LMFDB - Embedded newform 380.2.d.a.379.5

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			379.5
Root			$-0.846083 - 1.13320i$ of defining polynomial
Character	$\chi$	$=$	380.379
Dual form			380.2.d.a.379.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.846083 - 1.13320i) q^{2} -3.11095i q^{3} +(-0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(-3.52533 + 2.63212i) q^{6} +(2.65380 - 0.978437i) q^{8} -6.67802 q^{9} +O(q^{10})$	\(q+(-0.846083 - 1.13320i) q^{2} -3.11095i q^{3} +(-0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(-3.52533 + 2.63212i) q^{6} +(2.65380 - 0.978437i) q^{8} -6.67802 q^{9} +(-1.89190 - 2.53391i) q^{10} -2.92978i q^{11} +(5.96545 + 1.76791i) q^{12} -6.79166 q^{13} -6.95630i q^{15} +(-3.35410 - 2.17945i) q^{16} +(5.65017 + 7.56754i) q^{18} -4.35890i q^{19} +(-1.27073 + 4.28780i) q^{20} +(-3.32002 + 2.47884i) q^{22} +(-3.04387 - 8.25585i) q^{24} +5.00000 q^{25} +(5.74631 + 7.69631i) q^{26} +11.4422i q^{27} +(-7.88288 + 5.88561i) q^{30} +(0.368097 + 5.64487i) q^{32} -9.11440 q^{33} +(3.79503 - 12.8055i) q^{36} +12.1390 q^{37} +(-4.93951 + 3.68799i) q^{38} +21.1285i q^{39} +(5.93408 - 2.18785i) q^{40} +(5.61803 + 1.66495i) q^{44} -14.9325 q^{45} +(-6.78016 + 10.4345i) q^{48} -7.00000 q^{49} +(-4.23042 - 5.66600i) q^{50} +(3.85961 - 13.0234i) q^{52} +6.04380 q^{53} +(12.9663 - 9.68102i) q^{54} -6.55118i q^{55} -13.5603 q^{57} +(13.3392 + 3.95317i) q^{60} +15.5804 q^{61} +(6.08532 - 5.19315i) q^{64} -15.1866 q^{65} +(7.71154 + 10.3284i) q^{66} -16.0219i q^{67} +(-17.7222 + 6.53403i) q^{72} +(-10.2706 - 13.7559i) q^{74} -15.5548i q^{75} +(8.35847 + 2.47710i) q^{76} +(23.9429 - 17.8765i) q^{78} +(-7.50000 - 4.87340i) q^{80} +15.5619 q^{81} +(-2.86660 - 7.77505i) q^{88} +(12.6342 + 16.9215i) q^{90} -9.74679i q^{95} +(17.5609 - 1.14513i) q^{96} -2.99619 q^{97} +(5.92258 + 7.93240i) q^{98} +19.5651i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.846083	−	1.13320i	−0.598271	−	0.801294i
$2$	−0.846083	−	1.13320i	−0.598271	−	0.801294i
$3$		−	3.11095i		−	1.79611i	−0.439884	−	0.898055i	$-0.644980\pi$
$3$		−	3.11095i		−	1.79611i	0.439884	−	0.898055i	$-0.355020\pi$
$4$	−0.568286	+	1.91756i	−0.284143	+	0.958782i
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$	−3.52533	+	2.63212i	−1.43921	+	1.07456i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	2.65380	−	0.978437i	0.938260	−	0.345930i
$9$	−6.67802			−2.22601
$10$	−1.89190	−	2.53391i	−0.598271	−	0.801294i
$11$		−	2.92978i		−	0.883361i	−0.897172	−	0.441680i	$-0.854382\pi$
$11$		−	2.92978i		−	0.883361i	0.897172	−	0.441680i	$-0.145618\pi$
$12$	5.96545	+	1.76791i	1.72208	+	0.510352i
$13$	−6.79166			−1.88367			−0.941834	−	0.336079i	$-0.890899\pi$
$13$	−6.79166			−1.88367			−0.941834	+	0.336079i	$0.890899\pi$
$14$		0			0
$15$		−	6.95630i		−	1.79611i
$16$	−3.35410	−	2.17945i	−0.838525	−	0.544862i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	5.65017	+	7.56754i	1.33176	+	1.78369i
$19$		−	4.35890i		−	1.00000i
$19$		−	4.35890i		−	1.00000i
$20$	−1.27073	+	4.28780i	−0.284143	+	0.958782i
$21$		0			0
$22$	−3.32002	+	2.47884i	−0.707832	+	0.528489i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−3.04387	−	8.25585i	−0.621327	−	1.68522i
$25$	5.00000			1.00000
$26$	5.74631	+	7.69631i	1.12694	+	1.50937i
$27$			11.4422i			2.20204i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−7.88288	+	5.88561i	−1.43921	+	1.07456i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.368097	+	5.64487i	0.0650709	+	0.997881i
$33$	−9.11440			−1.58661
$34$		0			0
$35$		0			0
$36$	3.79503	−	12.8055i	0.632505	−	2.13426i
$37$	12.1390			1.99564			0.997820	−	0.0659893i	$-0.0210203\pi$
$37$	12.1390			1.99564			0.997820	+	0.0659893i	$0.0210203\pi$
$38$	−4.93951	+	3.68799i	−0.801294	+	0.598271i
$39$			21.1285i			3.38327i
$40$	5.93408	−	2.18785i	0.938260	−	0.345930i
$41$		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
$45$	−14.9325			−2.22601
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−6.78016	+	10.4345i	−0.978632	+	1.50608i
$49$	−7.00000			−1.00000
$50$	−4.23042	−	5.66600i	−0.598271	−	0.801294i
$51$		0			0
$52$	3.85961	−	13.0234i	0.535231	−	1.80603i
$53$	6.04380			0.830179			0.415090	−	0.909781i	$-0.363750\pi$
$53$	6.04380			0.830179			0.415090	+	0.909781i	$0.363750\pi$
$54$	12.9663	−	9.68102i	1.76448	−	1.31742i
$55$		−	6.55118i		−	0.883361i
$56$		0			0
$57$	−13.5603			−1.79611
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	13.3392	+	3.95317i	1.72208	+	0.510352i
$61$	15.5804			1.99486			0.997430	−	0.0716414i	$-0.0228237\pi$
$61$	15.5804			1.99486			0.997430	+	0.0716414i	$0.0228237\pi$
$62$		0			0
$63$		0			0
$64$	6.08532	−	5.19315i	0.760665	−	0.649144i
$65$	−15.1866			−1.88367
$66$	7.71154	+	10.3284i	0.949225	+	1.27134i
$67$		−	16.0219i		−	1.95739i	−0.205326	−	0.978694i	$-0.565826\pi$
$67$		−	16.0219i		−	1.95739i	0.205326	−	0.978694i	$-0.434174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−17.7222	+	6.53403i	−2.08858	+	0.770042i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−10.2706	−	13.7559i	−1.19393	−	1.59909i
$75$		−	15.5548i		−	1.79611i
$76$	8.35847	+	2.47710i	0.958782	+	0.284143i
$77$		0			0
$78$	23.9429	−	17.8765i	2.71099	−	2.02411i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−7.50000	−	4.87340i	−0.838525	−	0.544862i
$81$	15.5619			1.72910
$82$		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
$88$	−2.86660	−	7.77505i	−0.305581	−	0.828823i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	12.6342	+	16.9215i	1.33176	+	1.78369i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	9.74679i		−	1.00000i
$96$	17.5609	−	1.14513i	1.79230	−	0.116875i
$97$	−2.99619			−0.304217			−0.152109	−	0.988364i	$-0.548606\pi$
$97$	−2.99619			−0.304217			−0.152109	+	0.988364i	$0.548606\pi$
$98$	5.92258	+	7.93240i	0.598271	+	0.801294i
$99$			19.5651i			1.96637i
$100$	−2.84143	+	9.58782i	−0.284143	+	0.958782i
$101$	0.868264			0.0863955			0.0431977	−	0.999067i	$-0.486245\pi$
$101$	0.868264			0.0863955			0.0431977	+	0.999067i	$0.486245\pi$
$102$		0			0
$103$			11.1749i			1.10110i	0.834803	+	0.550548i	$0.185581\pi$
$103$			11.1749i			1.10110i	−0.834803	+	0.550548i	$0.814419\pi$
$104$	−18.0237	+	6.64521i	−1.76737	+	0.651617i
$105$		0			0
$106$	−5.11356	−	6.84883i	−0.496672	−	0.665217i
$107$			11.8033i			1.14107i	0.821274	+	0.570534i	$0.193264\pi$
$107$			11.8033i			1.14107i	−0.821274	+	0.570534i	$0.806736\pi$
$108$	−21.9411	−	6.50242i	−2.11128	−	0.625696i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−7.42380	+	5.54284i	−0.707832	+	0.528489i
$111$		−	37.7639i		−	3.58439i
$112$		0			0
$113$	13.6063			1.27997			0.639987	−	0.768386i	$-0.278940\pi$
$113$	13.6063			1.27997			0.639987	+	0.768386i	$0.278940\pi$
$114$	11.4732	+	15.3666i	1.07456	+	1.43921i
$115$		0			0
$116$		0			0
$117$	45.3549			4.19306
$118$		0			0
$119$		0			0
$120$	−6.80630	−	18.4606i	−0.621327	−	1.68522i
$121$	2.41641			0.219673
$122$	−13.1823	−	17.6557i	−1.19347	−	1.59847i
$123$		0			0
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$		−	10.9384i		−	0.970630i	−0.874339	−	0.485315i	$-0.838705\pi$
$127$		−	10.9384i		−	0.970630i	0.874339	−	0.485315i	$-0.161295\pi$
$128$	−11.0336	−	2.50205i	−0.975239	−	0.221152i
$129$		0			0
$130$	12.8491	+	17.2095i	1.12694	+	1.50937i
$131$			19.4936i			1.70316i	0.524222	+	0.851581i	$0.324356\pi$
$131$			19.4936i			1.70316i	−0.524222	+	0.851581i	$0.675644\pi$
$132$	5.17958	−	17.4774i	0.450825	−	1.52122i
$133$		0			0
$134$	−18.1560	+	13.5559i	−1.56844	+	1.17105i
$135$			25.5854i			2.20204i
$136$		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
$143$			19.8980i			1.66396i
$144$	22.3988	+	14.5544i	1.86656	+	1.21287i
$145$		0			0
$146$		0			0
$147$			21.7767i			1.79611i
$148$	−6.89843	+	23.2773i	−0.567047	+	1.91338i
$149$	−22.9364			−1.87902			−0.939512	−	0.342516i	$-0.888721\pi$
$149$	−22.9364			−1.87902			−0.939512	+	0.342516i	$0.888721\pi$
$150$	−17.6267	+	13.1606i	−1.43921	+	1.07456i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−4.26491	−	11.5677i	−0.345930	−	0.938260i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−40.5153	−	12.0070i	−3.24382	−	0.961333i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		−	18.8020i		−	1.49109i
$160$	0.823090	+	12.6223i	0.0650709	+	0.997881i
$161$		0			0
$162$	−13.1667	−	17.6348i	−1.03447	−	1.38552i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$	−20.3804			−1.58661
$166$		0			0
$167$			1.50536i			0.116488i	0.998302	+	0.0582442i	$0.0185502\pi$
$167$			1.50536i			0.116488i	−0.998302	+	0.0582442i	$0.981450\pi$
$168$		0			0
$169$	33.1266			2.54820
$170$		0			0
$171$			29.1088i			2.22601i
$172$		0			0
$173$	6.83765			0.519857			0.259928	−	0.965628i	$-0.416301\pi$
$173$	6.83765			0.519857			0.259928	+	0.965628i	$0.416301\pi$
$174$		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	$0.5\pi$
$180$	8.48594	−	28.6341i	0.632505	−	2.13426i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		−	48.4698i		−	3.58299i
$184$		0			0
$185$	27.1436			1.99564
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−11.0451	+	8.24660i	−0.801294	+	0.598271i
$191$		−	26.1534i		−	1.89239i	−0.323592	−	0.946197i	$-0.604891\pi$
$191$		−	26.1534i		−	1.89239i	0.323592	−	0.946197i	$-0.395109\pi$
$192$	−16.1557	−	18.9311i	−1.16593	−	1.36624i
$193$	−21.2818			−1.53190			−0.765950	−	0.642901i	$-0.777731\pi$
$193$	−21.2818			−1.53190			−0.765950	+	0.642901i	$0.777731\pi$
$194$	2.53503	+	3.39528i	0.182004	+	0.243767i
$195$			47.2448i			3.38327i
$196$	3.97800	−	13.4229i	0.284143	−	0.958782i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	22.1712	−	16.5537i	1.57564	−	1.17642i
$199$			8.71780i			0.617988i	0.951064	+	0.308994i	$0.0999924\pi$
$199$			8.71780i			0.617988i	−0.951064	+	0.308994i	$0.900008\pi$
$200$	13.2690	−	4.89218i	0.938260	−	0.345930i
$201$	−49.8434			−3.51568
$202$	−0.734624	−	0.983917i	−0.0516879	−	0.0692281i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	12.6634	−	9.45490i	0.882301	−	0.658754i
$207$		0			0
$208$	22.7799	+	14.8021i	1.57950	+	1.02634i
$209$	−12.7706			−0.883361
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−3.43461	+	11.5894i	−0.235890	+	0.795961i
$213$		0			0
$214$	13.3755	−	9.98658i	0.914331	−	0.682668i
$215$		0			0
$216$	11.1954	+	30.3652i	0.761753	+	2.06609i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	12.5623	+	3.72294i	0.846950	+	0.251001i
$221$		0			0
$222$	−42.7940	+	31.9514i	−2.87215	+	2.14444i
$223$		−	7.72727i		−	0.517456i	−0.965950	−	0.258728i	$-0.916697\pi$
$223$		−	7.72727i		−	0.517456i	0.965950	−	0.258728i	$-0.0833033\pi$
$224$		0			0
$225$	−33.3901			−2.22601
$226$	−11.5121	−	15.4187i	−0.765772	−	1.02564i
$227$			29.9345i			1.98682i	0.114602	+	0.993411i	$0.463441\pi$
$227$			29.9345i			1.98682i	−0.114602	+	0.993411i	$0.536559\pi$
$228$	7.70614	−	26.0028i	0.510352	−	1.72208i
$229$	6.48779			0.428725			0.214362	−	0.976754i	$-0.431233\pi$
$229$	6.48779			0.428725			0.214362	+	0.976754i	$0.431233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−38.3740	−	51.3962i	−2.50859	−	3.35987i
$235$		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	$-0.217137\pi$
$240$	−15.1609	+	23.3321i	−0.978632	+	1.50608i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−2.04448	−	2.73827i	−0.131424	−	0.176023i
$243$		−	14.0860i		−	0.903615i
$244$	−8.85410	+	29.8763i	−0.566826	+	1.91264i
$245$	−15.6525			−1.00000
$246$		0			0
$247$			29.6042i			1.88367i
$248$		0			0
$249$		0			0
$250$	−9.45950	−	12.6696i	−0.598271	−	0.801294i
$251$			26.1534i			1.65079i	0.564557	+	0.825394i	$0.309047\pi$
$251$			26.1534i			1.65079i	−0.564557	+	0.825394i	$0.690953\pi$
$252$		0			0
$253$		0			0
$254$	−12.3955	+	9.25484i	−0.777760	+	0.580700i
$255$		0			0
$256$	6.50000	+	14.6202i	0.406250	+	0.913762i
$257$	3.09902			0.193312			0.0966558	−	0.995318i	$-0.469185\pi$
$257$	3.09902			0.193312			0.0966558	+	0.995318i	$0.469185\pi$
$258$		0			0
$259$		0			0
$260$	8.63034	−	29.1213i	0.535231	−	1.80603i
$261$		0			0
$262$	22.0901	−	16.4932i	1.36473	−	1.01895i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−24.1878	+	8.91786i	−1.48866	+	0.548856i
$265$	13.5143			0.830179
$266$		0			0
$267$		0			0
$268$	30.7230	+	9.10502i	1.87671	+	0.556178i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	28.9934	−	21.6474i	1.76448	−	1.31742i
$271$			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			$\pi$
$278$	−24.8077	+	18.5222i	−1.48787	+	1.11089i
$279$		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
$285$	−30.3218			−1.79611
$286$	22.5485	−	16.8354i	1.33332	−	0.995499i
$287$		0			0
$288$	−2.45816	−	37.6965i	−0.144848	−	2.22129i
$289$	17.0000			1.00000
$290$		0			0
$291$			9.32101i			0.546407i
$292$		0			0
$293$	24.3294			1.42134			0.710670	−	0.703525i	$-0.248392\pi$
$293$	24.3294			1.42134			0.710670	+	0.703525i	$0.248392\pi$
$294$	24.6773	−	18.4249i	1.43921	−	1.07456i
$295$		0			0
$296$	32.2145	−	11.8773i	1.87243	−	0.690351i
$297$	33.5230			1.94520
$298$	19.4061	+	25.9915i	1.12417	+	1.50565i
$299$		0			0
$300$	29.8272	+	8.83955i	1.72208	+	0.510352i
$301$		0			0
$302$		0			0
$303$		−	2.70113i		−	0.155176i
$304$	−9.50000	+	14.6202i	−0.544862	+	0.838525i
$305$	34.8387			1.99486
$306$		0			0
$307$		−	25.7159i		−	1.46768i	−0.679320	−	0.733842i	$-0.737725\pi$
$307$		−	25.7159i		−	1.46768i	0.679320	−	0.733842i	$-0.262275\pi$
$308$		0			0
$309$	34.7646			1.97769
$310$		0			0
$311$			31.3726i			1.77898i	0.456955	+	0.889490i	$0.348940\pi$
$311$			31.3726i			1.77898i	−0.456955	+	0.889490i	$0.651060\pi$
$312$	20.6729	+	56.0709i	1.17037	+	3.17439i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	33.9123			1.90471			0.952353	−	0.304999i	$-0.0986561\pi$
$317$	33.9123			1.90471			0.952353	+	0.304999i	$0.0986561\pi$
$318$	−21.3064	+	15.9080i	−1.19480	+	0.892078i
$319$		0			0
$320$	13.6072	−	11.6122i	0.760665	−	0.649144i
$321$	36.7195			2.04948
$322$		0			0
$323$		0			0
$324$	−8.84363	+	29.8410i	−0.491313	+	1.65783i
$325$	−33.9583			−1.88367
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	17.2435	+	23.0951i	0.949225	+	1.27134i
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−81.0646			−4.44231
$334$	1.70588	−	1.27366i	0.0933414	−	0.0696917i
$335$		−	35.8261i		−	1.95739i
$336$		0			0
$337$	−27.0977			−1.47610			−0.738052	−	0.674744i	$-0.764254\pi$
$337$	−27.0977			−1.47610			−0.738052	+	0.674744i	$0.764254\pi$
$338$	−28.0279	−	37.5391i	−1.52452	−	2.04186i
$339$		−	42.3286i		−	2.29897i
$340$		0			0
$341$		0			0
$342$	32.9861	−	24.6285i	1.78369	−	1.33176i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−5.78522	−	7.74842i	−0.311015	−	0.416558i
$347$		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
$351$		−	77.7113i		−	4.14792i
$352$	16.5382	−	1.07844i	0.881489	−	0.0574811i
$353$		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	$-0.235109\pi$
$360$	−39.6279	+	14.6105i	−2.08858	+	0.770042i
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	7.51733i		−	0.394557i
$364$		0			0
$365$		0			0
$366$	−54.9260	+	41.0095i	−2.87103	+	2.14360i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−22.9658	−	30.7592i	−1.19393	−	1.59909i
$371$		0			0
$372$		0			0
$373$	−6.14663			−0.318260			−0.159130	−	0.987258i	$-0.550869\pi$
$373$	−6.14663			−0.318260			−0.159130	+	0.987258i	$0.550869\pi$
$374$		0			0
$375$		−	34.7815i		−	1.79611i
$376$		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
$381$	−34.0290			−1.74336
$382$	−29.6370	+	22.1280i	−1.51636	+	1.13216i
$383$			39.0001i			1.99281i	0.0847033	+	0.996406i	$0.473006\pi$
$383$			39.0001i			1.99281i	−0.0847033	+	0.996406i	$0.526994\pi$
$384$	−7.78376	+	34.3249i	−0.397213	+	1.75164i
$385$		0			0
$386$	18.0062	+	24.1166i	0.916491	+	1.22750i
$387$		0			0
$388$	1.70269	−	5.74539i	0.0864411	−	0.291678i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$	53.5379	−	39.9731i	2.71099	−	2.02411i
$391$		0			0
$392$	−18.5766	+	6.84906i	−0.938260	+	0.345930i
$393$	60.6436			3.05907
$394$		0			0
$395$		0			0
$396$	−37.5174	−	11.1186i	−1.88532	−	0.558730i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	9.87901	−	7.37598i	0.495190	−	0.369725i
$399$		0			0
$400$	−16.7705	−	10.8972i	−0.838525	−	0.544862i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	42.1717	+	56.4825i	2.10333	+	2.81709i
$403$		0			0
$404$	−0.493422	+	1.66495i	−0.0245487	+	0.0828344i
$405$	34.7976			1.72910
$406$		0			0
$407$		−	35.5646i		−	1.76287i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−21.4286	−	6.35054i	−1.05571	−	0.312869i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−2.49999	−	38.3380i	−0.122572	−	1.87968i
$417$	−68.1040			−3.33507
$418$	10.8050	+	14.4716i	0.528489	+	0.707832i
$419$			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
$424$	16.0390	−	5.91347i	0.778924	−	0.287184i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−22.6336	−	6.70765i	−1.09404	−	0.324227i
$429$	61.9019			2.98865
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	24.9376	−	38.3782i	1.19981	−	1.84647i
$433$	−13.6523			−0.656088			−0.328044	−	0.944662i	$-0.606389\pi$
$433$	−13.6523			−0.656088			−0.328044	+	0.944662i	$0.606389\pi$
$434$		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	$0.5\pi$
$440$	−6.40992	−	17.3855i	−0.305581	−	0.828823i
$441$	46.7462			2.22601
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	72.4146	+	21.4607i	3.43665	+	1.01848i
$445$		0			0
$446$	−8.75654	+	6.53791i	−0.414634	+	0.309579i
$447$			71.3541i			3.37493i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	28.2508	+	37.8377i	1.33176	+	1.78369i
$451$		0			0
$452$	−7.73228	+	26.0910i	−0.363696	+	1.22722i
$453$		0			0
$454$	33.9218	−	25.3271i	1.59203	−	1.18866i
$455$		0			0
$456$	−35.9864	+	13.2679i	−1.68522	+	0.621327i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−5.48921	−	7.35196i	−0.256494	−	0.343535i
$459$		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	$0.5\pi$
$468$	−25.7745	+	86.9709i	−1.19143	+	4.02023i
$469$		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
$477$	−40.3606			−1.84799
$478$	−22.0901	+	16.4932i	−1.01038	+	0.754381i
$479$			43.0918i			1.96891i	0.175630	+	0.984456i	$0.443804\pi$
$479$			43.0918i			1.96891i	−0.175630	+	0.984456i	$0.556196\pi$
$480$	39.2674	−	2.56059i	1.79230	−	0.116875i
$481$	−82.4440			−3.75912
$482$		0			0
$483$		0			0
$484$	−1.37321	+	4.63362i	−0.0624187	+	0.210619i
$485$	−6.69969			−0.304217
$486$	−15.9622	+	11.9179i	−0.724061	+	0.540607i
$487$			1.48091i			0.0671063i	0.999437	+	0.0335531i	$0.0106823\pi$
$487$			1.48091i			0.0671063i	−0.999437	+	0.0335531i	$0.989318\pi$
$488$	41.3472	−	15.2444i	1.87170	−	0.690082i
$489$		0			0
$490$	13.2433	+	17.7374i	0.598271	+	0.801294i
$491$			26.1534i			1.18029i	0.807299	+	0.590143i	$0.200929\pi$
$491$			26.1534i			1.18029i	−0.807299	+	0.590143i	$0.799071\pi$
$492$		0			0
$493$		0			0
$494$	33.5474	−	25.0476i	1.50937	−	1.12694i
$495$			43.7489i			1.96637i
$496$		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
$501$	4.68311			0.209226
$502$	29.6370	−	22.1280i	1.32277	−	0.987619i
$503$		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
$507$		−	103.055i		−	4.57685i
$508$	20.9752	+	6.21617i	0.930623	+	0.275798i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	11.0681	−	19.7357i	0.489144	−	0.872203i
$513$	49.8752			2.20204
$514$	−2.62203	−	3.51181i	−0.115653	−	0.154899i
$515$			24.9878i			1.10110i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	21.2716i		−	0.933719i
$520$	−40.3023	+	14.8591i	−1.76737	+	0.651617i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			12.5440i			0.548512i	0.961657	+	0.274256i	$0.0884316\pi$
$523$			12.5440i			0.548512i	−0.961657	+	0.274256i	$0.911568\pi$
$524$	−37.3802	−	11.0779i	−1.63296	−	0.483942i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	30.5706	+	19.8644i	1.33042	+	0.864486i
$529$	−23.0000			−1.00000
$530$	−11.4343	−	15.3145i	−0.496672	−	0.665217i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			26.3930i			1.14107i
$536$	−15.6764	−	42.5189i	−0.677118	−	1.83654i
$537$		0			0
$538$		0			0
$539$			20.5084i			0.883361i
$540$	−49.0617	−	14.5399i	−2.11128	−	0.625696i
$541$	−37.6485			−1.61864			−0.809318	−	0.587371i	$-0.800163\pi$
$541$	−37.6485			−1.61864			−0.809318	+	0.587371i	$0.800163\pi$
$542$	27.3439	−	20.4159i	1.17452	−	0.876936i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	31.2098i		−	1.33443i	−0.744864	−	0.667216i	$-0.767486\pi$
$547$		−	31.2098i		−	1.33443i	0.744864	−	0.667216i	$-0.232514\pi$
$548$		0			0
$549$	−104.046			−4.44058
$550$	−16.6001	+	12.3942i	−0.707832	+	0.528489i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		−	84.4426i		−	3.58439i
$556$	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
$563$		−	6.32213i		−	0.266446i	−0.991086	−	0.133223i	$-0.957467\pi$
$563$		−	6.32213i		−	0.266446i	0.991086	−	0.133223i	$-0.0425327\pi$
$564$		0			0
$565$	30.4246			1.27997
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	25.6548	+	34.3607i	1.07456	+	1.43921i
$571$			39.4704i			1.65178i	0.563829	+	0.825891i	$0.309328\pi$
$571$			39.4704i			1.65178i	−0.563829	+	0.825891i	$0.690672\pi$
$572$	−38.1558	−	11.3078i	−1.59537	−	0.472802i
$573$	−81.3620			−3.39894
$574$		0			0
$575$		0			0
$576$	−40.6379	+	34.6800i	−1.69325	+	1.44500i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−14.3834	−	19.2644i	−0.598271	−	0.801294i
$579$			66.2067i			2.75146i
$580$		0			0
$581$		0			0
$582$	10.5626	−	7.88635i	0.437832	−	0.326900i
$583$		−	17.7070i		−	0.733348i
$584$		0			0
$585$	101.417			4.19306
$586$	−20.5847	−	27.5701i	−0.850347	−	1.13891i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−41.7581	−	12.3754i	−1.72208	−	0.510352i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−40.7155	−	26.4564i	−1.67340	−	1.08735i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−28.3632	−	37.9882i	−1.16376	−	1.55868i
$595$		0			0
$596$	13.0344	−	43.9820i	0.533912	−	1.80157i
$597$	27.1207			1.10997
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−15.2194	−	41.2792i	−0.621327	−	1.68522i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			106.995i			4.35716i
$604$		0			0
$605$	5.40325			0.219673
$606$	−3.06092	+	2.28538i	−0.124341	+	0.0928372i
$607$		−	43.2187i		−	1.75419i	−0.480314	−	0.877097i	$-0.659477\pi$
$607$		−	43.2187i		−	1.75419i	0.480314	−	0.877097i	$-0.340523\pi$
$608$	24.6054	−	1.60450i	0.997881	−	0.0650709i
$609$		0			0
$610$	−29.4765	−	39.4793i	−1.19347	−	1.59847i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−29.1413	+	21.7578i	−1.17605	+	0.878073i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−29.4137	−	39.3953i	−1.18319	−	1.58471i
$619$			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
$622$	35.5515	−	26.5439i	1.42549	−	1.06431i
$623$		0			0
$624$	46.0486	−	70.8672i	1.84342	−	2.83696i
$625$	25.0000			1.00000
$626$		0			0
$627$			39.7287i			1.58661i
$628$		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
$634$	−28.6926	−	38.4294i	−1.13953	−	1.52623i
$635$		−	24.4591i		−	0.970630i
$636$	36.0540	+	10.6849i	1.42963	+	0.423684i
$637$	47.5416			1.88367
$638$		0			0
$639$		0			0
$640$	−24.6718	−	5.59475i	−0.975239	−	0.221152i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−31.0678	−	41.6106i	−1.22615	−	1.64224i
$643$		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	$0.5\pi$
$648$	41.2983	−	15.2264i	1.62235	−	0.598148i
$649$		0			0
$650$	28.7316	+	38.4816i	1.12694	+	1.50937i
$651$		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	$0.5\pi$
$660$	11.5819	−	39.0807i	0.450825	−	1.52122i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	68.5874	+	91.8624i	2.65771	+	3.55960i
$667$		0			0
$668$	−2.88663	−	0.855476i	−0.111687	−	0.0330994i
$669$	−24.0392			−0.929408
$670$	−40.5981	+	30.3118i	−1.56844	+	1.17105i
$671$		−	45.6470i		−	1.76218i
$672$		0			0
$673$	27.2356			1.04986			0.524928	−	0.851147i	$-0.324092\pi$
$673$	27.2356			1.04986			0.524928	+	0.851147i	$0.324092\pi$
$674$	22.9269	+	30.7071i	0.883110	+	1.18279i
$675$			57.2108i			2.20204i
$676$	−18.8254	+	63.5225i	−0.724054	+	2.44317i
$677$	−12.2418			−0.470492			−0.235246	−	0.971936i	$-0.575590\pi$
$677$	−12.2418			−0.470492			−0.235246	+	0.971936i	$0.575590\pi$
$678$	−47.9668	+	35.8135i	−1.84215	+	1.37541i
$679$		0			0
$680$		0			0
$681$	93.1248			3.56855
$682$		0			0
$683$		−	43.6536i		−	1.67036i	−0.549979	−	0.835179i	$-0.685364\pi$
$683$		−	43.6536i		−	1.67036i	0.549979	−	0.835179i	$-0.314636\pi$
$684$	−55.8180	−	16.5421i	−2.13426	−	0.632505i
$685$		0			0
$686$		0			0
$687$		−	20.1832i		−	0.770037i
$688$		0			0
$689$	−41.0474			−1.56378
$690$		0			0
$691$		−	52.5727i		−	1.99996i	−0.00630823	−	0.999980i	$-0.502008\pi$
$691$		−	52.5727i		−	1.99996i	0.00630823	−	0.999980i	$-0.497992\pi$
$692$	−3.88574	+	13.1116i	−0.147714	+	0.498429i
$693$		0			0
$694$		0			0
$695$		−	48.9513i		−	1.85683i
$696$		0			0
$697$		0			0
$698$	11.3514	+	15.2035i	0.429657	+	0.575460i
$699$		0			0
$700$		0			0
$701$	21.1999			0.800709			0.400354	−	0.916360i	$-0.368887\pi$
$701$	21.1999			0.800709			0.400354	+	0.916360i	$0.368887\pi$
$702$	−88.0624	+	65.7502i	−3.32370	+	2.48158i
$703$		−	52.9127i		−	1.99564i
$704$	−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
$715$			44.4934i			1.66396i
$716$		0			0
$717$	−60.6436			−2.26478
$718$	−28.9114	+	21.5862i	−1.07897	+	0.805590i
$719$		−	13.7940i		−	0.514429i	−0.966354	−	0.257214i	$-0.917195\pi$
$719$		−	13.7940i		−	0.514429i	0.966354	−	0.257214i	$-0.0828047\pi$
$720$	50.0852	+	32.5447i	1.86656	+	1.21287i
$721$		0			0
$722$	16.0756	+	21.5308i	0.598271	+	0.801294i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−8.51864	+	6.36029i	−0.316156	+	0.236052i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	2.86503			0.106112
$730$		0			0
$731$		0			0
$732$	92.9439	+	27.5447i	3.43530	+	1.01808i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$			48.6941i			1.79611i
$736$		0			0
$737$	−46.9406			−1.72908
$738$		0			0
$739$		−	8.71780i		−	0.320689i	−0.987061	−	0.160345i	$-0.948739\pi$
$739$		−	8.71780i		−	0.320689i	0.987061	−	0.160345i	$-0.0512606\pi$
$740$	−15.4254	+	52.0497i	−0.567047	+	1.91338i
$741$	92.0971			3.38327
$742$		0			0
$743$			54.4918i			1.99911i	0.0298377	+	0.999555i	$0.490501\pi$
$743$			54.4918i			1.99911i	−0.0298377	+	0.999555i	$0.509499\pi$
$744$		0			0
$745$	−51.2874			−1.87902
$746$	5.20056	+	6.96536i	0.190406	+	0.255020i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−39.4144	+	29.4281i	−1.43921	+	1.07456i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	81.3620			2.96499
$754$		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
$760$	−9.53662	−	25.8661i	−0.345930	−	0.938260i
$761$	−9.29755			−0.337036			−0.168518	−	0.985699i	$-0.553898\pi$
$761$	−9.29755			−0.337036			−0.168518	+	0.985699i	$0.553898\pi$
$762$	28.7914	+	38.5617i	1.04300	+	1.39694i
$763$		0			0
$764$	50.1508	+	14.8626i	1.81439	+	0.537710i
$765$		0			0
$766$	44.1949	−	32.9973i	1.59683	−	1.19224i
$767$		0			0
$768$	45.4827	−	20.2212i	1.64122	−	0.729669i
$769$	29.2192			1.05367			0.526836	−	0.849967i	$-0.323378\pi$
$769$	29.2192			1.05367			0.526836	+	0.849967i	$0.323378\pi$
$770$		0			0
$771$		−	9.64091i		−	0.347209i
$772$	12.0942	−	40.8093i	0.435279	−	1.46876i
$773$	47.4496			1.70665			0.853323	−	0.521383i	$-0.174584\pi$
$773$	47.4496			1.70665			0.853323	+	0.521383i	$0.174584\pi$
$774$		0			0
$775$		0			0
$776$	−7.95129	+	2.93158i	−0.285435	+	0.105238i
$777$		0			0
$778$	−5.07650	−	6.79920i	−0.182001	−	0.243763i
$779$		0			0
$780$	−90.5950	−	26.8486i	−3.24382	−	0.961333i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	23.4787	+	15.2561i	0.838525	+	0.544862i
$785$		0			0
$786$	−51.3096	−	68.7214i	−1.83015	−	2.45121i
$787$			49.8755i			1.77787i	0.458035	+	0.888934i	$0.348553\pi$
$787$			49.8755i			1.77787i	−0.458035	+	0.888934i	$0.651447\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	19.1432	+	51.9219i	0.680225	+	1.84497i
$793$	−105.817			−3.75765
$794$		0			0
$795$		−	42.0425i		−	1.49109i
$796$	−16.7169	−	4.95420i	−0.592516	−	0.175597i
$797$	48.5046			1.71812			0.859061	−	0.511873i	$-0.171048\pi$
$797$	48.5046			1.71812			0.859061	+	0.511873i	$0.171048\pi$
$798$		0			0
$799$		0			0
$800$	1.84048	+	28.2243i	0.0650709	+	0.997881i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	28.3253	−	95.5779i	0.998956	−	3.37077i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	2.30420	−	0.849541i	0.0810615	−	0.0298868i
$809$	−22.3607			−0.786160			−0.393080	−	0.919504i	$-0.628590\pi$
$809$	−22.3607			−0.786160			−0.393080	+	0.919504i	$0.628590\pi$
$810$	−29.4416	−	39.4326i	−1.03447	−	1.38552i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	75.0668			2.63271
$814$	−40.3018	+	30.0906i	−1.41258	+	1.05468i
$815$		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	$0.5\pi$
$824$	10.9339	+	29.6560i	0.380902	+	1.03311i
$825$	−45.5720			−1.58661
$826$		0			0
$827$		−	12.3436i		−	0.429228i	−0.976699	−	0.214614i	$-0.931151\pi$
$827$		−	12.3436i		−	0.429228i	0.976699	−	0.214614i	$-0.0688494\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−41.3294	+	35.2701i	−1.43284	+	1.22277i
$833$		0			0
$834$	57.6217	+	77.1755i	1.99528	+	2.67237i
$835$			3.36609i			0.116488i
$836$	7.25735	−	24.4884i	0.251001	−	0.846950i
$837$		0			0
$838$	22.0901	−	16.4932i	0.763091	−	0.569748i
$839$		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
$845$	74.0734			2.54820
$846$		0			0
$847$		0			0
$848$	−20.2715	−	13.1721i	−0.696126	−	0.452333i
$849$		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
$855$			65.0893i			2.22601i
$856$	11.5488	+	31.3236i	0.394729	+	1.07062i
$857$	−57.6474			−1.96920			−0.984599	−	0.174825i	$-0.944064\pi$
$857$	−57.6474			−1.96920			−0.984599	+	0.174825i	$0.944064\pi$
$858$	−52.3741	−	70.1472i	−1.78802	−	2.39479i
$859$		−	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
$863$			38.8368i			1.32202i	0.750377	+	0.661010i	$0.229872\pi$
$863$			38.8368i			1.32202i	−0.750377	+	0.661010i	$0.770128\pi$
$864$	−64.5894	+	4.21182i	−2.19738	+	0.143289i
$865$	15.2894			0.519857
$866$	11.5510	+	15.4708i	0.392518	+	0.525719i
$867$		−	52.8862i		−	1.79611i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			108.815i			3.68707i
$872$		0			0
$873$	20.0086			0.677190
$874$		0			0
$875$		0			0
$876$		0			0
$877$	42.4094			1.43206			0.716032	−	0.698068i	$-0.245956\pi$
$877$	42.4094			1.43206			0.716032	+	0.698068i	$0.245956\pi$
$878$		0			0
$879$		−	75.6877i		−	2.55288i
$880$	−14.2780	+	21.9733i	−0.481310	+	0.740721i
$881$	58.6434			1.97575			0.987874	−	0.155261i	$-0.0496217\pi$
$881$	58.6434			1.97575			0.987874	+	0.155261i	$0.0496217\pi$
$882$	−39.5512	−	52.9728i	−1.33176	−	1.78369i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			57.1313i			1.91828i	0.282929	+	0.959141i	$0.408694\pi$
$887$			57.1313i			1.91828i	−0.282929	+	0.959141i	$0.591306\pi$
$888$	−36.9496	−	100.218i	−1.23995	−	3.36309i
$889$		0			0
$890$		0			0
$891$		−	45.5930i		−	1.52742i
$892$	14.8175	+	4.39130i	0.496128	+	0.147032i
$893$		0			0
$894$	80.8585	−	60.3715i	2.70431	−	2.01912i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	18.9751	−	64.0277i	0.632505	−	2.13426i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	36.1085	−	13.3129i	1.20095	−	0.442781i
$905$		0			0
$906$		0			0
$907$			57.7597i			1.91788i	0.283610	+	0.958940i	$0.408468\pi$
$907$			57.7597i			1.91788i	−0.283610	+	0.958940i	$0.591532\pi$
$908$	−57.4013	−	17.0114i	−1.90493	−	0.564542i
$909$	−5.79829			−0.192317
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	45.4827	+	29.5540i	1.50608	+	0.978632i
$913$		0			0
$914$		0			0
$915$		−	108.382i		−	3.58299i
$916$	−3.68692	+	12.4407i	−0.121819	+	0.411054i
$917$		0			0
$918$		0			0
$919$			58.4808i			1.92910i	0.263896	+	0.964551i	$0.414993\pi$
$919$			58.4808i			1.92910i	−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	−80.0009			−2.63612
$922$	15.2295	+	20.3976i	0.501557	+	0.671759i
$923$		0			0
$924$		0			0
$925$	60.6950			1.99564
$926$		0			0
$927$		−	74.6263i		−	2.45105i
$928$		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
$933$	97.5988			3.19524
$934$		0			0
$935$		0			0
$936$	120.363	−	44.3769i	3.93418	−	1.45050i
$937$		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
$950$	−24.6975	+	18.4400i	−0.801294	+	0.598271i
$951$		−	105.500i		−	3.42106i
$952$		0			0
$953$	−39.3618			−1.27505			−0.637527	−	0.770428i	$-0.720043\pi$
$953$	−39.3618			−1.27505			−0.637527	+	0.770428i	$0.720043\pi$
$954$	34.1484	+	45.7367i	1.10560	+	1.48078i
$955$		−	58.4808i		−	1.89239i
$956$	37.3802	+	11.0779i	1.20896	+	0.358286i
$957$		0			0
$958$	48.8316	−	36.4592i	1.57768	−	1.17794i
$959$		0			0
$960$	−36.1251	−	42.3313i	−1.16593	−	1.36624i
$961$	−31.0000			−1.00000
$962$	69.7545	+	93.4256i	2.24898	+	3.01216i
$963$		−	78.8227i		−	2.54003i
$964$		0			0
$965$	−47.5876			−1.53190
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	6.41267	−	2.36430i	0.206111	−	0.0759916i
$969$		0			0
$970$	5.66849	+	7.59209i	0.182004	+	0.243767i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	27.0107	+	8.00486i	0.866370	+	0.256756i
$973$		0			0
$974$	1.67816	−	1.25297i	0.0537718	−	0.0401478i
$975$			105.643i			3.38327i
$976$	−52.2581	−	33.9566i	−1.67274	−	1.08692i
$977$	−54.3563			−1.73901			−0.869506	−	0.493923i	$-0.835562\pi$
$977$	−54.3563			−1.73901			−0.869506	+	0.493923i	$0.835562\pi$
$978$		0			0
$979$		0			0
$980$	8.89508	−	30.0146i	0.284143	−	0.958782i
$981$		0			0
$982$	29.6370	−	22.1280i	0.945756	−	0.706131i
$983$		−	44.4755i		−	1.41855i	−0.704933	−	0.709274i	$-0.749023\pi$
$983$		−	44.4755i		−	1.41855i	0.704933	−	0.709274i	$-0.250977\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−56.7679	−	16.8236i	−1.80603	−	0.535231i
$989$		0			0
$990$	49.5763	−	37.0152i	1.57564	−	1.17642i
$991$		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			$\pi$
$998$	4.88753	−	3.64918i	0.154712	−	0.115513i
$999$			138.896i			4.39449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.d.a.379.5	✓	16
4.3	odd	2	inner	380.2.d.a.379.6	yes	16
5.4	even	2	inner	380.2.d.a.379.12	yes	16
19.18	odd	2	inner	380.2.d.a.379.12	yes	16
20.19	odd	2	inner	380.2.d.a.379.11	yes	16
76.75	even	2	inner	380.2.d.a.379.11	yes	16
95.94	odd	2	CM	380.2.d.a.379.5	✓	16
380.379	even	2	inner	380.2.d.a.379.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.d.a.379.5	✓	16	1.1	even	1	trivial
380.2.d.a.379.5	✓	16	95.94	odd	2	CM
380.2.d.a.379.6	yes	16	4.3	odd	2	inner
380.2.d.a.379.6	yes	16	380.379	even	2	inner
380.2.d.a.379.11	yes	16	20.19	odd	2	inner
380.2.d.a.379.11	yes	16	76.75	even	2	inner
380.2.d.a.379.12	yes	16	5.4	even	2	inner
380.2.d.a.379.12	yes	16	19.18	odd	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)

\(f(q)\)	\(=\)	\(q+(-0.846083 - 1.13320i) q^{2} -3.11095i q^{3} +(-0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(-3.52533 + 2.63212i) q^{6} +(2.65380 - 0.978437i) q^{8} -6.67802 q^{9} +O(q^{10})\)	\(q+(-0.846083 - 1.13320i) q^{2} -3.11095i q^{3} +(-0.568286 + 1.91756i) q^{4} +2.23607 q^{5} +(-3.52533 + 2.63212i) q^{6} +(2.65380 - 0.978437i) q^{8} -6.67802 q^{9} +(-1.89190 - 2.53391i) q^{10} -2.92978i q^{11} +(5.96545 + 1.76791i) q^{12} -6.79166 q^{13} -6.95630i q^{15} +(-3.35410 - 2.17945i) q^{16} +(5.65017 + 7.56754i) q^{18} -4.35890i q^{19} +(-1.27073 + 4.28780i) q^{20} +(-3.32002 + 2.47884i) q^{22} +(-3.04387 - 8.25585i) q^{24} +5.00000 q^{25} +(5.74631 + 7.69631i) q^{26} +11.4422i q^{27} +(-7.88288 + 5.88561i) q^{30} +(0.368097 + 5.64487i) q^{32} -9.11440 q^{33} +(3.79503 - 12.8055i) q^{36} +12.1390 q^{37} +(-4.93951 + 3.68799i) q^{38} +21.1285i q^{39} +(5.93408 - 2.18785i) q^{40} +(5.61803 + 1.66495i) q^{44} -14.9325 q^{45} +(-6.78016 + 10.4345i) q^{48} -7.00000 q^{49} +(-4.23042 - 5.66600i) q^{50} +(3.85961 - 13.0234i) q^{52} +6.04380 q^{53} +(12.9663 - 9.68102i) q^{54} -6.55118i q^{55} -13.5603 q^{57} +(13.3392 + 3.95317i) q^{60} +15.5804 q^{61} +(6.08532 - 5.19315i) q^{64} -15.1866 q^{65} +(7.71154 + 10.3284i) q^{66} -16.0219i q^{67} +(-17.7222 + 6.53403i) q^{72} +(-10.2706 - 13.7559i) q^{74} -15.5548i q^{75} +(8.35847 + 2.47710i) q^{76} +(23.9429 - 17.8765i) q^{78} +(-7.50000 - 4.87340i) q^{80} +15.5619 q^{81} +(-2.86660 - 7.77505i) q^{88} +(12.6342 + 16.9215i) q^{90} -9.74679i q^{95} +(17.5609 - 1.14513i) q^{96} -2.99619 q^{97} +(5.92258 + 7.93240i) q^{98} +19.5651i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 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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