LMFDB - Embedded newform 1840.3.k.d.321.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.3
Root			$1.00527i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.30716 q^{3} -2.23607i q^{5} -1.47532i q^{7} +9.55167 q^{9} +O(q^{10})$	$q-4.30716 q^{3} -2.23607i q^{5} -1.47532i q^{7} +9.55167 q^{9} -6.04959i q^{11} -5.21324 q^{13} +9.63111i q^{15} -15.7063i q^{17} +4.82663i q^{19} +6.35447i q^{21} +(2.58013 - 22.8548i) q^{23} -5.00000 q^{25} -2.37613 q^{27} -23.4711 q^{29} +20.4887 q^{31} +26.0566i q^{33} -3.29893 q^{35} -15.5248i q^{37} +22.4543 q^{39} +20.3093 q^{41} -38.1696i q^{43} -21.3582i q^{45} +13.8273 q^{47} +46.8234 q^{49} +67.6497i q^{51} +38.2742i q^{53} -13.5273 q^{55} -20.7891i q^{57} +33.5696 q^{59} -100.567i q^{61} -14.0918i q^{63} +11.6572i q^{65} -32.4469i q^{67} +(-11.1130 + 98.4395i) q^{69} +24.1306 q^{71} +15.1818 q^{73} +21.5358 q^{75} -8.92511 q^{77} -11.2095i q^{79} -75.7306 q^{81} +44.1310i q^{83} -35.1204 q^{85} +101.094 q^{87} +111.039i q^{89} +7.69122i q^{91} -88.2480 q^{93} +10.7927 q^{95} -154.126i q^{97} -57.7837i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

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
$3$	−4.30716			−1.43572			−0.717861	−	0.696187i	$-0.754879\pi$
$3$	−4.30716			−1.43572			−0.717861	+	0.696187i	$0.754879\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	1.47532i		−	0.210761i	−0.994432	−	0.105380i	$-0.966394\pi$
$7$		−	1.47532i		−	0.210761i	0.994432	−	0.105380i	$-0.0336060\pi$
$8$		0			0
$9$	9.55167			1.06130
$10$		0			0
$11$		−	6.04959i		−	0.549963i	−0.961450	−	0.274981i	$-0.911328\pi$
$11$		−	6.04959i		−	0.549963i	0.961450	−	0.274981i	$-0.0886717\pi$
$12$		0			0
$13$	−5.21324			−0.401018			−0.200509	−	0.979692i	$-0.564260\pi$
$13$	−5.21324			−0.401018			−0.200509	+	0.979692i	$0.564260\pi$
$14$		0			0
$15$			9.63111i			0.642074i
$16$		0			0
$17$		−	15.7063i		−	0.923901i	−0.886906	−	0.461951i	$-0.847150\pi$
$17$		−	15.7063i		−	0.923901i	0.886906	−	0.461951i	$-0.152850\pi$
$18$		0			0
$19$			4.82663i			0.254033i	0.991901	+	0.127017i	$0.0405402\pi$
$19$			4.82663i			0.254033i	−0.991901	+	0.127017i	$0.959460\pi$
$20$		0			0
$21$			6.35447i			0.302594i
$22$		0			0
$23$	2.58013	−	22.8548i	0.112180	−	0.993688i
$23$	2.58013	−	22.8548i	0.112180	−	0.993688i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−2.37613			−0.0880047
$28$		0			0
$29$	−23.4711			−0.809349			−0.404674	−	0.914461i	$-0.632615\pi$
$29$	−23.4711			−0.809349			−0.404674	+	0.914461i	$0.632615\pi$
$30$		0			0
$31$	20.4887			0.660924			0.330462	−	0.943819i	$-0.392795\pi$
$31$	20.4887			0.660924			0.330462	+	0.943819i	$0.392795\pi$
$32$		0			0
$33$			26.0566i			0.789593i
$34$		0			0
$35$	−3.29893			−0.0942550
$36$		0			0
$37$		−	15.5248i		−	0.419589i	−0.977746	−	0.209794i	$-0.932721\pi$
$37$		−	15.5248i		−	0.419589i	0.977746	−	0.209794i	$-0.0672795\pi$
$38$		0			0
$39$	22.4543			0.575751
$40$		0			0
$41$	20.3093			0.495348			0.247674	−	0.968843i	$-0.420334\pi$
$41$	20.3093			0.495348			0.247674	+	0.968843i	$0.420334\pi$
$42$		0			0
$43$		−	38.1696i		−	0.887666i	−0.896109	−	0.443833i	$-0.853618\pi$
$43$		−	38.1696i		−	0.887666i	0.896109	−	0.443833i	$-0.146382\pi$
$44$		0			0
$45$		−	21.3582i		−	0.474626i
$46$		0			0
$47$	13.8273			0.294198			0.147099	−	0.989122i	$-0.453006\pi$
$47$	13.8273			0.294198			0.147099	+	0.989122i	$0.453006\pi$
$48$		0			0
$49$	46.8234			0.955580
$50$		0			0
$51$			67.6497i			1.32647i
$52$		0			0
$53$			38.2742i			0.722154i	0.932536	+	0.361077i	$0.117591\pi$
$53$			38.2742i			0.722154i	−0.932536	+	0.361077i	$0.882409\pi$
$54$		0			0
$55$	−13.5273			−0.245951
$56$		0			0
$57$		−	20.7891i		−	0.364721i
$58$		0			0
$59$	33.5696			0.568977			0.284488	−	0.958679i	$-0.408176\pi$
$59$	33.5696			0.568977			0.284488	+	0.958679i	$0.408176\pi$
$60$		0			0
$61$		−	100.567i		−	1.64864i	−0.566124	−	0.824320i	$-0.691558\pi$
$61$		−	100.567i		−	1.64864i	0.566124	−	0.824320i	$-0.308442\pi$
$62$		0			0
$63$		−	14.0918i		−	0.223680i
$64$		0			0
$65$			11.6572i			0.179341i
$66$		0			0
$67$		−	32.4469i		−	0.484282i	−0.970241	−	0.242141i	$-0.922150\pi$
$67$		−	32.4469i		−	0.484282i	0.970241	−	0.242141i	$-0.0778497\pi$
$68$		0			0
$69$	−11.1130	+	98.4395i	−0.161059	+	1.42666i
$70$		0			0
$71$	24.1306			0.339868			0.169934	−	0.985455i	$-0.445645\pi$
$71$	24.1306			0.339868			0.169934	+	0.985455i	$0.445645\pi$
$72$		0			0
$73$	15.1818			0.207969			0.103985	−	0.994579i	$-0.466841\pi$
$73$	15.1818			0.207969			0.103985	+	0.994579i	$0.466841\pi$
$74$		0			0
$75$	21.5358			0.287144
$76$		0			0
$77$	−8.92511			−0.115910
$78$		0			0
$79$		−	11.2095i		−	0.141893i	−0.997480	−	0.0709463i	$-0.977398\pi$
$79$		−	11.2095i		−	0.141893i	0.997480	−	0.0709463i	$-0.0226019\pi$
$80$		0			0
$81$	−75.7306			−0.934946
$82$		0			0
$83$			44.1310i			0.531698i	0.964015	+	0.265849i	$0.0856523\pi$
$83$			44.1310i			0.531698i	−0.964015	+	0.265849i	$0.914348\pi$
$84$		0			0
$85$	−35.1204			−0.413181
$86$		0			0
$87$	101.094			1.16200
$88$		0			0
$89$			111.039i			1.24763i	0.781572	+	0.623815i	$0.214418\pi$
$89$			111.039i			1.24763i	−0.781572	+	0.623815i	$0.785582\pi$
$90$		0			0
$91$			7.69122i			0.0845189i
$92$		0			0
$93$	−88.2480			−0.948903
$94$		0			0
$95$	10.7927			0.113607
$96$		0			0
$97$		−	154.126i		−	1.58893i	−0.607312	−	0.794463i	$-0.707752\pi$
$97$		−	154.126i		−	1.58893i	0.607312	−	0.794463i	$-0.292248\pi$
$98$		0			0
$99$		−	57.7837i		−	0.583673i
$100$		0			0
$101$	−58.8607			−0.582779			−0.291390	−	0.956604i	$-0.594118\pi$
$101$	−58.8607			−0.582779			−0.291390	+	0.956604i	$0.594118\pi$
$102$		0			0
$103$		−	54.2662i		−	0.526856i	−0.964679	−	0.263428i	$-0.915147\pi$
$103$		−	54.2662i		−	0.526856i	0.964679	−	0.263428i	$-0.0848531\pi$
$104$		0			0
$105$	14.2090			0.135324
$106$		0			0
$107$			119.124i			1.11331i	0.830745	+	0.556653i	$0.187915\pi$
$107$			119.124i			1.11331i	−0.830745	+	0.556653i	$0.812085\pi$
$108$		0			0
$109$			149.223i			1.36902i	0.729003	+	0.684510i	$0.239984\pi$
$109$			149.223i			1.36902i	−0.729003	+	0.684510i	$0.760016\pi$
$110$		0			0
$111$			66.8678i			0.602413i
$112$		0			0
$113$		−	35.3339i		−	0.312690i	−0.987703	−	0.156345i	$-0.950029\pi$
$113$		−	35.3339i		−	0.312690i	0.987703	−	0.156345i	$-0.0499711\pi$
$114$		0			0
$115$	−51.1049	−	5.76935i	−0.444391	−	0.0501682i
$116$		0			0
$117$	−49.7951			−0.425599
$118$		0			0
$119$	−23.1719			−0.194722
$120$		0			0
$121$	84.4025			0.697541
$122$		0			0
$123$	−87.4753			−0.711181
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−92.8398			−0.731022			−0.365511	−	0.930807i	$-0.619106\pi$
$127$	−92.8398			−0.731022			−0.365511	+	0.930807i	$0.619106\pi$
$128$		0			0
$129$			164.403i			1.27444i
$130$		0			0
$131$	−151.709			−1.15808			−0.579042	−	0.815297i	$-0.696573\pi$
$131$	−151.709			−1.15808			−0.579042	+	0.815297i	$0.696573\pi$
$132$		0			0
$133$	7.12085			0.0535402
$134$		0			0
$135$			5.31318i			0.0393569i
$136$		0			0
$137$		−	267.464i		−	1.95229i	−0.217121	−	0.976145i	$-0.569667\pi$
$137$		−	267.464i		−	1.95229i	0.217121	−	0.976145i	$-0.430333\pi$
$138$		0			0
$139$	−239.779			−1.72503			−0.862516	−	0.506030i	$-0.831112\pi$
$139$	−239.779			−1.72503			−0.862516	+	0.506030i	$0.831112\pi$
$140$		0			0
$141$	−59.5564			−0.422386
$142$		0			0
$143$			31.5380i			0.220545i
$144$		0			0
$145$			52.4830i			0.361952i
$146$		0			0
$147$	−201.676			−1.37195
$148$		0			0
$149$			201.647i			1.35333i	0.736290	+	0.676667i	$0.236576\pi$
$149$			201.647i			1.35333i	−0.736290	+	0.676667i	$0.763424\pi$
$150$		0			0
$151$	−120.787			−0.799912			−0.399956	−	0.916534i	$-0.630975\pi$
$151$	−120.787			−0.799912			−0.399956	+	0.916534i	$0.630975\pi$
$152$		0			0
$153$		−	150.022i		−	0.980533i
$154$		0			0
$155$		−	45.8140i		−	0.295574i
$156$		0			0
$157$		−	194.897i		−	1.24138i	−0.784055	−	0.620692i	$-0.786852\pi$
$157$		−	194.897i		−	1.24138i	0.784055	−	0.620692i	$-0.213148\pi$
$158$		0			0
$159$		−	164.853i		−	1.03681i
$160$		0			0
$161$	−33.7183	−	3.80653i	−0.209430	−	0.0236430i
$162$		0			0
$163$	−153.813			−0.943635			−0.471817	−	0.881696i	$-0.656402\pi$
$163$	−153.813			−0.943635			−0.471817	+	0.881696i	$0.656402\pi$
$164$		0			0
$165$	58.2643			0.353117
$166$		0			0
$167$	−48.4949			−0.290389			−0.145194	−	0.989403i	$-0.546381\pi$
$167$	−48.4949			−0.290389			−0.145194	+	0.989403i	$0.546381\pi$
$168$		0			0
$169$	−141.822			−0.839184
$170$		0			0
$171$			46.1024i			0.269605i
$172$		0			0
$173$	−111.269			−0.643174			−0.321587	−	0.946880i	$-0.604216\pi$
$173$	−111.269			−0.643174			−0.321587	+	0.946880i	$0.604216\pi$
$174$		0			0
$175$			7.37662i			0.0421521i
$176$		0			0
$177$	−144.590			−0.816892
$178$		0			0
$179$	−262.590			−1.46698			−0.733492	−	0.679698i	$-0.762111\pi$
$179$	−262.590			−1.46698			−0.733492	+	0.679698i	$0.762111\pi$
$180$		0			0
$181$			211.167i			1.16667i	0.812231	+	0.583335i	$0.198253\pi$
$181$			211.167i			1.16667i	−0.812231	+	0.583335i	$0.801747\pi$
$182$		0			0
$183$			433.159i			2.36699i
$184$		0			0
$185$	−34.7145			−0.187646
$186$		0			0
$187$	−95.0168			−0.508111
$188$		0			0
$189$			3.50556i			0.0185479i
$190$		0			0
$191$		−	72.8209i		−	0.381261i	−0.981662	−	0.190631i	$-0.938947\pi$
$191$		−	72.8209i		−	0.381261i	0.981662	−	0.190631i	$-0.0610533\pi$
$192$		0			0
$193$	198.550			1.02875			0.514377	−	0.857564i	$-0.328023\pi$
$193$	198.550			1.02875			0.514377	+	0.857564i	$0.328023\pi$
$194$		0			0
$195$		−	50.2093i		−	0.257484i
$196$		0			0
$197$	28.2055			0.143175			0.0715876	−	0.997434i	$-0.477193\pi$
$197$	28.2055			0.143175			0.0715876	+	0.997434i	$0.477193\pi$
$198$		0			0
$199$			214.499i			1.07788i	0.842343	+	0.538941i	$0.181175\pi$
$199$			214.499i			1.07788i	−0.842343	+	0.538941i	$0.818825\pi$
$200$		0			0
$201$			139.754i			0.695295i
$202$		0			0
$203$			34.6275i			0.170579i
$204$		0			0
$205$		−	45.4129i		−	0.221526i
$206$		0			0
$207$	24.6445	−	218.302i	0.119056	−	1.05460i
$208$		0			0
$209$	29.1991			0.139709
$210$		0			0
$211$	−240.262			−1.13868			−0.569340	−	0.822102i	$-0.692801\pi$
$211$	−240.262			−1.13868			−0.569340	+	0.822102i	$0.692801\pi$
$212$		0			0
$213$	−103.935			−0.487956
$214$		0			0
$215$	−85.3499			−0.396976
$216$		0			0
$217$		−	30.2274i		−	0.139297i
$218$		0			0
$219$	−65.3904			−0.298586
$220$		0			0
$221$			81.8808i			0.370501i
$222$		0			0
$223$	−257.402			−1.15427			−0.577136	−	0.816648i	$-0.695830\pi$
$223$	−257.402			−1.15427			−0.577136	+	0.816648i	$0.695830\pi$
$224$		0			0
$225$	−47.7583			−0.212259
$226$		0			0
$227$		−	106.401i		−	0.468727i	−0.972149	−	0.234363i	$-0.924699\pi$
$227$		−	106.401i		−	0.468727i	0.972149	−	0.234363i	$-0.0753005\pi$
$228$		0			0
$229$		−	328.507i		−	1.43453i	−0.696801	−	0.717265i	$-0.745394\pi$
$229$		−	328.507i		−	1.43453i	0.696801	−	0.717265i	$-0.254606\pi$
$230$		0			0
$231$	38.4419			0.166415
$232$		0			0
$233$	−149.213			−0.640400			−0.320200	−	0.947350i	$-0.603750\pi$
$233$	−149.213			−0.640400			−0.320200	+	0.947350i	$0.603750\pi$
$234$		0			0
$235$		−	30.9187i		−	0.131569i
$236$		0			0
$237$			48.2812i			0.203718i
$238$		0			0
$239$	343.168			1.43585			0.717925	−	0.696121i	$-0.245092\pi$
$239$	343.168			1.43585			0.717925	+	0.696121i	$0.245092\pi$
$240$		0			0
$241$			348.748i			1.44709i	0.690278	+	0.723544i	$0.257488\pi$
$241$			348.748i			1.44709i	−0.690278	+	0.723544i	$0.742512\pi$
$242$		0			0
$243$	347.570			1.43033
$244$		0			0
$245$		−	104.700i		−	0.427348i
$246$		0			0
$247$		−	25.1624i		−	0.101872i
$248$		0			0
$249$		−	190.079i		−	0.763371i
$250$		0			0
$251$		−	225.099i		−	0.896809i	−0.893831	−	0.448405i	$-0.851992\pi$
$251$		−	225.099i		−	0.896809i	0.893831	−	0.448405i	$-0.148008\pi$
$252$		0			0
$253$	−138.262	−	15.6087i	−0.546491	−	0.0616946i
$254$		0			0
$255$	151.269			0.593213
$256$		0			0
$257$	−393.632			−1.53164			−0.765821	−	0.643054i	$-0.777667\pi$
$257$	−393.632			−1.53164			−0.765821	+	0.643054i	$0.777667\pi$
$258$		0			0
$259$	−22.9041			−0.0884328
$260$		0			0
$261$	−224.188			−0.858959
$262$		0			0
$263$			132.720i			0.504640i	0.967644	+	0.252320i	$0.0811935\pi$
$263$			132.720i			0.504640i	−0.967644	+	0.252320i	$0.918806\pi$
$264$		0			0
$265$	85.5836			0.322957
$266$		0			0
$267$		−	478.263i		−	1.79125i
$268$		0			0
$269$	−438.116			−1.62868			−0.814341	−	0.580386i	$-0.802902\pi$
$269$	−438.116			−1.62868			−0.814341	+	0.580386i	$0.802902\pi$
$270$		0			0
$271$	204.698			0.755342			0.377671	−	0.925940i	$-0.376725\pi$
$271$	204.698			0.755342			0.377671	+	0.925940i	$0.376725\pi$
$272$		0			0
$273$		−	33.1274i		−	0.121346i
$274$		0			0
$275$			30.2479i			0.109993i
$276$		0			0
$277$	173.804			0.627452			0.313726	−	0.949514i	$-0.398423\pi$
$277$	173.804			0.627452			0.313726	+	0.949514i	$0.398423\pi$
$278$		0			0
$279$	195.701			0.701437
$280$		0			0
$281$			143.605i			0.511051i	0.966802	+	0.255526i	$0.0822485\pi$
$281$			143.605i			0.511051i	−0.966802	+	0.255526i	$0.917752\pi$
$282$		0			0
$283$			72.1714i			0.255023i	0.991837	+	0.127511i	$0.0406989\pi$
$283$			72.1714i			0.255023i	−0.991837	+	0.127511i	$0.959301\pi$
$284$		0			0
$285$	−46.4858			−0.163108
$286$		0			0
$287$		−	29.9627i		−	0.104400i
$288$		0			0
$289$	42.3114			0.146406
$290$		0			0
$291$			663.845i			2.28126i
$292$		0			0
$293$		−	124.199i		−	0.423889i	−0.977282	−	0.211944i	$-0.932020\pi$
$293$		−	124.199i		−	0.423889i	0.977282	−	0.211944i	$-0.0679796\pi$
$294$		0			0
$295$		−	75.0640i		−	0.254454i
$296$		0			0
$297$			14.3746i			0.0483993i
$298$		0			0
$299$	−13.4508	+	119.148i	−0.0449861	+	0.398487i
$300$		0			0
$301$	−56.3126			−0.187085
$302$		0			0
$303$	253.523			0.836709
$304$		0			0
$305$	−224.875			−0.737294
$306$		0			0
$307$	−219.717			−0.715690			−0.357845	−	0.933781i	$-0.616488\pi$
$307$	−219.717			−0.715690			−0.357845	+	0.933781i	$0.616488\pi$
$308$		0			0
$309$			233.733i			0.756418i
$310$		0			0
$311$	317.069			1.01951			0.509757	−	0.860319i	$-0.329735\pi$
$311$	317.069			1.01951			0.509757	+	0.860319i	$0.329735\pi$
$312$		0			0
$313$			484.654i			1.54841i	0.632932	+	0.774207i	$0.281851\pi$
$313$			484.654i			1.54841i	−0.632932	+	0.774207i	$0.718149\pi$
$314$		0			0
$315$	−31.5103			−0.100033
$316$		0			0
$317$	−521.910			−1.64640			−0.823202	−	0.567748i	$-0.807815\pi$
$317$	−521.910			−1.64640			−0.823202	+	0.567748i	$0.807815\pi$
$318$		0			0
$319$			141.991i			0.445111i
$320$		0			0
$321$		−	513.086i		−	1.59840i
$322$		0			0
$323$	75.8086			0.234702
$324$		0			0
$325$	26.0662			0.0802037
$326$		0			0
$327$		−	642.729i		−	1.96553i
$328$		0			0
$329$		−	20.3997i		−	0.0620053i
$330$		0			0
$331$	318.270			0.961540			0.480770	−	0.876847i	$-0.340357\pi$
$331$	318.270			0.961540			0.480770	+	0.876847i	$0.340357\pi$
$332$		0			0
$333$		−	148.288i		−	0.445308i
$334$		0			0
$335$	−72.5535			−0.216578
$336$		0			0
$337$		−	116.539i		−	0.345812i	−0.984938	−	0.172906i	$-0.944684\pi$
$337$		−	116.539i		−	0.345812i	0.984938	−	0.172906i	$-0.0553156\pi$
$338$		0			0
$339$			152.189i			0.448935i
$340$		0			0
$341$		−	123.948i		−	0.363484i
$342$		0			0
$343$		−	141.371i		−	0.412159i
$344$		0			0
$345$	220.117	+	24.8495i	0.638021	+	0.0720276i
$346$		0			0
$347$	100.715			0.290245			0.145122	−	0.989414i	$-0.453642\pi$
$347$	100.715			0.290245			0.145122	+	0.989414i	$0.453642\pi$
$348$		0			0
$349$	143.476			0.411106			0.205553	−	0.978646i	$-0.434101\pi$
$349$	143.476			0.411106			0.205553	+	0.978646i	$0.434101\pi$
$350$		0			0
$351$	12.3873			0.0352915
$352$		0			0
$353$	−274.394			−0.777320			−0.388660	−	0.921381i	$-0.627062\pi$
$353$	−274.394			−0.777320			−0.388660	+	0.921381i	$0.627062\pi$
$354$		0			0
$355$		−	53.9577i		−	0.151994i
$356$		0			0
$357$	99.8053			0.279567
$358$		0			0
$359$		−	476.871i		−	1.32833i	−0.747586	−	0.664166i	$-0.768787\pi$
$359$		−	476.871i		−	1.32833i	0.747586	−	0.664166i	$-0.231213\pi$
$360$		0			0
$361$	337.704			0.935467
$362$		0			0
$363$	−363.535			−1.00147
$364$		0			0
$365$		−	33.9475i		−	0.0930067i
$366$		0			0
$367$		−	14.0319i		−	0.0382340i	−0.999817	−	0.0191170i	$-0.993914\pi$
$367$		−	14.0319i		−	0.0382340i	0.999817	−	0.0191170i	$-0.00608551\pi$
$368$		0			0
$369$	193.987			0.525711
$370$		0			0
$371$	56.4668			0.152202
$372$		0			0
$373$			105.662i			0.283277i	0.989918	+	0.141639i	$0.0452371\pi$
$373$			105.662i			0.283277i	−0.989918	+	0.141639i	$0.954763\pi$
$374$		0			0
$375$		−	48.1556i		−	0.128415i
$376$		0			0
$377$	122.361			0.324564
$378$		0			0
$379$			339.983i			0.897053i	0.893770	+	0.448527i	$0.148051\pi$
$379$			339.983i			0.897053i	−0.893770	+	0.448527i	$0.851949\pi$
$380$		0			0
$381$	399.876			1.04954
$382$		0			0
$383$			699.796i			1.82714i	0.406676	+	0.913572i	$0.366688\pi$
$383$			699.796i			1.82714i	−0.406676	+	0.913572i	$0.633312\pi$
$384$		0			0
$385$			19.9571i			0.0518367i
$386$		0			0
$387$		−	364.584i		−	0.942077i
$388$		0			0
$389$			124.817i			0.320867i	0.987047	+	0.160433i	$0.0512892\pi$
$389$			124.817i			0.320867i	−0.987047	+	0.160433i	$0.948711\pi$
$390$		0			0
$391$	−358.965	−	40.5244i	−0.918070	−	0.103643i
$392$		0			0
$393$	653.436			1.66269
$394$		0			0
$395$	−25.0652			−0.0634563
$396$		0			0
$397$	−149.557			−0.376719			−0.188360	−	0.982100i	$-0.560317\pi$
$397$	−149.557			−0.376719			−0.188360	+	0.982100i	$0.560317\pi$
$398$		0			0
$399$	−30.6707			−0.0768688
$400$		0			0
$401$			426.696i			1.06408i	0.846719	+	0.532040i	$0.178574\pi$
$401$			426.696i			1.06408i	−0.846719	+	0.532040i	$0.821426\pi$
$402$		0			0
$403$	−106.812			−0.265043
$404$		0			0
$405$			169.339i			0.418121i
$406$		0			0
$407$	−93.9186			−0.230758
$408$		0			0
$409$	346.263			0.846610			0.423305	−	0.905987i	$-0.360870\pi$
$409$	346.263			0.846610			0.423305	+	0.905987i	$0.360870\pi$
$410$		0			0
$411$			1152.01i			2.80294i
$412$		0			0
$413$		−	49.5261i		−	0.119918i
$414$		0			0
$415$	98.6798			0.237783
$416$		0			0
$417$	1032.77			2.47667
$418$		0			0
$419$		−	158.433i		−	0.378121i	−0.981965	−	0.189060i	$-0.939456\pi$
$419$		−	158.433i		−	0.378121i	0.981965	−	0.189060i	$-0.0605442\pi$
$420$		0			0
$421$			18.0568i			0.0428902i	0.999770	+	0.0214451i	$0.00682671\pi$
$421$			18.0568i			0.0428902i	−0.999770	+	0.0214451i	$0.993173\pi$
$422$		0			0
$423$	132.074			0.312231
$424$		0			0
$425$			78.5316i			0.184780i
$426$		0			0
$427$	−148.369			−0.347469
$428$		0			0
$429$		−	135.839i		−	0.316641i
$430$		0			0
$431$		−	151.730i		−	0.352041i	−0.984387	−	0.176020i	$-0.943678\pi$
$431$		−	151.730i		−	0.352041i	0.984387	−	0.176020i	$-0.0563225\pi$
$432$		0			0
$433$		−	625.471i		−	1.44451i	−0.691629	−	0.722253i	$-0.743106\pi$
$433$		−	625.471i		−	1.44451i	0.691629	−	0.722253i	$-0.256894\pi$
$434$		0			0
$435$		−	226.053i		−	0.519662i
$436$		0			0
$437$	110.312	+	12.4533i	0.252430	+	0.0284973i
$438$		0			0
$439$	−272.842			−0.621509			−0.310755	−	0.950490i	$-0.600582\pi$
$439$	−272.842			−0.621509			−0.310755	+	0.950490i	$0.600582\pi$
$440$		0			0
$441$	447.242			1.01415
$442$		0			0
$443$	−5.57528			−0.0125853			−0.00629264	−	0.999980i	$-0.502003\pi$
$443$	−5.57528			−0.0125853			−0.00629264	+	0.999980i	$0.502003\pi$
$444$		0			0
$445$	248.291			0.557957
$446$		0			0
$447$		−	868.525i		−	1.94301i
$448$		0			0
$449$	455.331			1.01410			0.507050	−	0.861916i	$-0.330736\pi$
$449$	455.331			1.01410			0.507050	+	0.861916i	$0.330736\pi$
$450$		0			0
$451$		−	122.863i		−	0.272423i
$452$		0			0
$453$	520.248			1.14845
$454$		0			0
$455$	17.1981			0.0377980
$456$		0			0
$457$		−	597.785i		−	1.30806i	−0.756467	−	0.654032i	$-0.773076\pi$
$457$		−	597.785i		−	1.30806i	0.756467	−	0.654032i	$-0.226924\pi$
$458$		0			0
$459$			37.3202i			0.0813077i
$460$		0			0
$461$	−340.556			−0.738734			−0.369367	−	0.929284i	$-0.620425\pi$
$461$	−340.556			−0.738734			−0.369367	+	0.929284i	$0.620425\pi$
$462$		0			0
$463$	471.395			1.01813			0.509066	−	0.860727i	$-0.329991\pi$
$463$	471.395			1.01813			0.509066	+	0.860727i	$0.329991\pi$
$464$		0			0
$465$			197.329i			0.424362i
$466$		0			0
$467$		−	245.681i		−	0.526084i	−0.964784	−	0.263042i	$-0.915274\pi$
$467$		−	245.681i		−	0.526084i	0.964784	−	0.263042i	$-0.0847257\pi$
$468$		0			0
$469$	−47.8697			−0.102068
$470$		0			0
$471$			839.455i			1.78228i
$472$		0			0
$473$	−230.911			−0.488183
$474$		0			0
$475$		−	24.1332i		−	0.0508066i
$476$		0			0
$477$			365.582i			0.766420i
$478$		0			0
$479$			203.940i			0.425763i	0.977078	+	0.212881i	$0.0682848\pi$
$479$			203.940i			0.425763i	−0.977078	+	0.212881i	$0.931715\pi$
$480$		0			0
$481$			80.9345i			0.168263i
$482$		0			0
$483$	145.230	+	16.3954i	0.300684	+	0.0339448i
$484$		0			0
$485$	−344.636			−0.710589
$486$		0			0
$487$	681.456			1.39929			0.699647	−	0.714489i	$-0.253341\pi$
$487$	681.456			1.39929			0.699647	+	0.714489i	$0.253341\pi$
$488$		0			0
$489$	662.496			1.35480
$490$		0			0
$491$	329.666			0.671417			0.335709	−	0.941966i	$-0.391024\pi$
$491$	329.666			0.671417			0.335709	+	0.941966i	$0.391024\pi$
$492$		0			0
$493$			368.645i			0.747758i
$494$		0			0
$495$	−129.208			−0.261027
$496$		0			0
$497$		−	35.6005i		−	0.0716308i
$498$		0			0
$499$	993.708			1.99140			0.995700	−	0.0926377i	$-0.0295298\pi$
$499$	993.708			1.99140			0.995700	+	0.0926377i	$0.0295298\pi$
$500$		0			0
$501$	208.876			0.416918
$502$		0			0
$503$			880.515i			1.75053i	0.483646	+	0.875264i	$0.339312\pi$
$503$			880.515i			1.75053i	−0.483646	+	0.875264i	$0.660688\pi$
$504$		0			0
$505$			131.617i			0.260627i
$506$		0			0
$507$	610.851			1.20483
$508$		0			0
$509$	−689.907			−1.35542			−0.677708	−	0.735331i	$-0.737027\pi$
$509$	−689.907			−1.35542			−0.677708	+	0.735331i	$0.737027\pi$
$510$		0			0
$511$		−	22.3980i		−	0.0438318i
$512$		0			0
$513$		−	11.4687i		−	0.0223561i
$514$		0			0
$515$	−121.343			−0.235617
$516$		0			0
$517$		−	83.6494i		−	0.161798i
$518$		0			0
$519$	479.255			0.923419
$520$		0			0
$521$			408.559i			0.784183i	0.919926	+	0.392092i	$0.128248\pi$
$521$			408.559i			0.784183i	−0.919926	+	0.392092i	$0.871752\pi$
$522$		0			0
$523$		−	516.837i		−	0.988216i	−0.869400	−	0.494108i	$-0.835495\pi$
$523$		−	516.837i		−	0.988216i	0.869400	−	0.494108i	$-0.164505\pi$
$524$		0			0
$525$		−	31.7723i		−	0.0605187i
$526$		0			0
$527$		−	321.801i		−	0.610629i
$528$		0			0
$529$	−515.686	−	117.937i	−0.974831	−	0.222943i
$530$		0			0
$531$	320.646			0.603853
$532$		0			0
$533$	−105.877			−0.198644
$534$		0			0
$535$	266.369			0.497886
$536$		0			0
$537$	1131.02			2.10618
$538$		0			0
$539$		−	283.262i		−	0.525533i
$540$		0			0
$541$	−825.026			−1.52500			−0.762501	−	0.646987i	$-0.776029\pi$
$541$	−825.026			−1.52500			−0.762501	+	0.646987i	$0.776029\pi$
$542$		0			0
$543$		−	909.533i		−	1.67501i
$544$		0			0
$545$	333.673			0.612245
$546$		0			0
$547$	749.566			1.37032			0.685161	−	0.728392i	$-0.259732\pi$
$547$	749.566			1.37032			0.685161	+	0.728392i	$0.259732\pi$
$548$		0			0
$549$		−	960.583i		−	1.74970i
$550$		0			0
$551$		−	113.286i		−	0.205601i
$552$		0			0
$553$	−16.5377			−0.0299054
$554$		0			0
$555$	149.521			0.269407
$556$		0			0
$557$		−	406.149i		−	0.729173i	−0.931170	−	0.364586i	$-0.881210\pi$
$557$		−	406.149i		−	0.729173i	0.931170	−	0.364586i	$-0.118790\pi$
$558$		0			0
$559$			198.988i			0.355971i
$560$		0			0
$561$	409.253			0.729506
$562$		0			0
$563$		−	285.771i		−	0.507587i	−0.967258	−	0.253793i	$-0.918322\pi$
$563$		−	285.771i		−	0.507587i	0.967258	−	0.253793i	$-0.0816784\pi$
$564$		0			0
$565$	−79.0091			−0.139839
$566$		0			0
$567$			111.727i			0.197050i
$568$		0			0
$569$			315.624i			0.554699i	0.960769	+	0.277349i	$0.0894560\pi$
$569$			315.624i			0.554699i	−0.960769	+	0.277349i	$0.910544\pi$
$570$		0			0
$571$			383.907i			0.672342i	0.941801	+	0.336171i	$0.109132\pi$
$571$			383.907i			0.672342i	−0.941801	+	0.336171i	$0.890868\pi$
$572$		0			0
$573$			313.652i			0.547385i
$574$		0			0
$575$	−12.9007	+	114.274i	−0.0224359	+	0.198738i
$576$		0			0
$577$	−717.832			−1.24408			−0.622038	−	0.782987i	$-0.713695\pi$
$577$	−717.832			−1.24408			−0.622038	+	0.782987i	$0.713695\pi$
$578$		0			0
$579$	−855.186			−1.47701
$580$		0			0
$581$	65.1075			0.112061
$582$		0			0
$583$	231.543			0.397158
$584$		0			0
$585$			111.345i			0.190334i
$586$		0			0
$587$	296.393			0.504928			0.252464	−	0.967606i	$-0.418759\pi$
$587$	296.393			0.504928			0.252464	+	0.967606i	$0.418759\pi$
$588$		0			0
$589$			98.8912i			0.167897i
$590$		0			0
$591$	−121.486			−0.205560
$592$		0			0
$593$	654.259			1.10330			0.551652	−	0.834075i	$-0.313998\pi$
$593$	654.259			1.10330			0.551652	+	0.834075i	$0.313998\pi$
$594$		0			0
$595$			51.8140i			0.0870824i
$596$		0			0
$597$		−	923.881i		−	1.54754i
$598$		0			0
$599$	−368.673			−0.615480			−0.307740	−	0.951470i	$-0.599573\pi$
$599$	−368.673			−0.615480			−0.307740	+	0.951470i	$0.599573\pi$
$600$		0			0
$601$	148.316			0.246782			0.123391	−	0.992358i	$-0.460623\pi$
$601$	148.316			0.246782			0.123391	+	0.992358i	$0.460623\pi$
$602$		0			0
$603$		−	309.922i		−	0.513967i
$604$		0			0
$605$		−	188.730i		−	0.311950i
$606$		0			0
$607$	−939.932			−1.54849			−0.774244	−	0.632887i	$-0.781870\pi$
$607$	−939.932			−1.54849			−0.774244	+	0.632887i	$0.781870\pi$
$608$		0			0
$609$		−	149.146i		−	0.244904i
$610$		0			0
$611$	−72.0850			−0.117979
$612$		0			0
$613$			602.963i			0.983626i	0.870701	+	0.491813i	$0.163666\pi$
$613$			602.963i			0.983626i	−0.870701	+	0.491813i	$0.836334\pi$
$614$		0			0
$615$			195.601i			0.318050i
$616$		0			0
$617$		−	885.697i		−	1.43549i	−0.696306	−	0.717745i	$-0.745174\pi$
$617$		−	885.697i		−	1.43549i	0.696306	−	0.717745i	$-0.254826\pi$
$618$		0			0
$619$			85.9838i			0.138908i	0.997585	+	0.0694538i	$0.0221257\pi$
$619$			85.9838i			0.138908i	−0.997585	+	0.0694538i	$0.977874\pi$
$620$		0			0
$621$	−6.13072	+	54.3060i	−0.00987233	+	0.0874492i
$622$		0			0
$623$	163.819			0.262951
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−125.765			−0.200583
$628$		0			0
$629$	−243.837			−0.387659
$630$		0			0
$631$		−	645.385i		−	1.02280i	−0.859344	−	0.511398i	$-0.829128\pi$
$631$		−	645.385i		−	1.02280i	0.859344	−	0.511398i	$-0.170872\pi$
$632$		0			0
$633$	1034.85			1.63483
$634$		0			0
$635$			207.596i			0.326923i
$636$		0			0
$637$	−244.102			−0.383205
$638$		0			0
$639$	230.488			0.360701
$640$		0			0
$641$			140.119i			0.218594i	0.994009	+	0.109297i	$0.0348599\pi$
$641$			140.119i			0.218594i	−0.994009	+	0.109297i	$0.965140\pi$
$642$		0			0
$643$		−	1028.11i		−	1.59892i	−0.600718	−	0.799461i	$-0.705118\pi$
$643$		−	1028.11i		−	1.59892i	0.600718	−	0.799461i	$-0.294882\pi$
$644$		0			0
$645$	367.616			0.569948
$646$		0			0
$647$	−1248.29			−1.92934			−0.964672	−	0.263455i	$-0.915138\pi$
$647$	−1248.29			−1.92934			−0.964672	+	0.263455i	$0.915138\pi$
$648$		0			0
$649$		−	203.082i		−	0.312916i
$650$		0			0
$651$			130.194i			0.199992i
$652$		0			0
$653$	−457.504			−0.700618			−0.350309	−	0.936634i	$-0.613923\pi$
$653$	−457.504			−0.700618			−0.350309	+	0.936634i	$0.613923\pi$
$654$		0			0
$655$			339.232i			0.517911i
$656$		0			0
$657$	145.011			0.220717
$658$		0			0
$659$			173.014i			0.262540i	0.991347	+	0.131270i	$0.0419055\pi$
$659$			173.014i			0.262540i	−0.991347	+	0.131270i	$0.958095\pi$
$660$		0			0
$661$			582.673i			0.881502i	0.897629	+	0.440751i	$0.145288\pi$
$661$			582.673i			0.881502i	−0.897629	+	0.440751i	$0.854712\pi$
$662$		0			0
$663$		−	352.674i		−	0.531937i
$664$		0			0
$665$		−	15.9227i		−	0.0239439i
$666$		0			0
$667$	−60.5585	+	536.428i	−0.0907924	+	0.804240i
$668$		0			0
$669$	1108.67			1.65721
$670$		0			0
$671$	−608.389			−0.906690
$672$		0			0
$673$	−390.678			−0.580502			−0.290251	−	0.956951i	$-0.593739\pi$
$673$	−390.678			−0.580502			−0.290251	+	0.956951i	$0.593739\pi$
$674$		0			0
$675$	11.8806			0.0176009
$676$		0			0
$677$			952.947i			1.40760i	0.710397	+	0.703801i	$0.248515\pi$
$677$			952.947i			1.40760i	−0.710397	+	0.703801i	$0.751485\pi$
$678$		0			0
$679$	−227.386			−0.334883
$680$		0			0
$681$			458.286i			0.672961i
$682$		0			0
$683$	−299.732			−0.438847			−0.219423	−	0.975630i	$-0.570418\pi$
$683$	−299.732			−0.438847			−0.219423	+	0.975630i	$0.570418\pi$
$684$		0			0
$685$	−598.067			−0.873090
$686$		0			0
$687$			1414.93i			2.05958i
$688$		0			0
$689$		−	199.532i		−	0.289597i
$690$		0			0
$691$	−165.585			−0.239631			−0.119815	−	0.992796i	$-0.538230\pi$
$691$	−165.585			−0.239631			−0.119815	+	0.992796i	$0.538230\pi$
$692$		0			0
$693$	−85.2497			−0.123015
$694$		0			0
$695$			536.163i			0.771458i
$696$		0			0
$697$		−	318.984i		−	0.457652i
$698$		0			0
$699$	642.685			0.919436
$700$		0			0
$701$			831.750i			1.18652i	0.805011	+	0.593260i	$0.202159\pi$
$701$			831.750i			1.18652i	−0.805011	+	0.593260i	$0.797841\pi$
$702$		0			0
$703$	74.9324			0.106590
$704$		0			0
$705$			133.172i			0.188897i
$706$		0			0
$707$			86.8387i			0.122827i
$708$		0			0
$709$		−	835.784i		−	1.17882i	−0.807834	−	0.589410i	$-0.799360\pi$
$709$		−	835.784i		−	1.17882i	0.807834	−	0.589410i	$-0.200640\pi$
$710$		0			0
$711$		−	107.070i		−	0.150590i
$712$		0			0
$713$	52.8634	−	468.265i	0.0741422	−	0.656753i
$714$		0			0
$715$	70.5210			0.0986308
$716$		0			0
$717$	−1478.08			−2.06148
$718$		0			0
$719$	−881.656			−1.22623			−0.613113	−	0.789996i	$-0.710083\pi$
$719$	−881.656			−1.22623			−0.613113	+	0.789996i	$0.710083\pi$
$720$		0			0
$721$	−80.0602			−0.111041
$722$		0			0
$723$		−	1502.12i		−	2.07762i
$724$		0			0
$725$	117.356			0.161870
$726$		0			0
$727$		−	531.577i		−	0.731193i	−0.930774	−	0.365596i	$-0.880865\pi$
$727$		−	531.577i		−	0.731193i	0.930774	−	0.365596i	$-0.119135\pi$
$728$		0			0
$729$	−815.463			−1.11861
$730$		0			0
$731$	−599.505			−0.820116
$732$		0			0
$733$			18.4847i			0.0252178i	0.999921	+	0.0126089i	$0.00401364\pi$
$733$			18.4847i			0.0252178i	−0.999921	+	0.0126089i	$0.995986\pi$
$734$		0			0
$735$			450.962i			0.613553i
$736$		0			0
$737$	−196.290			−0.266337
$738$		0			0
$739$	960.371			1.29955			0.649777	−	0.760125i	$-0.274862\pi$
$739$	960.371			1.29955			0.649777	+	0.760125i	$0.274862\pi$
$740$		0			0
$741$			108.379i			0.146260i
$742$		0			0
$743$			248.966i			0.335082i	0.985865	+	0.167541i	$0.0535826\pi$
$743$			248.966i			0.335082i	−0.985865	+	0.167541i	$0.946417\pi$
$744$		0			0
$745$	450.896			0.605229
$746$		0			0
$747$			421.524i			0.564289i
$748$		0			0
$749$	175.746			0.234641
$750$		0			0
$751$			24.0757i			0.0320582i	0.999872	+	0.0160291i	$0.00510244\pi$
$751$			24.0757i			0.0320582i	−0.999872	+	0.0160291i	$0.994898\pi$
$752$		0			0
$753$			969.539i			1.28757i
$754$		0			0
$755$			270.087i			0.357731i
$756$		0			0
$757$		−	914.334i		−	1.20784i	−0.797045	−	0.603920i	$-0.793605\pi$
$757$		−	914.334i		−	1.20784i	0.797045	−	0.603920i	$-0.206395\pi$
$758$		0			0
$759$	595.518	+	67.2294i	0.784609	+	0.0885762i
$760$		0			0
$761$	313.770			0.412312			0.206156	−	0.978519i	$-0.433905\pi$
$761$	313.770			0.412312			0.206156	+	0.978519i	$0.433905\pi$
$762$		0			0
$763$	220.153			0.288536
$764$		0			0
$765$	−335.458			−0.438508
$766$		0			0
$767$	−175.007			−0.228170
$768$		0			0
$769$			309.335i			0.402257i	0.979565	+	0.201128i	$0.0644608\pi$
$769$			309.335i			0.402257i	−0.979565	+	0.201128i	$0.935539\pi$
$770$		0			0
$771$	1695.44			2.19901
$772$		0			0
$773$		−	233.356i		−	0.301884i	−0.988543	−	0.150942i	$-0.951769\pi$
$773$		−	233.356i		−	0.301884i	0.988543	−	0.150942i	$-0.0482306\pi$
$774$		0			0
$775$	−102.443			−0.132185
$776$		0			0
$777$	98.6518			0.126965
$778$		0			0
$779$			98.0253i			0.125835i
$780$		0			0
$781$		−	145.980i		−	0.186915i
$782$		0			0
$783$	55.7703			0.0712265
$784$		0			0
$785$	−435.803			−0.555164
$786$		0			0
$787$			690.751i			0.877702i	0.898560	+	0.438851i	$0.144614\pi$
$787$			690.751i			0.877702i	−0.898560	+	0.438851i	$0.855386\pi$
$788$		0			0
$789$		−	571.648i		−	0.724522i
$790$		0			0
$791$	−52.1290			−0.0659027
$792$		0			0
$793$			524.280i			0.661135i
$794$		0			0
$795$	−368.623			−0.463677
$796$		0			0
$797$			91.6732i			0.115023i	0.998345	+	0.0575114i	$0.0183166\pi$
$797$			91.6732i			0.115023i	−0.998345	+	0.0575114i	$0.981683\pi$
$798$		0			0
$799$		−	217.176i		−	0.271810i
$800$		0			0
$801$			1060.61i			1.32410i
$802$		0			0
$803$		−	91.8434i		−	0.114375i
$804$		0			0
$805$	−8.51166	+	75.3964i	−0.0105735	+	0.0936601i
$806$		0			0
$807$	1887.04			2.33833
$808$		0			0
$809$	1071.73			1.32476			0.662382	−	0.749166i	$-0.269546\pi$
$809$	1071.73			1.32476			0.662382	+	0.749166i	$0.269546\pi$
$810$		0			0
$811$	257.388			0.317371			0.158685	−	0.987329i	$-0.449274\pi$
$811$	257.388			0.317371			0.158685	+	0.987329i	$0.449274\pi$
$812$		0			0
$813$	−881.666			−1.08446
$814$		0			0
$815$			343.935i			0.422006i
$816$		0			0
$817$	184.231			0.225497
$818$		0			0
$819$			73.4640i			0.0896996i
$820$		0			0
$821$	−349.772			−0.426032			−0.213016	−	0.977049i	$-0.568329\pi$
$821$	−349.772			−0.426032			−0.213016	+	0.977049i	$0.568329\pi$
$822$		0			0
$823$	−1438.73			−1.74815			−0.874076	−	0.485789i	$-0.838532\pi$
$823$	−1438.73			−1.74815			−0.874076	+	0.485789i	$0.838532\pi$
$824$		0			0
$825$		−	130.283i		−	0.157919i
$826$		0			0
$827$		−	996.093i		−	1.20447i	−0.798320	−	0.602233i	$-0.794278\pi$
$827$		−	996.093i		−	1.20447i	0.798320	−	0.602233i	$-0.205722\pi$
$828$		0			0
$829$	−234.343			−0.282681			−0.141341	−	0.989961i	$-0.545141\pi$
$829$	−234.343			−0.282681			−0.141341	+	0.989961i	$0.545141\pi$
$830$		0			0
$831$	−748.603			−0.900846
$832$		0			0
$833$		−	735.424i		−	0.882862i
$834$		0			0
$835$			108.438i			0.129866i
$836$		0			0
$837$	−48.6836			−0.0581644
$838$		0			0
$839$		−	742.848i		−	0.885397i	−0.896670	−	0.442699i	$-0.854021\pi$
$839$		−	742.848i		−	0.885397i	0.896670	−	0.442699i	$-0.145979\pi$
$840$		0			0
$841$	−290.107			−0.344955
$842$		0			0
$843$		−	618.532i		−	0.733727i
$844$		0			0
$845$			317.124i			0.375295i
$846$		0			0
$847$		−	124.521i		−	0.147014i
$848$		0			0
$849$		−	310.854i		−	0.366142i
$850$		0			0
$851$	−354.816	−	40.0560i	−0.416940	−	0.0470693i
$852$		0			0
$853$	482.186			0.565283			0.282642	−	0.959226i	$-0.408789\pi$
$853$	482.186			0.565283			0.282642	+	0.959226i	$0.408789\pi$
$854$		0			0
$855$	103.088			0.120571
$856$		0			0
$857$	685.909			0.800361			0.400180	−	0.916436i	$-0.368947\pi$
$857$	685.909			0.800361			0.400180	+	0.916436i	$0.368947\pi$
$858$		0			0
$859$	−670.668			−0.780754			−0.390377	−	0.920655i	$-0.627655\pi$
$859$	−670.668			−0.780754			−0.390377	+	0.920655i	$0.627655\pi$
$860$		0			0
$861$			129.054i			0.149889i
$862$		0			0
$863$	−891.818			−1.03339			−0.516696	−	0.856169i	$-0.672838\pi$
$863$	−891.818			−1.03339			−0.516696	+	0.856169i	$0.672838\pi$
$864$		0			0
$865$			248.805i			0.287636i
$866$		0			0
$867$	−182.242			−0.210199
$868$		0			0
$869$	−67.8129			−0.0780356
$870$		0			0
$871$			169.154i			0.194206i
$872$		0			0
$873$		−	1472.16i		−	1.68632i
$874$		0			0
$875$	16.4946			0.0188510
$876$		0			0
$877$	181.676			0.207156			0.103578	−	0.994621i	$-0.466971\pi$
$877$	181.676			0.207156			0.103578	+	0.994621i	$0.466971\pi$
$878$		0			0
$879$			534.947i			0.608586i
$880$		0			0
$881$		−	316.887i		−	0.359691i	−0.983695	−	0.179845i	$-0.942440\pi$
$881$		−	316.887i		−	0.359691i	0.983695	−	0.179845i	$-0.0575597\pi$
$882$		0			0
$883$	1361.85			1.54230			0.771148	−	0.636656i	$-0.219683\pi$
$883$	1361.85			1.54230			0.771148	+	0.636656i	$0.219683\pi$
$884$		0			0
$885$			323.313i			0.365325i
$886$		0			0
$887$	−1628.63			−1.83611			−0.918057	−	0.396449i	$-0.870242\pi$
$887$	−1628.63			−1.83611			−0.918057	+	0.396449i	$0.870242\pi$
$888$		0			0
$889$			136.969i			0.154071i
$890$		0			0
$891$			458.139i			0.514185i
$892$		0			0
$893$			66.7392i			0.0747360i
$894$		0			0
$895$			587.169i			0.656055i
$896$		0			0
$897$	57.9350	−	513.189i	0.0645875	−	0.572117i
$898$		0			0
$899$	−480.891			−0.534918
$900$		0			0
$901$	601.146			0.667199
$902$		0			0
$903$	242.548			0.268602
$904$		0			0
$905$	472.185			0.521751
$906$		0			0
$907$			1551.21i			1.71026i	0.518413	+	0.855130i	$0.326523\pi$
$907$			1551.21i			1.71026i	−0.518413	+	0.855130i	$0.673477\pi$
$908$		0			0
$909$	−562.218			−0.618502
$910$		0			0
$911$		−	1672.31i		−	1.83569i	−0.396937	−	0.917846i	$-0.629927\pi$
$911$		−	1672.31i		−	1.83569i	0.396937	−	0.917846i	$-0.370073\pi$
$912$		0			0
$913$	266.974			0.292414
$914$		0			0
$915$	968.573			1.05855
$916$		0			0
$917$			223.820i			0.244079i
$918$		0			0
$919$		−	1670.00i		−	1.81719i	−0.417673	−	0.908597i	$-0.637154\pi$
$919$		−	1670.00i		−	1.81719i	0.417673	−	0.908597i	$-0.362846\pi$
$920$		0			0
$921$	946.357			1.02753
$922$		0			0
$923$	−125.799			−0.136293
$924$		0			0
$925$			77.6240i			0.0839178i
$926$		0			0
$927$		−	518.332i		−	0.559150i
$928$		0			0
$929$	−877.260			−0.944306			−0.472153	−	0.881517i	$-0.656523\pi$
$929$	−877.260			−0.944306			−0.472153	+	0.881517i	$0.656523\pi$
$930$		0			0
$931$			225.999i			0.242749i
$932$		0			0
$933$	−1365.67			−1.46374
$934$		0			0
$935$			212.464i			0.227234i
$936$		0			0
$937$			726.689i			0.775549i	0.921754	+	0.387774i	$0.126756\pi$
$937$			726.689i			0.775549i	−0.921754	+	0.387774i	$0.873244\pi$
$938$		0			0
$939$		−	2087.48i		−	2.22309i
$940$		0			0
$941$			289.839i			0.308011i	0.988070	+	0.154006i	$0.0492174\pi$
$941$			289.839i			0.308011i	−0.988070	+	0.154006i	$0.950783\pi$
$942$		0			0
$943$	52.4005	−	464.164i	0.0555679	−	0.492221i
$944$		0			0
$945$	7.83867			0.00829489
$946$		0			0
$947$	1420.59			1.50010			0.750050	−	0.661381i	$-0.230030\pi$
$947$	1420.59			1.50010			0.750050	+	0.661381i	$0.230030\pi$
$948$		0			0
$949$	−79.1462			−0.0833996
$950$		0			0
$951$	2247.95			2.36378
$952$		0			0
$953$		−	1465.45i		−	1.53772i	−0.639418	−	0.768860i	$-0.720824\pi$
$953$		−	1465.45i		−	1.53772i	0.639418	−	0.768860i	$-0.279176\pi$
$954$		0			0
$955$	−162.832			−0.170505
$956$		0			0
$957$		−	611.577i		−	0.639056i
$958$		0			0
$959$	−394.596			−0.411466
$960$		0			0
$961$	−541.215			−0.563179
$962$		0			0
$963$			1137.83i			1.18155i
$964$		0			0
$965$		−	443.970i		−	0.460073i
$966$		0			0
$967$	1536.47			1.58890			0.794451	−	0.607328i	$-0.207759\pi$
$967$	1536.47			1.58890			0.794451	+	0.607328i	$0.207759\pi$
$968$		0			0
$969$	−326.520			−0.336966
$970$		0			0
$971$			1693.09i			1.74366i	0.489812	+	0.871828i	$0.337065\pi$
$971$			1693.09i			1.74366i	−0.489812	+	0.871828i	$0.662935\pi$
$972$		0			0
$973$			353.752i			0.363569i
$974$		0			0
$975$	−112.271			−0.115150
$976$		0			0
$977$			1295.96i			1.32647i	0.748411	+	0.663235i	$0.230817\pi$
$977$			1295.96i			1.32647i	−0.748411	+	0.663235i	$0.769183\pi$
$978$		0			0
$979$	671.740			0.686150
$980$		0			0
$981$			1425.33i			1.45294i
$982$		0			0
$983$			1692.07i			1.72133i	0.509168	+	0.860667i	$0.329953\pi$
$983$			1692.07i			1.72133i	−0.509168	+	0.860667i	$0.670047\pi$
$984$		0			0
$985$		−	63.0695i		−	0.0640299i
$986$		0			0
$987$			87.8650i			0.0890223i
$988$		0			0
$989$	−872.361	−	98.4827i	−0.882063	−	0.0995780i
$990$		0			0
$991$	−45.3192			−0.0457308			−0.0228654	−	0.999739i	$-0.507279\pi$
$991$	−45.3192			−0.0457308			−0.0228654	+	0.999739i	$0.507279\pi$
$992$		0			0
$993$	−1370.84			−1.38050
$994$		0			0
$995$	479.633			0.482044
$996$		0			0
$997$	1358.89			1.36298			0.681488	−	0.731830i	$-0.261333\pi$
$997$	1358.89			1.36298			0.681488	+	0.731830i	$0.261333\pi$
$998$		0			0
$999$			36.8889i			0.0369258i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.3		16
4.3	odd	2		230.3.d.a.91.15	✓	16
12.11	even	2		2070.3.c.a.91.7		16
20.3	even	4		1150.3.c.c.1149.19		32
20.7	even	4		1150.3.c.c.1149.14		32
20.19	odd	2		1150.3.d.b.551.1		16
23.22	odd	2	inner	1840.3.k.d.321.4		16
92.91	even	2		230.3.d.a.91.16	yes	16
276.275	odd	2		2070.3.c.a.91.2		16
460.183	odd	4		1150.3.c.c.1149.13		32
460.367	odd	4		1150.3.c.c.1149.20		32
460.459	even	2		1150.3.d.b.551.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.15	✓	16	4.3	odd	2
230.3.d.a.91.16	yes	16	92.91	even	2
1150.3.c.c.1149.13		32	460.183	odd	4
1150.3.c.c.1149.14		32	20.7	even	4
1150.3.c.c.1149.19		32	20.3	even	4
1150.3.c.c.1149.20		32	460.367	odd	4
1150.3.d.b.551.1		16	20.19	odd	2
1150.3.d.b.551.2		16	460.459	even	2
1840.3.k.d.321.3		16	1.1	even	1	trivial
1840.3.k.d.321.4		16	23.22	odd	2	inner
2070.3.c.a.91.2		16	276.275	odd	2
2070.3.c.a.91.7		16	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.30716 q^{3} -2.23607i q^{5} -1.47532i q^{7} +9.55167 q^{9} +O(q^{10})\)	\(q-4.30716 q^{3} -2.23607i q^{5} -1.47532i q^{7} +9.55167 q^{9} -6.04959i q^{11} -5.21324 q^{13} +9.63111i q^{15} -15.7063i q^{17} +4.82663i q^{19} +6.35447i q^{21} +(2.58013 - 22.8548i) q^{23} -5.00000 q^{25} -2.37613 q^{27} -23.4711 q^{29} +20.4887 q^{31} +26.0566i q^{33} -3.29893 q^{35} -15.5248i q^{37} +22.4543 q^{39} +20.3093 q^{41} -38.1696i q^{43} -21.3582i q^{45} +13.8273 q^{47} +46.8234 q^{49} +67.6497i q^{51} +38.2742i q^{53} -13.5273 q^{55} -20.7891i q^{57} +33.5696 q^{59} -100.567i q^{61} -14.0918i q^{63} +11.6572i q^{65} -32.4469i q^{67} +(-11.1130 + 98.4395i) q^{69} +24.1306 q^{71} +15.1818 q^{73} +21.5358 q^{75} -8.92511 q^{77} -11.2095i q^{79} -75.7306 q^{81} +44.1310i q^{83} -35.1204 q^{85} +101.094 q^{87} +111.039i q^{89} +7.69122i q^{91} -88.2480 q^{93} +10.7927 q^{95} -154.126i q^{97} -57.7837i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more