LMFDB - Embedded newform 760.2.p.a.379.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(379,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("760.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.p (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:	$6.06863055362$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			379.3
Root			$0.874032i$ of defining polynomial
Character	$\chi$	$=$	760.379
Dual form			760.2.p.a.379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} +4.47214 q^{7} -2.82843i q^{8} -3.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} +4.47214 q^{7} -2.82843i q^{8} -3.00000 q^{9} -3.16228i q^{10} +2.00000 q^{11} -5.65685i q^{13} +6.32456i q^{14} +4.00000 q^{16} -4.24264i q^{18} +(3.00000 + 3.16228i) q^{19} +4.47214 q^{20} +2.82843i q^{22} +4.47214 q^{23} +5.00000 q^{25} +8.00000 q^{26} -8.94427 q^{28} +5.65685i q^{32} -10.0000 q^{35} +6.00000 q^{36} -11.3137i q^{37} +(-4.47214 + 4.24264i) q^{38} +6.32456i q^{40} +12.6491i q^{41} -4.00000 q^{44} +6.70820 q^{45} +6.32456i q^{46} +13.4164 q^{47} +13.0000 q^{49} +7.07107i q^{50} +11.3137i q^{52} +5.65685i q^{53} -4.47214 q^{55} -12.6491i q^{56} -6.32456i q^{59} -13.4164 q^{63} -8.00000 q^{64} +12.6491i q^{65} -14.1421i q^{70} +8.48528i q^{72} +16.0000 q^{74} +(-6.00000 - 6.32456i) q^{76} +8.94427 q^{77} -8.94427 q^{80} +9.00000 q^{81} -17.8885 q^{82} -5.65685i q^{88} -12.6491i q^{89} +9.48683i q^{90} -25.2982i q^{91} -8.94427 q^{92} +18.9737i q^{94} +(-6.70820 - 7.07107i) q^{95} +18.3848i q^{98} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 12 * q^9	$ 4 q - 8 q^{4} - 12 q^{9} + 8 q^{11} + 16 q^{16} + 12 q^{19} + 20 q^{25} + 32 q^{26} - 40 q^{35} + 24 q^{36} - 16 q^{44} + 52 q^{49} - 32 q^{64} + 64 q^{74} - 24 q^{76} + 36 q^{81} - 24 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 12 * q^9 + 8 * q^11 + 16 * q^16 + 12 * q^19 + 20 * q^25 + 32 * q^26 - 40 * q^35 + 24 * q^36 - 16 * q^44 + 52 * q^49 - 32 * q^64 + 64 * q^74 - 24 * q^76 + 36 * q^81 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−2.00000			−1.00000
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$		0			0
$7$	4.47214			1.69031			0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214			1.69031			0.845154	+	0.534522i	$0.179509\pi$
$8$		−	2.82843i		−	1.00000i
$9$	−3.00000			−1.00000
$10$		−	3.16228i		−	1.00000i
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	5.65685i		−	1.56893i	−0.620174	−	0.784465i	$-0.712938\pi$
$13$		−	5.65685i		−	1.56893i	0.620174	−	0.784465i	$-0.287062\pi$
$14$			6.32456i			1.69031i
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		−	4.24264i		−	1.00000i
$19$	3.00000	+	3.16228i	0.688247	+	0.725476i
$19$	3.00000	+	3.16228i	0.688247	+	0.725476i
$20$	4.47214			1.00000
$21$		0			0
$22$			2.82843i			0.603023i
$23$	4.47214			0.932505			0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214			0.932505			0.466252	+	0.884652i	$0.345604\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	8.00000			1.56893
$27$		0			0
$28$	−8.94427			−1.69031
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$			5.65685i			1.00000i
$33$		0			0
$34$		0			0
$35$	−10.0000			−1.69031
$36$	6.00000			1.00000
$37$		−	11.3137i		−	1.85996i	−0.367607	−	0.929981i	$-0.619823\pi$
$37$		−	11.3137i		−	1.85996i	0.367607	−	0.929981i	$-0.380177\pi$
$38$	−4.47214	+	4.24264i	−0.725476	+	0.688247i
$39$		0			0
$40$			6.32456i			1.00000i
$41$			12.6491i			1.97546i	0.156174	+	0.987730i	$0.450084\pi$
$41$			12.6491i			1.97546i	−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−4.00000			−0.603023
$45$	6.70820			1.00000
$46$			6.32456i			0.932505i
$47$	13.4164			1.95698			0.978492	−	0.206284i	$-0.0661372\pi$
$47$	13.4164			1.95698			0.978492	+	0.206284i	$0.0661372\pi$
$48$		0			0
$49$	13.0000			1.85714
$50$			7.07107i			1.00000i
$51$		0			0
$52$			11.3137i			1.56893i
$53$			5.65685i			0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$			5.65685i			0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$	−4.47214			−0.603023
$56$		−	12.6491i		−	1.69031i
$57$		0			0
$58$		0			0
$59$		−	6.32456i		−	0.823387i	−0.911322	−	0.411693i	$-0.864937\pi$
$59$		−	6.32456i		−	0.823387i	0.911322	−	0.411693i	$-0.135063\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−13.4164			−1.69031
$64$	−8.00000			−1.00000
$65$			12.6491i			1.56893i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		−	14.1421i		−	1.69031i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$			8.48528i			1.00000i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	16.0000			1.85996
$75$		0			0
$76$	−6.00000	−	6.32456i	−0.688247	−	0.725476i
$77$	8.94427			1.01929
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−8.94427			−1.00000
$81$	9.00000			1.00000
$82$	−17.8885			−1.97546
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		−	5.65685i		−	0.603023i
$89$		−	12.6491i		−	1.34080i	−0.741999	−	0.670402i	$-0.766122\pi$
$89$		−	12.6491i		−	1.34080i	0.741999	−	0.670402i	$-0.233878\pi$
$90$			9.48683i			1.00000i
$91$		−	25.2982i		−	2.65197i
$92$	−8.94427			−0.932505
$93$		0			0
$94$			18.9737i			1.95698i
$95$	−6.70820	−	7.07107i	−0.688247	−	0.725476i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$			18.3848i			1.85714i
$99$	−6.00000			−0.603023
$100$	−10.0000			−1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	19.7990i		−	1.95085i	−0.220326	−	0.975426i	$-0.570712\pi$
$103$		−	19.7990i		−	1.95085i	0.220326	−	0.975426i	$-0.429288\pi$
$104$	−16.0000			−1.56893
$105$		0			0
$106$	−8.00000			−0.777029
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		−	6.32456i		−	0.603023i
$111$		0			0
$112$	17.8885			1.69031
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−10.0000			−0.932505
$116$		0			0
$117$			16.9706i			1.56893i
$118$	8.94427			0.823387
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1803			−1.00000
$126$		−	18.9737i		−	1.69031i
$127$		−	2.82843i		−	0.250982i	−0.992095	−	0.125491i	$-0.959949\pi$
$127$		−	2.82843i		−	0.250982i	0.992095	−	0.125491i	$-0.0400507\pi$
$128$		−	11.3137i		−	1.00000i
$129$		0			0
$130$	−17.8885			−1.56893
$131$	−22.0000			−1.92215			−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000			−1.92215			−0.961074	+	0.276289i	$0.910895\pi$
$132$		0			0
$133$	13.4164	+	14.1421i	1.16335	+	1.22628i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$	20.0000			1.69031
$141$		0			0
$142$		0			0
$143$		−	11.3137i		−	0.946100i
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$			22.6274i			1.85996i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	8.94427	−	8.48528i	0.725476	−	0.688247i
$153$		0			0
$154$			12.6491i			1.01929i
$155$		0			0
$156$		0			0
$157$	22.3607			1.78458			0.892288	−	0.451466i	$-0.149099\pi$
$157$	22.3607			1.78458			0.892288	+	0.451466i	$0.149099\pi$
$158$		0			0
$159$		0			0
$160$		−	12.6491i		−	1.00000i
$161$	20.0000			1.57622
$162$			12.7279i			1.00000i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	25.2982i		−	1.97546i
$165$		0			0
$166$		0			0
$167$			25.4558i			1.96983i	0.173032	+	0.984916i	$0.444644\pi$
$167$			25.4558i			1.96983i	−0.173032	+	0.984916i	$0.555356\pi$
$168$		0			0
$169$	−19.0000			−1.46154
$170$		0			0
$171$	−9.00000	−	9.48683i	−0.688247	−	0.725476i
$172$		0			0
$173$		−	22.6274i		−	1.72033i	−0.510015	−	0.860165i	$-0.670360\pi$
$173$		−	22.6274i		−	1.72033i	0.510015	−	0.860165i	$-0.329640\pi$
$174$		0			0
$175$	22.3607			1.69031
$176$	8.00000			0.603023
$177$		0			0
$178$	17.8885			1.34080
$179$			6.32456i			0.472719i	0.971666	+	0.236360i	$0.0759544\pi$
$179$			6.32456i			0.472719i	−0.971666	+	0.236360i	$0.924046\pi$
$180$	−13.4164			−1.00000
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	35.7771			2.65197
$183$		0			0
$184$		−	12.6491i		−	0.932505i
$185$			25.2982i			1.85996i
$186$		0			0
$187$		0			0
$188$	−26.8328			−1.95698
$189$		0			0
$190$	10.0000	−	9.48683i	0.725476	−	0.688247i
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−26.0000			−1.85714
$197$	−22.3607			−1.59313			−0.796566	−	0.604551i	$-0.793352\pi$
$197$	−22.3607			−1.59313			−0.796566	+	0.604551i	$0.793352\pi$
$198$		−	8.48528i		−	0.603023i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	14.1421i		−	1.00000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	28.2843i		−	1.97546i
$206$	28.0000			1.95085
$207$	−13.4164			−0.932505
$208$		−	22.6274i		−	1.56893i
$209$	6.00000	+	6.32456i	0.415029	+	0.437479i
$210$		0			0
$211$			18.9737i			1.30620i	0.757271	+	0.653101i	$0.226532\pi$
$211$			18.9737i			1.30620i	−0.757271	+	0.653101i	$0.773468\pi$
$212$		−	11.3137i		−	0.777029i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	8.94427			0.603023
$221$		0			0
$222$		0			0
$223$			19.7990i			1.32584i	0.748691	+	0.662919i	$0.230683\pi$
$223$			19.7990i			1.32584i	−0.748691	+	0.662919i	$0.769317\pi$
$224$			25.2982i			1.69031i
$225$	−15.0000			−1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		−	14.1421i		−	0.932505i
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−24.0000			−1.56893
$235$	−30.0000			−1.95698
$236$			12.6491i			0.823387i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	25.2982i		−	1.62960i	−0.579741	−	0.814801i	$-0.696846\pi$
$241$		−	25.2982i		−	1.62960i	0.579741	−	0.814801i	$-0.303154\pi$
$242$		−	9.89949i		−	0.636364i
$243$		0			0
$244$		0			0
$245$	−29.0689			−1.85714
$246$		0			0
$247$	17.8885	−	16.9706i	1.13822	−	1.07981i
$248$		0			0
$249$		0			0
$250$		−	15.8114i		−	1.00000i
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$	26.8328			1.69031
$253$	8.94427			0.562322
$254$	4.00000			0.250982
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		−	50.5964i		−	3.14391i
$260$		−	25.2982i		−	1.56893i
$261$		0			0
$262$		−	31.1127i		−	1.92215i
$263$	−31.3050			−1.93035			−0.965173	−	0.261612i	$-0.915746\pi$
$263$	−31.3050			−1.93035			−0.965173	+	0.261612i	$0.915746\pi$
$264$		0			0
$265$		−	12.6491i		−	0.777029i
$266$	−20.0000	+	18.9737i	−1.22628	+	1.16335i
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0000			0.603023
$276$		0			0
$277$	−31.3050			−1.88093			−0.940466	−	0.339887i	$-0.889611\pi$
$277$	−31.3050			−1.88093			−0.940466	+	0.339887i	$0.889611\pi$
$278$			19.7990i			1.18746i
$279$		0			0
$280$			28.2843i			1.69031i
$281$			25.2982i			1.50917i	0.656205	+	0.754583i	$0.272161\pi$
$281$			25.2982i			1.50917i	−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	16.0000			0.946100
$287$			56.5685i			3.33914i
$288$		−	16.9706i		−	1.00000i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			33.9411i			1.98286i	0.130632	+	0.991431i	$0.458299\pi$
$293$			33.9411i			1.98286i	−0.130632	+	0.991431i	$0.541701\pi$
$294$		0			0
$295$			14.1421i			0.823387i
$296$	−32.0000			−1.85996
$297$		0			0
$298$		0			0
$299$		−	25.2982i		−	1.46303i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	12.0000	+	12.6491i	0.688247	+	0.725476i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−17.8885			−1.01929
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$			31.6228i			1.78458i
$315$	30.0000			1.69031
$316$		0			0
$317$			16.9706i			0.953162i	0.879131	+	0.476581i	$0.158124\pi$
$317$			16.9706i			0.953162i	−0.879131	+	0.476581i	$0.841876\pi$
$318$		0			0
$319$		0			0
$320$	17.8885			1.00000
$321$		0			0
$322$			28.2843i			1.57622i
$323$		0			0
$324$	−18.0000			−1.00000
$325$		−	28.2843i		−	1.56893i
$326$		0			0
$327$		0			0
$328$	35.7771			1.97546
$329$	60.0000			3.30791
$330$		0			0
$331$			31.6228i			1.73814i	0.494685	+	0.869072i	$0.335284\pi$
$331$			31.6228i			1.73814i	−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$			33.9411i			1.85996i
$334$	−36.0000			−1.96983
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		−	26.8701i		−	1.46154i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	13.4164	−	12.7279i	0.725476	−	0.688247i
$343$	26.8328			1.44884
$344$		0			0
$345$		0			0
$346$	32.0000			1.72033
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$			31.6228i			1.69031i
$351$		0			0
$352$			11.3137i			0.603023i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$			25.2982i			1.34080i
$357$		0			0
$358$	−8.94427			−0.472719
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		−	18.9737i		−	1.00000i
$361$	−1.00000	+	18.9737i	−0.0526316	+	0.998614i
$362$		0			0
$363$		0			0
$364$			50.5964i			2.65197i
$365$		0			0
$366$		0			0
$367$	−22.3607			−1.16722			−0.583609	−	0.812035i	$-0.698360\pi$
$367$	−22.3607			−1.16722			−0.583609	+	0.812035i	$0.698360\pi$
$368$	17.8885			0.932505
$369$		−	37.9473i		−	1.97546i
$370$	−35.7771			−1.85996
$371$			25.2982i			1.31342i
$372$		0			0
$373$			22.6274i			1.17160i	0.810454	+	0.585802i	$0.199220\pi$
$373$			22.6274i			1.17160i	−0.810454	+	0.585802i	$0.800780\pi$
$374$		0			0
$375$		0			0
$376$		−	37.9473i		−	1.95698i
$377$		0			0
$378$		0			0
$379$			18.9737i			0.974612i	0.873231	+	0.487306i	$0.162020\pi$
$379$			18.9737i			0.974612i	−0.873231	+	0.487306i	$0.837980\pi$
$380$	13.4164	+	14.1421i	0.688247	+	0.725476i
$381$		0			0
$382$		0			0
$383$		−	36.7696i		−	1.87884i	−0.342773	−	0.939418i	$-0.611366\pi$
$383$		−	36.7696i		−	1.87884i	0.342773	−	0.939418i	$-0.388634\pi$
$384$		0			0
$385$	−20.0000			−1.01929
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		−	36.7696i		−	1.85714i
$393$		0			0
$394$		−	31.6228i		−	1.59313i
$395$		0			0
$396$	12.0000			0.603023
$397$	−4.47214			−0.224450			−0.112225	−	0.993683i	$-0.535798\pi$
$397$	−4.47214			−0.224450			−0.112225	+	0.993683i	$0.535798\pi$
$398$		0			0
$399$		0			0
$400$	20.0000			1.00000
$401$			12.6491i			0.631666i	0.948815	+	0.315833i	$0.102284\pi$
$401$			12.6491i			0.631666i	−0.948815	+	0.315833i	$0.897716\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−20.1246			−1.00000
$406$		0			0
$407$		−	22.6274i		−	1.12160i
$408$		0			0
$409$			37.9473i			1.87637i	0.346128	+	0.938187i	$0.387496\pi$
$409$			37.9473i			1.87637i	−0.346128	+	0.938187i	$0.612504\pi$
$410$	40.0000			1.97546
$411$		0			0
$412$			39.5980i			1.95085i
$413$		−	28.2843i		−	1.39178i
$414$		−	18.9737i		−	0.932505i
$415$		0			0
$416$	32.0000			1.56893
$417$		0			0
$418$	−8.94427	+	8.48528i	−0.437479	+	0.415029i
$419$	26.0000			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	−26.8328			−1.30620
$423$	−40.2492			−1.95698
$424$	16.0000			0.777029
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.4164	+	14.1421i	0.641794	+	0.676510i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$			12.6491i			0.603023i
$441$	−39.0000			−1.85714
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$			28.2843i			1.34080i
$446$	−28.0000			−1.32584
$447$		0			0
$448$	−35.7771			−1.69031
$449$		−	25.2982i		−	1.19390i	−0.802280	−	0.596948i	$-0.796380\pi$
$449$		−	25.2982i		−	1.19390i	0.802280	−	0.596948i	$-0.203620\pi$
$450$		−	21.2132i		−	1.00000i
$451$			25.2982i			1.19125i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			56.5685i			2.65197i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$	20.0000			0.932505
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−22.3607			−1.03919			−0.519594	−	0.854413i	$-0.673917\pi$
$463$	−22.3607			−1.03919			−0.519594	+	0.854413i	$0.673917\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		−	33.9411i		−	1.56893i
$469$		0			0
$470$		−	42.4264i		−	1.95698i
$471$		0			0
$472$	−17.8885			−0.823387
$473$		0			0
$474$		0			0
$475$	15.0000	+	15.8114i	0.688247	+	0.725476i
$476$		0			0
$477$		−	16.9706i		−	0.777029i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−64.0000			−2.91815
$482$	35.7771			1.62960
$483$		0			0
$484$	14.0000			0.636364
$485$		0			0
$486$		0			0
$487$		−	31.1127i		−	1.40985i	−0.709281	−	0.704925i	$-0.750980\pi$
$487$		−	31.1127i		−	1.40985i	0.709281	−	0.704925i	$-0.249020\pi$
$488$		0			0
$489$		0			0
$490$		−	41.1096i		−	1.85714i
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$		0			0
$494$	24.0000	+	25.2982i	1.07981	+	1.13822i
$495$	13.4164			0.603023
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.00000			0.268597			0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000			0.268597			0.134298	+	0.990941i	$0.457122\pi$
$500$	22.3607			1.00000
$501$		0			0
$502$			2.82843i			0.126239i
$503$	−40.2492			−1.79462			−0.897312	−	0.441397i	$-0.854483\pi$
$503$	−40.2492			−1.79462			−0.897312	+	0.441397i	$0.854483\pi$
$504$			37.9473i			1.69031i
$505$		0			0
$506$			12.6491i			0.562322i
$507$		0			0
$508$			5.65685i			0.250982i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$		0			0
$515$			44.2719i			1.95085i
$516$		0			0
$517$	26.8328			1.18011
$518$	71.5542			3.14391
$519$		0			0
$520$	35.7771			1.56893
$521$		−	25.2982i		−	1.10834i	−0.832405	−	0.554168i	$-0.813037\pi$
$521$		−	25.2982i		−	1.10834i	0.832405	−	0.554168i	$-0.186963\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	44.0000			1.92215
$525$		0			0
$526$		−	44.2719i		−	1.93035i
$527$		0			0
$528$		0			0
$529$	−3.00000			−0.130435
$530$	17.8885			0.777029
$531$			18.9737i			0.823387i
$532$	−26.8328	−	28.2843i	−1.16335	−	1.22628i
$533$	71.5542			3.09936
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.0000			1.11990
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$			14.1421i			0.603023i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	44.2719i		−	1.88093i
$555$		0			0
$556$	−28.0000			−1.18746
$557$	−13.4164			−0.568471			−0.284236	−	0.958754i	$-0.591740\pi$
$557$	−13.4164			−0.568471			−0.284236	+	0.958754i	$0.591740\pi$
$558$		0			0
$559$		0			0
$560$	−40.0000			−1.69031
$561$		0			0
$562$	−35.7771			−1.50917
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	40.2492			1.69031
$568$		0			0
$569$		−	12.6491i		−	0.530278i	−0.964210	−	0.265139i	$-0.914582\pi$
$569$		−	12.6491i		−	0.530278i	0.964210	−	0.265139i	$-0.0854179\pi$
$570$		0			0
$571$	18.0000			0.753277			0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000			0.753277			0.376638	+	0.926360i	$0.377080\pi$
$572$			22.6274i			0.946100i
$573$		0			0
$574$	−80.0000			−3.33914
$575$	22.3607			0.932505
$576$	24.0000			1.00000
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			24.0416i			1.00000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			11.3137i			0.468566i
$584$		0			0
$585$		−	37.9473i		−	1.56893i
$586$	−48.0000			−1.98286
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	−20.0000			−0.823387
$591$		0			0
$592$		−	45.2548i		−	1.85996i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	35.7771			1.46303
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		−	25.2982i		−	1.03194i	−0.856608	−	0.515968i	$-0.827432\pi$
$601$		−	25.2982i		−	1.03194i	0.856608	−	0.515968i	$-0.172568\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.6525			0.636364
$606$		0			0
$607$		−	2.82843i		−	0.114802i	−0.998351	−	0.0574012i	$-0.981719\pi$
$607$		−	2.82843i		−	0.114802i	0.998351	−	0.0574012i	$-0.0182814\pi$
$608$	−17.8885	+	16.9706i	−0.725476	+	0.688247i
$609$		0			0
$610$		0			0
$611$		−	75.8947i		−	3.07037i
$612$		0			0
$613$	−49.1935			−1.98691			−0.993453	−	0.114239i	$-0.963557\pi$
$613$	−49.1935			−1.98691			−0.993453	+	0.114239i	$0.963557\pi$
$614$		0			0
$615$		0			0
$616$		−	25.2982i		−	1.01929i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−46.0000			−1.84890			−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000			−1.84890			−0.924448	+	0.381308i	$0.875474\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	56.5685i		−	2.26637i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$	−44.7214			−1.78458
$629$		0			0
$630$			42.4264i			1.69031i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−24.0000			−0.953162
$635$			6.32456i			0.250982i
$636$		0			0
$637$		−	73.5391i		−	2.91373i
$638$		0			0
$639$		0			0
$640$			25.2982i			1.00000i
$641$			50.5964i			1.99844i	0.0394976	+	0.999220i	$0.487424\pi$
$641$			50.5964i			1.99844i	−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−40.0000			−1.57622
$645$		0			0
$646$		0			0
$647$	40.2492			1.58236			0.791180	−	0.611583i	$-0.209467\pi$
$647$	40.2492			1.58236			0.791180	+	0.611583i	$0.209467\pi$
$648$		−	25.4558i		−	1.00000i
$649$		−	12.6491i		−	0.496521i
$650$	40.0000			1.56893
$651$		0			0
$652$		0			0
$653$	4.47214			0.175008			0.0875041	−	0.996164i	$-0.472111\pi$
$653$	4.47214			0.175008			0.0875041	+	0.996164i	$0.472111\pi$
$654$		0			0
$655$	49.1935			1.92215
$656$			50.5964i			1.97546i
$657$		0			0
$658$			84.8528i			3.30791i
$659$		−	44.2719i		−	1.72459i	−0.506408	−	0.862294i	$-0.669027\pi$
$659$		−	44.2719i		−	1.72459i	0.506408	−	0.862294i	$-0.330973\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−44.7214			−1.73814
$663$		0			0
$664$		0			0
$665$	−30.0000	−	31.6228i	−1.16335	−	1.22628i
$666$	−48.0000			−1.85996
$667$		0			0
$668$		−	50.9117i		−	1.96983i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	38.0000			1.46154
$677$		−	16.9706i		−	0.652232i	−0.945330	−	0.326116i	$-0.894260\pi$
$677$		−	16.9706i		−	0.652232i	0.945330	−	0.326116i	$-0.105740\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	18.0000	+	18.9737i	0.688247	+	0.725476i
$685$		0			0
$686$			37.9473i			1.44884i
$687$		0			0
$688$		0			0
$689$	32.0000			1.21910
$690$		0			0
$691$	42.0000			1.59776			0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000			1.59776			0.798878	+	0.601494i	$0.205427\pi$
$692$			45.2548i			1.72033i
$693$	−26.8328			−1.01929
$694$		0			0
$695$	−31.3050			−1.18746
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−44.7214			−1.69031
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	35.7771	−	33.9411i	1.34936	−	1.28011i
$704$	−16.0000			−0.603023
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	−35.7771			−1.34080
$713$		0			0
$714$		0			0
$715$			25.2982i			0.946100i
$716$		−	12.6491i		−	0.472719i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	26.8328			1.00000
$721$		−	88.5438i		−	3.29754i
$722$	−26.8328	−	1.41421i	−0.998614	−	0.0526316i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−4.47214			−0.165862			−0.0829312	−	0.996555i	$-0.526428\pi$
$727$	−4.47214			−0.165862			−0.0829312	+	0.996555i	$0.526428\pi$
$728$	−71.5542			−2.65197
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	49.1935			1.81700			0.908502	−	0.417881i	$-0.137227\pi$
$733$	49.1935			1.81700			0.908502	+	0.417881i	$0.137227\pi$
$734$		−	31.6228i		−	1.16722i
$735$		0			0
$736$			25.2982i			0.932505i
$737$		0			0
$738$	53.6656			1.97546
$739$	54.0000			1.98642			0.993211	−	0.116326i	$-0.0371118\pi$
$739$	54.0000			1.98642			0.993211	+	0.116326i	$0.0371118\pi$
$740$		−	50.5964i		−	1.85996i
$741$		0			0
$742$	−35.7771			−1.31342
$743$			36.7696i			1.34894i	0.738300	+	0.674472i	$0.235629\pi$
$743$			36.7696i			1.34894i	−0.738300	+	0.674472i	$0.764371\pi$
$744$		0			0
$745$		0			0
$746$	−32.0000			−1.17160
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	53.6656			1.95698
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−31.3050			−1.13780			−0.568899	−	0.822407i	$-0.692630\pi$
$757$	−31.3050			−1.13780			−0.568899	+	0.822407i	$0.692630\pi$
$758$	−26.8328			−0.974612
$759$		0			0
$760$	−20.0000	+	18.9737i	−0.725476	+	0.688247i
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	52.0000			1.87884
$767$	−35.7771			−1.29184
$768$		0			0
$769$	−54.0000			−1.94729			−0.973645	−	0.228069i	$-0.926759\pi$
$769$	−54.0000			−1.94729			−0.973645	+	0.228069i	$0.926759\pi$
$770$		−	28.2843i		−	1.01929i
$771$		0			0
$772$		0			0
$773$			50.9117i			1.83117i	0.402129	+	0.915583i	$0.368270\pi$
$773$			50.9117i			1.83117i	−0.402129	+	0.915583i	$0.631730\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−40.0000	+	37.9473i	−1.43315	+	1.35960i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	52.0000			1.85714
$785$	−50.0000			−1.78458
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	44.7214			1.59313
$789$		0			0
$790$		0			0
$791$		0			0
$792$			16.9706i			0.603023i
$793$		0			0
$794$		−	6.32456i		−	0.224450i
$795$		0			0
$796$		0			0
$797$		−	39.5980i		−	1.40263i	−0.712850	−	0.701316i	$-0.752596\pi$
$797$		−	39.5980i		−	1.40263i	0.712850	−	0.701316i	$-0.247404\pi$
$798$		0			0
$799$		0			0
$800$			28.2843i			1.00000i
$801$			37.9473i			1.34080i
$802$	−17.8885			−0.631666
$803$		0			0
$804$		0			0
$805$	−44.7214			−1.57622
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.0000			−0.914111			−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000			−0.914111			−0.457056	+	0.889438i	$0.651096\pi$
$810$		−	28.4605i		−	1.00000i
$811$			56.9210i			1.99877i	0.0351147	+	0.999383i	$0.488820\pi$
$811$			56.9210i			1.99877i	−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$	32.0000			1.12160
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−53.6656			−1.87637
$819$			75.8947i			2.65197i
$820$			56.5685i			1.97546i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	31.3050			1.09122			0.545611	−	0.838039i	$-0.316298\pi$
$823$	31.3050			1.09122			0.545611	+	0.838039i	$0.316298\pi$
$824$	−56.0000			−1.95085
$825$		0			0
$826$	40.0000			1.39178
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	26.8328			0.932505
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$			45.2548i			1.56893i
$833$		0			0
$834$		0			0
$835$		−	56.9210i		−	1.96983i
$836$	−12.0000	−	12.6491i	−0.415029	−	0.437479i
$837$		0			0
$838$			36.7696i			1.27018i
$839$		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
$844$		−	37.9473i		−	1.30620i
$845$	42.4853			1.46154
$846$		−	56.9210i		−	1.95698i
$847$	−31.3050			−1.07565
$848$			22.6274i			0.777029i
$849$		0			0
$850$		0			0
$851$		−	50.5964i		−	1.73442i
$852$		0			0
$853$	58.1378			1.99060			0.995300	−	0.0968435i	$-0.0308746\pi$
$853$	58.1378			1.99060			0.995300	+	0.0968435i	$0.0308746\pi$
$854$		0			0
$855$	20.1246	+	21.2132i	0.688247	+	0.725476i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.48528i		−	0.288842i	−0.989516	−	0.144421i	$-0.953868\pi$
$863$		−	8.48528i		−	0.288842i	0.989516	−	0.144421i	$-0.0461320\pi$
$864$		0			0
$865$			50.5964i			1.72033i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−20.0000	+	18.9737i	−0.676510	+	0.641794i
$875$	−50.0000			−1.69031
$876$		0			0
$877$		−	11.3137i		−	0.382037i	−0.981586	−	0.191018i	$-0.938821\pi$
$877$		−	11.3137i		−	0.382037i	0.981586	−	0.191018i	$-0.0611790\pi$
$878$		0			0
$879$		0			0
$880$	−17.8885			−0.603023
$881$	58.0000			1.95407			0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000			1.95407			0.977035	+	0.213080i	$0.0683494\pi$
$882$		−	55.1543i		−	1.85714i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	59.3970i		−	1.99436i	−0.0750798	−	0.997178i	$-0.523921\pi$
$887$		−	59.3970i		−	1.99436i	0.0750798	−	0.997178i	$-0.476079\pi$
$888$		0			0
$889$		−	12.6491i		−	0.424238i
$890$	−40.0000			−1.34080
$891$	18.0000			0.603023
$892$		−	39.5980i		−	1.32584i
$893$	40.2492	+	42.4264i	1.34689	+	1.41975i
$894$		0			0
$895$		−	14.1421i		−	0.472719i
$896$		−	50.5964i		−	1.69031i
$897$		0			0
$898$	35.7771			1.19390
$899$		0			0
$900$	30.0000			1.00000
$901$		0			0
$902$	−35.7771			−1.19125
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$	−80.0000			−2.65197
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−98.3870			−3.24902
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$			28.2843i			0.932505i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	56.5685i		−	1.85996i
$926$		−	31.6228i		−	1.03919i
$927$			59.3970i			1.95085i
$928$		0			0
$929$	−34.0000			−1.11550			−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000			−1.11550			−0.557752	+	0.830008i	$0.688336\pi$
$930$		0			0
$931$	39.0000	+	41.1096i	1.27817	+	1.34731i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	48.0000			1.56893
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	60.0000			1.95698
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$			56.5685i			1.84213i
$944$		−	25.2982i		−	0.823387i
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$	−22.3607	+	21.2132i	−0.725476	+	0.688247i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	24.0000			0.777029
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		−	90.5097i		−	2.91815i
$963$		0			0
$964$			50.5964i			1.62960i
$965$		0			0
$966$		0			0
$967$	−31.3050			−1.00670			−0.503350	−	0.864083i	$-0.667899\pi$
$967$	−31.3050			−1.00670			−0.503350	+	0.864083i	$0.667899\pi$
$968$			19.7990i			0.636364i
$969$		0			0
$970$		0			0
$971$		−	6.32456i		−	0.202965i	−0.994837	−	0.101482i	$-0.967641\pi$
$971$		−	6.32456i		−	0.202965i	0.994837	−	0.101482i	$-0.0323585\pi$
$972$		0			0
$973$	62.6099			2.00718
$974$	44.0000			1.40985
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		−	25.2982i		−	0.808535i
$980$	58.1378			1.85714
$981$		0			0
$982$		−	2.82843i		−	0.0902587i
$983$		−	48.0833i		−	1.53362i	−0.641875	−	0.766809i	$-0.721843\pi$
$983$		−	48.0833i		−	1.53362i	0.641875	−	0.766809i	$-0.278157\pi$
$984$		0			0
$985$	50.0000			1.59313
$986$		0			0
$987$		0			0
$988$	−35.7771	+	33.9411i	−1.13822	+	1.07981i
$989$		0			0
$990$			18.9737i			0.603023i
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	49.1935			1.55797			0.778987	−	0.627040i	$-0.215734\pi$
$997$	49.1935			1.55797			0.778987	+	0.627040i	$0.215734\pi$
$998$			8.48528i			0.268597i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	760.2.p.a.379.3	yes	4
5.4	even	2	inner	760.2.p.a.379.2	yes	4
8.3	odd	2	inner	760.2.p.a.379.2	yes	4
19.18	odd	2	inner	760.2.p.a.379.1	✓	4
40.19	odd	2	CM	760.2.p.a.379.3	yes	4
95.94	odd	2	inner	760.2.p.a.379.4	yes	4
152.75	even	2	inner	760.2.p.a.379.4	yes	4
760.379	even	2	inner	760.2.p.a.379.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.p.a.379.1	✓	4	19.18	odd	2	inner
760.2.p.a.379.1	✓	4	760.379	even	2	inner
760.2.p.a.379.2	yes	4	5.4	even	2	inner
760.2.p.a.379.2	yes	4	8.3	odd	2	inner
760.2.p.a.379.3	yes	4	1.1	even	1	trivial
760.2.p.a.379.3	yes	4	40.19	odd	2	CM
760.2.p.a.379.4	yes	4	95.94	odd	2	inner
760.2.p.a.379.4	yes	4	152.75	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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} +4.47214 q^{7} -2.82843i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} +4.47214 q^{7} -2.82843i q^{8} -3.00000 q^{9} -3.16228i q^{10} +2.00000 q^{11} -5.65685i q^{13} +6.32456i q^{14} +4.00000 q^{16} -4.24264i q^{18} +(3.00000 + 3.16228i) q^{19} +4.47214 q^{20} +2.82843i q^{22} +4.47214 q^{23} +5.00000 q^{25} +8.00000 q^{26} -8.94427 q^{28} +5.65685i q^{32} -10.0000 q^{35} +6.00000 q^{36} -11.3137i q^{37} +(-4.47214 + 4.24264i) q^{38} +6.32456i q^{40} +12.6491i q^{41} -4.00000 q^{44} +6.70820 q^{45} +6.32456i q^{46} +13.4164 q^{47} +13.0000 q^{49} +7.07107i q^{50} +11.3137i q^{52} +5.65685i q^{53} -4.47214 q^{55} -12.6491i q^{56} -6.32456i q^{59} -13.4164 q^{63} -8.00000 q^{64} +12.6491i q^{65} -14.1421i q^{70} +8.48528i q^{72} +16.0000 q^{74} +(-6.00000 - 6.32456i) q^{76} +8.94427 q^{77} -8.94427 q^{80} +9.00000 q^{81} -17.8885 q^{82} -5.65685i q^{88} -12.6491i q^{89} +9.48683i q^{90} -25.2982i q^{91} -8.94427 q^{92} +18.9737i q^{94} +(-6.70820 - 7.07107i) q^{95} +18.3848i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 12 * q^9	\( 4 q - 8 q^{4} - 12 q^{9} + 8 q^{11} + 16 q^{16} + 12 q^{19} + 20 q^{25} + 32 q^{26} - 40 q^{35} + 24 q^{36} - 16 q^{44} + 52 q^{49} - 32 q^{64} + 64 q^{74} - 24 q^{76} + 36 q^{81} - 24 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 12 * q^9 + 8 * q^11 + 16 * q^16 + 12 * q^19 + 20 * q^25 + 32 * q^26 - 40 * q^35 + 24 * q^36 - 16 * q^44 + 52 * q^49 - 32 * q^64 + 64 * q^74 - 24 * q^76 + 36 * q^81 - 24 * q^99

Introduction

L-functions

Modular forms

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Database

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