LMFDB - Embedded newform 1920.2.m.k.959.1

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:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 4x^{2} + 1$ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			959.1
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.k.959.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +(1.73205 - 1.41421i) q^{5} -2.00000 q^{7} +3.00000 q^{9} +5.65685i q^{11} +(-3.00000 + 2.44949i) q^{15} +3.46410 q^{21} +(1.00000 - 4.89898i) q^{25} -5.19615 q^{27} -10.3923 q^{29} -4.89898i q^{31} -9.79796i q^{33} +(-3.46410 + 2.82843i) q^{35} +(5.19615 - 4.24264i) q^{45} -3.00000 q^{49} +14.1421i q^{53} +(8.00000 + 9.79796i) q^{55} +11.3137i q^{59} -6.00000 q^{63} +9.79796i q^{73} +(-1.73205 + 8.48528i) q^{75} -11.3137i q^{77} -14.6969i q^{79} +9.00000 q^{81} -17.3205 q^{83} +18.0000 q^{87} +8.48528i q^{93} +19.5959i q^{97} +16.9706i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{7} + 12 q^{9} - 12 q^{15} + 4 q^{25} - 12 q^{49} + 32 q^{55} - 24 q^{63} + 36 q^{81} + 72 q^{87}+O(q^{100})$ 4 * q - 8 * q^7 + 12 * q^9 - 12 * q^15 + 4 * q^25 - 12 * q^49 + 32 * q^55 - 24 * q^63 + 36 * q^81 + 72 * 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
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$		0			0
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−3.00000	+	2.44949i	−0.774597	+	0.632456i
$16$		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
$21$	3.46410			0.755929
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$	−10.3923			−1.92980			−0.964901	−	0.262613i	$-0.915416\pi$
$29$	−10.3923			−1.92980			−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	4.89898i		−	0.879883i	−0.898027	−	0.439941i	$-0.854999\pi$
$31$		−	4.89898i		−	0.879883i	0.898027	−	0.439941i	$-0.145001\pi$
$32$		0			0
$33$		−	9.79796i		−	1.70561i
$34$		0			0
$35$	−3.46410	+	2.82843i	−0.585540	+	0.478091i
$36$		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
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	5.19615	−	4.24264i	0.774597	−	0.632456i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$			14.1421i			1.94257i	0.237915	+	0.971286i	$0.423536\pi$
$53$			14.1421i			1.94257i	−0.237915	+	0.971286i	$0.576464\pi$
$54$		0			0
$55$	8.00000	+	9.79796i	1.07872	+	1.32116i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			11.3137i			1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$			11.3137i			1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−6.00000			−0.755929
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			9.79796i			1.14676i	0.819288	+	0.573382i	$0.194369\pi$
$73$			9.79796i			1.14676i	−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$	−1.73205	+	8.48528i	−0.200000	+	0.979796i
$76$		0			0
$77$		−	11.3137i		−	1.28932i
$78$		0			0
$79$		−	14.6969i		−	1.65353i	−0.562544	−	0.826767i	$-0.690177\pi$
$79$		−	14.6969i		−	1.65353i	0.562544	−	0.826767i	$-0.309823\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−17.3205			−1.90117			−0.950586	−	0.310460i	$-0.899517\pi$
$83$	−17.3205			−1.90117			−0.950586	+	0.310460i	$0.899517\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	18.0000			1.92980
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			8.48528i			0.879883i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.5959i			1.98966i	0.101535	+	0.994832i	$0.467625\pi$
$97$			19.5959i			1.98966i	−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$			16.9706i			1.70561i
$100$		0			0
$101$	−3.46410			−0.344691			−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410			−0.344691			−0.172345	+	0.985037i	$0.555135\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$	6.00000	−	4.89898i	0.585540	−	0.478091i
$106$		0			0
$107$	−17.3205			−1.67444			−0.837218	−	0.546869i	$-0.815820\pi$
$107$	−17.3205			−1.67444			−0.837218	+	0.546869i	$0.815820\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		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
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.19615	−	9.89949i	−0.464758	−	0.885438i
$126$		0			0
$127$	−22.0000			−1.95218			−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000			−1.95218			−0.976092	+	0.217357i	$0.930256\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			22.6274i			1.97697i	0.151330	+	0.988483i	$0.451644\pi$
$131$			22.6274i			1.97697i	−0.151330	+	0.988483i	$0.548356\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−9.00000	+	7.34847i	−0.774597	+	0.632456i
$136$		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
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−18.0000	+	14.6969i	−1.49482	+	1.22051i
$146$		0			0
$147$	5.19615			0.428571
$148$		0			0
$149$	24.2487			1.98653			0.993266	−	0.115857i	$-0.0369614\pi$
$149$	24.2487			1.98653			0.993266	+	0.115857i	$0.0369614\pi$
$150$		0			0
$151$			24.4949i			1.99337i	0.0813788	+	0.996683i	$0.474068\pi$
$151$			24.4949i			1.99337i	−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.92820	−	8.48528i	−0.556487	−	0.681554i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		−	24.4949i		−	1.94257i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−13.8564	−	16.9706i	−1.07872	−	1.32116i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$			19.7990i			1.50529i	0.658427	+	0.752645i	$0.271222\pi$
$173$			19.7990i			1.50529i	−0.658427	+	0.752645i	$0.728778\pi$
$174$		0			0
$175$	−2.00000	+	9.79796i	−0.151186	+	0.740656i
$176$		0			0
$177$		−	19.5959i		−	1.47292i
$178$		0			0
$179$		−	11.3137i		−	0.845626i	−0.906217	−	0.422813i	$-0.861043\pi$
$179$		−	11.3137i		−	0.845626i	0.906217	−	0.422813i	$-0.138957\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.3923			0.755929
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.1421i		−	1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$		−	14.1421i		−	1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$		0			0
$199$		−	24.4949i		−	1.73640i	−0.496217	−	0.868199i	$-0.665278\pi$
$199$		−	24.4949i		−	1.73640i	0.496217	−	0.868199i	$-0.334722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20.7846			1.45879
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		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
$215$		0			0
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$		−	16.9706i		−	1.14676i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−26.0000			−1.74109			−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000			−1.74109			−0.870544	+	0.492090i	$0.836233\pi$
$224$		0			0
$225$	3.00000	−	14.6969i	0.200000	−	0.979796i
$226$		0			0
$227$	10.3923			0.689761			0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923			0.689761			0.344881	+	0.938647i	$0.387919\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$			19.5959i			1.28932i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			25.4558i			1.65353i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$	−15.5885			−1.00000
$244$		0			0
$245$	−5.19615	+	4.24264i	−0.331970	+	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	30.0000			1.90117
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		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
$261$	−31.1769			−1.92980
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	20.0000	+	24.4949i	1.22859	+	1.50471i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.3923			0.633630			0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923			0.633630			0.316815	+	0.948487i	$0.397387\pi$
$270$		0			0
$271$			24.4949i			1.48796i	0.668202	+	0.743980i	$0.267064\pi$
$271$			24.4949i			1.48796i	−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	27.7128	+	5.65685i	1.67115	+	0.341121i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		−	14.6969i		−	0.879883i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	33.9411i		−	1.98966i
$292$		0			0
$293$			14.1421i			0.826192i	0.910687	+	0.413096i	$0.135553\pi$
$293$			14.1421i			0.826192i	−0.910687	+	0.413096i	$0.864447\pi$
$294$		0			0
$295$	16.0000	+	19.5959i	0.931556	+	1.14092i
$296$		0			0
$297$		−	29.3939i		−	1.70561i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−24.2487			−1.37946
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	−10.3923	+	8.48528i	−0.585540	+	0.478091i
$316$		0			0
$317$		−	31.1127i		−	1.74746i	−0.486408	−	0.873732i	$-0.661693\pi$
$317$		−	31.1127i		−	1.74746i	0.486408	−	0.873732i	$-0.338307\pi$
$318$		0			0
$319$		−	58.7878i		−	3.29148i
$320$		0			0
$321$	30.0000			1.67444
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		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
$335$		0			0
$336$		0			0
$337$		−	29.3939i		−	1.60119i	−0.599208	−	0.800593i	$-0.704518\pi$
$337$		−	29.3939i		−	1.60119i	0.599208	−	0.800593i	$-0.295482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	27.7128			1.50073
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	24.2487			1.30174			0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487			1.30174			0.650870	+	0.759190i	$0.274404\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		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
$363$	36.3731			1.90909
$364$		0			0
$365$	13.8564	+	16.9706i	0.725277	+	0.888280i
$366$		0			0
$367$	38.0000			1.98358			0.991792	−	0.127862i	$-0.0408116\pi$
$367$	38.0000			1.98358			0.991792	+	0.127862i	$0.0408116\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	28.2843i		−	1.46845i
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	9.00000	+	17.1464i	0.464758	+	0.885438i
$376$		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
$381$	38.1051			1.95218
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	−16.0000	−	19.5959i	−0.815436	−	0.998700i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.2487			−1.22946			−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487			−1.22946			−0.614729	+	0.788738i	$0.710735\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	39.1918i		−	1.97697i
$394$		0			0
$395$	−20.7846	−	25.4558i	−1.04579	−	1.28082i
$396$		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
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	15.5885	−	12.7279i	0.774597	−	0.632456i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	22.6274i		−	1.11342i
$414$		0			0
$415$	−30.0000	+	24.4949i	−1.47264	+	1.20241i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	39.5980i		−	1.93449i	−0.253849	−	0.967244i	$-0.581697\pi$
$419$		−	39.5980i		−	1.93449i	0.253849	−	0.967244i	$-0.418303\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		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
$433$			39.1918i			1.88344i	0.336399	+	0.941720i	$0.390791\pi$
$433$			39.1918i			1.88344i	−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$	31.1769	−	25.4558i	1.49482	−	1.22051i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	24.4949i		−	1.16908i	−0.811366	−	0.584539i	$-0.801275\pi$
$439$		−	24.4949i		−	1.16908i	0.811366	−	0.584539i	$-0.198725\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		0			0
$443$	31.1769			1.48126			0.740630	−	0.671913i	$-0.234527\pi$
$443$	31.1769			1.48126			0.740630	+	0.671913i	$0.234527\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−42.0000			−1.98653
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	42.4264i		−	1.99337i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−38.1051			−1.77473			−0.887366	−	0.461065i	$-0.847467\pi$
$461$	−38.1051			−1.77473			−0.887366	+	0.461065i	$0.847467\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$		0			0
$465$	12.0000	+	14.6969i	0.556487	+	0.681554i
$466$		0			0
$467$	17.3205			0.801498			0.400749	−	0.916188i	$-0.368750\pi$
$467$	17.3205			0.801498			0.400749	+	0.916188i	$0.368750\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			42.4264i			1.94257i
$478$		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
$483$		0			0
$484$		0			0
$485$	27.7128	+	33.9411i	1.25837	+	1.54119i
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.6274i		−	1.02116i	−0.859830	−	0.510581i	$-0.829431\pi$
$491$		−	22.6274i		−	1.02116i	0.859830	−	0.510581i	$-0.170569\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	24.0000	+	29.3939i	1.07872	+	1.32116i
$496$		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
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−6.00000	+	4.89898i	−0.266996	+	0.218002i
$506$		0			0
$507$	22.5167			1.00000
$508$		0			0
$509$	−45.0333			−1.99607			−0.998033	−	0.0626839i	$-0.980034\pi$
$509$	−45.0333			−1.99607			−0.998033	+	0.0626839i	$0.980034\pi$
$510$		0			0
$511$		−	19.5959i		−	0.866872i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.2487	−	19.7990i	1.06853	−	0.872448i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	34.2929i		−	1.50529i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	3.46410	−	16.9706i	0.151186	−	0.740656i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$			33.9411i			1.47292i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−30.0000	+	24.4949i	−1.29701	+	1.05901i
$536$		0			0
$537$			19.5959i			0.845626i
$538$		0			0
$539$		−	16.9706i		−	0.730974i
$540$		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		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			29.3939i			1.24995i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	14.1421i		−	0.599222i	−0.954062	−	0.299611i	$-0.903143\pi$
$557$		−	14.1421i		−	0.599222i	0.954062	−	0.299611i	$-0.0968568\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−38.1051			−1.60594			−0.802970	−	0.596020i	$-0.796748\pi$
$563$	−38.1051			−1.60594			−0.802970	+	0.596020i	$0.796748\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−18.0000			−0.755929
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		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
$575$		0			0
$576$		0			0
$577$			29.3939i			1.22368i	0.790980	+	0.611842i	$0.209571\pi$
$577$			29.3939i			1.22368i	−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		−	16.9706i		−	0.705273i
$580$		0			0
$581$	34.6410			1.43715
$582$		0			0
$583$	−80.0000			−3.31326
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.3205			−0.714894			−0.357447	−	0.933933i	$-0.616353\pi$
$587$	−17.3205			−0.714894			−0.357447	+	0.933933i	$0.616353\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			24.4949i			1.00759i
$592$		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
$597$			42.4264i			1.73640i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−36.3731	+	29.6985i	−1.47878	+	1.20742i
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$	−36.0000			−1.45879
$610$		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
$615$		0			0
$616$		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
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	4.89898i		−	0.195025i	−0.995234	−	0.0975126i	$-0.968911\pi$
$631$		−	4.89898i		−	0.195025i	0.995234	−	0.0975126i	$-0.0310886\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−38.1051	+	31.1127i	−1.51216	+	1.23467i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−64.0000			−2.51222
$650$		0			0
$651$		−	16.9706i		−	0.665129i
$652$		0			0
$653$		−	48.0833i		−	1.88164i	−0.338902	−	0.940822i	$-0.610055\pi$
$653$		−	48.0833i		−	1.88164i	0.338902	−	0.940822i	$-0.389945\pi$
$654$		0			0
$655$	32.0000	+	39.1918i	1.25034	+	1.53135i
$656$		0			0
$657$			29.3939i			1.14676i
$658$		0			0
$659$			45.2548i			1.76288i	0.472298	+	0.881439i	$0.343425\pi$
$659$			45.2548i			1.76288i	−0.472298	+	0.881439i	$0.656575\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	45.0333			1.74109
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	39.1918i		−	1.51073i	−0.655302	−	0.755367i	$-0.727459\pi$
$673$		−	39.1918i		−	1.51073i	0.655302	−	0.755367i	$-0.272541\pi$
$674$		0			0
$675$	−5.19615	+	25.4558i	−0.200000	+	0.979796i
$676$		0			0
$677$			2.82843i			0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$			2.82843i			0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$		0			0
$679$		−	39.1918i		−	1.50404i
$680$		0			0
$681$	−18.0000			−0.689761
$682$		0			0
$683$	51.9615			1.98825			0.994126	−	0.108227i	$-0.0345173\pi$
$683$	51.9615			1.98825			0.994126	+	0.108227i	$0.0345173\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		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
$693$		−	33.9411i		−	1.28932i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.1051			−1.43921			−0.719605	−	0.694383i	$-0.755677\pi$
$701$	−38.1051			−1.43921			−0.719605	+	0.694383i	$0.755677\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.92820			0.260562
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		−	44.0908i		−	1.65353i
$712$		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
$721$	−28.0000			−1.04277
$722$		0			0
$723$	−17.3205			−0.644157
$724$		0			0
$725$	−10.3923	+	50.9117i	−0.385961	+	1.89081i
$726$		0			0
$727$	2.00000			0.0741759			0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000			0.0741759			0.0370879	+	0.999312i	$0.488192\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		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
$735$	9.00000	−	7.34847i	0.331970	−	0.271052i
$736$		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
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	42.0000	−	34.2929i	1.53876	−	1.25639i
$746$		0			0
$747$	−51.9615			−1.90117
$748$		0			0
$749$	34.6410			1.26576
$750$		0			0
$751$			53.8888i			1.96643i	0.182453	+	0.983215i	$0.441596\pi$
$751$			53.8888i			1.96643i	−0.182453	+	0.983215i	$0.558404\pi$
$752$		0			0
$753$			9.79796i			0.357057i
$754$		0			0
$755$	34.6410	+	42.4264i	1.26072	+	1.54406i
$756$		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
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	26.0000			0.937584			0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000			0.937584			0.468792	+	0.883309i	$0.344689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			19.7990i			0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$			19.7990i			0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$		0			0
$775$	−24.0000	−	4.89898i	−0.862105	−	0.175977i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	54.0000			1.92980
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−34.6410	−	42.4264i	−1.22859	−	1.50471i
$796$		0			0
$797$		−	53.7401i		−	1.90357i	−0.306762	−	0.951786i	$-0.599246\pi$
$797$		−	53.7401i		−	1.90357i	0.306762	−	0.951786i	$-0.400754\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−55.4256			−1.95593
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−18.0000			−0.633630
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		−	42.4264i		−	1.48796i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.1769			−1.08808			−0.544041	−	0.839059i	$-0.683106\pi$
$821$	−31.1769			−1.08808			−0.544041	+	0.839059i	$0.683106\pi$
$822$		0			0
$823$	46.0000			1.60346			0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000			1.60346			0.801730	+	0.597687i	$0.203913\pi$
$824$		0			0
$825$	−48.0000	−	9.79796i	−1.67115	−	0.341121i
$826$		0			0
$827$	−10.3923			−0.361376			−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923			−0.361376			−0.180688	+	0.983540i	$0.557832\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$			25.4558i			0.879883i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	79.0000			2.72414
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−22.5167	+	18.3848i	−0.774597	+	0.632456i
$846$		0			0
$847$	42.0000			1.44314
$848$		0			0
$849$		0			0
$850$		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
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	28.0000	+	34.2929i	0.952029	+	1.16599i
$866$		0			0
$867$	29.4449			1.00000
$868$		0			0
$869$	83.1384			2.82028
$870$		0			0
$871$		0			0
$872$		0			0
$873$			58.7878i			1.98966i
$874$		0			0
$875$	10.3923	+	19.7990i	0.351324	+	0.669328i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		−	24.4949i		−	0.826192i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−27.7128	−	33.9411i	−0.931556	−	1.14092i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	44.0000			1.47571
$890$		0			0
$891$			50.9117i			1.70561i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−16.0000	−	19.5959i	−0.534821	−	0.655019i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			50.9117i			1.69800i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−10.3923			−0.344691
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		−	97.9796i		−	3.24265i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	45.2548i		−	1.49445i
$918$		0			0
$919$		−	34.2929i		−	1.13122i	−0.824674	−	0.565608i	$-0.808641\pi$
$919$		−	34.2929i		−	1.13122i	0.824674	−	0.565608i	$-0.191359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	42.0000			1.37946
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	19.5959i		−	0.640171i	−0.947389	−	0.320085i	$-0.896288\pi$
$937$		−	19.5959i		−	0.640171i	0.947389	−	0.320085i	$-0.103712\pi$
$938$		0			0
$939$			16.9706i			0.553813i
$940$		0			0
$941$	38.1051			1.24219			0.621096	−	0.783735i	$-0.286688\pi$
$941$	38.1051			1.24219			0.621096	+	0.783735i	$0.286688\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	18.0000	−	14.6969i	0.585540	−	0.478091i
$946$		0			0
$947$	−24.2487			−0.787977			−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487			−0.787977			−0.393989	+	0.919115i	$0.628905\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$			53.8888i			1.74746i
$952$		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
$957$			101.823i			3.29148i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.00000			0.225806
$962$		0			0
$963$	−51.9615			−1.67444
$964$		0			0
$965$	13.8564	+	16.9706i	0.446054	+	0.546302i
$966$		0			0
$967$	−62.0000			−1.99379			−0.996893	−	0.0787703i	$-0.974901\pi$
$967$	−62.0000			−1.99379			−0.996893	+	0.0787703i	$0.974901\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			62.2254i			1.99691i	0.0555842	+	0.998454i	$0.482298\pi$
$971$			62.2254i			1.99691i	−0.0555842	+	0.998454i	$0.517702\pi$
$972$		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
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	−20.0000	−	24.4949i	−0.637253	−	0.780472i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			24.4949i			0.778106i	0.921215	+	0.389053i	$0.127198\pi$
$991$			24.4949i			0.778106i	−0.921215	+	0.389053i	$0.872802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−34.6410	−	42.4264i	−1.09819	−	1.34501i
$996$		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.k.959.1	✓	4
3.2	odd	2	inner	1920.2.m.k.959.4	yes	4
4.3	odd	2		1920.2.m.l.959.3	yes	4
5.4	even	2		1920.2.m.l.959.4	yes	4
8.3	odd	2		1920.2.m.l.959.2	yes	4
8.5	even	2	inner	1920.2.m.k.959.4	yes	4
12.11	even	2		1920.2.m.l.959.2	yes	4
15.14	odd	2		1920.2.m.l.959.1	yes	4
20.19	odd	2	inner	1920.2.m.k.959.2	yes	4
24.5	odd	2	CM	1920.2.m.k.959.1	✓	4
24.11	even	2		1920.2.m.l.959.3	yes	4
40.19	odd	2	inner	1920.2.m.k.959.3	yes	4
40.29	even	2		1920.2.m.l.959.1	yes	4
60.59	even	2	inner	1920.2.m.k.959.3	yes	4
120.29	odd	2		1920.2.m.l.959.4	yes	4
120.59	even	2	inner	1920.2.m.k.959.2	yes	4

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

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.k.959.1

Level	$1920$
Weight	$2$
Character	1920.959
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-24
Inner twists	$8$