LMFDB - Embedded newform 312.2.h.a.155.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,2,Mod(155,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("312.155");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	312.h (of order $2$, degree $1$, 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:	$2.49133254306$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.151613669376.21
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{4} + 16 $ x^8 + 5*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			155.8
Root			$1.19709 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	312.155
Dual form			312.2.h.a.155.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.19709 + 0.752986i) q^{2} -1.73205 q^{3} +(0.866025 + 1.80278i) q^{4} -4.11439i q^{5} +(-2.07341 - 1.30421i) q^{6} +(-0.320758 + 2.81018i) q^{8} +3.00000 q^{9} +(3.09808 - 4.92527i) q^{10} +6.54099 q^{11} +(-1.50000 - 3.12250i) q^{12} -3.60555i q^{13} +7.12633i q^{15} +(-2.50000 + 3.12250i) q^{16} +(3.59126 + 2.25896i) q^{18} +(7.41732 - 3.56317i) q^{20} +(7.83013 + 4.92527i) q^{22} +(0.555569 - 4.86738i) q^{24} -11.9282 q^{25} +(2.71493 - 4.31615i) q^{26} -5.19615 q^{27} +(-5.36603 + 8.53083i) q^{30} +(-5.34391 + 1.85543i) q^{32} -11.3293 q^{33} +(2.59808 + 5.40833i) q^{36} +6.24500i q^{39} +(11.5622 + 1.31972i) q^{40} +7.82403 q^{41} -4.00000 q^{43} +(5.66467 + 11.7919i) q^{44} -12.3432i q^{45} +9.33123i q^{47} +(4.33013 - 5.40833i) q^{48} -7.00000 q^{49} +(-14.2791 - 8.98177i) q^{50} +(6.50000 - 3.12250i) q^{52} +(-6.22024 - 3.91263i) q^{54} -26.9122i q^{55} +0.469622 q^{59} +(-12.8472 + 6.17158i) q^{60} +7.21110i q^{61} +(-7.79423 - 1.80278i) q^{64} -14.8346 q^{65} +(-13.5622 - 8.53083i) q^{66} -4.92144i q^{71} +(-0.962274 + 8.43054i) q^{72} +20.6603 q^{75} +(-4.70239 + 7.47579i) q^{78} +14.4222i q^{79} +(12.8472 + 10.2860i) q^{80} +9.00000 q^{81} +(9.36603 + 5.89138i) q^{82} -12.6124 q^{83} +(-4.78834 - 3.01194i) q^{86} +(-2.09808 + 18.3814i) q^{88} +4.31872 q^{89} +(9.29423 - 14.7758i) q^{90} +(-7.02628 + 11.1703i) q^{94} +(9.25592 - 3.21370i) q^{96} +(-8.37960 - 5.27090i) q^{98} +19.6230 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9} + 4 q^{10} - 12 q^{12} - 20 q^{16} + 28 q^{22} - 40 q^{25} - 36 q^{30} + 44 q^{40} - 32 q^{43} - 56 q^{49} + 52 q^{52} - 60 q^{66} + 96 q^{75} + 72 q^{81} + 68 q^{82} + 4 q^{88} + 12 q^{90}+ \cdots + 20 q^{94}+O(q^{100}) $ 8 * q + 24 * q^9 + 4 * q^10 - 12 * q^12 - 20 * q^16 + 28 * q^22 - 40 * q^25 - 36 * q^30 + 44 * q^40 - 32 * q^43 - 56 * q^49 + 52 * q^52 - 60 * q^66 + 96 * q^75 + 72 * q^81 + 68 * q^82 + 4 * q^88 + 12 * q^90 + 20 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	1.19709	+	0.752986i	0.846467	+	0.532441i
$2$	1.19709	+	0.752986i	0.846467	+	0.532441i
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$	0.866025	+	1.80278i	0.433013	+	0.901388i
$5$		−	4.11439i		−	1.84001i	−0.391905	−	0.920006i	$-0.628184\pi$
$5$		−	4.11439i		−	1.84001i	0.391905	−	0.920006i	$-0.371816\pi$
$6$	−2.07341	−	1.30421i	−0.846467	−	0.532441i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−0.320758	+	2.81018i	−0.113405	+	0.993549i
$9$	3.00000			1.00000
$10$	3.09808	−	4.92527i	0.979698	−	1.55751i
$11$	6.54099			1.97218			0.986092	−	0.166200i	$-0.0531498\pi$
$11$	6.54099			1.97218			0.986092	+	0.166200i	$0.0531498\pi$
$12$	−1.50000	−	3.12250i	−0.433013	−	0.901388i
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$		0			0
$15$			7.12633i			1.84001i
$16$	−2.50000	+	3.12250i	−0.625000	+	0.780625i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	3.59126	+	2.25896i	0.846467	+	0.532441i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	7.41732	−	3.56317i	1.65856	−	0.796748i
$21$		0			0
$22$	7.83013	+	4.92527i	1.66939	+	1.05007i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	0.555569	−	4.86738i	0.113405	−	0.993549i
$25$	−11.9282			−2.38564
$26$	2.71493	−	4.31615i	0.532441	−	0.846467i
$27$	−5.19615			−1.00000
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−5.36603	+	8.53083i	−0.979698	+	1.55751i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.34391	+	1.85543i	−0.944679	+	0.327997i
$33$	−11.3293			−1.97218
$34$		0			0
$35$		0			0
$36$	2.59808	+	5.40833i	0.433013	+	0.901388i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$			6.24500i			1.00000i
$40$	11.5622	+	1.31972i	1.82814	+	0.208667i
$41$	7.82403			1.22191			0.610954	−	0.791666i	$-0.290786\pi$
$41$	7.82403			1.22191			0.610954	+	0.791666i	$0.290786\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$	5.66467	+	11.7919i	0.853981	+	1.77770i
$45$		−	12.3432i		−	1.84001i
$46$		0			0
$47$			9.33123i			1.36110i	0.732702	+	0.680550i	$0.238259\pi$
$47$			9.33123i			1.36110i	−0.732702	+	0.680550i	$0.761741\pi$
$48$	4.33013	−	5.40833i	0.625000	−	0.780625i
$49$	−7.00000			−1.00000
$50$	−14.2791	−	8.98177i	−2.01937	−	1.27021i
$51$		0			0
$52$	6.50000	−	3.12250i	0.901388	−	0.433013i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−6.22024	−	3.91263i	−0.846467	−	0.532441i
$55$		−	26.9122i		−	3.62884i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.469622			0.0611396			0.0305698	−	0.999533i	$-0.490268\pi$
$59$	0.469622			0.0611396			0.0305698	+	0.999533i	$0.490268\pi$
$60$	−12.8472	+	6.17158i	−1.65856	+	0.796748i
$61$			7.21110i			0.923287i	0.887066	+	0.461644i	$0.152740\pi$
$61$			7.21110i			0.923287i	−0.887066	+	0.461644i	$0.847260\pi$
$62$		0			0
$63$		0			0
$64$	−7.79423	−	1.80278i	−0.974279	−	0.225347i
$65$	−14.8346			−1.84001
$66$	−13.5622	−	8.53083i	−1.66939	−	1.05007i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.92144i		−	0.584067i	−0.956408	−	0.292034i	$-0.905668\pi$
$71$		−	4.92144i		−	0.584067i	0.956408	−	0.292034i	$-0.0943319\pi$
$72$	−0.962274	+	8.43054i	−0.113405	+	0.993549i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	20.6603			2.38564
$76$		0			0
$77$		0			0
$78$	−4.70239	+	7.47579i	−0.532441	+	0.846467i
$79$			14.4222i			1.62262i	0.584613	+	0.811312i	$0.301246\pi$
$79$			14.4222i			1.62262i	−0.584613	+	0.811312i	$0.698754\pi$
$80$	12.8472	+	10.2860i	1.43636	+	1.15001i
$81$	9.00000			1.00000
$82$	9.36603	+	5.89138i	1.03430	+	0.650594i
$83$	−12.6124			−1.38439			−0.692194	−	0.721712i	$-0.743356\pi$
$83$	−12.6124			−1.38439			−0.692194	+	0.721712i	$0.743356\pi$
$84$		0			0
$85$		0			0
$86$	−4.78834	−	3.01194i	−0.516340	−	0.324786i
$87$		0			0
$88$	−2.09808	+	18.3814i	−0.223656	+	1.95946i
$89$	4.31872			0.457783			0.228892	−	0.973452i	$-0.426490\pi$
$89$	4.31872			0.457783			0.228892	+	0.973452i	$0.426490\pi$
$90$	9.29423	−	14.7758i	0.979698	−	1.55751i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−7.02628	+	11.1703i	−0.724705	+	1.15213i
$95$		0			0
$96$	9.25592	−	3.21370i	0.944679	−	0.327997i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−8.37960	−	5.27090i	−0.846467	−	0.532441i
$99$	19.6230			1.97218
$100$	−10.3301	−	21.5039i	−1.03301	−	2.15039i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			12.4900i			1.23068i	0.788263	+	0.615338i	$0.210980\pi$
$103$			12.4900i			1.23068i	−0.788263	+	0.615338i	$0.789020\pi$
$104$	10.1322	+	1.15651i	0.993549	+	0.113405i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−4.50000	−	9.36750i	−0.433013	−	0.901388i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	20.2645	−	32.2162i	1.93214	−	3.07169i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	10.8167i		−	1.00000i
$118$	0.562178	+	0.353619i	0.0517527	+	0.0325532i
$119$		0			0
$120$	−20.0263	−	2.28583i	−1.82814	−	0.208667i
$121$	31.7846			2.88951
$122$	−5.42986	+	8.63230i	−0.491596	+	0.781532i
$123$	−13.5516			−1.22191
$124$		0			0
$125$			28.5053i			2.54959i
$126$		0			0
$127$			14.4222i			1.27976i	0.768473	+	0.639882i	$0.221017\pi$
$127$			14.4222i			1.27976i	−0.768473	+	0.639882i	$0.778983\pi$
$128$	−7.97289	−	8.02702i	−0.704711	−	0.709495i
$129$	6.92820			0.609994
$130$	−17.7583	−	11.1703i	−1.55751	−	0.979698i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−9.81149	−	20.4242i	−0.853981	−	1.77770i
$133$		0			0
$134$		0			0
$135$			21.3790i			1.84001i
$136$		0			0
$137$	18.3400			1.56689			0.783444	−	0.621463i	$-0.213461\pi$
$137$	18.3400			1.56689			0.783444	+	0.621463i	$0.213461\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$		−	16.1622i		−	1.36110i
$142$	3.70577	−	5.89138i	0.310981	−	0.494394i
$143$		−	23.5839i		−	1.97218i
$144$	−7.50000	+	9.36750i	−0.625000	+	0.780625i
$145$		0			0
$146$		0			0
$147$	12.1244			1.00000
$148$		0			0
$149$		−	0.295400i		−	0.0242001i	−0.999927	−	0.0121001i	$-0.996148\pi$
$149$		−	0.295400i		−	0.0242001i	0.999927	−	0.0121001i	$-0.00385166\pi$
$150$	24.7321	+	15.5569i	2.01937	+	1.27021i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−11.2583	+	5.40833i	−0.901388	+	0.433013i
$157$		−	24.9800i		−	1.99362i	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$		−	24.9800i		−	1.99362i	0.0798087	−	0.996810i	$-0.474569\pi$
$158$	−10.8597	+	17.2646i	−0.863952	+	1.37350i
$159$		0			0
$160$	7.63397	+	21.9869i	0.603519	+	1.73822i
$161$		0			0
$162$	10.7738	+	6.77687i	0.846467	+	0.532441i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	6.77581	+	14.1050i	0.529102	+	1.10141i
$165$			46.6133i			3.62884i
$166$	−15.0981	−	9.49693i	−1.17184	−	0.737105i
$167$		−	11.5361i		−	0.892692i	−0.894860	−	0.446346i	$-0.852725\pi$
$167$		−	11.5361i		−	0.892692i	0.894860	−	0.446346i	$-0.147275\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	−3.46410	−	7.21110i	−0.264135	−	0.549841i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−16.3525	+	20.4242i	−1.23262	+	1.53954i
$177$	−0.813410			−0.0611396
$178$	5.16987	+	3.25193i	0.387498	+	0.243743i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	22.2520	−	10.6895i	1.65856	−	0.796748i
$181$			24.9800i			1.85675i	0.371647	+	0.928374i	$0.378793\pi$
$181$			24.9800i			1.85675i	−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	12.4900i		−	0.923287i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−16.8221	+	8.08108i	−1.22688	+	0.589373i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	13.5000	+	3.12250i	0.974279	+	0.225347i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	25.6944			1.84001
$196$	−6.06218	−	12.6194i	−0.433013	−	0.901388i
$197$		−	24.3909i		−	1.73778i	−0.495003	−	0.868891i	$-0.664833\pi$
$197$		−	24.3909i		−	1.73778i	0.495003	−	0.868891i	$-0.335167\pi$
$198$	23.4904	+	14.7758i	1.66939	+	1.05007i
$199$		−	14.4222i		−	1.02236i	−0.859473	−	0.511182i	$-0.829208\pi$
$199$		−	14.4222i		−	1.02236i	0.859473	−	0.511182i	$-0.170792\pi$
$200$	3.82607	−	33.5204i	0.270544	−	2.37025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	32.1911i		−	2.24832i
$206$	−9.40479	+	14.9516i	−0.655263	+	1.04173i
$207$		0			0
$208$	11.2583	+	9.01388i	0.780625	+	0.625000i
$209$		0			0
$210$		0			0
$211$	3.46410			0.238479			0.119239	−	0.992866i	$-0.461954\pi$
$211$	3.46410			0.238479			0.119239	+	0.992866i	$0.461954\pi$
$212$		0			0
$213$			8.52418i			0.584067i
$214$		0			0
$215$			16.4576i			1.12240i
$216$	1.66671	−	14.6021i	0.113405	−	0.993549i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	48.5167	−	23.3066i	3.27099	−	1.57133i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−35.7846			−2.38564
$226$		0			0
$227$	5.60175			0.371801			0.185901	−	0.982569i	$-0.440480\pi$
$227$	5.60175			0.371801			0.185901	+	0.982569i	$0.440480\pi$
$228$		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
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	8.14478	−	12.9485i	0.532441	−	0.846467i
$235$	38.3923			2.50444
$236$	0.406705	+	0.846624i	0.0264742	+	0.0551105i
$237$		−	24.9800i		−	1.62262i
$238$		0			0
$239$		−	25.7888i		−	1.66814i	−0.551660	−	0.834069i	$-0.686005\pi$
$239$		−	25.7888i		−	1.66814i	0.551660	−	0.834069i	$-0.313995\pi$
$240$	−22.2520	−	17.8158i	−1.43636	−	1.15001i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	38.0489	+	23.9334i	2.44587	+	1.53849i
$243$	−15.5885			−1.00000
$244$	−13.0000	+	6.24500i	−0.832240	+	0.399795i
$245$			28.8007i			1.84001i
$246$	−16.2224	−	10.2042i	−1.03430	−	0.650594i
$247$		0			0
$248$		0			0
$249$	21.8453			1.38439
$250$	−21.4641	+	34.1233i	−1.35751	+	2.15815i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−10.8597	+	17.2646i	−0.681399	+	1.08328i
$255$		0			0
$256$	−3.50000	−	15.6125i	−0.218750	−	0.975781i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	8.29365	+	5.21684i	0.516340	+	0.324786i
$259$		0			0
$260$	−12.8472	−	26.7435i	−0.796748	−	1.65856i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	3.63397	−	31.8375i	0.223656	−	1.95946i
$265$		0			0
$266$		0			0
$267$	−7.48024			−0.457783
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−16.0981	+	25.5925i	−0.979698	+	1.55751i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	21.9545	+	13.8097i	1.32632	+	0.834276i
$275$	−78.0223			−4.70492
$276$		0			0
$277$			24.9800i			1.50090i	0.660926	+	0.750451i	$0.270164\pi$
$277$			24.9800i			1.50090i	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−23.9417	−	15.0597i	−1.43593	−	0.903221i
$279$		0			0
$280$		0			0
$281$	−30.4827			−1.81845			−0.909223	−	0.416310i	$-0.863323\pi$
$281$	−30.4827			−1.81845			−0.909223	+	0.416310i	$0.863323\pi$
$282$	12.1699	−	19.3475i	0.724705	−	1.15213i
$283$	−17.3205			−1.02960			−0.514799	−	0.857311i	$-0.672133\pi$
$283$	−17.3205			−1.02960			−0.514799	+	0.857311i	$0.672133\pi$
$284$	8.87225	−	4.26209i	0.526471	−	0.252908i
$285$		0			0
$286$	17.7583	−	28.2319i	1.05007	−	1.66939i
$287$		0			0
$288$	−16.0317	+	5.56630i	−0.944679	+	0.327997i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			32.6197i			1.90566i	0.303498	+	0.952832i	$0.401846\pi$
$293$			32.6197i			1.90566i	−0.303498	+	0.952832i	$0.598154\pi$
$294$	14.5139	+	9.12947i	0.846467	+	0.532441i
$295$		−	1.93221i		−	0.112498i
$296$		0			0
$297$	−33.9880			−1.97218
$298$	0.222432	−	0.353619i	0.0128851	−	0.0204846i
$299$		0			0
$300$	17.8923	+	37.2458i	1.03301	+	2.15039i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	29.6693			1.69886
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		−	21.6333i		−	1.23068i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−17.5496	−	2.00313i	−0.993549	−	0.113405i
$313$	−34.6410			−1.95803			−0.979013	−	0.203798i	$-0.934671\pi$
$313$	−34.6410			−1.95803			−0.979013	+	0.203798i	$0.934671\pi$
$314$	18.8096	−	29.9032i	1.06149	−	1.68753i
$315$		0			0
$316$	−26.0000	+	12.4900i	−1.46261	+	0.702617i
$317$			16.7530i			0.940940i	0.882416	+	0.470470i	$0.155916\pi$
$317$			16.7530i			0.940940i	−0.882416	+	0.470470i	$0.844084\pi$
$318$		0			0
$319$		0			0
$320$	−7.41732	+	32.0685i	−0.414641	+	1.79268i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	7.79423	+	16.2250i	0.433013	+	0.901388i
$325$			43.0077i			2.38564i
$326$		0			0
$327$		0			0
$328$	−2.50962	+	21.9869i	−0.138571	+	1.21402i
$329$		0			0
$330$	−35.0991	+	55.8001i	−1.93214	+	3.07169i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−10.9226	−	22.7373i	−0.599457	−	1.24787i
$333$		0			0
$334$	8.68653	−	13.8097i	0.475306	−	0.755634i
$335$		0			0
$336$		0			0
$337$	6.92820			0.377403			0.188702	−	0.982034i	$-0.439572\pi$
$337$	6.92820			0.377403			0.188702	+	0.982034i	$0.439572\pi$
$338$	−15.5621	−	9.78881i	−0.846467	−	0.532441i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.28303	−	11.2407i	0.0691764	−	0.606059i
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$			18.7350i			1.00000i
$352$	−34.9545	+	12.1364i	−1.86308	+	0.646871i
$353$	37.4933			1.99557			0.997784	−	0.0665386i	$-0.0211956\pi$
$353$	37.4933			1.99557			0.997784	+	0.0665386i	$0.0211956\pi$
$354$	−0.973721	−	0.612486i	−0.0517527	−	0.0325532i
$355$	−20.2487			−1.07469
$356$	3.74012	+	7.78568i	0.198226	+	0.412640i
$357$		0			0
$358$		0			0
$359$		−	37.8366i		−	1.99694i	−0.0553230	−	0.998469i	$-0.517619\pi$
$359$		−	37.8366i		−	1.99694i	0.0553230	−	0.998469i	$-0.482381\pi$
$360$	34.6865	+	3.95917i	1.82814	+	0.208667i
$361$	19.0000			1.00000
$362$	−18.8096	+	29.9032i	−0.988609	+	1.57168i
$363$	−55.0526			−2.88951
$364$		0			0
$365$		0			0
$366$	9.40479	−	14.9516i	0.491596	−	0.781532i
$367$		−	37.4700i		−	1.95592i	−0.208798	−	0.977959i	$-0.566955\pi$
$367$		−	37.4700i		−	1.95592i	0.208798	−	0.977959i	$-0.433045\pi$
$368$		0			0
$369$	23.4721			1.22191
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	36.0555i		−	1.86688i	−0.358729	−	0.933442i	$-0.616790\pi$
$373$		−	36.0555i		−	1.86688i	0.358729	−	0.933442i	$-0.383210\pi$
$374$		0			0
$375$		−	49.3727i		−	2.54959i
$376$	−26.2224	−	2.99307i	−1.35232	−	0.154356i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		−	24.9800i		−	1.27976i
$382$		0			0
$383$		−	16.9692i		−	0.867086i	−0.901133	−	0.433543i	$-0.857263\pi$
$383$		−	16.9692i		−	0.867086i	0.901133	−	0.433543i	$-0.142737\pi$
$384$	13.8095	+	13.9032i	0.704711	+	0.709495i
$385$		0			0
$386$		0			0
$387$	−12.0000			−0.609994
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	30.7583	+	19.3475i	1.55751	+	0.979698i
$391$		0			0
$392$	2.24531	−	19.6713i	0.113405	−	0.993549i
$393$		0			0
$394$	18.3660	−	29.1980i	0.925267	−	1.47098i
$395$	59.3386			2.98565
$396$	16.9940	+	35.3758i	0.853981	+	1.77770i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	10.8597	−	17.2646i	0.544348	−	0.865397i
$399$		0			0
$400$	29.8205	−	37.2458i	1.49103	−	1.86229i
$401$	−25.3506			−1.26595			−0.632973	−	0.774173i	$-0.718166\pi$
$401$	−25.3506			−1.26595			−0.632973	+	0.774173i	$0.718166\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	37.0295i		−	1.84001i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	24.2394	−	38.5355i	1.19710	−	1.90313i
$411$	−31.7657			−1.56689
$412$	−22.5167	+	10.8167i	−1.10932	+	0.532898i
$413$		0			0
$414$		0			0
$415$			51.8922i			2.54729i
$416$	6.68986	+	19.2677i	0.327997	+	0.944679i
$417$	34.6410			1.69638
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	4.14682	+	2.60842i	0.201864	+	0.126976i
$423$			27.9937i			1.36110i
$424$		0			0
$425$		0			0
$426$	−6.41858	+	10.2042i	−0.310981	+	0.494394i
$427$		0			0
$428$		0			0
$429$			40.8485i			1.97218i
$430$	−12.3923	+	19.7011i	−0.597610	+	0.950071i
$431$			40.0415i			1.92873i	0.264578	+	0.964364i	$0.414767\pi$
$431$			40.0415i			1.92873i	−0.264578	+	0.964364i	$0.585233\pi$
$432$	12.9904	−	16.2250i	0.625000	−	0.780625i
$433$	−20.7846			−0.998845			−0.499422	−	0.866359i	$-0.666454\pi$
$433$	−20.7846			−0.998845			−0.499422	+	0.866359i	$0.666454\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	12.4900i		−	0.596115i	−0.954548	−	0.298057i	$-0.903661\pi$
$439$		−	12.4900i		−	0.596115i	0.954548	−	0.298057i	$-0.0963387\pi$
$440$	75.6281	+	8.63230i	3.60543	+	0.411529i
$441$	−21.0000			−1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		−	17.7689i		−	0.842326i
$446$		0			0
$447$			0.511648i			0.0242001i
$448$		0			0
$449$	40.9986			1.93484			0.967422	−	0.253168i	$-0.0814726\pi$
$449$	40.9986			1.93484			0.967422	+	0.253168i	$0.0814726\pi$
$450$	−42.8372	−	26.9453i	−2.01937	−	1.27021i
$451$	51.1769			2.40983
$452$		0			0
$453$		0			0
$454$	6.70577	+	4.21804i	0.314717	+	0.197962i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	24.9817i		−	1.16352i	−0.813362	−	0.581758i	$-0.802365\pi$
$461$		−	24.9817i		−	1.16352i	0.813362	−	0.581758i	$-0.197635\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	19.5000	−	9.36750i	0.901388	−	0.433013i
$469$		0			0
$470$	45.9589	+	28.9089i	2.11992	+	1.33347i
$471$			43.2666i			1.99362i
$472$	−0.150635	+	1.31972i	−0.00693354	+	0.0607452i
$473$	−26.1640			−1.20302
$474$	18.8096	−	29.9032i	0.863952	−	1.37350i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	19.4186	−	30.8714i	0.888185	−	1.41202i
$479$			33.4268i			1.52731i	0.645626	+	0.763654i	$0.276597\pi$
$479$			33.4268i			1.52731i	−0.645626	+	0.763654i	$0.723403\pi$
$480$	−13.2224	−	38.0825i	−0.603519	−	1.73822i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	27.5263	+	57.3005i	1.25119	+	2.60457i
$485$		0			0
$486$	−18.6607	−	11.7379i	−0.846467	−	0.532441i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−20.2645	−	2.31302i	−0.917331	−	0.104705i
$489$		0			0
$490$	−21.6865	+	34.4769i	−0.979698	+	1.55751i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−11.7360	−	24.4305i	−0.529102	−	1.10141i
$493$		0			0
$494$		0			0
$495$		−	80.7366i		−	3.62884i
$496$		0			0
$497$		0			0
$498$	26.1506	+	16.4492i	1.17184	+	0.737105i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−51.3887	+	24.6863i	−2.29817	+	1.10401i
$501$			19.9811i			0.892692i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	22.5167			1.00000
$508$	−26.0000	+	12.4900i	−1.15356	+	0.554154i
$509$			11.7524i			0.520915i	0.965485	+	0.260457i	$0.0838733\pi$
$509$			11.7524i			0.520915i	−0.965485	+	0.260457i	$0.916127\pi$
$510$		0			0
$511$		0			0
$512$	7.56619	−	21.3249i	0.334381	−	0.942438i
$513$		0			0
$514$		0			0
$515$	51.3887			2.26446
$516$	6.00000	+	12.4900i	0.264135	+	0.549841i
$517$			61.0355i			2.68434i
$518$		0			0
$519$		0			0
$520$	4.75833	−	41.6880i	0.208667	−	1.82814i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	44.0000			1.92399			0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000			1.92399			0.961993	+	0.273075i	$0.0880406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	28.3233	−	35.3758i	1.23262	−	1.53954i
$529$	−23.0000			−1.00000
$530$		0			0
$531$	1.40887			0.0611396
$532$		0			0
$533$		−	28.2099i		−	1.22191i
$534$	−8.95448	−	5.63251i	−0.387498	−	0.243743i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−45.7870			−1.97218
$540$	−38.5415	+	18.5148i	−1.65856	+	0.796748i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		−	43.2666i		−	1.85675i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$	15.8829	+	33.0628i	0.678482	+	1.41237i
$549$			21.6333i			0.923287i
$550$	−93.3993	−	58.7497i	−3.98256	−	2.50509i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−18.8096	+	29.9032i	−0.799142	+	1.27046i
$555$		0			0
$556$	−17.3205	−	36.0555i	−0.734553	−	1.52909i
$557$		−	3.52359i		−	0.149299i	−0.997210	−	0.0746496i	$-0.976216\pi$
$557$		−	3.52359i		−	0.149299i	0.997210	−	0.0746496i	$-0.0237838\pi$
$558$		0			0
$559$			14.4222i			0.609994i
$560$		0			0
$561$		0			0
$562$	−36.4904	−	22.9530i	−1.53925	−	0.968215i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	29.1367	−	13.9968i	1.22688	−	0.589373i
$565$		0			0
$566$	−20.7341	−	13.0421i	−0.871520	−	0.548200i
$567$		0			0
$568$	13.8301	+	1.57859i	0.580299	+	0.0662362i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	38.1051			1.59465			0.797325	−	0.603550i	$-0.206248\pi$
$571$	38.1051			1.59465			0.797325	+	0.603550i	$0.206248\pi$
$572$	42.5165	−	20.4242i	1.77770	−	0.853981i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−23.3827	−	5.40833i	−0.974279	−	0.225347i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	20.3504	+	12.8008i	0.846467	+	0.532441i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−44.5039			−1.84001
$586$	−24.5622	+	39.0486i	−1.01465	+	1.61308i
$587$	24.7551			1.02175			0.510876	−	0.859654i	$-0.329321\pi$
$587$	24.7551			1.02175			0.510876	+	0.859654i	$0.329321\pi$
$588$	10.5000	+	21.8575i	0.433013	+	0.901388i
$589$		0			0
$590$	1.45493	−	2.31302i	0.0598983	−	0.0952255i
$591$			42.2463i			1.73778i
$592$		0			0
$593$	−28.8559			−1.18497			−0.592484	−	0.805582i	$-0.701853\pi$
$593$	−28.8559			−1.18497			−0.592484	+	0.805582i	$0.701853\pi$
$594$	−40.6865	−	25.5925i	−1.66939	−	1.05007i
$595$		0			0
$596$	0.532540	−	0.255824i	0.0218137	−	0.0104790i
$597$			24.9800i			1.02236i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−6.62694	+	58.0590i	−0.270544	+	2.37025i
$601$	48.4974			1.97825			0.989126	−	0.147074i	$-0.0469854\pi$
$601$	48.4974			1.97825			0.989126	+	0.147074i	$0.0469854\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	130.774i		−	5.31673i
$606$		0			0
$607$			37.4700i			1.52086i	0.649420	+	0.760430i	$0.275012\pi$
$607$			37.4700i			1.52086i	−0.649420	+	0.760430i	$0.724988\pi$
$608$		0			0
$609$		0			0
$610$	35.5167	+	22.3405i	1.43803	+	0.904542i
$611$	33.6442			1.36110
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$			55.7566i			2.24832i
$616$		0			0
$617$	−16.4615			−0.662714			−0.331357	−	0.943506i	$-0.607506\pi$
$617$	−16.4615			−0.662714			−0.331357	+	0.943506i	$0.607506\pi$
$618$	16.2896	−	25.8969i	0.655263	−	1.04173i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−19.5000	−	15.6125i	−0.780625	−	0.625000i
$625$	57.6410			2.30564
$626$	−41.4682	−	26.0842i	−1.65740	−	1.04253i
$627$		0			0
$628$	45.0333	−	21.6333i	1.79703	−	0.863263i
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	−40.5290	−	4.62604i	−1.61216	−	0.184014i
$633$	−6.00000			−0.238479
$634$	−12.6147	+	20.0547i	−0.500995	+	0.796475i
$635$	59.3386			2.35478
$636$		0			0
$637$			25.2389i			1.00000i
$638$		0			0
$639$		−	14.7643i		−	0.584067i
$640$	−33.0263	+	32.8036i	−1.30548	+	1.29668i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		−	28.5053i		−	1.12240i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−2.88682	+	25.2916i	−0.113405	+	0.993549i
$649$	3.07180			0.120579
$650$	−32.3842	+	51.4839i	−1.27021	+	2.01937i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	−19.5601	+	24.4305i	−0.763692	+	0.953851i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$	−84.0333	+	40.3683i	−3.27099	+	1.57133i
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$	4.04552	−	35.4430i	0.156997	−	1.37546i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	20.7970	−	9.99057i	0.804662	−	0.386547i
$669$		0			0
$670$		0			0
$671$			47.1678i			1.82089i
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$	8.29365	+	5.21684i	0.319459	+	0.200945i
$675$	61.9808			2.38564
$676$	−11.2583	−	23.4361i	−0.433013	−	0.901388i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−9.70252			−0.371801
$682$		0			0
$683$	−51.8583			−1.98430			−0.992152	−	0.125038i	$-0.960095\pi$
$683$	−51.8583			−1.98430			−0.992152	+	0.125038i	$0.960095\pi$
$684$		0			0
$685$		−	75.4577i		−	2.88309i
$686$		0			0
$687$		0			0
$688$	10.0000	−	12.4900i	0.381246	−	0.476177i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			82.2878i			3.12135i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−14.1072	+	22.4274i	−0.532441	+	0.846467i
$703$		0			0
$704$	−50.9820	−	11.7919i	−1.92146	−	0.444426i
$705$	−66.4974			−2.50444
$706$	44.8827	+	28.2319i	1.68918	+	1.06252i
$707$		0			0
$708$	−0.704433	−	1.46640i	−0.0264742	−	0.0551105i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$	−24.2394	−	15.2470i	−0.909690	−	0.572209i
$711$			43.2666i			1.62262i
$712$	−1.38526	+	12.1364i	−0.0519149	+	0.454830i
$713$		0			0
$714$		0			0
$715$	−97.0333			−3.62884
$716$		0			0
$717$			44.6675i			1.66814i
$718$	28.4904	−	45.2936i	1.06325	−	1.69034i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	38.5415	+	30.8579i	1.43636	+	1.15001i
$721$		0			0
$722$	22.7446	+	14.3067i	0.846467	+	0.532441i
$723$		0			0
$724$	−45.0333	+	21.6333i	−1.67365	+	0.803996i
$725$		0			0
$726$	−65.9026	−	41.4538i	−2.44587	−	1.53849i
$727$		−	14.4222i		−	0.534890i	−0.963573	−	0.267445i	$-0.913821\pi$
$727$		−	14.4222i		−	0.534890i	0.963573	−	0.267445i	$-0.0861794\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	22.5167	−	10.8167i	0.832240	−	0.399795i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	28.2144	−	44.8548i	1.04141	−	1.65562i
$735$		−	49.8843i		−	1.84001i
$736$		0			0
$737$		0			0
$738$	28.0981	+	17.6741i	1.03430	+	0.650594i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			54.2941i			1.99186i	0.0901418	+	0.995929i	$0.471268\pi$
$743$			54.2941i			1.99186i	−0.0901418	+	0.995929i	$0.528732\pi$
$744$		0			0
$745$	−1.21539			−0.0445285
$746$	27.1493	−	43.1615i	0.994006	−	1.58026i
$747$	−37.8371			−1.38439
$748$		0			0
$749$		0			0
$750$	37.1769	−	59.1033i	1.35751	−	2.15815i
$751$		−	37.4700i		−	1.36730i	−0.729810	−	0.683650i	$-0.760392\pi$
$751$		−	37.4700i		−	1.36730i	0.729810	−	0.683650i	$-0.239608\pi$
$752$	−29.1367	−	23.3281i	−1.06251	−	0.850687i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	36.0555i		−	1.31046i	−0.755429	−	0.655230i	$-0.772572\pi$
$757$		−	36.0555i		−	1.31046i	0.755429	−	0.655230i	$-0.227428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−48.0092			−1.74033			−0.870167	−	0.492757i	$-0.835989\pi$
$761$	−48.0092			−1.74033			−0.870167	+	0.492757i	$0.835989\pi$
$762$	18.8096	−	29.9032i	0.681399	−	1.08328i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	12.7776	−	20.3136i	0.461672	−	0.733960i
$767$		−	1.69325i		−	0.0611396i
$768$	6.06218	+	27.0416i	0.218750	+	0.975781i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	12.9340i		−	0.465203i	−0.972572	−	0.232601i	$-0.925276\pi$
$773$		−	12.9340i		−	0.465203i	0.972572	−	0.232601i	$-0.0747237\pi$
$774$	−14.3650	−	9.03583i	−0.516340	−	0.324786i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	22.2520	+	46.3212i	0.796748	+	1.65856i
$781$		−	32.1911i		−	1.15189i
$782$		0			0
$783$		0			0
$784$	17.5000	−	21.8575i	0.625000	−	0.780625i
$785$	−102.777			−3.66828
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	43.9714	−	21.1232i	1.56642	−	0.752482i
$789$		0			0
$790$	71.0333	+	44.6811i	2.52725	+	1.58968i
$791$		0			0
$792$	−6.29423	+	55.1441i	−0.223656	+	1.95946i
$793$	26.0000			0.923287
$794$		0			0
$795$		0			0
$796$	26.0000	−	12.4900i	0.921546	−	0.442696i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	63.7432	−	22.1320i	2.25366	−	0.782484i
$801$	12.9562			0.457783
$802$	−30.3468	−	19.0886i	−1.07158	−	0.674042i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	27.8827	−	44.3275i	0.979698	−	1.55751i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	58.0333	−	27.8783i	2.02661	−	0.973553i
$821$		−	57.3061i		−	2.00000i	−0.00219583	−	0.999998i	$-0.500699\pi$
$821$		−	57.3061i		−	2.00000i	0.00219583	−	0.999998i	$-0.499301\pi$
$822$	−38.0263	−	23.9191i	−1.32632	−	0.834276i
$823$		−	12.4900i		−	0.435374i	−0.976019	−	0.217687i	$-0.930149\pi$
$823$		−	12.4900i		−	0.435374i	0.976019	−	0.217687i	$-0.0698512\pi$
$824$	−35.0991	−	4.00627i	−1.22274	−	0.139565i
$825$	135.139			4.70492
$826$		0			0
$827$	−27.5728			−0.958802			−0.479401	−	0.877596i	$-0.659146\pi$
$827$	−27.5728			−0.958802			−0.479401	+	0.877596i	$0.659146\pi$
$828$		0			0
$829$			50.4777i			1.75316i	0.481253	+	0.876582i	$0.340182\pi$
$829$			50.4777i			1.75316i	−0.481253	+	0.876582i	$0.659818\pi$
$830$	−39.0741	+	62.1194i	−1.35628	+	2.15619i
$831$		−	43.2666i		−	1.50090i
$832$	−6.50000	+	28.1025i	−0.225347	+	0.974279i
$833$		0			0
$834$	41.4682	+	26.0842i	1.43593	+	0.903221i
$835$	−47.4641			−1.64256
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	44.4512i		−	1.53463i	−0.641272	−	0.767314i	$-0.721593\pi$
$839$		−	44.4512i		−	1.53463i	0.641272	−	0.767314i	$-0.278407\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	52.7976			1.81845
$844$	3.00000	+	6.24500i	0.103264	+	0.214962i
$845$			53.4871i			1.84001i
$846$	−21.0788	+	33.5108i	−0.724705	+	1.15213i
$847$		0			0
$848$		0			0
$849$	30.0000			1.02960
$850$		0			0
$851$		0			0
$852$	−15.3672	+	7.38216i	−0.526471	+	0.252908i
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−30.7583	+	48.8991i	−1.05007	+	1.66939i
$859$	−10.3923			−0.354581			−0.177290	−	0.984159i	$-0.556733\pi$
$859$	−10.3923			−0.354581			−0.177290	+	0.984159i	$0.556733\pi$
$860$	−29.6693	+	14.2527i	−1.01171	+	0.486012i
$861$		0			0
$862$	−30.1506	+	47.9330i	−1.02693	+	1.63261i
$863$		−	47.6794i		−	1.62303i	−0.584334	−	0.811513i	$-0.698644\pi$
$863$		−	47.6794i		−	1.62303i	0.584334	−	0.811513i	$-0.301356\pi$
$864$	27.7678	−	9.64111i	0.944679	−	0.327997i
$865$		0			0
$866$	−24.8809	−	15.6505i	−0.845489	−	0.531826i
$867$	−29.4449			−1.00000
$868$		0			0
$869$			94.3356i			3.20011i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	9.40479	−	14.9516i	0.317396	−	0.504592i
$879$		−	56.4990i		−	1.90566i
$880$	84.0333	+	67.2805i	2.83276	+	2.26803i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−25.1388	−	15.8127i	−0.846467	−	0.532441i
$883$	−51.9615			−1.74864			−0.874322	−	0.485346i	$-0.838694\pi$
$883$	−51.9615			−1.74864			−0.874322	+	0.485346i	$0.838694\pi$
$884$		0			0
$885$			3.34668i			0.112498i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$	13.3797	−	21.2709i	0.448489	−	0.713001i
$891$	58.8690			1.97218
$892$		0			0
$893$		0			0
$894$	−0.385263	+	0.612486i	−0.0128851	+	0.0204846i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	49.0788	+	30.8714i	1.63778	+	1.03019i
$899$		0			0
$900$	−30.9904	−	64.5116i	−1.03301	−	2.15039i
$901$		0			0
$902$	61.2631	+	38.5355i	2.03984	+	1.28309i
$903$		0			0
$904$		0			0
$905$	102.777			3.41644
$906$		0			0
$907$	17.3205			0.575118			0.287559	−	0.957763i	$-0.407156\pi$
$907$	17.3205			0.575118			0.287559	+	0.957763i	$0.407156\pi$
$908$	4.85126	+	10.0987i	0.160995	+	0.335137i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−82.4974			−2.73027
$914$		0			0
$915$	−51.3887			−1.69886
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	14.4222i		−	0.475745i	−0.971296	−	0.237872i	$-0.923550\pi$
$919$		−	14.4222i		−	0.475745i	0.971296	−	0.237872i	$-0.0764500\pi$
$920$		0			0
$921$		0			0
$922$	18.8109	−	29.9053i	0.619504	−	0.984878i
$923$	−17.7445			−0.584067
$924$		0			0
$925$		0			0
$926$		0			0
$927$			37.4700i			1.23068i
$928$		0			0
$929$	13.2078			0.433335			0.216667	−	0.976245i	$-0.430481\pi$
$929$	13.2078			0.433335			0.216667	+	0.976245i	$0.430481\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	30.3967	+	3.46953i	0.993549	+	0.113405i
$937$	−34.6410			−1.13167			−0.565836	−	0.824518i	$-0.691447\pi$
$937$	−34.6410			−1.13167			−0.565836	+	0.824518i	$0.691447\pi$
$938$		0			0
$939$	60.0000			1.95803
$940$	33.2487	+	69.2127i	1.08445	+	2.25747i
$941$		−	21.1627i		−	0.689886i	−0.938624	−	0.344943i	$-0.887898\pi$
$941$		−	21.1627i		−	0.689886i	0.938624	−	0.344943i	$-0.112102\pi$
$942$	−32.5791	+	51.7938i	−1.06149	+	1.68753i
$943$		0			0
$944$	−1.17406	+	1.46640i	−0.0382123	+	0.0477271i
$945$		0			0
$946$	−31.3205	−	19.7011i	−1.01832	−	0.640538i
$947$	57.9297			1.88246			0.941231	−	0.337763i	$-0.109670\pi$
$947$	57.9297			1.88246			0.941231	+	0.337763i	$0.109670\pi$
$948$	45.0333	−	21.6333i	1.46261	−	0.702617i
$949$		0			0
$950$		0			0
$951$		−	29.0170i		−	0.940940i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	46.4914	−	22.3337i	1.50364	−	0.722325i
$957$		0			0
$958$	−25.1699	+	40.0147i	−0.813202	+	1.29282i
$959$		0			0
$960$	12.8472	−	55.5443i	0.414641	−	1.79268i
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−10.1952	+	89.3205i	−0.327685	+	2.87087i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−13.5000	−	28.1025i	−0.433013	−	0.901388i
$973$		0			0
$974$		0			0
$975$		−	74.4916i		−	2.38564i
$976$	−22.5167	−	18.0278i	−0.720741	−	0.577054i
$977$	51.5145			1.64810			0.824048	−	0.566520i	$-0.191711\pi$
$977$	51.5145			1.64810			0.824048	+	0.566520i	$0.191711\pi$
$978$		0			0
$979$	28.2487			0.902833
$980$	−51.9213	+	24.9422i	−1.65856	+	0.796748i
$981$		0			0
$982$		0			0
$983$			12.5594i			0.400583i	0.979736	+	0.200292i	$0.0641890\pi$
$983$			12.5594i			0.400583i	−0.979736	+	0.200292i	$0.935811\pi$
$984$	4.34679	−	38.0825i	0.138571	−	1.21402i
$985$	−100.354			−3.19754
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	60.7935	−	96.6486i	1.93214	−	3.07169i
$991$			62.4500i			1.98379i	0.127064	+	0.991894i	$0.459445\pi$
$991$			62.4500i			1.98379i	−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−59.3386			−1.88116
$996$	18.9186	+	39.3821i	0.599457	+	1.24787i
$997$			24.9800i			0.791124i	0.918439	+	0.395562i	$0.129450\pi$
$997$			24.9800i			0.791124i	−0.918439	+	0.395562i	$0.870550\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.2.h.a.155.8	yes	8
3.2	odd	2	inner	312.2.h.a.155.1	✓	8
4.3	odd	2		1248.2.h.a.623.5		8
8.3	odd	2	inner	312.2.h.a.155.7	yes	8
8.5	even	2		1248.2.h.a.623.7		8
12.11	even	2		1248.2.h.a.623.8		8
13.12	even	2	inner	312.2.h.a.155.1	✓	8
24.5	odd	2		1248.2.h.a.623.6		8
24.11	even	2	inner	312.2.h.a.155.2	yes	8
39.38	odd	2	CM	312.2.h.a.155.8	yes	8
52.51	odd	2		1248.2.h.a.623.8		8
104.51	odd	2	inner	312.2.h.a.155.2	yes	8
104.77	even	2		1248.2.h.a.623.6		8
156.155	even	2		1248.2.h.a.623.5		8
312.77	odd	2		1248.2.h.a.623.7		8
312.155	even	2	inner	312.2.h.a.155.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.h.a.155.1	✓	8	3.2	odd	2	inner
312.2.h.a.155.1	✓	8	13.12	even	2	inner
312.2.h.a.155.2	yes	8	24.11	even	2	inner
312.2.h.a.155.2	yes	8	104.51	odd	2	inner
312.2.h.a.155.7	yes	8	8.3	odd	2	inner
312.2.h.a.155.7	yes	8	312.155	even	2	inner
312.2.h.a.155.8	yes	8	1.1	even	1	trivial
312.2.h.a.155.8	yes	8	39.38	odd	2	CM
1248.2.h.a.623.5		8	4.3	odd	2
1248.2.h.a.623.5		8	156.155	even	2
1248.2.h.a.623.6		8	24.5	odd	2
1248.2.h.a.623.6		8	104.77	even	2
1248.2.h.a.623.7		8	8.5	even	2
1248.2.h.a.623.7		8	312.77	odd	2
1248.2.h.a.623.8		8	12.11	even	2
1248.2.h.a.623.8		8	52.51	odd	2

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.19709 + 0.752986i) q^{2} -1.73205 q^{3} +(0.866025 + 1.80278i) q^{4} -4.11439i q^{5} +(-2.07341 - 1.30421i) q^{6} +(-0.320758 + 2.81018i) q^{8} +3.00000 q^{9} +(3.09808 - 4.92527i) q^{10} +6.54099 q^{11} +(-1.50000 - 3.12250i) q^{12} -3.60555i q^{13} +7.12633i q^{15} +(-2.50000 + 3.12250i) q^{16} +(3.59126 + 2.25896i) q^{18} +(7.41732 - 3.56317i) q^{20} +(7.83013 + 4.92527i) q^{22} +(0.555569 - 4.86738i) q^{24} -11.9282 q^{25} +(2.71493 - 4.31615i) q^{26} -5.19615 q^{27} +(-5.36603 + 8.53083i) q^{30} +(-5.34391 + 1.85543i) q^{32} -11.3293 q^{33} +(2.59808 + 5.40833i) q^{36} +6.24500i q^{39} +(11.5622 + 1.31972i) q^{40} +7.82403 q^{41} -4.00000 q^{43} +(5.66467 + 11.7919i) q^{44} -12.3432i q^{45} +9.33123i q^{47} +(4.33013 - 5.40833i) q^{48} -7.00000 q^{49} +(-14.2791 - 8.98177i) q^{50} +(6.50000 - 3.12250i) q^{52} +(-6.22024 - 3.91263i) q^{54} -26.9122i q^{55} +0.469622 q^{59} +(-12.8472 + 6.17158i) q^{60} +7.21110i q^{61} +(-7.79423 - 1.80278i) q^{64} -14.8346 q^{65} +(-13.5622 - 8.53083i) q^{66} -4.92144i q^{71} +(-0.962274 + 8.43054i) q^{72} +20.6603 q^{75} +(-4.70239 + 7.47579i) q^{78} +14.4222i q^{79} +(12.8472 + 10.2860i) q^{80} +9.00000 q^{81} +(9.36603 + 5.89138i) q^{82} -12.6124 q^{83} +(-4.78834 - 3.01194i) q^{86} +(-2.09808 + 18.3814i) q^{88} +4.31872 q^{89} +(9.29423 - 14.7758i) q^{90} +(-7.02628 + 11.1703i) q^{94} +(9.25592 - 3.21370i) q^{96} +(-8.37960 - 5.27090i) q^{98} +19.6230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} + 4 q^{10} - 12 q^{12} - 20 q^{16} + 28 q^{22} - 40 q^{25} - 36 q^{30} + 44 q^{40} - 32 q^{43} - 56 q^{49} + 52 q^{52} - 60 q^{66} + 96 q^{75} + 72 q^{81} + 68 q^{82} + 4 q^{88} + 12 q^{90}+ \cdots + 20 q^{94}+O(q^{100}) \) 8 * q + 24 * q^9 + 4 * q^10 - 12 * q^12 - 20 * q^16 + 28 * q^22 - 40 * q^25 - 36 * q^30 + 44 * q^40 - 32 * q^43 - 56 * q^49 + 52 * q^52 - 60 * q^66 + 96 * q^75 + 72 * q^81 + 68 * q^82 + 4 * q^88 + 12 * q^90 + 20 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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