LMFDB - Embedded newform 1920.2.m.e.959.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(959,1920)]

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, 1]))

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

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

chi := DirichletCharacter("1920.959");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.m (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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1
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			959.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.e.959.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 - 1.61803i) q^{3} -2.23607 q^{5} -5.23607 q^{7} +(-2.23607 - 2.00000i) q^{9} +(-1.38197 + 3.61803i) q^{15} +(-3.23607 + 8.47214i) q^{21} +5.70820i q^{23} +5.00000 q^{25} +(-4.61803 + 2.38197i) q^{27} +6.00000 q^{29} +11.7082 q^{35} -12.0000i q^{41} +11.2361i q^{43} +(5.00000 + 4.47214i) q^{45} -13.7082i q^{47} +20.4164 q^{49} +8.00000i q^{61} +(11.7082 + 10.4721i) q^{63} +8.18034i q^{67} +(9.23607 + 3.52786i) q^{69} +(3.09017 - 8.09017i) q^{75} +(1.00000 + 8.94427i) q^{81} +4.29180 q^{83} +(3.70820 - 9.70820i) q^{87} +17.8885i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} - 12 q^{7} - 10 q^{15} - 4 q^{21} + 20 q^{25} - 14 q^{27} + 24 q^{29} + 20 q^{35} + 20 q^{45} + 28 q^{49} + 20 q^{63} + 28 q^{69} - 10 q^{75} + 4 q^{81} + 44 q^{83} - 12 q^{87}+O(q^{100})$ 4 * q - 2 * q^3 - 12 * q^7 - 10 * q^15 - 4 * q^21 + 20 * q^25 - 14 * q^27 + 24 * q^29 + 20 * q^35 + 20 * q^45 + 28 * q^49 + 20 * q^63 + 28 * q^69 - 10 * q^75 + 4 * q^81 + 44 * q^83 - 12 * q^87

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
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$4$		0			0
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$		0			0
$7$	−5.23607			−1.97905			−0.989524	−	0.144370i	$-0.953885\pi$
$7$	−5.23607			−1.97905			−0.989524	+	0.144370i	$0.953885\pi$
$8$		0			0
$9$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−1.38197	+	3.61803i	−0.356822	+	0.934172i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−3.23607	+	8.47214i	−0.706168	+	1.84877i
$22$		0			0
$23$			5.70820i			1.19024i	0.803636	+	0.595121i	$0.202896\pi$
$23$			5.70820i			1.19024i	−0.803636	+	0.595121i	$0.797104\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	−4.61803	+	2.38197i	−0.888741	+	0.458410i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	11.7082			1.97905
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	12.0000i		−	1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$		−	12.0000i		−	1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$		0			0
$43$			11.2361i			1.71348i	0.515745	+	0.856742i	$0.327515\pi$
$43$			11.2361i			1.71348i	−0.515745	+	0.856742i	$0.672485\pi$
$44$		0			0
$45$	5.00000	+	4.47214i	0.745356	+	0.666667i
$46$		0			0
$47$		−	13.7082i		−	1.99955i	−0.0212814	−	0.999774i	$-0.506775\pi$
$47$		−	13.7082i		−	1.99955i	0.0212814	−	0.999774i	$-0.493225\pi$
$48$		0			0
$49$	20.4164			2.91663
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$	11.7082	+	10.4721i	1.47510	+	1.31937i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.18034i			0.999388i	0.866202	+	0.499694i	$0.166554\pi$
$67$			8.18034i			0.999388i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		0			0
$69$	9.23607	+	3.52786i	1.11189	+	0.424705i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	3.09017	−	8.09017i	0.356822	−	0.934172i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	4.29180			0.471086			0.235543	−	0.971864i	$-0.424313\pi$
$83$	4.29180			0.471086			0.235543	+	0.971864i	$0.424313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.70820	−	9.70820i	0.397561	−	1.04083i
$88$		0			0
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−2.18034			−0.214835			−0.107418	−	0.994214i	$-0.534258\pi$
$103$	−2.18034			−0.214835			−0.107418	+	0.994214i	$0.534258\pi$
$104$		0			0
$105$	7.23607	−	18.9443i	0.706168	−	1.84877i
$106$		0			0
$107$	−19.7082			−1.90526			−0.952632	−	0.304125i	$-0.901636\pi$
$107$	−19.7082			−1.90526			−0.952632	+	0.304125i	$0.901636\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		−	12.7639i		−	1.19024i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−19.4164	−	7.41641i	−1.75072	−	0.668715i
$124$		0			0
$125$	−11.1803			−1.00000
$126$		0			0
$127$	−12.6525			−1.12273			−0.561363	−	0.827570i	$-0.689723\pi$
$127$	−12.6525			−1.12273			−0.561363	+	0.827570i	$0.689723\pi$
$128$		0			0
$129$	18.1803	+	6.94427i	1.60069	+	0.611409i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.3262	−	5.32624i	0.888741	−	0.458410i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−22.1803	−	8.47214i	−1.86792	−	0.713483i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−13.4164			−1.11417
$146$		0			0
$147$	12.6180	−	33.0344i	1.04072	−	2.72463i
$148$		0			0
$149$	4.47214			0.366372			0.183186	−	0.983078i	$-0.441359\pi$
$149$	4.47214			0.366372			0.183186	+	0.983078i	$0.441359\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	29.8885i		−	2.35555i
$162$		0			0
$163$			6.65248i			0.521062i	0.965465	+	0.260531i	$0.0838976\pi$
$163$			6.65248i			0.521062i	−0.965465	+	0.260531i	$0.916102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			10.2918i			0.796403i	0.917298	+	0.398202i	$0.130366\pi$
$167$			10.2918i			0.796403i	−0.917298	+	0.398202i	$0.869634\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−26.1803			−1.97905
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			26.8328i			1.99447i	0.0743294	+	0.997234i	$0.476318\pi$
$181$			26.8328i			1.99447i	−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	12.9443	+	4.94427i	0.956868	+	0.365491i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	24.1803	−	12.4721i	1.75886	−	0.907214i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	13.2361	+	5.05573i	0.933600	+	0.356604i
$202$		0			0
$203$	−31.4164			−2.20500
$204$		0			0
$205$			26.8328i			1.87409i
$206$		0			0
$207$	11.4164	−	12.7639i	0.793495	−	0.887155i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	25.1246i		−	1.71348i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	18.7639			1.25653			0.628263	−	0.778001i	$-0.283766\pi$
$223$	18.7639			1.25653			0.628263	+	0.778001i	$0.283766\pi$
$224$		0			0
$225$	−11.1803	−	10.0000i	−0.745356	−	0.666667i
$226$		0			0
$227$	27.1246			1.80032			0.900162	−	0.435556i	$-0.143448\pi$
$227$	27.1246			1.80032			0.900162	+	0.435556i	$0.143448\pi$
$228$		0			0
$229$		−	26.8328i		−	1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$		−	26.8328i		−	1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			30.6525i			1.99955i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−13.4164			−0.864227			−0.432113	−	0.901819i	$-0.642232\pi$
$241$	−13.4164			−0.864227			−0.432113	+	0.901819i	$0.642232\pi$
$242$		0			0
$243$	15.0902	+	3.90983i	0.968035	+	0.250816i
$244$		0			0
$245$	−45.6525			−2.91663
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.65248	−	6.94427i	0.168094	−	0.440075i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−13.4164	−	12.0000i	−0.830455	−	0.742781i
$262$		0			0
$263$			9.12461i			0.562648i	0.959613	+	0.281324i	$0.0907735\pi$
$263$			9.12461i			0.562648i	−0.959613	+	0.281324i	$0.909226\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	28.9443	+	11.0557i	1.77136	+	0.676600i
$268$		0			0
$269$	22.3607			1.36335			0.681677	−	0.731653i	$-0.261251\pi$
$269$	22.3607			1.36335			0.681677	+	0.731653i	$0.261251\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$			32.1803i			1.91292i	0.291859	+	0.956461i	$0.405726\pi$
$283$			32.1803i			1.91292i	−0.291859	+	0.956461i	$0.594274\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			62.8328i			3.70890i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	58.8328i		−	3.39107i
$302$		0			0
$303$	11.1246	−	29.1246i	0.639092	−	1.67317i
$304$		0			0
$305$		−	17.8885i		−	1.02430i
$306$		0			0
$307$			27.5967i			1.57503i	0.616296	+	0.787515i	$0.288633\pi$
$307$			27.5967i			1.57503i	−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$	−1.34752	+	3.52786i	−0.0766580	+	0.200693i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−26.1803	−	23.4164i	−1.47510	−	1.31937i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.1803	+	31.8885i	−0.679840	+	1.77984i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	25.8885	+	9.88854i	1.43164	+	0.546838i
$328$		0			0
$329$			71.7771i			3.95720i
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	18.2918i		−	0.999388i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−70.2492			−3.79310
$344$		0			0
$345$	−20.6525	−	7.88854i	−1.11189	−	0.424705i
$346$		0			0
$347$	3.12461			0.167738			0.0838690	−	0.996477i	$-0.473272\pi$
$347$	3.12461			0.167738			0.0838690	+	0.996477i	$0.473272\pi$
$348$		0			0
$349$		−	26.8328i		−	1.43633i	−0.695874	−	0.718164i	$-0.744983\pi$
$349$		−	26.8328i		−	1.43633i	0.695874	−	0.718164i	$-0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	6.79837	−	17.7984i	0.356822	−	0.934172i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−29.2361			−1.52611			−0.763055	−	0.646333i	$-0.776302\pi$
$367$	−29.2361			−1.52611			−0.763055	+	0.646333i	$0.776302\pi$
$368$		0			0
$369$	−24.0000	+	26.8328i	−1.24939	+	1.39686i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−6.90983	+	18.0902i	−0.356822	+	0.934172i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−7.81966	+	20.4721i	−0.400613	+	1.04882i
$382$		0			0
$383$		−	1.12461i		−	0.0574650i	−0.999587	−	0.0287325i	$-0.990853\pi$
$383$		−	1.12461i		−	0.0574650i	0.999587	−	0.0287325i	$-0.00914709\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.4721	−	25.1246i	1.14232	−	1.27716i
$388$		0			0
$389$	31.3050			1.58722			0.793612	−	0.608424i	$-0.208198\pi$
$389$	31.3050			1.58722			0.793612	+	0.608424i	$0.208198\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	35.7771i		−	1.78662i	−0.449439	−	0.893311i	$-0.648376\pi$
$401$		−	35.7771i		−	1.78662i	0.449439	−	0.893311i	$-0.351624\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.23607	−	20.0000i	−0.111111	−	0.993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	40.2492			1.99020			0.995098	−	0.0988936i	$-0.0315304\pi$
$409$	40.2492			1.99020			0.995098	+	0.0988936i	$0.0315304\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−9.59675			−0.471086
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	8.00000i		−	0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$421$		−	8.00000i		−	0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$		0			0
$423$	−27.4164	+	30.6525i	−1.33303	+	1.49037i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	41.8885i		−	2.02713i
$428$		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		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−8.29180	+	21.7082i	−0.397561	+	1.04083i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−45.6525	−	40.8328i	−2.17393	−	1.94442i
$442$		0			0
$443$	35.7082			1.69655			0.848274	−	0.529558i	$-0.177642\pi$
$443$	35.7082			1.69655			0.848274	+	0.529558i	$0.177642\pi$
$444$		0			0
$445$		−	40.0000i		−	1.89618i
$446$		0			0
$447$	2.76393	−	7.23607i	0.130729	−	0.342254i
$448$		0			0
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	20.0689			0.932680			0.466340	−	0.884606i	$-0.345572\pi$
$463$	20.0689			0.932680			0.466340	+	0.884606i	$0.345572\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−43.1246			−1.99557			−0.997785	−	0.0665285i	$-0.978808\pi$
$467$	−43.1246			−1.99557			−0.997785	+	0.0665285i	$0.978808\pi$
$468$		0			0
$469$		−	42.8328i		−	1.97784i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−48.3607	−	18.4721i	−2.20049	−	0.840511i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	11.3475			0.514205			0.257103	−	0.966384i	$-0.417232\pi$
$487$	11.3475			0.514205			0.257103	+	0.966384i	$0.417232\pi$
$488$		0			0
$489$	10.7639	+	4.11146i	0.486762	+	0.185926i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	16.6525	+	6.36068i	0.743978	+	0.284174i
$502$		0			0
$503$		−	37.7082i		−	1.68133i	−0.541559	−	0.840663i	$-0.682166\pi$
$503$		−	37.7082i		−	1.68133i	0.541559	−	0.840663i	$-0.317834\pi$
$504$		0			0
$505$	−40.2492			−1.79107
$506$		0			0
$507$	−8.03444	+	21.0344i	−0.356822	+	0.934172i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.87539			0.214835
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8885i			0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$			17.8885i			0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$			3.59675i			0.157275i	0.996903	+	0.0786374i	$0.0250569\pi$
$523$			3.59675i			0.157275i	−0.996903	+	0.0786374i	$0.974943\pi$
$524$		0			0
$525$	−16.1803	+	42.3607i	−0.706168	+	1.84877i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−9.58359			−0.416678
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	44.0689			1.90526
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$			26.8328i			1.15363i	0.816874	+	0.576816i	$0.195705\pi$
$541$			26.8328i			1.15363i	−0.816874	+	0.576816i	$0.804295\pi$
$542$		0			0
$543$	43.4164	+	16.5836i	1.86318	+	0.711670i
$544$		0			0
$545$		−	35.7771i		−	1.53252i
$546$		0			0
$547$			35.2361i			1.50659i	0.657685	+	0.753293i	$0.271536\pi$
$547$			35.2361i			1.50659i	−0.657685	+	0.753293i	$0.728464\pi$
$548$		0			0
$549$	16.0000	−	17.8885i	0.682863	−	0.763464i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		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
$563$	34.5410			1.45573			0.727865	−	0.685720i	$-0.240513\pi$
$563$	34.5410			1.45573			0.727865	+	0.685720i	$0.240513\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−5.23607	−	46.8328i	−0.219894	−	1.96679i
$568$		0			0
$569$		−	36.0000i		−	1.50920i	−0.656186	−	0.754599i	$-0.727831\pi$
$569$		−	36.0000i		−	1.50920i	0.656186	−	0.754599i	$-0.272169\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			28.5410i			1.19024i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−22.4721			−0.932301
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.5410			−1.09547			−0.547733	−	0.836653i	$-0.684509\pi$
$587$	−26.5410			−1.09547			−0.547733	+	0.836653i	$0.684509\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−40.2492			−1.64180			−0.820900	−	0.571072i	$-0.806528\pi$
$601$	−40.2492			−1.64180			−0.820900	+	0.571072i	$0.806528\pi$
$602$		0			0
$603$	16.3607	−	18.2918i	0.666258	−	0.744900i
$604$		0			0
$605$	−24.5967			−1.00000
$606$		0			0
$607$	21.8197			0.885633			0.442816	−	0.896612i	$-0.353979\pi$
$607$	21.8197			0.885633			0.442816	+	0.896612i	$0.353979\pi$
$608$		0			0
$609$	−19.4164	+	50.8328i	−0.786793	+	2.05985i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$	43.4164	+	16.5836i	1.75072	+	0.668715i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−13.5967	−	26.3607i	−0.545619	−	1.05782i
$622$		0			0
$623$		−	93.6656i		−	3.75263i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	28.2918			1.12273
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12.0000i			0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$			12.0000i			0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$		0			0
$643$		−	50.0689i		−	1.97452i	−0.159103	−	0.987262i	$-0.550860\pi$
$643$		−	50.0689i		−	1.97452i	0.159103	−	0.987262i	$-0.449140\pi$
$644$		0			0
$645$	−40.6525	−	15.5279i	−1.60069	−	0.611409i
$646$		0			0
$647$		−	20.5410i		−	0.807551i	−0.914858	−	0.403775i	$-0.867698\pi$
$647$		−	20.5410i		−	0.807551i	0.914858	−	0.403775i	$-0.132302\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			32.0000i			1.24466i	0.782757	+	0.622328i	$0.213813\pi$
$661$			32.0000i			1.24466i	−0.782757	+	0.622328i	$0.786187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			34.2492i			1.32614i
$668$		0			0
$669$	11.5967	−	30.3607i	0.448356	−	1.17381i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−23.0902	+	11.9098i	−0.888741	+	0.458410i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	16.7639	−	43.8885i	0.642395	−	1.68181i
$682$		0			0
$683$	51.1246			1.95623			0.978114	−	0.208068i	$-0.0667174\pi$
$683$	51.1246			1.95623			0.978114	+	0.208068i	$0.0667174\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−43.4164	−	16.5836i	−1.65644	−	0.632704i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.3607			−0.844551			−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607			−0.844551			−0.422276	+	0.906467i	$0.638769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	49.5967	+	18.9443i	1.86792	+	0.713483i
$706$		0			0
$707$	−94.2492			−3.54461
$708$		0			0
$709$			26.8328i			1.00773i	0.863783	+	0.503864i	$0.168089\pi$
$709$			26.8328i			1.00773i	−0.863783	+	0.503864i	$0.831911\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	11.4164			0.425169
$722$		0			0
$723$	−8.29180	+	21.7082i	−0.308375	+	0.807337i
$724$		0			0
$725$	30.0000			1.11417
$726$		0			0
$727$	41.0132			1.52109			0.760547	−	0.649283i	$-0.224931\pi$
$727$	41.0132			1.52109			0.760547	+	0.649283i	$0.224931\pi$
$728$		0			0
$729$	15.6525	−	22.0000i	0.579721	−	0.814815i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−28.2148	+	73.8673i	−1.04072	+	2.72463i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			52.5410i			1.92754i	0.266729	+	0.963772i	$0.414057\pi$
$743$			52.5410i			1.92754i	−0.266729	+	0.963772i	$0.585943\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	−9.59675	−	8.58359i	−0.351127	−	0.314057i
$748$		0			0
$749$	103.193			3.77061
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			35.7771i			1.29692i	0.761249	+	0.648459i	$0.224586\pi$
$761$			35.7771i			1.29692i	−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$		−	83.7771i		−	3.03293i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−27.7082	+	14.2918i	−0.990210	+	0.510747i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	2.06888i		−	0.0737477i	−0.999320	−	0.0368739i	$-0.988260\pi$
$787$		−	2.06888i		−	0.0737477i	0.999320	−	0.0368739i	$-0.0117400\pi$
$788$		0			0
$789$	14.7639	+	5.63932i	0.525610	+	0.200765i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	35.7771	−	40.0000i	1.26412	−	1.41333i
$802$		0			0
$803$		0			0
$804$		0			0
$805$			66.8328i			2.35555i
$806$		0			0
$807$	13.8197	−	36.1803i	0.486475	−	1.27361i
$808$		0			0
$809$			17.8885i			0.628928i	0.949269	+	0.314464i	$0.101825\pi$
$809$			17.8885i			0.628928i	−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	14.8754i		−	0.521062i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.3050			−1.09255			−0.546275	−	0.837606i	$-0.683955\pi$
$821$	−31.3050			−1.09255			−0.546275	+	0.837606i	$0.683955\pi$
$822$		0			0
$823$	−50.1803			−1.74918			−0.874588	−	0.484866i	$-0.838868\pi$
$823$	−50.1803			−1.74918			−0.874588	+	0.484866i	$0.838868\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.5410			0.366547			0.183274	−	0.983062i	$-0.441331\pi$
$827$	10.5410			0.366547			0.183274	+	0.983062i	$0.441331\pi$
$828$		0			0
$829$			56.0000i			1.94496i	0.232986	+	0.972480i	$0.425151\pi$
$829$			56.0000i			1.94496i	−0.232986	+	0.972480i	$0.574849\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	23.0132i		−	0.796403i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	19.4164	+	7.41641i	0.668737	+	0.255435i
$844$		0			0
$845$	29.0689			1.00000
$846$		0			0
$847$	−57.5967			−1.97905
$848$		0			0
$849$	52.0689	+	19.8885i	1.78700	+	0.682573i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	101.666	+	38.8328i	3.46476	+	1.32342i
$862$		0			0
$863$			34.2918i			1.16731i	0.812003	+	0.583653i	$0.198377\pi$
$863$			34.2918i			1.16731i	−0.812003	+	0.583653i	$0.801623\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−10.5066	+	27.5066i	−0.356822	+	0.934172i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	58.5410			1.97905
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.0000i			0.404290i	0.979356	+	0.202145i	$0.0647913\pi$
$881$			12.0000i			0.404290i	−0.979356	+	0.202145i	$0.935209\pi$
$882$		0			0
$883$			54.6525i			1.83920i	0.392853	+	0.919601i	$0.371488\pi$
$883$			54.6525i			1.83920i	−0.392853	+	0.919601i	$0.628512\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			57.1246i			1.91806i	0.283310	+	0.959028i	$0.408567\pi$
$887$			57.1246i			1.91806i	−0.283310	+	0.959028i	$0.591433\pi$
$888$		0			0
$889$	66.2492			2.22193
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−95.1935	−	36.3607i	−3.16784	−	1.21001i
$904$		0			0
$905$		−	60.0000i		−	1.99447i
$906$		0			0
$907$		−	45.4853i		−	1.51031i	−0.655544	−	0.755157i	$-0.727561\pi$
$907$		−	45.4853i		−	1.51031i	0.655544	−	0.755157i	$-0.272439\pi$
$908$		0			0
$909$	−40.2492	−	36.0000i	−1.33498	−	1.19404i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−28.9443	−	11.0557i	−0.956868	−	0.365491i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	44.6525	+	17.0557i	1.47135	+	0.562005i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	4.87539	+	4.36068i	0.160129	+	0.143224i
$928$		0			0
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	42.0000			1.36916			0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000			1.36916			0.684580	+	0.728937i	$0.259985\pi$
$942$		0			0
$943$	68.4984			2.23062
$944$		0			0
$945$	−54.0689	+	27.8885i	−1.75886	+	0.907214i
$946$		0			0
$947$	−36.2918			−1.17932			−0.589662	−	0.807650i	$-0.700739\pi$
$947$	−36.2918			−1.17932			−0.589662	+	0.807650i	$0.700739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	44.0689	+	39.4164i	1.42010	+	1.27018i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−3.93112			−0.126416			−0.0632081	−	0.998000i	$-0.520133\pi$
$967$	−3.93112			−0.126416			−0.0632081	+	0.998000i	$0.520133\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	32.0000	−	35.7771i	1.02168	−	1.14227i
$982$		0			0
$983$			4.54102i			0.144836i	0.997374	+	0.0724180i	$0.0230716\pi$
$983$			4.54102i			0.144836i	−0.997374	+	0.0724180i	$0.976928\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	116.138	+	44.3607i	3.69671	+	1.41202i
$988$		0			0
$989$	−64.1378			−2.03946
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.m.e.959.3	✓	4
3.2	odd	2		1920.2.m.q.959.1	yes	4
4.3	odd	2		1920.2.m.r.959.2	yes	4
5.4	even	2		1920.2.m.r.959.2	yes	4
8.3	odd	2		1920.2.m.f.959.3	yes	4
8.5	even	2		1920.2.m.q.959.2	yes	4
12.11	even	2		1920.2.m.f.959.4	yes	4
15.14	odd	2		1920.2.m.f.959.4	yes	4
20.19	odd	2	CM	1920.2.m.e.959.3	✓	4
24.5	odd	2	inner	1920.2.m.e.959.4	yes	4
24.11	even	2		1920.2.m.r.959.1	yes	4
40.19	odd	2		1920.2.m.q.959.2	yes	4
40.29	even	2		1920.2.m.f.959.3	yes	4
60.59	even	2		1920.2.m.q.959.1	yes	4
120.29	odd	2		1920.2.m.r.959.1	yes	4
120.59	even	2	inner	1920.2.m.e.959.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.m.e.959.3	✓	4	1.1	even	1	trivial
1920.2.m.e.959.3	✓	4	20.19	odd	2	CM
1920.2.m.e.959.4	yes	4	24.5	odd	2	inner
1920.2.m.e.959.4	yes	4	120.59	even	2	inner
1920.2.m.f.959.3	yes	4	8.3	odd	2
1920.2.m.f.959.3	yes	4	40.29	even	2
1920.2.m.f.959.4	yes	4	12.11	even	2
1920.2.m.f.959.4	yes	4	15.14	odd	2
1920.2.m.q.959.1	yes	4	3.2	odd	2
1920.2.m.q.959.1	yes	4	60.59	even	2
1920.2.m.q.959.2	yes	4	8.5	even	2
1920.2.m.q.959.2	yes	4	40.19	odd	2
1920.2.m.r.959.1	yes	4	24.11	even	2
1920.2.m.r.959.1	yes	4	120.29	odd	2
1920.2.m.r.959.2	yes	4	4.3	odd	2
1920.2.m.r.959.2	yes	4	5.4	even	2

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 $\F_{q}$

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Newspace parameters

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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1920.2.m.e.959.3

Level	$1920$
Weight	$2$
Character	1920.959
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-20
Inner twists	$4$