LMFDB - Embedded newform 1840.2.e.d.369.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not 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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.527896576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 2x^5 + 7x^4 - 10x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.8
Root			$-1.07037 - 1.07037i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.d.369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14073i q^{3} +(-2.07037 + 0.844739i) q^{5} +1.20647i q^{7} -6.86420 q^{9} +O(q^{10})$	$q+3.14073i q^{3} +(-2.07037 + 0.844739i) q^{5} +1.20647i q^{7} -6.86420 q^{9} -3.65773 q^{11} -0.859268i q^{13} +(-2.65310 - 6.50246i) q^{15} +6.72347i q^{17} -1.51699 q^{19} -3.78921 q^{21} -1.00000i q^{23} +(3.57283 - 3.49784i) q^{25} -12.1364i q^{27} -0.548747 q^{29} +5.99568 q^{31} -11.4879i q^{33} +(-1.01915 - 2.49784i) q^{35} +2.04100i q^{37} +2.69873 q^{39} -7.14998 q^{41} -10.0799i q^{43} +(14.2114 - 5.79846i) q^{45} +9.17040i q^{47} +5.54442 q^{49} -21.1166 q^{51} -5.37896i q^{53} +(7.57283 - 3.08982i) q^{55} -4.76447i q^{57} -0.582734 q^{59} -8.83244 q^{61} -8.28146i q^{63} +(0.725857 + 1.77900i) q^{65} +3.20647i q^{67} +3.14073 q^{69} +12.5784 q^{71} -8.62597i q^{73} +(10.9858 + 11.2213i) q^{75} -4.41294i q^{77} +0.0700619 q^{79} +17.5246 q^{81} +6.74197i q^{83} +(-5.67958 - 13.9200i) q^{85} -1.72347i q^{87} -4.96393 q^{89} +1.03668 q^{91} +18.8308i q^{93} +(3.14073 - 1.28146i) q^{95} +11.3380i q^{97} +25.1074 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 6 * q^5 - 8 * q^9	$ 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			3.14073i			1.81330i	0.421881	+	0.906651i	$0.361370\pi$
$3$			3.14073i			1.81330i	−0.421881	+	0.906651i	$0.638630\pi$
$4$		0			0
$5$	−2.07037	+	0.844739i	−0.925896	+	0.377779i
$5$	−2.07037	+	0.844739i	−0.925896	+	0.377779i
$6$		0			0
$7$			1.20647i			0.456004i	0.973661	+	0.228002i	$0.0732192\pi$
$7$			1.20647i			0.456004i	−0.973661	+	0.228002i	$0.926781\pi$
$8$		0			0
$9$	−6.86420			−2.28807
$10$		0			0
$11$	−3.65773			−1.10285			−0.551423	−	0.834226i	$-0.685915\pi$
$11$	−3.65773			−1.10285			−0.551423	+	0.834226i	$0.685915\pi$
$12$		0			0
$13$		−	0.859268i		−	0.238318i	−0.992875	−	0.119159i	$-0.961980\pi$
$13$		−	0.859268i		−	0.238318i	0.992875	−	0.119159i	$-0.0380198\pi$
$14$		0			0
$15$	−2.65310	−	6.50246i	−0.685027	−	1.67893i
$16$		0			0
$17$			6.72347i			1.63068i	0.578982	+	0.815340i	$0.303450\pi$
$17$			6.72347i			1.63068i	−0.578982	+	0.815340i	$0.696550\pi$
$18$		0			0
$19$	−1.51699			−0.348022			−0.174011	−	0.984744i	$-0.555673\pi$
$19$	−1.51699			−0.348022			−0.174011	+	0.984744i	$0.555673\pi$
$20$		0			0
$21$	−3.78921			−0.826873
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	3.57283	−	3.49784i	0.714566	−	0.699568i
$26$		0			0
$27$		−	12.1364i		−	2.33565i
$28$		0			0
$29$	−0.548747			−0.101900			−0.0509498	−	0.998701i	$-0.516225\pi$
$29$	−0.548747			−0.101900			−0.0509498	+	0.998701i	$0.516225\pi$
$30$		0			0
$31$	5.99568			1.07686			0.538428	−	0.842672i	$-0.319018\pi$
$31$	5.99568			1.07686			0.538428	+	0.842672i	$0.319018\pi$
$32$		0			0
$33$		−	11.4879i		−	1.99979i
$34$		0			0
$35$	−1.01915	−	2.49784i	−0.172269	−	0.422212i
$36$		0			0
$37$			2.04100i			0.335539i	0.985826	+	0.167770i	$0.0536565\pi$
$37$			2.04100i			0.335539i	−0.985826	+	0.167770i	$0.946344\pi$
$38$		0			0
$39$	2.69873			0.432143
$40$		0			0
$41$	−7.14998			−1.11664			−0.558320	−	0.829626i	$-0.688554\pi$
$41$	−7.14998			−1.11664			−0.558320	+	0.829626i	$0.688554\pi$
$42$		0			0
$43$		−	10.0799i		−	1.53717i	−0.639745	−	0.768587i	$-0.720960\pi$
$43$		−	10.0799i		−	1.53717i	0.639745	−	0.768587i	$-0.279040\pi$
$44$		0			0
$45$	14.2114	−	5.79846i	2.11851	−	0.864383i
$46$		0			0
$47$			9.17040i			1.33764i	0.743424	+	0.668820i	$0.233200\pi$
$47$			9.17040i			1.33764i	−0.743424	+	0.668820i	$0.766800\pi$
$48$		0			0
$49$	5.54442			0.792061
$50$		0			0
$51$	−21.1166			−2.95692
$52$		0			0
$53$		−	5.37896i		−	0.738857i	−0.929259	−	0.369428i	$-0.879554\pi$
$53$		−	5.37896i		−	0.738857i	0.929259	−	0.369428i	$-0.120446\pi$
$54$		0			0
$55$	7.57283	−	3.08982i	1.02112	−	0.416632i
$56$		0			0
$57$		−	4.76447i		−	0.631070i
$58$		0			0
$59$	−0.582734			−0.0758655			−0.0379327	−	0.999280i	$-0.512077\pi$
$59$	−0.582734			−0.0758655			−0.0379327	+	0.999280i	$0.512077\pi$
$60$		0			0
$61$	−8.83244			−1.13088			−0.565439	−	0.824790i	$-0.691293\pi$
$61$	−8.83244			−1.13088			−0.565439	+	0.824790i	$0.691293\pi$
$62$		0			0
$63$		−	8.28146i		−	1.04337i
$64$		0			0
$65$	0.725857	+	1.77900i	0.0900315	+	0.220658i
$66$		0			0
$67$			3.20647i			0.391733i	0.980631	+	0.195866i	$0.0627519\pi$
$67$			3.20647i			0.391733i	−0.980631	+	0.195866i	$0.937248\pi$
$68$		0			0
$69$	3.14073			0.378100
$70$		0			0
$71$	12.5784			1.49278			0.746391	−	0.665507i	$-0.231785\pi$
$71$	12.5784			1.49278			0.746391	+	0.665507i	$0.231785\pi$
$72$		0			0
$73$		−	8.62597i		−	1.00959i	−0.863238	−	0.504797i	$-0.831567\pi$
$73$		−	8.62597i		−	1.00959i	0.863238	−	0.504797i	$-0.168433\pi$
$74$		0			0
$75$	10.9858	+	11.2213i	1.26853	+	1.29572i
$76$		0			0
$77$		−	4.41294i		−	0.502902i
$78$		0			0
$79$	0.0700619			0.00788258			0.00394129	−	0.999992i	$-0.498745\pi$
$79$	0.0700619			0.00788258			0.00394129	+	0.999992i	$0.498745\pi$
$80$		0			0
$81$	17.5246			1.94718
$82$		0			0
$83$			6.74197i			0.740027i	0.929026	+	0.370014i	$0.120647\pi$
$83$			6.74197i			0.740027i	−0.929026	+	0.370014i	$0.879353\pi$
$84$		0			0
$85$	−5.67958	−	13.9200i	−0.616036	−	1.50984i
$86$		0			0
$87$		−	1.72347i		−	0.184775i
$88$		0			0
$89$	−4.96393			−0.526175			−0.263088	−	0.964772i	$-0.584741\pi$
$89$	−4.96393			−0.526175			−0.263088	+	0.964772i	$0.584741\pi$
$90$		0			0
$91$	1.03668			0.108674
$92$		0			0
$93$			18.8308i			1.95266i
$94$		0			0
$95$	3.14073	−	1.28146i	0.322232	−	0.131475i
$96$		0			0
$97$			11.3380i			1.15119i	0.817733	+	0.575597i	$0.195230\pi$
$97$			11.3380i			1.15119i	−0.817733	+	0.575597i	$0.804770\pi$
$98$		0			0
$99$	25.1074			2.52338
$100$		0			0
$101$	3.44693			0.342983			0.171491	−	0.985186i	$-0.445141\pi$
$101$	3.44693			0.342983			0.171491	+	0.985186i	$0.445141\pi$
$102$		0			0
$103$			9.13641i			0.900237i	0.892969	+	0.450119i	$0.148618\pi$
$103$			9.13641i			0.900237i	−0.892969	+	0.450119i	$0.851382\pi$
$104$		0			0
$105$	7.84504	−	3.20089i	0.765598	−	0.312375i
$106$		0			0
$107$		−	9.24539i		−	0.893786i	−0.894588	−	0.446893i	$-0.852531\pi$
$107$		−	9.24539i		−	0.893786i	0.894588	−	0.446893i	$-0.147469\pi$
$108$		0			0
$109$	−1.64847			−0.157895			−0.0789476	−	0.996879i	$-0.525156\pi$
$109$	−1.64847			−0.157895			−0.0789476	+	0.996879i	$0.525156\pi$
$110$		0			0
$111$	−6.41025			−0.608434
$112$		0			0
$113$		−	1.20647i		−	0.113495i	−0.998389	−	0.0567477i	$-0.981927\pi$
$113$		−	1.20647i		−	0.113495i	0.998389	−	0.0567477i	$-0.0180731\pi$
$114$		0			0
$115$	0.844739	+	2.07037i	0.0787723	+	0.193063i
$116$		0			0
$117$			5.89819i			0.545287i
$118$		0			0
$119$	−8.11167			−0.743596
$120$		0			0
$121$	2.37896			0.216269
$122$		0			0
$123$		−	22.4562i		−	2.02481i
$124$		0			0
$125$	−4.44231	+	10.2599i	−0.397332	+	0.917675i
$126$		0			0
$127$		−	15.3542i		−	1.36247i	−0.732066	−	0.681233i	$-0.761444\pi$
$127$		−	15.3542i		−	1.36247i	0.732066	−	0.681233i	$-0.238556\pi$
$128$		0			0
$129$	31.6583			2.78736
$130$		0			0
$131$	−13.0734			−1.14223			−0.571113	−	0.820872i	$-0.693488\pi$
$131$	−13.0734			−1.14223			−0.571113	+	0.820872i	$0.693488\pi$
$132$		0			0
$133$		−	1.83021i		−	0.158699i
$134$		0			0
$135$	10.2521	+	25.1268i	0.882361	+	2.16257i
$136$		0			0
$137$		−	13.1840i		−	1.12638i	−0.826327	−	0.563191i	$-0.809573\pi$
$137$		−	13.1840i		−	1.12638i	0.826327	−	0.563191i	$-0.190427\pi$
$138$		0			0
$139$	−14.1160			−1.19730			−0.598652	−	0.801010i	$-0.704297\pi$
$139$	−14.1160			−1.19730			−0.598652	+	0.801010i	$0.704297\pi$
$140$		0			0
$141$	−28.8018			−2.42555
$142$		0			0
$143$			3.14297i			0.262828i
$144$		0			0
$145$	1.13611	−	0.463548i	0.0943485	−	0.0384955i
$146$		0			0
$147$			17.4136i			1.43625i
$148$		0			0
$149$	6.54442			0.536140			0.268070	−	0.963399i	$-0.413614\pi$
$149$	6.54442			0.536140			0.268070	+	0.963399i	$0.413614\pi$
$150$		0			0
$151$	−2.61672			−0.212946			−0.106473	−	0.994316i	$-0.533956\pi$
$151$	−2.61672			−0.212946			−0.106473	+	0.994316i	$0.533956\pi$
$152$		0			0
$153$		−	46.1512i		−	3.73110i
$154$		0			0
$155$	−12.4132	+	5.06479i	−0.997056	+	0.406813i
$156$		0			0
$157$			8.81394i			0.703429i	0.936107	+	0.351715i	$0.114401\pi$
$157$			8.81394i			0.703429i	−0.936107	+	0.351715i	$0.885599\pi$
$158$		0			0
$159$	16.8939			1.33977
$160$		0			0
$161$	1.20647			0.0950833
$162$		0			0
$163$		−	10.1747i		−	0.796946i	−0.917180	−	0.398473i	$-0.869540\pi$
$163$		−	10.1747i		−	0.796946i	0.917180	−	0.398473i	$-0.130460\pi$
$164$		0			0
$165$	9.70431	+	23.7842i	0.755480	+	1.85160i
$166$		0			0
$167$			12.6647i			0.980027i	0.871715	+	0.490014i	$0.163008\pi$
$167$			12.6647i			0.980027i	−0.871715	+	0.490014i	$0.836992\pi$
$168$		0			0
$169$	12.2617			0.943205
$170$		0			0
$171$	10.4129			0.796298
$172$		0			0
$173$			15.6901i			1.19290i	0.802652	+	0.596448i	$0.203422\pi$
$173$			15.6901i			1.19290i	−0.802652	+	0.596448i	$0.796578\pi$
$174$		0			0
$175$	4.22005	+	4.31052i	0.319005	+	0.325845i
$176$		0			0
$177$		−	1.83021i		−	0.137567i
$178$		0			0
$179$	−2.39876			−0.179292			−0.0896460	−	0.995974i	$-0.528574\pi$
$179$	−2.39876			−0.179292			−0.0896460	+	0.995974i	$0.528574\pi$
$180$		0			0
$181$	−9.87122			−0.733722			−0.366861	−	0.930276i	$-0.619567\pi$
$181$	−9.87122			−0.733722			−0.366861	+	0.930276i	$0.619567\pi$
$182$		0			0
$183$		−	27.7403i		−	2.05063i
$184$		0			0
$185$	−1.72412	−	4.22563i	−0.126760	−	0.310674i
$186$		0			0
$187$		−	24.5926i		−	1.79839i
$188$		0			0
$189$	14.6422			1.06507
$190$		0			0
$191$	9.60617			0.695078			0.347539	−	0.937666i	$-0.387017\pi$
$191$	9.60617			0.695078			0.347539	+	0.937666i	$0.387017\pi$
$192$		0			0
$193$		−	24.9709i		−	1.79745i	−0.438515	−	0.898724i	$-0.644495\pi$
$193$		−	24.9709i		−	1.79745i	0.438515	−	0.898724i	$-0.355505\pi$
$194$		0			0
$195$	−5.58736	+	2.27972i	−0.400119	+	0.163254i
$196$		0			0
$197$		−	9.92292i		−	0.706979i	−0.935439	−	0.353489i	$-0.884995\pi$
$197$		−	9.92292i		−	0.706979i	0.935439	−	0.353489i	$-0.115005\pi$
$198$		0			0
$199$	7.01818			0.497506			0.248753	−	0.968567i	$-0.419979\pi$
$199$	7.01818			0.497506			0.248753	+	0.968567i	$0.419979\pi$
$200$		0			0
$201$	−10.0707			−0.710330
$202$		0			0
$203$		−	0.662047i		−	0.0464666i
$204$		0			0
$205$	14.8031	−	6.03987i	1.03389	−	0.421843i
$206$		0			0
$207$			6.86420i			0.477095i
$208$		0			0
$209$	5.54875			0.383815
$210$		0			0
$211$	−7.44693			−0.512668			−0.256334	−	0.966588i	$-0.582515\pi$
$211$	−7.44693			−0.512668			−0.256334	+	0.966588i	$0.582515\pi$
$212$		0			0
$213$			39.5054i			2.70687i
$214$		0			0
$215$	8.51491	+	20.8691i	0.580712	+	1.42326i
$216$		0			0
$217$			7.23362i			0.491050i
$218$		0			0
$219$	27.0919			1.83070
$220$		0			0
$221$	5.77726			0.388620
$222$		0			0
$223$		−	10.2180i		−	0.684245i	−0.939655	−	0.342123i	$-0.888854\pi$
$223$		−	10.2180i		−	0.684245i	0.939655	−	0.342123i	$-0.111146\pi$
$224$		0			0
$225$	−24.5246	+	24.0099i	−1.63497	+	1.60066i
$226$		0			0
$227$			5.83291i			0.387144i	0.981086	+	0.193572i	$0.0620072\pi$
$227$			5.83291i			0.387144i	−0.981086	+	0.193572i	$0.937993\pi$
$228$		0			0
$229$	−5.87061			−0.387941			−0.193970	−	0.981007i	$-0.562137\pi$
$229$	−5.87061			−0.387941			−0.193970	+	0.981007i	$0.562137\pi$
$230$		0			0
$231$	13.8599			0.911913
$232$		0			0
$233$			0.839462i			0.0549950i	0.999622	+	0.0274975i	$0.00875383\pi$
$233$			0.839462i			0.0549950i	−0.999622	+	0.0274975i	$0.991246\pi$
$234$		0			0
$235$	−7.74660	−	18.9861i	−0.505332	−	1.23852i
$236$		0			0
$237$			0.220046i			0.0142935i
$238$		0			0
$239$	−21.3296			−1.37970			−0.689850	−	0.723953i	$-0.742323\pi$
$239$	−21.3296			−1.37970			−0.689850	+	0.723953i	$0.742323\pi$
$240$		0			0
$241$	−20.7037			−1.33364			−0.666820	−	0.745219i	$-0.732345\pi$
$241$	−20.7037			−1.33364			−0.666820	+	0.745219i	$0.732345\pi$
$242$		0			0
$243$			18.6309i			1.19517i
$244$		0			0
$245$	−11.4790	+	4.68359i	−0.733366	+	0.299224i
$246$		0			0
$247$			1.30350i			0.0829400i
$248$		0			0
$249$	−21.1747			−1.34189
$250$		0			0
$251$	−23.7290			−1.49776			−0.748881	−	0.662705i	$-0.769408\pi$
$251$	−23.7290			−1.49776			−0.748881	+	0.662705i	$0.769408\pi$
$252$		0			0
$253$			3.65773i			0.229959i
$254$		0			0
$255$	43.7191	−	17.8380i	2.73780	−	1.11706i
$256$		0			0
$257$		−	18.2481i		−	1.13828i	−0.822239	−	0.569142i	$-0.807275\pi$
$257$		−	18.2481i		−	1.13828i	0.822239	−	0.569142i	$-0.192725\pi$
$258$		0			0
$259$	−2.46242			−0.153007
$260$		0			0
$261$	3.76670			0.233153
$262$		0			0
$263$			25.8718i			1.59532i	0.603104	+	0.797662i	$0.293930\pi$
$263$			25.8718i			1.59532i	−0.603104	+	0.797662i	$0.706070\pi$
$264$		0			0
$265$	4.54382	+	11.1364i	0.279124	+	0.684104i
$266$		0			0
$267$		−	15.5904i		−	0.954115i
$268$		0			0
$269$	−7.34497			−0.447831			−0.223915	−	0.974609i	$-0.571884\pi$
$269$	−7.34497			−0.447831			−0.223915	+	0.974609i	$0.571884\pi$
$270$		0			0
$271$	−30.2278			−1.83621			−0.918105	−	0.396338i	$-0.870281\pi$
$271$	−30.2278			−1.83621			−0.918105	+	0.396338i	$0.870281\pi$
$272$		0			0
$273$			3.25594i			0.197059i
$274$		0			0
$275$	−13.0684	+	12.7941i	−0.788056	+	0.771515i
$276$		0			0
$277$			2.50983i			0.150801i	0.997153	+	0.0754005i	$0.0240235\pi$
$277$			2.50983i			0.150801i	−0.997153	+	0.0754005i	$0.975976\pi$
$278$		0			0
$279$	−41.1555			−2.46392
$280$		0			0
$281$	−10.4836			−0.625400			−0.312700	−	0.949852i	$-0.601233\pi$
$281$	−10.4836			−0.625400			−0.312700	+	0.949852i	$0.601233\pi$
$282$		0			0
$283$			3.83946i			0.228232i	0.993467	+	0.114116i	$0.0364036\pi$
$283$			3.83946i			0.228232i	−0.993467	+	0.114116i	$0.963596\pi$
$284$		0			0
$285$	4.02474	+	9.86420i	0.238405	+	0.584305i
$286$		0			0
$287$		−	8.62626i		−	0.509192i
$288$		0			0
$289$	−28.2050			−1.65912
$290$		0			0
$291$	−35.6095			−2.08746
$292$		0			0
$293$		−	0.544425i		−	0.0318056i	−0.999874	−	0.0159028i	$-0.994938\pi$
$293$		−	0.544425i		−	0.0318056i	0.999874	−	0.0159028i	$-0.00506224\pi$
$294$		0			0
$295$	1.20647	−	0.492258i	0.0702435	−	0.0286604i
$296$		0			0
$297$			44.3917i			2.57587i
$298$		0			0
$299$	−0.859268			−0.0496927
$300$		0			0
$301$	12.1611			0.700957
$302$		0			0
$303$			10.8259i			0.621931i
$304$		0			0
$305$	18.2864	−	7.46111i	1.04708	−	0.427222i
$306$		0			0
$307$		−	16.1850i		−	0.923729i	−0.886950	−	0.461865i	$-0.847181\pi$
$307$		−	16.1850i		−	0.923729i	0.886950	−	0.461865i	$-0.152819\pi$
$308$		0			0
$309$	−28.6950			−1.63240
$310$		0			0
$311$	7.51923			0.426376			0.213188	−	0.977011i	$-0.431615\pi$
$311$	7.51923			0.426376			0.213188	+	0.977011i	$0.431615\pi$
$312$		0			0
$313$		−	25.2475i		−	1.42707i	−0.700619	−	0.713536i	$-0.747093\pi$
$313$		−	25.2475i		−	1.42707i	0.700619	−	0.713536i	$-0.252907\pi$
$314$		0			0
$315$	6.99568	+	17.1457i	0.394162	+	0.966049i
$316$		0			0
$317$		−	13.4173i		−	0.753589i	−0.926297	−	0.376794i	$-0.877026\pi$
$317$		−	13.4173i		−	0.753589i	0.926297	−	0.376794i	$-0.122974\pi$
$318$		0			0
$319$	2.00716			0.112380
$320$		0			0
$321$	29.0373			1.62070
$322$		0			0
$323$		−	10.1995i		−	0.567513i
$324$		0			0
$325$	−3.00558	−	3.07002i	−0.166720	−	0.170294i
$326$		0			0
$327$		−	5.17741i		−	0.286312i
$328$		0			0
$329$	−11.0638			−0.609969
$330$		0			0
$331$	24.3748			1.33976			0.669880	−	0.742470i	$-0.266346\pi$
$331$	24.3748			1.33976			0.669880	+	0.742470i	$0.266346\pi$
$332$		0			0
$333$		−	14.0099i		−	0.767736i
$334$		0			0
$335$	−2.70863	−	6.63857i	−0.147988	−	0.362704i
$336$		0			0
$337$			19.2454i			1.04836i	0.851607	+	0.524182i	$0.175629\pi$
$337$			19.2454i			1.04836i	−0.851607	+	0.524182i	$0.824371\pi$
$338$		0			0
$339$	3.78921			0.205801
$340$		0			0
$341$	−21.9305			−1.18761
$342$		0			0
$343$			15.1345i			0.817186i
$344$		0			0
$345$	−6.50246	+	2.65310i	−0.350081	+	0.142838i
$346$		0			0
$347$			7.51044i			0.403181i	0.979470	+	0.201591i	$0.0646111\pi$
$347$			7.51044i			0.403181i	−0.979470	+	0.201591i	$0.935389\pi$
$348$		0			0
$349$	−12.3208			−0.659520			−0.329760	−	0.944065i	$-0.606968\pi$
$349$	−12.3208			−0.659520			−0.329760	+	0.944065i	$0.606968\pi$
$350$		0			0
$351$	−10.4284			−0.556628
$352$		0			0
$353$			11.7333i			0.624502i	0.950000	+	0.312251i	$0.101083\pi$
$353$			11.7333i			0.624502i	−0.950000	+	0.312251i	$0.898917\pi$
$354$		0			0
$355$	−26.0419	+	10.6255i	−1.38216	+	0.563942i
$356$		0			0
$357$		−	25.4766i		−	1.34836i
$358$		0			0
$359$	−18.6879			−0.986307			−0.493154	−	0.869942i	$-0.664156\pi$
$359$	−18.6879			−0.986307			−0.493154	+	0.869942i	$0.664156\pi$
$360$		0			0
$361$	−16.6987			−0.878881
$362$		0			0
$363$			7.47167i			0.392161i
$364$		0			0
$365$	7.28670	+	17.8589i	0.381403	+	0.934779i
$366$		0			0
$367$			23.0103i			1.20113i	0.799576	+	0.600564i	$0.205057\pi$
$367$			23.0103i			1.20113i	−0.799576	+	0.600564i	$0.794943\pi$
$368$		0			0
$369$	49.0789			2.55495
$370$		0			0
$371$	6.48956			0.336921
$372$		0			0
$373$		−	35.7402i		−	1.85056i	−0.379290	−	0.925278i	$-0.623832\pi$
$373$		−	35.7402i		−	1.85056i	0.379290	−	0.925278i	$-0.376168\pi$
$374$		0			0
$375$	−32.2237	−	13.9521i	−1.66402	−	0.720483i
$376$		0			0
$377$			0.471520i			0.0242845i
$378$		0			0
$379$	9.65178			0.495779			0.247889	−	0.968788i	$-0.420263\pi$
$379$	9.65178			0.495779			0.247889	+	0.968788i	$0.420263\pi$
$380$		0			0
$381$	48.2235			2.47056
$382$		0			0
$383$		−	25.0954i		−	1.28232i	−0.767409	−	0.641158i	$-0.778454\pi$
$383$		−	25.0954i		−	1.28232i	0.767409	−	0.641158i	$-0.221546\pi$
$384$		0			0
$385$	3.72779	+	9.13641i	0.189986	+	0.465635i
$386$		0			0
$387$			69.1906i			3.51715i
$388$		0			0
$389$	−16.8259			−0.853106			−0.426553	−	0.904462i	$-0.640272\pi$
$389$	−16.8259			−0.853106			−0.426553	+	0.904462i	$0.640272\pi$
$390$		0			0
$391$	6.72347			0.340020
$392$		0			0
$393$		−	41.0599i		−	2.07120i
$394$		0			0
$395$	−0.145054	+	0.0591841i	−0.00729845	+	0.00297787i
$396$		0			0
$397$			27.8080i			1.39564i	0.716272	+	0.697822i	$0.245847\pi$
$397$			27.8080i			1.39564i	−0.716272	+	0.697822i	$0.754153\pi$
$398$		0			0
$399$	5.74820			0.287770
$400$		0			0
$401$	1.46483			0.0731500			0.0365750	−	0.999331i	$-0.488355\pi$
$401$	1.46483			0.0731500			0.0365750	+	0.999331i	$0.488355\pi$
$402$		0			0
$403$		−	5.15189i		−	0.256634i
$404$		0			0
$405$	−36.2824	+	14.8037i	−1.80289	+	0.735603i
$406$		0			0
$407$		−	7.46544i		−	0.370048i
$408$		0			0
$409$	−0.514906			−0.0254605			−0.0127302	−	0.999919i	$-0.504052\pi$
$409$	−0.514906			−0.0254605			−0.0127302	+	0.999919i	$0.504052\pi$
$410$		0			0
$411$	41.4073			2.04247
$412$		0			0
$413$		−	0.703052i		−	0.0345949i
$414$		0			0
$415$	−5.69521	−	13.9583i	−0.279567	−	0.685188i
$416$		0			0
$417$		−	44.3346i		−	2.17107i
$418$		0			0
$419$	−9.65387			−0.471622			−0.235811	−	0.971799i	$-0.575775\pi$
$419$	−9.65387			−0.471622			−0.235811	+	0.971799i	$0.575775\pi$
$420$		0			0
$421$	14.2788			0.695905			0.347952	−	0.937512i	$-0.386877\pi$
$421$	14.2788			0.695905			0.347952	+	0.937512i	$0.386877\pi$
$422$		0			0
$423$		−	62.9474i		−	3.06061i
$424$		0			0
$425$	23.5176	+	24.0218i	1.14077	+	1.16523i
$426$		0			0
$427$		−	10.6561i		−	0.515685i
$428$		0			0
$429$	−9.87122			−0.476587
$430$		0			0
$431$	−5.71198			−0.275136			−0.137568	−	0.990492i	$-0.543929\pi$
$431$	−5.71198			−0.275136			−0.137568	+	0.990492i	$0.543929\pi$
$432$		0			0
$433$			30.3302i			1.45758i	0.684738	+	0.728789i	$0.259917\pi$
$433$			30.3302i			1.45758i	−0.684738	+	0.728789i	$0.740083\pi$
$434$		0			0
$435$	1.45588	+	3.56821i	0.0698041	+	0.171082i
$436$		0			0
$437$			1.51699i			0.0725676i
$438$		0			0
$439$	−19.7579			−0.942994			−0.471497	−	0.881868i	$-0.656286\pi$
$439$	−19.7579			−0.942994			−0.471497	+	0.881868i	$0.656286\pi$
$440$		0			0
$441$	−38.0580			−1.81229
$442$		0			0
$443$			16.0643i			0.763236i	0.924320	+	0.381618i	$0.124633\pi$
$443$			16.0643i			0.763236i	−0.924320	+	0.381618i	$0.875367\pi$
$444$		0			0
$445$	10.2771	−	4.19322i	0.487183	−	0.198778i
$446$		0			0
$447$			20.5543i			0.972184i
$448$		0			0
$449$	−28.5881			−1.34916			−0.674579	−	0.738203i	$-0.735675\pi$
$449$	−28.5881			−1.34916			−0.674579	+	0.738203i	$0.735675\pi$
$450$		0			0
$451$	26.1527			1.23148
$452$		0			0
$453$		−	8.21842i		−	0.386135i
$454$		0			0
$455$	−2.14631	+	0.875727i	−0.100621	+	0.0410547i
$456$		0			0
$457$			10.1209i			0.473437i	0.971578	+	0.236718i	$0.0760719\pi$
$457$			10.1209i			0.473437i	−0.971578	+	0.236718i	$0.923928\pi$
$458$		0			0
$459$	81.5987			3.80870
$460$		0			0
$461$	15.6931			0.730901			0.365450	−	0.930831i	$-0.380915\pi$
$461$	15.6931			0.730901			0.365450	+	0.930831i	$0.380915\pi$
$462$		0			0
$463$			12.9124i			0.600089i	0.953925	+	0.300044i	$0.0970015\pi$
$463$			12.9124i			0.600089i	−0.953925	+	0.300044i	$0.902999\pi$
$464$		0			0
$465$	−15.9071	−	38.9867i	−0.737676	−	1.80796i
$466$		0			0
$467$			7.10196i			0.328640i	0.986407	+	0.164320i	$0.0525429\pi$
$467$			7.10196i			0.328640i	−0.986407	+	0.164320i	$0.947457\pi$
$468$		0			0
$469$	−3.86852			−0.178632
$470$		0			0
$471$	−27.6822			−1.27553
$472$		0			0
$473$			36.8696i			1.69527i
$474$		0			0
$475$	−5.41996	+	5.30620i	−0.248685	+	0.243465i
$476$		0			0
$477$			36.9222i			1.69055i
$478$		0			0
$479$	16.7758			0.766506			0.383253	−	0.923643i	$-0.374804\pi$
$479$	16.7758			0.766506			0.383253	+	0.923643i	$0.374804\pi$
$480$		0			0
$481$	1.75377			0.0799650
$482$		0			0
$483$			3.78921i			0.172415i
$484$		0			0
$485$	−9.57761	−	23.4737i	−0.434897	−	1.06589i
$486$		0			0
$487$		−	7.02488i		−	0.318328i	−0.987252	−	0.159164i	$-0.949120\pi$
$487$		−	7.02488i		−	0.318328i	0.987252	−	0.159164i	$-0.0508798\pi$
$488$		0			0
$489$	31.9561			1.44510
$490$		0			0
$491$	−11.1500			−0.503192			−0.251596	−	0.967832i	$-0.580955\pi$
$491$	−11.1500			−0.503192			−0.251596	+	0.967832i	$0.580955\pi$
$492$		0			0
$493$		−	3.68948i		−	0.166166i
$494$		0			0
$495$	−51.9814	+	21.2092i	−2.33639	+	0.953281i
$496$		0			0
$497$			15.1755i			0.680714i
$498$		0			0
$499$	14.8347			0.664091			0.332046	−	0.943263i	$-0.392261\pi$
$499$	14.8347			0.664091			0.332046	+	0.943263i	$0.392261\pi$
$500$		0			0
$501$	−39.7766			−1.77709
$502$		0			0
$503$		−	3.10196i		−	0.138310i	−0.997606	−	0.0691548i	$-0.977970\pi$
$503$		−	3.10196i		−	0.138310i	0.997606	−	0.0691548i	$-0.0220302\pi$
$504$		0			0
$505$	−7.13641	+	2.91176i	−0.317566	+	0.129572i
$506$		0			0
$507$			38.5106i			1.71032i
$508$		0			0
$509$	22.3353			0.989993			0.494996	−	0.868895i	$-0.335169\pi$
$509$	22.3353			0.989993			0.494996	+	0.868895i	$0.335169\pi$
$510$		0			0
$511$	10.4070			0.460378
$512$		0			0
$513$			18.4109i			0.812859i
$514$		0			0
$515$	−7.71788	−	18.9157i	−0.340091	−	0.833526i
$516$		0			0
$517$		−	33.5428i		−	1.47521i
$518$		0			0
$519$	−49.2784			−2.16308
$520$		0			0
$521$	−14.2147			−0.622758			−0.311379	−	0.950286i	$-0.600791\pi$
$521$	−14.2147			−0.622758			−0.311379	+	0.950286i	$0.600791\pi$
$522$		0			0
$523$			4.09494i			0.179059i	0.995984	+	0.0895297i	$0.0285364\pi$
$523$			4.09494i			0.179059i	−0.995984	+	0.0895297i	$0.971464\pi$
$524$		0			0
$525$	−13.5382	+	13.2540i	−0.590855	+	0.578453i
$526$		0			0
$527$			40.3117i			1.75601i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$			6.14375i			0.266115i
$534$		0			0
$535$	7.80994	+	19.1413i	0.337653	+	0.827552i
$536$		0			0
$537$		−	7.53387i		−	0.325111i
$538$		0			0
$539$	−20.2800			−0.873521
$540$		0			0
$541$	33.6902			1.44846			0.724228	−	0.689560i	$-0.242196\pi$
$541$	33.6902			1.44846			0.724228	+	0.689560i	$0.242196\pi$
$542$		0			0
$543$		−	31.0028i		−	1.33046i
$544$		0			0
$545$	3.41294	−	1.39253i	0.146194	−	0.0596495i
$546$		0			0
$547$			32.9358i			1.40823i	0.710084	+	0.704117i	$0.248657\pi$
$547$			32.9358i			1.40823i	−0.710084	+	0.704117i	$0.751343\pi$
$548$		0			0
$549$	60.6277			2.58753
$550$		0			0
$551$	0.832445			0.0354633
$552$		0			0
$553$			0.0845278i			0.00359449i
$554$		0			0
$555$	13.2716	−	5.41499i	0.563346	−	0.229853i
$556$		0			0
$557$			20.8337i			0.882751i	0.897323	+	0.441375i	$0.145509\pi$
$557$			20.8337i			0.882751i	−0.897323	+	0.441375i	$0.854491\pi$
$558$		0			0
$559$	−8.66135			−0.366336
$560$		0			0
$561$	77.2387			3.26102
$562$		0			0
$563$		−	30.4368i		−	1.28276i	−0.767223	−	0.641380i	$-0.778362\pi$
$563$		−	30.4368i		−	1.28276i	0.767223	−	0.641380i	$-0.221638\pi$
$564$		0			0
$565$	1.01915	+	2.49784i	0.0428762	+	0.105085i
$566$		0			0
$567$			21.1430i			0.887921i
$568$		0			0
$569$	−39.4801			−1.65509			−0.827545	−	0.561399i	$-0.810263\pi$
$569$	−39.4801			−1.65509			−0.827545	+	0.561399i	$0.810263\pi$
$570$		0			0
$571$	−27.0301			−1.13118			−0.565588	−	0.824688i	$-0.691351\pi$
$571$	−27.0301			−1.13118			−0.565588	+	0.824688i	$0.691351\pi$
$572$		0			0
$573$			30.1704i			1.26039i
$574$		0			0
$575$	−3.49784	−	3.57283i	−0.145870	−	0.148997i
$576$		0			0
$577$		−	4.68818i		−	0.195171i	−0.995227	−	0.0975857i	$-0.968888\pi$
$577$		−	4.68818i		−	0.195171i	0.995227	−	0.0975857i	$-0.0311120\pi$
$578$		0			0
$579$	78.4270			3.25932
$580$		0			0
$581$	−8.13400			−0.337455
$582$		0			0
$583$			19.6747i			0.814845i
$584$		0			0
$585$	−4.98243	−	12.2114i	−0.205998	−	0.504879i
$586$		0			0
$587$			9.61718i			0.396944i	0.980107	+	0.198472i	$0.0635978\pi$
$587$			9.61718i			0.396944i	−0.980107	+	0.198472i	$0.936402\pi$
$588$		0			0
$589$	−9.09541			−0.374770
$590$		0			0
$591$	31.1652			1.28197
$592$		0			0
$593$			8.68902i			0.356815i	0.983957	+	0.178408i	$0.0570946\pi$
$593$			8.68902i			0.356815i	−0.983957	+	0.178408i	$0.942905\pi$
$594$		0			0
$595$	16.7941	−	6.85225i	0.688492	−	0.280915i
$596$		0			0
$597$			22.0422i			0.902128i
$598$		0			0
$599$	11.7581			0.480421			0.240211	−	0.970721i	$-0.422783\pi$
$599$	11.7581			0.480421			0.240211	+	0.970721i	$0.422783\pi$
$600$		0			0
$601$	37.9388			1.54756			0.773778	−	0.633457i	$-0.218365\pi$
$601$	37.9388			1.54756			0.773778	+	0.633457i	$0.218365\pi$
$602$		0			0
$603$		−	22.0099i		−	0.896311i
$604$		0			0
$605$	−4.92531	+	2.00960i	−0.200242	+	0.0817018i
$606$		0			0
$607$			9.01418i			0.365874i	0.983125	+	0.182937i	$0.0585605\pi$
$607$			9.01418i			0.365874i	−0.983125	+	0.182937i	$0.941440\pi$
$608$		0			0
$609$	2.07931			0.0842580
$610$		0			0
$611$	7.87983			0.318784
$612$		0			0
$613$			42.5846i			1.71997i	0.510316	+	0.859987i	$0.329528\pi$
$613$			42.5846i			1.71997i	−0.510316	+	0.859987i	$0.670472\pi$
$614$		0			0
$615$	18.9696	+	46.4925i	0.764929	+	1.87476i
$616$		0			0
$617$			12.6190i			0.508020i	0.967202	+	0.254010i	$0.0817496\pi$
$617$			12.6190i			0.508020i	−0.967202	+	0.254010i	$0.918250\pi$
$618$		0			0
$619$	−26.0638			−1.04759			−0.523796	−	0.851844i	$-0.675485\pi$
$619$	−26.0638			−1.04759			−0.523796	+	0.851844i	$0.675485\pi$
$620$		0			0
$621$	−12.1364			−0.487017
$622$		0			0
$623$		−	5.98884i		−	0.239938i
$624$		0			0
$625$	0.530243	−	24.9944i	0.0212097	−	0.999775i
$626$		0			0
$627$			17.4271i			0.695972i
$628$		0			0
$629$	−13.7226			−0.547157
$630$		0			0
$631$	46.1083			1.83554			0.917771	−	0.397110i	$-0.129987\pi$
$631$	46.1083			1.83554			0.917771	+	0.397110i	$0.129987\pi$
$632$		0			0
$633$		−	23.3888i		−	0.929622i
$634$		0			0
$635$	12.9703	+	31.7889i	0.514711	+	1.26150i
$636$		0			0
$637$		−	4.76415i		−	0.188762i
$638$		0			0
$639$	−86.3407			−3.41559
$640$		0			0
$641$	19.3319			0.763563			0.381781	−	0.924253i	$-0.375311\pi$
$641$	19.3319			0.763563			0.381781	+	0.924253i	$0.375311\pi$
$642$		0			0
$643$			2.27891i			0.0898713i	0.998990	+	0.0449356i	$0.0143083\pi$
$643$			2.27891i			0.0898713i	−0.998990	+	0.0449356i	$0.985692\pi$
$644$		0			0
$645$	−65.5443	+	26.7430i	−2.58081	+	1.05301i
$646$		0			0
$647$		−	47.5054i		−	1.86763i	−0.357755	−	0.933815i	$-0.616458\pi$
$647$		−	47.5054i		−	1.86763i	0.357755	−	0.933815i	$-0.383542\pi$
$648$		0			0
$649$	2.13148			0.0836679
$650$		0			0
$651$	−22.7189			−0.890422
$652$		0			0
$653$		−	2.73226i		−	0.106921i	−0.998570	−	0.0534607i	$-0.982975\pi$
$653$		−	2.73226i		−	0.106921i	0.998570	−	0.0534607i	$-0.0170252\pi$
$654$		0			0
$655$	27.0667	−	11.0436i	1.05758	−	0.431509i
$656$		0			0
$657$			59.2104i			2.31002i
$658$		0			0
$659$	−12.1375			−0.472809			−0.236405	−	0.971655i	$-0.575969\pi$
$659$	−12.1375			−0.472809			−0.236405	+	0.971655i	$0.575969\pi$
$660$		0			0
$661$	41.1992			1.60246			0.801232	−	0.598354i	$-0.204178\pi$
$661$	41.1992			1.60246			0.801232	+	0.598354i	$0.204178\pi$
$662$		0			0
$663$			18.1448i			0.704686i
$664$		0			0
$665$	1.54605	+	3.78921i	0.0599533	+	0.146939i
$666$		0			0
$667$			0.548747i			0.0212476i
$668$		0			0
$669$	32.0919			1.24074
$670$		0			0
$671$	32.3067			1.24718
$672$		0			0
$673$		−	24.2175i		−	0.933516i	−0.884385	−	0.466758i	$-0.845422\pi$
$673$		−	24.2175i		−	0.933516i	0.884385	−	0.466758i	$-0.154578\pi$
$674$		0			0
$675$	−42.4512	−	43.3613i	−1.63395	−	1.66898i
$676$		0			0
$677$			48.1512i			1.85060i	0.379235	+	0.925300i	$0.376187\pi$
$677$			48.1512i			1.85060i	−0.379235	+	0.925300i	$0.623813\pi$
$678$		0			0
$679$	−13.6789			−0.524949
$680$		0			0
$681$	−18.3196			−0.702008
$682$		0			0
$683$			12.1351i			0.464337i	0.972676	+	0.232169i	$0.0745821\pi$
$683$			12.1351i			0.464337i	−0.972676	+	0.232169i	$0.925418\pi$
$684$		0			0
$685$	11.1370	+	27.2956i	0.425524	+	1.04291i
$686$		0			0
$687$		−	18.4380i		−	0.703454i
$688$		0			0
$689$	−4.62197			−0.176083
$690$		0			0
$691$	13.8289			0.526076			0.263038	−	0.964785i	$-0.415275\pi$
$691$	13.8289			0.526076			0.263038	+	0.964785i	$0.415275\pi$
$692$		0			0
$693$			30.2913i			1.15067i
$694$		0			0
$695$	29.2253	−	11.9243i	1.10858	−	0.452316i
$696$		0			0
$697$		−	48.0727i		−	1.82088i
$698$		0			0
$699$	−2.63653			−0.0997226
$700$		0			0
$701$	−15.2263			−0.575089			−0.287544	−	0.957767i	$-0.592839\pi$
$701$	−15.2263			−0.575089			−0.287544	+	0.957767i	$0.592839\pi$
$702$		0			0
$703$		−	3.09619i		−	0.116775i
$704$		0			0
$705$	59.6302	−	24.3300i	2.24580	−	0.916320i
$706$		0			0
$707$			4.15863i			0.156401i
$708$		0			0
$709$	−15.2317			−0.572037			−0.286019	−	0.958224i	$-0.592332\pi$
$709$	−15.2317			−0.572037			−0.286019	+	0.958224i	$0.592332\pi$
$710$		0			0
$711$	−0.480919			−0.0180359
$712$		0			0
$713$		−	5.99568i		−	0.224540i
$714$		0			0
$715$	−2.65499	−	6.50709i	−0.0992909	−	0.243351i
$716$		0			0
$717$		−	66.9907i		−	2.50181i
$718$		0			0
$719$	4.58943			0.171157			0.0855784	−	0.996331i	$-0.472726\pi$
$719$	4.58943			0.171157			0.0855784	+	0.996331i	$0.472726\pi$
$720$		0			0
$721$	−11.0228			−0.410511
$722$		0			0
$723$		−	65.0246i		−	2.41829i
$724$		0			0
$725$	−1.96058	+	1.91943i	−0.0728141	+	0.0712857i
$726$		0			0
$727$		−	21.2414i		−	0.787800i	−0.919153	−	0.393900i	$-0.871126\pi$
$727$		−	21.2414i		−	0.787800i	0.919153	−	0.393900i	$-0.128874\pi$
$728$		0			0
$729$	−5.94082			−0.220030
$730$		0			0
$731$	67.7720			2.50664
$732$		0			0
$733$		−	40.5297i		−	1.49700i	−0.663136	−	0.748499i	$-0.730775\pi$
$733$		−	40.5297i		−	1.49700i	0.663136	−	0.748499i	$-0.269225\pi$
$734$		0			0
$735$	−14.7099	−	36.0524i	−0.542583	−	1.32981i
$736$		0			0
$737$		−	11.7284i		−	0.432021i
$738$		0			0
$739$	30.6476			1.12739			0.563695	−	0.825983i	$-0.309379\pi$
$739$	30.6476			1.12739			0.563695	+	0.825983i	$0.309379\pi$
$740$		0			0
$741$	−4.09396			−0.150395
$742$		0			0
$743$			1.01357i			0.0371844i	0.999827	+	0.0185922i	$0.00591842\pi$
$743$			1.01357i			0.0371844i	−0.999827	+	0.0185922i	$0.994082\pi$
$744$		0			0
$745$	−13.5494	+	5.52833i	−0.496410	+	0.202542i
$746$		0			0
$747$		−	46.2782i		−	1.69323i
$748$		0			0
$749$	11.1543			0.407569
$750$		0			0
$751$	21.0644			0.768652			0.384326	−	0.923197i	$-0.374434\pi$
$751$	21.0644			0.768652			0.384326	+	0.923197i	$0.374434\pi$
$752$		0			0
$753$		−	74.5264i		−	2.71589i
$754$		0			0
$755$	5.41757	−	2.21045i	0.197166	−	0.0804464i
$756$		0			0
$757$		−	13.2109i		−	0.480160i	−0.970753	−	0.240080i	$-0.922826\pi$
$757$		−	13.2109i		−	0.480160i	0.970753	−	0.240080i	$-0.0771736\pi$
$758$		0			0
$759$	−11.4879			−0.416986
$760$		0			0
$761$	−11.9759			−0.434125			−0.217063	−	0.976158i	$-0.569648\pi$
$761$	−11.9759			−0.434125			−0.217063	+	0.976158i	$0.569648\pi$
$762$		0			0
$763$		−	1.98884i		−	0.0720008i
$764$		0			0
$765$	38.9857	+	95.5499i	1.40953	+	3.45461i
$766$		0			0
$767$			0.500724i			0.0180801i
$768$		0			0
$769$	−39.6569			−1.43006			−0.715031	−	0.699092i	$-0.753588\pi$
$769$	−39.6569			−1.43006			−0.715031	+	0.699092i	$0.753588\pi$
$770$		0			0
$771$	57.3123			2.06405
$772$		0			0
$773$		−	16.0305i		−	0.576575i	−0.957544	−	0.288288i	$-0.906914\pi$
$773$		−	16.0305i		−	0.576575i	0.957544	−	0.288288i	$-0.0930859\pi$
$774$		0			0
$775$	21.4215	−	20.9719i	0.769485	−	0.753334i
$776$		0			0
$777$		−	7.73379i		−	0.277448i
$778$		0			0
$779$	10.8465			0.388615
$780$		0			0
$781$	−46.0084			−1.64631
$782$		0			0
$783$			6.65981i			0.238002i
$784$		0			0
$785$	−7.44548	−	18.2481i	−0.265741	−	0.651302i
$786$		0			0
$787$			39.0352i			1.39145i	0.718306	+	0.695727i	$0.244918\pi$
$787$			39.0352i			1.39145i	−0.718306	+	0.695727i	$0.755082\pi$
$788$		0			0
$789$	−81.2565			−2.89281
$790$		0			0
$791$	1.45558			0.0517543
$792$		0			0
$793$			7.58944i			0.269509i
$794$		0			0
$795$	−34.9765	+	14.2709i	−1.24049	+	0.506137i
$796$		0			0
$797$		−	3.37687i		−	0.119615i	−0.998210	−	0.0598074i	$-0.980951\pi$
$797$		−	3.37687i		−	0.119615i	0.998210	−	0.0598074i	$-0.0190487\pi$
$798$		0			0
$799$	−61.6569			−2.18126
$800$		0			0
$801$	34.0734			1.20392
$802$		0			0
$803$			31.5514i			1.11343i
$804$		0			0
$805$	−2.49784	+	1.01915i	−0.0880373	+	0.0359205i
$806$		0			0
$807$		−	23.0686i		−	0.812053i
$808$		0			0
$809$	49.3406			1.73472			0.867361	−	0.497680i	$-0.165814\pi$
$809$	49.3406			1.73472			0.867361	+	0.497680i	$0.165814\pi$
$810$		0			0
$811$	−18.3582			−0.644645			−0.322322	−	0.946630i	$-0.604463\pi$
$811$	−18.3582			−0.644645			−0.322322	+	0.946630i	$0.604463\pi$
$812$		0			0
$813$		−	94.9375i		−	3.32960i
$814$		0			0
$815$	8.59499	+	21.0654i	0.301069	+	0.737889i
$816$		0			0
$817$			15.2912i			0.534971i
$818$		0			0
$819$	−7.11600			−0.248653
$820$		0			0
$821$	19.4610			0.679192			0.339596	−	0.940571i	$-0.389710\pi$
$821$	19.4610			0.679192			0.339596	+	0.940571i	$0.389710\pi$
$822$		0			0
$823$		−	47.2973i		−	1.64868i	−0.566094	−	0.824341i	$-0.691546\pi$
$823$		−	47.2973i		−	1.64868i	0.566094	−	0.824341i	$-0.308454\pi$
$824$		0			0
$825$	−40.1830	−	41.0445i	−1.39899	−	1.42898i
$826$		0			0
$827$		−	26.2901i		−	0.914197i	−0.889416	−	0.457098i	$-0.848889\pi$
$827$		−	26.2901i		−	0.914197i	0.889416	−	0.457098i	$-0.151111\pi$
$828$		0			0
$829$	20.0140			0.695116			0.347558	−	0.937658i	$-0.387011\pi$
$829$	20.0140			0.695116			0.347558	+	0.937658i	$0.387011\pi$
$830$		0			0
$831$	−7.88270			−0.273448
$832$		0			0
$833$			37.2778i			1.29160i
$834$		0			0
$835$	−10.6984	−	26.2207i	−0.370234	−	0.907403i
$836$		0			0
$837$		−	72.7660i		−	2.51516i
$838$		0			0
$839$	−40.7210			−1.40585			−0.702923	−	0.711266i	$-0.748122\pi$
$839$	−40.7210			−1.40585			−0.702923	+	0.711266i	$0.748122\pi$
$840$		0			0
$841$	−28.6989			−0.989616
$842$		0			0
$843$		−	32.9262i		−	1.13404i
$844$		0			0
$845$	−25.3861	+	10.3579i	−0.873309	+	0.356323i
$846$		0			0
$847$			2.87015i			0.0986194i
$848$		0			0
$849$	−12.0587			−0.413854
$850$		0			0
$851$	2.04100			0.0699647
$852$		0			0
$853$		−	5.34390i		−	0.182972i	−0.995806	−	0.0914858i	$-0.970838\pi$
$853$		−	5.34390i		−	0.182972i	0.995806	−	0.0914858i	$-0.0291616\pi$
$854$		0			0
$855$	−21.5586	+	8.79622i	−0.737289	+	0.300824i
$856$		0			0
$857$			48.4603i			1.65537i	0.561192	+	0.827686i	$0.310343\pi$
$857$			48.4603i			1.65537i	−0.561192	+	0.827686i	$0.689657\pi$
$858$		0			0
$859$	−51.3123			−1.75075			−0.875377	−	0.483440i	$-0.839387\pi$
$859$	−51.3123			−1.75075			−0.875377	+	0.483440i	$0.839387\pi$
$860$		0			0
$861$	27.0928			0.923319
$862$		0			0
$863$			24.5507i			0.835714i	0.908513	+	0.417857i	$0.137219\pi$
$863$			24.5507i			0.835714i	−0.908513	+	0.417857i	$0.862781\pi$
$864$		0			0
$865$	−13.2540	−	32.4842i	−0.450651	−	1.10450i
$866$		0			0
$867$		−	88.5843i		−	3.00848i
$868$		0			0
$869$	−0.256267			−0.00869327
$870$		0			0
$871$	2.75522			0.0933570
$872$		0			0
$873$		−	77.8260i		−	2.63401i
$874$		0			0
$875$	−12.3783	−	5.35952i	−0.418463	−	0.181185i
$876$		0			0
$877$			30.1400i			1.01776i	0.860838	+	0.508878i	$0.169940\pi$
$877$			30.1400i			1.01776i	−0.860838	+	0.508878i	$0.830060\pi$
$878$		0			0
$879$	1.70989			0.0576732
$880$		0			0
$881$	36.4729			1.22880			0.614401	−	0.788994i	$-0.289398\pi$
$881$	36.4729			1.22880			0.614401	+	0.788994i	$0.289398\pi$
$882$		0			0
$883$		−	2.17426i		−	0.0731696i	−0.999331	−	0.0365848i	$-0.988352\pi$
$883$		−	2.17426i		−	0.0731696i	0.999331	−	0.0365848i	$-0.0116479\pi$
$884$		0			0
$885$	1.54605	+	3.78921i	0.0519699	+	0.127373i
$886$		0			0
$887$			16.1463i			0.542139i	0.962560	+	0.271069i	$0.0873773\pi$
$887$			16.1463i			0.542139i	−0.962560	+	0.271069i	$0.912623\pi$
$888$		0			0
$889$	18.5244			0.621290
$890$		0			0
$891$	−64.1003			−2.14744
$892$		0			0
$893$		−	13.9114i		−	0.465528i
$894$		0			0
$895$	4.96632	−	2.02633i	0.166006	−	0.0677327i
$896$		0			0
$897$		−	2.69873i		−	0.0901080i
$898$		0			0
$899$	−3.29011			−0.109731
$900$		0			0
$901$	36.1652			1.20484
$902$		0			0
$903$			38.1949i			1.27105i
$904$		0			0
$905$	20.4370	−	8.33860i	0.679350	−	0.277185i
$906$		0			0
$907$			4.74475i			0.157547i	0.996893	+	0.0787734i	$0.0251004\pi$
$907$			4.74475i			0.157547i	−0.996893	+	0.0787734i	$0.974900\pi$
$908$		0			0
$909$	−23.6604			−0.784767
$910$		0			0
$911$	−38.8650			−1.28765			−0.643827	−	0.765171i	$-0.722654\pi$
$911$	−38.8650			−1.28765			−0.643827	+	0.765171i	$0.722654\pi$
$912$		0			0
$913$		−	24.6603i		−	0.816136i
$914$		0			0
$915$	23.4334	+	57.4327i	0.774683	+	1.89867i
$916$		0			0
$917$		−	15.7727i		−	0.520859i
$918$		0			0
$919$	40.5516			1.33767			0.668837	−	0.743409i	$-0.266793\pi$
$919$	40.5516			1.33767			0.668837	+	0.743409i	$0.266793\pi$
$920$		0			0
$921$	50.8329			1.67500
$922$		0			0
$923$		−	10.8082i		−	0.355757i
$924$		0			0
$925$	7.13911	+	7.29216i	0.234732	+	0.239765i
$926$		0			0
$927$		−	62.7141i		−	2.05980i
$928$		0			0
$929$	−19.4994			−0.639755			−0.319878	−	0.947459i	$-0.603642\pi$
$929$	−19.4994			−0.639755			−0.319878	+	0.947459i	$0.603642\pi$
$930$		0			0
$931$	−8.41086			−0.275655
$932$		0			0
$933$			23.6159i			0.773149i
$934$		0			0
$935$	20.7743	+	50.9157i	0.679393	+	1.66512i
$936$		0			0
$937$		−	24.9361i		−	0.814626i	−0.913289	−	0.407313i	$-0.866466\pi$
$937$		−	24.9361i		−	0.814626i	0.913289	−	0.407313i	$-0.133534\pi$
$938$		0			0
$939$	79.2956			2.58771
$940$		0			0
$941$	−41.9160			−1.36642			−0.683211	−	0.730221i	$-0.739417\pi$
$941$	−41.9160			−1.36642			−0.683211	+	0.730221i	$0.739417\pi$
$942$		0			0
$943$			7.14998i			0.232836i
$944$		0			0
$945$	−30.3148	+	12.3689i	−0.986141	+	0.402360i
$946$		0			0
$947$			29.3006i			0.952141i	0.879407	+	0.476070i	$0.157939\pi$
$947$			29.3006i			0.952141i	−0.879407	+	0.476070i	$0.842061\pi$
$948$		0			0
$949$	−7.41202			−0.240604
$950$		0			0
$951$	42.1400			1.36648
$952$		0			0
$953$			24.9759i			0.809048i	0.914527	+	0.404524i	$0.132563\pi$
$953$			24.9759i			0.809048i	−0.914527	+	0.404524i	$0.867437\pi$
$954$		0			0
$955$	−19.8883	+	8.11471i	−0.643570	+	0.262586i
$956$		0			0
$957$			6.30397i			0.203778i
$958$		0			0
$959$	15.9061			0.513635
$960$		0			0
$961$	4.94816			0.159618
$962$		0			0
$963$			63.4622i			2.04504i
$964$		0			0
$965$	21.0939	+	51.6990i	0.679038	+	1.66425i
$966$		0			0
$967$		−	0.405142i		−	0.0130285i	−0.999979	−	0.00651424i	$-0.997926\pi$
$967$		−	0.405142i		−	0.0130285i	0.999979	−	0.00651424i	$-0.00207356\pi$
$968$		0			0
$969$	32.0338			1.02907
$970$		0			0
$971$	−38.8032			−1.24525			−0.622627	−	0.782519i	$-0.713935\pi$
$971$	−38.8032			−1.24525			−0.622627	+	0.782519i	$0.713935\pi$
$972$		0			0
$973$		−	17.0306i		−	0.545975i
$974$		0			0
$975$	9.64211	−	9.43972i	0.308795	−	0.302313i
$976$		0			0
$977$		−	39.9940i		−	1.27952i	−0.768575	−	0.639760i	$-0.779034\pi$
$977$		−	39.9940i		−	1.27952i	0.768575	−	0.639760i	$-0.220966\pi$
$978$		0			0
$979$	18.1567			0.580290
$980$		0			0
$981$	11.3155			0.361275
$982$		0			0
$983$			21.3563i			0.681160i	0.940216	+	0.340580i	$0.110623\pi$
$983$			21.3563i			0.681160i	−0.940216	+	0.340580i	$0.889377\pi$
$984$		0			0
$985$	8.38228	+	20.5441i	0.267082	+	0.654589i
$986$		0			0
$987$		−	34.7485i		−	1.10606i
$988$		0			0
$989$	−10.0799			−0.320523
$990$		0			0
$991$	−4.39444			−0.139594			−0.0697970	−	0.997561i	$-0.522235\pi$
$991$	−4.39444			−0.139594			−0.0697970	+	0.997561i	$0.522235\pi$
$992$		0			0
$993$			76.5547i			2.42939i
$994$		0			0
$995$	−14.5302	+	5.92853i	−0.460638	+	0.187947i
$996$		0			0
$997$			28.9420i			0.916603i	0.888797	+	0.458302i	$0.151542\pi$
$997$			28.9420i			0.916603i	−0.888797	+	0.458302i	$0.848458\pi$
$998$		0			0
$999$	24.7705			0.783703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.d.369.8		8
4.3	odd	2		115.2.b.b.24.5	yes	8
5.2	odd	4		9200.2.a.cq.1.4		4
5.3	odd	4		9200.2.a.ck.1.1		4
5.4	even	2	inner	1840.2.e.d.369.1		8
12.11	even	2		1035.2.b.e.829.4		8
20.3	even	4		575.2.a.j.1.3		4
20.7	even	4		575.2.a.i.1.2		4
20.19	odd	2		115.2.b.b.24.4	✓	8
60.23	odd	4		5175.2.a.bw.1.2		4
60.47	odd	4		5175.2.a.bv.1.3		4
60.59	even	2		1035.2.b.e.829.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.4	✓	8	20.19	odd	2
115.2.b.b.24.5	yes	8	4.3	odd	2
575.2.a.i.1.2		4	20.7	even	4
575.2.a.j.1.3		4	20.3	even	4
1035.2.b.e.829.4		8	12.11	even	2
1035.2.b.e.829.5		8	60.59	even	2
1840.2.e.d.369.1		8	5.4	even	2	inner
1840.2.e.d.369.8		8	1.1	even	1	trivial
5175.2.a.bv.1.3		4	60.47	odd	4
5175.2.a.bw.1.2		4	60.23	odd	4
9200.2.a.ck.1.1		4	5.3	odd	4
9200.2.a.cq.1.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.14073i q^{3} +(-2.07037 + 0.844739i) q^{5} +1.20647i q^{7} -6.86420 q^{9} +O(q^{10})\)	\(q+3.14073i q^{3} +(-2.07037 + 0.844739i) q^{5} +1.20647i q^{7} -6.86420 q^{9} -3.65773 q^{11} -0.859268i q^{13} +(-2.65310 - 6.50246i) q^{15} +6.72347i q^{17} -1.51699 q^{19} -3.78921 q^{21} -1.00000i q^{23} +(3.57283 - 3.49784i) q^{25} -12.1364i q^{27} -0.548747 q^{29} +5.99568 q^{31} -11.4879i q^{33} +(-1.01915 - 2.49784i) q^{35} +2.04100i q^{37} +2.69873 q^{39} -7.14998 q^{41} -10.0799i q^{43} +(14.2114 - 5.79846i) q^{45} +9.17040i q^{47} +5.54442 q^{49} -21.1166 q^{51} -5.37896i q^{53} +(7.57283 - 3.08982i) q^{55} -4.76447i q^{57} -0.582734 q^{59} -8.83244 q^{61} -8.28146i q^{63} +(0.725857 + 1.77900i) q^{65} +3.20647i q^{67} +3.14073 q^{69} +12.5784 q^{71} -8.62597i q^{73} +(10.9858 + 11.2213i) q^{75} -4.41294i q^{77} +0.0700619 q^{79} +17.5246 q^{81} +6.74197i q^{83} +(-5.67958 - 13.9200i) q^{85} -1.72347i q^{87} -4.96393 q^{89} +1.03668 q^{91} +18.8308i q^{93} +(3.14073 - 1.28146i) q^{95} +11.3380i q^{97} +25.1074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 6 * q^5 - 8 * q^9	\( 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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