LMFDB - Embedded newform 1950.2.f.o.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(649,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1950.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.f (of order $2$, degree $1$, not 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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	1950.649
Dual form			1950.2.f.o.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +5.12311 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +5.12311 q^{7} +1.00000 q^{8} -1.00000 q^{9} -3.12311i q^{11} -1.00000i q^{12} +(3.56155 + 0.561553i) q^{13} +5.12311 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000 q^{18} +6.00000i q^{19} -5.12311i q^{21} -3.12311i q^{22} -3.12311i q^{23} -1.00000i q^{24} +(3.56155 + 0.561553i) q^{26} +1.00000i q^{27} +5.12311 q^{28} -2.00000 q^{29} -5.12311i q^{31} +1.00000 q^{32} -3.12311 q^{33} +2.00000i q^{34} -1.00000 q^{36} -3.12311 q^{37} +6.00000i q^{38} +(0.561553 - 3.56155i) q^{39} +9.12311i q^{41} -5.12311i q^{42} +10.2462i q^{43} -3.12311i q^{44} -3.12311i q^{46} -10.2462 q^{47} -1.00000i q^{48} +19.2462 q^{49} +2.00000 q^{51} +(3.56155 + 0.561553i) q^{52} -11.3693i q^{53} +1.00000i q^{54} +5.12311 q^{56} +6.00000 q^{57} -2.00000 q^{58} -7.12311i q^{59} +10.0000 q^{61} -5.12311i q^{62} -5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} -13.1231 q^{67} +2.00000i q^{68} -3.12311 q^{69} -6.24621i q^{71} -1.00000 q^{72} -4.87689 q^{73} -3.12311 q^{74} +6.00000i q^{76} -16.0000i q^{77} +(0.561553 - 3.56155i) q^{78} -8.00000 q^{79} +1.00000 q^{81} +9.12311i q^{82} +10.2462 q^{83} -5.12311i q^{84} +10.2462i q^{86} +2.00000i q^{87} -3.12311i q^{88} +5.12311i q^{89} +(18.2462 + 2.87689i) q^{91} -3.12311i q^{92} -5.12311 q^{93} -10.2462 q^{94} -1.00000i q^{96} +4.87689 q^{97} +19.2462 q^{98} +3.12311i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 - 4 * q^9	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} - 4 q^{9} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{18} + 6 q^{26} + 4 q^{28} - 8 q^{29} + 4 q^{32} + 4 q^{33} - 4 q^{36} + 4 q^{37} - 6 q^{39} - 8 q^{47} + 44 q^{49} + 8 q^{51} + 6 q^{52} + 4 q^{56} + 24 q^{57} - 8 q^{58} + 40 q^{61} - 4 q^{63} + 4 q^{64} + 4 q^{66} - 36 q^{67} + 4 q^{69} - 4 q^{72} - 36 q^{73} + 4 q^{74} - 6 q^{78} - 32 q^{79} + 4 q^{81} + 8 q^{83} + 40 q^{91} - 4 q^{93} - 8 q^{94} + 36 q^{97} + 44 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 - 4 * q^9 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^18 + 6 * q^26 + 4 * q^28 - 8 * q^29 + 4 * q^32 + 4 * q^33 - 4 * q^36 + 4 * q^37 - 6 * q^39 - 8 * q^47 + 44 * q^49 + 8 * q^51 + 6 * q^52 + 4 * q^56 + 24 * q^57 - 8 * q^58 + 40 * q^61 - 4 * q^63 + 4 * q^64 + 4 * q^66 - 36 * q^67 + 4 * q^69 - 4 * q^72 - 36 * q^73 + 4 * q^74 - 6 * q^78 - 32 * q^79 + 4 * q^81 + 8 * q^83 + 40 * q^91 - 4 * q^93 - 8 * q^94 + 36 * q^97 + 44 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		−	1.00000i		−	0.408248i
$7$	5.12311			1.93635			0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311			1.93635			0.968176	+	0.250270i	$0.0805195\pi$
$8$	1.00000			0.353553
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	3.12311i		−	0.941652i	−0.882226	−	0.470826i	$-0.843956\pi$
$11$		−	3.12311i		−	0.941652i	0.882226	−	0.470826i	$-0.156044\pi$
$12$		−	1.00000i		−	0.288675i
$13$	3.56155	+	0.561553i	0.987797	+	0.155747i
$13$	3.56155	+	0.561553i	0.987797	+	0.155747i
$14$	5.12311			1.36921
$15$		0			0
$16$	1.00000			0.250000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	−1.00000			−0.235702
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		−	5.12311i		−	1.11795i
$22$		−	3.12311i		−	0.665848i
$23$		−	3.12311i		−	0.651213i	−0.945505	−	0.325606i	$-0.894432\pi$
$23$		−	3.12311i		−	0.651213i	0.945505	−	0.325606i	$-0.105568\pi$
$24$		−	1.00000i		−	0.204124i
$25$		0			0
$26$	3.56155	+	0.561553i	0.698478	+	0.110130i
$27$			1.00000i			0.192450i
$28$	5.12311			0.968176
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		−	5.12311i		−	0.920137i	−0.887883	−	0.460068i	$-0.847825\pi$
$31$		−	5.12311i		−	0.920137i	0.887883	−	0.460068i	$-0.152175\pi$
$32$	1.00000			0.176777
$33$	−3.12311			−0.543663
$34$			2.00000i			0.342997i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	−3.12311			−0.513435			−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311			−0.513435			−0.256718	+	0.966486i	$0.582641\pi$
$38$			6.00000i			0.973329i
$39$	0.561553	−	3.56155i	0.0899204	−	0.570305i
$40$		0			0
$41$			9.12311i			1.42479i	0.701779	+	0.712395i	$0.252389\pi$
$41$			9.12311i			1.42479i	−0.701779	+	0.712395i	$0.747611\pi$
$42$		−	5.12311i		−	0.790512i
$43$			10.2462i			1.56253i	0.624198	+	0.781266i	$0.285426\pi$
$43$			10.2462i			1.56253i	−0.624198	+	0.781266i	$0.714574\pi$
$44$		−	3.12311i		−	0.470826i
$45$		0			0
$46$		−	3.12311i		−	0.460477i
$47$	−10.2462			−1.49456			−0.747282	−	0.664507i	$-0.768641\pi$
$47$	−10.2462			−1.49456			−0.747282	+	0.664507i	$0.768641\pi$
$48$		−	1.00000i		−	0.144338i
$49$	19.2462			2.74946
$50$		0			0
$51$	2.00000			0.280056
$52$	3.56155	+	0.561553i	0.493899	+	0.0778734i
$53$		−	11.3693i		−	1.56170i	−0.624721	−	0.780848i	$-0.714787\pi$
$53$		−	11.3693i		−	1.56170i	0.624721	−	0.780848i	$-0.285213\pi$
$54$			1.00000i			0.136083i
$55$		0			0
$56$	5.12311			0.684604
$57$	6.00000			0.794719
$58$	−2.00000			−0.262613
$59$		−	7.12311i		−	0.927349i	−0.886006	−	0.463675i	$-0.846531\pi$
$59$		−	7.12311i		−	0.927349i	0.886006	−	0.463675i	$-0.153469\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		−	5.12311i		−	0.650635i
$63$	−5.12311			−0.645451
$64$	1.00000			0.125000
$65$		0			0
$66$	−3.12311			−0.384428
$67$	−13.1231			−1.60324			−0.801621	−	0.597832i	$-0.796029\pi$
$67$	−13.1231			−1.60324			−0.801621	+	0.597832i	$0.796029\pi$
$68$			2.00000i			0.242536i
$69$	−3.12311			−0.375978
$70$		0			0
$71$		−	6.24621i		−	0.741289i	−0.928775	−	0.370644i	$-0.879137\pi$
$71$		−	6.24621i		−	0.741289i	0.928775	−	0.370644i	$-0.120863\pi$
$72$	−1.00000			−0.117851
$73$	−4.87689			−0.570797			−0.285399	−	0.958409i	$-0.592126\pi$
$73$	−4.87689			−0.570797			−0.285399	+	0.958409i	$0.592126\pi$
$74$	−3.12311			−0.363054
$75$		0			0
$76$			6.00000i			0.688247i
$77$		−	16.0000i		−	1.82337i
$78$	0.561553	−	3.56155i	0.0635833	−	0.403266i
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$			9.12311i			1.00748i
$83$	10.2462			1.12467			0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462			1.12467			0.562334	+	0.826910i	$0.309904\pi$
$84$		−	5.12311i		−	0.558977i
$85$		0			0
$86$			10.2462i			1.10488i
$87$			2.00000i			0.214423i
$88$		−	3.12311i		−	0.332924i
$89$			5.12311i			0.543048i	0.962432	+	0.271524i	$0.0875277\pi$
$89$			5.12311i			0.543048i	−0.962432	+	0.271524i	$0.912472\pi$
$90$		0			0
$91$	18.2462	+	2.87689i	1.91272	+	0.301580i
$92$		−	3.12311i		−	0.325606i
$93$	−5.12311			−0.531241
$94$	−10.2462			−1.05682
$95$		0			0
$96$		−	1.00000i		−	0.102062i
$97$	4.87689			0.495174			0.247587	−	0.968866i	$-0.420362\pi$
$97$	4.87689			0.495174			0.247587	+	0.968866i	$0.420362\pi$
$98$	19.2462			1.94416
$99$			3.12311i			0.313884i
$100$		0			0
$101$	4.24621			0.422514			0.211257	−	0.977431i	$-0.432244\pi$
$101$	4.24621			0.422514			0.211257	+	0.977431i	$0.432244\pi$
$102$	2.00000			0.198030
$103$			4.87689i			0.480535i	0.970707	+	0.240267i	$0.0772351\pi$
$103$			4.87689i			0.480535i	−0.970707	+	0.240267i	$0.922765\pi$
$104$	3.56155	+	0.561553i	0.349239	+	0.0550648i
$105$		0			0
$106$		−	11.3693i		−	1.10429i
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$			1.00000i			0.0962250i
$109$		−	11.1231i		−	1.06540i	−0.846304	−	0.532700i	$-0.821177\pi$
$109$		−	11.1231i		−	1.06540i	0.846304	−	0.532700i	$-0.178823\pi$
$110$		0			0
$111$			3.12311i			0.296432i
$112$	5.12311			0.484088
$113$			4.24621i			0.399450i	0.979852	+	0.199725i	$0.0640049\pi$
$113$			4.24621i			0.399450i	−0.979852	+	0.199725i	$0.935995\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$	−2.00000			−0.185695
$117$	−3.56155	−	0.561553i	−0.329266	−	0.0519156i
$118$		−	7.12311i		−	0.655735i
$119$			10.2462i			0.939269i
$120$		0			0
$121$	1.24621			0.113292
$122$	10.0000			0.905357
$123$	9.12311			0.822603
$124$		−	5.12311i		−	0.460068i
$125$		0			0
$126$	−5.12311			−0.456403
$127$		−	4.87689i		−	0.432754i	−0.976310	−	0.216377i	$-0.930576\pi$
$127$		−	4.87689i		−	0.432754i	0.976310	−	0.216377i	$-0.0694241\pi$
$128$	1.00000			0.0883883
$129$	10.2462			0.902129
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$	−3.12311			−0.271831
$133$			30.7386i			2.66538i
$134$	−13.1231			−1.13366
$135$		0			0
$136$			2.00000i			0.171499i
$137$	−22.4924			−1.92166			−0.960829	−	0.277143i	$-0.910612\pi$
$137$	−22.4924			−1.92166			−0.960829	+	0.277143i	$0.910612\pi$
$138$	−3.12311			−0.265856
$139$	−16.4924			−1.39887			−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924			−1.39887			−0.699435	+	0.714697i	$0.746565\pi$
$140$		0			0
$141$			10.2462i			0.862887i
$142$		−	6.24621i		−	0.524170i
$143$	1.75379	−	11.1231i	0.146659	−	0.930161i
$144$	−1.00000			−0.0833333
$145$		0			0
$146$	−4.87689			−0.403615
$147$		−	19.2462i		−	1.58740i
$148$	−3.12311			−0.256718
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		−	11.3693i		−	0.925222i	−0.886561	−	0.462611i	$-0.846913\pi$
$151$		−	11.3693i		−	0.925222i	0.886561	−	0.462611i	$-0.153087\pi$
$152$			6.00000i			0.486664i
$153$		−	2.00000i		−	0.161690i
$154$		−	16.0000i		−	1.28932i
$155$		0			0
$156$	0.561553	−	3.56155i	0.0449602	−	0.285152i
$157$		−	3.36932i		−	0.268901i	−0.990920	−	0.134450i	$-0.957073\pi$
$157$		−	3.36932i		−	0.268901i	0.990920	−	0.134450i	$-0.0429269\pi$
$158$	−8.00000			−0.636446
$159$	−11.3693			−0.901645
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$	1.00000			0.0785674
$163$	1.12311			0.0879684			0.0439842	−	0.999032i	$-0.485995\pi$
$163$	1.12311			0.0879684			0.0439842	+	0.999032i	$0.485995\pi$
$164$			9.12311i			0.712395i
$165$		0			0
$166$	10.2462			0.795260
$167$	5.75379			0.445242			0.222621	−	0.974905i	$-0.428539\pi$
$167$	5.75379			0.445242			0.222621	+	0.974905i	$0.428539\pi$
$168$		−	5.12311i		−	0.395256i
$169$	12.3693	+	4.00000i	0.951486	+	0.307692i
$170$		0			0
$171$		−	6.00000i		−	0.458831i
$172$			10.2462i			0.781266i
$173$		−	14.8769i		−	1.13107i	−0.824725	−	0.565535i	$-0.808670\pi$
$173$		−	14.8769i		−	1.13107i	0.824725	−	0.565535i	$-0.191330\pi$
$174$			2.00000i			0.151620i
$175$		0			0
$176$		−	3.12311i		−	0.235413i
$177$	−7.12311			−0.535405
$178$			5.12311i			0.383993i
$179$	16.4924			1.23270			0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924			1.23270			0.616351	+	0.787472i	$0.288610\pi$
$180$		0			0
$181$	−3.75379			−0.279017			−0.139508	−	0.990221i	$-0.544552\pi$
$181$	−3.75379			−0.279017			−0.139508	+	0.990221i	$0.544552\pi$
$182$	18.2462	+	2.87689i	1.35250	+	0.213250i
$183$		−	10.0000i		−	0.739221i
$184$		−	3.12311i		−	0.230238i
$185$		0			0
$186$	−5.12311			−0.375644
$187$	6.24621			0.456768
$188$	−10.2462			−0.747282
$189$			5.12311i			0.372651i
$190$		0			0
$191$	−16.4924			−1.19335			−0.596675	−	0.802483i	$-0.703512\pi$
$191$	−16.4924			−1.19335			−0.596675	+	0.802483i	$0.703512\pi$
$192$		−	1.00000i		−	0.0721688i
$193$	−16.8769			−1.21483			−0.607413	−	0.794386i	$-0.707793\pi$
$193$	−16.8769			−1.21483			−0.607413	+	0.794386i	$0.707793\pi$
$194$	4.87689			0.350141
$195$		0			0
$196$	19.2462			1.37473
$197$	0.246211			0.0175418			0.00877091	−	0.999962i	$-0.497208\pi$
$197$	0.246211			0.0175418			0.00877091	+	0.999962i	$0.497208\pi$
$198$			3.12311i			0.221949i
$199$	−1.75379			−0.124323			−0.0621614	−	0.998066i	$-0.519799\pi$
$199$	−1.75379			−0.124323			−0.0621614	+	0.998066i	$0.519799\pi$
$200$		0			0
$201$			13.1231i			0.925633i
$202$	4.24621			0.298762
$203$	−10.2462			−0.719143
$204$	2.00000			0.140028
$205$		0			0
$206$			4.87689i			0.339789i
$207$			3.12311i			0.217071i
$208$	3.56155	+	0.561553i	0.246949	+	0.0389367i
$209$	18.7386			1.29618
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		−	11.3693i		−	0.780848i
$213$	−6.24621			−0.427983
$214$		−	8.00000i		−	0.546869i
$215$		0			0
$216$			1.00000i			0.0680414i
$217$		−	26.2462i		−	1.78171i
$218$		−	11.1231i		−	0.753352i
$219$			4.87689i			0.329550i
$220$		0			0
$221$	−1.12311	+	7.12311i	−0.0755483	+	0.479152i
$222$			3.12311i			0.209609i
$223$	−15.3693			−1.02921			−0.514603	−	0.857429i	$-0.672061\pi$
$223$	−15.3693			−1.02921			−0.514603	+	0.857429i	$0.672061\pi$
$224$	5.12311			0.342302
$225$		0			0
$226$			4.24621i			0.282454i
$227$	−8.00000			−0.530979			−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000			−0.530979			−0.265489	+	0.964114i	$0.585534\pi$
$228$	6.00000			0.397360
$229$		−	3.12311i		−	0.206381i	−0.994662	−	0.103190i	$-0.967095\pi$
$229$		−	3.12311i		−	0.206381i	0.994662	−	0.103190i	$-0.0329051\pi$
$230$		0			0
$231$	−16.0000			−1.05272
$232$	−2.00000			−0.131306
$233$			24.2462i			1.58842i	0.607642	+	0.794211i	$0.292116\pi$
$233$			24.2462i			1.58842i	−0.607642	+	0.794211i	$0.707884\pi$
$234$	−3.56155	−	0.561553i	−0.232826	−	0.0367099i
$235$		0			0
$236$		−	7.12311i		−	0.463675i
$237$			8.00000i			0.519656i
$238$			10.2462i			0.664163i
$239$			28.4924i			1.84302i	0.388353	+	0.921511i	$0.373044\pi$
$239$			28.4924i			1.84302i	−0.388353	+	0.921511i	$0.626956\pi$
$240$		0			0
$241$			2.24621i			0.144691i	0.997380	+	0.0723456i	$0.0230485\pi$
$241$			2.24621i			0.144691i	−0.997380	+	0.0723456i	$0.976952\pi$
$242$	1.24621			0.0801095
$243$		−	1.00000i		−	0.0641500i
$244$	10.0000			0.640184
$245$		0			0
$246$	9.12311			0.581668
$247$	−3.36932	+	21.3693i	−0.214384	+	1.35970i
$248$		−	5.12311i		−	0.325318i
$249$		−	10.2462i		−	0.649327i
$250$		0			0
$251$	9.75379			0.615654			0.307827	−	0.951442i	$-0.400398\pi$
$251$	9.75379			0.615654			0.307827	+	0.951442i	$0.400398\pi$
$252$	−5.12311			−0.322725
$253$	−9.75379			−0.613215
$254$		−	4.87689i		−	0.306004i
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.24621i			0.264871i	0.991192	+	0.132436i	$0.0422798\pi$
$257$			4.24621i			0.264871i	−0.991192	+	0.132436i	$0.957720\pi$
$258$	10.2462			0.637901
$259$	−16.0000			−0.994192
$260$		0			0
$261$	2.00000			0.123797
$262$	4.00000			0.247121
$263$			2.63068i			0.162215i	0.996705	+	0.0811074i	$0.0258457\pi$
$263$			2.63068i			0.162215i	−0.996705	+	0.0811074i	$0.974154\pi$
$264$	−3.12311			−0.192214
$265$		0			0
$266$			30.7386i			1.88471i
$267$	5.12311			0.313529
$268$	−13.1231			−0.801621
$269$	−0.246211			−0.0150118			−0.00750588	−	0.999972i	$-0.502389\pi$
$269$	−0.246211			−0.0150118			−0.00750588	+	0.999972i	$0.502389\pi$
$270$		0			0
$271$			14.8769i			0.903707i	0.892092	+	0.451853i	$0.149237\pi$
$271$			14.8769i			0.903707i	−0.892092	+	0.451853i	$0.850763\pi$
$272$			2.00000i			0.121268i
$273$	2.87689	−	18.2462i	0.174118	−	1.10431i
$274$	−22.4924			−1.35882
$275$		0			0
$276$	−3.12311			−0.187989
$277$			27.8617i			1.67405i	0.547165	+	0.837025i	$0.315707\pi$
$277$			27.8617i			1.67405i	−0.547165	+	0.837025i	$0.684293\pi$
$278$	−16.4924			−0.989150
$279$			5.12311i			0.306712i
$280$		0			0
$281$			5.12311i			0.305619i	0.988256	+	0.152809i	$0.0488321\pi$
$281$			5.12311i			0.305619i	−0.988256	+	0.152809i	$0.951168\pi$
$282$			10.2462i			0.610153i
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		−	6.24621i		−	0.370644i
$285$		0			0
$286$	1.75379	−	11.1231i	0.103704	−	0.657723i
$287$			46.7386i			2.75889i
$288$	−1.00000			−0.0589256
$289$	13.0000			0.764706
$290$		0			0
$291$		−	4.87689i		−	0.285889i
$292$	−4.87689			−0.285399
$293$	−20.7386			−1.21156			−0.605782	−	0.795631i	$-0.707140\pi$
$293$	−20.7386			−1.21156			−0.605782	+	0.795631i	$0.707140\pi$
$294$		−	19.2462i		−	1.12246i
$295$		0			0
$296$	−3.12311			−0.181527
$297$	3.12311			0.181221
$298$			14.0000i			0.810998i
$299$	1.75379	−	11.1231i	0.101424	−	0.643266i
$300$		0			0
$301$			52.4924i			3.02561i
$302$		−	11.3693i		−	0.654231i
$303$		−	4.24621i		−	0.243938i
$304$			6.00000i			0.344124i
$305$		0			0
$306$		−	2.00000i		−	0.114332i
$307$	−22.8769			−1.30565			−0.652827	−	0.757507i	$-0.726417\pi$
$307$	−22.8769			−1.30565			−0.652827	+	0.757507i	$0.726417\pi$
$308$		−	16.0000i		−	0.911685i
$309$	4.87689			0.277437
$310$		0			0
$311$	−24.4924			−1.38884			−0.694419	−	0.719571i	$-0.744339\pi$
$311$	−24.4924			−1.38884			−0.694419	+	0.719571i	$0.744339\pi$
$312$	0.561553	−	3.56155i	0.0317917	−	0.201633i
$313$			0.246211i			0.0139167i	0.999976	+	0.00695834i	$0.00221493\pi$
$313$			0.246211i			0.0139167i	−0.999976	+	0.00695834i	$0.997785\pi$
$314$		−	3.36932i		−	0.190142i
$315$		0			0
$316$	−8.00000			−0.450035
$317$	−6.00000			−0.336994			−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000			−0.336994			−0.168497	+	0.985702i	$0.553891\pi$
$318$	−11.3693			−0.637560
$319$			6.24621i			0.349721i
$320$		0			0
$321$	−8.00000			−0.446516
$322$		−	16.0000i		−	0.891645i
$323$	−12.0000			−0.667698
$324$	1.00000			0.0555556
$325$		0			0
$326$	1.12311			0.0622031
$327$	−11.1231			−0.615109
$328$			9.12311i			0.503739i
$329$	−52.4924			−2.89400
$330$		0			0
$331$		−	24.2462i		−	1.33269i	−0.745643	−	0.666346i	$-0.767857\pi$
$331$		−	24.2462i		−	1.33269i	0.745643	−	0.666346i	$-0.232143\pi$
$332$	10.2462			0.562334
$333$	3.12311			0.171145
$334$	5.75379			0.314833
$335$		0			0
$336$		−	5.12311i		−	0.279488i
$337$			6.00000i			0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$			6.00000i			0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	12.3693	+	4.00000i	0.672802	+	0.217571i
$339$	4.24621			0.230623
$340$		0			0
$341$	−16.0000			−0.866449
$342$		−	6.00000i		−	0.324443i
$343$	62.7386			3.38757
$344$			10.2462i			0.552439i
$345$		0			0
$346$		−	14.8769i		−	0.799787i
$347$			2.24621i			0.120583i	0.998181	+	0.0602915i	$0.0192030\pi$
$347$			2.24621i			0.120583i	−0.998181	+	0.0602915i	$0.980797\pi$
$348$			2.00000i			0.107211i
$349$			5.36932i			0.287413i	0.989620	+	0.143706i	$0.0459021\pi$
$349$			5.36932i			0.287413i	−0.989620	+	0.143706i	$0.954098\pi$
$350$		0			0
$351$	−0.561553	+	3.56155i	−0.0299735	+	0.190102i
$352$		−	3.12311i		−	0.166462i
$353$	4.24621			0.226003			0.113002	−	0.993595i	$-0.463954\pi$
$353$	4.24621			0.226003			0.113002	+	0.993595i	$0.463954\pi$
$354$	−7.12311			−0.378589
$355$		0			0
$356$			5.12311i			0.271524i
$357$	10.2462			0.542287
$358$	16.4924			0.871652
$359$		−	34.2462i		−	1.80745i	−0.428118	−	0.903723i	$-0.640823\pi$
$359$		−	34.2462i		−	1.80745i	0.428118	−	0.903723i	$-0.359177\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$	−3.75379			−0.197295
$363$		−	1.24621i		−	0.0654091i
$364$	18.2462	+	2.87689i	0.956361	+	0.150790i
$365$		0			0
$366$		−	10.0000i		−	0.522708i
$367$			33.3693i			1.74186i	0.491403	+	0.870932i	$0.336484\pi$
$367$			33.3693i			1.74186i	−0.491403	+	0.870932i	$0.663516\pi$
$368$		−	3.12311i		−	0.162803i
$369$		−	9.12311i		−	0.474930i
$370$		0			0
$371$		−	58.2462i		−	3.02399i
$372$	−5.12311			−0.265621
$373$			1.12311i			0.0581522i	0.999577	+	0.0290761i	$0.00925652\pi$
$373$			1.12311i			0.0581522i	−0.999577	+	0.0290761i	$0.990743\pi$
$374$	6.24621			0.322984
$375$		0			0
$376$	−10.2462			−0.528408
$377$	−7.12311	−	1.12311i	−0.366859	−	0.0578429i
$378$			5.12311i			0.263504i
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	−4.87689			−0.249851
$382$	−16.4924			−0.843826
$383$	18.2462			0.932338			0.466169	−	0.884696i	$-0.345634\pi$
$383$	18.2462			0.932338			0.466169	+	0.884696i	$0.345634\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	−16.8769			−0.859011
$387$		−	10.2462i		−	0.520844i
$388$	4.87689			0.247587
$389$	−16.2462			−0.823716			−0.411858	−	0.911248i	$-0.635120\pi$
$389$	−16.2462			−0.823716			−0.411858	+	0.911248i	$0.635120\pi$
$390$		0			0
$391$	6.24621			0.315884
$392$	19.2462			0.972080
$393$		−	4.00000i		−	0.201773i
$394$	0.246211			0.0124039
$395$		0			0
$396$			3.12311i			0.156942i
$397$	−9.36932			−0.470233			−0.235116	−	0.971967i	$-0.575547\pi$
$397$	−9.36932			−0.470233			−0.235116	+	0.971967i	$0.575547\pi$
$398$	−1.75379			−0.0879095
$399$	30.7386			1.53886
$400$		0			0
$401$			23.3693i			1.16701i	0.812110	+	0.583504i	$0.198319\pi$
$401$			23.3693i			1.16701i	−0.812110	+	0.583504i	$0.801681\pi$
$402$			13.1231i			0.654521i
$403$	2.87689	−	18.2462i	0.143308	−	0.908909i
$404$	4.24621			0.211257
$405$		0			0
$406$	−10.2462			−0.508511
$407$			9.75379i			0.483477i
$408$	2.00000			0.0990148
$409$			24.4924i			1.21107i	0.795818	+	0.605536i	$0.207041\pi$
$409$			24.4924i			1.21107i	−0.795818	+	0.605536i	$0.792959\pi$
$410$		0			0
$411$			22.4924i			1.10947i
$412$			4.87689i			0.240267i
$413$		−	36.4924i		−	1.79567i
$414$			3.12311i			0.153492i
$415$		0			0
$416$	3.56155	+	0.561553i	0.174619	+	0.0275324i
$417$			16.4924i			0.807637i
$418$	18.7386			0.916537
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$		−	25.3693i		−	1.23642i	−0.786011	−	0.618212i	$-0.787857\pi$
$421$		−	25.3693i		−	1.23642i	0.786011	−	0.618212i	$-0.212143\pi$
$422$	4.00000			0.194717
$423$	10.2462			0.498188
$424$		−	11.3693i		−	0.552143i
$425$		0			0
$426$	−6.24621			−0.302630
$427$	51.2311			2.47924
$428$		−	8.00000i		−	0.386695i
$429$	−11.1231	−	1.75379i	−0.537029	−	0.0846737i
$430$		0			0
$431$			0.492423i			0.0237192i	0.999930	+	0.0118596i	$0.00377511\pi$
$431$			0.492423i			0.0237192i	−0.999930	+	0.0118596i	$0.996225\pi$
$432$			1.00000i			0.0481125i
$433$			18.0000i			0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$			18.0000i			0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$		−	26.2462i		−	1.25986i
$435$		0			0
$436$		−	11.1231i		−	0.532700i
$437$	18.7386			0.896390
$438$			4.87689i			0.233027i
$439$	3.50758			0.167408			0.0837038	−	0.996491i	$-0.473325\pi$
$439$	3.50758			0.167408			0.0837038	+	0.996491i	$0.473325\pi$
$440$		0			0
$441$	−19.2462			−0.916486
$442$	−1.12311	+	7.12311i	−0.0534207	+	0.338812i
$443$			36.4924i			1.73381i	0.498476	+	0.866904i	$0.333893\pi$
$443$			36.4924i			1.73381i	−0.498476	+	0.866904i	$0.666107\pi$
$444$			3.12311i			0.148216i
$445$		0			0
$446$	−15.3693			−0.727758
$447$	14.0000			0.662177
$448$	5.12311			0.242044
$449$			37.1231i			1.75195i	0.482359	+	0.875974i	$0.339780\pi$
$449$			37.1231i			1.75195i	−0.482359	+	0.875974i	$0.660220\pi$
$450$		0			0
$451$	28.4924			1.34166
$452$			4.24621i			0.199725i
$453$	−11.3693			−0.534177
$454$	−8.00000			−0.375459
$455$		0			0
$456$	6.00000			0.280976
$457$	−6.63068			−0.310170			−0.155085	−	0.987901i	$-0.549565\pi$
$457$	−6.63068			−0.310170			−0.155085	+	0.987901i	$0.549565\pi$
$458$		−	3.12311i		−	0.145933i
$459$	−2.00000			−0.0933520
$460$		0			0
$461$		−	14.4924i		−	0.674979i	−0.941329	−	0.337490i	$-0.890422\pi$
$461$		−	14.4924i		−	0.674979i	0.941329	−	0.337490i	$-0.109578\pi$
$462$	−16.0000			−0.744387
$463$	35.8617			1.66664			0.833318	−	0.552794i	$-0.186438\pi$
$463$	35.8617			1.66664			0.833318	+	0.552794i	$0.186438\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$			24.2462i			1.12318i
$467$		−	5.75379i		−	0.266254i	−0.991099	−	0.133127i	$-0.957498\pi$
$467$		−	5.75379i		−	0.266254i	0.991099	−	0.133127i	$-0.0425018\pi$
$468$	−3.56155	−	0.561553i	−0.164633	−	0.0259578i
$469$	−67.2311			−3.10444
$470$		0			0
$471$	−3.36932			−0.155250
$472$		−	7.12311i		−	0.327868i
$473$	32.0000			1.47136
$474$			8.00000i			0.367452i
$475$		0			0
$476$			10.2462i			0.469634i
$477$			11.3693i			0.520565i
$478$			28.4924i			1.30321i
$479$		−	20.4924i		−	0.936323i	−0.883643	−	0.468161i	$-0.844917\pi$
$479$		−	20.4924i		−	0.936323i	0.883643	−	0.468161i	$-0.155083\pi$
$480$		0			0
$481$	−11.1231	−	1.75379i	−0.507170	−	0.0799659i
$482$			2.24621i			0.102312i
$483$	−16.0000			−0.728025
$484$	1.24621			0.0566460
$485$		0			0
$486$		−	1.00000i		−	0.0453609i
$487$	7.36932			0.333936			0.166968	−	0.985962i	$-0.446602\pi$
$487$	7.36932			0.333936			0.166968	+	0.985962i	$0.446602\pi$
$488$	10.0000			0.452679
$489$		−	1.12311i		−	0.0507886i
$490$		0			0
$491$	−10.7386			−0.484628			−0.242314	−	0.970198i	$-0.577906\pi$
$491$	−10.7386			−0.484628			−0.242314	+	0.970198i	$0.577906\pi$
$492$	9.12311			0.411301
$493$		−	4.00000i		−	0.180151i
$494$	−3.36932	+	21.3693i	−0.151593	+	0.961451i
$495$		0			0
$496$		−	5.12311i		−	0.230034i
$497$		−	32.0000i		−	1.43540i
$498$		−	10.2462i		−	0.459144i
$499$		−	1.50758i		−	0.0674884i	−0.999431	−	0.0337442i	$-0.989257\pi$
$499$		−	1.50758i		−	0.0674884i	0.999431	−	0.0337442i	$-0.0107432\pi$
$500$		0			0
$501$		−	5.75379i		−	0.257060i
$502$	9.75379			0.435333
$503$		−	10.6307i		−	0.473999i	−0.971510	−	0.236999i	$-0.923836\pi$
$503$		−	10.6307i		−	0.473999i	0.971510	−	0.236999i	$-0.0761639\pi$
$504$	−5.12311			−0.228201
$505$		0			0
$506$	−9.75379			−0.433609
$507$	4.00000	−	12.3693i	0.177646	−	0.549341i
$508$		−	4.87689i		−	0.216377i
$509$		−	15.7538i		−	0.698274i	−0.937072	−	0.349137i	$-0.886475\pi$
$509$		−	15.7538i		−	0.698274i	0.937072	−	0.349137i	$-0.113525\pi$
$510$		0			0
$511$	−24.9848			−1.10526
$512$	1.00000			0.0441942
$513$	−6.00000			−0.264906
$514$			4.24621i			0.187292i
$515$		0			0
$516$	10.2462			0.451064
$517$			32.0000i			1.40736i
$518$	−16.0000			−0.703000
$519$	−14.8769			−0.653023
$520$		0			0
$521$	−16.2462			−0.711759			−0.355880	−	0.934532i	$-0.615819\pi$
$521$	−16.2462			−0.711759			−0.355880	+	0.934532i	$0.615819\pi$
$522$	2.00000			0.0875376
$523$		−	22.7386i		−	0.994291i	−0.867667	−	0.497146i	$-0.834382\pi$
$523$		−	22.7386i		−	0.994291i	0.867667	−	0.497146i	$-0.165618\pi$
$524$	4.00000			0.174741
$525$		0			0
$526$			2.63068i			0.114703i
$527$	10.2462			0.446332
$528$	−3.12311			−0.135916
$529$	13.2462			0.575922
$530$		0			0
$531$			7.12311i			0.309116i
$532$			30.7386i			1.33269i
$533$	−5.12311	+	32.4924i	−0.221906	+	1.40740i
$534$	5.12311			0.221698
$535$		0			0
$536$	−13.1231			−0.566832
$537$		−	16.4924i		−	0.711701i
$538$	−0.246211			−0.0106149
$539$		−	60.1080i		−	2.58903i
$540$		0			0
$541$			19.1231i			0.822167i	0.911598	+	0.411083i	$0.134849\pi$
$541$			19.1231i			0.822167i	−0.911598	+	0.411083i	$0.865151\pi$
$542$			14.8769i			0.639017i
$543$			3.75379i			0.161090i
$544$			2.00000i			0.0857493i
$545$		0			0
$546$	2.87689	−	18.2462i	0.123120	−	0.780866i
$547$		−	44.9848i		−	1.92341i	−0.274081	−	0.961707i	$-0.588374\pi$
$547$		−	44.9848i		−	1.92341i	0.274081	−	0.961707i	$-0.411626\pi$
$548$	−22.4924			−0.960829
$549$	−10.0000			−0.426790
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$	−3.12311			−0.132928
$553$	−40.9848			−1.74285
$554$			27.8617i			1.18373i
$555$		0			0
$556$	−16.4924			−0.699435
$557$	−28.2462			−1.19683			−0.598415	−	0.801186i	$-0.704203\pi$
$557$	−28.2462			−1.19683			−0.598415	+	0.801186i	$0.704203\pi$
$558$			5.12311i			0.216878i
$559$	−5.75379	+	36.4924i	−0.243359	+	1.54347i
$560$		0			0
$561$		−	6.24621i		−	0.263715i
$562$			5.12311i			0.216105i
$563$		−	32.9848i		−	1.39015i	−0.718939	−	0.695073i	$-0.755372\pi$
$563$		−	32.9848i		−	1.39015i	0.718939	−	0.695073i	$-0.244628\pi$
$564$			10.2462i			0.431443i
$565$		0			0
$566$			4.00000i			0.168133i
$567$	5.12311			0.215150
$568$		−	6.24621i		−	0.262085i
$569$	−36.7386			−1.54016			−0.770082	−	0.637945i	$-0.779785\pi$
$569$	−36.7386			−1.54016			−0.770082	+	0.637945i	$0.779785\pi$
$570$		0			0
$571$	24.4924			1.02498			0.512488	−	0.858694i	$-0.328724\pi$
$571$	24.4924			1.02498			0.512488	+	0.858694i	$0.328724\pi$
$572$	1.75379	−	11.1231i	0.0733296	−	0.465080i
$573$			16.4924i			0.688981i
$574$			46.7386i			1.95083i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$	−2.63068			−0.109517			−0.0547584	−	0.998500i	$-0.517439\pi$
$577$	−2.63068			−0.109517			−0.0547584	+	0.998500i	$0.517439\pi$
$578$	13.0000			0.540729
$579$			16.8769i			0.701380i
$580$		0			0
$581$	52.4924			2.17775
$582$		−	4.87689i		−	0.202154i
$583$	−35.5076			−1.47057
$584$	−4.87689			−0.201807
$585$		0			0
$586$	−20.7386			−0.856705
$587$	−16.4924			−0.680715			−0.340358	−	0.940296i	$-0.610548\pi$
$587$	−16.4924			−0.680715			−0.340358	+	0.940296i	$0.610548\pi$
$588$		−	19.2462i		−	0.793700i
$589$	30.7386			1.26656
$590$		0			0
$591$		−	0.246211i		−	0.0101278i
$592$	−3.12311			−0.128359
$593$	−38.4924			−1.58069			−0.790347	−	0.612659i	$-0.790100\pi$
$593$	−38.4924			−1.58069			−0.790347	+	0.612659i	$0.790100\pi$
$594$	3.12311			0.128143
$595$		0			0
$596$			14.0000i			0.573462i
$597$			1.75379i			0.0717778i
$598$	1.75379	−	11.1231i	0.0717178	−	0.454858i
$599$	3.50758			0.143316			0.0716579	−	0.997429i	$-0.477171\pi$
$599$	3.50758			0.143316			0.0716579	+	0.997429i	$0.477171\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$			52.4924i			2.13943i
$603$	13.1231			0.534414
$604$		−	11.3693i		−	0.462611i
$605$		0			0
$606$		−	4.24621i		−	0.172491i
$607$		−	9.36932i		−	0.380289i	−0.981756	−	0.190144i	$-0.939104\pi$
$607$		−	9.36932i		−	0.380289i	0.981756	−	0.190144i	$-0.0608956\pi$
$608$			6.00000i			0.243332i
$609$			10.2462i			0.415197i
$610$		0			0
$611$	−36.4924	−	5.75379i	−1.47633	−	0.232773i
$612$		−	2.00000i		−	0.0808452i
$613$	14.6307			0.590928			0.295464	−	0.955354i	$-0.404526\pi$
$613$	14.6307			0.590928			0.295464	+	0.955354i	$0.404526\pi$
$614$	−22.8769			−0.923236
$615$		0			0
$616$		−	16.0000i		−	0.644658i
$617$	−8.73863			−0.351804			−0.175902	−	0.984408i	$-0.556284\pi$
$617$	−8.73863			−0.351804			−0.175902	+	0.984408i	$0.556284\pi$
$618$	4.87689			0.196177
$619$			26.9848i			1.08461i	0.840181	+	0.542306i	$0.182449\pi$
$619$			26.9848i			1.08461i	−0.840181	+	0.542306i	$0.817551\pi$
$620$		0			0
$621$	3.12311			0.125326
$622$	−24.4924			−0.982057
$623$			26.2462i			1.05153i
$624$	0.561553	−	3.56155i	0.0224801	−	0.142576i
$625$		0			0
$626$			0.246211i			0.00984058i
$627$		−	18.7386i		−	0.748349i
$628$		−	3.36932i		−	0.134450i
$629$		−	6.24621i		−	0.249053i
$630$		0			0
$631$			5.61553i			0.223551i	0.993734	+	0.111775i	$0.0356537\pi$
$631$			5.61553i			0.223551i	−0.993734	+	0.111775i	$0.964346\pi$
$632$	−8.00000			−0.318223
$633$		−	4.00000i		−	0.158986i
$634$	−6.00000			−0.238290
$635$		0			0
$636$	−11.3693			−0.450823
$637$	68.5464	+	10.8078i	2.71591	+	0.428219i
$638$			6.24621i			0.247290i
$639$			6.24621i			0.247096i
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−8.00000			−0.315735
$643$	−7.36932			−0.290617			−0.145309	−	0.989386i	$-0.546418\pi$
$643$	−7.36932			−0.290617			−0.145309	+	0.989386i	$0.546418\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$	−12.0000			−0.472134
$647$		−	11.6155i		−	0.456654i	−0.973585	−	0.228327i	$-0.926675\pi$
$647$		−	11.6155i		−	0.456654i	0.973585	−	0.228327i	$-0.0733255\pi$
$648$	1.00000			0.0392837
$649$	−22.2462			−0.873240
$650$		0			0
$651$	−26.2462			−1.02867
$652$	1.12311			0.0439842
$653$		−	43.8617i		−	1.71644i	−0.513280	−	0.858221i	$-0.671570\pi$
$653$		−	43.8617i		−	1.71644i	0.513280	−	0.858221i	$-0.328430\pi$
$654$	−11.1231			−0.434948
$655$		0			0
$656$			9.12311i			0.356197i
$657$	4.87689			0.190266
$658$	−52.4924			−2.04637
$659$	38.2462			1.48986			0.744930	−	0.667142i	$-0.232483\pi$
$659$	38.2462			1.48986			0.744930	+	0.667142i	$0.232483\pi$
$660$		0			0
$661$			0.876894i			0.0341072i	0.999855	+	0.0170536i	$0.00542860\pi$
$661$			0.876894i			0.0341072i	−0.999855	+	0.0170536i	$0.994571\pi$
$662$		−	24.2462i		−	0.942356i
$663$	7.12311	+	1.12311i	0.276638	+	0.0436178i
$664$	10.2462			0.397630
$665$		0			0
$666$	3.12311			0.121018
$667$			6.24621i			0.241854i
$668$	5.75379			0.222621
$669$			15.3693i			0.594212i
$670$		0			0
$671$		−	31.2311i		−	1.20566i
$672$		−	5.12311i		−	0.197628i
$673$			38.9848i			1.50276i	0.659872	+	0.751378i	$0.270610\pi$
$673$			38.9848i			1.50276i	−0.659872	+	0.751378i	$0.729390\pi$
$674$			6.00000i			0.231111i
$675$		0			0
$676$	12.3693	+	4.00000i	0.475743	+	0.153846i
$677$			21.1231i			0.811827i	0.913912	+	0.405913i	$0.133047\pi$
$677$			21.1231i			0.811827i	−0.913912	+	0.405913i	$0.866953\pi$
$678$	4.24621			0.163075
$679$	24.9848			0.958830
$680$		0			0
$681$			8.00000i			0.306561i
$682$	−16.0000			−0.612672
$683$	48.9848			1.87435			0.937177	−	0.348856i	$-0.113430\pi$
$683$	48.9848			1.87435			0.937177	+	0.348856i	$0.113430\pi$
$684$		−	6.00000i		−	0.229416i
$685$		0			0
$686$	62.7386			2.39537
$687$	−3.12311			−0.119154
$688$			10.2462i			0.390633i
$689$	6.38447	−	40.4924i	0.243229	−	1.54264i
$690$		0			0
$691$		−	20.7386i		−	0.788935i	−0.918910	−	0.394467i	$-0.870929\pi$
$691$		−	20.7386i		−	0.788935i	0.918910	−	0.394467i	$-0.129071\pi$
$692$		−	14.8769i		−	0.565535i
$693$			16.0000i			0.607790i
$694$			2.24621i			0.0852650i
$695$		0			0
$696$			2.00000i			0.0758098i
$697$	−18.2462			−0.691125
$698$			5.36932i			0.203232i
$699$	24.2462			0.917076
$700$		0			0
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$	−0.561553	+	3.56155i	−0.0211944	+	0.134422i
$703$		−	18.7386i		−	0.706741i
$704$		−	3.12311i		−	0.117706i
$705$		0			0
$706$	4.24621			0.159808
$707$	21.7538			0.818135
$708$	−7.12311			−0.267703
$709$		−	39.6155i		−	1.48779i	−0.668295	−	0.743896i	$-0.732976\pi$
$709$		−	39.6155i		−	1.48779i	0.668295	−	0.743896i	$-0.267024\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$			5.12311i			0.191997i
$713$	−16.0000			−0.599205
$714$	10.2462			0.383455
$715$		0			0
$716$	16.4924			0.616351
$717$	28.4924			1.06407
$718$		−	34.2462i		−	1.27806i
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$			24.9848i			0.930484i
$722$	−17.0000			−0.632674
$723$	2.24621			0.0835375
$724$	−3.75379			−0.139508
$725$		0			0
$726$		−	1.24621i		−	0.0462512i
$727$		−	37.8617i		−	1.40421i	−0.712071	−	0.702107i	$-0.752243\pi$
$727$		−	37.8617i		−	1.40421i	0.712071	−	0.702107i	$-0.247757\pi$
$728$	18.2462	+	2.87689i	0.676250	+	0.106625i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−20.4924			−0.757940
$732$		−	10.0000i		−	0.369611i
$733$	10.6307			0.392653			0.196327	−	0.980539i	$-0.437099\pi$
$733$	10.6307			0.392653			0.196327	+	0.980539i	$0.437099\pi$
$734$			33.3693i			1.23168i
$735$		0			0
$736$		−	3.12311i		−	0.115119i
$737$			40.9848i			1.50970i
$738$		−	9.12311i		−	0.335826i
$739$		−	45.2311i		−	1.66385i	−0.554887	−	0.831926i	$-0.687239\pi$
$739$		−	45.2311i		−	1.66385i	0.554887	−	0.831926i	$-0.312761\pi$
$740$		0			0
$741$	21.3693	+	3.36932i	0.785021	+	0.123775i
$742$		−	58.2462i		−	2.13829i
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$	−5.12311			−0.187822
$745$		0			0
$746$			1.12311i			0.0411198i
$747$	−10.2462			−0.374889
$748$	6.24621			0.228384
$749$		−	40.9848i		−	1.49755i
$750$		0			0
$751$	−9.75379			−0.355921			−0.177960	−	0.984038i	$-0.556950\pi$
$751$	−9.75379			−0.355921			−0.177960	+	0.984038i	$0.556950\pi$
$752$	−10.2462			−0.373641
$753$		−	9.75379i		−	0.355448i
$754$	−7.12311	−	1.12311i	−0.259408	−	0.0409011i
$755$		0			0
$756$			5.12311i			0.186326i
$757$			5.12311i			0.186202i	0.995657	+	0.0931012i	$0.0296780\pi$
$757$			5.12311i			0.186202i	−0.995657	+	0.0931012i	$0.970322\pi$
$758$			6.00000i			0.217930i
$759$			9.75379i			0.354040i
$760$		0			0
$761$			5.12311i			0.185712i	0.995680	+	0.0928562i	$0.0295997\pi$
$761$			5.12311i			0.185712i	−0.995680	+	0.0928562i	$0.970400\pi$
$762$	−4.87689			−0.176671
$763$		−	56.9848i		−	2.06299i
$764$	−16.4924			−0.596675
$765$		0			0
$766$	18.2462			0.659262
$767$	4.00000	−	25.3693i	0.144432	−	0.916033i
$768$		−	1.00000i		−	0.0360844i
$769$		−	32.9848i		−	1.18946i	−0.803924	−	0.594732i	$-0.797258\pi$
$769$		−	32.9848i		−	1.18946i	0.803924	−	0.594732i	$-0.202742\pi$
$770$		0			0
$771$	4.24621			0.152924
$772$	−16.8769			−0.607413
$773$	0.246211			0.00885560			0.00442780	−	0.999990i	$-0.498591\pi$
$773$	0.246211			0.00885560			0.00442780	+	0.999990i	$0.498591\pi$
$774$		−	10.2462i		−	0.368292i
$775$		0			0
$776$	4.87689			0.175070
$777$			16.0000i			0.573997i
$778$	−16.2462			−0.582455
$779$	−54.7386			−1.96122
$780$		0			0
$781$	−19.5076			−0.698036
$782$	6.24621			0.223364
$783$		−	2.00000i		−	0.0714742i
$784$	19.2462			0.687365
$785$		0			0
$786$		−	4.00000i		−	0.142675i
$787$	53.6155			1.91119			0.955594	−	0.294688i	$-0.0952157\pi$
$787$	53.6155			1.91119			0.955594	+	0.294688i	$0.0952157\pi$
$788$	0.246211			0.00877091
$789$	2.63068			0.0936548
$790$		0			0
$791$			21.7538i			0.773476i
$792$			3.12311i			0.110975i
$793$	35.6155	+	5.61553i	1.26474	+	0.199413i
$794$	−9.36932			−0.332505
$795$		0			0
$796$	−1.75379			−0.0621614
$797$		−	31.8617i		−	1.12860i	−0.825570	−	0.564300i	$-0.809146\pi$
$797$		−	31.8617i		−	1.12860i	0.825570	−	0.564300i	$-0.190854\pi$
$798$	30.7386			1.08814
$799$		−	20.4924i		−	0.724970i
$800$		0			0
$801$		−	5.12311i		−	0.181016i
$802$			23.3693i			0.825199i
$803$			15.2311i			0.537492i
$804$			13.1231i			0.462816i
$805$		0			0
$806$	2.87689	−	18.2462i	0.101334	−	0.642695i
$807$			0.246211i			0.00866705i
$808$	4.24621			0.149381
$809$	46.4924			1.63459			0.817293	−	0.576222i	$-0.195474\pi$
$809$	46.4924			1.63459			0.817293	+	0.576222i	$0.195474\pi$
$810$		0			0
$811$		−	44.2462i		−	1.55369i	−0.629689	−	0.776847i	$-0.716818\pi$
$811$		−	44.2462i		−	1.55369i	0.629689	−	0.776847i	$-0.283182\pi$
$812$	−10.2462			−0.359572
$813$	14.8769			0.521755
$814$			9.75379i			0.341870i
$815$		0			0
$816$	2.00000			0.0700140
$817$	−61.4773			−2.15082
$818$			24.4924i			0.856357i
$819$	−18.2462	−	2.87689i	−0.637574	−	0.100527i
$820$		0			0
$821$			27.7538i			0.968614i	0.874898	+	0.484307i	$0.160928\pi$
$821$			27.7538i			0.968614i	−0.874898	+	0.484307i	$0.839072\pi$
$822$			22.4924i			0.784513i
$823$			51.1231i			1.78204i	0.453965	+	0.891020i	$0.350009\pi$
$823$			51.1231i			1.78204i	−0.453965	+	0.891020i	$0.649991\pi$
$824$			4.87689i			0.169895i
$825$		0			0
$826$		−	36.4924i		−	1.26973i
$827$	−50.7386			−1.76436			−0.882178	−	0.470917i	$-0.843923\pi$
$827$	−50.7386			−1.76436			−0.882178	+	0.470917i	$0.843923\pi$
$828$			3.12311i			0.108535i
$829$	7.75379			0.269300			0.134650	−	0.990893i	$-0.457009\pi$
$829$	7.75379			0.269300			0.134650	+	0.990893i	$0.457009\pi$
$830$		0			0
$831$	27.8617			0.966513
$832$	3.56155	+	0.561553i	0.123475	+	0.0194683i
$833$			38.4924i			1.33368i
$834$			16.4924i			0.571086i
$835$		0			0
$836$	18.7386			0.648089
$837$	5.12311			0.177080
$838$	28.0000			0.967244
$839$		−	2.73863i		−	0.0945481i	−0.998882	−	0.0472741i	$-0.984947\pi$
$839$		−	2.73863i		−	0.0945481i	0.998882	−	0.0472741i	$-0.0150534\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		−	25.3693i		−	0.874284i
$843$	5.12311			0.176449
$844$	4.00000			0.137686
$845$		0			0
$846$	10.2462			0.352272
$847$	6.38447			0.219373
$848$		−	11.3693i		−	0.390424i
$849$	4.00000			0.137280
$850$		0			0
$851$			9.75379i			0.334356i
$852$	−6.24621			−0.213992
$853$	−21.8617			−0.748532			−0.374266	−	0.927321i	$-0.622105\pi$
$853$	−21.8617			−0.748532			−0.374266	+	0.927321i	$0.622105\pi$
$854$	51.2311			1.75309
$855$		0			0
$856$		−	8.00000i		−	0.273434i
$857$			2.49242i			0.0851395i	0.999093	+	0.0425698i	$0.0135545\pi$
$857$			2.49242i			0.0851395i	−0.999093	+	0.0425698i	$0.986446\pi$
$858$	−11.1231	−	1.75379i	−0.379737	−	0.0598734i
$859$	12.0000			0.409435			0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000			0.409435			0.204717	+	0.978821i	$0.434372\pi$
$860$		0			0
$861$	46.7386			1.59285
$862$			0.492423i			0.0167720i
$863$	10.2462			0.348785			0.174393	−	0.984676i	$-0.444204\pi$
$863$	10.2462			0.348785			0.174393	+	0.984676i	$0.444204\pi$
$864$			1.00000i			0.0340207i
$865$		0			0
$866$			18.0000i			0.611665i
$867$		−	13.0000i		−	0.441503i
$868$		−	26.2462i		−	0.890854i
$869$			24.9848i			0.847553i
$870$		0			0
$871$	−46.7386	−	7.36932i	−1.58368	−	0.249700i
$872$		−	11.1231i		−	0.376676i
$873$	−4.87689			−0.165058
$874$	18.7386			0.633844
$875$		0			0
$876$			4.87689i			0.164775i
$877$	27.1231			0.915882			0.457941	−	0.888983i	$-0.348587\pi$
$877$	27.1231			0.915882			0.457941	+	0.888983i	$0.348587\pi$
$878$	3.50758			0.118375
$879$			20.7386i			0.699497i
$880$		0			0
$881$	11.7538			0.395995			0.197998	−	0.980203i	$-0.436556\pi$
$881$	11.7538			0.395995			0.197998	+	0.980203i	$0.436556\pi$
$882$	−19.2462			−0.648054
$883$			26.2462i			0.883255i	0.897198	+	0.441628i	$0.145599\pi$
$883$			26.2462i			0.883255i	−0.897198	+	0.441628i	$0.854401\pi$
$884$	−1.12311	+	7.12311i	−0.0377741	+	0.239576i
$885$		0			0
$886$			36.4924i			1.22599i
$887$		−	27.1231i		−	0.910705i	−0.890311	−	0.455352i	$-0.849513\pi$
$887$		−	27.1231i		−	0.910705i	0.890311	−	0.455352i	$-0.150487\pi$
$888$			3.12311i			0.104805i
$889$		−	24.9848i		−	0.837965i
$890$		0			0
$891$		−	3.12311i		−	0.104628i
$892$	−15.3693			−0.514603
$893$		−	61.4773i		−	2.05726i
$894$	14.0000			0.468230
$895$		0			0
$896$	5.12311			0.171151
$897$	−11.1231	−	1.75379i	−0.371390	−	0.0585573i
$898$			37.1231i			1.23881i
$899$			10.2462i			0.341730i
$900$		0			0
$901$	22.7386			0.757534
$902$	28.4924			0.948694
$903$	52.4924			1.74684
$904$			4.24621i			0.141227i
$905$		0			0
$906$	−11.3693			−0.377720
$907$			42.2462i			1.40276i	0.712786	+	0.701381i	$0.247433\pi$
$907$			42.2462i			1.40276i	−0.712786	+	0.701381i	$0.752567\pi$
$908$	−8.00000			−0.265489
$909$	−4.24621			−0.140838
$910$		0			0
$911$	4.00000			0.132526			0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000			0.132526			0.0662630	+	0.997802i	$0.478892\pi$
$912$	6.00000			0.198680
$913$		−	32.0000i		−	1.05905i
$914$	−6.63068			−0.219324
$915$		0			0
$916$		−	3.12311i		−	0.103190i
$917$	20.4924			0.676719
$918$	−2.00000			−0.0660098
$919$	38.2462			1.26163			0.630813	−	0.775935i	$-0.282721\pi$
$919$	38.2462			1.26163			0.630813	+	0.775935i	$0.282721\pi$
$920$		0			0
$921$			22.8769i			0.753819i
$922$		−	14.4924i		−	0.477283i
$923$	3.50758	−	22.2462i	0.115453	−	0.732243i
$924$	−16.0000			−0.526361
$925$		0			0
$926$	35.8617			1.17849
$927$		−	4.87689i		−	0.160178i
$928$	−2.00000			−0.0656532
$929$			46.1080i			1.51275i	0.654137	+	0.756376i	$0.273032\pi$
$929$			46.1080i			1.51275i	−0.654137	+	0.756376i	$0.726968\pi$
$930$		0			0
$931$			115.477i			3.78461i
$932$			24.2462i			0.794211i
$933$			24.4924i			0.801846i
$934$		−	5.75379i		−	0.188270i
$935$		0			0
$936$	−3.56155	−	0.561553i	−0.116413	−	0.0183549i
$937$		−	3.75379i		−	0.122631i	−0.998118	−	0.0613155i	$-0.980470\pi$
$937$		−	3.75379i		−	0.122631i	0.998118	−	0.0613155i	$-0.0195296\pi$
$938$	−67.2311			−2.19517
$939$	0.246211			0.00803480
$940$		0			0
$941$		−	14.0000i		−	0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$		−	14.0000i		−	0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$	−3.36932			−0.109778
$943$	28.4924			0.927841
$944$		−	7.12311i		−	0.231837i
$945$		0			0
$946$	32.0000			1.04041
$947$	24.4924			0.795897			0.397948	−	0.917408i	$-0.369722\pi$
$947$	24.4924			0.795897			0.397948	+	0.917408i	$0.369722\pi$
$948$			8.00000i			0.259828i
$949$	−17.3693	−	2.73863i	−0.563832	−	0.0888998i
$950$		0			0
$951$			6.00000i			0.194563i
$952$			10.2462i			0.332082i
$953$			42.9848i			1.39242i	0.717840	+	0.696208i	$0.245131\pi$
$953$			42.9848i			1.39242i	−0.717840	+	0.696208i	$0.754869\pi$
$954$			11.3693i			0.368095i
$955$		0			0
$956$			28.4924i			0.921511i
$957$	6.24621			0.201911
$958$		−	20.4924i		−	0.662080i
$959$	−115.231			−3.72100
$960$		0			0
$961$	4.75379			0.153348
$962$	−11.1231	−	1.75379i	−0.358623	−	0.0565444i
$963$			8.00000i			0.257796i
$964$			2.24621i			0.0723456i
$965$		0			0
$966$	−16.0000			−0.514792
$967$	6.38447			0.205311			0.102655	−	0.994717i	$-0.467266\pi$
$967$	6.38447			0.205311			0.102655	+	0.994717i	$0.467266\pi$
$968$	1.24621			0.0400547
$969$			12.0000i			0.385496i
$970$		0			0
$971$	54.2462			1.74084			0.870422	−	0.492307i	$-0.163846\pi$
$971$	54.2462			1.74084			0.870422	+	0.492307i	$0.163846\pi$
$972$		−	1.00000i		−	0.0320750i
$973$	−84.4924			−2.70870
$974$	7.36932			0.236128
$975$		0			0
$976$	10.0000			0.320092
$977$	−44.7386			−1.43132			−0.715658	−	0.698451i	$-0.753873\pi$
$977$	−44.7386			−1.43132			−0.715658	+	0.698451i	$0.753873\pi$
$978$		−	1.12311i		−	0.0359130i
$979$	16.0000			0.511362
$980$		0			0
$981$			11.1231i			0.355133i
$982$	−10.7386			−0.342684
$983$	−27.5076			−0.877355			−0.438678	−	0.898644i	$-0.644553\pi$
$983$	−27.5076			−0.877355			−0.438678	+	0.898644i	$0.644553\pi$
$984$	9.12311			0.290834
$985$		0			0
$986$		−	4.00000i		−	0.127386i
$987$			52.4924i			1.67085i
$988$	−3.36932	+	21.3693i	−0.107192	+	0.679849i
$989$	32.0000			1.01754
$990$		0			0
$991$	18.7386			0.595252			0.297626	−	0.954682i	$-0.403805\pi$
$991$	18.7386			0.595252			0.297626	+	0.954682i	$0.403805\pi$
$992$		−	5.12311i		−	0.162659i
$993$	−24.2462			−0.769430
$994$		−	32.0000i		−	1.01498i
$995$		0			0
$996$		−	10.2462i		−	0.324664i
$997$		−	52.3542i		−	1.65807i	−0.559195	−	0.829036i	$-0.688890\pi$
$997$		−	52.3542i		−	1.65807i	0.559195	−	0.829036i	$-0.311110\pi$
$998$		−	1.50758i		−	0.0477215i
$999$		−	3.12311i		−	0.0988107i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.f.o.649.2		4
5.2	odd	4		1950.2.b.h.1351.4		4
5.3	odd	4		390.2.b.d.181.1	✓	4
5.4	even	2		1950.2.f.l.649.3		4
13.12	even	2		1950.2.f.l.649.1		4
15.8	even	4		1170.2.b.f.181.3		4
20.3	even	4		3120.2.g.o.961.4		4
65.8	even	4		5070.2.a.bh.1.1		2
65.12	odd	4		1950.2.b.h.1351.1		4
65.18	even	4		5070.2.a.bd.1.2		2
65.38	odd	4		390.2.b.d.181.4	yes	4
65.64	even	2	inner	1950.2.f.o.649.4		4
195.38	even	4		1170.2.b.f.181.2		4
260.103	even	4		3120.2.g.o.961.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.1	✓	4	5.3	odd	4
390.2.b.d.181.4	yes	4	65.38	odd	4
1170.2.b.f.181.2		4	195.38	even	4
1170.2.b.f.181.3		4	15.8	even	4
1950.2.b.h.1351.1		4	65.12	odd	4
1950.2.b.h.1351.4		4	5.2	odd	4
1950.2.f.l.649.1		4	13.12	even	2
1950.2.f.l.649.3		4	5.4	even	2
1950.2.f.o.649.2		4	1.1	even	1	trivial
1950.2.f.o.649.4		4	65.64	even	2	inner
3120.2.g.o.961.1		4	260.103	even	4
3120.2.g.o.961.4		4	20.3	even	4
5070.2.a.bd.1.2		2	65.18	even	4
5070.2.a.bh.1.1		2	65.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +5.12311 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +5.12311 q^{7} +1.00000 q^{8} -1.00000 q^{9} -3.12311i q^{11} -1.00000i q^{12} +(3.56155 + 0.561553i) q^{13} +5.12311 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000 q^{18} +6.00000i q^{19} -5.12311i q^{21} -3.12311i q^{22} -3.12311i q^{23} -1.00000i q^{24} +(3.56155 + 0.561553i) q^{26} +1.00000i q^{27} +5.12311 q^{28} -2.00000 q^{29} -5.12311i q^{31} +1.00000 q^{32} -3.12311 q^{33} +2.00000i q^{34} -1.00000 q^{36} -3.12311 q^{37} +6.00000i q^{38} +(0.561553 - 3.56155i) q^{39} +9.12311i q^{41} -5.12311i q^{42} +10.2462i q^{43} -3.12311i q^{44} -3.12311i q^{46} -10.2462 q^{47} -1.00000i q^{48} +19.2462 q^{49} +2.00000 q^{51} +(3.56155 + 0.561553i) q^{52} -11.3693i q^{53} +1.00000i q^{54} +5.12311 q^{56} +6.00000 q^{57} -2.00000 q^{58} -7.12311i q^{59} +10.0000 q^{61} -5.12311i q^{62} -5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} -13.1231 q^{67} +2.00000i q^{68} -3.12311 q^{69} -6.24621i q^{71} -1.00000 q^{72} -4.87689 q^{73} -3.12311 q^{74} +6.00000i q^{76} -16.0000i q^{77} +(0.561553 - 3.56155i) q^{78} -8.00000 q^{79} +1.00000 q^{81} +9.12311i q^{82} +10.2462 q^{83} -5.12311i q^{84} +10.2462i q^{86} +2.00000i q^{87} -3.12311i q^{88} +5.12311i q^{89} +(18.2462 + 2.87689i) q^{91} -3.12311i q^{92} -5.12311 q^{93} -10.2462 q^{94} -1.00000i q^{96} +4.87689 q^{97} +19.2462 q^{98} +3.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 - 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} - 4 q^{9} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{18} + 6 q^{26} + 4 q^{28} - 8 q^{29} + 4 q^{32} + 4 q^{33} - 4 q^{36} + 4 q^{37} - 6 q^{39} - 8 q^{47} + 44 q^{49} + 8 q^{51} + 6 q^{52} + 4 q^{56} + 24 q^{57} - 8 q^{58} + 40 q^{61} - 4 q^{63} + 4 q^{64} + 4 q^{66} - 36 q^{67} + 4 q^{69} - 4 q^{72} - 36 q^{73} + 4 q^{74} - 6 q^{78} - 32 q^{79} + 4 q^{81} + 8 q^{83} + 40 q^{91} - 4 q^{93} - 8 q^{94} + 36 q^{97} + 44 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 - 4 * q^9 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^18 + 6 * q^26 + 4 * q^28 - 8 * q^29 + 4 * q^32 + 4 * q^33 - 4 * q^36 + 4 * q^37 - 6 * q^39 - 8 * q^47 + 44 * q^49 + 8 * q^51 + 6 * q^52 + 4 * q^56 + 24 * q^57 - 8 * q^58 + 40 * q^61 - 4 * q^63 + 4 * q^64 + 4 * q^66 - 36 * q^67 + 4 * q^69 - 4 * q^72 - 36 * q^73 + 4 * q^74 - 6 * q^78 - 32 * q^79 + 4 * q^81 + 8 * q^83 + 40 * q^91 - 4 * q^93 - 8 * q^94 + 36 * q^97 + 44 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more