LMFDB - Embedded newform 3456.2.i.k.2305.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1153,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("3456.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.i (of order $3$, degree $2$, 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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2305.6
Root			$-1.15879 + 1.28733i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.k.1153.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.74260 - 3.01828i) q^{5} +(-1.34988 - 2.33807i) q^{7} +O(q^{10})$	$q+(1.74260 - 3.01828i) q^{5} +(-1.34988 - 2.33807i) q^{7} +(2.84274 + 4.92377i) q^{11} +(1.76055 - 3.04937i) q^{13} -7.65970 q^{17} -2.02060 q^{19} +(0.0370909 - 0.0642434i) q^{23} +(-3.57334 - 6.18921i) q^{25} +(-2.46032 - 4.26140i) q^{29} +(3.72257 - 6.44769i) q^{31} -9.40925 q^{35} -5.00631 q^{37} +(-0.482053 + 0.834941i) q^{41} +(-0.255495 - 0.442530i) q^{43} +(2.83509 + 4.91053i) q^{47} +(-0.144369 + 0.250055i) q^{49} -10.4058 q^{53} +19.8151 q^{55} +(4.47636 - 7.75329i) q^{59} +(-1.46032 - 2.52935i) q^{61} +(-6.13589 - 10.6277i) q^{65} +(1.56829 - 2.71635i) q^{67} -8.19647 q^{71} +5.21796 q^{73} +(7.67474 - 13.2930i) q^{77} +(-0.716260 - 1.24060i) q^{79} +(-1.74052 - 3.01467i) q^{83} +(-13.3478 + 23.1191i) q^{85} +12.4058 q^{89} -9.50616 q^{91} +(-3.52110 + 6.09873i) q^{95} +(-3.50265 - 6.06677i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{5} - 6 q^{7}+O(q^{10}) $ 12 * q + 2 * q^5 - 6 * q^7	$ 12 q + 2 q^{5} - 6 q^{7} + 4 q^{11} + 10 q^{13} - 4 q^{17} - 4 q^{19} + 8 q^{23} - 14 q^{25} + 2 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{41} + 2 q^{43} - 14 q^{47} - 18 q^{49} - 24 q^{53} + 16 q^{55} + 6 q^{59} + 14 q^{61} + 8 q^{65} - 4 q^{67} - 28 q^{71} + 60 q^{73} - 2 q^{77} - 16 q^{79} + 24 q^{83} + 16 q^{85} + 48 q^{89} + 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) $ 12 * q + 2 * q^5 - 6 * q^7 + 4 * q^11 + 10 * q^13 - 4 * q^17 - 4 * q^19 + 8 * q^23 - 14 * q^25 + 2 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^41 + 2 * q^43 - 14 * q^47 - 18 * q^49 - 24 * q^53 + 16 * q^55 + 6 * q^59 + 14 * q^61 + 8 * q^65 - 4 * q^67 - 28 * q^71 + 60 * q^73 - 2 * q^77 - 16 * q^79 + 24 * q^83 + 16 * q^85 + 48 * q^89 + 52 * q^91 - 20 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.74260	−	3.01828i	0.779317	−	1.34982i	−0.153019	−	0.988223i	$-0.548900\pi$
$5$	1.74260	−	3.01828i	0.779317	−	1.34982i	0.932336	−	0.361593i	$-0.117767\pi$
$6$		0			0
$7$	−1.34988	−	2.33807i	−0.510208	−	0.883706i	−0.999930	−	0.0118274i	$-0.996235\pi$
$7$	−1.34988	−	2.33807i	−0.510208	−	0.883706i	0.489722	−	0.871879i	$-0.337098\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.84274	+	4.92377i	0.857119	+	1.48457i	0.874665	+	0.484727i	$0.161081\pi$
$11$	2.84274	+	4.92377i	0.857119	+	1.48457i	−0.0175468	+	0.999846i	$0.505586\pi$
$12$		0			0
$13$	1.76055	−	3.04937i	0.488289	−	0.845742i	−0.511620	−	0.859212i	$-0.670954\pi$
$13$	1.76055	−	3.04937i	0.488289	−	0.845742i	0.999909	+	0.0134701i	$0.00428779\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.65970			−1.85775			−0.928875	−	0.370392i	$-0.879223\pi$
$17$	−7.65970			−1.85775			−0.928875	+	0.370392i	$0.879223\pi$
$18$		0			0
$19$	−2.02060			−0.463557			−0.231778	−	0.972769i	$-0.574454\pi$
$19$	−2.02060			−0.463557			−0.231778	+	0.972769i	$0.574454\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.0370909	−	0.0642434i	0.00773399	−	0.0133957i	−0.862133	−	0.506683i	$-0.830872\pi$
$23$	0.0370909	−	0.0642434i	0.00773399	−	0.0133957i	0.869866	+	0.493287i	$0.164205\pi$
$24$		0			0
$25$	−3.57334	−	6.18921i	−0.714669	−	1.23784i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.46032	−	4.26140i	−0.456870	−	0.791321i	0.541924	−	0.840428i	$-0.317696\pi$
$29$	−2.46032	−	4.26140i	−0.456870	−	0.791321i	−0.998794	+	0.0491062i	$0.984363\pi$
$30$		0			0
$31$	3.72257	−	6.44769i	0.668594	−	1.15804i	−0.309704	−	0.950833i	$-0.600230\pi$
$31$	3.72257	−	6.44769i	0.668594	−	1.15804i	0.978297	−	0.207205i	$-0.0664368\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.40925			−1.59045
$36$		0			0
$37$	−5.00631			−0.823033			−0.411516	−	0.911402i	$-0.635001\pi$
$37$	−5.00631			−0.823033			−0.411516	+	0.911402i	$0.635001\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.482053	+	0.834941i	−0.0752841	+	0.130396i	−0.901210	−	0.433383i	$-0.857320\pi$
$41$	−0.482053	+	0.834941i	−0.0752841	+	0.130396i	0.825926	+	0.563779i	$0.190653\pi$
$42$		0			0
$43$	−0.255495	−	0.442530i	−0.0389626	−	0.0674852i	0.845887	−	0.533363i	$-0.179072\pi$
$43$	−0.255495	−	0.442530i	−0.0389626	−	0.0674852i	−0.884849	+	0.465878i	$0.845739\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.83509	+	4.91053i	0.413541	+	0.716274i	0.995274	−	0.0971059i	$-0.0309586\pi$
$47$	2.83509	+	4.91053i	0.413541	+	0.716274i	−0.581733	+	0.813380i	$0.697625\pi$
$48$		0			0
$49$	−0.144369	+	0.250055i	−0.0206242	+	0.0357221i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.4058			−1.42935			−0.714676	−	0.699455i	$-0.753426\pi$
$53$	−10.4058			−1.42935			−0.714676	+	0.699455i	$0.753426\pi$
$54$		0			0
$55$	19.8151			2.67187
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.47636	−	7.75329i	0.582773	−	1.00939i	−0.412376	−	0.911014i	$-0.635301\pi$
$59$	4.47636	−	7.75329i	0.582773	−	1.00939i	0.995149	−	0.0983788i	$-0.0313657\pi$
$60$		0			0
$61$	−1.46032	−	2.52935i	−0.186975	−	0.323849i	0.757266	−	0.653107i	$-0.226535\pi$
$61$	−1.46032	−	2.52935i	−0.186975	−	0.323849i	−0.944240	+	0.329258i	$0.893202\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.13589	−	10.6277i	−0.761064	−	1.31820i
$66$		0			0
$67$	1.56829	−	2.71635i	0.191597	−	0.331855i	−0.754183	−	0.656665i	$-0.771967\pi$
$67$	1.56829	−	2.71635i	0.191597	−	0.331855i	0.945780	+	0.324809i	$0.105300\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.19647			−0.972742			−0.486371	−	0.873752i	$-0.661680\pi$
$71$	−8.19647			−0.972742			−0.486371	+	0.873752i	$0.661680\pi$
$72$		0			0
$73$	5.21796			0.610716			0.305358	−	0.952238i	$-0.401224\pi$
$73$	5.21796			0.610716			0.305358	+	0.952238i	$0.401224\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.67474	−	13.2930i	0.874617	−	1.51488i
$78$		0			0
$79$	−0.716260	−	1.24060i	−0.0805855	−	0.139578i	0.822916	−	0.568163i	$-0.192346\pi$
$79$	−0.716260	−	1.24060i	−0.0805855	−	0.139578i	−0.903502	+	0.428585i	$0.859012\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.74052	−	3.01467i	−0.191047	−	0.330903i	0.754551	−	0.656242i	$-0.227855\pi$
$83$	−1.74052	−	3.01467i	−0.191047	−	0.330903i	−0.945597	+	0.325339i	$0.894522\pi$
$84$		0			0
$85$	−13.3478	+	23.1191i	−1.44778	+	2.50762i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.4058			1.31502			0.657509	−	0.753447i	$-0.271610\pi$
$89$	12.4058			1.31502			0.657509	+	0.753447i	$0.271610\pi$
$90$		0			0
$91$	−9.50616			−0.996516
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.52110	+	6.09873i	−0.361258	+	0.625717i
$96$		0			0
$97$	−3.50265	−	6.06677i	−0.355640	−	0.615987i	0.631587	−	0.775305i	$-0.282404\pi$
$97$	−3.50265	−	6.06677i	−0.355640	−	0.615987i	−0.987227	+	0.159318i	$0.949071\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.44237	+	7.69441i	0.442032	+	0.765623i	0.997840	−	0.0656882i	$-0.0209243\pi$
$101$	4.44237	+	7.69441i	0.442032	+	0.765623i	−0.555808	+	0.831311i	$0.687591\pi$
$102$		0			0
$103$	−2.65012	+	4.59014i	−0.261124	+	0.452280i	−0.966541	−	0.256513i	$-0.917426\pi$
$103$	−2.65012	+	4.59014i	−0.261124	+	0.452280i	0.705417	+	0.708792i	$0.250760\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.53993			−0.728912			−0.364456	−	0.931221i	$-0.618745\pi$
$107$	−7.53993			−0.728912			−0.364456	+	0.931221i	$0.618745\pi$
$108$		0			0
$109$	−13.8840			−1.32984			−0.664922	−	0.746913i	$-0.731535\pi$
$109$	−13.8840			−1.32984			−0.664922	+	0.746913i	$0.731535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.36549	+	2.36510i	−0.128454	+	0.222489i	−0.923078	−	0.384613i	$-0.874335\pi$
$113$	−1.36549	+	2.36510i	−0.128454	+	0.222489i	0.794624	+	0.607102i	$0.207668\pi$
$114$		0			0
$115$	−0.129270	−	0.223902i	−0.0120545	−	0.0208789i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.3397	+	17.9089i	0.947839	+	1.64171i
$120$		0			0
$121$	−10.6624	+	18.4677i	−0.969305	+	1.67889i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.48166			−0.669180
$126$		0			0
$127$	−5.89964			−0.523509			−0.261754	−	0.965135i	$-0.584301\pi$
$127$	−5.89964			−0.523509			−0.261754	+	0.965135i	$0.584301\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.566027	−	0.980388i	0.0494540	−	0.0856569i	−0.840239	−	0.542217i	$-0.817585\pi$
$131$	0.566027	−	0.980388i	0.0494540	−	0.0856569i	0.889693	+	0.456560i	$0.150919\pi$
$132$		0			0
$133$	2.72757	+	4.72429i	0.236510	+	0.409648i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.18182	−	5.51107i	−0.271841	−	0.470843i	0.697492	−	0.716592i	$-0.254299\pi$
$137$	−3.18182	−	5.51107i	−0.271841	−	0.470843i	−0.969333	+	0.245750i	$0.920966\pi$
$138$		0			0
$139$	−11.2001	+	19.3992i	−0.949983	+	1.64542i	−0.204531	+	0.978860i	$0.565567\pi$
$139$	−11.2001	+	19.3992i	−0.949983	+	1.64542i	−0.745453	+	0.666559i	$0.767767\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.0192			1.67409
$144$		0			0
$145$	−17.1495			−1.42418
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.939215	−	1.62677i	0.0769435	−	0.133270i	−0.824986	−	0.565153i	$-0.808817\pi$
$149$	0.939215	−	1.62677i	0.0769435	−	0.133270i	0.901930	+	0.431883i	$0.142151\pi$
$150$		0			0
$151$	0.183779	+	0.318315i	0.0149557	+	0.0259041i	0.873406	−	0.486992i	$-0.161906\pi$
$151$	0.183779	+	0.318315i	0.0149557	+	0.0259041i	−0.858451	+	0.512896i	$0.828573\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.9739	−	22.4715i	−1.04209	−	1.80496i
$156$		0			0
$157$	7.15363	−	12.3905i	0.570922	−	0.988866i	−0.425550	−	0.904935i	$-0.639919\pi$
$157$	7.15363	−	12.3905i	0.570922	−	0.988866i	0.996472	−	0.0839309i	$-0.0267475\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.200274			−0.0157838
$162$		0			0
$163$	3.02958			0.237295			0.118648	−	0.992936i	$-0.462144\pi$
$163$	3.02958			0.237295			0.118648	+	0.992936i	$0.462144\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.629519	−	1.09036i	0.0487137	−	0.0843745i	−0.840640	−	0.541594i	$-0.817821\pi$
$167$	0.629519	−	1.09036i	0.0487137	−	0.0843745i	0.889354	+	0.457219i	$0.151154\pi$
$168$		0			0
$169$	0.300915	+	0.521200i	0.0231473	+	0.0400923i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.14845	−	12.3815i	−0.543487	−	0.941347i	−0.998700	−	0.0509644i	$-0.983771\pi$
$173$	−7.14845	−	12.3815i	−0.543487	−	0.941347i	0.455214	−	0.890382i	$-0.349563\pi$
$174$		0			0
$175$	−9.64719	+	16.7094i	−0.729259	+	1.26311i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.7991			−0.956647			−0.478324	−	0.878184i	$-0.658755\pi$
$179$	−12.7991			−0.956647			−0.478324	+	0.878184i	$0.658755\pi$
$180$		0			0
$181$	−10.4986			−0.780355			−0.390177	−	0.920740i	$-0.627586\pi$
$181$	−10.4986			−0.780355			−0.390177	+	0.920740i	$0.627586\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.72403	+	15.1105i	−0.641403	+	1.11094i
$186$		0			0
$187$	−21.7746	−	37.7146i	−1.59231	−	2.75797i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.09526	+	7.09320i	0.296323	+	0.513246i	0.975292	−	0.220921i	$-0.0709062\pi$
$191$	4.09526	+	7.09320i	0.296323	+	0.513246i	−0.678969	+	0.734167i	$0.737573\pi$
$192$		0			0
$193$	−1.46146	+	2.53132i	−0.105198	+	0.182208i	−0.913819	−	0.406122i	$-0.866881\pi$
$193$	−1.46146	+	2.53132i	−0.105198	+	0.182208i	0.808621	+	0.588330i	$0.200214\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.44174			0.316461			0.158230	−	0.987402i	$-0.449421\pi$
$197$	4.44174			0.316461			0.158230	+	0.987402i	$0.449421\pi$
$198$		0			0
$199$	14.6898			1.04133			0.520665	−	0.853761i	$-0.325684\pi$
$199$	14.6898			1.04133			0.520665	+	0.853761i	$0.325684\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.64228	+	11.5048i	−0.466197	+	0.807477i
$204$		0			0
$205$	1.68006	+	2.90994i	0.117340	+	0.203239i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.74404	−	9.94896i	−0.397323	−	0.688184i
$210$		0			0
$211$	−2.04700	+	3.54551i	−0.140921	+	0.244083i	−0.927844	−	0.372969i	$-0.878340\pi$
$211$	−2.04700	+	3.54551i	−0.140921	+	0.244083i	0.786923	+	0.617052i	$0.211673\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.78091			−0.121457
$216$		0			0
$217$	−20.1002			−1.36449
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−13.4853	+	23.3572i	−0.907120	+	1.57118i
$222$		0			0
$223$	6.24612	+	10.8186i	0.418271	+	0.724467i	0.995766	−	0.0919276i	$-0.0293028\pi$
$223$	6.24612	+	10.8186i	0.418271	+	0.724467i	−0.577494	+	0.816395i	$0.695970\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.52721	−	7.84136i	−0.300482	−	0.520449i	0.675764	−	0.737118i	$-0.263814\pi$
$227$	−4.52721	−	7.84136i	−0.300482	−	0.520449i	−0.976245	+	0.216669i	$0.930481\pi$
$228$		0			0
$229$	13.1073	−	22.7024i	0.866152	−	1.50022i	0.000252919	−	1.00000i	$-0.499919\pi$
$229$	13.1073	−	22.7024i	0.866152	−	1.50022i	0.865899	−	0.500219i	$-0.166747\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.41840			−0.0929224			−0.0464612	−	0.998920i	$-0.514794\pi$
$233$	−1.41840			−0.0929224			−0.0464612	+	0.998920i	$0.514794\pi$
$234$		0			0
$235$	19.7618			1.28912
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.18200	−	14.1716i	0.529250	−	0.916688i	−0.470168	−	0.882577i	$-0.655807\pi$
$239$	8.18200	−	14.1716i	0.529250	−	0.916688i	0.999418	−	0.0341107i	$-0.0108599\pi$
$240$		0			0
$241$	−5.17500	−	8.96336i	−0.333351	−	0.577381i	0.649816	−	0.760092i	$-0.274846\pi$
$241$	−5.17500	−	8.96336i	−0.333351	−	0.577381i	−0.983167	+	0.182711i	$0.941513\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.503157	+	0.871493i	0.0321455	+	0.0556776i
$246$		0			0
$247$	−3.55737	+	6.16154i	−0.226350	+	0.392049i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	26.9624			1.70185			0.850927	−	0.525283i	$-0.176041\pi$
$251$	26.9624			1.70185			0.850927	+	0.525283i	$0.176041\pi$
$252$		0			0
$253$	0.421759			0.0265158
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.8380	−	25.7001i	0.925568	−	1.60313i	0.134922	−	0.990856i	$-0.456922\pi$
$257$	14.8380	−	25.7001i	0.925568	−	1.60313i	0.790646	−	0.612274i	$-0.209745\pi$
$258$		0			0
$259$	6.75794	+	11.7051i	0.419918	+	0.727319i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.46961	+	2.54544i	0.0906200	+	0.156959i	0.907772	−	0.419464i	$-0.137782\pi$
$263$	1.46961	+	2.54544i	0.0906200	+	0.156959i	−0.817152	+	0.576422i	$0.804448\pi$
$264$		0			0
$265$	−18.1333	+	31.4078i	−1.11392	+	1.92936i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.4251			−0.818541			−0.409270	−	0.912413i	$-0.634217\pi$
$269$	−13.4251			−0.818541			−0.409270	+	0.912413i	$0.634217\pi$
$270$		0			0
$271$	31.3320			1.90329			0.951643	−	0.307207i	$-0.0993944\pi$
$271$	31.3320			1.90329			0.951643	+	0.307207i	$0.0993944\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	20.3162	−	35.1887i	1.22511	−	2.12196i
$276$		0			0
$277$	4.78089	+	8.28075i	0.287256	+	0.497542i	0.973154	−	0.230156i	$-0.0739236\pi$
$277$	4.78089	+	8.28075i	0.287256	+	0.497542i	−0.685898	+	0.727698i	$0.740590\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.39152	−	5.87429i	−0.202321	−	0.350431i	0.746955	−	0.664875i	$-0.231515\pi$
$281$	−3.39152	−	5.87429i	−0.202321	−	0.350431i	−0.949276	+	0.314444i	$0.898182\pi$
$282$		0			0
$283$	−9.61895	+	16.6605i	−0.571787	+	0.990364i	0.424595	+	0.905383i	$0.360416\pi$
$283$	−9.61895	+	16.6605i	−0.571787	+	0.990364i	−0.996383	+	0.0849812i	$0.972917\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.60286			0.153642
$288$		0			0
$289$	41.6710			2.45124
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.63884	+	9.76676i	−0.329425	+	0.570580i	−0.982398	−	0.186801i	$-0.940188\pi$
$293$	−5.63884	+	9.76676i	−0.329425	+	0.570580i	0.652973	+	0.757381i	$0.273521\pi$
$294$		0			0
$295$	−15.6011	−	27.0218i	−0.908329	−	1.57327i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.130601	−	0.226207i	−0.00755285	−	0.0130819i
$300$		0			0
$301$	−0.689776	+	1.19473i	−0.0397580	+	0.0688629i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.1790			−0.582850
$306$		0			0
$307$	−22.2819			−1.27169			−0.635846	−	0.771816i	$-0.719349\pi$
$307$	−22.2819			−1.27169			−0.635846	+	0.771816i	$0.719349\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0604	−	20.8893i	0.683884	−	1.18452i	−0.289902	−	0.957056i	$-0.593623\pi$
$311$	12.0604	−	20.8893i	0.683884	−	1.18452i	0.973786	−	0.227466i	$-0.0730441\pi$
$312$		0			0
$313$	14.5297	+	25.1661i	0.821265	+	1.42247i	0.904741	+	0.425962i	$0.140064\pi$
$313$	14.5297	+	25.1661i	0.821265	+	1.42247i	−0.0834762	+	0.996510i	$0.526602\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.2862	−	28.2085i	−0.914724	−	1.58435i	−0.807305	−	0.590134i	$-0.799075\pi$
$317$	−16.2862	−	28.2085i	−0.914724	−	1.58435i	−0.107418	−	0.994214i	$-0.534258\pi$
$318$		0			0
$319$	13.9881	−	24.2281i	0.783183	−	1.35651i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.4772			0.861173
$324$		0			0
$325$	−25.1642			−1.39586
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.65409	−	13.2573i	0.421984	−	0.730897i
$330$		0			0
$331$	8.05963	+	13.9597i	0.442997	+	0.767293i	0.997910	−	0.0646148i	$-0.0205819\pi$
$331$	8.05963	+	13.9597i	0.442997	+	0.767293i	−0.554913	+	0.831908i	$0.687249\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.46581	−	9.46706i	−0.298629	−	0.517241i
$336$		0			0
$337$	12.6119	−	21.8445i	0.687015	−	1.18995i	−0.285783	−	0.958294i	$-0.592254\pi$
$337$	12.6119	−	21.8445i	0.687015	−	1.18995i	0.972799	−	0.231651i	$-0.0744128\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	42.3292			2.29226
$342$		0			0
$343$	−18.1188			−0.978325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.16774	+	3.75464i	−0.116371	+	0.201560i	−0.918327	−	0.395823i	$-0.870459\pi$
$347$	−2.16774	+	3.75464i	−0.116371	+	0.201560i	0.801956	+	0.597383i	$0.203793\pi$
$348$		0			0
$349$	−9.77744	−	16.9350i	−0.523374	−	0.906511i	−0.999630	−	0.0272042i	$-0.991340\pi$
$349$	−9.77744	−	16.9350i	−0.523374	−	0.906511i	0.476255	−	0.879307i	$-0.341994\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.13835	−	15.8281i	−0.486385	−	0.842444i	0.513492	−	0.858094i	$-0.328352\pi$
$353$	−9.13835	−	15.8281i	−0.486385	−	0.842444i	−0.999878	+	0.0156502i	$0.995018\pi$
$354$		0			0
$355$	−14.2832	+	24.7392i	−0.758074	+	1.31302i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.41976			−0.180488			−0.0902440	−	0.995920i	$-0.528765\pi$
$359$	−3.41976			−0.180488			−0.0902440	+	0.995920i	$0.528765\pi$
$360$		0			0
$361$	−14.9172			−0.785115
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.09285	−	15.7493i	0.475941	−	0.824355i
$366$		0			0
$367$	−5.46341	−	9.46291i	−0.285188	−	0.493960i	0.687467	−	0.726216i	$-0.258723\pi$
$367$	−5.46341	−	9.46291i	−0.285188	−	0.493960i	−0.972655	+	0.232256i	$0.925389\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.0467	+	24.3296i	0.729267	+	1.26313i
$372$		0			0
$373$	10.4603	−	18.1178i	0.541615	−	0.938104i	−0.457197	−	0.889366i	$-0.651147\pi$
$373$	10.4603	−	18.1178i	0.541615	−	0.938104i	0.998812	−	0.0487387i	$-0.0155202\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−17.3261			−0.892338
$378$		0			0
$379$	18.0219			0.925725			0.462862	−	0.886430i	$-0.346822\pi$
$379$	18.0219			0.925725			0.462862	+	0.886430i	$0.346822\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.4732	+	21.6042i	−0.637350	+	1.10392i	0.348662	+	0.937249i	$0.386636\pi$
$383$	−12.4732	+	21.6042i	−0.637350	+	1.10392i	−0.986012	+	0.166674i	$0.946697\pi$
$384$		0			0
$385$	−26.7481	−	46.3290i	−1.36321	−	2.36115i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.96347	+	12.0611i	0.353062	+	0.611522i	0.986784	−	0.162039i	$-0.0518071\pi$
$389$	6.96347	+	12.0611i	0.353062	+	0.611522i	−0.633722	+	0.773561i	$0.718474\pi$
$390$		0			0
$391$	−0.284105	+	0.492085i	−0.0143678	+	0.0248858i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.99263			−0.251206
$396$		0			0
$397$	−6.54840			−0.328655			−0.164327	−	0.986406i	$-0.552545\pi$
$397$	−6.54840			−0.328655			−0.164327	+	0.986406i	$0.552545\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.1738	−	21.0857i	0.607933	−	1.05297i	−0.383648	−	0.923480i	$-0.625332\pi$
$401$	12.1738	−	21.0857i	0.607933	−	1.05297i	0.991581	−	0.129491i	$-0.0413344\pi$
$402$		0			0
$403$	−13.1076	−	22.7030i	−0.652934	−	1.13092i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.2317	−	24.6499i	−0.705437	−	1.22185i
$408$		0			0
$409$	8.08792	−	14.0087i	0.399922	−	0.692685i	−0.593794	−	0.804617i	$-0.702371\pi$
$409$	8.08792	−	14.0087i	0.399922	−	0.692685i	0.993716	+	0.111932i	$0.0357040\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−24.1703			−1.18934
$414$		0			0
$415$	−12.1322			−0.595544
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.2436	+	29.8668i	−0.842405	+	1.45909i	0.0454518	+	0.998967i	$0.485527\pi$
$419$	−17.2436	+	29.8668i	−0.842405	+	1.45909i	−0.887856	+	0.460121i	$0.847806\pi$
$420$		0			0
$421$	15.6909	+	27.1774i	0.764728	+	1.32455i	0.940390	+	0.340097i	$0.110460\pi$
$421$	15.6909	+	27.1774i	0.764728	+	1.32455i	−0.175662	+	0.984451i	$0.556207\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	27.3708	+	47.4075i	1.32768	+	2.29960i
$426$		0			0
$427$	−3.94252	+	6.82864i	−0.190792	+	0.330461i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.1491			−0.585200			−0.292600	−	0.956235i	$-0.594520\pi$
$431$	−12.1491			−0.585200			−0.292600	+	0.956235i	$0.594520\pi$
$432$		0			0
$433$	2.40374			0.115517			0.0577583	−	0.998331i	$-0.481605\pi$
$433$	2.40374			0.115517			0.0577583	+	0.998331i	$0.481605\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.0749458	+	0.129810i	−0.00358514	+	0.00620965i
$438$		0			0
$439$	18.6941	+	32.3792i	0.892222	+	1.54537i	0.837206	+	0.546888i	$0.184188\pi$
$439$	18.6941	+	32.3792i	0.892222	+	1.54537i	0.0550162	+	0.998485i	$0.482479\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.22340	−	7.31515i	−0.200660	−	0.347553i	0.748081	−	0.663607i	$-0.230975\pi$
$443$	−4.22340	−	7.31515i	−0.200660	−	0.347553i	−0.948741	+	0.316054i	$0.897642\pi$
$444$		0			0
$445$	21.6185	−	37.4443i	1.02481	−	1.77503i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.6531			−0.880295			−0.440147	−	0.897926i	$-0.645074\pi$
$449$	−18.6531			−0.880295			−0.440147	+	0.897926i	$0.645074\pi$
$450$		0			0
$451$	−5.48141			−0.258109
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−16.5655	+	28.6922i	−0.776601	+	1.34511i
$456$		0			0
$457$	−8.91748	−	15.4455i	−0.417142	−	0.722511i	0.578509	−	0.815676i	$-0.303635\pi$
$457$	−8.91748	−	15.4455i	−0.417142	−	0.722511i	−0.995651	+	0.0931650i	$0.970302\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.0563	−	17.4181i	−0.468370	−	0.811240i	0.530977	−	0.847386i	$-0.321825\pi$
$461$	−10.0563	−	17.4181i	−0.468370	−	0.811240i	−0.999347	+	0.0361463i	$0.988492\pi$
$462$		0			0
$463$	−4.10747	+	7.11435i	−0.190890	+	0.330632i	−0.945546	−	0.325490i	$-0.894471\pi$
$463$	−4.10747	+	7.11435i	−0.190890	+	0.330632i	0.754655	+	0.656122i	$0.227804\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.3274			−0.894364			−0.447182	−	0.894443i	$-0.647572\pi$
$467$	−19.3274			−0.894364			−0.447182	+	0.894443i	$0.647572\pi$
$468$		0			0
$469$	−8.46802			−0.391017
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.45261	−	2.51600i	0.0667911	−	0.115686i
$474$		0			0
$475$	7.22029	+	12.5059i	0.331290	+	0.573811i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.1597	−	36.6498i	−0.966814	−	1.67457i	−0.704661	−	0.709544i	$-0.748901\pi$
$479$	−21.1597	−	36.6498i	−0.966814	−	1.67457i	−0.262152	−	0.965027i	$-0.584432\pi$
$480$		0			0
$481$	−8.81387	+	15.2661i	−0.401878	+	0.696073i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.4149			−1.10863
$486$		0			0
$487$	−1.36060			−0.0616547			−0.0308273	−	0.999525i	$-0.509814\pi$
$487$	−1.36060			−0.0616547			−0.0308273	+	0.999525i	$0.509814\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.6328	+	27.0768i	−0.705497	+	1.22196i	0.261015	+	0.965335i	$0.415943\pi$
$491$	−15.6328	+	27.0768i	−0.705497	+	1.22196i	−0.966512	+	0.256622i	$0.917390\pi$
$492$		0			0
$493$	18.8453	+	32.6410i	0.848750	+	1.47008i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.0643	+	19.1639i	0.496301	+	0.859618i
$498$		0			0
$499$	7.37981	−	12.7822i	0.330366	−	0.572210i	−0.652218	−	0.758032i	$-0.726161\pi$
$499$	7.37981	−	12.7822i	0.330366	−	0.572210i	0.982584	+	0.185821i	$0.0594946\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	44.6336			1.99011			0.995056	−	0.0993124i	$-0.0316643\pi$
$503$	44.6336			1.99011			0.995056	+	0.0993124i	$0.0316643\pi$
$504$		0			0
$505$	30.9652			1.37793
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.854549	−	1.48012i	0.0378772	−	0.0656053i	−0.846465	−	0.532444i	$-0.821274\pi$
$509$	0.854549	−	1.48012i	0.0378772	−	0.0656053i	0.884343	+	0.466839i	$0.154607\pi$
$510$		0			0
$511$	−7.04364	−	12.1999i	−0.311592	−	0.539694i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.23621	+	15.9976i	0.406996	+	0.704938i
$516$		0			0
$517$	−16.1189	+	27.9187i	−0.708907	+	1.22786i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.6018			0.464475			0.232237	−	0.972659i	$-0.425395\pi$
$521$	10.6018			0.464475			0.232237	+	0.972659i	$0.425395\pi$
$522$		0			0
$523$	27.1565			1.18747			0.593736	−	0.804660i	$-0.297652\pi$
$523$	27.1565			1.18747			0.593736	+	0.804660i	$0.297652\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−28.5138	+	49.3874i	−1.24208	+	2.15135i
$528$		0			0
$529$	11.4972	+	19.9138i	0.499880	+	0.865818i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.69736	+	2.93991i	0.0735208	+	0.127342i
$534$		0			0
$535$	−13.1391	+	22.7576i	−0.568053	+	0.983897i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.64162			−0.0707094
$540$		0			0
$541$	35.1225			1.51003			0.755017	−	0.655705i	$-0.227629\pi$
$541$	35.1225			1.51003			0.755017	+	0.655705i	$0.227629\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−24.1943	+	41.9057i	−1.03637	+	1.79504i
$546$		0			0
$547$	−13.0029	−	22.5218i	−0.555966	−	0.962961i	−0.997828	−	0.0658781i	$-0.979015\pi$
$547$	−13.0029	−	22.5218i	−0.555966	−	0.962961i	0.441862	−	0.897083i	$-0.354318\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.97131	+	8.61057i	0.211785	+	0.366822i
$552$		0			0
$553$	−1.93373	+	3.34932i	−0.0822307	+	0.142428i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	28.8670			1.22314			0.611568	−	0.791192i	$-0.290539\pi$
$557$	28.8670			1.22314			0.611568	+	0.791192i	$0.290539\pi$
$558$		0			0
$559$	−1.79925			−0.0761000
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.876491	−	1.51813i	0.0369397	−	0.0639815i	−0.846964	−	0.531649i	$-0.821572\pi$
$563$	0.876491	−	1.51813i	0.0369397	−	0.0639815i	0.883904	+	0.467668i	$0.154906\pi$
$564$		0			0
$565$	4.75901	+	8.24285i	0.200213	+	0.346779i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.3974	+	23.2049i	0.561647	+	0.972801i	0.997353	+	0.0727120i	$0.0231654\pi$
$569$	13.3974	+	23.2049i	0.561647	+	0.972801i	−0.435706	+	0.900089i	$0.643501\pi$
$570$		0			0
$571$	12.6759	−	21.9553i	0.530471	−	0.918803i	−0.468897	−	0.883253i	$-0.655348\pi$
$571$	12.6759	−	21.9553i	0.530471	−	0.918803i	0.999368	−	0.0355497i	$-0.0113182\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.530154			−0.0221090
$576$		0			0
$577$	33.8507			1.40922			0.704612	−	0.709593i	$-0.251121\pi$
$577$	33.8507			1.40922			0.704612	+	0.709593i	$0.251121\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.69900	+	8.13890i	−0.194947	+	0.337659i
$582$		0			0
$583$	−29.5811	−	51.2360i	−1.22513	−	2.12198i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.2279	−	19.4474i	−0.463427	−	0.802679i	0.535702	−	0.844407i	$-0.320047\pi$
$587$	−11.2279	−	19.4474i	−0.463427	−	0.802679i	−0.999129	+	0.0417284i	$0.986714\pi$
$588$		0			0
$589$	−7.52182	+	13.0282i	−0.309931	+	0.536817i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.35530			0.384176			0.192088	−	0.981378i	$-0.438474\pi$
$593$	9.35530			0.384176			0.192088	+	0.981378i	$0.438474\pi$
$594$		0			0
$595$	72.0721			2.95467
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.2259	−	21.1759i	0.499538	−	0.865226i	−0.500462	−	0.865759i	$-0.666836\pi$
$599$	12.2259	−	21.1759i	0.499538	−	0.865226i	1.00000		0.000533153i	$0.000169708\pi$
$600$		0			0
$601$	−6.64643	−	11.5120i	−0.271114	−	0.469583i	0.698034	−	0.716065i	$-0.254059\pi$
$601$	−6.64643	−	11.5120i	−0.271114	−	0.469583i	−0.969147	+	0.246482i	$0.920725\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	37.1605	+	64.3639i	1.51079	+	2.61677i
$606$		0			0
$607$	−18.1403	+	31.4198i	−0.736290	+	1.27529i	0.217864	+	0.975979i	$0.430091\pi$
$607$	−18.1403	+	31.4198i	−0.736290	+	1.27529i	−0.954155	+	0.299313i	$0.903242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	19.9653			0.807710
$612$		0			0
$613$	−6.73390			−0.271980			−0.135990	−	0.990710i	$-0.543421\pi$
$613$	−6.73390			−0.271980			−0.135990	+	0.990710i	$0.543421\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.553446	+	0.958597i	−0.0222809	+	0.0385917i	−0.876951	−	0.480580i	$-0.840426\pi$
$617$	−0.553446	+	0.958597i	−0.0222809	+	0.0385917i	0.854670	+	0.519172i	$0.173759\pi$
$618$		0			0
$619$	−7.60418	−	13.1708i	−0.305638	−	0.529380i	0.671765	−	0.740764i	$-0.265536\pi$
$619$	−7.60418	−	13.1708i	−0.305638	−	0.529380i	−0.977403	+	0.211384i	$0.932203\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.7464	−	29.0057i	−0.670932	−	1.16209i
$624$		0			0
$625$	4.82915	−	8.36432i	0.193166	−	0.334573i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	38.3469			1.52899
$630$		0			0
$631$	29.2163			1.16308			0.581541	−	0.813517i	$-0.302450\pi$
$631$	29.2163			1.16308			0.581541	+	0.813517i	$0.302450\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.2807	+	17.8068i	−0.407979	+	0.706641i
$636$		0			0
$637$	0.508339	+	0.880468i	0.0201411	+	0.0348854i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.4412	+	23.2809i	0.530897	+	0.919540i	0.999350	+	0.0360519i	$0.0114782\pi$
$641$	13.4412	+	23.2809i	0.530897	+	0.919540i	−0.468453	+	0.883488i	$0.655189\pi$
$642$		0			0
$643$	6.36207	−	11.0194i	0.250896	−	0.434564i	−0.712877	−	0.701289i	$-0.752608\pi$
$643$	6.36207	−	11.0194i	0.250896	−	0.434564i	0.963773	+	0.266725i	$0.0859416\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−47.3838			−1.86285			−0.931425	−	0.363933i	$-0.881434\pi$
$647$	−47.3838			−1.86285			−0.931425	+	0.363933i	$0.881434\pi$
$648$		0			0
$649$	50.9006			1.99802
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.13506	−	7.16213i	0.161817	−	0.280276i	−0.773703	−	0.633548i	$-0.781598\pi$
$653$	4.13506	−	7.16213i	0.161817	−	0.280276i	0.935521	+	0.353272i	$0.114931\pi$
$654$		0			0
$655$	−1.97272	−	3.41686i	−0.0770807	−	0.133508i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.00770	+	1.74539i	0.0392544	+	0.0679906i	0.884985	−	0.465619i	$-0.154168\pi$
$659$	1.00770	+	1.74539i	0.0392544	+	0.0679906i	−0.845731	+	0.533610i	$0.820835\pi$
$660$		0			0
$661$	−1.70432	+	2.95196i	−0.0662902	+	0.114818i	−0.897266	−	0.441491i	$-0.854450\pi$
$661$	−1.70432	+	2.95196i	−0.0662902	+	0.114818i	0.830975	+	0.556309i	$0.187783\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19.0123			0.737266
$666$		0			0
$667$	−0.365022			−0.0141337
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.30261	−	14.3805i	0.320519	−	0.555155i
$672$		0			0
$673$	7.56791	+	13.1080i	0.291722	+	0.505277i	0.974217	−	0.225613i	$-0.0724386\pi$
$673$	7.56791	+	13.1080i	0.291722	+	0.505277i	−0.682495	+	0.730890i	$0.739105\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.31110	+	16.1273i	0.357855	+	0.619822i	0.987602	−	0.156978i	$-0.0501750\pi$
$677$	9.31110	+	16.1273i	0.357855	+	0.619822i	−0.629748	+	0.776800i	$0.716842\pi$
$678$		0			0
$679$	−9.45634	+	16.3789i	−0.362901	+	0.628563i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	38.4656			1.47185			0.735923	−	0.677066i	$-0.236749\pi$
$683$	38.4656			1.47185			0.735923	+	0.677066i	$0.236749\pi$
$684$		0			0
$685$	−22.1786			−0.847401
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.3200	+	31.7312i	−0.697938	+	1.20886i
$690$		0			0
$691$	−11.2999	−	19.5720i	−0.429869	−	0.744555i	0.566992	−	0.823723i	$-0.308107\pi$
$691$	−11.2999	−	19.5720i	−0.429869	−	0.744555i	−0.996861	+	0.0791683i	$0.974774\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	39.0348	+	67.6103i	1.48068	+	2.56460i
$696$		0			0
$697$	3.69238	−	6.39540i	0.139859	−	0.242243i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	37.8883			1.43102			0.715510	−	0.698602i	$-0.246194\pi$
$701$	37.8883			1.43102			0.715510	+	0.698602i	$0.246194\pi$
$702$		0			0
$703$	10.1157			0.381523
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.9934	−	20.7731i	0.451057	−	0.781253i
$708$		0			0
$709$	7.75701	+	13.4355i	0.291321	+	0.504582i	0.974122	−	0.226022i	$-0.0725720\pi$
$709$	7.75701	+	13.4355i	0.291321	+	0.504582i	−0.682802	+	0.730604i	$0.739239\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.276147	−	0.478301i	−0.0103418	−	0.0179125i
$714$		0			0
$715$	34.8855	−	60.4235i	1.30464	−	2.25971i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.8701			1.11397			0.556983	−	0.830524i	$-0.311959\pi$
$719$	29.8701			1.11397			0.556983	+	0.830524i	$0.311959\pi$
$720$		0			0
$721$	14.3094			0.532910
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.5831	+	30.4549i	−0.653021	+	1.13107i
$726$		0			0
$727$	2.80759	+	4.86288i	0.104128	+	0.180354i	0.913381	−	0.407105i	$-0.133462\pi$
$727$	2.80759	+	4.86288i	0.104128	+	0.180354i	−0.809254	+	0.587459i	$0.800128\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.95701	+	3.38965i	0.0723828	+	0.125371i
$732$		0			0
$733$	6.02803	−	10.4409i	0.222650	−	0.385642i	−0.732962	−	0.680270i	$-0.761863\pi$
$733$	6.02803	−	10.4409i	0.222650	−	0.385642i	0.955612	+	0.294628i	$0.0951958\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	17.8329			0.656885
$738$		0			0
$739$	43.8187			1.61190			0.805948	−	0.591986i	$-0.201656\pi$
$739$	43.8187			1.61190			0.805948	+	0.591986i	$0.201656\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.0004	−	25.9814i	0.550310	−	0.953164i	−0.447942	−	0.894062i	$-0.647843\pi$
$743$	15.0004	−	25.9814i	0.550310	−	0.953164i	0.998252	−	0.0591018i	$-0.0188236\pi$
$744$		0			0
$745$	−3.27336	−	5.66963i	−0.119927	−	0.207719i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.1780	+	17.6288i	0.371897	+	0.644144i
$750$		0			0
$751$	−4.69583	+	8.13342i	−0.171353	+	0.296793i	−0.938893	−	0.344208i	$-0.888147\pi$
$751$	−4.69583	+	8.13342i	−0.171353	+	0.296793i	0.767540	+	0.641001i	$0.221481\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.28102			0.0466210
$756$		0			0
$757$	21.7285			0.789737			0.394868	−	0.918738i	$-0.370790\pi$
$757$	21.7285			0.789737			0.394868	+	0.918738i	$0.370790\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.78398	−	11.7502i	0.245919	−	0.425944i	−0.716471	−	0.697617i	$-0.754244\pi$
$761$	6.78398	−	11.7502i	0.245919	−	0.425944i	0.962390	+	0.271673i	$0.0875769\pi$
$762$		0			0
$763$	18.7417	+	32.4617i	0.678497	+	1.17519i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.7617	−	27.3001i	−0.569124	−	0.985751i
$768$		0			0
$769$	−21.3949	+	37.0571i	−0.771520	+	1.33631i	0.165209	+	0.986259i	$0.447170\pi$
$769$	−21.3949	+	37.0571i	−0.771520	+	1.33631i	−0.936730	+	0.350054i	$0.886163\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.4288			−0.447034			−0.223517	−	0.974700i	$-0.571754\pi$
$773$	−12.4288			−0.447034			−0.223517	+	0.974700i	$0.571754\pi$
$774$		0			0
$775$	−53.2081			−1.91129
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.974036	−	1.68708i	0.0348984	−	0.0604459i
$780$		0			0
$781$	−23.3004	−	40.3576i	−0.833756	−	1.44411i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.9319	−	43.1833i	−0.889858	−	1.54128i
$786$		0			0
$787$	−21.5079	+	37.2527i	−0.766673	+	1.32792i	0.172684	+	0.984977i	$0.444756\pi$
$787$	−21.5079	+	37.2527i	−0.766673	+	1.32792i	−0.939357	+	0.342940i	$0.888577\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.37300			0.262154
$792$		0			0
$793$	−10.2839			−0.365191
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.6697	−	23.6766i	0.484205	−	0.838668i	−0.515630	−	0.856811i	$-0.672442\pi$
$797$	13.6697	−	23.6766i	0.484205	−	0.838668i	0.999835	+	0.0181431i	$0.00577543\pi$
$798$		0			0
$799$	−21.7160	−	37.6132i	−0.768256	−	1.33066i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.8333	+	25.6921i	0.523456	+	0.906653i
$804$		0			0
$805$	−0.348998	+	0.604482i	−0.0123006	+	0.0213052i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.2692			−0.642311			−0.321156	−	0.947026i	$-0.604071\pi$
$809$	−18.2692			−0.642311			−0.321156	+	0.947026i	$0.604071\pi$
$810$		0			0
$811$	−33.6619			−1.18203			−0.591015	−	0.806660i	$-0.701273\pi$
$811$	−33.6619			−1.18203			−0.591015	+	0.806660i	$0.701273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.27936	−	9.14412i	0.184928	−	0.320305i
$816$		0			0
$817$	0.516252	+	0.894175i	0.0180614	+	0.0312832i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.5703	−	37.3609i	−0.752810	−	1.30391i	−0.946456	−	0.322834i	$-0.895364\pi$
$821$	−21.5703	−	37.3609i	−0.752810	−	1.30391i	0.193645	−	0.981072i	$-0.437969\pi$
$822$		0			0
$823$	8.45533	−	14.6451i	0.294734	−	0.510494i	−0.680189	−	0.733037i	$-0.738102\pi$
$823$	8.45533	−	14.6451i	0.294734	−	0.510494i	0.974923	+	0.222542i	$0.0714356\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.2376			−0.425542			−0.212771	−	0.977102i	$-0.568249\pi$
$827$	−12.2376			−0.425542			−0.212771	+	0.977102i	$0.568249\pi$
$828$		0			0
$829$	−11.4794			−0.398696			−0.199348	−	0.979929i	$-0.563882\pi$
$829$	−11.4794			−0.398696			−0.199348	+	0.979929i	$0.563882\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.10582	−	1.91534i	0.0383146	−	0.0663627i
$834$		0			0
$835$	−2.19401	−	3.80013i	−0.0759267	−	0.131509i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.5224	−	39.0100i	−0.777560	−	1.34677i	−0.933344	−	0.358983i	$-0.883124\pi$
$839$	−22.5224	−	39.0100i	−0.777560	−	1.34677i	0.155784	−	0.987791i	$-0.450210\pi$
$840$		0			0
$841$	2.39367	−	4.14596i	0.0825403	−	0.142964i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.09750			0.0721563
$846$		0			0
$847$	57.5717			1.97819
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.185689	+	0.321622i	−0.00636533	+	0.0110251i
$852$		0			0
$853$	6.84105	+	11.8490i	0.234233	+	0.405703i	0.959049	−	0.283239i	$-0.0914088\pi$
$853$	6.84105	+	11.8490i	0.234233	+	0.405703i	−0.724817	+	0.688942i	$0.758075\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.9348	+	25.8679i	0.510164	+	0.883631i	0.999931	+	0.0117768i	$0.00374876\pi$
$857$	14.9348	+	25.8679i	0.510164	+	0.883631i	−0.489766	+	0.871854i	$0.662918\pi$
$858$		0			0
$859$	3.77095	−	6.53148i	0.128663	−	0.222851i	−0.794496	−	0.607270i	$-0.792265\pi$
$859$	3.77095	−	6.53148i	0.128663	−	0.222851i	0.923159	+	0.384418i	$0.125598\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.0591			−0.580698			−0.290349	−	0.956921i	$-0.593771\pi$
$863$	−17.0591			−0.580698			−0.290349	+	0.956921i	$0.593771\pi$
$864$		0			0
$865$	−49.8277			−1.69419
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.07228	−	7.05340i	0.138143	−	0.239270i
$870$		0			0
$871$	−5.52210	−	9.56456i	−0.187109	−	0.324083i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.0994	+	17.4926i	0.341421	+	0.591358i
$876$		0			0
$877$	7.74247	−	13.4103i	0.261445	−	0.452835i	−0.705181	−	0.709027i	$-0.749134\pi$
$877$	7.74247	−	13.4103i	0.261445	−	0.452835i	0.966626	+	0.256192i	$0.0824678\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.5541			0.894631			0.447316	−	0.894376i	$-0.352380\pi$
$881$	26.5541			0.894631			0.447316	+	0.894376i	$0.352380\pi$
$882$		0			0
$883$	−32.2262			−1.08450			−0.542250	−	0.840217i	$-0.682427\pi$
$883$	−32.2262			−1.08450			−0.542250	+	0.840217i	$0.682427\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.4763	−	32.0018i	0.620373	−	1.07452i	−0.369044	−	0.929412i	$-0.620315\pi$
$887$	18.4763	−	32.0018i	0.620373	−	1.07452i	0.989416	−	0.145105i	$-0.0463519\pi$
$888$		0			0
$889$	7.96383	+	13.7938i	0.267098	+	0.462628i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.72858	−	9.92220i	−0.191700	−	0.332034i
$894$		0			0
$895$	−22.3037	+	38.6312i	−0.745531	+	1.29130i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.6349			−1.22184
$900$		0			0
$901$	79.7057			2.65538
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−18.2949	+	31.6877i	−0.608144	+	1.05334i
$906$		0			0
$907$	−0.591871	−	1.02515i	−0.0196528	−	0.0340396i	0.856032	−	0.516923i	$-0.172923\pi$
$907$	−0.591871	−	1.02515i	−0.0196528	−	0.0340396i	−0.875685	+	0.482884i	$0.839589\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.7752	+	46.3760i	0.887103	+	1.53651i	0.843285	+	0.537467i	$0.180619\pi$
$911$	26.7752	+	46.3760i	0.887103	+	1.53651i	0.0438179	+	0.999040i	$0.486048\pi$
$912$		0			0
$913$	9.89569	−	17.1398i	0.327500	−	0.567246i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.05628			−0.100927
$918$		0			0
$919$	−1.67908			−0.0553878			−0.0276939	−	0.999616i	$-0.508816\pi$
$919$	−1.67908			−0.0553878			−0.0276939	+	0.999616i	$0.508816\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.4303	+	24.9940i	−0.474980	+	0.822689i
$924$		0			0
$925$	17.8893	+	30.9851i	0.588196	+	1.01879i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.2756	+	24.7261i	0.468368	+	0.811237i	0.999346	−	0.0361482i	$-0.0115088\pi$
$929$	14.2756	+	24.7261i	0.468368	+	0.811237i	−0.530978	+	0.847385i	$0.678175\pi$
$930$		0			0
$931$	0.291712	−	0.505260i	0.00956047	−	0.0165592i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−151.778			−4.96366
$936$		0			0
$937$	33.8620			1.10622			0.553111	−	0.833108i	$-0.313441\pi$
$937$	33.8620			1.10622			0.553111	+	0.833108i	$0.313441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.28984	+	14.3584i	−0.270241	+	0.468071i	−0.968923	−	0.247361i	$-0.920437\pi$
$941$	−8.28984	+	14.3584i	−0.270241	+	0.468071i	0.698682	+	0.715432i	$0.253770\pi$
$942$		0			0
$943$	0.0357596	+	0.0619374i	0.00116449	+	0.00201696i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.8930	+	48.3121i	0.906401	+	1.56993i	0.819026	+	0.573756i	$0.194514\pi$
$947$	27.8930	+	48.3121i	0.906401	+	1.56993i	0.0873743	+	0.996176i	$0.472152\pi$
$948$		0			0
$949$	9.18649	−	15.9115i	0.298206	−	0.516508i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	44.5366			1.44268			0.721341	−	0.692580i	$-0.243526\pi$
$953$	44.5366			1.44268			0.721341	+	0.692580i	$0.243526\pi$
$954$		0			0
$955$	28.5457			0.923717
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.59017	+	14.8786i	−0.277391	+	0.480455i
$960$		0			0
$961$	−12.2151	−	21.1572i	−0.394035	−	0.682489i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.09348	+	8.82216i	0.163965	+	0.283996i
$966$		0			0
$967$	14.0863	−	24.3982i	0.452986	−	0.784595i	−0.545584	−	0.838056i	$-0.683692\pi$
$967$	14.0863	−	24.3982i	0.452986	−	0.784595i	0.998570	+	0.0534616i	$0.0170255\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.8598			0.990338			0.495169	−	0.868797i	$-0.335106\pi$
$971$	30.8598			0.990338			0.495169	+	0.868797i	$0.335106\pi$
$972$		0			0
$973$	60.4755			1.93876
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.6967	−	25.4555i	0.470190	−	0.814393i	−0.529229	−	0.848479i	$-0.677519\pi$
$977$	14.6967	−	25.4555i	0.470190	−	0.814393i	0.999419	+	0.0340864i	$0.0108521\pi$
$978$		0			0
$979$	35.2666	+	61.0836i	1.12713	+	1.95224i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.08964	−	12.2796i	−0.226124	−	0.391659i	0.730532	−	0.682879i	$-0.239272\pi$
$983$	−7.08964	−	12.2796i	−0.226124	−	0.391659i	−0.956656	+	0.291220i	$0.905939\pi$
$984$		0			0
$985$	7.74020	−	13.4064i	0.246623	−	0.427164i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.0379061			−0.00120535
$990$		0			0
$991$	−46.0022			−1.46131			−0.730654	−	0.682748i	$-0.760785\pi$
$991$	−46.0022			−1.46131			−0.730654	+	0.682748i	$0.760785\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25.5985	−	44.3378i	0.811526	−	1.40560i
$996$		0			0
$997$	−30.1749	−	52.2645i	−0.955650	−	1.65523i	−0.732873	−	0.680365i	$-0.761821\pi$
$997$	−30.1749	−	52.2645i	−0.955650	−	1.65523i	−0.222777	−	0.974869i	$-0.571512\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.k.2305.6		12
3.2	odd	2		1152.2.i.i.769.6	yes	12
4.3	odd	2		3456.2.i.l.2305.6		12
8.3	odd	2		3456.2.i.j.2305.1		12
8.5	even	2		3456.2.i.i.2305.1		12
9.2	odd	6		1152.2.i.i.385.6	✓	12
9.7	even	3	inner	3456.2.i.k.1153.6		12
12.11	even	2		1152.2.i.k.769.1	yes	12
24.5	odd	2		1152.2.i.l.769.1	yes	12
24.11	even	2		1152.2.i.j.769.6	yes	12
36.7	odd	6		3456.2.i.l.1153.6		12
36.11	even	6		1152.2.i.k.385.1	yes	12
72.11	even	6		1152.2.i.j.385.6	yes	12
72.29	odd	6		1152.2.i.l.385.1	yes	12
72.43	odd	6		3456.2.i.j.1153.1		12
72.61	even	6		3456.2.i.i.1153.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.6	✓	12	9.2	odd	6
1152.2.i.i.769.6	yes	12	3.2	odd	2
1152.2.i.j.385.6	yes	12	72.11	even	6
1152.2.i.j.769.6	yes	12	24.11	even	2
1152.2.i.k.385.1	yes	12	36.11	even	6
1152.2.i.k.769.1	yes	12	12.11	even	2
1152.2.i.l.385.1	yes	12	72.29	odd	6
1152.2.i.l.769.1	yes	12	24.5	odd	2
3456.2.i.i.1153.1		12	72.61	even	6
3456.2.i.i.2305.1		12	8.5	even	2
3456.2.i.j.1153.1		12	72.43	odd	6
3456.2.i.j.2305.1		12	8.3	odd	2
3456.2.i.k.1153.6		12	9.7	even	3	inner
3456.2.i.k.2305.6		12	1.1	even	1	trivial
3456.2.i.l.1153.6		12	36.7	odd	6
3456.2.i.l.2305.6		12	4.3	odd	2

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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.74260 - 3.01828i) q^{5} +(-1.34988 - 2.33807i) q^{7} +O(q^{10})\)	\(q+(1.74260 - 3.01828i) q^{5} +(-1.34988 - 2.33807i) q^{7} +(2.84274 + 4.92377i) q^{11} +(1.76055 - 3.04937i) q^{13} -7.65970 q^{17} -2.02060 q^{19} +(0.0370909 - 0.0642434i) q^{23} +(-3.57334 - 6.18921i) q^{25} +(-2.46032 - 4.26140i) q^{29} +(3.72257 - 6.44769i) q^{31} -9.40925 q^{35} -5.00631 q^{37} +(-0.482053 + 0.834941i) q^{41} +(-0.255495 - 0.442530i) q^{43} +(2.83509 + 4.91053i) q^{47} +(-0.144369 + 0.250055i) q^{49} -10.4058 q^{53} +19.8151 q^{55} +(4.47636 - 7.75329i) q^{59} +(-1.46032 - 2.52935i) q^{61} +(-6.13589 - 10.6277i) q^{65} +(1.56829 - 2.71635i) q^{67} -8.19647 q^{71} +5.21796 q^{73} +(7.67474 - 13.2930i) q^{77} +(-0.716260 - 1.24060i) q^{79} +(-1.74052 - 3.01467i) q^{83} +(-13.3478 + 23.1191i) q^{85} +12.4058 q^{89} -9.50616 q^{91} +(-3.52110 + 6.09873i) q^{95} +(-3.50265 - 6.06677i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 12 * q + 2 * q^5 - 6 * q^7	\( 12 q + 2 q^{5} - 6 q^{7} + 4 q^{11} + 10 q^{13} - 4 q^{17} - 4 q^{19} + 8 q^{23} - 14 q^{25} + 2 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{41} + 2 q^{43} - 14 q^{47} - 18 q^{49} - 24 q^{53} + 16 q^{55} + 6 q^{59} + 14 q^{61} + 8 q^{65} - 4 q^{67} - 28 q^{71} + 60 q^{73} - 2 q^{77} - 16 q^{79} + 24 q^{83} + 16 q^{85} + 48 q^{89} + 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) \) 12 * q + 2 * q^5 - 6 * q^7 + 4 * q^11 + 10 * q^13 - 4 * q^17 - 4 * q^19 + 8 * q^23 - 14 * q^25 + 2 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^41 + 2 * q^43 - 14 * q^47 - 18 * q^49 - 24 * q^53 + 16 * q^55 + 6 * q^59 + 14 * q^61 + 8 * q^65 - 4 * q^67 - 28 * q^71 + 60 * q^73 - 2 * q^77 - 16 * q^79 + 24 * q^83 + 16 * q^85 + 48 * q^89 + 52 * q^91 - 20 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more