LMFDB - Embedded newform 1840.3.c.b.1151.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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([1, 0, 0, 0]))

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

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.6
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.51

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.48524i q^{3} -2.23607 q^{5} -1.27380i q^{7} -11.1174 q^{9} +O(q^{10})$	$q-4.48524i q^{3} -2.23607 q^{5} -1.27380i q^{7} -11.1174 q^{9} +8.24171i q^{11} -18.3946 q^{13} +10.0293i q^{15} -2.71935 q^{17} +28.0997i q^{19} -5.71330 q^{21} -4.79583i q^{23} +5.00000 q^{25} +9.49715i q^{27} +45.6464 q^{29} -3.73087i q^{31} +36.9661 q^{33} +2.84830i q^{35} -18.6531 q^{37} +82.5041i q^{39} +8.29456 q^{41} +17.6012i q^{43} +24.8593 q^{45} -36.9085i q^{47} +47.3774 q^{49} +12.1970i q^{51} +53.0820 q^{53} -18.4290i q^{55} +126.034 q^{57} -17.7747i q^{59} +68.7826 q^{61} +14.1614i q^{63} +41.1315 q^{65} -19.5912i q^{67} -21.5105 q^{69} +114.683i q^{71} -130.007 q^{73} -22.4262i q^{75} +10.4983 q^{77} -61.0896i q^{79} -57.4597 q^{81} +101.488i q^{83} +6.08066 q^{85} -204.735i q^{87} +49.0728 q^{89} +23.4310i q^{91} -16.7339 q^{93} -62.8328i q^{95} -5.55033 q^{97} -91.6265i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

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
$3$	− 4.48524i	− 1.49508i	−0.664216	−	0.747541i	$-0.731234\pi$
$3$	− 4.48524i	− 1.49508i	0.664216	−	0.747541i	$-0.268766\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	− 1.27380i	− 0.181971i	−0.995852	−	0.0909856i	$-0.970998\pi$
$7$	− 1.27380i	− 0.181971i	0.995852	−	0.0909856i	$-0.0290017\pi$
$8$	0	0
$9$	−11.1174	−1.23527
$10$	0	0
$11$	8.24171i	0.749246i	0.927177	+	0.374623i	$0.122228\pi$
$11$	8.24171i	0.749246i	−0.927177	+	0.374623i	$0.877772\pi$
$12$	0	0
$13$	−18.3946	−1.41497	−0.707483	−	0.706731i	$-0.750169\pi$
$13$	−18.3946	−1.41497	−0.707483	+	0.706731i	$0.750169\pi$
$14$	0	0
$15$	10.0293i	0.668621i
$16$	0	0
$17$	−2.71935	−0.159962	−0.0799810	−	0.996796i	$-0.525486\pi$
$17$	−2.71935	−0.159962	−0.0799810	+	0.996796i	$0.525486\pi$
$18$	0	0
$19$	28.0997i	1.47893i	0.673195	+	0.739465i	$0.264922\pi$
$19$	28.0997i	1.47893i	−0.673195	+	0.739465i	$0.735078\pi$
$20$	0	0
$21$	−5.71330	−0.272062
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	9.49715i	0.351746i
$28$	0	0
$29$	45.6464	1.57401	0.787007	−	0.616945i	$-0.211630\pi$
$29$	45.6464	1.57401	0.787007	+	0.616945i	$0.211630\pi$
$30$	0	0
$31$	− 3.73087i	− 0.120351i	−0.998188	−	0.0601754i	$-0.980834\pi$
$31$	− 3.73087i	− 0.120351i	0.998188	−	0.0601754i	$-0.0191660\pi$
$32$	0	0
$33$	36.9661	1.12018
$34$	0	0
$35$	2.84830i	0.0813800i
$36$	0	0
$37$	−18.6531	−0.504139	−0.252069	−	0.967709i	$-0.581111\pi$
$37$	−18.6531	−0.504139	−0.252069	+	0.967709i	$0.581111\pi$
$38$	0	0
$39$	82.5041i	2.11549i
$40$	0	0
$41$	8.29456	0.202306	0.101153	−	0.994871i	$-0.467747\pi$
$41$	8.29456	0.202306	0.101153	+	0.994871i	$0.467747\pi$
$42$	0	0
$43$	17.6012i	0.409330i	0.978832	+	0.204665i	$0.0656106\pi$
$43$	17.6012i	0.409330i	−0.978832	+	0.204665i	$0.934389\pi$
$44$	0	0
$45$	24.8593	0.552429
$46$	0	0
$47$	− 36.9085i	− 0.785288i	−0.919691	−	0.392644i	$-0.871561\pi$
$47$	− 36.9085i	− 0.785288i	0.919691	−	0.392644i	$-0.128439\pi$
$48$	0	0
$49$	47.3774	0.966886
$50$	0	0
$51$	12.1970i	0.239156i
$52$	0	0
$53$	53.0820	1.00155	0.500774	−	0.865578i	$-0.333049\pi$
$53$	53.0820	1.00155	0.500774	+	0.865578i	$0.333049\pi$
$54$	0	0
$55$	− 18.4290i	− 0.335073i
$56$	0	0
$57$	126.034	2.21112
$58$	0	0
$59$	− 17.7747i	− 0.301267i	−0.988590	−	0.150633i	$-0.951869\pi$
$59$	− 17.7747i	− 0.301267i	0.988590	−	0.150633i	$-0.0481313\pi$
$60$	0	0
$61$	68.7826	1.12758	0.563792	−	0.825917i	$-0.309342\pi$
$61$	68.7826	1.12758	0.563792	+	0.825917i	$0.309342\pi$
$62$	0	0
$63$	14.1614i	0.224783i
$64$	0	0
$65$	41.1315	0.632792
$66$	0	0
$67$	− 19.5912i	− 0.292407i	−0.989255	−	0.146203i	$-0.953295\pi$
$67$	− 19.5912i	− 0.292407i	0.989255	−	0.146203i	$-0.0467054\pi$
$68$	0	0
$69$	−21.5105	−0.311746
$70$	0	0
$71$	114.683i	1.61525i	0.589696	+	0.807625i	$0.299248\pi$
$71$	114.683i	1.61525i	−0.589696	+	0.807625i	$0.700752\pi$
$72$	0	0
$73$	−130.007	−1.78091	−0.890457	−	0.455068i	$-0.849615\pi$
$73$	−130.007	−1.78091	−0.890457	+	0.455068i	$0.849615\pi$
$74$	0	0
$75$	− 22.4262i	− 0.299016i
$76$	0	0
$77$	10.4983	0.136341
$78$	0	0
$79$	− 61.0896i	− 0.773286i	−0.922230	−	0.386643i	$-0.873635\pi$
$79$	− 61.0896i	− 0.773286i	0.922230	−	0.386643i	$-0.126365\pi$
$80$	0	0
$81$	−57.4597	−0.709379
$82$	0	0
$83$	101.488i	1.22275i	0.791342	+	0.611374i	$0.209383\pi$
$83$	101.488i	1.22275i	−0.791342	+	0.611374i	$0.790617\pi$
$84$	0	0
$85$	6.08066	0.0715372
$86$	0	0
$87$	− 204.735i	− 2.35328i
$88$	0	0
$89$	49.0728	0.551380	0.275690	−	0.961247i	$-0.411094\pi$
$89$	49.0728	0.551380	0.275690	+	0.961247i	$0.411094\pi$
$90$	0	0
$91$	23.4310i	0.257483i
$92$	0	0
$93$	−16.7339	−0.179934
$94$	0	0
$95$	− 62.8328i	− 0.661398i
$96$	0	0
$97$	−5.55033	−0.0572199	−0.0286099	−	0.999591i	$-0.509108\pi$
$97$	−5.55033	−0.0572199	−0.0286099	+	0.999591i	$0.509108\pi$
$98$	0	0
$99$	− 91.6265i	− 0.925520i
$100$	0	0
$101$	100.618	0.996222	0.498111	−	0.867113i	$-0.334027\pi$
$101$	100.618	0.996222	0.498111	+	0.867113i	$0.334027\pi$
$102$	0	0
$103$	74.5916i	0.724190i	0.932141	+	0.362095i	$0.117938\pi$
$103$	74.5916i	0.724190i	−0.932141	+	0.362095i	$0.882062\pi$
$104$	0	0
$105$	12.7753	0.121670
$106$	0	0
$107$	6.28973i	0.0587825i	0.999568	+	0.0293912i	$0.00935687\pi$
$107$	6.28973i	0.0587825i	−0.999568	+	0.0293912i	$0.990643\pi$
$108$	0	0
$109$	194.282	1.78240	0.891202	−	0.453607i	$-0.149863\pi$
$109$	194.282	1.78240	0.891202	+	0.453607i	$0.149863\pi$
$110$	0	0
$111$	83.6639i	0.753729i
$112$	0	0
$113$	−16.0772	−0.142276	−0.0711382	−	0.997466i	$-0.522663\pi$
$113$	−16.0772	−0.142276	−0.0711382	+	0.997466i	$0.522663\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	204.500	1.74786
$118$	0	0
$119$	3.46391i	0.0291085i
$120$	0	0
$121$	53.0743	0.438630
$122$	0	0
$123$	− 37.2031i	− 0.302464i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 61.5905i	− 0.484965i	−0.970156	−	0.242482i	$-0.922038\pi$
$127$	− 61.5905i	− 0.484965i	0.970156	−	0.242482i	$-0.0779617\pi$
$128$	0	0
$129$	78.9457	0.611982
$130$	0	0
$131$	− 112.685i	− 0.860193i	−0.902783	−	0.430096i	$-0.858480\pi$
$131$	− 112.685i	− 0.860193i	0.902783	−	0.430096i	$-0.141520\pi$
$132$	0	0
$133$	35.7933	0.269123
$134$	0	0
$135$	− 21.2363i	− 0.157306i
$136$	0	0
$137$	−253.399	−1.84963	−0.924814	−	0.380420i	$-0.875779\pi$
$137$	−253.399	−1.84963	−0.924814	+	0.380420i	$0.875779\pi$
$138$	0	0
$139$	189.167i	1.36091i	0.732789	+	0.680456i	$0.238218\pi$
$139$	189.167i	1.36091i	−0.732789	+	0.680456i	$0.761782\pi$
$140$	0	0
$141$	−165.544	−1.17407
$142$	0	0
$143$	− 151.602i	− 1.06016i
$144$	0	0
$145$	−102.068	−0.703920
$146$	0	0
$147$	− 212.499i	− 1.44557i
$148$	0	0
$149$	103.119	0.692077	0.346039	−	0.938220i	$-0.387527\pi$
$149$	103.119	0.692077	0.346039	+	0.938220i	$0.387527\pi$
$150$	0	0
$151$	87.5410i	0.579742i	0.957066	+	0.289871i	$0.0936124\pi$
$151$	87.5410i	0.579742i	−0.957066	+	0.289871i	$0.906388\pi$
$152$	0	0
$153$	30.2322	0.197596
$154$	0	0
$155$	8.34249i	0.0538225i
$156$	0	0
$157$	2.84203	0.0181021	0.00905106	−	0.999959i	$-0.497119\pi$
$157$	2.84203	0.0181021	0.00905106	+	0.999959i	$0.497119\pi$
$158$	0	0
$159$	− 238.086i	− 1.49740i
$160$	0	0
$161$	−6.10892	−0.0379436
$162$	0	0
$163$	145.318i	0.891519i	0.895153	+	0.445760i	$0.147066\pi$
$163$	145.318i	0.891519i	−0.895153	+	0.445760i	$0.852934\pi$
$164$	0	0
$165$	−82.6586	−0.500961
$166$	0	0
$167$	− 51.1736i	− 0.306429i	−0.988193	−	0.153214i	$-0.951038\pi$
$167$	− 51.1736i	− 0.306429i	0.988193	−	0.153214i	$-0.0489625\pi$
$168$	0	0
$169$	169.360	1.00213
$170$	0	0
$171$	− 312.396i	− 1.82688i
$172$	0	0
$173$	240.591	1.39070	0.695351	−	0.718670i	$-0.255249\pi$
$173$	240.591	1.39070	0.695351	+	0.718670i	$0.255249\pi$
$174$	0	0
$175$	− 6.36899i	− 0.0363942i
$176$	0	0
$177$	−79.7241	−0.450418
$178$	0	0
$179$	− 81.1252i	− 0.453213i	−0.973986	−	0.226607i	$-0.927237\pi$
$179$	− 81.1252i	− 0.453213i	0.973986	−	0.226607i	$-0.0727632\pi$
$180$	0	0
$181$	88.0371	0.486393	0.243196	−	0.969977i	$-0.421804\pi$
$181$	88.0371	0.486393	0.243196	+	0.969977i	$0.421804\pi$
$182$	0	0
$183$	− 308.507i	− 1.68583i
$184$	0	0
$185$	41.7097	0.225458
$186$	0	0
$187$	− 22.4121i	− 0.119851i
$188$	0	0
$189$	12.0975	0.0640077
$190$	0	0
$191$	291.622i	1.52682i	0.645915	+	0.763409i	$0.276476\pi$
$191$	291.622i	1.52682i	−0.645915	+	0.763409i	$0.723524\pi$
$192$	0	0
$193$	70.8552	0.367125	0.183563	−	0.983008i	$-0.441237\pi$
$193$	70.8552	0.367125	0.183563	+	0.983008i	$0.441237\pi$
$194$	0	0
$195$	− 184.485i	− 0.946075i
$196$	0	0
$197$	44.4510	0.225639	0.112820	−	0.993615i	$-0.464012\pi$
$197$	44.4510	0.225639	0.112820	+	0.993615i	$0.464012\pi$
$198$	0	0
$199$	− 142.904i	− 0.718110i	−0.933316	−	0.359055i	$-0.883099\pi$
$199$	− 142.904i	− 0.718110i	0.933316	−	0.359055i	$-0.116901\pi$
$200$	0	0
$201$	−87.8715	−0.437172
$202$	0	0
$203$	− 58.1443i	− 0.286425i
$204$	0	0
$205$	−18.5472	−0.0904741
$206$	0	0
$207$	53.3173i	0.257571i
$208$	0	0
$209$	−231.589	−1.10808
$210$	0	0
$211$	− 226.012i	− 1.07115i	−0.844488	−	0.535574i	$-0.820095\pi$
$211$	− 226.012i	− 1.07115i	0.844488	−	0.535574i	$-0.179905\pi$
$212$	0	0
$213$	514.381	2.41493
$214$	0	0
$215$	− 39.3575i	− 0.183058i
$216$	0	0
$217$	−4.75238	−0.0219004
$218$	0	0
$219$	583.112i	2.66261i
$220$	0	0
$221$	50.0213	0.226341
$222$	0	0
$223$	− 264.882i	− 1.18781i	−0.804534	−	0.593906i	$-0.797585\pi$
$223$	− 264.882i	− 1.18781i	0.804534	−	0.593906i	$-0.202415\pi$
$224$	0	0
$225$	−55.5871	−0.247054
$226$	0	0
$227$	− 345.665i	− 1.52275i	−0.648309	−	0.761377i	$-0.724523\pi$
$227$	− 345.665i	− 1.52275i	0.648309	−	0.761377i	$-0.275477\pi$
$228$	0	0
$229$	347.018	1.51536	0.757680	−	0.652626i	$-0.226333\pi$
$229$	347.018	1.51536	0.757680	+	0.652626i	$0.226333\pi$
$230$	0	0
$231$	− 47.0873i	− 0.203841i
$232$	0	0
$233$	−250.362	−1.07451	−0.537257	−	0.843418i	$-0.680540\pi$
$233$	−250.362	−1.07451	−0.537257	+	0.843418i	$0.680540\pi$
$234$	0	0
$235$	82.5300i	0.351191i
$236$	0	0
$237$	−274.002	−1.15613
$238$	0	0
$239$	− 181.345i	− 0.758765i	−0.925240	−	0.379382i	$-0.876136\pi$
$239$	− 181.345i	− 0.758765i	0.925240	−	0.379382i	$-0.123864\pi$
$240$	0	0
$241$	155.317	0.644467	0.322234	−	0.946660i	$-0.395566\pi$
$241$	155.317	0.644467	0.322234	+	0.946660i	$0.395566\pi$
$242$	0	0
$243$	343.195i	1.41233i
$244$	0	0
$245$	−105.939	−0.432405
$246$	0	0
$247$	− 516.881i	− 2.09264i
$248$	0	0
$249$	455.199	1.82811
$250$	0	0
$251$	− 41.8270i	− 0.166642i	−0.996523	−	0.0833208i	$-0.973447\pi$
$251$	− 41.8270i	− 0.166642i	0.996523	−	0.0833208i	$-0.0265526\pi$
$252$	0	0
$253$	39.5258	0.156229
$254$	0	0
$255$	− 27.2732i	− 0.106954i
$256$	0	0
$257$	−148.188	−0.576608	−0.288304	−	0.957539i	$-0.593091\pi$
$257$	−148.188	−0.576608	−0.288304	+	0.957539i	$0.593091\pi$
$258$	0	0
$259$	23.7603i	0.0917388i
$260$	0	0
$261$	−507.470	−1.94433
$262$	0	0
$263$	426.018i	1.61984i	0.586540	+	0.809920i	$0.300490\pi$
$263$	426.018i	1.61984i	−0.586540	+	0.809920i	$0.699510\pi$
$264$	0	0
$265$	−118.695	−0.447906
$266$	0	0
$267$	− 220.104i	− 0.824358i
$268$	0	0
$269$	−331.879	−1.23375	−0.616875	−	0.787061i	$-0.711602\pi$
$269$	−331.879	−1.23375	−0.616875	+	0.787061i	$0.711602\pi$
$270$	0	0
$271$	− 270.708i	− 0.998921i	−0.866337	−	0.499461i	$-0.833532\pi$
$271$	− 270.708i	− 0.998921i	0.866337	−	0.499461i	$-0.166468\pi$
$272$	0	0
$273$	105.094	0.384958
$274$	0	0
$275$	41.2085i	0.149849i
$276$	0	0
$277$	278.484	1.00536	0.502678	−	0.864474i	$-0.332348\pi$
$277$	278.484	1.00536	0.502678	+	0.864474i	$0.332348\pi$
$278$	0	0
$279$	41.4777i	0.148666i
$280$	0	0
$281$	167.433	0.595846	0.297923	−	0.954590i	$-0.403706\pi$
$281$	167.433	0.595846	0.297923	+	0.954590i	$0.403706\pi$
$282$	0	0
$283$	262.536i	0.927691i	0.885916	+	0.463845i	$0.153531\pi$
$283$	262.536i	0.927691i	−0.885916	+	0.463845i	$0.846469\pi$
$284$	0	0
$285$	−281.820	−0.988844
$286$	0	0
$287$	− 10.5656i	− 0.0368139i
$288$	0	0
$289$	−281.605	−0.974412
$290$	0	0
$291$	24.8946i	0.0855483i
$292$	0	0
$293$	269.094	0.918410	0.459205	−	0.888330i	$-0.348134\pi$
$293$	269.094	0.918410	0.459205	+	0.888330i	$0.348134\pi$
$294$	0	0
$295$	39.7455i	0.134731i
$296$	0	0
$297$	−78.2727	−0.263545
$298$	0	0
$299$	88.2172i	0.295041i
$300$	0	0
$301$	22.4204	0.0744864
$302$	0	0
$303$	− 451.298i	− 1.48943i
$304$	0	0
$305$	−153.803	−0.504271
$306$	0	0
$307$	469.712i	1.53001i	0.644027	+	0.765003i	$0.277262\pi$
$307$	469.712i	1.53001i	−0.644027	+	0.765003i	$0.722738\pi$
$308$	0	0
$309$	334.562	1.08272
$310$	0	0
$311$	33.2718i	0.106983i	0.998568	+	0.0534917i	$0.0170351\pi$
$311$	33.2718i	0.106983i	−0.998568	+	0.0534917i	$0.982965\pi$
$312$	0	0
$313$	−111.430	−0.356005	−0.178003	−	0.984030i	$-0.556964\pi$
$313$	−111.430	−0.356005	−0.178003	+	0.984030i	$0.556964\pi$
$314$	0	0
$315$	− 31.6658i	− 0.100526i
$316$	0	0
$317$	176.335	0.556260	0.278130	−	0.960543i	$-0.410285\pi$
$317$	176.335	0.556260	0.278130	+	0.960543i	$0.410285\pi$
$318$	0	0
$319$	376.204i	1.17932i
$320$	0	0
$321$	28.2110	0.0878846
$322$	0	0
$323$	− 76.4129i	− 0.236573i
$324$	0	0
$325$	−91.9728	−0.282993
$326$	0	0
$327$	− 871.402i	− 2.66484i
$328$	0	0
$329$	−47.0140	−0.142900
$330$	0	0
$331$	414.594i	1.25255i	0.779602	+	0.626275i	$0.215421\pi$
$331$	414.594i	1.25255i	−0.779602	+	0.626275i	$0.784579\pi$
$332$	0	0
$333$	207.375	0.622747
$334$	0	0
$335$	43.8074i	0.130768i
$336$	0	0
$337$	223.662	0.663687	0.331843	−	0.943334i	$-0.392329\pi$
$337$	223.662	0.663687	0.331843	+	0.943334i	$0.392329\pi$
$338$	0	0
$339$	72.1103i	0.212715i
$340$	0	0
$341$	30.7488	0.0901723
$342$	0	0
$343$	− 122.765i	− 0.357917i
$344$	0	0
$345$	48.0989	0.139417
$346$	0	0
$347$	197.660i	0.569626i	0.958583	+	0.284813i	$0.0919315\pi$
$347$	197.660i	0.569626i	−0.958583	+	0.284813i	$0.908068\pi$
$348$	0	0
$349$	559.223	1.60236	0.801179	−	0.598425i	$-0.204207\pi$
$349$	559.223	1.60236	0.801179	+	0.598425i	$0.204207\pi$
$350$	0	0
$351$	− 174.696i	− 0.497709i
$352$	0	0
$353$	−2.31133	−0.00654768	−0.00327384	−	0.999995i	$-0.501042\pi$
$353$	−2.31133	−0.00654768	−0.00327384	+	0.999995i	$0.501042\pi$
$354$	0	0
$355$	− 256.439i	− 0.722362i
$356$	0	0
$357$	15.5365	0.0435196
$358$	0	0
$359$	572.221i	1.59393i	0.604025	+	0.796965i	$0.293563\pi$
$359$	572.221i	1.59393i	−0.604025	+	0.796965i	$0.706437\pi$
$360$	0	0
$361$	−428.592	−1.18723
$362$	0	0
$363$	− 238.051i	− 0.655788i
$364$	0	0
$365$	290.704	0.796449
$366$	0	0
$367$	299.745i	0.816744i	0.912816	+	0.408372i	$0.133903\pi$
$367$	299.745i	0.816744i	−0.912816	+	0.408372i	$0.866097\pi$
$368$	0	0
$369$	−92.2141	−0.249903
$370$	0	0
$371$	− 67.6158i	− 0.182253i
$372$	0	0
$373$	19.7320	0.0529008	0.0264504	−	0.999650i	$-0.491580\pi$
$373$	19.7320	0.0529008	0.0264504	+	0.999650i	$0.491580\pi$
$374$	0	0
$375$	50.1466i	0.133724i
$376$	0	0
$377$	−839.645	−2.22717
$378$	0	0
$379$	527.076i	1.39070i	0.718670	+	0.695351i	$0.244751\pi$
$379$	527.076i	1.39070i	−0.718670	+	0.695351i	$0.755249\pi$
$380$	0	0
$381$	−276.249	−0.725062
$382$	0	0
$383$	− 135.247i	− 0.353126i	−0.984289	−	0.176563i	$-0.943502\pi$
$383$	− 135.247i	− 0.353126i	0.984289	−	0.176563i	$-0.0564979\pi$
$384$	0	0
$385$	−23.4749	−0.0609736
$386$	0	0
$387$	− 195.680i	− 0.505633i
$388$	0	0
$389$	237.556	0.610684	0.305342	−	0.952243i	$-0.401229\pi$
$389$	237.556	0.610684	0.305342	+	0.952243i	$0.401229\pi$
$390$	0	0
$391$	13.0416i	0.0333544i
$392$	0	0
$393$	−505.421	−1.28606
$394$	0	0
$395$	136.600i	0.345824i
$396$	0	0
$397$	457.470	1.15232	0.576158	−	0.817338i	$-0.304551\pi$
$397$	457.470	1.15232	0.576158	+	0.817338i	$0.304551\pi$
$398$	0	0
$399$	− 160.542i	− 0.402360i
$400$	0	0
$401$	76.3007	0.190276	0.0951381	−	0.995464i	$-0.469671\pi$
$401$	76.3007	0.190276	0.0951381	+	0.995464i	$0.469671\pi$
$402$	0	0
$403$	68.6278i	0.170292i
$404$	0	0
$405$	128.484	0.317244
$406$	0	0
$407$	− 153.734i	− 0.377724i
$408$	0	0
$409$	−204.640	−0.500342	−0.250171	−	0.968202i	$-0.580487\pi$
$409$	−204.640	−0.500342	−0.250171	+	0.968202i	$0.580487\pi$
$410$	0	0
$411$	1136.56i	2.76534i
$412$	0	0
$413$	−22.6414	−0.0548219
$414$	0	0
$415$	− 226.934i	− 0.546829i
$416$	0	0
$417$	848.459	2.03467
$418$	0	0
$419$	652.899i	1.55823i	0.626881	+	0.779115i	$0.284331\pi$
$419$	652.899i	1.55823i	−0.626881	+	0.779115i	$0.715669\pi$
$420$	0	0
$421$	67.2452	0.159727	0.0798637	−	0.996806i	$-0.474552\pi$
$421$	67.2452	0.159727	0.0798637	+	0.996806i	$0.474552\pi$
$422$	0	0
$423$	410.328i	0.970042i
$424$	0	0
$425$	−13.5968	−0.0319924
$426$	0	0
$427$	− 87.6152i	− 0.205188i
$428$	0	0
$429$	−679.974	−1.58502
$430$	0	0
$431$	381.827i	0.885909i	0.896544	+	0.442954i	$0.146070\pi$
$431$	381.827i	0.885909i	−0.896544	+	0.442954i	$0.853930\pi$
$432$	0	0
$433$	443.969	1.02533	0.512666	−	0.858588i	$-0.328658\pi$
$433$	443.969	1.02533	0.512666	+	0.858588i	$0.328658\pi$
$434$	0	0
$435$	457.802i	1.05242i
$436$	0	0
$437$	134.761	0.308378
$438$	0	0
$439$	634.934i	1.44632i	0.690682	+	0.723159i	$0.257311\pi$
$439$	634.934i	1.44632i	−0.690682	+	0.723159i	$0.742689\pi$
$440$	0	0
$441$	−526.715	−1.19436
$442$	0	0
$443$	469.201i	1.05915i	0.848265	+	0.529573i	$0.177648\pi$
$443$	469.201i	1.05915i	−0.848265	+	0.529573i	$0.822352\pi$
$444$	0	0
$445$	−109.730	−0.246585
$446$	0	0
$447$	− 462.516i	− 1.03471i
$448$	0	0
$449$	588.084	1.30976	0.654882	−	0.755731i	$-0.272718\pi$
$449$	588.084	1.30976	0.654882	+	0.755731i	$0.272718\pi$
$450$	0	0
$451$	68.3613i	0.151577i
$452$	0	0
$453$	392.643	0.866761
$454$	0	0
$455$	− 52.3932i	− 0.115150i
$456$	0	0
$457$	721.922	1.57970	0.789849	−	0.613301i	$-0.210159\pi$
$457$	721.922	1.57970	0.789849	+	0.613301i	$0.210159\pi$
$458$	0	0
$459$	− 25.8261i	− 0.0562660i
$460$	0	0
$461$	−880.779	−1.91058	−0.955292	−	0.295664i	$-0.904459\pi$
$461$	−880.779	−1.91058	−0.955292	+	0.295664i	$0.904459\pi$
$462$	0	0
$463$	395.670i	0.854579i	0.904115	+	0.427290i	$0.140531\pi$
$463$	395.670i	0.854579i	−0.904115	+	0.427290i	$0.859469\pi$
$464$	0	0
$465$	37.4181	0.0804690
$466$	0	0
$467$	128.825i	0.275856i	0.990442	+	0.137928i	$0.0440442\pi$
$467$	128.825i	0.275856i	−0.990442	+	0.137928i	$0.955956\pi$
$468$	0	0
$469$	−24.9553	−0.0532096
$470$	0	0
$471$	− 12.7472i	− 0.0270641i
$472$	0	0
$473$	−145.064	−0.306689
$474$	0	0
$475$	140.498i	0.295786i
$476$	0	0
$477$	−590.135	−1.23718
$478$	0	0
$479$	− 19.8464i	− 0.0414329i	−0.999785	−	0.0207165i	$-0.993405\pi$
$479$	− 19.8464i	− 0.0414329i	0.999785	−	0.0207165i	$-0.00659473\pi$
$480$	0	0
$481$	343.116	0.713339
$482$	0	0
$483$	27.4000i	0.0567288i
$484$	0	0
$485$	12.4109	0.0255895
$486$	0	0
$487$	3.79884i	0.00780048i	0.999992	+	0.00390024i	$0.00124149\pi$
$487$	3.79884i	0.00780048i	−0.999992	+	0.00390024i	$0.998759\pi$
$488$	0	0
$489$	651.785	1.33289
$490$	0	0
$491$	214.126i	0.436101i	0.975938	+	0.218050i	$0.0699697\pi$
$491$	214.126i	0.436101i	−0.975938	+	0.218050i	$0.930030\pi$
$492$	0	0
$493$	−124.129	−0.251782
$494$	0	0
$495$	204.883i	0.413905i
$496$	0	0
$497$	146.083	0.293929
$498$	0	0
$499$	− 506.307i	− 1.01464i	−0.861757	−	0.507322i	$-0.830636\pi$
$499$	− 506.307i	− 1.01464i	0.861757	−	0.507322i	$-0.169364\pi$
$500$	0	0
$501$	−229.526	−0.458136
$502$	0	0
$503$	769.052i	1.52893i	0.644665	+	0.764465i	$0.276997\pi$
$503$	769.052i	1.52893i	−0.644665	+	0.764465i	$0.723003\pi$
$504$	0	0
$505$	−224.990	−0.445524
$506$	0	0
$507$	− 759.619i	− 1.49826i
$508$	0	0
$509$	−544.386	−1.06952	−0.534760	−	0.845004i	$-0.679598\pi$
$509$	−544.386	−1.06952	−0.534760	+	0.845004i	$0.679598\pi$
$510$	0	0
$511$	165.602i	0.324075i
$512$	0	0
$513$	−266.867	−0.520208
$514$	0	0
$515$	− 166.792i	− 0.323868i
$516$	0	0
$517$	304.189	0.588374
$518$	0	0
$519$	− 1079.11i	− 2.07921i
$520$	0	0
$521$	−66.4572	−0.127557	−0.0637785	−	0.997964i	$-0.520315\pi$
$521$	−66.4572	−0.127557	−0.0637785	+	0.997964i	$0.520315\pi$
$522$	0	0
$523$	− 463.775i	− 0.886759i	−0.896334	−	0.443379i	$-0.853779\pi$
$523$	− 463.775i	− 0.886759i	0.896334	−	0.443379i	$-0.146221\pi$
$524$	0	0
$525$	−28.5665	−0.0544124
$526$	0	0
$527$	10.1456i	0.0192515i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	197.609i	0.372146i
$532$	0	0
$533$	−152.575	−0.286256
$534$	0	0
$535$	− 14.0643i	− 0.0262883i
$536$	0	0
$537$	−363.866	−0.677591
$538$	0	0
$539$	390.471i	0.724436i
$540$	0	0
$541$	−561.724	−1.03831	−0.519153	−	0.854681i	$-0.673753\pi$
$541$	−561.724	−1.03831	−0.519153	+	0.854681i	$0.673753\pi$
$542$	0	0
$543$	− 394.868i	− 0.727197i
$544$	0	0
$545$	−434.428	−0.797115
$546$	0	0
$547$	− 184.781i	− 0.337807i	−0.985633	−	0.168904i	$-0.945977\pi$
$547$	− 184.781i	− 0.337807i	0.985633	−	0.168904i	$-0.0540227\pi$
$548$	0	0
$549$	−764.685	−1.39287
$550$	0	0
$551$	1282.65i	2.32786i
$552$	0	0
$553$	−77.8158	−0.140716
$554$	0	0
$555$	− 187.078i	− 0.337078i
$556$	0	0
$557$	−76.4442	−0.137243	−0.0686213	−	0.997643i	$-0.521860\pi$
$557$	−76.4442	−0.137243	−0.0686213	+	0.997643i	$0.521860\pi$
$558$	0	0
$559$	− 323.766i	− 0.579188i
$560$	0	0
$561$	−100.524	−0.179187
$562$	0	0
$563$	298.033i	0.529366i	0.964335	+	0.264683i	$0.0852673\pi$
$563$	298.033i	0.529366i	−0.964335	+	0.264683i	$0.914733\pi$
$564$	0	0
$565$	35.9498	0.0636279
$566$	0	0
$567$	73.1921i	0.129087i
$568$	0	0
$569$	−78.2507	−0.137523	−0.0687616	−	0.997633i	$-0.521905\pi$
$569$	−78.2507	−0.137523	−0.0687616	+	0.997633i	$0.521905\pi$
$570$	0	0
$571$	− 1042.23i	− 1.82528i	−0.408769	−	0.912638i	$-0.634042\pi$
$571$	− 1042.23i	− 1.82528i	0.408769	−	0.912638i	$-0.365958\pi$
$572$	0	0
$573$	1308.00	2.28272
$574$	0	0
$575$	− 23.9792i	− 0.0417029i
$576$	0	0
$577$	737.002	1.27730	0.638650	−	0.769497i	$-0.279493\pi$
$577$	737.002	1.27730	0.638650	+	0.769497i	$0.279493\pi$
$578$	0	0
$579$	− 317.803i	− 0.548883i
$580$	0	0
$581$	129.275	0.222505
$582$	0	0
$583$	437.487i	0.750406i
$584$	0	0
$585$	−457.276	−0.781668
$586$	0	0
$587$	415.635i	0.708066i	0.935233	+	0.354033i	$0.115190\pi$
$587$	415.635i	0.708066i	−0.935233	+	0.354033i	$0.884810\pi$
$588$	0	0
$589$	104.836	0.177990
$590$	0	0
$591$	− 199.373i	− 0.337349i
$592$	0	0
$593$	−306.241	−0.516426	−0.258213	−	0.966088i	$-0.583134\pi$
$593$	−306.241	−0.516426	−0.258213	+	0.966088i	$0.583134\pi$
$594$	0	0
$595$	− 7.74554i	− 0.0130177i
$596$	0	0
$597$	−640.959	−1.07363
$598$	0	0
$599$	444.262i	0.741673i	0.928698	+	0.370836i	$0.120929\pi$
$599$	444.262i	0.741673i	−0.928698	+	0.370836i	$0.879071\pi$
$600$	0	0
$601$	−769.191	−1.27985	−0.639926	−	0.768437i	$-0.721035\pi$
$601$	−769.191	−1.27985	−0.639926	+	0.768437i	$0.721035\pi$
$602$	0	0
$603$	217.804i	0.361201i
$604$	0	0
$605$	−118.678	−0.196161
$606$	0	0
$607$	− 605.918i	− 0.998217i	−0.866539	−	0.499109i	$-0.833661\pi$
$607$	− 605.918i	− 0.998217i	0.866539	−	0.499109i	$-0.166339\pi$
$608$	0	0
$609$	−260.791	−0.428229
$610$	0	0
$611$	678.916i	1.11116i
$612$	0	0
$613$	−835.562	−1.36307	−0.681535	−	0.731786i	$-0.738687\pi$
$613$	−835.562	−1.36307	−0.681535	+	0.731786i	$0.738687\pi$
$614$	0	0
$615$	83.1887i	0.135266i
$616$	0	0
$617$	1004.84	1.62858	0.814292	−	0.580456i	$-0.197126\pi$
$617$	1004.84	1.62858	0.814292	+	0.580456i	$0.197126\pi$
$618$	0	0
$619$	− 660.976i	− 1.06781i	−0.845544	−	0.533906i	$-0.820724\pi$
$619$	− 660.976i	− 1.06781i	0.845544	−	0.533906i	$-0.179276\pi$
$620$	0	0
$621$	45.5467	0.0733442
$622$	0	0
$623$	− 62.5089i	− 0.100335i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	1038.73i	1.65667i
$628$	0	0
$629$	50.7245	0.0806430
$630$	0	0
$631$	− 150.455i	− 0.238438i	−0.992868	−	0.119219i	$-0.961961\pi$
$631$	− 150.455i	− 0.238438i	0.992868	−	0.119219i	$-0.0380391\pi$
$632$	0	0
$633$	−1013.72	−1.60145
$634$	0	0
$635$	137.721i	0.216883i
$636$	0	0
$637$	−871.487	−1.36811
$638$	0	0
$639$	− 1274.98i	− 1.99527i
$640$	0	0
$641$	266.813	0.416245	0.208123	−	0.978103i	$-0.433265\pi$
$641$	266.813	0.416245	0.208123	+	0.978103i	$0.433265\pi$
$642$	0	0
$643$	421.678i	0.655797i	0.944713	+	0.327899i	$0.106340\pi$
$643$	421.678i	0.655797i	−0.944713	+	0.327899i	$0.893660\pi$
$644$	0	0
$645$	−176.528	−0.273687
$646$	0	0
$647$	231.315i	0.357519i	0.983893	+	0.178759i	$0.0572084\pi$
$647$	231.315i	0.357519i	−0.983893	+	0.178759i	$0.942792\pi$
$648$	0	0
$649$	146.494	0.225723
$650$	0	0
$651$	21.3156i	0.0327429i
$652$	0	0
$653$	−281.317	−0.430807	−0.215404	−	0.976525i	$-0.569107\pi$
$653$	−281.317	−0.430807	−0.215404	+	0.976525i	$0.569107\pi$
$654$	0	0
$655$	251.972i	0.384690i
$656$	0	0
$657$	1445.34	2.19991
$658$	0	0
$659$	− 874.082i	− 1.32638i	−0.748453	−	0.663188i	$-0.769203\pi$
$659$	− 874.082i	− 1.32638i	0.748453	−	0.663188i	$-0.230797\pi$
$660$	0	0
$661$	820.264	1.24094	0.620472	−	0.784229i	$-0.286941\pi$
$661$	820.264	1.24094	0.620472	+	0.784229i	$0.286941\pi$
$662$	0	0
$663$	− 224.358i	− 0.338398i
$664$	0	0
$665$	−80.0363	−0.120355
$666$	0	0
$667$	− 218.912i	− 0.328204i
$668$	0	0
$669$	−1188.06	−1.77588
$670$	0	0
$671$	566.886i	0.844838i
$672$	0	0
$673$	20.9878	0.0311854	0.0155927	−	0.999878i	$-0.495036\pi$
$673$	20.9878	0.0311854	0.0155927	+	0.999878i	$0.495036\pi$
$674$	0	0
$675$	47.4858i	0.0703493i
$676$	0	0
$677$	−1005.45	−1.48515	−0.742576	−	0.669761i	$-0.766396\pi$
$677$	−1005.45	−1.48515	−0.742576	+	0.669761i	$0.766396\pi$
$678$	0	0
$679$	7.07000i	0.0104124i
$680$	0	0
$681$	−1550.39	−2.27664
$682$	0	0
$683$	576.276i	0.843743i	0.906656	+	0.421871i	$0.138627\pi$
$683$	576.276i	0.843743i	−0.906656	+	0.421871i	$0.861373\pi$
$684$	0	0
$685$	566.617	0.827179
$686$	0	0
$687$	− 1556.46i	− 2.26559i
$688$	0	0
$689$	−976.420	−1.41716
$690$	0	0
$691$	684.223i	0.990193i	0.868838	+	0.495096i	$0.164867\pi$
$691$	684.223i	0.990193i	−0.868838	+	0.495096i	$0.835133\pi$
$692$	0	0
$693$	−116.714	−0.168418
$694$	0	0
$695$	− 422.990i	− 0.608618i
$696$	0	0
$697$	−22.5558	−0.0323613
$698$	0	0
$699$	1122.93i	1.60649i
$700$	0	0
$701$	−949.623	−1.35467	−0.677335	−	0.735675i	$-0.736865\pi$
$701$	−949.623	−1.35467	−0.677335	+	0.735675i	$0.736865\pi$
$702$	0	0
$703$	− 524.147i	− 0.745586i
$704$	0	0
$705$	370.167	0.525060
$706$	0	0
$707$	− 128.168i	− 0.181284i
$708$	0	0
$709$	−320.646	−0.452252	−0.226126	−	0.974098i	$-0.572606\pi$
$709$	−320.646	−0.452252	−0.226126	+	0.974098i	$0.572606\pi$
$710$	0	0
$711$	679.159i	0.955216i
$712$	0	0
$713$	−17.8926	−0.0250949
$714$	0	0
$715$	338.993i	0.474117i
$716$	0	0
$717$	−813.376	−1.13442
$718$	0	0
$719$	− 899.598i	− 1.25118i	−0.780152	−	0.625590i	$-0.784858\pi$
$719$	− 899.598i	− 1.25118i	0.780152	−	0.625590i	$-0.215142\pi$
$720$	0	0
$721$	95.0147	0.131782
$722$	0	0
$723$	− 696.633i	− 0.963531i
$724$	0	0
$725$	228.232	0.314803
$726$	0	0
$727$	913.821i	1.25698i	0.777820	+	0.628488i	$0.216326\pi$
$727$	913.821i	1.25698i	−0.777820	+	0.628488i	$0.783674\pi$
$728$	0	0
$729$	1022.18	1.40216
$730$	0	0
$731$	− 47.8639i	− 0.0654773i
$732$	0	0
$733$	−1015.23	−1.38503	−0.692516	−	0.721403i	$-0.743498\pi$
$733$	−1015.23	−1.38503	−0.692516	+	0.721403i	$0.743498\pi$
$734$	0	0
$735$	475.163i	0.646480i
$736$	0	0
$737$	161.465	0.219085
$738$	0	0
$739$	61.3070i	0.0829595i	0.999139	+	0.0414797i	$0.0132072\pi$
$739$	61.3070i	0.0829595i	−0.999139	+	0.0414797i	$0.986793\pi$
$740$	0	0
$741$	−2318.34	−3.12866
$742$	0	0
$743$	− 558.071i	− 0.751105i	−0.926801	−	0.375552i	$-0.877453\pi$
$743$	− 558.071i	− 0.751105i	0.926801	−	0.375552i	$-0.122547\pi$
$744$	0	0
$745$	−230.582	−0.309506
$746$	0	0
$747$	− 1128.29i	− 1.51042i
$748$	0	0
$749$	8.01185	0.0106967
$750$	0	0
$751$	− 227.403i	− 0.302801i	−0.988473	−	0.151400i	$-0.951622\pi$
$751$	− 227.403i	− 0.302801i	0.988473	−	0.151400i	$-0.0483783\pi$
$752$	0	0
$753$	−187.604	−0.249143
$754$	0	0
$755$	− 195.748i	− 0.259268i
$756$	0	0
$757$	555.504	0.733823	0.366912	−	0.930256i	$-0.380415\pi$
$757$	555.504	0.733823	0.366912	+	0.930256i	$0.380415\pi$
$758$	0	0
$759$	− 177.283i	− 0.233574i
$760$	0	0
$761$	−1215.02	−1.59661	−0.798307	−	0.602250i	$-0.794271\pi$
$761$	−1215.02	−1.59661	−0.798307	+	0.602250i	$0.794271\pi$
$762$	0	0
$763$	− 247.476i	− 0.324346i
$764$	0	0
$765$	−67.6013	−0.0883676
$766$	0	0
$767$	326.958i	0.426282i
$768$	0	0
$769$	951.657	1.23753	0.618763	−	0.785578i	$-0.287634\pi$
$769$	951.657	1.23753	0.618763	+	0.785578i	$0.287634\pi$
$770$	0	0
$771$	664.660i	0.862075i
$772$	0	0
$773$	278.122	0.359795	0.179898	−	0.983685i	$-0.442423\pi$
$773$	278.122	0.359795	0.179898	+	0.983685i	$0.442423\pi$
$774$	0	0
$775$	− 18.6544i	− 0.0240702i
$776$	0	0
$777$	106.571	0.137157
$778$	0	0
$779$	233.074i	0.299197i
$780$	0	0
$781$	−945.182	−1.21022
$782$	0	0
$783$	433.511i	0.553653i
$784$	0	0
$785$	−6.35498	−0.00809551
$786$	0	0
$787$	259.872i	0.330206i	0.986276	+	0.165103i	$0.0527956\pi$
$787$	259.872i	0.330206i	−0.986276	+	0.165103i	$0.947204\pi$
$788$	0	0
$789$	1910.79	2.42179
$790$	0	0
$791$	20.4791i	0.0258902i
$792$	0	0
$793$	−1265.23	−1.59549
$794$	0	0
$795$	532.376i	0.669656i
$796$	0	0
$797$	487.177	0.611263	0.305632	−	0.952150i	$-0.401132\pi$
$797$	487.177	0.611263	0.305632	+	0.952150i	$0.401132\pi$
$798$	0	0
$799$	100.367i	0.125616i
$800$	0	0
$801$	−545.563	−0.681103
$802$	0	0
$803$	− 1071.48i	− 1.33434i
$804$	0	0
$805$	13.6600	0.0169689
$806$	0	0
$807$	1488.56i	1.84456i
$808$	0	0
$809$	297.353	0.367556	0.183778	−	0.982968i	$-0.441167\pi$
$809$	297.353	0.367556	0.183778	+	0.982968i	$0.441167\pi$
$810$	0	0
$811$	− 864.257i	− 1.06567i	−0.846220	−	0.532834i	$-0.821127\pi$
$811$	− 864.257i	− 1.06567i	0.846220	−	0.532834i	$-0.178873\pi$
$812$	0	0
$813$	−1214.19	−1.49347
$814$	0	0
$815$	− 324.940i	− 0.398700i
$816$	0	0
$817$	−494.588	−0.605371
$818$	0	0
$819$	− 260.492i	− 0.318061i
$820$	0	0
$821$	−611.309	−0.744590	−0.372295	−	0.928114i	$-0.621429\pi$
$821$	−611.309	−0.744590	−0.372295	+	0.928114i	$0.621429\pi$
$822$	0	0
$823$	− 867.334i	− 1.05387i	−0.849906	−	0.526934i	$-0.823341\pi$
$823$	− 867.334i	− 1.05387i	0.849906	−	0.526934i	$-0.176659\pi$
$824$	0	0
$825$	184.830	0.224037
$826$	0	0
$827$	1591.23i	1.92410i	0.272872	+	0.962050i	$0.412026\pi$
$827$	1591.23i	1.92410i	−0.272872	+	0.962050i	$0.587974\pi$
$828$	0	0
$829$	764.680	0.922413	0.461206	−	0.887293i	$-0.347417\pi$
$829$	764.680	0.922413	0.461206	+	0.887293i	$0.347417\pi$
$830$	0	0
$831$	− 1249.07i	− 1.50309i
$832$	0	0
$833$	−128.836	−0.154665
$834$	0	0
$835$	114.428i	0.137039i
$836$	0	0
$837$	35.4327	0.0423329
$838$	0	0
$839$	− 1489.89i	− 1.77579i	−0.460048	−	0.887894i	$-0.652168\pi$
$839$	− 1489.89i	− 1.77579i	0.460048	−	0.887894i	$-0.347832\pi$
$840$	0	0
$841$	1242.59	1.47752
$842$	0	0
$843$	− 750.976i	− 0.890838i
$844$	0	0
$845$	−378.700	−0.448165
$846$	0	0
$847$	− 67.6059i	− 0.0798181i
$848$	0	0
$849$	1177.54	1.38697
$850$	0	0
$851$	89.4573i	0.105120i
$852$	0	0
$853$	5.02425	0.00589009	0.00294505	−	0.999996i	$-0.499063\pi$
$853$	5.02425	0.00589009	0.00294505	+	0.999996i	$0.499063\pi$
$854$	0	0
$855$	698.538i	0.817004i
$856$	0	0
$857$	1148.87	1.34058	0.670288	−	0.742101i	$-0.266171\pi$
$857$	1148.87	1.34058	0.670288	+	0.742101i	$0.266171\pi$
$858$	0	0
$859$	− 150.709i	− 0.175447i	−0.996145	−	0.0877233i	$-0.972041\pi$
$859$	− 150.709i	− 0.175447i	0.996145	−	0.0877233i	$-0.0279591\pi$
$860$	0	0
$861$	−47.3893	−0.0550398
$862$	0	0
$863$	258.876i	0.299973i	0.988688	+	0.149986i	$0.0479230\pi$
$863$	258.876i	0.299973i	−0.988688	+	0.149986i	$0.952077\pi$
$864$	0	0
$865$	−537.979	−0.621941
$866$	0	0
$867$	1263.07i	1.45683i
$868$	0	0
$869$	503.482	0.579381
$870$	0	0
$871$	360.372i	0.413745i
$872$	0	0
$873$	61.7053	0.0706819
$874$	0	0
$875$	14.2415i	0.0162760i
$876$	0	0
$877$	601.397	0.685743	0.342872	−	0.939382i	$-0.388600\pi$
$877$	601.397	0.685743	0.342872	+	0.939382i	$0.388600\pi$
$878$	0	0
$879$	− 1206.95i	− 1.37310i
$880$	0	0
$881$	−397.682	−0.451398	−0.225699	−	0.974197i	$-0.572467\pi$
$881$	−397.682	−0.451398	−0.225699	+	0.974197i	$0.572467\pi$
$882$	0	0
$883$	1149.21i	1.30148i	0.759301	+	0.650739i	$0.225541\pi$
$883$	1149.21i	1.30148i	−0.759301	+	0.650739i	$0.774459\pi$
$884$	0	0
$885$	178.268	0.201433
$886$	0	0
$887$	− 1442.43i	− 1.62618i	−0.582135	−	0.813092i	$-0.697783\pi$
$887$	− 1442.43i	− 1.62618i	0.582135	−	0.813092i	$-0.302217\pi$
$888$	0	0
$889$	−78.4539	−0.0882496
$890$	0	0
$891$	− 473.566i	− 0.531500i
$892$	0	0
$893$	1037.12	1.16139
$894$	0	0
$895$	181.401i	0.202683i
$896$	0	0
$897$	395.676	0.441110
$898$	0	0
$899$	− 170.301i	− 0.189434i
$900$	0	0
$901$	−144.349	−0.160210
$902$	0	0
$903$	− 100.561i	− 0.111363i
$904$	0	0
$905$	−196.857	−0.217521
$906$	0	0
$907$	125.657i	0.138541i	0.997598	+	0.0692706i	$0.0220672\pi$
$907$	125.657i	0.138541i	−0.997598	+	0.0692706i	$0.977933\pi$
$908$	0	0
$909$	−1118.62	−1.23060
$910$	0	0
$911$	287.551i	0.315643i	0.987468	+	0.157821i	$0.0504470\pi$
$911$	287.551i	0.315643i	−0.987468	+	0.157821i	$0.949553\pi$
$912$	0	0
$913$	−836.434	−0.916138
$914$	0	0
$915$	689.842i	0.753926i
$916$	0	0
$917$	−143.538	−0.156530
$918$	0	0
$919$	− 665.984i	− 0.724683i	−0.932045	−	0.362341i	$-0.881977\pi$
$919$	− 665.984i	− 0.724683i	0.932045	−	0.362341i	$-0.118023\pi$
$920$	0	0
$921$	2106.77	2.28748
$922$	0	0
$923$	− 2109.54i	− 2.28552i
$924$	0	0
$925$	−93.2657	−0.100828
$926$	0	0
$927$	− 829.266i	− 0.894570i
$928$	0	0
$929$	−126.674	−0.136355	−0.0681777	−	0.997673i	$-0.521718\pi$
$929$	−126.674	−0.136355	−0.0681777	+	0.997673i	$0.521718\pi$
$930$	0	0
$931$	1331.29i	1.42996i
$932$	0	0
$933$	149.232	0.159949
$934$	0	0
$935$	50.1150i	0.0535989i
$936$	0	0
$937$	812.762	0.867408	0.433704	−	0.901055i	$-0.357206\pi$
$937$	812.762	0.867408	0.433704	+	0.901055i	$0.357206\pi$
$938$	0	0
$939$	499.789i	0.532257i
$940$	0	0
$941$	642.308	0.682580	0.341290	−	0.939958i	$-0.389136\pi$
$941$	642.308	0.682580	0.341290	+	0.939958i	$0.389136\pi$
$942$	0	0
$943$	− 39.7793i	− 0.0421838i
$944$	0	0
$945$	−27.0507	−0.0286251
$946$	0	0
$947$	1061.86i	1.12128i	0.828059	+	0.560642i	$0.189445\pi$
$947$	1061.86i	1.12128i	−0.828059	+	0.560642i	$0.810555\pi$
$948$	0	0
$949$	2391.41	2.51993
$950$	0	0
$951$	− 790.904i	− 0.831655i
$952$	0	0
$953$	1651.46	1.73291	0.866455	−	0.499255i	$-0.166393\pi$
$953$	1651.46	1.73291	0.866455	+	0.499255i	$0.166393\pi$
$954$	0	0
$955$	− 652.088i	− 0.682814i
$956$	0	0
$957$	1687.37	1.76318
$958$	0	0
$959$	322.779i	0.336579i
$960$	0	0
$961$	947.081	0.985516
$962$	0	0
$963$	− 69.9255i	− 0.0726122i
$964$	0	0
$965$	−158.437	−0.164184
$966$	0	0
$967$	1165.15i	1.20491i	0.798153	+	0.602455i	$0.205811\pi$
$967$	1165.15i	1.20491i	−0.798153	+	0.602455i	$0.794189\pi$
$968$	0	0
$969$	−342.731	−0.353695
$970$	0	0
$971$	258.239i	0.265951i	0.991119	+	0.132976i	$0.0424532\pi$
$971$	258.239i	0.265951i	−0.991119	+	0.132976i	$0.957547\pi$
$972$	0	0
$973$	240.960	0.247647
$974$	0	0
$975$	412.520i	0.423098i
$976$	0	0
$977$	−881.665	−0.902421	−0.451210	−	0.892418i	$-0.649008\pi$
$977$	−881.665	−0.902421	−0.451210	+	0.892418i	$0.649008\pi$
$978$	0	0
$979$	404.444i	0.413119i
$980$	0	0
$981$	−2159.91	−2.20175
$982$	0	0
$983$	− 204.431i	− 0.207967i	−0.994579	−	0.103983i	$-0.966841\pi$
$983$	− 204.431i	− 0.207967i	0.994579	−	0.103983i	$-0.0331589\pi$
$984$	0	0
$985$	−99.3954	−0.100909
$986$	0	0
$987$	210.869i	0.213647i
$988$	0	0
$989$	84.4124	0.0853513
$990$	0	0
$991$	− 1170.77i	− 1.18140i	−0.806890	−	0.590701i	$-0.798851\pi$
$991$	− 1170.77i	− 1.18140i	0.806890	−	0.590701i	$-0.201149\pi$
$992$	0	0
$993$	1859.56	1.87266
$994$	0	0
$995$	319.543i	0.321148i
$996$	0	0
$997$	598.818	0.600620	0.300310	−	0.953842i	$-0.402910\pi$
$997$	598.818	0.600620	0.300310	+	0.953842i	$0.402910\pi$
$998$	0	0
$999$	− 177.152i	− 0.177329i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.6	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.51	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.6	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.51	yes	56	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.48524i q^{3} -2.23607 q^{5} -1.27380i q^{7} -11.1174 q^{9} +O(q^{10})\)	\(q-4.48524i q^{3} -2.23607 q^{5} -1.27380i q^{7} -11.1174 q^{9} +8.24171i q^{11} -18.3946 q^{13} +10.0293i q^{15} -2.71935 q^{17} +28.0997i q^{19} -5.71330 q^{21} -4.79583i q^{23} +5.00000 q^{25} +9.49715i q^{27} +45.6464 q^{29} -3.73087i q^{31} +36.9661 q^{33} +2.84830i q^{35} -18.6531 q^{37} +82.5041i q^{39} +8.29456 q^{41} +17.6012i q^{43} +24.8593 q^{45} -36.9085i q^{47} +47.3774 q^{49} +12.1970i q^{51} +53.0820 q^{53} -18.4290i q^{55} +126.034 q^{57} -17.7747i q^{59} +68.7826 q^{61} +14.1614i q^{63} +41.1315 q^{65} -19.5912i q^{67} -21.5105 q^{69} +114.683i q^{71} -130.007 q^{73} -22.4262i q^{75} +10.4983 q^{77} -61.0896i q^{79} -57.4597 q^{81} +101.488i q^{83} +6.08066 q^{85} -204.735i q^{87} +49.0728 q^{89} +23.4310i q^{91} -16.7339 q^{93} -62.8328i q^{95} -5.55033 q^{97} -91.6265i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more