LMFDB - Embedded newform 1840.3.k.d.321.12

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			321.12
Root			$-3.68124i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.36596 q^{3} +2.23607i q^{5} -1.16919i q^{7} +2.32968 q^{9} +O(q^{10})$	$q+3.36596 q^{3} +2.23607i q^{5} -1.16919i q^{7} +2.32968 q^{9} -10.6148i q^{11} -15.0913 q^{13} +7.52651i q^{15} +20.0887i q^{17} -22.5221i q^{19} -3.93543i q^{21} +(20.9683 - 9.45142i) q^{23} -5.00000 q^{25} -22.4520 q^{27} +32.5993 q^{29} +27.0975 q^{31} -35.7289i q^{33} +2.61438 q^{35} -53.0568i q^{37} -50.7968 q^{39} +9.43720 q^{41} -36.4382i q^{43} +5.20933i q^{45} +49.1365 q^{47} +47.6330 q^{49} +67.6176i q^{51} -104.253i q^{53} +23.7354 q^{55} -75.8083i q^{57} -53.5457 q^{59} -23.5166i q^{61} -2.72383i q^{63} -33.7453i q^{65} -59.4754i q^{67} +(70.5785 - 31.8131i) q^{69} -55.2130 q^{71} -8.77305 q^{73} -16.8298 q^{75} -12.4107 q^{77} -57.0848i q^{79} -96.5397 q^{81} +55.1788i q^{83} -44.9196 q^{85} +109.728 q^{87} +139.825i q^{89} +17.6446i q^{91} +91.2091 q^{93} +50.3608 q^{95} -19.8635i q^{97} -24.7291i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.36596			1.12199			0.560993	−	0.827820i	$-0.310419\pi$
$3$	3.36596			1.12199			0.560993	+	0.827820i	$0.310419\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	1.16919i		−	0.167026i	−0.996507	−	0.0835132i	$-0.973386\pi$
$7$		−	1.16919i		−	0.167026i	0.996507	−	0.0835132i	$-0.0266141\pi$
$8$		0			0
$9$	2.32968			0.258853
$10$		0			0
$11$		−	10.6148i		−	0.964981i	−0.875901	−	0.482490i	$-0.839732\pi$
$11$		−	10.6148i		−	0.964981i	0.875901	−	0.482490i	$-0.160268\pi$
$12$		0			0
$13$	−15.0913			−1.16087			−0.580436	−	0.814306i	$-0.697118\pi$
$13$	−15.0913			−1.16087			−0.580436	+	0.814306i	$0.697118\pi$
$14$		0			0
$15$			7.52651i			0.501768i
$16$		0			0
$17$			20.0887i			1.18169i	0.806787	+	0.590843i	$0.201205\pi$
$17$			20.0887i			1.18169i	−0.806787	+	0.590843i	$0.798795\pi$
$18$		0			0
$19$		−	22.5221i		−	1.18537i	−0.805434	−	0.592686i	$-0.798068\pi$
$19$		−	22.5221i		−	1.18537i	0.805434	−	0.592686i	$-0.201932\pi$
$20$		0			0
$21$		−	3.93543i		−	0.187401i
$22$		0			0
$23$	20.9683	−	9.45142i	0.911666	−	0.410931i
$23$	20.9683	−	9.45142i	0.911666	−	0.410931i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−22.4520			−0.831556
$28$		0			0
$29$	32.5993			1.12411			0.562056	−	0.827099i	$-0.310010\pi$
$29$	32.5993			1.12411			0.562056	+	0.827099i	$0.310010\pi$
$30$		0			0
$31$	27.0975			0.874113			0.437056	−	0.899434i	$-0.356021\pi$
$31$	27.0975			0.874113			0.437056	+	0.899434i	$0.356021\pi$
$32$		0			0
$33$		−	35.7289i		−	1.08270i
$34$		0			0
$35$	2.61438			0.0746965
$36$		0			0
$37$		−	53.0568i		−	1.43397i	−0.697089	−	0.716984i	$-0.745522\pi$
$37$		−	53.0568i		−	1.43397i	0.697089	−	0.716984i	$-0.254478\pi$
$38$		0			0
$39$	−50.7968			−1.30248
$40$		0			0
$41$	9.43720			0.230176			0.115088	−	0.993355i	$-0.463285\pi$
$41$	9.43720			0.230176			0.115088	+	0.993355i	$0.463285\pi$
$42$		0			0
$43$		−	36.4382i		−	0.847400i	−0.905802	−	0.423700i	$-0.860731\pi$
$43$		−	36.4382i		−	0.847400i	0.905802	−	0.423700i	$-0.139269\pi$
$44$		0			0
$45$			5.20933i			0.115763i
$46$		0			0
$47$	49.1365			1.04546			0.522728	−	0.852499i	$-0.324914\pi$
$47$	49.1365			1.04546			0.522728	+	0.852499i	$0.324914\pi$
$48$		0			0
$49$	47.6330			0.972102
$50$		0			0
$51$			67.6176i			1.32584i
$52$		0			0
$53$		−	104.253i		−	1.96703i	−0.180815	−	0.983517i	$-0.557874\pi$
$53$		−	104.253i		−	1.96703i	0.180815	−	0.983517i	$-0.442126\pi$
$54$		0			0
$55$	23.7354			0.431552
$56$		0			0
$57$		−	75.8083i		−	1.32997i
$58$		0			0
$59$	−53.5457			−0.907554			−0.453777	−	0.891115i	$-0.649924\pi$
$59$	−53.5457			−0.907554			−0.453777	+	0.891115i	$0.649924\pi$
$60$		0			0
$61$		−	23.5166i		−	0.385518i	−0.981246	−	0.192759i	$-0.938257\pi$
$61$		−	23.5166i		−	0.385518i	0.981246	−	0.192759i	$-0.0617435\pi$
$62$		0			0
$63$		−	2.72383i		−	0.0432354i
$64$		0			0
$65$		−	33.7453i		−	0.519158i
$66$		0			0
$67$		−	59.4754i		−	0.887692i	−0.896103	−	0.443846i	$-0.853614\pi$
$67$		−	59.4754i		−	0.887692i	0.896103	−	0.443846i	$-0.146386\pi$
$68$		0			0
$69$	70.5785	−	31.8131i	1.02288	−	0.461060i
$70$		0			0
$71$	−55.2130			−0.777648			−0.388824	−	0.921312i	$-0.627119\pi$
$71$	−55.2130			−0.777648			−0.388824	+	0.921312i	$0.627119\pi$
$72$		0			0
$73$	−8.77305			−0.120179			−0.0600894	−	0.998193i	$-0.519139\pi$
$73$	−8.77305			−0.120179			−0.0600894	+	0.998193i	$0.519139\pi$
$74$		0			0
$75$	−16.8298			−0.224397
$76$		0			0
$77$	−12.4107			−0.161177
$78$		0			0
$79$		−	57.0848i		−	0.722592i	−0.932451	−	0.361296i	$-0.882334\pi$
$79$		−	57.0848i		−	0.722592i	0.932451	−	0.361296i	$-0.117666\pi$
$80$		0			0
$81$	−96.5397			−1.19185
$82$		0			0
$83$			55.1788i			0.664805i	0.943138	+	0.332403i	$0.107859\pi$
$83$			55.1788i			0.664805i	−0.943138	+	0.332403i	$0.892141\pi$
$84$		0			0
$85$	−44.9196			−0.528466
$86$		0			0
$87$	109.728			1.26124
$88$		0			0
$89$			139.825i			1.57107i	0.618815	+	0.785536i	$0.287613\pi$
$89$			139.825i			1.57107i	−0.618815	+	0.785536i	$0.712387\pi$
$90$		0			0
$91$			17.6446i			0.193896i
$92$		0			0
$93$	91.2091			0.980743
$94$		0			0
$95$	50.3608			0.530114
$96$		0			0
$97$		−	19.8635i		−	0.204778i	−0.994744	−	0.102389i	$-0.967351\pi$
$97$		−	19.8635i		−	0.204778i	0.994744	−	0.102389i	$-0.0326487\pi$
$98$		0			0
$99$		−	24.7291i		−	0.249789i
$100$		0			0
$101$	86.5639			0.857068			0.428534	−	0.903526i	$-0.359030\pi$
$101$	86.5639			0.857068			0.428534	+	0.903526i	$0.359030\pi$
$102$		0			0
$103$		−	144.118i		−	1.39920i	−0.714535	−	0.699600i	$-0.753361\pi$
$103$		−	144.118i		−	1.39920i	0.714535	−	0.699600i	$-0.246639\pi$
$104$		0			0
$105$	8.79989			0.0838085
$106$		0			0
$107$			10.4544i			0.0977050i	0.998806	+	0.0488525i	$0.0155564\pi$
$107$			10.4544i			0.0977050i	−0.998806	+	0.0488525i	$0.984444\pi$
$108$		0			0
$109$		−	69.1221i		−	0.634148i	−0.948401	−	0.317074i	$-0.897300\pi$
$109$		−	69.1221i		−	0.634148i	0.948401	−	0.317074i	$-0.102700\pi$
$110$		0			0
$111$		−	178.587i		−	1.60889i
$112$		0			0
$113$			137.264i			1.21473i	0.794423	+	0.607364i	$0.207773\pi$
$113$			137.264i			1.21473i	−0.794423	+	0.607364i	$0.792227\pi$
$114$		0			0
$115$	21.1340	+	46.8866i	0.183774	+	0.407710i
$116$		0			0
$117$	−35.1580			−0.300496
$118$		0			0
$119$	23.4874			0.197373
$120$		0			0
$121$	8.32629			0.0688123
$122$		0			0
$123$	31.7652			0.258254
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−57.9697			−0.456455			−0.228227	−	0.973608i	$-0.573293\pi$
$127$	−57.9697			−0.456455			−0.228227	+	0.973608i	$0.573293\pi$
$128$		0			0
$129$		−	122.650i		−	0.950772i
$130$		0			0
$131$	95.0810			0.725809			0.362905	−	0.931826i	$-0.381785\pi$
$131$	95.0810			0.725809			0.362905	+	0.931826i	$0.381785\pi$
$132$		0			0
$133$	−26.3325			−0.197988
$134$		0			0
$135$		−	50.2042i		−	0.371883i
$136$		0			0
$137$		−	198.711i		−	1.45045i	−0.688514	−	0.725223i	$-0.741737\pi$
$137$		−	198.711i		−	1.45045i	0.688514	−	0.725223i	$-0.258263\pi$
$138$		0			0
$139$	0.129480			0.000931508			0.000465754	−	1.00000i	$-0.499852\pi$
$139$	0.129480			0.000931508			0.000465754		1.00000i	$0.499852\pi$
$140$		0			0
$141$	165.391			1.17299
$142$		0			0
$143$			160.191i			1.12022i
$144$		0			0
$145$			72.8942i			0.502718i
$146$		0			0
$147$	160.331			1.09069
$148$		0			0
$149$		−	63.8745i		−	0.428688i	−0.976758	−	0.214344i	$-0.931239\pi$
$149$		−	63.8745i		−	0.428688i	0.976758	−	0.214344i	$-0.0687613\pi$
$150$		0			0
$151$	35.2673			0.233558			0.116779	−	0.993158i	$-0.462743\pi$
$151$	35.2673			0.233558			0.116779	+	0.993158i	$0.462743\pi$
$152$		0			0
$153$			46.8002i			0.305884i
$154$		0			0
$155$			60.5919i			0.390915i
$156$		0			0
$157$			289.136i			1.84163i	0.390002	+	0.920814i	$0.372474\pi$
$157$			289.136i			1.84163i	−0.390002	+	0.920814i	$0.627526\pi$
$158$		0			0
$159$		−	350.911i		−	2.20699i
$160$		0			0
$161$	−11.0505	−	24.5159i	−0.0686364	−	0.152272i
$162$		0			0
$163$	46.2536			0.283764			0.141882	−	0.989884i	$-0.454685\pi$
$163$	46.2536			0.283764			0.141882	+	0.989884i	$0.454685\pi$
$164$		0			0
$165$	79.8923			0.484196
$166$		0			0
$167$	90.8735			0.544153			0.272076	−	0.962276i	$-0.412290\pi$
$167$	90.8735			0.544153			0.272076	+	0.962276i	$0.412290\pi$
$168$		0			0
$169$	58.7484			0.347624
$170$		0			0
$171$		−	52.4692i		−	0.306837i
$172$		0			0
$173$	−90.7893			−0.524794			−0.262397	−	0.964960i	$-0.584513\pi$
$173$	−90.7893			−0.524794			−0.262397	+	0.964960i	$0.584513\pi$
$174$		0			0
$175$			5.84593i			0.0334053i
$176$		0			0
$177$	−180.233			−1.01826
$178$		0			0
$179$	−301.636			−1.68511			−0.842557	−	0.538607i	$-0.818951\pi$
$179$	−301.636			−1.68511			−0.842557	+	0.538607i	$0.818951\pi$
$180$		0			0
$181$			248.547i			1.37319i	0.727040	+	0.686595i	$0.240895\pi$
$181$			248.547i			1.37319i	−0.727040	+	0.686595i	$0.759105\pi$
$182$		0			0
$183$		−	79.1558i		−	0.432546i
$184$		0			0
$185$	118.639			0.641290
$186$		0			0
$187$	213.237			1.14030
$188$		0			0
$189$			26.2506i			0.138892i
$190$		0			0
$191$		−	146.382i		−	0.766397i	−0.923666	−	0.383199i	$-0.874822\pi$
$191$		−	146.382i		−	0.766397i	0.923666	−	0.383199i	$-0.125178\pi$
$192$		0			0
$193$	−120.648			−0.625120			−0.312560	−	0.949898i	$-0.601187\pi$
$193$	−120.648			−0.625120			−0.312560	+	0.949898i	$0.601187\pi$
$194$		0			0
$195$		−	113.585i		−	0.582488i
$196$		0			0
$197$	31.6502			0.160661			0.0803305	−	0.996768i	$-0.474402\pi$
$197$	31.6502			0.160661			0.0803305	+	0.996768i	$0.474402\pi$
$198$		0			0
$199$		−	37.4496i		−	0.188189i	−0.995563	−	0.0940944i	$-0.970004\pi$
$199$		−	37.4496i		−	0.188189i	0.995563	−	0.0940944i	$-0.0299955\pi$
$200$		0			0
$201$		−	200.192i		−	0.995979i
$202$		0			0
$203$		−	38.1146i		−	0.187757i
$204$		0			0
$205$			21.1022i			0.102938i
$206$		0			0
$207$	48.8495	−	22.0188i	0.235988	−	0.106371i
$208$		0			0
$209$	−239.067			−1.14386
$210$		0			0
$211$	43.6572			0.206906			0.103453	−	0.994634i	$-0.467011\pi$
$211$	43.6572			0.206906			0.103453	+	0.994634i	$0.467011\pi$
$212$		0			0
$213$	−185.845			−0.872510
$214$		0			0
$215$	81.4783			0.378969
$216$		0			0
$217$		−	31.6820i		−	0.146000i
$218$		0			0
$219$	−29.5297			−0.134839
$220$		0			0
$221$		−	303.165i		−	1.37179i
$222$		0			0
$223$	−348.623			−1.56333			−0.781666	−	0.623697i	$-0.785630\pi$
$223$	−348.623			−1.56333			−0.781666	+	0.623697i	$0.785630\pi$
$224$		0			0
$225$	−11.6484			−0.0517707
$226$		0			0
$227$		−	175.793i		−	0.774421i	−0.921991	−	0.387210i	$-0.873439\pi$
$227$		−	175.793i		−	0.774421i	0.921991	−	0.387210i	$-0.126561\pi$
$228$		0			0
$229$			1.44148i			0.00629466i	0.999995	+	0.00314733i	$0.00100183\pi$
$229$			1.44148i			0.00629466i	−0.999995	+	0.00314733i	$0.998998\pi$
$230$		0			0
$231$	−41.7738			−0.180839
$232$		0			0
$233$	−113.416			−0.486763			−0.243381	−	0.969931i	$-0.578257\pi$
$233$	−113.416			−0.486763			−0.243381	+	0.969931i	$0.578257\pi$
$234$		0			0
$235$			109.872i			0.467542i
$236$		0			0
$237$		−	192.145i		−	0.810738i
$238$		0			0
$239$	−367.260			−1.53665			−0.768326	−	0.640059i	$-0.778910\pi$
$239$	−367.260			−1.53665			−0.768326	+	0.640059i	$0.778910\pi$
$240$		0			0
$241$			220.365i			0.914378i	0.889370	+	0.457189i	$0.151144\pi$
$241$			220.365i			0.914378i	−0.889370	+	0.457189i	$0.848856\pi$
$242$		0			0
$243$	−122.881			−0.505681
$244$		0			0
$245$			106.511i			0.434737i
$246$		0			0
$247$			339.888i			1.37606i
$248$		0			0
$249$			185.730i			0.745902i
$250$		0			0
$251$			440.210i			1.75382i	0.480650	+	0.876912i	$0.340401\pi$
$251$			440.210i			1.75382i	−0.480650	+	0.876912i	$0.659599\pi$
$252$		0			0
$253$	−100.325	−	222.574i	−0.396541	−	0.879740i
$254$		0			0
$255$	−151.198			−0.592932
$256$		0			0
$257$	491.519			1.91252			0.956262	−	0.292512i	$-0.0944913\pi$
$257$	491.519			1.91252			0.956262	+	0.292512i	$0.0944913\pi$
$258$		0			0
$259$	−62.0333			−0.239511
$260$		0			0
$261$	75.9459			0.290980
$262$		0			0
$263$			267.833i			1.01838i	0.860655	+	0.509189i	$0.170054\pi$
$263$			267.833i			1.01838i	−0.860655	+	0.509189i	$0.829946\pi$
$264$		0			0
$265$	233.116			0.879684
$266$		0			0
$267$			470.647i			1.76272i
$268$		0			0
$269$	95.6985			0.355756			0.177878	−	0.984053i	$-0.443077\pi$
$269$	95.6985			0.355756			0.177878	+	0.984053i	$0.443077\pi$
$270$		0			0
$271$	452.255			1.66884			0.834419	−	0.551131i	$-0.185803\pi$
$271$	452.255			1.66884			0.834419	+	0.551131i	$0.185803\pi$
$272$		0			0
$273$			59.3909i			0.217549i
$274$		0			0
$275$			53.0739i			0.192996i
$276$		0			0
$277$	−87.4783			−0.315806			−0.157903	−	0.987455i	$-0.550473\pi$
$277$	−87.4783			−0.315806			−0.157903	+	0.987455i	$0.550473\pi$
$278$		0			0
$279$	63.1285			0.226267
$280$		0			0
$281$		−	232.157i		−	0.826183i	−0.910690	−	0.413091i	$-0.864449\pi$
$281$		−	232.157i		−	0.826183i	0.910690	−	0.413091i	$-0.135551\pi$
$282$		0			0
$283$		−	239.367i		−	0.845820i	−0.906172	−	0.422910i	$-0.861009\pi$
$283$		−	239.367i		−	0.845820i	0.906172	−	0.422910i	$-0.138991\pi$
$284$		0			0
$285$	169.513			0.594781
$286$		0			0
$287$		−	11.0338i		−	0.0384454i
$288$		0			0
$289$	−114.554			−0.396382
$290$		0			0
$291$		−	66.8597i		−	0.229758i
$292$		0			0
$293$		−	581.830i		−	1.98577i	−0.119092	−	0.992883i	$-0.537998\pi$
$293$		−	581.830i		−	1.98577i	0.119092	−	0.992883i	$-0.462002\pi$
$294$		0			0
$295$		−	119.732i		−	0.405871i
$296$		0			0
$297$			238.323i			0.802436i
$298$		0			0
$299$	−316.440	+	142.635i	−1.05833	+	0.477039i
$300$		0			0
$301$	−42.6030			−0.141538
$302$		0			0
$303$	291.370			0.961619
$304$		0			0
$305$	52.5847			0.172409
$306$		0			0
$307$	−110.134			−0.358742			−0.179371	−	0.983782i	$-0.557406\pi$
$307$	−110.134			−0.358742			−0.179371	+	0.983782i	$0.557406\pi$
$308$		0			0
$309$		−	485.094i		−	1.56988i
$310$		0			0
$311$	298.100			0.958520			0.479260	−	0.877673i	$-0.340905\pi$
$311$	298.100			0.958520			0.479260	+	0.877673i	$0.340905\pi$
$312$		0			0
$313$		−	109.474i		−	0.349758i	−0.984590	−	0.174879i	$-0.944047\pi$
$313$		−	109.474i		−	0.349758i	0.984590	−	0.174879i	$-0.0559534\pi$
$314$		0			0
$315$	6.09067			0.0193355
$316$		0			0
$317$	474.434			1.49664			0.748318	−	0.663340i	$-0.230862\pi$
$317$	474.434			1.49664			0.748318	+	0.663340i	$0.230862\pi$
$318$		0			0
$319$		−	346.034i		−	1.08475i
$320$		0			0
$321$			35.1892i			0.109624i
$322$		0			0
$323$	452.438			1.40074
$324$		0			0
$325$	75.4567			0.232174
$326$		0			0
$327$		−	232.662i		−	0.711506i
$328$		0			0
$329$		−	57.4496i		−	0.174619i
$330$		0			0
$331$	−255.367			−0.771503			−0.385751	−	0.922603i	$-0.626058\pi$
$331$	−255.367			−0.771503			−0.385751	+	0.922603i	$0.626058\pi$
$332$		0			0
$333$		−	123.606i		−	0.371188i
$334$		0			0
$335$	132.991			0.396988
$336$		0			0
$337$			458.486i			1.36049i	0.732983	+	0.680247i	$0.238127\pi$
$337$			458.486i			1.36049i	−0.732983	+	0.680247i	$0.761873\pi$
$338$		0			0
$339$			462.026i			1.36291i
$340$		0			0
$341$		−	287.634i		−	0.843502i
$342$		0			0
$343$		−	112.982i		−	0.329393i
$344$		0			0
$345$	71.1363	+	157.818i	0.206192	+	0.457445i
$346$		0			0
$347$	−510.676			−1.47169			−0.735845	−	0.677151i	$-0.763215\pi$
$347$	−510.676			−1.47169			−0.735845	+	0.677151i	$0.763215\pi$
$348$		0			0
$349$	−591.870			−1.69590			−0.847951	−	0.530075i	$-0.822164\pi$
$349$	−591.870			−1.69590			−0.847951	+	0.530075i	$0.822164\pi$
$350$		0			0
$351$	338.831			0.965330
$352$		0			0
$353$	168.334			0.476866			0.238433	−	0.971159i	$-0.423366\pi$
$353$	168.334			0.476866			0.238433	+	0.971159i	$0.423366\pi$
$354$		0			0
$355$		−	123.460i		−	0.347775i
$356$		0			0
$357$	79.0575			0.221450
$358$		0			0
$359$		−	37.5719i		−	0.104657i	−0.998630	−	0.0523285i	$-0.983336\pi$
$359$		−	37.5719i		−	0.104657i	0.998630	−	0.0523285i	$-0.0166643\pi$
$360$		0			0
$361$	−146.243			−0.405105
$362$		0			0
$363$	28.0260			0.0772065
$364$		0			0
$365$		−	19.6171i		−	0.0537456i
$366$		0			0
$367$		−	427.036i		−	1.16359i	−0.813337	−	0.581793i	$-0.802351\pi$
$367$		−	427.036i		−	1.16359i	0.813337	−	0.581793i	$-0.197649\pi$
$368$		0			0
$369$	21.9857			0.0595818
$370$		0			0
$371$	−121.891			−0.328547
$372$		0			0
$373$			369.980i			0.991904i	0.868350	+	0.495952i	$0.165181\pi$
$373$			369.980i			0.991904i	−0.868350	+	0.495952i	$0.834819\pi$
$374$		0			0
$375$		−	37.6326i		−	0.100354i
$376$		0			0
$377$	−491.966			−1.30495
$378$		0			0
$379$		−	78.8009i		−	0.207918i	−0.994582	−	0.103959i	$-0.966849\pi$
$379$		−	78.8009i		−	0.207918i	0.994582	−	0.103959i	$-0.0331511\pi$
$380$		0			0
$381$	−195.124			−0.512136
$382$		0			0
$383$		−	254.083i		−	0.663403i	−0.943384	−	0.331702i	$-0.892377\pi$
$383$		−	254.083i		−	0.663403i	0.943384	−	0.331702i	$-0.107623\pi$
$384$		0			0
$385$		−	27.7511i		−	0.0720807i
$386$		0			0
$387$		−	84.8894i		−	0.219353i
$388$		0			0
$389$		−	322.363i		−	0.828698i	−0.910118	−	0.414349i	$-0.864009\pi$
$389$		−	322.363i		−	0.828698i	0.910118	−	0.414349i	$-0.135991\pi$
$390$		0			0
$391$	189.866	+	421.226i	0.485592	+	1.07730i
$392$		0			0
$393$	320.039			0.814348
$394$		0			0
$395$	127.645			0.323153
$396$		0			0
$397$	715.832			1.80310			0.901552	−	0.432671i	$-0.142429\pi$
$397$	715.832			1.80310			0.901552	+	0.432671i	$0.142429\pi$
$398$		0			0
$399$	−88.6340			−0.222140
$400$		0			0
$401$			441.435i			1.10083i	0.834890	+	0.550417i	$0.185531\pi$
$401$			441.435i			1.10083i	−0.834890	+	0.550417i	$0.814469\pi$
$402$		0			0
$403$	−408.937			−1.01473
$404$		0			0
$405$		−	215.869i		−	0.533011i
$406$		0			0
$407$	−563.187			−1.38375
$408$		0			0
$409$	−608.727			−1.48833			−0.744164	−	0.667996i	$-0.767152\pi$
$409$	−608.727			−1.48833			−0.744164	+	0.667996i	$0.767152\pi$
$410$		0			0
$411$		−	668.853i		−	1.62738i
$412$		0			0
$413$			62.6048i			0.151586i
$414$		0			0
$415$	−123.384			−0.297310
$416$		0			0
$417$	0.435823			0.00104514
$418$		0			0
$419$			287.454i			0.686047i	0.939327	+	0.343023i	$0.111451\pi$
$419$			287.454i			0.686047i	−0.939327	+	0.343023i	$0.888549\pi$
$420$		0			0
$421$		−	130.848i		−	0.310803i	−0.987851	−	0.155402i	$-0.950333\pi$
$421$		−	130.848i		−	0.310803i	0.987851	−	0.155402i	$-0.0496672\pi$
$422$		0			0
$423$	114.472			0.270620
$424$		0			0
$425$		−	100.443i		−	0.236337i
$426$		0			0
$427$	−27.4952			−0.0643917
$428$		0			0
$429$			539.197i			1.25687i
$430$		0			0
$431$			239.816i			0.556417i	0.960521	+	0.278209i	$0.0897406\pi$
$431$			239.816i			0.556417i	−0.960521	+	0.278209i	$0.910259\pi$
$432$		0			0
$433$		−	305.542i		−	0.705640i	−0.935691	−	0.352820i	$-0.885223\pi$
$433$		−	305.542i		−	0.705640i	0.935691	−	0.352820i	$-0.114777\pi$
$434$		0			0
$435$			245.359i			0.564043i
$436$		0			0
$437$	−212.865	−	472.250i	−0.487106	−	1.08066i
$438$		0			0
$439$	579.018			1.31895			0.659474	−	0.751727i	$-0.270779\pi$
$439$	579.018			1.31895			0.659474	+	0.751727i	$0.270779\pi$
$440$		0			0
$441$	110.970			0.251632
$442$		0			0
$443$	606.274			1.36856			0.684282	−	0.729217i	$-0.260116\pi$
$443$	606.274			1.36856			0.684282	+	0.729217i	$0.260116\pi$
$444$		0			0
$445$	−312.659			−0.702605
$446$		0			0
$447$		−	214.999i		−	0.480982i
$448$		0			0
$449$	−202.580			−0.451181			−0.225590	−	0.974222i	$-0.572431\pi$
$449$	−202.580			−0.451181			−0.225590	+	0.974222i	$0.572431\pi$
$450$		0			0
$451$		−	100.174i		−	0.222115i
$452$		0			0
$453$	118.708			0.262049
$454$		0			0
$455$	−39.4545			−0.0867131
$456$		0			0
$457$			373.760i			0.817857i	0.912567	+	0.408928i	$0.134097\pi$
$457$			373.760i			0.817857i	−0.912567	+	0.408928i	$0.865903\pi$
$458$		0			0
$459$		−	451.031i		−	0.982639i
$460$		0			0
$461$	709.377			1.53878			0.769390	−	0.638780i	$-0.220561\pi$
$461$	709.377			1.53878			0.769390	+	0.638780i	$0.220561\pi$
$462$		0			0
$463$	−266.488			−0.575569			−0.287785	−	0.957695i	$-0.592919\pi$
$463$	−266.488			−0.575569			−0.287785	+	0.957695i	$0.592919\pi$
$464$		0			0
$465$			203.950i			0.438602i
$466$		0			0
$467$		−	686.135i		−	1.46924i	−0.678479	−	0.734620i	$-0.737361\pi$
$467$		−	686.135i		−	1.46924i	0.678479	−	0.734620i	$-0.262639\pi$
$468$		0			0
$469$	−69.5378			−0.148268
$470$		0			0
$471$			973.219i			2.06628i
$472$		0			0
$473$	−386.784			−0.817725
$474$		0			0
$475$			112.610i			0.237074i
$476$		0			0
$477$		−	242.876i		−	0.509174i
$478$		0			0
$479$			176.517i			0.368511i	0.982878	+	0.184256i	$0.0589874\pi$
$479$			176.517i			0.368511i	−0.982878	+	0.184256i	$0.941013\pi$
$480$		0			0
$481$			800.698i			1.66465i
$482$		0			0
$483$	−37.1954	−	82.5194i	−0.0770091	−	0.170848i
$484$		0			0
$485$	44.4161			0.0915796
$486$		0			0
$487$	−524.802			−1.07762			−0.538811	−	0.842427i	$-0.681126\pi$
$487$	−524.802			−1.07762			−0.538811	+	0.842427i	$0.681126\pi$
$488$		0			0
$489$	155.688			0.318380
$490$		0			0
$491$	551.047			1.12229			0.561147	−	0.827716i	$-0.310360\pi$
$491$	551.047			1.12229			0.561147	+	0.827716i	$0.310360\pi$
$492$		0			0
$493$			654.876i			1.32835i
$494$		0			0
$495$	55.2959			0.111709
$496$		0			0
$497$			64.5542i			0.129888i
$498$		0			0
$499$	736.739			1.47643			0.738216	−	0.674565i	$-0.235669\pi$
$499$	736.739			1.47643			0.738216	+	0.674565i	$0.235669\pi$
$500$		0			0
$501$	305.876			0.610532
$502$		0			0
$503$		−	699.308i		−	1.39027i	−0.718877	−	0.695137i	$-0.755344\pi$
$503$		−	699.308i		−	1.39027i	0.718877	−	0.695137i	$-0.244656\pi$
$504$		0			0
$505$			193.563i			0.383292i
$506$		0			0
$507$	197.745			0.390029
$508$		0			0
$509$	−316.428			−0.621666			−0.310833	−	0.950464i	$-0.600608\pi$
$509$	−316.428			−0.621666			−0.310833	+	0.950464i	$0.600608\pi$
$510$		0			0
$511$			10.2573i			0.0200730i
$512$		0			0
$513$			505.666i			0.985703i
$514$		0			0
$515$	322.257			0.625741
$516$		0			0
$517$		−	521.573i		−	1.00885i
$518$		0			0
$519$	−305.593			−0.588811
$520$		0			0
$521$			798.875i			1.53335i	0.642035	+	0.766675i	$0.278090\pi$
$521$			798.875i			1.53335i	−0.642035	+	0.766675i	$0.721910\pi$
$522$		0			0
$523$			819.091i			1.56614i	0.621934	+	0.783070i	$0.286347\pi$
$523$			819.091i			1.56614i	−0.621934	+	0.783070i	$0.713653\pi$
$524$		0			0
$525$			19.6772i			0.0374803i
$526$		0			0
$527$			544.353i			1.03293i
$528$		0			0
$529$	350.341	−	396.361i	0.662271	−	0.749265i
$530$		0			0
$531$	−124.744			−0.234924
$532$		0			0
$533$	−142.420			−0.267204
$534$		0			0
$535$	−23.3768			−0.0436950
$536$		0			0
$537$	−1015.29			−1.89068
$538$		0			0
$539$		−	505.614i		−	0.938060i
$540$		0			0
$541$	289.616			0.535335			0.267668	−	0.963511i	$-0.413747\pi$
$541$	289.616			0.535335			0.267668	+	0.963511i	$0.413747\pi$
$542$		0			0
$543$			836.600i			1.54070i
$544$		0			0
$545$	154.562			0.283600
$546$		0			0
$547$	−996.117			−1.82106			−0.910528	−	0.413448i	$-0.864324\pi$
$547$	−996.117			−1.82106			−0.910528	+	0.413448i	$0.864324\pi$
$548$		0			0
$549$		−	54.7861i		−	0.0997926i
$550$		0			0
$551$		−	734.202i		−	1.33249i
$552$		0			0
$553$	−66.7427			−0.120692
$554$		0			0
$555$	399.333			0.719519
$556$		0			0
$557$			937.756i			1.68358i	0.539802	+	0.841792i	$0.318499\pi$
$557$			937.756i			1.68358i	−0.539802	+	0.841792i	$0.681501\pi$
$558$		0			0
$559$			549.901i			0.983723i
$560$		0			0
$561$	717.747			1.27941
$562$		0			0
$563$			785.252i			1.39476i	0.716700	+	0.697382i	$0.245652\pi$
$563$			785.252i			1.39476i	−0.716700	+	0.697382i	$0.754348\pi$
$564$		0			0
$565$	−306.932			−0.543243
$566$		0			0
$567$			112.873i			0.199070i
$568$		0			0
$569$			403.851i			0.709755i	0.934913	+	0.354877i	$0.115477\pi$
$569$			403.851i			0.709755i	−0.934913	+	0.354877i	$0.884523\pi$
$570$		0			0
$571$			335.474i			0.587519i	0.955879	+	0.293760i	$0.0949065\pi$
$571$			335.474i			0.587519i	−0.955879	+	0.293760i	$0.905093\pi$
$572$		0			0
$573$		−	492.715i		−	0.859887i
$574$		0			0
$575$	−104.842	+	47.2571i	−0.182333	+	0.0821863i
$576$		0			0
$577$	−152.703			−0.264649			−0.132325	−	0.991206i	$-0.542244\pi$
$577$	−152.703			−0.264649			−0.132325	+	0.991206i	$0.542244\pi$
$578$		0			0
$579$	−406.097			−0.701376
$580$		0			0
$581$	64.5143			0.111040
$582$		0			0
$583$	−1106.62			−1.89815
$584$		0			0
$585$		−	78.6157i		−	0.134386i
$586$		0			0
$587$	−129.262			−0.220207			−0.110104	−	0.993920i	$-0.535118\pi$
$587$	−129.262			−0.220207			−0.110104	+	0.993920i	$0.535118\pi$
$588$		0			0
$589$		−	610.291i		−	1.03615i
$590$		0			0
$591$	106.533			0.180260
$592$		0			0
$593$	−879.853			−1.48373			−0.741866	−	0.670548i	$-0.766059\pi$
$593$	−879.853			−1.48373			−0.741866	+	0.670548i	$0.766059\pi$
$594$		0			0
$595$			52.5194i			0.0882678i
$596$		0			0
$597$		−	126.054i		−	0.211145i
$598$		0			0
$599$	709.708			1.18482			0.592411	−	0.805636i	$-0.298176\pi$
$599$	709.708			1.18482			0.592411	+	0.805636i	$0.298176\pi$
$600$		0			0
$601$	238.861			0.397439			0.198719	−	0.980056i	$-0.436322\pi$
$601$	238.861			0.397439			0.198719	+	0.980056i	$0.436322\pi$
$602$		0			0
$603$		−	138.559i		−	0.229782i
$604$		0			0
$605$			18.6182i			0.0307738i
$606$		0			0
$607$	1082.53			1.78341			0.891703	−	0.452621i	$-0.149511\pi$
$607$	1082.53			1.78341			0.891703	+	0.452621i	$0.149511\pi$
$608$		0			0
$609$		−	128.292i		−	0.210660i
$610$		0			0
$611$	−741.535			−1.21364
$612$		0			0
$613$			54.0902i			0.0882384i	0.999026	+	0.0441192i	$0.0140481\pi$
$613$			54.0902i			0.0882384i	−0.999026	+	0.0441192i	$0.985952\pi$
$614$		0			0
$615$			71.0292i			0.115495i
$616$		0			0
$617$		−	454.342i		−	0.736373i	−0.929752	−	0.368186i	$-0.879979\pi$
$617$		−	454.342i		−	0.736373i	0.929752	−	0.368186i	$-0.120021\pi$
$618$		0			0
$619$			153.676i			0.248264i	0.992266	+	0.124132i	$0.0396147\pi$
$619$			153.676i			0.248264i	−0.992266	+	0.124132i	$0.960385\pi$
$620$		0			0
$621$	−470.781	+	212.204i	−0.758102	+	0.341713i
$622$		0			0
$623$	163.482			0.262411
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−804.689			−1.28340
$628$		0			0
$629$	1065.84			1.69450
$630$		0			0
$631$			1241.15i			1.96696i	0.181027	+	0.983478i	$0.442058\pi$
$631$			1241.15i			1.96696i	−0.181027	+	0.983478i	$0.557942\pi$
$632$		0			0
$633$	146.948			0.232146
$634$		0			0
$635$		−	129.624i		−	0.204133i
$636$		0			0
$637$	−718.846			−1.12849
$638$		0			0
$639$	−128.629			−0.201297
$640$		0			0
$641$			545.978i			0.851759i	0.904780	+	0.425880i	$0.140035\pi$
$641$			545.978i			0.851759i	−0.904780	+	0.425880i	$0.859965\pi$
$642$		0			0
$643$		−	539.118i		−	0.838441i	−0.907884	−	0.419221i	$-0.862303\pi$
$643$		−	539.118i		−	0.838441i	0.907884	−	0.419221i	$-0.137697\pi$
$644$		0			0
$645$	274.253			0.425198
$646$		0			0
$647$	−652.715			−1.00883			−0.504417	−	0.863460i	$-0.668292\pi$
$647$	−652.715			−1.00883			−0.504417	+	0.863460i	$0.668292\pi$
$648$		0			0
$649$			568.376i			0.875772i
$650$		0			0
$651$		−	106.640i		−	0.163810i
$652$		0			0
$653$	374.815			0.573989			0.286995	−	0.957932i	$-0.407344\pi$
$653$	374.815			0.573989			0.286995	+	0.957932i	$0.407344\pi$
$654$		0			0
$655$			212.608i			0.324592i
$656$		0			0
$657$	−20.4384			−0.0311087
$658$		0			0
$659$			440.090i			0.667815i	0.942606	+	0.333907i	$0.108367\pi$
$659$			440.090i			0.667815i	−0.942606	+	0.333907i	$0.891633\pi$
$660$		0			0
$661$		−	72.6725i		−	0.109943i	−0.998488	−	0.0549716i	$-0.982493\pi$
$661$		−	72.6725i		−	0.109943i	0.998488	−	0.0549716i	$-0.0175068\pi$
$662$		0			0
$663$		−	1020.44i		−	1.53913i
$664$		0			0
$665$		−	58.8812i		−	0.0885431i
$666$		0			0
$667$	683.552	−	308.110i	1.02482	−	0.461933i
$668$		0			0
$669$	−1173.45			−1.75404
$670$		0			0
$671$	−249.623			−0.372017
$672$		0			0
$673$	−1052.88			−1.56445			−0.782227	−	0.622993i	$-0.785916\pi$
$673$	−1052.88			−1.56445			−0.782227	+	0.622993i	$0.785916\pi$
$674$		0			0
$675$	112.260			0.166311
$676$		0			0
$677$		−	393.587i		−	0.581369i	−0.956819	−	0.290684i	$-0.906117\pi$
$677$		−	393.587i		−	0.581369i	0.956819	−	0.290684i	$-0.0938830\pi$
$678$		0			0
$679$	−23.2241			−0.0342034
$680$		0			0
$681$		−	591.714i		−	0.868889i
$682$		0			0
$683$	645.240			0.944714			0.472357	−	0.881407i	$-0.343403\pi$
$683$	645.240			0.944714			0.472357	+	0.881407i	$0.343403\pi$
$684$		0			0
$685$	444.331			0.648659
$686$		0			0
$687$			4.85195i			0.00706252i
$688$		0			0
$689$			1573.31i			2.28347i
$690$		0			0
$691$	−548.216			−0.793366			−0.396683	−	0.917956i	$-0.629839\pi$
$691$	−548.216			−0.793366			−0.396683	+	0.917956i	$0.629839\pi$
$692$		0			0
$693$	−28.9129			−0.0417213
$694$		0			0
$695$			0.289525i			0.000416583i
$696$		0			0
$697$			189.581i			0.271995i
$698$		0			0
$699$	−381.753			−0.546141
$700$		0			0
$701$		−	92.2064i		−	0.131536i	−0.997835	−	0.0657678i	$-0.979050\pi$
$701$		−	92.2064i		−	0.131536i	0.997835	−	0.0657678i	$-0.0209496\pi$
$702$		0			0
$703$	−1194.95			−1.69978
$704$		0			0
$705$			369.826i			0.524576i
$706$		0			0
$707$		−	101.209i		−	0.143153i
$708$		0			0
$709$		−	922.612i		−	1.30129i	−0.759384	−	0.650643i	$-0.774499\pi$
$709$		−	922.612i		−	1.30129i	0.759384	−	0.650643i	$-0.225501\pi$
$710$		0			0
$711$		−	132.989i		−	0.187045i
$712$		0			0
$713$	568.189	−	256.110i	0.796899	−	0.359201i
$714$		0			0
$715$	−358.199			−0.500977
$716$		0			0
$717$	−1236.18			−1.72410
$718$		0			0
$719$	−606.887			−0.844071			−0.422035	−	0.906579i	$-0.638684\pi$
$719$	−606.887			−0.844071			−0.422035	+	0.906579i	$0.638684\pi$
$720$		0			0
$721$	−168.500			−0.233704
$722$		0			0
$723$			741.740i			1.02592i
$724$		0			0
$725$	−162.996			−0.224823
$726$		0			0
$727$			787.544i			1.08328i	0.840611	+	0.541640i	$0.182196\pi$
$727$			787.544i			1.08328i	−0.840611	+	0.541640i	$0.817804\pi$
$728$		0			0
$729$	455.246			0.624481
$730$		0			0
$731$	731.995			1.00136
$732$		0			0
$733$			415.085i			0.566282i	0.959078	+	0.283141i	$0.0913765\pi$
$733$			415.085i			0.566282i	−0.959078	+	0.283141i	$0.908624\pi$
$734$		0			0
$735$			358.510i			0.487769i
$736$		0			0
$737$	−631.319			−0.856606
$738$		0			0
$739$	−639.964			−0.865987			−0.432993	−	0.901397i	$-0.642543\pi$
$739$	−639.964			−0.865987			−0.432993	+	0.901397i	$0.642543\pi$
$740$		0			0
$741$			1144.05i			1.54393i
$742$		0			0
$743$			743.739i			1.00099i	0.865738	+	0.500497i	$0.166849\pi$
$743$			743.739i			1.00099i	−0.865738	+	0.500497i	$0.833151\pi$
$744$		0			0
$745$	142.828			0.191715
$746$		0			0
$747$			128.549i			0.172087i
$748$		0			0
$749$	12.2232			0.0163193
$750$		0			0
$751$			517.047i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$751$			517.047i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$752$		0			0
$753$			1481.73i			1.96777i
$754$		0			0
$755$			78.8601i			0.104450i
$756$		0			0
$757$		−	911.823i		−	1.20452i	−0.798299	−	0.602261i	$-0.794267\pi$
$757$		−	911.823i		−	1.20452i	0.798299	−	0.602261i	$-0.205733\pi$
$758$		0			0
$759$	−337.689	−	749.176i	−0.444914	−	0.987057i
$760$		0			0
$761$	954.073			1.25371			0.626855	−	0.779136i	$-0.284342\pi$
$761$	954.073			1.25371			0.626855	+	0.779136i	$0.284342\pi$
$762$		0			0
$763$	−80.8166			−0.105920
$764$		0			0
$765$	−104.648			−0.136795
$766$		0			0
$767$	808.076			1.05355
$768$		0			0
$769$		−	276.277i		−	0.359268i	−0.983734	−	0.179634i	$-0.942509\pi$
$769$		−	276.277i		−	0.359268i	0.983734	−	0.179634i	$-0.0574913\pi$
$770$		0			0
$771$	1654.43			2.14583
$772$		0			0
$773$		−	40.8440i		−	0.0528383i	−0.999651	−	0.0264192i	$-0.991590\pi$
$773$		−	40.8440i		−	0.0528383i	0.999651	−	0.0264192i	$-0.00841046\pi$
$774$		0			0
$775$	−135.488			−0.174823
$776$		0			0
$777$	−208.801			−0.268728
$778$		0			0
$779$		−	212.545i		−	0.272843i
$780$		0			0
$781$			586.074i			0.750415i
$782$		0			0
$783$	−731.919			−0.934763
$784$		0			0
$785$	−646.527			−0.823601
$786$		0			0
$787$			1364.69i			1.73404i	0.498274	+	0.867019i	$0.333967\pi$
$787$			1364.69i			1.73404i	−0.498274	+	0.867019i	$0.666033\pi$
$788$		0			0
$789$			901.515i			1.14261i
$790$		0			0
$791$	160.487			0.202892
$792$		0			0
$793$			354.897i			0.447537i
$794$		0			0
$795$	784.660			0.986994
$796$		0			0
$797$			20.2192i			0.0253692i	0.999920	+	0.0126846i	$0.00403774\pi$
$797$			20.2192i			0.0253692i	−0.999920	+	0.0126846i	$0.995962\pi$
$798$		0			0
$799$			987.086i			1.23540i
$800$		0			0
$801$			325.749i			0.406678i
$802$		0			0
$803$			93.1240i			0.115970i
$804$		0			0
$805$	54.8191	−	24.7096i	0.0680983	−	0.0306951i
$806$		0			0
$807$	322.117			0.399154
$808$		0			0
$809$	−1032.11			−1.27579			−0.637894	−	0.770124i	$-0.720194\pi$
$809$	−1032.11			−1.27579			−0.637894	+	0.770124i	$0.720194\pi$
$810$		0			0
$811$	70.2781			0.0866561			0.0433281	−	0.999061i	$-0.486204\pi$
$811$	70.2781			0.0866561			0.0433281	+	0.999061i	$0.486204\pi$
$812$		0			0
$813$	1522.27			1.87241
$814$		0			0
$815$			103.426i			0.126903i
$816$		0			0
$817$	−820.663			−1.00448
$818$		0			0
$819$			41.1062i			0.0501907i
$820$		0			0
$821$	84.7400			0.103216			0.0516078	−	0.998667i	$-0.483565\pi$
$821$	84.7400			0.103216			0.0516078	+	0.998667i	$0.483565\pi$
$822$		0			0
$823$	−46.0292			−0.0559286			−0.0279643	−	0.999609i	$-0.508902\pi$
$823$	−46.0292			−0.0559286			−0.0279643	+	0.999609i	$0.508902\pi$
$824$		0			0
$825$			178.645i			0.216539i
$826$		0			0
$827$		−	1283.77i		−	1.55232i	−0.630536	−	0.776160i	$-0.717165\pi$
$827$		−	1283.77i		−	1.55232i	0.630536	−	0.776160i	$-0.282835\pi$
$828$		0			0
$829$	−483.453			−0.583177			−0.291588	−	0.956544i	$-0.594184\pi$
$829$	−483.453			−0.583177			−0.291588	+	0.956544i	$0.594184\pi$
$830$		0			0
$831$	−294.449			−0.354330
$832$		0			0
$833$			956.883i			1.14872i
$834$		0			0
$835$			203.199i			0.243352i
$836$		0			0
$837$	−608.394			−0.726874
$838$		0			0
$839$			1060.74i			1.26429i	0.774852	+	0.632143i	$0.217824\pi$
$839$			1060.74i			1.26429i	−0.774852	+	0.632143i	$0.782176\pi$
$840$		0			0
$841$	221.712			0.263629
$842$		0			0
$843$		−	781.432i		−	0.926966i
$844$		0			0
$845$			131.365i			0.155462i
$846$		0			0
$847$		−	9.73498i		−	0.0114935i
$848$		0			0
$849$		−	805.700i		−	0.948999i
$850$		0			0
$851$	−501.463	−	1112.51i	−0.589263	−	1.30730i
$852$		0			0
$853$	1653.13			1.93802			0.969010	−	0.247023i	$-0.0794522\pi$
$853$	1653.13			1.93802			0.969010	+	0.247023i	$0.0794522\pi$
$854$		0			0
$855$	117.325			0.137222
$856$		0			0
$857$	1453.42			1.69593			0.847967	−	0.530048i	$-0.177826\pi$
$857$	1453.42			1.69593			0.847967	+	0.530048i	$0.177826\pi$
$858$		0			0
$859$	−936.399			−1.09010			−0.545052	−	0.838402i	$-0.683490\pi$
$859$	−936.399			−1.09010			−0.545052	+	0.838402i	$0.683490\pi$
$860$		0			0
$861$		−	37.1394i		−	0.0431352i
$862$		0			0
$863$	−961.830			−1.11452			−0.557259	−	0.830339i	$-0.688147\pi$
$863$	−961.830			−1.11452			−0.557259	+	0.830339i	$0.688147\pi$
$864$		0			0
$865$		−	203.011i		−	0.234695i
$866$		0			0
$867$	−385.585			−0.444735
$868$		0			0
$869$	−605.943			−0.697287
$870$		0			0
$871$			897.563i			1.03050i
$872$		0			0
$873$		−	46.2756i		−	0.0530075i
$874$		0			0
$875$	−13.0719			−0.0149393
$876$		0			0
$877$	−1244.55			−1.41910			−0.709549	−	0.704657i	$-0.751101\pi$
$877$	−1244.55			−1.41910			−0.709549	+	0.704657i	$0.751101\pi$
$878$		0			0
$879$		−	1958.41i		−	2.22800i
$880$		0			0
$881$		−	961.719i		−	1.09162i	−0.837908	−	0.545811i	$-0.816222\pi$
$881$		−	961.719i		−	1.09162i	0.837908	−	0.545811i	$-0.183778\pi$
$882$		0			0
$883$	−42.4945			−0.0481251			−0.0240626	−	0.999710i	$-0.507660\pi$
$883$	−42.4945			−0.0481251			−0.0240626	+	0.999710i	$0.507660\pi$
$884$		0			0
$885$		−	403.012i		−	0.455381i
$886$		0			0
$887$	551.962			0.622279			0.311140	−	0.950364i	$-0.399289\pi$
$887$	551.962			0.622279			0.311140	+	0.950364i	$0.399289\pi$
$888$		0			0
$889$			67.7774i			0.0762400i
$890$		0			0
$891$			1024.75i			1.15011i
$892$		0			0
$893$		−	1106.65i		−	1.23925i
$894$		0			0
$895$		−	674.478i		−	0.753606i
$896$		0			0
$897$	−1065.12	+	480.102i	−1.18743	+	0.535231i
$898$		0			0
$899$	883.359			0.982601
$900$		0			0
$901$	2094.30			2.32442
$902$		0			0
$903$	−143.400			−0.158804
$904$		0			0
$905$	−555.769			−0.614109
$906$		0			0
$907$		−	237.524i		−	0.261878i	−0.991390	−	0.130939i	$-0.958201\pi$
$907$		−	237.524i		−	0.261878i	0.991390	−	0.130939i	$-0.0417993\pi$
$908$		0			0
$909$	201.666			0.221855
$910$		0			0
$911$			402.885i			0.442245i	0.975246	+	0.221122i	$0.0709720\pi$
$911$			402.885i			0.442245i	−0.975246	+	0.221122i	$0.929028\pi$
$912$		0			0
$913$	585.712			0.641524
$914$		0			0
$915$	176.998			0.193440
$916$		0			0
$917$		−	111.167i		−	0.121229i
$918$		0			0
$919$			1338.53i			1.45650i	0.685310	+	0.728251i	$0.259667\pi$
$919$			1338.53i			1.45650i	−0.685310	+	0.728251i	$0.740333\pi$
$920$		0			0
$921$	−370.706			−0.402503
$922$		0			0
$923$	833.238			0.902750
$924$		0			0
$925$			265.284i			0.286794i
$926$		0			0
$927$		−	335.748i		−	0.362188i
$928$		0			0
$929$	621.771			0.669291			0.334646	−	0.942344i	$-0.391383\pi$
$929$	621.771			0.669291			0.334646	+	0.942344i	$0.391383\pi$
$930$		0			0
$931$		−	1072.79i		−	1.15230i
$932$		0			0
$933$	1003.39			1.07545
$934$		0			0
$935$			476.812i			0.509960i
$936$		0			0
$937$		−	915.580i		−	0.977140i	−0.872525	−	0.488570i	$-0.837519\pi$
$937$		−	915.580i		−	0.977140i	0.872525	−	0.488570i	$-0.162481\pi$
$938$		0			0
$939$		−	368.486i		−	0.392424i
$940$		0			0
$941$		−	834.255i		−	0.886563i	−0.896383	−	0.443281i	$-0.853814\pi$
$941$		−	834.255i		−	0.886563i	0.896383	−	0.443281i	$-0.146186\pi$
$942$		0			0
$943$	197.882	−	89.1950i	0.209843	−	0.0945864i
$944$		0			0
$945$	−58.6981			−0.0621144
$946$		0			0
$947$	−918.683			−0.970099			−0.485049	−	0.874487i	$-0.661198\pi$
$947$	−918.683			−0.970099			−0.485049	+	0.874487i	$0.661198\pi$
$948$		0			0
$949$	132.397			0.139512
$950$		0			0
$951$	1596.92			1.67921
$952$		0			0
$953$		−	789.341i		−	0.828270i	−0.910215	−	0.414135i	$-0.864084\pi$
$953$		−	789.341i		−	0.828270i	0.910215	−	0.414135i	$-0.135916\pi$
$954$		0			0
$955$	327.320			0.342743
$956$		0			0
$957$		−	1164.74i		−	1.21707i
$958$		0			0
$959$	−232.330			−0.242263
$960$		0			0
$961$	−226.725			−0.235927
$962$		0			0
$963$			24.3555i			0.0252913i
$964$		0			0
$965$		−	269.778i		−	0.279562i
$966$		0			0
$967$	−875.476			−0.905353			−0.452677	−	0.891675i	$-0.649531\pi$
$967$	−875.476			−0.905353			−0.452677	+	0.891675i	$0.649531\pi$
$968$		0			0
$969$	1522.89			1.57161
$970$		0			0
$971$		−	1232.49i		−	1.26930i	−0.772801	−	0.634648i	$-0.781145\pi$
$971$		−	1232.49i		−	1.26930i	0.772801	−	0.634648i	$-0.218855\pi$
$972$		0			0
$973$		−	0.151386i		−	0.000155587i
$974$		0			0
$975$	253.984			0.260497
$976$		0			0
$977$			879.183i			0.899880i	0.893059	+	0.449940i	$0.148555\pi$
$977$			879.183i			0.899880i	−0.893059	+	0.449940i	$0.851445\pi$
$978$		0			0
$979$	1484.22			1.51606
$980$		0			0
$981$		−	161.033i		−	0.164151i
$982$		0			0
$983$		−	247.090i		−	0.251363i	−0.992071	−	0.125682i	$-0.959888\pi$
$983$		−	247.090i		−	0.251363i	0.992071	−	0.125682i	$-0.0401118\pi$
$984$		0			0
$985$			70.7721i			0.0718498i
$986$		0			0
$987$		−	193.373i		−	0.195920i
$988$		0			0
$989$	−344.393	−	764.048i	−0.348223	−	0.772546i
$990$		0			0
$991$	1750.79			1.76669			0.883347	−	0.468721i	$-0.155285\pi$
$991$	1750.79			1.76669			0.883347	+	0.468721i	$0.155285\pi$
$992$		0			0
$993$	−859.556			−0.865615
$994$		0			0
$995$	83.7398			0.0841606
$996$		0			0
$997$	−456.156			−0.457529			−0.228765	−	0.973482i	$-0.573469\pi$
$997$	−456.156			−0.457529			−0.228765	+	0.973482i	$0.573469\pi$
$998$		0			0
$999$			1191.23i			1.19243i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.12		16
4.3	odd	2		230.3.d.a.91.12	yes	16
12.11	even	2		2070.3.c.a.91.3		16
20.3	even	4		1150.3.c.c.1149.15		32
20.7	even	4		1150.3.c.c.1149.18		32
20.19	odd	2		1150.3.d.b.551.5		16
23.22	odd	2	inner	1840.3.k.d.321.11		16
92.91	even	2		230.3.d.a.91.11	✓	16
276.275	odd	2		2070.3.c.a.91.6		16
460.183	odd	4		1150.3.c.c.1149.17		32
460.367	odd	4		1150.3.c.c.1149.16		32
460.459	even	2		1150.3.d.b.551.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.11	✓	16	92.91	even	2
230.3.d.a.91.12	yes	16	4.3	odd	2
1150.3.c.c.1149.15		32	20.3	even	4
1150.3.c.c.1149.16		32	460.367	odd	4
1150.3.c.c.1149.17		32	460.183	odd	4
1150.3.c.c.1149.18		32	20.7	even	4
1150.3.d.b.551.5		16	20.19	odd	2
1150.3.d.b.551.6		16	460.459	even	2
1840.3.k.d.321.11		16	23.22	odd	2	inner
1840.3.k.d.321.12		16	1.1	even	1	trivial
2070.3.c.a.91.3		16	12.11	even	2
2070.3.c.a.91.6		16	276.275	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.36596 q^{3} +2.23607i q^{5} -1.16919i q^{7} +2.32968 q^{9} +O(q^{10})\)	\(q+3.36596 q^{3} +2.23607i q^{5} -1.16919i q^{7} +2.32968 q^{9} -10.6148i q^{11} -15.0913 q^{13} +7.52651i q^{15} +20.0887i q^{17} -22.5221i q^{19} -3.93543i q^{21} +(20.9683 - 9.45142i) q^{23} -5.00000 q^{25} -22.4520 q^{27} +32.5993 q^{29} +27.0975 q^{31} -35.7289i q^{33} +2.61438 q^{35} -53.0568i q^{37} -50.7968 q^{39} +9.43720 q^{41} -36.4382i q^{43} +5.20933i q^{45} +49.1365 q^{47} +47.6330 q^{49} +67.6176i q^{51} -104.253i q^{53} +23.7354 q^{55} -75.8083i q^{57} -53.5457 q^{59} -23.5166i q^{61} -2.72383i q^{63} -33.7453i q^{65} -59.4754i q^{67} +(70.5785 - 31.8131i) q^{69} -55.2130 q^{71} -8.77305 q^{73} -16.8298 q^{75} -12.4107 q^{77} -57.0848i q^{79} -96.5397 q^{81} +55.1788i q^{83} -44.9196 q^{85} +109.728 q^{87} +139.825i q^{89} +17.6446i q^{91} +91.2091 q^{93} +50.3608 q^{95} -19.8635i q^{97} -24.7291i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 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

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