LMFDB - Embedded newform 1512.2.t.a.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(289,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))

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

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

chi := DirichletCharacter("1512.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.t (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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.361
Dual form			1512.2.t.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q-1.00000 q^{5} +(-2.50000 + 0.866025i) q^{7} +3.00000 q^{11} +(-0.500000 + 0.866025i) q^{13} +(1.50000 - 2.59808i) q^{17} +(-2.50000 - 4.33013i) q^{19} -1.00000 q^{23} -4.00000 q^{25} +(4.50000 + 7.79423i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(2.50000 - 0.866025i) q^{35} +(-2.50000 - 4.33013i) q^{37} +(3.50000 - 6.06218i) q^{41} +(-1.50000 - 2.59808i) q^{43} +(4.00000 - 6.92820i) q^{47} +(5.50000 - 4.33013i) q^{49} +(4.50000 - 7.79423i) q^{53} -3.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(0.500000 - 0.866025i) q^{65} +(-6.00000 - 10.3923i) q^{67} -8.00000 q^{71} +(6.50000 - 11.2583i) q^{73} +(-7.50000 + 2.59808i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-6.50000 - 11.2583i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(-4.50000 - 7.79423i) q^{89} +(0.500000 - 2.59808i) q^{91} +(2.50000 + 4.33013i) q^{95} +(8.50000 + 14.7224i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 5 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 5 * q^7	$ 2 q - 2 q^{5} - 5 q^{7} + 6 q^{11} - q^{13} + 3 q^{17} - 5 q^{19} - 2 q^{23} - 8 q^{25} + 9 q^{29} - 4 q^{31} + 5 q^{35} - 5 q^{37} + 7 q^{41} - 3 q^{43} + 8 q^{47} + 11 q^{49} + 9 q^{53} - 6 q^{55} - 4 q^{59} - 2 q^{61} + q^{65} - 12 q^{67} - 16 q^{71} + 13 q^{73} - 15 q^{77} - 8 q^{79} - 13 q^{83} - 3 q^{85} - 9 q^{89} + q^{91} + 5 q^{95} + 17 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 5 * q^7 + 6 * q^11 - q^13 + 3 * q^17 - 5 * q^19 - 2 * q^23 - 8 * q^25 + 9 * q^29 - 4 * q^31 + 5 * q^35 - 5 * q^37 + 7 * q^41 - 3 * q^43 + 8 * q^47 + 11 * q^49 + 9 * q^53 - 6 * q^55 - 4 * q^59 - 2 * q^61 + q^65 - 12 * q^67 - 16 * q^71 + 13 * q^73 - 15 * q^77 - 8 * q^79 - 13 * q^83 - 3 * q^85 - 9 * q^89 + q^91 + 5 * q^95 + 17 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	−0.996199	−	0.0871106i	$-0.972237\pi$
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	0.422659	−	0.906289i	$-0.361097\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	$0.148230\pi$
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.50000	−	0.866025i	0.422577	−	0.146385i
$36$		0			0
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	0.584002	−	0.811752i	$-0.301486\pi$
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	−0.994999	+	0.0998840i	$0.968153\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.50000	−	6.06218i	0.546608	−	0.946753i	−0.451896	−	0.892071i	$-0.649252\pi$
$41$	3.50000	−	6.06218i	0.546608	−	0.946753i	0.998504	−	0.0546823i	$-0.0174146\pi$
$42$		0			0
$43$	−1.50000	−	2.59808i	−0.228748	−	0.396203i	0.728689	−	0.684844i	$-0.240130\pi$
$43$	−1.50000	−	2.59808i	−0.228748	−	0.396203i	−0.957437	+	0.288641i	$0.906796\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	−0.371706	−	0.928351i	$-0.621227\pi$
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	0.989828	−	0.142269i	$-0.0454398\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.500000	−	0.866025i	0.0620174	−	0.107417i
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	6.50000	−	11.2583i	0.760767	−	1.31769i	−0.181688	−	0.983356i	$-0.558156\pi$
$73$	6.50000	−	11.2583i	0.760767	−	1.31769i	0.942455	−	0.334332i	$-0.108511\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.50000	+	2.59808i	−0.854704	+	0.296078i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.50000	−	11.2583i	−0.713468	−	1.23576i	−0.963548	−	0.267537i	$-0.913790\pi$
$83$	−6.50000	−	11.2583i	−0.713468	−	1.23576i	0.250080	−	0.968225i	$-0.419543\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	0.522654	−	0.852545i	$-0.324942\pi$
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	−0.999653	+	0.0263586i	$0.991609\pi$
$90$		0			0
$91$	0.500000	−	2.59808i	0.0524142	−	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.50000	+	4.33013i	0.256495	+	0.444262i
$96$		0			0
$97$	8.50000	+	14.7224i	0.863044	+	1.49484i	0.868976	+	0.494854i	$0.164778\pi$
$97$	8.50000	+	14.7224i	0.863044	+	1.49484i	−0.00593185	+	0.999982i	$0.501888\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.00000			0.696526			0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000			0.696526			0.348263	+	0.937397i	$0.386772\pi$
$102$		0			0
$103$	9.00000			0.886796			0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000			0.886796			0.443398	+	0.896325i	$0.353773\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.50000	−	6.06218i	−0.338358	−	0.586053i	0.645766	−	0.763535i	$-0.276538\pi$
$107$	−3.50000	−	6.06218i	−0.338358	−	0.586053i	−0.984124	+	0.177482i	$0.943205\pi$
$108$		0			0
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	−0.960558	−	0.278078i	$-0.910303\pi$
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	0.721102	+	0.692829i	$0.243636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.500000	+	0.866025i	−0.0470360	+	0.0814688i	−0.888585	−	0.458712i	$-0.848311\pi$
$113$	−0.500000	+	0.866025i	−0.0470360	+	0.0814688i	0.841549	+	0.540181i	$0.181644\pi$
$114$		0			0
$115$	1.00000			0.0932505
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.50000	+	7.79423i	−0.137505	+	0.714496i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.00000			−0.262111			−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000			−0.262111			−0.131056	+	0.991375i	$0.541837\pi$
$132$		0			0
$133$	10.0000	+	8.66025i	0.867110	+	0.750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.0000			−0.939793			−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000			−0.939793			−0.469897	+	0.882721i	$0.655709\pi$
$138$		0			0
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	−0.952355	−	0.304991i	$-0.901346\pi$
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	0.740308	+	0.672268i	$0.234680\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.50000	+	2.59808i	−0.125436	+	0.217262i
$144$		0			0
$145$	−4.50000	−	7.79423i	−0.373705	−	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.0000			1.22885			0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000			1.22885			0.614424	+	0.788976i	$0.289388\pi$
$150$		0			0
$151$	−17.0000			−1.38344			−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000			−1.38344			−0.691720	+	0.722166i	$0.743147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	+	3.46410i	0.160644	+	0.278243i
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.50000	−	0.866025i	0.197028	−	0.0682524i
$162$		0			0
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	0.313304	−	0.949653i	$-0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	+	0.866025i	−0.0386912	+	0.0670151i	−0.884723	−	0.466118i	$-0.845652\pi$
$167$	−0.500000	+	0.866025i	−0.0386912	+	0.0670151i	0.846031	+	0.533133i	$0.178986\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	−0.901533	−	0.432710i	$-0.857557\pi$
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	0.825505	+	0.564396i	$0.190891\pi$
$174$		0			0
$175$	10.0000	−	3.46410i	0.755929	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5000	+	18.1865i	−0.784807	+	1.35933i	0.144308	+	0.989533i	$0.453905\pi$
$179$	−10.5000	+	18.1865i	−0.784807	+	1.35933i	−0.929114	+	0.369792i	$0.879429\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.50000	+	4.33013i	0.183804	+	0.318357i
$186$		0			0
$187$	4.50000	−	7.79423i	0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	−0.784552	−	0.620063i	$-0.787107\pi$
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	0.929267	+	0.369410i	$0.120440\pi$
$192$		0			0
$193$	9.00000	+	15.5885i	0.647834	+	1.12208i	0.983639	+	0.180150i	$0.0576584\pi$
$193$	9.00000	+	15.5885i	0.647834	+	1.12208i	−0.335805	+	0.941932i	$0.609008\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	−10.5000	+	18.1865i	−0.744325	+	1.28921i	0.206184	+	0.978513i	$0.433895\pi$
$199$	−10.5000	+	18.1865i	−0.744325	+	1.28921i	−0.950509	+	0.310696i	$0.899438\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−18.0000	−	15.5885i	−1.26335	−	1.09410i
$204$		0			0
$205$	−3.50000	+	6.06218i	−0.244451	+	0.423401i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.50000	−	12.9904i	−0.518786	−	0.898563i
$210$		0			0
$211$	13.5000	−	23.3827i	0.929378	−	1.60973i	0.145014	−	0.989430i	$-0.453677\pi$
$211$	13.5000	−	23.3827i	0.929378	−	1.60973i	0.784364	−	0.620301i	$-0.212990\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.50000	+	2.59808i	0.102299	+	0.177187i
$216$		0			0
$217$	8.00000	+	6.92820i	0.543075	+	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.50000	+	2.59808i	0.100901	+	0.174766i
$222$		0			0
$223$	−6.50000	−	11.2583i	−0.435272	−	0.753914i	0.562046	−	0.827106i	$-0.310015\pi$
$223$	−6.50000	−	11.2583i	−0.435272	−	0.753914i	−0.997318	+	0.0731927i	$0.976681\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.0000			0.995585			0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000			0.995585			0.497792	+	0.867296i	$0.334144\pi$
$228$		0			0
$229$	21.0000			1.38772			0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000			1.38772			0.693860	+	0.720110i	$0.255909\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.50000	+	2.59808i	0.0982683	+	0.170206i	0.910968	−	0.412477i	$-0.135336\pi$
$233$	1.50000	+	2.59808i	0.0982683	+	0.170206i	−0.812700	+	0.582683i	$0.802003\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.50000	+	7.79423i	−0.291081	+	0.504167i	−0.974066	−	0.226266i	$-0.927348\pi$
$239$	−4.50000	+	7.79423i	−0.291081	+	0.504167i	0.682985	+	0.730433i	$0.260682\pi$
$240$		0			0
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.50000	+	4.33013i	−0.351382	+	0.276642i
$246$		0			0
$247$	5.00000			0.318142
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$	−3.00000			−0.188608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.0000			1.06043			0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000			1.06043			0.530215	+	0.847863i	$0.322111\pi$
$258$		0			0
$259$	10.0000	+	8.66025i	0.621370	+	0.538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0000			−0.678289			−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000			−0.678289			−0.339145	+	0.940734i	$0.610138\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.50000	+	6.06218i	−0.213399	+	0.369618i	−0.952776	−	0.303674i	$-0.901787\pi$
$269$	−3.50000	+	6.06218i	−0.213399	+	0.369618i	0.739377	+	0.673291i	$0.235120\pi$
$270$		0			0
$271$	−15.5000	−	26.8468i	−0.941558	−	1.63083i	−0.762501	−	0.646988i	$-0.776029\pi$
$271$	−15.5000	−	26.8468i	−0.941558	−	1.63083i	−0.179057	−	0.983839i	$-0.557305\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.0000			−0.723627
$276$		0			0
$277$	−19.0000			−1.14160			−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000			−1.14160			−0.570800	+	0.821089i	$0.693367\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.5000	+	23.3827i	0.805342	+	1.39489i	0.916060	+	0.401042i	$0.131352\pi$
$281$	13.5000	+	23.3827i	0.805342	+	1.39489i	−0.110717	+	0.993852i	$0.535315\pi$
$282$		0			0
$283$	12.0000	+	20.7846i	0.713326	+	1.23552i	0.963602	+	0.267342i	$0.0861454\pi$
$283$	12.0000	+	20.7846i	0.713326	+	1.23552i	−0.250276	+	0.968175i	$0.580521\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.50000	+	18.1865i	−0.206598	+	1.07352i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.500000	−	0.866025i	0.0292103	−	0.0505937i	−0.851051	−	0.525084i	$-0.824034\pi$
$293$	0.500000	−	0.866025i	0.0292103	−	0.0505937i	0.880261	+	0.474490i	$0.157367\pi$
$294$		0			0
$295$	2.00000	+	3.46410i	0.116445	+	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.500000	−	0.866025i	0.0289157	−	0.0500835i
$300$		0			0
$301$	6.00000	+	5.19615i	0.345834	+	0.299501i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.00000	−	1.73205i	0.0572598	−	0.0991769i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	−0.938269	−	0.345907i	$-0.887571\pi$
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	0.768699	+	0.639611i	$0.220905\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.00000	−	12.1244i	0.393159	−	0.680972i	−0.599705	−	0.800221i	$-0.704715\pi$
$317$	7.00000	−	12.1244i	0.393159	−	0.680972i	0.992864	+	0.119249i	$0.0380488\pi$
$318$		0			0
$319$	13.5000	+	23.3827i	0.755855	+	1.30918i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.0000			−0.834622
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	20.7846i	−0.220527	+	1.14589i
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.00000	+	10.3923i	0.327815	+	0.567792i
$336$		0			0
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	−0.994728	−	0.102552i	$-0.967299\pi$
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	0.586177	+	0.810183i	$0.300632\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000	−	10.3923i	−0.324918	−	0.562775i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.0000	−	24.2487i	−0.751559	−	1.30174i	−0.947067	−	0.321037i	$-0.895969\pi$
$347$	−14.0000	−	24.2487i	−0.751559	−	1.30174i	0.195507	−	0.980702i	$-0.437365\pi$
$348$		0			0
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	0.543693	−	0.839284i	$-0.317025\pi$
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	−0.998688	+	0.0512103i	$0.983692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−31.0000			−1.64996			−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000			−1.64996			−0.824982	+	0.565159i	$0.808815\pi$
$354$		0			0
$355$	8.00000			0.424596
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.5000	+	23.3827i	0.712503	+	1.23409i	0.963915	+	0.266211i	$0.0857717\pi$
$359$	13.5000	+	23.3827i	0.712503	+	1.23409i	−0.251412	+	0.967880i	$0.580895\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.50000	+	11.2583i	−0.340226	+	0.589288i
$366$		0			0
$367$	11.0000			0.574195			0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000			0.574195			0.287098	+	0.957901i	$0.407310\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.50000	+	23.3827i	−0.233628	+	1.21397i
$372$		0			0
$373$	13.0000			0.673114			0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000			0.673114			0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.00000			−0.463524
$378$		0			0
$379$	32.0000			1.64373			0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000			1.64373			0.821865	+	0.569683i	$0.192934\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	35.0000			1.78842			0.894208	−	0.447651i	$-0.147739\pi$
$383$	35.0000			1.78842			0.894208	+	0.447651i	$0.147739\pi$
$384$		0			0
$385$	7.50000	−	2.59808i	0.382235	−	0.132410i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−37.0000			−1.87597			−0.937987	−	0.346670i	$-0.887312\pi$
$389$	−37.0000			−1.87597			−0.937987	+	0.346670i	$0.887312\pi$
$390$		0			0
$391$	−1.50000	+	2.59808i	−0.0758583	+	0.131390i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000	−	6.92820i	0.201262	−	0.348596i
$396$		0			0
$397$	−12.5000	−	21.6506i	−0.627357	−	1.08661i	−0.988080	−	0.153941i	$-0.950803\pi$
$397$	−12.5000	−	21.6506i	−0.627357	−	1.08661i	0.360723	−	0.932673i	$-0.382530\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.0000			0.649189			0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000			0.649189			0.324595	+	0.945853i	$0.394772\pi$
$402$		0			0
$403$	4.00000			0.199254
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.50000	−	12.9904i	−0.371761	−	0.643909i
$408$		0			0
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000	+	6.92820i	0.393654	+	0.340915i
$414$		0			0
$415$	6.50000	+	11.2583i	0.319072	+	0.552650i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.5000	−	18.1865i	0.512959	−	0.888470i	−0.486928	−	0.873442i	$-0.661883\pi$
$419$	10.5000	−	18.1865i	0.512959	−	0.888470i	0.999887	−	0.0150285i	$-0.00478389\pi$
$420$		0			0
$421$	1.50000	+	2.59808i	0.0731055	+	0.126622i	0.900261	−	0.435351i	$-0.143376\pi$
$421$	1.50000	+	2.59808i	0.0731055	+	0.126622i	−0.827155	+	0.561973i	$0.810042\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000	+	10.3923i	−0.291043	+	0.504101i
$426$		0			0
$427$	1.00000	−	5.19615i	0.0483934	−	0.251459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	−0.988169	−	0.153370i	$-0.950987\pi$
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	0.626907	+	0.779094i	$0.284321\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.50000	+	4.33013i	0.119591	+	0.207138i
$438$		0			0
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	0.423556	+	0.905870i	$0.360782\pi$
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	−0.996284	+	0.0861252i	$0.972552\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.00000	−	13.8564i	0.380091	−	0.658338i	−0.610984	−	0.791643i	$-0.709226\pi$
$443$	8.00000	−	13.8564i	0.380091	−	0.658338i	0.991075	+	0.133306i	$0.0425592\pi$
$444$		0			0
$445$	4.50000	+	7.79423i	0.213320	+	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	10.5000	−	18.1865i	0.494426	−	0.856370i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.500000	+	2.59808i	−0.0234404	+	0.121800i
$456$		0			0
$457$	13.0000	−	22.5167i	0.608114	−	1.05328i	−0.383437	−	0.923567i	$-0.625260\pi$
$457$	13.0000	−	22.5167i	0.608114	−	1.05328i	0.991551	−	0.129718i	$-0.0414071\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.5000	−	23.3827i	−0.628758	−	1.08904i	−0.987801	−	0.155719i	$-0.950230\pi$
$461$	−13.5000	−	23.3827i	−0.628758	−	1.08904i	0.359044	−	0.933321i	$-0.383103\pi$
$462$		0			0
$463$	0.500000	−	0.866025i	0.0232370	−	0.0402476i	−0.854173	−	0.519989i	$-0.825936\pi$
$463$	0.500000	−	0.866025i	0.0232370	−	0.0402476i	0.877410	+	0.479741i	$0.159269\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.50000	+	11.2583i	0.300784	+	0.520973i	0.976314	−	0.216359i	$-0.0694183\pi$
$467$	6.50000	+	11.2583i	0.300784	+	0.520973i	−0.675530	+	0.737333i	$0.736085\pi$
$468$		0			0
$469$	24.0000	+	20.7846i	1.10822	+	0.959744i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.50000	−	7.79423i	−0.206910	−	0.358379i
$474$		0			0
$475$	10.0000	+	17.3205i	0.458831	+	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.0000			0.959514			0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000			0.959514			0.479757	+	0.877401i	$0.340725\pi$
$480$		0			0
$481$	5.00000			0.227980
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.50000	−	14.7224i	−0.385965	−	0.668511i
$486$		0			0
$487$	−18.5000	+	32.0429i	−0.838315	+	1.45200i	0.0529875	+	0.998595i	$0.483126\pi$
$487$	−18.5000	+	32.0429i	−0.838315	+	1.45200i	−0.891303	+	0.453409i	$0.850208\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.50000	−	7.79423i	0.203082	−	0.351749i	−0.746438	−	0.665455i	$-0.768237\pi$
$491$	4.50000	−	7.79423i	0.203082	−	0.351749i	0.949520	+	0.313707i	$0.101571\pi$
$492$		0			0
$493$	27.0000			1.21602
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.0000	−	6.92820i	0.897123	−	0.310772i
$498$		0			0
$499$	−29.0000			−1.29822			−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000			−1.29822			−0.649109	+	0.760695i	$0.724858\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−7.00000			−0.311496
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−33.0000			−1.46270			−0.731350	−	0.682003i	$-0.761109\pi$
$509$	−33.0000			−1.46270			−0.731350	+	0.682003i	$0.761109\pi$
$510$		0			0
$511$	−6.50000	+	33.7750i	−0.287543	+	1.49412i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.00000			−0.396587
$516$		0			0
$517$	12.0000	−	20.7846i	0.527759	−	0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	−0.831295	−	0.555831i	$-0.812400\pi$
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	0.897011	+	0.442007i	$0.145733\pi$
$522$		0			0
$523$	−16.5000	−	28.5788i	−0.721495	−	1.24967i	−0.960401	−	0.278623i	$-0.910122\pi$
$523$	−16.5000	−	28.5788i	−0.721495	−	1.24967i	0.238906	−	0.971043i	$-0.423211\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.50000	+	6.06218i	0.151602	+	0.262582i
$534$		0			0
$535$	3.50000	+	6.06218i	0.151318	+	0.262091i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.5000	−	12.9904i	0.710705	−	0.559535i
$540$		0			0
$541$	5.50000	+	9.52628i	0.236463	+	0.409567i	0.959697	−	0.281037i	$-0.0906783\pi$
$541$	5.50000	+	9.52628i	0.236463	+	0.409567i	−0.723234	+	0.690604i	$0.757345\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.50000	−	4.33013i	0.107088	−	0.185482i
$546$		0			0
$547$	2.50000	+	4.33013i	0.106892	+	0.185143i	0.914510	−	0.404564i	$-0.132577\pi$
$547$	2.50000	+	4.33013i	0.106892	+	0.185143i	−0.807617	+	0.589707i	$0.799243\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.5000	−	38.9711i	0.958532	−	1.66023i
$552$		0			0
$553$	4.00000	−	20.7846i	0.170097	−	0.883852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5000	+	33.7750i	−0.826242	+	1.43109i	0.0747252	+	0.997204i	$0.476192\pi$
$557$	−19.5000	+	33.7750i	−0.826242	+	1.43109i	−0.900967	+	0.433888i	$0.857141\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	0.711445	−	0.702742i	$-0.248041\pi$
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	−0.964315	+	0.264758i	$0.914708\pi$
$564$		0			0
$565$	0.500000	−	0.866025i	0.0210352	−	0.0364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.0294311	+	0.999567i	$0.509370\pi$
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.850935	+	0.525271i	$0.823964\pi$
$570$		0			0
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	0.821138	−	0.570730i	$-0.193340\pi$
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	−0.904835	+	0.425762i	$0.860006\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−1.50000	+	2.59808i	−0.0624458	+	0.108159i	−0.895558	−	0.444945i	$-0.853223\pi$
$577$	−1.50000	+	2.59808i	−0.0624458	+	0.108159i	0.833112	+	0.553104i	$0.186557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	26.0000	+	22.5167i	1.07866	+	0.934148i
$582$		0			0
$583$	13.5000	−	23.3827i	0.559113	−	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.50000	+	12.9904i	0.309558	+	0.536170i	0.978266	−	0.207355i	$-0.0664855\pi$
$587$	7.50000	+	12.9904i	0.309558	+	0.536170i	−0.668708	+	0.743525i	$0.733152\pi$
$588$		0			0
$589$	−10.0000	+	17.3205i	−0.412043	+	0.713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	0.565792	−	0.824548i	$-0.308570\pi$
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	−0.996976	+	0.0777165i	$0.975237\pi$
$594$		0			0
$595$	1.50000	−	7.79423i	0.0614940	−	0.319532i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	−11.5000	−	19.9186i	−0.469095	−	0.812496i	0.530281	−	0.847822i	$-0.322086\pi$
$601$	−11.5000	−	19.9186i	−0.469095	−	0.812496i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.00000			0.0813116
$606$		0			0
$607$	−23.0000			−0.933541			−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000			−0.933541			−0.466771	+	0.884378i	$0.654583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000	+	6.92820i	0.161823	+	0.280285i
$612$		0			0
$613$	−2.50000	+	4.33013i	−0.100974	+	0.174892i	−0.912086	−	0.409998i	$-0.865529\pi$
$613$	−2.50000	+	4.33013i	−0.100974	+	0.174892i	0.811112	+	0.584891i	$0.198863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	−0.834251	−	0.551385i	$-0.814100\pi$
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	0.894639	+	0.446790i	$0.147433\pi$
$618$		0			0
$619$	21.0000			0.844061			0.422031	−	0.906582i	$-0.361317\pi$
$619$	21.0000			0.844061			0.422031	+	0.906582i	$0.361317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	+	15.5885i	0.721155	+	0.624538i
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000			−0.598089
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000			−0.317470
$636$		0			0
$637$	1.00000	+	6.92820i	0.0396214	+	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.00000			0.197488			0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000			0.197488			0.0987441	+	0.995113i	$0.468517\pi$
$642$		0			0
$643$	−18.5000	+	32.0429i	−0.729569	+	1.26365i	0.227497	+	0.973779i	$0.426946\pi$
$643$	−18.5000	+	32.0429i	−0.729569	+	1.26365i	−0.957066	+	0.289871i	$0.906387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.5000	−	35.5070i	0.805938	−	1.39593i	−0.109718	−	0.993963i	$-0.534995\pi$
$647$	20.5000	−	35.5070i	0.805938	−	1.39593i	0.915656	−	0.401963i	$-0.131672\pi$
$648$		0			0
$649$	−6.00000	−	10.3923i	−0.235521	−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000			0.117399			0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000			0.117399			0.0586995	+	0.998276i	$0.481305\pi$
$654$		0			0
$655$	3.00000			0.117220
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.50000	+	12.9904i	0.292159	+	0.506033i	0.974320	−	0.225168i	$-0.0722932\pi$
$659$	7.50000	+	12.9904i	0.292159	+	0.506033i	−0.682161	+	0.731202i	$0.738960\pi$
$660$		0			0
$661$	−19.0000	−	32.9090i	−0.739014	−	1.28001i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−19.0000	−	32.9090i	−0.739014	−	1.28001i	0.213925	−	0.976850i	$-0.431375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.0000	−	8.66025i	−0.387783	−	0.335830i
$666$		0			0
$667$	−4.50000	−	7.79423i	−0.174241	−	0.301794i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.00000	+	5.19615i	−0.115814	+	0.200595i
$672$		0			0
$673$	−17.5000	−	30.3109i	−0.674575	−	1.16840i	−0.976593	−	0.215096i	$-0.930993\pi$
$673$	−17.5000	−	30.3109i	−0.674575	−	1.16840i	0.302017	−	0.953302i	$-0.402340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.0000	−	32.9090i	0.730229	−	1.26479i	−0.226556	−	0.973998i	$-0.572747\pi$
$677$	19.0000	−	32.9090i	0.730229	−	1.26479i	0.956785	−	0.290796i	$-0.0939201\pi$
$678$		0			0
$679$	−34.0000	−	29.4449i	−1.30480	−	1.12999i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.5000	−	26.8468i	0.593091	−	1.02726i	−0.400722	−	0.916200i	$-0.631241\pi$
$683$	15.5000	−	26.8468i	0.593091	−	1.02726i	0.993813	−	0.111064i	$-0.0354259\pi$
$684$		0			0
$685$	11.0000			0.420288
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.50000	+	7.79423i	0.171436	+	0.296936i
$690$		0			0
$691$	6.00000	−	10.3923i	0.228251	−	0.395342i	−0.729039	−	0.684472i	$-0.760033\pi$
$691$	6.00000	−	10.3923i	0.228251	−	0.395342i	0.957290	+	0.289130i	$0.0933661\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.50000	−	4.33013i	0.0948304	−	0.164251i
$696$		0			0
$697$	−10.5000	−	18.1865i	−0.397716	−	0.688864i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$	−12.5000	+	21.6506i	−0.471446	+	0.816569i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.5000	+	6.06218i	−0.658155	+	0.227992i
$708$		0			0
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	−0.646055	−	0.763291i	$-0.723582\pi$
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	0.984057	+	0.177854i	$0.0569156\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.00000	+	3.46410i	0.0749006	+	0.129732i
$714$		0			0
$715$	1.50000	−	2.59808i	0.0560968	−	0.0971625i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.50000	+	16.4545i	0.354290	+	0.613649i	0.986996	−	0.160743i	$-0.0513892\pi$
$719$	9.50000	+	16.4545i	0.354290	+	0.613649i	−0.632706	+	0.774392i	$0.718056\pi$
$720$		0			0
$721$	−22.5000	+	7.79423i	−0.837944	+	0.290272i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	−	31.1769i	−0.668503	−	1.15788i
$726$		0			0
$727$	−8.50000	−	14.7224i	−0.315248	−	0.546025i	0.664243	−	0.747517i	$-0.268754\pi$
$727$	−8.50000	−	14.7224i	−0.315248	−	0.546025i	−0.979490	+	0.201492i	$0.935421\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.00000			−0.332877
$732$		0			0
$733$	−43.0000			−1.58824			−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000			−1.58824			−0.794121	+	0.607760i	$0.792068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18.0000	−	31.1769i	−0.663039	−	1.14842i
$738$		0			0
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	−0.666264	−	0.745716i	$-0.732107\pi$
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	0.978941	+	0.204143i	$0.0654407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.500000	+	0.866025i	−0.0183432	+	0.0317714i	−0.875051	−	0.484030i	$-0.839172\pi$
$743$	−0.500000	+	0.866025i	−0.0183432	+	0.0317714i	0.856708	+	0.515802i	$0.172506\pi$
$744$		0			0
$745$	−15.0000			−0.549557
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.0000	+	12.1244i	0.511549	+	0.443014i
$750$		0			0
$751$	−15.0000			−0.547358			−0.273679	−	0.961821i	$-0.588241\pi$
$751$	−15.0000			−0.547358			−0.273679	+	0.961821i	$0.588241\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.0000			0.618693
$756$		0			0
$757$	14.0000			0.508839			0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000			0.508839			0.254419	+	0.967094i	$0.418116\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.0000			1.63125			0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000			1.63125			0.815624	+	0.578582i	$0.196394\pi$
$762$		0			0
$763$	2.50000	−	12.9904i	0.0905061	−	0.470283i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.00000			0.144432
$768$		0			0
$769$	20.5000	−	35.5070i	0.739249	−	1.28042i	−0.213585	−	0.976924i	$-0.568514\pi$
$769$	20.5000	−	35.5070i	0.739249	−	1.28042i	0.952834	−	0.303492i	$-0.0981526\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.50000	+	6.06218i	−0.125886	+	0.218041i	−0.922079	−	0.387002i	$-0.873511\pi$
$773$	−3.50000	+	6.06218i	−0.125886	+	0.218041i	0.796193	+	0.605043i	$0.206844\pi$
$774$		0			0
$775$	8.00000	+	13.8564i	0.287368	+	0.497737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.0000			−1.25401
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.00000	+	12.1244i	0.249841	+	0.432737i
$786$		0			0
$787$	−10.0000	−	17.3205i	−0.356462	−	0.617409i	0.630905	−	0.775860i	$-0.282684\pi$
$787$	−10.0000	−	17.3205i	−0.356462	−	0.617409i	−0.987367	+	0.158450i	$0.949350\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.500000	−	2.59808i	0.0177780	−	0.0923770i
$792$		0			0
$793$	−1.00000	−	1.73205i	−0.0355110	−	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.5000	+	33.7750i	−0.690725	+	1.19637i	0.280875	+	0.959744i	$0.409375\pi$
$797$	−19.5000	+	33.7750i	−0.690725	+	1.19637i	−0.971601	+	0.236627i	$0.923958\pi$
$798$		0			0
$799$	−12.0000	−	20.7846i	−0.424529	−	0.735307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.5000	−	33.7750i	0.688140	−	1.19189i
$804$		0			0
$805$	−2.50000	+	0.866025i	−0.0881134	+	0.0305234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.5000	−	47.6314i	0.966849	−	1.67463i	0.262284	−	0.964991i	$-0.415524\pi$
$809$	27.5000	−	47.6314i	0.966849	−	1.67463i	0.704564	−	0.709640i	$-0.251142\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.50000	+	14.7224i	0.297742	+	0.515704i
$816$		0			0
$817$	−7.50000	+	12.9904i	−0.262392	+	0.454476i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.0000	−	36.3731i	0.732905	−	1.26943i	−0.222731	−	0.974880i	$-0.571497\pi$
$821$	21.0000	−	36.3731i	0.732905	−	1.26943i	0.955636	−	0.294549i	$-0.0951694\pi$
$822$		0			0
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	0.969448	+	0.245295i	$0.0788849\pi$
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	−0.272292	+	0.962215i	$0.587782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	40.0000			1.39094			0.695468	−	0.718557i	$-0.255197\pi$
$827$	40.0000			1.39094			0.695468	+	0.718557i	$0.255197\pi$
$828$		0			0
$829$	−2.50000	+	4.33013i	−0.0868286	+	0.150392i	−0.906169	−	0.422916i	$-0.861007\pi$
$829$	−2.50000	+	4.33013i	−0.0868286	+	0.150392i	0.819340	+	0.573307i	$0.194340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.00000	−	20.7846i	−0.103944	−	0.720144i
$834$		0			0
$835$	0.500000	−	0.866025i	0.0173032	−	0.0299700i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.5000	+	38.9711i	0.776786	+	1.34543i	0.933785	+	0.357834i	$0.116485\pi$
$839$	22.5000	+	38.9711i	0.776786	+	1.34543i	−0.156999	+	0.987599i	$0.550182\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.00000	−	10.3923i	−0.206406	−	0.357506i
$846$		0			0
$847$	5.00000	−	1.73205i	0.171802	−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.50000	+	4.33013i	0.0856989	+	0.148435i
$852$		0			0
$853$	−14.5000	−	25.1147i	−0.496471	−	0.859912i	0.503521	−	0.863983i	$-0.332038\pi$
$853$	−14.5000	−	25.1147i	−0.496471	−	0.859912i	−0.999992	+	0.00407068i	$0.998704\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.00000			−0.239115			−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000			−0.239115			−0.119558	+	0.992827i	$0.538148\pi$
$858$		0			0
$859$	19.0000			0.648272			0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000			0.648272			0.324136	+	0.946011i	$0.394927\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.50000	+	9.52628i	0.187222	+	0.324278i	0.944323	−	0.329020i	$-0.106718\pi$
$863$	5.50000	+	9.52628i	0.187222	+	0.324278i	−0.757101	+	0.653298i	$0.773385\pi$
$864$		0			0
$865$	1.00000	−	1.73205i	0.0340010	−	0.0588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.0000	+	20.7846i	−0.407072	+	0.705070i
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.5000	+	7.79423i	−0.760639	+	0.263493i
$876$		0			0
$877$	−23.0000			−0.776655			−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000			−0.776655			−0.388327	+	0.921521i	$0.626947\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	52.0000			1.74994			0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000			1.74994			0.874970	+	0.484178i	$0.160881\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.0000			0.906571			0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000			0.906571			0.453286	+	0.891365i	$0.350252\pi$
$888$		0			0
$889$	−20.0000	+	6.92820i	−0.670778	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−40.0000			−1.33855
$894$		0			0
$895$	10.5000	−	18.1865i	0.350976	−	0.607909i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.0000	−	31.1769i	0.600334	−	1.03981i
$900$		0			0
$901$	−13.5000	−	23.3827i	−0.449750	−	0.778990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.0000			0.598340
$906$		0			0
$907$	−7.00000			−0.232431			−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000			−0.232431			−0.116216	+	0.993224i	$0.537076\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−13.5000	−	23.3827i	−0.447275	−	0.774703i	0.550933	−	0.834550i	$-0.314272\pi$
$911$	−13.5000	−	23.3827i	−0.447275	−	0.774703i	−0.998208	+	0.0598468i	$0.980939\pi$
$912$		0			0
$913$	−19.5000	−	33.7750i	−0.645356	−	1.11779i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.50000	−	2.59808i	0.247672	−	0.0857960i
$918$		0			0
$919$	6.50000	+	11.2583i	0.214415	+	0.371378i	0.953092	−	0.302682i	$-0.0978821\pi$
$919$	6.50000	+	11.2583i	0.214415	+	0.371378i	−0.738676	+	0.674060i	$0.764549\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.00000	−	6.92820i	0.131662	−	0.228045i
$924$		0			0
$925$	10.0000	+	17.3205i	0.328798	+	0.569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.00000	+	15.5885i	−0.295280	+	0.511441i	−0.975050	−	0.221985i	$-0.928746\pi$
$929$	−9.00000	+	15.5885i	−0.295280	+	0.511441i	0.679770	+	0.733426i	$0.262080\pi$
$930$		0			0
$931$	−32.5000	−	12.9904i	−1.06514	−	0.425743i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.50000	+	7.79423i	−0.147166	+	0.254899i
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.0000	−	43.3013i	−0.814977	−	1.41158i	−0.909345	−	0.416044i	$-0.863416\pi$
$941$	−25.0000	−	43.3013i	−0.814977	−	1.41158i	0.0943679	−	0.995537i	$-0.469917\pi$
$942$		0			0
$943$	−3.50000	+	6.06218i	−0.113976	+	0.197412i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.0000	+	31.1769i	−0.584921	+	1.01311i	0.409964	+	0.912102i	$0.365541\pi$
$947$	−18.0000	+	31.1769i	−0.584921	+	1.01311i	−0.994885	+	0.101012i	$0.967792\pi$
$948$		0			0
$949$	6.50000	+	11.2583i	0.210999	+	0.365461i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	−2.00000	+	3.46410i	−0.0647185	+	0.112096i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	27.5000	−	9.52628i	0.888021	−	0.307620i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.00000	−	15.5885i	−0.289720	−	0.501810i
$966$		0			0
$967$	−13.5000	+	23.3827i	−0.434131	+	0.751936i	−0.997224	−	0.0744567i	$-0.976278\pi$
$967$	−13.5000	+	23.3827i	−0.434131	+	0.751936i	0.563094	+	0.826393i	$0.309611\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.5000	−	37.2391i	−0.689968	−	1.19506i	−0.971848	−	0.235610i	$-0.924291\pi$
$971$	−21.5000	−	37.2391i	−0.689968	−	1.19506i	0.281880	−	0.959450i	$-0.409042\pi$
$972$		0			0
$973$	2.50000	−	12.9904i	0.0801463	−	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	0.986586	−	0.163242i	$-0.0521952\pi$
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	−0.634665	+	0.772787i	$0.718862\pi$
$978$		0			0
$979$	−13.5000	−	23.3827i	−0.431462	−	0.747314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	41.0000			1.30770			0.653848	−	0.756626i	$-0.273153\pi$
$983$	41.0000			1.30770			0.653848	+	0.756626i	$0.273153\pi$
$984$		0			0
$985$	−22.0000			−0.700978
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.50000	+	2.59808i	0.0476972	+	0.0826140i
$990$		0			0
$991$	7.50000	−	12.9904i	0.238245	−	0.412653i	−0.721966	−	0.691929i	$-0.756761\pi$
$991$	7.50000	−	12.9904i	0.238245	−	0.412653i	0.960211	+	0.279276i	$0.0900944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.5000	−	18.1865i	0.332872	−	0.576552i
$996$		0			0
$997$	13.0000			0.411714			0.205857	−	0.978582i	$-0.434002\pi$
$997$	13.0000			0.411714			0.205857	+	0.978582i	$0.434002\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.t.a.361.1		2
3.2	odd	2		504.2.t.a.193.1	yes	2
4.3	odd	2		3024.2.t.c.1873.1		2
7.2	even	3		1512.2.q.b.793.1		2
9.2	odd	6		504.2.q.a.25.1	✓	2
9.7	even	3		1512.2.q.b.1369.1		2
12.11	even	2		1008.2.t.e.193.1		2
21.2	odd	6		504.2.q.a.121.1	yes	2
28.23	odd	6		3024.2.q.d.2305.1		2
36.7	odd	6		3024.2.q.d.2881.1		2
36.11	even	6		1008.2.q.b.529.1		2
63.2	odd	6		504.2.t.a.457.1	yes	2
63.16	even	3	inner	1512.2.t.a.289.1		2
84.23	even	6		1008.2.q.b.625.1		2
252.79	odd	6		3024.2.t.c.289.1		2
252.191	even	6		1008.2.t.e.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.q.a.25.1	✓	2	9.2	odd	6
504.2.q.a.121.1	yes	2	21.2	odd	6
504.2.t.a.193.1	yes	2	3.2	odd	2
504.2.t.a.457.1	yes	2	63.2	odd	6
1008.2.q.b.529.1		2	36.11	even	6
1008.2.q.b.625.1		2	84.23	even	6
1008.2.t.e.193.1		2	12.11	even	2
1008.2.t.e.961.1		2	252.191	even	6
1512.2.q.b.793.1		2	7.2	even	3
1512.2.q.b.1369.1		2	9.7	even	3
1512.2.t.a.289.1		2	63.16	even	3	inner
1512.2.t.a.361.1		2	1.1	even	1	trivial
3024.2.q.d.2305.1		2	28.23	odd	6
3024.2.q.d.2881.1		2	36.7	odd	6
3024.2.t.c.289.1		2	252.79	odd	6
3024.2.t.c.1873.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +(-2.50000 + 0.866025i) q^{7} +3.00000 q^{11} +(-0.500000 + 0.866025i) q^{13} +(1.50000 - 2.59808i) q^{17} +(-2.50000 - 4.33013i) q^{19} -1.00000 q^{23} -4.00000 q^{25} +(4.50000 + 7.79423i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(2.50000 - 0.866025i) q^{35} +(-2.50000 - 4.33013i) q^{37} +(3.50000 - 6.06218i) q^{41} +(-1.50000 - 2.59808i) q^{43} +(4.00000 - 6.92820i) q^{47} +(5.50000 - 4.33013i) q^{49} +(4.50000 - 7.79423i) q^{53} -3.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(0.500000 - 0.866025i) q^{65} +(-6.00000 - 10.3923i) q^{67} -8.00000 q^{71} +(6.50000 - 11.2583i) q^{73} +(-7.50000 + 2.59808i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-6.50000 - 11.2583i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(-4.50000 - 7.79423i) q^{89} +(0.500000 - 2.59808i) q^{91} +(2.50000 + 4.33013i) q^{95} +(8.50000 + 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 5 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 5 * q^7	\( 2 q - 2 q^{5} - 5 q^{7} + 6 q^{11} - q^{13} + 3 q^{17} - 5 q^{19} - 2 q^{23} - 8 q^{25} + 9 q^{29} - 4 q^{31} + 5 q^{35} - 5 q^{37} + 7 q^{41} - 3 q^{43} + 8 q^{47} + 11 q^{49} + 9 q^{53} - 6 q^{55} - 4 q^{59} - 2 q^{61} + q^{65} - 12 q^{67} - 16 q^{71} + 13 q^{73} - 15 q^{77} - 8 q^{79} - 13 q^{83} - 3 q^{85} - 9 q^{89} + q^{91} + 5 q^{95} + 17 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 5 * q^7 + 6 * q^11 - q^13 + 3 * q^17 - 5 * q^19 - 2 * q^23 - 8 * q^25 + 9 * q^29 - 4 * q^31 + 5 * q^35 - 5 * q^37 + 7 * q^41 - 3 * q^43 + 8 * q^47 + 11 * q^49 + 9 * q^53 - 6 * q^55 - 4 * q^59 - 2 * q^61 + q^65 - 12 * q^67 - 16 * q^71 + 13 * q^73 - 15 * q^77 - 8 * q^79 - 13 * q^83 - 3 * q^85 - 9 * q^89 + q^91 + 5 * q^95 + 17 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more