
$600$		0			0
$601$			1.22631i			0.0500224i	0.999687	+	0.0250112i	$0.00796214\pi$
$601$			1.22631i			0.0500224i	−0.999687	+	0.0250112i	$0.992038\pi$
$602$	−13.3693			−0.544893
$603$		0			0
$604$			9.48274i			0.385847i
$605$		0			0
$606$		0			0
$607$		−	28.8066i		−	1.16922i	−0.811314	−	0.584611i	$-0.801247\pi$
$607$		−	28.8066i		−	1.16922i	0.811314	−	0.584611i	$-0.198753\pi$
$608$			6.07263i			0.246278i
$609$		0			0
$610$		0			0
$611$	−36.0863			−1.45989
$612$		0			0
$613$		−	27.4901i		−	1.11032i	−0.831745	−	0.555158i	$-0.812658\pi$
$613$		−	27.4901i		−	1.11032i	0.831745	−	0.555158i	$-0.187342\pi$
$614$			10.6937i			0.431561i
$615$		0			0
$616$	4.56155	−	1.09190i	0.183790	−	0.0439937i
$617$			3.88884i			0.156559i	0.996931	+	0.0782795i	$0.0249426\pi$
$617$			3.88884i			0.156559i	−0.996931	+	0.0782795i	$0.975057\pi$
$618$		0			0
$619$	−22.7386			−0.913943			−0.456971	−	0.889481i	$-0.651066\pi$
$619$	−22.7386			−0.913943			−0.456971	+	0.889481i	$0.651066\pi$
$620$		0			0
$621$		0			0
$622$			4.41674i			0.177095i
$623$	−16.2462			−0.650891
$624$		0			0
$625$		0			0
$626$	16.4990			0.659431
$627$		0			0
$628$	21.1315			0.843238
$629$	−3.08835			−0.123140
$630$		0			0
$631$	22.2462			0.885608			0.442804	−	0.896619i	$-0.353984\pi$
$631$	22.2462			0.885608			0.442804	+	0.896619i	$0.353984\pi$
$632$			6.07263i			0.241556i
$633$		0			0
$634$		−	27.2219i		−	1.08112i
$635$		0			0
$636$		0			0
$637$			18.1129i			0.717660i
$638$	4.24621	+	17.7392i	0.168109	+	0.702300i
$639$		0			0
$640$		0			0
$641$		−	31.6350i		−	1.24951i	−0.780822	−	0.624754i	$-0.785199\pi$
$641$		−	31.6350i		−	1.24951i	0.780822	−	0.624754i	$-0.214801\pi$
$642$		0			0
$643$	−14.0877			−0.555563			−0.277782	−	0.960644i	$-0.589599\pi$
$643$	−14.0877			−0.555563			−0.277782	+	0.960644i	$0.589599\pi$
$644$	10.9993			0.433434
$645$		0			0
$646$			12.1453i			0.477849i
$647$			13.3716i			0.525691i	0.964838	+	0.262846i	$0.0846610\pi$
$647$			13.3716i			0.525691i	−0.964838	+	0.262846i	$0.915339\pi$
$648$		0			0
$649$	−2.56155	+	0.613157i	−0.100550	+	0.0240685i
$650$		0			0
$651$		0			0
$652$	7.91096			0.309817
$653$		−	16.0341i		−	0.627463i	−0.949512	−	0.313732i	$-0.898421\pi$
$653$		−	16.0341i		−	0.627463i	0.949512	−	0.313732i	$-0.101579\pi$
$654$		0			0
$655$		0			0
$656$	12.5435			0.489741
$657$		0			0
$658$	−14.0877			−0.549194
$659$	−23.5428			−0.917097			−0.458549	−	0.888669i	$-0.651630\pi$
$659$	−23.5428			−0.917097			−0.458549	+	0.888669i	$0.651630\pi$
$660$		0			0
$661$	−48.7386			−1.89571			−0.947857	−	0.318697i	$-0.896755\pi$
$661$	−48.7386			−1.89571			−0.947857	+	0.318697i	$0.896755\pi$
$662$	−7.36932			−0.286417
$663$		0			0
$664$	17.3693			0.674060
$665$		0			0
$666$		0			0
$667$			42.7746i			1.65624i
$668$	−7.36932			−0.285127
$669$		0			0
$670$		0			0
$671$	−32.1308	+	7.69113i	−1.24040	+	0.296913i
$672$		0			0
$673$			34.2893i			1.32176i	0.750493	+	0.660878i	$0.229816\pi$
$673$			34.2893i			1.32176i	−0.750493	+	0.660878i	$0.770184\pi$
$674$			16.0786i			0.619326i
$675$		0			0
$676$	−0.123106			−0.00473483
$677$	−23.6155			−0.907618			−0.453809	−	0.891099i	$-0.649935\pi$
$677$	−23.6155			−0.907618			−0.453809	+	0.891099i	$0.649935\pi$
$678$		0			0
$679$		−	15.5554i		−	0.596961i
$680$		0			0
$681$		0			0
$682$	−4.82262	−	20.1472i	−0.184668	−	0.771476i
$683$		−	51.0337i		−	1.95275i	−0.216082	−	0.976375i	$-0.569328\pi$
$683$		−	51.0337i		−	1.95275i	0.216082	−	0.976375i	$-0.430672\pi$
$684$		0			0
$685$		0			0
$686$			16.9706i			0.647939i
$687$		0			0
$688$			9.45353i			0.360413i
$689$	−21.9986			−0.838081
$690$		0			0
$691$	−25.6155			−0.974461			−0.487230	−	0.873274i	$-0.661993\pi$
$691$	−25.6155			−0.974461			−0.487230	+	0.873274i	$0.661993\pi$
$692$	18.0000			0.684257
$693$		0			0
$694$	−23.6155			−0.896433
$695$		0			0
$696$		0			0
$697$	25.0870			0.950237
$698$		−	29.8844i		−	1.13114i
$699$		0			0
$700$		0			0
$701$	39.8517			1.50518			0.752588	−	0.658491i	$-0.228805\pi$
$701$	39.8517			1.50518			0.752588	+	0.658491i	$0.228805\pi$
$702$		0			0
$703$		−	9.37720i		−	0.353668i
$704$	−0.772087	−	3.22550i	−0.0290991	−	0.121566i
$705$		0			0
$706$			16.0341i			0.603452i
$707$		−	23.3331i		−	0.877530i
$708$		0			0
$709$	−22.0000			−0.826227			−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000			−0.826227			−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$		0			0
$712$			11.4878i			0.430524i
$713$		−	48.5811i		−	1.81938i
$714$		0			0
$715$		0			0
$716$			10.8677i			0.406147i
$717$		0			0
$718$	−21.9986			−0.820981
$719$		−	26.6960i		−	0.995591i	−0.867294	−	0.497796i	$-0.834143\pi$
$719$		−	26.6960i		−	0.995591i	0.867294	−	0.497796i	$-0.165857\pi$
$720$		0			0
$721$			5.59390i			0.208328i
$722$	−17.8769			−0.665309
$723$		0			0
$724$	−9.12311			−0.339058
$725$		0			0
$726$		0			0
$727$	38.3075			1.42075			0.710373	−	0.703825i	$-0.248526\pi$
$727$	38.3075			1.42075			0.710373	+	0.703825i	$0.248526\pi$
$728$	−5.12311			−0.189875
$729$		0			0
$730$		0			0
$731$			18.9071i			0.699303i
$732$		0			0
$733$		−	4.31897i		−	0.159525i	−0.996814	−	0.0797625i	$-0.974584\pi$
$733$		−	4.31897i		−	0.159525i	0.996814	−	0.0797625i	$-0.0254162\pi$
$734$	7.04383			0.259992
$735$		0			0
$736$		−	7.77769i		−	0.286689i
$737$	2.38447	+	9.96148i	0.0878332	+	0.366936i
$738$		0			0
$739$		−	25.0381i		−	0.921042i	−0.887649	−	0.460521i	$-0.847663\pi$
$739$		−	25.0381i		−	0.921042i	0.887649	−	0.460521i	$-0.152337\pi$
$740$		0			0
$741$		0			0
$742$	−8.58800			−0.315275
$743$	−14.6307			−0.536748			−0.268374	−	0.963315i	$-0.586486\pi$
$743$	−14.6307			−0.536748			−0.268374	+	0.963315i	$0.586486\pi$
$744$		0			0
$745$		0			0
$746$		−	24.1180i		−	0.883021i
$747$		0			0
$748$	−1.54417	−	6.45101i	−0.0564606	−	0.235872i
$749$			15.7304i			0.574778i
$750$		0			0
$751$	10.2462			0.373890			0.186945	−	0.982370i	$-0.440141\pi$
$751$	10.2462			0.373890			0.186945	+	0.982370i	$0.440141\pi$
$752$			9.96148i			0.363258i
$753$		0			0
$754$		−	19.9230i		−	0.725551i
$755$		0			0
$756$		0			0
$757$	−43.8071			−1.59220			−0.796099	−	0.605166i	$-0.793107\pi$
$757$	−43.8071			−1.59220			−0.796099	+	0.605166i	$0.793107\pi$
$758$	−0.492423			−0.0178856
$759$		0			0
$760$		0			0
$761$	−21.8085			−0.790558			−0.395279	−	0.918561i	$-0.629352\pi$
$761$	−21.8085			−0.790558			−0.395279	+	0.918561i	$0.629352\pi$
$762$		0			0
$763$	3.08835			0.111806
$764$		−	20.1472i		−	0.728900i
$765$		0			0
$766$			22.1067i			0.798749i
$767$	2.87689			0.103879
$768$		0			0
$769$		−	45.4398i		−	1.63860i	−0.573364	−	0.819301i	$-0.694362\pi$
$769$		−	45.4398i		−	1.63860i	0.573364	−	0.819301i	$-0.305638\pi$
$770$		0			0
$771$		0			0
$772$			17.3188i			0.623316i
$773$			42.7773i			1.53859i	0.638893	+	0.769296i	$0.279393\pi$
$773$			42.7773i			1.53859i	−0.638893	+	0.769296i	$0.720607\pi$
$774$		0			0
$775$		0			0
$776$	−10.9993			−0.394852
$777$		0			0
$778$		−	36.1495i		−	1.29602i
$779$			76.1720i			2.72915i
$780$		0			0
$781$		0			0
$782$		−	15.5554i		−	0.556259i
$783$		0			0
$784$	5.00000			0.178571
$785$		0			0
$786$		0			0
$787$		−	4.14488i		−	0.147749i	−0.997268	−	0.0738744i	$-0.976464\pi$
$787$		−	4.14488i		−	0.147749i	0.997268	−	0.0738744i	$-0.0235364\pi$
$788$	−0.876894			−0.0312381
$789$		0			0
$790$		0			0
$791$	−11.6763			−0.415163
$792$		0			0
$793$	36.0863			1.28146
$794$	23.5428			0.835503
$795$		0			0
$796$	16.4924			0.584558
$797$		−	10.4402i		−	0.369811i	−0.982756	−	0.184906i	$-0.940802\pi$
$797$		−	10.4402i		−	0.369811i	0.982756	−	0.184906i	$-0.0591980\pi$
$798$		0			0
$799$			19.9230i			0.704824i
$800$		0			0
$801$		0			0
$802$			9.89949i			0.349563i
$803$	13.1231	−	3.14127i	0.463104	−	0.110853i
$804$		0			0
$805$		0			0
$806$			22.6274i			0.797017i
$807$		0			0
$808$	−16.4990			−0.580432
$809$	21.8085			0.766747			0.383373	−	0.923593i	$-0.374762\pi$
$809$	21.8085			0.766747			0.383373	+	0.923593i	$0.374762\pi$
$810$		0			0
$811$			22.8543i			0.802524i	0.915963	+	0.401262i	$0.131428\pi$
$811$			22.8543i			0.802524i	−0.915963	+	0.401262i	$0.868572\pi$
$812$		−	7.77769i		−	0.272943i
$813$		0			0
$814$	1.19224	+	4.98074i	0.0417878	+	0.174575i
$815$		0			0
$816$		0			0
$817$	−57.4079			−2.00845
$818$			18.9655i			0.663112i
$819$		0			0
$820$		0			0
$821$	5.49966			0.191939			0.0959697	−	0.995384i	$-0.469405\pi$
$821$	5.49966			0.191939			0.0959697	+	0.995384i	$0.469405\pi$
$822$		0			0
$823$	10.1322			0.353185			0.176593	−	0.984284i	$-0.443492\pi$
$823$	10.1322			0.353185			0.176593	+	0.984284i	$0.443492\pi$
$824$	3.95548			0.137796
$825$		0			0
$826$	1.12311			0.0390778
$827$	−29.3693			−1.02127			−0.510636	−	0.859797i	$-0.670590\pi$
$827$	−29.3693			−1.02127			−0.510636	+	0.859797i	$0.670590\pi$
$828$		0			0
$829$	32.7386			1.13706			0.568530	−	0.822663i	$-0.307512\pi$
$829$	32.7386			1.13706			0.568530	+	0.822663i	$0.307512\pi$
$830$		0			0
$831$		0			0
$832$			3.62258i			0.125590i
$833$	10.0000			0.346479
$834$		0			0
$835$		0			0
$836$	19.5873	−	4.68860i	0.677441	−	0.162159i
$837$		0			0
$838$			13.3480i			0.461098i
$839$		−	2.82843i		−	0.0976481i	−0.998807	−	0.0488241i	$-0.984453\pi$
$839$		−	2.82843i		−	0.0976481i	0.998807	−	0.0488241i	$-0.0155474\pi$
$840$		0			0
$841$	1.24621			0.0429728
$842$	−14.8769			−0.512692
$843$		0			0
$844$		−	3.88884i		−	0.133860i
$845$		0			0
$846$		0			0
$847$	−7.04383	−	13.8703i	−0.242029	−	0.476588i
$848$			6.07263i			0.208535i
$849$		0			0
$850$		0			0
$851$			12.0101i			0.411701i
$852$		0			0
$853$		−	9.62763i		−	0.329644i	−0.986323	−	0.164822i	$-0.947295\pi$
$853$		−	9.62763i		−	0.329644i	0.986323	−	0.164822i	$-0.0527049\pi$
$854$	14.0877			0.482070
$855$		0			0
$856$	11.1231			0.380180
$857$	−26.4924			−0.904964			−0.452482	−	0.891774i	$-0.649461\pi$
$857$	−26.4924			−0.904964			−0.452482	+	0.891774i	$0.649461\pi$
$858$		0			0
$859$	12.9848			0.443037			0.221519	−	0.975156i	$-0.428899\pi$
$859$	12.9848			0.443037			0.221519	+	0.975156i	$0.428899\pi$
$860$		0			0
$861$		0			0
$862$	36.7633			1.25216
$863$		−	16.5129i		−	0.562104i	−0.959693	−	0.281052i	$-0.909317\pi$
$863$		−	16.5129i		−	0.562104i	0.959693	−	0.281052i	$-0.0906833\pi$
$864$		0			0
$865$		0			0
$866$	−5.49966			−0.186886
$867$		0			0
$868$			8.83348i			0.299828i
$869$	19.5873	−	4.68860i	0.664454	−	0.159050i
$870$		0			0
$871$		−	11.1878i		−	0.379084i
$872$		−	2.18379i		−	0.0739525i
$873$		0			0
$874$	47.2311			1.59761
$875$		0			0
$876$		0			0
$877$		−	40.3921i		−	1.36395i	−0.731378	−	0.681973i	$-0.761122\pi$
$877$		−	40.3921i		−	1.36395i	0.731378	−	0.681973i	$-0.238878\pi$
$878$			6.07263i			0.204942i
$879$		0			0
$880$		0			0
$881$		−	12.1842i		−	0.410496i	−0.978710	−	0.205248i	$-0.934200\pi$
$881$		−	12.1842i		−	0.410496i	0.978710	−	0.205248i	$-0.0658001\pi$
$882$		0			0
$883$	47.0856			1.58456			0.792278	−	0.610160i	$-0.208895\pi$
$883$	47.0856			1.58456			0.792278	+	0.610160i	$0.208895\pi$
$884$			7.24517i			0.243681i
$885$		0			0
$886$		−	12.1453i		−	0.408028i
$887$	10.0000			0.335767			0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000			0.335767			0.167884	+	0.985807i	$0.446307\pi$
$888$		0			0
$889$	−14.0000			−0.469545
$890$		0			0
$891$		0			0
$892$	3.95548			0.132439
$893$	−60.4924			−2.02430
$894$		0			0
$895$		0			0
$896$			1.41421i			0.0472456i
$897$		0			0
$898$		−	21.9096i		−	0.731133i
$899$	−34.3520			−1.14570
$900$		0			0
$901$			12.1453i			0.404618i
$902$	−9.68466	−	40.4591i	−0.322464	−	1.34714i
$903$		0			0
$904$			8.25643i			0.274605i
$905$		0			0
$906$		0			0
$907$	6.17669			0.205094			0.102547	−	0.994728i	$-0.467301\pi$
$907$	6.17669			0.205094			0.102547	+	0.994728i	$0.467301\pi$
$908$	5.36932			0.178187
$909$		0			0
$910$		0			0
$911$		−	10.4218i		−	0.345289i	−0.984984	−	0.172645i	$-0.944769\pi$
$911$		−	10.4218i		−	0.345289i	0.984984	−	0.172645i	$-0.0552312\pi$
$912$		0			0
$913$	−13.4106	−	56.0248i	−0.443827	−	1.85415i
$914$		−	12.0101i		−	0.397259i
$915$		0			0
$916$	−20.2462			−0.668954
$917$		−	2.18379i		−	0.0721151i
$918$		0			0
$919$		−	8.52526i		−	0.281222i	−0.990065	−	0.140611i	$-0.955093\pi$
$919$		−	8.52526i		−	0.281222i	0.990065	−	0.140611i	$-0.0449068\pi$
$920$		0			0
$921$		0			0
$922$	14.7647			0.486249
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−22.8658			−0.751415
$927$		0			0
$928$	−5.49966			−0.180535
$929$		−	24.7380i		−	0.811628i	−0.913956	−	0.405814i	$-0.866988\pi$
$929$		−	24.7380i		−	0.811628i	0.913956	−	0.405814i	$-0.133012\pi$
$930$		0			0
$931$			30.3632i			0.995113i
$932$	14.4924			0.474715
$933$		0			0
$934$		−	12.1453i		−	0.397405i
$935$		0			0
$936$		0			0
$937$			47.1913i			1.54167i	0.637032	+	0.770837i	$0.280162\pi$
$937$			47.1913i			1.54167i	−0.637032	+	0.770837i	$0.719838\pi$
$938$		−	4.36758i		−	0.142607i
$939$		0			0
$940$		0			0
$941$	−2.41131			−0.0786064			−0.0393032	−	0.999227i	$-0.512514\pi$
$941$	−2.41131			−0.0786064			−0.0393032	+	0.999227i	$0.512514\pi$
$942$		0			0
$943$		−	97.5593i		−	3.17697i
$944$		−	0.794156i		−	0.0258476i
$945$		0			0
$946$	30.4924	−	7.29895i	0.991394	−	0.237309i
$947$			15.5554i			0.505482i	0.967534	+	0.252741i	$0.0813320\pi$
$947$			15.5554i			0.505482i	−0.967534	+	0.252741i	$0.918668\pi$
$948$		0			0
$949$	−14.7386			−0.478436
$950$		0			0
$951$		0			0
$952$			2.82843i			0.0916698i
$953$	−48.2462			−1.56285			−0.781424	−	0.624000i	$-0.785506\pi$
$953$	−48.2462			−1.56285			−0.781424	+	0.624000i	$0.785506\pi$
$954$		0			0
$955$		0			0
$956$	−16.4990			−0.533615
$957$		0			0
$958$	27.4983			0.888429
$959$	−0.677039			−0.0218627
$960$		0			0
$961$	8.01515			0.258553
$962$		−	5.59390i		−	0.180354i
$963$		0			0
$964$		−	2.18379i		−	0.0703352i
$965$		0			0
$966$		0			0
$967$		−	16.7965i		−	0.540138i	−0.962841	−	0.270069i	$-0.912954\pi$
$967$		−	16.7965i		−	0.540138i	0.962841	−	0.270069i	$-0.0870465\pi$
$968$	−9.80776	+	4.98074i	−0.315233	+	0.160087i
$969$		0			0
$970$		0			0
$971$			7.14740i			0.229371i	0.993402	+	0.114685i	$0.0365860\pi$
$971$			7.14740i			0.229371i	−0.993402	+	0.114685i	$0.963414\pi$
$972$		0			0
$973$	27.4983			0.881554
$974$	−41.3958			−1.32641
$975$		0			0
$976$		−	9.96148i		−	0.318859i
$977$			59.2901i			1.89686i	0.316988	+	0.948430i	$0.397329\pi$
$977$			59.2901i			1.89686i	−0.316988	+	0.948430i	$0.602671\pi$
$978$		0			0
$979$	37.0540	−	8.86958i	1.18425	−	0.283473i
$980$		0			0
$981$		0			0
$982$	43.1301			1.37634
$983$		−	13.3716i		−	0.426487i	−0.976999	−	0.213244i	$-0.931597\pi$
$983$		−	13.3716i		−	0.426487i	0.976999	−	0.213244i	$-0.0684028\pi$
$984$		0			0
$985$		0			0
$986$	−10.9993			−0.350289
$987$		0			0
$988$	−21.9986			−0.699869
$989$	73.5266			2.33801
$990$		0			0
$991$	−12.4924			−0.396835			−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924			−0.396835			−0.198417	+	0.980118i	$0.563580\pi$
$992$	6.24621			0.198317
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	48.3337i		−	1.53074i	−0.643589	−	0.765372i	$-0.722555\pi$
$997$		−	48.3337i		−	1.53074i	0.643589	−	0.765372i	$-0.277445\pi$
$998$	14.8769			0.470920
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.d.n.4751.6		8
3.2	odd	2		4950.2.d.g.4751.7		8
5.2	odd	4		990.2.f.c.989.10	yes	16
5.3	odd	4		990.2.f.c.989.8	yes	16
5.4	even	2		4950.2.d.g.4751.2		8
11.10	odd	2		4950.2.d.g.4751.3		8
15.2	even	4		990.2.f.c.989.7	yes	16
15.8	even	4		990.2.f.c.989.9	yes	16
15.14	odd	2	inner	4950.2.d.n.4751.3		8
33.32	even	2	inner	4950.2.d.n.4751.2		8
55.32	even	4		990.2.f.c.989.2	yes	16
55.43	even	4		990.2.f.c.989.16	yes	16
55.54	odd	2	inner	4950.2.d.n.4751.7		8
165.32	odd	4		990.2.f.c.989.15	yes	16
165.98	odd	4		990.2.f.c.989.1	✓	16
165.164	even	2		4950.2.d.g.4751.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.f.c.989.1	✓	16	165.98	odd	4
990.2.f.c.989.2	yes	16	55.32	even	4
990.2.f.c.989.7	yes	16	15.2	even	4
990.2.f.c.989.8	yes	16	5.3	odd	4
990.2.f.c.989.9	yes	16	15.8	even	4
990.2.f.c.989.10	yes	16	5.2	odd	4
990.2.f.c.989.15	yes	16	165.32	odd	4
990.2.f.c.989.16	yes	16	55.43	even	4
4950.2.d.g.4751.2		8	5.4	even	2
4950.2.d.g.4751.3		8	11.10	odd	2
4950.2.d.g.4751.6		8	165.164	even	2
4950.2.d.g.4751.7		8	3.2	odd	2
4950.2.d.n.4751.2		8	33.32	even	2	inner
4950.2.d.n.4751.3		8	15.14	odd	2	inner
4950.2.d.n.4751.6		8	1.1	even	1	trivial
4950.2.d.n.4751.7		8	55.54	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:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.41421i q^{7} +1.00000 q^{8} +(-0.772087 - 3.22550i) q^{11} +3.62258i q^{13} +1.41421i q^{14} +1.00000 q^{16} +2.00000 q^{17} +6.07263i q^{19} +(-0.772087 - 3.22550i) q^{22} -7.77769i q^{23} +3.62258i q^{26} +1.41421i q^{28} -5.49966 q^{29} +6.24621 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.54417 q^{37} +6.07263i q^{38} +12.5435 q^{41} +9.45353i q^{43} +(-0.772087 - 3.22550i) q^{44} -7.77769i q^{46} +9.96148i q^{47} +5.00000 q^{49} +3.62258i q^{52} +6.07263i q^{53} +1.41421i q^{56} -5.49966 q^{58} -0.794156i q^{59} -9.96148i q^{61} +6.24621 q^{62} +1.00000 q^{64} -3.08835 q^{67} +2.00000 q^{68} +4.06854i q^{73} -1.54417 q^{74} +6.07263i q^{76} +(4.56155 - 1.09190i) q^{77} +6.07263i q^{79} +12.5435 q^{82} +17.3693 q^{83} +9.45353i q^{86} +(-0.772087 - 3.22550i) q^{88} +11.4878i q^{89} -5.12311 q^{91} -7.77769i q^{92} +9.96148i q^{94} -10.9993 q^{97} +5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{16} + 16 q^{17} - 16 q^{31} + 8 q^{32} + 16 q^{34} + 40 q^{49} - 16 q^{62} + 8 q^{64} + 16 q^{68} + 20 q^{77} + 40 q^{83} - 8 q^{91} + 40 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 + 8 * q^16 + 16 * q^17 - 16 * q^31 + 8 * q^32 + 16 * q^34 + 40 * q^49 - 16 * q^62 + 8 * q^64 + 16 * q^68 + 20 * q^77 + 40 * q^83 - 8 * q^91 + 40 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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