LMFDB - Embedded newform 2352.2.bl.k.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(31,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2352.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.bl (of order $6$, 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:	$18.7808145554$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.31
Dual form			2352.2.bl.k.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(1.50000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(1.50000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-4.50000 - 2.59808i) q^{11} +6.92820i q^{13} -1.73205i q^{15} +(-3.00000 - 1.73205i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(-6.00000 + 3.46410i) q^{23} +(-1.00000 + 1.73205i) q^{25} -1.00000 q^{27} -9.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-4.50000 + 2.59808i) q^{33} +(1.00000 + 1.73205i) q^{37} +(6.00000 + 3.46410i) q^{39} -3.46410i q^{41} -3.46410i q^{43} +(-1.50000 - 0.866025i) q^{45} +(-3.00000 + 1.73205i) q^{51} +(-4.50000 + 7.79423i) q^{53} -9.00000 q^{55} -2.00000 q^{57} +(1.50000 - 2.59808i) q^{59} +(6.00000 - 3.46410i) q^{61} +(6.00000 + 10.3923i) q^{65} +6.92820i q^{69} -6.92820i q^{71} +(-6.00000 - 3.46410i) q^{73} +(1.00000 + 1.73205i) q^{75} +(1.50000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.0000 q^{83} -6.00000 q^{85} +(-4.50000 + 7.79423i) q^{87} +(9.00000 - 5.19615i) q^{89} +(0.500000 + 0.866025i) q^{93} +(-3.00000 - 1.73205i) q^{95} +8.66025i q^{97} +5.19615i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 3 q^{5} - q^{9}+O(q^{10}) $ 2 * q + q^3 + 3 * q^5 - q^9	$ 2 q + q^{3} + 3 q^{5} - q^{9} - 9 q^{11} - 6 q^{17} - 2 q^{19} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 18 q^{29} - q^{31} - 9 q^{33} + 2 q^{37} + 12 q^{39} - 3 q^{45} - 6 q^{51} - 9 q^{53} - 18 q^{55} - 4 q^{57} + 3 q^{59} + 12 q^{61} + 12 q^{65} - 12 q^{73} + 2 q^{75} + 3 q^{79} - q^{81} - 30 q^{83} - 12 q^{85} - 9 q^{87} + 18 q^{89} + q^{93} - 6 q^{95}+O(q^{100}) $ 2 * q + q^3 + 3 * q^5 - q^9 - 9 * q^11 - 6 * q^17 - 2 * q^19 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 18 * q^29 - q^31 - 9 * q^33 + 2 * q^37 + 12 * q^39 - 3 * q^45 - 6 * q^51 - 9 * q^53 - 18 * q^55 - 4 * q^57 + 3 * q^59 + 12 * q^61 + 12 * q^65 - 12 * q^73 + 2 * q^75 + 3 * q^79 - q^81 - 30 * q^83 - 12 * q^85 - 9 * q^87 + 18 * q^89 + q^93 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	0.796387	+	0.604787i	$0.206742\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.989191	+	0.146631i	$0.953157\pi$
$12$		0			0
$13$			6.92820i			1.92154i	0.277350	+	0.960769i	$0.410544\pi$
$13$			6.92820i			1.92154i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		−	1.73205i		−	0.447214i
$16$		0			0
$17$	−3.00000	−	1.73205i	−0.727607	−	0.420084i	0.0899392	−	0.995947i	$-0.471333\pi$
$17$	−3.00000	−	1.73205i	−0.727607	−	0.420084i	−0.817546	+	0.575863i	$0.804666\pi$
$18$		0			0
$19$	−1.00000	−	1.73205i	−0.229416	−	0.397360i	0.728219	−	0.685344i	$-0.240348\pi$
$19$	−1.00000	−	1.73205i	−0.229416	−	0.397360i	−0.957635	+	0.287984i	$0.907015\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.00000	+	3.46410i	−1.25109	+	0.722315i	−0.971325	−	0.237754i	$-0.923589\pi$
$23$	−6.00000	+	3.46410i	−1.25109	+	0.722315i	−0.279761	+	0.960070i	$0.590255\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$	−4.50000	+	2.59808i	−0.783349	+	0.452267i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$	6.00000	+	3.46410i	0.960769	+	0.554700i
$40$		0			0
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$	−1.50000	−	0.866025i	−0.223607	−	0.129099i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−3.00000	+	1.73205i	−0.420084	+	0.242536i
$52$		0			0
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	0.371706	+	0.928351i	$0.378773\pi$
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	−0.989828	+	0.142269i	$0.954560\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$		0			0
$57$	−2.00000			−0.264906
$58$		0			0
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	10.3923i	0.744208	+	1.28901i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$			6.92820i			0.834058i
$70$		0			0
$71$		−	6.92820i		−	0.822226i	−0.911584	−	0.411113i	$-0.865140\pi$
$71$		−	6.92820i		−	0.822226i	0.911584	−	0.411113i	$-0.134860\pi$
$72$		0			0
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	0.105937	−	0.994373i	$-0.466216\pi$
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	−0.808184	+	0.588930i	$0.799549\pi$
$74$		0			0
$75$	1.00000	+	1.73205i	0.115470	+	0.200000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.50000	−	0.866025i	0.168763	−	0.0974355i	−0.413239	−	0.910622i	$-0.635603\pi$
$79$	1.50000	−	0.866025i	0.168763	−	0.0974355i	0.582003	+	0.813187i	$0.302269\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−15.0000			−1.64646			−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000			−1.64646			−0.823232	+	0.567705i	$0.807831\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$	−4.50000	+	7.79423i	−0.482451	+	0.835629i
$88$		0			0
$89$	9.00000	−	5.19615i	0.953998	−	0.550791i	0.0596775	−	0.998218i	$-0.480993\pi$
$89$	9.00000	−	5.19615i	0.953998	−	0.550791i	0.894321	+	0.447427i	$0.147659\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.500000	+	0.866025i	0.0518476	+	0.0898027i
$94$		0			0
$95$	−3.00000	−	1.73205i	−0.307794	−	0.177705i
$96$		0			0
$97$			8.66025i			0.879316i	0.898165	+	0.439658i	$0.144900\pi$
$97$			8.66025i			0.879316i	−0.898165	+	0.439658i	$0.855100\pi$
$98$		0			0
$99$			5.19615i			0.522233i
$100$		0			0
$101$	12.0000	+	6.92820i	1.19404	+	0.689382i	0.959221	−	0.282656i	$-0.0912155\pi$
$101$	12.0000	+	6.92820i	1.19404	+	0.689382i	0.234823	+	0.972038i	$0.424549\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.50000	−	4.33013i	0.725052	−	0.418609i	−0.0915571	−	0.995800i	$-0.529184\pi$
$107$	7.50000	−	4.33013i	0.725052	−	0.418609i	0.816609	+	0.577191i	$0.195851\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−6.00000	+	10.3923i	−0.559503	+	0.969087i
$116$		0			0
$117$	6.00000	−	3.46410i	0.554700	−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.00000	+	13.8564i	0.727273	+	1.25967i
$122$		0			0
$123$	−3.00000	−	1.73205i	−0.270501	−	0.156174i
$124$		0			0
$125$			12.1244i			1.08444i
$126$		0			0
$127$		−	15.5885i		−	1.38325i	−0.722256	−	0.691626i	$-0.756895\pi$
$127$		−	15.5885i		−	1.38325i	0.722256	−	0.691626i	$-0.243105\pi$
$128$		0			0
$129$	−3.00000	−	1.73205i	−0.264135	−	0.152499i
$130$		0			0
$131$	1.50000	+	2.59808i	0.131056	+	0.226995i	0.924084	−	0.382190i	$-0.124830\pi$
$131$	1.50000	+	2.59808i	0.131056	+	0.226995i	−0.793028	+	0.609185i	$0.791497\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.50000	+	0.866025i	−0.129099	+	0.0745356i
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	22.0000			1.86602			0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000			1.86602			0.933008	+	0.359856i	$0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.0000	−	31.1769i	1.50524	−	2.60714i
$144$		0			0
$145$	−13.5000	+	7.79423i	−1.12111	+	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	1.50000	+	0.866025i	0.122068	+	0.0704761i	0.559791	−	0.828634i	$-0.310881\pi$
$151$	1.50000	+	0.866025i	0.122068	+	0.0704761i	−0.437723	+	0.899110i	$0.644215\pi$
$152$		0			0
$153$			3.46410i			0.280056i
$154$		0			0
$155$			1.73205i			0.139122i
$156$		0			0
$157$	−6.00000	−	3.46410i	−0.478852	−	0.276465i	0.241086	−	0.970504i	$-0.422496\pi$
$157$	−6.00000	−	3.46410i	−0.478852	−	0.276465i	−0.719938	+	0.694038i	$0.755830\pi$
$158$		0			0
$159$	4.50000	+	7.79423i	0.356873	+	0.618123i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−18.0000	+	10.3923i	−1.40987	+	0.813988i	−0.995375	−	0.0960641i	$-0.969375\pi$
$163$	−18.0000	+	10.3923i	−1.40987	+	0.813988i	−0.414494	+	0.910052i	$0.636041\pi$
$164$		0			0
$165$	−4.50000	+	7.79423i	−0.350325	+	0.606780i
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	−35.0000			−2.69231
$170$		0			0
$171$	−1.00000	+	1.73205i	−0.0764719	+	0.132453i
$172$		0			0
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.881184	−	0.472773i	$-0.843253\pi$
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.0311588	+	0.999514i	$0.509920\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.50000	−	2.59808i	−0.112747	−	0.195283i
$178$		0			0
$179$	−15.0000	−	8.66025i	−1.12115	−	0.647298i	−0.179458	−	0.983766i	$-0.557434\pi$
$179$	−15.0000	−	8.66025i	−1.12115	−	0.647298i	−0.941695	+	0.336468i	$0.890768\pi$
$180$		0			0
$181$		−	3.46410i		−	0.257485i	−0.991678	−	0.128742i	$-0.958906\pi$
$181$		−	3.46410i		−	0.257485i	0.991678	−	0.128742i	$-0.0410940\pi$
$182$		0			0
$183$		−	6.92820i		−	0.512148i
$184$		0			0
$185$	3.00000	+	1.73205i	0.220564	+	0.127343i
$186$		0			0
$187$	9.00000	+	15.5885i	0.658145	+	1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	0.289980	+	0.957033i	$0.406351\pi$
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	−0.973805	+	0.227387i	$0.926982\pi$
$194$		0			0
$195$	12.0000			0.859338
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	−0.429745	−	0.902950i	$-0.641397\pi$
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	0.996850	−	0.0793045i	$-0.0252700\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$		0			0
$207$	6.00000	+	3.46410i	0.417029	+	0.240772i
$208$		0			0
$209$			10.3923i			0.718851i
$210$		0			0
$211$			13.8564i			0.953914i	0.878927	+	0.476957i	$0.158260\pi$
$211$			13.8564i			0.953914i	−0.878927	+	0.476957i	$0.841740\pi$
$212$		0			0
$213$	−6.00000	−	3.46410i	−0.411113	−	0.237356i
$214$		0			0
$215$	−3.00000	−	5.19615i	−0.204598	−	0.354375i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.00000	+	3.46410i	−0.405442	+	0.234082i
$220$		0			0
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$		0			0
$225$	2.00000			0.133333
$226$		0			0
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	−0.975833	−	0.218517i	$-0.929878\pi$
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	0.677158	+	0.735838i	$0.263211\pi$
$228$		0			0
$229$	3.00000	−	1.73205i	0.198246	−	0.114457i	−0.397591	−	0.917563i	$-0.630154\pi$
$229$	3.00000	−	1.73205i	0.198246	−	0.114457i	0.595837	+	0.803105i	$0.296820\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	−0.994283	−	0.106773i	$-0.965948\pi$
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	0.404674	−	0.914461i	$-0.367385\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	1.73205i		−	0.112509i
$238$		0			0
$239$			13.8564i			0.896296i	0.893959	+	0.448148i	$0.147916\pi$
$239$			13.8564i			0.896296i	−0.893959	+	0.448148i	$0.852084\pi$
$240$		0			0
$241$	4.50000	+	2.59808i	0.289870	+	0.167357i	0.637883	−	0.770133i	$-0.279810\pi$
$241$	4.50000	+	2.59808i	0.289870	+	0.167357i	−0.348013	+	0.937490i	$0.613143\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000	−	6.92820i	0.763542	−	0.440831i
$248$		0			0
$249$	−7.50000	+	12.9904i	−0.475293	+	0.823232i
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$	−3.00000	+	5.19615i	−0.187867	+	0.325396i
$256$		0			0
$257$	3.00000	−	1.73205i	0.187135	−	0.108042i	−0.403506	−	0.914977i	$-0.632208\pi$
$257$	3.00000	−	1.73205i	0.187135	−	0.108042i	0.590641	+	0.806935i	$0.298875\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	4.50000	+	7.79423i	0.278543	+	0.482451i
$262$		0			0
$263$	−15.0000	−	8.66025i	−0.924940	−	0.534014i	−0.0397320	−	0.999210i	$-0.512650\pi$
$263$	−15.0000	−	8.66025i	−0.924940	−	0.534014i	−0.885208	+	0.465196i	$0.845984\pi$
$264$		0			0
$265$			15.5885i			0.957591i
$266$		0			0
$267$		−	10.3923i		−	0.635999i
$268$		0			0
$269$	−1.50000	−	0.866025i	−0.0914566	−	0.0528025i	0.453574	−	0.891219i	$-0.350149\pi$
$269$	−1.50000	−	0.866025i	−0.0914566	−	0.0528025i	−0.545031	+	0.838416i	$0.683482\pi$
$270$		0			0
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	0.999820	+	0.0189696i	$0.00603859\pi$
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	−0.483482	+	0.875354i	$0.660628\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.00000	−	5.19615i	0.542720	−	0.313340i
$276$		0			0
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	0.0477206	+	0.998861i	$0.484804\pi$
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	−0.888899	+	0.458103i	$0.848529\pi$
$278$		0			0
$279$	1.00000			0.0598684
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		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
$285$	−3.00000	+	1.73205i	−0.177705	+	0.102598i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.50000	−	4.33013i	−0.147059	−	0.254713i
$290$		0			0
$291$	7.50000	+	4.33013i	0.439658	+	0.253837i
$292$		0			0
$293$		−	5.19615i		−	0.303562i	−0.988414	−	0.151781i	$-0.951499\pi$
$293$		−	5.19615i		−	0.303562i	0.988414	−	0.151781i	$-0.0485009\pi$
$294$		0			0
$295$		−	5.19615i		−	0.302532i
$296$		0			0
$297$	4.50000	+	2.59808i	0.261116	+	0.150756i
$298$		0			0
$299$	−24.0000	−	41.5692i	−1.38796	−	2.40401i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	12.0000	−	6.92820i	0.689382	−	0.398015i
$304$		0			0
$305$	6.00000	−	10.3923i	0.343559	−	0.595062i
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	−0.768345	−	0.640036i	$-0.778920\pi$
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	0.938460	+	0.345389i	$0.112253\pi$
$312$		0			0
$313$	16.5000	−	9.52628i	0.932635	−	0.538457i	0.0449911	−	0.998987i	$-0.485674\pi$
$313$	16.5000	−	9.52628i	0.932635	−	0.538457i	0.887644	+	0.460530i	$0.152341\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	0.943750	+	0.330661i	$0.107272\pi$
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	−0.185514	+	0.982642i	$0.559395\pi$
$318$		0			0
$319$	40.5000	+	23.3827i	2.26756	+	1.30918i
$320$		0			0
$321$		−	8.66025i		−	0.483368i
$322$		0			0
$323$			6.92820i			0.385496i
$324$		0			0
$325$	−12.0000	−	6.92820i	−0.665640	−	0.384308i
$326$		0			0
$327$	−2.00000	−	3.46410i	−0.110600	−	0.191565i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$	1.00000	−	1.73205i	0.0547997	−	0.0949158i
$334$		0			0
$335$		0			0
$336$		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	+	5.19615i	−0.162938	+	0.282216i
$340$		0			0
$341$	4.50000	−	2.59808i	0.243689	−	0.140694i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	6.00000	+	10.3923i	0.323029	+	0.559503i
$346$		0			0
$347$	−27.0000	−	15.5885i	−1.44944	−	0.836832i	−0.450988	−	0.892530i	$-0.648928\pi$
$347$	−27.0000	−	15.5885i	−1.44944	−	0.836832i	−0.998448	+	0.0556976i	$0.982262\pi$
$348$		0			0
$349$		−	20.7846i		−	1.11257i	−0.830990	−	0.556287i	$-0.812225\pi$
$349$		−	20.7846i		−	1.11257i	0.830990	−	0.556287i	$-0.187775\pi$
$350$		0			0
$351$		−	6.92820i		−	0.369800i
$352$		0			0
$353$	6.00000	+	3.46410i	0.319348	+	0.184376i	0.651102	−	0.758990i	$-0.274307\pi$
$353$	6.00000	+	3.46410i	0.319348	+	0.184376i	−0.331754	+	0.943366i	$0.607640\pi$
$354$		0			0
$355$	−6.00000	−	10.3923i	−0.318447	−	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.0000	−	13.8564i	1.26667	−	0.731313i	0.292315	−	0.956322i	$-0.405574\pi$
$359$	24.0000	−	13.8564i	1.26667	−	0.731313i	0.974357	+	0.225009i	$0.0722411\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$	16.0000			0.839782
$364$		0			0
$365$	−12.0000			−0.628109
$366$		0			0
$367$	−11.5000	+	19.9186i	−0.600295	+	1.03974i	0.392481	+	0.919760i	$0.371617\pi$
$367$	−11.5000	+	19.9186i	−0.600295	+	1.03974i	−0.992776	+	0.119982i	$0.961716\pi$
$368$		0			0
$369$	−3.00000	+	1.73205i	−0.156174	+	0.0901670i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$		0			0
$375$	10.5000	+	6.06218i	0.542218	+	0.313050i
$376$		0			0
$377$		−	62.3538i		−	3.21139i
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$	−13.5000	−	7.79423i	−0.691626	−	0.399310i
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.00000	+	1.73205i	−0.152499	+	0.0880451i
$388$		0			0
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	3.00000			0.151330
$394$		0			0
$395$	1.50000	−	2.59808i	0.0754732	−	0.130723i
$396$		0			0
$397$	−18.0000	+	10.3923i	−0.903394	+	0.521575i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−18.0000	+	10.3923i	−0.903394	+	0.521575i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.828932	−	0.559350i	$-0.811051\pi$
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.0699455	−	0.997551i	$-0.522283\pi$
$402$		0			0
$403$	−6.00000	−	3.46410i	−0.298881	−	0.172559i
$404$		0			0
$405$			1.73205i			0.0860663i
$406$		0			0
$407$		−	10.3923i		−	0.515127i
$408$		0			0
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	0.673830	−	0.738886i	$-0.264648\pi$
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	−0.302979	+	0.952997i	$0.597981\pi$
$410$		0			0
$411$	−6.00000	−	10.3923i	−0.295958	−	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−22.5000	+	12.9904i	−1.10448	+	0.637673i
$416$		0			0
$417$	11.0000	−	19.0526i	0.538672	−	0.933008i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	−	3.46410i	0.291043	−	0.168034i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−18.0000	−	31.1769i	−0.869048	−	1.50524i
$430$		0			0
$431$	21.0000	+	12.1244i	1.01153	+	0.584010i	0.911641	−	0.410988i	$-0.134816\pi$
$431$	21.0000	+	12.1244i	1.01153	+	0.584010i	0.0998939	+	0.994998i	$0.468150\pi$
$432$		0			0
$433$		−	6.92820i		−	0.332948i	−0.986046	−	0.166474i	$-0.946762\pi$
$433$		−	6.92820i		−	0.332948i	0.986046	−	0.166474i	$-0.0532382\pi$
$434$		0			0
$435$			15.5885i			0.747409i
$436$		0			0
$437$	12.0000	+	6.92820i	0.574038	+	0.331421i
$438$		0			0
$439$	12.5000	+	21.6506i	0.596592	+	1.03333i	0.993320	+	0.115392i	$0.0368124\pi$
$439$	12.5000	+	21.6506i	0.596592	+	1.03333i	−0.396728	+	0.917936i	$0.629854\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.5000	+	11.2583i	−0.926473	+	0.534899i	−0.885694	−	0.464269i	$-0.846317\pi$
$443$	−19.5000	+	11.2583i	−0.926473	+	0.534899i	−0.0407786	+	0.999168i	$0.512984\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$		0			0
$447$	6.00000			0.283790
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−9.00000	+	15.5885i	−0.423793	+	0.734032i
$452$		0			0
$453$	1.50000	−	0.866025i	0.0704761	−	0.0406894i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	0.595818	−	0.803120i	$-0.296828\pi$
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	−0.993431	+	0.114433i	$0.963495\pi$
$458$		0			0
$459$	3.00000	+	1.73205i	0.140028	+	0.0808452i
$460$		0			0
$461$		−	20.7846i		−	0.968036i	−0.875058	−	0.484018i	$-0.839177\pi$
$461$		−	20.7846i		−	0.968036i	0.875058	−	0.484018i	$-0.160823\pi$
$462$		0			0
$463$		−	10.3923i		−	0.482971i	−0.970404	−	0.241486i	$-0.922365\pi$
$463$		−	10.3923i		−	0.482971i	0.970404	−	0.241486i	$-0.0776347\pi$
$464$		0			0
$465$	1.50000	+	0.866025i	0.0695608	+	0.0401610i
$466$		0			0
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	0.895696	+	0.444667i	$0.146678\pi$
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	−0.0627555	+	0.998029i	$0.519989\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−6.00000	+	3.46410i	−0.276465	+	0.159617i
$472$		0			0
$473$	−9.00000	+	15.5885i	−0.413820	+	0.716758i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$	9.00000			0.412082
$478$		0			0
$479$	21.0000	−	36.3731i	0.959514	−	1.66193i	0.235833	−	0.971794i	$-0.424218\pi$
$479$	21.0000	−	36.3731i	0.959514	−	1.66193i	0.723681	−	0.690134i	$-0.242449\pi$
$480$		0			0
$481$	−12.0000	+	6.92820i	−0.547153	+	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.50000	+	12.9904i	0.340557	+	0.589863i
$486$		0			0
$487$	13.5000	+	7.79423i	0.611743	+	0.353190i	0.773647	−	0.633616i	$-0.218430\pi$
$487$	13.5000	+	7.79423i	0.611743	+	0.353190i	−0.161904	+	0.986807i	$0.551764\pi$
$488$		0			0
$489$			20.7846i			0.939913i
$490$		0			0
$491$			19.0526i			0.859830i	0.902869	+	0.429915i	$0.141456\pi$
$491$			19.0526i			0.859830i	−0.902869	+	0.429915i	$0.858544\pi$
$492$		0			0
$493$	27.0000	+	15.5885i	1.21602	+	0.702069i
$494$		0			0
$495$	4.50000	+	7.79423i	0.202260	+	0.350325i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−15.0000	+	8.66025i	−0.671492	+	0.387686i	−0.796642	−	0.604452i	$-0.793392\pi$
$499$	−15.0000	+	8.66025i	−0.671492	+	0.387686i	0.125150	+	0.992138i	$0.460059\pi$
$500$		0			0
$501$	−6.00000	+	10.3923i	−0.268060	+	0.464294i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$	−17.5000	+	30.3109i	−0.777202	+	1.34615i
$508$		0			0
$509$	16.5000	−	9.52628i	0.731350	−	0.422245i	−0.0875661	−	0.996159i	$-0.527909\pi$
$509$	16.5000	−	9.52628i	0.731350	−	0.422245i	0.818916	+	0.573914i	$0.194576\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.00000	+	1.73205i	0.0441511	+	0.0764719i
$514$		0			0
$515$	−6.00000	−	3.46410i	−0.264392	−	0.152647i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			13.8564i			0.608229i
$520$		0			0
$521$	27.0000	+	15.5885i	1.18289	+	0.682943i	0.956682	−	0.291136i	$-0.0940332\pi$
$521$	27.0000	+	15.5885i	1.18289	+	0.682943i	0.226210	+	0.974079i	$0.427367\pi$
$522$		0			0
$523$	−20.0000	−	34.6410i	−0.874539	−	1.51475i	−0.857253	−	0.514895i	$-0.827831\pi$
$523$	−20.0000	−	34.6410i	−0.874539	−	1.51475i	−0.0172859	−	0.999851i	$-0.505503\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.00000	−	1.73205i	0.130682	−	0.0754493i
$528$		0			0
$529$	12.5000	−	21.6506i	0.543478	−	0.941332i
$530$		0			0
$531$	−3.00000			−0.130189
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$	7.50000	−	12.9904i	0.324253	−	0.561623i
$536$		0			0
$537$	−15.0000	+	8.66025i	−0.647298	+	0.373718i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	0.738296	−	0.674477i	$-0.235631\pi$
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	−0.953262	+	0.302144i	$0.902298\pi$
$542$		0			0
$543$	−3.00000	−	1.73205i	−0.128742	−	0.0743294i
$544$		0			0
$545$		−	6.92820i		−	0.296772i
$546$		0			0
$547$			24.2487i			1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$			24.2487i			1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	−6.00000	−	3.46410i	−0.256074	−	0.147844i
$550$		0			0
$551$	9.00000	+	15.5885i	0.383413	+	0.664091i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	3.00000	−	1.73205i	0.127343	−	0.0735215i
$556$		0			0
$557$	−1.50000	+	2.59808i	−0.0635570	+	0.110084i	−0.896053	−	0.443947i	$-0.853578\pi$
$557$	−1.50000	+	2.59808i	−0.0635570	+	0.110084i	0.832496	+	0.554031i	$0.186911\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	18.0000			0.759961
$562$		0			0
$563$	−4.50000	+	7.79423i	−0.189652	+	0.328488i	−0.945134	−	0.326682i	$-0.894069\pi$
$563$	−4.50000	+	7.79423i	−0.189652	+	0.328488i	0.755482	+	0.655169i	$0.227403\pi$
$564$		0			0
$565$	−9.00000	+	5.19615i	−0.378633	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	0.613369	−	0.789797i	$-0.289814\pi$
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	−0.990668	+	0.136295i	$0.956481\pi$
$570$		0			0
$571$	−24.0000	−	13.8564i	−1.00437	−	0.579873i	−0.0948308	−	0.995493i	$-0.530231\pi$
$571$	−24.0000	−	13.8564i	−1.00437	−	0.579873i	−0.909538	+	0.415621i	$0.863564\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	13.8564i		−	0.577852i
$576$		0			0
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.0477669	−	0.998859i	$-0.515210\pi$
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.888920	+	0.458062i	$0.848544\pi$
$578$		0			0
$579$	9.50000	+	16.4545i	0.394807	+	0.683825i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	40.5000	−	23.3827i	1.67734	−	0.968412i
$584$		0			0
$585$	6.00000	−	10.3923i	0.248069	−	0.429669i
$586$		0			0
$587$	21.0000			0.866763			0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000			0.866763			0.433381	+	0.901211i	$0.357320\pi$
$588$		0			0
$589$	2.00000			0.0824086
$590$		0			0
$591$	3.00000	−	5.19615i	0.123404	−	0.213741i
$592$		0			0
$593$	18.0000	−	10.3923i	0.739171	−	0.426761i	−0.0825966	−	0.996583i	$-0.526321\pi$
$593$	18.0000	−	10.3923i	0.739171	−	0.426761i	0.821768	+	0.569822i	$0.192988\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.00000	−	13.8564i	−0.327418	−	0.567105i
$598$		0			0
$599$	−24.0000	−	13.8564i	−0.980613	−	0.566157i	−0.0781581	−	0.996941i	$-0.524904\pi$
$599$	−24.0000	−	13.8564i	−0.980613	−	0.566157i	−0.902455	+	0.430784i	$0.858237\pi$
$600$		0			0
$601$		−	15.5885i		−	0.635866i	−0.948113	−	0.317933i	$-0.897011\pi$
$601$		−	15.5885i		−	0.635866i	0.948113	−	0.317933i	$-0.102989\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.0000	+	13.8564i	0.975739	+	0.563343i
$606$		0			0
$607$	−11.5000	−	19.9186i	−0.466771	−	0.808470i	0.532509	−	0.846424i	$-0.321249\pi$
$607$	−11.5000	−	19.9186i	−0.466771	−	0.808470i	−0.999279	+	0.0379540i	$0.987916\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$		0			0
$615$	−6.00000			−0.241943
$616$		0			0
$617$	24.0000			0.966204			0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000			0.966204			0.483102	+	0.875564i	$0.339510\pi$
$618$		0			0
$619$	−8.00000	+	13.8564i	−0.321547	+	0.556936i	−0.980807	−	0.194979i	$-0.937536\pi$
$619$	−8.00000	+	13.8564i	−0.321547	+	0.556936i	0.659260	+	0.751915i	$0.270870\pi$
$620$		0			0
$621$	6.00000	−	3.46410i	0.240772	−	0.139010i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$	9.00000	+	5.19615i	0.359425	+	0.207514i
$628$		0			0
$629$		−	6.92820i		−	0.276246i
$630$		0			0
$631$			25.9808i			1.03428i	0.855901	+	0.517139i	$0.173003\pi$
$631$			25.9808i			1.03428i	−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	12.0000	+	6.92820i	0.476957	+	0.275371i
$634$		0			0
$635$	−13.5000	−	23.3827i	−0.535731	−	0.927914i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−6.00000	+	3.46410i	−0.237356	+	0.137038i
$640$		0			0
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	−0.999556	−	0.0297987i	$-0.990513\pi$
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	0.525584	+	0.850741i	$0.323847\pi$
$642$		0			0
$643$	32.0000			1.26196			0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000			1.26196			0.630978	+	0.775800i	$0.282654\pi$
$644$		0			0
$645$	−6.00000			−0.236250
$646$		0			0
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	−0.723645	−	0.690172i	$-0.757535\pi$
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	0.959529	+	0.281609i	$0.0908680\pi$
$648$		0			0
$649$	−13.5000	+	7.79423i	−0.529921	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.5000	+	18.1865i	0.410897	+	0.711694i	0.994988	−	0.0999939i	$-0.0318823\pi$
$653$	10.5000	+	18.1865i	0.410897	+	0.711694i	−0.584091	+	0.811688i	$0.698549\pi$
$654$		0			0
$655$	4.50000	+	2.59808i	0.175830	+	0.101515i
$656$		0			0
$657$			6.92820i			0.270295i
$658$		0			0
$659$			17.3205i			0.674711i	0.941377	+	0.337356i	$0.109532\pi$
$659$			17.3205i			0.674711i	−0.941377	+	0.337356i	$0.890468\pi$
$660$		0			0
$661$	−33.0000	−	19.0526i	−1.28355	−	0.741059i	−0.306055	−	0.952014i	$-0.599009\pi$
$661$	−33.0000	−	19.0526i	−1.28355	−	0.741059i	−0.977496	+	0.210955i	$0.932343\pi$
$662$		0			0
$663$	−12.0000	−	20.7846i	−0.466041	−	0.807207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	54.0000	−	31.1769i	2.09089	−	1.20717i
$668$		0			0
$669$	−0.500000	+	0.866025i	−0.0193311	+	0.0334825i
$670$		0			0
$671$	−36.0000			−1.38976
$672$		0			0
$673$	−11.0000			−0.424019			−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000			−0.424019			−0.212009	+	0.977268i	$0.568001\pi$
$674$		0			0
$675$	1.00000	−	1.73205i	0.0384900	−	0.0666667i
$676$		0			0
$677$	−22.5000	+	12.9904i	−0.864745	+	0.499261i	−0.865598	−	0.500739i	$-0.833062\pi$
$677$	−22.5000	+	12.9904i	−0.864745	+	0.499261i	0.000853228		1.00000i	$0.499728\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	4.50000	+	7.79423i	0.172440	+	0.298675i
$682$		0			0
$683$	7.50000	+	4.33013i	0.286980	+	0.165688i	0.636579	−	0.771212i	$-0.280349\pi$
$683$	7.50000	+	4.33013i	0.286980	+	0.165688i	−0.349599	+	0.936899i	$0.613682\pi$
$684$		0			0
$685$		−	20.7846i		−	0.794139i
$686$		0			0
$687$		−	3.46410i		−	0.132164i
$688$		0			0
$689$	−54.0000	−	31.1769i	−2.05724	−	1.18775i
$690$		0			0
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	0.967901	−	0.251330i	$-0.0808679\pi$
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	−0.701609	+	0.712562i	$0.747535\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.0000	−	19.0526i	1.25176	−	0.722705i
$696$		0			0
$697$	−6.00000	+	10.3923i	−0.227266	+	0.393637i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	−27.0000			−1.01978			−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000			−1.01978			−0.509888	+	0.860241i	$0.670313\pi$
$702$		0			0
$703$	2.00000	−	3.46410i	0.0754314	−	0.130651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	0.704118	−	0.710083i	$-0.251342\pi$
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	−0.967009	+	0.254743i	$0.918009\pi$
$710$		0			0
$711$	−1.50000	−	0.866025i	−0.0562544	−	0.0324785i
$712$		0			0
$713$		−	6.92820i		−	0.259463i
$714$		0			0
$715$		−	62.3538i		−	2.33190i
$716$		0			0
$717$	12.0000	+	6.92820i	0.448148	+	0.258738i
$718$		0			0
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	0.804648	−	0.593753i	$-0.202354\pi$
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	−0.916529	+	0.399969i	$0.869021\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	4.50000	−	2.59808i	0.167357	−	0.0966235i
$724$		0			0
$725$	9.00000	−	15.5885i	0.334252	−	0.578941i
$726$		0			0
$727$	−17.0000			−0.630495			−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000			−0.630495			−0.315248	+	0.949009i	$0.602088\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−6.00000	+	10.3923i	−0.221918	+	0.384373i
$732$		0			0
$733$	27.0000	−	15.5885i	0.997268	−	0.575773i	0.0898290	−	0.995957i	$-0.471368\pi$
$733$	27.0000	−	15.5885i	0.997268	−	0.575773i	0.907439	+	0.420184i	$0.138035\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	27.0000	+	15.5885i	0.993211	+	0.573431i	0.906233	−	0.422780i	$-0.138946\pi$
$739$	27.0000	+	15.5885i	0.993211	+	0.573431i	0.0869785	+	0.996210i	$0.472279\pi$
$740$		0			0
$741$		−	13.8564i		−	0.509028i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	9.00000	+	5.19615i	0.329734	+	0.190372i
$746$		0			0
$747$	7.50000	+	12.9904i	0.274411	+	0.475293i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.5000	+	6.06218i	−0.383150	+	0.221212i	−0.679188	−	0.733964i	$-0.737668\pi$
$751$	−10.5000	+	6.06218i	−0.383150	+	0.221212i	0.296038	+	0.955176i	$0.404335\pi$
$752$		0			0
$753$	−10.5000	+	18.1865i	−0.382641	+	0.662754i
$754$		0			0
$755$	3.00000			0.109181
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$		0			0
$759$	18.0000	−	31.1769i	0.653359	−	1.13165i
$760$		0			0
$761$	−12.0000	+	6.92820i	−0.435000	+	0.251147i	−0.701474	−	0.712695i	$-0.747474\pi$
$761$	−12.0000	+	6.92820i	−0.435000	+	0.251147i	0.266475	+	0.963842i	$0.414141\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.00000	+	5.19615i	0.108465	+	0.187867i
$766$		0			0
$767$	18.0000	+	10.3923i	0.649942	+	0.375244i
$768$		0			0
$769$			5.19615i			0.187378i	0.995602	+	0.0936890i	$0.0298659\pi$
$769$			5.19615i			0.187378i	−0.995602	+	0.0936890i	$0.970134\pi$
$770$		0			0
$771$		−	3.46410i		−	0.124757i
$772$		0			0
$773$	24.0000	+	13.8564i	0.863220	+	0.498380i	0.865089	−	0.501618i	$-0.167262\pi$
$773$	24.0000	+	13.8564i	0.863220	+	0.498380i	−0.00186926	+	0.999998i	$0.500595\pi$
$774$		0			0
$775$	−1.00000	−	1.73205i	−0.0359211	−	0.0622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000	+	3.46410i	−0.214972	+	0.124114i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$		0			0
$783$	9.00000			0.321634
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$	−15.0000	+	8.66025i	−0.534014	+	0.308313i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	24.0000	+	41.5692i	0.852265	+	1.47617i
$794$		0			0
$795$	13.5000	+	7.79423i	0.478796	+	0.276433i
$796$		0			0
$797$			8.66025i			0.306762i	0.988167	+	0.153381i	$0.0490162\pi$
$797$			8.66025i			0.306762i	−0.988167	+	0.153381i	$0.950984\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−9.00000	−	5.19615i	−0.317999	−	0.183597i
$802$		0			0
$803$	18.0000	+	31.1769i	0.635206	+	1.10021i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.50000	+	0.866025i	−0.0528025	+	0.0304855i
$808$		0			0
$809$	12.0000	−	20.7846i	0.421898	−	0.730748i	−0.574228	−	0.818696i	$-0.694698\pi$
$809$	12.0000	−	20.7846i	0.421898	−	0.730748i	0.996125	+	0.0879478i	$0.0280309\pi$
$810$		0			0
$811$	−8.00000			−0.280918			−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000			−0.280918			−0.140459	+	0.990086i	$0.544858\pi$
$812$		0			0
$813$	17.0000			0.596216
$814$		0			0
$815$	−18.0000	+	31.1769i	−0.630512	+	1.09208i
$816$		0			0
$817$	−6.00000	+	3.46410i	−0.209913	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.5000	−	38.9711i	−0.785255	−	1.36010i	−0.928846	−	0.370465i	$-0.879198\pi$
$821$	−22.5000	−	38.9711i	−0.785255	−	1.36010i	0.143591	−	0.989637i	$-0.454135\pi$
$822$		0			0
$823$	−3.00000	−	1.73205i	−0.104573	−	0.0603755i	0.446801	−	0.894633i	$-0.352563\pi$
$823$	−3.00000	−	1.73205i	−0.104573	−	0.0603755i	−0.551375	+	0.834258i	$0.685896\pi$
$824$		0			0
$825$		−	10.3923i		−	0.361814i
$826$		0			0
$827$			36.3731i			1.26482i	0.774636	+	0.632408i	$0.217933\pi$
$827$			36.3731i			1.26482i	−0.774636	+	0.632408i	$0.782067\pi$
$828$		0			0
$829$	9.00000	+	5.19615i	0.312583	+	0.180470i	0.648082	−	0.761571i	$-0.275572\pi$
$829$	9.00000	+	5.19615i	0.312583	+	0.180470i	−0.335499	+	0.942041i	$0.608905\pi$
$830$		0			0
$831$	14.0000	+	24.2487i	0.485655	+	0.841178i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−18.0000	+	10.3923i	−0.622916	+	0.359641i
$836$		0			0
$837$	0.500000	−	0.866025i	0.0172825	−	0.0299342i
$838$		0			0
$839$	−18.0000			−0.621429			−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000			−0.621429			−0.310715	+	0.950503i	$0.600568\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−52.5000	+	30.3109i	−1.80605	+	1.04273i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	7.00000	+	12.1244i	0.240239	+	0.416107i
$850$		0			0
$851$	−12.0000	−	6.92820i	−0.411355	−	0.237496i
$852$		0			0
$853$			6.92820i			0.237217i	0.992941	+	0.118609i	$0.0378434\pi$
$853$			6.92820i			0.237217i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$			3.46410i			0.118470i
$856$		0			0
$857$	−42.0000	−	24.2487i	−1.43469	−	0.828320i	−0.437219	−	0.899355i	$-0.644036\pi$
$857$	−42.0000	−	24.2487i	−1.43469	−	0.828320i	−0.997474	+	0.0710349i	$0.977370\pi$
$858$		0			0
$859$	5.00000	+	8.66025i	0.170598	+	0.295484i	0.938629	−	0.344928i	$-0.112097\pi$
$859$	5.00000	+	8.66025i	0.170598	+	0.295484i	−0.768031	+	0.640412i	$0.778763\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.0000	+	15.5885i	−0.919091	+	0.530637i	−0.883345	−	0.468724i	$-0.844714\pi$
$863$	−27.0000	+	15.5885i	−0.919091	+	0.530637i	−0.0357458	+	0.999361i	$0.511381\pi$
$864$		0			0
$865$	−12.0000	+	20.7846i	−0.408012	+	0.706698i
$866$		0			0
$867$	−5.00000			−0.169809
$868$		0			0
$869$	−9.00000			−0.305304
$870$		0			0
$871$		0			0
$872$		0			0
$873$	7.50000	−	4.33013i	0.253837	−	0.146553i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.00000	−	6.92820i	−0.135070	−	0.233949i	0.790554	−	0.612392i	$-0.209793\pi$
$877$	−4.00000	−	6.92820i	−0.135070	−	0.233949i	−0.925624	+	0.378444i	$0.876459\pi$
$878$		0			0
$879$	−4.50000	−	2.59808i	−0.151781	−	0.0876309i
$880$		0			0
$881$			55.4256i			1.86734i	0.358139	+	0.933668i	$0.383411\pi$
$881$			55.4256i			1.86734i	−0.358139	+	0.933668i	$0.616589\pi$
$882$		0			0
$883$			31.1769i			1.04919i	0.851353	+	0.524593i	$0.175783\pi$
$883$			31.1769i			1.04919i	−0.851353	+	0.524593i	$0.824217\pi$
$884$		0			0
$885$	−4.50000	−	2.59808i	−0.151266	−	0.0873334i
$886$		0			0
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	0.747539	−	0.664218i	$-0.231235\pi$
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	−0.948999	+	0.315279i	$0.897902\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	4.50000	−	2.59808i	0.150756	−	0.0870388i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−30.0000			−1.00279
$896$		0			0
$897$	−48.0000			−1.60267
$898$		0			0
$899$	4.50000	−	7.79423i	0.150083	−	0.259952i
$900$		0			0
$901$	27.0000	−	15.5885i	0.899500	−	0.519327i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.00000	−	5.19615i	−0.0997234	−	0.172726i
$906$		0			0
$907$	−48.0000	−	27.7128i	−1.59381	−	0.920189i	−0.992644	−	0.121067i	$-0.961368\pi$
$907$	−48.0000	−	27.7128i	−1.59381	−	0.920189i	−0.601170	−	0.799122i	$-0.705298\pi$
$908$		0			0
$909$		−	13.8564i		−	0.459588i
$910$		0			0
$911$			20.7846i			0.688625i	0.938855	+	0.344312i	$0.111888\pi$
$911$			20.7846i			0.688625i	−0.938855	+	0.344312i	$0.888112\pi$
$912$		0			0
$913$	67.5000	+	38.9711i	2.23392	+	1.28976i
$914$		0			0
$915$	−6.00000	−	10.3923i	−0.198354	−	0.343559i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−3.00000	+	1.73205i	−0.0989609	+	0.0571351i	−0.548664	−	0.836043i	$-0.684863\pi$
$919$	−3.00000	+	1.73205i	−0.0989609	+	0.0571351i	0.449703	+	0.893178i	$0.351530\pi$
$920$		0			0
$921$	−8.00000	+	13.8564i	−0.263609	+	0.456584i
$922$		0			0
$923$	48.0000			1.57994
$924$		0			0
$925$	−4.00000			−0.131519
$926$		0			0
$927$	−2.00000	+	3.46410i	−0.0656886	+	0.113776i
$928$		0			0
$929$	51.0000	−	29.4449i	1.67326	−	0.966055i	0.707459	−	0.706754i	$-0.249841\pi$
$929$	51.0000	−	29.4449i	1.67326	−	0.966055i	0.965797	−	0.259300i	$-0.0834919\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−3.00000	−	5.19615i	−0.0982156	−	0.170114i
$934$		0			0
$935$	27.0000	+	15.5885i	0.882994	+	0.509797i
$936$		0			0
$937$			50.2295i			1.64093i	0.571700	+	0.820463i	$0.306284\pi$
$937$			50.2295i			1.64093i	−0.571700	+	0.820463i	$0.693716\pi$
$938$		0			0
$939$		−	19.0526i		−	0.621757i
$940$		0			0
$941$	13.5000	+	7.79423i	0.440087	+	0.254085i	0.703635	−	0.710562i	$-0.251559\pi$
$941$	13.5000	+	7.79423i	0.440087	+	0.254085i	−0.263547	+	0.964646i	$0.584893\pi$
$942$		0			0
$943$	12.0000	+	20.7846i	0.390774	+	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.0000	+	12.1244i	−0.682408	+	0.393989i	−0.800762	−	0.598983i	$-0.795572\pi$
$947$	−21.0000	+	12.1244i	−0.682408	+	0.393989i	0.118354	+	0.992972i	$0.462238\pi$
$948$		0			0
$949$	24.0000	−	41.5692i	0.779073	−	1.34939i
$950$		0			0
$951$	27.0000			0.875535
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	40.5000	−	23.3827i	1.30918	−	0.755855i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$	−7.50000	−	4.33013i	−0.241684	−	0.139536i
$964$		0			0
$965$			32.9090i			1.05938i
$966$		0			0
$967$		−	46.7654i		−	1.50387i	−0.659236	−	0.751936i	$-0.729120\pi$
$967$		−	46.7654i		−	1.50387i	0.659236	−	0.751936i	$-0.270880\pi$
$968$		0			0
$969$	6.00000	+	3.46410i	0.192748	+	0.111283i
$970$		0			0
$971$	−10.5000	−	18.1865i	−0.336961	−	0.583634i	0.646899	−	0.762576i	$-0.276066\pi$
$971$	−10.5000	−	18.1865i	−0.336961	−	0.583634i	−0.983860	+	0.178942i	$0.942732\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−12.0000	+	6.92820i	−0.384308	+	0.221880i
$976$		0			0
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	−0.685381	−	0.728184i	$-0.740364\pi$
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	0.973317	+	0.229465i	$0.0736978\pi$
$978$		0			0
$979$	−54.0000			−1.72585
$980$		0			0
$981$	−4.00000			−0.127710
$982$		0			0
$983$	−27.0000	+	46.7654i	−0.861166	+	1.49158i	0.00963785	+	0.999954i	$0.496932\pi$
$983$	−27.0000	+	46.7654i	−0.861166	+	1.49158i	−0.870804	+	0.491630i	$0.836401\pi$
$984$		0			0
$985$	9.00000	−	5.19615i	0.286764	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000	+	20.7846i	0.381578	+	0.660912i
$990$		0			0
$991$	−40.5000	−	23.3827i	−1.28652	−	0.742775i	−0.308492	−	0.951227i	$-0.599824\pi$
$991$	−40.5000	−	23.3827i	−1.28652	−	0.742775i	−0.978033	+	0.208451i	$0.933158\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	27.7128i		−	0.878555i
$996$		0			0
$997$	−24.0000	−	13.8564i	−0.760088	−	0.438837i	0.0692396	−	0.997600i	$-0.477943\pi$
$997$	−24.0000	−	13.8564i	−0.760088	−	0.438837i	−0.829327	+	0.558763i	$0.811276\pi$
$998$		0			0
$999$	−1.00000	−	1.73205i	−0.0316386	−	0.0547997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.bl.k.31.1		2
4.3	odd	2		2352.2.bl.e.31.1		2
7.2	even	3		336.2.bl.f.271.1	yes	2
7.3	odd	6		2352.2.b.f.1567.1		2
7.4	even	3		2352.2.b.b.1567.2		2
7.5	odd	6		2352.2.bl.e.607.1		2
7.6	odd	2		336.2.bl.b.31.1	✓	2
21.2	odd	6		1008.2.cs.l.271.1		2
21.11	odd	6		7056.2.b.j.1567.1		2
21.17	even	6		7056.2.b.f.1567.2		2
21.20	even	2		1008.2.cs.k.703.1		2
28.3	even	6		2352.2.b.b.1567.1		2
28.11	odd	6		2352.2.b.f.1567.2		2
28.19	even	6	inner	2352.2.bl.k.607.1		2
28.23	odd	6		336.2.bl.b.271.1	yes	2
28.27	even	2		336.2.bl.f.31.1	yes	2
56.13	odd	2		1344.2.bl.g.703.1		2
56.27	even	2		1344.2.bl.c.703.1		2
56.37	even	6		1344.2.bl.c.1279.1		2
56.51	odd	6		1344.2.bl.g.1279.1		2
84.11	even	6		7056.2.b.f.1567.1		2
84.23	even	6		1008.2.cs.k.271.1		2
84.59	odd	6		7056.2.b.j.1567.2		2
84.83	odd	2		1008.2.cs.l.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.b.31.1	✓	2	7.6	odd	2
336.2.bl.b.271.1	yes	2	28.23	odd	6
336.2.bl.f.31.1	yes	2	28.27	even	2
336.2.bl.f.271.1	yes	2	7.2	even	3
1008.2.cs.k.271.1		2	84.23	even	6
1008.2.cs.k.703.1		2	21.20	even	2
1008.2.cs.l.271.1		2	21.2	odd	6
1008.2.cs.l.703.1		2	84.83	odd	2
1344.2.bl.c.703.1		2	56.27	even	2
1344.2.bl.c.1279.1		2	56.37	even	6
1344.2.bl.g.703.1		2	56.13	odd	2
1344.2.bl.g.1279.1		2	56.51	odd	6
2352.2.b.b.1567.1		2	28.3	even	6
2352.2.b.b.1567.2		2	7.4	even	3
2352.2.b.f.1567.1		2	7.3	odd	6
2352.2.b.f.1567.2		2	28.11	odd	6
2352.2.bl.e.31.1		2	4.3	odd	2
2352.2.bl.e.607.1		2	7.5	odd	6
2352.2.bl.k.31.1		2	1.1	even	1	trivial
2352.2.bl.k.607.1		2	28.19	even	6	inner
7056.2.b.f.1567.1		2	84.11	even	6
7056.2.b.f.1567.2		2	21.17	even	6
7056.2.b.j.1567.1		2	21.11	odd	6
7056.2.b.j.1567.2		2	84.59	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(1.50000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(1.50000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-4.50000 - 2.59808i) q^{11} +6.92820i q^{13} -1.73205i q^{15} +(-3.00000 - 1.73205i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(-6.00000 + 3.46410i) q^{23} +(-1.00000 + 1.73205i) q^{25} -1.00000 q^{27} -9.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-4.50000 + 2.59808i) q^{33} +(1.00000 + 1.73205i) q^{37} +(6.00000 + 3.46410i) q^{39} -3.46410i q^{41} -3.46410i q^{43} +(-1.50000 - 0.866025i) q^{45} +(-3.00000 + 1.73205i) q^{51} +(-4.50000 + 7.79423i) q^{53} -9.00000 q^{55} -2.00000 q^{57} +(1.50000 - 2.59808i) q^{59} +(6.00000 - 3.46410i) q^{61} +(6.00000 + 10.3923i) q^{65} +6.92820i q^{69} -6.92820i q^{71} +(-6.00000 - 3.46410i) q^{73} +(1.00000 + 1.73205i) q^{75} +(1.50000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.0000 q^{83} -6.00000 q^{85} +(-4.50000 + 7.79423i) q^{87} +(9.00000 - 5.19615i) q^{89} +(0.500000 + 0.866025i) q^{93} +(-3.00000 - 1.73205i) q^{95} +8.66025i q^{97} +5.19615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} - q^{9}+O(q^{10}) \) 2 * q + q^3 + 3 * q^5 - q^9	\( 2 q + q^{3} + 3 q^{5} - q^{9} - 9 q^{11} - 6 q^{17} - 2 q^{19} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 18 q^{29} - q^{31} - 9 q^{33} + 2 q^{37} + 12 q^{39} - 3 q^{45} - 6 q^{51} - 9 q^{53} - 18 q^{55} - 4 q^{57} + 3 q^{59} + 12 q^{61} + 12 q^{65} - 12 q^{73} + 2 q^{75} + 3 q^{79} - q^{81} - 30 q^{83} - 12 q^{85} - 9 q^{87} + 18 q^{89} + q^{93} - 6 q^{95}+O(q^{100}) \) 2 * q + q^3 + 3 * q^5 - q^9 - 9 * q^11 - 6 * q^17 - 2 * q^19 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 18 * q^29 - q^31 - 9 * q^33 + 2 * q^37 + 12 * q^39 - 3 * q^45 - 6 * q^51 - 9 * q^53 - 18 * q^55 - 4 * q^57 + 3 * q^59 + 12 * q^61 + 12 * q^65 - 12 * q^73 + 2 * q^75 + 3 * q^79 - q^81 - 30 * q^83 - 12 * q^85 - 9 * q^87 + 18 * q^89 + q^93 - 6 * q^95

Introduction

L-functions

Modular forms

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