LMFDB - Embedded newform 4950.2.f.c.4949.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(4949,4950)] 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("4950.4949"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-8,0,0,0,0,0,0,0,0,-8,0,0,8,0,0,0,0,0,-12,-8,0,0,0,0,0, -8] 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:	8.0.4328587264.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 109x^{4} + 260x^{2} + 196 $ x^8 + 18x^6 + 109x^4 + 260*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 990)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4949.2
Root			$2.87996i$ of defining polynomial
Character	$\chi$	$=$	4950.4949
Dual form			4950.2.f.c.4949.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.65867 q^{7} +1.00000i q^{8} +(2.74355 - 1.86359i) q^{11} -7.17414 q^{13} +2.65867i q^{14} +1.00000 q^{16} -4.82843i q^{17} -0.755547i q^{19} +(-1.86359 - 2.74355i) q^{22} +7.83280 q^{23} +7.17414i q^{26} +2.65867 q^{28} +3.75992 q^{29} +5.75992 q^{31} -1.00000i q^{32} -4.82843 q^{34} -10.4186i q^{37} -0.755547 q^{38} -3.92968 q^{41} +2.07288 q^{43} +(-2.74355 + 1.86359i) q^{44} -7.83280i q^{46} -8.07288 q^{47} +0.0685055 q^{49} +7.17414 q^{52} -3.55741 q^{53} -2.65867i q^{56} -3.75992i q^{58} -8.41859i q^{59} +9.72973i q^{61} -5.75992i q^{62} -1.00000 q^{64} +13.4631i q^{67} +4.82843i q^{68} +2.17595i q^{71} -7.75992 q^{73} -10.4186 q^{74} +0.755547i q^{76} +(-7.29417 + 4.95465i) q^{77} -4.58835i q^{79} +3.92968i q^{82} +6.72536i q^{83} -2.07288i q^{86} +(1.86359 + 2.74355i) q^{88} +10.4452i q^{89} +19.0736 q^{91} -7.83280 q^{92} +8.07288i q^{94} +15.3600i q^{97} -0.0685055i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{13} + 8 q^{16} - 12 q^{22} - 8 q^{23} - 8 q^{29} + 8 q^{31} - 16 q^{34} - 16 q^{38} + 8 q^{41} - 16 q^{43} - 32 q^{47} + 16 q^{49} + 8 q^{52} - 24 q^{53} - 8 q^{64} - 24 q^{73} - 24 q^{74}+ \cdots + 8 q^{92}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^13 + 8 * q^16 - 12 * q^22 - 8 * q^23 - 8 * q^29 + 8 * q^31 - 16 * q^34 - 16 * q^38 + 8 * q^41 - 16 * q^43 - 32 * q^47 + 16 * q^49 + 8 * q^52 - 24 * q^53 - 8 * q^64 - 24 * q^73 - 24 * q^74 - 28 * q^77 + 12 * q^88 + 24 * q^91 + 8 * q^92

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.65867			−1.00488			−0.502441	−	0.864612i	$-0.667564\pi$
$7$	−2.65867			−1.00488			−0.502441	+	0.864612i	$0.667564\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$		0			0
$11$	2.74355	−	1.86359i	0.827210	−	0.561892i
$11$	2.74355	−	1.86359i	0.827210	−	0.561892i
$12$		0			0
$13$	−7.17414			−1.98975			−0.994874	−	0.101127i	$-0.967755\pi$
$13$	−7.17414			−1.98975			−0.994874	+	0.101127i	$0.967755\pi$
$14$			2.65867i			0.710558i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	4.82843i		−	1.17107i	−0.810649	−	0.585533i	$-0.800885\pi$
$17$		−	4.82843i		−	1.17107i	0.810649	−	0.585533i	$-0.199115\pi$
$18$		0			0
$19$		−	0.755547i		−	0.173334i	−0.996237	−	0.0866672i	$-0.972378\pi$
$19$		−	0.755547i		−	0.173334i	0.996237	−	0.0866672i	$-0.0276217\pi$
$20$		0			0
$21$		0			0
$22$	−1.86359	−	2.74355i	−0.397318	−	0.584926i
$23$	7.83280			1.63325			0.816626	−	0.577167i	$-0.195842\pi$
$23$	7.83280			1.63325			0.816626	+	0.577167i	$0.195842\pi$
$24$		0			0
$25$		0			0
$26$			7.17414i			1.40696i
$27$		0			0
$28$	2.65867			0.502441
$29$	3.75992			0.698200			0.349100	−	0.937085i	$-0.386487\pi$
$29$	3.75992			0.698200			0.349100	+	0.937085i	$0.386487\pi$
$30$		0			0
$31$	5.75992			1.03451			0.517256	−	0.855831i	$-0.326953\pi$
$31$	5.75992			1.03451			0.517256	+	0.855831i	$0.326953\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−4.82843			−0.828068
$35$		0			0
$36$		0			0
$37$		−	10.4186i		−	1.71281i	−0.516309	−	0.856403i	$-0.672694\pi$
$37$		−	10.4186i		−	1.71281i	0.516309	−	0.856403i	$-0.327306\pi$
$38$	−0.755547			−0.122566
$39$		0			0
$40$		0			0
$41$	−3.92968			−0.613713			−0.306857	−	0.951756i	$-0.599277\pi$
$41$	−3.92968			−0.613713			−0.306857	+	0.951756i	$0.599277\pi$
$42$		0			0
$43$	2.07288			0.316111			0.158056	−	0.987430i	$-0.449477\pi$
$43$	2.07288			0.316111			0.158056	+	0.987430i	$0.449477\pi$
$44$	−2.74355	+	1.86359i	−0.413605	+	0.280946i
$45$		0			0
$46$		−	7.83280i		−	1.15488i
$47$	−8.07288			−1.17755			−0.588775	−	0.808297i	$-0.700390\pi$
$47$	−8.07288			−1.17755			−0.588775	+	0.808297i	$0.700390\pi$
$48$		0			0
$49$	0.0685055			0.00978649
$50$		0			0
$51$		0			0
$52$	7.17414			0.994874
$53$	−3.55741			−0.488648			−0.244324	−	0.969694i	$-0.578566\pi$
$53$	−3.55741			−0.488648			−0.244324	+	0.969694i	$0.578566\pi$
$54$		0			0
$55$		0			0
$56$		−	2.65867i		−	0.355279i
$57$		0			0
$58$		−	3.75992i		−	0.493702i
$59$		−	8.41859i		−	1.09601i	−0.836476	−	0.548003i	$-0.815388\pi$
$59$		−	8.41859i		−	1.09601i	0.836476	−	0.548003i	$-0.184612\pi$
$60$		0			0
$61$			9.72973i			1.24576i	0.782315	+	0.622882i	$0.214038\pi$
$61$			9.72973i			1.24576i	−0.782315	+	0.622882i	$0.785962\pi$
$62$		−	5.75992i		−	0.731511i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$			13.4631i			1.64478i	0.568925	+	0.822389i	$0.307359\pi$
$67$			13.4631i			1.64478i	−0.568925	+	0.822389i	$0.692641\pi$
$68$			4.82843i			0.585533i
$69$		0			0
$70$		0			0
$71$			2.17595i			0.258237i	0.991629	+	0.129119i	$0.0412148\pi$
$71$			2.17595i			0.258237i	−0.991629	+	0.129119i	$0.958785\pi$
$72$		0			0
$73$	−7.75992			−0.908230			−0.454115	−	0.890943i	$-0.650045\pi$
$73$	−7.75992			−0.908230			−0.454115	+	0.890943i	$0.650045\pi$
$74$	−10.4186			−1.21114
$75$		0			0
$76$			0.755547i			0.0866672i
$77$	−7.29417	+	4.95465i	−0.831248	+	0.564635i
$78$		0			0
$79$		−	4.58835i		−	0.516230i	−0.966114	−	0.258115i	$-0.916899\pi$
$79$		−	4.58835i		−	0.516230i	0.966114	−	0.258115i	$-0.0831013\pi$
$80$		0			0
$81$		0			0
$82$			3.92968i			0.433961i
$83$			6.72536i			0.738204i	0.929389	+	0.369102i	$0.120335\pi$
$83$			6.72536i			0.738204i	−0.929389	+	0.369102i	$0.879665\pi$
$84$		0			0
$85$		0			0
$86$		−	2.07288i		−	0.223524i
$87$		0			0
$88$	1.86359	+	2.74355i	0.198659	+	0.292463i
$89$			10.4452i			1.10718i	0.832788	+	0.553592i	$0.186743\pi$
$89$			10.4452i			1.10718i	−0.832788	+	0.553592i	$0.813257\pi$
$90$		0			0
$91$	19.0736			1.99946
$92$	−7.83280			−0.816626
$93$		0			0
$94$			8.07288i			0.832654i
$95$		0			0
$96$		0			0
$97$			15.3600i			1.55957i	0.626045	+	0.779787i	$0.284673\pi$
$97$			15.3600i			1.55957i	−0.626045	+	0.779787i	$0.715327\pi$
$98$		−	0.0685055i		−	0.00692010i
$99$		0			0
$100$		0			0
$101$	−8.20251			−0.816180			−0.408090	−	0.912942i	$-0.633805\pi$
$101$	−8.20251			−0.816180			−0.408090	+	0.912942i	$0.633805\pi$
$102$		0			0
$103$		−	12.6550i		−	1.24694i	−0.781848	−	0.623469i	$-0.785723\pi$
$103$		−	12.6550i		−	1.24694i	0.781848	−	0.623469i	$-0.214277\pi$
$104$		−	7.17414i		−	0.703482i
$105$		0			0
$106$			3.55741i			0.345526i
$107$			16.2915i			1.57496i	0.616340	+	0.787480i	$0.288615\pi$
$107$			16.2915i			1.57496i	−0.616340	+	0.787480i	$0.711385\pi$
$108$		0			0
$109$		−	2.41603i		−	0.231413i	−0.993283	−	0.115707i	$-0.963087\pi$
$109$		−	2.41603i		−	0.231413i	0.993283	−	0.115707i	$-0.0369132\pi$
$110$		0			0
$111$		0			0
$112$	−2.65867			−0.251220
$113$	7.19814			0.677144			0.338572	−	0.940941i	$-0.390056\pi$
$113$	7.19814			0.677144			0.338572	+	0.940941i	$0.390056\pi$
$114$		0			0
$115$		0			0
$116$	−3.75992			−0.349100
$117$		0			0
$118$	−8.41859			−0.774994
$119$			12.8372i			1.17678i
$120$		0			0
$121$	4.05410	−	10.2257i	0.368554	−	0.929606i
$122$	9.72973			0.880889
$123$		0			0
$124$	−5.75992			−0.517256
$125$		0			0
$126$		0			0
$127$	−17.4871			−1.55173			−0.775864	−	0.630900i	$-0.782686\pi$
$127$	−17.4871			−1.55173			−0.775864	+	0.630900i	$0.782686\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	−18.4150			−1.60892			−0.804461	−	0.594005i	$-0.797546\pi$
$131$	−18.4150			−1.60892			−0.804461	+	0.594005i	$0.797546\pi$
$132$		0			0
$133$			2.00875i			0.174181i
$134$	13.4631			1.16303
$135$		0			0
$136$	4.82843			0.414034
$137$	−21.0008			−1.79422			−0.897108	−	0.441812i	$-0.854336\pi$
$137$	−21.0008			−1.79422			−0.897108	+	0.441812i	$0.854336\pi$
$138$		0			0
$139$		−	7.00437i		−	0.594103i	−0.954861	−	0.297052i	$-0.903997\pi$
$139$		−	7.00437i		−	0.594103i	0.954861	−	0.297052i	$-0.0960034\pi$
$140$		0			0
$141$		0			0
$142$	2.17595			0.182601
$143$	−19.6826	+	13.3696i	−1.64594	+	1.11802i
$144$		0			0
$145$		0			0
$146$			7.75992i			0.642216i
$147$		0			0
$148$			10.4186i			0.856403i
$149$	−16.8336			−1.37906			−0.689529	−	0.724258i	$-0.742182\pi$
$149$	−16.8336			−1.37906			−0.689529	+	0.724258i	$0.742182\pi$
$150$		0			0
$151$			8.52285i			0.693580i	0.937943	+	0.346790i	$0.112728\pi$
$151$			8.52285i			0.693580i	−0.937943	+	0.346790i	$0.887272\pi$
$152$	0.755547			0.0612830
$153$		0			0
$154$	4.95465	+	7.29417i	0.399257	+	0.587781i
$155$		0			0
$156$		0			0
$157$		−	4.26920i		−	0.340720i	−0.985382	−	0.170360i	$-0.945507\pi$
$157$		−	4.26920i		−	0.340720i	0.985382	−	0.170360i	$-0.0544930\pi$
$158$	−4.58835			−0.365029
$159$		0			0
$160$		0			0
$161$	−20.8248			−1.64122
$162$		0			0
$163$		−	19.3792i		−	1.51790i	−0.651151	−	0.758948i	$-0.725713\pi$
$163$		−	19.3792i		−	1.51790i	0.651151	−	0.758948i	$-0.274287\pi$
$164$	3.92968			0.306857
$165$		0			0
$166$	6.72536			0.521989
$167$		−	16.7909i		−	1.29932i	−0.760227	−	0.649658i	$-0.774912\pi$
$167$		−	16.7909i		−	1.29932i	0.760227	−	0.649658i	$-0.225088\pi$
$168$		0			0
$169$	38.4682			2.95909
$170$		0			0
$171$		0			0
$172$	−2.07288			−0.158056
$173$		−	4.09944i		−	0.311675i	−0.987783	−	0.155837i	$-0.950192\pi$
$173$		−	4.09944i		−	0.311675i	0.987783	−	0.155837i	$-0.0498076\pi$
$174$		0			0
$175$		0			0
$176$	2.74355	−	1.86359i	0.206803	−	0.140473i
$177$		0			0
$178$	10.4452			0.782897
$179$		−	1.38403i		−	0.103447i	−0.998661	−	0.0517235i	$-0.983529\pi$
$179$		−	1.38403i		−	0.103447i	0.998661	−	0.0517235i	$-0.0164714\pi$
$180$		0			0
$181$	−18.2113			−1.35363			−0.676816	−	0.736152i	$-0.736641\pi$
$181$	−18.2113			−1.35363			−0.676816	+	0.736152i	$0.736641\pi$
$182$		−	19.0736i		−	1.41383i
$183$		0			0
$184$			7.83280i			0.577442i
$185$		0			0
$186$		0			0
$187$	−8.99819	−	13.2470i	−0.658013	−	0.968718i
$188$	8.07288			0.588775
$189$		0			0
$190$		0			0
$191$			10.0302i			0.725759i	0.931836	+	0.362879i	$0.118206\pi$
$191$			10.0302i			0.725759i	−0.931836	+	0.362879i	$0.881794\pi$
$192$		0			0
$193$	−12.0514			−0.867482			−0.433741	−	0.901038i	$-0.642807\pi$
$193$	−12.0514			−0.867482			−0.433741	+	0.901038i	$0.642807\pi$
$194$	15.3600			1.10279
$195$		0			0
$196$	−0.0685055			−0.00489325
$197$			0.183327i			0.0130615i	0.999979	+	0.00653074i	$0.00207881\pi$
$197$			0.183327i			0.0130615i	−0.999979	+	0.00653074i	$0.997921\pi$
$198$		0			0
$199$	−8.77168			−0.621808			−0.310904	−	0.950441i	$-0.600632\pi$
$199$	−8.77168			−0.621808			−0.310904	+	0.950441i	$0.600632\pi$
$200$		0			0
$201$		0			0
$202$			8.20251i			0.577127i
$203$	−9.99638			−0.701608
$204$		0			0
$205$		0			0
$206$	−12.6550			−0.881719
$207$		0			0
$208$	−7.17414			−0.497437
$209$	−1.40803	−	2.07288i	−0.0973953	−	0.143384i
$210$		0			0
$211$			0.462341i			0.0318289i	0.999873	+	0.0159144i	$0.00506594\pi$
$211$			0.462341i			0.0318289i	−0.999873	+	0.0159144i	$0.994934\pi$
$212$	3.55741			0.244324
$213$		0			0
$214$	16.2915			1.11367
$215$		0			0
$216$		0			0
$217$	−15.3137			−1.03956
$218$	−2.41603			−0.163634
$219$		0			0
$220$		0			0
$221$			34.6398i			2.33012i
$222$		0			0
$223$			6.00994i			0.402455i	0.979544	+	0.201228i	$0.0644931\pi$
$223$			6.00994i			0.402455i	−0.979544	+	0.201228i	$0.935507\pi$
$224$			2.65867i			0.177640i
$225$		0			0
$226$		−	7.19814i		−	0.478813i
$227$			4.63104i			0.307373i	0.988120	+	0.153686i	$0.0491146\pi$
$227$			4.63104i			0.307373i	−0.988120	+	0.153686i	$0.950885\pi$
$228$		0			0
$229$	−1.32096			−0.0872912			−0.0436456	−	0.999047i	$-0.513897\pi$
$229$	−1.32096			−0.0872912			−0.0436456	+	0.999047i	$0.513897\pi$
$230$		0			0
$231$		0			0
$232$			3.75992i			0.246851i
$233$		−	15.3173i		−	1.00347i	−0.865021	−	0.501736i	$-0.832695\pi$
$233$		−	15.3173i		−	1.00347i	0.865021	−	0.501736i	$-0.167305\pi$
$234$		0			0
$235$		0			0
$236$			8.41859i			0.548003i
$237$		0			0
$238$	12.8372			0.832110
$239$	9.73711			0.629842			0.314921	−	0.949118i	$-0.398022\pi$
$239$	9.73711			0.629842			0.314921	+	0.949118i	$0.398022\pi$
$240$		0			0
$241$			11.8402i			0.762693i	0.924432	+	0.381347i	$0.124540\pi$
$241$			11.8402i			0.762693i	−0.924432	+	0.381347i	$0.875460\pi$
$242$	−10.2257	−	4.05410i	−0.657331	−	0.260607i
$243$		0			0
$244$		−	9.72973i		−	0.622882i
$245$		0			0
$246$		0			0
$247$			5.42040i			0.344892i
$248$			5.75992i			0.365755i
$249$		0			0
$250$		0			0
$251$			21.7890i			1.37531i	0.726036	+	0.687656i	$0.241360\pi$
$251$			21.7890i			1.37531i	−0.726036	+	0.687656i	$0.758640\pi$
$252$		0			0
$253$	21.4897	−	14.5971i	1.35104	−	0.917712i
$254$			17.4871i			1.09724i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−13.2069			−0.823823			−0.411911	−	0.911224i	$-0.635139\pi$
$257$	−13.2069			−0.823823			−0.411911	+	0.911224i	$0.635139\pi$
$258$		0			0
$259$			27.6995i			1.72117i
$260$		0			0
$261$		0			0
$262$			18.4150i			1.13768i
$263$			19.9057i			1.22744i	0.789525	+	0.613718i	$0.210327\pi$
$263$			19.9057i			1.22744i	−0.789525	+	0.613718i	$0.789673\pi$
$264$		0			0
$265$		0			0
$266$	2.00875			0.123164
$267$		0			0
$268$		−	13.4631i		−	0.822389i
$269$			1.03456i			0.0630784i	0.999503	+	0.0315392i	$0.0100409\pi$
$269$			1.03456i			0.0630784i	−0.999503	+	0.0315392i	$0.989959\pi$
$270$		0			0
$271$		−	3.56254i		−	0.216409i	−0.994129	−	0.108204i	$-0.965490\pi$
$271$		−	3.56254i		−	0.216409i	0.994129	−	0.108204i	$-0.0345101\pi$
$272$		−	4.82843i		−	0.292766i
$273$		0			0
$274$			21.0008i			1.26870i
$275$		0			0
$276$		0			0
$277$	−11.3148			−0.679839			−0.339919	−	0.940455i	$-0.610400\pi$
$277$	−11.3148			−0.679839			−0.339919	+	0.940455i	$0.610400\pi$
$278$	−7.00437			−0.420094
$279$		0			0
$280$		0			0
$281$	7.52979			0.449189			0.224595	−	0.974452i	$-0.427894\pi$
$281$	7.52979			0.449189			0.224595	+	0.974452i	$0.427894\pi$
$282$		0			0
$283$	−2.06775			−0.122915			−0.0614576	−	0.998110i	$-0.519575\pi$
$283$	−2.06775			−0.122915			−0.0614576	+	0.998110i	$0.519575\pi$
$284$		−	2.17595i		−	0.129119i
$285$		0			0
$286$	13.3696	+	19.6826i	0.790562	+	1.16386i
$287$	10.4477			0.616709
$288$		0			0
$289$	−6.31371			−0.371395
$290$		0			0
$291$		0			0
$292$	7.75992			0.454115
$293$		−	17.5965i		−	1.02800i	−0.857791	−	0.513999i	$-0.828164\pi$
$293$		−	17.5965i		−	1.02800i	0.857791	−	0.513999i	$-0.171836\pi$
$294$		0			0
$295$		0			0
$296$	10.4186			0.605568
$297$		0			0
$298$			16.8336i			0.975141i
$299$	−56.1936			−3.24976
$300$		0			0
$301$	−5.51109			−0.317654
$302$	8.52285			0.490435
$303$		0			0
$304$		−	0.755547i		−	0.0433336i
$305$		0			0
$306$		0			0
$307$	−22.3644			−1.27640			−0.638202	−	0.769869i	$-0.720321\pi$
$307$	−22.3644			−1.27640			−0.638202	+	0.769869i	$0.720321\pi$
$308$	7.29417	−	4.95465i	0.415624	−	0.282318i
$309$		0			0
$310$		0			0
$311$			6.37483i			0.361484i	0.983531	+	0.180742i	$0.0578499\pi$
$311$			6.37483i			0.361484i	−0.983531	+	0.180742i	$0.942150\pi$
$312$		0			0
$313$			25.9676i			1.46777i	0.679272	+	0.733887i	$0.262296\pi$
$313$			25.9676i			1.46777i	−0.679272	+	0.733887i	$0.737704\pi$
$314$	−4.26920			−0.240925
$315$		0			0
$316$			4.58835i			0.258115i
$317$	7.03094			0.394897			0.197448	−	0.980313i	$-0.436735\pi$
$317$	7.03094			0.394897			0.197448	+	0.980313i	$0.436735\pi$
$318$		0			0
$319$	10.3155	−	7.00694i	0.577558	−	0.392313i
$320$		0			0
$321$		0			0
$322$			20.8248i			1.16052i
$323$	−3.64811			−0.202986
$324$		0			0
$325$		0			0
$326$	−19.3792			−1.07332
$327$		0			0
$328$		−	3.92968i		−	0.216980i
$329$	21.4631			1.18330
$330$		0			0
$331$	13.0118			0.715191			0.357595	−	0.933877i	$-0.383597\pi$
$331$	13.0118			0.715191			0.357595	+	0.933877i	$0.383597\pi$
$332$		−	6.72536i		−	0.369102i
$333$		0			0
$334$	−16.7909			−0.918755
$335$		0			0
$336$		0			0
$337$	33.7083			1.83621			0.918104	−	0.396340i	$-0.129720\pi$
$337$	33.7083			1.83621			0.918104	+	0.396340i	$0.129720\pi$
$338$		−	38.4682i		−	2.09240i
$339$		0			0
$340$		0			0
$341$	15.8026	−	10.7341i	0.855760	−	0.581285i
$342$		0			0
$343$	18.4285			0.995047
$344$			2.07288i			0.111762i
$345$		0			0
$346$	−4.09944			−0.220387
$347$			5.70679i			0.306357i	0.988199	+	0.153178i	$0.0489509\pi$
$347$			5.70679i			0.306357i	−0.988199	+	0.153178i	$0.951049\pi$
$348$		0			0
$349$		−	13.3100i		−	0.712465i	−0.934397	−	0.356233i	$-0.884061\pi$
$349$		−	13.3100i		−	0.712465i	0.934397	−	0.356233i	$-0.115939\pi$
$350$		0			0
$351$		0			0
$352$	−1.86359	−	2.74355i	−0.0993295	−	0.146232i
$353$	26.2577			1.39756			0.698778	−	0.715338i	$-0.253727\pi$
$353$	26.2577			1.39756			0.698778	+	0.715338i	$0.253727\pi$
$354$		0			0
$355$		0			0
$356$		−	10.4452i		−	0.553592i
$357$		0			0
$358$	−1.38403			−0.0731480
$359$	−18.8408			−0.994379			−0.497190	−	0.867642i	$-0.665635\pi$
$359$	−18.8408			−0.994379			−0.497190	+	0.867642i	$0.665635\pi$
$360$		0			0
$361$	18.4291			0.969955
$362$			18.2113i			0.957163i
$363$		0			0
$364$	−19.0736			−0.999730
$365$		0			0
$366$		0			0
$367$			7.39277i			0.385900i	0.981209	+	0.192950i	$0.0618055\pi$
$367$			7.39277i			0.385900i	−0.981209	+	0.192950i	$0.938195\pi$
$368$	7.83280			0.408313
$369$		0			0
$370$		0			0
$371$	9.45797			0.491033
$372$		0			0
$373$	−6.96800			−0.360789			−0.180395	−	0.983594i	$-0.557738\pi$
$373$	−6.96800			−0.360789			−0.180395	+	0.983594i	$0.557738\pi$
$374$	−13.2470	+	8.99819i	−0.684987	+	0.465285i
$375$		0			0
$376$		−	8.07288i		−	0.416327i
$377$	−26.9742			−1.38924
$378$		0			0
$379$	6.89393			0.354117			0.177059	−	0.984200i	$-0.443342\pi$
$379$	6.89393			0.354117			0.177059	+	0.984200i	$0.443342\pi$
$380$		0			0
$381$		0			0
$382$	10.0302			0.513189
$383$	−16.4248			−0.839267			−0.419633	−	0.907694i	$-0.637841\pi$
$383$	−16.4248			−0.839267			−0.419633	+	0.907694i	$0.637841\pi$
$384$		0			0
$385$		0			0
$386$			12.0514i			0.613402i
$387$		0			0
$388$		−	15.3600i		−	0.779787i
$389$			3.17670i			0.161065i	0.996752	+	0.0805325i	$0.0256621\pi$
$389$			3.17670i			0.161065i	−0.996752	+	0.0805325i	$0.974338\pi$
$390$		0			0
$391$		−	37.8201i		−	1.91265i
$392$			0.0685055i			0.00346005i
$393$		0			0
$394$	0.183327			0.00923586
$395$		0			0
$396$		0			0
$397$			5.73230i			0.287696i	0.989600	+	0.143848i	$0.0459476\pi$
$397$			5.73230i			0.287696i	−0.989600	+	0.143848i	$0.954052\pi$
$398$			8.77168i			0.439684i
$399$		0			0
$400$		0			0
$401$			14.5378i			0.725982i	0.931793	+	0.362991i	$0.118244\pi$
$401$			14.5378i			0.725982i	−0.931793	+	0.362991i	$0.881756\pi$
$402$		0			0
$403$	−41.3225			−2.05842
$404$	8.20251			0.408090
$405$		0			0
$406$			9.99638i			0.496112i
$407$	−19.4159	−	28.5839i	−0.962412	−	1.41685i
$408$		0			0
$409$			14.8408i			0.733830i	0.930254	+	0.366915i	$0.119586\pi$
$409$			14.8408i			0.733830i	−0.930254	+	0.366915i	$0.880414\pi$
$410$		0			0
$411$		0			0
$412$			12.6550i			0.623469i
$413$			22.3822i			1.10136i
$414$		0			0
$415$		0			0
$416$			7.17414i			0.351741i
$417$		0			0
$418$	−2.07288	+	1.40803i	−0.101388	+	0.0688689i
$419$		−	4.61235i		−	0.225328i	−0.993633	−	0.112664i	$-0.964062\pi$
$419$		−	4.61235i		−	0.225328i	0.993633	−	0.112664i	$-0.0359384\pi$
$420$		0			0
$421$	18.4373			0.898578			0.449289	−	0.893386i	$-0.351677\pi$
$421$	18.4373			0.898578			0.449289	+	0.893386i	$0.351677\pi$
$422$	0.462341			0.0225064
$423$		0			0
$424$		−	3.55741i		−	0.172763i
$425$		0			0
$426$		0			0
$427$		−	25.8681i		−	1.25185i
$428$		−	16.2915i		−	0.787480i
$429$		0			0
$430$		0			0
$431$	−24.3519			−1.17299			−0.586495	−	0.809953i	$-0.699493\pi$
$431$	−24.3519			−1.17299			−0.586495	+	0.809953i	$0.699493\pi$
$432$		0			0
$433$			28.7909i			1.38360i	0.722089	+	0.691800i	$0.243182\pi$
$433$			28.7909i			1.38360i	−0.722089	+	0.691800i	$0.756818\pi$
$434$			15.3137i			0.735082i
$435$		0			0
$436$			2.41603i			0.115707i
$437$		−	5.91805i		−	0.283099i
$438$		0			0
$439$		−	12.5081i		−	0.596979i	−0.954413	−	0.298489i	$-0.903517\pi$
$439$		−	12.5081i		−	0.596979i	0.954413	−	0.298489i	$-0.0964827\pi$
$440$		0			0
$441$		0			0
$442$	34.6398			1.64765
$443$	10.6347			0.505268			0.252634	−	0.967562i	$-0.418703\pi$
$443$	10.6347			0.505268			0.252634	+	0.967562i	$0.418703\pi$
$444$		0			0
$445$		0			0
$446$	6.00994			0.284579
$447$		0			0
$448$	2.65867			0.125610
$449$		−	36.6799i		−	1.73103i	−0.500881	−	0.865516i	$-0.666991\pi$
$449$		−	36.6799i		−	1.73103i	0.500881	−	0.865516i	$-0.333009\pi$
$450$		0			0
$451$	−10.7813	+	7.32330i	−0.507670	+	0.344841i
$452$	−7.19814			−0.338572
$453$		0			0
$454$	4.63104			0.217346
$455$		0			0
$456$		0			0
$457$	−5.41165			−0.253146			−0.126573	−	0.991957i	$-0.540398\pi$
$457$	−5.41165			−0.253146			−0.126573	+	0.991957i	$0.540398\pi$
$458$			1.32096i			0.0617242i
$459$		0			0
$460$		0			0
$461$	31.9859			1.48973			0.744867	−	0.667213i	$-0.232513\pi$
$461$	31.9859			1.48973			0.744867	+	0.667213i	$0.232513\pi$
$462$		0			0
$463$			2.17851i			0.101244i	0.998718	+	0.0506220i	$0.0161204\pi$
$463$			2.17851i			0.101244i	−0.998718	+	0.0506220i	$0.983880\pi$
$464$	3.75992			0.174550
$465$		0			0
$466$	−15.3173			−0.709562
$467$	−6.62229			−0.306443			−0.153222	−	0.988192i	$-0.548965\pi$
$467$	−6.62229			−0.306443			−0.153222	+	0.988192i	$0.548965\pi$
$468$		0			0
$469$		−	35.7939i		−	1.65281i
$470$		0			0
$471$		0			0
$472$	8.41859			0.387497
$473$	5.68704	−	3.86299i	0.261490	−	0.177620i
$474$		0			0
$475$		0			0
$476$		−	12.8372i		−	0.588391i
$477$		0			0
$478$		−	9.73711i		−	0.445365i
$479$	8.89755			0.406539			0.203270	−	0.979123i	$-0.434843\pi$
$479$	8.89755			0.406539			0.203270	+	0.979123i	$0.434843\pi$
$480$		0			0
$481$			74.7444i			3.40805i
$482$	11.8402			0.539305
$483$		0			0
$484$	−4.05410	+	10.2257i	−0.184277	+	0.464803i
$485$		0			0
$486$		0			0
$487$		−	32.0718i		−	1.45331i	−0.687001	−	0.726656i	$-0.741073\pi$
$487$		−	32.0718i		−	1.45331i	0.687001	−	0.726656i	$-0.258927\pi$
$488$	−9.72973			−0.440444
$489$		0			0
$490$		0			0
$491$	−20.0063			−0.902872			−0.451436	−	0.892303i	$-0.649088\pi$
$491$	−20.0063			−0.902872			−0.451436	+	0.892303i	$0.649088\pi$
$492$		0			0
$493$		−	18.1545i		−	0.817638i
$494$	5.42040			0.243875
$495$		0			0
$496$	5.75992			0.258628
$497$		−	5.78512i		−	0.259498i
$498$		0			0
$499$	4.91912			0.220210			0.110105	−	0.993920i	$-0.464881\pi$
$499$	4.91912			0.220210			0.110105	+	0.993920i	$0.464881\pi$
$500$		0			0
$501$		0			0
$502$	21.7890			0.972493
$503$		−	2.13701i		−	0.0952846i	−0.998864	−	0.0476423i	$-0.984829\pi$
$503$		−	2.13701i		−	0.0952846i	0.998864	−	0.0476423i	$-0.0151708\pi$
$504$		0			0
$505$		0			0
$506$	−14.5971	−	21.4897i	−0.648920	−	0.955332i
$507$		0			0
$508$	17.4871			0.775864
$509$			22.4853i			0.996643i	0.866992	+	0.498321i	$0.166050\pi$
$509$			22.4853i			0.996643i	−0.866992	+	0.498321i	$0.833950\pi$
$510$		0			0
$511$	20.6310			0.912664
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			13.2069i			0.582531i
$515$		0			0
$516$		0			0
$517$	−22.1483	+	15.0445i	−0.974082	+	0.661657i
$518$	27.6995			1.21705
$519$		0			0
$520$		0			0
$521$		−	0.436403i		−	0.0191192i	−0.999954	−	0.00955959i	$-0.996957\pi$
$521$		−	0.436403i		−	0.0191192i	0.999954	−	0.00955959i	$-0.00304296\pi$
$522$		0			0
$523$	24.3680			1.06554			0.532770	−	0.846260i	$-0.321151\pi$
$523$	24.3680			1.06554			0.532770	+	0.846260i	$0.321151\pi$
$524$	18.4150			0.804461
$525$		0			0
$526$	19.9057			0.867929
$527$		−	27.8114i		−	1.21148i
$528$		0			0
$529$	38.3528			1.66751
$530$		0			0
$531$		0			0
$532$		−	2.00875i		−	0.0870903i
$533$	28.1921			1.22113
$534$		0			0
$535$		0			0
$536$	−13.4631			−0.581517
$537$		0			0
$538$	1.03456			0.0446031
$539$	0.187948	−	0.127666i	0.00809549	−	0.00549896i
$540$		0			0
$541$		−	13.3175i		−	0.572563i	−0.958146	−	0.286281i	$-0.907581\pi$
$541$		−	13.3175i		−	0.572563i	0.958146	−	0.286281i	$-0.0924192\pi$
$542$	−3.56254			−0.153024
$543$		0			0
$544$	−4.82843			−0.207017
$545$		0			0
$546$		0			0
$547$	−40.3659			−1.72592			−0.862961	−	0.505271i	$-0.831392\pi$
$547$	−40.3659			−1.72592			−0.862961	+	0.505271i	$0.831392\pi$
$548$	21.0008			0.897108
$549$		0			0
$550$		0			0
$551$		−	2.84080i		−	0.121022i
$552$		0			0
$553$			12.1989i			0.518749i
$554$			11.3148i			0.480719i
$555$		0			0
$556$			7.00437i			0.297052i
$557$			41.8592i			1.77363i	0.462124	+	0.886815i	$0.347087\pi$
$557$			41.8592i			1.77363i	−0.462124	+	0.886815i	$0.652913\pi$
$558$		0			0
$559$	−14.8711			−0.628981
$560$		0			0
$561$		0			0
$562$		−	7.52979i		−	0.317625i
$563$			16.4477i			0.693189i	0.938015	+	0.346594i	$0.112662\pi$
$563$			16.4477i			0.693189i	−0.938015	+	0.346594i	$0.887338\pi$
$564$		0			0
$565$		0			0
$566$			2.06775i			0.0869142i
$567$		0			0
$568$	−2.17595			−0.0913007
$569$	37.3176			1.56444			0.782218	−	0.623004i	$-0.214088\pi$
$569$	37.3176			1.56444			0.782218	+	0.623004i	$0.214088\pi$
$570$		0			0
$571$			23.2547i			0.973179i	0.873631	+	0.486589i	$0.161759\pi$
$571$			23.2547i			0.973179i	−0.873631	+	0.486589i	$0.838241\pi$
$572$	19.6826	−	13.3696i	0.822970	−	0.559012i
$573$		0			0
$574$		−	10.4477i		−	0.436079i
$575$		0			0
$576$		0			0
$577$		−	4.24520i		−	0.176730i	−0.996088	−	0.0883651i	$-0.971836\pi$
$577$		−	4.24520i		−	0.176730i	0.996088	−	0.0883651i	$-0.0281642\pi$
$578$			6.31371i			0.262616i
$579$		0			0
$580$		0			0
$581$		−	17.8805i		−	0.741808i
$582$		0			0
$583$	−9.75992	+	6.62954i	−0.404215	+	0.274567i
$584$		−	7.75992i		−	0.321108i
$585$		0			0
$586$	−17.5965			−0.726904
$587$	9.43084			0.389252			0.194626	−	0.980878i	$-0.437651\pi$
$587$	9.43084			0.389252			0.194626	+	0.980878i	$0.437651\pi$
$588$		0			0
$589$		−	4.35189i		−	0.179317i
$590$		0			0
$591$		0			0
$592$		−	10.4186i		−	0.428201i
$593$		−	4.00875i		−	0.164620i	−0.996607	−	0.0823098i	$-0.973770\pi$
$593$		−	4.00875i		−	0.164620i	0.996607	−	0.0823098i	$-0.0262297\pi$
$594$		0			0
$595$		0			0
$596$	16.8336			0.689529
$597$		0			0
$598$			56.1936i			2.29793i
$599$		−	10.6612i		−	0.435606i	−0.975993	−	0.217803i	$-0.930111\pi$
$599$		−	10.6612i		−	0.435606i	0.975993	−	0.217803i	$-0.0698890\pi$
$600$		0			0
$601$		−	5.01688i		−	0.204643i	−0.994751	−	0.102321i	$-0.967373\pi$
$601$		−	5.01688i		−	0.204643i	0.994751	−	0.102321i	$-0.0326270\pi$
$602$			5.51109i			0.224615i
$603$		0			0
$604$		−	8.52285i		−	0.346790i
$605$		0			0
$606$		0			0
$607$	1.96000			0.0795540			0.0397770	−	0.999209i	$-0.487335\pi$
$607$	1.96000			0.0795540			0.0397770	+	0.999209i	$0.487335\pi$
$608$	−0.755547			−0.0306415
$609$		0			0
$610$		0			0
$611$	57.9159			2.34303
$612$		0			0
$613$	26.0828			1.05348			0.526738	−	0.850028i	$-0.323415\pi$
$613$	26.0828			1.05348			0.526738	+	0.850028i	$0.323415\pi$
$614$			22.3644i			0.902554i
$615$		0			0
$616$	−4.95465	−	7.29417i	−0.199629	−	0.293891i
$617$	35.5699			1.43199			0.715995	−	0.698105i	$-0.245973\pi$
$617$	35.5699			1.43199			0.715995	+	0.698105i	$0.245973\pi$
$618$		0			0
$619$	−28.0854			−1.12885			−0.564423	−	0.825485i	$-0.690901\pi$
$619$	−28.0854			−1.12885			−0.564423	+	0.825485i	$0.690901\pi$
$620$		0			0
$621$		0			0
$622$	6.37483			0.255608
$623$		−	27.7702i		−	1.11259i
$624$		0			0
$625$		0			0
$626$	25.9676			1.03787
$627$		0			0
$628$			4.26920i			0.170360i
$629$	−50.3054			−2.00581
$630$		0			0
$631$	−42.2503			−1.68196			−0.840979	−	0.541067i	$-0.818021\pi$
$631$	−42.2503			−1.68196			−0.840979	+	0.541067i	$0.818021\pi$
$632$	4.58835			0.182515
$633$		0			0
$634$		−	7.03094i		−	0.279234i
$635$		0			0
$636$		0			0
$637$	−0.491467			−0.0194726
$638$	−7.00694	−	10.3155i	−0.277407	−	0.408395i
$639$		0			0
$640$		0			0
$641$		−	32.3133i		−	1.27630i	−0.769913	−	0.638149i	$-0.779701\pi$
$641$		−	32.3133i		−	1.27630i	0.769913	−	0.638149i	$-0.220299\pi$
$642$		0			0
$643$			17.6692i			0.696806i	0.937345	+	0.348403i	$0.113276\pi$
$643$			17.6692i			0.696806i	−0.937345	+	0.348403i	$0.886724\pi$
$644$	20.8248			0.820612
$645$		0			0
$646$			3.64811i			0.143533i
$647$	−36.5102			−1.43536			−0.717681	−	0.696372i	$-0.754796\pi$
$647$	−36.5102			−1.43536			−0.717681	+	0.696372i	$0.754796\pi$
$648$		0			0
$649$	−15.6888	−	23.0968i	−0.615838	−	0.906628i
$650$		0			0
$651$		0			0
$652$			19.3792i			0.758948i
$653$	−35.6671			−1.39576			−0.697881	−	0.716214i	$-0.745873\pi$
$653$	−35.6671			−1.39576			−0.697881	+	0.716214i	$0.745873\pi$
$654$		0			0
$655$		0			0
$656$	−3.92968			−0.153428
$657$		0			0
$658$		−	21.4631i		−	0.836718i
$659$	−28.8183			−1.12260			−0.561301	−	0.827612i	$-0.689699\pi$
$659$	−28.8183			−1.12260			−0.561301	+	0.827612i	$0.689699\pi$
$660$		0			0
$661$	−42.6413			−1.65855			−0.829277	−	0.558838i	$-0.811247\pi$
$661$	−42.6413			−1.65855			−0.829277	+	0.558838i	$0.811247\pi$
$662$		−	13.0118i		−	0.505716i
$663$		0			0
$664$	−6.72536			−0.260995
$665$		0			0
$666$		0			0
$667$	29.4507			1.14034
$668$			16.7909i			0.649658i
$669$		0			0
$670$		0			0
$671$	18.1322	+	26.6940i	0.699986	+	1.03051i
$672$		0			0
$673$	6.50809			0.250868			0.125434	−	0.992102i	$-0.459968\pi$
$673$	6.50809			0.250868			0.125434	+	0.992102i	$0.459968\pi$
$674$		−	33.7083i		−	1.29839i
$675$		0			0
$676$	−38.4682			−1.47955
$677$		−	19.4970i		−	0.749332i	−0.927160	−	0.374666i	$-0.877757\pi$
$677$		−	19.4970i		−	0.749332i	0.927160	−	0.374666i	$-0.122243\pi$
$678$		0			0
$679$		−	40.8372i		−	1.56719i
$680$		0			0
$681$		0			0
$682$	−10.7341	−	15.8026i	−0.411030	−	0.605113i
$683$	−10.1494			−0.388355			−0.194178	−	0.980966i	$-0.562204\pi$
$683$	−10.1494			−0.388355			−0.194178	+	0.980966i	$0.562204\pi$
$684$		0			0
$685$		0			0
$686$		−	18.4285i		−	0.703605i
$687$		0			0
$688$	2.07288			0.0790278
$689$	25.5213			0.972286
$690$		0			0
$691$	−1.56978			−0.0597173			−0.0298587	−	0.999554i	$-0.509506\pi$
$691$	−1.56978			−0.0597173			−0.0298587	+	0.999554i	$0.509506\pi$
$692$			4.09944i			0.155837i
$693$		0			0
$694$	5.70679			0.216627
$695$		0			0
$696$		0			0
$697$			18.9742i			0.718699i
$698$	−13.3100			−0.503789
$699$		0			0
$700$		0			0
$701$	5.33139			0.201364			0.100682	−	0.994919i	$-0.467898\pi$
$701$	5.33139			0.201364			0.100682	+	0.994919i	$0.467898\pi$
$702$		0			0
$703$	−7.87174			−0.296888
$704$	−2.74355	+	1.86359i	−0.103401	+	0.0702365i
$705$		0			0
$706$		−	26.2577i		−	0.988222i
$707$	21.8077			0.820164
$708$		0			0
$709$	−34.0854			−1.28010			−0.640052	−	0.768332i	$-0.721087\pi$
$709$	−34.0854			−1.28010			−0.640052	+	0.768332i	$0.721087\pi$
$710$		0			0
$711$		0			0
$712$	−10.4452			−0.391449
$713$	45.1163			1.68962
$714$		0			0
$715$		0			0
$716$			1.38403i			0.0517235i
$717$		0			0
$718$			18.8408i			0.703132i
$719$			12.3181i			0.459387i	0.973263	+	0.229693i	$0.0737724\pi$
$719$			12.3181i			0.459387i	−0.973263	+	0.229693i	$0.926228\pi$
$720$		0			0
$721$			33.6455i			1.25303i
$722$		−	18.4291i		−	0.685862i
$723$		0			0
$724$	18.2113			0.676816
$725$		0			0
$726$		0			0
$727$		−	50.7065i		−	1.88060i	−0.340348	−	0.940300i	$-0.610545\pi$
$727$		−	50.7065i		−	1.88060i	0.340348	−	0.940300i	$-0.389455\pi$
$728$			19.0736i			0.706916i
$729$		0			0
$730$		0			0
$731$		−	10.0087i		−	0.370187i
$732$		0			0
$733$	40.3683			1.49104			0.745519	−	0.666484i	$-0.232202\pi$
$733$	40.3683			1.49104			0.745519	+	0.666484i	$0.232202\pi$
$734$	7.39277			0.272872
$735$		0			0
$736$		−	7.83280i		−	0.288721i
$737$	25.0896	+	36.9366i	0.924188	+	1.36058i
$738$		0			0
$739$		−	19.6039i		−	0.721140i	−0.932732	−	0.360570i	$-0.882582\pi$
$739$		−	19.6039i		−	0.721140i	0.932732	−	0.360570i	$-0.117418\pi$
$740$		0			0
$741$		0			0
$742$		−	9.45797i		−	0.347213i
$743$			22.2076i			0.814719i	0.913268	+	0.407360i	$0.133550\pi$
$743$			22.2076i			0.814719i	−0.913268	+	0.407360i	$0.866450\pi$
$744$		0			0
$745$		0			0
$746$			6.96800i			0.255117i
$747$		0			0
$748$	8.99819	+	13.2470i	0.329006	+	0.484359i
$749$		−	43.3137i		−	1.58265i
$750$		0			0
$751$	−7.98956			−0.291543			−0.145772	−	0.989318i	$-0.546566\pi$
$751$	−7.98956			−0.291543			−0.145772	+	0.989318i	$0.546566\pi$
$752$	−8.07288			−0.294388
$753$		0			0
$754$			26.9742i			0.982342i
$755$		0			0
$756$		0			0
$757$		−	13.6668i		−	0.496728i	−0.968667	−	0.248364i	$-0.920107\pi$
$757$		−	13.6668i		−	0.496728i	0.968667	−	0.248364i	$-0.0798929\pi$
$758$		−	6.89393i		−	0.250399i
$759$		0			0
$760$		0			0
$761$	−29.7395			−1.07806			−0.539029	−	0.842287i	$-0.681209\pi$
$761$	−29.7395			−1.07806			−0.539029	+	0.842287i	$0.681209\pi$
$762$		0			0
$763$			6.42340i			0.232543i
$764$		−	10.0302i		−	0.362879i
$765$		0			0
$766$			16.4248i			0.593451i
$767$			60.3961i			2.18078i
$768$		0			0
$769$			42.0478i			1.51628i	0.652090	+	0.758142i	$0.273892\pi$
$769$			42.0478i			1.51628i	−0.652090	+	0.758142i	$0.726108\pi$
$770$		0			0
$771$		0			0
$772$	12.0514			0.433741
$773$	−33.1854			−1.19360			−0.596799	−	0.802391i	$-0.703561\pi$
$773$	−33.1854			−1.19360			−0.596799	+	0.802391i	$0.703561\pi$
$774$		0			0
$775$		0			0
$776$	−15.3600			−0.551393
$777$		0			0
$778$	3.17670			0.113890
$779$			2.96906i			0.106378i
$780$		0			0
$781$	4.05506	+	5.96981i	0.145102	+	0.213617i
$782$	−37.8201			−1.35244
$783$		0			0
$784$	0.0685055			0.00244662
$785$		0			0
$786$		0			0
$787$	30.4447			1.08523			0.542617	−	0.839980i	$-0.317433\pi$
$787$	30.4447			1.08523			0.542617	+	0.839980i	$0.317433\pi$
$788$		−	0.183327i		−	0.00653074i
$789$		0			0
$790$		0			0
$791$	−19.1374			−0.680449
$792$		0			0
$793$		−	69.8024i		−	2.47876i
$794$	5.73230			0.203432
$795$		0			0
$796$	8.77168			0.310904
$797$	0.296830			0.0105143			0.00525713	−	0.999986i	$-0.498327\pi$
$797$	0.296830			0.0105143			0.00525713	+	0.999986i	$0.498327\pi$
$798$		0			0
$799$			38.9793i			1.37899i
$800$		0			0
$801$		0			0
$802$	14.5378			0.513347
$803$	−21.2897	+	14.4613i	−0.751297	+	0.510327i
$804$		0			0
$805$		0			0
$806$			41.3225i			1.45552i
$807$		0			0
$808$		−	8.20251i		−	0.288563i
$809$	−22.8559			−0.803569			−0.401785	−	0.915734i	$-0.631610\pi$
$809$	−22.8559			−0.803569			−0.401785	+	0.915734i	$0.631610\pi$
$810$		0			0
$811$		−	53.9909i		−	1.89588i	−0.318452	−	0.947939i	$-0.603163\pi$
$811$		−	53.9909i		−	1.89588i	0.318452	−	0.947939i	$-0.396837\pi$
$812$	9.99638			0.350804
$813$		0			0
$814$	−28.5839	+	19.4159i	−1.00186	+	0.680528i
$815$		0			0
$816$		0			0
$817$		−	1.56616i		−	0.0547930i
$818$	14.8408			0.518896
$819$		0			0
$820$		0			0
$821$	20.8991			0.729382			0.364691	−	0.931129i	$-0.381175\pi$
$821$	20.8991			0.729382			0.364691	+	0.931129i	$0.381175\pi$
$822$		0			0
$823$		−	10.9504i		−	0.381706i	−0.981619	−	0.190853i	$-0.938875\pi$
$823$		−	10.9504i		−	0.381706i	0.981619	−	0.190853i	$-0.0611254\pi$
$824$	12.6550			0.440859
$825$		0			0
$826$	22.3822			0.778777
$827$		−	1.34014i		−	0.0466013i	−0.999729	−	0.0233006i	$-0.992583\pi$
$827$		−	1.34014i		−	0.0466013i	0.999729	−	0.0233006i	$-0.00741749\pi$
$828$		0			0
$829$	−53.8487			−1.87024			−0.935122	−	0.354326i	$-0.884710\pi$
$829$	−53.8487			−1.87024			−0.935122	+	0.354326i	$0.884710\pi$
$830$		0			0
$831$		0			0
$832$	7.17414			0.248718
$833$		−	0.330774i		−	0.0114606i
$834$		0			0
$835$		0			0
$836$	1.40803	+	2.07288i	0.0486976	+	0.0716920i
$837$		0			0
$838$	−4.61235			−0.159331
$839$		−	7.34752i		−	0.253665i	−0.991924	−	0.126832i	$-0.959519\pi$
$839$		−	7.34752i		−	0.253665i	0.991924	−	0.126832i	$-0.0404810\pi$
$840$		0			0
$841$	−14.8630			−0.512517
$842$		−	18.4373i		−	0.635391i
$843$		0			0
$844$		−	0.462341i		−	0.0159144i
$845$		0			0
$846$		0			0
$847$	−10.7785	+	27.1866i	−0.370353	+	0.934144i
$848$	−3.55741			−0.122162
$849$		0			0
$850$		0			0
$851$		−	81.6067i		−	2.79744i
$852$		0			0
$853$	14.2087			0.486497			0.243248	−	0.969964i	$-0.421787\pi$
$853$	14.2087			0.486497			0.243248	+	0.969964i	$0.421787\pi$
$854$	−25.8681			−0.885189
$855$		0			0
$856$	−16.2915			−0.556833
$857$			39.1112i			1.33601i	0.744155	+	0.668007i	$0.232852\pi$
$857$			39.1112i			1.33601i	−0.744155	+	0.668007i	$0.767148\pi$
$858$		0			0
$859$	37.2090			1.26955			0.634777	−	0.772696i	$-0.281092\pi$
$859$	37.2090			1.26955			0.634777	+	0.772696i	$0.281092\pi$
$860$		0			0
$861$		0			0
$862$			24.3519i			0.829429i
$863$	−7.42053			−0.252598			−0.126299	−	0.991992i	$-0.540310\pi$
$863$	−7.42053			−0.252598			−0.126299	+	0.991992i	$0.540310\pi$
$864$		0			0
$865$		0			0
$866$	28.7909			0.978353
$867$		0			0
$868$	15.3137			0.519781
$869$	−8.55078	−	12.5883i	−0.290065	−	0.427030i
$870$		0			0
$871$		−	96.5860i		−	3.27269i
$872$	2.41603			0.0818169
$873$		0			0
$874$	−5.91805			−0.200181
$875$		0			0
$876$		0			0
$877$	2.68373			0.0906231			0.0453115	−	0.998973i	$-0.485572\pi$
$877$	2.68373			0.0906231			0.0453115	+	0.998973i	$0.485572\pi$
$878$	−12.5081			−0.422128
$879$		0			0
$880$		0			0
$881$			12.1168i			0.408224i	0.978948	+	0.204112i	$0.0654307\pi$
$881$			12.1168i			0.408224i	−0.978948	+	0.204112i	$0.934569\pi$
$882$		0			0
$883$		−	34.3483i		−	1.15591i	−0.816068	−	0.577956i	$-0.803851\pi$
$883$		−	34.3483i		−	1.15591i	0.816068	−	0.577956i	$-0.196149\pi$
$884$		−	34.6398i		−	1.16506i
$885$		0			0
$886$		−	10.6347i		−	0.357279i
$887$			26.5008i			0.889811i	0.895577	+	0.444906i	$0.146763\pi$
$887$			26.5008i			0.889811i	−0.895577	+	0.444906i	$0.853237\pi$
$888$		0			0
$889$	46.4923			1.55930
$890$		0			0
$891$		0			0
$892$		−	6.00994i		−	0.201228i
$893$			6.09944i			0.204110i
$894$		0			0
$895$		0			0
$896$		−	2.65867i		−	0.0888198i
$897$		0			0
$898$	−36.6799			−1.22402
$899$	21.6569			0.722297
$900$		0			0
$901$			17.1767i			0.572239i
$902$	7.32330	+	10.7813i	0.243839	+	0.358977i
$903$		0			0
$904$			7.19814i			0.239406i
$905$		0			0
$906$		0			0
$907$			32.6434i			1.08391i	0.840409	+	0.541953i	$0.182315\pi$
$907$			32.6434i			1.08391i	−0.840409	+	0.541953i	$0.817685\pi$
$908$		−	4.63104i		−	0.153686i
$909$		0			0
$910$		0			0
$911$		−	0.901494i		−	0.0298678i	−0.999888	−	0.0149339i	$-0.995246\pi$
$911$		−	0.901494i		−	0.0298678i	0.999888	−	0.0149339i	$-0.00475379\pi$
$912$		0			0
$913$	12.5333	+	18.4513i	0.414791	+	0.610650i
$914$			5.41165i			0.179002i
$915$		0			0
$916$	1.32096			0.0436456
$917$	48.9592			1.61678
$918$		0			0
$919$		−	8.95500i		−	0.295398i	−0.989032	−	0.147699i	$-0.952813\pi$
$919$		−	8.95500i		−	0.295398i	0.989032	−	0.147699i	$-0.0471867\pi$
$920$		0			0
$921$		0			0
$922$		−	31.9859i		−	1.05340i
$923$		−	15.6105i		−	0.513827i
$924$		0			0
$925$		0			0
$926$	2.17851			0.0715903
$927$		0			0
$928$		−	3.75992i		−	0.123425i
$929$		−	13.9973i		−	0.459235i	−0.973281	−	0.229617i	$-0.926252\pi$
$929$		−	13.9973i		−	0.459235i	0.973281	−	0.229617i	$-0.0737475\pi$
$930$		0			0
$931$		−	0.0517591i		−	0.00169634i
$932$			15.3173i			0.501736i
$933$		0			0
$934$			6.62229i			0.216688i
$935$		0			0
$936$		0			0
$937$	44.1245			1.44148			0.720742	−	0.693204i	$-0.243801\pi$
$937$	44.1245			1.44148			0.720742	+	0.693204i	$0.243801\pi$
$938$	−35.7939			−1.16871
$939$		0			0
$940$		0			0
$941$	−14.8643			−0.484563			−0.242281	−	0.970206i	$-0.577896\pi$
$941$	−14.8643			−0.484563			−0.242281	+	0.970206i	$0.577896\pi$
$942$		0			0
$943$	−30.7804			−1.00235
$944$		−	8.41859i		−	0.274002i
$945$		0			0
$946$	−3.86299	−	5.68704i	−0.125597	−	0.184902i
$947$	−6.54203			−0.212588			−0.106294	−	0.994335i	$-0.533898\pi$
$947$	−6.54203			−0.212588			−0.106294	+	0.994335i	$0.533898\pi$
$948$		0			0
$949$	55.6707			1.80715
$950$		0			0
$951$		0			0
$952$	−12.8372			−0.416055
$953$		−	26.2915i		−	0.851666i	−0.904802	−	0.425833i	$-0.859981\pi$
$953$		−	26.2915i		−	0.851666i	0.904802	−	0.425833i	$-0.140019\pi$
$954$		0			0
$955$		0			0
$956$	−9.73711			−0.314921
$957$		0			0
$958$		−	8.89755i		−	0.287467i
$959$	55.8340			1.80297
$960$		0			0
$961$	2.17670			0.0702161
$962$	74.7444			2.40985
$963$		0			0
$964$		−	11.8402i		−	0.381347i
$965$		0			0
$966$		0			0
$967$	−33.2538			−1.06937			−0.534685	−	0.845051i	$-0.679570\pi$
$967$	−33.2538			−1.06937			−0.534685	+	0.845051i	$0.679570\pi$
$968$	10.2257	+	4.05410i	0.328665	+	0.130304i
$969$		0			0
$970$		0			0
$971$		−	21.7839i		−	0.699079i	−0.936922	−	0.349540i	$-0.886338\pi$
$971$		−	21.7839i		−	0.699079i	0.936922	−	0.349540i	$-0.113662\pi$
$972$		0			0
$973$			18.6223i			0.597003i
$974$	−32.0718			−1.02765
$975$		0			0
$976$			9.72973i			0.311441i
$977$	1.71417			0.0548413			0.0274206	−	0.999624i	$-0.491271\pi$
$977$	1.71417			0.0548413			0.0274206	+	0.999624i	$0.491271\pi$
$978$		0			0
$979$	19.4654	+	28.6568i	0.622118	+	0.915874i
$980$		0			0
$981$		0			0
$982$			20.0063i			0.638427i
$983$	−58.7929			−1.87520			−0.937602	−	0.347711i	$-0.886959\pi$
$983$	−58.7929			−1.87520			−0.937602	+	0.347711i	$0.886959\pi$
$984$		0			0
$985$		0			0
$986$	−18.1545			−0.578157
$987$		0			0
$988$		−	5.42040i		−	0.172446i
$989$	16.2365			0.516289
$990$		0			0
$991$	−1.10076			−0.0349668			−0.0174834	−	0.999847i	$-0.505565\pi$
$991$	−1.10076			−0.0349668			−0.0174834	+	0.999847i	$0.505565\pi$
$992$		−	5.75992i		−	0.182878i
$993$		0			0
$994$	−5.78512			−0.183493
$995$		0			0
$996$		0			0
$997$	3.88031			0.122891			0.0614453	−	0.998110i	$-0.480429\pi$
$997$	3.88031			0.122891			0.0614453	+	0.998110i	$0.480429\pi$
$998$		−	4.91912i		−	0.155712i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.f.c.4949.2		8
3.2	odd	2		4950.2.f.d.4949.6		8
5.2	odd	4		990.2.d.b.791.6	yes	8
5.3	odd	4		4950.2.d.h.4751.6		8
5.4	even	2		4950.2.f.f.4949.7		8
11.10	odd	2		4950.2.f.e.4949.7		8
15.2	even	4		990.2.d.a.791.2	✓	8
15.8	even	4		4950.2.d.m.4751.6		8
15.14	odd	2		4950.2.f.e.4949.3		8
20.7	even	4		7920.2.f.b.3761.7		8
33.32	even	2		4950.2.f.f.4949.3		8
55.32	even	4		990.2.d.a.791.7	yes	8
55.43	even	4		4950.2.d.m.4751.3		8
55.54	odd	2		4950.2.f.d.4949.2		8
60.47	odd	4		7920.2.f.a.3761.3		8
165.32	odd	4		990.2.d.b.791.3	yes	8
165.98	odd	4		4950.2.d.h.4751.3		8
165.164	even	2	inner	4950.2.f.c.4949.6		8
220.87	odd	4		7920.2.f.a.3761.6		8
660.527	even	4		7920.2.f.b.3761.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.d.a.791.2	✓	8	15.2	even	4
990.2.d.a.791.7	yes	8	55.32	even	4
990.2.d.b.791.3	yes	8	165.32	odd	4
990.2.d.b.791.6	yes	8	5.2	odd	4
4950.2.d.h.4751.3		8	165.98	odd	4
4950.2.d.h.4751.6		8	5.3	odd	4
4950.2.d.m.4751.3		8	55.43	even	4
4950.2.d.m.4751.6		8	15.8	even	4
4950.2.f.c.4949.2		8	1.1	even	1	trivial
4950.2.f.c.4949.6		8	165.164	even	2	inner
4950.2.f.d.4949.2		8	55.54	odd	2
4950.2.f.d.4949.6		8	3.2	odd	2
4950.2.f.e.4949.3		8	15.14	odd	2
4950.2.f.e.4949.7		8	11.10	odd	2
4950.2.f.f.4949.3		8	33.32	even	2
4950.2.f.f.4949.7		8	5.4	even	2
7920.2.f.a.3761.3		8	60.47	odd	4
7920.2.f.a.3761.6		8	220.87	odd	4
7920.2.f.b.3761.2		8	660.527	even	4
7920.2.f.b.3761.7		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.65867 q^{7} +1.00000i q^{8} +(2.74355 - 1.86359i) q^{11} -7.17414 q^{13} +2.65867i q^{14} +1.00000 q^{16} -4.82843i q^{17} -0.755547i q^{19} +(-1.86359 - 2.74355i) q^{22} +7.83280 q^{23} +7.17414i q^{26} +2.65867 q^{28} +3.75992 q^{29} +5.75992 q^{31} -1.00000i q^{32} -4.82843 q^{34} -10.4186i q^{37} -0.755547 q^{38} -3.92968 q^{41} +2.07288 q^{43} +(-2.74355 + 1.86359i) q^{44} -7.83280i q^{46} -8.07288 q^{47} +0.0685055 q^{49} +7.17414 q^{52} -3.55741 q^{53} -2.65867i q^{56} -3.75992i q^{58} -8.41859i q^{59} +9.72973i q^{61} -5.75992i q^{62} -1.00000 q^{64} +13.4631i q^{67} +4.82843i q^{68} +2.17595i q^{71} -7.75992 q^{73} -10.4186 q^{74} +0.755547i q^{76} +(-7.29417 + 4.95465i) q^{77} -4.58835i q^{79} +3.92968i q^{82} +6.72536i q^{83} -2.07288i q^{86} +(1.86359 + 2.74355i) q^{88} +10.4452i q^{89} +19.0736 q^{91} -7.83280 q^{92} +8.07288i q^{94} +15.3600i q^{97} -0.0685055i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{13} + 8 q^{16} - 12 q^{22} - 8 q^{23} - 8 q^{29} + 8 q^{31} - 16 q^{34} - 16 q^{38} + 8 q^{41} - 16 q^{43} - 32 q^{47} + 16 q^{49} + 8 q^{52} - 24 q^{53} - 8 q^{64} - 24 q^{73} - 24 q^{74}+ \cdots + 8 q^{92}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^13 + 8 * q^16 - 12 * q^22 - 8 * q^23 - 8 * q^29 + 8 * q^31 - 16 * q^34 - 16 * q^38 + 8 * q^41 - 16 * q^43 - 32 * q^47 + 16 * q^49 + 8 * q^52 - 24 * q^53 - 8 * q^64 - 24 * q^73 - 24 * q^74 - 28 * q^77 + 12 * q^88 + 24 * q^91 + 8 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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