LMFDB - Embedded newform 2028.2.q.h.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(361,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2028.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,0,0,-2,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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 $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			361.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2028.361
Dual form			2028.2.q.h.1837.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{3} -4.00000i q^{5} +(-1.73205 - 1.00000i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.46410 - 2.00000i) q^{11} +(3.46410 - 2.00000i) q^{15} +(1.00000 - 1.73205i) q^{17} +(1.73205 + 1.00000i) q^{19} -2.00000i q^{21} -11.0000 q^{25} -1.00000 q^{27} +(3.00000 + 5.19615i) q^{29} -10.0000i q^{31} +(3.46410 + 2.00000i) q^{33} +(-4.00000 + 6.92820i) q^{35} +(-8.66025 + 5.00000i) q^{37} +(6.92820 - 4.00000i) q^{41} +(2.00000 - 3.46410i) q^{43} +(3.46410 + 2.00000i) q^{45} +4.00000i q^{47} +(-1.50000 - 2.59808i) q^{49} +2.00000 q^{51} -10.0000 q^{53} +(-8.00000 - 13.8564i) q^{55} +2.00000i q^{57} +(-6.92820 - 4.00000i) q^{59} +(7.00000 - 12.1244i) q^{61} +(1.73205 - 1.00000i) q^{63} +(1.73205 - 1.00000i) q^{67} +(-13.8564 - 8.00000i) q^{71} +10.0000i q^{73} +(-5.50000 - 9.52628i) q^{75} -8.00000 q^{77} -16.0000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(-6.92820 - 4.00000i) q^{85} +(-3.00000 + 5.19615i) q^{87} +(3.46410 - 2.00000i) q^{89} +(8.66025 - 5.00000i) q^{93} +(4.00000 - 6.92820i) q^{95} +(1.73205 + 1.00000i) q^{97} +4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{9} + 4 q^{17} - 44 q^{25} - 4 q^{27} + 12 q^{29} - 16 q^{35} + 8 q^{43} - 6 q^{49} + 8 q^{51} - 40 q^{53} - 32 q^{55} + 28 q^{61} - 22 q^{75} - 32 q^{77} - 64 q^{79} - 2 q^{81} - 12 q^{87}+ \cdots + 16 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^9 + 4 * q^17 - 44 * q^25 - 4 * q^27 + 12 * q^29 - 16 * q^35 + 8 * q^43 - 6 * q^49 + 8 * q^51 - 40 * q^53 - 32 * q^55 + 28 * q^61 - 22 * q^75 - 32 * q^77 - 64 * q^79 - 2 * q^81 - 12 * q^87 + 16 * q^95

Character values

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

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$		−	4.00000i		−	1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$		−	4.00000i		−	1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$		0			0
$7$	−1.73205	−	1.00000i	−0.654654	−	0.377964i	0.135583	−	0.990766i	$-0.456709\pi$
$7$	−1.73205	−	1.00000i	−0.654654	−	0.377964i	−0.790237	+	0.612801i	$0.790043\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	3.46410	−	2.00000i	1.04447	−	0.603023i	0.123371	−	0.992361i	$-0.460630\pi$
$11$	3.46410	−	2.00000i	1.04447	−	0.603023i	0.921095	+	0.389338i	$0.127296\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	3.46410	−	2.00000i	0.894427	−	0.516398i
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	1.73205	+	1.00000i	0.397360	+	0.229416i	0.685344	−	0.728219i	$-0.259652\pi$
$19$	1.73205	+	1.00000i	0.397360	+	0.229416i	−0.287984	+	0.957635i	$0.592985\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−11.0000			−2.20000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	$0.0214140\pi$
$29$	3.00000	+	5.19615i	0.557086	+	0.964901i	−0.440652	+	0.897678i	$0.645253\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$		0			0
$33$	3.46410	+	2.00000i	0.603023	+	0.348155i
$34$		0			0
$35$	−4.00000	+	6.92820i	−0.676123	+	1.17108i
$36$		0			0
$37$	−8.66025	+	5.00000i	−1.42374	+	0.821995i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	−8.66025	+	5.00000i	−1.42374	+	0.821995i	−0.427121	+	0.904194i	$0.640472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.92820	−	4.00000i	1.08200	−	0.624695i	0.150567	−	0.988600i	$-0.451890\pi$
$41$	6.92820	−	4.00000i	1.08200	−	0.624695i	0.931436	+	0.363905i	$0.118557\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$		0			0
$45$	3.46410	+	2.00000i	0.516398	+	0.298142i
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	−1.50000	−	2.59808i	−0.214286	−	0.371154i
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	−8.00000	−	13.8564i	−1.07872	−	1.86840i
$56$		0			0
$57$			2.00000i			0.264906i
$58$		0			0
$59$	−6.92820	−	4.00000i	−0.901975	−	0.520756i	−0.0241347	−	0.999709i	$-0.507683\pi$
$59$	−6.92820	−	4.00000i	−0.901975	−	0.520756i	−0.877841	+	0.478953i	$0.841016\pi$
$60$		0			0
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$	1.73205	−	1.00000i	0.218218	−	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.73205	−	1.00000i	0.211604	−	0.122169i	−0.390453	−	0.920623i	$-0.627682\pi$
$67$	1.73205	−	1.00000i	0.211604	−	0.122169i	0.602056	+	0.798454i	$0.294348\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.8564	−	8.00000i	−1.64445	−	0.949425i	−0.979223	−	0.202787i	$-0.935000\pi$
$71$	−13.8564	−	8.00000i	−1.64445	−	0.949425i	−0.665230	−	0.746639i	$-0.731667\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$	−5.50000	−	9.52628i	−0.635085	−	1.10000i
$76$		0			0
$77$	−8.00000			−0.911685
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−6.92820	−	4.00000i	−0.751469	−	0.433861i
$86$		0			0
$87$	−3.00000	+	5.19615i	−0.321634	+	0.557086i
$88$		0			0
$89$	3.46410	−	2.00000i	0.367194	−	0.212000i	−0.305038	−	0.952340i	$-0.598669\pi$
$89$	3.46410	−	2.00000i	0.367194	−	0.212000i	0.672232	+	0.740341i	$0.265336\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	8.66025	−	5.00000i	0.898027	−	0.518476i
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	1.73205	+	1.00000i	0.175863	+	0.101535i	0.585348	−	0.810782i	$-0.300958\pi$
$97$	1.73205	+	1.00000i	0.175863	+	0.101535i	−0.409484	+	0.912317i	$0.634291\pi$
$98$		0			0
$99$			4.00000i			0.402015i
$100$		0			0
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	0.999996	−	0.00286291i	$-0.000911295\pi$
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	−0.502477	+	0.864590i	$0.667578\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$	−8.00000			−0.780720
$106$		0			0
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	−0.995474	−	0.0950377i	$-0.969703\pi$
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	0.415432	−	0.909624i	$-0.363630\pi$
$108$		0			0
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$	−8.66025	−	5.00000i	−0.821995	−	0.474579i
$112$		0			0
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	−0.971930	−	0.235269i	$-0.924403\pi$
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	0.689714	+	0.724082i	$0.257736\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.46410	+	2.00000i	−0.317554	+	0.183340i
$120$		0			0
$121$	2.50000	−	4.33013i	0.227273	−	0.393648i
$122$		0			0
$123$	6.92820	+	4.00000i	0.624695	+	0.360668i
$124$		0			0
$125$			24.0000i			2.14663i
$126$		0			0
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	0.999284	+	0.0378419i	$0.0120483\pi$
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	−0.466870	+	0.884326i	$0.654618\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$		0			0
$135$			4.00000i			0.344265i
$136$		0			0
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	0.765855	−	0.643013i	$-0.222316\pi$
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	−0.173939	+	0.984757i	$0.555649\pi$
$138$		0			0
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	−3.46410	+	2.00000i	−0.291730	+	0.168430i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	20.7846	−	12.0000i	1.72607	−	0.996546i
$146$		0			0
$147$	1.50000	−	2.59808i	0.123718	−	0.214286i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$	1.00000	+	1.73205i	0.0808452	+	0.140028i
$154$		0			0
$155$	−40.0000			−3.21288
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−5.00000	−	8.66025i	−0.396526	−	0.686803i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.73205	+	1.00000i	0.135665	+	0.0783260i	0.566296	−	0.824202i	$-0.308376\pi$
$163$	1.73205	+	1.00000i	0.135665	+	0.0783260i	−0.430632	+	0.902528i	$0.641709\pi$
$164$		0			0
$165$	8.00000	−	13.8564i	0.622799	−	1.07872i
$166$		0			0
$167$	10.3923	−	6.00000i	0.804181	−	0.464294i	−0.0407502	−	0.999169i	$-0.512975\pi$
$167$	10.3923	−	6.00000i	0.804181	−	0.464294i	0.844931	+	0.534875i	$0.179641\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−1.73205	+	1.00000i	−0.132453	+	0.0764719i
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$	19.0526	+	11.0000i	1.44024	+	0.831522i
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$		0			0
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	14.0000			1.03491
$184$		0			0
$185$	20.0000	+	34.6410i	1.47043	+	2.54686i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$	1.73205	+	1.00000i	0.125988	+	0.0727393i
$190$		0			0
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	−0.684244	−	0.729253i	$-0.739868\pi$
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	0.973674	+	0.227946i	$0.0732010\pi$
$192$		0			0
$193$	12.1244	−	7.00000i	0.872730	−	0.503871i	0.00447566	−	0.999990i	$-0.498575\pi$
$193$	12.1244	−	7.00000i	0.872730	−	0.503871i	0.868255	+	0.496119i	$0.165242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.3923	+	6.00000i	−0.740421	+	0.427482i	−0.822222	−	0.569166i	$-0.807266\pi$
$197$	−10.3923	+	6.00000i	−0.740421	+	0.427482i	0.0818013	+	0.996649i	$0.473933\pi$
$198$		0			0
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	−0.786389	−	0.617731i	$-0.788052\pi$
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	0.928166	+	0.372168i	$0.121385\pi$
$200$		0			0
$201$	1.73205	+	1.00000i	0.122169	+	0.0705346i
$202$		0			0
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$	−16.0000	−	27.7128i	−1.11749	−	1.93555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$		−	16.0000i		−	1.09630i
$214$		0			0
$215$	−13.8564	−	8.00000i	−0.944999	−	0.545595i
$216$		0			0
$217$	−10.0000	+	17.3205i	−0.678844	+	1.17579i
$218$		0			0
$219$	−8.66025	+	5.00000i	−0.585206	+	0.337869i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	12.1244	−	7.00000i	0.811907	−	0.468755i	−0.0357107	−	0.999362i	$-0.511370\pi$
$223$	12.1244	−	7.00000i	0.811907	−	0.468755i	0.847618	+	0.530607i	$0.178036\pi$
$224$		0			0
$225$	5.50000	−	9.52628i	0.366667	−	0.635085i
$226$		0			0
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	0.610535	−	0.791989i	$-0.290954\pi$
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	−0.380615	+	0.924734i	$0.624288\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$		0			0
$231$	−4.00000	−	6.92820i	−0.263181	−	0.455842i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	16.0000			1.04372
$236$		0			0
$237$	−8.00000	−	13.8564i	−0.519656	−	0.900070i
$238$		0			0
$239$		−	24.0000i		−	1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$		−	24.0000i		−	1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$		0			0
$241$	−1.73205	−	1.00000i	−0.111571	−	0.0644157i	0.443176	−	0.896435i	$-0.353852\pi$
$241$	−1.73205	−	1.00000i	−0.111571	−	0.0644157i	−0.554747	+	0.832019i	$0.687185\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−10.3923	+	6.00000i	−0.663940	+	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.0000	+	17.3205i	−0.631194	+	1.09326i	0.356113	+	0.934443i	$0.384102\pi$
$251$	−10.0000	+	17.3205i	−0.631194	+	1.09326i	−0.987308	+	0.158818i	$0.949232\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	8.00000i		−	0.500979i
$256$		0			0
$257$	15.0000	+	25.9808i	0.935674	+	1.62064i	0.773427	+	0.633885i	$0.218541\pi$
$257$	15.0000	+	25.9808i	0.935674	+	1.62064i	0.162247	+	0.986750i	$0.448126\pi$
$258$		0			0
$259$	20.0000			1.24274
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	8.00000	+	13.8564i	0.493301	+	0.854423i	0.999970	−	0.00771799i	$-0.00245674\pi$
$263$	8.00000	+	13.8564i	0.493301	+	0.854423i	−0.506669	+	0.862141i	$0.669123\pi$
$264$		0			0
$265$			40.0000i			2.45718i
$266$		0			0
$267$	3.46410	+	2.00000i	0.212000	+	0.122398i
$268$		0			0
$269$	7.00000	−	12.1244i	0.426798	−	0.739235i	−0.569789	−	0.821791i	$-0.692975\pi$
$269$	7.00000	−	12.1244i	0.426798	−	0.739235i	0.996586	+	0.0825561i	$0.0263084\pi$
$270$		0			0
$271$	8.66025	−	5.00000i	0.526073	−	0.303728i	−0.213343	−	0.976977i	$-0.568435\pi$
$271$	8.66025	−	5.00000i	0.526073	−	0.303728i	0.739416	+	0.673249i	$0.235102\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−38.1051	+	22.0000i	−2.29783	+	1.32665i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$	8.66025	+	5.00000i	0.518476	+	0.299342i
$280$		0			0
$281$			8.00000i			0.477240i	0.971113	+	0.238620i	$0.0766950\pi$
$281$			8.00000i			0.477240i	−0.971113	+	0.238620i	$0.923305\pi$
$282$		0			0
$283$	−8.00000	−	13.8564i	−0.475551	−	0.823678i	0.524057	−	0.851683i	$-0.324418\pi$
$283$	−8.00000	−	13.8564i	−0.475551	−	0.823678i	−0.999608	+	0.0280052i	$0.991084\pi$
$284$		0			0
$285$	8.00000			0.473879
$286$		0			0
$287$	−16.0000			−0.944450
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$			2.00000i			0.117242i
$292$		0			0
$293$	6.92820	+	4.00000i	0.404750	+	0.233682i	0.688531	−	0.725206i	$-0.258256\pi$
$293$	6.92820	+	4.00000i	0.404750	+	0.233682i	−0.283782	+	0.958889i	$0.591589\pi$
$294$		0			0
$295$	−16.0000	+	27.7128i	−0.931556	+	1.61350i
$296$		0			0
$297$	−3.46410	+	2.00000i	−0.201008	+	0.116052i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−6.92820	+	4.00000i	−0.399335	+	0.230556i
$302$		0			0
$303$	−5.00000	+	8.66025i	−0.287242	+	0.497519i
$304$		0			0
$305$	−48.4974	−	28.0000i	−2.77695	−	1.60328i
$306$		0			0
$307$		−	22.0000i		−	1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$		−	22.0000i		−	1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$		0			0
$309$	4.00000	+	6.92820i	0.227552	+	0.394132i
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$		0			0
$315$	−4.00000	−	6.92820i	−0.225374	−	0.390360i
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$	20.7846	+	12.0000i	1.16371	+	0.671871i
$320$		0			0
$321$	6.00000	−	10.3923i	0.334887	−	0.580042i
$322$		0			0
$323$	3.46410	−	2.00000i	0.192748	−	0.111283i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.73205	−	1.00000i	0.0957826	−	0.0553001i
$328$		0			0
$329$	4.00000	−	6.92820i	0.220527	−	0.381964i
$330$		0			0
$331$	−1.73205	−	1.00000i	−0.0952021	−	0.0549650i	0.451643	−	0.892199i	$-0.350838\pi$
$331$	−1.73205	−	1.00000i	−0.0952021	−	0.0549650i	−0.546845	+	0.837234i	$0.684171\pi$
$332$		0			0
$333$		−	10.0000i		−	0.547997i
$334$		0			0
$335$	−4.00000	−	6.92820i	−0.218543	−	0.378528i
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−20.0000	−	34.6410i	−1.08306	−	1.87592i
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−15.5885	+	9.00000i	−0.834431	+	0.481759i	−0.855367	−	0.518022i	$-0.826669\pi$
$349$	−15.5885	+	9.00000i	−0.834431	+	0.481759i	0.0209364	+	0.999781i	$0.493335\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.8564	+	8.00000i	−0.737502	+	0.425797i	−0.821160	−	0.570697i	$-0.806673\pi$
$353$	−13.8564	+	8.00000i	−0.737502	+	0.425797i	0.0836583	+	0.996495i	$0.473340\pi$
$354$		0			0
$355$	−32.0000	+	55.4256i	−1.69838	+	2.94169i
$356$		0			0
$357$	−3.46410	−	2.00000i	−0.183340	−	0.105851i
$358$		0			0
$359$			12.0000i			0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$			12.0000i			0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$		0			0
$361$	−7.50000	−	12.9904i	−0.394737	−	0.683704i
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	40.0000			2.09370
$366$		0			0
$367$	−2.00000	−	3.46410i	−0.104399	−	0.180825i	0.809093	−	0.587680i	$-0.199959\pi$
$367$	−2.00000	−	3.46410i	−0.104399	−	0.180825i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$			8.00000i			0.416463i
$370$		0			0
$371$	17.3205	+	10.0000i	0.899236	+	0.519174i
$372$		0			0
$373$	9.00000	−	15.5885i	0.466002	−	0.807140i	−0.533244	−	0.845962i	$-0.679027\pi$
$373$	9.00000	−	15.5885i	0.466002	−	0.807140i	0.999246	+	0.0388219i	$0.0123605\pi$
$374$		0			0
$375$	−20.7846	+	12.0000i	−1.07331	+	0.619677i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	5.19615	−	3.00000i	0.266908	−	0.154100i	−0.360573	−	0.932731i	$-0.617419\pi$
$379$	5.19615	−	3.00000i	0.266908	−	0.154100i	0.627482	+	0.778631i	$0.284086\pi$
$380$		0			0
$381$	−6.00000	+	10.3923i	−0.307389	+	0.532414i
$382$		0			0
$383$	−10.3923	−	6.00000i	−0.531022	−	0.306586i	0.210411	−	0.977613i	$-0.432520\pi$
$383$	−10.3923	−	6.00000i	−0.531022	−	0.306586i	−0.741433	+	0.671027i	$0.765853\pi$
$384$		0			0
$385$			32.0000i			1.63087i
$386$		0			0
$387$	2.00000	+	3.46410i	0.101666	+	0.176090i
$388$		0			0
$389$	26.0000			1.31825			0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000			1.31825			0.659126	+	0.752032i	$0.270926\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	2.00000	+	3.46410i	0.100887	+	0.174741i
$394$		0			0
$395$			64.0000i			3.22019i
$396$		0			0
$397$	15.5885	+	9.00000i	0.782362	+	0.451697i	0.837267	−	0.546795i	$-0.184152\pi$
$397$	15.5885	+	9.00000i	0.782362	+	0.451697i	−0.0549046	+	0.998492i	$0.517485\pi$
$398$		0			0
$399$	2.00000	−	3.46410i	0.100125	−	0.173422i
$400$		0			0
$401$	10.3923	−	6.00000i	0.518967	−	0.299626i	−0.217545	−	0.976050i	$-0.569805\pi$
$401$	10.3923	−	6.00000i	0.518967	−	0.299626i	0.736512	+	0.676425i	$0.236472\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−3.46410	+	2.00000i	−0.172133	+	0.0993808i
$406$		0			0
$407$	−20.0000	+	34.6410i	−0.991363	+	1.71709i
$408$		0			0
$409$	−12.1244	−	7.00000i	−0.599511	−	0.346128i	0.169338	−	0.985558i	$-0.445837\pi$
$409$	−12.1244	−	7.00000i	−0.599511	−	0.346128i	−0.768849	+	0.639430i	$0.779170\pi$
$410$		0			0
$411$			8.00000i			0.394611i
$412$		0			0
$413$	8.00000	+	13.8564i	0.393654	+	0.681829i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	0.999913	−	0.0131919i	$-0.00419923\pi$
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	−0.511381	+	0.859354i	$0.670866\pi$
$420$		0			0
$421$		−	2.00000i		−	0.0974740i	−0.998812	−	0.0487370i	$-0.984480\pi$
$421$		−	2.00000i		−	0.0974740i	0.998812	−	0.0487370i	$-0.0155196\pi$
$422$		0			0
$423$	−3.46410	−	2.00000i	−0.168430	−	0.0972433i
$424$		0			0
$425$	−11.0000	+	19.0526i	−0.533578	+	0.924185i
$426$		0			0
$427$	−24.2487	+	14.0000i	−1.17348	+	0.677507i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.3205	+	10.0000i	−0.834300	+	0.481683i	−0.855323	−	0.518096i	$-0.826641\pi$
$431$	−17.3205	+	10.0000i	−0.834300	+	0.481683i	0.0210230	+	0.999779i	$0.493308\pi$
$432$		0			0
$433$	−9.00000	+	15.5885i	−0.432512	+	0.749133i	−0.997089	−	0.0762473i	$-0.975706\pi$
$433$	−9.00000	+	15.5885i	−0.432512	+	0.749133i	0.564577	+	0.825381i	$0.309039\pi$
$434$		0			0
$435$	20.7846	+	12.0000i	0.996546	+	0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	8.00000	+	13.8564i	0.381819	+	0.661330i	0.991322	−	0.131453i	$-0.0419644\pi$
$439$	8.00000	+	13.8564i	0.381819	+	0.661330i	−0.609503	+	0.792784i	$0.708631\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	−8.00000	−	13.8564i	−0.379236	−	0.656857i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.3923	+	6.00000i	0.490443	+	0.283158i	0.724758	−	0.689003i	$-0.241951\pi$
$449$	10.3923	+	6.00000i	0.490443	+	0.283158i	−0.234315	+	0.972161i	$0.575285\pi$
$450$		0			0
$451$	16.0000	−	27.7128i	0.753411	−	1.30495i
$452$		0			0
$453$	15.5885	−	9.00000i	0.732410	−	0.422857i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.5885	+	9.00000i	−0.729197	+	0.421002i	−0.818128	−	0.575036i	$-0.804988\pi$
$457$	−15.5885	+	9.00000i	−0.729197	+	0.421002i	0.0889312	+	0.996038i	$0.471655\pi$
$458$		0			0
$459$	−1.00000	+	1.73205i	−0.0466760	+	0.0808452i
$460$		0			0
$461$	−10.3923	−	6.00000i	−0.484018	−	0.279448i	0.238071	−	0.971248i	$-0.423485\pi$
$461$	−10.3923	−	6.00000i	−0.484018	−	0.279448i	−0.722089	+	0.691800i	$0.756818\pi$
$462$		0			0
$463$		−	22.0000i		−	1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$		−	22.0000i		−	1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$		0			0
$465$	−20.0000	−	34.6410i	−0.927478	−	1.60644i
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	1.00000	+	1.73205i	0.0460776	+	0.0798087i
$472$		0			0
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$	−19.0526	−	11.0000i	−0.874191	−	0.504715i
$476$		0			0
$477$	5.00000	−	8.66025i	0.228934	−	0.396526i
$478$		0			0
$479$	13.8564	−	8.00000i	0.633115	−	0.365529i	−0.148842	−	0.988861i	$-0.547555\pi$
$479$	13.8564	−	8.00000i	0.633115	−	0.365529i	0.781958	+	0.623332i	$0.214221\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	−	6.92820i	0.181631	−	0.314594i
$486$		0			0
$487$	22.5167	+	13.0000i	1.02033	+	0.589086i	0.914199	−	0.405266i	$-0.132821\pi$
$487$	22.5167	+	13.0000i	1.02033	+	0.589086i	0.106129	+	0.994352i	$0.466154\pi$
$488$		0			0
$489$			2.00000i			0.0904431i
$490$		0			0
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	0.969061	−	0.246822i	$-0.0793863\pi$
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	−0.698285	+	0.715820i	$0.746053\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$	16.0000			0.719147
$496$		0			0
$497$	16.0000	+	27.7128i	0.717698	+	1.24309i
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$	10.3923	+	6.00000i	0.464294	+	0.268060i
$502$		0			0
$503$	−8.00000	+	13.8564i	−0.356702	+	0.617827i	−0.987408	−	0.158196i	$-0.949432\pi$
$503$	−8.00000	+	13.8564i	−0.356702	+	0.617827i	0.630705	+	0.776022i	$0.282766\pi$
$504$		0			0
$505$	34.6410	−	20.0000i	1.54150	−	0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.3923	−	6.00000i	0.460631	−	0.265945i	−0.251679	−	0.967811i	$-0.580983\pi$
$509$	10.3923	−	6.00000i	0.460631	−	0.265945i	0.712309	+	0.701866i	$0.247649\pi$
$510$		0			0
$511$	10.0000	−	17.3205i	0.442374	−	0.766214i
$512$		0			0
$513$	−1.73205	−	1.00000i	−0.0764719	−	0.0441511i
$514$		0			0
$515$		−	32.0000i		−	1.41009i
$516$		0			0
$517$	8.00000	+	13.8564i	0.351840	+	0.609404i
$518$		0			0
$519$	2.00000			0.0877903
$520$		0			0
$521$	−2.00000			−0.0876216			−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000			−0.0876216			−0.0438108	+	0.999040i	$0.513950\pi$
$522$		0			0
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	0.636401	−	0.771358i	$-0.280422\pi$
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	−0.986216	+	0.165460i	$0.947089\pi$
$524$		0			0
$525$			22.0000i			0.960159i
$526$		0			0
$527$	−17.3205	−	10.0000i	−0.754493	−	0.435607i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$	6.92820	−	4.00000i	0.300658	−	0.173585i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−41.5692	+	24.0000i	−1.79719	+	1.03761i
$536$		0			0
$537$	−6.00000	+	10.3923i	−0.258919	+	0.448461i
$538$		0			0
$539$	−10.3923	−	6.00000i	−0.447628	−	0.258438i
$540$		0			0
$541$		−	38.0000i		−	1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$		−	38.0000i		−	1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$		0			0
$543$	−1.00000	−	1.73205i	−0.0429141	−	0.0743294i
$544$		0			0
$545$	−8.00000			−0.342682
$546$		0			0
$547$	32.0000			1.36822			0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000			1.36822			0.684111	+	0.729378i	$0.260191\pi$
$548$		0			0
$549$	7.00000	+	12.1244i	0.298753	+	0.517455i
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$	27.7128	+	16.0000i	1.17847	+	0.680389i
$554$		0			0
$555$	−20.0000	+	34.6410i	−0.848953	+	1.47043i
$556$		0			0
$557$	−20.7846	+	12.0000i	−0.880672	+	0.508456i	−0.870880	−	0.491496i	$-0.836450\pi$
$557$	−20.7846	+	12.0000i	−0.880672	+	0.508456i	−0.00979220	+	0.999952i	$0.503117\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	6.92820	−	4.00000i	0.292509	−	0.168880i
$562$		0			0
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	−0.404198	−	0.914671i	$-0.632449\pi$
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	0.994228	−	0.107290i	$-0.0342173\pi$
$564$		0			0
$565$	20.7846	+	12.0000i	0.874415	+	0.504844i
$566$		0			0
$567$			2.00000i			0.0839921i
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	8.00000			0.334790			0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000			0.334790			0.167395	+	0.985890i	$0.446465\pi$
$572$		0			0
$573$	8.00000			0.334205
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	30.0000i		−	1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$		−	30.0000i		−	1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$		0			0
$579$	12.1244	+	7.00000i	0.503871	+	0.290910i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−34.6410	+	20.0000i	−1.43468	+	0.828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.92820	+	4.00000i	−0.285958	+	0.165098i	−0.636117	−	0.771592i	$-0.719461\pi$
$587$	−6.92820	+	4.00000i	−0.285958	+	0.165098i	0.350160	+	0.936690i	$0.386127\pi$
$588$		0			0
$589$	10.0000	−	17.3205i	0.412043	−	0.713679i
$590$		0			0
$591$	−10.3923	−	6.00000i	−0.427482	−	0.246807i
$592$		0			0
$593$			36.0000i			1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$			36.0000i			1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$		0			0
$595$	8.00000	+	13.8564i	0.327968	+	0.568057i
$596$		0			0
$597$	4.00000			0.163709
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	0.998302	−	0.0582563i	$-0.0185541\pi$
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	−0.549602	+	0.835426i	$0.685221\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$	−17.3205	−	10.0000i	−0.704179	−	0.406558i
$606$		0			0
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	−0.822578	−	0.568652i	$-0.807465\pi$
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	0.903756	+	0.428048i	$0.140799\pi$
$608$		0			0
$609$	10.3923	−	6.00000i	0.421117	−	0.243132i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.1244	+	7.00000i	−0.489698	+	0.282727i	−0.724449	−	0.689328i	$-0.757906\pi$
$613$	−12.1244	+	7.00000i	−0.489698	+	0.282727i	0.234751	+	0.972056i	$0.424572\pi$
$614$		0			0
$615$	16.0000	−	27.7128i	0.645182	−	1.11749i
$616$		0			0
$617$	41.5692	+	24.0000i	1.67351	+	0.966204i	0.965647	+	0.259858i	$0.0836757\pi$
$617$	41.5692	+	24.0000i	1.67351	+	0.966204i	0.707867	+	0.706346i	$0.249658\pi$
$618$		0			0
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.00000			−0.320513
$624$		0			0
$625$	41.0000			1.64000
$626$		0			0
$627$	4.00000	+	6.92820i	0.159745	+	0.276686i
$628$		0			0
$629$			20.0000i			0.797452i
$630$		0			0
$631$	−25.9808	−	15.0000i	−1.03428	−	0.597141i	−0.116071	−	0.993241i	$-0.537030\pi$
$631$	−25.9808	−	15.0000i	−1.03428	−	0.597141i	−0.918207	+	0.396100i	$0.870363\pi$
$632$		0			0
$633$	−6.00000	+	10.3923i	−0.238479	+	0.413057i
$634$		0			0
$635$	41.5692	−	24.0000i	1.64962	−	0.952411i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	13.8564	−	8.00000i	0.548151	−	0.316475i
$640$		0			0
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	−0.486409	−	0.873731i	$-0.661693\pi$
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	0.999878	−	0.0156233i	$-0.00497325\pi$
$642$		0			0
$643$	−22.5167	−	13.0000i	−0.887970	−	0.512670i	−0.0146923	−	0.999892i	$-0.504677\pi$
$643$	−22.5167	−	13.0000i	−0.887970	−	0.512670i	−0.873278	+	0.487222i	$0.838010\pi$
$644$		0			0
$645$		−	16.0000i		−	0.629999i
$646$		0			0
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	0.527710	−	0.849425i	$-0.323051\pi$
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	−0.999478	+	0.0322975i	$0.989718\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$	−20.0000			−0.783862
$652$		0			0
$653$	23.0000	+	39.8372i	0.900060	+	1.55895i	0.827415	+	0.561591i	$0.189811\pi$
$653$	23.0000	+	39.8372i	0.900060	+	1.55895i	0.0726446	+	0.997358i	$0.476856\pi$
$654$		0			0
$655$		−	16.0000i		−	0.625172i
$656$		0			0
$657$	−8.66025	−	5.00000i	−0.337869	−	0.195069i
$658$		0			0
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	−0.725175	−	0.688565i	$-0.758241\pi$
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	0.958902	+	0.283738i	$0.0915745\pi$
$660$		0			0
$661$	−8.66025	+	5.00000i	−0.336845	+	0.194477i	−0.658876	−	0.752252i	$-0.728968\pi$
$661$	−8.66025	+	5.00000i	−0.336845	+	0.194477i	0.322031	+	0.946729i	$0.395634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13.8564	+	8.00000i	−0.537328	+	0.310227i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	12.1244	+	7.00000i	0.468755	+	0.270636i
$670$		0			0
$671$		−	56.0000i		−	2.16186i
$672$		0			0
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	0.996328	−	0.0856156i	$-0.0272857\pi$
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	−0.572309	+	0.820038i	$0.693952\pi$
$674$		0			0
$675$	11.0000			0.423390
$676$		0			0
$677$	38.0000			1.46046			0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000			1.46046			0.730229	+	0.683202i	$0.239413\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$		0			0
$681$			4.00000i			0.153280i
$682$		0			0
$683$	17.3205	+	10.0000i	0.662751	+	0.382639i	0.793324	−	0.608799i	$-0.208349\pi$
$683$	17.3205	+	10.0000i	0.662751	+	0.382639i	−0.130573	+	0.991439i	$0.541682\pi$
$684$		0			0
$685$	16.0000	−	27.7128i	0.611329	−	1.05885i
$686$		0			0
$687$	−1.73205	+	1.00000i	−0.0660819	+	0.0381524i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	8.66025	−	5.00000i	0.329452	−	0.190209i	−0.326146	−	0.945319i	$-0.605750\pi$
$691$	8.66025	−	5.00000i	0.329452	−	0.190209i	0.655598	+	0.755110i	$0.272417\pi$
$692$		0			0
$693$	4.00000	−	6.92820i	0.151947	−	0.263181i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	16.0000i		−	0.606043i
$698$		0			0
$699$	9.00000	+	15.5885i	0.340411	+	0.589610i
$700$		0			0
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$	8.00000	+	13.8564i	0.301297	+	0.521862i
$706$		0			0
$707$		−	20.0000i		−	0.752177i
$708$		0			0
$709$	5.19615	+	3.00000i	0.195146	+	0.112667i	0.594389	−	0.804178i	$-0.297394\pi$
$709$	5.19615	+	3.00000i	0.195146	+	0.112667i	−0.399244	+	0.916845i	$0.630727\pi$
$710$		0			0
$711$	8.00000	−	13.8564i	0.300023	−	0.519656i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	20.7846	−	12.0000i	0.776215	−	0.448148i
$718$		0			0
$719$	−24.0000	+	41.5692i	−0.895049	+	1.55027i	−0.0613050	+	0.998119i	$0.519526\pi$
$719$	−24.0000	+	41.5692i	−0.895049	+	1.55027i	−0.833744	+	0.552151i	$0.813807\pi$
$720$		0			0
$721$	−13.8564	−	8.00000i	−0.516040	−	0.297936i
$722$		0			0
$723$		−	2.00000i		−	0.0743808i
$724$		0			0
$725$	−33.0000	−	57.1577i	−1.22559	−	2.12278i
$726$		0			0
$727$	−44.0000			−1.63187			−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000			−1.63187			−0.815935	+	0.578144i	$0.803777\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−4.00000	−	6.92820i	−0.147945	−	0.256249i
$732$		0			0
$733$			22.0000i			0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$			22.0000i			0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$		0			0
$735$	−10.3923	−	6.00000i	−0.383326	−	0.221313i
$736$		0			0
$737$	4.00000	−	6.92820i	0.147342	−	0.255204i
$738$		0			0
$739$	−19.0526	+	11.0000i	−0.700860	+	0.404642i	−0.807668	−	0.589638i	$-0.799270\pi$
$739$	−19.0526	+	11.0000i	−0.700860	+	0.404642i	0.106808	+	0.994280i	$0.465937\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	41.5692	−	24.0000i	1.52503	−	0.880475i	0.525467	−	0.850814i	$-0.323891\pi$
$743$	41.5692	−	24.0000i	1.52503	−	0.880475i	0.999560	−	0.0296605i	$-0.00944260\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			24.0000i			0.876941i
$750$		0			0
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	0.929731	−	0.368238i	$-0.120039\pi$
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	−0.783769	+	0.621052i	$0.786706\pi$
$752$		0			0
$753$	−20.0000			−0.728841
$754$		0			0
$755$	−72.0000			−2.62035
$756$		0			0
$757$	−9.00000	−	15.5885i	−0.327111	−	0.566572i	0.654827	−	0.755779i	$-0.272742\pi$
$757$	−9.00000	−	15.5885i	−0.327111	−	0.566572i	−0.981937	+	0.189207i	$0.939408\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	34.6410	+	20.0000i	1.25574	+	0.724999i	0.972243	−	0.233975i	$-0.0751733\pi$
$761$	34.6410	+	20.0000i	1.25574	+	0.724999i	0.283493	+	0.958974i	$0.408507\pi$
$762$		0			0
$763$	−2.00000	+	3.46410i	−0.0724049	+	0.125409i
$764$		0			0
$765$	6.92820	−	4.00000i	0.250490	−	0.144620i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−19.0526	+	11.0000i	−0.687053	+	0.396670i	−0.802507	−	0.596643i	$-0.796501\pi$
$769$	−19.0526	+	11.0000i	−0.687053	+	0.396670i	0.115454	+	0.993313i	$0.463168\pi$
$770$		0			0
$771$	−15.0000	+	25.9808i	−0.540212	+	0.935674i
$772$		0			0
$773$	41.5692	+	24.0000i	1.49514	+	0.863220i	0.999984	−	0.00558380i	$-0.00177739\pi$
$773$	41.5692	+	24.0000i	1.49514	+	0.863220i	0.495156	+	0.868804i	$0.335111\pi$
$774$		0			0
$775$			110.000i			3.95132i
$776$		0			0
$777$	10.0000	+	17.3205i	0.358748	+	0.621370i
$778$		0			0
$779$	16.0000			0.573259
$780$		0			0
$781$	−64.0000			−2.29010
$782$		0			0
$783$	−3.00000	−	5.19615i	−0.107211	−	0.185695i
$784$		0			0
$785$		−	8.00000i		−	0.285532i
$786$		0			0
$787$	−43.3013	−	25.0000i	−1.54352	−	0.891154i	−0.998613	−	0.0526599i	$-0.983230\pi$
$787$	−43.3013	−	25.0000i	−1.54352	−	0.891154i	−0.544911	−	0.838494i	$-0.683437\pi$
$788$		0			0
$789$	−8.00000	+	13.8564i	−0.284808	+	0.493301i
$790$		0			0
$791$	10.3923	−	6.00000i	0.369508	−	0.213335i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−34.6410	+	20.0000i	−1.22859	+	0.709327i
$796$		0			0
$797$	−13.0000	+	22.5167i	−0.460484	+	0.797581i	−0.998985	−	0.0450436i	$-0.985657\pi$
$797$	−13.0000	+	22.5167i	−0.460484	+	0.797581i	0.538501	+	0.842625i	$0.318991\pi$
$798$		0			0
$799$	6.92820	+	4.00000i	0.245102	+	0.141510i
$800$		0			0
$801$			4.00000i			0.141333i
$802$		0			0
$803$	20.0000	+	34.6410i	0.705785	+	1.22245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.0000			0.492823
$808$		0			0
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	−0.978461	−	0.206430i	$-0.933815\pi$
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	0.310457	−	0.950587i	$-0.399518\pi$
$810$		0			0
$811$			6.00000i			0.210688i	0.994436	+	0.105344i	$0.0335944\pi$
$811$			6.00000i			0.210688i	−0.994436	+	0.105344i	$0.966406\pi$
$812$		0			0
$813$	8.66025	+	5.00000i	0.303728	+	0.175358i
$814$		0			0
$815$	4.00000	−	6.92820i	0.140114	−	0.242684i
$816$		0			0
$817$	6.92820	−	4.00000i	0.242387	−	0.139942i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.7846	−	12.0000i	0.725388	−	0.418803i	−0.0913446	−	0.995819i	$-0.529116\pi$
$821$	20.7846	−	12.0000i	0.725388	−	0.418803i	0.816733	+	0.577016i	$0.195783\pi$
$822$		0			0
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	16.0000	−	27.7128i	0.557725	−	0.966008i	0.997686	−	0.0679910i	$-0.0216589\pi$
$824$		0			0
$825$	−38.1051	−	22.0000i	−1.32665	−	0.765942i
$826$		0			0
$827$			20.0000i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$			20.0000i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$		0			0
$829$	−7.00000	−	12.1244i	−0.243120	−	0.421096i	0.718481	−	0.695546i	$-0.244838\pi$
$829$	−7.00000	−	12.1244i	−0.243120	−	0.421096i	−0.961601	+	0.274450i	$0.911504\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$	−24.0000	−	41.5692i	−0.830554	−	1.43856i
$836$		0			0
$837$			10.0000i			0.345651i
$838$		0			0
$839$	−13.8564	−	8.00000i	−0.478376	−	0.276191i	0.241363	−	0.970435i	$-0.422405\pi$
$839$	−13.8564	−	8.00000i	−0.478376	−	0.276191i	−0.719740	+	0.694244i	$0.755739\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$	−6.92820	+	4.00000i	−0.238620	+	0.137767i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−8.66025	+	5.00000i	−0.297570	+	0.171802i
$848$		0			0
$849$	8.00000	−	13.8564i	0.274559	−	0.475551i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$	4.00000	+	6.92820i	0.136797	+	0.236940i
$856$		0			0
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$	52.0000			1.77422			0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000			1.77422			0.887109	+	0.461561i	$0.152710\pi$
$860$		0			0
$861$	−8.00000	−	13.8564i	−0.272639	−	0.472225i
$862$		0			0
$863$			4.00000i			0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$			4.00000i			0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$		0			0
$865$	−6.92820	−	4.00000i	−0.235566	−	0.136004i
$866$		0			0
$867$	−6.50000	+	11.2583i	−0.220752	+	0.382353i
$868$		0			0
$869$	−55.4256	+	32.0000i	−1.88019	+	1.08553i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.73205	+	1.00000i	−0.0586210	+	0.0338449i
$874$		0			0
$875$	24.0000	−	41.5692i	0.811348	−	1.40530i
$876$		0			0
$877$	−8.66025	−	5.00000i	−0.292436	−	0.168838i	0.346604	−	0.938012i	$-0.387335\pi$
$877$	−8.66025	−	5.00000i	−0.292436	−	0.168838i	−0.639040	+	0.769174i	$0.720668\pi$
$878$		0			0
$879$			8.00000i			0.269833i
$880$		0			0
$881$	7.00000	+	12.1244i	0.235836	+	0.408480i	0.959515	−	0.281656i	$-0.0908839\pi$
$881$	7.00000	+	12.1244i	0.235836	+	0.408480i	−0.723679	+	0.690136i	$0.757551\pi$
$882$		0			0
$883$	32.0000			1.07689			0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000			1.07689			0.538443	+	0.842662i	$0.319013\pi$
$884$		0			0
$885$	−32.0000			−1.07567
$886$		0			0
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	−0.915722	−	0.401813i	$-0.868380\pi$
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	0.109881	−	0.993945i	$-0.464953\pi$
$888$		0			0
$889$		−	24.0000i		−	0.804934i
$890$		0			0
$891$	−3.46410	−	2.00000i	−0.116052	−	0.0670025i
$892$		0			0
$893$	−4.00000	+	6.92820i	−0.133855	+	0.231843i
$894$		0			0
$895$	41.5692	−	24.0000i	1.38951	−	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	51.9615	−	30.0000i	1.73301	−	1.00056i
$900$		0			0
$901$	−10.0000	+	17.3205i	−0.333148	+	0.577030i
$902$		0			0
$903$	−6.92820	−	4.00000i	−0.230556	−	0.133112i
$904$		0			0
$905$			8.00000i			0.265929i
$906$		0			0
$907$	−28.0000	−	48.4974i	−0.929725	−	1.61033i	−0.783781	−	0.621038i	$-0.786711\pi$
$907$	−28.0000	−	48.4974i	−0.929725	−	1.61033i	−0.145944	−	0.989293i	$-0.546622\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	−56.0000			−1.85536			−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000			−1.85536			−0.927681	+	0.373373i	$0.878201\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	56.0000i		−	1.85130i
$916$		0			0
$917$	−6.92820	−	4.00000i	−0.228789	−	0.132092i
$918$		0			0
$919$	−28.0000	+	48.4974i	−0.923635	+	1.59978i	−0.129893	+	0.991528i	$0.541463\pi$
$919$	−28.0000	+	48.4974i	−0.923635	+	1.59978i	−0.793742	+	0.608254i	$0.791870\pi$
$920$		0			0
$921$	19.0526	−	11.0000i	0.627803	−	0.362462i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	95.2628	−	55.0000i	3.13222	−	1.80839i
$926$		0			0
$927$	−4.00000	+	6.92820i	−0.131377	+	0.227552i
$928$		0			0
$929$	−13.8564	−	8.00000i	−0.454614	−	0.262471i	0.255163	−	0.966898i	$-0.417871\pi$
$929$	−13.8564	−	8.00000i	−0.454614	−	0.262471i	−0.709777	+	0.704427i	$0.751204\pi$
$930$		0			0
$931$		−	6.00000i		−	0.196642i
$932$		0			0
$933$	−12.0000	−	20.7846i	−0.392862	−	0.680458i
$934$		0			0
$935$	−32.0000			−1.04651
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−3.00000	−	5.19615i	−0.0979013	−	0.169570i
$940$		0			0
$941$			8.00000i			0.260793i	0.991462	+	0.130396i	$0.0416250\pi$
$941$			8.00000i			0.260793i	−0.991462	+	0.130396i	$0.958375\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	4.00000	−	6.92820i	0.130120	−	0.225374i
$946$		0			0
$947$	−6.92820	+	4.00000i	−0.225136	+	0.129983i	−0.608326	−	0.793687i	$-0.708159\pi$
$947$	−6.92820	+	4.00000i	−0.225136	+	0.129983i	0.383190	+	0.923670i	$0.374825\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	10.3923	−	6.00000i	0.336994	−	0.194563i
$952$		0			0
$953$	−21.0000	+	36.3731i	−0.680257	+	1.17824i	0.294646	+	0.955607i	$0.404798\pi$
$953$	−21.0000	+	36.3731i	−0.680257	+	1.17824i	−0.974902	+	0.222633i	$0.928535\pi$
$954$		0			0
$955$	−27.7128	−	16.0000i	−0.896766	−	0.517748i
$956$		0			0
$957$			24.0000i			0.775810i
$958$		0			0
$959$	−8.00000	−	13.8564i	−0.258333	−	0.447447i
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$	12.0000			0.386695
$964$		0			0
$965$	−28.0000	−	48.4974i	−0.901352	−	1.56119i
$966$		0			0
$967$			22.0000i			0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$			22.0000i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$		0			0
$969$	3.46410	+	2.00000i	0.111283	+	0.0642493i
$970$		0			0
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	−0.549058	−	0.835784i	$-0.685013\pi$
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	0.998339	+	0.0576061i	$0.0183467\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10.3923	+	6.00000i	−0.332479	+	0.191957i	−0.656941	−	0.753942i	$-0.728150\pi$
$977$	−10.3923	+	6.00000i	−0.332479	+	0.191957i	0.324462	+	0.945899i	$0.394817\pi$
$978$		0			0
$979$	8.00000	−	13.8564i	0.255681	−	0.442853i
$980$		0			0
$981$	1.73205	+	1.00000i	0.0553001	+	0.0319275i
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	24.0000	+	41.5692i	0.764704	+	1.32451i
$986$		0			0
$987$	8.00000			0.254643
$988$		0			0
$989$		0			0
$990$		0			0
$991$	18.0000	+	31.1769i	0.571789	+	0.990367i	0.996382	+	0.0849833i	$0.0270837\pi$
$991$	18.0000	+	31.1769i	0.571789	+	0.990367i	−0.424594	+	0.905384i	$0.639583\pi$
$992$		0			0
$993$		−	2.00000i		−	0.0634681i
$994$		0			0
$995$	−13.8564	−	8.00000i	−0.439278	−	0.253617i
$996$		0			0
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	−0.985961	−	0.166978i	$-0.946599\pi$
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	0.637587	+	0.770378i	$0.279933\pi$
$998$		0			0
$999$	8.66025	−	5.00000i	0.273998	−	0.158193i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.q.h.361.1		4
13.2	odd	12		156.2.a.a.1.1	✓	1
13.3	even	3		2028.2.b.a.337.1		2
13.4	even	6	inner	2028.2.q.h.1837.2		4
13.5	odd	4		2028.2.i.e.2005.1		2
13.6	odd	12		2028.2.i.e.529.1		2
13.7	odd	12		2028.2.i.g.529.1		2
13.8	odd	4		2028.2.i.g.2005.1		2
13.9	even	3	inner	2028.2.q.h.1837.1		4
13.10	even	6		2028.2.b.a.337.2		2
13.11	odd	12		2028.2.a.c.1.1		1
13.12	even	2	inner	2028.2.q.h.361.2		4
39.2	even	12		468.2.a.d.1.1		1
39.11	even	12		6084.2.a.b.1.1		1
39.23	odd	6		6084.2.b.j.4393.1		2
39.29	odd	6		6084.2.b.j.4393.2		2
52.11	even	12		8112.2.a.bi.1.1		1
52.15	even	12		624.2.a.e.1.1		1
65.2	even	12		3900.2.h.b.1249.2		2
65.28	even	12		3900.2.h.b.1249.1		2
65.54	odd	12		3900.2.a.m.1.1		1
91.41	even	12		7644.2.a.k.1.1		1
104.67	even	12		2496.2.a.o.1.1		1
104.93	odd	12		2496.2.a.bc.1.1		1
117.2	even	12		4212.2.i.b.1405.1		2
117.41	even	12		4212.2.i.b.2809.1		2
117.67	odd	12		4212.2.i.l.2809.1		2
117.106	odd	12		4212.2.i.l.1405.1		2
156.119	odd	12		1872.2.a.s.1.1		1
312.197	even	12		7488.2.a.c.1.1		1
312.275	odd	12		7488.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.a.1.1	✓	1	13.2	odd	12
468.2.a.d.1.1		1	39.2	even	12
624.2.a.e.1.1		1	52.15	even	12
1872.2.a.s.1.1		1	156.119	odd	12
2028.2.a.c.1.1		1	13.11	odd	12
2028.2.b.a.337.1		2	13.3	even	3
2028.2.b.a.337.2		2	13.10	even	6
2028.2.i.e.529.1		2	13.6	odd	12
2028.2.i.e.2005.1		2	13.5	odd	4
2028.2.i.g.529.1		2	13.7	odd	12
2028.2.i.g.2005.1		2	13.8	odd	4
2028.2.q.h.361.1		4	1.1	even	1	trivial
2028.2.q.h.361.2		4	13.12	even	2	inner
2028.2.q.h.1837.1		4	13.9	even	3	inner
2028.2.q.h.1837.2		4	13.4	even	6	inner
2496.2.a.o.1.1		1	104.67	even	12
2496.2.a.bc.1.1		1	104.93	odd	12
3900.2.a.m.1.1		1	65.54	odd	12
3900.2.h.b.1249.1		2	65.28	even	12
3900.2.h.b.1249.2		2	65.2	even	12
4212.2.i.b.1405.1		2	117.2	even	12
4212.2.i.b.2809.1		2	117.41	even	12
4212.2.i.l.1405.1		2	117.106	odd	12
4212.2.i.l.2809.1		2	117.67	odd	12
6084.2.a.b.1.1		1	39.11	even	12
6084.2.b.j.4393.1		2	39.23	odd	6
6084.2.b.j.4393.2		2	39.29	odd	6
7488.2.a.c.1.1		1	312.197	even	12
7488.2.a.d.1.1		1	312.275	odd	12
7644.2.a.k.1.1		1	91.41	even	12
8112.2.a.bi.1.1		1	52.11	even	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} -4.00000i q^{5} +(-1.73205 - 1.00000i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.46410 - 2.00000i) q^{11} +(3.46410 - 2.00000i) q^{15} +(1.00000 - 1.73205i) q^{17} +(1.73205 + 1.00000i) q^{19} -2.00000i q^{21} -11.0000 q^{25} -1.00000 q^{27} +(3.00000 + 5.19615i) q^{29} -10.0000i q^{31} +(3.46410 + 2.00000i) q^{33} +(-4.00000 + 6.92820i) q^{35} +(-8.66025 + 5.00000i) q^{37} +(6.92820 - 4.00000i) q^{41} +(2.00000 - 3.46410i) q^{43} +(3.46410 + 2.00000i) q^{45} +4.00000i q^{47} +(-1.50000 - 2.59808i) q^{49} +2.00000 q^{51} -10.0000 q^{53} +(-8.00000 - 13.8564i) q^{55} +2.00000i q^{57} +(-6.92820 - 4.00000i) q^{59} +(7.00000 - 12.1244i) q^{61} +(1.73205 - 1.00000i) q^{63} +(1.73205 - 1.00000i) q^{67} +(-13.8564 - 8.00000i) q^{71} +10.0000i q^{73} +(-5.50000 - 9.52628i) q^{75} -8.00000 q^{77} -16.0000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(-6.92820 - 4.00000i) q^{85} +(-3.00000 + 5.19615i) q^{87} +(3.46410 - 2.00000i) q^{89} +(8.66025 - 5.00000i) q^{93} +(4.00000 - 6.92820i) q^{95} +(1.73205 + 1.00000i) q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{9} + 4 q^{17} - 44 q^{25} - 4 q^{27} + 12 q^{29} - 16 q^{35} + 8 q^{43} - 6 q^{49} + 8 q^{51} - 40 q^{53} - 32 q^{55} + 28 q^{61} - 22 q^{75} - 32 q^{77} - 64 q^{79} - 2 q^{81} - 12 q^{87}+ \cdots + 16 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^9 + 4 * q^17 - 44 * q^25 - 4 * q^27 + 12 * q^29 - 16 * q^35 + 8 * q^43 - 6 * q^49 + 8 * q^51 - 40 * q^53 - 32 * q^55 + 28 * q^61 - 22 * q^75 - 32 * q^77 - 64 * q^79 - 2 * q^81 - 12 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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