LMFDB - Embedded newform 312.2.h.b.155.9

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 $ x^12 - 16x^9 + 92x^6 - 68*x^3 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			155.9
Root			$0.803519 - 0.227114i$ of defining polynomial
Character	$\chi$	$=$	312.155
Dual form			312.2.h.b.155.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} -5.72660 q^{10} +(-0.223462 + 3.45689i) q^{12} -3.60555 q^{13} -4.22937i q^{14} +(6.99902 + 0.452435i) q^{15} +4.00000 q^{16} +6.14129i q^{17} +(0.546228 - 4.20733i) q^{18} -8.09864i q^{20} +(-0.334145 + 5.16911i) q^{21} +(-4.88878 - 0.316023i) q^{24} -11.3970 q^{25} -5.09902i q^{26} +(-1.00000 + 5.09902i) q^{27} +5.98123 q^{28} +(-0.639839 + 9.89811i) q^{30} -7.21110 q^{31} +5.65685i q^{32} -8.68510 q^{34} -12.1100i q^{35} +(5.95006 + 0.772483i) q^{36} +11.6757 q^{37} +(-0.402852 + 6.23199i) q^{39} +11.4532 q^{40} +(-7.31023 - 0.472552i) q^{42} +12.1236 q^{43} +(1.56402 - 12.0469i) q^{45} +4.99739i q^{47} +(0.446924 - 6.91377i) q^{48} +1.94378 q^{49} -16.1178i q^{50} +(10.6149 + 0.686173i) q^{51} +7.21110 q^{52} +(-7.21110 - 1.41421i) q^{54} +8.45874i q^{56} +(-13.9980 - 0.904869i) q^{60} -10.1980i q^{62} +(8.89718 + 1.15510i) q^{63} -8.00000 q^{64} -14.6000i q^{65} -12.2826i q^{68} +17.1261 q^{70} +3.10125i q^{71} +(-1.09246 + 8.41466i) q^{72} +16.5119i q^{74} +(-1.27340 + 19.6990i) q^{75} +(-8.81337 - 0.569719i) q^{78} +16.1973i q^{80} +(8.70163 + 2.29816i) q^{81} +(0.668289 - 10.3382i) q^{84} -24.8680 q^{85} +17.1453i q^{86} +(17.0368 + 2.21185i) q^{90} +10.7828 q^{91} +(-0.805704 + 12.4640i) q^{93} -7.06738 q^{94} +(9.77755 + 0.632046i) q^{96} +2.74893i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 24 q^{4}+O(q^{10}) $ 12 * q - 24 * q^4	$ 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} - 12 q^{27} + 24 q^{30} - 48 q^{42} + 84 q^{49} + 60 q^{51} - 96 q^{64} - 84 q^{75} + 96 q^{90}+O(q^{100}) $ 12 * q - 24 * q^4 + 48 * q^16 - 60 * q^25 - 12 * q^27 + 24 * q^30 - 48 * q^42 + 84 * q^49 + 60 * q^51 - 96 * q^64 - 84 * q^75 + 96 * q^90

Display 100 coefficients 10 coefficients

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	0.111731	−	1.72844i	0.0645079	−	0.997917i
$3$	0.111731	−	1.72844i	0.0645079	−	0.997917i
$4$	−2.00000			−1.00000
$5$			4.04932i			1.81091i	0.424441	+	0.905455i	$0.360470\pi$
$5$			4.04932i			1.81091i	−0.424441	+	0.905455i	$0.639530\pi$
$6$	2.44439	+	0.158012i	0.997917	+	0.0645079i
$7$	−2.99062			−1.13035			−0.565173	−	0.824972i	$-0.691191\pi$
$7$	−2.99062			−1.13035			−0.565173	+	0.824972i	$0.691191\pi$
$8$		−	2.82843i		−	1.00000i
$9$	−2.97503	−	0.386242i	−0.991677	−	0.128747i
$10$	−5.72660			−1.81091
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.223462	+	3.45689i	−0.0645079	+	0.997917i
$13$	−3.60555			−1.00000
$13$	−3.60555			−1.00000
$14$		−	4.22937i		−	1.13035i
$15$	6.99902	+	0.452435i	1.80714	+	0.116818i
$16$	4.00000			1.00000
$17$			6.14129i			1.48948i	0.667354	+	0.744741i	$0.267427\pi$
$17$			6.14129i			1.48948i	−0.667354	+	0.744741i	$0.732573\pi$
$18$	0.546228	−	4.20733i	0.128747	−	0.991677i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		−	8.09864i		−	1.81091i
$21$	−0.334145	+	5.16911i	−0.0729163	+	1.12799i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−4.88878	−	0.316023i	−0.997917	−	0.0645079i
$25$	−11.3970			−2.27940
$26$		−	5.09902i		−	1.00000i
$27$	−1.00000	+	5.09902i	−0.192450	+	0.981307i
$28$	5.98123			1.13035
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−0.639839	+	9.89811i	−0.116818	+	1.80714i
$31$	−7.21110			−1.29515			−0.647576	−	0.762001i	$-0.724217\pi$
$31$	−7.21110			−1.29515			−0.647576	+	0.762001i	$0.724217\pi$
$32$			5.65685i			1.00000i
$33$		0			0
$34$	−8.68510			−1.48948
$35$		−	12.1100i		−	2.04696i
$36$	5.95006	+	0.772483i	0.991677	+	0.128747i
$37$	11.6757			1.91948			0.959738	−	0.280898i	$-0.0906323\pi$
$37$	11.6757			1.91948			0.959738	+	0.280898i	$0.0906323\pi$
$38$		0			0
$39$	−0.402852	+	6.23199i	−0.0645079	+	0.997917i
$40$	11.4532			1.81091
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−7.31023	−	0.472552i	−1.12799	−	0.0729163i
$43$	12.1236			1.84883			0.924415	−	0.381388i	$-0.124554\pi$
$43$	12.1236			1.84883			0.924415	+	0.381388i	$0.124554\pi$
$44$		0			0
$45$	1.56402	−	12.0469i	0.233150	−	1.79584i
$46$		0			0
$47$			4.99739i			0.728944i	0.931214	+	0.364472i	$0.118751\pi$
$47$			4.99739i			0.728944i	−0.931214	+	0.364472i	$0.881249\pi$
$48$	0.446924	−	6.91377i	0.0645079	−	0.997917i
$49$	1.94378			0.277683
$50$		−	16.1178i		−	2.27940i
$51$	10.6149	+	0.686173i	1.48638	+	0.0960834i
$52$	7.21110			1.00000
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−7.21110	−	1.41421i	−0.981307	−	0.192450i
$55$		0			0
$56$			8.45874i			1.13035i
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−13.9980	−	0.904869i	−1.80714	−	0.116818i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	10.1980i		−	1.29515i
$63$	8.89718	+	1.15510i	1.12094	+	0.145529i
$64$	−8.00000			−1.00000
$65$		−	14.6000i		−	1.81091i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		−	12.2826i		−	1.48948i
$69$		0			0
$70$	17.1261			2.04696
$71$			3.10125i			0.368051i	0.982921	+	0.184025i	$0.0589129\pi$
$71$			3.10125i			0.368051i	−0.982921	+	0.184025i	$0.941087\pi$
$72$	−1.09246	+	8.41466i	−0.128747	+	0.991677i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$			16.5119i			1.91948i
$75$	−1.27340	+	19.6990i	−0.147039	+	2.27465i
$76$		0			0
$77$		0			0
$78$	−8.81337	−	0.569719i	−0.997917	−	0.0645079i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			16.1973i			1.81091i
$81$	8.70163	+	2.29816i	0.966848	+	0.255351i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	0.668289	−	10.3382i	0.0729163	−	1.12799i
$85$	−24.8680			−2.69732
$86$			17.1453i			1.84883i
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	17.0368	+	2.21185i	1.79584	+	0.233150i
$91$	10.7828			1.13035
$92$		0			0
$93$	−0.805704	+	12.4640i	−0.0835476	+	1.29245i
$94$	−7.06738			−0.728944
$95$		0			0
$96$	9.77755	+	0.632046i	0.997917	+	0.0645079i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			2.74893i			0.277683i
$99$		0			0
$100$	22.7940			2.27940
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−0.970395	+	15.0117i	−0.0960834	+	1.48638i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			10.1980i			1.00000i
$105$	−20.9314	−	1.35306i	−2.04269	−	0.132045i
$106$		0			0
$107$		−	10.1980i		−	0.985882i	−0.870063	−	0.492941i	$-0.835922\pi$
$107$		−	10.1980i		−	0.985882i	0.870063	−	0.492941i	$-0.164078\pi$
$108$	2.00000	−	10.1980i	0.192450	−	0.981307i
$109$	0.286752			0.0274659			0.0137329	−	0.999906i	$-0.495629\pi$
$109$	0.286752			0.0274659			0.0137329	+	0.999906i	$0.495629\pi$
$110$		0			0
$111$	1.30454	−	20.1808i	0.123821	−	1.91548i
$112$	−11.9625			−1.13035
$113$			20.3961i			1.91870i	0.282216	+	0.959351i	$0.408930\pi$
$113$			20.3961i			1.91870i	−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.7266	+	1.39261i	0.991677	+	0.128747i
$118$		0			0
$119$		−	18.3662i		−	1.68363i
$120$	1.27968	−	19.7962i	0.116818	−	1.80714i
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	14.4222			1.29515
$125$		−	25.9035i		−	2.31688i
$126$	−1.63356	+	12.5825i	−0.145529	+	1.12094i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	11.3137i		−	1.00000i
$129$	1.35458	−	20.9549i	0.119264	−	1.84498i
$130$	20.6476			1.81091
$131$			18.8294i			1.64513i	0.568669	+	0.822566i	$0.307458\pi$
$131$			18.8294i			1.64513i	−0.568669	+	0.822566i	$0.692542\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−20.6476	−	4.04932i	−1.77706	−	0.348510i
$136$	17.3702			1.48948
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	−23.5768			−1.99976			−0.999879	−	0.0155623i	$-0.995046\pi$
$139$	−23.5768			−1.99976			−0.999879	+	0.0155623i	$0.995046\pi$
$140$			24.2199i			2.04696i
$141$	8.63770	+	0.558363i	0.727426	+	0.0470227i
$142$	−4.38583			−0.368051
$143$		0			0
$144$	−11.9001	−	1.54497i	−0.991677	−	0.128747i
$145$		0			0
$146$		0			0
$147$	0.217181	−	3.35972i	0.0179128	−	0.277105i
$148$	−23.3514			−1.91948
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$	−27.8587	−	1.80086i	−2.27465	−	0.147039i
$151$	−14.3796			−1.17019			−0.585097	−	0.810964i	$-0.698943\pi$
$151$	−14.3796			−1.17019			−0.585097	+	0.810964i	$0.698943\pi$
$152$		0			0
$153$	2.37202	−	18.2705i	0.191767	−	1.47709i
$154$		0			0
$155$		−	29.2001i		−	2.34541i
$156$	0.805704	−	12.4640i	0.0645079	−	0.997917i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	−22.9064			−1.81091
$161$		0			0
$162$	−3.25009	+	12.3060i	−0.255351	+	0.966848i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.1421i		−	1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$		−	14.1421i		−	1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$	14.6205	+	0.945104i	1.12799	+	0.0729163i
$169$	13.0000			1.00000
$170$		−	35.1687i		−	2.69732i
$171$		0			0
$172$	−24.2472			−1.84883
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	34.0840			2.57651
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.1945i			1.06095i	0.847702	+	0.530473i	$0.177986\pi$
$179$			14.1945i			1.06095i	−0.847702	+	0.530473i	$0.822014\pi$
$180$	−3.12803	+	24.0937i	−0.233150	+	1.79584i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$			15.2492i			1.13035i
$183$		0			0
$184$		0			0
$185$			47.2787i			3.47600i
$186$	−17.6267	−	1.13944i	−1.29245	−	0.0835476i
$187$		0			0
$188$		−	9.99478i		−	0.728944i
$189$	2.99062	−	15.2492i	0.217535	−	1.10922i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−0.893848	+	13.8275i	−0.0645079	+	0.997917i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	−25.2353	−	1.63128i	−1.80714	−	0.116818i
$196$	−3.88757			−0.277683
$197$			5.94546i			0.423596i	0.977313	+	0.211798i	$0.0679319\pi$
$197$			5.94546i			0.423596i	−0.977313	+	0.211798i	$0.932068\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			32.2356i			2.27940i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−21.2297	−	1.37235i	−1.48638	−	0.0960834i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−14.4222			−1.00000
$209$		0			0
$210$	1.91351	−	29.6014i	0.132045	−	2.04269i
$211$	13.4644			0.926925			0.463463	−	0.886117i	$-0.346607\pi$
$211$	13.4644			0.926925			0.463463	+	0.886117i	$0.346607\pi$
$212$		0			0
$213$	5.36034	+	0.346506i	0.367284	+	0.0237422i
$214$	14.4222			0.985882
$215$			49.0923i			3.34807i
$216$	14.4222	+	2.82843i	0.981307	+	0.192450i
$217$	21.5656			1.46397
$218$			0.405529i			0.0274659i
$219$		0			0
$220$		0			0
$221$		−	22.1427i		−	1.48948i
$222$	28.5400	+	1.84490i	1.91548	+	0.123821i
$223$	8.39834			0.562395			0.281197	−	0.959650i	$-0.409268\pi$
$223$	8.39834			0.562395			0.281197	+	0.959650i	$0.409268\pi$
$224$		−	16.9175i		−	1.13035i
$225$	33.9064	+	4.40199i	2.26043	+	0.293466i
$226$	−28.8444			−1.91870
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−6.26798			−0.414200			−0.207100	−	0.978320i	$-0.566403\pi$
$229$	−6.26798			−0.414200			−0.207100	+	0.978320i	$0.566403\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.96513i		−	0.652837i	−0.945225	−	0.326419i	$-0.894158\pi$
$233$		−	9.96513i		−	0.652837i	0.945225	−	0.326419i	$-0.105842\pi$
$234$	−1.96945	+	15.1697i	−0.128747	+	0.991677i
$235$	−20.2360			−1.32005
$236$		0			0
$237$		0			0
$238$	25.9738			1.68363
$239$		−	19.2985i		−	1.24832i	−0.781297	−	0.624159i	$-0.785442\pi$
$239$		−	19.2985i		−	1.24832i	0.781297	−	0.624159i	$-0.214558\pi$
$240$	27.9961	+	1.80974i	1.80714	+	0.116818i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	15.5563i		−	1.00000i
$243$	4.94449	−	14.7835i	0.317189	−	0.948362i
$244$		0			0
$245$			7.87100i			0.502860i
$246$		0			0
$247$		0			0
$248$			20.3961i			1.29515i
$249$		0			0
$250$	36.6330			2.31688
$251$		−	10.1980i		−	0.643695i	−0.946792	−	0.321847i	$-0.895696\pi$
$251$		−	10.1980i		−	0.643695i	0.946792	−	0.321847i	$-0.104304\pi$
$252$	−17.7944	−	2.31020i	−1.12094	−	0.145529i
$253$		0			0
$254$		0			0
$255$	−2.77853	+	42.9830i	−0.173998	+	2.69170i
$256$	16.0000			1.00000
$257$			10.7762i			0.672200i	0.941826	+	0.336100i	$0.109108\pi$
$257$			10.7762i			0.672200i	−0.941826	+	0.336100i	$0.890892\pi$
$258$	29.6348	+	1.91567i	1.84498	+	0.119264i
$259$	−34.9176			−2.16967
$260$			29.2001i			1.81091i
$261$		0			0
$262$	−26.6288			−1.64513
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	5.72660	−	29.2001i	0.348510	−	1.77706i
$271$	−32.3233			−1.96350			−0.981749	−	0.190179i	$-0.939093\pi$
$271$	−32.3233			−1.96350			−0.981749	+	0.190179i	$0.939093\pi$
$272$			24.5652i			1.48948i
$273$	1.20478	−	18.6375i	0.0729163	−	1.12799i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	33.3426i		−	1.99976i
$279$	21.4533	+	2.78523i	1.28437	+	0.166747i
$280$	−34.2521			−2.04696
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−0.789645	+	12.2156i	−0.0470227	+	0.727426i
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		−	6.20250i		−	0.368051i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.18491	−	16.8293i	0.128747	−	0.991677i
$289$	−20.7154			−1.21856
$290$		0			0
$291$		0			0
$292$		0			0
$293$			30.2414i			1.76672i	0.468695	+	0.883360i	$0.344724\pi$
$293$			30.2414i			1.76672i	−0.468695	+	0.883360i	$0.655276\pi$
$294$	4.75136	+	0.307140i	0.277105	+	0.0179128i
$295$		0			0
$296$		−	33.0239i		−	1.91948i
$297$		0			0
$298$	−16.0000			−0.926855
$299$		0			0
$300$	2.54679	−	39.3981i	0.147039	−	2.27465i
$301$	−36.2570			−2.08982
$302$		−	20.3358i		−	1.17019i
$303$		0			0
$304$		0			0
$305$		0			0
$306$	25.8384	+	3.35454i	1.47709	+	0.191767i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	41.2951			2.34541
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	17.6267	+	1.13944i	0.997917	+	0.0645079i
$313$	30.6442			1.73211			0.866055	−	0.499948i	$-0.166648\pi$
$313$	30.6442			1.73211			0.866055	+	0.499948i	$0.166648\pi$
$314$		0			0
$315$	−4.67737	+	36.0275i	−0.263540	+	2.02992i
$316$		0			0
$317$			28.2843i			1.58860i	0.607524	+	0.794301i	$0.292163\pi$
$317$			28.2843i			1.58860i	−0.607524	+	0.794301i	$0.707837\pi$
$318$		0			0
$319$		0			0
$320$		−	32.3946i		−	1.81091i
$321$	−17.6267	−	1.13944i	−0.983828	−	0.0635972i
$322$		0			0
$323$		0			0
$324$	−17.4033	−	4.59632i	−0.966848	−	0.255351i
$325$	41.0924			2.27940
$326$		0			0
$327$	0.0320391	−	0.495635i	0.00177177	−	0.0274087i
$328$		0			0
$329$		−	14.9453i		−	0.823959i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−34.7356	−	4.50964i	−1.90350	−	0.247127i
$334$	20.0000			1.09435
$335$		0			0
$336$	−1.33658	+	20.6764i	−0.0729163	+	1.12799i
$337$	27.9626			1.52322			0.761611	−	0.648035i	$-0.224409\pi$
$337$	27.9626			1.52322			0.761611	+	0.648035i	$0.224409\pi$
$338$			18.3848i			1.00000i
$339$	35.2535	+	2.27887i	1.91471	+	0.123771i
$340$	49.7361			2.69732
$341$		0			0
$342$		0			0
$343$	15.1212			0.816468
$344$		−	34.2907i		−	1.84883i
$345$		0			0
$346$		0			0
$347$		−	27.2881i		−	1.46490i	−0.680819	−	0.732452i	$-0.738376\pi$
$347$		−	27.2881i		−	1.46490i	0.680819	−	0.732452i	$-0.261624\pi$
$348$		0			0
$349$	29.6194			1.58549			0.792745	−	0.609553i	$-0.208651\pi$
$349$	29.6194			1.58549			0.792745	+	0.609553i	$0.208651\pi$
$350$			48.2021i			2.57651i
$351$	3.60555	−	18.3848i	0.192450	−	0.981307i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−12.5580			−0.666507
$356$		0			0
$357$	−31.7450	−	2.05208i	−1.68012	−	0.108608i
$358$	−20.0741			−1.06095
$359$		−	31.1127i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$359$		−	31.1127i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$360$	−34.0737	−	4.42370i	−1.79584	−	0.233150i
$361$	19.0000			1.00000
$362$		0			0
$363$	−1.22904	+	19.0129i	−0.0645079	+	0.997917i
$364$	−21.5656			−1.13035
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$	−66.8622			−3.47600
$371$		0			0
$372$	1.61141	−	24.9280i	0.0835476	−	1.29245i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−44.7726	−	2.89422i	−2.31205	−	0.149457i
$376$	14.1348			0.728944
$377$		0			0
$378$	21.5656	+	4.22937i	1.10922	+	0.217535i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			27.3972i			1.39993i	0.714177	+	0.699965i	$0.246801\pi$
$383$			27.3972i			1.39993i	−0.714177	+	0.699965i	$0.753199\pi$
$384$	−19.5551	−	1.26409i	−0.997917	−	0.0645079i
$385$		0			0
$386$		0			0
$387$	−36.0681	−	4.68263i	−1.83344	−	0.238032i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	2.30697	−	35.6881i	0.116818	−	1.80714i
$391$		0			0
$392$		−	5.49785i		−	0.277683i
$393$	32.5455	+	2.10383i	1.64171	+	0.106124i
$394$	−8.40815			−0.423596
$395$		0			0
$396$		0			0
$397$	36.0555			1.80957			0.904787	−	0.425864i	$-0.140030\pi$
$397$	36.0555			1.80957			0.904787	+	0.425864i	$0.140030\pi$
$398$		0			0
$399$		0			0
$400$	−45.5880			−2.27940
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	26.0000			1.29515
$404$		0			0
$405$	−9.30599	+	35.2357i	−0.462418	+	1.75088i
$406$		0			0
$407$		0			0
$408$	1.94079	−	30.0234i	0.0960834	−	1.48638i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	20.3961i		−	1.00000i
$417$	−2.63426	+	40.7512i	−0.129000	+	1.99559i
$418$		0			0
$419$			34.9358i			1.70673i	0.521317	+	0.853363i	$0.325441\pi$
$419$			34.9358i			1.70673i	−0.521317	+	0.853363i	$0.674559\pi$
$420$	41.8628	+	2.70612i	2.04269	+	0.132045i
$421$	41.0084			1.99863			0.999313	−	0.0370678i	$-0.0118017\pi$
$421$	41.0084			1.99863			0.999313	+	0.0370678i	$0.0118017\pi$
$422$			19.0415i			0.926925i
$423$	1.93020	−	14.8674i	0.0938495	−	0.722877i
$424$		0			0
$425$		−	69.9922i		−	3.39512i
$426$	−0.490033	+	7.58066i	−0.0237422	+	0.367284i
$427$		0			0
$428$			20.3961i			0.985882i
$429$		0			0
$430$	−69.4270			−3.34807
$431$		−	23.0908i		−	1.11224i	−0.831100	−	0.556122i	$-0.812289\pi$
$431$		−	23.0908i		−	1.11224i	0.831100	−	0.556122i	$-0.187711\pi$
$432$	−4.00000	+	20.3961i	−0.192450	+	0.981307i
$433$	−40.7566			−1.95864			−0.979319	−	0.202324i	$-0.935151\pi$
$433$	−40.7566			−1.95864			−0.979319	+	0.202324i	$0.935151\pi$
$434$			30.4984i			1.46397i
$435$		0			0
$436$	−0.573504			−0.0274659
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−5.78282	−	0.750770i	−0.275372	−	0.0357509i
$442$	31.3146			1.48948
$443$			39.5707i			1.88006i	0.341090	+	0.940031i	$0.389204\pi$
$443$			39.5707i			1.88006i	−0.341090	+	0.940031i	$0.610796\pi$
$444$	−2.60908	+	40.3616i	−0.123821	+	1.91548i
$445$		0			0
$446$			11.8771i			0.562395i
$447$	19.5551	+	1.26409i	0.924925	+	0.0597895i
$448$	23.9249			1.13035
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−6.22535	+	47.9509i	−0.293466	+	2.26043i
$451$		0			0
$452$		−	40.7922i		−	1.91870i
$453$	−1.60664	+	24.8543i	−0.0754868	+	1.16776i
$454$		0			0
$455$			43.6631i			2.04696i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		−	8.86427i		−	0.414200i
$459$	−31.3146	−	6.14129i	−1.46164	−	0.286651i
$460$		0			0
$461$		−	18.3505i		−	0.854666i	−0.904094	−	0.427333i	$-0.859453\pi$
$461$		−	18.3505i		−	0.854666i	0.904094	−	0.427333i	$-0.140547\pi$
$462$		0			0
$463$	−7.21110			−0.335128			−0.167564	−	0.985861i	$-0.553590\pi$
$463$	−7.21110			−0.335128			−0.167564	+	0.985861i	$0.553590\pi$
$464$		0			0
$465$	−50.4706	−	3.26255i	−2.34052	−	0.151297i
$466$	14.0928			0.652837
$467$		−	10.1980i		−	0.471909i	−0.971764	−	0.235954i	$-0.924178\pi$
$467$		−	10.1980i		−	0.471909i	0.971764	−	0.235954i	$-0.0758216\pi$
$468$	−21.4533	−	2.78523i	−0.991677	−	0.128747i
$469$		0			0
$470$		−	28.6181i		−	1.32005i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$			36.7325i			1.68363i
$477$		0			0
$478$	27.2922			1.24832
$479$			31.1894i			1.42508i	0.701630	+	0.712541i	$0.252456\pi$
$479$			31.1894i			1.42508i	−0.701630	+	0.712541i	$0.747544\pi$
$480$	−2.55936	+	39.5924i	−0.116818	+	1.80714i
$481$	−42.0974			−1.91948
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$	20.9070	+	6.99256i	0.948362	+	0.317189i
$487$	36.0555			1.63383			0.816916	−	0.576757i	$-0.195682\pi$
$487$	36.0555			1.63383			0.816916	+	0.576757i	$0.195682\pi$
$488$		0			0
$489$		0			0
$490$	−11.1313			−0.502860
$491$		−	43.3946i		−	1.95837i	−0.202972	−	0.979185i	$-0.565060\pi$
$491$		−	43.3946i		−	1.95837i	0.202972	−	0.979185i	$-0.434940\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−28.8444			−1.29515
$497$		−	9.27465i		−	0.416025i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			51.8069i			2.31688i
$501$	−24.4439	−	1.58012i	−1.09207	−	0.0705944i
$502$	14.4222			0.643695
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	3.26712	−	25.1650i	0.145529	−	1.12094i
$505$		0			0
$506$		0			0
$507$	1.45250	−	22.4698i	0.0645079	−	0.997917i
$508$		0			0
$509$		−	39.5980i		−	1.75515i	−0.479440	−	0.877575i	$-0.659160\pi$
$509$		−	39.5980i		−	1.75515i	0.479440	−	0.877575i	$-0.340840\pi$
$510$	−60.7872	−	3.92944i	−2.69170	−	0.173998i
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$	−15.2398			−0.672200
$515$		0			0
$516$	−2.70916	+	41.9099i	−0.119264	+	1.84498i
$517$		0			0
$518$		−	49.3809i		−	2.16967i
$519$		0			0
$520$	−41.2951			−1.81091
$521$			22.2477i			0.974690i	0.873210	+	0.487345i	$0.162035\pi$
$521$			22.2477i			0.974690i	−0.873210	+	0.487345i	$0.837965\pi$
$522$		0			0
$523$	−34.0000			−1.48672			−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000			−1.48672			−0.743358	+	0.668894i	$0.766768\pi$
$524$		−	37.6588i		−	1.64513i
$525$	3.80824	−	58.9123i	0.166205	−	2.57114i
$526$		0			0
$527$		−	44.2855i		−	1.92911i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	41.2951			1.78534
$536$		0			0
$537$	24.5344	+	1.58597i	1.05874	+	0.0684395i
$538$		0			0
$539$		0			0
$540$	41.2951	+	8.09864i	1.77706	+	0.348510i
$541$	−11.1022			−0.477321			−0.238661	−	0.971103i	$-0.576708\pi$
$541$	−11.1022			−0.477321			−0.238661	+	0.971103i	$0.576708\pi$
$542$		−	45.7120i		−	1.96350i
$543$		0			0
$544$	−34.7404			−1.48948
$545$			1.16115i			0.0497382i
$546$	26.3574	+	1.70381i	1.12799	+	0.0729163i
$547$	−20.8953			−0.893416			−0.446708	−	0.894680i	$-0.647404\pi$
$547$	−20.8953			−0.893416			−0.446708	+	0.894680i	$0.647404\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	81.7185	+	5.28250i	3.46876	+	0.224229i
$556$	47.1536			1.99976
$557$		−	46.4387i		−	1.96767i	−0.179080	−	0.983834i	$-0.557312\pi$
$557$		−	46.4387i		−	1.96767i	0.179080	−	0.983834i	$-0.442688\pi$
$558$	−3.93891	+	30.3395i	−0.166747	+	1.28437i
$559$	−43.7122			−1.84883
$560$		−	48.4398i		−	2.04696i
$561$		0			0
$562$		0			0
$563$			2.72298i			0.114760i	0.998352	+	0.0573799i	$0.0182746\pi$
$563$			2.72298i			0.114760i	−0.998352	+	0.0573799i	$0.981725\pi$
$564$	−17.2754	−	1.11673i	−0.727426	−	0.0470227i
$565$	−82.5902			−3.47460
$566$			19.7990i			0.832214i
$567$	−26.0232	−	6.87292i	−1.09287	−	0.288636i
$568$	8.77166			0.368051
$569$			47.6239i			1.99650i	0.0591437	+	0.998249i	$0.481163\pi$
$569$			47.6239i			1.99650i	−0.0591437	+	0.998249i	$0.518837\pi$
$570$		0			0
$571$	46.4832			1.94526			0.972631	−	0.232356i	$-0.0746434\pi$
$571$	46.4832			1.94526			0.972631	+	0.232356i	$0.0746434\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.8003	+	3.08993i	0.991677	+	0.128747i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	29.2961i		−	1.21856i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−5.63914	+	43.4356i	−0.233150	+	1.79584i
$586$	−42.7678			−1.76672
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.434362	+	6.71944i	−0.0179128	+	0.277105i
$589$		0			0
$590$		0			0
$591$	10.2764	+	0.664292i	0.422714	+	0.0273253i
$592$	46.7028			1.91948
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	74.3708			3.04890
$596$		−	22.6274i		−	0.926855i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	55.7173	+	3.60171i	2.27465	+	0.147039i
$601$	−39.4158			−1.60781			−0.803903	−	0.594761i	$-0.797247\pi$
$601$	−39.4158			−1.60781			−0.803903	+	0.594761i	$0.797247\pi$
$602$		−	51.2751i		−	2.08982i
$603$		0			0
$604$	28.7592			1.17019
$605$		−	44.5425i		−	1.81091i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	18.0183i		−	0.728944i
$612$	−4.74404	+	36.5411i	−0.191767	+	1.47709i
$613$	36.0555			1.45627			0.728134	−	0.685435i	$-0.240388\pi$
$613$	36.0555			1.45627			0.728134	+	0.685435i	$0.240388\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$			58.4001i			2.34541i
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.61141	+	24.9280i	−0.0645079	+	0.997917i
$625$	47.9064			1.91626
$626$			43.3374i			1.73211i
$627$		0			0
$628$		0			0
$629$			71.7039i			2.85902i
$630$	−50.9506	−	6.61480i	−2.02992	−	0.263540i
$631$	44.2857			1.76299			0.881494	−	0.472196i	$-0.156539\pi$
$631$	44.2857			1.76299			0.881494	+	0.472196i	$0.156539\pi$
$632$		0			0
$633$	1.50439	−	23.2724i	0.0597940	−	0.924994i
$634$	−40.0000			−1.58860
$635$		0			0
$636$		0			0
$637$	−7.00841			−0.277683
$638$		0			0
$639$	1.19783	−	9.22632i	0.0473855	−	0.364988i
$640$	45.8128			1.81091
$641$		−	40.7922i		−	1.61119i	−0.592464	−	0.805597i	$-0.701845\pi$
$641$		−	40.7922i		−	1.61119i	0.592464	−	0.805597i	$-0.298155\pi$
$642$	1.61141	−	24.9280i	0.0635972	−	0.983828i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$	84.8532	+	5.48513i	3.34109	+	0.215977i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	6.50018	−	24.6119i	0.255351	−	0.966848i
$649$		0			0
$650$			58.1135i			2.27940i
$651$	2.40955	−	37.2750i	0.0944378	−	1.46092i
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	0.700933	+	0.0453101i	0.0274087	+	0.00177177i
$655$	−76.2463			−2.97919
$656$		0			0
$657$		0			0
$658$	21.1358			0.823959
$659$			50.9902i			1.98630i	0.116863	+	0.993148i	$0.462716\pi$
$659$			50.9902i			1.98630i	−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	−7.21110			−0.280479			−0.140240	−	0.990118i	$-0.544787\pi$
$661$	−7.21110			−0.280479			−0.140240	+	0.990118i	$0.544787\pi$
$662$		0			0
$663$	−38.2725	−	2.47403i	−1.48638	−	0.0960834i
$664$		0			0
$665$		0			0
$666$	6.37760	−	49.1236i	0.247127	−	1.90350i
$667$		0			0
$668$			28.2843i			1.09435i
$669$	0.938356	−	14.5161i	0.0362789	−	0.561223i
$670$		0			0
$671$		0			0
$672$	−29.2409	−	1.89021i	−1.12799	−	0.0729163i
$673$	26.6219			1.02620			0.513099	−	0.858330i	$-0.328498\pi$
$673$	26.6219			1.02620			0.513099	+	0.858330i	$0.328498\pi$
$674$			39.5451i			1.52322i
$675$	11.3970	−	58.1135i	0.438670	−	2.23679i
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−3.22282	+	49.8559i	−0.123771	+	1.91471i
$679$		0			0
$680$			70.3375i			2.69732i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$			21.3846i			0.816468i
$687$	−0.700328	+	10.8339i	−0.0267192	+	0.413337i
$688$	48.4944			1.84883
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	38.5913			1.46490
$695$		−	95.4700i		−	3.62138i
$696$		0			0
$697$		0			0
$698$			41.8882i			1.58549i
$699$	−17.2242	−	1.11341i	−0.651478	−	0.0421132i
$700$	−68.1680			−2.57651
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	26.0000	+	5.09902i	0.981307	+	0.192450i
$703$		0			0
$704$		0			0
$705$	−2.26099	+	34.9768i	−0.0851539	+	1.31730i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−50.4777			−1.89573			−0.947865	−	0.318671i	$-0.896763\pi$
$709$	−50.4777			−1.89573			−0.947865	+	0.318671i	$0.896763\pi$
$710$		−	17.7596i		−	0.666507i
$711$		0			0
$712$		0			0
$713$		0			0
$714$	2.90208	−	44.8942i	0.108608	−	1.68012i
$715$		0			0
$716$		−	28.3890i		−	1.06095i
$717$	−33.3564	−	2.15624i	−1.24572	−	0.0805264i
$718$	44.0000			1.64207
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	6.25606	−	48.1874i	0.233150	−	1.79584i
$721$		0			0
$722$			26.8701i			1.00000i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−26.8883	−	1.73813i	−0.997917	−	0.0645079i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		−	30.4984i		−	1.13035i
$729$	−25.0000	−	10.1980i	−0.925926	−	0.377705i
$730$		0			0
$731$			74.4545i			2.75380i
$732$		0			0
$733$	5.12098			0.189148			0.0945738	−	0.995518i	$-0.469851\pi$
$733$	5.12098			0.189148			0.0945738	+	0.995518i	$0.469851\pi$
$734$		0			0
$735$	13.6046	+	0.879435i	0.501812	+	0.0324385i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		−	94.5574i		−	3.47600i
$741$		0			0
$742$		0			0
$743$			53.5892i			1.96600i	0.183611	+	0.982999i	$0.441221\pi$
$743$			53.5892i			1.96600i	−0.183611	+	0.982999i	$0.558779\pi$
$744$	35.2535	+	2.27887i	1.29245	+	0.0835476i
$745$	−45.8128			−1.67845
$746$		0			0
$747$		0			0
$748$		0			0
$749$			30.4984i			1.11439i
$750$	4.09304	−	63.3181i	0.149457	−	2.31205i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			19.9896i			0.728944i
$753$	−17.6267	−	1.13944i	−0.642354	−	0.0415234i
$754$		0			0
$755$		−	58.2275i		−	2.11912i
$756$	−5.98123	+	30.4984i	−0.217535	+	1.10922i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−0.857565			−0.0310459
$764$		0			0
$765$	73.9832	+	9.60507i	2.67487	+	0.347272i
$766$	−38.7454			−1.39993
$767$		0			0
$768$	1.78770	−	27.6551i	0.0645079	−	0.997917i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	18.6260	+	1.20403i	0.670800	+	0.0433622i
$772$		0			0
$773$			32.1375i			1.15591i	0.816070	+	0.577953i	$0.196148\pi$
$773$			32.1375i			1.15591i	−0.816070	+	0.577953i	$0.803852\pi$
$774$	6.62225	−	51.0080i	0.238032	−	1.83344i
$775$	82.1849			2.95217
$776$		0			0
$777$	−3.90138	+	60.3530i	−0.139961	+	2.16515i
$778$		0			0
$779$		0			0
$780$	50.4706	+	3.26255i	1.80714	+	0.116818i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	7.77513			0.277683
$785$		0			0
$786$	−2.97526	+	46.0264i	−0.106124	+	1.64171i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	11.8909i		−	0.423596i
$789$		0			0
$790$		0			0
$791$		−	60.9968i		−	2.16880i
$792$		0			0
$793$		0			0
$794$			50.9902i			1.80957i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−30.6904			−1.08575
$800$		−	64.4711i		−	2.27940i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$			36.7696i			1.29515i
$807$		0			0
$808$		0			0
$809$		−	56.0827i		−	1.97176i	−0.167449	−	0.985881i	$-0.553553\pi$
$809$		−	56.0827i		−	1.97176i	0.167449	−	0.985881i	$-0.446447\pi$
$810$	−49.8308	−	13.1607i	−1.75088	−	0.462418i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−3.61151	+	55.8689i	−0.126661	+	1.95941i
$814$		0			0
$815$		0			0
$816$	42.4595	+	2.74469i	1.48638	+	0.0960834i
$817$		0			0
$818$		0			0
$819$	−32.0792	−	4.16477i	−1.12094	−	0.145529i
$820$		0			0
$821$			7.84160i			0.273674i	0.990594	+	0.136837i	$0.0436936\pi$
$821$			7.84160i			0.273674i	−0.990594	+	0.136837i	$0.956306\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	28.8444			1.00000
$833$			11.9373i			0.413604i
$834$	−57.6308	−	3.72541i	−1.99559	−	0.129000i
$835$	57.2660			1.98177
$836$		0			0
$837$	7.21110	−	36.7696i	0.249252	−	1.27094i
$838$	−49.4067			−1.70673
$839$			53.7401i			1.85531i	0.373432	+	0.927657i	$0.378181\pi$
$839$			53.7401i			1.85531i	−0.373432	+	0.927657i	$0.621819\pi$
$840$	−3.82703	+	59.2029i	−0.132045	+	2.04269i
$841$	−29.0000			−1.00000
$842$			57.9946i			1.99863i
$843$		0			0
$844$	−26.9287			−0.926925
$845$			52.6412i			1.81091i
$846$	21.0257	+	2.72971i	0.722877	+	0.0938495i
$847$	32.8968			1.13035
$848$		0			0
$849$	1.56423	−	24.1982i	0.0536844	−	0.830480i
$850$	98.9839			3.39512
$851$		0			0
$852$	−10.7207	−	0.693012i	−0.367284	−	0.0237422i
$853$	52.3973			1.79405			0.897025	−	0.441980i	$-0.145724\pi$
$853$	52.3973			1.79405			0.897025	+	0.441980i	$0.145724\pi$
$854$		0			0
$855$		0			0
$856$	−28.8444			−0.985882
$857$		−	40.7922i		−	1.39343i	−0.717346	−	0.696717i	$-0.754643\pi$
$857$		−	40.7922i		−	1.39343i	0.717346	−	0.696717i	$-0.245357\pi$
$858$		0			0
$859$	50.0000			1.70598			0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000			1.70598			0.852989	+	0.521929i	$0.174787\pi$
$860$		−	98.1846i		−	3.34807i
$861$		0			0
$862$	32.6553			1.11224
$863$			55.4854i			1.88874i	0.328880	+	0.944372i	$0.393329\pi$
$863$			55.4854i			1.88874i	−0.328880	+	0.944372i	$0.606671\pi$
$864$	−28.8444	−	5.65685i	−0.981307	−	0.192450i
$865$		0			0
$866$		−	57.6385i		−	1.95864i
$867$	−2.31456	+	35.8055i	−0.0786065	+	1.21602i
$868$	−43.1313			−1.46397
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	0.811057i		−	0.0274659i
$873$		0			0
$874$		0			0
$875$			77.4673i			2.61887i
$876$		0			0
$877$	−40.4349			−1.36539			−0.682694	−	0.730704i	$-0.739192\pi$
$877$	−40.4349			−1.36539			−0.682694	+	0.730704i	$0.739192\pi$
$878$		0			0
$879$	52.2705	+	3.37890i	1.76304	+	0.113967i
$880$		0			0
$881$		−	39.9762i		−	1.34683i	−0.739263	−	0.673417i	$-0.764826\pi$
$881$		−	39.9762i		−	1.34683i	0.739263	−	0.673417i	$-0.235174\pi$
$882$	1.06175	−	8.17814i	0.0357509	−	0.275372i
$883$	−59.2772			−1.99484			−0.997418	−	0.0718139i	$-0.977121\pi$
$883$	−59.2772			−1.99484			−0.997418	+	0.0718139i	$0.977121\pi$
$884$			44.2855i			1.48948i
$885$		0			0
$886$	−55.9614			−1.88006
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$	−57.0799	−	3.68979i	−1.91548	−	0.123821i
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−16.7967			−0.562395
$893$		0			0
$894$	−1.78770	+	27.6551i	−0.0597895	+	0.924925i
$895$	−57.4781			−1.92128
$896$			33.8350i			1.13035i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−67.8128	−	8.80398i	−2.26043	−	0.293466i
$901$		0			0
$902$		0			0
$903$	−4.05103	+	62.6682i	−0.134810	+	2.08547i
$904$	57.6888			1.91870
$905$		0			0
$906$	−35.1493	−	2.27214i	−1.16776	−	0.0754868i
$907$	−57.9364			−1.92375			−0.961874	−	0.273495i	$-0.911820\pi$
$907$	−57.9364			−1.92375			−0.961874	+	0.273495i	$0.911820\pi$
$908$		0			0
$909$		0			0
$910$	−61.7489			−2.04696
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	12.5360			0.414200
$917$		−	56.3115i		−	1.85957i
$918$	8.68510	−	44.2855i	0.286651	−	1.46164i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	25.9515			0.854666
$923$		−	11.1817i		−	0.368051i
$924$		0			0
$925$	−133.068			−4.37525
$926$		−	10.1980i		−	0.335128i
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$	4.61395	−	71.3763i	0.151297	−	2.34052i
$931$		0			0
$932$			19.9303i			0.652837i
$933$		0			0
$934$	14.4222			0.471909
$935$		0			0
$936$	3.93891	−	30.3395i	0.128747	−	0.991677i
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	3.42391	−	52.9667i	0.111735	−	1.72850i
$940$	40.4720			1.32005
$941$		−	48.3348i		−	1.57567i	−0.615887	−	0.787835i	$-0.711202\pi$
$941$		−	48.3348i		−	1.57567i	0.615887	−	0.787835i	$-0.288798\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	61.7489	+	12.1100i	2.00869	+	0.393937i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	48.8878	+	3.16023i	1.58529	+	0.102478i
$952$	−51.9476			−1.68363
$953$			52.2588i			1.69283i	0.532524	+	0.846415i	$0.321244\pi$
$953$			52.2588i			1.69283i	−0.532524	+	0.846415i	$0.678756\pi$
$954$		0			0
$955$		0			0
$956$			38.5971i			1.24832i
$957$		0			0
$958$	−44.1085			−1.42508
$959$		0			0
$960$	−55.9922	−	3.61948i	−1.80714	−	0.116818i
$961$	21.0000			0.677419
$962$		−	59.5347i		−	1.91948i
$963$	−3.93891	+	30.3395i	−0.126929	+	0.977677i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−61.6559			−1.98272			−0.991360	−	0.131166i	$-0.958128\pi$
$967$	−61.6559			−1.98272			−0.991360	+	0.131166i	$0.958128\pi$
$968$			31.1127i			1.00000i
$969$		0			0
$970$		0			0
$971$			60.3120i			1.93551i	0.251902	+	0.967753i	$0.418944\pi$
$971$			60.3120i			1.93551i	−0.251902	+	0.967753i	$0.581056\pi$
$972$	−9.88897	+	29.5670i	−0.317189	+	0.948362i
$973$	70.5091			2.26042
$974$			50.9902i			1.63383i
$975$	4.59130	−	71.0259i	0.147039	−	2.27465i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		−	15.7420i		−	0.502860i
$981$	−0.853097	−	0.110756i	−0.0272373	−	0.00353615i
$982$	61.3692			1.95837
$983$			25.5010i			0.813357i	0.913571	+	0.406678i	$0.133313\pi$
$983$			25.5010i			0.813357i	−0.913571	+	0.406678i	$0.866687\pi$
$984$		0			0
$985$	−24.0751			−0.767095
$986$		0			0
$987$	−25.8321	−	1.66985i	−0.822243	−	0.0531519i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		−	40.7922i		−	1.29515i
$993$		0			0
$994$	13.1163			0.416025
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−11.6757	+	59.5347i	−0.369403	+	1.88359i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.2.h.b.155.9	yes	12
3.2	odd	2	inner	312.2.h.b.155.4	yes	12
4.3	odd	2		1248.2.h.b.623.8		12
8.3	odd	2	inner	312.2.h.b.155.3	✓	12
8.5	even	2		1248.2.h.b.623.7		12
12.11	even	2		1248.2.h.b.623.5		12
13.12	even	2	inner	312.2.h.b.155.3	✓	12
24.5	odd	2		1248.2.h.b.623.6		12
24.11	even	2	inner	312.2.h.b.155.10	yes	12
39.38	odd	2	inner	312.2.h.b.155.10	yes	12
52.51	odd	2		1248.2.h.b.623.7		12
104.51	odd	2	CM	312.2.h.b.155.9	yes	12
104.77	even	2		1248.2.h.b.623.8		12
156.155	even	2		1248.2.h.b.623.6		12
312.77	odd	2		1248.2.h.b.623.5		12
312.155	even	2	inner	312.2.h.b.155.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.h.b.155.3	✓	12	8.3	odd	2	inner
312.2.h.b.155.3	✓	12	13.12	even	2	inner
312.2.h.b.155.4	yes	12	3.2	odd	2	inner
312.2.h.b.155.4	yes	12	312.155	even	2	inner
312.2.h.b.155.9	yes	12	1.1	even	1	trivial
312.2.h.b.155.9	yes	12	104.51	odd	2	CM
312.2.h.b.155.10	yes	12	24.11	even	2	inner
312.2.h.b.155.10	yes	12	39.38	odd	2	inner
1248.2.h.b.623.5		12	12.11	even	2
1248.2.h.b.623.5		12	312.77	odd	2
1248.2.h.b.623.6		12	24.5	odd	2
1248.2.h.b.623.6		12	156.155	even	2
1248.2.h.b.623.7		12	8.5	even	2
1248.2.h.b.623.7		12	52.51	odd	2
1248.2.h.b.623.8		12	4.3	odd	2
1248.2.h.b.623.8		12	104.77	even	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} -5.72660 q^{10} +(-0.223462 + 3.45689i) q^{12} -3.60555 q^{13} -4.22937i q^{14} +(6.99902 + 0.452435i) q^{15} +4.00000 q^{16} +6.14129i q^{17} +(0.546228 - 4.20733i) q^{18} -8.09864i q^{20} +(-0.334145 + 5.16911i) q^{21} +(-4.88878 - 0.316023i) q^{24} -11.3970 q^{25} -5.09902i q^{26} +(-1.00000 + 5.09902i) q^{27} +5.98123 q^{28} +(-0.639839 + 9.89811i) q^{30} -7.21110 q^{31} +5.65685i q^{32} -8.68510 q^{34} -12.1100i q^{35} +(5.95006 + 0.772483i) q^{36} +11.6757 q^{37} +(-0.402852 + 6.23199i) q^{39} +11.4532 q^{40} +(-7.31023 - 0.472552i) q^{42} +12.1236 q^{43} +(1.56402 - 12.0469i) q^{45} +4.99739i q^{47} +(0.446924 - 6.91377i) q^{48} +1.94378 q^{49} -16.1178i q^{50} +(10.6149 + 0.686173i) q^{51} +7.21110 q^{52} +(-7.21110 - 1.41421i) q^{54} +8.45874i q^{56} +(-13.9980 - 0.904869i) q^{60} -10.1980i q^{62} +(8.89718 + 1.15510i) q^{63} -8.00000 q^{64} -14.6000i q^{65} -12.2826i q^{68} +17.1261 q^{70} +3.10125i q^{71} +(-1.09246 + 8.41466i) q^{72} +16.5119i q^{74} +(-1.27340 + 19.6990i) q^{75} +(-8.81337 - 0.569719i) q^{78} +16.1973i q^{80} +(8.70163 + 2.29816i) q^{81} +(0.668289 - 10.3382i) q^{84} -24.8680 q^{85} +17.1453i q^{86} +(17.0368 + 2.21185i) q^{90} +10.7828 q^{91} +(-0.805704 + 12.4640i) q^{93} -7.06738 q^{94} +(9.77755 + 0.632046i) q^{96} +2.74893i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{4}+O(q^{10}) \) 12 * q - 24 * q^4	\( 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} - 12 q^{27} + 24 q^{30} - 48 q^{42} + 84 q^{49} + 60 q^{51} - 96 q^{64} - 84 q^{75} + 96 q^{90}+O(q^{100}) \) 12 * q - 24 * q^4 + 48 * q^16 - 60 * q^25 - 12 * q^27 + 24 * q^30 - 48 * q^42 + 84 * q^49 + 60 * q^51 - 96 * q^64 - 84 * q^75 + 96 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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