LMFDB - Embedded newform 380.2.d.a.379.8

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			379.8
Root			$-0.203022 + 1.39956i$ of defining polynomial
Character	$\chi$	$=$	380.379
Dual form			380.2.d.a.379.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.203022 + 1.39956i) q^{2} -1.12228i q^{3} +(-1.91756 - 0.568286i) q^{4} -2.23607 q^{5} +(1.57071 + 0.227849i) q^{6} +(1.18466 - 2.56838i) q^{8} +1.74048 q^{9} +O(q^{10})$	\(q+(-0.203022 + 1.39956i) q^{2} -1.12228i q^{3} +(-1.91756 - 0.568286i) q^{4} -2.23607 q^{5} +(1.57071 + 0.227849i) q^{6} +(1.18466 - 2.56838i) q^{8} +1.74048 q^{9} +(0.453972 - 3.12952i) q^{10} +5.95117i q^{11} +(-0.637778 + 2.15205i) q^{12} +6.51610 q^{13} +2.50950i q^{15} +(3.35410 + 2.17945i) q^{16} +(-0.353356 + 2.43591i) q^{18} +4.35890i q^{19} +(4.28780 + 1.27073i) q^{20} +(-8.32905 - 1.20822i) q^{22} +(-2.88245 - 1.32953i) q^{24} +5.00000 q^{25} +(-1.32291 + 9.11971i) q^{26} -5.32016i q^{27} +(-3.51221 - 0.509485i) q^{30} +(-3.73124 + 4.25181i) q^{32} +6.67890 q^{33} +(-3.33748 - 0.989090i) q^{36} +8.01591 q^{37} +(-6.10056 - 0.884954i) q^{38} -7.31292i q^{39} +(-2.64898 + 5.74307i) q^{40} +(3.38197 - 11.4117i) q^{44} -3.89183 q^{45} +(2.44596 - 3.76426i) q^{48} -7.00000 q^{49} +(-1.01511 + 6.99782i) q^{50} +(-12.4950 - 3.70301i) q^{52} -5.09315 q^{53} +(7.44591 + 1.08011i) q^{54} -13.3072i q^{55} +4.89192 q^{57} +(1.42612 - 4.81213i) q^{60} +1.11908 q^{61} +(-5.19315 - 6.08532i) q^{64} -14.5704 q^{65} +(-1.35597 + 9.34756i) q^{66} +13.7060i q^{67} +(2.06188 - 4.47021i) q^{72} +(-1.62741 + 11.2188i) q^{74} -5.61142i q^{75} +(2.47710 - 8.35847i) q^{76} +(10.2349 + 1.48469i) q^{78} +(-7.50000 - 4.87340i) q^{80} -0.749299 q^{81} +(15.2849 + 7.05012i) q^{88} +(0.790128 - 5.44687i) q^{90} -9.74679i q^{95} +(4.77173 + 4.18751i) q^{96} +11.6477 q^{97} +(1.42116 - 9.79695i) q^{98} +10.3579i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.203022	+	1.39956i	−0.143559	+	0.989642i
$2$	−0.203022	+	1.39956i	−0.143559	+	0.989642i
$3$		−	1.12228i		−	0.647951i	−0.946065	−	0.323976i	$-0.894980\pi$
$3$		−	1.12228i		−	0.647951i	0.946065	−	0.323976i	$-0.105020\pi$
$4$	−1.91756	−	0.568286i	−0.958782	−	0.284143i
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$	1.57071	+	0.227849i	0.641239	+	0.0930189i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	1.18466	−	2.56838i	0.418841	−	0.908060i
$9$	1.74048			0.580159
$10$	0.453972	−	3.12952i	0.143559	−	0.989642i
$11$			5.95117i			1.79434i	0.441680	+	0.897172i	$0.354382\pi$
$11$			5.95117i			1.79434i	−0.441680	+	0.897172i	$0.645618\pi$
$12$	−0.637778	+	2.15205i	−0.184111	+	0.621244i
$13$	6.51610			1.80724			0.903621	−	0.428333i	$-0.140899\pi$
$13$	6.51610			1.80724			0.903621	+	0.428333i	$0.140899\pi$
$14$		0			0
$15$			2.50950i			0.647951i
$16$	3.35410	+	2.17945i	0.838525	+	0.544862i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.353356	+	2.43591i	−0.0832868	+	0.574150i
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$	4.28780	+	1.27073i	0.958782	+	0.284143i
$21$		0			0
$22$	−8.32905	−	1.20822i	−1.77576	−	0.257593i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−2.88245	−	1.32953i	−0.588378	−	0.271389i
$25$	5.00000			1.00000
$26$	−1.32291	+	9.11971i	−0.259445	+	1.78852i
$27$		−	5.32016i		−	1.02387i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−3.51221	−	0.509485i	−0.641239	−	0.0930189i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−3.73124	+	4.25181i	−0.659596	+	0.751620i
$33$	6.67890			1.16265
$34$		0			0
$35$		0			0
$36$	−3.33748	−	0.989090i	−0.556246	−	0.164848i
$37$	8.01591			1.31781			0.658904	−	0.752227i	$-0.271020\pi$
$37$	8.01591			1.31781			0.658904	+	0.752227i	$0.271020\pi$
$38$	−6.10056	−	0.884954i	−0.989642	−	0.143559i
$39$		−	7.31292i		−	1.17100i
$40$	−2.64898	+	5.74307i	−0.418841	+	0.908060i
$41$		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	$0.5\pi$
$44$	3.38197	−	11.4117i	0.509851	−	1.72039i
$45$	−3.89183			−0.580159
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	2.44596	−	3.76426i	0.353044	−	0.543323i
$49$	−7.00000			−1.00000
$50$	−1.01511	+	6.99782i	−0.143559	+	0.989642i
$51$		0			0
$52$	−12.4950	−	3.70301i	−1.73275	−	0.513515i
$53$	−5.09315			−0.699599			−0.349799	−	0.936825i	$-0.613750\pi$
$53$	−5.09315			−0.699599			−0.349799	+	0.936825i	$0.613750\pi$
$54$	7.44591	+	1.08011i	1.01326	+	0.146985i
$55$		−	13.3072i		−	1.79434i
$56$		0			0
$57$	4.89192			0.647951
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.42612	−	4.81213i	0.184111	−	0.621244i
$61$	1.11908			0.143283			0.0716414	−	0.997430i	$-0.477176\pi$
$61$	1.11908			0.143283			0.0716414	+	0.997430i	$0.477176\pi$
$62$		0			0
$63$		0			0
$64$	−5.19315	−	6.08532i	−0.649144	−	0.760665i
$65$	−14.5704			−1.80724
$66$	−1.35597	+	9.34756i	−0.166908	+	1.15060i
$67$			13.7060i			1.67446i	0.546853	+	0.837229i	$0.315826\pi$
$67$			13.7060i			1.67446i	−0.546853	+	0.837229i	$0.684174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	2.06188	−	4.47021i	0.242995	−	0.526819i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−1.62741	+	11.2188i	−0.189183	+	1.30416i
$75$		−	5.61142i		−	0.647951i
$76$	2.47710	−	8.35847i	0.284143	−	0.958782i
$77$		0			0
$78$	10.2349	+	1.48469i	1.15887	+	0.168108i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−7.50000	−	4.87340i	−0.838525	−	0.544862i
$81$	−0.749299			−0.0832555
$82$		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
$88$	15.2849	+	7.05012i	1.62937	+	0.751545i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.790128	−	5.44687i	0.0832868	−	0.574150i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	9.74679i		−	1.00000i
$96$	4.77173	+	4.18751i	0.487013	+	0.427386i
$97$	11.6477			1.18264			0.591322	−	0.806436i	$-0.298606\pi$
$97$	11.6477			1.18264			0.591322	+	0.806436i	$0.298606\pi$
$98$	1.42116	−	9.79695i	0.143559	−	0.989642i
$99$			10.3579i			1.04101i
$100$	−9.58782	−	2.84143i	−0.958782	−	0.284143i
$101$	20.0810			1.99813			0.999067	−	0.0431977i	$-0.0137545\pi$
$101$	20.0810			1.99813			0.999067	+	0.0431977i	$0.0137545\pi$
$102$		0			0
$103$		−	19.8835i		−	1.95918i	−0.200999	−	0.979591i	$-0.564419\pi$
$103$		−	19.8835i		−	1.95918i	0.200999	−	0.979591i	$-0.435581\pi$
$104$	7.71938	−	16.7358i	0.756947	−	1.64108i
$105$		0			0
$106$	1.03402	−	7.12820i	0.100433	−	0.692352i
$107$			3.66801i			0.354600i	0.984157	+	0.177300i	$0.0567363\pi$
$107$			3.66801i			0.354600i	−0.984157	+	0.177300i	$0.943264\pi$
$108$	−3.02337	+	10.2018i	−0.290924	+	0.981664i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	18.6243	+	2.70166i	1.77576	+	0.257593i
$111$		−	8.99613i		−	0.853875i
$112$		0			0
$113$	−21.1725			−1.99174			−0.995870	−	0.0907914i	$-0.971060\pi$
$113$	−21.1725			−1.99174			−0.995870	+	0.0907914i	$0.971060\pi$
$114$	−0.993170	+	6.84656i	−0.0930189	+	0.641239i
$115$		0			0
$116$		0			0
$117$	11.3411			1.04849
$118$		0			0
$119$		0			0
$120$	6.44536	+	2.97291i	0.588378	+	0.271389i
$121$	−24.4164			−2.21967
$122$	−0.227197	+	1.56622i	−0.0205695	+	0.141799i
$123$		0			0
$124$		0			0
$125$	−11.1803			−1.00000
$126$		0			0
$127$		−	21.6693i		−	1.92284i	−0.275082	−	0.961421i	$-0.588705\pi$
$127$		−	21.6693i		−	1.92284i	0.275082	−	0.961421i	$-0.411295\pi$
$128$	9.57113	−	6.03270i	0.845976	−	0.533220i
$129$		0			0
$130$	2.95813	−	20.3923i	0.259445	−	1.78852i
$131$			19.4936i			1.70316i	0.524222	+	0.851581i	$0.324356\pi$
$131$			19.4936i			1.70316i	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−12.8072	−	3.79553i	−1.11473	−	0.330358i
$133$		0			0
$134$	−19.1825	−	2.78263i	−1.65711	−	0.240383i
$135$			11.8962i			1.02387i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	8.76093i		−	0.743092i	−0.928414	−	0.371546i	$-0.878828\pi$
$139$		−	8.76093i		−	0.743092i	0.928414	−	0.371546i	$-0.121172\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			38.7784i			3.24281i
$144$	5.83774	+	3.79328i	0.486479	+	0.316107i
$145$		0			0
$146$		0			0
$147$			7.85599i			0.647951i
$148$	−15.3710	−	4.55533i	−1.26349	−	0.374446i
$149$	8.36188			0.685032			0.342516	−	0.939512i	$-0.388721\pi$
$149$	8.36188			0.685032			0.342516	+	0.939512i	$0.388721\pi$
$150$	7.85355	+	1.13924i	0.641239	+	0.0930189i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	11.1953	+	5.16382i	0.908060	+	0.418841i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.15583	+	14.0230i	−0.332733	+	1.12274i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$			5.71597i			0.453306i
$160$	8.34330	−	9.50733i	0.659596	−	0.751620i
$161$		0			0
$162$	0.152125	−	1.04869i	0.0119520	−	0.0823931i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$	−14.9345			−1.16265
$166$		0			0
$167$		−	17.1802i		−	1.32944i	−0.747091	−	0.664721i	$-0.768550\pi$
$167$		−	17.1802i		−	1.32944i	0.747091	−	0.664721i	$-0.231450\pi$
$168$		0			0
$169$	29.4596			2.26612
$170$		0			0
$171$			7.58657i			0.580159i
$172$		0			0
$173$	−22.7967			−1.73320			−0.866599	−	0.499005i	$-0.833699\pi$
$173$	−22.7967			−1.73320			−0.866599	+	0.499005i	$0.833699\pi$
$174$		0			0
$175$		0			0
$176$	−12.9703	+	19.9608i	−0.977671	+	1.50460i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	7.46283	+	2.21167i	0.556246	+	0.164848i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		−	1.25592i		−	0.0928403i
$184$		0			0
$185$	−17.9241			−1.31781
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	13.6413	+	1.97882i	0.989642	+	0.143559i
$191$			26.1534i			1.89239i	0.323592	+	0.946197i	$0.395109\pi$
$191$			26.1534i			1.89239i	−0.323592	+	0.946197i	$0.604891\pi$
$192$	−6.82946	+	5.82819i	−0.492874	+	0.420614i
$193$	−27.6795			−1.99242			−0.996208	−	0.0870089i	$-0.972269\pi$
$193$	−27.6795			−1.99242			−0.996208	+	0.0870089i	$0.972269\pi$
$194$	−2.36474	+	16.3017i	−0.169779	+	1.17039i
$195$			16.3522i			1.17100i
$196$	13.4229	+	3.97800i	0.958782	+	0.284143i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−14.4965	−	2.10288i	−1.03022	−	0.149445i
$199$		−	8.71780i		−	0.617988i	−0.951064	−	0.308994i	$-0.900008\pi$
$199$		−	8.71780i		−	0.617988i	0.951064	−	0.308994i	$-0.0999924\pi$
$200$	5.92331	−	12.8419i	0.418841	−	0.908060i
$201$	15.3821			1.08497
$202$	−4.07689	+	28.1046i	−0.286849	+	1.97744i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	27.8283	+	4.03680i	1.93889	+	0.281257i
$207$		0			0
$208$	21.8557	+	14.2015i	1.51542	+	0.984698i
$209$	−25.9405			−1.79434
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	9.76645	+	2.89437i	0.670762	+	0.198786i
$213$		0			0
$214$	−5.13362	−	0.744688i	−0.350927	−	0.0509058i
$215$		0			0
$216$	−13.6642	−	6.30259i	−0.929731	−	0.428837i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−7.56231	+	25.5174i	−0.509851	+	1.72039i
$221$		0			0
$222$	12.5907	+	1.82642i	0.845030	+	0.122581i
$223$			14.9356i			1.00016i	0.865978	+	0.500082i	$0.166697\pi$
$223$			14.9356i			1.00016i	−0.865978	+	0.500082i	$0.833303\pi$
$224$		0			0
$225$	8.70239			0.580159
$226$	4.29849	−	29.6323i	0.285931	−	1.97111i
$227$		−	18.7250i		−	1.24282i	−0.783484	−	0.621412i	$-0.786559\pi$
$227$		−	18.7250i		−	1.24282i	0.783484	−	0.621412i	$-0.213441\pi$
$228$	−9.38057	−	2.78001i	−0.621244	−	0.184111i
$229$	−29.5619			−1.95351			−0.976754	−	0.214362i	$-0.931233\pi$
$229$	−29.5619			−1.95351			−0.976754	+	0.214362i	$0.931233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−2.30250	+	15.8727i	−0.150519	+	1.03763i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.4936i		−	1.26094i	−0.776215	−	0.630468i	$-0.782863\pi$
$239$		−	19.4936i		−	1.26094i	0.776215	−	0.630468i	$-0.217137\pi$
$240$	−5.46934	+	8.41713i	−0.353044	+	0.543323i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	4.95708	−	34.1723i	0.318653	−	2.19668i
$243$		−	15.1196i		−	0.969920i
$244$	−2.14590	−	0.635955i	−0.137377	−	0.0407128i
$245$	15.6525			1.00000
$246$		0			0
$247$			28.4030i			1.80724i
$248$		0			0
$249$		0			0
$250$	2.26986	−	15.6476i	0.143559	−	0.989642i
$251$		−	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
$254$	30.3276	+	4.39936i	1.90292	+	0.276040i
$255$		0			0
$256$	6.50000	+	14.6202i	0.406250	+	0.913762i
$257$	24.7568			1.54428			0.772142	−	0.635450i	$-0.219185\pi$
$257$	24.7568			1.54428			0.772142	+	0.635450i	$0.219185\pi$
$258$		0			0
$259$		0			0
$260$	27.9398	+	8.28018i	1.73275	+	0.513515i
$261$		0			0
$262$	−27.2825	−	3.95764i	−1.68552	−	0.244504i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	7.91224	−	17.1540i	0.486965	−	1.05575i
$265$	11.3886			0.699599
$266$		0			0
$267$		0			0
$268$	7.78894	−	26.2822i	0.475785	−	1.60544i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−16.6496	−	2.41520i	−1.01326	−	0.146985i
$271$		−	22.3998i		−	1.36069i	−0.732892	−	0.680345i	$-0.761830\pi$
$271$		−	22.3998i		−	1.36069i	0.732892	−	0.680345i	$-0.238170\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			29.7558i			1.79434i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	12.2615	+	1.77866i	0.735395	+	0.106677i
$279$		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
$285$	−10.9387			−0.647951
$286$	−54.2729	−	7.87289i	−3.20922	−	0.465534i
$287$		0			0
$288$	−6.49414	+	7.40018i	−0.382671	+	0.436060i
$289$	17.0000			1.00000
$290$		0			0
$291$		−	13.0720i		−	0.766295i
$292$		0			0
$293$	34.2340			1.99997			0.999987	−	0.00505234i	$-0.00160822\pi$
$293$	34.2340			1.99997			0.999987	+	0.00505234i	$0.00160822\pi$
$294$	−10.9950	−	1.59494i	−0.641239	−	0.0930189i
$295$		0			0
$296$	9.49614	−	20.5879i	0.551952	−	1.19665i
$297$	31.6612			1.83717
$298$	−1.69765	+	11.7030i	−0.0983422	+	0.677936i
$299$		0			0
$300$	−3.18889	+	10.7603i	−0.184111	+	0.621244i
$301$		0			0
$302$		0			0
$303$		−	22.5366i		−	1.29469i
$304$	−9.50000	+	14.6202i	−0.544862	+	0.838525i
$305$	−2.50233			−0.143283
$306$		0			0
$307$			1.35100i			0.0771055i	0.999257	+	0.0385528i	$0.0122748\pi$
$307$			1.35100i			0.0771055i	−0.999257	+	0.0385528i	$0.987725\pi$
$308$		0			0
$309$	−22.3150			−1.26945
$310$		0			0
$311$			16.1170i			0.913910i	0.889490	+	0.456955i	$0.151060\pi$
$311$			16.1170i			0.913910i	−0.889490	+	0.456955i	$0.848940\pi$
$312$	−18.7824	−	8.66333i	−1.06334	−	0.490465i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.2999			−0.915496			−0.457748	−	0.889082i	$-0.651344\pi$
$317$	−16.2999			−0.915496			−0.457748	+	0.889082i	$0.651344\pi$
$318$	−7.99987	−	1.16047i	−0.448610	−	0.0650759i
$319$		0			0
$320$	11.6122	+	13.6072i	0.649144	+	0.760665i
$321$	4.11655			0.229763
$322$		0			0
$323$		0			0
$324$	1.43683	+	0.425816i	0.0798239	+	0.0236565i
$325$	32.5805			1.80724
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	3.03203	−	20.9018i	0.166908	−	1.15060i
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	13.9515			0.764539
$334$	24.0448	+	3.48796i	1.31567	+	0.190853i
$335$		−	30.6476i		−	1.67446i
$336$		0			0
$337$	1.64356			0.0895307			0.0447653	−	0.998998i	$-0.485746\pi$
$337$	1.64356			0.0895307			0.0447653	+	0.998998i	$0.485746\pi$
$338$	−5.98096	+	41.2306i	−0.325321	+	2.24265i
$339$			23.7615i			1.29055i
$340$		0			0
$341$		0			0
$342$	−10.6179	−	1.54024i	−0.574150	−	0.0832868i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	4.62823	−	31.9054i	0.248815	−	1.71525i
$347$		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	$-0.383090\pi$
$349$	13.4164			0.718164			0.359082	+	0.933306i	$0.383090\pi$
$350$		0			0
$351$		−	34.6667i		−	1.85037i
$352$	−25.3032	−	22.2052i	−1.34867	−	1.18354i
$353$		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
$359$		−	28.0193i		−	1.47880i	−0.673265	−	0.739401i	$-0.735109\pi$
$359$		−	28.0193i		−	1.47880i	0.673265	−	0.739401i	$-0.264891\pi$
$360$	−4.61050	+	9.99569i	−0.242995	+	0.526819i
$361$	−19.0000			−1.00000
$362$		0			0
$363$			27.4021i			1.43824i
$364$		0			0
$365$		0			0
$366$	1.75774	+	0.254980i	0.0918786	+	0.0133280i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	3.63900	−	25.0860i	0.189183	−	1.30416i
$371$		0			0
$372$		0			0
$373$	−31.3113			−1.62124			−0.810619	−	0.585575i	$-0.800869\pi$
$373$	−31.3113			−1.62124			−0.810619	+	0.585575i	$0.800869\pi$
$374$		0			0
$375$			12.5475i			0.647951i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−5.53897	+	18.6901i	−0.284143	+	0.958782i
$381$	−24.3191			−1.24591
$382$	−36.6034	−	5.30972i	−1.87279	−	0.271669i
$383$		−	29.9215i		−	1.52892i	−0.644671	−	0.764460i	$-0.723006\pi$
$383$		−	29.9215i		−	1.52892i	0.644671	−	0.764460i	$-0.276994\pi$
$384$	−6.77040	−	10.7415i	−0.345501	−	0.548151i
$385$		0			0
$386$	5.61956	−	38.7393i	0.286028	−	1.97178i
$387$		0			0
$388$	−22.3352	−	6.61922i	−1.13390	−	0.336040i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$	−22.8859	−	3.31986i	−1.15887	−	0.168108i
$391$		0			0
$392$	−8.29263	+	17.9787i	−0.418841	+	0.908060i
$393$	21.8773			1.10357
$394$		0			0
$395$		0			0
$396$	5.88624	−	19.8619i	0.295795	−	0.998098i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	12.2011	+	1.76991i	0.611587	+	0.0887175i
$399$		0			0
$400$	16.7705	+	10.8972i	0.838525	+	0.544862i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−3.12290	+	21.5282i	−0.155756	+	1.07373i
$403$		0			0
$404$	−38.5066	−	11.4117i	−1.91577	−	0.567756i
$405$	1.67548			0.0832555
$406$		0			0
$407$			47.7041i			2.36460i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−11.2995	+	38.1279i	−0.556688	+	1.87843i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−24.3131	+	27.7052i	−1.19205	+	1.35836i
$417$	−9.83225			−0.481487
$418$	5.26651	−	36.3055i	0.257593	−	1.77576i
$419$			19.4936i			0.952324i	0.879358	+	0.476162i	$0.157972\pi$
$419$			19.4936i			0.952324i	−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−6.03366	+	13.0812i	−0.293021	+	0.635277i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	2.08448	−	7.03364i	0.100757	−	0.339984i
$429$	43.5204			2.10118
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	11.5950	−	17.8444i	0.557866	−	0.858538i
$433$	37.4530			1.79988			0.899939	−	0.436015i	$-0.143611\pi$
$433$	37.4530			1.79988			0.899939	+	0.436015i	$0.143611\pi$
$434$		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	$0.5\pi$
$440$	−34.1780	−	15.7645i	−1.62937	−	0.751545i
$441$	−12.1833			−0.580159
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−5.11238	+	17.2507i	−0.242623	+	0.818680i
$445$		0			0
$446$	−20.9034	−	3.03227i	−0.989803	−	0.143582i
$447$		−	9.38441i		−	0.443867i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.76678	+	12.1796i	−0.0832868	+	0.574150i
$451$		0			0
$452$	40.5996	+	12.0320i	1.90964	+	0.565939i
$453$		0			0
$454$	26.2069	+	3.80160i	1.22995	+	0.178418i
$455$		0			0
$456$	5.79527	−	12.5643i	0.271389	−	0.588378i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	6.00174	−	41.3739i	0.280443	−	1.93327i
$459$		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	$0.5\pi$
$468$	−21.7474	−	6.44501i	−1.00527	−	0.297921i
$469$		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
$477$	−8.86453			−0.405879
$478$	27.2825	+	3.95764i	1.24787	+	0.181018i
$479$		−	7.68770i		−	0.351260i	−0.984456	−	0.175630i	$-0.943804\pi$
$479$		−	7.68770i		−	0.351260i	0.984456	−	0.175630i	$-0.0561962\pi$
$480$	−10.6699	−	9.36356i	−0.487013	−	0.427386i
$481$	52.2325			2.38160
$482$		0			0
$483$		0			0
$484$	46.8200	+	13.8755i	2.12818	+	0.630705i
$485$	−26.0450			−1.18264
$486$	21.1608	+	3.06961i	0.959874	+	0.139240i
$487$		−	32.2386i		−	1.46087i	−0.682983	−	0.730434i	$-0.739318\pi$
$487$		−	32.2386i		−	1.46087i	0.682983	−	0.730434i	$-0.260682\pi$
$488$	1.32573	−	2.87421i	0.0600128	−	0.130109i
$489$		0			0
$490$	−3.17780	+	21.9067i	−0.143559	+	0.989642i
$491$		−	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
$494$	−39.7519	−	5.76645i	−1.78852	−	0.259445i
$495$		−	23.1609i		−	1.04101i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			44.4679i			1.99066i	0.0965389	+	0.995329i	$0.469223\pi$
$499$			44.4679i			1.99066i	−0.0965389	+	0.995329i	$0.530777\pi$
$500$	21.4390	+	6.35363i	0.958782	+	0.284143i
$501$	−19.2811			−0.861414
$502$	36.6034	+	5.30972i	1.63369	+	0.236985i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−44.9025			−1.99813
$506$		0			0
$507$		−	33.0620i		−	1.46834i
$508$	−12.3144	+	41.5523i	−0.546362	+	1.84359i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−21.7816	+	6.12895i	−0.962618	+	0.270864i
$513$	23.1901			1.02387
$514$	−5.02618	+	34.6487i	−0.221695	+	1.52829i
$515$			44.4609i			1.95918i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			25.5843i			1.12303i
$520$	−17.2610	+	37.4224i	−0.756947	+	1.64108i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			39.9718i			1.74784i	0.486066	+	0.873922i	$0.338432\pi$
$523$			39.9718i			1.74784i	−0.486066	+	0.873922i	$0.661568\pi$
$524$	11.0779	−	37.3802i	0.483942	−	1.63296i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	22.4017	+	14.5563i	0.974910	+	0.633483i
$529$	−23.0000			−1.00000
$530$	−2.31215	+	15.9391i	−0.100433	+	0.692352i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	8.20192i		−	0.354600i
$536$	35.2023	+	16.2370i	1.52051	+	0.701332i
$537$		0			0
$538$		0			0
$539$		−	41.6582i		−	1.79434i
$540$	6.76047	−	22.8118i	0.290924	−	0.981664i
$541$	27.3238			1.17474			0.587371	−	0.809318i	$-0.300163\pi$
$541$	27.3238			1.17474			0.587371	+	0.809318i	$0.300163\pi$
$542$	31.3500	+	4.54766i	1.34660	+	0.195339i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	46.7055i		−	1.99698i	−0.0549052	−	0.998492i	$-0.517486\pi$
$547$		−	46.7055i		−	1.99698i	0.0549052	−	0.998492i	$-0.482514\pi$
$548$		0			0
$549$	1.94773			0.0831269
$550$	−41.6452	−	6.04110i	−1.77576	−	0.257593i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$			20.1160i			0.853875i
$556$	−4.97871	+	16.7996i	−0.211144	+	0.712463i
$557$		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
$563$		−	37.7272i		−	1.59001i	−0.606601	−	0.795007i	$-0.707467\pi$
$563$		−	37.7272i		−	1.59001i	0.606601	−	0.795007i	$-0.292533\pi$
$564$		0			0
$565$	47.3431			1.99174
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	2.22080	−	15.3094i	0.0930189	−	0.641239i
$571$		−	26.9461i		−	1.12766i	−0.825891	−	0.563829i	$-0.809328\pi$
$571$		−	26.9461i		−	1.12766i	0.825891	−	0.563829i	$-0.190672\pi$
$572$	22.0372	−	74.3601i	0.921423	−	3.10915i
$573$	29.3515			1.22618
$574$		0			0
$575$		0			0
$576$	−9.03857	−	10.5914i	−0.376607	−	0.441307i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−3.45138	+	23.7926i	−0.143559	+	0.989642i
$579$			31.0643i			1.29099i
$580$		0			0
$581$		0			0
$582$	18.2951	+	2.65391i	0.758358	+	0.110008i
$583$		−	30.3102i		−	1.25532i
$584$		0			0
$585$	−25.3595			−1.04849
$586$	−6.95028	+	47.9128i	−0.287113	+	1.97926i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	4.46445	−	15.0644i	0.184111	−	0.621244i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	26.8862	+	17.4703i	1.10502	+	0.718024i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−6.42793	+	44.3119i	−0.263741	+	1.81814i
$595$		0			0
$596$	−16.0344	−	4.75194i	−0.656796	−	0.194647i
$597$	−9.78385			−0.400426
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−14.4123	−	6.64763i	−0.588378	−	0.271389i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			23.8550i			0.971452i
$604$		0			0
$605$	54.5967			2.21967
$606$	31.5414	+	4.57543i	1.28128	+	0.185864i
$607$			47.2956i			1.91967i	0.280568	+	0.959834i	$0.409477\pi$
$607$			47.2956i			1.91967i	−0.280568	+	0.959834i	$0.590523\pi$
$608$	−18.5332	−	16.2641i	−0.751620	−	0.659596i
$609$		0			0
$610$	0.508029	−	3.50217i	0.0205695	−	0.141799i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−1.89081	−	0.274283i	−0.0763069	−	0.0110692i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	4.53044	−	31.2313i	0.182241	−	1.25631i
$619$			35.3754i			1.42186i	0.703265	+	0.710928i	$0.251725\pi$
$619$			35.3754i			1.42186i	−0.703265	+	0.710928i	$0.748275\pi$
$620$		0			0
$621$		0			0
$622$	−22.5568	−	3.27211i	−0.904444	−	0.131200i
$623$		0			0
$624$	15.9381	−	24.5283i	0.638036	−	0.981917i
$625$	25.0000			1.00000
$626$		0			0
$627$			29.1127i			1.16265i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			49.0142i			1.95123i	0.219499	+	0.975613i	$0.429558\pi$
$631$			49.0142i			1.95123i	−0.219499	+	0.975613i	$0.570442\pi$
$632$		0			0
$633$		0			0
$634$	3.30925	−	22.8128i	0.131427	−	0.906013i
$635$			48.4541i			1.92284i
$636$	3.24830	−	10.9607i	0.128804	−	0.434621i
$637$	−45.6127			−1.80724
$638$		0			0
$639$		0			0
$640$	−21.4017	+	13.4895i	−0.845976	+	0.533220i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−0.835751	+	5.76138i	−0.0329845	+	0.227383i
$643$		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	$0.5\pi$
$648$	−0.887666	+	1.92449i	−0.0348708	+	0.0756009i
$649$		0			0
$650$	−6.61457	+	45.5985i	−0.259445	+	1.78852i
$651$		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	$0.5\pi$
$660$	28.6378	+	8.48706i	1.11473	+	0.330358i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.83247	+	19.5261i	−0.109756	+	0.756620i
$667$		0			0
$668$	−9.76326	+	32.9441i	−0.377752	+	1.27465i
$669$	16.7620			0.648057
$670$	42.8933	+	6.22215i	1.65711	+	0.240383i
$671$			6.65980i			0.257099i
$672$		0			0
$673$	−50.4853			−1.94606			−0.973032	−	0.230671i	$-0.925908\pi$
$673$	−50.4853			−1.94606			−0.973032	+	0.230671i	$0.925908\pi$
$674$	−0.333680	+	2.30028i	−0.0128529	+	0.0886033i
$675$		−	26.6008i		−	1.02387i
$676$	−56.4906	−	16.7415i	−2.17272	−	0.643903i
$677$	−44.4204			−1.70721			−0.853607	−	0.520918i	$-0.825590\pi$
$677$	−44.4204			−1.70721			−0.853607	+	0.520918i	$0.825590\pi$
$678$	−33.2558	−	4.82413i	−1.27718	−	0.185269i
$679$		0			0
$680$		0			0
$681$	−21.0148			−0.805289
$682$		0			0
$683$		−	51.1946i		−	1.95891i	−0.201667	−	0.979454i	$-0.564636\pi$
$683$		−	51.1946i		−	1.95891i	0.201667	−	0.979454i	$-0.435364\pi$
$684$	4.31134	−	14.5477i	0.164848	−	0.556246i
$685$		0			0
$686$		0			0
$687$			33.1769i			1.26578i
$688$		0			0
$689$	−33.1875			−1.26434
$690$		0			0
$691$			0.331647i			0.0126165i	0.999980	+	0.00630823i	$0.00200798\pi$
$691$			0.331647i			0.0126165i	−0.999980	+	0.00630823i	$0.997992\pi$
$692$	43.7141	+	12.9550i	1.66176	+	0.492476i
$693$		0			0
$694$		0			0
$695$			19.5900i			0.743092i
$696$		0			0
$697$		0			0
$698$	−2.72383	+	18.7771i	−0.103099	+	0.710725i
$699$		0			0
$700$		0			0
$701$	−48.5239			−1.83272			−0.916360	−	0.400354i	$-0.868887\pi$
$701$	−48.5239			−1.83272			−0.916360	+	0.400354i	$0.868887\pi$
$702$	48.5183	+	7.03812i	1.83121	+	0.265637i
$703$			34.9406i			1.31781i
$704$	36.2148	−	30.9053i	1.36490	−	1.16479i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−40.2492			−1.51159			−0.755796	−	0.654808i	$-0.772750\pi$
$709$	−40.2492			−1.51159			−0.755796	+	0.654808i	$0.772750\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		−	86.7112i		−	3.24281i
$716$		0			0
$717$	−21.8773			−0.817024
$718$	39.2149	+	5.68855i	1.46349	+	0.212295i
$719$		−	51.8240i		−	1.93271i	−0.257214	−	0.966354i	$-0.582805\pi$
$719$		−	51.8240i		−	1.93271i	0.257214	−	0.966354i	$-0.417195\pi$
$720$	−13.0536	−	8.48204i	−0.486479	−	0.316107i
$721$		0			0
$722$	3.85743	−	26.5917i	0.143559	−	0.989642i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−38.3511	−	5.56325i	−1.42334	−	0.206472i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−19.2163			−0.711716
$730$		0			0
$731$		0			0
$732$	−0.713722	+	2.40831i	−0.0263799	+	0.0890136i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		−	17.5665i		−	0.647951i
$736$		0			0
$737$	−81.5669			−3.00455
$738$		0			0
$739$			8.71780i			0.320689i	0.987061	+	0.160345i	$0.0512606\pi$
$739$			8.71780i			0.320689i	−0.987061	+	0.160345i	$0.948739\pi$
$740$	34.3707	+	10.1860i	1.26349	+	0.374446i
$741$	31.8763			1.17100
$742$		0			0
$743$			37.3813i			1.37139i	0.727890	+	0.685693i	$0.240501\pi$
$743$			37.3813i			1.37139i	−0.727890	+	0.685693i	$0.759499\pi$
$744$		0			0
$745$	−18.6977			−0.685032
$746$	6.35689	−	43.8222i	0.232742	−	1.60444i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−17.5611	−	2.54743i	−0.641239	−	0.0930189i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−29.3515			−1.06963
$754$		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
$760$	−25.0335	−	11.5467i	−0.908060	−	0.418841i
$761$	54.3834			1.97140			0.985699	−	0.168518i	$-0.0538981\pi$
$761$	54.3834			1.97140			0.985699	+	0.168518i	$0.0538981\pi$
$762$	4.93733	−	34.0362i	0.178861	−	1.23300i
$763$		0			0
$764$	14.8626	−	50.1508i	0.537710	−	1.81439i
$765$		0			0
$766$	41.8771	+	6.07474i	1.51308	+	0.219489i
$767$		0			0
$768$	16.4080	−	7.29485i	0.592073	−	0.263230i
$769$	47.1406			1.69993			0.849967	−	0.526836i	$-0.176622\pi$
$769$	47.1406			1.69993			0.849967	+	0.526836i	$0.176622\pi$
$770$		0			0
$771$		−	27.7841i		−	1.00062i
$772$	53.0772	+	15.7299i	1.91029	+	0.566131i
$773$	−13.0516			−0.469433			−0.234717	−	0.972064i	$-0.575416\pi$
$773$	−13.0516			−0.469433			−0.234717	+	0.972064i	$0.575416\pi$
$774$		0			0
$775$		0			0
$776$	13.7986	−	29.9157i	0.495340	−	1.07391i
$777$		0			0
$778$	−1.21813	+	8.39739i	−0.0436722	+	0.301061i
$779$		0			0
$780$	9.29272	−	31.3564i	0.332733	−	1.12274i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−23.4787	−	15.2561i	−0.838525	−	0.544862i
$785$		0			0
$786$	−4.44159	+	30.6188i	−0.158426	+	1.09214i
$787$			53.4392i			1.90490i	0.304692	+	0.952451i	$0.401447\pi$
$787$			53.4392i			1.90490i	−0.304692	+	0.952451i	$0.598553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	26.6030	+	12.2706i	0.945296	+	0.436016i
$793$	7.29201			0.258947
$794$		0			0
$795$		−	12.7813i		−	0.453306i
$796$	−4.95420	+	16.7169i	−0.175597	+	0.592516i
$797$	13.8614			0.490997			0.245499	−	0.969397i	$-0.421048\pi$
$797$	13.8614			0.490997			0.245499	+	0.969397i	$0.421048\pi$
$798$		0			0
$799$		0			0
$800$	−18.6562	+	21.2590i	−0.659596	+	0.751620i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−29.4961	−	8.74140i	−1.04025	−	0.308286i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	23.7892	−	51.5756i	0.836900	−	1.81442i
$809$	22.3607			0.786160			0.393080	−	0.919504i	$-0.371410\pi$
$809$	22.3607			0.786160			0.393080	+	0.919504i	$0.371410\pi$
$810$	−0.340161	+	2.34495i	−0.0119520	+	0.0823931i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	−25.1389			−0.881661
$814$	−66.7649	−	9.68499i	−2.34011	−	0.339459i
$815$		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	$0.5\pi$
$824$	−51.0685	−	23.5553i	−1.77905	−	0.820586i
$825$	33.3945			1.16265
$826$		0			0
$827$			30.9935i			1.07775i	0.842386	+	0.538875i	$0.181151\pi$
$827$			30.9935i			1.07775i	−0.842386	+	0.538875i	$0.818849\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−33.8391	−	39.6526i	−1.17316	−	1.37471i
$833$		0			0
$834$	1.99617	−	13.7609i	0.0691216	−	0.476500i
$835$			38.4161i			1.32944i
$836$	49.7426	+	14.7416i	1.72039	+	0.509851i
$837$		0			0
$838$	−27.2825	−	3.95764i	−0.942460	−	0.136714i
$839$		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
$845$	−65.8736			−2.26612
$846$		0			0
$847$		0			0
$848$	−17.0830	−	11.1003i	−0.586631	−	0.381185i
$849$		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
$855$		−	16.9641i		−	0.580159i
$856$	9.42084	+	4.34535i	0.321998	+	0.148521i
$857$	−33.5250			−1.14519			−0.572597	−	0.819837i	$-0.694064\pi$
$857$	−33.5250			−1.14519			−0.572597	+	0.819837i	$0.694064\pi$
$858$	−8.83562	+	60.9096i	−0.301643	+	2.07942i
$859$		−	58.4808i		−	1.99534i	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$		−	58.4808i		−	1.99534i	0.0682391	−	0.997669i	$-0.478262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	3.71278i		−	0.126385i	−0.998001	−	0.0631923i	$-0.979872\pi$
$863$		−	3.71278i		−	0.126385i	0.998001	−	0.0631923i	$-0.0201281\pi$
$864$	22.6203	+	19.8508i	0.769558	+	0.675338i
$865$	50.9749			1.73320
$866$	−7.60381	+	52.4180i	−0.258388	+	1.78124i
$867$		−	19.0788i		−	0.647951i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			89.3098i			3.02615i
$872$		0			0
$873$	20.2726			0.686122
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.752363			0.0254055			0.0127028	−	0.999919i	$-0.495956\pi$
$877$	0.752363			0.0254055			0.0127028	+	0.999919i	$0.495956\pi$
$878$		0			0
$879$		−	38.4203i		−	1.29589i
$880$	29.0024	−	44.6338i	0.977671	−	1.50460i
$881$	9.21678			0.310521			0.155261	−	0.987874i	$-0.450378\pi$
$881$	9.21678			0.310521			0.155261	+	0.987874i	$0.450378\pi$
$882$	2.47349	−	17.0514i	0.0832868	−	0.574150i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	52.3146i		−	1.75655i	−0.478154	−	0.878276i	$-0.658694\pi$
$887$		−	52.3146i		−	1.75655i	0.478154	−	0.878276i	$-0.341306\pi$
$888$	−23.1055	−	10.6574i	−0.775369	−	0.357638i
$889$		0			0
$890$		0			0
$891$		−	4.45921i		−	0.149389i
$892$	8.48771	−	28.6400i	0.284189	−	0.958938i
$893$		0			0
$894$	13.1341	+	1.90524i	0.439270	+	0.0637209i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−16.6874	−	4.94545i	−0.556246	−	0.164848i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−25.0822	+	54.3790i	−0.834223	+	1.80862i
$905$		0			0
$906$		0			0
$907$		−	28.7630i		−	0.955061i	−0.878615	−	0.477531i	$-0.841532\pi$
$907$		−	28.7630i		−	0.955061i	0.878615	−	0.477531i	$-0.158468\pi$
$908$	−10.6412	+	35.9064i	−0.353140	+	1.19160i
$909$	34.9505			1.15924
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	16.4080	+	10.6617i	0.543323	+	0.353044i
$913$		0			0
$914$		0			0
$915$			2.80832i			0.0928403i
$916$	56.6869	+	16.7996i	1.87299	+	0.555076i
$917$		0			0
$918$		0			0
$919$			58.4808i			1.92910i	0.263896	+	0.964551i	$0.414993\pi$
$919$			58.4808i			1.92910i	−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	1.51620			0.0499606
$922$	3.65440	−	25.1922i	0.120351	−	0.829660i
$923$		0			0
$924$		0			0
$925$	40.0796			1.31781
$926$		0			0
$927$		−	34.6069i		−	1.13664i
$928$		0			0
$929$	−31.3050			−1.02708			−0.513541	−	0.858065i	$-0.671667\pi$
$929$	−31.3050			−1.02708			−0.513541	+	0.858065i	$0.671667\pi$
$930$		0			0
$931$		−	30.5123i		−	1.00000i
$932$		0			0
$933$	18.0878			0.592169
$934$		0			0
$935$		0			0
$936$	13.4354	−	29.1283i	0.439150	−	0.952090i
$937$		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
$950$	−30.5028	−	4.42477i	−0.989642	−	0.143559i
$951$			18.2932i			0.593197i
$952$		0			0
$953$	5.80217			0.187951			0.0939754	−	0.995575i	$-0.470043\pi$
$953$	5.80217			0.187951			0.0939754	+	0.995575i	$0.470043\pi$
$954$	1.79970	−	12.4065i	0.0582673	−	0.401675i
$955$		−	58.4808i		−	1.89239i
$956$	−11.0779	+	37.3802i	−0.358286	+	1.20896i
$957$		0			0
$958$	10.7594	+	1.56077i	0.347621	+	0.0504263i
$959$		0			0
$960$	15.2711	−	13.0322i	0.492874	−	0.420614i
$961$	−31.0000			−1.00000
$962$	−10.6044	+	73.1028i	−0.341899	+	2.35693i
$963$			6.38409i			0.205724i
$964$		0			0
$965$	61.8933			1.99242
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−28.9252	+	62.7106i	−0.929691	+	2.01560i
$969$		0			0
$970$	5.28772	−	36.4517i	0.169779	−	1.17039i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−8.59224	+	28.9927i	−0.275596	+	0.929942i
$973$		0			0
$974$	45.1200	+	6.54515i	1.44574	+	0.209720i
$975$		−	36.5646i		−	1.17100i
$976$	3.75349	+	2.43897i	0.120146	+	0.0780695i
$977$	60.2691			1.92818			0.964090	−	0.265577i	$-0.0855625\pi$
$977$	60.2691			1.92818			0.964090	+	0.265577i	$0.0855625\pi$
$978$		0			0
$979$		0			0
$980$	−30.0146	−	8.89508i	−0.958782	−	0.284143i
$981$		0			0
$982$	36.6034	+	5.30972i	1.16806	+	0.169440i
$983$			0.192493i			0.00613957i	0.999995	+	0.00306979i	$0.000977145\pi$
$983$			0.192493i			0.00613957i	−0.999995	+	0.00306979i	$0.999023\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	16.1410	−	54.4646i	0.513515	−	1.73275i
$989$		0			0
$990$	32.4152	+	4.70219i	1.03022	+	0.149445i
$991$		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			$\pi$
$998$	−62.2358	−	9.02799i	−1.97004	−	0.285776i
$999$		−	42.6460i		−	1.34926i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.d.a.379.8	yes	16
4.3	odd	2	inner	380.2.d.a.379.7	✓	16
5.4	even	2	inner	380.2.d.a.379.9	yes	16
19.18	odd	2	inner	380.2.d.a.379.9	yes	16
20.19	odd	2	inner	380.2.d.a.379.10	yes	16
76.75	even	2	inner	380.2.d.a.379.10	yes	16
95.94	odd	2	CM	380.2.d.a.379.8	yes	16
380.379	even	2	inner	380.2.d.a.379.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.d.a.379.7	✓	16	4.3	odd	2	inner
380.2.d.a.379.7	✓	16	380.379	even	2	inner
380.2.d.a.379.8	yes	16	1.1	even	1	trivial
380.2.d.a.379.8	yes	16	95.94	odd	2	CM
380.2.d.a.379.9	yes	16	5.4	even	2	inner
380.2.d.a.379.9	yes	16	19.18	odd	2	inner
380.2.d.a.379.10	yes	16	20.19	odd	2	inner
380.2.d.a.379.10	yes	16	76.75	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+(-0.203022 + 1.39956i) q^{2} -1.12228i q^{3} +(-1.91756 - 0.568286i) q^{4} -2.23607 q^{5} +(1.57071 + 0.227849i) q^{6} +(1.18466 - 2.56838i) q^{8} +1.74048 q^{9} +O(q^{10})\)	\(q+(-0.203022 + 1.39956i) q^{2} -1.12228i q^{3} +(-1.91756 - 0.568286i) q^{4} -2.23607 q^{5} +(1.57071 + 0.227849i) q^{6} +(1.18466 - 2.56838i) q^{8} +1.74048 q^{9} +(0.453972 - 3.12952i) q^{10} +5.95117i q^{11} +(-0.637778 + 2.15205i) q^{12} +6.51610 q^{13} +2.50950i q^{15} +(3.35410 + 2.17945i) q^{16} +(-0.353356 + 2.43591i) q^{18} +4.35890i q^{19} +(4.28780 + 1.27073i) q^{20} +(-8.32905 - 1.20822i) q^{22} +(-2.88245 - 1.32953i) q^{24} +5.00000 q^{25} +(-1.32291 + 9.11971i) q^{26} -5.32016i q^{27} +(-3.51221 - 0.509485i) q^{30} +(-3.73124 + 4.25181i) q^{32} +6.67890 q^{33} +(-3.33748 - 0.989090i) q^{36} +8.01591 q^{37} +(-6.10056 - 0.884954i) q^{38} -7.31292i q^{39} +(-2.64898 + 5.74307i) q^{40} +(3.38197 - 11.4117i) q^{44} -3.89183 q^{45} +(2.44596 - 3.76426i) q^{48} -7.00000 q^{49} +(-1.01511 + 6.99782i) q^{50} +(-12.4950 - 3.70301i) q^{52} -5.09315 q^{53} +(7.44591 + 1.08011i) q^{54} -13.3072i q^{55} +4.89192 q^{57} +(1.42612 - 4.81213i) q^{60} +1.11908 q^{61} +(-5.19315 - 6.08532i) q^{64} -14.5704 q^{65} +(-1.35597 + 9.34756i) q^{66} +13.7060i q^{67} +(2.06188 - 4.47021i) q^{72} +(-1.62741 + 11.2188i) q^{74} -5.61142i q^{75} +(2.47710 - 8.35847i) q^{76} +(10.2349 + 1.48469i) q^{78} +(-7.50000 - 4.87340i) q^{80} -0.749299 q^{81} +(15.2849 + 7.05012i) q^{88} +(0.790128 - 5.44687i) q^{90} -9.74679i q^{95} +(4.77173 + 4.18751i) q^{96} +11.6477 q^{97} +(1.42116 - 9.79695i) q^{98} +10.3579i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 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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