LMFDB - Embedded newform 2028.2.i.h.529.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,2,Mod(529,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, 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("2028.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.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:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			529.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2028.529
Dual form			2028.2.i.h.2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} -1.73205 q^{5} +(-1.73205 - 3.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} -1.73205 q^{5} +(-1.73205 - 3.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.73205 - 3.00000i) q^{11} +(-0.866025 + 1.50000i) q^{15} +(1.50000 + 2.59808i) q^{17} +(-1.73205 - 3.00000i) q^{19} -3.46410 q^{21} +(3.00000 - 5.19615i) q^{23} -2.00000 q^{25} -1.00000 q^{27} +(-4.50000 + 7.79423i) q^{29} +(-1.73205 - 3.00000i) q^{33} +(3.00000 + 5.19615i) q^{35} +(-2.59808 + 4.50000i) q^{37} +(4.33013 - 7.50000i) q^{41} +(1.00000 + 1.73205i) q^{43} +(0.866025 + 1.50000i) q^{45} -3.46410 q^{47} +(-2.50000 + 4.33013i) q^{49} +3.00000 q^{51} -9.00000 q^{53} +(-3.00000 + 5.19615i) q^{55} -3.46410 q^{57} +(-6.92820 - 12.0000i) q^{59} +(5.50000 + 9.52628i) q^{61} +(-1.73205 + 3.00000i) q^{63} +(-5.19615 + 9.00000i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(5.19615 + 9.00000i) q^{71} -5.19615 q^{73} +(-1.00000 + 1.73205i) q^{75} -12.0000 q^{77} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410 q^{83} +(-2.59808 - 4.50000i) q^{85} +(4.50000 + 7.79423i) q^{87} +(-3.46410 + 6.00000i) q^{89} +(3.00000 + 5.19615i) q^{95} +(3.46410 + 6.00000i) q^{97} -3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^9	$ 4 q + 2 q^{3} - 2 q^{9} + 6 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} - 18 q^{29} + 12 q^{35} + 4 q^{43} - 10 q^{49} + 12 q^{51} - 36 q^{53} - 12 q^{55} + 22 q^{61} - 12 q^{69} - 4 q^{75} - 48 q^{77} - 32 q^{79} - 2 q^{81} + 18 q^{87} + 12 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^9 + 6 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 - 18 * q^29 + 12 * q^35 + 4 * q^43 - 10 * q^49 + 12 * q^51 - 36 * q^53 - 12 * q^55 + 22 * q^61 - 12 * q^69 - 4 * q^75 - 48 * q^77 - 32 * q^79 - 2 * q^81 + 18 * q^87 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$		0			0
$7$	−1.73205	−	3.00000i	−0.654654	−	1.13389i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	−1.73205	−	3.00000i	−0.654654	−	1.13389i	0.327327	−	0.944911i	$-0.393852\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	1.73205	−	3.00000i	0.522233	−	0.904534i	−0.477432	−	0.878668i	$-0.658432\pi$
$11$	1.73205	−	3.00000i	0.522233	−	0.904534i	0.999665	−	0.0258656i	$-0.00823419\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−0.866025	+	1.50000i	−0.223607	+	0.387298i
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−1.73205	−	3.00000i	−0.397360	−	0.688247i	0.596040	−	0.802955i	$-0.296740\pi$
$19$	−1.73205	−	3.00000i	−0.397360	−	0.688247i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$	−3.46410			−0.755929
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	−1.73205	−	3.00000i	−0.301511	−	0.522233i
$34$		0			0
$35$	3.00000	+	5.19615i	0.507093	+	0.878310i
$36$		0			0
$37$	−2.59808	+	4.50000i	−0.427121	+	0.739795i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	−2.59808	+	4.50000i	−0.427121	+	0.739795i	0.569495	+	0.821995i	$0.307139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.33013	−	7.50000i	0.676252	−	1.17130i	−0.299849	−	0.953987i	$-0.596936\pi$
$41$	4.33013	−	7.50000i	0.676252	−	1.17130i	0.976101	−	0.217317i	$-0.0697304\pi$
$42$		0			0
$43$	1.00000	+	1.73205i	0.152499	+	0.264135i	0.932145	−	0.362084i	$-0.117935\pi$
$43$	1.00000	+	1.73205i	0.152499	+	0.264135i	−0.779647	+	0.626219i	$0.784601\pi$
$44$		0			0
$45$	0.866025	+	1.50000i	0.129099	+	0.223607i
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		0			0
$49$	−2.50000	+	4.33013i	−0.357143	+	0.618590i
$50$		0			0
$51$	3.00000			0.420084
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−3.00000	+	5.19615i	−0.404520	+	0.700649i
$56$		0			0
$57$	−3.46410			−0.458831
$58$		0			0
$59$	−6.92820	−	12.0000i	−0.901975	−	1.56227i	−0.824927	−	0.565240i	$-0.808784\pi$
$59$	−6.92820	−	12.0000i	−0.901975	−	1.56227i	−0.0770484	−	0.997027i	$-0.524550\pi$
$60$		0			0
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	0.966978	+	0.254858i	$0.0820288\pi$
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	−0.262776	+	0.964857i	$0.584638\pi$
$62$		0			0
$63$	−1.73205	+	3.00000i	−0.218218	+	0.377964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.19615	+	9.00000i	−0.634811	+	1.09952i	0.351744	+	0.936096i	$0.385589\pi$
$67$	−5.19615	+	9.00000i	−0.634811	+	1.09952i	−0.986555	+	0.163429i	$0.947745\pi$
$68$		0			0
$69$	−3.00000	−	5.19615i	−0.361158	−	0.625543i
$70$		0			0
$71$	5.19615	+	9.00000i	0.616670	+	1.06810i	0.990089	+	0.140441i	$0.0448520\pi$
$71$	5.19615	+	9.00000i	0.616670	+	1.06810i	−0.373419	+	0.927663i	$0.621815\pi$
$72$		0			0
$73$	−5.19615			−0.608164			−0.304082	−	0.952646i	$-0.598350\pi$
$73$	−5.19615			−0.608164			−0.304082	+	0.952646i	$0.598350\pi$
$74$		0			0
$75$	−1.00000	+	1.73205i	−0.115470	+	0.200000i
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$		0			0
$85$	−2.59808	−	4.50000i	−0.281801	−	0.488094i
$86$		0			0
$87$	4.50000	+	7.79423i	0.482451	+	0.835629i
$88$		0			0
$89$	−3.46410	+	6.00000i	−0.367194	+	0.635999i	−0.989126	−	0.147073i	$-0.953015\pi$
$89$	−3.46410	+	6.00000i	−0.367194	+	0.635999i	0.621932	+	0.783072i	$0.286348\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000	+	5.19615i	0.307794	+	0.533114i
$96$		0			0
$97$	3.46410	+	6.00000i	0.351726	+	0.609208i	0.986552	−	0.163448i	$-0.0522615\pi$
$97$	3.46410	+	6.00000i	0.351726	+	0.609208i	−0.634826	+	0.772655i	$0.718928\pi$
$98$		0			0
$99$	−3.46410			−0.348155
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$	6.00000			0.585540
$106$		0			0
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.00813215	−	0.999967i	$-0.497411\pi$
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.861931	−	0.507026i	$-0.169255\pi$
$108$		0			0
$109$	−13.8564			−1.32720			−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564			−1.32720			−0.663602	+	0.748086i	$0.730973\pi$
$110$		0			0
$111$	2.59808	+	4.50000i	0.246598	+	0.427121i
$112$		0			0
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	0.786806	−	0.617200i	$-0.211733\pi$
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	−0.927914	+	0.372794i	$0.878400\pi$
$114$		0			0
$115$	−5.19615	+	9.00000i	−0.484544	+	0.839254i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.19615	−	9.00000i	0.476331	−	0.825029i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$	−4.33013	−	7.50000i	−0.390434	−	0.676252i
$124$		0			0
$125$	12.1244			1.08444
$126$		0			0
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	0.964899	−	0.262620i	$-0.0845865\pi$
$128$		0			0
$129$	2.00000			0.176090
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−6.00000	+	10.3923i	−0.520266	+	0.901127i
$134$		0			0
$135$	1.73205			0.149071
$136$		0			0
$137$	−7.79423	−	13.5000i	−0.665906	−	1.15338i	−0.979039	−	0.203674i	$-0.934712\pi$
$137$	−7.79423	−	13.5000i	−0.665906	−	1.15338i	0.313133	−	0.949709i	$-0.398621\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$	−1.73205	+	3.00000i	−0.145865	+	0.252646i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	7.79423	−	13.5000i	0.647275	−	1.12111i
$146$		0			0
$147$	2.50000	+	4.33013i	0.206197	+	0.357143i
$148$		0			0
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	0.632335	−	0.774695i	$-0.282097\pi$
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	−0.987073	+	0.160271i	$0.948763\pi$
$150$		0			0
$151$	−17.3205			−1.40952			−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205			−1.40952			−0.704761	+	0.709444i	$0.748946\pi$
$152$		0			0
$153$	1.50000	−	2.59808i	0.121268	−	0.210042i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	23.0000			1.83560			0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000			1.83560			0.917800	+	0.397043i	$0.129964\pi$
$158$		0			0
$159$	−4.50000	+	7.79423i	−0.356873	+	0.618123i
$160$		0			0
$161$	−20.7846			−1.63806
$162$		0			0
$163$	6.92820	+	12.0000i	0.542659	+	0.939913i	0.998750	+	0.0499796i	$0.0159156\pi$
$163$	6.92820	+	12.0000i	0.542659	+	0.939913i	−0.456091	+	0.889933i	$0.650751\pi$
$164$		0			0
$165$	3.00000	+	5.19615i	0.233550	+	0.404520i
$166$		0			0
$167$	3.46410	−	6.00000i	0.268060	−	0.464294i	−0.700301	−	0.713848i	$-0.746951\pi$
$167$	3.46410	−	6.00000i	0.268060	−	0.464294i	0.968361	+	0.249554i	$0.0802840\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−1.73205	+	3.00000i	−0.132453	+	0.229416i
$172$		0			0
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$	3.46410	+	6.00000i	0.261861	+	0.453557i
$176$		0			0
$177$	−13.8564			−1.04151
$178$		0			0
$179$	9.00000	−	15.5885i	0.672692	−	1.16514i	−0.304446	−	0.952529i	$-0.598471\pi$
$179$	9.00000	−	15.5885i	0.672692	−	1.16514i	0.977138	−	0.212607i	$-0.0681952\pi$
$180$		0			0
$181$	13.0000			0.966282			0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000			0.966282			0.483141	+	0.875542i	$0.339496\pi$
$182$		0			0
$183$	11.0000			0.813143
$184$		0			0
$185$	4.50000	−	7.79423i	0.330847	−	0.573043i
$186$		0			0
$187$	10.3923			0.759961
$188$		0			0
$189$	1.73205	+	3.00000i	0.125988	+	0.218218i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	0.866025	−	1.50000i	0.0623379	−	0.107972i	−0.833172	−	0.553014i	$-0.813478\pi$
$193$	0.866025	−	1.50000i	0.0623379	−	0.107972i	0.895510	+	0.445041i	$0.146811\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.46410	+	6.00000i	−0.246807	+	0.427482i	−0.962638	−	0.270791i	$-0.912715\pi$
$197$	−3.46410	+	6.00000i	−0.246807	+	0.427482i	0.715831	+	0.698273i	$0.246048\pi$
$198$		0			0
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	0.987022	−	0.160585i	$-0.0513380\pi$
$199$	5.00000	+	8.66025i	0.354441	+	0.613909i	−0.632581	+	0.774494i	$0.718005\pi$
$200$		0			0
$201$	5.19615	+	9.00000i	0.366508	+	0.634811i
$202$		0			0
$203$	31.1769			2.18819
$204$		0			0
$205$	−7.50000	+	12.9904i	−0.523823	+	0.907288i
$206$		0			0
$207$	−6.00000			−0.417029
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$	10.3923			0.712069
$214$		0			0
$215$	−1.73205	−	3.00000i	−0.118125	−	0.204598i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.59808	+	4.50000i	−0.175562	+	0.304082i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−6.92820	+	12.0000i	−0.463947	+	0.803579i	−0.999153	−	0.0411418i	$-0.986900\pi$
$223$	−6.92820	+	12.0000i	−0.463947	+	0.803579i	0.535207	+	0.844721i	$0.320234\pi$
$224$		0			0
$225$	1.00000	+	1.73205i	0.0666667	+	0.115470i
$226$		0			0
$227$	5.19615	+	9.00000i	0.344881	+	0.597351i	0.985332	−	0.170648i	$-0.0545860\pi$
$227$	5.19615	+	9.00000i	0.344881	+	0.597351i	−0.640451	+	0.767999i	$0.721253\pi$
$228$		0			0
$229$	−6.92820			−0.457829			−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820			−0.457829			−0.228914	+	0.973447i	$0.573518\pi$
$230$		0			0
$231$	−6.00000	+	10.3923i	−0.394771	+	0.683763i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$		0			0
$237$	−4.00000	+	6.92820i	−0.259828	+	0.450035i
$238$		0			0
$239$	10.3923			0.672222			0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923			0.672222			0.336111	+	0.941822i	$0.390888\pi$
$240$		0			0
$241$	−12.9904	−	22.5000i	−0.836784	−	1.44935i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	−12.9904	−	22.5000i	−0.836784	−	1.44935i	0.0557856	−	0.998443i	$-0.482234\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	4.33013	−	7.50000i	0.276642	−	0.479157i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.73205	+	3.00000i	−0.109764	+	0.190117i
$250$		0			0
$251$	−12.0000	−	20.7846i	−0.757433	−	1.31191i	−0.944156	−	0.329500i	$-0.893120\pi$
$251$	−12.0000	−	20.7846i	−0.757433	−	1.31191i	0.186722	−	0.982413i	$-0.440214\pi$
$252$		0			0
$253$	−10.3923	−	18.0000i	−0.653359	−	1.13165i
$254$		0			0
$255$	−5.19615			−0.325396
$256$		0			0
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	−0.909010	−	0.416775i	$-0.863160\pi$
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	0.815442	+	0.578838i	$0.196494\pi$
$258$		0			0
$259$	18.0000			1.11847
$260$		0			0
$261$	9.00000			0.557086
$262$		0			0
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	−0.943572	−	0.331166i	$-0.892558\pi$
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	0.758585	+	0.651575i	$0.225891\pi$
$264$		0			0
$265$	15.5885			0.957591
$266$		0			0
$267$	3.46410	+	6.00000i	0.212000	+	0.367194i
$268$		0			0
$269$	−15.0000	−	25.9808i	−0.914566	−	1.58408i	−0.807535	−	0.589819i	$-0.799199\pi$
$269$	−15.0000	−	25.9808i	−0.914566	−	1.58408i	−0.107031	−	0.994256i	$-0.534134\pi$
$270$		0			0
$271$	3.46410	−	6.00000i	0.210429	−	0.364474i	−0.741420	−	0.671042i	$-0.765847\pi$
$271$	3.46410	−	6.00000i	0.210429	−	0.364474i	0.951849	+	0.306568i	$0.0991805\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.46410	+	6.00000i	−0.208893	+	0.361814i
$276$		0			0
$277$	−9.50000	−	16.4545i	−0.570800	−	0.988654i	−0.996484	−	0.0837823i	$-0.973300\pi$
$277$	−9.50000	−	16.4545i	−0.570800	−	0.988654i	0.425684	−	0.904872i	$-0.360033\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	29.4449			1.75653			0.878267	−	0.478171i	$-0.158700\pi$
$281$	29.4449			1.75653			0.878267	+	0.478171i	$0.158700\pi$
$282$		0			0
$283$	−7.00000	+	12.1244i	−0.416107	+	0.720718i	−0.995544	−	0.0942988i	$-0.969939\pi$
$283$	−7.00000	+	12.1244i	−0.416107	+	0.720718i	0.579437	+	0.815017i	$0.303272\pi$
$284$		0			0
$285$	6.00000			0.355409
$286$		0			0
$287$	−30.0000			−1.77084
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$	6.92820			0.406138
$292$		0			0
$293$	−0.866025	−	1.50000i	−0.0505937	−	0.0876309i	0.839619	−	0.543175i	$-0.182778\pi$
$293$	−0.866025	−	1.50000i	−0.0505937	−	0.0876309i	−0.890213	+	0.455544i	$0.849445\pi$
$294$		0			0
$295$	12.0000	+	20.7846i	0.698667	+	1.21013i
$296$		0			0
$297$	−1.73205	+	3.00000i	−0.100504	+	0.174078i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	3.46410	−	6.00000i	0.199667	−	0.345834i
$302$		0			0
$303$	−1.50000	−	2.59808i	−0.0861727	−	0.149256i
$304$		0			0
$305$	−9.52628	−	16.5000i	−0.545473	−	0.944787i
$306$		0			0
$307$	−17.3205			−0.988534			−0.494267	−	0.869310i	$-0.664563\pi$
$307$	−17.3205			−0.988534			−0.494267	+	0.869310i	$0.664563\pi$
$308$		0			0
$309$	−7.00000	+	12.1244i	−0.398216	+	0.689730i
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	3.00000	−	5.19615i	0.169031	−	0.292770i
$316$		0			0
$317$	1.73205			0.0972817			0.0486408	−	0.998816i	$-0.484511\pi$
$317$	1.73205			0.0972817			0.0486408	+	0.998816i	$0.484511\pi$
$318$		0			0
$319$	15.5885	+	27.0000i	0.872786	+	1.51171i
$320$		0			0
$321$	−9.00000	−	15.5885i	−0.502331	−	0.870063i
$322$		0			0
$323$	5.19615	−	9.00000i	0.289122	−	0.500773i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.92820	+	12.0000i	−0.383131	+	0.663602i
$328$		0			0
$329$	6.00000	+	10.3923i	0.330791	+	0.572946i
$330$		0			0
$331$	6.92820	+	12.0000i	0.380808	+	0.659580i	0.991178	−	0.132537i	$-0.0423124\pi$
$331$	6.92820	+	12.0000i	0.380808	+	0.659580i	−0.610370	+	0.792117i	$0.708979\pi$
$332$		0			0
$333$	5.19615			0.284747
$334$		0			0
$335$	9.00000	−	15.5885i	0.491723	−	0.851688i
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$		0			0
$339$	−3.00000			−0.162938
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.92820			−0.374088
$344$		0			0
$345$	5.19615	+	9.00000i	0.279751	+	0.484544i
$346$		0			0
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	0.516667	−	0.856186i	$-0.327172\pi$
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	−0.999813	+	0.0193540i	$0.993839\pi$
$348$		0			0
$349$	−6.92820	+	12.0000i	−0.370858	+	0.642345i	−0.989698	−	0.143172i	$-0.954270\pi$
$349$	−6.92820	+	12.0000i	−0.370858	+	0.642345i	0.618840	+	0.785517i	$0.287603\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.9904	−	22.5000i	0.691408	−	1.19755i	−0.279968	−	0.960009i	$-0.590324\pi$
$353$	12.9904	−	22.5000i	0.691408	−	1.19755i	0.971377	−	0.237545i	$-0.0763427\pi$
$354$		0			0
$355$	−9.00000	−	15.5885i	−0.477670	−	0.827349i
$356$		0			0
$357$	−5.19615	−	9.00000i	−0.275010	−	0.476331i
$358$		0			0
$359$	3.46410			0.182828			0.0914141	−	0.995813i	$-0.470861\pi$
$359$	3.46410			0.182828			0.0914141	+	0.995813i	$0.470861\pi$
$360$		0			0
$361$	3.50000	−	6.06218i	0.184211	−	0.319062i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	9.00000			0.471082
$366$		0			0
$367$	−1.00000	+	1.73205i	−0.0521996	+	0.0904123i	−0.890945	−	0.454112i	$-0.849957\pi$
$367$	−1.00000	+	1.73205i	−0.0521996	+	0.0904123i	0.838745	+	0.544524i	$0.183290\pi$
$368$		0			0
$369$	−8.66025			−0.450835
$370$		0			0
$371$	15.5885	+	27.0000i	0.809312	+	1.40177i
$372$		0			0
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	−0.917926	−	0.396751i	$-0.870138\pi$
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	0.115367	−	0.993323i	$-0.463196\pi$
$374$		0			0
$375$	6.06218	−	10.5000i	0.313050	−	0.542218i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.92820	+	12.0000i	−0.355878	+	0.616399i	−0.987268	−	0.159067i	$-0.949151\pi$
$379$	−6.92820	+	12.0000i	−0.355878	+	0.616399i	0.631390	+	0.775466i	$0.282485\pi$
$380$		0			0
$381$	−8.00000	−	13.8564i	−0.409852	−	0.709885i
$382$		0			0
$383$	6.92820	+	12.0000i	0.354015	+	0.613171i	0.986949	−	0.161034i	$-0.0514830\pi$
$383$	6.92820	+	12.0000i	0.354015	+	0.613171i	−0.632934	+	0.774206i	$0.718150\pi$
$384$		0			0
$385$	20.7846			1.05928
$386$		0			0
$387$	1.00000	−	1.73205i	0.0508329	−	0.0880451i
$388$		0			0
$389$	−21.0000			−1.06474			−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000			−1.06474			−0.532371	+	0.846511i	$0.678699\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	−6.00000	+	10.3923i	−0.302660	+	0.524222i
$394$		0			0
$395$	13.8564			0.697191
$396$		0			0
$397$	13.8564	+	24.0000i	0.695433	+	1.20453i	0.970034	+	0.242967i	$0.0781208\pi$
$397$	13.8564	+	24.0000i	0.695433	+	1.20453i	−0.274601	+	0.961558i	$0.588546\pi$
$398$		0			0
$399$	6.00000	+	10.3923i	0.300376	+	0.520266i
$400$		0			0
$401$	18.1865	−	31.5000i	0.908192	−	1.57303i	0.0916181	−	0.995794i	$-0.470796\pi$
$401$	18.1865	−	31.5000i	0.908192	−	1.57303i	0.816574	−	0.577241i	$-0.195871\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.866025	−	1.50000i	0.0430331	−	0.0745356i
$406$		0			0
$407$	9.00000	+	15.5885i	0.446113	+	0.772691i
$408$		0			0
$409$	−6.06218	−	10.5000i	−0.299755	−	0.519192i	0.676324	−	0.736604i	$-0.263572\pi$
$409$	−6.06218	−	10.5000i	−0.299755	−	0.519192i	−0.976080	+	0.217412i	$0.930238\pi$
$410$		0			0
$411$	−15.5885			−0.768922
$412$		0			0
$413$	−24.0000	+	41.5692i	−1.18096	+	2.04549i
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$	4.00000			0.195881
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	12.1244			0.590905			0.295452	−	0.955357i	$-0.404530\pi$
$421$	12.1244			0.590905			0.295452	+	0.955357i	$0.404530\pi$
$422$		0			0
$423$	1.73205	+	3.00000i	0.0842152	+	0.145865i
$424$		0			0
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$		0			0
$427$	19.0526	−	33.0000i	0.922018	−	1.59698i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.5885	−	27.0000i	0.750870	−	1.30054i	−0.196532	−	0.980497i	$-0.562968\pi$
$431$	15.5885	−	27.0000i	0.750870	−	1.30054i	0.947402	−	0.320047i	$-0.103699\pi$
$432$		0			0
$433$	−14.5000	−	25.1147i	−0.696826	−	1.20694i	−0.969561	−	0.244848i	$-0.921262\pi$
$433$	−14.5000	−	25.1147i	−0.696826	−	1.20694i	0.272736	−	0.962089i	$-0.412071\pi$
$434$		0			0
$435$	−7.79423	−	13.5000i	−0.373705	−	0.647275i
$436$		0			0
$437$	−20.7846			−0.994263
$438$		0			0
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	−0.960323	−	0.278889i	$-0.910034\pi$
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	0.721686	+	0.692220i	$0.243367\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		0			0
$447$	−8.66025			−0.409616
$448$		0			0
$449$	3.46410	+	6.00000i	0.163481	+	0.283158i	0.936115	−	0.351694i	$-0.114394\pi$
$449$	3.46410	+	6.00000i	0.163481	+	0.283158i	−0.772634	+	0.634852i	$0.781061\pi$
$450$		0			0
$451$	−15.0000	−	25.9808i	−0.706322	−	1.22339i
$452$		0			0
$453$	−8.66025	+	15.0000i	−0.406894	+	0.704761i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.33013	+	7.50000i	−0.202555	+	0.350835i	−0.949351	−	0.314218i	$-0.898258\pi$
$457$	−4.33013	+	7.50000i	−0.202555	+	0.350835i	0.746796	+	0.665053i	$0.231591\pi$
$458$		0			0
$459$	−1.50000	−	2.59808i	−0.0700140	−	0.121268i
$460$		0			0
$461$	2.59808	+	4.50000i	0.121004	+	0.209586i	0.920164	−	0.391533i	$-0.128055\pi$
$461$	2.59808	+	4.50000i	0.121004	+	0.209586i	−0.799160	+	0.601119i	$0.794722\pi$
$462$		0			0
$463$	3.46410			0.160990			0.0804952	−	0.996755i	$-0.474350\pi$
$463$	3.46410			0.160990			0.0804952	+	0.996755i	$0.474350\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$	11.5000	−	19.9186i	0.529892	−	0.917800i
$472$		0			0
$473$	6.92820			0.318559
$474$		0			0
$475$	3.46410	+	6.00000i	0.158944	+	0.275299i
$476$		0			0
$477$	4.50000	+	7.79423i	0.206041	+	0.356873i
$478$		0			0
$479$	3.46410	−	6.00000i	0.158279	−	0.274147i	−0.775969	−	0.630771i	$-0.782739\pi$
$479$	3.46410	−	6.00000i	0.158279	−	0.274147i	0.934248	+	0.356624i	$0.116072\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−10.3923	+	18.0000i	−0.472866	+	0.819028i
$484$		0			0
$485$	−6.00000	−	10.3923i	−0.272446	−	0.471890i
$486$		0			0
$487$	−8.66025	−	15.0000i	−0.392434	−	0.679715i	0.600336	−	0.799748i	$-0.295033\pi$
$487$	−8.66025	−	15.0000i	−0.392434	−	0.679715i	−0.992770	+	0.120033i	$0.961700\pi$
$488$		0			0
$489$	13.8564			0.626608
$490$		0			0
$491$	15.0000	−	25.9808i	0.676941	−	1.17250i	−0.298957	−	0.954267i	$-0.596639\pi$
$491$	15.0000	−	25.9808i	0.676941	−	1.17250i	0.975898	−	0.218229i	$-0.0700279\pi$
$492$		0			0
$493$	−27.0000			−1.21602
$494$		0			0
$495$	6.00000			0.269680
$496$		0			0
$497$	18.0000	−	31.1769i	0.807410	−	1.39848i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−3.46410	−	6.00000i	−0.154765	−	0.268060i
$502$		0			0
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	0.791361	−	0.611349i	$-0.209373\pi$
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	−0.925124	+	0.379664i	$0.876040\pi$
$504$		0			0
$505$	−2.59808	+	4.50000i	−0.115613	+	0.200247i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.866025	+	1.50000i	−0.0383859	+	0.0664863i	−0.884580	−	0.466388i	$-0.845555\pi$
$509$	−0.866025	+	1.50000i	−0.0383859	+	0.0664863i	0.846194	+	0.532875i	$0.178888\pi$
$510$		0			0
$511$	9.00000	+	15.5885i	0.398137	+	0.689593i
$512$		0			0
$513$	1.73205	+	3.00000i	0.0764719	+	0.132453i
$514$		0			0
$515$	24.2487			1.06853
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$	18.0000			0.790112
$520$		0			0
$521$	−9.00000			−0.394297			−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000			−0.394297			−0.197149	+	0.980374i	$0.563168\pi$
$522$		0			0
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	−0.735756	−	0.677247i	$-0.763173\pi$
$523$	5.00000	−	8.66025i	0.218635	−	0.378686i	0.954391	+	0.298560i	$0.0965063\pi$
$524$		0			0
$525$	6.92820			0.302372
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	−6.92820	+	12.0000i	−0.300658	+	0.520756i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−15.5885	+	27.0000i	−0.673948	+	1.16731i
$536$		0			0
$537$	−9.00000	−	15.5885i	−0.388379	−	0.672692i
$538$		0			0
$539$	8.66025	+	15.0000i	0.373024	+	0.646096i
$540$		0			0
$541$	1.73205			0.0744667			0.0372333	−	0.999307i	$-0.488146\pi$
$541$	1.73205			0.0744667			0.0372333	+	0.999307i	$0.488146\pi$
$542$		0			0
$543$	6.50000	−	11.2583i	0.278942	−	0.483141i
$544$		0			0
$545$	24.0000			1.02805
$546$		0			0
$547$	−34.0000			−1.45374			−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000			−1.45374			−0.726868	+	0.686778i	$0.759025\pi$
$548$		0			0
$549$	5.50000	−	9.52628i	0.234734	−	0.406572i
$550$		0			0
$551$	31.1769			1.32818
$552$		0			0
$553$	13.8564	+	24.0000i	0.589234	+	1.02058i
$554$		0			0
$555$	−4.50000	−	7.79423i	−0.191014	−	0.330847i
$556$		0			0
$557$	0.866025	−	1.50000i	0.0366947	−	0.0635570i	−0.847095	−	0.531442i	$-0.821650\pi$
$557$	0.866025	−	1.50000i	0.0366947	−	0.0635570i	0.883789	+	0.467885i	$0.154984\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.19615	−	9.00000i	0.219382	−	0.379980i
$562$		0			0
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	0.999978	+	0.00664037i	$0.00211371\pi$
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	−0.494238	+	0.869326i	$0.664553\pi$
$564$		0			0
$565$	2.59808	+	4.50000i	0.109302	+	0.189316i
$566$		0			0
$567$	3.46410			0.145479
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	26.0000			1.08807			0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000			1.08807			0.544033	+	0.839064i	$0.316897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000	+	10.3923i	−0.250217	+	0.433389i
$576$		0			0
$577$	−8.66025			−0.360531			−0.180266	−	0.983618i	$-0.557696\pi$
$577$	−8.66025			−0.360531			−0.180266	+	0.983618i	$0.557696\pi$
$578$		0			0
$579$	−0.866025	−	1.50000i	−0.0359908	−	0.0623379i
$580$		0			0
$581$	6.00000	+	10.3923i	0.248922	+	0.431145i
$582$		0			0
$583$	−15.5885	+	27.0000i	−0.645608	+	1.11823i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.3923	+	18.0000i	−0.428936	+	0.742940i	−0.996779	−	0.0801976i	$-0.974445\pi$
$587$	−10.3923	+	18.0000i	−0.428936	+	0.742940i	0.567843	+	0.823137i	$0.307778\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	3.46410	+	6.00000i	0.142494	+	0.246807i
$592$		0			0
$593$	12.1244			0.497888			0.248944	−	0.968518i	$-0.419917\pi$
$593$	12.1244			0.497888			0.248944	+	0.968518i	$0.419917\pi$
$594$		0			0
$595$	−9.00000	+	15.5885i	−0.368964	+	0.639064i
$596$		0			0
$597$	10.0000			0.409273
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−18.5000	+	32.0429i	−0.754631	+	1.30706i	0.190927	+	0.981604i	$0.438851\pi$
$601$	−18.5000	+	32.0429i	−0.754631	+	1.30706i	−0.945558	+	0.325455i	$0.894483\pi$
$602$		0			0
$603$	10.3923			0.423207
$604$		0			0
$605$	0.866025	+	1.50000i	0.0352089	+	0.0609837i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$	15.5885	−	27.0000i	0.631676	−	1.09410i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.59808	−	4.50000i	0.104935	−	0.181753i	−0.808776	−	0.588116i	$-0.799870\pi$
$613$	2.59808	−	4.50000i	0.104935	−	0.181753i	0.913712	+	0.406363i	$0.133203\pi$
$614$		0			0
$615$	7.50000	+	12.9904i	0.302429	+	0.523823i
$616$		0			0
$617$	−12.9904	−	22.5000i	−0.522973	−	0.905816i	−0.999643	−	0.0267333i	$-0.991490\pi$
$617$	−12.9904	−	22.5000i	−0.522973	−	0.905816i	0.476670	−	0.879083i	$-0.341844\pi$
$618$		0			0
$619$	13.8564			0.556936			0.278468	−	0.960446i	$-0.410173\pi$
$619$	13.8564			0.556936			0.278468	+	0.960446i	$0.410173\pi$
$620$		0			0
$621$	−3.00000	+	5.19615i	−0.120386	+	0.208514i
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−6.00000	+	10.3923i	−0.239617	+	0.415029i
$628$		0			0
$629$	−15.5885			−0.621552
$630$		0			0
$631$	−13.8564	−	24.0000i	−0.551615	−	0.955425i	−0.998158	−	0.0606630i	$-0.980679\pi$
$631$	−13.8564	−	24.0000i	−0.551615	−	0.955425i	0.446543	−	0.894762i	$-0.352655\pi$
$632$		0			0
$633$	−4.00000	−	6.92820i	−0.158986	−	0.275371i
$634$		0			0
$635$	−13.8564	+	24.0000i	−0.549875	+	0.952411i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5.19615	−	9.00000i	0.205557	−	0.356034i
$640$		0			0
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	0.580674	−	0.814136i	$-0.302789\pi$
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	−0.995400	+	0.0958109i	$0.969456\pi$
$642$		0			0
$643$	−3.46410	−	6.00000i	−0.136611	−	0.236617i	0.789601	−	0.613621i	$-0.210288\pi$
$643$	−3.46410	−	6.00000i	−0.136611	−	0.236617i	−0.926212	+	0.377004i	$0.876954\pi$
$644$		0			0
$645$	−3.46410			−0.136399
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$	20.7846			0.812122
$656$		0			0
$657$	2.59808	+	4.50000i	0.101361	+	0.175562i
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	−6.06218	+	10.5000i	−0.235791	+	0.408403i	−0.959502	−	0.281701i	$-0.909102\pi$
$661$	−6.06218	+	10.5000i	−0.235791	+	0.408403i	0.723711	+	0.690103i	$0.242435\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.3923	−	18.0000i	0.402996	−	0.698010i
$666$		0			0
$667$	27.0000	+	46.7654i	1.04544	+	1.81076i
$668$		0			0
$669$	6.92820	+	12.0000i	0.267860	+	0.463947i
$670$		0			0
$671$	38.1051			1.47103
$672$		0			0
$673$	2.50000	−	4.33013i	0.0963679	−	0.166914i	−0.813811	−	0.581130i	$-0.802611\pi$
$673$	2.50000	−	4.33013i	0.0963679	−	0.166914i	0.910179	+	0.414216i	$0.135944\pi$
$674$		0			0
$675$	2.00000			0.0769800
$676$		0			0
$677$	−30.0000			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$678$		0			0
$679$	12.0000	−	20.7846i	0.460518	−	0.797640i
$680$		0			0
$681$	10.3923			0.398234
$682$		0			0
$683$	6.92820	+	12.0000i	0.265100	+	0.459167i	0.967590	−	0.252527i	$-0.0812616\pi$
$683$	6.92820	+	12.0000i	0.265100	+	0.459167i	−0.702490	+	0.711694i	$0.747928\pi$
$684$		0			0
$685$	13.5000	+	23.3827i	0.515808	+	0.893407i
$686$		0			0
$687$	−3.46410	+	6.00000i	−0.132164	+	0.228914i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.19615	−	9.00000i	0.197671	−	0.342376i	−0.750102	−	0.661322i	$-0.769996\pi$
$691$	5.19615	−	9.00000i	0.197671	−	0.342376i	0.947773	+	0.318946i	$0.103329\pi$
$692$		0			0
$693$	6.00000	+	10.3923i	0.227921	+	0.394771i
$694$		0			0
$695$	−3.46410	−	6.00000i	−0.131401	−	0.227593i
$696$		0			0
$697$	25.9808			0.984092
$698$		0			0
$699$	−9.00000	+	15.5885i	−0.340411	+	0.589610i
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$		0			0
$705$	3.00000	−	5.19615i	0.112987	−	0.195698i
$706$		0			0
$707$	−10.3923			−0.390843
$708$		0			0
$709$	−7.79423	−	13.5000i	−0.292718	−	0.507003i	0.681733	−	0.731601i	$-0.261227\pi$
$709$	−7.79423	−	13.5000i	−0.292718	−	0.507003i	−0.974452	+	0.224598i	$0.927893\pi$
$710$		0			0
$711$	4.00000	+	6.92820i	0.150012	+	0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	5.19615	−	9.00000i	0.194054	−	0.336111i
$718$		0			0
$719$	−18.0000	−	31.1769i	−0.671287	−	1.16270i	−0.977539	−	0.210752i	$-0.932409\pi$
$719$	−18.0000	−	31.1769i	−0.671287	−	1.16270i	0.306253	−	0.951950i	$-0.400925\pi$
$720$		0			0
$721$	24.2487	+	42.0000i	0.903069	+	1.56416i
$722$		0			0
$723$	−25.9808			−0.966235
$724$		0			0
$725$	9.00000	−	15.5885i	0.334252	−	0.578941i
$726$		0			0
$727$	−10.0000			−0.370879			−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000			−0.370879			−0.185440	+	0.982656i	$0.559371\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−3.00000	+	5.19615i	−0.110959	+	0.192187i
$732$		0			0
$733$	36.3731			1.34347			0.671735	−	0.740792i	$-0.265549\pi$
$733$	36.3731			1.34347			0.671735	+	0.740792i	$0.265549\pi$
$734$		0			0
$735$	−4.33013	−	7.50000i	−0.159719	−	0.276642i
$736$		0			0
$737$	18.0000	+	31.1769i	0.663039	+	1.14842i
$738$		0			0
$739$	13.8564	−	24.0000i	0.509716	−	0.882854i	−0.490221	−	0.871598i	$-0.663084\pi$
$739$	13.8564	−	24.0000i	0.509716	−	0.882854i	0.999937	−	0.0112558i	$-0.00358291\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.7846	+	36.0000i	−0.762513	+	1.32071i	0.179038	+	0.983842i	$0.442702\pi$
$743$	−20.7846	+	36.0000i	−0.762513	+	1.32071i	−0.941551	+	0.336870i	$0.890632\pi$
$744$		0			0
$745$	7.50000	+	12.9904i	0.274779	+	0.475931i
$746$		0			0
$747$	1.73205	+	3.00000i	0.0633724	+	0.109764i
$748$		0			0
$749$	−62.3538			−2.27836
$750$		0			0
$751$	13.0000	−	22.5167i	0.474377	−	0.821645i	−0.525193	−	0.850983i	$-0.676007\pi$
$751$	13.0000	−	22.5167i	0.474377	−	0.821645i	0.999570	+	0.0293387i	$0.00934013\pi$
$752$		0			0
$753$	−24.0000			−0.874609
$754$		0			0
$755$	30.0000			1.09181
$756$		0			0
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	−0.847280	−	0.531146i	$-0.821762\pi$
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	0.883626	+	0.468193i	$0.155095\pi$
$758$		0			0
$759$	−20.7846			−0.754434
$760$		0			0
$761$	−10.3923	−	18.0000i	−0.376721	−	0.652499i	0.613862	−	0.789413i	$-0.289615\pi$
$761$	−10.3923	−	18.0000i	−0.376721	−	0.652499i	−0.990583	+	0.136914i	$0.956282\pi$
$762$		0			0
$763$	24.0000	+	41.5692i	0.868858	+	1.50491i
$764$		0			0
$765$	−2.59808	+	4.50000i	−0.0939336	+	0.162698i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	17.3205	−	30.0000i	0.624593	−	1.08183i	−0.364026	−	0.931389i	$-0.618598\pi$
$769$	17.3205	−	30.0000i	0.624593	−	1.08183i	0.988619	−	0.150439i	$-0.0480687\pi$
$770$		0			0
$771$	1.50000	+	2.59808i	0.0540212	+	0.0935674i
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.00000	−	15.5885i	0.322873	−	0.559233i
$778$		0			0
$779$	−30.0000			−1.07486
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	4.50000	−	7.79423i	0.160817	−	0.278543i
$784$		0			0
$785$	−39.8372			−1.42185
$786$		0			0
$787$	20.7846	+	36.0000i	0.740891	+	1.28326i	0.952090	+	0.305818i	$0.0989300\pi$
$787$	20.7846	+	36.0000i	0.740891	+	1.28326i	−0.211199	+	0.977443i	$0.567737\pi$
$788$		0			0
$789$	3.00000	+	5.19615i	0.106803	+	0.184988i
$790$		0			0
$791$	−5.19615	+	9.00000i	−0.184754	+	0.320003i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	7.79423	−	13.5000i	0.276433	−	0.478796i
$796$		0			0
$797$	9.00000	+	15.5885i	0.318796	+	0.552171i	0.980237	−	0.197826i	$-0.0633881\pi$
$797$	9.00000	+	15.5885i	0.318796	+	0.552171i	−0.661441	+	0.749997i	$0.730055\pi$
$798$		0			0
$799$	−5.19615	−	9.00000i	−0.183827	−	0.318397i
$800$		0			0
$801$	6.92820			0.244796
$802$		0			0
$803$	−9.00000	+	15.5885i	−0.317603	+	0.550105i
$804$		0			0
$805$	36.0000			1.26883
$806$		0			0
$807$	−30.0000			−1.05605
$808$		0			0
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	−0.620274	−	0.784385i	$-0.712979\pi$
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	0.989434	+	0.144981i	$0.0463120\pi$
$810$		0			0
$811$	−6.92820			−0.243282			−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820			−0.243282			−0.121641	+	0.992574i	$0.538816\pi$
$812$		0			0
$813$	−3.46410	−	6.00000i	−0.121491	−	0.210429i
$814$		0			0
$815$	−12.0000	−	20.7846i	−0.420342	−	0.728053i
$816$		0			0
$817$	3.46410	−	6.00000i	0.121194	−	0.209913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.8564	+	24.0000i	−0.483592	+	0.837606i	−0.999822	−	0.0188439i	$-0.994001\pi$
$821$	−13.8564	+	24.0000i	−0.483592	+	0.837606i	0.516231	+	0.856450i	$0.327335\pi$
$822$		0			0
$823$	−20.0000	−	34.6410i	−0.697156	−	1.20751i	−0.969448	−	0.245295i	$-0.921115\pi$
$823$	−20.0000	−	34.6410i	−0.697156	−	1.20751i	0.272292	−	0.962215i	$-0.412218\pi$
$824$		0			0
$825$	3.46410	+	6.00000i	0.120605	+	0.208893i
$826$		0			0
$827$	−27.7128			−0.963669			−0.481834	−	0.876262i	$-0.660029\pi$
$827$	−27.7128			−0.963669			−0.481834	+	0.876262i	$0.660029\pi$
$828$		0			0
$829$	−5.50000	+	9.52628i	−0.191023	+	0.330861i	−0.945589	−	0.325362i	$-0.894514\pi$
$829$	−5.50000	+	9.52628i	−0.191023	+	0.330861i	0.754567	+	0.656223i	$0.227847\pi$
$830$		0			0
$831$	−19.0000			−0.659103
$832$		0			0
$833$	−15.0000			−0.519719
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.7846	−	36.0000i	−0.717564	−	1.24286i	−0.961962	−	0.273183i	$-0.911924\pi$
$839$	−20.7846	−	36.0000i	−0.717564	−	1.24286i	0.244398	−	0.969675i	$-0.421410\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$		0			0
$843$	14.7224	−	25.5000i	0.507067	−	0.878267i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.73205	+	3.00000i	−0.0595140	+	0.103081i
$848$		0			0
$849$	7.00000	+	12.1244i	0.240239	+	0.416107i
$850$		0			0
$851$	15.5885	+	27.0000i	0.534365	+	0.925548i
$852$		0			0
$853$	−43.3013			−1.48261			−0.741304	−	0.671170i	$-0.765792\pi$
$853$	−43.3013			−1.48261			−0.741304	+	0.671170i	$0.765792\pi$
$854$		0			0
$855$	3.00000	−	5.19615i	0.102598	−	0.177705i
$856$		0			0
$857$	3.00000			0.102478			0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000			0.102478			0.0512390	+	0.998686i	$0.483683\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$	−15.0000	+	25.9808i	−0.511199	+	0.885422i
$862$		0			0
$863$	−38.1051			−1.29711			−0.648557	−	0.761166i	$-0.724627\pi$
$863$	−38.1051			−1.29711			−0.648557	+	0.761166i	$0.724627\pi$
$864$		0			0
$865$	−15.5885	−	27.0000i	−0.530023	−	0.918028i
$866$		0			0
$867$	−4.00000	−	6.92820i	−0.135847	−	0.235294i
$868$		0			0
$869$	−13.8564	+	24.0000i	−0.470046	+	0.814144i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	3.46410	−	6.00000i	0.117242	−	0.203069i
$874$		0			0
$875$	−21.0000	−	36.3731i	−0.709930	−	1.22963i
$876$		0			0
$877$	−18.1865	−	31.5000i	−0.614116	−	1.06368i	−0.990539	−	0.137232i	$-0.956180\pi$
$877$	−18.1865	−	31.5000i	−0.614116	−	1.06368i	0.376423	−	0.926448i	$-0.377154\pi$
$878$		0			0
$879$	−1.73205			−0.0584206
$880$		0			0
$881$	4.50000	−	7.79423i	0.151609	−	0.262594i	−0.780210	−	0.625517i	$-0.784888\pi$
$881$	4.50000	−	7.79423i	0.151609	−	0.262594i	0.931819	+	0.362923i	$0.118221\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$	24.0000			0.806751
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−55.4256			−1.85892
$890$		0			0
$891$	1.73205	+	3.00000i	0.0580259	+	0.100504i
$892$		0			0
$893$	6.00000	+	10.3923i	0.200782	+	0.347765i
$894$		0			0
$895$	−15.5885	+	27.0000i	−0.521065	+	0.902510i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−13.5000	−	23.3827i	−0.449750	−	0.778990i
$902$		0			0
$903$	−3.46410	−	6.00000i	−0.115278	−	0.199667i
$904$		0			0
$905$	−22.5167			−0.748479
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$		0			0
$909$	−3.00000			−0.0995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−6.00000	+	10.3923i	−0.198571	+	0.343935i
$914$		0			0
$915$	−19.0526			−0.629858
$916$		0			0
$917$	20.7846	+	36.0000i	0.686368	+	1.18882i
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$	−8.66025	+	15.0000i	−0.285365	+	0.494267i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	5.19615	−	9.00000i	0.170848	−	0.295918i
$926$		0			0
$927$	7.00000	+	12.1244i	0.229910	+	0.398216i
$928$		0			0
$929$	−28.5788	−	49.5000i	−0.937641	−	1.62404i	−0.769855	−	0.638219i	$-0.779671\pi$
$929$	−28.5788	−	49.5000i	−0.937641	−	1.62404i	−0.167786	−	0.985823i	$-0.553662\pi$
$930$		0			0
$931$	17.3205			0.567657
$932$		0			0
$933$	9.00000	−	15.5885i	0.294647	−	0.510343i
$934$		0			0
$935$	−18.0000			−0.588663
$936$		0			0
$937$	−41.0000			−1.33941			−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000			−1.33941			−0.669706	+	0.742627i	$0.733580\pi$
$938$		0			0
$939$	−7.00000	+	12.1244i	−0.228436	+	0.395663i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−25.9808	−	45.0000i	−0.846050	−	1.46540i
$944$		0			0
$945$	−3.00000	−	5.19615i	−0.0975900	−	0.169031i
$946$		0			0
$947$	−6.92820	+	12.0000i	−0.225136	+	0.389948i	−0.956360	−	0.292190i	$-0.905616\pi$
$947$	−6.92820	+	12.0000i	−0.225136	+	0.389948i	0.731224	+	0.682137i	$0.238949\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.866025	−	1.50000i	0.0280828	−	0.0486408i
$952$		0			0
$953$	27.0000	+	46.7654i	0.874616	+	1.51488i	0.857171	+	0.515031i	$0.172220\pi$
$953$	27.0000	+	46.7654i	0.874616	+	1.51488i	0.0174443	+	0.999848i	$0.494447\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	31.1769			1.00781
$958$		0			0
$959$	−27.0000	+	46.7654i	−0.871875	+	1.51013i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−18.0000			−0.580042
$964$		0			0
$965$	−1.50000	+	2.59808i	−0.0482867	+	0.0836350i
$966$		0			0
$967$	−45.0333			−1.44817			−0.724087	−	0.689709i	$-0.757739\pi$
$967$	−45.0333			−1.44817			−0.724087	+	0.689709i	$0.757739\pi$
$968$		0			0
$969$	−5.19615	−	9.00000i	−0.166924	−	0.289122i
$970$		0			0
$971$	12.0000	+	20.7846i	0.385098	+	0.667010i	0.991783	−	0.127933i	$-0.0408342\pi$
$971$	12.0000	+	20.7846i	0.385098	+	0.667010i	−0.606685	+	0.794943i	$0.707501\pi$
$972$		0			0
$973$	6.92820	−	12.0000i	0.222108	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.33013	−	7.50000i	0.138533	−	0.239946i	−0.788409	−	0.615152i	$-0.789095\pi$
$977$	4.33013	−	7.50000i	0.138533	−	0.239946i	0.926942	+	0.375206i	$0.122428\pi$
$978$		0			0
$979$	12.0000	+	20.7846i	0.383522	+	0.664279i
$980$		0			0
$981$	6.92820	+	12.0000i	0.221201	+	0.383131i
$982$		0			0
$983$	−20.7846			−0.662926			−0.331463	−	0.943468i	$-0.607542\pi$
$983$	−20.7846			−0.662926			−0.331463	+	0.943468i	$0.607542\pi$
$984$		0			0
$985$	6.00000	−	10.3923i	0.191176	−	0.331126i
$986$		0			0
$987$	12.0000			0.381964
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	−1.00000	+	1.73205i	−0.0317660	+	0.0550204i	−0.881471	−	0.472237i	$-0.843446\pi$
$991$	−1.00000	+	1.73205i	−0.0317660	+	0.0550204i	0.849705	+	0.527258i	$0.176780\pi$
$992$		0			0
$993$	13.8564			0.439720
$994$		0			0
$995$	−8.66025	−	15.0000i	−0.274549	−	0.475532i
$996$		0			0
$997$	27.5000	+	47.6314i	0.870934	+	1.50850i	0.861032	+	0.508551i	$0.169818\pi$
$997$	27.5000	+	47.6314i	0.870934	+	1.50850i	0.00990158	+	0.999951i	$0.496848\pi$
$998$		0			0
$999$	2.59808	−	4.50000i	0.0821995	−	0.142374i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.i.h.529.1		4
13.2	odd	12		2028.2.q.a.361.1		2
13.3	even	3	inner	2028.2.i.h.2005.1		4
13.4	even	6		2028.2.a.h.1.2		2
13.5	odd	4		156.2.q.a.121.1	yes	2
13.6	odd	12		2028.2.b.b.337.2		2
13.7	odd	12		2028.2.b.b.337.1		2
13.8	odd	4		2028.2.q.a.1837.1		2
13.9	even	3		2028.2.a.h.1.1		2
13.10	even	6	inner	2028.2.i.h.2005.2		4
13.11	odd	12		156.2.q.a.49.1	✓	2
13.12	even	2	inner	2028.2.i.h.529.2		4
39.5	even	4		468.2.t.c.433.1		2
39.11	even	12		468.2.t.c.361.1		2
39.17	odd	6		6084.2.a.u.1.1		2
39.20	even	12		6084.2.b.c.4393.2		2
39.32	even	12		6084.2.b.c.4393.1		2
39.35	odd	6		6084.2.a.u.1.2		2
52.11	even	12		624.2.bv.a.49.1		2
52.31	even	4		624.2.bv.a.433.1		2
52.35	odd	6		8112.2.a.bt.1.1		2
52.43	odd	6		8112.2.a.bt.1.2		2
65.18	even	4		3900.2.bw.e.2149.2		4
65.24	odd	12		3900.2.cd.a.2701.1		2
65.37	even	12		3900.2.bw.e.49.2		4
65.44	odd	4		3900.2.cd.a.901.1		2
65.57	even	4		3900.2.bw.e.2149.1		4
65.63	even	12		3900.2.bw.e.49.1		4
156.11	odd	12		1872.2.by.b.1297.1		2
156.83	odd	4		1872.2.by.b.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.a.49.1	✓	2	13.11	odd	12
156.2.q.a.121.1	yes	2	13.5	odd	4
468.2.t.c.361.1		2	39.11	even	12
468.2.t.c.433.1		2	39.5	even	4
624.2.bv.a.49.1		2	52.11	even	12
624.2.bv.a.433.1		2	52.31	even	4
1872.2.by.b.433.1		2	156.83	odd	4
1872.2.by.b.1297.1		2	156.11	odd	12
2028.2.a.h.1.1		2	13.9	even	3
2028.2.a.h.1.2		2	13.4	even	6
2028.2.b.b.337.1		2	13.7	odd	12
2028.2.b.b.337.2		2	13.6	odd	12
2028.2.i.h.529.1		4	1.1	even	1	trivial
2028.2.i.h.529.2		4	13.12	even	2	inner
2028.2.i.h.2005.1		4	13.3	even	3	inner
2028.2.i.h.2005.2		4	13.10	even	6	inner
2028.2.q.a.361.1		2	13.2	odd	12
2028.2.q.a.1837.1		2	13.8	odd	4
3900.2.bw.e.49.1		4	65.63	even	12
3900.2.bw.e.49.2		4	65.37	even	12
3900.2.bw.e.2149.1		4	65.57	even	4
3900.2.bw.e.2149.2		4	65.18	even	4
3900.2.cd.a.901.1		2	65.44	odd	4
3900.2.cd.a.2701.1		2	65.24	odd	12
6084.2.a.u.1.1		2	39.17	odd	6
6084.2.a.u.1.2		2	39.35	odd	6
6084.2.b.c.4393.1		2	39.32	even	12
6084.2.b.c.4393.2		2	39.20	even	12
8112.2.a.bt.1.1		2	52.35	odd	6
8112.2.a.bt.1.2		2	52.43	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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} -1.73205 q^{5} +(-1.73205 - 3.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} -1.73205 q^{5} +(-1.73205 - 3.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.73205 - 3.00000i) q^{11} +(-0.866025 + 1.50000i) q^{15} +(1.50000 + 2.59808i) q^{17} +(-1.73205 - 3.00000i) q^{19} -3.46410 q^{21} +(3.00000 - 5.19615i) q^{23} -2.00000 q^{25} -1.00000 q^{27} +(-4.50000 + 7.79423i) q^{29} +(-1.73205 - 3.00000i) q^{33} +(3.00000 + 5.19615i) q^{35} +(-2.59808 + 4.50000i) q^{37} +(4.33013 - 7.50000i) q^{41} +(1.00000 + 1.73205i) q^{43} +(0.866025 + 1.50000i) q^{45} -3.46410 q^{47} +(-2.50000 + 4.33013i) q^{49} +3.00000 q^{51} -9.00000 q^{53} +(-3.00000 + 5.19615i) q^{55} -3.46410 q^{57} +(-6.92820 - 12.0000i) q^{59} +(5.50000 + 9.52628i) q^{61} +(-1.73205 + 3.00000i) q^{63} +(-5.19615 + 9.00000i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(5.19615 + 9.00000i) q^{71} -5.19615 q^{73} +(-1.00000 + 1.73205i) q^{75} -12.0000 q^{77} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410 q^{83} +(-2.59808 - 4.50000i) q^{85} +(4.50000 + 7.79423i) q^{87} +(-3.46410 + 6.00000i) q^{89} +(3.00000 + 5.19615i) q^{95} +(3.46410 + 6.00000i) q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 2 * q^9	\( 4 q + 2 q^{3} - 2 q^{9} + 6 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} - 18 q^{29} + 12 q^{35} + 4 q^{43} - 10 q^{49} + 12 q^{51} - 36 q^{53} - 12 q^{55} + 22 q^{61} - 12 q^{69} - 4 q^{75} - 48 q^{77} - 32 q^{79} - 2 q^{81} + 18 q^{87} + 12 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^9 + 6 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 - 18 * q^29 + 12 * q^35 + 4 * q^43 - 10 * q^49 + 12 * q^51 - 36 * q^53 - 12 * q^55 + 22 * q^61 - 12 * q^69 - 4 * q^75 - 48 * q^77 - 32 * q^79 - 2 * q^81 + 18 * q^87 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more