LMFDB - Embedded newform 1920.2.b.f.191.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(191,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("1920.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.b (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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3288334336.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 $ x^8 + 8x^6 - 8x^5 + 14x^4 + 8x^3 - 16*x^2 + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.3
Root			$-0.707107 - 0.179070i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.f.191.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.13705 - 1.30656i) q^{3} +1.00000 q^{5} -1.08239i q^{7} +(-0.414214 + 2.97127i) q^{9} -2.16478i q^{11} +1.74053i q^{13} +(-1.13705 - 1.30656i) q^{15} +4.20201i q^{17} +3.21608 q^{19} +(-1.41421 + 1.23074i) q^{21} +5.49019 q^{23} +1.00000 q^{25} +(4.35313 - 2.83730i) q^{27} +0.828427 q^{29} +3.69552i q^{31} +(-2.82843 + 2.46148i) q^{33} -1.08239i q^{35} -10.1445i q^{37} +(2.27411 - 1.97908i) q^{39} +2.46148i q^{41} +8.70626 q^{43} +(-0.414214 + 2.97127i) q^{45} -10.0384 q^{47} +5.82843 q^{49} +(5.49019 - 4.77791i) q^{51} +2.00000 q^{53} -2.16478i q^{55} +(-3.65685 - 4.20201i) q^{57} -10.4525i q^{59} +10.8655i q^{61} +(3.21608 + 0.448342i) q^{63} +1.74053i q^{65} +4.15804 q^{67} +(-6.24264 - 7.17327i) q^{69} -1.88393 q^{71} +3.65685 q^{73} +(-1.13705 - 1.30656i) q^{75} -2.34315 q^{77} +11.9832i q^{79} +(-8.65685 - 2.46148i) q^{81} -10.0042i q^{83} +4.20201i q^{85} +(-0.941967 - 1.08239i) q^{87} +3.48106i q^{89} +1.88393 q^{91} +(4.82843 - 4.20201i) q^{93} +3.21608 q^{95} +9.31371 q^{97} +(6.43215 + 0.896683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 8 q^{9} + 8 q^{25} - 16 q^{29} + 8 q^{45} + 24 q^{49} + 16 q^{53} + 16 q^{57} - 16 q^{69} - 16 q^{73} - 64 q^{77} - 24 q^{81} + 16 q^{93} - 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^9 + 8 * q^25 - 16 * q^29 + 8 * q^45 + 24 * q^49 + 16 * q^53 + 16 * q^57 - 16 * q^69 - 16 * q^73 - 64 * q^77 - 24 * q^81 + 16 * q^93 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.13705	−	1.30656i	−0.656479	−	0.754344i
$3$	−1.13705	−	1.30656i	−0.656479	−	0.754344i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	1.08239i		−	0.409106i	−0.978856	−	0.204553i	$-0.934426\pi$
$7$		−	1.08239i		−	0.409106i	0.978856	−	0.204553i	$-0.0655740\pi$
$8$		0			0
$9$	−0.414214	+	2.97127i	−0.138071	+	0.990422i
$10$		0			0
$11$		−	2.16478i		−	0.652707i	−0.945248	−	0.326354i	$-0.894180\pi$
$11$		−	2.16478i		−	0.652707i	0.945248	−	0.326354i	$-0.105820\pi$
$12$		0			0
$13$			1.74053i			0.482736i	0.970434	+	0.241368i	$0.0775960\pi$
$13$			1.74053i			0.482736i	−0.970434	+	0.241368i	$0.922404\pi$
$14$		0			0
$15$	−1.13705	−	1.30656i	−0.293586	−	0.337353i
$16$		0			0
$17$			4.20201i			1.01914i	0.860430	+	0.509568i	$0.170195\pi$
$17$			4.20201i			1.01914i	−0.860430	+	0.509568i	$0.829805\pi$
$18$		0			0
$19$	3.21608			0.737818			0.368909	−	0.929465i	$-0.379731\pi$
$19$	3.21608			0.737818			0.368909	+	0.929465i	$0.379731\pi$
$20$		0			0
$21$	−1.41421	+	1.23074i	−0.308607	+	0.268569i
$22$		0			0
$23$	5.49019			1.14478			0.572391	−	0.819981i	$-0.306016\pi$
$23$	5.49019			1.14478			0.572391	+	0.819981i	$0.306016\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.35313	−	2.83730i	0.837760	−	0.546038i
$28$		0			0
$29$	0.828427			0.153835			0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427			0.153835			0.0769175	+	0.997037i	$0.475492\pi$
$30$		0			0
$31$			3.69552i			0.663735i	0.943326	+	0.331867i	$0.107679\pi$
$31$			3.69552i			0.663735i	−0.943326	+	0.331867i	$0.892321\pi$
$32$		0			0
$33$	−2.82843	+	2.46148i	−0.492366	+	0.428488i
$34$		0			0
$35$		−	1.08239i		−	0.182958i
$36$		0			0
$37$		−	10.1445i		−	1.66775i	−0.551952	−	0.833876i	$-0.686117\pi$
$37$		−	10.1445i		−	1.66775i	0.551952	−	0.833876i	$-0.313883\pi$
$38$		0			0
$39$	2.27411	−	1.97908i	0.364149	−	0.316906i
$40$		0			0
$41$			2.46148i			0.384418i	0.981354	+	0.192209i	$0.0615652\pi$
$41$			2.46148i			0.384418i	−0.981354	+	0.192209i	$0.938435\pi$
$42$		0			0
$43$	8.70626			1.32769			0.663846	−	0.747869i	$-0.268923\pi$
$43$	8.70626			1.32769			0.663846	+	0.747869i	$0.268923\pi$
$44$		0			0
$45$	−0.414214	+	2.97127i	−0.0617473	+	0.442930i
$46$		0			0
$47$	−10.0384			−1.46425			−0.732126	−	0.681169i	$-0.761472\pi$
$47$	−10.0384			−1.46425			−0.732126	+	0.681169i	$0.761472\pi$
$48$		0			0
$49$	5.82843			0.832632
$50$		0			0
$51$	5.49019	−	4.77791i	0.768780	−	0.669041i
$52$		0			0
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$		−	2.16478i		−	0.291899i
$56$		0			0
$57$	−3.65685	−	4.20201i	−0.484362	−	0.556569i
$58$		0			0
$59$		−	10.4525i		−	1.36080i	−0.732841	−	0.680400i	$-0.761806\pi$
$59$		−	10.4525i		−	1.36080i	0.732841	−	0.680400i	$-0.238194\pi$
$60$		0			0
$61$			10.8655i			1.39118i	0.718437	+	0.695592i	$0.244858\pi$
$61$			10.8655i			1.39118i	−0.718437	+	0.695592i	$0.755142\pi$
$62$		0			0
$63$	3.21608	+	0.448342i	0.405188	+	0.0564857i
$64$		0			0
$65$			1.74053i			0.215886i
$66$		0			0
$67$	4.15804			0.507986			0.253993	−	0.967206i	$-0.418256\pi$
$67$	4.15804			0.507986			0.253993	+	0.967206i	$0.418256\pi$
$68$		0			0
$69$	−6.24264	−	7.17327i	−0.751526	−	0.863561i
$70$		0			0
$71$	−1.88393			−0.223582			−0.111791	−	0.993732i	$-0.535659\pi$
$71$	−1.88393			−0.223582			−0.111791	+	0.993732i	$0.535659\pi$
$72$		0			0
$73$	3.65685			0.428002			0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685			0.428002			0.214001	+	0.976833i	$0.431350\pi$
$74$		0			0
$75$	−1.13705	−	1.30656i	−0.131296	−	0.150869i
$76$		0			0
$77$	−2.34315			−0.267026
$78$		0			0
$79$			11.9832i			1.34822i	0.738631	+	0.674110i	$0.235473\pi$
$79$			11.9832i			1.34822i	−0.738631	+	0.674110i	$0.764527\pi$
$80$		0			0
$81$	−8.65685	−	2.46148i	−0.961873	−	0.273498i
$82$		0			0
$83$		−	10.0042i		−	1.09810i	−0.835790	−	0.549050i	$-0.814990\pi$
$83$		−	10.0042i		−	1.09810i	0.835790	−	0.549050i	$-0.185010\pi$
$84$		0			0
$85$			4.20201i			0.455772i
$86$		0			0
$87$	−0.941967	−	1.08239i	−0.100989	−	0.116045i
$88$		0			0
$89$			3.48106i			0.368991i	0.982833	+	0.184496i	$0.0590651\pi$
$89$			3.48106i			0.368991i	−0.982833	+	0.184496i	$0.940935\pi$
$90$		0			0
$91$	1.88393			0.197490
$92$		0			0
$93$	4.82843	−	4.20201i	0.500685	−	0.435728i
$94$		0			0
$95$	3.21608			0.329962
$96$		0			0
$97$	9.31371			0.945664			0.472832	−	0.881153i	$-0.343232\pi$
$97$	9.31371			0.945664			0.472832	+	0.881153i	$0.343232\pi$
$98$		0			0
$99$	6.43215	+	0.896683i	0.646456	+	0.0901200i
$100$		0			0
$101$	1.51472			0.150720			0.0753601	−	0.997156i	$-0.475989\pi$
$101$	1.51472			0.150720			0.0753601	+	0.997156i	$0.475989\pi$
$102$		0			0
$103$		−	19.8226i		−	1.95318i	−0.215108	−	0.976590i	$-0.569011\pi$
$103$		−	19.8226i		−	1.95318i	0.215108	−	0.976590i	$-0.430989\pi$
$104$		0			0
$105$	−1.41421	+	1.23074i	−0.138013	+	0.120108i
$106$		0			0
$107$		−	9.10748i		−	0.880453i	−0.897887	−	0.440227i	$-0.854898\pi$
$107$		−	9.10748i		−	0.880453i	0.897887	−	0.440227i	$-0.145102\pi$
$108$		0			0
$109$		−	2.46148i		−	0.235767i	−0.993027	−	0.117883i	$-0.962389\pi$
$109$		−	2.46148i		−	0.235767i	0.993027	−	0.117883i	$-0.0376109\pi$
$110$		0			0
$111$	−13.2545	+	11.5349i	−1.25806	+	1.09484i
$112$		0			0
$113$		−	12.6060i		−	1.18587i	−0.805249	−	0.592937i	$-0.797968\pi$
$113$		−	12.6060i		−	1.18587i	0.805249	−	0.592937i	$-0.202032\pi$
$114$		0			0
$115$	5.49019			0.511962
$116$		0			0
$117$	−5.17157	−	0.720950i	−0.478112	−	0.0666519i
$118$		0			0
$119$	4.54822			0.416935
$120$		0			0
$121$	6.31371			0.573973
$122$		0			0
$123$	3.21608	−	2.79884i	0.289984	−	0.252362i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			8.47343i			0.751895i	0.926641	+	0.375948i	$0.122683\pi$
$127$			8.47343i			0.751895i	−0.926641	+	0.375948i	$0.877317\pi$
$128$		0			0
$129$	−9.89949	−	11.3753i	−0.871602	−	1.00154i
$130$		0			0
$131$		−	11.3492i		−	0.991583i	−0.868442	−	0.495792i	$-0.834878\pi$
$131$		−	11.3492i		−	0.991583i	0.868442	−	0.495792i	$-0.165122\pi$
$132$		0			0
$133$		−	3.48106i		−	0.301846i
$134$		0			0
$135$	4.35313	−	2.83730i	0.374658	−	0.244196i
$136$		0			0
$137$		−	16.0871i		−	1.37441i	−0.726463	−	0.687206i	$-0.758837\pi$
$137$		−	16.0871i		−	1.37441i	0.726463	−	0.687206i	$-0.241163\pi$
$138$		0			0
$139$	3.21608			0.272784			0.136392	−	0.990655i	$-0.456449\pi$
$139$	3.21608			0.272784			0.136392	+	0.990655i	$0.456449\pi$
$140$		0			0
$141$	11.4142	+	13.1158i	0.961250	+	1.10455i
$142$		0			0
$143$	3.76787			0.315085
$144$		0			0
$145$	0.828427			0.0687971
$146$		0			0
$147$	−6.62724	−	7.61521i	−0.546606	−	0.628092i
$148$		0			0
$149$	13.3137			1.09070			0.545351	−	0.838208i	$-0.316396\pi$
$149$	13.3137			1.09070			0.545351	+	0.838208i	$0.316396\pi$
$150$		0			0
$151$			4.96362i			0.403934i	0.979392	+	0.201967i	$0.0647333\pi$
$151$			4.96362i			0.403934i	−0.979392	+	0.201967i	$0.935267\pi$
$152$		0			0
$153$	−12.4853	−	1.74053i	−1.00938	−	0.140713i
$154$		0			0
$155$			3.69552i			0.296831i
$156$		0			0
$157$		−	15.0675i		−	1.20252i	−0.799055	−	0.601259i	$-0.794666\pi$
$157$		−	15.0675i		−	1.20252i	0.799055	−	0.601259i	$-0.205334\pi$
$158$		0			0
$159$	−2.27411	−	2.61313i	−0.180349	−	0.207234i
$160$		0			0
$161$		−	5.94253i		−	0.468337i
$162$		0			0
$163$	−2.27411			−0.178122			−0.0890610	−	0.996026i	$-0.528387\pi$
$163$	−2.27411			−0.178122			−0.0890610	+	0.996026i	$0.528387\pi$
$164$		0			0
$165$	−2.82843	+	2.46148i	−0.220193	+	0.191626i
$166$		0			0
$167$	8.15447			0.631012			0.315506	−	0.948924i	$-0.397826\pi$
$167$	8.15447			0.631012			0.315506	+	0.948924i	$0.397826\pi$
$168$		0			0
$169$	9.97056			0.766966
$170$		0			0
$171$	−1.33214	+	9.55582i	−0.101871	+	0.730752i
$172$		0			0
$173$	−21.3137			−1.62045			−0.810226	−	0.586118i	$-0.800655\pi$
$173$	−21.3137			−1.62045			−0.810226	+	0.586118i	$0.800655\pi$
$174$		0			0
$175$		−	1.08239i		−	0.0818212i
$176$		0			0
$177$	−13.6569	+	11.8851i	−1.02651	+	0.893336i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			23.7701i			1.76682i	0.468601	+	0.883410i	$0.344758\pi$
$181$			23.7701i			1.76682i	−0.468601	+	0.883410i	$0.655242\pi$
$182$		0			0
$183$	14.1964	−	12.3547i	1.04943	−	0.913282i
$184$		0			0
$185$		−	10.1445i		−	0.745841i
$186$		0			0
$187$	9.09644			0.665197
$188$		0			0
$189$	−3.07107	−	4.71179i	−0.223387	−	0.342733i
$190$		0			0
$191$	−17.4125			−1.25993			−0.629963	−	0.776625i	$-0.716930\pi$
$191$	−17.4125			−1.25993			−0.629963	+	0.776625i	$0.716930\pi$
$192$		0			0
$193$	9.31371			0.670415			0.335208	−	0.942144i	$-0.391194\pi$
$193$	9.31371			0.670415			0.335208	+	0.942144i	$0.391194\pi$
$194$		0			0
$195$	2.27411	−	1.97908i	0.162852	−	0.141725i
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		−	10.1899i		−	0.722341i	−0.932500	−	0.361170i	$-0.882377\pi$
$199$		−	10.1899i		−	0.722341i	0.932500	−	0.361170i	$-0.117623\pi$
$200$		0			0
$201$	−4.72792	−	5.43275i	−0.333482	−	0.383196i
$202$		0			0
$203$		−	0.896683i		−	0.0629348i
$204$		0			0
$205$			2.46148i			0.171917i
$206$		0			0
$207$	−2.27411	+	16.3128i	−0.158062	+	1.13382i
$208$		0			0
$209$		−	6.96211i		−	0.481579i
$210$		0			0
$211$	16.8607			1.16074			0.580370	−	0.814353i	$-0.302908\pi$
$211$	16.8607			1.16074			0.580370	+	0.814353i	$0.302908\pi$
$212$		0			0
$213$	2.14214	+	2.46148i	0.146777	+	0.168658i
$214$		0			0
$215$	8.70626			0.593762
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$	−4.15804	−	4.77791i	−0.280975	−	0.322861i
$220$		0			0
$221$	−7.31371			−0.491973
$222$		0			0
$223$		−	25.4203i		−	1.70227i	−0.524948	−	0.851134i	$-0.675915\pi$
$223$		−	25.4203i		−	1.70227i	0.524948	−	0.851134i	$-0.324085\pi$
$224$		0			0
$225$	−0.414214	+	2.97127i	−0.0276142	+	0.198084i
$226$		0			0
$227$			19.1886i			1.27359i	0.771033	+	0.636795i	$0.219740\pi$
$227$			19.1886i			1.27359i	−0.771033	+	0.636795i	$0.780260\pi$
$228$		0			0
$229$		−	8.40401i		−	0.555353i	−0.960675	−	0.277676i	$-0.910436\pi$
$229$		−	8.40401i		−	0.555353i	0.960675	−	0.277676i	$-0.0895643\pi$
$230$		0			0
$231$	2.66428	+	3.06147i	0.175297	+	0.201430i
$232$		0			0
$233$			19.5681i			1.28195i	0.767561	+	0.640975i	$0.221470\pi$
$233$			19.5681i			1.28195i	−0.767561	+	0.640975i	$0.778530\pi$
$234$		0			0
$235$	−10.0384			−0.654833
$236$		0			0
$237$	15.6569	−	13.6256i	1.01702	−	0.885078i
$238$		0			0
$239$	−24.6250			−1.59286			−0.796430	−	0.604730i	$-0.793281\pi$
$239$	−24.6250			−1.59286			−0.796430	+	0.604730i	$0.793281\pi$
$240$		0			0
$241$	20.1421			1.29747			0.648735	−	0.761015i	$-0.275299\pi$
$241$	20.1421			1.29747			0.648735	+	0.761015i	$0.275299\pi$
$242$		0			0
$243$	6.62724	+	14.1096i	0.425138	+	0.905129i
$244$		0			0
$245$	5.82843			0.372365
$246$		0			0
$247$			5.59767i			0.356171i
$248$		0			0
$249$	−13.0711	+	11.3753i	−0.828345	+	0.720879i
$250$		0			0
$251$			18.7402i			1.18287i	0.806352	+	0.591436i	$0.201439\pi$
$251$			18.7402i			1.18287i	−0.806352	+	0.591436i	$0.798561\pi$
$252$		0			0
$253$		−	11.8851i		−	0.747208i
$254$		0			0
$255$	5.49019	−	4.77791i	0.343809	−	0.299204i
$256$		0			0
$257$			19.5681i			1.22063i	0.792160	+	0.610313i	$0.208956\pi$
$257$			19.5681i			1.22063i	−0.792160	+	0.610313i	$0.791044\pi$
$258$		0			0
$259$	−10.9804			−0.682287
$260$		0			0
$261$	−0.343146	+	2.46148i	−0.0212402	+	0.152362i
$262$		0			0
$263$	29.3349			1.80887			0.904433	−	0.426617i	$-0.140295\pi$
$263$	29.3349			1.80887			0.904433	+	0.426617i	$0.140295\pi$
$264$		0			0
$265$	2.00000			0.122859
$266$		0			0
$267$	4.54822	−	3.95815i	0.278346	−	0.242235i
$268$		0			0
$269$	−2.68629			−0.163786			−0.0818930	−	0.996641i	$-0.526097\pi$
$269$	−2.68629			−0.163786			−0.0818930	+	0.996641i	$0.526097\pi$
$270$		0			0
$271$			31.9916i			1.94335i	0.236323	+	0.971674i	$0.424058\pi$
$271$			31.9916i			1.94335i	−0.236323	+	0.971674i	$0.575942\pi$
$272$		0			0
$273$	−2.14214	−	2.46148i	−0.129648	−	0.148975i
$274$		0			0
$275$		−	2.16478i		−	0.130541i
$276$		0			0
$277$			1.74053i			0.104578i	0.998632	+	0.0522891i	$0.0166517\pi$
$277$			1.74053i			0.104578i	−0.998632	+	0.0522891i	$0.983348\pi$
$278$		0			0
$279$	−10.9804	−	1.53073i	−0.657378	−	0.0916426i
$280$		0			0
$281$			9.42359i			0.562164i	0.959684	+	0.281082i	$0.0906933\pi$
$281$			9.42359i			0.562164i	−0.959684	+	0.281082i	$0.909307\pi$
$282$		0			0
$283$	−19.6866			−1.17025			−0.585124	−	0.810944i	$-0.698954\pi$
$283$	−19.6866			−1.17025			−0.585124	+	0.810944i	$0.698954\pi$
$284$		0			0
$285$	−3.65685	−	4.20201i	−0.216613	−	0.248905i
$286$		0			0
$287$	2.66428			0.157268
$288$		0			0
$289$	−0.656854			−0.0386385
$290$		0			0
$291$	−10.5902	−	12.1689i	−0.620808	−	0.713356i
$292$		0			0
$293$	−5.31371			−0.310430			−0.155215	−	0.987881i	$-0.549607\pi$
$293$	−5.31371			−0.310430			−0.155215	+	0.987881i	$0.549607\pi$
$294$		0			0
$295$		−	10.4525i		−	0.608568i
$296$		0			0
$297$	−6.14214	−	9.42359i	−0.356403	−	0.546812i
$298$		0			0
$299$			9.55582i			0.552627i
$300$		0			0
$301$		−	9.42359i		−	0.543167i
$302$		0			0
$303$	−1.72232	−	1.97908i	−0.0989446	−	0.113695i
$304$		0			0
$305$			10.8655i			0.622156i
$306$		0			0
$307$	−28.7831			−1.64274			−0.821368	−	0.570398i	$-0.806789\pi$
$307$	−28.7831			−1.64274			−0.821368	+	0.570398i	$0.806789\pi$
$308$		0			0
$309$	−25.8995	+	22.5394i	−1.47337	+	1.28222i
$310$		0			0
$311$	−23.8447			−1.35211			−0.676054	−	0.736852i	$-0.736311\pi$
$311$	−23.8447			−1.35211			−0.676054	+	0.736852i	$0.736311\pi$
$312$		0			0
$313$	29.3137			1.65691			0.828454	−	0.560057i	$-0.189221\pi$
$313$	29.3137			1.65691			0.828454	+	0.560057i	$0.189221\pi$
$314$		0			0
$315$	3.21608	+	0.448342i	0.181205	+	0.0252612i
$316$		0			0
$317$	−17.3137			−0.972435			−0.486217	−	0.873838i	$-0.661624\pi$
$317$	−17.3137			−0.972435			−0.486217	+	0.873838i	$0.661624\pi$
$318$		0			0
$319$		−	1.79337i		−	0.100409i
$320$		0			0
$321$	−11.8995	+	10.3557i	−0.664165	+	0.577999i
$322$		0			0
$323$			13.5140i			0.751937i
$324$		0			0
$325$			1.74053i			0.0965471i
$326$		0			0
$327$	−3.21608	+	2.79884i	−0.177849	+	0.154776i
$328$		0			0
$329$			10.8655i			0.599034i
$330$		0			0
$331$	−29.7250			−1.63384			−0.816918	−	0.576754i	$-0.804319\pi$
$331$	−29.7250			−1.63384			−0.816918	+	0.576754i	$0.804319\pi$
$332$		0			0
$333$	30.1421	+	4.20201i	1.65178	+	0.230269i
$334$		0			0
$335$	4.15804			0.227178
$336$		0			0
$337$	18.9706			1.03339			0.516696	−	0.856169i	$-0.327162\pi$
$337$	18.9706			1.03339			0.516696	+	0.856169i	$0.327162\pi$
$338$		0			0
$339$	−16.4706	+	14.3337i	−0.894558	+	0.778501i
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$		−	13.8854i		−	0.749741i
$344$		0			0
$345$	−6.24264	−	7.17327i	−0.336092	−	0.386196i
$346$		0			0
$347$			0.819760i			0.0440070i	0.999758	+	0.0220035i	$0.00700450\pi$
$347$			0.819760i			0.0440070i	−0.999758	+	0.0220035i	$0.992996\pi$
$348$		0			0
$349$		−	3.48106i		−	0.186337i	−0.995650	−	0.0931683i	$-0.970301\pi$
$349$		−	3.48106i		−	0.186337i	0.995650	−	0.0931683i	$-0.0296995\pi$
$350$		0			0
$351$	4.93839	+	7.57675i	0.263592	+	0.404417i
$352$		0			0
$353$			9.12496i			0.485673i	0.970067	+	0.242836i	$0.0780778\pi$
$353$			9.12496i			0.485673i	−0.970067	+	0.242836i	$0.921922\pi$
$354$		0			0
$355$	−1.88393			−0.0999888
$356$		0			0
$357$	−5.17157	−	5.94253i	−0.273709	−	0.314512i
$358$		0			0
$359$	2.66428			0.140616			0.0703078	−	0.997525i	$-0.477602\pi$
$359$	2.66428			0.140616			0.0703078	+	0.997525i	$0.477602\pi$
$360$		0			0
$361$	−8.65685			−0.455624
$362$		0			0
$363$	−7.17903	−	8.24926i	−0.376801	−	0.432974i
$364$		0			0
$365$	3.65685			0.191408
$366$		0			0
$367$			25.9456i			1.35435i	0.735824	+	0.677173i	$0.236795\pi$
$367$			25.9456i			1.35435i	−0.735824	+	0.677173i	$0.763205\pi$
$368$		0			0
$369$	−7.31371	−	1.01958i	−0.380736	−	0.0530771i
$370$		0			0
$371$		−	2.16478i		−	0.112390i
$372$		0			0
$373$			17.1067i			0.885749i	0.896584	+	0.442874i	$0.146041\pi$
$373$			17.1067i			0.885749i	−0.896584	+	0.442874i	$0.853959\pi$
$374$		0			0
$375$	−1.13705	−	1.30656i	−0.0587172	−	0.0674706i
$376$		0			0
$377$			1.44190i			0.0742617i
$378$		0			0
$379$	14.9768			0.769306			0.384653	−	0.923061i	$-0.374321\pi$
$379$	14.9768			0.769306			0.384653	+	0.923061i	$0.374321\pi$
$380$		0			0
$381$	11.0711	−	9.63475i	0.567188	−	0.493603i
$382$		0			0
$383$	33.8831			1.73134			0.865672	−	0.500611i	$-0.166891\pi$
$383$	33.8831			1.73134			0.865672	+	0.500611i	$0.166891\pi$
$384$		0			0
$385$	−2.34315			−0.119418
$386$		0			0
$387$	−3.60625	+	25.8686i	−0.183316	+	1.31498i
$388$		0			0
$389$	25.3137			1.28346			0.641728	−	0.766932i	$-0.278218\pi$
$389$	25.3137			1.28346			0.641728	+	0.766932i	$0.278218\pi$
$390$		0			0
$391$			23.0698i			1.16669i
$392$		0			0
$393$	−14.8284	+	12.9046i	−0.747995	+	0.650953i
$394$		0			0
$395$			11.9832i			0.602942i
$396$		0			0
$397$			26.9526i			1.35271i	0.736576	+	0.676355i	$0.236442\pi$
$397$			26.9526i			1.35271i	−0.736576	+	0.676355i	$0.763558\pi$
$398$		0			0
$399$	−4.54822	+	3.95815i	−0.227696	+	0.198155i
$400$		0			0
$401$			11.8851i			0.593512i	0.954953	+	0.296756i	$0.0959048\pi$
$401$			11.8851i			0.593512i	−0.954953	+	0.296756i	$0.904095\pi$
$402$		0			0
$403$	−6.43215			−0.320408
$404$		0			0
$405$	−8.65685	−	2.46148i	−0.430163	−	0.122312i
$406$		0			0
$407$	−21.9607			−1.08855
$408$		0			0
$409$	12.1421			0.600390			0.300195	−	0.953878i	$-0.402948\pi$
$409$	12.1421			0.600390			0.300195	+	0.953878i	$0.402948\pi$
$410$		0			0
$411$	−21.0188	+	18.2919i	−1.03678	+	0.902272i
$412$		0			0
$413$	−11.3137			−0.556711
$414$		0			0
$415$		−	10.0042i		−	0.491085i
$416$		0			0
$417$	−3.65685	−	4.20201i	−0.179077	−	0.205773i
$418$		0			0
$419$			16.0502i			0.784102i	0.919943	+	0.392051i	$0.128234\pi$
$419$			16.0502i			0.784102i	−0.919943	+	0.392051i	$0.871766\pi$
$420$		0			0
$421$		−	17.8276i		−	0.868864i	−0.900705	−	0.434432i	$-0.856949\pi$
$421$		−	17.8276i		−	0.868864i	0.900705	−	0.434432i	$-0.143051\pi$
$422$		0			0
$423$	4.15804	−	29.8268i	0.202171	−	1.45023i
$424$		0			0
$425$			4.20201i			0.203827i
$426$		0			0
$427$	11.7607			0.569141
$428$		0			0
$429$	−4.28427	−	4.92296i	−0.206847	−	0.237683i
$430$		0			0
$431$	−32.9411			−1.58672			−0.793359	−	0.608754i	$-0.791670\pi$
$431$	−32.9411			−1.58672			−0.793359	+	0.608754i	$0.791670\pi$
$432$		0			0
$433$	−21.3137			−1.02427			−0.512136	−	0.858905i	$-0.671146\pi$
$433$	−21.3137			−1.02427			−0.512136	+	0.858905i	$0.671146\pi$
$434$		0			0
$435$	−0.941967	−	1.08239i	−0.0451639	−	0.0518967i
$436$		0			0
$437$	17.6569			0.844642
$438$		0			0
$439$			8.92177i			0.425813i	0.977073	+	0.212906i	$0.0682929\pi$
$439$			8.92177i			0.425813i	−0.977073	+	0.212906i	$0.931707\pi$
$440$		0			0
$441$	−2.41421	+	17.3178i	−0.114963	+	0.824658i
$442$		0			0
$443$			15.6018i			0.741265i	0.928780	+	0.370633i	$0.120859\pi$
$443$			15.6018i			0.741265i	−0.928780	+	0.370633i	$0.879141\pi$
$444$		0			0
$445$			3.48106i			0.165018i
$446$		0			0
$447$	−15.1384	−	17.3952i	−0.716022	−	0.822765i
$448$		0			0
$449$		−	27.6735i		−	1.30599i	−0.757361	−	0.652997i	$-0.773511\pi$
$449$		−	27.6735i		−	1.30599i	0.757361	−	0.652997i	$-0.226489\pi$
$450$		0			0
$451$	5.32857			0.250913
$452$		0			0
$453$	6.48528	−	5.64391i	0.304705	−	0.265174i
$454$		0			0
$455$	1.88393			0.0883202
$456$		0			0
$457$	7.65685			0.358173			0.179086	−	0.983833i	$-0.442686\pi$
$457$	7.65685			0.358173			0.179086	+	0.983833i	$0.442686\pi$
$458$		0			0
$459$	11.9223	+	18.2919i	0.556487	+	0.853792i
$460$		0			0
$461$	−36.8284			−1.71527			−0.857635	−	0.514258i	$-0.828067\pi$
$461$	−36.8284			−1.71527			−0.857635	+	0.514258i	$0.828067\pi$
$462$		0			0
$463$		−	17.1326i		−	0.796218i	−0.917338	−	0.398109i	$-0.869667\pi$
$463$		−	17.1326i		−	0.796218i	0.917338	−	0.398109i	$-0.130333\pi$
$464$		0			0
$465$	4.82843	−	4.20201i	0.223913	−	0.194863i
$466$		0			0
$467$			28.2191i			1.30583i	0.757433	+	0.652913i	$0.226453\pi$
$467$			28.2191i			1.30583i	−0.757433	+	0.652913i	$0.773547\pi$
$468$		0			0
$469$		−	4.50063i		−	0.207820i
$470$		0			0
$471$	−19.6866	+	17.1326i	−0.907112	+	0.789427i
$472$		0			0
$473$		−	18.8472i		−	0.866594i
$474$		0			0
$475$	3.21608			0.147564
$476$		0			0
$477$	−0.828427	+	5.94253i	−0.0379311	+	0.272090i
$478$		0			0
$479$	33.7215			1.54077			0.770386	−	0.637577i	$-0.220063\pi$
$479$	33.7215			1.54077			0.770386	+	0.637577i	$0.220063\pi$
$480$		0			0
$481$	17.6569			0.805083
$482$		0			0
$483$	−7.76429	+	6.75699i	−0.353288	+	0.307453i
$484$		0			0
$485$	9.31371			0.422914
$486$		0			0
$487$		−	0.185709i		−	0.00841528i	−0.999991	−	0.00420764i	$-0.998661\pi$
$487$		−	0.185709i		−	0.00841528i	0.999991	−	0.00420764i	$-0.00133934\pi$
$488$		0			0
$489$	2.58579	+	2.97127i	0.116933	+	0.134365i
$490$		0			0
$491$			20.0083i			0.902963i	0.892280	+	0.451482i	$0.149104\pi$
$491$			20.0083i			0.902963i	−0.892280	+	0.451482i	$0.850896\pi$
$492$		0			0
$493$			3.48106i			0.156779i
$494$		0			0
$495$	6.43215	+	0.896683i	0.289104	+	0.0403029i
$496$		0			0
$497$			2.03916i			0.0914686i
$498$		0			0
$499$	−34.2733			−1.53428			−0.767141	−	0.641479i	$-0.778321\pi$
$499$	−34.2733			−1.53428			−0.767141	+	0.641479i	$0.778321\pi$
$500$		0			0
$501$	−9.27208	−	10.6543i	−0.414246	−	0.476000i
$502$		0			0
$503$	4.70983			0.210001			0.105001	−	0.994472i	$-0.466516\pi$
$503$	4.70983			0.210001			0.105001	+	0.994472i	$0.466516\pi$
$504$		0			0
$505$	1.51472			0.0674041
$506$		0			0
$507$	−11.3371	−	13.0272i	−0.503497	−	0.578557i
$508$		0			0
$509$	33.1127			1.46769			0.733847	−	0.679314i	$-0.237723\pi$
$509$	33.1127			1.46769			0.733847	+	0.679314i	$0.237723\pi$
$510$		0			0
$511$		−	3.95815i		−	0.175098i
$512$		0			0
$513$	14.0000	−	9.12496i	0.618115	−	0.402877i
$514$		0			0
$515$		−	19.8226i		−	0.873489i
$516$		0			0
$517$			21.7310i			0.955727i
$518$		0			0
$519$	24.2349	+	27.8477i	1.06379	+	1.22238i
$520$		0			0
$521$		−	33.6160i		−	1.47275i	−0.676576	−	0.736373i	$-0.736537\pi$
$521$		−	33.6160i		−	1.47275i	0.676576	−	0.736373i	$-0.263463\pi$
$522$		0			0
$523$	9.48661			0.414821			0.207410	−	0.978254i	$-0.433496\pi$
$523$	9.48661			0.414821			0.207410	+	0.978254i	$0.433496\pi$
$524$		0			0
$525$	−1.41421	+	1.23074i	−0.0617213	+	0.0537139i
$526$		0			0
$527$	−15.5286			−0.676436
$528$		0			0
$529$	7.14214			0.310528
$530$		0			0
$531$	31.0572	+	4.32957i	1.34777	+	0.187887i
$532$		0			0
$533$	−4.28427			−0.185572
$534$		0			0
$535$		−	9.10748i		−	0.393751i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	12.6173i		−	0.543465i
$540$		0			0
$541$		−	21.7310i		−	0.934288i	−0.884181	−	0.467144i	$-0.845283\pi$
$541$		−	21.7310i		−	0.934288i	0.884181	−	0.467144i	$-0.154717\pi$
$542$		0			0
$543$	31.0572	−	27.0279i	1.33279	−	1.15988i
$544$		0			0
$545$		−	2.46148i		−	0.105438i
$546$		0			0
$547$	−11.3705			−0.486169			−0.243085	−	0.970005i	$-0.578159\pi$
$547$	−11.3705			−0.486169			−0.243085	+	0.970005i	$0.578159\pi$
$548$		0			0
$549$	−32.2843	−	4.50063i	−1.37786	−	0.192082i
$550$		0			0
$551$	2.66428			0.113502
$552$		0			0
$553$	12.9706			0.551564
$554$		0			0
$555$	−13.2545	+	11.5349i	−0.562621	+	0.489629i
$556$		0			0
$557$	−15.6569			−0.663402			−0.331701	−	0.943385i	$-0.607623\pi$
$557$	−15.6569			−0.663402			−0.331701	+	0.943385i	$0.607623\pi$
$558$		0			0
$559$			15.1535i			0.640924i
$560$		0			0
$561$	−10.3431	−	11.8851i	−0.436688	−	0.501788i
$562$		0			0
$563$			11.7975i			0.497207i	0.968605	+	0.248603i	$0.0799715\pi$
$563$			11.7975i			0.497207i	−0.968605	+	0.248603i	$0.920028\pi$
$564$		0			0
$565$		−	12.6060i		−	0.530339i
$566$		0			0
$567$	−2.66428	+	9.37011i	−0.111889	+	0.393508i
$568$		0			0
$569$			17.8276i			0.747372i	0.927555	+	0.373686i	$0.121906\pi$
$569$			17.8276i			0.747372i	−0.927555	+	0.373686i	$0.878094\pi$
$570$		0			0
$571$	−35.0536			−1.46695			−0.733474	−	0.679718i	$-0.762102\pi$
$571$	−35.0536			−1.46695			−0.733474	+	0.679718i	$0.762102\pi$
$572$		0			0
$573$	19.7990	+	22.7506i	0.827115	+	0.950418i
$574$		0			0
$575$	5.49019			0.228957
$576$		0			0
$577$	−25.3137			−1.05382			−0.526912	−	0.849920i	$-0.676650\pi$
$577$	−25.3137			−1.05382			−0.526912	+	0.849920i	$0.676650\pi$
$578$		0			0
$579$	−10.5902	−	12.1689i	−0.440113	−	0.505724i
$580$		0			0
$581$	−10.8284			−0.449239
$582$		0			0
$583$		−	4.32957i		−	0.179312i
$584$		0			0
$585$	−5.17157	−	0.720950i	−0.213818	−	0.0298076i
$586$		0			0
$587$		−	22.6215i		−	0.933687i	−0.884340	−	0.466844i	$-0.845391\pi$
$587$		−	22.6215i		−	0.933687i	0.884340	−	0.466844i	$-0.154609\pi$
$588$		0			0
$589$			11.8851i			0.489716i
$590$		0			0
$591$	−2.27411	−	2.61313i	−0.0935444	−	0.107490i
$592$		0			0
$593$		−	7.68306i		−	0.315506i	−0.987479	−	0.157753i	$-0.949575\pi$
$593$		−	7.68306i		−	0.315506i	0.987479	−	0.157753i	$-0.0504249\pi$
$594$		0			0
$595$	4.54822			0.186459
$596$		0			0
$597$	−13.3137	+	11.5864i	−0.544894	+	0.474201i
$598$		0			0
$599$	−11.7607			−0.480530			−0.240265	−	0.970707i	$-0.577234\pi$
$599$	−11.7607			−0.480530			−0.240265	+	0.970707i	$0.577234\pi$
$600$		0			0
$601$	0.142136			0.00579783			0.00289892	−	0.999996i	$-0.499077\pi$
$601$	0.142136			0.00579783			0.00289892	+	0.999996i	$0.499077\pi$
$602$		0			0
$603$	−1.72232	+	12.3547i	−0.0701382	+	0.503120i
$604$		0			0
$605$	6.31371			0.256689
$606$		0			0
$607$		−	31.1718i		−	1.26522i	−0.774469	−	0.632612i	$-0.781983\pi$
$607$		−	31.1718i		−	1.26522i	0.774469	−	0.632612i	$-0.218017\pi$
$608$		0			0
$609$	−1.17157	+	1.01958i	−0.0474745	+	0.0413154i
$610$		0			0
$611$		−	17.4721i		−	0.706846i
$612$		0			0
$613$		−	22.0296i		−	0.889767i	−0.895588	−	0.444884i	$-0.853245\pi$
$613$		−	22.0296i		−	0.889767i	0.895588	−	0.444884i	$-0.146755\pi$
$614$		0			0
$615$	3.21608	−	2.79884i	0.129685	−	0.112860i
$616$		0			0
$617$		−	29.4140i		−	1.18416i	−0.805877	−	0.592082i	$-0.798306\pi$
$617$		−	29.4140i		−	1.18416i	0.805877	−	0.592082i	$-0.201694\pi$
$618$		0			0
$619$	−18.7447			−0.753412			−0.376706	−	0.926333i	$-0.622943\pi$
$619$	−18.7447			−0.753412			−0.376706	+	0.926333i	$0.622943\pi$
$620$		0			0
$621$	23.8995	−	15.5773i	0.959054	−	0.625095i
$622$		0			0
$623$	3.76787			0.150956
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−9.09644	+	7.91630i	−0.363277	+	0.316147i
$628$		0			0
$629$	42.6274			1.69967
$630$		0			0
$631$			30.7235i			1.22308i	0.791213	+	0.611541i	$0.209450\pi$
$631$			30.7235i			1.22308i	−0.791213	+	0.611541i	$0.790550\pi$
$632$		0			0
$633$	−19.1716	−	22.0296i	−0.762002	−	0.875598i
$634$		0			0
$635$			8.47343i			0.336258i
$636$		0			0
$637$			10.1445i			0.401941i
$638$		0			0
$639$	0.780351	−	5.59767i	0.0308702	−	0.221440i
$640$		0			0
$641$		−	26.2316i		−	1.03609i	−0.855354	−	0.518043i	$-0.826661\pi$
$641$		−	26.2316i		−	1.03609i	0.855354	−	0.518043i	$-0.173339\pi$
$642$		0			0
$643$	4.15804			0.163977			0.0819886	−	0.996633i	$-0.473873\pi$
$643$	4.15804			0.163977			0.0819886	+	0.996633i	$0.473873\pi$
$644$		0			0
$645$	−9.89949	−	11.3753i	−0.389792	−	0.447901i
$646$		0			0
$647$	−25.5670			−1.00514			−0.502571	−	0.864536i	$-0.667612\pi$
$647$	−25.5670			−1.00514			−0.502571	+	0.864536i	$0.667612\pi$
$648$		0			0
$649$	−22.6274			−0.888204
$650$		0			0
$651$	−4.54822	−	5.22625i	−0.178259	−	0.204833i
$652$		0			0
$653$	−27.6569			−1.08230			−0.541148	−	0.840927i	$-0.682010\pi$
$653$	−27.6569			−1.08230			−0.541148	+	0.840927i	$0.682010\pi$
$654$		0			0
$655$		−	11.3492i		−	0.443449i
$656$		0			0
$657$	−1.51472	+	10.8655i	−0.0590948	+	0.423903i
$658$		0			0
$659$			7.91630i			0.308375i	0.988042	+	0.154188i	$0.0492760\pi$
$659$			7.91630i			0.308375i	−0.988042	+	0.154188i	$0.950724\pi$
$660$		0			0
$661$		−	46.5207i		−	1.80945i	−0.426001	−	0.904723i	$-0.640078\pi$
$661$		−	46.5207i		−	1.80945i	0.426001	−	0.904723i	$-0.359922\pi$
$662$		0			0
$663$	8.31609	+	9.55582i	0.322970	+	0.371117i
$664$		0			0
$665$		−	3.48106i		−	0.134990i
$666$		0			0
$667$	4.54822			0.176108
$668$		0			0
$669$	−33.2132	+	28.9043i	−1.28410	+	1.11750i
$670$		0			0
$671$	23.5214			0.908035
$672$		0			0
$673$	−14.6863			−0.566115			−0.283057	−	0.959103i	$-0.591349\pi$
$673$	−14.6863			−0.566115			−0.283057	+	0.959103i	$0.591349\pi$
$674$		0			0
$675$	4.35313	−	2.83730i	0.167552	−	0.109208i
$676$		0			0
$677$	9.02944			0.347029			0.173515	−	0.984831i	$-0.444488\pi$
$677$	9.02944			0.347029			0.173515	+	0.984831i	$0.444488\pi$
$678$		0			0
$679$		−	10.0811i		−	0.386877i
$680$		0			0
$681$	25.0711	−	21.8184i	0.960725	−	0.836085i
$682$		0			0
$683$		−	21.7248i		−	0.831275i	−0.909530	−	0.415638i	$-0.863558\pi$
$683$		−	21.7248i		−	0.831275i	0.909530	−	0.415638i	$-0.136442\pi$
$684$		0			0
$685$		−	16.0871i		−	0.614655i
$686$		0			0
$687$	−10.9804	+	9.55582i	−0.418927	+	0.364577i
$688$		0			0
$689$			3.48106i			0.132618i
$690$		0			0
$691$	32.3893			1.23215			0.616074	−	0.787688i	$-0.288722\pi$
$691$	32.3893			1.23215			0.616074	+	0.787688i	$0.288722\pi$
$692$		0			0
$693$	0.970563	−	6.96211i	0.0368686	−	0.264469i
$694$		0			0
$695$	3.21608			0.121993
$696$		0			0
$697$	−10.3431			−0.391775
$698$		0			0
$699$	25.5670	−	22.2500i	0.967032	−	0.841574i
$700$		0			0
$701$	1.31371			0.0496181			0.0248090	−	0.999692i	$-0.492102\pi$
$701$	1.31371			0.0496181			0.0248090	+	0.999692i	$0.492102\pi$
$702$		0			0
$703$		−	32.6256i		−	1.23050i
$704$		0			0
$705$	11.4142	+	13.1158i	0.429884	+	0.493970i
$706$		0			0
$707$		−	1.63952i		−	0.0616605i
$708$		0			0
$709$			34.2133i			1.28491i	0.766324	+	0.642454i	$0.222084\pi$
$709$			34.2133i			1.28491i	−0.766324	+	0.642454i	$0.777916\pi$
$710$		0			0
$711$	−35.6054	−	4.96362i	−1.33531	−	0.186150i
$712$		0			0
$713$			20.2891i			0.759832i
$714$		0			0
$715$	3.76787			0.140910
$716$		0			0
$717$	28.0000	+	32.1741i	1.04568	+	1.20157i
$718$		0			0
$719$	9.09644			0.339240			0.169620	−	0.985510i	$-0.445746\pi$
$719$	9.09644			0.339240			0.169620	+	0.985510i	$0.445746\pi$
$720$		0			0
$721$	−21.4558			−0.799057
$722$		0			0
$723$	−22.9027	−	26.3170i	−0.851761	−	0.978739i
$724$		0			0
$725$	0.828427			0.0307670
$726$		0			0
$727$			37.2947i			1.38319i	0.722288	+	0.691593i	$0.243091\pi$
$727$			37.2947i			1.38319i	−0.722288	+	0.691593i	$0.756909\pi$
$728$		0			0
$729$	10.8995	−	24.7022i	0.403685	−	0.914898i
$730$		0			0
$731$			36.5838i			1.35310i
$732$		0			0
$733$			36.7985i			1.35918i	0.733591	+	0.679591i	$0.237843\pi$
$733$			36.7985i			1.35918i	−0.733591	+	0.679591i	$0.762157\pi$
$734$		0			0
$735$	−6.62724	−	7.61521i	−0.244449	−	0.280891i
$736$		0			0
$737$		−	9.00127i		−	0.331566i
$738$		0			0
$739$	31.6090			1.16275			0.581377	−	0.813634i	$-0.302514\pi$
$739$	31.6090			1.16275			0.581377	+	0.813634i	$0.302514\pi$
$740$		0			0
$741$	7.31371	−	6.36486i	0.268676	−	0.233819i
$742$		0			0
$743$	11.9223			0.437388			0.218694	−	0.975793i	$-0.429820\pi$
$743$	11.9223			0.437388			0.218694	+	0.975793i	$0.429820\pi$
$744$		0			0
$745$	13.3137			0.487777
$746$		0			0
$747$	29.7250	+	4.14386i	1.08758	+	0.151616i
$748$		0			0
$749$	−9.85786			−0.360199
$750$		0			0
$751$			24.6005i			0.897686i	0.893611	+	0.448843i	$0.148164\pi$
$751$			24.6005i			0.897686i	−0.893611	+	0.448843i	$0.851836\pi$
$752$		0			0
$753$	24.4853	−	21.3087i	0.892293	−	0.776531i
$754$		0			0
$755$			4.96362i			0.180645i
$756$		0			0
$757$			1.74053i			0.0632606i	0.999500	+	0.0316303i	$0.0100699\pi$
$757$			1.74053i			0.0632606i	−0.999500	+	0.0316303i	$0.989930\pi$
$758$		0			0
$759$	−15.5286	+	13.5140i	−0.563652	+	0.490526i
$760$		0			0
$761$			28.6931i			1.04012i	0.854129	+	0.520062i	$0.174091\pi$
$761$			28.6931i			1.04012i	−0.854129	+	0.520062i	$0.825909\pi$
$762$		0			0
$763$	−2.66428			−0.0964536
$764$		0			0
$765$	−12.4853	−	1.74053i	−0.451406	−	0.0629289i
$766$		0			0
$767$	18.1929			0.656907
$768$		0			0
$769$	12.3431			0.445105			0.222553	−	0.974921i	$-0.428561\pi$
$769$	12.3431			0.445105			0.222553	+	0.974921i	$0.428561\pi$
$770$		0			0
$771$	25.5670	−	22.2500i	0.920773	−	0.801315i
$772$		0			0
$773$	−28.6274			−1.02966			−0.514828	−	0.857293i	$-0.672144\pi$
$773$	−28.6274			−1.02966			−0.514828	+	0.857293i	$0.672144\pi$
$774$		0			0
$775$			3.69552i			0.132747i
$776$		0			0
$777$	12.4853	+	14.3465i	0.447907	+	0.514679i
$778$		0			0
$779$			7.91630i			0.283631i
$780$		0			0
$781$			4.07831i			0.145933i
$782$		0			0
$783$	3.60625	−	2.35049i	0.128877	−	0.0839998i
$784$		0			0
$785$		−	15.0675i		−	0.537782i
$786$		0			0
$787$	−15.9188			−0.567443			−0.283721	−	0.958907i	$-0.591569\pi$
$787$	−15.9188			−0.567443			−0.283721	+	0.958907i	$0.591569\pi$
$788$		0			0
$789$	−33.3553	−	38.3278i	−1.18748	−	1.36451i
$790$		0			0
$791$	−13.6447			−0.485148
$792$		0			0
$793$	−18.9117			−0.671574
$794$		0			0
$795$	−2.27411	−	2.61313i	−0.0806543	−	0.0926780i
$796$		0			0
$797$	26.0000			0.920967			0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000			0.920967			0.460484	+	0.887668i	$0.347676\pi$
$798$		0			0
$799$		−	42.1814i		−	1.49227i
$800$		0			0
$801$	−10.3431	−	1.44190i	−0.365457	−	0.0509470i
$802$		0			0
$803$		−	7.91630i		−	0.279360i
$804$		0			0
$805$		−	5.94253i		−	0.209447i
$806$		0			0
$807$	3.05446	+	3.50981i	0.107522	+	0.123551i
$808$		0			0
$809$		−	1.44190i		−	0.0506945i	−0.999679	−	0.0253473i	$-0.991931\pi$
$809$		−	1.44190i		−	0.0506945i	0.999679	−	0.0253473i	$-0.00806915\pi$
$810$		0			0
$811$	2.43573			0.0855299			0.0427649	−	0.999085i	$-0.486383\pi$
$811$	2.43573			0.0855299			0.0427649	+	0.999085i	$0.486383\pi$
$812$		0			0
$813$	41.7990	−	36.3762i	1.46595	−	1.27577i
$814$		0			0
$815$	−2.27411			−0.0796586
$816$		0			0
$817$	28.0000			0.979596
$818$		0			0
$819$	−0.780351	+	5.59767i	−0.0272677	+	0.195598i
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$			11.5349i			0.402081i	0.979583	+	0.201041i	$0.0644323\pi$
$823$			11.5349i			0.402081i	−0.979583	+	0.201041i	$0.935568\pi$
$824$		0			0
$825$	−2.82843	+	2.46148i	−0.0984732	+	0.0856977i
$826$		0			0
$827$		−	51.8142i		−	1.80176i	−0.434073	−	0.900878i	$-0.642924\pi$
$827$		−	51.8142i		−	1.80176i	0.434073	−	0.900878i	$-0.357076\pi$
$828$		0			0
$829$			26.2316i			0.911062i	0.890220	+	0.455531i	$0.150551\pi$
$829$			26.2316i			0.911062i	−0.890220	+	0.455531i	$0.849449\pi$
$830$		0			0
$831$	2.27411	−	1.97908i	0.0788880	−	0.0686534i
$832$		0			0
$833$			24.4911i			0.848566i
$834$		0			0
$835$	8.15447			0.282197
$836$		0			0
$837$	10.4853	+	16.0871i	0.362424	+	0.556051i
$838$		0			0
$839$	−46.5858			−1.60832			−0.804160	−	0.594413i	$-0.797384\pi$
$839$	−46.5858			−1.60832			−0.804160	+	0.594413i	$0.797384\pi$
$840$		0			0
$841$	−28.3137			−0.976335
$842$		0			0
$843$	12.3125	−	10.7151i	0.424065	−	0.369049i
$844$		0			0
$845$	9.97056			0.342998
$846$		0			0
$847$		−	6.83391i		−	0.234816i
$848$		0			0
$849$	22.3848	+	25.7218i	0.768244	+	0.882771i
$850$		0			0
$851$		−	55.6954i		−	1.90921i
$852$		0			0
$853$		−	22.0296i		−	0.754279i	−0.926156	−	0.377140i	$-0.876908\pi$
$853$		−	22.0296i		−	0.754279i	0.926156	−	0.377140i	$-0.123092\pi$
$854$		0			0
$855$	−1.33214	+	9.55582i	−0.0455583	+	0.326802i
$856$		0			0
$857$			21.6073i			0.738091i	0.929411	+	0.369045i	$0.120315\pi$
$857$			21.6073i			0.738091i	−0.929411	+	0.369045i	$0.879685\pi$
$858$		0			0
$859$	36.9375			1.26029			0.630146	−	0.776476i	$-0.282995\pi$
$859$	36.9375			1.26029			0.630146	+	0.776476i	$0.282995\pi$
$860$		0			0
$861$	−3.02944	−	3.48106i	−0.103243	−	0.118634i
$862$		0			0
$863$	28.2313			0.961004			0.480502	−	0.876994i	$-0.340455\pi$
$863$	28.2313			0.961004			0.480502	+	0.876994i	$0.340455\pi$
$864$		0			0
$865$	−21.3137			−0.724688
$866$		0			0
$867$	0.746879	+	0.858221i	0.0253653	+	0.0291467i
$868$		0			0
$869$	25.9411			0.879992
$870$		0			0
$871$			7.23719i			0.245223i
$872$		0			0
$873$	−3.85786	+	27.6735i	−0.130569	+	0.936607i
$874$		0			0
$875$		−	1.08239i		−	0.0365915i
$876$		0			0
$877$		−	12.1837i		−	0.411414i	−0.978614	−	0.205707i	$-0.934051\pi$
$877$		−	12.1837i		−	0.411414i	0.978614	−	0.205707i	$-0.0659494\pi$
$878$		0			0
$879$	6.04198	+	6.94269i	0.203791	+	0.234171i
$880$		0			0
$881$			43.0396i			1.45004i	0.688727	+	0.725021i	$0.258170\pi$
$881$			43.0396i			1.45004i	−0.688727	+	0.725021i	$0.741830\pi$
$882$		0			0
$883$	32.5509			1.09543			0.547713	−	0.836666i	$-0.315499\pi$
$883$	32.5509			1.09543			0.547713	+	0.836666i	$0.315499\pi$
$884$		0			0
$885$	−13.6569	+	11.8851i	−0.459070	+	0.399512i
$886$		0			0
$887$	−3.60625			−0.121086			−0.0605430	−	0.998166i	$-0.519283\pi$
$887$	−3.60625			−0.121086			−0.0605430	+	0.998166i	$0.519283\pi$
$888$		0			0
$889$	9.17157			0.307605
$890$		0			0
$891$	−5.32857	+	18.7402i	−0.178514	+	0.627821i
$892$		0			0
$893$	−32.2843			−1.08035
$894$		0			0
$895$		0			0
$896$		0			0
$897$	12.4853	−	10.8655i	0.416871	−	0.362788i
$898$		0			0
$899$			3.06147i			0.102106i
$900$		0			0
$901$			8.40401i			0.279978i
$902$		0			0
$903$	−12.3125	+	10.7151i	−0.409735	+	0.356577i
$904$		0			0
$905$			23.7701i			0.790146i
$906$		0			0
$907$	44.3117			1.47134			0.735672	−	0.677338i	$-0.236866\pi$
$907$	44.3117			1.47134			0.735672	+	0.677338i	$0.236866\pi$
$908$		0			0
$909$	−0.627417	+	4.50063i	−0.0208101	+	0.149277i
$910$		0			0
$911$	−10.9804			−0.363796			−0.181898	−	0.983317i	$-0.558224\pi$
$911$	−10.9804			−0.363796			−0.181898	+	0.983317i	$0.558224\pi$
$912$		0			0
$913$	−21.6569			−0.716737
$914$		0			0
$915$	14.1964	−	12.3547i	0.469320	−	0.408432i
$916$		0			0
$917$	−12.2843			−0.405662
$918$		0			0
$919$			16.3128i			0.538110i	0.963125	+	0.269055i	$0.0867113\pi$
$919$			16.3128i			0.538110i	−0.963125	+	0.269055i	$0.913289\pi$
$920$		0			0
$921$	32.7279	+	37.6069i	1.07842	+	1.23919i
$922$		0			0
$923$		−	3.27904i		−	0.107931i
$924$		0			0
$925$		−	10.1445i		−	0.333550i
$926$		0			0
$927$	58.8983	+	8.21080i	1.93447	+	0.269678i
$928$		0			0
$929$			44.4815i			1.45939i	0.683772	+	0.729696i	$0.260338\pi$
$929$			44.4815i			1.45939i	−0.683772	+	0.729696i	$0.739662\pi$
$930$		0			0
$931$	18.7447			0.614332
$932$		0			0
$933$	27.1127	+	31.1546i	0.887630	+	1.01995i
$934$		0			0
$935$	9.09644			0.297485
$936$		0			0
$937$	−44.9117			−1.46720			−0.733600	−	0.679581i	$-0.762162\pi$
$937$	−44.9117			−1.46720			−0.733600	+	0.679581i	$0.762162\pi$
$938$		0			0
$939$	−33.3313	−	38.3002i	−1.08773	−	1.24988i
$940$		0			0
$941$	−37.1127			−1.20984			−0.604920	−	0.796286i	$-0.706795\pi$
$941$	−37.1127			−1.20984			−0.604920	+	0.796286i	$0.706795\pi$
$942$		0			0
$943$			13.5140i			0.440075i
$944$		0			0
$945$	−3.07107	−	4.71179i	−0.0999018	−	0.153275i
$946$		0			0
$947$			10.3756i			0.337161i	0.985688	+	0.168581i	$0.0539183\pi$
$947$			10.3756i			0.337161i	−0.985688	+	0.168581i	$0.946082\pi$
$948$		0			0
$949$			6.36486i			0.206612i
$950$		0			0
$951$	19.6866	+	22.6215i	0.638383	+	0.733551i
$952$		0			0
$953$		−	53.1842i		−	1.72280i	−0.507923	−	0.861402i	$-0.669587\pi$
$953$		−	53.1842i		−	1.72280i	0.507923	−	0.861402i	$-0.330413\pi$
$954$		0			0
$955$	−17.4125			−0.563456
$956$		0			0
$957$	−2.34315	+	2.03916i	−0.0757431	+	0.0659165i
$958$		0			0
$959$	−17.4125			−0.562280
$960$		0			0
$961$	17.3431			0.559456
$962$		0			0
$963$	27.0608	+	3.77244i	0.872021	+	0.121565i
$964$		0			0
$965$	9.31371			0.299819
$966$		0			0
$967$			14.0711i			0.452496i	0.974070	+	0.226248i	$0.0726460\pi$
$967$			14.0711i			0.452496i	−0.974070	+	0.226248i	$0.927354\pi$
$968$		0			0
$969$	17.6569	−	15.3661i	0.567220	−	0.493631i
$970$		0			0
$971$		−	2.16478i		−	0.0694712i	−0.999397	−	0.0347356i	$-0.988941\pi$
$971$		−	2.16478i		−	0.0694712i	0.999397	−	0.0347356i	$-0.0110589\pi$
$972$		0			0
$973$		−	3.48106i		−	0.111598i
$974$		0			0
$975$	2.27411	−	1.97908i	0.0728298	−	0.0633811i
$976$		0			0
$977$		−	14.6452i		−	0.468541i	−0.972172	−	0.234270i	$-0.924730\pi$
$977$		−	14.6452i		−	0.468541i	0.972172	−	0.234270i	$-0.0752701\pi$
$978$		0			0
$979$	7.53574			0.240843
$980$		0			0
$981$	7.31371	+	1.01958i	0.233509	+	0.0325526i
$982$		0			0
$983$	−28.2313			−0.900438			−0.450219	−	0.892918i	$-0.648654\pi$
$983$	−28.2313			−0.900438			−0.450219	+	0.892918i	$0.648654\pi$
$984$		0			0
$985$	2.00000			0.0637253
$986$		0			0
$987$	14.1964	−	12.3547i	0.451878	−	0.393253i
$988$		0			0
$989$	47.7990			1.51992
$990$		0			0
$991$		−	52.3713i		−	1.66363i	−0.555053	−	0.831815i	$-0.687302\pi$
$991$		−	52.3713i		−	1.66363i	0.555053	−	0.831815i	$-0.312698\pi$
$992$		0			0
$993$	33.7990	+	38.8376i	1.07258	+	1.23247i
$994$		0			0
$995$		−	10.1899i		−	0.323041i
$996$		0			0
$997$			22.0296i			0.697685i	0.937181	+	0.348842i	$0.113425\pi$
$997$			22.0296i			0.697685i	−0.937181	+	0.348842i	$0.886575\pi$
$998$		0			0
$999$	−28.7831	−	44.1605i	−0.910656	−	1.39718i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.f.191.3	yes	8
3.2	odd	2		1920.2.b.d.191.4	yes	8
4.3	odd	2	inner	1920.2.b.f.191.6	yes	8
8.3	odd	2		1920.2.b.d.191.3	✓	8
8.5	even	2		1920.2.b.d.191.6	yes	8
12.11	even	2		1920.2.b.d.191.5	yes	8
24.5	odd	2	inner	1920.2.b.f.191.5	yes	8
24.11	even	2	inner	1920.2.b.f.191.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.d.191.3	✓	8	8.3	odd	2
1920.2.b.d.191.4	yes	8	3.2	odd	2
1920.2.b.d.191.5	yes	8	12.11	even	2
1920.2.b.d.191.6	yes	8	8.5	even	2
1920.2.b.f.191.3	yes	8	1.1	even	1	trivial
1920.2.b.f.191.4	yes	8	24.11	even	2	inner
1920.2.b.f.191.5	yes	8	24.5	odd	2	inner
1920.2.b.f.191.6	yes	8	4.3	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.13705 - 1.30656i) q^{3} +1.00000 q^{5} -1.08239i q^{7} +(-0.414214 + 2.97127i) q^{9} -2.16478i q^{11} +1.74053i q^{13} +(-1.13705 - 1.30656i) q^{15} +4.20201i q^{17} +3.21608 q^{19} +(-1.41421 + 1.23074i) q^{21} +5.49019 q^{23} +1.00000 q^{25} +(4.35313 - 2.83730i) q^{27} +0.828427 q^{29} +3.69552i q^{31} +(-2.82843 + 2.46148i) q^{33} -1.08239i q^{35} -10.1445i q^{37} +(2.27411 - 1.97908i) q^{39} +2.46148i q^{41} +8.70626 q^{43} +(-0.414214 + 2.97127i) q^{45} -10.0384 q^{47} +5.82843 q^{49} +(5.49019 - 4.77791i) q^{51} +2.00000 q^{53} -2.16478i q^{55} +(-3.65685 - 4.20201i) q^{57} -10.4525i q^{59} +10.8655i q^{61} +(3.21608 + 0.448342i) q^{63} +1.74053i q^{65} +4.15804 q^{67} +(-6.24264 - 7.17327i) q^{69} -1.88393 q^{71} +3.65685 q^{73} +(-1.13705 - 1.30656i) q^{75} -2.34315 q^{77} +11.9832i q^{79} +(-8.65685 - 2.46148i) q^{81} -10.0042i q^{83} +4.20201i q^{85} +(-0.941967 - 1.08239i) q^{87} +3.48106i q^{89} +1.88393 q^{91} +(4.82843 - 4.20201i) q^{93} +3.21608 q^{95} +9.31371 q^{97} +(6.43215 + 0.896683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 8 q^{9} + 8 q^{25} - 16 q^{29} + 8 q^{45} + 24 q^{49} + 16 q^{53} + 16 q^{57} - 16 q^{69} - 16 q^{73} - 64 q^{77} - 24 q^{81} + 16 q^{93} - 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^9 + 8 * q^25 - 16 * q^29 + 8 * q^45 + 24 * q^49 + 16 * q^53 + 16 * q^57 - 16 * q^69 - 16 * q^73 - 64 * q^77 - 24 * q^81 + 16 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more