LMFDB - Embedded newform 155.2.f.a.92.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [155,2,Mod(92,155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(155, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

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

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

chi := DirichletCharacter("155.92");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 155 = 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	155.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.23768123133$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.1931902335935778816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{6} + 64 $ x^12 - 15*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			92.1
Root			$0.633359 - 1.26446i$ of defining polynomial
Character	$\chi$	$=$	155.92
Dual form			155.2.f.a.123.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.89782 - 1.89782i) q^{2} +5.20343i q^{4} +(2.23507 + 0.0667460i) q^{5} +(-3.43053 - 3.43053i) q^{7} +(6.07952 - 6.07952i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.89782 - 1.89782i) q^{2} +5.20343i q^{4} +(2.23507 + 0.0667460i) q^{5} +(-3.43053 - 3.43053i) q^{7} +(6.07952 - 6.07952i) q^{8} -3.00000i q^{9} +(-4.11509 - 4.36843i) q^{10} +13.0210i q^{14} -12.6688 q^{16} +(-5.69345 + 5.69345i) q^{18} -5.19133i q^{19} +(-0.347308 + 11.6300i) q^{20} +(4.99109 + 0.298364i) q^{25} +(17.8505 - 17.8505i) q^{28} -5.56776 q^{31} +(11.8840 + 11.8840i) q^{32} +(-7.43850 - 7.89644i) q^{35} +15.6103 q^{36} +(-9.85219 + 9.85219i) q^{38} +(13.9939 - 13.1824i) q^{40} +9.71520 q^{41} +(0.200238 - 6.70521i) q^{45} +(9.56776 + 9.56776i) q^{47} +16.5370i q^{49} +(-8.90594 - 10.0384i) q^{50} -41.7119 q^{56} -3.13016i q^{59} +(10.5666 + 10.5666i) q^{62} +(-10.2916 + 10.2916i) q^{63} -19.7698i q^{64} +(-0.432236 - 0.432236i) q^{67} +(-0.869101 + 29.1029i) q^{70} +15.9439 q^{71} +(-18.2386 - 18.2386i) q^{72} +27.0127 q^{76} +(-28.3156 - 0.845590i) q^{80} -9.00000 q^{81} +(-18.4377 - 18.4377i) q^{82} +(-13.1053 + 12.3453i) q^{90} -36.3157i q^{94} +(0.346500 - 11.6030i) q^{95} +(-3.91782 - 3.91782i) q^{97} +(31.3842 - 31.3842i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{8}+O(q^{10}) $ 12 * q - 6 * q^8	$ 12 q - 6 q^{8} - 48 q^{16} + 54 q^{28} - 12 q^{32} - 24 q^{35} + 72 q^{36} - 66 q^{38} + 36 q^{40} + 48 q^{47} - 72 q^{67} + 6 q^{70} + 18 q^{72} - 84 q^{80} - 108 q^{81} - 42 q^{82} - 54 q^{90} + 96 q^{95} + 114 q^{98}+O(q^{100}) $ 12 * q - 6 * q^8 - 48 * q^16 + 54 * q^28 - 12 * q^32 - 24 * q^35 + 72 * q^36 - 66 * q^38 + 36 * q^40 + 48 * q^47 - 72 * q^67 + 6 * q^70 + 18 * q^72 - 84 * q^80 - 108 * q^81 - 42 * q^82 - 54 * q^90 + 96 * q^95 + 114 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/155\mathbb{Z}\right)^\times$.

$n$	$32$	$96$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

$2$	−1.89782	−	1.89782i	−1.34196	−	1.34196i	−0.894108	−	0.447852i	$-0.852189\pi$
$2$	−1.89782	−	1.89782i	−1.34196	−	1.34196i	−0.447852	−	0.894108i	$-0.647811\pi$
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$			5.20343i			2.60171i
$5$	2.23507	+	0.0667460i	0.999554	+	0.0298497i
$5$	2.23507	+	0.0667460i	0.999554	+	0.0298497i
$6$		0			0
$7$	−3.43053	−	3.43053i	−1.29662	−	1.29662i	−0.930614	−	0.366003i	$-0.880726\pi$
$7$	−3.43053	−	3.43053i	−1.29662	−	1.29662i	−0.366003	−	0.930614i	$-0.619274\pi$
$8$	6.07952	−	6.07952i	2.14943	−	2.14943i
$9$		−	3.00000i		−	1.00000i
$10$	−4.11509	−	4.36843i	−1.30130	−	1.38142i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$			13.0210i			3.48002i
$15$		0			0
$16$	−12.6688			−3.16720
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−5.69345	+	5.69345i	−1.34196	+	1.34196i
$19$		−	5.19133i		−	1.19097i	−0.803366	−	0.595486i	$-0.796959\pi$
$19$		−	5.19133i		−	1.19097i	0.803366	−	0.595486i	$-0.203041\pi$
$20$	−0.347308	+	11.6300i	−0.0776603	+	2.60055i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	4.99109	+	0.298364i	0.998218	+	0.0596728i
$26$		0			0
$27$		0			0
$28$	17.8505	−	17.8505i	3.37342	−	3.37342i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−5.56776			−1.00000
$31$	−5.56776			−1.00000
$32$	11.8840	+	11.8840i	2.10082	+	2.10082i
$33$		0			0
$34$		0			0
$35$	−7.43850	−	7.89644i	−1.25734	−	1.33474i
$36$	15.6103			2.60171
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	−9.85219	+	9.85219i	−1.59824	+	1.59824i
$39$		0			0
$40$	13.9939	−	13.1824i	2.21264	−	2.08432i
$41$	9.71520			1.51726			0.758629	−	0.651522i	$-0.225869\pi$
$41$	9.71520			1.51726			0.758629	+	0.651522i	$0.225869\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	0.200238	−	6.70521i	0.0298497	−	0.999554i
$46$		0			0
$47$	9.56776	+	9.56776i	1.39560	+	1.39560i	0.812142	+	0.583460i	$0.198301\pi$
$47$	9.56776	+	9.56776i	1.39560	+	1.39560i	0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$			16.5370i			2.36243i
$50$	−8.90594	−	10.0384i	−1.25949	−	1.41965i
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$	−41.7119			−5.57399
$57$		0			0
$58$		0			0
$59$		−	3.13016i		−	0.407513i	−0.979022	−	0.203756i	$-0.934685\pi$
$59$		−	3.13016i		−	0.407513i	0.979022	−	0.203756i	$-0.0653150\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	10.5666	+	10.5666i	1.34196	+	1.34196i
$63$	−10.2916	+	10.2916i	−1.29662	+	1.29662i
$64$		−	19.7698i		−	2.47123i
$65$		0			0
$66$		0			0
$67$	−0.432236	−	0.432236i	−0.0528060	−	0.0528060i	0.680211	−	0.733017i	$-0.261888\pi$
$67$	−0.432236	−	0.432236i	−0.0528060	−	0.0528060i	−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$		0			0
$70$	−0.869101	+	29.1029i	−0.103877	+	3.47846i
$71$	15.9439			1.89219			0.946094	−	0.323891i	$-0.104991\pi$
$71$	15.9439			1.89219			0.946094	+	0.323891i	$0.104991\pi$
$72$	−18.2386	−	18.2386i	−2.14943	−	2.14943i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	27.0127			3.09857
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−28.3156	−	0.845590i	−3.16578	−	0.0945399i
$81$	−9.00000			−1.00000
$82$	−18.4377	−	18.4377i	−2.03610	−	2.03610i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−13.1053	+	12.3453i	−1.38142	+	1.30130i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	36.3157i		−	3.74568i
$95$	0.346500	−	11.6030i	0.0355502	−	1.19044i
$96$		0			0
$97$	−3.91782	−	3.91782i	−0.397794	−	0.397794i	0.479660	−	0.877454i	$-0.340760\pi$
$97$	−3.91782	−	3.91782i	−0.397794	−	0.397794i	−0.877454	+	0.479660i	$0.840760\pi$
$98$	31.3842	−	31.3842i	3.17029	−	3.17029i
$99$		0			0
$100$	−1.55251	+	25.9708i	−0.155251	+	2.59708i
$101$	3.27669			0.326043			0.163021	−	0.986623i	$-0.447876\pi$
$101$	3.27669			0.326043			0.163021	+	0.986623i	$0.447876\pi$
$102$		0			0
$103$	−7.71345	+	7.71345i	−0.760029	+	0.760029i	−0.976327	−	0.216298i	$-0.930602\pi$
$103$	−7.71345	+	7.71345i	−0.760029	+	0.760029i	0.216298	+	0.976327i	$0.430602\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.122180	+	0.122180i	0.0118116	+	0.0118116i	0.712988	−	0.701176i	$-0.247341\pi$
$107$	0.122180	+	0.122180i	0.0118116	+	0.0118116i	−0.701176	+	0.712988i	$0.747341\pi$
$108$		0			0
$109$			20.0979i			1.92503i	0.271237	+	0.962513i	$0.412567\pi$
$109$			20.0979i			1.92503i	−0.271237	+	0.962513i	$0.587433\pi$
$110$		0			0
$111$		0			0
$112$	43.4606	+	43.4606i	4.10664	+	4.10664i
$113$	−3.98058	+	3.98058i	−0.374461	+	0.374461i	−0.869099	−	0.494638i	$-0.835301\pi$
$113$	−3.98058	+	3.98058i	−0.374461	+	0.374461i	0.494638	+	0.869099i	$0.335301\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−5.94048	+	5.94048i	−0.546866	+	0.546866i
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		−	28.9714i		−	2.60171i
$125$	11.1355	+	1.00000i	0.995992	+	0.0894427i
$126$	39.0631			3.48002
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−13.7515	+	13.7515i	−1.21547	+	1.21547i
$129$		0			0
$130$		0			0
$131$	−11.1355			−0.972916			−0.486458	−	0.873704i	$-0.661711\pi$
$131$	−11.1355			−0.972916			−0.486458	+	0.873704i	$0.661711\pi$
$132$		0			0
$133$	−17.8090	+	17.8090i	−1.54423	+	1.54423i
$134$			1.64061i			0.141727i
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	41.0886	−	38.7057i	3.47262	−	3.27123i
$141$		0			0
$142$	−30.2586	−	30.2586i	−2.53924	−	2.53924i
$143$		0			0
$144$			38.0064i			3.16720i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	22.2711i		−	1.82452i	−0.409616	−	0.912258i	$-0.634337\pi$
$149$		−	22.2711i		−	1.82452i	0.409616	−	0.912258i	$-0.365663\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−31.5608	−	31.5608i	−2.55992	−	2.55992i
$153$		0			0
$154$		0			0
$155$	−12.4444	−	0.371626i	−0.999554	−	0.0298497i
$156$		0			0
$157$	2.88048	+	2.88048i	0.229887	+	0.229887i	0.812645	−	0.582758i	$-0.198027\pi$
$157$	2.88048	+	2.88048i	0.229887	+	0.229887i	−0.582758	+	0.812645i	$0.698027\pi$
$158$		0			0
$159$		0			0
$160$	25.7684	+	27.3548i	2.03717	+	2.16259i
$161$		0			0
$162$	17.0804	+	17.0804i	1.34196	+	1.34196i
$163$	15.3047	−	15.3047i	1.19876	−	1.19876i	0.224220	−	0.974539i	$-0.428017\pi$
$163$	15.3047	−	15.3047i	1.19876	−	1.19876i	0.974539	−	0.224220i	$-0.0719834\pi$
$164$			50.5523i			3.94747i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−15.5740			−1.19097
$172$		0			0
$173$	−4.13553	+	4.13553i	−0.314418	+	0.314418i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	−4.13553	+	4.13553i	−0.314418	+	0.314418i	0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$	−16.0985	−	18.1456i	−1.21693	−	1.37168i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	34.8901	+	1.04192i	2.60055	+	0.0776603i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−49.7851	+	49.7851i	−3.63096	+	3.63096i
$189$		0			0
$190$	−22.6780	+	21.3628i	−1.64523	+	1.54982i
$191$	−14.2391			−1.03030			−0.515151	−	0.857099i	$-0.672264\pi$
$191$	−14.2391			−1.03030			−0.515151	+	0.857099i	$0.672264\pi$
$192$		0			0
$193$	17.7027	−	17.7027i	1.27427	−	1.27427i	0.330440	−	0.943827i	$-0.392803\pi$
$193$	17.7027	−	17.7027i	1.27427	−	1.27427i	0.943827	−	0.330440i	$-0.107197\pi$
$194$			14.8706i			1.06765i
$195$		0			0
$196$	−86.0491			−6.14637
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	32.1573	−	28.5295i	2.27387	−	2.01734i
$201$		0			0
$202$	−6.21856	−	6.21856i	−0.437536	−	0.437536i
$203$		0			0
$204$		0			0
$205$	21.7142	+	0.648450i	1.51658	+	0.0452897i
$206$	29.2775			2.03986
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−28.7576			−1.97975			−0.989876	−	0.141933i	$-0.954668\pi$
$211$	−28.7576			−1.97975			−0.989876	+	0.141933i	$0.954668\pi$
$212$		0			0
$213$		0			0
$214$		−	0.463753i		−	0.0317015i
$215$		0			0
$216$		0			0
$217$	19.1004	+	19.1004i	1.29662	+	1.29662i
$218$	38.1421	−	38.1421i	2.58331	−	2.58331i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		−	81.5368i		−	5.44791i
$225$	0.895092	−	14.9733i	0.0596728	−	0.998218i
$226$	15.1088			1.00502
$227$	19.5678	+	19.5678i	1.29876	+	1.29876i	0.929213	+	0.369546i	$0.120487\pi$
$227$	19.5678	+	19.5678i	1.29876	+	1.29876i	0.369546	+	0.929213i	$0.379513\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.2647	−	11.2647i	0.737977	−	0.737977i	−0.234209	−	0.972186i	$-0.575250\pi$
$233$	11.2647	−	11.2647i	0.737977	−	0.737977i	0.972186	+	0.234209i	$0.0752501\pi$
$234$		0			0
$235$	20.7460	+	22.0232i	1.35332	+	1.43664i
$236$	16.2876			1.06023
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−20.8760	−	20.8760i	−1.34196	−	1.34196i
$243$		0			0
$244$		0			0
$245$	−1.10378	+	36.9614i	−0.0705178	+	2.36138i
$246$		0			0
$247$		0			0
$248$	−33.8493	+	33.8493i	−2.14943	+	2.14943i
$249$		0			0
$250$	−19.2354	−	23.0310i	−1.21655	−	1.45661i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−53.5515	−	53.5515i	−3.37342	−	3.37342i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.6560			0.790997
$257$	−9.74153	−	9.74153i	−0.607660	−	0.607660i	0.334674	−	0.942334i	$-0.391374\pi$
$257$	−9.74153	−	9.74153i	−0.607660	−	0.607660i	−0.942334	+	0.334674i	$0.891374\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	21.1332	+	21.1332i	1.30561	+	1.30561i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$		0			0
$266$	67.5964			4.14460
$267$		0			0
$268$	2.24911	−	2.24911i	0.137386	−	0.137386i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$			16.7033i			1.00000i
$280$	−93.2291	−	2.78410i	−5.57150	−	0.166382i
$281$	−16.0904			−0.959872			−0.479936	−	0.877303i	$-0.659340\pi$
$281$	−16.0904			−0.959872			−0.479936	+	0.877303i	$0.659340\pi$
$282$		0			0
$283$	18.7033	−	18.7033i	1.11180	−	1.11180i	0.118888	−	0.992908i	$-0.462067\pi$
$283$	18.7033	−	18.7033i	1.11180	−	1.11180i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			82.9627i			4.92293i
$285$		0			0
$286$		0			0
$287$	−33.3282	−	33.3282i	−1.96730	−	1.96730i
$288$	35.6520	−	35.6520i	2.10082	−	2.10082i
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.1355	+	24.1355i	−1.41001	+	1.41001i	−0.650545	+	0.759468i	$0.725459\pi$
$293$	−24.1355	+	24.1355i	−1.41001	+	1.41001i	−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$	0.208926	−	6.99614i	0.0121641	−	0.407331i
$296$		0			0
$297$		0			0
$298$	−42.2664	+	42.2664i	−2.44843	+	2.44843i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$			65.7678i			3.77204i
$305$		0			0
$306$		0			0
$307$	15.0604	+	15.0604i	0.859540	+	0.859540i	0.991284	−	0.131744i	$-0.0420575\pi$
$307$	15.0604	+	15.0604i	0.859540	+	0.859540i	−0.131744	+	0.991284i	$0.542058\pi$
$308$		0			0
$309$		0			0
$310$	22.9118	+	24.3224i	1.30130	+	1.38142i
$311$	35.0044			1.98492			0.992458	−	0.122585i	$-0.0391184\pi$
$311$	35.0044			1.98492			0.992458	+	0.122585i	$0.0391184\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		−	10.9332i		−	0.616998i
$315$	−23.6893	+	22.3155i	−1.33474	+	1.25734i
$316$		0			0
$317$	24.5637	+	24.5637i	1.37964	+	1.37964i	0.845223	+	0.534413i	$0.179467\pi$
$317$	24.5637	+	24.5637i	1.37964	+	1.37964i	0.534413	+	0.845223i	$0.320533\pi$
$318$		0			0
$319$		0			0
$320$	1.31955	−	44.1869i	0.0737653	−	2.47012i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	46.8308i		−	2.60171i
$325$		0			0
$326$	−58.0912			−3.21737
$327$		0			0
$328$	59.0637	−	59.0637i	3.26125	−	3.26125i
$329$		−	65.6449i		−	3.61912i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.937228	−	0.994928i	−0.0512062	−	0.0543587i
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	−24.6716	+	24.6716i	−1.34196	+	1.34196i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	29.5566	+	29.5566i	1.59824	+	1.59824i
$343$	32.7170	−	32.7170i	1.76655	−	1.76655i
$344$		0			0
$345$		0			0
$346$	15.6970			0.843874
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	22.2711i		−	1.19214i	−0.802932	−	0.596071i	$-0.796728\pi$
$349$		−	22.2711i		−	1.19214i	0.802932	−	0.596071i	$-0.203272\pi$
$350$	−3.88501	+	64.9891i	−0.207662	+	3.47381i
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$	35.6357	+	1.06419i	1.89135	+	0.0564813i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3.85641i		−	0.203533i	−0.994808	−	0.101767i	$-0.967550\pi$
$359$		−	3.85641i		−	0.203533i	0.994808	−	0.101767i	$-0.0324495\pi$
$360$	−39.5471	−	41.9818i	−2.08432	−	2.21264i
$361$	−7.94988			−0.418415
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		−	29.1456i		−	1.51726i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−26.6916	+	26.6916i	−1.38204	+	1.38204i	−0.541050	+	0.840991i	$0.681973\pi$
$373$	−26.6916	+	26.6916i	−1.38204	+	1.38204i	−0.840991	+	0.541050i	$0.818027\pi$
$374$		0			0
$375$		0			0
$376$	116.335			5.99951
$377$		0			0
$378$		0			0
$379$			33.4066i			1.71598i	0.513665	+	0.857991i	$0.328287\pi$
$379$			33.4066i			1.71598i	−0.513665	+	0.857991i	$0.671713\pi$
$380$	60.3753	+	1.80299i	3.09719	+	0.0924913i
$381$		0			0
$382$	27.0232	+	27.0232i	1.38262	+	1.38262i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−67.1929			−3.42003
$387$		0			0
$388$	20.3861	−	20.3861i	1.03495	−	1.03495i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	100.537	+	100.537i	5.07789	+	5.07789i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.8560	−	18.8560i	−0.946355	−	0.946355i	0.0522772	−	0.998633i	$-0.483352\pi$
$397$	−18.8560	−	18.8560i	−0.946355	−	0.946355i	−0.998633	+	0.0522772i	$0.983352\pi$
$398$		0			0
$399$		0			0
$400$	−63.2310	−	3.77991i	−3.16155	−	0.188995i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$			17.0500i			0.848269i
$405$	−20.1156	−	0.600714i	−0.999554	−	0.0298497i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	−39.9789	−	42.4402i	−1.97442	−	2.09597i
$411$		0			0
$412$	−40.1364	−	40.1364i	−1.97738	−	1.97738i
$413$	−10.7381	+	10.7381i	−0.528388	+	0.528388i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	22.4972i		−	1.09906i	−0.835473	−	0.549531i	$-0.814806\pi$
$419$		−	22.4972i		−	1.09906i	0.835473	−	0.549531i	$-0.185194\pi$
$420$		0			0
$421$	−39.5282			−1.92649			−0.963244	−	0.268627i	$-0.913430\pi$
$421$	−39.5282			−1.92649			−0.963244	+	0.268627i	$0.913430\pi$
$422$	54.5766	+	54.5766i	2.65675	+	2.65675i
$423$	28.7033	−	28.7033i	1.39560	−	1.39560i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.635757	+	0.635757i	−0.0307305	+	0.0307305i
$429$		0			0
$430$		0			0
$431$	−11.1355			−0.536380			−0.268190	−	0.963366i	$-0.586425\pi$
$431$	−11.1355			−0.536380			−0.268190	+	0.963366i	$0.586425\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		−	72.4980i		−	3.48002i
$435$		0			0
$436$	−104.578			−5.00836
$437$		0			0
$438$		0			0
$439$			22.2042i			1.05975i	0.848076	+	0.529874i	$0.177761\pi$
$439$			22.2042i			1.05975i	−0.848076	+	0.529874i	$0.822239\pi$
$440$		0			0
$441$	49.6110			2.36243
$442$		0			0
$443$	11.3917	−	11.3917i	0.541235	−	0.541235i	−0.382656	−	0.923891i	$-0.624991\pi$
$443$	11.3917	−	11.3917i	0.541235	−	0.541235i	0.923891	+	0.382656i	$0.124991\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−67.8208	+	67.8208i	−3.20423	+	3.20423i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−30.1153	+	26.7178i	−1.41965	+	1.25949i
$451$		0			0
$452$	−20.7126	−	20.7126i	−0.974240	−	0.974240i
$453$		0			0
$454$		−	74.2721i		−	3.48576i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−42.7568			−1.98067
$467$	18.2527	+	18.2527i	0.844636	+	0.844636i	0.989458	−	0.144822i	$-0.0462610\pi$
$467$	18.2527	+	18.2527i	0.844636	+	0.844636i	−0.144822	+	0.989458i	$0.546261\pi$
$468$		0			0
$469$			2.96559i			0.136938i
$470$	2.42393	−	81.1683i	0.111808	−	3.74401i
$471$		0			0
$472$	−19.0299	−	19.0299i	−0.875921	−	0.875921i
$473$		0			0
$474$		0			0
$475$	1.54891	−	25.9104i	0.0710687	−	1.18885i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			41.5713i			1.89944i	0.313101	+	0.949720i	$0.398632\pi$
$479$			41.5713i			1.89944i	−0.313101	+	0.949720i	$0.601368\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$			57.2377i			2.60171i
$485$	−8.49510	−	9.01810i	−0.385743	−	0.409491i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$	72.2408	−	68.0513i	3.26351	−	3.07424i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	70.5368			3.16720
$497$	−54.6958	−	54.6958i	−2.45344	−	2.45344i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−5.20343	+	57.9429i	−0.232704	+	2.59128i
$501$		0			0
$502$		0			0
$503$	−22.6516	+	22.6516i	−1.00999	+	1.00999i	−0.0100368	+	0.999950i	$0.503195\pi$
$503$	−22.6516	+	22.6516i	−1.00999	+	1.00999i	−0.999950	+	0.0100368i	$0.996805\pi$
$504$			125.136i			5.57399i
$505$	7.32363	+	0.218706i	0.325897	+	0.00973227i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	3.48424	+	3.48424i	0.153983	+	0.153983i
$513$		0			0
$514$			36.9753i			1.63091i
$515$	−17.7550	+	16.7253i	−0.782377	+	0.737004i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	44.5421			1.95143			0.975713	−	0.219054i	$-0.0702971\pi$
$521$	44.5421			1.95143			0.975713	+	0.219054i	$0.0702971\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		−	57.9429i		−	2.53125i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$	−9.39049			−0.407513
$532$	−92.6677	−	92.6677i	−4.01766	−	4.01766i
$533$		0			0
$534$		0			0
$535$	0.264927	+	0.281237i	0.0114538	+	0.0121589i
$536$	−5.25557			−0.227006
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−41.4247			−1.78099			−0.890494	−	0.454994i	$-0.849641\pi$
$541$	−41.4247			−1.78099			−0.890494	+	0.454994i	$0.849641\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.34145	+	44.9201i	−0.0574614	+	1.92417i
$546$		0			0
$547$	−25.1138	−	25.1138i	−1.07379	−	1.07379i	−0.997051	−	0.0767364i	$-0.975550\pi$
$547$	−25.1138	−	25.1138i	−1.07379	−	1.07379i	−0.0767364	−	0.997051i	$-0.524450\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$	31.6998	−	31.6998i	1.34196	−	1.34196i
$559$		0			0
$560$	94.2367	+	100.038i	3.98223	+	4.22739i
$561$		0			0
$562$	30.5366	+	30.5366i	1.28811	+	1.28811i
$563$	2.33042	−	2.33042i	0.0982157	−	0.0982157i	−0.656292	−	0.754507i	$-0.727876\pi$
$563$	2.33042	−	2.33042i	0.0982157	−	0.0982157i	0.754507	+	0.656292i	$0.227876\pi$
$564$		0			0
$565$	−9.16256	+	8.63119i	−0.385472	+	0.363117i
$566$	−70.9909			−2.98397
$567$	30.8747	+	30.8747i	1.29662	+	1.29662i
$568$	96.9310	−	96.9310i	4.06713	−	4.06713i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$			126.502i			5.28008i
$575$		0			0
$576$	−59.3094			−2.47123
$577$	−13.2711	−	13.2711i	−0.552481	−	0.552481i	0.374675	−	0.927156i	$-0.377754\pi$
$577$	−13.2711	−	13.2711i	−0.552481	−	0.552481i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	32.2629	−	32.2629i	1.34196	−	1.34196i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	91.6097			3.78436
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$			28.9041i			1.19097i
$590$	−13.6739	+	12.8809i	−0.562946	+	0.530298i
$591$		0			0
$592$		0			0
$593$	34.2829	−	34.2829i	1.40783	−	1.40783i	0.636806	−	0.771024i	$-0.280255\pi$
$593$	34.2829	−	34.2829i	1.40783	−	1.40783i	0.771024	−	0.636806i	$-0.219745\pi$
$594$		0			0
$595$		0			0
$596$	115.886			4.74687
$597$		0			0
$598$		0			0
$599$			45.3870i			1.85446i	0.374488	+	0.927232i	$0.377819\pi$
$599$			45.3870i			1.85446i	−0.374488	+	0.927232i	$0.622181\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−1.29671	+	1.29671i	−0.0528060	+	0.0528060i
$604$		0			0
$605$	24.5858	+	0.734206i	0.999554	+	0.0298497i
$606$		0			0
$607$	29.5678	+	29.5678i	1.20012	+	1.20012i	0.974130	+	0.225989i	$0.0725612\pi$
$607$	29.5678	+	29.5678i	1.20012	+	1.20012i	0.225989	+	0.974130i	$0.427439\pi$
$608$	61.6938	−	61.6938i	2.50201	−	2.50201i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		−	57.1637i		−	2.30694i
$615$		0			0
$616$		0			0
$617$	−33.2711	−	33.2711i	−1.33944	−	1.33944i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	−33.2711	−	33.2711i	−1.33944	−	1.33944i	−0.442843	−	0.896599i	$-0.646030\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	1.93373	−	64.7533i	0.0776603	−	2.60055i
$621$		0			0
$622$	−66.4319	−	66.4319i	−2.66368	−	2.66368i
$623$		0			0
$624$		0			0
$625$	24.8220	+	2.97832i	0.992878	+	0.119133i
$626$		0			0
$627$		0			0
$628$	−14.9883	+	14.9883i	−0.598100	+	0.598100i
$629$		0			0
$630$	87.3088	+	2.60730i	3.47846	+	0.103877i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	93.2350i		−	3.70283i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	47.8316i		−	1.89219i
$640$	−31.6534	+	29.8177i	−1.25121	+	1.17865i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−54.7157	+	54.7157i	−2.14943	+	2.14943i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	79.6370	+	79.6370i	3.11883	+	3.11883i
$653$	−34.1355	+	34.1355i	−1.33583	+	1.33583i	−0.435767	+	0.900060i	$0.643523\pi$
$653$	−34.1355	+	34.1355i	−1.33583	+	1.33583i	−0.900060	+	0.435767i	$0.856477\pi$
$654$		0			0
$655$	−24.8887	−	0.743252i	−0.972482	−	0.0290412i
$656$	−123.080			−4.80546
$657$		0			0
$658$	−124.582	+	124.582i	−4.85672	+	4.85672i
$659$			44.0521i			1.71603i	0.513627	+	0.858013i	$0.328301\pi$
$659$			44.0521i			1.71603i	−0.513627	+	0.858013i	$0.671699\pi$
$660$		0			0
$661$	47.9781			1.86613			0.933066	−	0.359705i	$-0.117123\pi$
$661$	47.9781			1.86613			0.933066	+	0.359705i	$0.117123\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−40.9930	+	38.6157i	−1.58964	+	1.49745i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−0.109504	+	3.66688i	−0.00423051	+	0.141664i
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	67.6445			2.60171
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			26.8803i			1.03157i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−31.9748	+	31.9748i	−1.22348	+	1.22348i	−0.257098	+	0.966385i	$0.582766\pi$
$683$	−31.9748	+	31.9748i	−1.22348	+	1.22348i	−0.966385	+	0.257098i	$0.917234\pi$
$684$		−	81.0381i		−	3.09857i
$685$		0			0
$686$	−124.182			−4.74128
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	33.6695			1.28085			0.640423	−	0.768022i	$-0.278759\pi$
$691$	33.6695			1.28085			0.640423	+	0.768022i	$0.278759\pi$
$692$	−21.5189	−	21.5189i	−0.818026	−	0.818026i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−42.2664	+	42.2664i	−1.59981	+	1.59981i
$699$		0			0
$700$	94.4193	−	83.7674i	3.56871	−	3.16611i
$701$	−40.8632			−1.54338			−0.771690	−	0.635999i	$-0.780588\pi$
$701$	−40.8632			−1.54338			−0.771690	+	0.635999i	$0.780588\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.2408	−	11.2408i	−0.422752	−	0.422752i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$	−65.6104	−	69.6497i	−2.46231	−	2.61391i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−7.31876	+	7.31876i	−0.273134	+	0.273134i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−2.53677	+	84.9469i	−0.0945399	+	3.16578i
$721$	52.9224			1.97093
$722$	15.0874	+	15.0874i	0.561496	+	0.561496i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	38.0785	+	38.0785i	1.41225	+	1.41225i	0.743331	+	0.668924i	$0.233245\pi$
$727$	38.0785	+	38.0785i	1.41225	+	1.41225i	0.668924	+	0.743331i	$0.266755\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−38.2858	+	38.2858i	−1.41412	+	1.41412i	−0.698907	+	0.715213i	$0.746330\pi$
$733$	−38.2858	+	38.2858i	−1.41412	+	1.41412i	−0.715213	+	0.698907i	$0.753670\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−55.3130	+	55.3130i	−2.03610	+	2.03610i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	1.48650	−	49.7774i	0.0544613	−	1.82370i
$746$	101.312			3.70929
$747$		0			0
$748$		0			0
$749$		−	0.838286i		−	0.0306303i
$750$		0			0
$751$	−3.42321			−0.124915			−0.0624574	−	0.998048i	$-0.519894\pi$
$751$	−3.42321			−0.124915			−0.0624574	+	0.998048i	$0.519894\pi$
$752$	−121.212	−	121.212i	−4.42014	−	4.42014i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	63.3996	−	63.3996i	2.30278	−	2.30278i
$759$		0			0
$760$	−68.4340	−	72.6471i	−2.48236	−	2.63519i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	68.9462	−	68.9462i	2.49602	−	2.49602i
$764$		−	74.0919i		−	2.68055i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			54.2384i			1.95589i	0.208866	+	0.977944i	$0.433023\pi$
$769$			54.2384i			1.95589i	−0.208866	+	0.977944i	$0.566977\pi$
$770$		0			0
$771$		0			0
$772$	92.1146	+	92.1146i	3.31528	+	3.31528i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$	−27.7892	−	1.66122i	−0.998218	−	0.0596728i
$776$	−47.6369			−1.71006
$777$		0			0
$778$		0			0
$779$		−	50.4348i		−	1.80701i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	209.504i		−	7.48228i
$785$	6.24581	+	6.63033i	0.222922	+	0.236647i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	27.3109			0.971065
$792$		0			0
$793$		0			0
$794$			71.5705i			2.53994i
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	55.7684	+	62.8599i	1.97171	+	2.22243i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	19.9207	−	19.9207i	0.700807	−	0.700807i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	37.0358	+	39.3159i	1.30130	+	1.38142i
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	35.2287	−	33.1856i	1.23401	−	1.16244i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−3.37416	+	112.988i	−0.117831	+	3.94571i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$			93.7881i			3.26726i
$825$		0			0
$826$	40.7579			1.41815
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−42.6957	+	42.6957i	−1.47490	+	1.47490i
$839$		−	16.0000i		−	0.552381i	−0.961103	−	0.276191i	$-0.910928\pi$
$839$		−	16.0000i		−	0.552381i	0.961103	−	0.276191i	$-0.0890721\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	75.0174	+	75.0174i	2.58527	+	2.58527i
$843$		0			0
$844$		−	149.638i		−	5.15075i
$845$	0.867697	−	29.0559i	0.0298497	−	0.999554i
$846$	−108.947			−3.74568
$847$	−37.7358	−	37.7358i	−1.29662	−	1.29662i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	15.8645	−	15.8645i	0.543189	−	0.543189i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$	15.8645	−	15.8645i	0.543189	−	0.543189i	0.924462	+	0.381273i	$0.124514\pi$
$854$		0			0
$855$	−34.8090	−	1.03950i	−1.19044	−	0.0355502i
$856$	1.48560			0.0507766
$857$	−3.27106	−	3.27106i	−0.111737	−	0.111737i	0.649028	−	0.760765i	$-0.275176\pi$
$857$	−3.27106	−	3.27106i	−0.111737	−	0.111737i	−0.760765	+	0.649028i	$0.775176\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	21.1332	+	21.1332i	0.719800	+	0.719800i
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	−9.51923	+	8.96717i	−0.323664	+	0.304893i
$866$		0			0
$867$		0			0
$868$	−99.3873	+	99.3873i	−3.37342	+	3.37342i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	122.185	+	122.185i	4.13771	+	4.13771i
$873$	−11.7534	+	11.7534i	−0.397794	+	0.397794i
$874$		0			0
$875$	−34.7702	−	41.6312i	−1.17545	−	1.40739i
$876$		0			0
$877$	−41.8742	−	41.8742i	−1.41399	−	1.41399i	−0.719557	−	0.694433i	$-0.755655\pi$
$877$	−41.8742	−	41.8742i	−1.41399	−	1.41399i	−0.694433	−	0.719557i	$-0.744345\pi$
$878$	42.1395	−	42.1395i	1.42214	−	1.42214i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−94.1527	−	94.1527i	−3.17029	−	3.17029i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	−43.2387			−1.45263
$887$	−7.95781	−	7.95781i	−0.267197	−	0.267197i	0.560773	−	0.827970i	$-0.310504\pi$
$887$	−7.95781	−	7.95781i	−0.267197	−	0.267197i	−0.827970	+	0.560773i	$0.810504\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	49.6694	−	49.6694i	1.66212	−	1.66212i
$894$		0			0
$895$		0			0
$896$	94.3495			3.15200
$897$		0			0
$898$		0			0
$899$		0			0
$900$	77.9123	+	4.65754i	2.59708	+	0.155251i
$901$		0			0
$902$		0			0
$903$		0			0
$904$			48.4000i			1.60976i
$905$		0			0
$906$		0			0
$907$	−16.0525	−	16.0525i	−0.533015	−	0.533015i	0.388453	−	0.921469i	$-0.373010\pi$
$907$	−16.0525	−	16.0525i	−0.533015	−	0.533015i	−0.921469	+	0.388453i	$0.873010\pi$
$908$	−101.819	+	101.819i	−3.37900	+	3.37900i
$909$		−	9.83006i		−	0.326043i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	38.2007	+	38.2007i	1.26150	+	1.26150i
$918$		0			0
$919$			24.0000i			0.791687i	0.918318	+	0.395843i	$0.129548\pi$
$919$			24.0000i			0.791687i	−0.918318	+	0.395843i	$0.870452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	23.1404	+	23.1404i	0.760029	+	0.760029i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	85.8491			2.81359
$932$	58.6152	+	58.6152i	1.92000	+	1.92000i
$933$		0			0
$934$		−	69.2807i		−	2.26693i
$935$		0			0
$936$		0			0
$937$	−43.2711	−	43.2711i	−1.41360	−	1.41360i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$	−43.2711	−	43.2711i	−1.41360	−	1.41360i	−0.686040	−	0.727564i	$-0.740653\pi$
$938$	5.62815	−	5.62815i	0.183766	−	0.183766i
$939$		0			0
$940$	−114.596	+	107.950i	−3.73772	+	3.52095i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			39.6554i			1.29067i
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		0			0
$950$	−52.1127	+	46.2337i	−1.69076	+	1.50002i
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	−31.8253	−	0.950400i	−1.02984	−	0.0307542i
$956$		0			0
$957$		0			0
$958$	78.8947	−	78.8947i	2.54897	−	2.54897i
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	0.366541	−	0.366541i	0.0118116	−	0.0118116i
$964$		0			0
$965$	40.7483	−	38.3852i	1.31174	−	1.23566i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	66.8747	−	66.8747i	2.14943	−	2.14943i
$969$		0			0
$970$	−0.992553	+	33.2369i	−0.0318689	+	1.06717i
$971$	−28.0000			−0.898563			−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000			−0.898563			−0.449281	+	0.893390i	$0.648320\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.9360	−	26.9360i	−0.861759	−	0.861759i	0.129784	−	0.991542i	$-0.458572\pi$
$977$	−26.9360	−	26.9360i	−0.861759	−	0.861759i	−0.991542	+	0.129784i	$0.958572\pi$
$978$		0			0
$979$		0			0
$980$	−192.326	−	5.74343i	−6.14363	−	0.183467i
$981$	60.2936			1.92503
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−66.1674	−	66.1674i	−2.10082	−	2.10082i
$993$		0			0
$994$			207.605i			6.58485i
$995$		0			0
$996$		0			0
$997$	−18.8028	−	18.8028i	−0.595490	−	0.595490i	0.343619	−	0.939109i	$-0.388347\pi$
$997$	−18.8028	−	18.8028i	−0.595490	−	0.595490i	−0.939109	+	0.343619i	$0.888347\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	155.2.f.a.92.1	✓	12
5.2	odd	4		775.2.f.e.743.6		12
5.3	odd	4	inner	155.2.f.a.123.1	yes	12
5.4	even	2		775.2.f.e.557.6		12
31.30	odd	2	CM	155.2.f.a.92.1	✓	12
155.92	even	4		775.2.f.e.743.6		12
155.123	even	4	inner	155.2.f.a.123.1	yes	12
155.154	odd	2		775.2.f.e.557.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.f.a.92.1	✓	12	1.1	even	1	trivial
155.2.f.a.92.1	✓	12	31.30	odd	2	CM
155.2.f.a.123.1	yes	12	5.3	odd	4	inner
155.2.f.a.123.1	yes	12	155.123	even	4	inner
775.2.f.e.557.6		12	5.4	even	2
775.2.f.e.557.6		12	155.154	odd	2
775.2.f.e.743.6		12	5.2	odd	4
775.2.f.e.743.6		12	155.92	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 155 = 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	155.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.89782 - 1.89782i) q^{2} +5.20343i q^{4} +(2.23507 + 0.0667460i) q^{5} +(-3.43053 - 3.43053i) q^{7} +(6.07952 - 6.07952i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.89782 - 1.89782i) q^{2} +5.20343i q^{4} +(2.23507 + 0.0667460i) q^{5} +(-3.43053 - 3.43053i) q^{7} +(6.07952 - 6.07952i) q^{8} -3.00000i q^{9} +(-4.11509 - 4.36843i) q^{10} +13.0210i q^{14} -12.6688 q^{16} +(-5.69345 + 5.69345i) q^{18} -5.19133i q^{19} +(-0.347308 + 11.6300i) q^{20} +(4.99109 + 0.298364i) q^{25} +(17.8505 - 17.8505i) q^{28} -5.56776 q^{31} +(11.8840 + 11.8840i) q^{32} +(-7.43850 - 7.89644i) q^{35} +15.6103 q^{36} +(-9.85219 + 9.85219i) q^{38} +(13.9939 - 13.1824i) q^{40} +9.71520 q^{41} +(0.200238 - 6.70521i) q^{45} +(9.56776 + 9.56776i) q^{47} +16.5370i q^{49} +(-8.90594 - 10.0384i) q^{50} -41.7119 q^{56} -3.13016i q^{59} +(10.5666 + 10.5666i) q^{62} +(-10.2916 + 10.2916i) q^{63} -19.7698i q^{64} +(-0.432236 - 0.432236i) q^{67} +(-0.869101 + 29.1029i) q^{70} +15.9439 q^{71} +(-18.2386 - 18.2386i) q^{72} +27.0127 q^{76} +(-28.3156 - 0.845590i) q^{80} -9.00000 q^{81} +(-18.4377 - 18.4377i) q^{82} +(-13.1053 + 12.3453i) q^{90} -36.3157i q^{94} +(0.346500 - 11.6030i) q^{95} +(-3.91782 - 3.91782i) q^{97} +(31.3842 - 31.3842i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{8}+O(q^{10}) \) 12 * q - 6 * q^8	\( 12 q - 6 q^{8} - 48 q^{16} + 54 q^{28} - 12 q^{32} - 24 q^{35} + 72 q^{36} - 66 q^{38} + 36 q^{40} + 48 q^{47} - 72 q^{67} + 6 q^{70} + 18 q^{72} - 84 q^{80} - 108 q^{81} - 42 q^{82} - 54 q^{90} + 96 q^{95} + 114 q^{98}+O(q^{100}) \) 12 * q - 6 * q^8 - 48 * q^16 + 54 * q^28 - 12 * q^32 - 24 * q^35 + 72 * q^36 - 66 * q^38 + 36 * q^40 + 48 * q^47 - 72 * q^67 + 6 * q^70 + 18 * q^72 - 84 * q^80 - 108 * q^81 - 42 * q^82 - 54 * q^90 + 96 * q^95 + 114 * q^98

Introduction

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