LMFDB - Embedded newform 216.2.i.b.73.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(73,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, 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("216.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.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:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			73.2
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	216.73
Dual form			216.2.i.b.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.18614 + 2.05446i) q^{5} +(-2.18614 + 3.78651i) q^{7} +O(q^{10})$	$q+(1.18614 + 2.05446i) q^{5} +(-2.18614 + 3.78651i) q^{7} +(0.500000 - 0.866025i) q^{11} +(0.186141 + 0.322405i) q^{13} +5.37228 q^{17} +0.627719 q^{19} +(-0.186141 - 0.322405i) q^{23} +(-0.313859 + 0.543620i) q^{25} +(-2.18614 + 3.78651i) q^{29} +(-3.18614 - 5.51856i) q^{31} -10.3723 q^{35} +8.74456 q^{37} +(-5.87228 - 10.1711i) q^{41} +(0.872281 - 1.51084i) q^{43} +(2.18614 - 3.78651i) q^{47} +(-6.05842 - 10.4935i) q^{49} -0.744563 q^{53} +2.37228 q^{55} +(3.50000 + 6.06218i) q^{59} +(-1.18614 + 2.05446i) q^{61} +(-0.441578 + 0.764836i) q^{65} +(1.87228 + 3.24289i) q^{67} +4.00000 q^{71} -12.1168 q^{73} +(2.18614 + 3.78651i) q^{77} +(3.18614 - 5.51856i) q^{79} +(4.81386 - 8.33785i) q^{83} +(6.37228 + 11.0371i) q^{85} -6.00000 q^{89} -1.62772 q^{91} +(0.744563 + 1.28962i) q^{95} +(-0.872281 + 1.51084i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} - 3 q^{7}+O(q^{10}) $ 4 * q - q^5 - 3 * q^7	$ 4 q - q^{5} - 3 q^{7} + 2 q^{11} - 5 q^{13} + 10 q^{17} + 14 q^{19} + 5 q^{23} - 7 q^{25} - 3 q^{29} - 7 q^{31} - 30 q^{35} + 12 q^{37} - 12 q^{41} - 8 q^{43} + 3 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 14 q^{59} + q^{61} - 19 q^{65} - 4 q^{67} + 16 q^{71} - 14 q^{73} + 3 q^{77} + 7 q^{79} + 25 q^{83} + 14 q^{85} - 24 q^{89} - 18 q^{91} - 20 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - q^5 - 3 * q^7 + 2 * q^11 - 5 * q^13 + 10 * q^17 + 14 * q^19 + 5 * q^23 - 7 * q^25 - 3 * q^29 - 7 * q^31 - 30 * q^35 + 12 * q^37 - 12 * q^41 - 8 * q^43 + 3 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 14 * q^59 + q^61 - 19 * q^65 - 4 * q^67 + 16 * q^71 - 14 * q^73 + 3 * q^77 + 7 * q^79 + 25 * q^83 + 14 * q^85 - 24 * q^89 - 18 * q^91 - 20 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	1.18614	+	2.05446i	0.530458	+	0.918781i	0.999368	+	0.0355348i	$0.0113134\pi$
$5$	1.18614	+	2.05446i	0.530458	+	0.918781i	−0.468910	+	0.883246i	$0.655353\pi$
$6$		0			0
$7$	−2.18614	+	3.78651i	−0.826284	+	1.43117i	0.0746509	+	0.997210i	$0.476216\pi$
$7$	−2.18614	+	3.78651i	−0.826284	+	1.43117i	−0.900934	+	0.433955i	$0.857118\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i	−0.780750	−	0.624844i	$-0.785163\pi$
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i	0.931505	+	0.363727i	$0.118496\pi$
$12$		0			0
$13$	0.186141	+	0.322405i	0.0516261	+	0.0894191i	0.890684	−	0.454624i	$-0.150226\pi$
$13$	0.186141	+	0.322405i	0.0516261	+	0.0894191i	−0.839057	+	0.544043i	$0.816893\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.37228			1.30297			0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228			1.30297			0.651485	+	0.758662i	$0.274146\pi$
$18$		0			0
$19$	0.627719			0.144009			0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719			0.144009			0.0720043	+	0.997404i	$0.477060\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.186141	−	0.322405i	−0.0388130	−	0.0672261i	0.845966	−	0.533236i	$-0.179024\pi$
$23$	−0.186141	−	0.322405i	−0.0388130	−	0.0672261i	−0.884779	+	0.466010i	$0.845691\pi$
$24$		0			0
$25$	−0.313859	+	0.543620i	−0.0627719	+	0.108724i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.18614	+	3.78651i	−0.405956	+	0.703137i	−0.994432	−	0.105378i	$-0.966395\pi$
$29$	−2.18614	+	3.78651i	−0.405956	+	0.703137i	0.588476	+	0.808515i	$0.299728\pi$
$30$		0			0
$31$	−3.18614	−	5.51856i	−0.572248	−	0.991162i	−0.996335	−	0.0855407i	$-0.972738\pi$
$31$	−3.18614	−	5.51856i	−0.572248	−	0.991162i	0.424087	−	0.905621i	$-0.360595\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.3723			−1.75324
$36$		0			0
$37$	8.74456			1.43760			0.718799	−	0.695218i	$-0.244692\pi$
$37$	8.74456			1.43760			0.718799	+	0.695218i	$0.244692\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.87228	−	10.1711i	−0.917096	−	1.58846i	−0.803803	−	0.594896i	$-0.797193\pi$
$41$	−5.87228	−	10.1711i	−0.917096	−	1.58846i	−0.113293	−	0.993562i	$-0.536140\pi$
$42$		0			0
$43$	0.872281	−	1.51084i	0.133022	−	0.230400i	−0.791818	−	0.610757i	$-0.790865\pi$
$43$	0.872281	−	1.51084i	0.133022	−	0.230400i	0.924840	+	0.380356i	$0.124199\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.18614	−	3.78651i	0.318881	−	0.552319i	−0.661374	−	0.750057i	$-0.730026\pi$
$47$	2.18614	−	3.78651i	0.318881	−	0.552319i	0.980255	+	0.197738i	$0.0633595\pi$
$48$		0			0
$49$	−6.05842	−	10.4935i	−0.865489	−	1.49907i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.744563			−0.102274			−0.0511368	−	0.998692i	$-0.516284\pi$
$53$	−0.744563			−0.102274			−0.0511368	+	0.998692i	$0.516284\pi$
$54$		0			0
$55$	2.37228			0.319878
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.50000	+	6.06218i	0.455661	+	0.789228i	0.998726	−	0.0504625i	$-0.0160695\pi$
$59$	3.50000	+	6.06218i	0.455661	+	0.789228i	−0.543065	+	0.839691i	$0.682736\pi$
$60$		0			0
$61$	−1.18614	+	2.05446i	−0.151870	+	0.263046i	−0.931915	−	0.362677i	$-0.881863\pi$
$61$	−1.18614	+	2.05446i	−0.151870	+	0.263046i	0.780045	+	0.625723i	$0.215196\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.441578	+	0.764836i	−0.0547710	+	0.0948662i
$66$		0			0
$67$	1.87228	+	3.24289i	0.228736	+	0.396182i	0.957434	−	0.288653i	$-0.0932076\pi$
$67$	1.87228	+	3.24289i	0.228736	+	0.396182i	−0.728698	+	0.684835i	$0.759874\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	−12.1168			−1.41817			−0.709085	−	0.705123i	$-0.750892\pi$
$73$	−12.1168			−1.41817			−0.709085	+	0.705123i	$0.750892\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.18614	+	3.78651i	0.249134	+	0.431512i
$78$		0			0
$79$	3.18614	−	5.51856i	0.358469	−	0.620886i	−0.629236	−	0.777214i	$-0.716632\pi$
$79$	3.18614	−	5.51856i	0.358469	−	0.620886i	0.987705	+	0.156328i	$0.0499656\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.81386	−	8.33785i	0.528390	−	0.915198i	−0.471062	−	0.882100i	$-0.656129\pi$
$83$	4.81386	−	8.33785i	0.528390	−	0.915198i	0.999452	−	0.0330979i	$-0.0105373\pi$
$84$		0			0
$85$	6.37228	+	11.0371i	0.691171	+	1.19714i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−1.62772			−0.170631
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.744563	+	1.28962i	0.0763905	+	0.132312i
$96$		0			0
$97$	−0.872281	+	1.51084i	−0.0885667	+	0.153402i	−0.906906	−	0.421334i	$-0.861562\pi$
$97$	−0.872281	+	1.51084i	−0.0885667	+	0.153402i	0.818339	+	0.574736i	$0.194895\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.55842	+	6.16337i	−0.354076	+	0.613278i	−0.986959	−	0.160969i	$-0.948538\pi$
$101$	−3.55842	+	6.16337i	−0.354076	+	0.613278i	0.632883	+	0.774247i	$0.281871\pi$
$102$		0			0
$103$	6.18614	+	10.7147i	0.609539	+	1.05575i	0.991316	+	0.131498i	$0.0419786\pi$
$103$	6.18614	+	10.7147i	0.609539	+	1.05575i	−0.381778	+	0.924254i	$0.624688\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.8614			1.24336			0.621680	−	0.783272i	$-0.286451\pi$
$107$	12.8614			1.24336			0.621680	+	0.783272i	$0.286451\pi$
$108$		0			0
$109$	−4.74456			−0.454447			−0.227223	−	0.973843i	$-0.572965\pi$
$109$	−4.74456			−0.454447			−0.227223	+	0.973843i	$0.572965\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.18614	−	3.78651i	−0.205655	−	0.356205i	0.744686	−	0.667415i	$-0.232599\pi$
$113$	−2.18614	−	3.78651i	−0.205655	−	0.356205i	−0.950341	+	0.311210i	$0.899266\pi$
$114$		0			0
$115$	0.441578	−	0.764836i	0.0411774	−	0.0713213i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.7446	+	20.3422i	−1.07662	+	1.86476i
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.3723			0.927725
$126$		0			0
$127$	6.74456			0.598483			0.299242	−	0.954177i	$-0.403266\pi$
$127$	6.74456			0.598483			0.299242	+	0.954177i	$0.403266\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.18614	+	15.9109i	0.802597	+	1.39014i	0.917901	+	0.396808i	$0.129882\pi$
$131$	9.18614	+	15.9109i	0.802597	+	1.39014i	−0.115305	+	0.993330i	$0.536784\pi$
$132$		0			0
$133$	−1.37228	+	2.37686i	−0.118992	+	0.206100i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.87228	−	15.3672i	0.758010	−	1.31291i	−0.185854	−	0.982577i	$-0.559505\pi$
$137$	8.87228	−	15.3672i	0.758010	−	1.31291i	0.943864	−	0.330335i	$-0.107162\pi$
$138$		0			0
$139$	−2.87228	−	4.97494i	−0.243624	−	0.421969i	0.718120	−	0.695919i	$-0.245003\pi$
$139$	−2.87228	−	4.97494i	−0.243624	−	0.421969i	−0.961744	+	0.273951i	$0.911670\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.372281			0.0311317
$144$		0			0
$145$	−10.3723			−0.861371
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.18614	−	7.25061i	−0.342942	−	0.593993i	0.642036	−	0.766675i	$-0.278090\pi$
$149$	−4.18614	−	7.25061i	−0.342942	−	0.593993i	−0.984978	+	0.172682i	$0.944757\pi$
$150$		0			0
$151$	−0.186141	+	0.322405i	−0.0151479	+	0.0262370i	−0.873500	−	0.486824i	$-0.838155\pi$
$151$	−0.186141	+	0.322405i	−0.0151479	+	0.0262370i	0.858352	+	0.513061i	$0.171489\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.55842	−	13.0916i	0.607107	−	1.05154i
$156$		0			0
$157$	1.55842	+	2.69927i	0.124376	+	0.215425i	0.921489	−	0.388405i	$-0.126974\pi$
$157$	1.55842	+	2.69927i	0.124376	+	0.215425i	−0.797113	+	0.603830i	$0.793641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.62772			0.128282
$162$		0			0
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.55842	−	16.5557i	−0.739653	−	1.28112i	−0.952652	−	0.304064i	$-0.901656\pi$
$167$	−9.55842	−	16.5557i	−0.739653	−	1.28112i	0.212999	−	0.977052i	$-0.431677\pi$
$168$		0			0
$169$	6.43070	−	11.1383i	0.494669	−	0.856793i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.18614	+	14.1788i	−0.622381	+	1.07800i	0.366660	+	0.930355i	$0.380501\pi$
$173$	−8.18614	+	14.1788i	−0.622381	+	1.07800i	−0.989041	+	0.147640i	$0.952832\pi$
$174$		0			0
$175$	−1.37228	−	2.37686i	−0.103735	−	0.179674i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−22.9783			−1.71748			−0.858738	−	0.512416i	$-0.828751\pi$
$179$	−22.9783			−1.71748			−0.858738	+	0.512416i	$0.828751\pi$
$180$		0			0
$181$	−0.510875			−0.0379730			−0.0189865	−	0.999820i	$-0.506044\pi$
$181$	−0.510875			−0.0379730			−0.0189865	+	0.999820i	$0.506044\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.3723	+	17.9653i	0.762585	+	1.32084i
$186$		0			0
$187$	2.68614	−	4.65253i	0.196430	−	0.340227i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.93070	−	8.54023i	0.356773	−	0.617949i	−0.630647	−	0.776070i	$-0.717210\pi$
$191$	4.93070	−	8.54023i	0.356773	−	0.617949i	0.987420	+	0.158121i	$0.0505436\pi$
$192$		0			0
$193$	−0.872281	−	1.51084i	−0.0627882	−	0.108752i	0.832923	−	0.553390i	$-0.186666\pi$
$193$	−0.872281	−	1.51084i	−0.0627882	−	0.108752i	−0.895711	+	0.444637i	$0.853333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.2554			−1.08690			−0.543452	−	0.839440i	$-0.682883\pi$
$197$	−15.2554			−1.08690			−0.543452	+	0.839440i	$0.682883\pi$
$198$		0			0
$199$	−16.2337			−1.15078			−0.575388	−	0.817881i	$-0.695149\pi$
$199$	−16.2337			−1.15078			−0.575388	+	0.817881i	$0.695149\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.55842	−	16.5557i	−0.670870	−	1.16198i
$204$		0			0
$205$	13.9307	−	24.1287i	0.972963	−	1.68522i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.313859	−	0.543620i	0.0217101	−	0.0376030i
$210$		0			0
$211$	4.81386	+	8.33785i	0.331400	+	0.574001i	0.982787	−	0.184745i	$-0.0591458\pi$
$211$	4.81386	+	8.33785i	0.331400	+	0.574001i	−0.651387	+	0.758746i	$0.725812\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.13859			0.282250
$216$		0			0
$217$	27.8614			1.89136
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.00000	+	1.73205i	0.0672673	+	0.116510i
$222$		0			0
$223$	7.30298	−	12.6491i	0.489044	−	0.847049i	−0.510877	−	0.859654i	$-0.670679\pi$
$223$	7.30298	−	12.6491i	0.489044	−	0.847049i	0.999921	+	0.0126050i	$0.00401240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.50000	+	12.9904i	−0.497792	+	0.862202i	−0.999997	−	0.00254715i	$-0.999189\pi$
$227$	−7.50000	+	12.9904i	−0.497792	+	0.862202i	0.502204	+	0.864749i	$0.332523\pi$
$228$		0			0
$229$	−11.3030	−	19.5773i	−0.746922	−	1.29371i	−0.949291	−	0.314398i	$-0.898197\pi$
$229$	−11.3030	−	19.5773i	−0.746922	−	1.29371i	0.202369	−	0.979309i	$-0.435136\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.37228			−0.351950			−0.175975	−	0.984395i	$-0.556308\pi$
$233$	−5.37228			−0.351950			−0.175975	+	0.984395i	$0.556308\pi$
$234$		0			0
$235$	10.3723			0.676613
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.93070	−	12.0043i	−0.448310	−	0.776496i	0.549966	−	0.835187i	$-0.314641\pi$
$239$	−6.93070	−	12.0043i	−0.448310	−	0.776496i	−0.998276	+	0.0586913i	$0.981307\pi$
$240$		0			0
$241$	−2.87228	+	4.97494i	−0.185020	+	0.320464i	−0.943583	−	0.331135i	$-0.892568\pi$
$241$	−2.87228	+	4.97494i	−0.185020	+	0.320464i	0.758563	+	0.651599i	$0.225902\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.3723	−	24.8935i	0.918211	−	1.59039i
$246$		0			0
$247$	0.116844	+	0.202380i	0.00743460	+	0.0128771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.88316			−0.623819			−0.311910	−	0.950112i	$-0.600969\pi$
$251$	−9.88316			−0.623819			−0.311910	+	0.950112i	$0.600969\pi$
$252$		0			0
$253$	−0.372281			−0.0234051
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.24456	+	3.88770i	0.140012	+	0.242508i	0.927501	−	0.373821i	$-0.121953\pi$
$257$	2.24456	+	3.88770i	0.140012	+	0.242508i	−0.787489	+	0.616329i	$0.788619\pi$
$258$		0			0
$259$	−19.1168	+	33.1113i	−1.18786	+	2.05744i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.5584	+	18.2877i	−0.651060	+	1.12767i	0.331806	+	0.943348i	$0.392342\pi$
$263$	−10.5584	+	18.2877i	−0.651060	+	1.12767i	−0.982866	+	0.184322i	$0.940991\pi$
$264$		0			0
$265$	−0.883156	−	1.52967i	−0.0542518	−	0.0939669i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.7446			0.777050			0.388525	−	0.921438i	$-0.372985\pi$
$269$	12.7446			0.777050			0.388525	+	0.921438i	$0.372985\pi$
$270$		0			0
$271$	−21.4891			−1.30537			−0.652686	−	0.757629i	$-0.726358\pi$
$271$	−21.4891			−1.30537			−0.652686	+	0.757629i	$0.726358\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.313859	+	0.543620i	0.0189264	+	0.0327815i
$276$		0			0
$277$	−14.5584	+	25.2159i	−0.874731	+	1.51508i	−0.0176816	+	0.999844i	$0.505629\pi$
$277$	−14.5584	+	25.2159i	−0.874731	+	1.51508i	−0.857049	+	0.515235i	$0.827705\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.93070	+	12.0043i	−0.413451	+	0.716118i	−0.995264	−	0.0972040i	$-0.969010\pi$
$281$	−6.93070	+	12.0043i	−0.413451	+	0.716118i	0.581813	+	0.813322i	$0.302343\pi$
$282$		0			0
$283$	−11.9307	−	20.6646i	−0.709207	−	1.22838i	−0.965152	−	0.261691i	$-0.915720\pi$
$283$	−11.9307	−	20.6646i	−0.709207	−	1.22838i	0.255945	−	0.966691i	$-0.417613\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	51.3505			3.03113
$288$		0			0
$289$	11.8614			0.697730
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.81386	+	16.9981i	0.573332	+	0.993040i	0.996221	+	0.0868582i	$0.0276827\pi$
$293$	9.81386	+	16.9981i	0.573332	+	0.993040i	−0.422889	+	0.906182i	$0.638984\pi$
$294$		0			0
$295$	−8.30298	+	14.3812i	−0.483418	+	0.837305i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.0692967	−	0.120025i	0.00400753	−	0.00694125i
$300$		0			0
$301$	3.81386	+	6.60580i	0.219827	+	0.380752i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.62772			−0.322242
$306$		0			0
$307$	31.3723			1.79051			0.895255	−	0.445553i	$-0.146993\pi$
$307$	31.3723			1.79051			0.895255	+	0.445553i	$0.146993\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.44158	−	9.42509i	−0.308564	−	0.534448i	0.669485	−	0.742826i	$-0.266515\pi$
$311$	−5.44158	−	9.42509i	−0.308564	−	0.534448i	−0.978048	+	0.208378i	$0.933182\pi$
$312$		0			0
$313$	−5.61684	+	9.72866i	−0.317483	+	0.549896i	−0.979962	−	0.199184i	$-0.936171\pi$
$313$	−5.61684	+	9.72866i	−0.317483	+	0.549896i	0.662479	+	0.749080i	$0.269504\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.3030	−	26.5055i	0.859501	−	1.48870i	−0.0129041	−	0.999917i	$-0.504108\pi$
$317$	15.3030	−	26.5055i	0.859501	−	1.48870i	0.872405	−	0.488783i	$-0.162559\pi$
$318$		0			0
$319$	2.18614	+	3.78651i	0.122400	+	0.212004i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.37228			0.187639
$324$		0			0
$325$	−0.233688			−0.0129627
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.55842	+	16.5557i	0.526973	+	0.912744i
$330$		0			0
$331$	15.9307	−	27.5928i	0.875631	−	1.51664i	0.0195412	−	0.999809i	$-0.493779\pi$
$331$	15.9307	−	27.5928i	0.875631	−	1.51664i	0.856089	−	0.516828i	$-0.172887\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.44158	+	7.69304i	−0.242669	+	0.420316i
$336$		0			0
$337$	9.98913	+	17.3017i	0.544142	+	0.942482i	0.998660	+	0.0517446i	$0.0164782\pi$
$337$	9.98913	+	17.3017i	0.544142	+	0.942482i	−0.454518	+	0.890738i	$0.650188\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.37228			−0.345078
$342$		0			0
$343$	22.3723			1.20799
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.3614	+	24.8747i	0.770961	+	1.33534i	0.937037	+	0.349230i	$0.113557\pi$
$347$	14.3614	+	24.8747i	0.770961	+	1.33534i	−0.166076	+	0.986113i	$0.553110\pi$
$348$		0			0
$349$	−8.44158	+	14.6212i	−0.451867	+	0.782657i	−0.998502	−	0.0547140i	$-0.982575\pi$
$349$	−8.44158	+	14.6212i	−0.451867	+	0.782657i	0.546635	+	0.837371i	$0.315909\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.9891	+	20.7658i	−0.638117	+	1.10525i	0.347729	+	0.937595i	$0.386953\pi$
$353$	−11.9891	+	20.7658i	−0.638117	+	1.10525i	−0.985846	+	0.167656i	$0.946380\pi$
$354$		0			0
$355$	4.74456	+	8.21782i	0.251815	+	0.436157i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.2337			−1.06789			−0.533947	−	0.845518i	$-0.679292\pi$
$359$	−20.2337			−1.06789			−0.533947	+	0.845518i	$0.679292\pi$
$360$		0			0
$361$	−18.6060			−0.979262
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.3723	−	24.8935i	−0.752280	−	1.30299i
$366$		0			0
$367$	−8.81386	+	15.2661i	−0.460080	+	0.796881i	−0.998964	−	0.0454981i	$-0.985513\pi$
$367$	−8.81386	+	15.2661i	−0.460080	+	0.796881i	0.538885	+	0.842380i	$0.318846\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.62772	−	2.81929i	0.0845069	−	0.146370i
$372$		0			0
$373$	−10.5584	−	18.2877i	−0.546694	−	0.946902i	−0.998498	−	0.0547851i	$-0.982553\pi$
$373$	−10.5584	−	18.2877i	−0.546694	−	0.946902i	0.451804	−	0.892117i	$-0.350781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.62772			−0.0838318
$378$		0			0
$379$	−5.88316			−0.302197			−0.151099	−	0.988519i	$-0.548281\pi$
$379$	−5.88316			−0.302197			−0.151099	+	0.988519i	$0.548281\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.6753	+	18.4901i	0.545481	+	0.944800i	0.998576	+	0.0533383i	$0.0169862\pi$
$383$	10.6753	+	18.4901i	0.545481	+	0.944800i	−0.453096	+	0.891462i	$0.649680\pi$
$384$		0			0
$385$	−5.18614	+	8.98266i	−0.264310	+	0.457799i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.30298	−	9.18504i	0.268872	−	0.465700i	−0.699699	−	0.714438i	$-0.746682\pi$
$389$	5.30298	−	9.18504i	0.268872	−	0.465700i	0.968571	+	0.248738i	$0.0800158\pi$
$390$		0			0
$391$	−1.00000	−	1.73205i	−0.0505722	−	0.0875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.1168			0.760611
$396$		0			0
$397$	−18.2337			−0.915123			−0.457561	−	0.889178i	$-0.651277\pi$
$397$	−18.2337			−0.915123			−0.457561	+	0.889178i	$0.651277\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.61684	−	14.9248i	−0.430305	−	0.745310i	0.566595	−	0.823997i	$-0.308261\pi$
$401$	−8.61684	−	14.9248i	−0.430305	−	0.745310i	−0.996899	+	0.0786871i	$0.974927\pi$
$402$		0			0
$403$	1.18614	−	2.05446i	0.0590859	−	0.102340i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.37228	−	7.57301i	0.216726	−	0.375380i
$408$		0			0
$409$	−2.87228	−	4.97494i	−0.142025	−	0.245995i	0.786234	−	0.617929i	$-0.212028\pi$
$409$	−2.87228	−	4.97494i	−0.142025	−	0.245995i	−0.928259	+	0.371934i	$0.878695\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−30.6060			−1.50602
$414$		0			0
$415$	22.8397			1.12115
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.30298	−	10.9171i	−0.307921	−	0.533335i	0.669986	−	0.742373i	$-0.266300\pi$
$419$	−6.30298	−	10.9171i	−0.307921	−	0.533335i	−0.977907	+	0.209039i	$0.932967\pi$
$420$		0			0
$421$	17.5584	−	30.4121i	0.855745	−	1.48219i	−0.0202069	−	0.999796i	$-0.506432\pi$
$421$	17.5584	−	30.4121i	0.855745	−	1.48219i	0.875952	−	0.482398i	$-0.160234\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.68614	+	2.92048i	−0.0817898	+	0.141664i
$426$		0			0
$427$	−5.18614	−	8.98266i	−0.250975	−	0.434701i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.25544			0.0604723			0.0302361	−	0.999543i	$-0.490374\pi$
$431$	1.25544			0.0604723			0.0302361	+	0.999543i	$0.490374\pi$
$432$		0			0
$433$	32.1168			1.54344			0.771719	−	0.635964i	$-0.219397\pi$
$433$	32.1168			1.54344			0.771719	+	0.635964i	$0.219397\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.116844	−	0.202380i	−0.00558941	−	0.00968113i
$438$		0			0
$439$	−3.55842	+	6.16337i	−0.169834	+	0.294161i	−0.938361	−	0.345656i	$-0.887657\pi$
$439$	−3.55842	+	6.16337i	−0.169834	+	0.294161i	0.768527	+	0.639817i	$0.220990\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.3614	−	26.6067i	0.729842	−	1.26412i	−0.227107	−	0.973870i	$-0.572927\pi$
$443$	15.3614	−	26.6067i	0.729842	−	1.26412i	0.956950	−	0.290254i	$-0.0937399\pi$
$444$		0			0
$445$	−7.11684	−	12.3267i	−0.337371	−	0.584343i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.8832			0.749572			0.374786	−	0.927111i	$-0.377716\pi$
$449$	15.8832			0.749572			0.374786	+	0.927111i	$0.377716\pi$
$450$		0			0
$451$	−11.7446			−0.553030
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.93070	−	3.34408i	−0.0905128	−	0.156773i
$456$		0			0
$457$	1.87228	−	3.24289i	0.0875816	−	0.151696i	−0.818907	−	0.573927i	$-0.805419\pi$
$457$	1.87228	−	3.24289i	0.0875816	−	0.151696i	0.906488	+	0.422231i	$0.138753\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.44158	+	5.96099i	−0.160290	+	0.277631i	−0.934973	−	0.354720i	$-0.884576\pi$
$461$	−3.44158	+	5.96099i	−0.160290	+	0.277631i	0.774682	+	0.632350i	$0.217910\pi$
$462$		0			0
$463$	−20.6753	−	35.8106i	−0.960861	−	1.66426i	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−20.6753	−	35.8106i	−0.960861	−	1.66426i	−0.240515	−	0.970645i	$-0.577316\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.37228			−0.341148			−0.170574	−	0.985345i	$-0.554562\pi$
$467$	−7.37228			−0.341148			−0.170574	+	0.985345i	$0.554562\pi$
$468$		0			0
$469$	−16.3723			−0.756002
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.872281	−	1.51084i	−0.0401075	−	0.0694683i
$474$		0			0
$475$	−0.197015	+	0.341241i	−0.00903969	+	0.0156572i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.30298	+	9.18504i	−0.242300	+	0.419675i	−0.961369	−	0.275263i	$-0.911235\pi$
$479$	−5.30298	+	9.18504i	−0.242300	+	0.419675i	0.719069	+	0.694938i	$0.244568\pi$
$480$		0			0
$481$	1.62772	+	2.81929i	0.0742176	+	0.128549i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.13859			−0.187924
$486$		0			0
$487$	6.74456			0.305625			0.152813	−	0.988255i	$-0.451167\pi$
$487$	6.74456			0.305625			0.152813	+	0.988255i	$0.451167\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.127719	+	0.221215i	0.00576386	+	0.00998330i	0.868893	−	0.495000i	$-0.164832\pi$
$491$	0.127719	+	0.221215i	0.00576386	+	0.00998330i	−0.863129	+	0.504983i	$0.831499\pi$
$492$		0			0
$493$	−11.7446	+	20.3422i	−0.528948	+	0.916166i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.74456	+	15.1460i	−0.392247	+	0.679392i
$498$		0			0
$499$	9.98913	+	17.3017i	0.447175	+	0.774529i	0.998201	−	0.0599587i	$-0.0190969\pi$
$499$	9.98913	+	17.3017i	0.447175	+	0.774529i	−0.551026	+	0.834488i	$0.685764\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.51087			−0.290306			−0.145153	−	0.989409i	$-0.546367\pi$
$503$	−6.51087			−0.290306			−0.145153	+	0.989409i	$0.546367\pi$
$504$		0			0
$505$	−16.8832			−0.751291
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.5584	−	26.9480i	−0.689615	−	1.19445i	−0.971962	−	0.235136i	$-0.924446\pi$
$509$	−15.5584	−	26.9480i	−0.689615	−	1.19445i	0.282348	−	0.959312i	$-0.408887\pi$
$510$		0			0
$511$	26.4891	−	45.8805i	1.17181	−	2.02963i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.6753	+	25.4183i	−0.646669	+	1.12006i
$516$		0			0
$517$	−2.18614	−	3.78651i	−0.0961464	−	0.166530i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.1168			−0.530849			−0.265424	−	0.964132i	$-0.585512\pi$
$521$	−12.1168			−0.530849			−0.265424	+	0.964132i	$0.585512\pi$
$522$		0			0
$523$	−13.4891			−0.589838			−0.294919	−	0.955522i	$-0.595293\pi$
$523$	−13.4891			−0.589838			−0.294919	+	0.955522i	$0.595293\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.1168	−	29.6472i	−0.745621	−	1.29145i
$528$		0			0
$529$	11.4307	−	19.7986i	0.496987	−	0.860807i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.18614	−	3.78651i	0.0946923	−	0.164012i
$534$		0			0
$535$	15.2554	+	26.4232i	0.659550	+	1.14237i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.1168			−0.521909
$540$		0			0
$541$	2.23369			0.0960337			0.0480169	−	0.998847i	$-0.484710\pi$
$541$	2.23369			0.0960337			0.0480169	+	0.998847i	$0.484710\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.62772	−	9.74749i	−0.241065	−	0.417537i
$546$		0			0
$547$	8.12772	−	14.0776i	0.347516	−	0.601916i	−0.638291	−	0.769795i	$-0.720359\pi$
$547$	8.12772	−	14.0776i	0.347516	−	0.601916i	0.985808	+	0.167879i	$0.0536918\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.37228	+	2.37686i	−0.0584611	+	0.101258i
$552$		0			0
$553$	13.9307	+	24.1287i	0.592394	+	1.02606i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.0000			0.593199			0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000			0.593199			0.296600	+	0.955002i	$0.404147\pi$
$558$		0			0
$559$	0.649468			0.0274696
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.87228	+	8.43904i	0.205342	+	0.355663i	0.950242	−	0.311514i	$-0.100836\pi$
$563$	4.87228	+	8.43904i	0.205342	+	0.355663i	−0.744900	+	0.667177i	$0.767503\pi$
$564$		0			0
$565$	5.18614	−	8.98266i	0.218183	−	0.377903i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.6168	−	23.5851i	0.570848	−	0.988737i	−0.425631	−	0.904897i	$-0.639948\pi$
$569$	13.6168	−	23.5851i	0.570848	−	0.988737i	0.996479	−	0.0838407i	$-0.0267187\pi$
$570$		0			0
$571$	13.2446	+	22.9403i	0.554268	+	0.960020i	0.997960	+	0.0638407i	$0.0203349\pi$
$571$	13.2446	+	22.9403i	0.554268	+	0.960020i	−0.443692	+	0.896179i	$0.646332\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.233688			0.00974546
$576$		0			0
$577$	14.8614			0.618688			0.309344	−	0.950950i	$-0.399890\pi$
$577$	14.8614			0.618688			0.309344	+	0.950950i	$0.399890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21.0475	+	36.4554i	0.873199	+	1.51243i
$582$		0			0
$583$	−0.372281	+	0.644810i	−0.0154183	+	0.0267053i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.61684	−	7.99661i	0.190558	−	0.330055i	−0.754878	−	0.655866i	$-0.772304\pi$
$587$	4.61684	−	7.99661i	0.190558	−	0.330055i	0.945435	+	0.325810i	$0.105637\pi$
$588$		0			0
$589$	−2.00000	−	3.46410i	−0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26.0000			1.06769			0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000			1.06769			0.533846	+	0.845582i	$0.320746\pi$
$594$		0			0
$595$	−55.7228			−2.28441
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.9307	+	34.5210i	0.814346	+	1.41049i	0.909796	+	0.415055i	$0.136238\pi$
$599$	19.9307	+	34.5210i	0.814346	+	1.41049i	−0.0954498	+	0.995434i	$0.530429\pi$
$600$		0			0
$601$	−2.98913	+	5.17732i	−0.121929	+	0.211187i	−0.920528	−	0.390676i	$-0.872241\pi$
$601$	−2.98913	+	5.17732i	−0.121929	+	0.211187i	0.798599	+	0.601863i	$0.205575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.8614	+	20.5446i	−0.482235	+	0.835255i
$606$		0			0
$607$	5.55842	+	9.62747i	0.225609	+	0.390767i	0.956502	−	0.291725i	$-0.0942294\pi$
$607$	5.55842	+	9.62747i	0.225609	+	0.390767i	−0.730893	+	0.682492i	$0.760896\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.62772			0.0658504
$612$		0			0
$613$	−12.7446			−0.514748			−0.257374	−	0.966312i	$-0.582857\pi$
$613$	−12.7446			−0.514748			−0.257374	+	0.966312i	$0.582857\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.9891	+	32.8901i	0.764473	+	1.32411i	0.940525	+	0.339726i	$0.110334\pi$
$617$	18.9891	+	32.8901i	0.764473	+	1.32411i	−0.176051	+	0.984381i	$0.556332\pi$
$618$		0			0
$619$	−10.6168	+	18.3889i	−0.426727	+	0.739113i	−0.996580	−	0.0826338i	$-0.973667\pi$
$619$	−10.6168	+	18.3889i	−0.426727	+	0.739113i	0.569853	+	0.821747i	$0.307000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.1168	−	22.7190i	0.525515	−	0.910219i
$624$		0			0
$625$	13.8723	+	24.0275i	0.554891	+	0.961100i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	46.9783			1.87315
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	13.8564i	0.317470	+	0.549875i
$636$		0			0
$637$	2.25544	−	3.90653i	0.0893637	−	0.154782i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.61684	−	2.80046i	0.0638615	−	0.110611i	−0.832327	−	0.554285i	$-0.812992\pi$
$641$	1.61684	−	2.80046i	0.0638615	−	0.110611i	0.896188	+	0.443674i	$0.146325\pi$
$642$		0			0
$643$	−1.50000	−	2.59808i	−0.0591542	−	0.102458i	0.834932	−	0.550353i	$-0.185507\pi$
$643$	−1.50000	−	2.59808i	−0.0591542	−	0.102458i	−0.894086	+	0.447895i	$0.852174\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−33.7228			−1.32578			−0.662890	−	0.748717i	$-0.730670\pi$
$647$	−33.7228			−1.32578			−0.662890	+	0.748717i	$0.730670\pi$
$648$		0			0
$649$	7.00000			0.274774
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.4416	−	19.8174i	−0.447744	−	0.775515i	0.550495	−	0.834838i	$-0.314439\pi$
$653$	−11.4416	−	19.8174i	−0.447744	−	0.775515i	−0.998239	+	0.0593237i	$0.981106\pi$
$654$		0			0
$655$	−21.7921	+	37.7450i	−0.851488	+	1.47482i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.55842	−	16.5557i	0.372343	−	0.644917i	−0.617582	−	0.786506i	$-0.711888\pi$
$659$	9.55842	−	16.5557i	0.372343	−	0.644917i	0.989926	+	0.141589i	$0.0452211\pi$
$660$		0			0
$661$	23.0475	+	39.9195i	0.896446	+	1.55269i	0.832005	+	0.554768i	$0.187193\pi$
$661$	23.0475	+	39.9195i	0.896446	+	1.55269i	0.0644406	+	0.997922i	$0.479474\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.51087			−0.252481
$666$		0			0
$667$	1.62772			0.0630255
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.18614	+	2.05446i	0.0457905	+	0.0793114i
$672$		0			0
$673$	0.186141	−	0.322405i	0.00717520	−	0.0124278i	−0.862416	−	0.506201i	$-0.831049\pi$
$673$	0.186141	−	0.322405i	0.00717520	−	0.0124278i	0.869591	+	0.493773i	$0.164383\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.3030	+	17.8453i	−0.395976	+	0.685850i	−0.993225	−	0.116205i	$-0.962927\pi$
$677$	−10.3030	+	17.8453i	−0.395976	+	0.685850i	0.597249	+	0.802056i	$0.296260\pi$
$678$		0			0
$679$	−3.81386	−	6.60580i	−0.146362	−	0.253507i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.3723			0.588204			0.294102	−	0.955774i	$-0.404979\pi$
$683$	15.3723			0.588204			0.294102	+	0.955774i	$0.404979\pi$
$684$		0			0
$685$	42.0951			1.60837
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.138593	−	0.240051i	−0.00527999	−	0.00914521i
$690$		0			0
$691$	−7.55842	+	13.0916i	−0.287536	+	0.498027i	−0.973221	−	0.229872i	$-0.926169\pi$
$691$	−7.55842	+	13.0916i	−0.287536	+	0.498027i	0.685685	+	0.727898i	$0.259503\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.81386	−	11.8020i	0.258464	−	0.447674i
$696$		0			0
$697$	−31.5475	−	54.6420i	−1.19495	−	2.06971i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	35.4891			1.34041			0.670203	−	0.742178i	$-0.266207\pi$
$701$	35.4891			1.34041			0.670203	+	0.742178i	$0.266207\pi$
$702$		0			0
$703$	5.48913			0.207026
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.5584	−	26.9480i	−0.585135	−	1.01348i
$708$		0			0
$709$	14.3030	−	24.7735i	0.537160	−	0.930388i	−0.461896	−	0.886934i	$-0.652831\pi$
$709$	14.3030	−	24.7735i	0.537160	−	0.930388i	0.999055	−	0.0434539i	$-0.0138361\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.18614	+	2.05446i	−0.0444213	+	0.0769400i
$714$		0			0
$715$	0.441578	+	0.764836i	0.0165141	+	0.0286032i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−45.4891			−1.69646			−0.848229	−	0.529630i	$-0.822331\pi$
$719$	−45.4891			−1.69646			−0.848229	+	0.529630i	$0.822331\pi$
$720$		0			0
$721$	−54.0951			−2.01461
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.37228	−	2.37686i	−0.0509652	−	0.0882744i
$726$		0			0
$727$	−5.44158	+	9.42509i	−0.201817	+	0.349557i	−0.949114	−	0.314933i	$-0.898018\pi$
$727$	−5.44158	+	9.42509i	−0.201817	+	0.349557i	0.747297	+	0.664490i	$0.231351\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.68614	−	8.11663i	0.173323	−	0.300205i
$732$		0			0
$733$	18.9307	+	32.7889i	0.699221	+	1.21109i	0.968737	+	0.248091i	$0.0798032\pi$
$733$	18.9307	+	32.7889i	0.699221	+	1.21109i	−0.269515	+	0.962996i	$0.586863\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.74456			0.137933
$738$		0			0
$739$	42.1168			1.54929			0.774647	−	0.632394i	$-0.217928\pi$
$739$	42.1168			1.54929			0.774647	+	0.632394i	$0.217928\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.8139	+	30.8545i	0.653527	+	1.13194i	0.982261	+	0.187520i	$0.0600448\pi$
$743$	17.8139	+	30.8545i	0.653527	+	1.13194i	−0.328734	+	0.944423i	$0.606622\pi$
$744$		0			0
$745$	9.93070	−	17.2005i	0.363833	−	0.630177i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−28.1168	+	48.6998i	−1.02737	+	1.77945i
$750$		0			0
$751$	18.8139	+	32.5866i	0.686527	+	1.18910i	0.972954	+	0.230998i	$0.0741992\pi$
$751$	18.8139	+	32.5866i	0.686527	+	1.18910i	−0.286427	+	0.958102i	$0.592467\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.883156			−0.0321413
$756$		0			0
$757$	8.51087			0.309333			0.154667	−	0.987967i	$-0.450570\pi$
$757$	8.51087			0.309333			0.154667	+	0.987967i	$0.450570\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.67527	−	16.7581i	−0.350728	−	0.607479i	0.635649	−	0.771978i	$-0.280733\pi$
$761$	−9.67527	−	16.7581i	−0.350728	−	0.607479i	−0.986377	+	0.164499i	$0.947399\pi$
$762$		0			0
$763$	10.3723	−	17.9653i	0.375502	−	0.650388i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.30298	+	2.25684i	−0.0470480	+	0.0814896i
$768$		0			0
$769$	−16.5584	−	28.6800i	−0.597112	−	1.03423i	−0.993245	−	0.116035i	$-0.962981\pi$
$769$	−16.5584	−	28.6800i	−0.597112	−	1.03423i	0.396133	−	0.918193i	$-0.370352\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28.9783			−1.04228			−0.521138	−	0.853473i	$-0.674492\pi$
$773$	−28.9783			−1.04228			−0.521138	+	0.853473i	$0.674492\pi$
$774$		0			0
$775$	4.00000			0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.68614	−	6.38458i	−0.132070	−	0.228751i
$780$		0			0
$781$	2.00000	−	3.46410i	0.0715656	−	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.69702	+	6.40342i	−0.131952	+	0.228548i
$786$		0			0
$787$	−21.4198	−	37.1002i	−0.763534	−	1.32248i	−0.941018	−	0.338357i	$-0.890129\pi$
$787$	−21.4198	−	37.1002i	−0.763534	−	1.32248i	0.177484	−	0.984124i	$-0.443204\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	19.1168			0.679717
$792$		0			0
$793$	−0.883156			−0.0313618
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.69702	+	2.93932i	0.0601114	+	0.104116i	0.894515	−	0.447038i	$-0.147521\pi$
$797$	1.69702	+	2.93932i	0.0601114	+	0.104116i	−0.834404	+	0.551154i	$0.814188\pi$
$798$		0			0
$799$	11.7446	−	20.3422i	0.415493	−	0.719655i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.05842	+	10.4935i	−0.213797	+	0.370307i
$804$		0			0
$805$	1.93070	+	3.34408i	0.0680484	+	0.117863i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.1386			−0.461928			−0.230964	−	0.972962i	$-0.574188\pi$
$809$	−13.1386			−0.461928			−0.230964	+	0.972962i	$0.574188\pi$
$810$		0			0
$811$	10.3505			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$811$	10.3505			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14.2337	−	24.6535i	−0.498584	−	0.863573i
$816$		0			0
$817$	0.547547	−	0.948380i	0.0191563	−	0.0331796i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.81386	−	16.9981i	0.342506	−	0.593238i	−0.642391	−	0.766377i	$-0.722058\pi$
$821$	9.81386	−	16.9981i	0.342506	−	0.593238i	0.984897	+	0.173139i	$0.0553909\pi$
$822$		0			0
$823$	−12.4416	−	21.5494i	−0.433686	−	0.751166i	0.563501	−	0.826115i	$-0.309454\pi$
$823$	−12.4416	−	21.5494i	−0.433686	−	0.751166i	−0.997187	+	0.0749488i	$0.976121\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.9783			−1.07722			−0.538610	−	0.842555i	$-0.681050\pi$
$827$	−30.9783			−1.07722			−0.538610	+	0.842555i	$0.681050\pi$
$828$		0			0
$829$	15.4891			0.537960			0.268980	−	0.963146i	$-0.413314\pi$
$829$	15.4891			0.537960			0.268980	+	0.963146i	$0.413314\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−32.5475	−	56.3740i	−1.12771	−	1.95324i
$834$		0			0
$835$	22.6753	−	39.2747i	0.784710	−	1.35916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.0475	−	26.0631i	0.519499	−	0.899799i	−0.480244	−	0.877135i	$-0.659452\pi$
$839$	15.0475	−	26.0631i	0.519499	−	0.899799i	0.999743	−	0.0226638i	$-0.00721474\pi$
$840$		0			0
$841$	4.94158	+	8.55906i	0.170399	+	0.295140i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	30.5109			1.04961
$846$		0			0
$847$	−43.7228			−1.50233
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.62772	−	2.81929i	−0.0557975	−	0.0966441i
$852$		0			0
$853$	−16.5584	+	28.6800i	−0.566950	+	0.981985i	0.429916	+	0.902869i	$0.358543\pi$
$853$	−16.5584	+	28.6800i	−0.566950	+	0.981985i	−0.996865	+	0.0791165i	$0.974790\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.5584	−	32.1441i	0.633944	−	1.09802i	−0.352794	−	0.935701i	$-0.614768\pi$
$857$	18.5584	−	32.1441i	0.633944	−	1.09802i	0.986738	−	0.162322i	$-0.0518982\pi$
$858$		0			0
$859$	1.24456	+	2.15565i	0.0424639	+	0.0735497i	0.886476	−	0.462774i	$-0.153146\pi$
$859$	1.24456	+	2.15565i	0.0424639	+	0.0735497i	−0.844012	+	0.536324i	$0.819813\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.9783			0.646027			0.323014	−	0.946394i	$-0.395304\pi$
$863$	18.9783			0.646027			0.323014	+	0.946394i	$0.395304\pi$
$864$		0			0
$865$	−38.8397			−1.32059
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.18614	−	5.51856i	−0.108082	−	0.187204i
$870$		0			0
$871$	−0.697015	+	1.20727i	−0.0236175	+	0.0409066i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6753	+	39.2747i	−0.766564	+	1.32773i
$876$		0			0
$877$	−13.9307	−	24.1287i	−0.470406	−	0.814768i	0.529021	−	0.848609i	$-0.322559\pi$
$877$	−13.9307	−	24.1287i	−0.470406	−	0.814768i	−0.999427	+	0.0338410i	$0.989226\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.9783			−0.841539			−0.420769	−	0.907168i	$-0.638240\pi$
$881$	−24.9783			−0.841539			−0.420769	+	0.907168i	$0.638240\pi$
$882$		0			0
$883$	31.8397			1.07149			0.535745	−	0.844380i	$-0.320031\pi$
$883$	31.8397			1.07149			0.535745	+	0.844380i	$0.320031\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.93070	+	10.2723i	0.199134	+	0.344909i	0.948248	−	0.317531i	$-0.102854\pi$
$887$	5.93070	+	10.2723i	0.199134	+	0.344909i	−0.749114	+	0.662441i	$0.769521\pi$
$888$		0			0
$889$	−14.7446	+	25.5383i	−0.494517	+	0.856528i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.37228	−	2.37686i	0.0459216	−	0.0795386i
$894$		0			0
$895$	−27.2554	−	47.2078i	−0.911049	−	1.57798i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	27.8614			0.929230
$900$		0			0
$901$	−4.00000			−0.133259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.605969	−	1.04957i	−0.0201431	−	0.0348889i
$906$		0			0
$907$	−18.5000	+	32.0429i	−0.614282	+	1.06397i	0.376228	+	0.926527i	$0.377221\pi$
$907$	−18.5000	+	32.0429i	−0.614282	+	1.06397i	−0.990510	+	0.137441i	$0.956112\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.441578	+	0.764836i	−0.0146301	+	0.0253401i	−0.873248	−	0.487276i	$-0.837990\pi$
$911$	−0.441578	+	0.764836i	−0.0146301	+	0.0253401i	0.858618	+	0.512617i	$0.171324\pi$
$912$		0			0
$913$	−4.81386	−	8.33785i	−0.159315	−	0.275943i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−80.3288			−2.65269
$918$		0			0
$919$	36.2337			1.19524			0.597620	−	0.801780i	$-0.296113\pi$
$919$	36.2337			1.19524			0.597620	+	0.801780i	$0.296113\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.744563	+	1.28962i	0.0245076	+	0.0424484i
$924$		0			0
$925$	−2.74456	+	4.75372i	−0.0902407	+	0.156301i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.7921	−	32.5489i	0.616549	−	1.06789i	−0.373561	−	0.927605i	$-0.621864\pi$
$929$	18.7921	−	32.5489i	0.616549	−	1.06789i	0.990111	−	0.140289i	$-0.0448032\pi$
$930$		0			0
$931$	−3.80298	−	6.58696i	−0.124638	−	0.215879i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.7446			0.416792
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.4198	+	23.2438i	0.437474	+	0.757727i	0.997494	−	0.0707520i	$-0.0225399\pi$
$941$	13.4198	+	23.2438i	0.437474	+	0.757727i	−0.560020	+	0.828479i	$0.689207\pi$
$942$		0			0
$943$	−2.18614	+	3.78651i	−0.0711905	+	0.123306i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.8723	+	46.5442i	−0.873232	+	1.51248i	−0.0145974	+	0.999893i	$0.504647\pi$
$947$	−26.8723	+	46.5442i	−0.873232	+	1.51248i	−0.858634	+	0.512588i	$0.828687\pi$
$948$		0			0
$949$	−2.25544	−	3.90653i	−0.0732146	−	0.126811i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.88316			0.255360			0.127680	−	0.991815i	$-0.459247\pi$
$953$	7.88316			0.255360			0.127680	+	0.991815i	$0.459247\pi$
$954$		0			0
$955$	23.3940			0.757013
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.7921	+	67.1899i	1.25266	+	2.16968i
$960$		0			0
$961$	−4.80298	+	8.31901i	−0.154935	+	0.268355i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.06930	−	3.58413i	0.0666130	−	0.115377i
$966$		0			0
$967$	4.18614	+	7.25061i	0.134617	+	0.233164i	0.925451	−	0.378867i	$-0.123686\pi$
$967$	4.18614	+	7.25061i	0.134617	+	0.233164i	−0.790834	+	0.612031i	$0.790353\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.48913			0.176154			0.0880772	−	0.996114i	$-0.471928\pi$
$971$	5.48913			0.176154			0.0880772	+	0.996114i	$0.471928\pi$
$972$		0			0
$973$	25.1168			0.805209
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.127719	+	0.221215i	0.00408608	+	0.00707730i	0.868061	−	0.496457i	$-0.165366\pi$
$977$	0.127719	+	0.221215i	0.00408608	+	0.00707730i	−0.863975	+	0.503534i	$0.832033\pi$
$978$		0			0
$979$	−3.00000	+	5.19615i	−0.0958804	+	0.166070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.5584	−	47.7326i	0.878977	−	1.52243i	0.0265123	−	0.999648i	$-0.491560\pi$
$983$	27.5584	−	47.7326i	0.878977	−	1.52243i	0.852465	−	0.522785i	$-0.175107\pi$
$984$		0			0
$985$	−18.0951	−	31.3416i	−0.576558	−	0.998627i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.649468			−0.0206519
$990$		0			0
$991$	50.9783			1.61938			0.809689	−	0.586860i	$-0.199636\pi$
$991$	50.9783			1.61938			0.809689	+	0.586860i	$0.199636\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−19.2554	−	33.3514i	−0.610438	−	1.05731i
$996$		0			0
$997$	−24.6753	+	42.7388i	−0.781474	+	1.35355i	0.149610	+	0.988745i	$0.452198\pi$
$997$	−24.6753	+	42.7388i	−0.781474	+	1.35355i	−0.931083	+	0.364807i	$0.881135\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.i.b.73.2		4
3.2	odd	2		72.2.i.b.25.2	✓	4
4.3	odd	2		432.2.i.d.289.2		4
8.3	odd	2		1728.2.i.j.1153.1		4
8.5	even	2		1728.2.i.i.1153.1		4
9.2	odd	6		648.2.a.f.1.2		2
9.4	even	3	inner	216.2.i.b.145.2		4
9.5	odd	6		72.2.i.b.49.2	yes	4
9.7	even	3		648.2.a.g.1.1		2
12.11	even	2		144.2.i.d.97.1		4
24.5	odd	2		576.2.i.j.385.1		4
24.11	even	2		576.2.i.l.385.2		4
36.7	odd	6		1296.2.a.p.1.1		2
36.11	even	6		1296.2.a.n.1.2		2
36.23	even	6		144.2.i.d.49.1		4
36.31	odd	6		432.2.i.d.145.2		4
72.5	odd	6		576.2.i.j.193.1		4
72.11	even	6		5184.2.a.bs.1.1		2
72.13	even	6		1728.2.i.i.577.1		4
72.29	odd	6		5184.2.a.bt.1.1		2
72.43	odd	6		5184.2.a.bo.1.2		2
72.59	even	6		576.2.i.l.193.2		4
72.61	even	6		5184.2.a.bp.1.2		2
72.67	odd	6		1728.2.i.j.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.b.25.2	✓	4	3.2	odd	2
72.2.i.b.49.2	yes	4	9.5	odd	6
144.2.i.d.49.1		4	36.23	even	6
144.2.i.d.97.1		4	12.11	even	2
216.2.i.b.73.2		4	1.1	even	1	trivial
216.2.i.b.145.2		4	9.4	even	3	inner
432.2.i.d.145.2		4	36.31	odd	6
432.2.i.d.289.2		4	4.3	odd	2
576.2.i.j.193.1		4	72.5	odd	6
576.2.i.j.385.1		4	24.5	odd	2
576.2.i.l.193.2		4	72.59	even	6
576.2.i.l.385.2		4	24.11	even	2
648.2.a.f.1.2		2	9.2	odd	6
648.2.a.g.1.1		2	9.7	even	3
1296.2.a.n.1.2		2	36.11	even	6
1296.2.a.p.1.1		2	36.7	odd	6
1728.2.i.i.577.1		4	72.13	even	6
1728.2.i.i.1153.1		4	8.5	even	2
1728.2.i.j.577.1		4	72.67	odd	6
1728.2.i.j.1153.1		4	8.3	odd	2
5184.2.a.bo.1.2		2	72.43	odd	6
5184.2.a.bp.1.2		2	72.61	even	6
5184.2.a.bs.1.1		2	72.11	even	6
5184.2.a.bt.1.1		2	72.29	odd	6

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.18614 + 2.05446i) q^{5} +(-2.18614 + 3.78651i) q^{7} +O(q^{10})\)	\(q+(1.18614 + 2.05446i) q^{5} +(-2.18614 + 3.78651i) q^{7} +(0.500000 - 0.866025i) q^{11} +(0.186141 + 0.322405i) q^{13} +5.37228 q^{17} +0.627719 q^{19} +(-0.186141 - 0.322405i) q^{23} +(-0.313859 + 0.543620i) q^{25} +(-2.18614 + 3.78651i) q^{29} +(-3.18614 - 5.51856i) q^{31} -10.3723 q^{35} +8.74456 q^{37} +(-5.87228 - 10.1711i) q^{41} +(0.872281 - 1.51084i) q^{43} +(2.18614 - 3.78651i) q^{47} +(-6.05842 - 10.4935i) q^{49} -0.744563 q^{53} +2.37228 q^{55} +(3.50000 + 6.06218i) q^{59} +(-1.18614 + 2.05446i) q^{61} +(-0.441578 + 0.764836i) q^{65} +(1.87228 + 3.24289i) q^{67} +4.00000 q^{71} -12.1168 q^{73} +(2.18614 + 3.78651i) q^{77} +(3.18614 - 5.51856i) q^{79} +(4.81386 - 8.33785i) q^{83} +(6.37228 + 11.0371i) q^{85} -6.00000 q^{89} -1.62772 q^{91} +(0.744563 + 1.28962i) q^{95} +(-0.872281 + 1.51084i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} - 3 q^{7}+O(q^{10}) \) 4 * q - q^5 - 3 * q^7	\( 4 q - q^{5} - 3 q^{7} + 2 q^{11} - 5 q^{13} + 10 q^{17} + 14 q^{19} + 5 q^{23} - 7 q^{25} - 3 q^{29} - 7 q^{31} - 30 q^{35} + 12 q^{37} - 12 q^{41} - 8 q^{43} + 3 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 14 q^{59} + q^{61} - 19 q^{65} - 4 q^{67} + 16 q^{71} - 14 q^{73} + 3 q^{77} + 7 q^{79} + 25 q^{83} + 14 q^{85} - 24 q^{89} - 18 q^{91} - 20 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - q^5 - 3 * q^7 + 2 * q^11 - 5 * q^13 + 10 * q^17 + 14 * q^19 + 5 * q^23 - 7 * q^25 - 3 * q^29 - 7 * q^31 - 30 * q^35 + 12 * q^37 - 12 * q^41 - 8 * q^43 + 3 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 14 * q^59 + q^61 - 19 * q^65 - 4 * q^67 + 16 * q^71 - 14 * q^73 + 3 * q^77 + 7 * q^79 + 25 * q^83 + 14 * q^85 - 24 * q^89 - 18 * q^91 - 20 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more