LMFDB - Embedded newform 2352.2.q.bd.961.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,2,Mod(961,2352)] 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("2352.961"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,-4,0,0,0,-2,0,-4,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.961
Dual form			2352.2.q.bd.1537.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{3} +(-1.70711 + 2.95680i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +2.58579 q^{13} -3.41421 q^{15} +(1.12132 + 1.94218i) q^{17} +(-1.41421 + 2.44949i) q^{19} +(-3.82843 + 6.63103i) q^{23} +(-3.32843 - 5.76500i) q^{25} -1.00000 q^{27} -6.82843 q^{29} +(-0.585786 - 1.01461i) q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{37} +(1.29289 + 2.23936i) q^{39} +6.24264 q^{41} -5.65685 q^{43} +(-1.70711 - 2.95680i) q^{45} +(-1.41421 + 2.44949i) q^{47} +(-1.12132 + 1.94218i) q^{51} +(1.00000 + 1.73205i) q^{53} +6.82843 q^{55} -2.82843 q^{57} +(-0.585786 - 1.01461i) q^{59} +(-6.12132 + 10.6024i) q^{61} +(-4.41421 + 7.64564i) q^{65} +(-2.82843 - 4.89898i) q^{67} -7.65685 q^{69} -9.31371 q^{71} +(-6.94975 - 12.0373i) q^{73} +(3.32843 - 5.76500i) q^{75} +(6.82843 - 11.8272i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.31371 q^{83} -7.65685 q^{85} +(-3.41421 - 5.91359i) q^{87} +(7.12132 - 12.3345i) q^{89} +(0.585786 - 1.01461i) q^{93} +(-4.82843 - 8.36308i) q^{95} +2.58579 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9} - 4 q^{11} + 16 q^{13} - 8 q^{15} - 4 q^{17} - 4 q^{23} - 2 q^{25} - 4 q^{27} - 16 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 4 q^{45} + 4 q^{51} + 4 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^9 - 4 * q^11 + 16 * q^13 - 8 * q^15 - 4 * q^17 - 4 * q^23 - 2 * q^25 - 4 * q^27 - 16 * q^29 - 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^39 + 8 * q^41 - 4 * q^45 + 4 * q^51 + 4 * q^53 + 16 * q^55 - 8 * q^59 - 16 * q^61 - 12 * q^65 - 8 * q^69 + 8 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^79 - 2 * q^81 + 16 * q^83 - 8 * q^85 - 8 * q^87 + 20 * q^89 + 8 * q^93 - 8 * q^95 + 16 * q^97 + 8 * q^99

Character values

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

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$		0			0
$5$	−1.70711	+	2.95680i	−0.763441	+	1.32232i	0.177625	+	0.984098i	$0.443158\pi$
$5$	−1.70711	+	2.95680i	−0.763441	+	1.32232i	−0.941067	+	0.338221i	$0.890175\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	2.58579			0.717168			0.358584	−	0.933497i	$-0.383260\pi$
$13$	2.58579			0.717168			0.358584	+	0.933497i	$0.383260\pi$
$14$		0			0
$15$	−3.41421			−0.881546
$16$		0			0
$17$	1.12132	+	1.94218i	0.271960	+	0.471049i	0.969364	−	0.245630i	$-0.0789948\pi$
$17$	1.12132	+	1.94218i	0.271960	+	0.471049i	−0.697404	+	0.716679i	$0.745661\pi$
$18$		0			0
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	−0.981399	−	0.191977i	$-0.938510\pi$
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	0.656957	+	0.753928i	$0.271843\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.82843	+	6.63103i	−0.798282	+	1.38267i	0.122452	+	0.992474i	$0.460924\pi$
$23$	−3.82843	+	6.63103i	−0.798282	+	1.38267i	−0.920734	+	0.390191i	$0.872409\pi$
$24$		0			0
$25$	−3.32843	−	5.76500i	−0.665685	−	1.15300i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−6.82843			−1.26801			−0.634004	−	0.773330i	$-0.718590\pi$
$29$	−6.82843			−1.26801			−0.634004	+	0.773330i	$0.718590\pi$
$30$		0			0
$31$	−0.585786	−	1.01461i	−0.105210	−	0.182230i	0.808614	−	0.588340i	$-0.200218\pi$
$31$	−0.585786	−	1.01461i	−0.105210	−	0.182230i	−0.913824	+	0.406110i	$0.866885\pi$
$32$		0			0
$33$	1.00000	−	1.73205i	0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	$-0.726690\pi$
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	0.982274	+	0.187453i	$0.0600231\pi$
$38$		0			0
$39$	1.29289	+	2.23936i	0.207029	+	0.358584i
$40$		0			0
$41$	6.24264			0.974937			0.487468	−	0.873141i	$-0.337920\pi$
$41$	6.24264			0.974937			0.487468	+	0.873141i	$0.337920\pi$
$42$		0			0
$43$	−5.65685			−0.862662			−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685			−0.862662			−0.431331	+	0.902194i	$0.641956\pi$
$44$		0			0
$45$	−1.70711	−	2.95680i	−0.254480	−	0.440773i
$46$		0			0
$47$	−1.41421	+	2.44949i	−0.206284	+	0.357295i	−0.950541	−	0.310599i	$-0.899470\pi$
$47$	−1.41421	+	2.44949i	−0.206284	+	0.357295i	0.744257	+	0.667893i	$0.232804\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−1.12132	+	1.94218i	−0.157016	+	0.271960i
$52$		0			0
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	0.926497	−	0.376303i	$-0.122805\pi$
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	−0.789136	+	0.614218i	$0.789471\pi$
$54$		0			0
$55$	6.82843			0.920745
$56$		0			0
$57$	−2.82843			−0.374634
$58$		0			0
$59$	−0.585786	−	1.01461i	−0.0762629	−	0.132091i	0.825372	−	0.564589i	$-0.190965\pi$
$59$	−0.585786	−	1.01461i	−0.0762629	−	0.132091i	−0.901635	+	0.432498i	$0.857632\pi$
$60$		0			0
$61$	−6.12132	+	10.6024i	−0.783755	+	1.35750i	0.145985	+	0.989287i	$0.453365\pi$
$61$	−6.12132	+	10.6024i	−0.783755	+	1.35750i	−0.929740	+	0.368216i	$0.879969\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.41421	+	7.64564i	−0.547516	+	0.948325i
$66$		0			0
$67$	−2.82843	−	4.89898i	−0.345547	−	0.598506i	0.639906	−	0.768453i	$-0.278973\pi$
$67$	−2.82843	−	4.89898i	−0.345547	−	0.598506i	−0.985453	+	0.169948i	$0.945640\pi$
$68$		0			0
$69$	−7.65685			−0.921777
$70$		0			0
$71$	−9.31371			−1.10533			−0.552667	−	0.833402i	$-0.686390\pi$
$71$	−9.31371			−1.10533			−0.552667	+	0.833402i	$0.686390\pi$
$72$		0			0
$73$	−6.94975	−	12.0373i	−0.813406	−	1.40886i	−0.910467	−	0.413583i	$-0.864277\pi$
$73$	−6.94975	−	12.0373i	−0.813406	−	1.40886i	0.0970601	−	0.995279i	$-0.469056\pi$
$74$		0			0
$75$	3.32843	−	5.76500i	0.384334	−	0.665685i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.82843	−	11.8272i	0.768258	−	1.33066i	−0.170249	−	0.985401i	$-0.554457\pi$
$79$	6.82843	−	11.8272i	0.768258	−	1.33066i	0.938507	−	0.345261i	$-0.112210\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−7.31371			−0.802784			−0.401392	−	0.915906i	$-0.631473\pi$
$83$	−7.31371			−0.802784			−0.401392	+	0.915906i	$0.631473\pi$
$84$		0			0
$85$	−7.65685			−0.830502
$86$		0			0
$87$	−3.41421	−	5.91359i	−0.366042	−	0.634004i
$88$		0			0
$89$	7.12132	−	12.3345i	0.754858	−	1.30745i	−0.190586	−	0.981670i	$-0.561039\pi$
$89$	7.12132	−	12.3345i	0.754858	−	1.30745i	0.945445	−	0.325783i	$-0.105628\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.585786	−	1.01461i	0.0607432	−	0.105210i
$94$		0			0
$95$	−4.82843	−	8.36308i	−0.495386	−	0.858034i
$96$		0			0
$97$	2.58579			0.262547			0.131273	−	0.991346i	$-0.458093\pi$
$97$	2.58579			0.262547			0.131273	+	0.991346i	$0.458093\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−1.46447	−	2.53653i	−0.145720	−	0.252394i	0.783921	−	0.620860i	$-0.213216\pi$
$101$	−1.46447	−	2.53653i	−0.145720	−	0.252394i	−0.929641	+	0.368466i	$0.879883\pi$
$102$		0			0
$103$	2.24264	−	3.88437i	0.220974	−	0.382738i	−0.734130	−	0.679009i	$-0.762410\pi$
$103$	2.24264	−	3.88437i	0.220974	−	0.382738i	0.955104	+	0.296271i	$0.0957431\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.171573	+	0.297173i	−0.0165866	+	0.0287288i	−0.874200	−	0.485567i	$-0.838613\pi$
$107$	−0.171573	+	0.297173i	−0.0165866	+	0.0287288i	0.857613	+	0.514296i	$0.171947\pi$
$108$		0			0
$109$	2.82843	+	4.89898i	0.270914	+	0.469237i	0.969096	−	0.246683i	$-0.0793407\pi$
$109$	2.82843	+	4.89898i	0.270914	+	0.469237i	−0.698182	+	0.715920i	$0.746007\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$	−5.31371			−0.499872			−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371			−0.499872			−0.249936	+	0.968262i	$0.580410\pi$
$114$		0			0
$115$	−13.0711	−	22.6398i	−1.21888	−	2.11117i
$116$		0			0
$117$	−1.29289	+	2.23936i	−0.119528	+	0.207029i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	3.12132	+	5.40629i	0.281440	+	0.487468i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	1.65685			0.147022			0.0735110	−	0.997294i	$-0.476580\pi$
$127$	1.65685			0.147022			0.0735110	+	0.997294i	$0.476580\pi$
$128$		0			0
$129$	−2.82843	−	4.89898i	−0.249029	−	0.431331i
$130$		0			0
$131$	−7.65685	+	13.2621i	−0.668982	+	1.15871i	0.309207	+	0.950995i	$0.399937\pi$
$131$	−7.65685	+	13.2621i	−0.668982	+	1.15871i	−0.978189	+	0.207717i	$0.933397\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.70711	−	2.95680i	0.146924	−	0.254480i
$136$		0			0
$137$	−7.07107	−	12.2474i	−0.604122	−	1.04637i	−0.992190	−	0.124739i	$-0.960191\pi$
$137$	−7.07107	−	12.2474i	−0.604122	−	1.04637i	0.388067	−	0.921631i	$-0.373143\pi$
$138$		0			0
$139$	17.6569			1.49763			0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569			1.49763			0.748817	+	0.662776i	$0.230622\pi$
$140$		0			0
$141$	−2.82843			−0.238197
$142$		0			0
$143$	−2.58579	−	4.47871i	−0.216234	−	0.374529i
$144$		0			0
$145$	11.6569	−	20.1903i	0.968049	−	1.67671i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.65685	+	14.9941i	−0.709197	+	1.22837i	0.255958	+	0.966688i	$0.417609\pi$
$149$	−8.65685	+	14.9941i	−0.709197	+	1.22837i	−0.965155	+	0.261678i	$0.915724\pi$
$150$		0			0
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	0.999909	−	0.0134886i	$-0.00429367\pi$
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	−0.511636	+	0.859202i	$0.670960\pi$
$152$		0			0
$153$	−2.24264			−0.181307
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−5.87868	−	10.1822i	−0.469170	−	0.812626i	0.530209	−	0.847867i	$-0.322113\pi$
$157$	−5.87868	−	10.1822i	−0.469170	−	0.812626i	−0.999379	+	0.0352411i	$0.988780\pi$
$158$		0			0
$159$	−1.00000	+	1.73205i	−0.0793052	+	0.137361i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.65685	+	9.79796i	−0.443079	+	0.767435i	−0.997916	−	0.0645236i	$-0.979447\pi$
$163$	−5.65685	+	9.79796i	−0.443079	+	0.767435i	0.554837	+	0.831959i	$0.312781\pi$
$164$		0			0
$165$	3.41421	+	5.91359i	0.265796	+	0.460372i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	−6.31371			−0.485670
$170$		0			0
$171$	−1.41421	−	2.44949i	−0.108148	−	0.187317i
$172$		0			0
$173$	−10.5355	+	18.2481i	−0.801002	+	1.38738i	0.117956	+	0.993019i	$0.462366\pi$
$173$	−10.5355	+	18.2481i	−0.801002	+	1.38738i	−0.918957	+	0.394357i	$0.870967\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.585786	−	1.01461i	0.0440304	−	0.0762629i
$178$		0			0
$179$	−9.82843	−	17.0233i	−0.734611	−	1.27238i	−0.954894	−	0.296948i	$-0.904031\pi$
$179$	−9.82843	−	17.0233i	−0.734611	−	1.27238i	0.220283	−	0.975436i	$-0.429302\pi$
$180$		0			0
$181$	−2.58579			−0.192200			−0.0961000	−	0.995372i	$-0.530637\pi$
$181$	−2.58579			−0.192200			−0.0961000	+	0.995372i	$0.530637\pi$
$182$		0			0
$183$	−12.2426			−0.905002
$184$		0			0
$185$	6.82843	+	11.8272i	0.502036	+	0.869552i
$186$		0			0
$187$	2.24264	−	3.88437i	0.163998	−	0.284053i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	0.331611	+	0.943416i	$0.392408\pi$
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	−0.982828	+	0.184525i	$0.940925\pi$
$192$		0			0
$193$	−2.65685	−	4.60181i	−0.191245	−	0.331245i	0.754418	−	0.656394i	$-0.227919\pi$
$193$	−2.65685	−	4.60181i	−0.191245	−	0.331245i	−0.945663	+	0.325149i	$0.894586\pi$
$194$		0			0
$195$	−8.82843			−0.632217
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	10.8284	+	18.7554i	0.767607	+	1.32953i	0.938857	+	0.344307i	$0.111886\pi$
$199$	10.8284	+	18.7554i	0.767607	+	1.32953i	−0.171250	+	0.985228i	$0.554781\pi$
$200$		0			0
$201$	2.82843	−	4.89898i	0.199502	−	0.345547i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−10.6569	+	18.4582i	−0.744307	+	1.28918i
$206$		0			0
$207$	−3.82843	−	6.63103i	−0.266094	−	0.460888i
$208$		0			0
$209$	5.65685			0.391293
$210$		0			0
$211$	−12.9706			−0.892930			−0.446465	−	0.894801i	$-0.647317\pi$
$211$	−12.9706			−0.892930			−0.446465	+	0.894801i	$0.647317\pi$
$212$		0			0
$213$	−4.65685	−	8.06591i	−0.319082	−	0.552667i
$214$		0			0
$215$	9.65685	−	16.7262i	0.658592	−	1.14071i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.94975	−	12.0373i	0.469620	−	0.813406i
$220$		0			0
$221$	2.89949	+	5.02207i	0.195041	+	0.337821i
$222$		0			0
$223$	−24.9706			−1.67215			−0.836076	−	0.548613i	$-0.815156\pi$
$223$	−24.9706			−1.67215			−0.836076	+	0.548613i	$0.815156\pi$
$224$		0			0
$225$	6.65685			0.443790
$226$		0			0
$227$	11.8995	+	20.6105i	0.789797	+	1.36797i	0.926091	+	0.377300i	$0.123148\pi$
$227$	11.8995	+	20.6105i	0.789797	+	1.36797i	−0.136294	+	0.990668i	$0.543519\pi$
$228$		0			0
$229$	0.121320	−	0.210133i	0.00801707	−	0.0138860i	−0.861989	−	0.506927i	$-0.830781\pi$
$229$	0.121320	−	0.210133i	0.00801707	−	0.0138860i	0.870006	+	0.493041i	$0.164115\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.07107	−	5.31925i	0.201192	−	0.348475i	−0.747721	−	0.664014i	$-0.768852\pi$
$233$	3.07107	−	5.31925i	0.201192	−	0.348475i	0.948913	+	0.315538i	$0.102185\pi$
$234$		0			0
$235$	−4.82843	−	8.36308i	−0.314972	−	0.545547i
$236$		0			0
$237$	13.6569			0.887108
$238$		0			0
$239$	15.6569			1.01276			0.506379	−	0.862311i	$-0.330984\pi$
$239$	15.6569			1.01276			0.506379	+	0.862311i	$0.330984\pi$
$240$		0			0
$241$	8.12132	+	14.0665i	0.523140	+	0.906105i	0.999637	+	0.0269294i	$0.00857294\pi$
$241$	8.12132	+	14.0665i	0.523140	+	0.906105i	−0.476497	+	0.879176i	$0.658094\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.65685	+	6.33386i	−0.232680	+	0.403014i
$248$		0			0
$249$	−3.65685	−	6.33386i	−0.231744	−	0.401392i
$250$		0			0
$251$	12.4853			0.788064			0.394032	−	0.919097i	$-0.371080\pi$
$251$	12.4853			0.788064			0.394032	+	0.919097i	$0.371080\pi$
$252$		0			0
$253$	15.3137			0.962765
$254$		0			0
$255$	−3.82843	−	6.63103i	−0.239745	−	0.415251i
$256$		0			0
$257$	−11.6066	+	20.1032i	−0.724000	+	1.25400i	0.235384	+	0.971902i	$0.424365\pi$
$257$	−11.6066	+	20.1032i	−0.724000	+	1.25400i	−0.959384	+	0.282102i	$0.908968\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.41421	−	5.91359i	0.211335	−	0.366042i
$262$		0			0
$263$	2.65685	+	4.60181i	0.163829	+	0.283760i	0.936239	−	0.351365i	$-0.114282\pi$
$263$	2.65685	+	4.60181i	0.163829	+	0.283760i	−0.772410	+	0.635124i	$0.780949\pi$
$264$		0			0
$265$	−6.82843			−0.419467
$266$		0			0
$267$	14.2426			0.871635
$268$		0			0
$269$	7.36396	+	12.7548i	0.448989	+	0.777671i	0.998320	−	0.0579332i	$-0.0184510\pi$
$269$	7.36396	+	12.7548i	0.448989	+	0.777671i	−0.549332	+	0.835604i	$0.685118\pi$
$270$		0			0
$271$	−5.07107	+	8.78335i	−0.308045	+	0.533550i	−0.977935	−	0.208911i	$-0.933008\pi$
$271$	−5.07107	+	8.78335i	−0.308045	+	0.533550i	0.669889	+	0.742461i	$0.266342\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.65685	+	11.5300i	−0.401423	+	0.695286i
$276$		0			0
$277$	4.65685	+	8.06591i	0.279803	+	0.484633i	0.971336	−	0.237712i	$-0.0763974\pi$
$277$	4.65685	+	8.06591i	0.279803	+	0.484633i	−0.691532	+	0.722345i	$0.743064\pi$
$278$		0			0
$279$	1.17157			0.0701402
$280$		0			0
$281$	0.485281			0.0289495			0.0144747	−	0.999895i	$-0.495392\pi$
$281$	0.485281			0.0289495			0.0144747	+	0.999895i	$0.495392\pi$
$282$		0			0
$283$	−4.24264	−	7.34847i	−0.252199	−	0.436821i	0.711932	−	0.702248i	$-0.247820\pi$
$283$	−4.24264	−	7.34847i	−0.252199	−	0.436821i	−0.964131	+	0.265427i	$0.914487\pi$
$284$		0			0
$285$	4.82843	−	8.36308i	0.286011	−	0.495386i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.98528	−	10.3668i	0.352075	−	0.609812i
$290$		0			0
$291$	1.29289	+	2.23936i	0.0757907	+	0.131273i
$292$		0			0
$293$	16.5858			0.968952			0.484476	−	0.874805i	$-0.339010\pi$
$293$	16.5858			0.968952			0.484476	+	0.874805i	$0.339010\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	1.00000	+	1.73205i	0.0580259	+	0.100504i
$298$		0			0
$299$	−9.89949	+	17.1464i	−0.572503	+	0.991604i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.46447	−	2.53653i	0.0841314	−	0.145720i
$304$		0			0
$305$	−20.8995	−	36.1990i	−1.19670	−	2.07275i
$306$		0			0
$307$	−30.1421			−1.72030			−0.860151	−	0.510039i	$-0.829631\pi$
$307$	−30.1421			−1.72030			−0.860151	+	0.510039i	$0.829631\pi$
$308$		0			0
$309$	4.48528			0.255159
$310$		0			0
$311$	3.07107	+	5.31925i	0.174144	+	0.301627i	0.939865	−	0.341547i	$-0.110951\pi$
$311$	3.07107	+	5.31925i	0.174144	+	0.301627i	−0.765721	+	0.643173i	$0.777617\pi$
$312$		0			0
$313$	−0.949747	+	1.64501i	−0.0536829	+	0.0929815i	−0.891618	−	0.452788i	$-0.850429\pi$
$313$	−0.949747	+	1.64501i	−0.0536829	+	0.0929815i	0.837935	+	0.545770i	$0.183763\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	−0.971589	−	0.236675i	$-0.923942\pi$
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	0.690761	+	0.723083i	$0.257276\pi$
$318$		0			0
$319$	6.82843	+	11.8272i	0.382319	+	0.662195i
$320$		0			0
$321$	−0.343146			−0.0191525
$322$		0			0
$323$	−6.34315			−0.352942
$324$		0			0
$325$	−8.60660	−	14.9071i	−0.477408	−	0.826896i
$326$		0			0
$327$	−2.82843	+	4.89898i	−0.156412	+	0.270914i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$	2.00000	+	3.46410i	0.109599	+	0.189832i
$334$		0			0
$335$	19.3137			1.05522
$336$		0			0
$337$	−29.6569			−1.61551			−0.807756	−	0.589517i	$-0.799318\pi$
$337$	−29.6569			−1.61551			−0.807756	+	0.589517i	$0.799318\pi$
$338$		0			0
$339$	−2.65685	−	4.60181i	−0.144301	−	0.249936i
$340$		0			0
$341$	−1.17157	+	2.02922i	−0.0634442	+	0.109889i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	13.0711	−	22.6398i	0.703723	−	1.21888i
$346$		0			0
$347$	16.6569	+	28.8505i	0.894187	+	1.54878i	0.834808	+	0.550541i	$0.185579\pi$
$347$	16.6569	+	28.8505i	0.894187	+	1.54878i	0.0593789	+	0.998236i	$0.481088\pi$
$348$		0			0
$349$	−9.89949			−0.529908			−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949			−0.529908			−0.264954	+	0.964261i	$0.585357\pi$
$350$		0			0
$351$	−2.58579			−0.138019
$352$		0			0
$353$	7.36396	+	12.7548i	0.391944	+	0.678867i	0.992706	−	0.120561i	$-0.0384693\pi$
$353$	7.36396	+	12.7548i	0.391944	+	0.678867i	−0.600762	+	0.799428i	$0.705136\pi$
$354$		0			0
$355$	15.8995	−	27.5387i	0.843858	−	1.46160i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.171573	+	0.297173i	−0.00905527	+	0.0156842i	−0.870518	−	0.492137i	$-0.836216\pi$
$359$	−0.171573	+	0.297173i	−0.00905527	+	0.0156842i	0.861462	+	0.507822i	$0.169549\pi$
$360$		0			0
$361$	5.50000	+	9.52628i	0.289474	+	0.501383i
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$	47.4558			2.48395
$366$		0			0
$367$	−1.65685	−	2.86976i	−0.0864871	−	0.149800i	0.819537	−	0.573027i	$-0.194231\pi$
$367$	−1.65685	−	2.86976i	−0.0864871	−	0.149800i	−0.906024	+	0.423226i	$0.860897\pi$
$368$		0			0
$369$	−3.12132	+	5.40629i	−0.162489	+	0.281440i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.34315	−	9.25460i	0.276658	−	0.479185i	−0.693894	−	0.720077i	$-0.744107\pi$
$373$	5.34315	−	9.25460i	0.276658	−	0.479185i	0.970552	+	0.240892i	$0.0774399\pi$
$374$		0			0
$375$	2.82843	+	4.89898i	0.146059	+	0.252982i
$376$		0			0
$377$	−17.6569			−0.909374
$378$		0			0
$379$	−8.68629			−0.446185			−0.223092	−	0.974797i	$-0.571615\pi$
$379$	−8.68629			−0.446185			−0.223092	+	0.974797i	$0.571615\pi$
$380$		0			0
$381$	0.828427	+	1.43488i	0.0424416	+	0.0735110i
$382$		0			0
$383$	−9.17157	+	15.8856i	−0.468645	+	0.811718i	−0.999358	−	0.0358343i	$-0.988591\pi$
$383$	−9.17157	+	15.8856i	−0.468645	+	0.811718i	0.530712	+	0.847552i	$0.321924\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.82843	−	4.89898i	0.143777	−	0.249029i
$388$		0			0
$389$	9.07107	+	15.7116i	0.459921	+	0.796607i	0.998956	−	0.0456762i	$-0.0145442\pi$
$389$	9.07107	+	15.7116i	0.459921	+	0.796607i	−0.539035	+	0.842283i	$0.681211\pi$
$390$		0			0
$391$	−17.1716			−0.868404
$392$		0			0
$393$	−15.3137			−0.772474
$394$		0			0
$395$	23.3137	+	40.3805i	1.17304	+	2.03176i
$396$		0			0
$397$	1.19239	−	2.06528i	0.0598442	−	0.103653i	−0.834551	−	0.550931i	$-0.814273\pi$
$397$	1.19239	−	2.06528i	0.0598442	−	0.103653i	0.894395	+	0.447277i	$0.147606\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.07107	−	5.31925i	0.153362	−	0.265630i	−0.779100	−	0.626900i	$-0.784323\pi$
$401$	3.07107	−	5.31925i	0.153362	−	0.265630i	0.932461	+	0.361270i	$0.117657\pi$
$402$		0			0
$403$	−1.51472	−	2.62357i	−0.0754535	−	0.130689i
$404$		0			0
$405$	3.41421			0.169654
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	10.7071	+	18.5453i	0.529432	+	0.917004i	0.999411	+	0.0343258i	$0.0109284\pi$
$409$	10.7071	+	18.5453i	0.529432	+	0.917004i	−0.469978	+	0.882678i	$0.655738\pi$
$410$		0			0
$411$	7.07107	−	12.2474i	0.348790	−	0.604122i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	12.4853	−	21.6251i	0.612878	−	1.06154i
$416$		0			0
$417$	8.82843	+	15.2913i	0.432330	+	0.748817i
$418$		0			0
$419$	−33.1716			−1.62054			−0.810269	−	0.586059i	$-0.800679\pi$
$419$	−33.1716			−1.62054			−0.810269	+	0.586059i	$0.800679\pi$
$420$		0			0
$421$	16.6274			0.810371			0.405185	−	0.914235i	$-0.367207\pi$
$421$	16.6274			0.810371			0.405185	+	0.914235i	$0.367207\pi$
$422$		0			0
$423$	−1.41421	−	2.44949i	−0.0687614	−	0.119098i
$424$		0			0
$425$	7.46447	−	12.9288i	0.362080	−	0.627141i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.58579	−	4.47871i	0.124843	−	0.216234i
$430$		0			0
$431$	−13.4853	−	23.3572i	−0.649563	−	1.12508i	−0.983227	−	0.182384i	$-0.941618\pi$
$431$	−13.4853	−	23.3572i	−0.649563	−	1.12508i	0.333664	−	0.942692i	$-0.391715\pi$
$432$		0			0
$433$	−20.2426			−0.972799			−0.486400	−	0.873736i	$-0.661690\pi$
$433$	−20.2426			−0.972799			−0.486400	+	0.873736i	$0.661690\pi$
$434$		0			0
$435$	23.3137			1.11781
$436$		0			0
$437$	−10.8284	−	18.7554i	−0.517994	−	0.897192i
$438$		0			0
$439$	6.34315	−	10.9867i	0.302742	−	0.524364i	−0.674014	−	0.738718i	$-0.735431\pi$
$439$	6.34315	−	10.9867i	0.302742	−	0.524364i	0.976756	+	0.214354i	$0.0687647\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.4853	−	30.2854i	0.830751	−	1.43890i	−0.0666929	−	0.997774i	$-0.521245\pi$
$443$	17.4853	−	30.2854i	0.830751	−	1.43890i	0.897444	−	0.441129i	$-0.145422\pi$
$444$		0			0
$445$	24.3137	+	42.1126i	1.15258	+	1.99633i
$446$		0			0
$447$	−17.3137			−0.818910
$448$		0			0
$449$	−5.31371			−0.250769			−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371			−0.250769			−0.125385	+	0.992108i	$0.540017\pi$
$450$		0			0
$451$	−6.24264	−	10.8126i	−0.293954	−	0.509144i
$452$		0			0
$453$	−6.00000	+	10.3923i	−0.281905	+	0.488273i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	−0.575036	−	0.818128i	$-0.695012\pi$
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	0.996038	+	0.0889312i	$0.0283451\pi$
$458$		0			0
$459$	−1.12132	−	1.94218i	−0.0523388	−	0.0906534i
$460$		0			0
$461$	−16.5858			−0.772477			−0.386239	−	0.922399i	$-0.626226\pi$
$461$	−16.5858			−0.772477			−0.386239	+	0.922399i	$0.626226\pi$
$462$		0			0
$463$	26.6274			1.23748			0.618741	−	0.785595i	$-0.287643\pi$
$463$	26.6274			1.23748			0.618741	+	0.785595i	$0.287643\pi$
$464$		0			0
$465$	2.00000	+	3.46410i	0.0927478	+	0.160644i
$466$		0			0
$467$	0.100505	−	0.174080i	0.00465082	−	0.00805546i	−0.863691	−	0.504022i	$-0.831853\pi$
$467$	0.100505	−	0.174080i	0.00465082	−	0.00805546i	0.868341	+	0.495967i	$0.165186\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	5.87868	−	10.1822i	0.270875	−	0.469170i
$472$		0			0
$473$	5.65685	+	9.79796i	0.260102	+	0.450511i
$474$		0			0
$475$	18.8284			0.863907
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	0.928932	+	1.60896i	0.0424440	+	0.0735152i	0.886467	−	0.462792i	$-0.153152\pi$
$479$	0.928932	+	1.60896i	0.0424440	+	0.0735152i	−0.844023	+	0.536307i	$0.819819\pi$
$480$		0			0
$481$	5.17157	−	8.95743i	0.235803	−	0.408424i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.41421	+	7.64564i	−0.200439	+	0.347171i
$486$		0			0
$487$	13.3137	+	23.0600i	0.603302	+	1.04495i	0.992317	+	0.123718i	$0.0394819\pi$
$487$	13.3137	+	23.0600i	0.603302	+	1.04495i	−0.389016	+	0.921231i	$0.627185\pi$
$488$		0			0
$489$	−11.3137			−0.511624
$490$		0			0
$491$	−5.02944			−0.226975			−0.113488	−	0.993539i	$-0.536202\pi$
$491$	−5.02944			−0.226975			−0.113488	+	0.993539i	$0.536202\pi$
$492$		0			0
$493$	−7.65685	−	13.2621i	−0.344847	−	0.597293i
$494$		0			0
$495$	−3.41421	+	5.91359i	−0.153457	+	0.265796i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.65685	−	2.86976i	0.0741710	−	0.128468i	−0.826554	−	0.562857i	$-0.809702\pi$
$499$	1.65685	−	2.86976i	0.0741710	−	0.128468i	0.900725	+	0.434389i	$0.143036\pi$
$500$		0			0
$501$	9.89949	+	17.1464i	0.442277	+	0.766046i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$	−3.15685	−	5.46783i	−0.140201	−	0.242835i
$508$		0			0
$509$	−2.77817	+	4.81194i	−0.123140	+	0.213285i	−0.921005	−	0.389552i	$-0.872630\pi$
$509$	−2.77817	+	4.81194i	−0.123140	+	0.213285i	0.797864	+	0.602837i	$0.205963\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.41421	−	2.44949i	0.0624391	−	0.108148i
$514$		0			0
$515$	7.65685	+	13.2621i	0.337401	+	0.584396i
$516$		0			0
$517$	5.65685			0.248788
$518$		0			0
$519$	−21.0711			−0.924917
$520$		0			0
$521$	17.7071	+	30.6696i	0.775762	+	1.34366i	0.934365	+	0.356317i	$0.115968\pi$
$521$	17.7071	+	30.6696i	0.775762	+	1.34366i	−0.158603	+	0.987343i	$0.550699\pi$
$522$		0			0
$523$	−12.8284	+	22.2195i	−0.560948	+	0.971590i	0.436466	+	0.899721i	$0.356230\pi$
$523$	−12.8284	+	22.2195i	−0.560948	+	0.971590i	−0.997414	+	0.0718696i	$0.977103\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.31371	−	2.27541i	0.0572260	−	0.0991184i
$528$		0			0
$529$	−17.8137	−	30.8542i	−0.774509	−	1.34149i
$530$		0			0
$531$	1.17157			0.0508419
$532$		0			0
$533$	16.1421			0.699194
$534$		0			0
$535$	−0.585786	−	1.01461i	−0.0253258	−	0.0438655i
$536$		0			0
$537$	9.82843	−	17.0233i	0.424128	−	0.734611i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.65685	+	14.9941i	−0.372187	+	0.644647i	−0.989902	−	0.141755i	$-0.954725\pi$
$541$	−8.65685	+	14.9941i	−0.372187	+	0.644647i	0.617715	+	0.786402i	$0.288059\pi$
$542$		0			0
$543$	−1.29289	−	2.23936i	−0.0554834	−	0.0961000i
$544$		0			0
$545$	−19.3137			−0.827308
$546$		0			0
$547$	36.9706			1.58075			0.790374	−	0.612625i	$-0.209886\pi$
$547$	36.9706			1.58075			0.790374	+	0.612625i	$0.209886\pi$
$548$		0			0
$549$	−6.12132	−	10.6024i	−0.261252	−	0.452501i
$550$		0			0
$551$	9.65685	−	16.7262i	0.411396	−	0.712558i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−6.82843	+	11.8272i	−0.289851	+	0.502036i
$556$		0			0
$557$	−13.0000	−	22.5167i	−0.550828	−	0.954062i	−0.998215	−	0.0597213i	$-0.980979\pi$
$557$	−13.0000	−	22.5167i	−0.550828	−	0.954062i	0.447387	−	0.894340i	$-0.352355\pi$
$558$		0			0
$559$	−14.6274			−0.618674
$560$		0			0
$561$	4.48528			0.189369
$562$		0			0
$563$	−0.585786	−	1.01461i	−0.0246880	−	0.0427608i	0.853417	−	0.521228i	$-0.174526\pi$
$563$	−0.585786	−	1.01461i	−0.0246880	−	0.0427608i	−0.878105	+	0.478467i	$0.841193\pi$
$564$		0			0
$565$	9.07107	−	15.7116i	0.381623	−	0.660990i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.24264	−	14.2767i	0.345549	−	0.598509i	−0.639904	−	0.768455i	$-0.721026\pi$
$569$	8.24264	−	14.2767i	0.345549	−	0.598509i	0.985453	+	0.169946i	$0.0543592\pi$
$570$		0			0
$571$	11.1716	+	19.3497i	0.467516	+	0.809761i	0.999311	−	0.0371118i	$-0.0118158\pi$
$571$	11.1716	+	19.3497i	0.467516	+	0.809761i	−0.531795	+	0.846873i	$0.678482\pi$
$572$		0			0
$573$	−18.0000			−0.751961
$574$		0			0
$575$	50.9706			2.12562
$576$		0			0
$577$	16.9497	+	29.3578i	0.705627	+	1.22218i	0.966465	+	0.256799i	$0.0826680\pi$
$577$	16.9497	+	29.3578i	0.705627	+	1.22218i	−0.260837	+	0.965383i	$0.583999\pi$
$578$		0			0
$579$	2.65685	−	4.60181i	0.110415	−	0.191245i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.00000	−	3.46410i	0.0828315	−	0.143468i
$584$		0			0
$585$	−4.41421	−	7.64564i	−0.182505	−	0.316108i
$586$		0			0
$587$	22.8284			0.942230			0.471115	−	0.882072i	$-0.343852\pi$
$587$	22.8284			0.942230			0.471115	+	0.882072i	$0.343852\pi$
$588$		0			0
$589$	3.31371			0.136539
$590$		0			0
$591$	1.00000	+	1.73205i	0.0411345	+	0.0712470i
$592$		0			0
$593$	3.46447	−	6.00063i	0.142269	−	0.246416i	−0.786082	−	0.618122i	$-0.787894\pi$
$593$	3.46447	−	6.00063i	0.142269	−	0.246416i	0.928351	+	0.371706i	$0.121227\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.8284	+	18.7554i	−0.443178	+	0.767607i
$598$		0			0
$599$	−1.00000	−	1.73205i	−0.0408589	−	0.0707697i	0.844873	−	0.534967i	$-0.179676\pi$
$599$	−1.00000	−	1.73205i	−0.0408589	−	0.0707697i	−0.885732	+	0.464198i	$0.846343\pi$
$600$		0			0
$601$	−15.0711			−0.614762			−0.307381	−	0.951587i	$-0.599453\pi$
$601$	−15.0711			−0.614762			−0.307381	+	0.951587i	$0.599453\pi$
$602$		0			0
$603$	5.65685			0.230365
$604$		0			0
$605$	11.9497	+	20.6976i	0.485826	+	0.841476i
$606$		0			0
$607$	−9.17157	+	15.8856i	−0.372263	+	0.644778i	−0.989913	−	0.141675i	$-0.954751\pi$
$607$	−9.17157	+	15.8856i	−0.372263	+	0.644778i	0.617651	+	0.786453i	$0.288085\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.65685	+	6.33386i	−0.147940	+	0.256240i
$612$		0			0
$613$	−2.34315	−	4.05845i	−0.0946388	−	0.163919i	0.814819	−	0.579715i	$-0.196836\pi$
$613$	−2.34315	−	4.05845i	−0.0946388	−	0.163919i	−0.909458	+	0.415796i	$0.863503\pi$
$614$		0			0
$615$	−21.3137			−0.859452
$616$		0			0
$617$	−24.4853			−0.985740			−0.492870	−	0.870103i	$-0.664052\pi$
$617$	−24.4853			−0.985740			−0.492870	+	0.870103i	$0.664052\pi$
$618$		0			0
$619$	14.4853	+	25.0892i	0.582213	+	1.00842i	0.995217	+	0.0976926i	$0.0311462\pi$
$619$	14.4853	+	25.0892i	0.582213	+	1.00842i	−0.413004	+	0.910729i	$0.635520\pi$
$620$		0			0
$621$	3.82843	−	6.63103i	0.153629	−	0.266094i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.98528	−	12.0989i	0.279411	−	0.483954i
$626$		0			0
$627$	2.82843	+	4.89898i	0.112956	+	0.195646i
$628$		0			0
$629$	8.97056			0.357680
$630$		0			0
$631$	−23.3137			−0.928104			−0.464052	−	0.885808i	$-0.653605\pi$
$631$	−23.3137			−0.928104			−0.464052	+	0.885808i	$0.653605\pi$
$632$		0			0
$633$	−6.48528	−	11.2328i	−0.257767	−	0.446465i
$634$		0			0
$635$	−2.82843	+	4.89898i	−0.112243	+	0.194410i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	4.65685	−	8.06591i	0.184222	−	0.319082i
$640$		0			0
$641$	5.41421	+	9.37769i	0.213849	+	0.370397i	0.952916	−	0.303235i	$-0.0980668\pi$
$641$	5.41421	+	9.37769i	0.213849	+	0.370397i	−0.739067	+	0.673632i	$0.764733\pi$
$642$		0			0
$643$	34.4264			1.35764			0.678822	−	0.734302i	$-0.262491\pi$
$643$	34.4264			1.35764			0.678822	+	0.734302i	$0.262491\pi$
$644$		0			0
$645$	19.3137			0.760477
$646$		0			0
$647$	13.4142	+	23.2341i	0.527367	+	0.913427i	0.999491	+	0.0318946i	$0.0101541\pi$
$647$	13.4142	+	23.2341i	0.527367	+	0.913427i	−0.472124	+	0.881532i	$0.656513\pi$
$648$		0			0
$649$	−1.17157	+	2.02922i	−0.0459883	+	0.0796540i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.2426	+	31.5972i	−0.713890	+	1.23649i	0.249497	+	0.968376i	$0.419735\pi$
$653$	−18.2426	+	31.5972i	−0.713890	+	1.23649i	−0.963386	+	0.268118i	$0.913598\pi$
$654$		0			0
$655$	−26.1421	−	45.2795i	−1.02146	−	1.76922i
$656$		0			0
$657$	13.8995			0.542271
$658$		0			0
$659$	−9.31371			−0.362811			−0.181405	−	0.983408i	$-0.558065\pi$
$659$	−9.31371			−0.362811			−0.181405	+	0.983408i	$0.558065\pi$
$660$		0			0
$661$	11.7782	+	20.4004i	0.458118	+	0.793483i	0.998862	−	0.0477040i	$-0.0151904\pi$
$661$	11.7782	+	20.4004i	0.458118	+	0.793483i	−0.540744	+	0.841187i	$0.681857\pi$
$662$		0			0
$663$	−2.89949	+	5.02207i	−0.112607	+	0.195041i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	26.1421	−	45.2795i	1.01223	−	1.75323i
$668$		0			0
$669$	−12.4853	−	21.6251i	−0.482709	−	0.836076i
$670$		0			0
$671$	24.4853			0.945244
$672$		0			0
$673$	23.3137			0.898677			0.449339	−	0.893361i	$-0.351660\pi$
$673$	23.3137			0.898677			0.449339	+	0.893361i	$0.351660\pi$
$674$		0			0
$675$	3.32843	+	5.76500i	0.128111	+	0.221895i
$676$		0			0
$677$	−15.7071	+	27.2055i	−0.603673	+	1.04559i	0.388587	+	0.921412i	$0.372963\pi$
$677$	−15.7071	+	27.2055i	−0.603673	+	1.04559i	−0.992260	+	0.124180i	$0.960370\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−11.8995	+	20.6105i	−0.455990	+	0.789797i
$682$		0			0
$683$	−9.82843	−	17.0233i	−0.376074	−	0.651380i	0.614413	−	0.788985i	$-0.289393\pi$
$683$	−9.82843	−	17.0233i	−0.376074	−	0.651380i	−0.990487	+	0.137605i	$0.956060\pi$
$684$		0			0
$685$	48.2843			1.84485
$686$		0			0
$687$	0.242641			0.00925732
$688$		0			0
$689$	2.58579	+	4.47871i	0.0985106	+	0.170625i
$690$		0			0
$691$	−0.343146	+	0.594346i	−0.0130539	+	0.0226100i	−0.872479	−	0.488652i	$-0.837489\pi$
$691$	−0.343146	+	0.594346i	−0.0130539	+	0.0226100i	0.859425	+	0.511262i	$0.170822\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−30.1421	+	52.2077i	−1.14336	+	1.98035i
$696$		0			0
$697$	7.00000	+	12.1244i	0.265144	+	0.459243i
$698$		0			0
$699$	6.14214			0.232317
$700$		0			0
$701$	−17.1716			−0.648561			−0.324281	−	0.945961i	$-0.605122\pi$
$701$	−17.1716			−0.648561			−0.324281	+	0.945961i	$0.605122\pi$
$702$		0			0
$703$	5.65685	+	9.79796i	0.213352	+	0.369537i
$704$		0			0
$705$	4.82843	−	8.36308i	0.181849	−	0.314972i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	18.1421	−	31.4231i	0.681342	−	1.18012i	−0.293229	−	0.956042i	$-0.594730\pi$
$709$	18.1421	−	31.4231i	0.681342	−	1.18012i	0.974571	−	0.224077i	$-0.0719368\pi$
$710$		0			0
$711$	6.82843	+	11.8272i	0.256086	+	0.443554i
$712$		0			0
$713$	8.97056			0.335950
$714$		0			0
$715$	17.6569			0.660329
$716$		0			0
$717$	7.82843	+	13.5592i	0.292358	+	0.506379i
$718$		0			0
$719$	20.9706	−	36.3221i	0.782070	−	1.35459i	−0.148664	−	0.988888i	$-0.547497\pi$
$719$	20.9706	−	36.3221i	0.782070	−	1.35459i	0.930734	−	0.365697i	$-0.119169\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−8.12132	+	14.0665i	−0.302035	+	0.523140i
$724$		0			0
$725$	22.7279	+	39.3659i	0.844094	+	1.46201i
$726$		0			0
$727$	−12.4853			−0.463053			−0.231527	−	0.972829i	$-0.574372\pi$
$727$	−12.4853			−0.463053			−0.231527	+	0.972829i	$0.574372\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−6.34315	−	10.9867i	−0.234610	−	0.406356i
$732$		0			0
$733$	−24.8492	+	43.0402i	−0.917828	+	1.58972i	−0.115120	+	0.993352i	$0.536725\pi$
$733$	−24.8492	+	43.0402i	−0.917828	+	1.58972i	−0.802708	+	0.596373i	$0.796608\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.65685	+	9.79796i	−0.208373	+	0.360912i
$738$		0			0
$739$	2.34315	+	4.05845i	0.0861940	+	0.149292i	0.905899	−	0.423493i	$-0.139196\pi$
$739$	2.34315	+	4.05845i	0.0861940	+	0.149292i	−0.819705	+	0.572785i	$0.805863\pi$
$740$		0			0
$741$	−7.31371			−0.268676
$742$		0			0
$743$	50.9706			1.86993			0.934964	−	0.354742i	$-0.115431\pi$
$743$	50.9706			1.86993			0.934964	+	0.354742i	$0.115431\pi$
$744$		0			0
$745$	−29.5563	−	51.1931i	−1.08286	−	1.87557i
$746$		0			0
$747$	3.65685	−	6.33386i	0.133797	−	0.231744i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.82843	−	11.8272i	0.249173	−	0.431580i	−0.714124	−	0.700020i	$-0.753175\pi$
$751$	6.82843	−	11.8272i	0.249173	−	0.431580i	0.963297	+	0.268440i	$0.0865079\pi$
$752$		0			0
$753$	6.24264	+	10.8126i	0.227494	+	0.394032i
$754$		0			0
$755$	−40.9706			−1.49107
$756$		0			0
$757$	26.3431			0.957458			0.478729	−	0.877963i	$-0.341098\pi$
$757$	26.3431			0.957458			0.478729	+	0.877963i	$0.341098\pi$
$758$		0			0
$759$	7.65685	+	13.2621i	0.277926	+	0.481382i
$760$		0			0
$761$	9.26346	−	16.0448i	0.335800	−	0.581623i	−0.647838	−	0.761778i	$-0.724327\pi$
$761$	9.26346	−	16.0448i	0.335800	−	0.581623i	0.983638	+	0.180155i	$0.0576600\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.82843	−	6.63103i	0.138417	−	0.239745i
$766$		0			0
$767$	−1.51472	−	2.62357i	−0.0546933	−	0.0947316i
$768$		0			0
$769$	29.6985			1.07095			0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985			1.07095			0.535477	+	0.844550i	$0.320132\pi$
$770$		0			0
$771$	−23.2132			−0.836003
$772$		0			0
$773$	4.77817	+	8.27604i	0.171859	+	0.297669i	0.939070	−	0.343727i	$-0.111689\pi$
$773$	4.77817	+	8.27604i	0.171859	+	0.297669i	−0.767211	+	0.641395i	$0.778356\pi$
$774$		0			0
$775$	−3.89949	+	6.75412i	−0.140074	+	0.242615i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.82843	+	15.2913i	−0.316311	+	0.547867i
$780$		0			0
$781$	9.31371	+	16.1318i	0.333271	+	0.577242i
$782$		0			0
$783$	6.82843			0.244028
$784$		0			0
$785$	40.1421			1.43273
$786$		0			0
$787$	12.3431	+	21.3790i	0.439986	+	0.762077i	0.997688	−	0.0679637i	$-0.0216502\pi$
$787$	12.3431	+	21.3790i	0.439986	+	0.762077i	−0.557702	+	0.830041i	$0.688317\pi$
$788$		0			0
$789$	−2.65685	+	4.60181i	−0.0945865	+	0.163829i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−15.8284	+	27.4156i	−0.562084	+	0.973558i
$794$		0			0
$795$	−3.41421	−	5.91359i	−0.121090	−	0.209733i
$796$		0			0
$797$	−8.38478			−0.297004			−0.148502	−	0.988912i	$-0.547445\pi$
$797$	−8.38478			−0.297004			−0.148502	+	0.988912i	$0.547445\pi$
$798$		0			0
$799$	−6.34315			−0.224404
$800$		0			0
$801$	7.12132	+	12.3345i	0.251619	+	0.435818i
$802$		0			0
$803$	−13.8995	+	24.0746i	−0.490503	+	0.849575i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−7.36396	+	12.7548i	−0.259224	+	0.448989i
$808$		0			0
$809$	−9.97056	−	17.2695i	−0.350546	−	0.607164i	0.635799	−	0.771855i	$-0.280671\pi$
$809$	−9.97056	−	17.2695i	−0.350546	−	0.607164i	−0.986345	+	0.164691i	$0.947337\pi$
$810$		0			0
$811$	17.6569			0.620016			0.310008	−	0.950734i	$-0.399668\pi$
$811$	17.6569			0.620016			0.310008	+	0.950734i	$0.399668\pi$
$812$		0			0
$813$	−10.1421			−0.355700
$814$		0			0
$815$	−19.3137	−	33.4523i	−0.676530	−	1.17178i
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.34315	−	9.25460i	0.186477	−	0.322988i	−0.757596	−	0.652724i	$-0.773626\pi$
$821$	5.34315	−	9.25460i	0.186477	−	0.322988i	0.944073	+	0.329736i	$0.106960\pi$
$822$		0			0
$823$	4.48528	+	7.76874i	0.156347	+	0.270801i	0.933549	−	0.358451i	$-0.116695\pi$
$823$	4.48528	+	7.76874i	0.156347	+	0.270801i	−0.777202	+	0.629252i	$0.783361\pi$
$824$		0			0
$825$	−13.3137			−0.463524
$826$		0			0
$827$	−47.6569			−1.65719			−0.828596	−	0.559848i	$-0.810860\pi$
$827$	−47.6569			−1.65719			−0.828596	+	0.559848i	$0.810860\pi$
$828$		0			0
$829$	0.363961	+	0.630399i	0.0126409	+	0.0218947i	0.872277	−	0.489013i	$-0.162643\pi$
$829$	0.363961	+	0.630399i	0.0126409	+	0.0218947i	−0.859636	+	0.510907i	$0.829310\pi$
$830$		0			0
$831$	−4.65685	+	8.06591i	−0.161544	+	0.279803i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−33.7990	+	58.5416i	−1.16966	+	2.02591i
$836$		0			0
$837$	0.585786	+	1.01461i	0.0202477	+	0.0350701i
$838$		0			0
$839$	−50.8284			−1.75479			−0.877396	−	0.479767i	$-0.840721\pi$
$839$	−50.8284			−1.75479			−0.877396	+	0.479767i	$0.840721\pi$
$840$		0			0
$841$	17.6274			0.607842
$842$		0			0
$843$	0.242641	+	0.420266i	0.00835699	+	0.0144747i
$844$		0			0
$845$	10.7782	−	18.6683i	0.370780	−	0.642211i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	4.24264	−	7.34847i	0.145607	−	0.252199i
$850$		0			0
$851$	15.3137	+	26.5241i	0.524947	+	0.909235i
$852$		0			0
$853$	−49.4975			−1.69476			−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975			−1.69476			−0.847381	+	0.530986i	$0.821822\pi$
$854$		0			0
$855$	9.65685			0.330257
$856$		0			0
$857$	−7.70711	−	13.3491i	−0.263270	−	0.455997i	0.703839	−	0.710359i	$-0.251468\pi$
$857$	−7.70711	−	13.3491i	−0.263270	−	0.455997i	−0.967109	+	0.254363i	$0.918134\pi$
$858$		0			0
$859$	28.7279	−	49.7582i	0.980184	−	1.69773i	0.318542	−	0.947909i	$-0.396807\pi$
$859$	28.7279	−	49.7582i	0.980184	−	1.69773i	0.661642	−	0.749820i	$-0.269860\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.65685	−	14.9941i	0.294683	−	0.510405i	−0.680228	−	0.733000i	$-0.738119\pi$
$863$	8.65685	−	14.9941i	0.294683	−	0.510405i	0.974911	+	0.222595i	$0.0714527\pi$
$864$		0			0
$865$	−35.9706	−	62.3028i	−1.22304	−	2.11836i
$866$		0			0
$867$	11.9706			0.406542
$868$		0			0
$869$	−27.3137			−0.926554
$870$		0			0
$871$	−7.31371	−	12.6677i	−0.247816	−	0.429229i
$872$		0			0
$873$	−1.29289	+	2.23936i	−0.0437578	+	0.0757907i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.65685	−	9.79796i	0.191018	−	0.330854i	−0.754570	−	0.656220i	$-0.772154\pi$
$877$	5.65685	−	9.79796i	0.191018	−	0.330854i	0.945588	+	0.325366i	$0.105488\pi$
$878$		0			0
$879$	8.29289	+	14.3637i	0.279712	+	0.484476i
$880$		0			0
$881$	21.7574			0.733024			0.366512	−	0.930413i	$-0.380552\pi$
$881$	21.7574			0.733024			0.366512	+	0.930413i	$0.380552\pi$
$882$		0			0
$883$	4.68629			0.157706			0.0788531	−	0.996886i	$-0.474874\pi$
$883$	4.68629			0.157706			0.0788531	+	0.996886i	$0.474874\pi$
$884$		0			0
$885$	2.00000	+	3.46410i	0.0672293	+	0.116445i
$886$		0			0
$887$	−1.41421	+	2.44949i	−0.0474846	+	0.0822458i	−0.888791	−	0.458313i	$-0.848454\pi$
$887$	−1.41421	+	2.44949i	−0.0474846	+	0.0822458i	0.841306	+	0.540559i	$0.181787\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.00000	+	1.73205i	−0.0335013	+	0.0580259i
$892$		0			0
$893$	−4.00000	−	6.92820i	−0.133855	−	0.231843i
$894$		0			0
$895$	67.1127			2.24333
$896$		0			0
$897$	−19.7990			−0.661069
$898$		0			0
$899$	4.00000	+	6.92820i	0.133407	+	0.231069i
$900$		0			0
$901$	−2.24264	+	3.88437i	−0.0747132	+	0.129407i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.41421	−	7.64564i	0.146733	−	0.254150i
$906$		0			0
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	0.702094	−	0.712084i	$-0.252248\pi$
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	−0.967730	+	0.251990i	$0.918915\pi$
$908$		0			0
$909$	2.92893			0.0971465
$910$		0			0
$911$	1.02944			0.0341068			0.0170534	−	0.999855i	$-0.494571\pi$
$911$	1.02944			0.0341068			0.0170534	+	0.999855i	$0.494571\pi$
$912$		0			0
$913$	7.31371	+	12.6677i	0.242048	+	0.419240i
$914$		0			0
$915$	20.8995	−	36.1990i	0.690916	−	1.19670i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−4.14214	+	7.17439i	−0.136636	+	0.236661i	−0.926221	−	0.376980i	$-0.876963\pi$
$919$	−4.14214	+	7.17439i	−0.136636	+	0.236661i	0.789585	+	0.613641i	$0.210296\pi$
$920$		0			0
$921$	−15.0711	−	26.1039i	−0.496609	−	0.860151i
$922$		0			0
$923$	−24.0833			−0.792710
$924$		0			0
$925$	−26.6274			−0.875504
$926$		0			0
$927$	2.24264	+	3.88437i	0.0736580	+	0.127579i
$928$		0			0
$929$	19.6066	−	33.9596i	0.643272	−	1.11418i	−0.341426	−	0.939909i	$-0.610910\pi$
$929$	19.6066	−	33.9596i	0.643272	−	1.11418i	0.984698	−	0.174271i	$-0.0557568\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−3.07107	+	5.31925i	−0.100542	+	0.174144i
$934$		0			0
$935$	7.65685	+	13.2621i	0.250406	+	0.433716i
$936$		0			0
$937$	30.5858			0.999194			0.499597	−	0.866258i	$-0.333481\pi$
$937$	30.5858			0.999194			0.499597	+	0.866258i	$0.333481\pi$
$938$		0			0
$939$	−1.89949			−0.0619877
$940$		0			0
$941$	−17.6066	−	30.4955i	−0.573959	−	0.994126i	−0.996154	−	0.0876208i	$-0.972074\pi$
$941$	−17.6066	−	30.4955i	−0.573959	−	0.994126i	0.422195	−	0.906505i	$-0.361260\pi$
$942$		0			0
$943$	−23.8995	+	41.3951i	−0.778275	+	1.34801i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.3431	−	26.5751i	0.498585	−	0.863575i	−0.501413	−	0.865208i	$-0.667186\pi$
$947$	15.3431	−	26.5751i	0.498585	−	0.863575i	0.999999	+	0.00163285i	$0.000519752\pi$
$948$		0			0
$949$	−17.9706	−	31.1259i	−0.583349	−	1.01039i
$950$		0			0
$951$	−10.0000			−0.324272
$952$		0			0
$953$	2.00000			0.0647864			0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000			0.0647864			0.0323932	+	0.999475i	$0.489687\pi$
$954$		0			0
$955$	−30.7279	−	53.2223i	−0.994332	−	1.72223i
$956$		0			0
$957$	−6.82843	+	11.8272i	−0.220732	+	0.382319i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	14.8137	−	25.6581i	0.477862	−	0.827681i
$962$		0			0
$963$	−0.171573	−	0.297173i	−0.00552886	−	0.00957626i
$964$		0			0
$965$	18.1421			0.584016
$966$		0			0
$967$	−33.6569			−1.08233			−0.541166	−	0.840916i	$-0.682017\pi$
$967$	−33.6569			−1.08233			−0.541166	+	0.840916i	$0.682017\pi$
$968$		0			0
$969$	−3.17157	−	5.49333i	−0.101886	−	0.176471i
$970$		0			0
$971$	−25.3137	+	43.8446i	−0.812356	+	1.40704i	0.0988557	+	0.995102i	$0.468482\pi$
$971$	−25.3137	+	43.8446i	−0.812356	+	1.40704i	−0.911211	+	0.411939i	$0.864852\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	8.60660	−	14.9071i	0.275632	−	0.477408i
$976$		0			0
$977$	10.5858	+	18.3351i	0.338669	+	0.586592i	0.984183	−	0.177157i	$-0.0566899\pi$
$977$	10.5858	+	18.3351i	0.338669	+	0.586592i	−0.645513	+	0.763749i	$0.723357\pi$
$978$		0			0
$979$	−28.4853			−0.910394
$980$		0			0
$981$	−5.65685			−0.180609
$982$		0			0
$983$	−26.6274	−	46.1200i	−0.849283	−	1.47100i	−0.881849	−	0.471531i	$-0.843701\pi$
$983$	−26.6274	−	46.1200i	−0.849283	−	1.47100i	0.0325667	−	0.999470i	$-0.489632\pi$
$984$		0			0
$985$	−3.41421	+	5.91359i	−0.108786	+	0.188423i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.6569	−	37.5108i	0.688648	−	1.19277i
$990$		0			0
$991$	−6.48528	−	11.2328i	−0.206012	−	0.356823i	0.744443	−	0.667686i	$-0.232715\pi$
$991$	−6.48528	−	11.2328i	−0.206012	−	0.356823i	−0.950455	+	0.310863i	$0.899382\pi$
$992$		0			0
$993$	−4.00000			−0.126936
$994$		0			0
$995$	−73.9411			−2.34409
$996$		0			0
$997$	−13.1924	−	22.8499i	−0.417807	−	0.723663i	0.577911	−	0.816099i	$-0.303868\pi$
$997$	−13.1924	−	22.8499i	−0.417807	−	0.723663i	−0.995719	+	0.0924363i	$0.970535\pi$
$998$		0			0
$999$	−2.00000	+	3.46410i	−0.0632772	+	0.109599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.q.bd.961.1		4
4.3	odd	2		147.2.e.d.79.1		4
7.2	even	3		2352.2.a.bc.1.2		2
7.3	odd	6		2352.2.q.bb.1537.2		4
7.4	even	3	inner	2352.2.q.bd.1537.1		4
7.5	odd	6		2352.2.a.be.1.1		2
7.6	odd	2		2352.2.q.bb.961.2		4
12.11	even	2		441.2.e.g.226.2		4
21.2	odd	6		7056.2.a.cf.1.1		2
21.5	even	6		7056.2.a.cv.1.2		2
28.3	even	6		147.2.e.e.67.1		4
28.11	odd	6		147.2.e.d.67.1		4
28.19	even	6		147.2.a.d.1.2	✓	2
28.23	odd	6		147.2.a.e.1.2	yes	2
28.27	even	2		147.2.e.e.79.1		4
56.5	odd	6		9408.2.a.dq.1.2		2
56.19	even	6		9408.2.a.ef.1.2		2
56.37	even	6		9408.2.a.dt.1.1		2
56.51	odd	6		9408.2.a.di.1.1		2
84.11	even	6		441.2.e.g.361.2		4
84.23	even	6		441.2.a.i.1.1		2
84.47	odd	6		441.2.a.j.1.1		2
84.59	odd	6		441.2.e.f.361.2		4
84.83	odd	2		441.2.e.f.226.2		4
140.19	even	6		3675.2.a.bf.1.1		2
140.79	odd	6		3675.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.2	✓	2	28.19	even	6
147.2.a.e.1.2	yes	2	28.23	odd	6
147.2.e.d.67.1		4	28.11	odd	6
147.2.e.d.79.1		4	4.3	odd	2
147.2.e.e.67.1		4	28.3	even	6
147.2.e.e.79.1		4	28.27	even	2
441.2.a.i.1.1		2	84.23	even	6
441.2.a.j.1.1		2	84.47	odd	6
441.2.e.f.226.2		4	84.83	odd	2
441.2.e.f.361.2		4	84.59	odd	6
441.2.e.g.226.2		4	12.11	even	2
441.2.e.g.361.2		4	84.11	even	6
2352.2.a.bc.1.2		2	7.2	even	3
2352.2.a.be.1.1		2	7.5	odd	6
2352.2.q.bb.961.2		4	7.6	odd	2
2352.2.q.bb.1537.2		4	7.3	odd	6
2352.2.q.bd.961.1		4	1.1	even	1	trivial
2352.2.q.bd.1537.1		4	7.4	even	3	inner
3675.2.a.bd.1.1		2	140.79	odd	6
3675.2.a.bf.1.1		2	140.19	even	6
7056.2.a.cf.1.1		2	21.2	odd	6
7056.2.a.cv.1.2		2	21.5	even	6
9408.2.a.di.1.1		2	56.51	odd	6
9408.2.a.dq.1.2		2	56.5	odd	6
9408.2.a.dt.1.1		2	56.37	even	6
9408.2.a.ef.1.2		2	56.19	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(-1.70711 + 2.95680i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +2.58579 q^{13} -3.41421 q^{15} +(1.12132 + 1.94218i) q^{17} +(-1.41421 + 2.44949i) q^{19} +(-3.82843 + 6.63103i) q^{23} +(-3.32843 - 5.76500i) q^{25} -1.00000 q^{27} -6.82843 q^{29} +(-0.585786 - 1.01461i) q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{37} +(1.29289 + 2.23936i) q^{39} +6.24264 q^{41} -5.65685 q^{43} +(-1.70711 - 2.95680i) q^{45} +(-1.41421 + 2.44949i) q^{47} +(-1.12132 + 1.94218i) q^{51} +(1.00000 + 1.73205i) q^{53} +6.82843 q^{55} -2.82843 q^{57} +(-0.585786 - 1.01461i) q^{59} +(-6.12132 + 10.6024i) q^{61} +(-4.41421 + 7.64564i) q^{65} +(-2.82843 - 4.89898i) q^{67} -7.65685 q^{69} -9.31371 q^{71} +(-6.94975 - 12.0373i) q^{73} +(3.32843 - 5.76500i) q^{75} +(6.82843 - 11.8272i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.31371 q^{83} -7.65685 q^{85} +(-3.41421 - 5.91359i) q^{87} +(7.12132 - 12.3345i) q^{89} +(0.585786 - 1.01461i) q^{93} +(-4.82843 - 8.36308i) q^{95} +2.58579 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9} - 4 q^{11} + 16 q^{13} - 8 q^{15} - 4 q^{17} - 4 q^{23} - 2 q^{25} - 4 q^{27} - 16 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 4 q^{45} + 4 q^{51} + 4 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^9 - 4 * q^11 + 16 * q^13 - 8 * q^15 - 4 * q^17 - 4 * q^23 - 2 * q^25 - 4 * q^27 - 16 * q^29 - 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^39 + 8 * q^41 - 4 * q^45 + 4 * q^51 + 4 * q^53 + 16 * q^55 - 8 * q^59 - 16 * q^61 - 12 * q^65 - 8 * q^69 + 8 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^79 - 2 * q^81 + 16 * q^83 - 8 * q^85 - 8 * q^87 + 20 * q^89 + 8 * q^93 - 8 * q^95 + 16 * q^97 + 8 * q^99

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