LMFDB - Embedded newform 135.4.b.b.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(109,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("135.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.b (of order $2$, degree $1$, 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:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.218111416576.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 52x^{4} - 79x^{2} + 64 $ x^8 - 12x^6 + 52x^4 - 79*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.2
Root			$-2.30468 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	135.109
Dual form			135.4.b.b.109.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.56155i q^{2} -4.68466 q^{4} +(10.7966 + 2.90388i) q^{5} +6.06288i q^{7} -11.8078i q^{8} +O(q^{10})$	\(q-3.56155i q^{2} -4.68466 q^{4} +(10.7966 + 2.90388i) q^{5} +6.06288i q^{7} -11.8078i q^{8} +(10.3423 - 38.4528i) q^{10} +61.3752 q^{11} -70.8427i q^{13} +21.5933 q^{14} -79.5312 q^{16} +51.5464i q^{17} -15.7926 q^{19} +(-50.5786 - 13.6037i) q^{20} -218.591i q^{22} -114.885i q^{23} +(108.135 + 62.7043i) q^{25} -252.310 q^{26} -28.4025i q^{28} -39.7819 q^{29} -240.170 q^{31} +188.793i q^{32} +183.585 q^{34} +(-17.6059 + 65.4588i) q^{35} +129.560i q^{37} +56.2462i q^{38} +(34.2884 - 127.484i) q^{40} +262.524 q^{41} +127.648i q^{43} -287.522 q^{44} -409.170 q^{46} +85.2140i q^{47} +306.241 q^{49} +(223.325 - 385.128i) q^{50} +331.874i q^{52} -328.184i q^{53} +(662.646 + 178.226i) q^{55} +71.5891 q^{56} +141.685i q^{58} +687.580 q^{59} -455.685 q^{61} +855.380i q^{62} +36.1449 q^{64} +(205.719 - 764.864i) q^{65} +955.421i q^{67} -241.477i q^{68} +(233.135 + 62.7043i) q^{70} -160.201 q^{71} +702.364i q^{73} +461.434 q^{74} +73.9830 q^{76} +372.111i q^{77} -65.3608 q^{79} +(-858.670 - 230.949i) q^{80} -934.993i q^{82} +996.623i q^{83} +(-149.685 + 556.528i) q^{85} +454.624 q^{86} -724.704i q^{88} -1337.71 q^{89} +429.511 q^{91} +538.199i q^{92} +303.494 q^{94} +(-170.507 - 45.8599i) q^{95} -1404.40i q^{97} -1090.70i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{4}+O(q^{10}) $ 8 * q + 12 * q^4	$ 8 q + 12 q^{4} + 58 q^{10} - 92 q^{16} + 220 q^{19} + 494 q^{25} - 536 q^{31} + 776 q^{34} + 398 q^{40} - 1888 q^{46} - 964 q^{49} + 774 q^{55} - 3596 q^{61} + 1724 q^{64} + 1494 q^{70} + 2472 q^{76} - 1364 q^{79} - 1148 q^{85} - 720 q^{91} + 152 q^{94}+O(q^{100}) $ 8 * q + 12 * q^4 + 58 * q^10 - 92 * q^16 + 220 * q^19 + 494 * q^25 - 536 * q^31 + 776 * q^34 + 398 * q^40 - 1888 * q^46 - 964 * q^49 + 774 * q^55 - 3596 * q^61 + 1724 * q^64 + 1494 * q^70 + 2472 * q^76 - 1364 * q^79 - 1148 * q^85 - 720 * q^91 + 152 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	3.56155i		−	1.25920i	−0.776920	−	0.629600i	$-0.783219\pi$
$2$		−	3.56155i		−	1.25920i	0.776920	−	0.629600i	$-0.216781\pi$
$3$		0			0
$3$		0			0
$4$	−4.68466			−0.585582
$5$	10.7966	+	2.90388i	0.965681	+	0.259731i
$5$	10.7966	+	2.90388i	0.965681	+	0.259731i
$6$		0			0
$7$			6.06288i			0.327365i	0.986513	+	0.163682i	$0.0523373\pi$
$7$			6.06288i			0.327365i	−0.986513	+	0.163682i	$0.947663\pi$
$8$		−	11.8078i		−	0.521834i
$9$		0			0
$10$	10.3423	−	38.4528i	0.327053	−	1.21598i
$11$	61.3752			1.68230			0.841151	−	0.540800i	$-0.181878\pi$
$11$	61.3752			1.68230			0.841151	+	0.540800i	$0.181878\pi$
$12$		0			0
$13$		−	70.8427i		−	1.51140i	−0.654916	−	0.755702i	$-0.727296\pi$
$13$		−	70.8427i		−	1.51140i	0.654916	−	0.755702i	$-0.272704\pi$
$14$	21.5933			0.412218
$15$		0			0
$16$	−79.5312			−1.24268
$17$			51.5464i			0.735402i	0.929944	+	0.367701i	$0.119855\pi$
$17$			51.5464i			0.735402i	−0.929944	+	0.367701i	$0.880145\pi$
$18$		0			0
$19$	−15.7926			−0.190688			−0.0953440	−	0.995444i	$-0.530395\pi$
$19$	−15.7926			−0.190688			−0.0953440	+	0.995444i	$0.530395\pi$
$20$	−50.5786	−	13.6037i	−0.565486	−	0.152094i
$21$		0			0
$22$		−	218.591i		−	2.11835i
$23$		−	114.885i		−	1.04153i	−0.853699	−	0.520767i	$-0.825646\pi$
$23$		−	114.885i		−	1.04153i	0.853699	−	0.520767i	$-0.174354\pi$
$24$		0			0
$25$	108.135	+	62.7043i	0.865080	+	0.501635i
$26$	−252.310			−1.90316
$27$		0			0
$28$		−	28.4025i		−	0.191699i
$29$	−39.7819			−0.254735			−0.127368	−	0.991856i	$-0.540653\pi$
$29$	−39.7819			−0.254735			−0.127368	+	0.991856i	$0.540653\pi$
$30$		0			0
$31$	−240.170			−1.39148			−0.695740	−	0.718294i	$-0.744923\pi$
$31$	−240.170			−1.39148			−0.695740	+	0.718294i	$0.744923\pi$
$32$			188.793i			1.04294i
$33$		0			0
$34$	183.585			0.926018
$35$	−17.6059	+	65.4588i	−0.0850269	+	0.316130i
$36$		0			0
$37$			129.560i			0.575662i	0.957681	+	0.287831i	$0.0929341\pi$
$37$			129.560i			0.575662i	−0.957681	+	0.287831i	$0.907066\pi$
$38$			56.2462i			0.240114i
$39$		0			0
$40$	34.2884	−	127.484i	0.135537	−	0.503926i
$41$	262.524			0.999985			0.499992	−	0.866030i	$-0.333336\pi$
$41$	262.524			0.999985			0.499992	+	0.866030i	$0.333336\pi$
$42$		0			0
$43$			127.648i			0.452700i	0.974046	+	0.226350i	$0.0726793\pi$
$43$			127.648i			0.452700i	−0.974046	+	0.226350i	$0.927321\pi$
$44$	−287.522			−0.985126
$45$		0			0
$46$	−409.170			−1.31150
$47$			85.2140i			0.264463i	0.991219	+	0.132231i	$0.0422142\pi$
$47$			85.2140i			0.264463i	−0.991219	+	0.132231i	$0.957786\pi$
$48$		0			0
$49$	306.241			0.892832
$50$	223.325	−	385.128i	0.631658	−	1.08931i
$51$		0			0
$52$			331.874i			0.885051i
$53$		−	328.184i		−	0.850557i	−0.905063	−	0.425278i	$-0.860176\pi$
$53$		−	328.184i		−	0.850557i	0.905063	−	0.425278i	$-0.139824\pi$
$54$		0			0
$55$	662.646	+	178.226i	1.62457	+	0.436946i
$56$	71.5891			0.170830
$57$		0			0
$58$			141.685i			0.320762i
$59$	687.580			1.51721			0.758605	−	0.651551i	$-0.225881\pi$
$59$	687.580			1.51721			0.758605	+	0.651551i	$0.225881\pi$
$60$		0			0
$61$	−455.685			−0.956466			−0.478233	−	0.878233i	$-0.658723\pi$
$61$	−455.685			−0.956466			−0.478233	+	0.878233i	$0.658723\pi$
$62$			855.380i			1.75215i
$63$		0			0
$64$	36.1449			0.0705955
$65$	205.719	−	764.864i	0.392558	−	1.45953i
$66$		0			0
$67$			955.421i			1.74214i	0.491161	+	0.871069i	$0.336573\pi$
$67$			955.421i			1.74214i	−0.491161	+	0.871069i	$0.663427\pi$
$68$		−	241.477i		−	0.430639i
$69$		0			0
$70$	233.135	+	62.7043i	0.398071	+	0.107066i
$71$	−160.201			−0.267780			−0.133890	−	0.990996i	$-0.542747\pi$
$71$	−160.201			−0.267780			−0.133890	+	0.990996i	$0.542747\pi$
$72$		0			0
$73$			702.364i			1.12610i	0.826422	+	0.563052i	$0.190373\pi$
$73$			702.364i			1.12610i	−0.826422	+	0.563052i	$0.809627\pi$
$74$	461.434			0.724873
$75$		0			0
$76$	73.9830			0.111664
$77$			372.111i			0.550727i
$78$		0			0
$79$	−65.3608			−0.0930844			−0.0465422	−	0.998916i	$-0.514820\pi$
$79$	−65.3608			−0.0930844			−0.0465422	+	0.998916i	$0.514820\pi$
$80$	−858.670	−	230.949i	−1.20003	−	0.322762i
$81$		0			0
$82$		−	934.993i		−	1.25918i
$83$			996.623i			1.31800i	0.752145	+	0.658998i	$0.229019\pi$
$83$			996.623i			1.31800i	−0.752145	+	0.658998i	$0.770981\pi$
$84$		0			0
$85$	−149.685	+	556.528i	−0.191007	+	0.710164i
$86$	454.624			0.570040
$87$		0			0
$88$		−	724.704i		−	0.877883i
$89$	−1337.71			−1.59322			−0.796612	−	0.604491i	$-0.793377\pi$
$89$	−1337.71			−1.59322			−0.796612	+	0.604491i	$0.793377\pi$
$90$		0			0
$91$	429.511			0.494780
$92$			538.199i			0.609903i
$93$		0			0
$94$	303.494			0.333011
$95$	−170.507	−	45.8599i	−0.184144	−	0.0495276i
$96$		0			0
$97$		−	1404.40i		−	1.47006i	−0.678037	−	0.735028i	$-0.737169\pi$
$97$		−	1404.40i		−	1.47006i	0.678037	−	0.735028i	$-0.262831\pi$
$98$		−	1090.70i		−	1.12425i
$99$		0			0
$100$	−506.575	−	293.748i	−0.506575	−	0.293748i
$101$	−1110.40			−1.09395			−0.546974	−	0.837150i	$-0.684220\pi$
$101$	−1110.40			−1.09395			−0.546974	+	0.837150i	$0.684220\pi$
$102$		0			0
$103$			509.937i			0.487821i	0.969798	+	0.243911i	$0.0784303\pi$
$103$			509.937i			0.487821i	−0.969798	+	0.243911i	$0.921570\pi$
$104$	−836.494			−0.788702
$105$		0			0
$106$	−1168.84			−1.07102
$107$			1748.35i			1.57962i	0.613353	+	0.789809i	$0.289820\pi$
$107$			1748.35i			1.57962i	−0.613353	+	0.789809i	$0.710180\pi$
$108$		0			0
$109$	−1357.35			−1.19276			−0.596381	−	0.802702i	$-0.703395\pi$
$109$	−1357.35			−1.19276			−0.596381	+	0.802702i	$0.703395\pi$
$110$	634.763	−	2360.05i	0.550202	−	2.04565i
$111$		0			0
$112$		−	482.189i		−	0.406809i
$113$		−	1153.33i		−	0.960139i	−0.877230	−	0.480070i	$-0.840611\pi$
$113$		−	1153.33i		−	0.960139i	0.877230	−	0.480070i	$-0.159389\pi$
$114$		0			0
$115$	333.614	−	1240.38i	0.270519	−	1.00579i
$116$	186.365			0.149168
$117$		0			0
$118$		−	2448.85i		−	1.91047i
$119$	−312.520			−0.240745
$120$		0			0
$121$	2435.92			1.83014
$122$			1622.94i			1.20438i
$123$		0			0
$124$	1125.12			0.814826
$125$	985.408	+	991.007i	0.705101	+	0.709107i
$126$		0			0
$127$			1978.79i			1.38259i	0.722571	+	0.691297i	$0.242960\pi$
$127$			1978.79i			1.38259i	−0.722571	+	0.691297i	$0.757040\pi$
$128$			1381.61i			0.954048i
$129$		0			0
$130$	−2724.10	−	732.679i	−1.83784	−	0.494309i
$131$	−160.293			−0.106908			−0.0534538	−	0.998570i	$-0.517023\pi$
$131$	−160.293			−0.106908			−0.0534538	+	0.998570i	$0.517023\pi$
$132$		0			0
$133$		−	95.7488i		−	0.0624246i
$134$	3402.78			2.19370
$135$		0			0
$136$	608.648			0.383758
$137$			661.330i			0.412418i	0.978508	+	0.206209i	$0.0661127\pi$
$137$			661.330i			0.412418i	−0.978508	+	0.206209i	$0.933887\pi$
$138$		0			0
$139$	−2126.16			−1.29740			−0.648699	−	0.761045i	$-0.724687\pi$
$139$	−2126.16			−1.29740			−0.648699	+	0.761045i	$0.724687\pi$
$140$	82.4776	−	306.652i	0.0497902	−	0.185120i
$141$		0			0
$142$			570.566i			0.337189i
$143$		−	4347.99i		−	2.54264i
$144$		0			0
$145$	−429.511	−	115.522i	−0.245993	−	0.0661626i
$146$	2501.51			1.41799
$147$		0			0
$148$		−	606.943i		−	0.337097i
$149$	670.649			0.368736			0.184368	−	0.982857i	$-0.440976\pi$
$149$	670.649			0.368736			0.184368	+	0.982857i	$0.440976\pi$
$150$		0			0
$151$	1440.42			0.776291			0.388146	−	0.921598i	$-0.373116\pi$
$151$	1440.42			0.776291			0.388146	+	0.921598i	$0.373116\pi$
$152$			186.475i			0.0995076i
$153$		0			0
$154$	1325.29			0.693475
$155$	−2593.03	−	697.427i	−1.34373	−	0.361411i
$156$		0			0
$157$		−	3037.28i		−	1.54396i	−0.635648	−	0.771979i	$-0.719267\pi$
$157$		−	3037.28i		−	1.54396i	0.635648	−	0.771979i	$-0.280733\pi$
$158$			232.786i			0.117212i
$159$		0			0
$160$	−548.231	+	2038.33i	−0.270884	+	1.00715i
$161$	696.537			0.340961
$162$		0			0
$163$			1243.27i			0.597426i	0.954343	+	0.298713i	$0.0965573\pi$
$163$			1243.27i			0.597426i	−0.954343	+	0.298713i	$0.903443\pi$
$164$	−1229.84			−0.585573
$165$		0			0
$166$	3549.53			1.65962
$167$		−	165.425i		−	0.0766526i	−0.999265	−	0.0383263i	$-0.987797\pi$
$167$		−	165.425i		−	0.0766526i	0.999265	−	0.0383263i	$-0.0122026\pi$
$168$		0			0
$169$	−2821.69			−1.28434
$170$	1982.10	+	533.110i	0.894238	+	0.240516i
$171$		0			0
$172$		−	597.986i		−	0.265093i
$173$			1931.11i			0.848666i	0.905506	+	0.424333i	$0.139491\pi$
$173$			1931.11i			0.848666i	−0.905506	+	0.424333i	$0.860509\pi$
$174$		0			0
$175$	−380.169	+	655.610i	−0.164218	+	0.283197i
$176$	−4881.25			−2.09056
$177$		0			0
$178$			4764.32i			2.00619i
$179$	623.874			0.260506			0.130253	−	0.991481i	$-0.458421\pi$
$179$	623.874			0.260506			0.130253	+	0.991481i	$0.458421\pi$
$180$		0			0
$181$	1139.12			0.467792			0.233896	−	0.972262i	$-0.424853\pi$
$181$	1139.12			0.467792			0.233896	+	0.972262i	$0.424853\pi$
$182$		−	1529.73i		−	0.623027i
$183$		0			0
$184$	−1356.54			−0.543508
$185$	−376.226	+	1398.81i	−0.149517	+	0.555906i
$186$		0			0
$187$			3163.67i			1.23717i
$188$		−	399.199i		−	0.154865i
$189$		0			0
$190$	−163.332	+	607.270i	−0.0623651	+	0.231874i
$191$	−468.059			−0.177317			−0.0886586	−	0.996062i	$-0.528258\pi$
$191$	−468.059			−0.177317			−0.0886586	+	0.996062i	$0.528258\pi$
$192$		0			0
$193$		−	2479.17i		−	0.924634i	−0.886715	−	0.462317i	$-0.847018\pi$
$193$		−	2479.17i		−	0.924634i	0.886715	−	0.462317i	$-0.152982\pi$
$194$	−5001.85			−1.85109
$195$		0			0
$196$	−1434.64			−0.522827
$197$			2401.87i			0.868660i	0.900754	+	0.434330i	$0.143015\pi$
$197$			2401.87i			0.868660i	−0.900754	+	0.434330i	$0.856985\pi$
$198$		0			0
$199$	1039.02			0.370121			0.185061	−	0.982727i	$-0.440752\pi$
$199$	1039.02			0.370121			0.185061	+	0.982727i	$0.440752\pi$
$200$	740.398	−	1276.83i	0.261770	−	0.451428i
$201$		0			0
$202$			3954.74i			1.37750i
$203$		−	241.193i		−	0.0833914i
$204$		0			0
$205$	2834.38	+	762.339i	0.965666	+	0.259727i
$206$	1816.17			0.614264
$207$		0			0
$208$			5634.21i			1.87818i
$209$	−969.275			−0.320795
$210$		0			0
$211$	3311.99			1.08060			0.540301	−	0.841472i	$-0.318310\pi$
$211$	3311.99			1.08060			0.540301	+	0.841472i	$0.318310\pi$
$212$			1537.43i			0.498071i
$213$		0			0
$214$	6226.83			1.98905
$215$	−370.674	+	1378.17i	−0.117580	+	0.437164i
$216$		0			0
$217$		−	1456.13i		−	0.455522i
$218$			4834.29i			1.50192i
$219$		0			0
$220$	−3104.27	−	834.930i	−0.951318	−	0.255868i
$221$	3651.69			1.11149
$222$		0			0
$223$		−	3985.66i		−	1.19686i	−0.801176	−	0.598429i	$-0.795792\pi$
$223$		−	3985.66i		−	1.19686i	0.801176	−	0.598429i	$-0.204208\pi$
$224$	−1144.63			−0.341423
$225$		0			0
$226$	−4107.63			−1.20901
$227$		−	4430.88i		−	1.29554i	−0.761836	−	0.647770i	$-0.775702\pi$
$227$		−	4430.88i		−	1.29554i	0.761836	−	0.647770i	$-0.224298\pi$
$228$		0			0
$229$	−4653.72			−1.34291			−0.671456	−	0.741045i	$-0.734331\pi$
$229$	−4653.72			−1.34291			−0.671456	+	0.741045i	$0.734331\pi$
$230$	−4417.67	−	1188.18i	−1.26649	−	0.340637i
$231$		0			0
$232$			469.736i			0.132930i
$233$		−	2296.46i		−	0.645692i	−0.946452	−	0.322846i	$-0.895360\pi$
$233$		−	2296.46i		−	0.645692i	0.946452	−	0.322846i	$-0.104640\pi$
$234$		0			0
$235$	−247.452	+	920.025i	−0.0686892	+	0.255387i
$236$	−3221.08			−0.888451
$237$		0			0
$238$			1113.06i			0.303146i
$239$	3418.55			0.925220			0.462610	−	0.886562i	$-0.346913\pi$
$239$	3418.55			0.925220			0.462610	+	0.886562i	$0.346913\pi$
$240$		0			0
$241$	−5904.16			−1.57809			−0.789047	−	0.614333i	$-0.789425\pi$
$241$	−5904.16			−1.57809			−0.789047	+	0.614333i	$0.789425\pi$
$242$		−	8675.65i		−	2.30451i
$243$		0			0
$244$	2134.73			0.560090
$245$	3306.38	+	889.289i	0.862191	+	0.231896i
$246$		0			0
$247$			1118.79i			0.288206i
$248$			2835.88i			0.726122i
$249$		0			0
$250$	3529.53	−	3509.58i	0.892907	−	0.887862i
$251$	2391.39			0.601368			0.300684	−	0.953724i	$-0.402785\pi$
$251$	2391.39			0.601368			0.300684	+	0.953724i	$0.402785\pi$
$252$		0			0
$253$		−	7051.12i		−	1.75217i
$254$	7047.57			1.74096
$255$		0			0
$256$	5209.83			1.27193
$257$		−	3580.72i		−	0.869102i	−0.900647	−	0.434551i	$-0.856907\pi$
$257$		−	3580.72i		−	0.869102i	0.900647	−	0.434551i	$-0.143093\pi$
$258$		0			0
$259$	−785.505			−0.188451
$260$	−963.723	+	3583.12i	−0.229875	+	0.854677i
$261$		0			0
$262$			570.893i			0.134618i
$263$			1703.58i			0.399419i	0.979855	+	0.199709i	$0.0639998\pi$
$263$			1703.58i			0.399419i	−0.979855	+	0.199709i	$0.936000\pi$
$264$		0			0
$265$	953.007	−	3543.28i	0.220916	−	0.821366i
$266$	−341.014			−0.0786050
$267$		0			0
$268$		−	4475.82i		−	1.02017i
$269$	4087.14			0.926385			0.463192	−	0.886258i	$-0.346704\pi$
$269$	4087.14			0.926385			0.463192	+	0.886258i	$0.346704\pi$
$270$		0			0
$271$	−4572.37			−1.02492			−0.512458	−	0.858713i	$-0.671265\pi$
$271$	−4572.37			−1.02492			−0.512458	+	0.858713i	$0.671265\pi$
$272$		−	4099.55i		−	0.913866i
$273$		0			0
$274$	2355.36			0.519316
$275$	6636.81	+	3848.49i	1.45533	+	0.843901i
$276$		0			0
$277$		−	8334.59i		−	1.80786i	−0.427682	−	0.903929i	$-0.640670\pi$
$277$		−	8334.59i		−	1.80786i	0.427682	−	0.903929i	$-0.359330\pi$
$278$			7572.43i			1.63368i
$279$		0			0
$280$	772.922	+	207.886i	0.164968	+	0.0443699i
$281$	−2490.50			−0.528721			−0.264360	−	0.964424i	$-0.585161\pi$
$281$	−2490.50			−0.528721			−0.264360	+	0.964424i	$0.585161\pi$
$282$		0			0
$283$			7132.72i			1.49822i	0.662445	+	0.749110i	$0.269519\pi$
$283$			7132.72i			1.49822i	−0.662445	+	0.749110i	$0.730481\pi$
$284$	750.489			0.156807
$285$		0			0
$286$	−15485.6			−3.20169
$287$			1591.65i			0.327360i
$288$		0			0
$289$	2255.97			0.459184
$290$	−411.438	+	1529.73i	−0.0833119	+	0.309754i
$291$		0			0
$292$		−	3290.34i		−	0.659426i
$293$		−	4915.24i		−	0.980039i	−0.871711	−	0.490020i	$-0.836990\pi$
$293$		−	4915.24i		−	0.980039i	0.871711	−	0.490020i	$-0.163010\pi$
$294$		0			0
$295$	7423.56	+	1996.65i	1.46514	+	0.394066i
$296$	1529.81			0.300400
$297$		0			0
$298$		−	2388.55i		−	0.464313i
$299$	−8138.80			−1.57418
$300$		0			0
$301$	−773.914			−0.148198
$302$		−	5130.14i		−	0.977505i
$303$		0			0
$304$	1256.01			0.236963
$305$	−4919.86	−	1323.25i	−0.923641	−	0.248424i
$306$		0			0
$307$		−	5798.84i		−	1.07804i	−0.842294	−	0.539018i	$-0.818795\pi$
$307$		−	5798.84i		−	1.07804i	0.842294	−	0.539018i	$-0.181205\pi$
$308$		−	1743.21i		−	0.322496i
$309$		0			0
$310$	−2483.92	+	9235.23i	−0.455088	+	1.69202i
$311$	−60.3935			−0.0110116			−0.00550579	−	0.999985i	$-0.501753\pi$
$311$	−60.3935			−0.0110116			−0.00550579	+	0.999985i	$0.501753\pi$
$312$		0			0
$313$			605.307i			0.109310i	0.998505	+	0.0546549i	$0.0174059\pi$
$313$			605.307i			0.109310i	−0.998505	+	0.0546549i	$0.982594\pi$
$314$	−10817.4			−1.94415
$315$		0			0
$316$	306.193			0.0545086
$317$			3144.61i			0.557157i	0.960414	+	0.278578i	$0.0898632\pi$
$317$			3144.61i			0.557157i	−0.960414	+	0.278578i	$0.910137\pi$
$318$		0			0
$319$	−2441.63			−0.428542
$320$	390.244	+	104.961i	0.0681727	+	0.0183359i
$321$		0			0
$322$		−	2480.75i		−	0.429338i
$323$		−	814.052i		−	0.140232i
$324$		0			0
$325$	4442.15	−	7660.57i	0.758172	−	1.30748i
$326$	4427.97			0.752278
$327$		0			0
$328$		−	3099.82i		−	0.521826i
$329$	−516.643			−0.0865758
$330$		0			0
$331$	−2122.94			−0.352529			−0.176265	−	0.984343i	$-0.556401\pi$
$331$	−2122.94			−0.352529			−0.176265	+	0.984343i	$0.556401\pi$
$332$		−	4668.84i		−	0.771795i
$333$		0			0
$334$	−589.170			−0.0965209
$335$	−2774.43	+	10315.3i	−0.452487	+	1.68235i
$336$		0			0
$337$			1486.39i			0.240263i	0.992758	+	0.120132i	$0.0383317\pi$
$337$			1486.39i			0.240263i	−0.992758	+	0.120132i	$0.961668\pi$
$338$			10049.6i			1.61724i
$339$		0			0
$340$	701.221	−	2607.14i	0.111850	−	0.415859i
$341$	−14740.5			−2.34089
$342$		0			0
$343$			3936.28i			0.619647i
$344$	1507.24			0.236235
$345$		0			0
$346$	6877.74			1.06864
$347$		−	7649.36i		−	1.18340i	−0.806159	−	0.591699i	$-0.798457\pi$
$347$		−	7649.36i		−	1.18340i	0.806159	−	0.591699i	$-0.201543\pi$
$348$		0			0
$349$	1123.61			0.172337			0.0861683	−	0.996281i	$-0.472538\pi$
$349$	1123.61			0.172337			0.0861683	+	0.996281i	$0.472538\pi$
$350$	2334.99	+	1353.99i	0.356601	+	0.206783i
$351$		0			0
$352$			11587.2i			1.75454i
$353$		−	12564.5i		−	1.89446i	−0.320560	−	0.947228i	$-0.603871\pi$
$353$		−	12564.5i		−	1.89446i	0.320560	−	0.947228i	$-0.396129\pi$
$354$		0			0
$355$	−1729.64	−	465.206i	−0.258590	−	0.0695509i
$356$	6266.71			0.932964
$357$		0			0
$358$		−	2221.96i		−	0.328029i
$359$	−2491.02			−0.366214			−0.183107	−	0.983093i	$-0.558616\pi$
$359$	−2491.02			−0.366214			−0.183107	+	0.983093i	$0.558616\pi$
$360$		0			0
$361$	−6609.59			−0.963638
$362$		−	4057.04i		−	0.589043i
$363$		0			0
$364$	−2012.11			−0.289735
$365$	−2039.58	+	7583.18i	−0.292484	+	1.08746i
$366$		0			0
$367$			12962.0i			1.84363i	0.387629	+	0.921816i	$0.373294\pi$
$367$			12962.0i			1.84363i	−0.387629	+	0.921816i	$0.626706\pi$
$368$			9136.98i			1.29429i
$369$		0			0
$370$	4981.93	+	1339.95i	0.699996	+	0.188272i
$371$	1989.74			0.278442
$372$		0			0
$373$		−	5045.73i		−	0.700424i	−0.936670	−	0.350212i	$-0.886110\pi$
$373$		−	5045.73i		−	0.700424i	0.936670	−	0.350212i	$-0.113890\pi$
$374$	11267.6			1.55784
$375$		0			0
$376$	1006.19			0.138006
$377$			2818.26i			0.385008i
$378$		0			0
$379$	−357.793			−0.0484923			−0.0242461	−	0.999706i	$-0.507719\pi$
$379$	−357.793			−0.0484923			−0.0242461	+	0.999706i	$0.507719\pi$
$380$	798.768	+	214.838i	0.107831	+	0.0290025i
$381$		0			0
$382$			1667.02i			0.223278i
$383$		−	886.550i		−	0.118278i	−0.998250	−	0.0591391i	$-0.981164\pi$
$383$		−	886.550i		−	0.118278i	0.998250	−	0.0591391i	$-0.0188356\pi$
$384$		0			0
$385$	−1080.57	+	4017.55i	−0.143041	+	0.531826i
$386$	−8829.69			−1.16430
$387$		0			0
$388$			6579.14i			0.860838i
$389$	8284.92			1.07985			0.539926	−	0.841713i	$-0.318452\pi$
$389$	8284.92			1.07985			0.539926	+	0.841713i	$0.318452\pi$
$390$		0			0
$391$	5921.93			0.765946
$392$		−	3616.03i		−	0.465911i
$393$		0			0
$394$	8554.38			1.09382
$395$	−705.677	−	189.800i	−0.0898898	−	0.0241769i
$396$		0			0
$397$			4474.56i			0.565672i	0.959168	+	0.282836i	$0.0912753\pi$
$397$			4474.56i			0.565672i	−0.959168	+	0.282836i	$0.908725\pi$
$398$		−	3700.52i		−	0.466056i
$399$		0			0
$400$	−8600.11	−	4986.95i	−1.07501	−	0.623369i
$401$	10558.3			1.31486			0.657428	−	0.753517i	$-0.271644\pi$
$401$	10558.3			1.31486			0.657428	+	0.753517i	$0.271644\pi$
$402$		0			0
$403$			17014.3i			2.10309i
$404$	5201.83			0.640596
$405$		0			0
$406$	−859.023			−0.105006
$407$			7951.75i			0.968437i
$408$		0			0
$409$	5232.44			0.632586			0.316293	−	0.948662i	$-0.397562\pi$
$409$	5232.44			0.632586			0.316293	+	0.948662i	$0.397562\pi$
$410$	2715.11	−	10094.8i	0.327048	−	1.21597i
$411$		0			0
$412$		−	2388.88i		−	0.285659i
$413$			4168.72i			0.496681i
$414$		0			0
$415$	−2894.08	+	10760.2i	−0.342324	+	1.27276i
$416$	13374.6			1.57631
$417$		0			0
$418$			3452.12i			0.403945i
$419$	−3903.81			−0.455164			−0.227582	−	0.973759i	$-0.573082\pi$
$419$	−3903.81			−0.455164			−0.227582	+	0.973759i	$0.573082\pi$
$420$		0			0
$421$	6274.83			0.726405			0.363203	−	0.931710i	$-0.381683\pi$
$421$	6274.83			0.726405			0.363203	+	0.931710i	$0.381683\pi$
$422$		−	11795.8i		−	1.36069i
$423$		0			0
$424$	−3875.12			−0.443850
$425$	−3232.18	+	5573.97i	−0.368903	+	0.636181i
$426$		0			0
$427$		−	2762.76i		−	0.313114i
$428$		−	8190.41i		−	0.924996i
$429$		0			0
$430$	4908.42	+	1320.18i	0.550477	+	0.148057i
$431$	8040.40			0.898590			0.449295	−	0.893383i	$-0.351675\pi$
$431$	8040.40			0.898590			0.449295	+	0.893383i	$0.351675\pi$
$432$		0			0
$433$		−	3121.18i		−	0.346407i	−0.984886	−	0.173204i	$-0.944588\pi$
$433$		−	3121.18i		−	0.346407i	0.984886	−	0.173204i	$-0.0554119\pi$
$434$	−5186.07			−0.573593
$435$		0			0
$436$	6358.74			0.698460
$437$			1814.34i			0.198608i
$438$		0			0
$439$	2373.05			0.257994			0.128997	−	0.991645i	$-0.458824\pi$
$439$	2373.05			0.257994			0.128997	+	0.991645i	$0.458824\pi$
$440$	2104.46	−	7824.37i	0.228014	−	0.847755i
$441$		0			0
$442$		−	13005.7i		−	1.39959i
$443$		−	13152.7i		−	1.41062i	−0.708898	−	0.705311i	$-0.750807\pi$
$443$		−	13152.7i		−	1.41062i	0.708898	−	0.705311i	$-0.249193\pi$
$444$		0			0
$445$	−14442.8	−	3884.55i	−1.53855	−	0.413810i
$446$	−14195.1			−1.50708
$447$		0			0
$448$			219.142i			0.0231105i
$449$	−9265.76			−0.973893			−0.486947	−	0.873432i	$-0.661889\pi$
$449$	−9265.76			−0.973893			−0.486947	+	0.873432i	$0.661889\pi$
$450$		0			0
$451$	16112.5			1.68228
$452$			5402.94i			0.562241i
$453$		0			0
$454$	−15780.8			−1.63134
$455$	4637.28	+	1247.25i	0.477800	+	0.128510i
$456$		0			0
$457$			6314.89i			0.646385i	0.946333	+	0.323193i	$0.104756\pi$
$457$			6314.89i			0.646385i	−0.946333	+	0.323193i	$0.895244\pi$
$458$			16574.5i			1.69099i
$459$		0			0
$460$	−1562.87	+	5810.74i	−0.158411	+	0.588972i
$461$	−4870.70			−0.492084			−0.246042	−	0.969259i	$-0.579130\pi$
$461$	−4870.70			−0.492084			−0.246042	+	0.969259i	$0.579130\pi$
$462$		0			0
$463$		−	4892.25i		−	0.491063i	−0.969389	−	0.245531i	$-0.921038\pi$
$463$		−	4892.25i		−	0.491063i	0.969389	−	0.245531i	$-0.0789624\pi$
$464$	3163.91			0.316553
$465$		0			0
$466$	−8178.97			−0.813055
$467$			6854.09i			0.679164i	0.940577	+	0.339582i	$0.110286\pi$
$467$			6854.09i			0.679164i	−0.940577	+	0.339582i	$0.889714\pi$
$468$		0			0
$469$	−5792.61			−0.570315
$470$	3276.72	+	881.312i	0.321583	+	0.0864933i
$471$		0			0
$472$		−	8118.79i		−	0.791732i
$473$			7834.41i			0.761579i
$474$		0			0
$475$	−1707.73	−	990.265i	−0.164960	−	0.0956558i
$476$	1464.05			0.140976
$477$		0			0
$478$		−	12175.3i		−	1.16504i
$479$	−8572.26			−0.817696			−0.408848	−	0.912602i	$-0.634069\pi$
$479$	−8572.26			−0.817696			−0.408848	+	0.912602i	$0.634069\pi$
$480$		0			0
$481$	9178.36			0.870057
$482$			21028.0i			1.98713i
$483$		0			0
$484$	−11411.4			−1.07170
$485$	4078.22	−	15162.8i	0.381819	−	1.41960i
$486$		0			0
$487$		−	5413.38i		−	0.503704i	−0.967766	−	0.251852i	$-0.918960\pi$
$487$		−	5413.38i		−	0.503704i	0.967766	−	0.251852i	$-0.0810396\pi$
$488$			5380.62i			0.499117i
$489$		0			0
$490$	3167.25	−	11775.8i	0.292004	−	1.08567i
$491$	−10314.3			−0.948025			−0.474012	−	0.880518i	$-0.657195\pi$
$491$	−10314.3			−0.948025			−0.474012	+	0.880518i	$0.657195\pi$
$492$		0			0
$493$		−	2050.62i		−	0.187333i
$494$	3984.64			0.362909
$495$		0			0
$496$	19101.1			1.72916
$497$		−	971.282i		−	0.0876619i
$498$		0			0
$499$	13527.2			1.21355			0.606774	−	0.794874i	$-0.292463\pi$
$499$	13527.2			1.21355			0.606774	+	0.794874i	$0.292463\pi$
$500$	−4616.30	−	4642.53i	−0.412894	−	0.415241i
$501$		0			0
$502$		−	8517.08i		−	0.757243i
$503$			245.421i			0.0217550i	0.999941	+	0.0108775i	$0.00346249\pi$
$503$			245.421i			0.0217550i	−0.999941	+	0.0108775i	$0.996538\pi$
$504$		0			0
$505$	−11988.6	−	3224.46i	−1.05640	−	0.284132i
$506$	−25112.9			−2.20634
$507$		0			0
$508$		−	9269.96i		−	0.809622i
$509$	−4680.90			−0.407617			−0.203809	−	0.979011i	$-0.565332\pi$
$509$	−4680.90			−0.407617			−0.203809	+	0.979011i	$0.565332\pi$
$510$		0			0
$511$	−4258.35			−0.368647
$512$		−	7502.22i		−	0.647567i
$513$		0			0
$514$	−12752.9			−1.09437
$515$	−1480.80	+	5505.60i	−0.126702	+	0.471079i
$516$		0			0
$517$			5230.03i			0.444906i
$518$			2797.62i			0.237298i
$519$		0			0
$520$	−9031.33	−	2429.08i	−0.761635	−	0.204850i
$521$	10212.1			0.858735			0.429368	−	0.903130i	$-0.358736\pi$
$521$	10212.1			0.858735			0.429368	+	0.903130i	$0.358736\pi$
$522$		0			0
$523$			8159.59i			0.682207i	0.940026	+	0.341103i	$0.110801\pi$
$523$			8159.59i			0.682207i	−0.940026	+	0.341103i	$0.889199\pi$
$524$	750.919			0.0626032
$525$		0			0
$526$	6067.38			0.502947
$527$		−	12379.9i		−	1.02330i
$528$		0			0
$529$	−1031.66			−0.0847913
$530$	−12619.6	−	3394.18i	−1.03426	−	0.278177i
$531$		0			0
$532$			448.550i			0.0365547i
$533$		−	18597.9i		−	1.51138i
$534$		0			0
$535$	−5076.99	+	18876.3i	−0.410276	+	1.52541i
$536$	11281.4			0.909108
$537$		0			0
$538$		−	14556.6i		−	1.16650i
$539$	18795.6			1.50201
$540$		0			0
$541$	−7889.11			−0.626949			−0.313474	−	0.949597i	$-0.601493\pi$
$541$	−7889.11			−0.626949			−0.313474	+	0.949597i	$0.601493\pi$
$542$			16284.8i			1.29057i
$543$		0			0
$544$	−9731.58			−0.766982
$545$	−14654.9	−	3941.60i	−1.15183	−	0.309797i
$546$		0			0
$547$		−	1394.62i		−	0.109012i	−0.998513	−	0.0545060i	$-0.982642\pi$
$547$		−	1394.62i		−	0.109012i	0.998513	−	0.0545060i	$-0.0173584\pi$
$548$		−	3098.11i		−	0.241505i
$549$		0			0
$550$	13706.6	−	23637.3i	1.06264	−	1.83254i
$551$	628.261			0.0485750
$552$		0			0
$553$		−	396.275i		−	0.0304726i
$554$	−29684.1			−2.27645
$555$		0			0
$556$	9960.33			0.759734
$557$			19617.8i			1.49234i	0.665756	+	0.746169i	$0.268109\pi$
$557$			19617.8i			1.49234i	−0.665756	+	0.746169i	$0.731891\pi$
$558$		0			0
$559$	9042.92			0.684212
$560$	1400.22	−	5206.02i	0.105661	−	0.392847i
$561$		0			0
$562$			8870.03i			0.665765i
$563$			1339.24i			0.100253i	0.998743	+	0.0501263i	$0.0159624\pi$
$563$			1339.24i			0.100253i	−0.998743	+	0.0501263i	$0.984038\pi$
$564$		0			0
$565$	3349.12	−	12452.0i	0.249378	−	0.927188i
$566$	25403.6			1.88656
$567$		0			0
$568$			1891.62i			0.139737i
$569$	26030.2			1.91783			0.958914	−	0.283696i	$-0.0915606\pi$
$569$	26030.2			1.91783			0.958914	+	0.283696i	$0.0915606\pi$
$570$		0			0
$571$	−10020.8			−0.734426			−0.367213	−	0.930137i	$-0.619688\pi$
$571$	−10020.8			−0.734426			−0.367213	+	0.930137i	$0.619688\pi$
$572$			20368.8i			1.48892i
$573$		0			0
$574$	5668.76			0.412211
$575$	7203.81	−	12423.1i	0.522469	−	0.901009i
$576$		0			0
$577$			11192.7i			0.807550i	0.914858	+	0.403775i	$0.132302\pi$
$577$			11192.7i			0.807550i	−0.914858	+	0.403775i	$0.867698\pi$
$578$		−	8034.75i		−	0.578204i
$579$		0			0
$580$	2012.11	+	541.181i	0.144049	+	0.0387437i
$581$	−6042.41			−0.431465
$582$		0			0
$583$		−	20142.3i		−	1.43089i
$584$	8293.35			0.587639
$585$		0			0
$586$	−17505.9			−1.23406
$587$			11087.1i			0.779578i	0.920904	+	0.389789i	$0.127452\pi$
$587$			11087.1i			0.779578i	−0.920904	+	0.389789i	$0.872548\pi$
$588$		0			0
$589$	3792.92			0.265339
$590$	7111.18	−	26439.4i	0.496208	−	1.84490i
$591$		0			0
$592$		−	10304.0i		−	0.715361i
$593$		−	14525.9i		−	1.00592i	−0.864311	−	0.502958i	$-0.832245\pi$
$593$		−	14525.9i		−	1.00592i	0.864311	−	0.502958i	$-0.167755\pi$
$594$		0			0
$595$	−3374.16	−	907.521i	−0.232483	−	0.0625289i
$596$	−3141.76			−0.215925
$597$		0			0
$598$			28986.8i			1.98220i
$599$	16502.5			1.12567			0.562833	−	0.826570i	$-0.309711\pi$
$599$	16502.5			1.12567			0.562833	+	0.826570i	$0.309711\pi$
$600$		0			0
$601$	15945.4			1.08224			0.541119	−	0.840946i	$-0.318001\pi$
$601$	15945.4			1.08224			0.541119	+	0.840946i	$0.318001\pi$
$602$			2756.34i			0.186611i
$603$		0			0
$604$	−6747.89			−0.454582
$605$	26299.7	+	7073.62i	1.76733	+	0.475345i
$606$		0			0
$607$			935.699i			0.0625681i	0.999511	+	0.0312841i	$0.00995965\pi$
$607$			935.699i			0.0625681i	−0.999511	+	0.0312841i	$0.990040\pi$
$608$		−	2981.53i		−	0.198877i
$609$		0			0
$610$	−4712.84	+	17522.4i	−0.312815	+	1.16305i
$611$	6036.80			0.399710
$612$		0			0
$613$			375.796i			0.0247606i	0.999923	+	0.0123803i	$0.00394087\pi$
$613$			375.796i			0.0247606i	−0.999923	+	0.0123803i	$0.996059\pi$
$614$	−20652.9			−1.35746
$615$		0			0
$616$	4393.80			0.287388
$617$			14932.2i			0.974306i	0.873317	+	0.487153i	$0.161965\pi$
$617$			14932.2i			0.974306i	−0.873317	+	0.487153i	$0.838035\pi$
$618$		0			0
$619$	−11320.8			−0.735090			−0.367545	−	0.930006i	$-0.619802\pi$
$619$	−11320.8			−0.735090			−0.367545	+	0.930006i	$0.619802\pi$
$620$	12147.5	+	3267.21i	0.786862	+	0.211636i
$621$		0			0
$622$			215.095i			0.0138658i
$623$		−	8110.38i		−	0.521566i
$624$		0			0
$625$	7761.33	+	13561.1i	0.496725	+	0.867908i
$626$	2155.83			0.137643
$627$		0			0
$628$			14228.6i			0.904115i
$629$	−6678.34			−0.423343
$630$		0			0
$631$	−15521.7			−0.979253			−0.489627	−	0.871932i	$-0.662867\pi$
$631$	−15521.7			−0.979253			−0.489627	+	0.871932i	$0.662867\pi$
$632$			771.765i			0.0485746i
$633$		0			0
$634$	11199.7			0.701571
$635$	−5746.18	+	21364.3i	−0.359102	+	1.33514i
$636$		0			0
$637$		−	21695.0i		−	1.34943i
$638$			8695.98i			0.539619i
$639$		0			0
$640$	−4012.03	+	14916.7i	−0.247796	+	0.921306i
$641$	20110.1			1.23916			0.619579	−	0.784934i	$-0.287303\pi$
$641$	20110.1			1.23916			0.619579	+	0.784934i	$0.287303\pi$
$642$		0			0
$643$			16073.4i			0.985808i	0.870084	+	0.492904i	$0.164065\pi$
$643$			16073.4i			0.985808i	−0.870084	+	0.492904i	$0.835935\pi$
$644$	−3263.04			−0.199661
$645$		0			0
$646$	−2899.29			−0.176581
$647$		−	3841.57i		−	0.233427i	−0.993166	−	0.116714i	$-0.962764\pi$
$647$		−	3841.57i		−	0.233427i	0.993166	−	0.116714i	$-0.0372360\pi$
$648$		0			0
$649$	42200.4			2.55240
$650$	−27283.5	−	15820.9i	−1.64638	−	0.954690i
$651$		0			0
$652$		−	5824.30i		−	0.349842i
$653$		−	14694.4i		−	0.880609i	−0.897849	−	0.440304i	$-0.854871\pi$
$653$		−	14694.4i		−	0.880609i	0.897849	−	0.440304i	$-0.145129\pi$
$654$		0			0
$655$	−1730.63	−	465.473i	−0.103239	−	0.0277672i
$656$	−20878.9			−1.24266
$657$		0			0
$658$			1840.05i			0.109016i
$659$	−8601.76			−0.508463			−0.254231	−	0.967143i	$-0.581823\pi$
$659$	−8601.76			−0.508463			−0.254231	+	0.967143i	$0.581823\pi$
$660$		0			0
$661$	−21709.0			−1.27743			−0.638717	−	0.769442i	$-0.720534\pi$
$661$	−21709.0			−1.27743			−0.638717	+	0.769442i	$0.720534\pi$
$662$			7560.96i			0.443905i
$663$		0			0
$664$	11767.9			0.687775
$665$	278.043	−	1033.77i	0.0162136	−	0.0602822i
$666$		0			0
$667$			4570.36i			0.265315i
$668$			774.960i			0.0448864i
$669$		0			0
$670$	36738.6	+	9881.28i	2.11841	+	0.569772i
$671$	−27967.7			−1.60907
$672$		0			0
$673$			13192.3i			0.755611i	0.925885	+	0.377806i	$0.123321\pi$
$673$			13192.3i			0.755611i	−0.925885	+	0.377806i	$0.876679\pi$
$674$	5293.85			0.302539
$675$		0			0
$676$	13218.7			0.752086
$677$			4885.74i			0.277362i	0.990337	+	0.138681i	$0.0442863\pi$
$677$			4885.74i			0.277362i	−0.990337	+	0.138681i	$0.955714\pi$
$678$		0			0
$679$	8514.72			0.481245
$680$	6571.35	+	1767.44i	0.370588	+	0.0996739i
$681$		0			0
$682$			52499.1i			2.94765i
$683$		−	17180.4i		−	0.962502i	−0.876583	−	0.481251i	$-0.840182\pi$
$683$		−	17180.4i		−	0.962502i	0.876583	−	0.481251i	$-0.159818\pi$
$684$		0			0
$685$	−1920.43	+	7140.15i	−0.107118	+	0.398264i
$686$	14019.3			0.780259
$687$		0			0
$688$		−	10152.0i		−	0.562560i
$689$	−23249.4			−1.28553
$690$		0			0
$691$	−21428.3			−1.17970			−0.589850	−	0.807513i	$-0.700813\pi$
$691$	−21428.3			−1.17970			−0.589850	+	0.807513i	$0.700813\pi$
$692$		−	9046.57i		−	0.496964i
$693$		0			0
$694$	−27243.6			−1.49013
$695$	−22955.4	−	6174.11i	−1.25287	−	0.336975i
$696$		0			0
$697$			13532.2i			0.735391i
$698$		−	4001.80i		−	0.217006i
$699$		0			0
$700$	1780.96	−	3071.31i	0.0961630	−	0.165835i
$701$	−9809.73			−0.528543			−0.264271	−	0.964448i	$-0.585131\pi$
$701$	−9809.73			−0.528543			−0.264271	+	0.964448i	$0.585131\pi$
$702$		0			0
$703$		−	2046.09i		−	0.109772i
$704$	2218.40			0.118763
$705$		0			0
$706$	−44749.3			−2.38550
$707$		−	6732.21i		−	0.358120i
$708$		0			0
$709$	−13541.9			−0.717315			−0.358658	−	0.933469i	$-0.616765\pi$
$709$	−13541.9			−0.717315			−0.358658	+	0.933469i	$0.616765\pi$
$710$	−1656.86	+	6160.19i	−0.0875784	+	0.325617i
$711$		0			0
$712$			15795.4i			0.831399i
$713$			27592.1i			1.44927i
$714$		0			0
$715$	12626.0	−	46943.7i	0.660402	−	2.45538i
$716$	−2922.64			−0.152548
$717$		0			0
$718$			8871.89i			0.461137i
$719$	−344.757			−0.0178822			−0.00894109	−	0.999960i	$-0.502846\pi$
$719$	−344.757			−0.0178822			−0.00894109	+	0.999960i	$0.502846\pi$
$720$		0			0
$721$	−3091.69			−0.159696
$722$			23540.4i			1.21341i
$723$		0			0
$724$	−5336.40			−0.273930
$725$	−4301.82	−	2494.50i	−0.220366	−	0.127784i
$726$		0			0
$727$		−	10425.2i		−	0.531841i	−0.963995	−	0.265921i	$-0.914324\pi$
$727$		−	10425.2i		−	0.531841i	0.963995	−	0.265921i	$-0.0856759\pi$
$728$		−	5071.57i		−	0.258193i
$729$		0			0
$730$	27007.9	+	7264.08i	1.36932	+	0.368296i
$731$	−6579.79			−0.332917
$732$		0			0
$733$		−	12660.4i		−	0.637956i	−0.947762	−	0.318978i	$-0.896660\pi$
$733$		−	12660.4i		−	0.637956i	0.947762	−	0.318978i	$-0.103340\pi$
$734$	46165.0			2.32150
$735$		0			0
$736$	21689.5			1.08626
$737$			58639.2i			2.93080i
$738$		0			0
$739$	35663.9			1.77526			0.887630	−	0.460557i	$-0.152350\pi$
$739$	35663.9			1.77526			0.887630	+	0.460557i	$0.152350\pi$
$740$	1762.49	−	6552.95i	0.0875547	−	0.325528i
$741$		0			0
$742$		−	7086.56i		−	0.350614i
$743$			9880.98i			0.487884i	0.969790	+	0.243942i	$0.0784407\pi$
$743$			9880.98i			0.487884i	−0.969790	+	0.243942i	$0.921559\pi$
$744$		0			0
$745$	7240.76	+	1947.49i	0.356082	+	0.0957723i
$746$	−17970.6			−0.881973
$747$		0			0
$748$		−	14820.7i		−	0.724464i
$749$	−10600.0			−0.517112
$750$		0			0
$751$	−7951.18			−0.386342			−0.193171	−	0.981165i	$-0.561877\pi$
$751$	−7951.18			−0.386342			−0.193171	+	0.981165i	$0.561877\pi$
$752$		−	6777.18i		−	0.328641i
$753$		0			0
$754$	10037.4			0.484801
$755$	15551.7	+	4182.82i	0.749650	+	0.201627i
$756$		0			0
$757$		−	26539.3i		−	1.27422i	−0.770771	−	0.637112i	$-0.780129\pi$
$757$		−	26539.3i		−	1.27422i	0.770771	−	0.637112i	$-0.219871\pi$
$758$			1274.30i			0.0610614i
$759$		0			0
$760$	−541.503	+	2013.31i	−0.0258452	+	0.0960926i
$761$	6411.70			0.305419			0.152710	−	0.988271i	$-0.451200\pi$
$761$	6411.70			0.305419			0.152710	+	0.988271i	$0.451200\pi$
$762$		0			0
$763$		−	8229.49i		−	0.390468i
$764$	2192.70			0.103834
$765$		0			0
$766$	−3157.49			−0.148936
$767$		−	48710.1i		−	2.29311i
$768$		0			0
$769$	−21232.0			−0.995636			−0.497818	−	0.867282i	$-0.665865\pi$
$769$	−21232.0			−0.995636			−0.497818	+	0.867282i	$0.665865\pi$
$770$	14308.7	+	3848.49i	0.669675	+	0.180117i
$771$		0			0
$772$			11614.1i			0.541450i
$773$		−	12948.2i		−	0.602478i	−0.953549	−	0.301239i	$-0.902600\pi$
$773$		−	12948.2i		−	0.602478i	0.953549	−	0.301239i	$-0.0974003\pi$
$774$		0			0
$775$	−25970.8	−	15059.7i	−1.20374	−	0.698015i
$776$	−16582.8			−0.767126
$777$		0			0
$778$		−	29507.2i		−	1.35975i
$779$	−4145.94			−0.190685
$780$		0			0
$781$	−9832.39			−0.450487
$782$		−	21091.3i		−	0.964478i
$783$		0			0
$784$	−24355.8			−1.10950
$785$	8819.91	−	32792.4i	0.401014	−	1.49097i
$786$		0			0
$787$		−	16388.9i		−	0.742315i	−0.928570	−	0.371158i	$-0.878961\pi$
$787$		−	16388.9i		−	0.742315i	0.928570	−	0.371158i	$-0.121039\pi$
$788$		−	11251.9i		−	0.508672i
$789$		0			0
$790$	−675.983	+	2513.31i	−0.0304435	+	0.113189i
$791$	6992.48			0.314316
$792$		0			0
$793$			32281.9i			1.44561i
$794$	15936.4			0.712294
$795$		0			0
$796$	−4867.45			−0.216737
$797$		−	22291.8i		−	0.990734i	−0.868684	−	0.495367i	$-0.835034\pi$
$797$		−	22291.8i		−	0.990734i	0.868684	−	0.495367i	$-0.164966\pi$
$798$		0			0
$799$	−4392.48			−0.194486
$800$	−11838.1	+	20415.1i	−0.523176	+	0.902227i
$801$		0			0
$802$		−	37604.0i		−	1.65567i
$803$			43107.8i			1.89445i
$804$		0			0
$805$	7520.26	+	2022.66i	0.329260	+	0.0885583i
$806$	60597.4			2.64821
$807$		0			0
$808$			13111.3i			0.570859i
$809$	−36536.8			−1.58784			−0.793921	−	0.608020i	$-0.791964\pi$
$809$	−36536.8			−1.58784			−0.793921	+	0.608020i	$0.791964\pi$
$810$		0			0
$811$	−22132.8			−0.958309			−0.479155	−	0.877730i	$-0.659057\pi$
$811$	−22132.8			−0.958309			−0.479155	+	0.877730i	$0.659057\pi$
$812$			1129.91i			0.0488325i
$813$		0			0
$814$	28320.6			1.21945
$815$	−3610.31	+	13423.1i	−0.155170	+	0.576923i
$816$		0			0
$817$		−	2015.89i		−	0.0863245i
$818$		−	18635.6i		−	0.796552i
$819$		0			0
$820$	−13278.1	−	3571.30i	−0.565477	−	0.152092i
$821$	28628.6			1.21698			0.608492	−	0.793560i	$-0.291775\pi$
$821$	28628.6			1.21698			0.608492	+	0.793560i	$0.291775\pi$
$822$		0			0
$823$		−	42695.0i		−	1.80833i	−0.427185	−	0.904164i	$-0.640495\pi$
$823$		−	42695.0i		−	1.80833i	0.427185	−	0.904164i	$-0.359505\pi$
$824$	6021.21			0.254562
$825$		0			0
$826$	14847.1			0.625421
$827$		−	25905.2i		−	1.08925i	−0.838678	−	0.544627i	$-0.816671\pi$
$827$		−	25905.2i		−	1.08925i	0.838678	−	0.544627i	$-0.183329\pi$
$828$		0			0
$829$	9952.93			0.416984			0.208492	−	0.978024i	$-0.433144\pi$
$829$	9952.93			0.416984			0.208492	+	0.978024i	$0.433144\pi$
$830$	38323.0	+	10307.4i	1.60266	+	0.431054i
$831$		0			0
$832$		−	2560.60i		−	0.106698i
$833$			15785.6i			0.656591i
$834$		0			0
$835$	480.375	−	1786.04i	0.0199091	−	0.0740220i
$836$	4540.72			0.187852
$837$		0			0
$838$			13903.6i			0.573142i
$839$	16474.0			0.677886			0.338943	−	0.940807i	$-0.389931\pi$
$839$	16474.0			0.677886			0.338943	+	0.940807i	$0.389931\pi$
$840$		0			0
$841$	−22806.4			−0.935110
$842$		−	22348.1i		−	0.914689i
$843$		0			0
$844$	−15515.5			−0.632781
$845$	−30464.8	−	8193.86i	−1.24026	−	0.333583i
$846$		0			0
$847$			14768.7i			0.599124i
$848$			26100.9i			1.05697i
$849$		0			0
$850$	19852.0	+	11511.6i	0.801079	+	0.464523i
$851$	14884.5			0.599571
$852$		0			0
$853$		−	22391.0i		−	0.898772i	−0.893338	−	0.449386i	$-0.851643\pi$
$853$		−	22391.0i		−	0.898772i	0.893338	−	0.449386i	$-0.148357\pi$
$854$	−9839.73			−0.394272
$855$		0			0
$856$	20644.1			0.824299
$857$		−	15349.4i		−	0.611814i	−0.952061	−	0.305907i	$-0.901040\pi$
$857$		−	15349.4i		−	0.611814i	0.952061	−	0.305907i	$-0.0989596\pi$
$858$		0			0
$859$	5218.87			0.207294			0.103647	−	0.994614i	$-0.466949\pi$
$859$	5218.87			0.207294			0.103647	+	0.994614i	$0.466949\pi$
$860$	1736.48	−	6456.25i	0.0688530	−	0.255996i
$861$		0			0
$862$		−	28636.3i		−	1.13150i
$863$			12101.4i			0.477330i	0.971102	+	0.238665i	$0.0767097\pi$
$863$			12101.4i			0.477330i	−0.971102	+	0.238665i	$0.923290\pi$
$864$		0			0
$865$	−5607.70	+	20849.5i	−0.220425	+	0.819541i
$866$	−11116.2			−0.436196
$867$		0			0
$868$			6821.45i			0.266746i
$869$	−4011.53			−0.156596
$870$		0			0
$871$	67684.6			2.63307
$872$			16027.3i			0.622424i
$873$		0			0
$874$	6461.87			0.250087
$875$	−6008.36	+	5974.42i	−0.232137	+	0.230825i
$876$		0			0
$877$		−	24268.8i		−	0.934435i	−0.884142	−	0.467218i	$-0.845256\pi$
$877$		−	24268.8i		−	0.934435i	0.884142	−	0.467218i	$-0.154744\pi$
$878$		−	8451.75i		−	0.324866i
$879$		0			0
$880$	−52701.1	−	14174.6i	−2.01881	−	0.542982i
$881$	−24286.6			−0.928758			−0.464379	−	0.885637i	$-0.653722\pi$
$881$	−24286.6			−0.928758			−0.464379	+	0.885637i	$0.653722\pi$
$882$		0			0
$883$		−	48922.2i		−	1.86451i	−0.361801	−	0.932255i	$-0.617838\pi$
$883$		−	48922.2i		−	1.86451i	0.361801	−	0.932255i	$-0.382162\pi$
$884$	−17106.9			−0.650868
$885$		0			0
$886$	−46844.2			−1.77626
$887$			1678.06i			0.0635215i	0.999495	+	0.0317608i	$0.0101115\pi$
$887$			1678.06i			0.0635215i	−0.999495	+	0.0317608i	$0.989889\pi$
$888$		0			0
$889$	−11997.2			−0.452613
$890$	−13835.0	+	51438.7i	−0.521069	+	1.93734i
$891$		0			0
$892$			18671.4i			0.700859i
$893$		−	1345.75i		−	0.0504299i
$894$		0			0
$895$	6735.75	+	1811.66i	0.251566	+	0.0676615i
$896$	−8376.54			−0.312322
$897$		0			0
$898$			33000.5i			1.22633i
$899$	9554.45			0.354459
$900$		0			0
$901$	16916.7			0.625501
$902$		−	57385.4i		−	2.11832i
$903$		0			0
$904$	−13618.2			−0.501034
$905$	12298.7	+	3307.88i	0.451737	+	0.121500i
$906$		0			0
$907$		−	14710.6i		−	0.538541i	−0.963065	−	0.269271i	$-0.913217\pi$
$907$		−	14710.6i		−	0.538541i	0.963065	−	0.269271i	$-0.0867826\pi$
$908$			20757.2i			0.758646i
$909$		0			0
$910$	4442.15	−	16515.9i	0.161820	−	0.601645i
$911$	−30257.7			−1.10042			−0.550209	−	0.835027i	$-0.685452\pi$
$911$	−30257.7			−1.10042			−0.550209	+	0.835027i	$0.685452\pi$
$912$		0			0
$913$			61168.0i			2.21727i
$914$	22490.8			0.813928
$915$		0			0
$916$	21801.1			0.786385
$917$		−	971.839i		−	0.0349978i
$918$		0			0
$919$	−51369.1			−1.84386			−0.921931	−	0.387355i	$-0.873389\pi$
$919$	−51369.1			−1.84386			−0.921931	+	0.387355i	$0.873389\pi$
$920$	−14646.1	−	3939.23i	−0.524855	−	0.141166i
$921$		0			0
$922$			17347.2i			0.619632i
$923$			11349.1i			0.404724i
$924$		0			0
$925$	−8123.96	+	14009.9i	−0.288772	+	0.497993i
$926$	−17424.0			−0.618346
$927$		0			0
$928$		−	7510.54i		−	0.265674i
$929$	40968.5			1.44686			0.723430	−	0.690398i	$-0.242564\pi$
$929$	40968.5			1.44686			0.723430	+	0.690398i	$0.242564\pi$
$930$		0			0
$931$	−4836.35			−0.170252
$932$			10758.1i			0.378106i
$933$		0			0
$934$	24411.2			0.855202
$935$	−9186.93	+	34157.0i	−0.321331	+	1.19471i
$936$		0			0
$937$			26835.8i			0.935632i	0.883826	+	0.467816i	$0.154959\pi$
$937$			26835.8i			0.935632i	−0.883826	+	0.467816i	$0.845041\pi$
$938$			20630.7i			0.718140i
$939$		0			0
$940$	1159.23	−	4310.00i	0.0402232	−	0.149550i
$941$	−41221.5			−1.42803			−0.714017	−	0.700128i	$-0.753126\pi$
$941$	−41221.5			−1.42803			−0.714017	+	0.700128i	$0.753126\pi$
$942$		0			0
$943$		−	30160.2i		−	1.04152i
$944$	−54684.1			−1.88540
$945$		0			0
$946$	27902.7			0.958979
$947$			14488.5i			0.497164i	0.968611	+	0.248582i	$0.0799645\pi$
$947$			14488.5i			0.497164i	−0.968611	+	0.248582i	$0.920035\pi$
$948$		0			0
$949$	49757.4			1.70200
$950$	−3526.88	+	6082.18i	−0.120450	+	0.207718i
$951$		0			0
$952$			3690.16i			0.125629i
$953$		−	32207.2i		−	1.09475i	−0.836889	−	0.547373i	$-0.815628\pi$
$953$		−	32207.2i		−	1.09475i	0.836889	−	0.547373i	$-0.184372\pi$
$954$		0			0
$955$	−5053.47	−	1359.19i	−0.171232	−	0.0460548i
$956$	−16014.7			−0.541792
$957$		0			0
$958$			30530.5i			1.02964i
$959$	−4009.57			−0.135011
$960$		0			0
$961$	27890.8			0.936217
$962$		−	32689.2i		−	1.09557i
$963$		0			0
$964$	27659.0			0.924104
$965$	7199.21	−	26766.7i	0.240156	−	0.892902i
$966$		0			0
$967$			20552.1i			0.683464i	0.939797	+	0.341732i	$0.111014\pi$
$967$			20552.1i			0.683464i	−0.939797	+	0.341732i	$0.888986\pi$
$968$		−	28762.7i		−	0.955030i
$969$		0			0
$970$	−54003.2	−	14524.8i	−1.78756	−	0.480786i
$971$	23686.9			0.782852			0.391426	−	0.920210i	$-0.371982\pi$
$971$	23686.9			0.782852			0.391426	+	0.920210i	$0.371982\pi$
$972$		0			0
$973$		−	12890.7i		−	0.424723i
$974$	−19280.0			−0.634263
$975$		0			0
$976$	36241.2			1.18858
$977$		−	12597.1i		−	0.412506i	−0.978499	−	0.206253i	$-0.933873\pi$
$977$		−	12597.1i		−	0.412506i	0.978499	−	0.206253i	$-0.0661269\pi$
$978$		0			0
$979$	−82102.2			−2.68028
$980$	−15489.3	−	4166.02i	−0.504884	−	0.135794i
$981$		0			0
$982$			36735.1i			1.19375i
$983$			51076.9i			1.65728i	0.559785	+	0.828638i	$0.310884\pi$
$983$			51076.9i			1.65728i	−0.559785	+	0.828638i	$0.689116\pi$
$984$		0			0
$985$	−6974.74	+	25932.1i	−0.225618	+	0.838848i
$986$	−7303.38			−0.235889
$987$		0			0
$988$		−	5241.16i		−	0.168769i
$989$	14664.9			0.471502
$990$		0			0
$991$	27888.6			0.893957			0.446979	−	0.894545i	$-0.352500\pi$
$991$	27888.6			0.893957			0.446979	+	0.894545i	$0.352500\pi$
$992$		−	45342.4i		−	1.45123i
$993$		0			0
$994$	−3459.27			−0.110384
$995$	11217.9	+	3017.19i	0.357419	+	0.0961320i
$996$		0			0
$997$			54930.0i			1.74488i	0.488717	+	0.872442i	$0.337465\pi$
$997$			54930.0i			1.74488i	−0.488717	+	0.872442i	$0.662535\pi$
$998$		−	48177.8i		−	1.52810i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.b.b.109.2	yes	8
3.2	odd	2	inner	135.4.b.b.109.7	yes	8
5.2	odd	4		675.4.a.ba.1.3		4
5.3	odd	4		675.4.a.t.1.2		4
5.4	even	2	inner	135.4.b.b.109.8	yes	8
15.2	even	4		675.4.a.t.1.1		4
15.8	even	4		675.4.a.ba.1.4		4
15.14	odd	2	inner	135.4.b.b.109.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.b.109.1	✓	8	15.14	odd	2	inner
135.4.b.b.109.2	yes	8	1.1	even	1	trivial
135.4.b.b.109.7	yes	8	3.2	odd	2	inner
135.4.b.b.109.8	yes	8	5.4	even	2	inner
675.4.a.t.1.1		4	15.2	even	4
675.4.a.t.1.2		4	5.3	odd	4
675.4.a.ba.1.3		4	5.2	odd	4
675.4.a.ba.1.4		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.56155i q^{2} -4.68466 q^{4} +(10.7966 + 2.90388i) q^{5} +6.06288i q^{7} -11.8078i q^{8} +O(q^{10})\)	\(q-3.56155i q^{2} -4.68466 q^{4} +(10.7966 + 2.90388i) q^{5} +6.06288i q^{7} -11.8078i q^{8} +(10.3423 - 38.4528i) q^{10} +61.3752 q^{11} -70.8427i q^{13} +21.5933 q^{14} -79.5312 q^{16} +51.5464i q^{17} -15.7926 q^{19} +(-50.5786 - 13.6037i) q^{20} -218.591i q^{22} -114.885i q^{23} +(108.135 + 62.7043i) q^{25} -252.310 q^{26} -28.4025i q^{28} -39.7819 q^{29} -240.170 q^{31} +188.793i q^{32} +183.585 q^{34} +(-17.6059 + 65.4588i) q^{35} +129.560i q^{37} +56.2462i q^{38} +(34.2884 - 127.484i) q^{40} +262.524 q^{41} +127.648i q^{43} -287.522 q^{44} -409.170 q^{46} +85.2140i q^{47} +306.241 q^{49} +(223.325 - 385.128i) q^{50} +331.874i q^{52} -328.184i q^{53} +(662.646 + 178.226i) q^{55} +71.5891 q^{56} +141.685i q^{58} +687.580 q^{59} -455.685 q^{61} +855.380i q^{62} +36.1449 q^{64} +(205.719 - 764.864i) q^{65} +955.421i q^{67} -241.477i q^{68} +(233.135 + 62.7043i) q^{70} -160.201 q^{71} +702.364i q^{73} +461.434 q^{74} +73.9830 q^{76} +372.111i q^{77} -65.3608 q^{79} +(-858.670 - 230.949i) q^{80} -934.993i q^{82} +996.623i q^{83} +(-149.685 + 556.528i) q^{85} +454.624 q^{86} -724.704i q^{88} -1337.71 q^{89} +429.511 q^{91} +538.199i q^{92} +303.494 q^{94} +(-170.507 - 45.8599i) q^{95} -1404.40i q^{97} -1090.70i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{4}+O(q^{10}) \) 8 * q + 12 * q^4	\( 8 q + 12 q^{4} + 58 q^{10} - 92 q^{16} + 220 q^{19} + 494 q^{25} - 536 q^{31} + 776 q^{34} + 398 q^{40} - 1888 q^{46} - 964 q^{49} + 774 q^{55} - 3596 q^{61} + 1724 q^{64} + 1494 q^{70} + 2472 q^{76} - 1364 q^{79} - 1148 q^{85} - 720 q^{91} + 152 q^{94}+O(q^{100}) \) 8 * q + 12 * q^4 + 58 * q^10 - 92 * q^16 + 220 * q^19 + 494 * q^25 - 536 * q^31 + 776 * q^34 + 398 * q^40 - 1888 * q^46 - 964 * q^49 + 774 * q^55 - 3596 * q^61 + 1724 * q^64 + 1494 * q^70 + 2472 * q^76 - 1364 * q^79 - 1148 * q^85 - 720 * q^91 + 152 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more