LMFDB - Embedded newform 3750.2.c.k.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1249,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3750.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.c (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:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 $ x^16 - 4x^15 - 24x^14 + 94x^13 + 262x^12 - 936x^11 - 1584x^10 + 4642x^9 + 6259x^8 - 11958x^7 - 15752x^6 + 14670x^5 + 18271x^4 - 10440x^3 + 1135x^2 + 21080*x + 11105
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$2.32349 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.k.1249.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.329315i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.329315i q^{7} +1.00000i q^{8} -1.00000 q^{9} -5.03339 q^{11} -1.00000i q^{12} -0.482152i q^{13} +0.329315 q^{14} +1.00000 q^{16} -6.78610i q^{17} +1.00000i q^{18} -5.44012 q^{19} -0.329315 q^{21} +5.03339i q^{22} +6.50055i q^{23} -1.00000 q^{24} -0.482152 q^{26} -1.00000i q^{27} -0.329315i q^{28} +6.02216 q^{29} +1.31869 q^{31} -1.00000i q^{32} -5.03339i q^{33} -6.78610 q^{34} +1.00000 q^{36} -0.780139i q^{37} +5.44012i q^{38} +0.482152 q^{39} +12.5205 q^{41} +0.329315i q^{42} -2.47582i q^{43} +5.03339 q^{44} +6.50055 q^{46} +4.38040i q^{47} +1.00000i q^{48} +6.89155 q^{49} +6.78610 q^{51} +0.482152i q^{52} -1.68591i q^{53} -1.00000 q^{54} -0.329315 q^{56} -5.44012i q^{57} -6.02216i q^{58} -1.01583 q^{59} +4.18808 q^{61} -1.31869i q^{62} -0.329315i q^{63} -1.00000 q^{64} -5.03339 q^{66} -3.14433i q^{67} +6.78610i q^{68} -6.50055 q^{69} +5.71189 q^{71} -1.00000i q^{72} +2.94269i q^{73} -0.780139 q^{74} +5.44012 q^{76} -1.65757i q^{77} -0.482152i q^{78} -8.48510 q^{79} +1.00000 q^{81} -12.5205i q^{82} +17.1955i q^{83} +0.329315 q^{84} -2.47582 q^{86} +6.02216i q^{87} -5.03339i q^{88} -3.45233 q^{89} +0.158780 q^{91} -6.50055i q^{92} +1.31869i q^{93} +4.38040 q^{94} +1.00000 q^{96} +9.51004i q^{97} -6.89155i q^{98} +5.03339 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$.

$n$	$2501$	$3127$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0.329315i	0.124469i	0.998062	+	0.0622347i	$0.0198227\pi$
$7$	0.329315i	0.124469i	−0.998062	+	0.0622347i	$0.980177\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.03339	−1.51762	−0.758812	−	0.651309i	$-0.774220\pi$
$11$	−5.03339	−1.51762	−0.758812	+	0.651309i	$0.774220\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 0.482152i	− 0.133725i	−0.997762	−	0.0668625i	$-0.978701\pi$
$13$	− 0.482152i	− 0.133725i	0.997762	−	0.0668625i	$-0.0212989\pi$
$14$	0.329315	0.0880131
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.78610i	− 1.64587i	−0.568135	−	0.822935i	$-0.692335\pi$
$17$	− 6.78610i	− 1.64587i	0.568135	−	0.822935i	$-0.307665\pi$
$18$	1.00000i	0.235702i
$19$	−5.44012	−1.24805	−0.624025	−	0.781404i	$-0.714504\pi$
$19$	−5.44012	−1.24805	−0.624025	+	0.781404i	$0.714504\pi$
$20$	0	0
$21$	−0.329315	−0.0718624
$22$	5.03339i	1.07312i
$23$	6.50055i	1.35546i	0.735312	+	0.677729i	$0.237036\pi$
$23$	6.50055i	1.35546i	−0.735312	+	0.677729i	$0.762964\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.482152	−0.0945578
$27$	− 1.00000i	− 0.192450i
$28$	− 0.329315i	− 0.0622347i
$29$	6.02216	1.11829	0.559144	−	0.829071i	$-0.311130\pi$
$29$	6.02216	1.11829	0.559144	+	0.829071i	$0.311130\pi$
$30$	0	0
$31$	1.31869	0.236844	0.118422	−	0.992963i	$-0.462216\pi$
$31$	1.31869	0.236844	0.118422	+	0.992963i	$0.462216\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 5.03339i	− 0.876201i
$34$	−6.78610	−1.16381
$35$	0	0
$36$	1.00000	0.166667
$37$	− 0.780139i	− 0.128254i	−0.997942	−	0.0641270i	$-0.979574\pi$
$37$	− 0.780139i	− 0.128254i	0.997942	−	0.0641270i	$-0.0204263\pi$
$38$	5.44012i	0.882504i
$39$	0.482152	0.0772061
$40$	0	0
$41$	12.5205	1.95538	0.977689	−	0.210059i	$-0.0673657\pi$
$41$	12.5205	1.95538	0.977689	+	0.210059i	$0.0673657\pi$
$42$	0.329315i	0.0508144i
$43$	− 2.47582i	− 0.377559i	−0.982020	−	0.188779i	$-0.939547\pi$
$43$	− 2.47582i	− 0.377559i	0.982020	−	0.188779i	$-0.0604531\pi$
$44$	5.03339	0.758812
$45$	0	0
$46$	6.50055	0.958454
$47$	4.38040i	0.638946i	0.947595	+	0.319473i	$0.103506\pi$
$47$	4.38040i	0.638946i	−0.947595	+	0.319473i	$0.896494\pi$
$48$	1.00000i	0.144338i
$49$	6.89155	0.984507
$50$	0	0
$51$	6.78610	0.950244
$52$	0.482152i	0.0668625i
$53$	− 1.68591i	− 0.231578i	−0.993274	−	0.115789i	$-0.963060\pi$
$53$	− 1.68591i	− 0.231578i	0.993274	−	0.115789i	$-0.0369396\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−0.329315	−0.0440066
$57$	− 5.44012i	− 0.720562i
$58$	− 6.02216i	− 0.790749i
$59$	−1.01583	−0.132250	−0.0661250	−	0.997811i	$-0.521064\pi$
$59$	−1.01583	−0.132250	−0.0661250	+	0.997811i	$0.521064\pi$
$60$	0	0
$61$	4.18808	0.536229	0.268114	−	0.963387i	$-0.413599\pi$
$61$	4.18808	0.536229	0.268114	+	0.963387i	$0.413599\pi$
$62$	− 1.31869i	− 0.167474i
$63$	− 0.329315i	− 0.0414898i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−5.03339	−0.619568
$67$	− 3.14433i	− 0.384141i	−0.981381	−	0.192070i	$-0.938480\pi$
$67$	− 3.14433i	− 0.384141i	0.981381	−	0.192070i	$-0.0615202\pi$
$68$	6.78610i	0.822935i
$69$	−6.50055	−0.782574
$70$	0	0
$71$	5.71189	0.677876	0.338938	−	0.940809i	$-0.389932\pi$
$71$	5.71189	0.677876	0.338938	+	0.940809i	$0.389932\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.94269i	0.344415i	0.985061	+	0.172208i	$0.0550900\pi$
$73$	2.94269i	0.344415i	−0.985061	+	0.172208i	$0.944910\pi$
$74$	−0.780139	−0.0906893
$75$	0	0
$76$	5.44012	0.624025
$77$	− 1.65757i	− 0.188898i
$78$	− 0.482152i	− 0.0545930i
$79$	−8.48510	−0.954648	−0.477324	−	0.878727i	$-0.658393\pi$
$79$	−8.48510	−0.954648	−0.477324	+	0.878727i	$0.658393\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 12.5205i	− 1.38266i
$83$	17.1955i	1.88745i	0.330730	+	0.943725i	$0.392705\pi$
$83$	17.1955i	1.88745i	−0.330730	+	0.943725i	$0.607295\pi$
$84$	0.329315	0.0359312
$85$	0	0
$86$	−2.47582	−0.266974
$87$	6.02216i	0.645644i
$88$	− 5.03339i	− 0.536561i
$89$	−3.45233	−0.365946	−0.182973	−	0.983118i	$-0.558572\pi$
$89$	−3.45233	−0.365946	−0.182973	+	0.983118i	$0.558572\pi$
$90$	0	0
$91$	0.158780	0.0166447
$92$	− 6.50055i	− 0.677729i
$93$	1.31869i	0.136742i
$94$	4.38040	0.451803
$95$	0	0
$96$	1.00000	0.102062
$97$	9.51004i	0.965598i	0.875731	+	0.482799i	$0.160380\pi$
$97$	9.51004i	0.965598i	−0.875731	+	0.482799i	$0.839620\pi$
$98$	− 6.89155i	− 0.696152i
$99$	5.03339	0.505875
$100$	0	0
$101$	19.2435	1.91480	0.957400	−	0.288765i	$-0.0932447\pi$
$101$	19.2435	1.91480	0.957400	+	0.288765i	$0.0932447\pi$
$102$	− 6.78610i	− 0.671924i
$103$	5.10689i	0.503196i	0.967832	+	0.251598i	$0.0809562\pi$
$103$	5.10689i	0.503196i	−0.967832	+	0.251598i	$0.919044\pi$
$104$	0.482152	0.0472789
$105$	0	0
$106$	−1.68591	−0.163750
$107$	15.3340i	1.48239i	0.671290	+	0.741194i	$0.265740\pi$
$107$	15.3340i	1.48239i	−0.671290	+	0.741194i	$0.734260\pi$
$108$	1.00000i	0.0962250i
$109$	9.76027	0.934865	0.467432	−	0.884029i	$-0.345179\pi$
$109$	9.76027	0.934865	0.467432	+	0.884029i	$0.345179\pi$
$110$	0	0
$111$	0.780139	0.0740475
$112$	0.329315i	0.0311173i
$113$	1.51566i	0.142581i	0.997456	+	0.0712905i	$0.0227117\pi$
$113$	1.51566i	0.142581i	−0.997456	+	0.0712905i	$0.977288\pi$
$114$	−5.44012	−0.509514
$115$	0	0
$116$	−6.02216	−0.559144
$117$	0.482152i	0.0445750i
$118$	1.01583i	0.0935149i
$119$	2.23476	0.204860
$120$	0	0
$121$	14.3350	1.30319
$122$	− 4.18808i	− 0.379171i
$123$	12.5205i	1.12894i
$124$	−1.31869	−0.118422
$125$	0	0
$126$	−0.329315	−0.0293377
$127$	− 12.6056i	− 1.11856i	−0.828977	−	0.559282i	$-0.811077\pi$
$127$	− 12.6056i	− 1.11856i	0.828977	−	0.559282i	$-0.188923\pi$
$128$	1.00000i	0.0883883i
$129$	2.47582	0.217984
$130$	0	0
$131$	14.0559	1.22807	0.614036	−	0.789278i	$-0.289545\pi$
$131$	14.0559	1.22807	0.614036	+	0.789278i	$0.289545\pi$
$132$	5.03339i	0.438101i
$133$	− 1.79151i	− 0.155344i
$134$	−3.14433	−0.271629
$135$	0	0
$136$	6.78610	0.581903
$137$	6.68863i	0.571448i	0.958312	+	0.285724i	$0.0922341\pi$
$137$	6.68863i	0.571448i	−0.958312	+	0.285724i	$0.907766\pi$
$138$	6.50055i	0.553364i
$139$	3.07799	0.261072	0.130536	−	0.991444i	$-0.458330\pi$
$139$	3.07799	0.261072	0.130536	+	0.991444i	$0.458330\pi$
$140$	0	0
$141$	−4.38040	−0.368896
$142$	− 5.71189i	− 0.479331i
$143$	2.42686i	0.202944i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.94269	0.243538
$147$	6.89155i	0.568406i
$148$	0.780139i	0.0641270i
$149$	−5.06465	−0.414912	−0.207456	−	0.978244i	$-0.566518\pi$
$149$	−5.06465	−0.414912	−0.207456	+	0.978244i	$0.566518\pi$
$150$	0	0
$151$	−16.9581	−1.38003	−0.690015	−	0.723795i	$-0.742396\pi$
$151$	−16.9581	−1.38003	−0.690015	+	0.723795i	$0.742396\pi$
$152$	− 5.44012i	− 0.441252i
$153$	6.78610i	0.548623i
$154$	−1.65757	−0.133571
$155$	0	0
$156$	−0.482152	−0.0386031
$157$	22.8284i	1.82190i	0.412513	+	0.910952i	$0.364651\pi$
$157$	22.8284i	1.82190i	−0.412513	+	0.910952i	$0.635349\pi$
$158$	8.48510i	0.675038i
$159$	1.68591	0.133701
$160$	0	0
$161$	−2.14073	−0.168713
$162$	− 1.00000i	− 0.0785674i
$163$	7.95823i	0.623337i	0.950191	+	0.311668i	$0.100888\pi$
$163$	7.95823i	0.623337i	−0.950191	+	0.311668i	$0.899112\pi$
$164$	−12.5205	−0.977689
$165$	0	0
$166$	17.1955	1.33463
$167$	− 13.9503i	− 1.07951i	−0.841823	−	0.539754i	$-0.818517\pi$
$167$	− 13.9503i	− 1.07951i	0.841823	−	0.539754i	$-0.181483\pi$
$168$	− 0.329315i	− 0.0254072i
$169$	12.7675	0.982118
$170$	0	0
$171$	5.44012	0.416017
$172$	2.47582i	0.188779i
$173$	− 19.5009i	− 1.48262i	−0.671161	−	0.741312i	$-0.734204\pi$
$173$	− 19.5009i	− 1.48262i	0.671161	−	0.741312i	$-0.265796\pi$
$174$	6.02216	0.456539
$175$	0	0
$176$	−5.03339	−0.379406
$177$	− 1.01583i	− 0.0763546i
$178$	3.45233i	0.258763i
$179$	18.9911	1.41946	0.709731	−	0.704473i	$-0.248817\pi$
$179$	18.9911	1.41946	0.709731	+	0.704473i	$0.248817\pi$
$180$	0	0
$181$	−18.9484	−1.40842	−0.704211	−	0.709991i	$-0.748699\pi$
$181$	−18.9484	−1.40842	−0.704211	+	0.709991i	$0.748699\pi$
$182$	− 0.158780i	− 0.0117695i
$183$	4.18808i	0.309592i
$184$	−6.50055	−0.479227
$185$	0	0
$186$	1.31869	0.0966913
$187$	34.1571i	2.49781i
$188$	− 4.38040i	− 0.319473i
$189$	0.329315	0.0239541
$190$	0	0
$191$	−12.0933	−0.875040	−0.437520	−	0.899209i	$-0.644143\pi$
$191$	−12.0933	−0.875040	−0.437520	+	0.899209i	$0.644143\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 11.0357i	− 0.794368i	−0.917739	−	0.397184i	$-0.869987\pi$
$193$	− 11.0357i	− 0.794368i	0.917739	−	0.397184i	$-0.130013\pi$
$194$	9.51004	0.682781
$195$	0	0
$196$	−6.89155	−0.492254
$197$	20.6212i	1.46920i	0.678502	+	0.734599i	$0.262630\pi$
$197$	20.6212i	1.46920i	−0.678502	+	0.734599i	$0.737370\pi$
$198$	− 5.03339i	− 0.357708i
$199$	−18.5313	−1.31365	−0.656826	−	0.754042i	$-0.728101\pi$
$199$	−18.5313	−1.31365	−0.656826	+	0.754042i	$0.728101\pi$
$200$	0	0
$201$	3.14433	0.221784
$202$	− 19.2435i	− 1.35397i
$203$	1.98319i	0.139193i
$204$	−6.78610	−0.475122
$205$	0	0
$206$	5.10689	0.355814
$207$	− 6.50055i	− 0.451819i
$208$	− 0.482152i	− 0.0334312i
$209$	27.3823	1.89407
$210$	0	0
$211$	15.7638	1.08523	0.542613	−	0.839983i	$-0.317435\pi$
$211$	15.7638	1.08523	0.542613	+	0.839983i	$0.317435\pi$
$212$	1.68591i	0.115789i
$213$	5.71189i	0.391372i
$214$	15.3340	1.04821
$215$	0	0
$216$	1.00000	0.0680414
$217$	0.434265i	0.0294799i
$218$	− 9.76027i	− 0.661049i
$219$	−2.94269	−0.198848
$220$	0	0
$221$	−3.27193	−0.220094
$222$	− 0.780139i	− 0.0523595i
$223$	− 22.1782i	− 1.48516i	−0.669755	−	0.742582i	$-0.733601\pi$
$223$	− 22.1782i	− 1.48516i	0.669755	−	0.742582i	$-0.266399\pi$
$224$	0.329315	0.0220033
$225$	0	0
$226$	1.51566	0.100820
$227$	− 11.0969i	− 0.736528i	−0.929721	−	0.368264i	$-0.879952\pi$
$227$	− 11.0969i	− 0.736528i	0.929721	−	0.368264i	$-0.120048\pi$
$228$	5.44012i	0.360281i
$229$	−1.79879	−0.118867	−0.0594337	−	0.998232i	$-0.518929\pi$
$229$	−1.79879	−0.118867	−0.0594337	+	0.998232i	$0.518929\pi$
$230$	0	0
$231$	1.65757	0.109060
$232$	6.02216i	0.395374i
$233$	3.70579i	0.242774i	0.992605	+	0.121387i	$0.0387343\pi$
$233$	3.70579i	0.242774i	−0.992605	+	0.121387i	$0.961266\pi$
$234$	0.482152	0.0315193
$235$	0	0
$236$	1.01583	0.0661250
$237$	− 8.48510i	− 0.551167i
$238$	− 2.23476i	− 0.144858i
$239$	16.8733	1.09145	0.545723	−	0.837966i	$-0.316255\pi$
$239$	16.8733	1.09145	0.545723	+	0.837966i	$0.316255\pi$
$240$	0	0
$241$	20.6256	1.32861	0.664306	−	0.747461i	$-0.268727\pi$
$241$	20.6256	1.32861	0.664306	+	0.747461i	$0.268727\pi$
$242$	− 14.3350i	− 0.921491i
$243$	1.00000i	0.0641500i
$244$	−4.18808	−0.268114
$245$	0	0
$246$	12.5205	0.798279
$247$	2.62297i	0.166895i
$248$	1.31869i	0.0837371i
$249$	−17.1955	−1.08972
$250$	0	0
$251$	8.69615	0.548896	0.274448	−	0.961602i	$-0.411505\pi$
$251$	8.69615	0.548896	0.274448	+	0.961602i	$0.411505\pi$
$252$	0.329315i	0.0207449i
$253$	− 32.7198i	− 2.05708i
$254$	−12.6056	−0.790945
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 7.10714i	− 0.443331i	−0.975123	−	0.221666i	$-0.928851\pi$
$257$	− 7.10714i	− 0.443331i	0.975123	−	0.221666i	$-0.0711493\pi$
$258$	− 2.47582i	− 0.154138i
$259$	0.256911	0.0159637
$260$	0	0
$261$	−6.02216	−0.372763
$262$	− 14.0559i	− 0.868378i
$263$	− 15.8609i	− 0.978028i	−0.872276	−	0.489014i	$-0.837357\pi$
$263$	− 15.8609i	− 0.978028i	0.872276	−	0.489014i	$-0.162643\pi$
$264$	5.03339	0.309784
$265$	0	0
$266$	−1.79151	−0.109845
$267$	− 3.45233i	− 0.211279i
$268$	3.14433i	0.192070i
$269$	−4.51935	−0.275550	−0.137775	−	0.990464i	$-0.543995\pi$
$269$	−4.51935	−0.275550	−0.137775	+	0.990464i	$0.543995\pi$
$270$	0	0
$271$	11.1314	0.676186	0.338093	−	0.941113i	$-0.390218\pi$
$271$	11.1314	0.676186	0.338093	+	0.941113i	$0.390218\pi$
$272$	− 6.78610i	− 0.411468i
$273$	0.158780i	0.00960980i
$274$	6.68863	0.404075
$275$	0	0
$276$	6.50055	0.391287
$277$	22.8338i	1.37195i	0.727625	+	0.685975i	$0.240624\pi$
$277$	22.8338i	1.37195i	−0.727625	+	0.685975i	$0.759376\pi$
$278$	− 3.07799i	− 0.184605i
$279$	−1.31869	−0.0789481
$280$	0	0
$281$	28.7548	1.71537	0.857683	−	0.514179i	$-0.171903\pi$
$281$	28.7548	1.71537	0.857683	+	0.514179i	$0.171903\pi$
$282$	4.38040i	0.260849i
$283$	− 19.3471i	− 1.15007i	−0.818130	−	0.575034i	$-0.804989\pi$
$283$	− 19.3471i	− 1.15007i	0.818130	−	0.575034i	$-0.195011\pi$
$284$	−5.71189	−0.338938
$285$	0	0
$286$	2.42686	0.143503
$287$	4.12320i	0.243385i
$288$	1.00000i	0.0589256i
$289$	−29.0511	−1.70889
$290$	0	0
$291$	−9.51004	−0.557488
$292$	− 2.94269i	− 0.172208i
$293$	15.0301i	0.878069i	0.898470	+	0.439035i	$0.144679\pi$
$293$	15.0301i	0.878069i	−0.898470	+	0.439035i	$0.855321\pi$
$294$	6.89155	0.401923
$295$	0	0
$296$	0.780139	0.0453446
$297$	5.03339i	0.292067i
$298$	5.06465i	0.293387i
$299$	3.13425	0.181259
$300$	0	0
$301$	0.815324	0.0469945
$302$	16.9581i	0.975829i
$303$	19.2435i	1.10551i
$304$	−5.44012	−0.312012
$305$	0	0
$306$	6.78610	0.387935
$307$	− 19.7061i	− 1.12468i	−0.826905	−	0.562342i	$-0.809900\pi$
$307$	− 19.7061i	− 1.12468i	0.826905	−	0.562342i	$-0.190100\pi$
$308$	1.65757i	0.0944489i
$309$	−5.10689	−0.290521
$310$	0	0
$311$	28.2393	1.60130	0.800651	−	0.599130i	$-0.204487\pi$
$311$	28.2393	1.60130	0.800651	+	0.599130i	$0.204487\pi$
$312$	0.482152i	0.0272965i
$313$	− 11.2478i	− 0.635764i	−0.948130	−	0.317882i	$-0.897028\pi$
$313$	− 11.2478i	− 0.635764i	0.948130	−	0.317882i	$-0.102972\pi$
$314$	22.8284	1.28828
$315$	0	0
$316$	8.48510	0.477324
$317$	23.2917i	1.30819i	0.756412	+	0.654095i	$0.226950\pi$
$317$	23.2917i	1.30819i	−0.756412	+	0.654095i	$0.773050\pi$
$318$	− 1.68591i	− 0.0945412i
$319$	−30.3119	−1.69714
$320$	0	0
$321$	−15.3340	−0.855858
$322$	2.14073i	0.119298i
$323$	36.9172i	2.05413i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	7.95823	0.440766
$327$	9.76027i	0.539744i
$328$	12.5205i	0.691330i
$329$	−1.44253	−0.0795292
$330$	0	0
$331$	0.129334	0.00710883	0.00355441	−	0.999994i	$-0.498869\pi$
$331$	0.129334	0.00710883	0.00355441	+	0.999994i	$0.498869\pi$
$332$	− 17.1955i	− 0.943725i
$333$	0.780139i	0.0427513i
$334$	−13.9503	−0.763327
$335$	0	0
$336$	−0.329315	−0.0179656
$337$	− 0.538069i	− 0.0293105i	−0.999893	−	0.0146552i	$-0.995335\pi$
$337$	− 0.538069i	− 0.0293105i	0.999893	−	0.0146552i	$-0.00466507\pi$
$338$	− 12.7675i	− 0.694462i
$339$	−1.51566	−0.0823192
$340$	0	0
$341$	−6.63750	−0.359441
$342$	− 5.44012i	− 0.294168i
$343$	4.57470i	0.247010i
$344$	2.47582	0.133487
$345$	0	0
$346$	−19.5009	−1.04837
$347$	22.7156i	1.21944i	0.792618	+	0.609719i	$0.208718\pi$
$347$	22.7156i	1.21944i	−0.792618	+	0.609719i	$0.791282\pi$
$348$	− 6.02216i	− 0.322822i
$349$	−6.84350	−0.366324	−0.183162	−	0.983083i	$-0.558633\pi$
$349$	−6.84350	−0.366324	−0.183162	+	0.983083i	$0.558633\pi$
$350$	0	0
$351$	−0.482152	−0.0257354
$352$	5.03339i	0.268281i
$353$	− 16.7789i	− 0.893049i	−0.894771	−	0.446524i	$-0.852662\pi$
$353$	− 16.7789i	− 0.893049i	0.894771	−	0.446524i	$-0.147338\pi$
$354$	−1.01583	−0.0539909
$355$	0	0
$356$	3.45233	0.182973
$357$	2.23476i	0.118276i
$358$	− 18.9911i	− 1.00371i
$359$	7.46364	0.393916	0.196958	−	0.980412i	$-0.436894\pi$
$359$	7.46364	0.393916	0.196958	+	0.980412i	$0.436894\pi$
$360$	0	0
$361$	10.5949	0.557628
$362$	18.9484i	0.995905i
$363$	14.3350i	0.752395i
$364$	−0.158780	−0.00832233
$365$	0	0
$366$	4.18808	0.218914
$367$	− 6.15474i	− 0.321275i	−0.987013	−	0.160637i	$-0.948645\pi$
$367$	− 6.15474i	− 0.321275i	0.987013	−	0.160637i	$-0.0513550\pi$
$368$	6.50055i	0.338865i
$369$	−12.5205	−0.651792
$370$	0	0
$371$	0.555196	0.0288243
$372$	− 1.31869i	− 0.0683711i
$373$	− 9.16928i	− 0.474767i	−0.971416	−	0.237384i	$-0.923710\pi$
$373$	− 9.16928i	− 0.474767i	0.971416	−	0.237384i	$-0.0762899\pi$
$374$	34.1571	1.76622
$375$	0	0
$376$	−4.38040	−0.225902
$377$	− 2.90360i	− 0.149543i
$378$	− 0.329315i	− 0.0169381i
$379$	34.6321	1.77893	0.889465	−	0.457004i	$-0.151077\pi$
$379$	34.6321	1.77893	0.889465	+	0.457004i	$0.151077\pi$
$380$	0	0
$381$	12.6056	0.645804
$382$	12.0933i	0.618746i
$383$	− 9.52044i	− 0.486472i	−0.969967	−	0.243236i	$-0.921791\pi$
$383$	− 9.52044i	− 0.486472i	0.969967	−	0.243236i	$-0.0782089\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−11.0357	−0.561703
$387$	2.47582i	0.125853i
$388$	− 9.51004i	− 0.482799i
$389$	17.4454	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$389$	17.4454	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$390$	0	0
$391$	44.1134	2.23091
$392$	6.89155i	0.348076i
$393$	14.0559i	0.709028i
$394$	20.6212	1.03888
$395$	0	0
$396$	−5.03339	−0.252937
$397$	13.5551i	0.680313i	0.940369	+	0.340157i	$0.110480\pi$
$397$	13.5551i	0.680313i	−0.940369	+	0.340157i	$0.889520\pi$
$398$	18.5313i	0.928892i
$399$	1.79151	0.0896879
$400$	0	0
$401$	14.1105	0.704642	0.352321	−	0.935879i	$-0.385392\pi$
$401$	14.1105	0.704642	0.352321	+	0.935879i	$0.385392\pi$
$402$	− 3.14433i	− 0.156825i
$403$	− 0.635811i	− 0.0316720i
$404$	−19.2435	−0.957400
$405$	0	0
$406$	1.98319	0.0984240
$407$	3.92674i	0.194641i
$408$	6.78610i	0.335962i
$409$	16.5637	0.819024	0.409512	−	0.912305i	$-0.365699\pi$
$409$	16.5637	0.819024	0.409512	+	0.912305i	$0.365699\pi$
$410$	0	0
$411$	−6.68863	−0.329926
$412$	− 5.10689i	− 0.251598i
$413$	− 0.334529i	− 0.0164611i
$414$	−6.50055	−0.319485
$415$	0	0
$416$	−0.482152	−0.0236395
$417$	3.07799i	0.150730i
$418$	− 27.3823i	− 1.33931i
$419$	−16.9705	−0.829062	−0.414531	−	0.910035i	$-0.636054\pi$
$419$	−16.9705	−0.829062	−0.414531	+	0.910035i	$0.636054\pi$
$420$	0	0
$421$	−7.06240	−0.344200	−0.172100	−	0.985079i	$-0.555055\pi$
$421$	−7.06240	−0.344200	−0.172100	+	0.985079i	$0.555055\pi$
$422$	− 15.7638i	− 0.767371i
$423$	− 4.38040i	− 0.212982i
$424$	1.68591	0.0818751
$425$	0	0
$426$	5.71189	0.276742
$427$	1.37920i	0.0667440i
$428$	− 15.3340i	− 0.741194i
$429$	−2.42686	−0.117170
$430$	0	0
$431$	−17.8582	−0.860199	−0.430099	−	0.902782i	$-0.641521\pi$
$431$	−17.8582	−0.860199	−0.430099	+	0.902782i	$0.641521\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 24.1797i	− 1.16200i	−0.813903	−	0.581000i	$-0.802661\pi$
$433$	− 24.1797i	− 1.16200i	0.813903	−	0.581000i	$-0.197339\pi$
$434$	0.434265	0.0208454
$435$	0	0
$436$	−9.76027	−0.467432
$437$	− 35.3638i	− 1.69168i
$438$	2.94269i	0.140607i
$439$	18.9103	0.902537	0.451269	−	0.892388i	$-0.350972\pi$
$439$	18.9103	0.902537	0.451269	+	0.892388i	$0.350972\pi$
$440$	0	0
$441$	−6.89155	−0.328169
$442$	3.27193i	0.155630i
$443$	4.05769i	0.192787i	0.995343	+	0.0963933i	$0.0307307\pi$
$443$	4.05769i	0.192787i	−0.995343	+	0.0963933i	$0.969269\pi$
$444$	−0.780139	−0.0370237
$445$	0	0
$446$	−22.1782	−1.05017
$447$	− 5.06465i	− 0.239550i
$448$	− 0.329315i	− 0.0155587i
$449$	6.26150	0.295498	0.147749	−	0.989025i	$-0.452797\pi$
$449$	6.26150	0.295498	0.147749	+	0.989025i	$0.452797\pi$
$450$	0	0
$451$	−63.0207	−2.96753
$452$	− 1.51566i	− 0.0712905i
$453$	− 16.9581i	− 0.796761i
$454$	−11.0969	−0.520804
$455$	0	0
$456$	5.44012	0.254757
$457$	− 1.94229i	− 0.0908564i	−0.998968	−	0.0454282i	$-0.985535\pi$
$457$	− 1.94229i	− 0.0908564i	0.998968	−	0.0454282i	$-0.0144652\pi$
$458$	1.79879i	0.0840520i
$459$	−6.78610	−0.316748
$460$	0	0
$461$	3.12391	0.145495	0.0727475	−	0.997350i	$-0.476823\pi$
$461$	3.12391	0.145495	0.0727475	+	0.997350i	$0.476823\pi$
$462$	− 1.65757i	− 0.0771172i
$463$	− 8.26641i	− 0.384173i	−0.981378	−	0.192086i	$-0.938475\pi$
$463$	− 8.26641i	− 0.384173i	0.981378	−	0.192086i	$-0.0615253\pi$
$464$	6.02216	0.279572
$465$	0	0
$466$	3.70579	0.171667
$467$	30.6192i	1.41689i	0.705766	+	0.708445i	$0.250603\pi$
$467$	30.6192i	1.41689i	−0.705766	+	0.708445i	$0.749397\pi$
$468$	− 0.482152i	− 0.0222875i
$469$	1.03547	0.0478137
$470$	0	0
$471$	−22.8284	−1.05188
$472$	− 1.01583i	− 0.0467575i
$473$	12.4618i	0.572993i
$474$	−8.48510	−0.389734
$475$	0	0
$476$	−2.23476	−0.102430
$477$	1.68591i	0.0771925i
$478$	− 16.8733i	− 0.771769i
$479$	1.17289	0.0535907	0.0267953	−	0.999641i	$-0.491470\pi$
$479$	1.17289	0.0535907	0.0267953	+	0.999641i	$0.491470\pi$
$480$	0	0
$481$	−0.376146	−0.0171508
$482$	− 20.6256i	− 0.939471i
$483$	− 2.14073i	− 0.0974065i
$484$	−14.3350	−0.651593
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 11.7556i	− 0.532695i	−0.963877	−	0.266348i	$-0.914183\pi$
$487$	− 11.7556i	− 0.532695i	0.963877	−	0.266348i	$-0.0858169\pi$
$488$	4.18808i	0.189585i
$489$	−7.95823	−0.359884
$490$	0	0
$491$	4.13052	0.186408	0.0932038	−	0.995647i	$-0.470289\pi$
$491$	4.13052	0.186408	0.0932038	+	0.995647i	$0.470289\pi$
$492$	− 12.5205i	− 0.564469i
$493$	− 40.8670i	− 1.84056i
$494$	2.62297	0.118013
$495$	0	0
$496$	1.31869	0.0592111
$497$	1.88101i	0.0843748i
$498$	17.1955i	0.770549i
$499$	−9.59154	−0.429376	−0.214688	−	0.976683i	$-0.568873\pi$
$499$	−9.59154	−0.429376	−0.214688	+	0.976683i	$0.568873\pi$
$500$	0	0
$501$	13.9503	0.623254
$502$	− 8.69615i	− 0.388128i
$503$	16.9453i	0.755555i	0.925896	+	0.377777i	$0.123312\pi$
$503$	16.9453i	0.755555i	−0.925896	+	0.377777i	$0.876688\pi$
$504$	0.329315	0.0146689
$505$	0	0
$506$	−32.7198	−1.45457
$507$	12.7675i	0.567026i
$508$	12.6056i	0.559282i
$509$	−34.3000	−1.52032	−0.760161	−	0.649735i	$-0.774880\pi$
$509$	−34.3000	−1.52032	−0.760161	+	0.649735i	$0.774880\pi$
$510$	0	0
$511$	−0.969070	−0.0428691
$512$	− 1.00000i	− 0.0441942i
$513$	5.44012i	0.240187i
$514$	−7.10714	−0.313483
$515$	0	0
$516$	−2.47582	−0.108992
$517$	− 22.0482i	− 0.969681i
$518$	− 0.256911i	− 0.0112880i
$519$	19.5009	0.855993
$520$	0	0
$521$	21.2832	0.932436	0.466218	−	0.884670i	$-0.345616\pi$
$521$	21.2832	0.932436	0.466218	+	0.884670i	$0.345616\pi$
$522$	6.02216i	0.263583i
$523$	− 41.3492i	− 1.80807i	−0.427454	−	0.904037i	$-0.640589\pi$
$523$	− 41.3492i	− 1.80807i	0.427454	−	0.904037i	$-0.359411\pi$
$524$	−14.0559	−0.614036
$525$	0	0
$526$	−15.8609	−0.691570
$527$	− 8.94878i	− 0.389815i
$528$	− 5.03339i	− 0.219050i
$529$	−19.2571	−0.837267
$530$	0	0
$531$	1.01583	0.0440833
$532$	1.79151i	0.0776720i
$533$	− 6.03680i	− 0.261483i
$534$	−3.45233	−0.149397
$535$	0	0
$536$	3.14433	0.135814
$537$	18.9911i	0.819526i
$538$	4.51935i	0.194843i
$539$	−34.6879	−1.49411
$540$	0	0
$541$	−35.8217	−1.54010	−0.770048	−	0.637987i	$-0.779768\pi$
$541$	−35.8217	−1.54010	−0.770048	+	0.637987i	$0.779768\pi$
$542$	− 11.1314i	− 0.478136i
$543$	− 18.9484i	− 0.813153i
$544$	−6.78610	−0.290951
$545$	0	0
$546$	0.158780	0.00679515
$547$	− 8.36253i	− 0.357556i	−0.983889	−	0.178778i	$-0.942786\pi$
$547$	− 8.36253i	− 0.357556i	0.983889	−	0.178778i	$-0.0572144\pi$
$548$	− 6.68863i	− 0.285724i
$549$	−4.18808	−0.178743
$550$	0	0
$551$	−32.7613	−1.39568
$552$	− 6.50055i	− 0.276682i
$553$	− 2.79427i	− 0.118824i
$554$	22.8338	0.970115
$555$	0	0
$556$	−3.07799	−0.130536
$557$	28.1467i	1.19262i	0.802756	+	0.596308i	$0.203366\pi$
$557$	28.1467i	1.19262i	−0.802756	+	0.596308i	$0.796634\pi$
$558$	1.31869i	0.0558247i
$559$	−1.19372	−0.0504890
$560$	0	0
$561$	−34.1571	−1.44211
$562$	− 28.7548i	− 1.21295i
$563$	− 3.45072i	− 0.145431i	−0.997353	−	0.0727153i	$-0.976834\pi$
$563$	− 3.45072i	− 0.145431i	0.997353	−	0.0727153i	$-0.0231665\pi$
$564$	4.38040	0.184448
$565$	0	0
$566$	−19.3471	−0.813220
$567$	0.329315i	0.0138299i
$568$	5.71189i	0.239665i
$569$	−17.4040	−0.729614	−0.364807	−	0.931083i	$-0.618865\pi$
$569$	−17.4040	−0.729614	−0.364807	+	0.931083i	$0.618865\pi$
$570$	0	0
$571$	−37.2213	−1.55766	−0.778831	−	0.627234i	$-0.784187\pi$
$571$	−37.2213	−1.55766	−0.778831	+	0.627234i	$0.784187\pi$
$572$	− 2.42686i	− 0.101472i
$573$	− 12.0933i	− 0.505204i
$574$	4.12320	0.172099
$575$	0	0
$576$	1.00000	0.0416667
$577$	20.1041i	0.836943i	0.908230	+	0.418472i	$0.137434\pi$
$577$	20.1041i	0.836943i	−0.908230	+	0.418472i	$0.862566\pi$
$578$	29.0511i	1.20837i
$579$	11.0357	0.458629
$580$	0	0
$581$	−5.66273	−0.234930
$582$	9.51004i	0.394204i
$583$	8.48585i	0.351448i
$584$	−2.94269	−0.121769
$585$	0	0
$586$	15.0301	0.620889
$587$	6.11838i	0.252533i	0.991996	+	0.126266i	$0.0402994\pi$
$587$	6.11838i	0.252533i	−0.991996	+	0.126266i	$0.959701\pi$
$588$	− 6.89155i	− 0.284203i
$589$	−7.17385	−0.295593
$590$	0	0
$591$	−20.6212	−0.848242
$592$	− 0.780139i	− 0.0320635i
$593$	24.9458i	1.02440i	0.858866	+	0.512201i	$0.171170\pi$
$593$	24.9458i	1.02440i	−0.858866	+	0.512201i	$0.828830\pi$
$594$	5.03339	0.206523
$595$	0	0
$596$	5.06465	0.207456
$597$	− 18.5313i	− 0.758437i
$598$	− 3.13425i	− 0.128169i
$599$	0.941228	0.0384575	0.0192288	−	0.999815i	$-0.493879\pi$
$599$	0.941228	0.0384575	0.0192288	+	0.999815i	$0.493879\pi$
$600$	0	0
$601$	10.2333	0.417426	0.208713	−	0.977977i	$-0.433073\pi$
$601$	10.2333	0.417426	0.208713	+	0.977977i	$0.433073\pi$
$602$	− 0.815324i	− 0.0332301i
$603$	3.14433i	0.128047i
$604$	16.9581	0.690015
$605$	0	0
$606$	19.2435	0.781714
$607$	− 6.80623i	− 0.276256i	−0.990414	−	0.138128i	$-0.955891\pi$
$607$	− 6.80623i	− 0.276256i	0.990414	−	0.138128i	$-0.0441086\pi$
$608$	5.44012i	0.220626i
$609$	−1.98319	−0.0803629
$610$	0	0
$611$	2.11202	0.0854431
$612$	− 6.78610i	− 0.274312i
$613$	− 1.13944i	− 0.0460215i	−0.999735	−	0.0230107i	$-0.992675\pi$
$613$	− 1.13944i	− 0.0460215i	0.999735	−	0.0230107i	$-0.00732519\pi$
$614$	−19.7061	−0.795272
$615$	0	0
$616$	1.65757	0.0667854
$617$	14.3577i	0.578020i	0.957326	+	0.289010i	$0.0933261\pi$
$617$	14.3577i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$618$	5.10689i	0.205429i
$619$	−0.476518	−0.0191529	−0.00957644	−	0.999954i	$-0.503048\pi$
$619$	−0.476518	−0.0191529	−0.00957644	+	0.999954i	$0.503048\pi$
$620$	0	0
$621$	6.50055	0.260858
$622$	− 28.2393i	− 1.13229i
$623$	− 1.13690i	− 0.0455491i
$624$	0.482152	0.0193015
$625$	0	0
$626$	−11.2478	−0.449553
$627$	27.3823i	1.09354i
$628$	− 22.8284i	− 0.910952i
$629$	−5.29410	−0.211089
$630$	0	0
$631$	−17.0525	−0.678850	−0.339425	−	0.940633i	$-0.610232\pi$
$631$	−17.0525	−0.678850	−0.339425	+	0.940633i	$0.610232\pi$
$632$	− 8.48510i	− 0.337519i
$633$	15.7638i	0.626556i
$634$	23.2917	0.925030
$635$	0	0
$636$	−1.68591	−0.0668507
$637$	− 3.32278i	− 0.131653i
$638$	30.3119i	1.20006i
$639$	−5.71189	−0.225959
$640$	0	0
$641$	−32.7254	−1.29258	−0.646288	−	0.763094i	$-0.723680\pi$
$641$	−32.7254	−1.29258	−0.646288	+	0.763094i	$0.723680\pi$
$642$	15.3340i	0.605183i
$643$	− 12.5844i	− 0.496281i	−0.968724	−	0.248141i	$-0.920180\pi$
$643$	− 12.5844i	− 0.496281i	0.968724	−	0.248141i	$-0.0798195\pi$
$644$	2.14073	0.0843565
$645$	0	0
$646$	36.9172	1.45249
$647$	− 25.6397i	− 1.00800i	−0.863703	−	0.504000i	$-0.831861\pi$
$647$	− 25.6397i	− 1.00800i	0.863703	−	0.504000i	$-0.168139\pi$
$648$	1.00000i	0.0392837i
$649$	5.11308	0.200706
$650$	0	0
$651$	−0.434265	−0.0170202
$652$	− 7.95823i	− 0.311668i
$653$	30.0568i	1.17621i	0.808783	+	0.588107i	$0.200127\pi$
$653$	30.0568i	1.17621i	−0.808783	+	0.588107i	$0.799873\pi$
$654$	9.76027	0.381657
$655$	0	0
$656$	12.5205	0.488844
$657$	− 2.94269i	− 0.114805i
$658$	1.44253i	0.0562357i
$659$	8.04829	0.313517	0.156759	−	0.987637i	$-0.449896\pi$
$659$	8.04829	0.313517	0.156759	+	0.987637i	$0.449896\pi$
$660$	0	0
$661$	−0.728568	−0.0283380	−0.0141690	−	0.999900i	$-0.504510\pi$
$661$	−0.728568	−0.0283380	−0.0141690	+	0.999900i	$0.504510\pi$
$662$	− 0.129334i	− 0.00502670i
$663$	− 3.27193i	− 0.127071i
$664$	−17.1955	−0.667315
$665$	0	0
$666$	0.780139	0.0302298
$667$	39.1474i	1.51579i
$668$	13.9503i	0.539754i
$669$	22.1782	0.857460
$670$	0	0
$671$	−21.0803	−0.813794
$672$	0.329315i	0.0127036i
$673$	7.40821i	0.285566i	0.989754	+	0.142783i	$0.0456050\pi$
$673$	7.40821i	0.285566i	−0.989754	+	0.142783i	$0.954395\pi$
$674$	−0.538069	−0.0207256
$675$	0	0
$676$	−12.7675	−0.491059
$677$	− 9.79504i	− 0.376454i	−0.982126	−	0.188227i	$-0.939726\pi$
$677$	− 9.79504i	− 0.376454i	0.982126	−	0.188227i	$-0.0602740\pi$
$678$	1.51566i	0.0582084i
$679$	−3.13180	−0.120187
$680$	0	0
$681$	11.0969	0.425235
$682$	6.63750i	0.254163i
$683$	33.5847i	1.28508i	0.766250	+	0.642542i	$0.222120\pi$
$683$	33.5847i	1.28508i	−0.766250	+	0.642542i	$0.777880\pi$
$684$	−5.44012	−0.208008
$685$	0	0
$686$	4.57470	0.174663
$687$	− 1.79879i	− 0.0686281i
$688$	− 2.47582i	− 0.0943897i
$689$	−0.812865	−0.0309677
$690$	0	0
$691$	26.1110	0.993309	0.496655	−	0.867948i	$-0.334562\pi$
$691$	26.1110	0.993309	0.496655	+	0.867948i	$0.334562\pi$
$692$	19.5009i	0.741312i
$693$	1.65757i	0.0629659i
$694$	22.7156	0.862273
$695$	0	0
$696$	−6.02216	−0.228270
$697$	− 84.9655i	− 3.21830i
$698$	6.84350i	0.259030i
$699$	−3.70579	−0.140166
$700$	0	0
$701$	41.8212	1.57956	0.789782	−	0.613388i	$-0.210194\pi$
$701$	41.8212	1.57956	0.789782	+	0.613388i	$0.210194\pi$
$702$	0.482152i	0.0181977i
$703$	4.24405i	0.160067i
$704$	5.03339	0.189703
$705$	0	0
$706$	−16.7789	−0.631481
$707$	6.33717i	0.238334i
$708$	1.01583i	0.0381773i
$709$	8.86903	0.333083	0.166542	−	0.986034i	$-0.446740\pi$
$709$	8.86903	0.333083	0.166542	+	0.986034i	$0.446740\pi$
$710$	0	0
$711$	8.48510	0.318216
$712$	− 3.45233i	− 0.129382i
$713$	8.57223i	0.321033i
$714$	2.23476	0.0836339
$715$	0	0
$716$	−18.9911	−0.709731
$717$	16.8733i	0.630146i
$718$	− 7.46364i	− 0.278541i
$719$	2.65509	0.0990182	0.0495091	−	0.998774i	$-0.484234\pi$
$719$	2.65509	0.0990182	0.0495091	+	0.998774i	$0.484234\pi$
$720$	0	0
$721$	−1.68177	−0.0626325
$722$	− 10.5949i	− 0.394303i
$723$	20.6256i	0.767075i
$724$	18.9484	0.704211
$725$	0	0
$726$	14.3350	0.532023
$727$	7.03681i	0.260981i	0.991450	+	0.130490i	$0.0416552\pi$
$727$	7.03681i	0.260981i	−0.991450	+	0.130490i	$0.958345\pi$
$728$	0.158780i	0.00588477i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.8011	−0.621413
$732$	− 4.18808i	− 0.154796i
$733$	20.2601i	0.748324i	0.927363	+	0.374162i	$0.122070\pi$
$733$	20.2601i	0.748324i	−0.927363	+	0.374162i	$0.877930\pi$
$734$	−6.15474	−0.227176
$735$	0	0
$736$	6.50055	0.239613
$737$	15.8266i	0.582982i
$738$	12.5205i	0.460887i
$739$	−17.1512	−0.630919	−0.315459	−	0.948939i	$-0.602159\pi$
$739$	−17.1512	−0.630919	−0.315459	+	0.948939i	$0.602159\pi$
$740$	0	0
$741$	−2.62297	−0.0963571
$742$	− 0.555196i	− 0.0203819i
$743$	− 9.94252i	− 0.364756i	−0.983229	−	0.182378i	$-0.941621\pi$
$743$	− 9.94252i	− 0.364756i	0.983229	−	0.182378i	$-0.0583794\pi$
$744$	−1.31869	−0.0483456
$745$	0	0
$746$	−9.16928	−0.335711
$747$	− 17.1955i	− 0.629150i
$748$	− 34.1571i	− 1.24891i
$749$	−5.04970	−0.184512
$750$	0	0
$751$	−17.6941	−0.645667	−0.322833	−	0.946456i	$-0.604635\pi$
$751$	−17.6941	−0.645667	−0.322833	+	0.946456i	$0.604635\pi$
$752$	4.38040i	0.159737i
$753$	8.69615i	0.316905i
$754$	−2.90360	−0.105743
$755$	0	0
$756$	−0.329315	−0.0119771
$757$	− 37.1272i	− 1.34941i	−0.738087	−	0.674706i	$-0.764271\pi$
$757$	− 37.1272i	− 1.34941i	0.738087	−	0.674706i	$-0.235729\pi$
$758$	− 34.6321i	− 1.25789i
$759$	32.7198	1.18765
$760$	0	0
$761$	−49.5640	−1.79669	−0.898346	−	0.439289i	$-0.855230\pi$
$761$	−49.5640	−1.79669	−0.898346	+	0.439289i	$0.855230\pi$
$762$	− 12.6056i	− 0.456652i
$763$	3.21420i	0.116362i
$764$	12.0933	0.437520
$765$	0	0
$766$	−9.52044	−0.343987
$767$	0.489786i	0.0176851i
$768$	1.00000i	0.0360844i
$769$	−34.3645	−1.23922	−0.619608	−	0.784911i	$-0.712709\pi$
$769$	−34.3645	−1.23922	−0.619608	+	0.784911i	$0.712709\pi$
$770$	0	0
$771$	7.10714	0.255957
$772$	11.0357i	0.397184i
$773$	− 22.9926i	− 0.826985i	−0.910507	−	0.413493i	$-0.864309\pi$
$773$	− 22.9926i	− 0.826985i	0.910507	−	0.413493i	$-0.135691\pi$
$774$	2.47582	0.0889915
$775$	0	0
$776$	−9.51004	−0.341390
$777$	0.256911i	0.00921664i
$778$	− 17.4454i	− 0.625447i
$779$	−68.1132	−2.44041
$780$	0	0
$781$	−28.7502	−1.02876
$782$	− 44.1134i	− 1.57749i
$783$	− 6.02216i	− 0.215215i
$784$	6.89155	0.246127
$785$	0	0
$786$	14.0559	0.501358
$787$	8.41183i	0.299849i	0.988697	+	0.149925i	$0.0479031\pi$
$787$	8.41183i	0.299849i	−0.988697	+	0.149925i	$0.952097\pi$
$788$	− 20.6212i	− 0.734599i
$789$	15.8609	0.564665
$790$	0	0
$791$	−0.499128	−0.0177470
$792$	5.03339i	0.178854i
$793$	− 2.01929i	− 0.0717072i
$794$	13.5551	0.481054
$795$	0	0
$796$	18.5313	0.656826
$797$	− 38.2745i	− 1.35575i	−0.735177	−	0.677875i	$-0.762901\pi$
$797$	− 38.2745i	− 1.35575i	0.735177	−	0.677875i	$-0.237099\pi$
$798$	− 1.79151i	− 0.0634189i
$799$	29.7258	1.05162
$800$	0	0
$801$	3.45233	0.121982
$802$	− 14.1105i	− 0.498257i
$803$	− 14.8117i	− 0.522693i
$804$	−3.14433	−0.110892
$805$	0	0
$806$	−0.635811	−0.0223955
$807$	− 4.51935i	− 0.159089i
$808$	19.2435i	0.676984i
$809$	30.8014	1.08292	0.541461	−	0.840726i	$-0.317872\pi$
$809$	30.8014	1.08292	0.541461	+	0.840726i	$0.317872\pi$
$810$	0	0
$811$	−0.787233	−0.0276435	−0.0138217	−	0.999904i	$-0.504400\pi$
$811$	−0.787233	−0.0276435	−0.0138217	+	0.999904i	$0.504400\pi$
$812$	− 1.98319i	− 0.0695963i
$813$	11.1314i	0.390396i
$814$	3.92674	0.137632
$815$	0	0
$816$	6.78610	0.237561
$817$	13.4688i	0.471212i
$818$	− 16.5637i	− 0.579138i
$819$	−0.158780	−0.00554822
$820$	0	0
$821$	17.4710	0.609743	0.304872	−	0.952393i	$-0.401386\pi$
$821$	17.4710	0.609743	0.304872	+	0.952393i	$0.401386\pi$
$822$	6.68863i	0.233293i
$823$	37.4268i	1.30462i	0.757953	+	0.652309i	$0.226199\pi$
$823$	37.4268i	1.30462i	−0.757953	+	0.652309i	$0.773801\pi$
$824$	−5.10689	−0.177907
$825$	0	0
$826$	−0.334529	−0.0116397
$827$	11.7918i	0.410041i	0.978758	+	0.205020i	$0.0657260\pi$
$827$	11.7918i	0.410041i	−0.978758	+	0.205020i	$0.934274\pi$
$828$	6.50055i	0.225910i
$829$	35.0287	1.21660	0.608298	−	0.793708i	$-0.291852\pi$
$829$	35.0287	1.21660	0.608298	+	0.793708i	$0.291852\pi$
$830$	0	0
$831$	−22.8338	−0.792096
$832$	0.482152i	0.0167156i
$833$	− 46.7667i	− 1.62037i
$834$	3.07799	0.106582
$835$	0	0
$836$	−27.3823	−0.947036
$837$	− 1.31869i	− 0.0455807i
$838$	16.9705i	0.586235i
$839$	−0.498658	−0.0172156	−0.00860779	−	0.999963i	$-0.502740\pi$
$839$	−0.498658	−0.0172156	−0.00860779	+	0.999963i	$0.502740\pi$
$840$	0	0
$841$	7.26647	0.250568
$842$	7.06240i	0.243386i
$843$	28.7548i	0.990367i
$844$	−15.7638	−0.542613
$845$	0	0
$846$	−4.38040	−0.150601
$847$	4.72074i	0.162207i
$848$	− 1.68591i	− 0.0578944i
$849$	19.3471	0.663992
$850$	0	0
$851$	5.07133	0.173843
$852$	− 5.71189i	− 0.195686i
$853$	21.3584i	0.731298i	0.930753	+	0.365649i	$0.119153\pi$
$853$	21.3584i	0.731298i	−0.930753	+	0.365649i	$0.880847\pi$
$854$	1.37920	0.0471952
$855$	0	0
$856$	−15.3340	−0.524104
$857$	33.4831i	1.14376i	0.820337	+	0.571880i	$0.193786\pi$
$857$	33.4831i	1.14376i	−0.820337	+	0.571880i	$0.806214\pi$
$858$	2.42686i	0.0828517i
$859$	−45.4873	−1.55201	−0.776003	−	0.630729i	$-0.782756\pi$
$859$	−45.4873	−1.55201	−0.776003	+	0.630729i	$0.782756\pi$
$860$	0	0
$861$	−4.12320	−0.140518
$862$	17.8582i	0.608252i
$863$	14.4689i	0.492529i	0.969203	+	0.246264i	$0.0792031\pi$
$863$	14.4689i	0.492529i	−0.969203	+	0.246264i	$0.920797\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−24.1797	−0.821659
$867$	− 29.0511i	− 0.986627i
$868$	− 0.434265i	− 0.0147399i
$869$	42.7088	1.44880
$870$	0	0
$871$	−1.51604	−0.0513692
$872$	9.76027i	0.330525i
$873$	− 9.51004i	− 0.321866i
$874$	−35.3638	−1.19620
$875$	0	0
$876$	2.94269	0.0994241
$877$	45.1991i	1.52626i	0.646243	+	0.763132i	$0.276339\pi$
$877$	45.1991i	1.52626i	−0.646243	+	0.763132i	$0.723661\pi$
$878$	− 18.9103i	− 0.638190i
$879$	−15.0301	−0.506954
$880$	0	0
$881$	13.9752	0.470836	0.235418	−	0.971894i	$-0.424354\pi$
$881$	13.9752	0.470836	0.235418	+	0.971894i	$0.424354\pi$
$882$	6.89155i	0.232051i
$883$	− 11.7349i	− 0.394911i	−0.980312	−	0.197455i	$-0.936732\pi$
$883$	− 11.7349i	− 0.394911i	0.980312	−	0.197455i	$-0.0632678\pi$
$884$	3.27193	0.110047
$885$	0	0
$886$	4.05769	0.136321
$887$	− 19.7512i	− 0.663180i	−0.943424	−	0.331590i	$-0.892415\pi$
$887$	− 19.7512i	− 0.663180i	0.943424	−	0.331590i	$-0.107585\pi$
$888$	0.780139i	0.0261797i
$889$	4.15121	0.139227
$890$	0	0
$891$	−5.03339	−0.168625
$892$	22.1782i	0.742582i
$893$	− 23.8299i	− 0.797437i
$894$	−5.06465	−0.169387
$895$	0	0
$896$	−0.329315	−0.0110016
$897$	3.13425i	0.104650i
$898$	− 6.26150i	− 0.208949i
$899$	7.94139	0.264860
$900$	0	0
$901$	−11.4408	−0.381147
$902$	63.0207i	2.09836i
$903$	0.815324i	0.0271323i
$904$	−1.51566	−0.0504100
$905$	0	0
$906$	−16.9581	−0.563395
$907$	21.0111i	0.697661i	0.937186	+	0.348831i	$0.113421\pi$
$907$	21.0111i	0.697661i	−0.937186	+	0.348831i	$0.886579\pi$
$908$	11.0969i	0.368264i
$909$	−19.2435	−0.638267
$910$	0	0
$911$	24.8288	0.822614	0.411307	−	0.911497i	$-0.365072\pi$
$911$	24.8288	0.822614	0.411307	+	0.911497i	$0.365072\pi$
$912$	− 5.44012i	− 0.180140i
$913$	− 86.5517i	− 2.86444i
$914$	−1.94229	−0.0642452
$915$	0	0
$916$	1.79879	0.0594337
$917$	4.62883i	0.152857i
$918$	6.78610i	0.223975i
$919$	36.0292	1.18849	0.594247	−	0.804283i	$-0.297450\pi$
$919$	36.0292	1.18849	0.594247	+	0.804283i	$0.297450\pi$
$920$	0	0
$921$	19.7061	0.649337
$922$	− 3.12391i	− 0.102881i
$923$	− 2.75400i	− 0.0906490i
$924$	−1.65757	−0.0545301
$925$	0	0
$926$	−8.26641	−0.271651
$927$	− 5.10689i	− 0.167732i
$928$	− 6.02216i	− 0.197687i
$929$	56.9634	1.86891	0.934454	−	0.356084i	$-0.115888\pi$
$929$	56.9634	1.86891	0.934454	+	0.356084i	$0.115888\pi$
$930$	0	0
$931$	−37.4909	−1.22871
$932$	− 3.70579i	− 0.121387i
$933$	28.2393i	0.924513i
$934$	30.6192	1.00189
$935$	0	0
$936$	−0.482152	−0.0157596
$937$	− 41.7570i	− 1.36414i	−0.731286	−	0.682071i	$-0.761080\pi$
$937$	− 41.7570i	− 1.36414i	0.731286	−	0.682071i	$-0.238920\pi$
$938$	− 1.03547i	− 0.0338094i
$939$	11.2478	0.367059
$940$	0	0
$941$	−22.8232	−0.744014	−0.372007	−	0.928230i	$-0.621330\pi$
$941$	−22.8232	−0.744014	−0.372007	+	0.928230i	$0.621330\pi$
$942$	22.8284i	0.743789i
$943$	81.3903i	2.65043i
$944$	−1.01583	−0.0330625
$945$	0	0
$946$	12.4618	0.405167
$947$	8.46744i	0.275155i	0.990491	+	0.137577i	$0.0439316\pi$
$947$	8.46744i	0.275155i	−0.990491	+	0.137577i	$0.956068\pi$
$948$	8.48510i	0.275583i
$949$	1.41882	0.0460569
$950$	0	0
$951$	−23.2917	−0.755284
$952$	2.23476i	0.0724291i
$953$	24.8878i	0.806195i	0.915157	+	0.403097i	$0.132066\pi$
$953$	24.8878i	0.806195i	−0.915157	+	0.403097i	$0.867934\pi$
$954$	1.68591	0.0545834
$955$	0	0
$956$	−16.8733	−0.545723
$957$	− 30.3119i	− 0.979845i
$958$	− 1.17289i	− 0.0378943i
$959$	−2.20267	−0.0711278
$960$	0	0
$961$	−29.2610	−0.943905
$962$	0.376146i	0.0121274i
$963$	− 15.3340i	− 0.494130i
$964$	−20.6256	−0.664306
$965$	0	0
$966$	−2.14073	−0.0688768
$967$	22.1247i	0.711483i	0.934584	+	0.355742i	$0.115772\pi$
$967$	22.1247i	0.711483i	−0.934584	+	0.355742i	$0.884228\pi$
$968$	14.3350i	0.460746i
$969$	−36.9172	−1.18595
$970$	0	0
$971$	−17.0050	−0.545716	−0.272858	−	0.962054i	$-0.587969\pi$
$971$	−17.0050	−0.545716	−0.272858	+	0.962054i	$0.587969\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	1.01363i	0.0324954i
$974$	−11.7556	−0.376672
$975$	0	0
$976$	4.18808	0.134057
$977$	− 41.3030i	− 1.32140i	−0.750650	−	0.660701i	$-0.770259\pi$
$977$	− 41.3030i	− 1.32140i	0.750650	−	0.660701i	$-0.229741\pi$
$978$	7.95823i	0.254476i
$979$	17.3769	0.555369
$980$	0	0
$981$	−9.76027	−0.311622
$982$	− 4.13052i	− 0.131810i
$983$	61.9160i	1.97481i	0.158199	+	0.987407i	$0.449431\pi$
$983$	61.9160i	1.97481i	−0.158199	+	0.987407i	$0.550569\pi$
$984$	−12.5205	−0.399140
$985$	0	0
$986$	−40.8670	−1.30147
$987$	− 1.44253i	− 0.0459162i
$988$	− 2.62297i	− 0.0834477i
$989$	16.0942	0.511765
$990$	0	0
$991$	4.93098	0.156638	0.0783188	−	0.996928i	$-0.475045\pi$
$991$	4.93098	0.156638	0.0783188	+	0.996928i	$0.475045\pi$
$992$	− 1.31869i	− 0.0418686i
$993$	0.129334i	0.00410428i
$994$	1.88101	0.0596620
$995$	0	0
$996$	17.1955	0.544860
$997$	49.5596i	1.56957i	0.619769	+	0.784784i	$0.287226\pi$
$997$	49.5596i	1.56957i	−0.619769	+	0.784784i	$0.712774\pi$
$998$	9.59154i	0.303615i
$999$	−0.780139	−0.0246825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.k.1249.5		16
5.2	odd	4		3750.2.a.v.1.4		8
5.3	odd	4		3750.2.a.u.1.5		8
5.4	even	2	inner	3750.2.c.k.1249.12		16
25.2	odd	20		750.2.g.f.601.2		16
25.9	even	10		750.2.h.d.349.1		16
25.11	even	5		750.2.h.d.649.2		16
25.12	odd	20		750.2.g.f.151.2		16
25.13	odd	20		750.2.g.g.151.3		16
25.14	even	10		150.2.h.b.79.4	yes	16
25.16	even	5		150.2.h.b.19.4	✓	16
25.23	odd	20		750.2.g.g.601.3		16
75.14	odd	10		450.2.l.c.379.1		16
75.41	odd	10		450.2.l.c.19.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.b.19.4	✓	16	25.16	even	5
150.2.h.b.79.4	yes	16	25.14	even	10
450.2.l.c.19.1		16	75.41	odd	10
450.2.l.c.379.1		16	75.14	odd	10
750.2.g.f.151.2		16	25.12	odd	20
750.2.g.f.601.2		16	25.2	odd	20
750.2.g.g.151.3		16	25.13	odd	20
750.2.g.g.601.3		16	25.23	odd	20
750.2.h.d.349.1		16	25.9	even	10
750.2.h.d.649.2		16	25.11	even	5
3750.2.a.u.1.5		8	5.3	odd	4
3750.2.a.v.1.4		8	5.2	odd	4
3750.2.c.k.1249.5		16	1.1	even	1	trivial
3750.2.c.k.1249.12		16	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.329315i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.329315i q^{7} +1.00000i q^{8} -1.00000 q^{9} -5.03339 q^{11} -1.00000i q^{12} -0.482152i q^{13} +0.329315 q^{14} +1.00000 q^{16} -6.78610i q^{17} +1.00000i q^{18} -5.44012 q^{19} -0.329315 q^{21} +5.03339i q^{22} +6.50055i q^{23} -1.00000 q^{24} -0.482152 q^{26} -1.00000i q^{27} -0.329315i q^{28} +6.02216 q^{29} +1.31869 q^{31} -1.00000i q^{32} -5.03339i q^{33} -6.78610 q^{34} +1.00000 q^{36} -0.780139i q^{37} +5.44012i q^{38} +0.482152 q^{39} +12.5205 q^{41} +0.329315i q^{42} -2.47582i q^{43} +5.03339 q^{44} +6.50055 q^{46} +4.38040i q^{47} +1.00000i q^{48} +6.89155 q^{49} +6.78610 q^{51} +0.482152i q^{52} -1.68591i q^{53} -1.00000 q^{54} -0.329315 q^{56} -5.44012i q^{57} -6.02216i q^{58} -1.01583 q^{59} +4.18808 q^{61} -1.31869i q^{62} -0.329315i q^{63} -1.00000 q^{64} -5.03339 q^{66} -3.14433i q^{67} +6.78610i q^{68} -6.50055 q^{69} +5.71189 q^{71} -1.00000i q^{72} +2.94269i q^{73} -0.780139 q^{74} +5.44012 q^{76} -1.65757i q^{77} -0.482152i q^{78} -8.48510 q^{79} +1.00000 q^{81} -12.5205i q^{82} +17.1955i q^{83} +0.329315 q^{84} -2.47582 q^{86} +6.02216i q^{87} -5.03339i q^{88} -3.45233 q^{89} +0.158780 q^{91} -6.50055i q^{92} +1.31869i q^{93} +4.38040 q^{94} +1.00000 q^{96} +9.51004i q^{97} -6.89155i q^{98} +5.03339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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