LMFDB - Embedded newform 380.2.d.a.379.16

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			379.16
Root			$1.39956 + 0.203022i$ of defining polynomial
Character	$\chi$	$=$	380.379
Dual form			380.2.d.a.379.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.39956 + 0.203022i) q^{2} -3.27727i q^{3} +(1.91756 + 0.568286i) q^{4} -2.23607 q^{5} +(0.665359 - 4.58675i) q^{6} +(2.56838 + 1.18466i) q^{8} -7.74048 q^{9} +O(q^{10})$	\(q+(1.39956 + 0.203022i) q^{2} -3.27727i q^{3} +(1.91756 + 0.568286i) q^{4} -2.23607 q^{5} +(0.665359 - 4.58675i) q^{6} +(2.56838 + 1.18466i) q^{8} -7.74048 q^{9} +(-3.12952 - 0.453972i) q^{10} -5.95117i q^{11} +(1.86243 - 6.28437i) q^{12} +3.08876 q^{13} +7.32819i q^{15} +(3.35410 + 2.17945i) q^{16} +(-10.8333 - 1.57149i) q^{18} +4.35890i q^{19} +(-4.28780 - 1.27073i) q^{20} +(1.20822 - 8.32905i) q^{22} +(3.88245 - 8.41727i) q^{24} +5.00000 q^{25} +(4.32291 + 0.627087i) q^{26} +15.5358i q^{27} +(-1.48779 + 10.2563i) q^{30} +(4.25181 + 3.73124i) q^{32} -19.5036 q^{33} +(-14.8429 - 4.39881i) q^{36} +9.15124 q^{37} +(-0.884954 + 6.10056i) q^{38} -10.1227i q^{39} +(-5.74307 - 2.64898i) q^{40} +(3.38197 - 11.4117i) q^{44} +17.3082 q^{45} +(7.14264 - 10.9923i) q^{48} -7.00000 q^{49} +(6.99782 + 1.01511i) q^{50} +(5.92289 + 1.75530i) q^{52} +13.6404 q^{53} +(-3.15412 + 21.7434i) q^{54} +13.3072i q^{55} +14.2853 q^{57} +(-4.16451 + 14.0523i) q^{60} -1.11908 q^{61} +(5.19315 + 6.08532i) q^{64} -6.90667 q^{65} +(-27.2965 - 3.95966i) q^{66} +8.95237i q^{67} +(-19.8805 - 9.16985i) q^{72} +(12.8077 + 1.85791i) q^{74} -16.3863i q^{75} +(-2.47710 + 8.35847i) q^{76} +(2.05513 - 14.1673i) q^{78} +(-7.50000 - 4.87340i) q^{80} +27.6936 q^{81} +(7.05012 - 15.2849i) q^{88} +(24.2240 + 3.51396i) q^{90} -9.74679i q^{95} +(12.2283 - 13.9343i) q^{96} -15.8849 q^{97} +(-9.79695 - 1.42116i) q^{98} +46.0649i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.39956	+	0.203022i	0.989642	+	0.143559i
$2$	1.39956	+	0.203022i	0.989642	+	0.143559i
$3$		−	3.27727i		−	1.89213i	−0.323976	−	0.946065i	$-0.605020\pi$
$3$		−	3.27727i		−	1.89213i	0.323976	−	0.946065i	$-0.394980\pi$
$4$	1.91756	+	0.568286i	0.958782	+	0.284143i
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$	0.665359	−	4.58675i	0.271632	−	1.87253i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	2.56838	+	1.18466i	0.908060	+	0.418841i
$9$	−7.74048			−2.58016
$10$	−3.12952	−	0.453972i	−0.989642	−	0.143559i
$11$		−	5.95117i		−	1.79434i	−0.441680	−	0.897172i	$-0.645618\pi$
$11$		−	5.95117i		−	1.79434i	0.441680	−	0.897172i	$-0.354382\pi$
$12$	1.86243	−	6.28437i	0.537636	−	1.81414i
$13$	3.08876			0.856667			0.428333	−	0.903621i	$-0.359101\pi$
$13$	3.08876			0.856667			0.428333	+	0.903621i	$0.359101\pi$
$14$		0			0
$15$			7.32819i			1.89213i
$16$	3.35410	+	2.17945i	0.838525	+	0.544862i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−10.8333	−	1.57149i	−2.55343	−	0.370404i
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$	−4.28780	−	1.27073i	−0.958782	−	0.284143i
$21$		0			0
$22$	1.20822	−	8.32905i	0.257593	−	1.77576i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	3.88245	−	8.41727i	0.792502	−	1.71817i
$25$	5.00000			1.00000
$26$	4.32291	+	0.627087i	0.847793	+	0.122982i
$27$			15.5358i			2.98987i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−1.48779	+	10.2563i	−0.271632	+	1.87253i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	4.25181	+	3.73124i	0.751620	+	0.659596i
$33$	−19.5036			−3.39514
$34$		0			0
$35$		0			0
$36$	−14.8429	−	4.39881i	−2.47381	−	0.733134i
$37$	9.15124			1.50445			0.752227	−	0.658904i	$-0.228980\pi$
$37$	9.15124			1.50445			0.752227	+	0.658904i	$0.228980\pi$
$38$	−0.884954	+	6.10056i	−0.143559	+	0.989642i
$39$		−	10.1227i		−	1.62093i
$40$	−5.74307	−	2.64898i	−0.908060	−	0.418841i
$41$		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
$45$	17.3082			2.58016
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	7.14264	−	10.9923i	1.03095	−	1.58660i
$49$	−7.00000			−1.00000
$50$	6.99782	+	1.01511i	0.989642	+	0.143559i
$51$		0			0
$52$	5.92289	+	1.75530i	0.821357	+	0.243416i
$53$	13.6404			1.87365			0.936825	−	0.349799i	$-0.113750\pi$
$53$	13.6404			1.87365			0.936825	+	0.349799i	$0.113750\pi$
$54$	−3.15412	+	21.7434i	−0.429221	+	2.95890i
$55$			13.3072i			1.79434i
$56$		0			0
$57$	14.2853			1.89213
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−4.16451	+	14.0523i	−0.537636	+	1.81414i
$61$	−1.11908			−0.143283			−0.0716414	−	0.997430i	$-0.522824\pi$
$61$	−1.11908			−0.143283			−0.0716414	+	0.997430i	$0.522824\pi$
$62$		0			0
$63$		0			0
$64$	5.19315	+	6.08532i	0.649144	+	0.760665i
$65$	−6.90667			−0.856667
$66$	−27.2965	−	3.95966i	−3.35997	−	0.487401i
$67$			8.95237i			1.09371i	0.837229	+	0.546853i	$0.184174\pi$
$67$			8.95237i			1.09371i	−0.837229	+	0.546853i	$0.815826\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−19.8805	−	9.16985i	−2.34294	−	1.08068i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	12.8077	+	1.85791i	1.48887	+	0.215977i
$75$		−	16.3863i		−	1.89213i
$76$	−2.47710	+	8.35847i	−0.284143	+	0.958782i
$77$		0			0
$78$	2.05513	−	14.1673i	0.232698	−	1.60414i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−7.50000	−	4.87340i	−0.838525	−	0.544862i
$81$	27.6936			3.07706
$82$		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
$88$	7.05012	−	15.2849i	0.751545	−	1.62937i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	24.2240	+	3.51396i	2.55343	+	0.370404i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	9.74679i		−	1.00000i
$96$	12.2283	−	13.9343i	1.24804	−	1.42216i
$97$	−15.8849			−1.61287			−0.806436	−	0.591322i	$-0.798606\pi$
$97$	−15.8849			−1.61287			−0.806436	+	0.591322i	$0.798606\pi$
$98$	−9.79695	−	1.42116i	−0.989642	−	0.143559i
$99$			46.0649i			4.62970i
$100$	9.58782	+	2.84143i	0.958782	+	0.284143i
$101$	−20.0810			−1.99813			−0.999067	−	0.0431977i	$-0.986245\pi$
$101$	−20.0810			−1.99813			−0.999067	+	0.0431977i	$0.986245\pi$
$102$		0			0
$103$			4.07983i			0.401998i	0.979591	+	0.200999i	$0.0644188\pi$
$103$			4.07983i			0.401998i	−0.979591	+	0.200999i	$0.935581\pi$
$104$	7.93310	+	3.65913i	0.777904	+	0.358807i
$105$		0			0
$106$	19.0906	+	2.76930i	1.85424	+	0.268978i
$107$		−	20.3604i		−	1.96831i	−0.177300	−	0.984157i	$-0.556736\pi$
$107$		−	20.3604i		−	1.96831i	0.177300	−	0.984157i	$-0.443264\pi$
$108$	−8.82879	+	29.7909i	−0.849550	+	2.86663i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−2.70166	+	18.6243i	−0.257593	+	1.77576i
$111$		−	29.9910i		−	2.84662i
$112$		0			0
$113$	1.93025			0.181583			0.0907914	−	0.995870i	$-0.471060\pi$
$113$	1.93025			0.181583			0.0907914	+	0.995870i	$0.471060\pi$
$114$	19.9932	+	2.90023i	1.87253	+	0.271632i
$115$		0			0
$116$		0			0
$117$	−23.9085			−2.21034
$118$		0			0
$119$		0			0
$120$	−8.68143	+	18.8216i	−0.792502	+	1.71817i
$121$	−24.4164			−2.21967
$122$	−1.56622	−	0.227197i	−0.141799	−	0.0205695i
$123$		0			0
$124$		0			0
$125$	−11.1803			−1.00000
$126$		0			0
$127$			6.20003i			0.550163i	0.961421	+	0.275082i	$0.0887049\pi$
$127$			6.20003i			0.550163i	−0.961421	+	0.275082i	$0.911295\pi$
$128$	6.03270	+	9.57113i	0.533220	+	0.845976i
$129$		0			0
$130$	−9.66633	−	1.40221i	−0.847793	−	0.122982i
$131$			19.4936i			1.70316i	0.524222	+	0.851581i	$0.324356\pi$
$131$			19.4936i			1.70316i	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−37.3993	−	11.0836i	−3.25519	−	0.964704i
$133$		0			0
$134$	−1.81753	+	12.5294i	−0.157011	+	1.08238i
$135$		−	34.7391i		−	2.98987i
$136$		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	$0.121172\pi$
$139$			8.76093i			0.743092i	−0.928414	+	0.371546i	$0.878828\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	18.3817i		−	1.53716i
$144$	−25.9624	−	16.8700i	−2.16353	−	1.40583i
$145$		0			0
$146$		0			0
$147$			22.9409i			1.89213i
$148$	17.5481	+	5.20052i	1.44244	+	0.427480i
$149$	−8.36188			−0.685032			−0.342516	−	0.939512i	$-0.611279\pi$
$149$	−8.36188			−0.685032			−0.342516	+	0.939512i	$0.611279\pi$
$150$	3.32679	−	22.9337i	0.271632	−	1.87253i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−5.16382	+	11.1953i	−0.418841	+	0.908060i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	5.75258	−	19.4109i	0.460575	−	1.55411i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		−	44.7031i		−	3.54519i
$160$	−9.50733	−	8.34330i	−0.751620	−	0.659596i
$161$		0			0
$162$	38.7590	+	5.62242i	3.04519	+	0.441739i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$	43.6113			3.39514
$166$		0			0
$167$			19.3091i			1.49418i	0.664721	+	0.747091i	$0.268550\pi$
$167$			19.3091i			1.49418i	−0.664721	+	0.747091i	$0.731450\pi$
$168$		0			0
$169$	−3.45959			−0.266122
$170$		0			0
$171$		−	33.7400i		−	2.58016i
$172$		0			0
$173$	13.1268			0.998010			0.499005	−	0.866599i	$-0.333699\pi$
$173$	13.1268			0.998010			0.499005	+	0.866599i	$0.333699\pi$
$174$		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	$0.5\pi$
$180$	33.1896	+	9.83603i	2.47381	+	0.733134i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$			3.66751i			0.271110i
$184$		0			0
$185$	−20.4628			−1.50445
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.97882	−	13.6413i	0.143559	−	0.989642i
$191$			26.1534i			1.89239i	0.323592	+	0.946197i	$0.395109\pi$
$191$			26.1534i			1.89239i	−0.323592	+	0.946197i	$0.604891\pi$
$192$	19.9432	−	17.0194i	1.43928	−	1.22827i
$193$	−2.41753			−0.174018			−0.0870089	−	0.996208i	$-0.527731\pi$
$193$	−2.41753			−0.174018			−0.0870089	+	0.996208i	$0.527731\pi$
$194$	−22.2320	−	3.22500i	−1.59617	−	0.231541i
$195$			22.6350i			1.62093i
$196$	−13.4229	−	3.97800i	−0.958782	−	0.284143i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−9.35220	+	64.4708i	−0.664632	+	4.58174i
$199$		−	8.71780i		−	0.617988i	−0.951064	−	0.308994i	$-0.900008\pi$
$199$		−	8.71780i		−	0.617988i	0.951064	−	0.308994i	$-0.0999924\pi$
$200$	12.8419	+	5.92331i	0.908060	+	0.418841i
$201$	29.3393			2.06944
$202$	−28.1046	−	4.07689i	−1.97744	−	0.286849i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.828298	+	5.70999i	−0.0577102	+	0.397834i
$207$		0			0
$208$	10.3600	+	6.73179i	0.718337	+	0.466766i
$209$	25.9405			1.79434
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	26.1563	+	7.75163i	1.79642	+	0.532384i
$213$		0			0
$214$	4.13362	−	28.4957i	0.282568	−	1.94793i
$215$		0			0
$216$	−18.4047	+	39.9019i	−1.25228	+	2.71498i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−7.56231	+	25.5174i	−0.509851	+	1.72039i
$221$		0			0
$222$	6.08885	−	41.9744i	0.408657	−	2.81714i
$223$		−	25.8636i		−	1.73196i	−0.500082	−	0.865978i	$-0.666697\pi$
$223$		−	25.8636i		−	1.73196i	0.500082	−	0.865978i	$-0.333303\pi$
$224$		0			0
$225$	−38.7024			−2.58016
$226$	2.70151	+	0.391884i	0.179702	+	0.0260677i
$227$		−	23.6088i		−	1.56697i	−0.621412	−	0.783484i	$-0.713441\pi$
$227$		−	23.6088i		−	1.56697i	0.621412	−	0.783484i	$-0.286559\pi$
$228$	27.3929	+	8.11812i	1.81414	+	0.537636i
$229$	29.5619			1.95351			0.976754	−	0.214362i	$-0.0687674\pi$
$229$	29.5619			1.95351			0.976754	+	0.214362i	$0.0687674\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−33.4614	−	4.85395i	−2.18744	−	0.317313i
$235$		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	$-0.217137\pi$
$240$	−15.9714	+	24.5795i	−1.03095	+	1.58660i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−34.1723	−	4.95708i	−2.19668	−	0.318653i
$243$		−	44.1518i		−	2.83234i
$244$	−2.14590	−	0.635955i	−0.137377	−	0.0407128i
$245$	15.6525			1.00000
$246$		0			0
$247$			13.4636i			0.856667i
$248$		0			0
$249$		0			0
$250$	−15.6476	−	2.26986i	−0.989642	−	0.143559i
$251$		−	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
$254$	−1.25874	+	8.67734i	−0.0789806	+	0.544465i
$255$		0			0
$256$	6.50000	+	14.6202i	0.406250	+	0.913762i
$257$	−20.3741			−1.27090			−0.635450	−	0.772142i	$-0.719185\pi$
$257$	−20.3741			−1.27090			−0.635450	+	0.772142i	$0.719185\pi$
$258$		0			0
$259$		0			0
$260$	−13.2440	−	3.92496i	−0.821357	−	0.243416i
$261$		0			0
$262$	−3.95764	+	27.2825i	−0.244504	+	1.68552i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−50.0926	−	23.1051i	−3.08299	−	1.42202i
$265$	−30.5008			−1.87365
$266$		0			0
$267$		0			0
$268$	−5.08751	+	17.1667i	−0.310769	+	1.04863i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	7.05282	−	48.6197i	0.429221	−	2.95890i
$271$			22.3998i			1.36069i	0.732892	+	0.680345i	$0.238170\pi$
$271$			22.3998i			1.36069i	−0.732892	+	0.680345i	$0.761830\pi$
$272$		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			$\pi$
$278$	−1.77866	+	12.2615i	−0.106677	+	0.735395i
$279$		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
$285$	−31.9428			−1.89213
$286$	3.73190	−	25.7264i	0.220672	−	1.52123i
$287$		0			0
$288$	−32.9110	−	28.8816i	−1.93930	−	1.70186i
$289$	17.0000			1.00000
$290$		0			0
$291$			52.0592i			3.05176i
$292$		0			0
$293$	0.172964			0.0101047			0.00505234	−	0.999987i	$-0.498392\pi$
$293$	0.172964			0.0101047			0.00505234	+	0.999987i	$0.498392\pi$
$294$	−4.65751	+	32.1072i	−0.271632	+	1.87253i
$295$		0			0
$296$	23.5039	+	10.8411i	1.36613	+	0.630127i
$297$	92.4562			5.36486
$298$	−11.7030	−	1.69765i	−0.677936	−	0.0983422i
$299$		0			0
$300$	9.31213	−	31.4218i	0.537636	−	1.81414i
$301$		0			0
$302$		0			0
$303$			65.8108i			3.78073i
$304$	−9.50000	+	14.6202i	−0.544862	+	0.838525i
$305$	2.50233			0.143283
$306$		0			0
$307$			35.0168i			1.99851i	0.0385528	+	0.999257i	$0.487725\pi$
$307$			35.0168i			1.99851i	−0.0385528	+	0.999257i	$0.512275\pi$
$308$		0			0
$309$	13.3707			0.760633
$310$		0			0
$311$		−	16.1170i		−	0.913910i	−0.889490	−	0.456955i	$-0.848940\pi$
$311$		−	16.1170i		−	0.913910i	0.889490	−	0.456955i	$-0.151060\pi$
$312$	11.9919	−	25.9989i	0.678910	−	1.47190i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.6593			−1.77816			−0.889082	−	0.457748i	$-0.848656\pi$
$317$	−31.6593			−1.77816			−0.889082	+	0.457748i	$0.848656\pi$
$318$	9.07574	−	62.5650i	0.508942	−	3.50847i
$319$		0			0
$320$	−11.6122	−	13.6072i	−0.649144	−	0.760665i
$321$	−66.7265			−3.72431
$322$		0			0
$323$		0			0
$324$	53.1042	+	15.7379i	2.95023	+	0.874326i
$325$	15.4438			0.856667
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	61.0369	+	8.85407i	3.35997	+	0.487401i
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−70.8350			−3.88173
$334$	−3.92018	+	27.0243i	−0.214503	+	1.47871i
$335$		−	20.0181i		−	1.09371i
$336$		0			0
$337$	36.6783			1.99800			0.998998	−	0.0447653i	$-0.0142540\pi$
$337$	36.6783			1.99800			0.998998	+	0.0447653i	$0.0142540\pi$
$338$	−4.84191	−	0.702373i	−0.263365	−	0.0382041i
$339$		−	6.32595i		−	0.343578i
$340$		0			0
$341$		0			0
$342$	6.84997	−	47.2213i	0.370404	−	2.55343i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	18.3718	+	2.66503i	0.987672	+	0.143273i
$347$		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
$351$			47.9863i			2.56132i
$352$	22.2052	−	25.3032i	1.18354	−	1.34867i
$353$		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	$0.264891\pi$
$359$			28.0193i			1.47880i	−0.673265	+	0.739401i	$0.735109\pi$
$360$	44.4541	+	20.5044i	2.34294	+	1.08068i
$361$	−19.0000			−1.00000
$362$		0			0
$363$			80.0191i			4.19991i
$364$		0			0
$365$		0			0
$366$	−0.744586	+	5.13292i	−0.0389201	+	0.268302i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−28.6390	−	4.15440i	−1.48887	−	0.215977i
$371$		0			0
$372$		0			0
$373$	22.6186			1.17115			0.585575	−	0.810619i	$-0.300869\pi$
$373$	22.6186			1.17115			0.585575	+	0.810619i	$0.300869\pi$
$374$		0			0
$375$			36.6410i			1.89213i
$376$		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
$381$	20.3191			1.04098
$382$	−5.30972	+	36.6034i	−0.271669	+	1.87279i
$383$		−	25.2329i		−	1.28934i	−0.764460	−	0.644671i	$-0.776994\pi$
$383$		−	25.2329i		−	1.28934i	0.764460	−	0.644671i	$-0.223006\pi$
$384$	31.3671	−	19.7708i	1.60070	−	1.00892i
$385$		0			0
$386$	−3.38349	−	0.490813i	−0.172215	−	0.0249817i
$387$		0			0
$388$	−30.4604	−	9.02719i	−1.54639	−	0.458286i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.59541	+	31.6791i	−0.232698	+	1.60414i
$391$		0			0
$392$	−17.9787	−	8.29263i	−0.908060	−	0.418841i
$393$	63.8857			3.22261
$394$		0			0
$395$		0			0
$396$	−26.1780	+	88.3324i	−1.31550	+	4.43887i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	1.76991	−	12.2011i	0.0887175	−	0.611587i
$399$		0			0
$400$	16.7705	+	10.8972i	0.838525	+	0.544862i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	41.0623	+	5.95654i	2.04800	+	0.297085i
$403$		0			0
$404$	−38.5066	−	11.4117i	−1.91577	−	0.567756i
$405$	−61.9247			−3.07706
$406$		0			0
$407$		−	54.4606i		−	2.69951i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−2.31851	+	7.82334i	−0.114225	+	0.385428i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	13.1328	+	11.5249i	0.643888	+	0.565054i
$417$	28.7119			1.40603
$418$	36.3055	+	5.26651i	1.77576	+	0.257593i
$419$			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
$424$	35.0337	+	16.1592i	1.70139	+	0.784761i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	11.5705	−	39.0424i	0.559283	−	1.88718i
$429$	−60.2418			−2.90850
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−33.8595	+	52.1087i	−1.62907	+	2.50708i
$433$	−18.1458			−0.872030			−0.436015	−	0.899939i	$-0.643611\pi$
$433$	−18.1458			−0.872030			−0.436015	+	0.899939i	$0.643611\pi$
$434$		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	$0.5\pi$
$440$	−15.7645	+	34.1780i	−0.751545	+	1.62937i
$441$	54.1833			2.58016
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	17.0435	−	57.5097i	0.808848	−	2.72929i
$445$		0			0
$446$	5.25090	−	36.1978i	0.248637	−	1.71402i
$447$			27.4041i			1.29617i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−54.1665	−	7.85745i	−2.55343	−	0.370404i
$451$		0			0
$452$	3.70138	+	1.09693i	0.174098	+	0.0515955i
$453$		0			0
$454$	4.79311	−	33.0420i	0.224952	−	1.55074i
$455$		0			0
$456$	36.6900	+	16.9232i	1.71817	+	0.792502i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	41.3739	+	6.00174i	1.93327	+	0.280443i
$459$		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	$0.5\pi$
$468$	−45.8460	−	13.5868i	−2.11923	−	0.628052i
$469$		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
$477$	−105.583			−4.83431
$478$	3.95764	−	27.2825i	0.181018	−	1.24787i
$479$			7.68770i			0.351260i	0.984456	+	0.175630i	$0.0561962\pi$
$479$			7.68770i			0.351260i	−0.984456	+	0.175630i	$0.943804\pi$
$480$	−27.3432	+	31.1581i	−1.24804	+	1.42216i
$481$	28.2659			1.28882
$482$		0			0
$483$		0			0
$484$	−46.8200	−	13.8755i	−2.12818	−	0.630705i
$485$	35.5198			1.61287
$486$	8.96380	−	61.7933i	0.406606	−	2.80300i
$487$			30.1442i			1.36597i	0.730434	+	0.682983i	$0.239318\pi$
$487$			30.1442i			1.36597i	−0.730434	+	0.682983i	$0.760682\pi$
$488$	−2.87421	−	1.32573i	−0.130109	−	0.0600128i
$489$		0			0
$490$	21.9067	+	3.17780i	0.989642	+	0.143559i
$491$		−	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
$494$	−2.73341	+	18.8431i	−0.122982	+	0.847793i
$495$		−	103.004i		−	4.62970i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	44.4679i		−	1.99066i	−0.0965389	−	0.995329i	$-0.530777\pi$
$499$		−	44.4679i		−	1.99066i	0.0965389	−	0.995329i	$-0.469223\pi$
$500$	−21.4390	−	6.35363i	−0.958782	−	0.284143i
$501$	63.2811			2.82719
$502$	5.30972	−	36.6034i	0.236985	−	1.63369i
$503$		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
$507$			11.3380i			0.503538i
$508$	−3.52339	+	11.8889i	−0.156325	+	0.527487i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	6.12895	+	21.7816i	0.270864	+	0.962618i
$513$	−67.7190			−2.98987
$514$	−28.5148	−	4.13639i	−1.25774	−	0.182449i
$515$		−	9.12278i		−	0.401998i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	43.0199i		−	1.88837i
$520$	−17.7390	−	8.18207i	−0.777904	−	0.358807i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	22.2319i		−	0.972131i	−0.873922	−	0.486066i	$-0.838432\pi$
$523$		−	22.2319i		−	0.972131i	0.873922	−	0.486066i	$-0.161568\pi$
$524$	−11.0779	+	37.3802i	−0.483942	+	1.63296i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−65.4170	−	42.5070i	−2.84691	−	1.84988i
$529$	−23.0000			−1.00000
$530$	−42.6879	−	6.19235i	−1.85424	−	0.268978i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			45.5272i			1.96831i
$536$	−10.6055	+	22.9931i	−0.458089	+	0.993150i
$537$		0			0
$538$		0			0
$539$			41.6582i			1.79434i
$540$	19.7418	−	66.6145i	0.849550	−	2.86663i
$541$	−27.3238			−1.17474			−0.587371	−	0.809318i	$-0.699837\pi$
$541$	−27.3238			−1.17474			−0.587371	+	0.809318i	$0.699837\pi$
$542$	−4.54766	+	31.3500i	−0.195339	+	1.34660i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			2.56825i			0.109810i	0.998492	+	0.0549052i	$0.0174857\pi$
$547$			2.56825i			0.109810i	−0.998492	+	0.0549052i	$0.982514\pi$
$548$		0			0
$549$	8.66218			0.369693
$550$	6.04110	−	41.6452i	0.257593	−	1.77576i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$			67.0620i			2.84662i
$556$	−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
$563$			28.7864i			1.21320i	0.795007	+	0.606601i	$0.207467\pi$
$563$			28.7864i			1.21320i	−0.795007	+	0.606601i	$0.792533\pi$
$564$		0			0
$565$	−4.31617			−0.181583
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	−44.7061	−	6.48511i	−1.87253	−	0.271632i
$571$			26.9461i			1.12766i	0.825891	+	0.563829i	$0.190672\pi$
$571$			26.9461i			1.12766i	−0.825891	+	0.563829i	$0.809328\pi$
$572$	10.4461	−	35.2481i	0.436772	−	1.47380i
$573$	85.7117			3.58066
$574$		0			0
$575$		0			0
$576$	−40.1975	−	47.1033i	−1.67490	−	1.96264i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	23.7926	+	3.45138i	0.989642	+	0.143559i
$579$			7.92290i			0.329264i
$580$		0			0
$581$		0			0
$582$	−10.5692	+	72.8602i	−0.438107	+	3.02015i
$583$		−	81.1762i		−	3.36197i
$584$		0			0
$585$	53.4609			2.21034
$586$	0.242075	+	0.0351156i	0.0100000	+	0.00145061i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−13.0370	+	43.9906i	−0.537636	+	1.81414i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	30.6942	+	19.9447i	1.26152	+	0.819720i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	129.399	+	18.7707i	5.30929	+	0.770171i
$595$		0			0
$596$	−16.0344	−	4.75194i	−0.656796	−	0.194647i
$597$	−28.5706			−1.16931
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	19.4123	−	42.0863i	0.792502	−	1.71817i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	69.2956i		−	2.82194i
$604$		0			0
$605$	54.5967			2.21967
$606$	−13.3611	+	92.1064i	−0.542756	+	3.74157i
$607$			13.8249i			0.561136i	0.959834	+	0.280568i	$0.0905228\pi$
$607$			13.8249i			0.561136i	−0.959834	+	0.280568i	$0.909477\pi$
$608$	−16.2641	+	18.5332i	−0.659596	+	0.751620i
$609$		0			0
$610$	3.50217	+	0.508029i	0.141799	+	0.0205695i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−7.10919	+	49.0083i	−0.286904	+	1.97781i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	18.7132	+	2.71455i	0.752754	+	0.109195i
$619$		−	35.3754i		−	1.42186i	−0.703265	−	0.710928i	$-0.748275\pi$
$619$		−	35.3754i		−	1.42186i	0.703265	−	0.710928i	$-0.251725\pi$
$620$		0			0
$621$		0			0
$622$	3.27211	−	22.5568i	0.131200	−	0.904444i
$623$		0			0
$624$	22.0619	−	33.9525i	0.883181	−	1.35919i
$625$	25.0000			1.00000
$626$		0			0
$627$		−	85.0141i		−	3.39514i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	49.0142i		−	1.95123i	−0.219499	−	0.975613i	$-0.570442\pi$
$631$		−	49.0142i		−	1.95123i	0.219499	−	0.975613i	$-0.429558\pi$
$632$		0			0
$633$		0			0
$634$	−44.3093	−	6.42755i	−1.75975	−	0.255271i
$635$		−	13.8637i		−	0.550163i
$636$	25.4042	−	85.7211i	1.00734	−	3.39906i
$637$	−21.6213			−0.856667
$638$		0			0
$639$		0			0
$640$	−13.4895	−	21.4017i	−0.533220	−	0.845976i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−93.3880	−	13.5470i	−3.68573	−	0.534656i
$643$		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	$0.5\pi$
$648$	71.1276	+	32.8075i	2.79416	+	1.28880i
$649$		0			0
$650$	21.6146	+	3.13543i	0.847793	+	0.122982i
$651$		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	$0.5\pi$
$660$	83.6275	+	24.7837i	3.25519	+	0.964704i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−99.1381	−	14.3811i	−3.84152	−	0.557256i
$667$		0			0
$668$	−10.9731	+	37.0264i	−0.424562	+	1.43260i
$669$	−84.7620			−3.27709
$670$	4.06412	−	28.0166i	0.157011	−	1.08238i
$671$			6.65980i			0.257099i
$672$		0			0
$673$	11.9683			0.461343			0.230671	−	0.973032i	$-0.425908\pi$
$673$	11.9683			0.461343			0.230671	+	0.973032i	$0.425908\pi$
$674$	51.3337	+	7.44652i	1.97730	+	0.286829i
$675$			77.6791i			2.98987i
$676$	−6.63398	−	1.96603i	−0.255153	−	0.0756167i
$677$	27.1078			1.04184			0.520918	−	0.853607i	$-0.325590\pi$
$677$	27.1078			1.04184			0.520918	+	0.853607i	$0.325590\pi$
$678$	1.28431	−	8.85357i	0.0493236	−	0.340019i
$679$		0			0
$680$		0			0
$681$	−77.3722			−2.96491
$682$		0			0
$683$		−	10.5408i		−	0.403333i	−0.979454	−	0.201667i	$-0.935364\pi$
$683$		−	10.5408i		−	0.403333i	0.979454	−	0.201667i	$-0.0646357\pi$
$684$	19.1739	−	64.6985i	0.733134	−	2.47381i
$685$		0			0
$686$		0			0
$687$		−	96.8824i		−	3.69629i
$688$		0			0
$689$	42.1318			1.60509
$690$		0			0
$691$		−	0.331647i		−	0.0126165i	−0.999980	−	0.00630823i	$-0.997992\pi$
$691$		−	0.331647i		−	0.0126165i	0.999980	−	0.00630823i	$-0.00200798\pi$
$692$	25.1714	+	7.45976i	0.956874	+	0.283578i
$693$		0			0
$694$		0			0
$695$		−	19.5900i		−	0.743092i
$696$		0			0
$697$		0			0
$698$	18.7771	+	2.72383i	0.710725	+	0.103099i
$699$		0			0
$700$		0			0
$701$	48.5239			1.83272			0.916360	−	0.400354i	$-0.131113\pi$
$701$	48.5239			1.83272			0.916360	+	0.400354i	$0.131113\pi$
$702$	−9.74230	+	67.1600i	−0.367699	+	2.53479i
$703$			39.8893i			1.50445i
$704$	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
$715$			41.1027i			1.53716i
$716$		0			0
$717$	−63.8857			−2.38585
$718$	−5.68855	+	39.2149i	−0.212295	+	1.46349i
$719$			51.8240i			1.93271i	0.257214	+	0.966354i	$0.417195\pi$
$719$			51.8240i			1.93271i	−0.257214	+	0.966354i	$0.582805\pi$
$720$	58.0536	+	37.7224i	2.16353	+	1.40583i
$721$		0			0
$722$	−26.5917	−	3.85743i	−0.989642	−	0.143559i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−16.2457	+	111.992i	−0.602933	+	4.15641i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−61.6165			−2.28209
$730$		0			0
$731$		0			0
$732$	−2.08419	+	7.03268i	−0.0770340	+	0.259935i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		−	51.2973i		−	1.89213i
$736$		0			0
$737$	53.2771			1.96249
$738$		0			0
$739$			8.71780i			0.320689i	0.987061	+	0.160345i	$0.0512606\pi$
$739$			8.71780i			0.320689i	−0.987061	+	0.160345i	$0.948739\pi$
$740$	−39.2387	−	11.6287i	−1.44244	−	0.427480i
$741$	44.1237			1.62093
$742$		0			0
$743$			39.6817i			1.45578i	0.685693	+	0.727890i	$0.259499\pi$
$743$			39.6817i			1.45578i	−0.685693	+	0.727890i	$0.740501\pi$
$744$		0			0
$745$	18.6977			0.685032
$746$	31.6563	+	4.59209i	1.15902	+	0.168128i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−7.43894	+	51.2814i	−0.271632	+	1.87253i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−85.7117			−3.12351
$754$		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
$760$	11.5467	−	25.0335i	0.418841	−	0.908060i
$761$	−54.3834			−1.97140			−0.985699	−	0.168518i	$-0.946102\pi$
$761$	−54.3834			−1.97140			−0.985699	+	0.168518i	$0.946102\pi$
$762$	28.4380	+	4.12524i	1.03020	+	0.149442i
$763$		0			0
$764$	−14.8626	+	50.1508i	−0.537710	+	1.81439i
$765$		0			0
$766$	5.12285	−	35.3151i	0.185096	−	1.27599i
$767$		0			0
$768$	47.9143	−	21.3022i	1.72896	−	0.768678i
$769$	−47.1406			−1.69993			−0.849967	−	0.526836i	$-0.823378\pi$
$769$	−47.1406			−1.69993			−0.849967	+	0.526836i	$0.823378\pi$
$770$		0			0
$771$			66.7713i			2.40471i
$772$	−4.63577	−	1.37385i	−0.166845	−	0.0494459i
$773$	−54.0523			−1.94413			−0.972064	−	0.234717i	$-0.924584\pi$
$773$	−54.0523			−1.94413			−0.972064	+	0.234717i	$0.924584\pi$
$774$		0			0
$775$		0			0
$776$	−40.7986	−	18.8183i	−1.46458	−	0.675537i
$777$		0			0
$778$	8.39739	+	1.21813i	0.301061	+	0.0436722i
$779$		0			0
$780$	−12.8632	+	43.4041i	−0.460575	+	1.55411i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−23.4787	−	15.2561i	−0.838525	−	0.544862i
$785$		0			0
$786$	89.4122	+	12.9702i	3.18923	+	0.462633i
$787$			17.0954i			0.609383i	0.952451	+	0.304692i	$0.0985534\pi$
$787$			17.0954i			0.609383i	−0.952451	+	0.304692i	$0.901447\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−54.5713	+	118.312i	−1.93911	+	4.20404i
$793$	−3.45655			−0.122746
$794$		0			0
$795$			99.9593i			3.54519i
$796$	4.95420	−	16.7169i	0.175597	−	0.592516i
$797$	54.7345			1.93879			0.969397	−	0.245499i	$-0.0789517\pi$
$797$	54.7345			1.93879			0.969397	+	0.245499i	$0.0789517\pi$
$798$		0			0
$799$		0			0
$800$	21.2590	+	18.6562i	0.751620	+	0.659596i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	56.2600	+	16.6731i	1.98414	+	0.588016i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−51.5756	−	23.7892i	−1.81442	−	0.836900i
$809$	22.3607			0.786160			0.393080	−	0.919504i	$-0.371410\pi$
$809$	22.3607			0.786160			0.393080	+	0.919504i	$0.371410\pi$
$810$	−86.6676	−	12.5721i	−3.04519	−	0.441739i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	73.4101			2.57460
$814$	11.0567	−	76.2211i	0.387538	−	2.67155i
$815$		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	$0.5\pi$
$824$	−4.83322	+	10.4786i	−0.168373	+	0.365038i
$825$	−97.5178			−3.39514
$826$		0			0
$827$		−	48.4500i		−	1.68477i	−0.538875	−	0.842386i	$-0.681151\pi$
$827$		−	48.4500i		−	1.68477i	0.538875	−	0.842386i	$-0.318849\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	16.0404	+	18.7961i	0.556100	+	0.651637i
$833$		0			0
$834$	40.1842	+	5.82916i	1.39146	+	0.201847i
$835$		−	43.1764i		−	1.49418i
$836$	49.7426	+	14.7416i	1.72039	+	0.509851i
$837$		0			0
$838$	−3.95764	+	27.2825i	−0.136714	+	0.942460i
$839$		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
$845$	7.73587			0.266122
$846$		0			0
$847$		0			0
$848$	45.7512	+	29.7285i	1.57110	+	1.02088i
$849$		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
$855$			75.4449i			2.58016i
$856$	24.1202	−	52.2932i	0.824411	−	1.78735i
$857$	−48.0008			−1.63967			−0.819837	−	0.572597i	$-0.805936\pi$
$857$	−48.0008			−1.63967			−0.819837	+	0.572597i	$0.805936\pi$
$858$	−84.3123	−	12.2304i	−2.87837	−	0.417540i
$859$		−	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
$863$			58.6363i			1.99600i	0.0631923	+	0.998001i	$0.479872\pi$
$863$			58.6363i			1.99600i	−0.0631923	+	0.998001i	$0.520128\pi$
$864$	−57.9678	+	66.0553i	−1.97211	+	2.24725i
$865$	−29.3524			−0.998010
$866$	−25.3962	−	3.68400i	−0.862998	−	0.125187i
$867$		−	55.7135i		−	1.89213i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			27.6517i			0.936942i
$872$		0			0
$873$	122.957			4.16147
$874$		0			0
$875$		0			0
$876$		0			0
$877$	59.2236			1.99984			0.999919	−	0.0127028i	$-0.00404352\pi$
$877$	59.2236			1.99984			0.999919	+	0.0127028i	$0.00404352\pi$
$878$		0			0
$879$		−	0.566850i		−	0.0191194i
$880$	−29.0024	+	44.6338i	−0.977671	+	1.50460i
$881$	−9.21678			−0.310521			−0.155261	−	0.987874i	$-0.549622\pi$
$881$	−9.21678			−0.310521			−0.155261	+	0.987874i	$0.549622\pi$
$882$	75.8331	+	11.0004i	2.55343	+	0.370404i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	28.4813i		−	0.956308i	−0.878276	−	0.478154i	$-0.841306\pi$
$887$		−	28.4813i		−	0.956308i	0.878276	−	0.478154i	$-0.158694\pi$
$888$	35.5292	−	77.0284i	1.19228	−	2.58490i
$889$		0			0
$890$		0			0
$891$		−	164.809i		−	5.52131i
$892$	14.6979	−	49.5952i	0.492123	−	1.66057i
$893$		0			0
$894$	−5.56365	+	38.3538i	−0.186076	+	1.28274i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−74.2143	−	21.9940i	−2.47381	−	0.733134i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	4.95762	+	2.28669i	0.164888	+	0.0760543i
$905$		0			0
$906$		0			0
$907$		−	52.9215i		−	1.75723i	−0.477531	−	0.878615i	$-0.658468\pi$
$907$		−	52.9215i		−	1.75723i	0.477531	−	0.878615i	$-0.341532\pi$
$908$	13.4165	−	45.2713i	0.445243	−	1.50238i
$909$	155.436			5.15550
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	47.9143	+	31.1340i	1.58660	+	1.03095i
$913$		0			0
$914$		0			0
$915$		−	8.20080i		−	0.271110i
$916$	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
$921$	114.759			3.78145
$922$	−25.1922	−	3.65440i	−0.829660	−	0.120351i
$923$		0			0
$924$		0			0
$925$	45.7562			1.50445
$926$		0			0
$927$		−	31.5799i		−	1.03722i
$928$		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
$933$	−52.8196			−1.72924
$934$		0			0
$935$		0			0
$936$	−61.4060	−	28.3234i	−2.00712	−	0.925780i
$937$		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
$950$	−4.42477	+	30.5028i	−0.143559	+	0.989642i
$951$			103.756i			3.36452i
$952$		0			0
$953$	−61.4682			−1.99115			−0.995575	−	0.0939754i	$-0.970043\pi$
$953$	−61.4682			−1.99115			−0.995575	+	0.0939754i	$0.970043\pi$
$954$	−147.770	−	21.4357i	−4.78424	−	0.694007i
$955$		−	58.4808i		−	1.89239i
$956$	11.0779	−	37.3802i	0.358286	−	1.20896i
$957$		0			0
$958$	−1.56077	+	10.7594i	−0.0504263	+	0.347621i
$959$		0			0
$960$	−44.5944	+	38.0564i	−1.43928	+	1.22827i
$961$	−31.0000			−1.00000
$962$	39.5600	+	5.73862i	1.27547	+	0.185020i
$963$			157.599i			5.07856i
$964$		0			0
$965$	5.40577			0.174018
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−62.7106	−	28.9252i	−2.01560	−	0.929691i
$969$		0			0
$970$	49.7123	+	7.21132i	1.59617	+	0.231541i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	25.0908	−	84.6639i	0.804789	−	2.71559i
$973$		0			0
$974$	−6.11996	+	42.1888i	−0.196096	+	1.35182i
$975$		−	50.6134i		−	1.62093i
$976$	−3.75349	−	2.43897i	−0.120146	−	0.0780695i
$977$	16.6023			0.531154			0.265577	−	0.964090i	$-0.414438\pi$
$977$	16.6023			0.531154			0.265577	+	0.964090i	$0.414438\pi$
$978$		0			0
$979$		0			0
$980$	30.0146	+	8.89508i	0.958782	+	0.284143i
$981$		0			0
$982$	5.30972	−	36.6034i	0.169440	−	1.16806i
$983$			62.7054i			1.99999i	0.00306979	+	0.999995i	$0.499023\pi$
$983$			62.7054i			1.99999i	−0.00306979	+	0.999995i	$0.500977\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−7.65116	+	25.8173i	−0.243416	+	0.821357i
$989$		0			0
$990$	20.9122	−	144.161i	0.664632	−	4.58174i
$991$		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			$\pi$
$998$	9.02799	−	62.2358i	0.285776	−	1.97004i
$999$			142.172i			4.49812i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.d.a.379.16	yes	16
4.3	odd	2	inner	380.2.d.a.379.15	yes	16
5.4	even	2	inner	380.2.d.a.379.1	✓	16
19.18	odd	2	inner	380.2.d.a.379.1	✓	16
20.19	odd	2	inner	380.2.d.a.379.2	yes	16
76.75	even	2	inner	380.2.d.a.379.2	yes	16
95.94	odd	2	CM	380.2.d.a.379.16	yes	16
380.379	even	2	inner	380.2.d.a.379.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.d.a.379.1	✓	16	5.4	even	2	inner
380.2.d.a.379.1	✓	16	19.18	odd	2	inner
380.2.d.a.379.2	yes	16	20.19	odd	2	inner
380.2.d.a.379.2	yes	16	76.75	even	2	inner
380.2.d.a.379.15	yes	16	4.3	odd	2	inner
380.2.d.a.379.15	yes	16	380.379	even	2	inner
380.2.d.a.379.16	yes	16	1.1	even	1	trivial
380.2.d.a.379.16	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)

\(f(q)\)	\(=\)	\(q+(1.39956 + 0.203022i) q^{2} -3.27727i q^{3} +(1.91756 + 0.568286i) q^{4} -2.23607 q^{5} +(0.665359 - 4.58675i) q^{6} +(2.56838 + 1.18466i) q^{8} -7.74048 q^{9} +O(q^{10})\)	\(q+(1.39956 + 0.203022i) q^{2} -3.27727i q^{3} +(1.91756 + 0.568286i) q^{4} -2.23607 q^{5} +(0.665359 - 4.58675i) q^{6} +(2.56838 + 1.18466i) q^{8} -7.74048 q^{9} +(-3.12952 - 0.453972i) q^{10} -5.95117i q^{11} +(1.86243 - 6.28437i) q^{12} +3.08876 q^{13} +7.32819i q^{15} +(3.35410 + 2.17945i) q^{16} +(-10.8333 - 1.57149i) q^{18} +4.35890i q^{19} +(-4.28780 - 1.27073i) q^{20} +(1.20822 - 8.32905i) q^{22} +(3.88245 - 8.41727i) q^{24} +5.00000 q^{25} +(4.32291 + 0.627087i) q^{26} +15.5358i q^{27} +(-1.48779 + 10.2563i) q^{30} +(4.25181 + 3.73124i) q^{32} -19.5036 q^{33} +(-14.8429 - 4.39881i) q^{36} +9.15124 q^{37} +(-0.884954 + 6.10056i) q^{38} -10.1227i q^{39} +(-5.74307 - 2.64898i) q^{40} +(3.38197 - 11.4117i) q^{44} +17.3082 q^{45} +(7.14264 - 10.9923i) q^{48} -7.00000 q^{49} +(6.99782 + 1.01511i) q^{50} +(5.92289 + 1.75530i) q^{52} +13.6404 q^{53} +(-3.15412 + 21.7434i) q^{54} +13.3072i q^{55} +14.2853 q^{57} +(-4.16451 + 14.0523i) q^{60} -1.11908 q^{61} +(5.19315 + 6.08532i) q^{64} -6.90667 q^{65} +(-27.2965 - 3.95966i) q^{66} +8.95237i q^{67} +(-19.8805 - 9.16985i) q^{72} +(12.8077 + 1.85791i) q^{74} -16.3863i q^{75} +(-2.47710 + 8.35847i) q^{76} +(2.05513 - 14.1673i) q^{78} +(-7.50000 - 4.87340i) q^{80} +27.6936 q^{81} +(7.05012 - 15.2849i) q^{88} +(24.2240 + 3.51396i) q^{90} -9.74679i q^{95} +(12.2283 - 13.9343i) q^{96} -15.8849 q^{97} +(-9.79695 - 1.42116i) q^{98} +46.0649i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 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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