LMFDB - Embedded newform 380.2.d.a.379.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(379,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("380.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	380.d (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:	$3.03431527681$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 13x^{8} + 256 $ x^16 - 13*x^8 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			379.14
Root			$1.13320 + 0.846083i$ of defining polynomial
Character	$\chi$	$=$	380.379
Dual form			380.2.d.a.379.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 - 1.72677i) q^{6} +(-0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +O(q^{10})$	\(q+(1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 - 1.72677i) q^{6} +(-0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +(2.53391 + 1.89190i) q^{10} -2.92978i q^{11} +(2.92199 - 0.865956i) q^{12} -2.42350 q^{13} -3.40733i q^{15} +(-3.35410 + 2.17945i) q^{16} +(0.768338 + 0.573665i) q^{18} +4.35890i q^{19} +(1.27073 + 4.28780i) q^{20} +(2.47884 - 3.32002i) q^{22} +(4.04387 + 1.49094i) q^{24} +5.00000 q^{25} +(-2.74631 - 2.05048i) q^{26} -5.60458i q^{27} +(2.88288 - 3.86118i) q^{30} +(-5.64487 - 0.368097i) q^{32} -4.46440 q^{33} +(0.385312 + 1.30016i) q^{36} -0.802795 q^{37} +(-3.68799 + 4.93951i) q^{38} +3.69294i q^{39} +(-2.18785 + 5.93408i) q^{40} +(5.61803 - 1.66495i) q^{44} +1.51611 q^{45} +(3.32105 + 5.11099i) q^{48} -7.00000 q^{49} +(5.66600 + 4.23042i) q^{50} +(-1.37724 - 4.64722i) q^{52} -13.2466 q^{53} +(4.74195 - 6.35112i) q^{54} -6.55118i q^{55} +6.64210 q^{57} +(6.53377 - 1.93634i) q^{60} -15.5804 q^{61} +(-6.08532 - 5.19315i) q^{64} -5.41911 q^{65} +(-5.05906 - 3.77726i) q^{66} +3.36134i q^{67} +(-0.663404 + 1.79934i) q^{72} +(-0.909728 - 0.679231i) q^{74} -7.61901i q^{75} +(-8.35847 + 2.47710i) q^{76} +(-3.12453 + 4.18484i) q^{78} +(-7.50000 + 4.87340i) q^{80} -6.50621 q^{81} +(7.77505 + 2.86660i) q^{88} +(1.71806 + 1.28275i) q^{90} +9.74679i q^{95} +(-0.560907 + 8.60166i) q^{96} +19.4685 q^{97} +(-7.93240 - 5.92258i) q^{98} -1.98646i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} + 8 q^{24} + 80 q^{25} + 24 q^{26} - 40 q^{30} - 56 q^{36} + 72 q^{44} - 112 q^{49} + 88 q^{54} - 104 q^{66} - 120 q^{80} + 144 q^{81} + 136 q^{96}+O(q^{100}) $ 16 * q - 48 * q^9 + 8 * q^24 + 80 * q^25 + 24 * q^26 - 40 * q^30 - 56 * q^36 + 72 * q^44 - 112 * q^49 + 88 * q^54 - 104 * q^66 - 120 * q^80 + 144 * q^81 + 136 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.13320	+	0.846083i	0.801294	+	0.598271i
$2$	1.13320	+	0.846083i	0.801294	+	0.598271i
$3$		−	1.52380i		−	0.879768i	−0.898055	−	0.439884i	$-0.855020\pi$
$3$		−	1.52380i		−	0.879768i	0.898055	−	0.439884i	$-0.144980\pi$
$4$	0.568286	+	1.91756i	0.284143	+	0.958782i
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$	1.28926	−	1.72677i	0.526340	−	0.704953i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−0.978437	+	2.65380i	−0.345930	+	0.938260i
$9$	0.678024			0.226008
$10$	2.53391	+	1.89190i	0.801294	+	0.598271i
$11$		−	2.92978i		−	0.883361i	−0.897172	−	0.441680i	$-0.854382\pi$
$11$		−	2.92978i		−	0.883361i	0.897172	−	0.441680i	$-0.145618\pi$
$12$	2.92199	−	0.865956i	0.843506	−	0.249980i
$13$	−2.42350			−0.672158			−0.336079	−	0.941834i	$-0.609101\pi$
$13$	−2.42350			−0.672158			−0.336079	+	0.941834i	$0.609101\pi$
$14$		0			0
$15$		−	3.40733i		−	0.879768i
$16$	−3.35410	+	2.17945i	−0.838525	+	0.544862i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.768338	+	0.573665i	0.181099	+	0.135214i
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$	1.27073	+	4.28780i	0.284143	+	0.958782i
$21$		0			0
$22$	2.47884	−	3.32002i	0.528489	−	0.707832i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	4.04387	+	1.49094i	0.825452	+	0.304338i
$25$	5.00000			1.00000
$26$	−2.74631	−	2.05048i	−0.538596	−	0.402133i
$27$		−	5.60458i		−	1.07860i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	2.88288	−	3.86118i	0.526340	−	0.704953i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.64487	−	0.368097i	−0.997881	−	0.0650709i
$33$	−4.46440			−0.777153
$34$		0			0
$35$		0			0
$36$	0.385312	+	1.30016i	0.0642186	+	0.216693i
$37$	−0.802795			−0.131979			−0.0659893	−	0.997820i	$-0.521020\pi$
$37$	−0.802795			−0.131979			−0.0659893	+	0.997820i	$0.521020\pi$
$38$	−3.68799	+	4.93951i	−0.598271	+	0.801294i
$39$			3.69294i			0.591343i
$40$	−2.18785	+	5.93408i	−0.345930	+	0.938260i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	5.61803	−	1.66495i	0.846950	−	0.251001i
$45$	1.51611			0.226008
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	3.32105	+	5.11099i	0.479353	+	0.737708i
$49$	−7.00000			−1.00000
$50$	5.66600	+	4.23042i	0.801294	+	0.598271i
$51$		0			0
$52$	−1.37724	−	4.64722i	−0.190989	−	0.644453i
$53$	−13.2466			−1.81956			−0.909781	−	0.415090i	$-0.863750\pi$
$53$	−13.2466			−1.81956			−0.909781	+	0.415090i	$0.863750\pi$
$54$	4.74195	−	6.35112i	0.645297	−	0.864278i
$55$		−	6.55118i		−	0.883361i
$56$		0			0
$57$	6.64210			0.879768
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	6.53377	−	1.93634i	0.843506	−	0.249980i
$61$	−15.5804			−1.99486			−0.997430	−	0.0716414i	$-0.977176\pi$
$61$	−15.5804			−1.99486			−0.997430	+	0.0716414i	$0.977176\pi$
$62$		0			0
$63$		0			0
$64$	−6.08532	−	5.19315i	−0.760665	−	0.649144i
$65$	−5.41911			−0.672158
$66$	−5.05906	−	3.77726i	−0.622728	−	0.464948i
$67$			3.36134i			0.410653i	0.978694	+	0.205326i	$0.0658256\pi$
$67$			3.36134i			0.410653i	−0.978694	+	0.205326i	$0.934174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−0.663404	+	1.79934i	−0.0781829	+	0.212055i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−0.909728	−	0.679231i	−0.105754	−	0.0789591i
$75$		−	7.61901i		−	0.879768i
$76$	−8.35847	+	2.47710i	−0.958782	+	0.284143i
$77$		0			0
$78$	−3.12453	+	4.18484i	−0.353784	+	0.473839i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−7.50000	+	4.87340i	−0.838525	+	0.544862i
$81$	−6.50621			−0.722912
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	7.77505	+	2.86660i	0.828823	+	0.305581i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	1.71806	+	1.28275i	0.181099	+	0.135214i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			9.74679i			1.00000i
$96$	−0.560907	+	8.60166i	−0.0572473	+	0.877904i
$97$	19.4685			1.97673			0.988364	−	0.152109i	$-0.0486063\pi$
$97$	19.4685			1.97673			0.988364	+	0.152109i	$0.0486063\pi$
$98$	−7.93240	−	5.92258i	−0.801294	−	0.598271i
$99$		−	1.98646i		−	0.199647i
$100$	2.84143	+	9.58782i	0.284143	+	0.958782i
$101$	−0.868264			−0.0863955			−0.0431977	−	0.999067i	$-0.513755\pi$
$101$	−0.868264			−0.0863955			−0.0431977	+	0.999067i	$0.513755\pi$
$102$		0			0
$103$		−	16.9447i		−	1.66961i	−0.550548	−	0.834803i	$-0.685581\pi$
$103$		−	16.9447i		−	1.66961i	0.550548	−	0.834803i	$-0.314419\pi$
$104$	2.37124	−	6.43149i	0.232519	−	0.630659i
$105$		0			0
$106$	−15.0111	−	11.2077i	−1.45800	−	1.08859i
$107$			16.9906i			1.64255i	0.570534	+	0.821274i	$0.306736\pi$
$107$			16.9906i			1.64255i	−0.570534	+	0.821274i	$0.693264\pi$
$108$	10.7471	−	3.18501i	1.03414	−	0.306477i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	5.54284	−	7.42380i	0.528489	−	0.707832i
$111$			1.22330i			0.116111i
$112$		0			0
$113$	16.3361			1.53677			0.768386	−	0.639987i	$-0.221060\pi$
$113$	16.3361			1.53677			0.768386	+	0.639987i	$0.221060\pi$
$114$	7.52683	+	5.61977i	0.704953	+	0.526340i
$115$		0			0
$116$		0			0
$117$	−1.64319			−0.151913
$118$		0			0
$119$		0			0
$120$	9.04237	+	3.33385i	0.825452	+	0.304338i
$121$	2.41641			0.219673
$122$	−17.6557	−	13.1823i	−1.59847	−	1.19347i
$123$		0			0
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$			19.7066i			1.74868i	0.485315	+	0.874339i	$0.338705\pi$
$127$			19.7066i			1.74868i	−0.485315	+	0.874339i	$0.661295\pi$
$128$	−2.50205	−	11.0336i	−0.221152	−	0.975239i
$129$		0			0
$130$	−6.14094	−	4.58502i	−0.538596	−	0.402133i
$131$		−	19.4936i		−	1.70316i	−0.524222	−	0.851581i	$-0.675644\pi$
$131$		−	19.4936i		−	1.70316i	0.524222	−	0.851581i	$-0.324356\pi$
$132$	−2.53706	−	8.56078i	−0.220823	−	0.745120i
$133$		0			0
$134$	−2.84397	+	3.80907i	−0.245682	+	0.329054i
$135$		−	12.5322i		−	1.07860i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	21.8917i		−	1.85683i	−0.371546	−	0.928414i	$-0.621172\pi$
$139$		−	21.8917i		−	1.85683i	0.371546	−	0.928414i	$-0.378828\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.10031i			0.593758i
$144$	−2.27416	+	1.47772i	−0.189514	+	0.123143i
$145$		0			0
$146$		0			0
$147$			10.6666i			0.879768i
$148$	−0.456217	−	1.53941i	−0.0375008	−	0.126539i
$149$	22.9364			1.87902			0.939512	−	0.342516i	$-0.111279\pi$
$149$	22.9364			1.87902			0.939512	+	0.342516i	$0.111279\pi$
$150$	6.44632	−	8.63387i	0.526340	−	0.704953i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−11.5677	−	4.26491i	−0.938260	−	0.345930i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−7.08144	+	2.09864i	−0.566969	+	0.168026i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$			20.1852i			1.60079i
$160$	−12.6223	−	0.823090i	−0.997881	−	0.0650709i
$161$		0			0
$162$	−7.37284	−	5.50480i	−0.579265	−	0.432498i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$	−9.98271			−0.777153
$166$		0			0
$167$			25.8018i			1.99660i	0.0582442	+	0.998302i	$0.481450\pi$
$167$			25.8018i			1.99660i	−0.0582442	+	0.998302i	$0.518550\pi$
$168$		0			0
$169$	−7.12665			−0.548204
$170$		0			0
$171$			2.95544i			0.226008i
$172$		0			0
$173$	25.4017			1.93126			0.965628	−	0.259928i	$-0.0836990\pi$
$173$	25.4017			1.93126			0.965628	+	0.259928i	$0.0836990\pi$
$174$		0			0
$175$		0			0
$176$	6.38530	+	9.82677i	0.481310	+	0.740721i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.861583	+	2.90724i	0.0642186	+	0.216693i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$			23.7414i			1.75501i
$184$		0			0
$185$	−1.79510			−0.131979
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−8.24660	+	11.0451i	−0.598271	+	0.801294i
$191$			26.1534i			1.89239i	0.323592	+	0.946197i	$0.395109\pi$
$191$			26.1534i			1.89239i	−0.323592	+	0.946197i	$0.604891\pi$
$192$	−7.91334	+	9.27283i	−0.571096	+	0.669209i
$193$	−17.8629			−1.28580			−0.642901	−	0.765950i	$-0.722269\pi$
$193$	−17.8629			−1.28580			−0.642901	+	0.765950i	$0.722269\pi$
$194$	22.0617	+	16.4720i	1.58394	+	1.18262i
$195$			8.25766i			0.591343i
$196$	−3.97800	−	13.4229i	−0.284143	−	0.958782i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	1.68071	−	2.25106i	0.119443	−	0.159976i
$199$		−	8.71780i		−	0.617988i	−0.951064	−	0.308994i	$-0.900008\pi$
$199$		−	8.71780i		−	0.617988i	0.951064	−	0.308994i	$-0.0999924\pi$
$200$	−4.89218	+	13.2690i	−0.345930	+	0.938260i
$201$	5.12202			0.361279
$202$	−0.983917	−	0.734624i	−0.0692281	−	0.0516879i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	14.3366	−	19.2017i	0.998878	−	1.33785i
$207$		0			0
$208$	8.12866	−	5.28189i	0.563621	−	0.366234i
$209$	12.7706			0.883361
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−7.52786	−	25.4012i	−0.517016	−	1.74456i
$213$		0			0
$214$	−14.3755	+	19.2538i	−0.982689	+	1.31616i
$215$		0			0
$216$	14.8735	+	5.48373i	1.01201	+	0.373121i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	12.5623	−	3.72294i	0.846950	−	0.251001i
$221$		0			0
$222$	−1.03501	+	1.38625i	−0.0694657	+	0.0930387i
$223$		−	28.8494i		−	1.93190i	−0.258728	−	0.965950i	$-0.583303\pi$
$223$		−	28.8494i		−	1.93190i	0.258728	−	0.965950i	$-0.416697\pi$
$224$		0			0
$225$	3.39012			0.226008
$226$	18.5121	+	13.8217i	1.23140	+	0.919406i
$227$			3.45331i			0.229205i	0.993411	+	0.114602i	$0.0365594\pi$
$227$			3.45331i			0.229205i	−0.993411	+	0.114602i	$0.963441\pi$
$228$	3.77461	+	12.7367i	0.249980	+	0.843506i
$229$	−6.48779			−0.428725			−0.214362	−	0.976754i	$-0.568767\pi$
$229$	−6.48779			−0.428725			−0.214362	+	0.976754i	$0.568767\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−1.86207	−	1.39028i	−0.121727	−	0.0908853i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			19.4936i			1.26094i	0.776215	+	0.630468i	$0.217137\pi$
$239$			19.4936i			1.26094i	−0.776215	+	0.630468i	$0.782863\pi$
$240$	7.42610	+	11.4285i	0.479353	+	0.737708i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	2.73827	+	2.04448i	0.176023	+	0.131424i
$243$		−	6.89957i		−	0.442608i
$244$	−8.85410	−	29.8763i	−0.566826	−	1.91264i
$245$	−15.6525			−1.00000
$246$		0			0
$247$		−	10.5638i		−	0.672158i
$248$		0			0
$249$		0			0
$250$	12.6696	+	9.45950i	0.801294	+	0.598271i
$251$		−	26.1534i		−	1.65079i	−0.564557	−	0.825394i	$-0.690953\pi$
$251$		−	26.1534i		−	1.65079i	0.564557	−	0.825394i	$-0.309047\pi$
$252$		0			0
$253$		0			0
$254$	−16.6734	+	22.3315i	−1.04618	+	1.40121i
$255$		0			0
$256$	6.50000	−	14.6202i	0.406250	−	0.913762i
$257$	31.9123			1.99064			0.995318	−	0.0966558i	$-0.0308146\pi$
$257$	31.9123			1.99064			0.995318	+	0.0966558i	$0.0308146\pi$
$258$		0			0
$259$		0			0
$260$	−3.07960	−	10.3915i	−0.190989	−	0.644453i
$261$		0			0
$262$	16.4932	−	22.0901i	1.01895	−	1.36473i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	4.36814	−	11.8476i	0.268840	−	0.729172i
$265$	−29.6203			−1.81956
$266$		0			0
$267$		0			0
$268$	−6.44558	+	1.91020i	−0.393727	+	0.116684i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	10.6033	−	14.2015i	0.645297	−	0.864278i
$271$			24.1298i			1.46578i	0.680345	+	0.732892i	$0.261830\pi$
$271$			24.1298i			1.46578i	−0.680345	+	0.732892i	$0.738170\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	14.6489i		−	0.883361i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	18.5222	−	24.8077i	1.11089	−	1.48787i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$	14.8522			0.879768
$286$	−6.00746	+	8.04608i	−0.355228	+	0.475774i
$287$		0			0
$288$	−3.82736	−	0.249579i	−0.225529	−	0.0147066i
$289$	17.0000			1.00000
$290$		0			0
$291$		−	29.6662i		−	1.73906i
$292$		0			0
$293$	24.0848			1.40705			0.703525	−	0.710670i	$-0.251608\pi$
$293$	24.0848			1.40705			0.703525	+	0.710670i	$0.251608\pi$
$294$	−9.02485	+	12.0874i	−0.526340	+	0.704953i
$295$		0			0
$296$	0.785484	−	2.13046i	0.0456553	−	0.123830i
$297$	−16.4202			−0.952796
$298$	25.9915	+	19.4061i	1.50565	+	1.12417i
$299$		0			0
$300$	14.6099	−	4.32978i	0.843506	−	0.249980i
$301$		0			0
$302$		0			0
$303$			1.32306i			0.0760080i
$304$	−9.50000	−	14.6202i	−0.544862	−	0.838525i
$305$	−34.8387			−1.99486
$306$		0			0
$307$		−	23.8053i		−	1.35864i	−0.733842	−	0.679320i	$-0.762275\pi$
$307$		−	23.8053i		−	1.35864i	0.733842	−	0.679320i	$-0.237725\pi$
$308$		0			0
$309$	−25.8203			−1.46887
$310$		0			0
$311$			31.3726i			1.77898i	0.456955	+	0.889490i	$0.348940\pi$
$311$			31.3726i			1.77898i	−0.456955	+	0.889490i	$0.651060\pi$
$312$	−9.80032	−	3.61330i	−0.554834	−	0.204563i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.8607			−0.609998			−0.304999	−	0.952353i	$-0.598656\pi$
$317$	−10.8607			−0.609998			−0.304999	+	0.952353i	$0.598656\pi$
$318$	−17.0784	+	22.8739i	−0.957708	+	1.28270i
$319$		0			0
$320$	−13.6072	−	11.6122i	−0.760665	−	0.649144i
$321$	25.8904			1.44506
$322$		0			0
$323$		0			0
$324$	−3.69739	−	12.4761i	−0.205410	−	0.693115i
$325$	−12.1175			−0.672158
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−11.3124	−	8.44620i	−0.622728	−	0.464948i
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−0.544315			−0.0298283
$334$	−21.8305	+	29.2386i	−1.19451	+	1.59987i
$335$			7.51618i			0.410653i
$336$		0			0
$337$	24.7733			1.34949			0.674744	−	0.738052i	$-0.264254\pi$
$337$	24.7733			1.34949			0.674744	+	0.738052i	$0.264254\pi$
$338$	−8.07592	−	6.02974i	−0.439272	−	0.327975i
$339$		−	24.8930i		−	1.35200i
$340$		0			0
$341$		0			0
$342$	−2.50055	+	3.34911i	−0.135214	+	0.181099i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	28.7852	+	21.4920i	1.54750	+	1.15541i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−13.4164			−0.718164			−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164			−0.718164			−0.359082	+	0.933306i	$0.616910\pi$
$350$		0			0
$351$			13.5827i			0.724991i
$352$	−1.07844	+	16.5382i	−0.0574811	+	0.881489i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	25.5131i		−	1.34653i	−0.739401	−	0.673265i	$-0.764891\pi$
$359$		−	25.5131i		−	1.34653i	0.739401	−	0.673265i	$-0.235109\pi$
$360$	−1.48342	+	4.02345i	−0.0781829	+	0.212055i
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	3.68213i		−	0.193262i
$364$		0			0
$365$		0			0
$366$	−20.0872	+	26.9038i	−1.04997	+	1.40628i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−2.03421	−	1.51881i	−0.105754	−	0.0789591i
$371$		0			0
$372$		0			0
$373$	−38.1342			−1.97452			−0.987258	−	0.159130i	$-0.949131\pi$
$373$	−38.1342			−1.97452			−0.987258	+	0.159130i	$0.949131\pi$
$374$		0			0
$375$		−	17.0366i		−	0.879768i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−18.6901	+	5.53897i	−0.958782	+	0.284143i
$381$	30.0290			1.53843
$382$	−22.1280	+	29.6370i	−1.13216	+	1.51636i
$383$		−	3.31535i		−	0.169407i	−0.996406	−	0.0847033i	$-0.973006\pi$
$383$		−	3.31535i		−	0.169407i	0.996406	−	0.0847033i	$-0.0269942\pi$
$384$	−16.8130	+	3.81263i	−0.857984	+	0.194562i
$385$		0			0
$386$	−20.2423	−	15.1135i	−1.03030	−	0.769258i
$387$		0			0
$388$	11.0637	+	37.3321i	0.561673	+	1.89525i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$	−6.98666	+	9.35758i	−0.353784	+	0.473839i
$391$		0			0
$392$	6.84906	−	18.5766i	0.345930	−	0.938260i
$393$	−29.7044			−1.49839
$394$		0			0
$395$		0			0
$396$	3.80916	−	1.12888i	0.191418	−	0.0567282i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	7.37598	−	9.87901i	0.369725	−	0.495190i
$399$		0			0
$400$	−16.7705	+	10.8972i	−0.838525	+	0.544862i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	5.80427	+	4.33365i	0.289491	+	0.216143i
$403$		0			0
$404$	−0.493422	−	1.66495i	−0.0245487	−	0.0828344i
$405$	−14.5483			−0.722912
$406$		0			0
$407$			2.35201i			0.116585i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	32.4925	−	9.62941i	1.60079	−	0.474407i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	13.6803	+	0.892083i	0.670733	+	0.0437379i
$417$	−33.3586			−1.63358
$418$	14.4716	+	10.8050i	0.707832	+	0.528489i
$419$		−	19.4936i		−	0.952324i	−0.879358	−	0.476162i	$-0.842028\pi$
$419$		−	19.4936i		−	0.952324i	0.879358	−	0.476162i	$-0.157972\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	12.9610	−	35.1539i	0.629440	−	1.70722i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−32.5807	+	9.65555i	−1.57485	+	0.466719i
$429$	10.8195			0.522369
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	12.2149	+	18.7983i	0.587690	+	0.904436i
$433$	−39.3143			−1.88932			−0.944662	−	0.328044i	$-0.893611\pi$
$433$	−39.3143			−1.88932			−0.944662	+	0.328044i	$0.893611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	17.3855	+	6.40992i	0.828823	+	0.305581i
$441$	−4.74617			−0.226008
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−2.34576	+	0.695185i	−0.111325	+	0.0329920i
$445$		0			0
$446$	24.4090	−	32.6922i	1.15580	−	1.54802i
$447$		−	34.9506i		−	1.65311i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	3.84169	+	2.86833i	0.181099	+	0.135214i
$451$		0			0
$452$	9.28358	+	31.3255i	0.436663	+	1.47343i
$453$		0			0
$454$	−2.92179	+	3.91330i	−0.137126	+	0.183660i
$455$		0			0
$456$	−6.49888	+	17.6268i	−0.304338	+	0.825452i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−7.35196	−	5.48921i	−0.343535	−	0.256494i
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		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			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−0.933803	−	3.15093i	−0.0431651	−	0.145652i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			21.7945i			1.00000i
$476$		0			0
$477$	−8.98152			−0.411236
$478$	−16.4932	+	22.0901i	−0.754381	+	1.01038i
$479$			43.0918i			1.96891i	0.175630	+	0.984456i	$0.443804\pi$
$479$			43.0918i			1.96891i	−0.175630	+	0.984456i	$0.556196\pi$
$480$	−1.25423	+	19.2339i	−0.0572473	+	0.877904i
$481$	1.94557			0.0887105
$482$		0			0
$483$		0			0
$484$	1.37321	+	4.63362i	0.0624187	+	0.210619i
$485$	43.5329			1.97673
$486$	5.83761	−	7.81860i	0.264800	−	0.354659i
$487$		−	44.1113i		−	1.99887i	−0.0335531	−	0.999437i	$-0.510682\pi$
$487$		−	44.1113i		−	1.99887i	0.0335531	−	0.999437i	$-0.489318\pi$
$488$	15.2444	−	41.3472i	0.690082	−	1.87170i
$489$		0			0
$490$	−17.7374	−	13.2433i	−0.801294	−	0.598271i
$491$		−	26.1534i		−	1.18029i	−0.807299	−	0.590143i	$-0.799071\pi$
$491$		−	26.1534i		−	1.18029i	0.807299	−	0.590143i	$-0.200929\pi$
$492$		0			0
$493$		0			0
$494$	8.93785	−	11.9709i	0.402133	−	0.538596i
$495$		−	4.44186i		−	0.199647i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			4.31303i			0.193078i	0.995329	+	0.0965389i	$0.0307772\pi$
$499$			4.31303i			0.193078i	−0.995329	+	0.0965389i	$0.969223\pi$
$500$	6.35363	+	21.4390i	0.284143	+	0.958782i
$501$	39.3169			1.75655
$502$	22.1280	−	29.6370i	0.987619	−	1.32277i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−1.94150			−0.0863955
$506$		0			0
$507$			10.8596i			0.482292i
$508$	−37.7887	+	11.1990i	−1.67660	+	0.496875i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	19.7357	−	11.0681i	0.872203	−	0.489144i
$513$	24.4298			1.07860
$514$	36.1631	+	27.0005i	1.59508	+	1.19094i
$515$		−	37.8894i		−	1.66961i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	38.7072i		−	1.69906i
$520$	5.30226	−	14.3812i	0.232519	−	0.630659i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	43.9846i		−	1.92331i	−0.274256	−	0.961657i	$-0.588432\pi$
$523$		−	43.9846i		−	1.92331i	0.274256	−	0.961657i	$-0.411568\pi$
$524$	37.3802	−	11.0779i	1.63296	−	0.483942i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	14.9741	−	9.72994i	0.651662	−	0.423441i
$529$	−23.0000			−1.00000
$530$	−33.5657	−	25.0612i	−1.45800	−	1.08859i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			37.9922i			1.64255i
$536$	−8.92033	−	3.28886i	−0.385299	−	0.142057i
$537$		0			0
$538$		0			0
$539$			20.5084i			0.883361i
$540$	24.0314	−	7.12189i	1.03414	−	0.306477i
$541$	37.6485			1.61864			0.809318	−	0.587371i	$-0.199837\pi$
$541$	37.6485			1.61864			0.809318	+	0.587371i	$0.199837\pi$
$542$	−20.4159	+	27.3439i	−0.876936	+	1.17452i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			34.8418i			1.48973i	0.667216	+	0.744864i	$0.267486\pi$
$547$			34.8418i			1.48973i	−0.667216	+	0.744864i	$0.732514\pi$
$548$		0			0
$549$	−10.5639			−0.450855
$550$	12.3942	−	16.6001i	0.528489	−	0.707832i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$			2.73539i			0.116111i
$556$	41.9787	−	12.4407i	1.78029	−	0.527605i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			47.0322i			1.98217i	0.133223	+	0.991086i	$0.457467\pi$
$563$			47.0322i			1.98217i	−0.133223	+	0.991086i	$0.542533\pi$
$564$		0			0
$565$	36.5286			1.53677
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	16.8305	+	12.5662i	0.704953	+	0.526340i
$571$			39.4704i			1.65178i	0.563829	+	0.825891i	$0.309328\pi$
$571$			39.4704i			1.65178i	−0.563829	+	0.825891i	$0.690672\pi$
$572$	−13.6153	+	4.03501i	−0.569284	+	0.168712i
$573$	39.8526			1.66487
$574$		0			0
$575$		0			0
$576$	−4.12600	−	3.52109i	−0.171917	−	0.146712i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	19.2644	+	14.3834i	0.801294	+	0.598271i
$579$			27.2196i			1.13121i
$580$		0			0
$581$		0			0
$582$	25.1001	−	33.6177i	1.04043	−	1.39350i
$583$			38.8096i			1.60733i
$584$		0			0
$585$	−3.67429			−0.151913
$586$	27.2929	+	20.3778i	1.12746	+	0.841798i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−20.4539	+	6.06169i	−0.843506	+	0.249980i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	2.69266	−	1.74965i	0.110667	−	0.0719102i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−18.6074	−	13.8928i	−0.763469	−	0.570030i
$595$		0			0
$596$	13.0344	+	43.9820i	0.533912	+	1.80157i
$597$	−13.2842			−0.543686
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	20.2194	+	7.45472i	0.825452	+	0.304338i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			2.27907i			0.0928109i
$604$		0			0
$605$	5.40325			0.219673
$606$	−1.11942	+	1.49930i	−0.0454734	+	0.0609047i
$607$			23.6673i			0.960628i	0.877097	+	0.480314i	$0.159477\pi$
$607$			23.6673i			0.960628i	−0.877097	+	0.480314i	$0.840523\pi$
$608$	1.60450	−	24.6054i	0.0650709	−	0.997881i
$609$		0			0
$610$	−39.4793	−	29.4765i	−1.59847	−	1.19347i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	20.1413	−	26.9762i	0.812835	−	1.08867i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−29.2596	−	21.8461i	−1.17699	−	0.878781i
$619$			34.9941i			1.40653i	0.710928	+	0.703265i	$0.248275\pi$
$619$			34.9941i			1.40653i	−0.710928	+	0.703265i	$0.751725\pi$
$620$		0			0
$621$		0			0
$622$	−26.5439	+	35.5515i	−1.06431	+	1.42549i
$623$		0			0
$624$	−8.04857	−	12.3865i	−0.322201	−	0.495856i
$625$	25.0000			1.00000
$626$		0			0
$627$		−	19.4599i		−	0.777153i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	11.0275i		−	0.438997i	−0.975613	−	0.219499i	$-0.929558\pi$
$631$		−	11.0275i		−	0.438997i	0.975613	−	0.219499i	$-0.0704421\pi$
$632$		0			0
$633$		0			0
$634$	−12.3074	−	9.18906i	−0.488788	−	0.364944i
$635$			44.0653i			1.74868i
$636$	−38.7064	+	11.4710i	−1.53481	+	0.454854i
$637$	16.9645			0.672158
$638$		0			0
$639$		0			0
$640$	−5.59475	−	24.6718i	−0.221152	−	0.975239i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	29.3390	+	21.9054i	1.15792	+	0.864539i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	6.36591	−	17.2662i	0.250077	−	0.678280i
$649$		0			0
$650$	−13.7316	−	10.2524i	−0.538596	−	0.402133i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		−	43.5890i		−	1.70316i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	−5.67303	−	19.1425i	−0.220823	−	0.745120i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.616818	−	0.460536i	−0.0239012	−	0.0178454i
$667$		0			0
$668$	−49.4766	+	14.6628i	−1.91431	+	0.567321i
$669$	−43.9608			−1.69962
$670$	−6.35932	+	8.51734i	−0.245682	+	0.329054i
$671$			45.6470i			1.76218i
$672$		0			0
$673$	44.1613			1.70229			0.851147	−	0.524928i	$-0.175908\pi$
$673$	44.1613			1.70229			0.851147	+	0.524928i	$0.175908\pi$
$674$	28.0731	+	20.9603i	1.08134	+	0.807360i
$675$		−	28.0229i		−	1.07860i
$676$	−4.04998	−	13.6658i	−0.155768	−	0.525608i
$677$	−50.5780			−1.94387			−0.971936	−	0.235246i	$-0.924410\pi$
$677$	−50.5780			−1.94387			−0.971936	+	0.235246i	$0.924410\pi$
$678$	21.0616	−	28.2088i	0.808864	−	1.08335i
$679$		0			0
$680$		0			0
$681$	5.26217			0.201647
$682$		0			0
$683$			28.7466i			1.09996i	0.835179	+	0.549979i	$0.185364\pi$
$683$			28.7466i			1.09996i	−0.835179	+	0.549979i	$0.814636\pi$
$684$	−5.66725	+	1.67954i	−0.216693	+	0.0642186i
$685$		0			0
$686$		0			0
$687$			9.88611i			0.377178i
$688$		0			0
$689$	32.1031			1.22303
$690$		0			0
$691$		−	52.5727i		−	1.99996i	−0.00630823	−	0.999980i	$-0.502008\pi$
$691$		−	52.5727i		−	1.99996i	0.00630823	−	0.999980i	$-0.497992\pi$
$692$	14.4354	+	48.7094i	0.548753	+	1.85165i
$693$		0			0
$694$		0			0
$695$		−	48.9513i		−	1.85683i
$696$		0			0
$697$		0			0
$698$	−15.2035	−	11.3514i	−0.575460	−	0.429657i
$699$		0			0
$700$		0			0
$701$	−21.1999			−0.800709			−0.400354	−	0.916360i	$-0.631113\pi$
$701$	−21.1999			−0.800709			−0.400354	+	0.916360i	$0.631113\pi$
$702$	−11.4921	+	15.3919i	−0.433741	+	0.580931i
$703$		−	3.49930i		−	0.131979i
$704$	−15.2148	+	17.8286i	−0.573429	+	0.671942i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.2492			1.51159			0.755796	−	0.654808i	$-0.227250\pi$
$709$	40.2492			1.51159			0.755796	+	0.654808i	$0.227250\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$			15.8768i			0.593758i
$716$		0			0
$717$	29.7044			1.10933
$718$	21.5862	−	28.9114i	0.805590	−	1.07897i
$719$		−	13.7940i		−	0.514429i	−0.966354	−	0.257214i	$-0.917195\pi$
$719$		−	13.7940i		−	0.514429i	0.966354	−	0.257214i	$-0.0828047\pi$
$720$	−5.08518	+	3.30428i	−0.189514	+	0.123143i
$721$		0			0
$722$	−21.5308	−	16.0756i	−0.801294	−	0.598271i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	3.11539	−	4.17259i	0.115623	−	0.154859i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−30.0322			−1.11230
$730$		0			0
$731$		0			0
$732$	−45.5257	+	13.4919i	−1.68268	+	0.498675i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$			23.8513i			0.879768i
$736$		0			0
$737$	9.84797			0.362755
$738$		0			0
$739$			8.71780i			0.320689i	0.987061	+	0.160345i	$0.0512606\pi$
$739$			8.71780i			0.320689i	−0.987061	+	0.160345i	$0.948739\pi$
$740$	−1.02013	−	3.44223i	−0.0375008	−	0.126539i
$741$	−16.0971			−0.591343
$742$		0			0
$743$			1.62663i			0.0596754i	0.999555	+	0.0298377i	$0.00949905\pi$
$743$			1.62663i			0.0596754i	−0.999555	+	0.0298377i	$0.990501\pi$
$744$		0			0
$745$	51.2874			1.87902
$746$	−43.2137	−	32.2647i	−1.58217	−	1.18130i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	14.4144	−	19.3059i	0.526340	−	0.704953i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−39.8526			−1.45231
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−25.8661	−	9.53662i	−0.938260	−	0.345930i
$761$	9.29755			0.337036			0.168518	−	0.985699i	$-0.446102\pi$
$761$	9.29755			0.337036			0.168518	+	0.985699i	$0.446102\pi$
$762$	34.0289	+	25.4070i	1.23274	+	0.920399i
$763$		0			0
$764$	−50.1508	+	14.8626i	−1.81439	+	0.537710i
$765$		0			0
$766$	2.80506	−	3.75696i	0.101351	−	0.135744i
$767$		0			0
$768$	−22.2783	−	9.90472i	−0.803899	−	0.357406i
$769$	−29.2192			−1.05367			−0.526836	−	0.849967i	$-0.676622\pi$
$769$	−29.2192			−1.05367			−0.526836	+	0.849967i	$0.676622\pi$
$770$		0			0
$771$		−	48.6281i		−	1.75130i
$772$	−10.1512	−	34.2533i	−0.365351	−	1.23280i
$773$	−28.9919			−1.04277			−0.521383	−	0.853323i	$-0.674584\pi$
$773$	−28.9919			−1.04277			−0.521383	+	0.853323i	$0.674584\pi$
$774$		0			0
$775$		0			0
$776$	−19.0487	+	51.6656i	−0.683809	+	1.85469i
$777$		0			0
$778$	6.79920	+	5.07650i	0.243763	+	0.182001i
$779$		0			0
$780$	−15.8346	+	4.69271i	−0.566969	+	0.168026i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	23.4787	−	15.2561i	0.838525	−	0.544862i
$785$		0			0
$786$	−33.6610	−	25.1324i	−1.20065	−	0.896443i
$787$		−	25.6990i		−	0.916070i	−0.888934	−	0.458035i	$-0.848553\pi$
$787$		−	25.6990i		−	0.916070i	0.888934	−	0.458035i	$-0.151447\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	5.27167	+	1.94363i	0.187321	+	0.0690637i
$793$	37.7590			1.34086
$794$		0			0
$795$			45.1355i			1.60079i
$796$	16.7169	−	4.95420i	0.592516	−	0.175597i
$797$	−28.9016			−1.02375			−0.511873	−	0.859061i	$-0.671048\pi$
$797$	−28.9016			−1.02375			−0.511873	+	0.859061i	$0.671048\pi$
$798$		0			0
$799$		0			0
$800$	−28.2243	−	1.84048i	−0.997881	−	0.0650709i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	2.91077	+	9.82180i	0.102655	+	0.346388i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.849541	−	2.30420i	0.0298868	−	0.0810615i
$809$	−22.3607			−0.786160			−0.393080	−	0.919504i	$-0.628590\pi$
$809$	−22.3607			−0.786160			−0.393080	+	0.919504i	$0.628590\pi$
$810$	−16.4862	−	12.3091i	−0.579265	−	0.432498i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	36.7691			1.28955
$814$	−1.99000	+	2.66530i	−0.0697493	+	0.0934187i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−42.0000			−1.46581			−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000			−1.46581			−0.732905	+	0.680331i	$0.761836\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$	44.9678	+	16.5793i	1.56653	+	0.577567i
$825$	−22.3220			−0.777153
$826$		0			0
$827$		−	56.1750i		−	1.95340i	−0.214614	−	0.976699i	$-0.568849\pi$
$827$		−	56.1750i		−	1.95340i	0.214614	−	0.976699i	$-0.431151\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	14.7478	+	12.5856i	0.511287	+	0.436327i
$833$		0			0
$834$	−37.8020	−	28.2242i	−1.30898	−	0.977323i
$835$			57.6946i			1.99660i
$836$	7.25735	+	24.4884i	0.251001	+	0.846950i
$837$		0			0
$838$	16.4932	−	22.0901i	0.569748	−	0.763091i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.9357			−0.548204
$846$		0			0
$847$		0			0
$848$	44.4305	−	28.8703i	1.52575	−	0.991410i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$			6.60857i			0.226008i
$856$	−45.0898	−	16.6243i	−1.54114	−	0.568206i
$857$	10.2359			0.349651			0.174825	−	0.984599i	$-0.444064\pi$
$857$	10.2359			0.349651			0.174825	+	0.984599i	$0.444064\pi$
$858$	12.2606	+	9.15418i	0.418571	+	0.312519i
$859$			58.4808i			1.99534i	0.0682391	+	0.997669i	$0.478262\pi$
$859$			58.4808i			1.99534i	−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44.0875i			1.50075i	0.661010	+	0.750377i	$0.270128\pi$
$863$			44.0875i			1.50075i	−0.661010	+	0.750377i	$0.729872\pi$
$864$	−2.06303	+	31.6371i	−0.0701857	+	1.07632i
$865$	56.7999			1.93126
$866$	−44.5510	−	33.2632i	−1.51390	−	1.13033i
$867$		−	25.9047i		−	0.879768i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	8.14620i		−	0.276024i
$872$		0			0
$873$	13.2001			0.446757
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−41.3454			−1.39614			−0.698068	−	0.716032i	$-0.745956\pi$
$877$	−41.3454			−1.39614			−0.698068	+	0.716032i	$0.745956\pi$
$878$		0			0
$879$		−	36.7005i		−	1.23788i
$880$	14.2780	+	21.9733i	0.481310	+	0.740721i
$881$	−58.6434			−1.97575			−0.987874	−	0.155261i	$-0.950378\pi$
$881$	−58.6434			−1.97575			−0.987874	+	0.155261i	$0.950378\pi$
$882$	−5.37836	−	4.01566i	−0.181099	−	0.135214i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.8527i		−	0.565858i	−0.959141	−	0.282929i	$-0.908694\pi$
$887$		−	16.8527i		−	0.565858i	0.959141	−	0.282929i	$-0.0913060\pi$
$888$	−3.24640	−	1.19692i	−0.108942	−	0.0401661i
$889$		0			0
$890$		0			0
$891$			19.0617i			0.638592i
$892$	55.3206	−	16.3947i	1.85227	−	0.548936i
$893$		0			0
$894$	29.5711	−	39.6060i	0.989005	−	1.32462i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.92656	+	6.50078i	0.0642186	+	0.216693i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−15.9838	+	43.3528i	−0.531615	+	1.44189i
$905$		0			0
$906$		0			0
$907$			17.0826i			0.567219i	0.958940	+	0.283610i	$0.0915320\pi$
$907$			17.0826i			0.567219i	−0.958940	+	0.283610i	$0.908468\pi$
$908$	−6.62195	+	1.96247i	−0.219757	+	0.0651269i
$909$	−0.588704			−0.0195261
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−22.2783	+	14.4761i	−0.737708	+	0.479353i
$913$		0			0
$914$		0			0
$915$			53.0874i			1.75501i
$916$	−3.68692	−	12.4407i	−0.121819	−	0.411054i
$917$		0			0
$918$		0			0
$919$		−	58.4808i		−	1.92910i	−0.263896	−	0.964551i	$-0.585007\pi$
$919$		−	58.4808i		−	1.92910i	0.263896	−	0.964551i	$-0.414993\pi$
$920$		0			0
$921$	−36.2746			−1.19529
$922$	−20.3976	−	15.2295i	−0.671759	−	0.501557i
$923$		0			0
$924$		0			0
$925$	−4.01397			−0.131979
$926$		0			0
$927$		−	11.4889i		−	0.377345i
$928$		0			0
$929$	31.3050			1.02708			0.513541	−	0.858065i	$-0.328333\pi$
$929$	31.3050			1.02708			0.513541	+	0.858065i	$0.328333\pi$
$930$		0			0
$931$		−	30.5123i		−	1.00000i
$932$		0			0
$933$	47.8057			1.56509
$934$		0			0
$935$		0			0
$936$	1.60776	−	4.36071i	0.0525513	−	0.142534i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	−18.4400	+	24.6975i	−0.598271	+	0.801294i
$951$			16.5496i			0.536657i
$952$		0			0
$953$	47.5673			1.54086			0.770428	−	0.637527i	$-0.220043\pi$
$953$	47.5673			1.54086			0.770428	+	0.637527i	$0.220043\pi$
$954$	−10.1779	−	7.59912i	−0.329521	−	0.246030i
$955$			58.4808i			1.89239i
$956$	−37.3802	+	11.0779i	−1.20896	+	0.358286i
$957$		0			0
$958$	−36.4592	+	48.8316i	−1.17794	+	1.57768i
$959$		0			0
$960$	−17.6948	+	20.7347i	−0.571096	+	0.669209i
$961$	−31.0000			−1.00000
$962$	2.20472	+	1.64612i	0.0710832	+	0.0530729i
$963$			11.5201i			0.371229i
$964$		0			0
$965$	−39.9427			−1.28580
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−2.36430	+	6.41267i	−0.0759916	+	0.206111i
$969$		0			0
$970$	49.3315	+	36.8325i	1.58394	+	1.18262i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	13.2304	−	3.92093i	0.424364	−	0.125764i
$973$		0			0
$974$	37.3218	−	49.9869i	1.19587	−	1.60168i
$975$			18.4647i			0.591343i
$976$	52.2581	−	33.9566i	1.67274	−	1.08692i
$977$	−30.8771			−0.987846			−0.493923	−	0.869506i	$-0.664438\pi$
$977$	−30.8771			−0.987846			−0.493923	+	0.869506i	$0.664438\pi$
$978$		0			0
$979$		0			0
$980$	−8.89508	−	30.0146i	−0.284143	−	0.958782i
$981$		0			0
$982$	22.1280	−	29.6370i	0.706131	−	0.945756i
$983$		−	44.2033i		−	1.40987i	−0.709274	−	0.704933i	$-0.750977\pi$
$983$		−	44.2033i		−	1.40987i	0.709274	−	0.704933i	$-0.249023\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	20.2567	−	6.00325i	0.644453	−	0.190989i
$989$		0			0
$990$	3.75818	−	5.03352i	0.119443	−	0.159976i
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	19.4936i		−	0.617988i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−3.64918	+	4.88753i	−0.115513	+	0.154712i
$999$			4.49933i			0.142353i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.d.a.379.14	yes	16
4.3	odd	2	inner	380.2.d.a.379.13	yes	16
5.4	even	2	inner	380.2.d.a.379.3	✓	16
19.18	odd	2	inner	380.2.d.a.379.3	✓	16
20.19	odd	2	inner	380.2.d.a.379.4	yes	16
76.75	even	2	inner	380.2.d.a.379.4	yes	16
95.94	odd	2	CM	380.2.d.a.379.14	yes	16
380.379	even	2	inner	380.2.d.a.379.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.d.a.379.3	✓	16	5.4	even	2	inner
380.2.d.a.379.3	✓	16	19.18	odd	2	inner
380.2.d.a.379.4	yes	16	20.19	odd	2	inner
380.2.d.a.379.4	yes	16	76.75	even	2	inner
380.2.d.a.379.13	yes	16	4.3	odd	2	inner
380.2.d.a.379.13	yes	16	380.379	even	2	inner
380.2.d.a.379.14	yes	16	1.1	even	1	trivial
380.2.d.a.379.14	yes	16	95.94	odd	2	CM

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:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 - 1.72677i) q^{6} +(-0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +O(q^{10})\)	\(q+(1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 - 1.72677i) q^{6} +(-0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +(2.53391 + 1.89190i) q^{10} -2.92978i q^{11} +(2.92199 - 0.865956i) q^{12} -2.42350 q^{13} -3.40733i q^{15} +(-3.35410 + 2.17945i) q^{16} +(0.768338 + 0.573665i) q^{18} +4.35890i q^{19} +(1.27073 + 4.28780i) q^{20} +(2.47884 - 3.32002i) q^{22} +(4.04387 + 1.49094i) q^{24} +5.00000 q^{25} +(-2.74631 - 2.05048i) q^{26} -5.60458i q^{27} +(2.88288 - 3.86118i) q^{30} +(-5.64487 - 0.368097i) q^{32} -4.46440 q^{33} +(0.385312 + 1.30016i) q^{36} -0.802795 q^{37} +(-3.68799 + 4.93951i) q^{38} +3.69294i q^{39} +(-2.18785 + 5.93408i) q^{40} +(5.61803 - 1.66495i) q^{44} +1.51611 q^{45} +(3.32105 + 5.11099i) q^{48} -7.00000 q^{49} +(5.66600 + 4.23042i) q^{50} +(-1.37724 - 4.64722i) q^{52} -13.2466 q^{53} +(4.74195 - 6.35112i) q^{54} -6.55118i q^{55} +6.64210 q^{57} +(6.53377 - 1.93634i) q^{60} -15.5804 q^{61} +(-6.08532 - 5.19315i) q^{64} -5.41911 q^{65} +(-5.05906 - 3.77726i) q^{66} +3.36134i q^{67} +(-0.663404 + 1.79934i) q^{72} +(-0.909728 - 0.679231i) q^{74} -7.61901i q^{75} +(-8.35847 + 2.47710i) q^{76} +(-3.12453 + 4.18484i) q^{78} +(-7.50000 + 4.87340i) q^{80} -6.50621 q^{81} +(7.77505 + 2.86660i) q^{88} +(1.71806 + 1.28275i) q^{90} +9.74679i q^{95} +(-0.560907 + 8.60166i) q^{96} +19.4685 q^{97} +(-7.93240 - 5.92258i) q^{98} -1.98646i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} + 8 q^{24} + 80 q^{25} + 24 q^{26} - 40 q^{30} - 56 q^{36} + 72 q^{44} - 112 q^{49} + 88 q^{54} - 104 q^{66} - 120 q^{80} + 144 q^{81} + 136 q^{96}+O(q^{100}) \) 16 * q - 48 * q^9 + 8 * q^24 + 80 * q^25 + 24 * q^26 - 40 * q^30 - 56 * q^36 + 72 * q^44 - 112 * q^49 + 88 * q^54 - 104 * q^66 - 120 * q^80 + 144 * q^81 + 136 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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