LMFDB - Embedded newform 2320.2.j.c.289.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(289,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,3,0,0,0,6,0,0,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.4
Root			$-0.780776 + 1.35234i$ of defining polynomial
Character	$\chi$	$=$	2320.289
Dual form			2320.2.j.c.289.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +(-0.280776 + 2.21837i) q^{5} -3.46410i q^{7} +3.56155 q^{9} -0.972638i q^{11} -4.43674i q^{13} +(-0.719224 + 5.68247i) q^{15} -2.00000 q^{17} -3.46410i q^{19} -8.87348i q^{21} -3.46410i q^{23} +(-4.84233 - 1.24573i) q^{25} +1.43845 q^{27} +(4.12311 - 3.46410i) q^{29} -7.90084i q^{31} -2.49146i q^{33} +(7.68466 + 0.972638i) q^{35} +7.12311 q^{37} -11.3649i q^{39} +8.87348i q^{41} -5.43845 q^{43} +(-1.00000 + 7.90084i) q^{45} +11.6847 q^{47} -5.00000 q^{49} -5.12311 q^{51} +4.43674i q^{53} +(2.15767 + 0.273094i) q^{55} -8.87348i q^{57} +1.12311 q^{59} -1.94528i q^{61} -12.3376i q^{63} +(9.84233 + 1.24573i) q^{65} +12.3376i q^{67} -8.87348i q^{69} +2.24621 q^{71} -7.12311 q^{73} +(-12.4039 - 3.19101i) q^{75} -3.36932 q^{77} +14.8290i q^{79} -7.00000 q^{81} -10.3923i q^{83} +(0.561553 - 4.43674i) q^{85} +(10.5616 - 8.87348i) q^{87} +8.87348i q^{89} -15.3693 q^{91} -20.2384i q^{93} +(7.68466 + 0.972638i) q^{95} +8.24621 q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 3 q^{5} + 6 q^{9} - 7 q^{15} - 8 q^{17} - 7 q^{25} + 14 q^{27} + 6 q^{35} + 12 q^{37} - 30 q^{43} - 4 q^{45} + 22 q^{47} - 20 q^{49} - 4 q^{51} + 21 q^{55} - 12 q^{59} + 27 q^{65} - 24 q^{71}+ \cdots + 6 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 3 * q^5 + 6 * q^9 - 7 * q^15 - 8 * q^17 - 7 * q^25 + 14 * q^27 + 6 * q^35 + 12 * q^37 - 30 * q^43 - 4 * q^45 + 22 * q^47 - 20 * q^49 - 4 * q^51 + 21 * q^55 - 12 * q^59 + 27 * q^65 - 24 * q^71 - 12 * q^73 - 29 * q^75 + 36 * q^77 - 28 * q^81 - 6 * q^85 + 34 * q^87 - 12 * q^91 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.56155			1.47891			0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155			1.47891			0.739457	+	0.673204i	$0.235083\pi$
$4$		0			0
$5$	−0.280776	+	2.21837i	−0.125567	+	0.992085i
$5$	−0.280776	+	2.21837i	−0.125567	+	0.992085i
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$		0			0
$9$	3.56155			1.18718
$10$		0			0
$11$		−	0.972638i		−	0.293261i	−0.989191	−	0.146631i	$-0.953157\pi$
$11$		−	0.972638i		−	0.293261i	0.989191	−	0.146631i	$-0.0468429\pi$
$12$		0			0
$13$		−	4.43674i		−	1.23053i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$		−	4.43674i		−	1.23053i	0.788320	−	0.615265i	$-0.210951\pi$
$14$		0			0
$15$	−0.719224	+	5.68247i	−0.185703	+	1.46721i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		−	8.87348i		−	1.93635i
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	−4.84233	−	1.24573i	−0.968466	−	0.249146i
$26$		0			0
$27$	1.43845			0.276829
$28$		0			0
$29$	4.12311	−	3.46410i	0.765641	−	0.643268i
$29$	4.12311	−	3.46410i	0.765641	−	0.643268i
$30$		0			0
$31$		−	7.90084i		−	1.41903i	−0.704689	−	0.709516i	$-0.748913\pi$
$31$		−	7.90084i		−	1.41903i	0.704689	−	0.709516i	$-0.251087\pi$
$32$		0			0
$33$		−	2.49146i		−	0.433708i
$34$		0			0
$35$	7.68466	+	0.972638i	1.29894	+	0.164406i
$36$		0			0
$37$	7.12311			1.17103			0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311			1.17103			0.585516	+	0.810661i	$0.300892\pi$
$38$		0			0
$39$		−	11.3649i		−	1.81985i
$40$		0			0
$41$			8.87348i			1.38580i	0.721031	+	0.692902i	$0.243668\pi$
$41$			8.87348i			1.38580i	−0.721031	+	0.692902i	$0.756332\pi$
$42$		0			0
$43$	−5.43845			−0.829355			−0.414678	−	0.909968i	$-0.636106\pi$
$43$	−5.43845			−0.829355			−0.414678	+	0.909968i	$0.636106\pi$
$44$		0			0
$45$	−1.00000	+	7.90084i	−0.149071	+	1.17779i
$46$		0			0
$47$	11.6847			1.70438			0.852191	−	0.523230i	$-0.175273\pi$
$47$	11.6847			1.70438			0.852191	+	0.523230i	$0.175273\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	−5.12311			−0.717378
$52$		0			0
$53$			4.43674i			0.609433i	0.952443	+	0.304717i	$0.0985617\pi$
$53$			4.43674i			0.609433i	−0.952443	+	0.304717i	$0.901438\pi$
$54$		0			0
$55$	2.15767	+	0.273094i	0.290940	+	0.0368240i
$56$		0			0
$57$		−	8.87348i		−	1.17532i
$58$		0			0
$59$	1.12311			0.146216			0.0731079	−	0.997324i	$-0.476708\pi$
$59$	1.12311			0.146216			0.0731079	+	0.997324i	$0.476708\pi$
$60$		0			0
$61$		−	1.94528i		−	0.249067i	−0.992215	−	0.124534i	$-0.960257\pi$
$61$		−	1.94528i		−	0.249067i	0.992215	−	0.124534i	$-0.0397434\pi$
$62$		0			0
$63$		−	12.3376i		−	1.55439i
$64$		0			0
$65$	9.84233	+	1.24573i	1.22079	+	0.154514i
$66$		0			0
$67$			12.3376i			1.50728i	0.657290	+	0.753638i	$0.271703\pi$
$67$			12.3376i			1.50728i	−0.657290	+	0.753638i	$0.728297\pi$
$68$		0			0
$69$		−	8.87348i		−	1.06824i
$70$		0			0
$71$	2.24621			0.266576			0.133288	−	0.991077i	$-0.457446\pi$
$71$	2.24621			0.266576			0.133288	+	0.991077i	$0.457446\pi$
$72$		0			0
$73$	−7.12311			−0.833696			−0.416848	−	0.908976i	$-0.636865\pi$
$73$	−7.12311			−0.833696			−0.416848	+	0.908976i	$0.636865\pi$
$74$		0			0
$75$	−12.4039	−	3.19101i	−1.43228	−	0.368466i
$76$		0			0
$77$	−3.36932			−0.383969
$78$		0			0
$79$			14.8290i			1.66840i	0.551464	+	0.834199i	$0.314070\pi$
$79$			14.8290i			1.66840i	−0.551464	+	0.834199i	$0.685930\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$		−	10.3923i		−	1.14070i	−0.821401	−	0.570352i	$-0.806807\pi$
$83$		−	10.3923i		−	1.14070i	0.821401	−	0.570352i	$-0.193193\pi$
$84$		0			0
$85$	0.561553	−	4.43674i	0.0609090	−	0.481232i
$86$		0			0
$87$	10.5616	−	8.87348i	1.13232	−	0.951337i
$88$		0			0
$89$			8.87348i			0.940587i	0.882510	+	0.470293i	$0.155852\pi$
$89$			8.87348i			0.940587i	−0.882510	+	0.470293i	$0.844148\pi$
$90$		0			0
$91$	−15.3693			−1.61114
$92$		0			0
$93$		−	20.2384i		−	2.09863i
$94$		0			0
$95$	7.68466	+	0.972638i	0.788429	+	0.0997906i
$96$		0			0
$97$	8.24621			0.837276			0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621			0.837276			0.418638	+	0.908153i	$0.362508\pi$
$98$		0			0
$99$		−	3.46410i		−	0.348155i
$100$		0			0
$101$			6.92820i			0.689382i	0.938716	+	0.344691i	$0.112016\pi$
$101$			6.92820i			0.689382i	−0.938716	+	0.344691i	$0.887984\pi$
$102$		0			0
$103$			1.51883i			0.149654i	0.997197	+	0.0748272i	$0.0238405\pi$
$103$			1.51883i			0.149654i	−0.997197	+	0.0748272i	$0.976159\pi$
$104$		0			0
$105$	19.6847	+	2.49146i	1.92103	+	0.243142i
$106$		0			0
$107$		−	14.2829i		−	1.38078i	−0.723439	−	0.690388i	$-0.757440\pi$
$107$		−	14.2829i		−	1.38078i	0.723439	−	0.690388i	$-0.242560\pi$
$108$		0			0
$109$	−5.68466			−0.544492			−0.272246	−	0.962228i	$-0.587766\pi$
$109$	−5.68466			−0.544492			−0.272246	+	0.962228i	$0.587766\pi$
$110$		0			0
$111$	18.2462			1.73185
$112$		0			0
$113$	5.36932			0.505103			0.252551	−	0.967583i	$-0.418730\pi$
$113$	5.36932			0.505103			0.252551	+	0.967583i	$0.418730\pi$
$114$		0			0
$115$	7.68466	+	0.972638i	0.716598	+	0.0906990i
$116$		0			0
$117$		−	15.8017i		−	1.46087i
$118$		0			0
$119$			6.92820i			0.635107i
$120$		0			0
$121$	10.0540			0.913998
$122$		0			0
$123$			22.7299i			2.04948i
$124$		0			0
$125$	4.12311	−	10.3923i	0.368782	−	0.929516i
$126$		0			0
$127$	2.24621			0.199319			0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621			0.199319			0.0996595	+	0.995022i	$0.468225\pi$
$128$		0			0
$129$	−13.9309			−1.22654
$130$		0			0
$131$			14.2829i			1.24790i	0.781465	+	0.623949i	$0.214473\pi$
$131$			14.2829i			1.24790i	−0.781465	+	0.623949i	$0.785527\pi$
$132$		0			0
$133$	−12.0000			−1.04053
$134$		0			0
$135$	−0.403882	+	3.19101i	−0.0347606	+	0.274638i
$136$		0			0
$137$	13.3693			1.14222			0.571109	−	0.820874i	$-0.306513\pi$
$137$	13.3693			1.14222			0.571109	+	0.820874i	$0.306513\pi$
$138$		0			0
$139$	−19.3693			−1.64288			−0.821442	−	0.570292i	$-0.806830\pi$
$139$	−19.3693			−1.64288			−0.821442	+	0.570292i	$0.806830\pi$
$140$		0			0
$141$	29.9309			2.52063
$142$		0			0
$143$	−4.31534			−0.360867
$144$		0			0
$145$	6.52699	+	10.1192i	0.542037	+	0.840355i
$146$		0			0
$147$	−12.8078			−1.05637
$148$		0			0
$149$	−13.6847			−1.12109			−0.560545	−	0.828124i	$-0.689408\pi$
$149$	−13.6847			−1.12109			−0.560545	+	0.828124i	$0.689408\pi$
$150$		0			0
$151$	−15.3693			−1.25074			−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693			−1.25074			−0.625369	+	0.780329i	$0.715051\pi$
$152$		0			0
$153$	−7.12311			−0.575869
$154$		0			0
$155$	17.5270	+	2.21837i	1.40780	+	0.178184i
$156$		0			0
$157$	22.4924			1.79509			0.897545	−	0.440922i	$-0.145349\pi$
$157$	22.4924			1.79509			0.897545	+	0.440922i	$0.145349\pi$
$158$		0			0
$159$			11.3649i			0.901299i
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	5.43845			0.425972			0.212986	−	0.977055i	$-0.431681\pi$
$163$	5.43845			0.425972			0.212986	+	0.977055i	$0.431681\pi$
$164$		0			0
$165$	5.52699	+	0.699544i	0.430275	+	0.0544594i
$166$		0			0
$167$		−	8.44703i		−	0.653651i	−0.945085	−	0.326825i	$-0.894021\pi$
$167$		−	8.44703i		−	0.653651i	0.945085	−	0.326825i	$-0.105979\pi$
$168$		0			0
$169$	−6.68466			−0.514204
$170$		0			0
$171$		−	12.3376i		−	0.943478i
$172$		0			0
$173$		−	10.8188i		−	0.822535i	−0.911515	−	0.411267i	$-0.865086\pi$
$173$		−	10.8188i		−	0.822535i	0.911515	−	0.411267i	$-0.134914\pi$
$174$		0			0
$175$	−4.31534	+	16.7743i	−0.326209	+	1.26802i
$176$		0			0
$177$	2.87689			0.216241
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−13.6847			−1.01717			−0.508586	−	0.861011i	$-0.669832\pi$
$181$	−13.6847			−1.01717			−0.508586	+	0.861011i	$0.669832\pi$
$182$		0			0
$183$		−	4.98293i		−	0.368349i
$184$		0			0
$185$	−2.00000	+	15.8017i	−0.147043	+	1.16176i
$186$		0			0
$187$			1.94528i			0.142253i
$188$		0			0
$189$		−	4.98293i		−	0.362455i
$190$		0			0
$191$			3.46410i			0.250654i	0.992116	+	0.125327i	$0.0399979\pi$
$191$			3.46410i			0.250654i	−0.992116	+	0.125327i	$0.960002\pi$
$192$		0			0
$193$	8.24621			0.593575			0.296788	−	0.954944i	$-0.404085\pi$
$193$	8.24621			0.593575			0.296788	+	0.954944i	$0.404085\pi$
$194$		0			0
$195$	25.2116	+	3.19101i	1.80544	+	0.228513i
$196$		0			0
$197$			10.8188i			0.770804i	0.922749	+	0.385402i	$0.125937\pi$
$197$			10.8188i			0.770804i	−0.922749	+	0.385402i	$0.874063\pi$
$198$		0			0
$199$	15.3693			1.08950			0.544751	−	0.838598i	$-0.316624\pi$
$199$	15.3693			1.08950			0.544751	+	0.838598i	$0.316624\pi$
$200$		0			0
$201$			31.6034i			2.22913i
$202$		0			0
$203$	−12.0000	−	14.2829i	−0.842235	−	1.00246i
$204$		0			0
$205$	−19.6847	−	2.49146i	−1.37484	−	0.174011i
$206$		0			0
$207$		−	12.3376i		−	0.857521i
$208$		0			0
$209$	−3.36932			−0.233061
$210$		0			0
$211$		−	14.8290i		−	1.02087i	−0.859915	−	0.510437i	$-0.829484\pi$
$211$		−	14.8290i		−	1.02087i	0.859915	−	0.510437i	$-0.170516\pi$
$212$		0			0
$213$	5.75379			0.394243
$214$		0			0
$215$	1.52699	−	12.0645i	0.104140	−	0.822791i
$216$		0			0
$217$	−27.3693			−1.85795
$218$		0			0
$219$	−18.2462			−1.23296
$220$		0			0
$221$			8.87348i			0.596895i
$222$		0			0
$223$		−	12.3376i		−	0.826186i	−0.910689	−	0.413093i	$-0.864449\pi$
$223$		−	12.3376i		−	0.826186i	0.910689	−	0.413093i	$-0.135551\pi$
$224$		0			0
$225$	−17.2462	−	4.43674i	−1.14975	−	0.295783i
$226$		0			0
$227$			12.3376i			0.818874i	0.912338	+	0.409437i	$0.134275\pi$
$227$			12.3376i			0.818874i	−0.912338	+	0.409437i	$0.865725\pi$
$228$		0			0
$229$		−	15.8017i		−	1.04420i	−0.852883	−	0.522102i	$-0.825148\pi$
$229$		−	15.8017i		−	1.04420i	0.852883	−	0.522102i	$-0.174852\pi$
$230$		0			0
$231$	−8.63068			−0.567857
$232$		0			0
$233$			2.49146i			0.163221i	0.996664	+	0.0816106i	$0.0260064\pi$
$233$			2.49146i			0.163221i	−0.996664	+	0.0816106i	$0.973994\pi$
$234$		0			0
$235$	−3.28078	+	25.9209i	−0.214014	+	1.69089i
$236$		0			0
$237$			37.9854i			2.46742i
$238$		0			0
$239$	−13.1231			−0.848863			−0.424432	−	0.905460i	$-0.639526\pi$
$239$	−13.1231			−0.848863			−0.424432	+	0.905460i	$0.639526\pi$
$240$		0			0
$241$	−1.68466			−0.108518			−0.0542592	−	0.998527i	$-0.517280\pi$
$241$	−1.68466			−0.108518			−0.0542592	+	0.998527i	$0.517280\pi$
$242$		0			0
$243$	−22.2462			−1.42710
$244$		0			0
$245$	1.40388	−	11.0918i	0.0896907	−	0.708632i
$246$		0			0
$247$	−15.3693			−0.977926
$248$		0			0
$249$		−	26.6204i		−	1.68700i
$250$		0			0
$251$			16.7743i			1.05879i	0.848377	+	0.529393i	$0.177580\pi$
$251$			16.7743i			1.05879i	−0.848377	+	0.529393i	$0.822420\pi$
$252$		0			0
$253$	−3.36932			−0.211827
$254$		0			0
$255$	1.43845	−	11.3649i	0.0900791	−	0.711700i
$256$		0			0
$257$		−	11.3649i		−	0.708926i	−0.935070	−	0.354463i	$-0.884664\pi$
$257$		−	11.3649i		−	0.708926i	0.935070	−	0.354463i	$-0.115336\pi$
$258$		0			0
$259$		−	24.6752i		−	1.53324i
$260$		0			0
$261$	14.6847	−	12.3376i	0.908958	−	0.763677i
$262$		0			0
$263$	−3.68466			−0.227206			−0.113603	−	0.993526i	$-0.536239\pi$
$263$	−3.68466			−0.227206			−0.113603	+	0.993526i	$0.536239\pi$
$264$		0			0
$265$	−9.84233	−	1.24573i	−0.604609	−	0.0765247i
$266$		0			0
$267$			22.7299i			1.39105i
$268$		0			0
$269$			15.8017i			0.963446i	0.876324	+	0.481723i	$0.159989\pi$
$269$			15.8017i			0.963446i	−0.876324	+	0.481723i	$0.840011\pi$
$270$		0			0
$271$			14.8290i			0.900800i	0.892827	+	0.450400i	$0.148719\pi$
$271$			14.8290i			0.900800i	−0.892827	+	0.450400i	$0.851281\pi$
$272$		0			0
$273$	−39.3693			−2.38274
$274$		0			0
$275$	−1.21165	+	4.70983i	−0.0730650	+	0.284014i
$276$		0			0
$277$			10.8188i			0.650036i	0.945708	+	0.325018i	$0.105370\pi$
$277$			10.8188i			0.650036i	−0.945708	+	0.325018i	$0.894630\pi$
$278$		0			0
$279$		−	28.1393i		−	1.68465i
$280$		0			0
$281$	13.6847			0.816358			0.408179	−	0.912902i	$-0.366164\pi$
$281$	13.6847			0.816358			0.408179	+	0.912902i	$0.366164\pi$
$282$		0			0
$283$		−	14.2829i		−	0.849028i	−0.905421	−	0.424514i	$-0.860445\pi$
$283$		−	14.2829i		−	0.849028i	0.905421	−	0.424514i	$-0.139555\pi$
$284$		0			0
$285$	19.6847	+	2.49146i	1.16602	+	0.147582i
$286$		0			0
$287$	30.7386			1.81444
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$	21.1231			1.23826
$292$		0			0
$293$	1.36932			0.0799963			0.0399982	−	0.999200i	$-0.487265\pi$
$293$	1.36932			0.0799963			0.0399982	+	0.999200i	$0.487265\pi$
$294$		0			0
$295$	−0.315342	+	2.49146i	−0.0183599	+	0.145059i
$296$		0			0
$297$		−	1.39909i		−	0.0811833i
$298$		0			0
$299$	−15.3693			−0.888831
$300$		0			0
$301$			18.8393i			1.08588i
$302$		0			0
$303$			17.7470i			1.01954i
$304$		0			0
$305$	4.31534	+	0.546188i	0.247096	+	0.0312746i
$306$		0			0
$307$	−3.19224			−0.182191			−0.0910953	−	0.995842i	$-0.529037\pi$
$307$	−3.19224			−0.182191			−0.0910953	+	0.995842i	$0.529037\pi$
$308$		0			0
$309$			3.89055i			0.221326i
$310$		0			0
$311$		−	0.426450i		−	0.0241818i	−0.999927	−	0.0120909i	$-0.996151\pi$
$311$		−	0.426450i		−	0.0241818i	0.999927	−	0.0120909i	$-0.00384874\pi$
$312$		0			0
$313$			2.49146i			0.140826i	0.997518	+	0.0704129i	$0.0224317\pi$
$313$			2.49146i			0.140826i	−0.997518	+	0.0704129i	$0.977568\pi$
$314$		0			0
$315$	27.3693	+	3.46410i	1.54209	+	0.195180i
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$	−3.36932	−	4.01029i	−0.188646	−	0.224533i
$320$		0			0
$321$		−	36.5863i		−	2.04205i
$322$		0			0
$323$			6.92820i			0.385496i
$324$		0			0
$325$	−5.52699	+	21.4842i	−0.306582	+	1.19173i
$326$		0			0
$327$	−14.5616			−0.805256
$328$		0			0
$329$		−	40.4768i		−	2.23156i
$330$		0			0
$331$		−	5.95557i		−	0.327347i	−0.986515	−	0.163674i	$-0.947666\pi$
$331$		−	5.95557i		−	0.327347i	0.986515	−	0.163674i	$-0.0523344\pi$
$332$		0			0
$333$	25.3693			1.39023
$334$		0			0
$335$	−27.3693	−	3.46410i	−1.49535	−	0.189264i
$336$		0			0
$337$	3.75379			0.204482			0.102241	−	0.994760i	$-0.467399\pi$
$337$	3.75379			0.204482			0.102241	+	0.994760i	$0.467399\pi$
$338$		0			0
$339$	13.7538			0.747003
$340$		0			0
$341$	−7.68466			−0.416148
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$	19.6847	+	2.49146i	1.05979	+	0.134136i
$346$		0			0
$347$			26.1940i			1.40617i	0.711107	+	0.703083i	$0.248194\pi$
$347$			26.1940i			1.40617i	−0.711107	+	0.703083i	$0.751806\pi$
$348$		0			0
$349$	10.3153			0.552168			0.276084	−	0.961134i	$-0.410963\pi$
$349$	10.3153			0.552168			0.276084	+	0.961134i	$0.410963\pi$
$350$		0			0
$351$		−	6.38202i		−	0.340647i
$352$		0			0
$353$			17.7470i			0.944575i	0.881444	+	0.472288i	$0.156572\pi$
$353$			17.7470i			0.944575i	−0.881444	+	0.472288i	$0.843428\pi$
$354$		0			0
$355$	−0.630683	+	4.98293i	−0.0334732	+	0.264466i
$356$		0			0
$357$			17.7470i			0.939269i
$358$		0			0
$359$		−	21.7572i		−	1.14830i	−0.818749	−	0.574152i	$-0.805332\pi$
$359$		−	21.7572i		−	1.14830i	0.818749	−	0.574152i	$-0.194668\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	25.7538			1.35172
$364$		0			0
$365$	2.00000	−	15.8017i	0.104685	−	0.827098i
$366$		0			0
$367$	28.4924			1.48729			0.743646	−	0.668573i	$-0.233095\pi$
$367$	28.4924			1.48729			0.743646	+	0.668573i	$0.233095\pi$
$368$		0			0
$369$			31.6034i			1.64521i
$370$		0			0
$371$	15.3693			0.797935
$372$		0			0
$373$		−	36.0401i		−	1.86609i	−0.359765	−	0.933043i	$-0.617143\pi$
$373$		−	36.0401i		−	1.86609i	0.359765	−	0.933043i	$-0.382857\pi$
$374$		0			0
$375$	10.5616	−	26.6204i	0.545396	−	1.37467i
$376$		0			0
$377$	−15.3693	−	18.2931i	−0.791560	−	0.942145i
$378$		0			0
$379$			0.426450i			0.0219053i	0.999940	+	0.0109526i	$0.00348640\pi$
$379$			0.426450i			0.0219053i	−0.999940	+	0.0109526i	$0.996514\pi$
$380$		0			0
$381$	5.75379			0.294776
$382$		0			0
$383$		−	17.3205i		−	0.885037i	−0.896759	−	0.442518i	$-0.854085\pi$
$383$		−	17.3205i		−	0.885037i	0.896759	−	0.442518i	$-0.145915\pi$
$384$		0			0
$385$	0.946025	−	7.47439i	0.0482139	−	0.380930i
$386$		0			0
$387$	−19.3693			−0.984598
$388$		0			0
$389$		−	10.8188i		−	0.548533i	−0.961654	−	0.274266i	$-0.911565\pi$
$389$		−	10.8188i		−	0.548533i	0.961654	−	0.274266i	$-0.0884350\pi$
$390$		0			0
$391$			6.92820i			0.350374i
$392$		0			0
$393$			36.5863i			1.84553i
$394$		0			0
$395$	−32.8963	−	4.16365i	−1.65519	−	0.209496i
$396$		0			0
$397$			0.546188i			0.0274124i	0.999906	+	0.0137062i	$0.00436295\pi$
$397$			0.546188i			0.0274124i	−0.999906	+	0.0137062i	$0.995637\pi$
$398$		0			0
$399$	−30.7386			−1.53886
$400$		0			0
$401$	15.9309			0.795550			0.397775	−	0.917483i	$-0.369783\pi$
$401$	15.9309			0.795550			0.397775	+	0.917483i	$0.369783\pi$
$402$		0			0
$403$	−35.0540			−1.74616
$404$		0			0
$405$	1.96543	−	15.5286i	0.0976632	−	0.771622i
$406$		0			0
$407$		−	6.92820i		−	0.343418i
$408$		0			0
$409$			3.89055i			0.192375i	0.995363	+	0.0961877i	$0.0306649\pi$
$409$			3.89055i			0.192375i	−0.995363	+	0.0961877i	$0.969335\pi$
$410$		0			0
$411$	34.2462			1.68924
$412$		0			0
$413$		−	3.89055i		−	0.191442i
$414$		0			0
$415$	23.0540	+	2.91791i	1.13168	+	0.143235i
$416$		0			0
$417$	−49.6155			−2.42968
$418$		0			0
$419$	−14.2462			−0.695973			−0.347986	−	0.937500i	$-0.613134\pi$
$419$	−14.2462			−0.695973			−0.347986	+	0.937500i	$0.613134\pi$
$420$		0			0
$421$			37.4392i			1.82467i	0.409439	+	0.912337i	$0.365724\pi$
$421$			37.4392i			1.82467i	−0.409439	+	0.912337i	$0.634276\pi$
$422$		0			0
$423$	41.6155			2.02342
$424$		0			0
$425$	9.68466	+	2.49146i	0.469775	+	0.120854i
$426$		0			0
$427$	−6.73863			−0.326105
$428$		0			0
$429$	−11.0540			−0.533691
$430$		0			0
$431$	−8.63068			−0.415725			−0.207863	−	0.978158i	$-0.566651\pi$
$431$	−8.63068			−0.415725			−0.207863	+	0.978158i	$0.566651\pi$
$432$		0			0
$433$	16.8769			0.811052			0.405526	−	0.914084i	$-0.367088\pi$
$433$	16.8769			0.811052			0.405526	+	0.914084i	$0.367088\pi$
$434$		0			0
$435$	16.7192	+	25.9209i	0.801625	+	1.24281i
$436$		0			0
$437$	−12.0000			−0.574038
$438$		0			0
$439$	−15.3693			−0.733537			−0.366769	−	0.930312i	$-0.619536\pi$
$439$	−15.3693			−0.733537			−0.366769	+	0.930312i	$0.619536\pi$
$440$		0			0
$441$	−17.8078			−0.847989
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$	−19.6847	−	2.49146i	−0.933142	−	0.118107i
$446$		0			0
$447$	−35.0540			−1.65800
$448$		0			0
$449$			17.7470i			0.837531i	0.908094	+	0.418765i	$0.137537\pi$
$449$			17.7470i			0.837531i	−0.908094	+	0.418765i	$0.862463\pi$
$450$		0			0
$451$	8.63068			0.406403
$452$		0			0
$453$	−39.3693			−1.84973
$454$		0			0
$455$	4.31534	−	34.0948i	0.202306	−	1.59839i
$456$		0			0
$457$			3.89055i			0.181992i	0.995851	+	0.0909962i	$0.0290051\pi$
$457$			3.89055i			0.181992i	−0.995851	+	0.0909962i	$0.970995\pi$
$458$		0			0
$459$	−2.87689			−0.134282
$460$		0			0
$461$		−	34.6410i		−	1.61339i	−0.590966	−	0.806696i	$-0.701253\pi$
$461$		−	34.6410i		−	1.61339i	0.590966	−	0.806696i	$-0.298747\pi$
$462$		0			0
$463$		−	7.35465i		−	0.341800i	−0.985288	−	0.170900i	$-0.945333\pi$
$463$		−	7.35465i		−	0.341800i	0.985288	−	0.170900i	$-0.0546675\pi$
$464$		0			0
$465$	44.8963	+	5.68247i	2.08202	+	0.263518i
$466$		0			0
$467$	8.31534			0.384788			0.192394	−	0.981318i	$-0.438375\pi$
$467$	8.31534			0.384788			0.192394	+	0.981318i	$0.438375\pi$
$468$		0			0
$469$	42.7386			1.97349
$470$		0			0
$471$	57.6155			2.65478
$472$		0			0
$473$			5.28964i			0.243218i
$474$		0			0
$475$	−4.31534	+	16.7743i	−0.198001	+	0.769659i
$476$		0			0
$477$			15.8017i			0.723509i
$478$		0			0
$479$		−	2.91791i		−	0.133323i	−0.997776	−	0.0666614i	$-0.978765\pi$
$479$		−	2.91791i		−	0.133323i	0.997776	−	0.0666614i	$-0.0212347\pi$
$480$		0			0
$481$		−	31.6034i		−	1.44099i
$482$		0			0
$483$	−30.7386			−1.39866
$484$		0			0
$485$	−2.31534	+	18.2931i	−0.105134	+	0.830649i
$486$		0			0
$487$			32.0298i			1.45141i	0.688006	+	0.725705i	$0.258486\pi$
$487$			32.0298i			1.45141i	−0.688006	+	0.725705i	$0.741514\pi$
$488$		0			0
$489$	13.9309			0.629976
$490$		0			0
$491$			29.5384i			1.33305i	0.745484	+	0.666524i	$0.232218\pi$
$491$			29.5384i			1.33305i	−0.745484	+	0.666524i	$0.767782\pi$
$492$		0			0
$493$	−8.24621	+	6.92820i	−0.371391	+	0.312031i
$494$		0			0
$495$	7.68466	+	0.972638i	0.345400	+	0.0437168i
$496$		0			0
$497$		−	7.78110i		−	0.349030i
$498$		0			0
$499$	4.63068			0.207298			0.103649	−	0.994614i	$-0.466948\pi$
$499$	4.63068			0.207298			0.103649	+	0.994614i	$0.466948\pi$
$500$		0			0
$501$		−	21.6375i		−	0.966693i
$502$		0			0
$503$	−4.94602			−0.220532			−0.110266	−	0.993902i	$-0.535170\pi$
$503$	−4.94602			−0.220532			−0.110266	+	0.993902i	$0.535170\pi$
$504$		0			0
$505$	−15.3693	−	1.94528i	−0.683926	−	0.0865636i
$506$		0			0
$507$	−17.1231			−0.760464
$508$		0			0
$509$	−29.0540			−1.28779			−0.643897	−	0.765112i	$-0.722684\pi$
$509$	−29.0540			−1.28779			−0.643897	+	0.765112i	$0.722684\pi$
$510$		0			0
$511$			24.6752i			1.09156i
$512$		0			0
$513$		−	4.98293i		−	0.220002i
$514$		0			0
$515$	−3.36932	−	0.426450i	−0.148470	−	0.0187916i
$516$		0			0
$517$		−	11.3649i		−	0.499830i
$518$		0			0
$519$		−	27.7128i		−	1.21646i
$520$		0			0
$521$	22.3153			0.977653			0.488826	−	0.872381i	$-0.337425\pi$
$521$	22.3153			0.977653			0.488826	+	0.872381i	$0.337425\pi$
$522$		0			0
$523$			26.1940i			1.14538i	0.819771	+	0.572692i	$0.194101\pi$
$523$			26.1940i			1.14538i	−0.819771	+	0.572692i	$0.805899\pi$
$524$		0			0
$525$	−11.0540	+	42.9683i	−0.482435	+	1.87529i
$526$		0			0
$527$			15.8017i			0.688332i
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$	39.3693			1.70527
$534$		0			0
$535$	31.6847	+	4.01029i	1.36985	+	0.173380i
$536$		0			0
$537$	30.7386			1.32647
$538$		0			0
$539$			4.86319i			0.209472i
$540$		0			0
$541$			38.5316i			1.65660i	0.560284	+	0.828301i	$0.310692\pi$
$541$			38.5316i			1.65660i	−0.560284	+	0.828301i	$0.689308\pi$
$542$		0			0
$543$	−35.0540			−1.50431
$544$		0			0
$545$	1.59612	−	12.6107i	0.0683702	−	0.540182i
$546$		0			0
$547$			7.35465i			0.314462i	0.987562	+	0.157231i	$0.0502568\pi$
$547$			7.35465i			0.314462i	−0.987562	+	0.157231i	$0.949743\pi$
$548$		0			0
$549$		−	6.92820i		−	0.295689i
$550$		0			0
$551$	−12.0000	−	14.2829i	−0.511217	−	0.608470i
$552$		0			0
$553$	51.3693			2.18445
$554$		0			0
$555$	−5.12311	+	40.4768i	−0.217464	+	1.71815i
$556$		0			0
$557$			6.92820i			0.293557i	0.989169	+	0.146779i	$0.0468905\pi$
$557$			6.92820i			0.293557i	−0.989169	+	0.146779i	$0.953109\pi$
$558$		0			0
$559$			24.1290i			1.02055i
$560$		0			0
$561$			4.98293i			0.210379i
$562$		0			0
$563$	15.6847			0.661030			0.330515	−	0.943801i	$-0.392778\pi$
$563$	15.6847			0.661030			0.330515	+	0.943801i	$0.392778\pi$
$564$		0			0
$565$	−1.50758	+	11.9111i	−0.0634243	+	0.501105i
$566$		0			0
$567$			24.2487i			1.01835i
$568$		0			0
$569$			3.89055i			0.163100i	0.996669	+	0.0815502i	$0.0259871\pi$
$569$			3.89055i			0.163100i	−0.996669	+	0.0815502i	$0.974013\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$			8.87348i			0.370695i
$574$		0			0
$575$	−4.31534	+	16.7743i	−0.179962	+	0.699538i
$576$		0			0
$577$	−44.2462			−1.84199			−0.920997	−	0.389570i	$-0.872624\pi$
$577$	−44.2462			−1.84199			−0.920997	+	0.389570i	$0.872624\pi$
$578$		0			0
$579$	21.1231			0.877846
$580$		0			0
$581$	−36.0000			−1.49353
$582$		0			0
$583$	4.31534			0.178723
$584$		0			0
$585$	35.0540	+	4.43674i	1.44930	+	0.183437i
$586$		0			0
$587$			13.4300i			0.554314i	0.960825	+	0.277157i	$0.0893921\pi$
$587$			13.4300i			0.554314i	−0.960825	+	0.277157i	$0.910608\pi$
$588$		0			0
$589$	−27.3693			−1.12773
$590$		0			0
$591$			27.7128i			1.13995i
$592$		0			0
$593$			6.38202i			0.262078i	0.991377	+	0.131039i	$0.0418313\pi$
$593$			6.38202i			0.262078i	−0.991377	+	0.131039i	$0.958169\pi$
$594$		0			0
$595$	−15.3693	−	1.94528i	−0.630081	−	0.0797485i
$596$		0			0
$597$	39.3693			1.61128
$598$		0			0
$599$			0.972638i			0.0397409i	0.999803	+	0.0198705i	$0.00632538\pi$
$599$			0.972638i			0.0397409i	−0.999803	+	0.0198705i	$0.993675\pi$
$600$		0			0
$601$		−	8.87348i		−	0.361957i	−0.983487	−	0.180978i	$-0.942074\pi$
$601$		−	8.87348i		−	0.361957i	0.983487	−	0.180978i	$-0.0579264\pi$
$602$		0			0
$603$			43.9409i			1.78941i
$604$		0			0
$605$	−2.82292	+	22.3034i	−0.114768	+	0.906764i
$606$		0			0
$607$	−0.177081			−0.00718749			−0.00359375	−	0.999994i	$-0.501144\pi$
$607$	−0.177081			−0.00718749			−0.00359375	+	0.999994i	$0.501144\pi$
$608$		0			0
$609$	−30.7386	−	36.5863i	−1.24559	−	1.48255i
$610$		0			0
$611$		−	51.8418i		−	2.09729i
$612$		0			0
$613$			14.4026i			0.581715i	0.956766	+	0.290858i	$0.0939406\pi$
$613$			14.4026i			0.581715i	−0.956766	+	0.290858i	$0.906059\pi$
$614$		0			0
$615$	−50.4233	−	6.38202i	−2.03326	−	0.257348i
$616$		0			0
$617$	13.3693			0.538228			0.269114	−	0.963108i	$-0.413269\pi$
$617$	13.3693			0.538228			0.269114	+	0.963108i	$0.413269\pi$
$618$		0			0
$619$		−	10.9385i		−	0.439655i	−0.975539	−	0.219828i	$-0.929451\pi$
$619$		−	10.9385i		−	0.439655i	0.975539	−	0.219828i	$-0.0705495\pi$
$620$		0			0
$621$		−	4.98293i		−	0.199958i
$622$		0			0
$623$	30.7386			1.23152
$624$		0			0
$625$	21.8963	+	12.0645i	0.875852	+	0.482579i
$626$		0			0
$627$	−8.63068			−0.344676
$628$		0			0
$629$	−14.2462			−0.568034
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		−	37.9854i		−	1.50978i
$634$		0			0
$635$	−0.630683	+	4.98293i	−0.0250279	+	0.197741i
$636$		0			0
$637$			22.1837i			0.878950i
$638$		0			0
$639$	8.00000			0.316475
$640$		0			0
$641$			4.98293i			0.196814i	0.995146	+	0.0984069i	$0.0313747\pi$
$641$			4.98293i			0.196814i	−0.995146	+	0.0984069i	$0.968625\pi$
$642$		0			0
$643$			35.0675i			1.38293i	0.722412	+	0.691463i	$0.243033\pi$
$643$			35.0675i			1.38293i	−0.722412	+	0.691463i	$0.756967\pi$
$644$		0			0
$645$	3.91146	−	30.9038i	0.154014	−	1.21684i
$646$		0			0
$647$		−	48.9239i		−	1.92340i	−0.274111	−	0.961698i	$-0.588384\pi$
$647$		−	48.9239i		−	1.92340i	0.274111	−	0.961698i	$-0.411616\pi$
$648$		0			0
$649$		−	1.09238i		−	0.0428795i
$650$		0			0
$651$	−70.1080			−2.74775
$652$		0			0
$653$	−28.7386			−1.12463			−0.562315	−	0.826923i	$-0.690089\pi$
$653$	−28.7386			−1.12463			−0.562315	+	0.826923i	$0.690089\pi$
$654$		0			0
$655$	−31.6847	−	4.01029i	−1.23802	−	0.156695i
$656$		0			0
$657$	−25.3693			−0.989751
$658$		0			0
$659$			35.6137i			1.38731i	0.720307	+	0.693655i	$0.244001\pi$
$659$			35.6137i			1.38731i	−0.720307	+	0.693655i	$0.755999\pi$
$660$		0			0
$661$	−24.7386			−0.962221			−0.481111	−	0.876660i	$-0.659766\pi$
$661$	−24.7386			−0.962221			−0.481111	+	0.876660i	$0.659766\pi$
$662$		0			0
$663$			22.7299i			0.882756i
$664$		0			0
$665$	3.36932	−	26.6204i	0.130657	−	1.03230i
$666$		0			0
$667$	−12.0000	−	14.2829i	−0.464642	−	0.553034i
$668$		0			0
$669$		−	31.6034i		−	1.22186i
$670$		0			0
$671$	−1.89205			−0.0730418
$672$		0			0
$673$			29.1119i			1.12218i	0.827754	+	0.561091i	$0.189618\pi$
$673$			29.1119i			1.12218i	−0.827754	+	0.561091i	$0.810382\pi$
$674$		0			0
$675$	−6.96543	−	1.79192i	−0.268100	−	0.0689710i
$676$		0			0
$677$	−13.3693			−0.513825			−0.256912	−	0.966435i	$-0.582705\pi$
$677$	−13.3693			−0.513825			−0.256912	+	0.966435i	$0.582705\pi$
$678$		0			0
$679$		−	28.5657i		−	1.09625i
$680$		0			0
$681$			31.6034i			1.21104i
$682$		0			0
$683$			16.2281i			0.620952i	0.950581	+	0.310476i	$0.100488\pi$
$683$			16.2281i			0.620952i	−0.950581	+	0.310476i	$0.899512\pi$
$684$		0			0
$685$	−3.75379	+	29.6581i	−0.143425	+	1.13318i
$686$		0			0
$687$		−	40.4768i		−	1.54429i
$688$		0			0
$689$	19.6847			0.749926
$690$		0			0
$691$	−18.7386			−0.712851			−0.356426	−	0.934324i	$-0.616005\pi$
$691$	−18.7386			−0.712851			−0.356426	+	0.934324i	$0.616005\pi$
$692$		0			0
$693$	−12.0000			−0.455842
$694$		0			0
$695$	5.43845	−	42.9683i	0.206292	−	1.62988i
$696$		0			0
$697$		−	17.7470i		−	0.672214i
$698$		0			0
$699$			6.38202i			0.241390i
$700$		0			0
$701$	−50.8078			−1.91898			−0.959491	−	0.281738i	$-0.909089\pi$
$701$	−50.8078			−1.91898			−0.959491	+	0.281738i	$0.909089\pi$
$702$		0			0
$703$		−	24.6752i		−	0.930641i
$704$		0			0
$705$	−8.40388	+	66.3977i	−0.316509	+	2.50068i
$706$		0			0
$707$	24.0000			0.902613
$708$		0			0
$709$	−6.31534			−0.237178			−0.118589	−	0.992943i	$-0.537837\pi$
$709$	−6.31534			−0.237178			−0.118589	+	0.992943i	$0.537837\pi$
$710$		0			0
$711$			52.8144i			1.98070i
$712$		0			0
$713$	−27.3693			−1.02499
$714$		0			0
$715$	1.21165	−	9.57302i	0.0453130	−	0.358011i
$716$		0			0
$717$	−33.6155			−1.25539
$718$		0			0
$719$	−26.2462			−0.978819			−0.489409	−	0.872054i	$-0.662788\pi$
$719$	−26.2462			−0.978819			−0.489409	+	0.872054i	$0.662788\pi$
$720$		0			0
$721$	5.26137			0.195944
$722$		0			0
$723$	−4.31534			−0.160489
$724$		0			0
$725$	−24.2808	+	11.6380i	−0.901765	+	0.432226i
$726$		0			0
$727$	4.49242			0.166615			0.0833074	−	0.996524i	$-0.473452\pi$
$727$	4.49242			0.166615			0.0833074	+	0.996524i	$0.473452\pi$
$728$		0			0
$729$	−35.9848			−1.33277
$730$		0			0
$731$	10.8769			0.402296
$732$		0			0
$733$	−16.8769			−0.623362			−0.311681	−	0.950187i	$-0.600892\pi$
$733$	−16.8769			−0.623362			−0.311681	+	0.950187i	$0.600892\pi$
$734$		0			0
$735$	3.59612	−	28.4124i	0.132645	−	1.04801i
$736$		0			0
$737$	12.0000			0.442026
$738$		0			0
$739$		−	27.5931i		−	1.01503i	−0.861644	−	0.507514i	$-0.830565\pi$
$739$		−	27.5931i		−	1.01503i	0.861644	−	0.507514i	$-0.169435\pi$
$740$		0			0
$741$	−39.3693			−1.44627
$742$		0			0
$743$	32.0000			1.17397			0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000			1.17397			0.586983	+	0.809599i	$0.300316\pi$
$744$		0			0
$745$	3.84233	−	30.3576i	0.140772	−	1.11222i
$746$		0			0
$747$		−	37.0127i		−	1.35423i
$748$		0			0
$749$	−49.4773			−1.80786
$750$		0			0
$751$		−	6.50175i		−	0.237252i	−0.992939	−	0.118626i	$-0.962151\pi$
$751$		−	6.50175i		−	0.237252i	0.992939	−	0.118626i	$-0.0378490\pi$
$752$		0			0
$753$			42.9683i			1.56585i
$754$		0			0
$755$	4.31534	−	34.0948i	0.157051	−	1.24084i
$756$		0			0
$757$	15.7538			0.572581			0.286291	−	0.958143i	$-0.407578\pi$
$757$	15.7538			0.572581			0.286291	+	0.958143i	$0.407578\pi$
$758$		0			0
$759$	−8.63068			−0.313274
$760$		0			0
$761$	−1.50758			−0.0546496			−0.0273248	−	0.999627i	$-0.508699\pi$
$761$	−1.50758			−0.0546496			−0.0273248	+	0.999627i	$0.508699\pi$
$762$		0			0
$763$			19.6922i			0.712907i
$764$		0			0
$765$	2.00000	−	15.8017i	0.0723102	−	0.571311i
$766$		0			0
$767$		−	4.98293i		−	0.179923i
$768$		0			0
$769$			32.6957i			1.17904i	0.807754	+	0.589519i	$0.200683\pi$
$769$			32.6957i			1.17904i	−0.807754	+	0.589519i	$0.799317\pi$
$770$		0			0
$771$		−	29.1119i		−	1.04844i
$772$		0			0
$773$	26.0000			0.935155			0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000			0.935155			0.467578	+	0.883952i	$0.345127\pi$
$774$		0			0
$775$	−9.84233	+	38.2585i	−0.353547	+	1.37428i
$776$		0			0
$777$		−	63.2067i		−	2.26753i
$778$		0			0
$779$	30.7386			1.10133
$780$		0			0
$781$		−	2.18475i		−	0.0781765i
$782$		0			0
$783$	5.93087	−	4.98293i	0.211952	−	0.178075i
$784$		0			0
$785$	−6.31534	+	49.8965i	−0.225404	+	1.78088i
$786$		0			0
$787$			12.3376i			0.439787i	0.975524	+	0.219894i	$0.0705710\pi$
$787$			12.3376i			0.439787i	−0.975524	+	0.219894i	$0.929429\pi$
$788$		0			0
$789$	−9.43845			−0.336018
$790$		0			0
$791$		−	18.5999i		−	0.661335i
$792$		0			0
$793$	−8.63068			−0.306485
$794$		0			0
$795$	−25.2116	−	3.19101i	−0.894165	−	0.113173i
$796$		0			0
$797$	34.0000			1.20434			0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000			1.20434			0.602171	+	0.798367i	$0.294303\pi$
$798$		0			0
$799$	−23.3693			−0.826747
$800$		0			0
$801$			31.6034i			1.11665i
$802$		0			0
$803$			6.92820i			0.244491i
$804$		0			0
$805$	3.36932	−	26.6204i	0.118753	−	0.938247i
$806$		0			0
$807$			40.4768i			1.42485i
$808$		0			0
$809$		−	31.6034i		−	1.11112i	−0.831478	−	0.555558i	$-0.812505\pi$
$809$		−	31.6034i		−	1.11112i	0.831478	−	0.555558i	$-0.187495\pi$
$810$		0			0
$811$	−4.63068			−0.162605			−0.0813026	−	0.996689i	$-0.525908\pi$
$811$	−4.63068			−0.162605			−0.0813026	+	0.996689i	$0.525908\pi$
$812$		0			0
$813$			37.9854i			1.33221i
$814$		0			0
$815$	−1.52699	+	12.0645i	−0.0534880	+	0.422601i
$816$		0			0
$817$			18.8393i			0.659105i
$818$		0			0
$819$	−54.7386			−1.91272
$820$		0			0
$821$	47.7926			1.66797			0.833987	−	0.551784i	$-0.186053\pi$
$821$	47.7926			1.66797			0.833987	+	0.551784i	$0.186053\pi$
$822$		0			0
$823$	21.7538			0.758289			0.379145	−	0.925337i	$-0.376218\pi$
$823$	21.7538			0.758289			0.379145	+	0.925337i	$0.376218\pi$
$824$		0			0
$825$	−3.10370	+	12.0645i	−0.108057	+	0.420032i
$826$		0			0
$827$	15.0540			0.523478			0.261739	−	0.965139i	$-0.415704\pi$
$827$	15.0540			0.523478			0.261739	+	0.965139i	$0.415704\pi$
$828$		0			0
$829$			23.5828i			0.819064i	0.912296	+	0.409532i	$0.134308\pi$
$829$			23.5828i			0.819064i	−0.912296	+	0.409532i	$0.865692\pi$
$830$		0			0
$831$			27.7128i			0.961347i
$832$		0			0
$833$	10.0000			0.346479
$834$		0			0
$835$	18.7386	+	2.37173i	0.648477	+	0.0820770i
$836$		0			0
$837$		−	11.3649i		−	0.392830i
$838$		0			0
$839$			18.7196i			0.646272i	0.946352	+	0.323136i	$0.104737\pi$
$839$			18.7196i			0.646272i	−0.946352	+	0.323136i	$0.895263\pi$
$840$		0			0
$841$	5.00000	−	28.5657i	0.172414	−	0.985025i
$842$		0			0
$843$	35.0540			1.20732
$844$		0			0
$845$	1.87689	−	14.8290i	0.0645671	−	0.510135i
$846$		0			0
$847$		−	34.8280i		−	1.19670i
$848$		0			0
$849$		−	36.5863i		−	1.25564i
$850$		0			0
$851$		−	24.6752i		−	0.845854i
$852$		0			0
$853$	37.8617			1.29636			0.648181	−	0.761487i	$-0.275530\pi$
$853$	37.8617			1.29636			0.648181	+	0.761487i	$0.275530\pi$
$854$		0			0
$855$	27.3693	+	3.46410i	0.936011	+	0.118470i
$856$		0			0
$857$		−	20.2384i		−	0.691331i	−0.938358	−	0.345666i	$-0.887653\pi$
$857$		−	20.2384i		−	0.691331i	0.938358	−	0.345666i	$-0.112347\pi$
$858$		0			0
$859$			52.2682i			1.78337i	0.452657	+	0.891685i	$0.350476\pi$
$859$			52.2682i			1.78337i	−0.452657	+	0.891685i	$0.649524\pi$
$860$		0			0
$861$	78.7386			2.68341
$862$		0			0
$863$			15.3752i			0.523379i	0.965152	+	0.261689i	$0.0842796\pi$
$863$			15.3752i			0.523379i	−0.965152	+	0.261689i	$0.915720\pi$
$864$		0			0
$865$	24.0000	+	3.03765i	0.816024	+	0.103283i
$866$		0			0
$867$	−33.3002			−1.13093
$868$		0			0
$869$	14.4233			0.489277
$870$		0			0
$871$	54.7386			1.85475
$872$		0			0
$873$	29.3693			0.994001
$874$		0			0
$875$	−36.0000	−	14.2829i	−1.21702	−	0.482849i
$876$		0			0
$877$			41.0230i			1.38525i	0.721298	+	0.692625i	$0.243546\pi$
$877$			41.0230i			1.38525i	−0.721298	+	0.692625i	$0.756454\pi$
$878$		0			0
$879$	3.50758			0.118308
$880$		0			0
$881$		−	45.4598i		−	1.53158i	−0.643092	−	0.765789i	$-0.722349\pi$
$881$		−	45.4598i		−	1.53158i	0.643092	−	0.765789i	$-0.277651\pi$
$882$		0			0
$883$		−	35.9204i		−	1.20882i	−0.796675	−	0.604408i	$-0.793410\pi$
$883$		−	35.9204i		−	1.20882i	0.796675	−	0.604408i	$-0.206590\pi$
$884$		0			0
$885$	−0.807764	+	6.38202i	−0.0271527	+	0.214529i
$886$		0			0
$887$	3.68466			0.123719			0.0618594	−	0.998085i	$-0.480297\pi$
$887$	3.68466			0.123719			0.0618594	+	0.998085i	$0.480297\pi$
$888$		0			0
$889$		−	7.78110i		−	0.260970i
$890$		0			0
$891$			6.80847i			0.228092i
$892$		0			0
$893$		−	40.4768i		−	1.35451i
$894$		0			0
$895$	−3.36932	+	26.6204i	−0.112624	+	0.889823i
$896$		0			0
$897$	−39.3693			−1.31450
$898$		0			0
$899$	−27.3693	−	32.5760i	−0.912818	−	1.08647i
$900$		0			0
$901$		−	8.87348i		−	0.295618i
$902$		0			0
$903$			48.2579i			1.60592i
$904$		0			0
$905$	3.84233	−	30.3576i	0.127723	−	1.00912i
$906$		0			0
$907$	40.4924			1.34453			0.672264	−	0.740311i	$-0.265322\pi$
$907$	40.4924			1.34453			0.672264	+	0.740311i	$0.265322\pi$
$908$		0			0
$909$			24.6752i			0.818423i
$910$		0			0
$911$			9.84612i			0.326216i	0.986608	+	0.163108i	$0.0521520\pi$
$911$			9.84612i			0.326216i	−0.986608	+	0.163108i	$0.947848\pi$
$912$		0			0
$913$	−10.1080			−0.334524
$914$		0			0
$915$	11.0540	+	1.39909i	0.365433	+	0.0462524i
$916$		0			0
$917$	49.4773			1.63388
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$	−8.17708			−0.269444
$922$		0			0
$923$		−	9.96585i		−	0.328030i
$924$		0			0
$925$	−34.4924	−	8.87348i	−1.13410	−	0.291758i
$926$		0			0
$927$			5.40938i			0.177667i
$928$		0			0
$929$	48.7386			1.59906			0.799531	−	0.600624i	$-0.205081\pi$
$929$	48.7386			1.59906			0.799531	+	0.600624i	$0.205081\pi$
$930$		0			0
$931$			17.3205i			0.567657i
$932$		0			0
$933$		−	1.09238i		−	0.0357628i
$934$		0			0
$935$	−4.31534	−	0.546188i	−0.141127	−	0.0178622i
$936$		0			0
$937$		−	31.6034i		−	1.03244i	−0.856457	−	0.516218i	$-0.827339\pi$
$937$		−	31.6034i		−	1.03244i	0.856457	−	0.516218i	$-0.172661\pi$
$938$		0			0
$939$			6.38202i			0.208269i
$940$		0			0
$941$	32.4233			1.05697			0.528485	−	0.848943i	$-0.322760\pi$
$941$	32.4233			1.05697			0.528485	+	0.848943i	$0.322760\pi$
$942$		0			0
$943$	30.7386			1.00099
$944$		0			0
$945$	11.0540	+	1.39909i	0.359586	+	0.0455123i
$946$		0			0
$947$	−23.6847			−0.769648			−0.384824	−	0.922990i	$-0.625738\pi$
$947$	−23.6847			−0.769648			−0.384824	+	0.922990i	$0.625738\pi$
$948$		0			0
$949$			31.6034i			1.02589i
$950$		0			0
$951$	5.12311			0.166128
$952$		0			0
$953$			10.2726i			0.332761i	0.986062	+	0.166381i	$0.0532080\pi$
$953$			10.2726i			0.332761i	−0.986062	+	0.166381i	$0.946792\pi$
$954$		0			0
$955$	−7.68466	−	0.972638i	−0.248670	−	0.0314738i
$956$		0			0
$957$	−8.63068	−	10.2726i	−0.278990	−	0.332065i
$958$		0			0
$959$		−	46.3127i		−	1.49551i
$960$		0			0
$961$	−31.4233			−1.01365
$962$		0			0
$963$		−	50.8691i		−	1.63924i
$964$		0			0
$965$	−2.31534	+	18.2931i	−0.0745335	+	0.588877i
$966$		0			0
$967$	−45.9309			−1.47704			−0.738519	−	0.674233i	$-0.764474\pi$
$967$	−45.9309			−1.47704			−0.738519	+	0.674233i	$0.764474\pi$
$968$		0			0
$969$			17.7470i			0.570114i
$970$		0			0
$971$			45.8862i			1.47256i	0.676678	+	0.736279i	$0.263419\pi$
$971$			45.8862i			1.47256i	−0.676678	+	0.736279i	$0.736581\pi$
$972$		0			0
$973$			67.0973i			2.15104i
$974$		0			0
$975$	−14.1577	+	55.0328i	−0.453408	+	1.76246i
$976$		0			0
$977$			6.38202i			0.204179i	0.994775	+	0.102089i	$0.0325528\pi$
$977$			6.38202i			0.204179i	−0.994775	+	0.102089i	$0.967447\pi$
$978$		0			0
$979$	8.63068			0.275838
$980$		0			0
$981$	−20.2462			−0.646412
$982$		0			0
$983$	−27.0540			−0.862888			−0.431444	−	0.902140i	$-0.641996\pi$
$983$	−27.0540			−0.862888			−0.431444	+	0.902140i	$0.641996\pi$
$984$		0			0
$985$	−24.0000	−	3.03765i	−0.764704	−	0.0967876i
$986$		0			0
$987$		−	103.684i		−	3.30028i
$988$		0			0
$989$			18.8393i			0.599056i
$990$		0			0
$991$	30.7386			0.976445			0.488222	−	0.872719i	$-0.337645\pi$
$991$	30.7386			0.976445			0.488222	+	0.872719i	$0.337645\pi$
$992$		0			0
$993$		−	15.2555i		−	0.484118i
$994$		0			0
$995$	−4.31534	+	34.0948i	−0.136806	+	1.08088i
$996$		0			0
$997$	59.6155			1.88804			0.944021	−	0.329884i	$-0.107010\pi$
$997$	59.6155			1.88804			0.944021	+	0.329884i	$0.107010\pi$
$998$		0			0
$999$	10.2462			0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.j.c.289.4		4
4.3	odd	2		145.2.d.c.144.2	yes	4
5.4	even	2		2320.2.j.a.289.1		4
12.11	even	2		1305.2.f.e.289.3		4
20.3	even	4		725.2.c.f.376.2		8
20.7	even	4		725.2.c.f.376.7		8
20.19	odd	2		145.2.d.a.144.3	✓	4
29.28	even	2		2320.2.j.a.289.2		4
60.59	even	2		1305.2.f.i.289.4		4
116.115	odd	2		145.2.d.a.144.4	yes	4
145.144	even	2	inner	2320.2.j.c.289.3		4
348.347	even	2		1305.2.f.i.289.3		4
580.347	even	4		725.2.c.f.376.1		8
580.463	even	4		725.2.c.f.376.8		8
580.579	odd	2		145.2.d.c.144.1	yes	4
1740.1739	even	2		1305.2.f.e.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.d.a.144.3	✓	4	20.19	odd	2
145.2.d.a.144.4	yes	4	116.115	odd	2
145.2.d.c.144.1	yes	4	580.579	odd	2
145.2.d.c.144.2	yes	4	4.3	odd	2
725.2.c.f.376.1		8	580.347	even	4
725.2.c.f.376.2		8	20.3	even	4
725.2.c.f.376.7		8	20.7	even	4
725.2.c.f.376.8		8	580.463	even	4
1305.2.f.e.289.3		4	12.11	even	2
1305.2.f.e.289.4		4	1740.1739	even	2
1305.2.f.i.289.3		4	348.347	even	2
1305.2.f.i.289.4		4	60.59	even	2
2320.2.j.a.289.1		4	5.4	even	2
2320.2.j.a.289.2		4	29.28	even	2
2320.2.j.c.289.3		4	145.144	even	2	inner
2320.2.j.c.289.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +(-0.280776 + 2.21837i) q^{5} -3.46410i q^{7} +3.56155 q^{9} -0.972638i q^{11} -4.43674i q^{13} +(-0.719224 + 5.68247i) q^{15} -2.00000 q^{17} -3.46410i q^{19} -8.87348i q^{21} -3.46410i q^{23} +(-4.84233 - 1.24573i) q^{25} +1.43845 q^{27} +(4.12311 - 3.46410i) q^{29} -7.90084i q^{31} -2.49146i q^{33} +(7.68466 + 0.972638i) q^{35} +7.12311 q^{37} -11.3649i q^{39} +8.87348i q^{41} -5.43845 q^{43} +(-1.00000 + 7.90084i) q^{45} +11.6847 q^{47} -5.00000 q^{49} -5.12311 q^{51} +4.43674i q^{53} +(2.15767 + 0.273094i) q^{55} -8.87348i q^{57} +1.12311 q^{59} -1.94528i q^{61} -12.3376i q^{63} +(9.84233 + 1.24573i) q^{65} +12.3376i q^{67} -8.87348i q^{69} +2.24621 q^{71} -7.12311 q^{73} +(-12.4039 - 3.19101i) q^{75} -3.36932 q^{77} +14.8290i q^{79} -7.00000 q^{81} -10.3923i q^{83} +(0.561553 - 4.43674i) q^{85} +(10.5616 - 8.87348i) q^{87} +8.87348i q^{89} -15.3693 q^{91} -20.2384i q^{93} +(7.68466 + 0.972638i) q^{95} +8.24621 q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 3 q^{5} + 6 q^{9} - 7 q^{15} - 8 q^{17} - 7 q^{25} + 14 q^{27} + 6 q^{35} + 12 q^{37} - 30 q^{43} - 4 q^{45} + 22 q^{47} - 20 q^{49} - 4 q^{51} + 21 q^{55} - 12 q^{59} + 27 q^{65} - 24 q^{71}+ \cdots + 6 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 3 * q^5 + 6 * q^9 - 7 * q^15 - 8 * q^17 - 7 * q^25 + 14 * q^27 + 6 * q^35 + 12 * q^37 - 30 * q^43 - 4 * q^45 + 22 * q^47 - 20 * q^49 - 4 * q^51 + 21 * q^55 - 12 * q^59 + 27 * q^65 - 24 * q^71 - 12 * q^73 - 29 * q^75 + 36 * q^77 - 28 * q^81 - 6 * q^85 + 34 * q^87 - 12 * q^91 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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