LMFDB - Embedded newform 1840.2.m.g.1839.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1839.4
Character	$\chi$	$=$	1840.1839
Dual form			1840.2.m.g.1839.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} +O(q^{10})$	$q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} -2.28639 q^{11} +4.92586i q^{13} +(-5.06519 + 5.32446i) q^{15} -5.97341 q^{17} +0.377877 q^{19} +8.97791i q^{21} +(-0.582385 + 4.76034i) q^{23} +(-0.249397 - 4.99378i) q^{25} -15.7793 q^{27} +3.70400 q^{29} -2.89324i q^{31} +7.51426 q^{33} +(-4.42567 - 4.21016i) q^{35} -7.23320 q^{37} -16.1889i q^{39} +0.451624 q^{41} +0.359476i q^{43} +(12.0232 - 12.6387i) q^{45} +12.1733 q^{47} -0.462395 q^{49} +19.6317 q^{51} -9.69298 q^{53} +(-3.52378 + 3.70416i) q^{55} -1.24190 q^{57} +10.7427i q^{59} +6.73249i q^{61} -21.3109i q^{63} +(7.98034 + 7.59174i) q^{65} -7.06480i q^{67} +(1.91402 - 15.6449i) q^{69} +7.34667i q^{71} -1.24543i q^{73} +(0.819647 + 16.4121i) q^{75} +6.24581i q^{77} -13.6417 q^{79} +28.4553 q^{81} +15.1841i q^{83} +(-9.20623 + 9.67748i) q^{85} -12.1733 q^{87} -12.1526i q^{89} +13.4562 q^{91} +9.50868i q^{93} +(0.582385 - 0.612196i) q^{95} +4.34220 q^{97} -17.8366 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 80 q^{9}+O(q^{10}) $ 40 * q + 80 * q^9	$ 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) $ 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

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
$3$	−3.28652			−1.89747			−0.948737	−	0.316068i	$-0.897637\pi$
$3$	−3.28652			−1.89747			−0.948737	+	0.316068i	$0.897637\pi$
$4$		0			0
$5$	1.54120	−	1.62009i	0.689246	−	0.724527i
$5$	1.54120	−	1.62009i	0.689246	−	0.724527i
$6$		0			0
$7$		−	2.73174i		−	1.03250i	−0.856438	−	0.516250i	$-0.827328\pi$
$7$		−	2.73174i		−	1.03250i	0.856438	−	0.516250i	$-0.172672\pi$
$8$		0			0
$9$	7.80121			2.60040
$10$		0			0
$11$	−2.28639			−0.689372			−0.344686	−	0.938718i	$-0.612015\pi$
$11$	−2.28639			−0.689372			−0.344686	+	0.938718i	$0.612015\pi$
$12$		0			0
$13$			4.92586i			1.36619i	0.730331	+	0.683094i	$0.239366\pi$
$13$			4.92586i			1.36619i	−0.730331	+	0.683094i	$0.760634\pi$
$14$		0			0
$15$	−5.06519	+	5.32446i	−1.30783	+	1.37477i
$16$		0			0
$17$	−5.97341			−1.44876			−0.724382	−	0.689398i	$-0.757875\pi$
$17$	−5.97341			−1.44876			−0.724382	+	0.689398i	$0.757875\pi$
$18$		0			0
$19$	0.377877			0.0866910			0.0433455	−	0.999060i	$-0.486198\pi$
$19$	0.377877			0.0866910			0.0433455	+	0.999060i	$0.486198\pi$
$20$		0			0
$21$			8.97791i			1.95914i
$22$		0			0
$23$	−0.582385	+	4.76034i	−0.121436	+	0.992599i
$23$	−0.582385	+	4.76034i	−0.121436	+	0.992599i
$24$		0			0
$25$	−0.249397	−	4.99378i	−0.0498794	−	0.998755i
$26$		0			0
$27$	−15.7793			−3.03672
$28$		0			0
$29$	3.70400			0.687815			0.343907	−	0.939004i	$-0.388249\pi$
$29$	3.70400			0.687815			0.343907	+	0.939004i	$0.388249\pi$
$30$		0			0
$31$		−	2.89324i		−	0.519641i	−0.965657	−	0.259820i	$-0.916337\pi$
$31$		−	2.89324i		−	0.519641i	0.965657	−	0.259820i	$-0.0836634\pi$
$32$		0			0
$33$	7.51426			1.30806
$34$		0			0
$35$	−4.42567	−	4.21016i	−0.748074	−	0.711647i
$36$		0			0
$37$	−7.23320			−1.18913			−0.594566	−	0.804047i	$-0.702676\pi$
$37$	−7.23320			−1.18913			−0.594566	+	0.804047i	$0.702676\pi$
$38$		0			0
$39$		−	16.1889i		−	2.59230i
$40$		0			0
$41$	0.451624			0.0705318			0.0352659	−	0.999378i	$-0.488772\pi$
$41$	0.451624			0.0705318			0.0352659	+	0.999378i	$0.488772\pi$
$42$		0			0
$43$			0.359476i			0.0548196i	0.999624	+	0.0274098i	$0.00872591\pi$
$43$			0.359476i			0.0548196i	−0.999624	+	0.0274098i	$0.991274\pi$
$44$		0			0
$45$	12.0232	−	12.6387i	1.79232	−	1.88406i
$46$		0			0
$47$	12.1733			1.77565			0.887826	−	0.460179i	$-0.152215\pi$
$47$	12.1733			1.77565			0.887826	+	0.460179i	$0.152215\pi$
$48$		0			0
$49$	−0.462395			−0.0660565
$50$		0			0
$51$	19.6317			2.74899
$52$		0			0
$53$	−9.69298			−1.33143			−0.665716	−	0.746205i	$-0.731874\pi$
$53$	−9.69298			−1.33143			−0.665716	+	0.746205i	$0.731874\pi$
$54$		0			0
$55$	−3.52378	+	3.70416i	−0.475147	+	0.499468i
$56$		0			0
$57$	−1.24190			−0.164494
$58$		0			0
$59$			10.7427i			1.39858i	0.714838	+	0.699290i	$0.246500\pi$
$59$			10.7427i			1.39858i	−0.714838	+	0.699290i	$0.753500\pi$
$60$		0			0
$61$			6.73249i			0.862007i	0.902350	+	0.431003i	$0.141840\pi$
$61$			6.73249i			0.862007i	−0.902350	+	0.431003i	$0.858160\pi$
$62$		0			0
$63$		−	21.3109i		−	2.68492i
$64$		0			0
$65$	7.98034	+	7.59174i	0.989840	+	0.941639i
$66$		0			0
$67$		−	7.06480i		−	0.863103i	−0.902088	−	0.431551i	$-0.857966\pi$
$67$		−	7.06480i		−	0.863103i	0.902088	−	0.431551i	$-0.142034\pi$
$68$		0			0
$69$	1.91402	−	15.6449i	0.230421	−	1.88343i
$70$		0			0
$71$			7.34667i			0.871889i	0.899974	+	0.435945i	$0.143586\pi$
$71$			7.34667i			0.871889i	−0.899974	+	0.435945i	$0.856414\pi$
$72$		0			0
$73$		−	1.24543i		−	0.145767i	−0.997340	−	0.0728834i	$-0.976780\pi$
$73$		−	1.24543i		−	0.145767i	0.997340	−	0.0728834i	$-0.0232201\pi$
$74$		0			0
$75$	0.819647	+	16.4121i	0.0946447	+	1.89511i
$76$		0			0
$77$			6.24581i			0.711776i
$78$		0			0
$79$	−13.6417			−1.53481			−0.767405	−	0.641163i	$-0.778452\pi$
$79$	−13.6417			−1.53481			−0.767405	+	0.641163i	$0.778452\pi$
$80$		0			0
$81$	28.4553			3.16170
$82$		0			0
$83$			15.1841i			1.66667i	0.552768	+	0.833335i	$0.313571\pi$
$83$			15.1841i			1.66667i	−0.552768	+	0.833335i	$0.686429\pi$
$84$		0			0
$85$	−9.20623	+	9.67748i	−0.998556	+	1.04967i
$86$		0			0
$87$	−12.1733			−1.30511
$88$		0			0
$89$		−	12.1526i		−	1.28818i	−0.764951	−	0.644089i	$-0.777237\pi$
$89$		−	12.1526i		−	1.28818i	0.764951	−	0.644089i	$-0.222763\pi$
$90$		0			0
$91$	13.4562			1.41059
$92$		0			0
$93$			9.50868i			0.986004i
$94$		0			0
$95$	0.582385	−	0.612196i	0.0597514	−	0.0628100i
$96$		0			0
$97$	4.34220			0.440883			0.220442	−	0.975400i	$-0.429250\pi$
$97$	4.34220			0.440883			0.220442	+	0.975400i	$0.429250\pi$
$98$		0			0
$99$	−17.8366			−1.79264
$100$		0			0
$101$	3.11922			0.310374			0.155187	−	0.987885i	$-0.450402\pi$
$101$	3.11922			0.310374			0.155187	+	0.987885i	$0.450402\pi$
$102$		0			0
$103$			3.38893i			0.333921i	0.985964	+	0.166960i	$0.0533952\pi$
$103$			3.38893i			0.333921i	−0.985964	+	0.166960i	$0.946605\pi$
$104$		0			0
$105$	14.5450	+	13.8368i	1.41945	+	1.35033i
$106$		0			0
$107$			7.94818i			0.768380i	0.923254	+	0.384190i	$0.125519\pi$
$107$			7.94818i			0.768380i	−0.923254	+	0.384190i	$0.874481\pi$
$108$		0			0
$109$			14.1195i			1.35240i	0.736718	+	0.676200i	$0.236375\pi$
$109$			14.1195i			1.35240i	−0.736718	+	0.676200i	$0.763625\pi$
$110$		0			0
$111$	23.7721			2.25634
$112$		0			0
$113$	0.322974			0.0303828			0.0151914	−	0.999885i	$-0.495164\pi$
$113$	0.322974			0.0303828			0.0151914	+	0.999885i	$0.495164\pi$
$114$		0			0
$115$	6.81462	+	8.28016i	0.635466	+	0.772129i
$116$		0			0
$117$			38.4277i			3.55264i
$118$		0			0
$119$			16.3178i			1.49585i
$120$		0			0
$121$	−5.77244			−0.524767
$122$		0			0
$123$	−1.48427			−0.133832
$124$		0			0
$125$	−8.47475	−	7.29237i	−0.758005	−	0.652249i
$126$		0			0
$127$	10.5340			0.934738			0.467369	−	0.884062i	$-0.345202\pi$
$127$	10.5340			0.934738			0.467369	+	0.884062i	$0.345202\pi$
$128$		0			0
$129$		−	1.18143i		−	0.104019i
$130$		0			0
$131$			11.3581i			0.992362i	0.868219	+	0.496181i	$0.165265\pi$
$131$			11.3581i			0.992362i	−0.868219	+	0.496181i	$0.834735\pi$
$132$		0			0
$133$		−	1.03226i		−	0.0895084i
$134$		0			0
$135$	−24.3190	+	25.5639i	−2.09305	+	2.20019i
$136$		0			0
$137$	19.0252			1.62543			0.812716	−	0.582660i	$-0.197988\pi$
$137$	19.0252			1.62543			0.812716	+	0.582660i	$0.197988\pi$
$138$		0			0
$139$			9.12171i			0.773693i	0.922144	+	0.386847i	$0.126436\pi$
$139$			9.12171i			0.773693i	−0.922144	+	0.386847i	$0.873564\pi$
$140$		0			0
$141$	−40.0076			−3.36925
$142$		0			0
$143$		−	11.2624i		−	0.941810i
$144$		0			0
$145$	5.70860	−	6.00081i	0.474074	−	0.498341i
$146$		0			0
$147$	1.51967			0.125340
$148$		0			0
$149$		−	5.95344i		−	0.487725i	−0.969810	−	0.243863i	$-0.921585\pi$
$149$		−	5.95344i		−	0.487725i	0.969810	−	0.243863i	$-0.0784146\pi$
$150$		0			0
$151$			9.94683i			0.809462i	0.914436	+	0.404731i	$0.132635\pi$
$151$			9.94683i			0.809462i	−0.914436	+	0.404731i	$0.867365\pi$
$152$		0			0
$153$	−46.5998			−3.76737
$154$		0			0
$155$	−4.68731	−	4.45906i	−0.376494	−	0.358160i
$156$		0			0
$157$	16.2147			1.29407			0.647036	−	0.762460i	$-0.276008\pi$
$157$	16.2147			1.29407			0.647036	+	0.762460i	$0.276008\pi$
$158$		0			0
$159$	31.8562			2.52636
$160$		0			0
$161$	13.0040	+	1.59092i	1.02486	+	0.125382i
$162$		0			0
$163$	14.1754			1.11030			0.555151	−	0.831750i	$-0.312661\pi$
$163$	14.1754			1.11030			0.555151	+	0.831750i	$0.312661\pi$
$164$		0			0
$165$	11.5810	−	12.1738i	0.901578	−	0.947728i
$166$		0			0
$167$	8.85134			0.684937			0.342468	−	0.939529i	$-0.388737\pi$
$167$	8.85134			0.684937			0.342468	+	0.939529i	$0.388737\pi$
$168$		0			0
$169$	−11.2641			−0.866467
$170$		0			0
$171$	2.94790			0.225432
$172$		0			0
$173$			24.2863i			1.84645i	0.384260	+	0.923225i	$0.374457\pi$
$173$			24.2863i			1.84645i	−0.384260	+	0.923225i	$0.625543\pi$
$174$		0			0
$175$	−13.6417	+	0.681287i	−1.03121	+	0.0515004i
$176$		0			0
$177$		−	35.3061i		−	2.65377i
$178$		0			0
$179$		−	14.8358i		−	1.10888i	−0.832223	−	0.554441i	$-0.812932\pi$
$179$		−	14.8358i		−	1.10888i	0.832223	−	0.554441i	$-0.187068\pi$
$180$		0			0
$181$			3.42685i			0.254716i	0.991857	+	0.127358i	$0.0406497\pi$
$181$			3.42685i			0.254716i	−0.991857	+	0.127358i	$0.959350\pi$
$182$		0			0
$183$		−	22.1265i		−	1.63563i
$184$		0			0
$185$	−11.1478	+	11.7185i	−0.819604	+	0.861558i
$186$		0			0
$187$	13.6575			0.998737
$188$		0			0
$189$			43.1049i			3.13542i
$190$		0			0
$191$	9.37462			0.678324			0.339162	−	0.940728i	$-0.389857\pi$
$191$	9.37462			0.678324			0.339162	+	0.940728i	$0.389857\pi$
$192$		0			0
$193$			1.72975i			0.124510i	0.998060	+	0.0622550i	$0.0198292\pi$
$193$			1.72975i			0.124510i	−0.998060	+	0.0622550i	$0.980171\pi$
$194$		0			0
$195$	−26.2276	−	24.9504i	−1.87819	−	1.78673i
$196$		0			0
$197$		−	15.8158i		−	1.12683i	−0.826174	−	0.563415i	$-0.809487\pi$
$197$		−	15.8158i		−	1.12683i	0.826174	−	0.563415i	$-0.190513\pi$
$198$		0			0
$199$	−19.9550			−1.41458			−0.707288	−	0.706926i	$-0.750082\pi$
$199$	−19.9550			−1.41458			−0.707288	+	0.706926i	$0.750082\pi$
$200$		0			0
$201$			23.2186i			1.63771i
$202$		0			0
$203$		−	10.1183i		−	0.710169i
$204$		0			0
$205$	0.696043	−	0.731672i	0.0486138	−	0.0511022i
$206$		0			0
$207$	−4.54331	+	37.1364i	−0.315782	+	2.58116i
$208$		0			0
$209$	−0.863973			−0.0597623
$210$		0			0
$211$		−	19.0013i		−	1.30810i	−0.756449	−	0.654052i	$-0.773068\pi$
$211$		−	19.0013i		−	1.30810i	0.756449	−	0.654052i	$-0.226932\pi$
$212$		0			0
$213$		−	24.1450i		−	1.65439i
$214$		0			0
$215$	0.582385	+	0.554025i	0.0397183	+	0.0377842i
$216$		0			0
$217$	−7.90357			−0.536529
$218$		0			0
$219$			4.09314i			0.276588i
$220$		0			0
$221$		−	29.4242i		−	1.97928i
$222$		0			0
$223$	−1.87457			−0.125531			−0.0627653	−	0.998028i	$-0.519992\pi$
$223$	−1.87457			−0.125531			−0.0627653	+	0.998028i	$0.519992\pi$
$224$		0			0
$225$	−1.94560	−	38.9575i	−0.129706	−	2.59717i
$226$		0			0
$227$		−	5.18877i		−	0.344391i	−0.985063	−	0.172195i	$-0.944914\pi$
$227$		−	5.18877i		−	0.344391i	0.985063	−	0.172195i	$-0.0550860\pi$
$228$		0			0
$229$			15.8023i			1.04425i	0.852870	+	0.522124i	$0.174860\pi$
$229$			15.8023i			1.04425i	−0.852870	+	0.522124i	$0.825140\pi$
$230$		0			0
$231$		−	20.5270i		−	1.35058i
$232$		0			0
$233$			14.6477i			0.959603i	0.877377	+	0.479802i	$0.159291\pi$
$233$			14.6477i			0.959603i	−0.877377	+	0.479802i	$0.840709\pi$
$234$		0			0
$235$	18.7614	−	19.7218i	1.22386	−	1.28651i
$236$		0			0
$237$	44.8337			2.91226
$238$		0			0
$239$			9.60182i			0.621090i	0.950559	+	0.310545i	$0.100512\pi$
$239$			9.60182i			0.621090i	−0.950559	+	0.310545i	$0.899488\pi$
$240$		0			0
$241$			18.8553i			1.21458i	0.794481	+	0.607289i	$0.207743\pi$
$241$			18.8553i			1.21458i	−0.794481	+	0.607289i	$0.792257\pi$
$242$		0			0
$243$	−46.1810			−2.96251
$244$		0			0
$245$	−0.712644	+	0.749123i	−0.0455292	+	0.0478597i
$246$		0			0
$247$			1.86137i			0.118436i
$248$		0			0
$249$		−	49.9028i		−	3.16246i
$250$		0			0
$251$	24.9057			1.57203			0.786017	−	0.618205i	$-0.212140\pi$
$251$	24.9057			1.57203			0.786017	+	0.618205i	$0.212140\pi$
$252$		0			0
$253$	1.33156	−	10.8840i	0.0837143	−	0.684270i
$254$		0			0
$255$	30.2565	−	31.8052i	1.89473	−	1.99172i
$256$		0			0
$257$			16.5764i			1.03401i	0.855983	+	0.517004i	$0.172953\pi$
$257$			16.5764i			1.03401i	−0.855983	+	0.517004i	$0.827047\pi$
$258$		0			0
$259$			19.7592i			1.22778i
$260$		0			0
$261$	28.8957			1.78860
$262$		0			0
$263$		−	17.7754i		−	1.09608i	−0.836453	−	0.548039i	$-0.815374\pi$
$263$		−	17.7754i		−	1.09608i	0.836453	−	0.548039i	$-0.184626\pi$
$264$		0			0
$265$	−14.9388	+	15.7035i	−0.917685	+	0.964659i
$266$		0			0
$267$			39.9399i			2.44428i
$268$		0			0
$269$	−18.2884			−1.11506			−0.557532	−	0.830155i	$-0.688252\pi$
$269$	−18.2884			−1.11506			−0.557532	+	0.830155i	$0.688252\pi$
$270$		0			0
$271$			8.44481i			0.512986i	0.966546	+	0.256493i	$0.0825670\pi$
$271$			8.44481i			0.512986i	−0.966546	+	0.256493i	$0.917433\pi$
$272$		0			0
$273$	−44.2239			−2.67655
$274$		0			0
$275$	0.570218	+	11.4177i	0.0343854	+	0.688513i
$276$		0			0
$277$		−	27.9612i		−	1.68003i	−0.542565	−	0.840014i	$-0.682547\pi$
$277$		−	27.9612i		−	1.68003i	0.542565	−	0.840014i	$-0.317453\pi$
$278$		0			0
$279$		−	22.5708i		−	1.35128i
$280$		0			0
$281$			28.3900i			1.69360i	0.531909	+	0.846802i	$0.321475\pi$
$281$			28.3900i			1.69360i	−0.531909	+	0.846802i	$0.678525\pi$
$282$		0			0
$283$			9.11525i			0.541845i	0.962601	+	0.270923i	$0.0873288\pi$
$283$			9.11525i			0.541845i	−0.962601	+	0.270923i	$0.912671\pi$
$284$		0			0
$285$	−1.91402	+	2.01199i	−0.113377	+	0.119180i
$286$		0			0
$287$		−	1.23372i		−	0.0728241i
$288$		0			0
$289$	18.6816			1.09892
$290$		0			0
$291$	−14.2707			−0.836564
$292$		0			0
$293$	−5.14021			−0.300294			−0.150147	−	0.988664i	$-0.547975\pi$
$293$	−5.14021			−0.300294			−0.150147	+	0.988664i	$0.547975\pi$
$294$		0			0
$295$	17.4042	+	16.5567i	1.01331	+	0.963966i
$296$		0			0
$297$	36.0775			2.09343
$298$		0			0
$299$	−23.4488	−	2.86874i	−1.35608	−	0.165904i
$300$		0			0
$301$	0.981995			0.0566013
$302$		0			0
$303$	−10.2514			−0.588927
$304$		0			0
$305$	10.9072	+	10.3761i	0.624547	+	0.594135i
$306$		0			0
$307$	−16.0988			−0.918806			−0.459403	−	0.888228i	$-0.651937\pi$
$307$	−16.0988			−0.918806			−0.459403	+	0.888228i	$0.651937\pi$
$308$		0			0
$309$		−	11.1378i		−	0.633606i
$310$		0			0
$311$			13.6114i			0.771834i	0.922534	+	0.385917i	$0.126115\pi$
$311$			13.6114i			0.771834i	−0.922534	+	0.385917i	$0.873885\pi$
$312$		0			0
$313$	16.7024			0.944073			0.472037	−	0.881579i	$-0.343519\pi$
$313$	16.7024			0.944073			0.472037	+	0.881579i	$0.343519\pi$
$314$		0			0
$315$	−34.5256	−	32.8443i	−1.94530	−	1.85057i
$316$		0			0
$317$			24.1290i			1.35522i	0.735423	+	0.677609i	$0.236984\pi$
$317$			24.1290i			1.35522i	−0.735423	+	0.677609i	$0.763016\pi$
$318$		0			0
$319$	−8.46877			−0.474160
$320$		0			0
$321$		−	26.1219i		−	1.45798i
$322$		0			0
$323$	−2.25722			−0.125595
$324$		0			0
$325$	24.5986	−	1.22849i	1.36449	−	0.0681445i
$326$		0			0
$327$		−	46.4039i		−	2.56614i
$328$		0			0
$329$		−	33.2542i		−	1.83336i
$330$		0			0
$331$		−	9.49579i		−	0.521936i	−0.965348	−	0.260968i	$-0.915958\pi$
$331$		−	9.49579i		−	0.521936i	0.965348	−	0.260968i	$-0.0840416\pi$
$332$		0			0
$333$	−56.4278			−3.09222
$334$		0			0
$335$	−11.4456	−	10.8883i	−0.625341	−	0.594890i
$336$		0			0
$337$	−9.95485			−0.542275			−0.271138	−	0.962541i	$-0.587400\pi$
$337$	−9.95485			−0.542275			−0.271138	+	0.962541i	$0.587400\pi$
$338$		0			0
$339$	−1.06146			−0.0576506
$340$		0			0
$341$			6.61506i			0.358226i
$342$		0			0
$343$		−	17.8590i		−	0.964297i
$344$		0			0
$345$	−22.3964	−	27.2129i	−1.20578	−	1.46509i
$346$		0			0
$347$	−28.5546			−1.53289			−0.766446	−	0.642309i	$-0.777977\pi$
$347$	−28.5546			−1.53289			−0.766446	+	0.642309i	$0.777977\pi$
$348$		0			0
$349$	11.7060			0.626610			0.313305	−	0.949652i	$-0.398564\pi$
$349$	11.7060			0.626610			0.313305	+	0.949652i	$0.398564\pi$
$350$		0			0
$351$		−	77.7265i		−	4.14873i
$352$		0			0
$353$		−	10.6128i		−	0.564864i	−0.959287	−	0.282432i	$-0.908859\pi$
$353$		−	10.6128i		−	0.564864i	0.959287	−	0.282432i	$-0.0911411\pi$
$354$		0			0
$355$	11.9023	+	11.3227i	0.631707	+	0.600946i
$356$		0			0
$357$		−	53.6288i		−	2.83833i
$358$		0			0
$359$	16.7030			0.881548			0.440774	−	0.897618i	$-0.354704\pi$
$359$	16.7030			0.881548			0.440774	+	0.897618i	$0.354704\pi$
$360$		0			0
$361$	−18.8572			−0.992485
$362$		0			0
$363$	18.9712			0.995731
$364$		0			0
$365$	−2.01771	−	1.91946i	−0.105612	−	0.100469i
$366$		0			0
$367$		−	1.96080i		−	0.102353i	−0.998690	−	0.0511764i	$-0.983703\pi$
$367$		−	1.96080i		−	0.102353i	0.998690	−	0.0511764i	$-0.0162971\pi$
$368$		0			0
$369$	3.52321			0.183411
$370$		0			0
$371$			26.4787i			1.37470i
$372$		0			0
$373$	−31.3228			−1.62183			−0.810916	−	0.585163i	$-0.801031\pi$
$373$	−31.3228			−1.62183			−0.810916	+	0.585163i	$0.801031\pi$
$374$		0			0
$375$	27.8524	+	23.9665i	1.43829	+	1.23763i
$376$		0			0
$377$			18.2454i			0.939684i
$378$		0			0
$379$	−0.853853			−0.0438595			−0.0219297	−	0.999760i	$-0.506981\pi$
$379$	−0.853853			−0.0438595			−0.0219297	+	0.999760i	$0.506981\pi$
$380$		0			0
$381$	−34.6201			−1.77364
$382$		0			0
$383$		−	2.33128i		−	0.119123i	−0.998225	−	0.0595614i	$-0.981030\pi$
$383$		−	2.33128i		−	0.119123i	0.998225	−	0.0595614i	$-0.0189702\pi$
$384$		0			0
$385$	10.1188	+	9.62605i	0.515701	+	0.490589i
$386$		0			0
$387$			2.80435i			0.142553i
$388$		0			0
$389$			10.4152i			0.528074i	0.964513	+	0.264037i	$0.0850541\pi$
$389$			10.4152i			0.528074i	−0.964513	+	0.264037i	$0.914946\pi$
$390$		0			0
$391$	3.47882	−	28.4355i	0.175932	−	1.43804i
$392$		0			0
$393$		−	37.3286i		−	1.88298i
$394$		0			0
$395$	−21.0246	+	22.1008i	−1.05786	+	1.11201i
$396$		0			0
$397$		−	9.63854i		−	0.483745i	−0.970308	−	0.241872i	$-0.922238\pi$
$397$		−	9.63854i		−	0.483745i	0.970308	−	0.241872i	$-0.0777615\pi$
$398$		0			0
$399$			3.39255i			0.169840i
$400$		0			0
$401$			18.4820i			0.922947i	0.887154	+	0.461473i	$0.152679\pi$
$401$			18.4820i			0.922947i	−0.887154	+	0.461473i	$0.847321\pi$
$402$		0			0
$403$	14.2517			0.709927
$404$		0			0
$405$	43.8553	−	46.1002i	2.17919	−	2.29074i
$406$		0			0
$407$	16.5379			0.819753
$408$		0			0
$409$	−26.0001			−1.28562			−0.642811	−	0.766025i	$-0.722232\pi$
$409$	−26.0001			−1.28562			−0.642811	+	0.766025i	$0.722232\pi$
$410$		0			0
$411$	−62.5267			−3.08421
$412$		0			0
$413$	29.3462			1.44403
$414$		0			0
$415$	24.5996	+	23.4017i	1.20755	+	1.14875i
$416$		0			0
$417$		−	29.9787i		−	1.46806i
$418$		0			0
$419$	−16.0811			−0.785614			−0.392807	−	0.919621i	$-0.628496\pi$
$419$	−16.0811			−0.785614			−0.392807	+	0.919621i	$0.628496\pi$
$420$		0			0
$421$		−	23.5031i		−	1.14547i	−0.819740	−	0.572735i	$-0.805882\pi$
$421$		−	23.5031i		−	1.14547i	0.819740	−	0.572735i	$-0.194118\pi$
$422$		0			0
$423$	94.9662			4.61741
$424$		0			0
$425$	1.48975	+	29.8299i	0.0722635	+	1.44696i
$426$		0			0
$427$	18.3914			0.890022
$428$		0			0
$429$			37.0141i			1.78706i
$430$		0			0
$431$	−6.61906			−0.318829			−0.159415	−	0.987212i	$-0.550961\pi$
$431$	−6.61906			−0.318829			−0.159415	+	0.987212i	$0.550961\pi$
$432$		0			0
$433$	20.3334			0.977163			0.488581	−	0.872518i	$-0.337515\pi$
$433$	20.3334			0.977163			0.488581	+	0.872518i	$0.337515\pi$
$434$		0			0
$435$	−18.7614	+	19.7218i	−0.899542	+	0.945588i
$436$		0			0
$437$	−0.220070	+	1.79882i	−0.0105274	+	0.0860494i
$438$		0			0
$439$		−	15.1971i		−	0.725317i	−0.931922	−	0.362659i	$-0.881869\pi$
$439$		−	15.1971i		−	0.725317i	0.931922	−	0.362659i	$-0.118131\pi$
$440$		0			0
$441$	−3.60724			−0.171774
$442$		0			0
$443$	−8.20593			−0.389875			−0.194938	−	0.980816i	$-0.562450\pi$
$443$	−8.20593			−0.389875			−0.194938	+	0.980816i	$0.562450\pi$
$444$		0			0
$445$	−19.6884	−	18.7297i	−0.933320	−	0.887872i
$446$		0			0
$447$			19.5661i			0.925445i
$448$		0			0
$449$	−23.0265			−1.08669			−0.543343	−	0.839511i	$-0.682842\pi$
$449$	−23.0265			−1.08669			−0.543343	+	0.839511i	$0.682842\pi$
$450$		0			0
$451$	−1.03259			−0.0486226
$452$		0			0
$453$		−	32.6905i		−	1.53593i
$454$		0			0
$455$	20.7386	−	21.8002i	0.972243	−	1.02201i
$456$		0			0
$457$	−38.0523			−1.78001			−0.890005	−	0.455951i	$-0.849299\pi$
$457$	−38.0523			−1.78001			−0.890005	+	0.455951i	$0.849299\pi$
$458$		0			0
$459$	94.2561			4.39950
$460$		0			0
$461$	28.7920			1.34098			0.670489	−	0.741919i	$-0.266084\pi$
$461$	28.7920			1.34098			0.670489	+	0.741919i	$0.266084\pi$
$462$		0			0
$463$	1.34881			0.0626843			0.0313421	−	0.999509i	$-0.490022\pi$
$463$	1.34881			0.0626843			0.0313421	+	0.999509i	$0.490022\pi$
$464$		0			0
$465$	15.4049	+	14.6548i	0.714387	+	0.679600i
$466$		0			0
$467$		−	8.23187i		−	0.380926i	−0.981694	−	0.190463i	$-0.939001\pi$
$467$		−	8.23187i		−	0.380926i	0.981694	−	0.190463i	$-0.0609988\pi$
$468$		0			0
$469$	−19.2992			−0.891154
$470$		0			0
$471$	−53.2898			−2.45547
$472$		0			0
$473$		−	0.821902i		−	0.0377911i
$474$		0			0
$475$	−0.0942413	−	1.88703i	−0.00432409	−	0.0865831i
$476$		0			0
$477$	−75.6170			−3.46226
$478$		0			0
$479$	−2.78805			−0.127389			−0.0636946	−	0.997969i	$-0.520288\pi$
$479$	−2.78805			−0.127389			−0.0636946	+	0.997969i	$0.520288\pi$
$480$		0			0
$481$		−	35.6297i		−	1.62458i
$482$		0			0
$483$	−42.7379	−	5.22860i	−1.94464	−	0.237910i
$484$		0			0
$485$	6.69220	−	7.03476i	0.303877	−	0.319432i
$486$		0			0
$487$	−1.35672			−0.0614787			−0.0307394	−	0.999527i	$-0.509786\pi$
$487$	−1.35672			−0.0614787			−0.0307394	+	0.999527i	$0.509786\pi$
$488$		0			0
$489$	−46.5877			−2.10677
$490$		0			0
$491$			0.539939i			0.0243671i	0.999926	+	0.0121835i	$0.00387824\pi$
$491$			0.539939i			0.0243671i	−0.999926	+	0.0121835i	$0.996122\pi$
$492$		0			0
$493$	−22.1255			−0.996482
$494$		0			0
$495$	−27.4898	+	28.8969i	−1.23557	+	1.29882i
$496$		0			0
$497$	20.0692			0.900226
$498$		0			0
$499$			23.4902i			1.05157i	0.850619	+	0.525783i	$0.176228\pi$
$499$			23.4902i			1.05157i	−0.850619	+	0.525783i	$0.823772\pi$
$500$		0			0
$501$	−29.0901			−1.29965
$502$		0			0
$503$			17.7835i			0.792929i	0.918050	+	0.396464i	$0.129763\pi$
$503$			17.7835i			0.792929i	−0.918050	+	0.396464i	$0.870237\pi$
$504$		0			0
$505$	4.80735	−	5.05343i	0.213924	−	0.224875i
$506$		0			0
$507$	37.0196			1.64410
$508$		0			0
$509$	29.3504			1.30094			0.650468	−	0.759534i	$-0.274573\pi$
$509$	29.3504			1.30094			0.650468	+	0.759534i	$0.274573\pi$
$510$		0			0
$511$	−3.40219			−0.150504
$512$		0			0
$513$	−5.96263			−0.263256
$514$		0			0
$515$	5.49037	+	5.22302i	0.241935	+	0.230154i
$516$		0			0
$517$	−27.8328			−1.22408
$518$		0			0
$519$		−	79.8173i		−	3.50359i
$520$		0			0
$521$			15.4545i			0.677074i	0.940953	+	0.338537i	$0.109932\pi$
$521$			15.4545i			0.677074i	−0.940953	+	0.338537i	$0.890068\pi$
$522$		0			0
$523$		−	31.5873i		−	1.38122i	−0.723229	−	0.690608i	$-0.757343\pi$
$523$		−	31.5873i		−	1.38122i	0.723229	−	0.690608i	$-0.242657\pi$
$524$		0			0
$525$	44.8337	−	2.23906i	1.95670	−	0.0977207i
$526$		0			0
$527$			17.2825i			0.752837i
$528$		0			0
$529$	−22.3217	−	5.54470i	−0.970507	−	0.241074i
$530$		0			0
$531$			83.8060i			3.63687i
$532$		0			0
$533$			2.22463i			0.0963596i
$534$		0			0
$535$	12.8768	+	12.2497i	0.556712	+	0.529603i
$536$		0			0
$537$			48.7582i			2.10407i
$538$		0			0
$539$	1.05721			0.0455375
$540$		0			0
$541$	−20.7309			−0.891292			−0.445646	−	0.895209i	$-0.647026\pi$
$541$	−20.7309			−0.891292			−0.445646	+	0.895209i	$0.647026\pi$
$542$		0			0
$543$		−	11.2624i		−	0.483316i
$544$		0			0
$545$	22.8749	+	21.7610i	0.979851	+	0.932137i
$546$		0			0
$547$	−9.88064			−0.422466			−0.211233	−	0.977436i	$-0.567748\pi$
$547$	−9.88064			−0.422466			−0.211233	+	0.977436i	$0.567748\pi$
$548$		0			0
$549$			52.5216i			2.24157i
$550$		0			0
$551$	1.39966			0.0596273
$552$		0			0
$553$			37.2655i			1.58469i
$554$		0			0
$555$	36.6375	−	38.5129i	1.55518	−	1.63478i
$556$		0			0
$557$	−13.3481			−0.565576			−0.282788	−	0.959182i	$-0.591259\pi$
$557$	−13.3481			−0.565576			−0.282788	+	0.959182i	$0.591259\pi$
$558$		0			0
$559$	−1.77073			−0.0748939
$560$		0			0
$561$	−44.8857			−1.89508
$562$		0			0
$563$		−	24.9591i		−	1.05190i	−0.850516	−	0.525950i	$-0.823710\pi$
$563$		−	24.9591i		−	1.05190i	0.850516	−	0.525950i	$-0.176290\pi$
$564$		0			0
$565$	0.497768	−	0.523247i	0.0209412	−	0.0220132i
$566$		0			0
$567$		−	77.7324i		−	3.26445i
$568$		0			0
$569$			33.7602i			1.41530i	0.706562	+	0.707651i	$0.250245\pi$
$569$			33.7602i			1.41530i	−0.706562	+	0.707651i	$0.749755\pi$
$570$		0			0
$571$	−38.3767			−1.60602			−0.803008	−	0.595968i	$-0.796768\pi$
$571$	−38.3767			−1.60602			−0.803008	+	0.595968i	$0.796768\pi$
$572$		0			0
$573$	−30.8099			−1.28710
$574$		0			0
$575$	23.9173	+	1.72109i	0.997421	+	0.0717743i
$576$		0			0
$577$			25.7420i			1.07166i	0.844327	+	0.535828i	$0.180000\pi$
$577$			25.7420i			1.07166i	−0.844327	+	0.535828i	$0.820000\pi$
$578$		0			0
$579$		−	5.68485i		−	0.236254i
$580$		0			0
$581$	41.4790			1.72084
$582$		0			0
$583$	22.1619			0.917852
$584$		0			0
$585$	62.2563	+	59.2248i	2.57398	+	2.44864i
$586$		0			0
$587$	16.8165			0.694091			0.347045	−	0.937848i	$-0.387185\pi$
$587$	16.8165			0.694091			0.347045	+	0.937848i	$0.387185\pi$
$588$		0			0
$589$		−	1.09329i		−	0.0450482i
$590$		0			0
$591$			51.9790i			2.13813i
$592$		0			0
$593$			26.0486i			1.06969i	0.844951	+	0.534843i	$0.179629\pi$
$593$			26.0486i			1.06969i	−0.844951	+	0.534843i	$0.820371\pi$
$594$		0			0
$595$	26.4363	+	25.1490i	1.08378	+	1.03101i
$596$		0			0
$597$	65.5827			2.68412
$598$		0			0
$599$			20.6703i			0.844567i	0.906464	+	0.422283i	$0.138771\pi$
$599$			20.6703i			0.844567i	−0.906464	+	0.422283i	$0.861229\pi$
$600$		0			0
$601$	8.32601			0.339625			0.169813	−	0.985476i	$-0.445684\pi$
$601$	8.32601			0.339625			0.169813	+	0.985476i	$0.445684\pi$
$602$		0			0
$603$		−	55.1140i		−	2.24442i
$604$		0			0
$605$	−8.89649	+	9.35188i	−0.361694	+	0.380208i
$606$		0			0
$607$	−15.9501			−0.647396			−0.323698	−	0.946160i	$-0.604926\pi$
$607$	−15.9501			−0.647396			−0.323698	+	0.946160i	$0.604926\pi$
$608$		0			0
$609$			33.2542i			1.34753i
$610$		0			0
$611$			59.9637i			2.42587i
$612$		0			0
$613$	23.1501			0.935023			0.467512	−	0.883987i	$-0.345151\pi$
$613$	23.1501			0.935023			0.467512	+	0.883987i	$0.345151\pi$
$614$		0			0
$615$	−2.28756	+	2.40465i	−0.0922433	+	0.0969650i
$616$		0			0
$617$	−2.88371			−0.116094			−0.0580468	−	0.998314i	$-0.518487\pi$
$617$	−2.88371			−0.116094			−0.0580468	+	0.998314i	$0.518487\pi$
$618$		0			0
$619$	−6.49465			−0.261042			−0.130521	−	0.991446i	$-0.541665\pi$
$619$	−6.49465			−0.261042			−0.130521	+	0.991446i	$0.541665\pi$
$620$		0			0
$621$	9.18961	−	75.1147i	0.368766	−	3.01425i
$622$		0			0
$623$	−33.1978			−1.33004
$624$		0			0
$625$	−24.8756	+	2.49086i	−0.995024	+	0.0996345i
$626$		0			0
$627$	2.83947			0.113397
$628$		0			0
$629$	43.2069			1.72277
$630$		0			0
$631$	28.6830			1.14185			0.570927	−	0.821001i	$-0.306584\pi$
$631$	28.6830			1.14185			0.570927	+	0.821001i	$0.306584\pi$
$632$		0			0
$633$			62.4482i			2.48209i
$634$		0			0
$635$	16.2350	−	17.0660i	0.644265	−	0.677243i
$636$		0			0
$637$		−	2.27769i		−	0.0902455i
$638$		0			0
$639$			57.3129i			2.26726i
$640$		0			0
$641$		−	31.5938i		−	1.24788i	−0.781473	−	0.623939i	$-0.785531\pi$
$641$		−	31.5938i		−	1.24788i	0.781473	−	0.623939i	$-0.214469\pi$
$642$		0			0
$643$			47.3512i			1.86735i	0.358120	+	0.933675i	$0.383418\pi$
$643$			47.3512i			1.86735i	−0.358120	+	0.933675i	$0.616582\pi$
$644$		0			0
$645$	−1.91402	−	1.82082i	−0.0753644	−	0.0716945i
$646$		0			0
$647$	21.8763			0.860045			0.430022	−	0.902818i	$-0.358506\pi$
$647$	21.8763			0.860045			0.430022	+	0.902818i	$0.358506\pi$
$648$		0			0
$649$		−	24.5619i		−	0.964141i
$650$		0			0
$651$	25.9752			1.01805
$652$		0			0
$653$		−	0.359578i		−	0.0140714i	−0.999975	−	0.00703568i	$-0.997760\pi$
$653$		−	0.359578i		−	0.0140714i	0.999975	−	0.00703568i	$-0.00223954\pi$
$654$		0			0
$655$	18.4012	+	17.5051i	0.718993	+	0.683982i
$656$		0			0
$657$		−	9.71588i		−	0.379052i
$658$		0			0
$659$	−1.91275			−0.0745104			−0.0372552	−	0.999306i	$-0.511861\pi$
$659$	−1.91275			−0.0745104			−0.0372552	+	0.999306i	$0.511861\pi$
$660$		0			0
$661$			4.73207i			0.184056i	0.995756	+	0.0920280i	$0.0293349\pi$
$661$			4.73207i			0.184056i	−0.995756	+	0.0920280i	$0.970665\pi$
$662$		0			0
$663$			96.7031i			3.75564i
$664$		0			0
$665$	−1.67236	−	1.59092i	−0.0648513	−	0.0616933i
$666$		0			0
$667$	−2.15715	+	17.6323i	−0.0835252	+	0.682725i
$668$		0			0
$669$	6.16082			0.238191
$670$		0			0
$671$		−	15.3931i		−	0.594243i
$672$		0			0
$673$		−	42.7608i		−	1.64831i	−0.566366	−	0.824154i	$-0.691651\pi$
$673$		−	42.7608i		−	1.64831i	0.566366	−	0.824154i	$-0.308349\pi$
$674$		0			0
$675$	3.93530	+	78.7982i	0.151470	+	3.03294i
$676$		0			0
$677$	−1.30895			−0.0503072			−0.0251536	−	0.999684i	$-0.508007\pi$
$677$	−1.30895			−0.0503072			−0.0251536	+	0.999684i	$0.508007\pi$
$678$		0			0
$679$		−	11.8617i		−	0.455212i
$680$		0			0
$681$			17.0530i			0.653472i
$682$		0			0
$683$	16.4311			0.628718			0.314359	−	0.949304i	$-0.398210\pi$
$683$	16.4311			0.628718			0.314359	+	0.949304i	$0.398210\pi$
$684$		0			0
$685$	29.3217	−	30.8226i	1.12032	−	1.17767i
$686$		0			0
$687$		−	51.9347i		−	1.98143i
$688$		0			0
$689$		−	47.7462i		−	1.81899i
$690$		0			0
$691$		−	50.1615i		−	1.90823i	−0.299433	−	0.954117i	$-0.596797\pi$
$691$		−	50.1615i		−	1.90823i	0.299433	−	0.954117i	$-0.403203\pi$
$692$		0			0
$693$			48.7249i			1.85091i
$694$		0			0
$695$	14.7780	+	14.0584i	0.560562	+	0.533265i
$696$		0			0
$697$	−2.69773			−0.102184
$698$		0			0
$699$		−	48.1400i		−	1.82082i
$700$		0			0
$701$			0.199633i			0.00754004i	0.999993	+	0.00377002i	$0.00120004\pi$
$701$			0.199633i			0.00754004i	−0.999993	+	0.00377002i	$0.998800\pi$
$702$		0			0
$703$	−2.73326			−0.103087
$704$		0			0
$705$	−61.6598	+	64.8161i	−2.32224	+	2.44111i
$706$		0			0
$707$		−	8.52090i		−	0.320461i
$708$		0			0
$709$		−	40.1194i		−	1.50671i	−0.657611	−	0.753357i	$-0.728433\pi$
$709$		−	40.1194i		−	1.50671i	0.657611	−	0.753357i	$-0.271567\pi$
$710$		0			0
$711$	−106.422			−3.99113
$712$		0			0
$713$	13.7728	+	1.68498i	0.515795	+	0.0631029i
$714$		0			0
$715$	−18.2461	−	17.3576i	−0.682367	−	0.649139i
$716$		0			0
$717$		−	31.5566i		−	1.17850i
$718$		0			0
$719$			1.40670i			0.0524610i	0.999656	+	0.0262305i	$0.00835039\pi$
$719$			1.40670i			0.0524610i	−0.999656	+	0.0262305i	$0.991650\pi$
$720$		0			0
$721$	9.25766			0.344773
$722$		0			0
$723$		−	61.9684i		−	2.30463i
$724$		0			0
$725$	−0.923765	−	18.4969i	−0.0343078	−	0.686959i
$726$		0			0
$727$		−	36.3686i		−	1.34884i	−0.738349	−	0.674419i	$-0.764394\pi$
$727$		−	36.3686i		−	1.34884i	0.738349	−	0.674419i	$-0.235606\pi$
$728$		0			0
$729$	66.4089			2.45959
$730$		0			0
$731$		−	2.14730i		−	0.0794208i
$732$		0			0
$733$	35.2005			1.30016			0.650081	−	0.759865i	$-0.274735\pi$
$733$	35.2005			1.30016			0.650081	+	0.759865i	$0.274735\pi$
$734$		0			0
$735$	2.34212	−	2.46201i	0.0863904	−	0.0908125i
$736$		0			0
$737$			16.1529i			0.594998i
$738$		0			0
$739$		−	6.59702i		−	0.242675i	−0.992611	−	0.121338i	$-0.961282\pi$
$739$		−	6.59702i		−	0.242675i	0.992611	−	0.121338i	$-0.0387184\pi$
$740$		0			0
$741$		−	6.11743i		−	0.224729i
$742$		0			0
$743$			21.9968i			0.806985i	0.914983	+	0.403493i	$0.132204\pi$
$743$			21.9968i			0.806985i	−0.914983	+	0.403493i	$0.867796\pi$
$744$		0			0
$745$	−9.64513	−	9.17546i	−0.353370	−	0.336163i
$746$		0			0
$747$			118.454i			4.33402i
$748$		0			0
$749$	21.7124			0.793352
$750$		0			0
$751$	−10.8072			−0.394359			−0.197179	−	0.980367i	$-0.563178\pi$
$751$	−10.8072			−0.394359			−0.197179	+	0.980367i	$0.563178\pi$
$752$		0			0
$753$	−81.8531			−2.98289
$754$		0			0
$755$	16.1148	+	15.3301i	0.586477	+	0.557919i
$756$		0			0
$757$	9.47219			0.344273			0.172136	−	0.985073i	$-0.444933\pi$
$757$	9.47219			0.344273			0.172136	+	0.985073i	$0.444933\pi$
$758$		0			0
$759$	−4.37619	+	35.7704i	−0.158846	+	1.29838i
$760$		0			0
$761$	−48.3960			−1.75436			−0.877178	−	0.480166i	$-0.840576\pi$
$761$	−48.3960			−1.75436			−0.877178	+	0.480166i	$0.840576\pi$
$762$		0			0
$763$	38.5707			1.39635
$764$		0			0
$765$	−71.8197	+	75.4960i	−2.59665	+	2.72956i
$766$		0			0
$767$	−52.9170			−1.91072
$768$		0			0
$769$		−	5.14793i		−	0.185639i	−0.995683	−	0.0928195i	$-0.970412\pi$
$769$		−	5.14793i		−	0.185639i	0.995683	−	0.0928195i	$-0.0295880\pi$
$770$		0			0
$771$		−	54.4786i		−	1.96200i
$772$		0			0
$773$	−35.7974			−1.28754			−0.643771	−	0.765218i	$-0.722631\pi$
$773$	−35.7974			−1.28754			−0.643771	+	0.765218i	$0.722631\pi$
$774$		0			0
$775$	−14.4482	+	0.721564i	−0.518994	+	0.0259193i
$776$		0			0
$777$		−	64.9391i		−	2.32968i
$778$		0			0
$779$	0.170658			0.00611447
$780$		0			0
$781$		−	16.7973i		−	0.601056i
$782$		0			0
$783$	−58.4464			−2.08870
$784$		0			0
$785$	24.9901	−	26.2693i	0.891934	−	0.937590i
$786$		0			0
$787$			29.7273i			1.05966i	0.848103	+	0.529832i	$0.177745\pi$
$787$			29.7273i			1.05966i	−0.848103	+	0.529832i	$0.822255\pi$
$788$		0			0
$789$			58.4192i			2.07978i
$790$		0			0
$791$		−	0.882280i		−	0.0313703i
$792$		0			0
$793$	−33.1633			−1.17766
$794$		0			0
$795$	49.0967	−	51.6099i	1.74128	−	1.83041i
$796$		0			0
$797$	22.5547			0.798927			0.399464	−	0.916749i	$-0.369196\pi$
$797$	22.5547			0.798927			0.399464	+	0.916749i	$0.369196\pi$
$798$		0			0
$799$	−72.7159			−2.57250
$800$		0			0
$801$		−	94.8054i		−	3.34978i
$802$		0			0
$803$			2.84754i			0.100487i
$804$		0			0
$805$	22.6192	−	18.6157i	0.797223	−	0.656119i
$806$		0			0
$807$	60.1052			2.11580
$808$		0			0
$809$	25.2281			0.886972			0.443486	−	0.896281i	$-0.353742\pi$
$809$	25.2281			0.886972			0.443486	+	0.896281i	$0.353742\pi$
$810$		0			0
$811$		−	22.6698i		−	0.796043i	−0.917376	−	0.398022i	$-0.869697\pi$
$811$		−	22.6698i		−	0.796043i	0.917376	−	0.398022i	$-0.130303\pi$
$812$		0			0
$813$		−	27.7540i		−	0.973377i
$814$		0			0
$815$	21.8471	−	22.9654i	0.765271	−	0.804444i
$816$		0			0
$817$			0.135838i			0.00475237i
$818$		0			0
$819$	104.974			3.66810
$820$		0			0
$821$	−6.09527			−0.212726			−0.106363	−	0.994327i	$-0.533921\pi$
$821$	−6.09527			−0.212726			−0.106363	+	0.994327i	$0.533921\pi$
$822$		0			0
$823$	−54.0738			−1.88490			−0.942448	−	0.334354i	$-0.891482\pi$
$823$	−54.0738			−1.88490			−0.942448	+	0.334354i	$0.891482\pi$
$824$		0			0
$825$	−1.87403	−	37.5245i	−0.0652454	−	1.30644i
$826$		0			0
$827$			15.7729i			0.548476i	0.961662	+	0.274238i	$0.0884256\pi$
$827$			15.7729i			0.548476i	−0.961662	+	0.274238i	$0.911574\pi$
$828$		0			0
$829$	−47.7096			−1.65702			−0.828511	−	0.559972i	$-0.810812\pi$
$829$	−47.7096			−1.65702			−0.828511	+	0.559972i	$0.810812\pi$
$830$		0			0
$831$			91.8951i			3.18781i
$832$		0			0
$833$	2.76208			0.0957003
$834$		0			0
$835$	13.6417	−	14.3400i	0.472090	−	0.496255i
$836$		0			0
$837$			45.6532i			1.57801i
$838$		0			0
$839$	−33.8696			−1.16931			−0.584654	−	0.811283i	$-0.698770\pi$
$839$	−33.8696			−1.16931			−0.584654	+	0.811283i	$0.698770\pi$
$840$		0			0
$841$	−15.2804			−0.526911
$842$		0			0
$843$		−	93.3042i		−	3.21357i
$844$		0			0
$845$	−17.3602	+	18.2488i	−0.597209	+	0.627779i
$846$		0			0
$847$			15.7688i			0.541822i
$848$		0			0
$849$		−	29.9575i		−	1.02814i
$850$		0			0
$851$	4.21251	−	34.4325i	0.144403	−	1.18033i
$852$		0			0
$853$		−	33.1776i		−	1.13598i	−0.823036	−	0.567990i	$-0.807721\pi$
$853$		−	33.1776i		−	1.13598i	0.823036	−	0.567990i	$-0.192279\pi$
$854$		0			0
$855$	4.54331	−	4.77587i	0.155378	−	0.163331i
$856$		0			0
$857$			38.9386i			1.33012i	0.746792	+	0.665058i	$0.231593\pi$
$857$			38.9386i			1.33012i	−0.746792	+	0.665058i	$0.768407\pi$
$858$		0			0
$859$		−	23.5373i		−	0.803081i	−0.915841	−	0.401540i	$-0.868475\pi$
$859$		−	23.5373i		−	0.803081i	0.915841	−	0.401540i	$-0.131525\pi$
$860$		0			0
$861$			4.05464i			0.138182i
$862$		0			0
$863$	2.47024			0.0840881			0.0420440	−	0.999116i	$-0.486613\pi$
$863$	2.47024			0.0840881			0.0420440	+	0.999116i	$0.486613\pi$
$864$		0			0
$865$	39.3460	+	37.4300i	1.33780	+	1.27266i
$866$		0			0
$867$	−61.3976			−2.08517
$868$		0			0
$869$	31.1902			1.05805
$870$		0			0
$871$	34.8002			1.17916
$872$		0			0
$873$	33.8744			1.14647
$874$		0			0
$875$	−19.9208	+	23.1508i	−0.673447	+	0.782640i
$876$		0			0
$877$			41.1951i			1.39106i	0.718497	+	0.695530i	$0.244830\pi$
$877$			41.1951i			1.39106i	−0.718497	+	0.695530i	$0.755170\pi$
$878$		0			0
$879$	16.8934			0.569800
$880$		0			0
$881$			24.8457i			0.837073i	0.908200	+	0.418536i	$0.137457\pi$
$881$			24.8457i			0.837073i	−0.908200	+	0.418536i	$0.862543\pi$
$882$		0			0
$883$	−38.8271			−1.30664			−0.653318	−	0.757084i	$-0.726623\pi$
$883$	−38.8271			−1.30664			−0.653318	+	0.757084i	$0.726623\pi$
$884$		0			0
$885$	−57.1991	−	54.4138i	−1.92273	−	1.82910i
$886$		0			0
$887$	−25.0801			−0.842107			−0.421053	−	0.907036i	$-0.638340\pi$
$887$	−25.0801			−0.842107			−0.421053	+	0.907036i	$0.638340\pi$
$888$		0			0
$889$		−	28.7760i		−	0.965117i
$890$		0			0
$891$	−65.0598			−2.17958
$892$		0			0
$893$	4.60000			0.153933
$894$		0			0
$895$	−24.0354	−	22.8650i	−0.803415	−	0.764293i
$896$		0			0
$897$	77.0648	+	9.42818i	2.57312	+	0.314798i
$898$		0			0
$899$		−	10.7165i		−	0.357417i
$900$		0			0
$901$	57.9001			1.92893
$902$		0			0
$903$	−3.22735			−0.107399
$904$		0			0
$905$	5.55181	+	5.28147i	0.184549	+	0.175562i
$906$		0			0
$907$		−	32.5720i		−	1.08154i	−0.841171	−	0.540769i	$-0.818133\pi$
$907$		−	32.5720i		−	1.08154i	0.841171	−	0.540769i	$-0.181867\pi$
$908$		0			0
$909$	24.3337			0.807098
$910$		0			0
$911$	−2.27302			−0.0753085			−0.0376543	−	0.999291i	$-0.511989\pi$
$911$	−2.27302			−0.0753085			−0.0376543	+	0.999291i	$0.511989\pi$
$912$		0			0
$913$		−	34.7167i		−	1.14895i
$914$		0			0
$915$	−35.8469	−	34.1013i	−1.18506	−	1.12735i
$916$		0			0
$917$	31.0274			1.02461
$918$		0			0
$919$	41.7571			1.37744			0.688719	−	0.725028i	$-0.258173\pi$
$919$	41.7571			1.37744			0.688719	+	0.725028i	$0.258173\pi$
$920$		0			0
$921$	52.9090			1.74341
$922$		0			0
$923$	−36.1886			−1.19116
$924$		0			0
$925$	1.80394	+	36.1210i	0.0593131	+	1.18765i
$926$		0			0
$927$			26.4377i			0.868329i
$928$		0			0
$929$	41.6236			1.36563			0.682813	−	0.730593i	$-0.260756\pi$
$929$	41.6236			1.36563			0.682813	+	0.730593i	$0.260756\pi$
$930$		0			0
$931$	−0.174729			−0.00572650
$932$		0			0
$933$		−	44.7343i		−	1.46453i
$934$		0			0
$935$	21.0490	−	22.1265i	0.688376	−	0.723612i
$936$		0			0
$937$	−13.9992			−0.457334			−0.228667	−	0.973505i	$-0.573437\pi$
$937$	−13.9992			−0.457334			−0.228667	+	0.973505i	$0.573437\pi$
$938$		0			0
$939$	−54.8926			−1.79135
$940$		0			0
$941$		−	35.7963i		−	1.16693i	−0.812139	−	0.583463i	$-0.801697\pi$
$941$		−	35.7963i		−	1.16693i	0.812139	−	0.583463i	$-0.198303\pi$
$942$		0			0
$943$	−0.263019	+	2.14988i	−0.00856507	+	0.0700098i
$944$		0			0
$945$	69.8338	+	66.4333i	2.27170	+	2.16107i
$946$		0			0
$947$	−31.8042			−1.03350			−0.516749	−	0.856137i	$-0.672858\pi$
$947$	−31.8042			−1.03350			−0.516749	+	0.856137i	$0.672858\pi$
$948$		0			0
$949$	6.13482			0.199145
$950$		0			0
$951$		−	79.3003i		−	2.57149i
$952$		0			0
$953$	−4.67822			−0.151542			−0.0757712	−	0.997125i	$-0.524142\pi$
$953$	−4.67822			−0.151542			−0.0757712	+	0.997125i	$0.524142\pi$
$954$		0			0
$955$	14.4482	−	15.1878i	0.467532	−	0.491464i
$956$		0			0
$957$	27.8328			0.899706
$958$		0			0
$959$		−	51.9719i		−	1.67826i
$960$		0			0
$961$	22.6292			0.729973
$962$		0			0
$963$			62.0055i			1.99810i
$964$		0			0
$965$	2.80235	+	2.66589i	0.0902109	+	0.0858180i
$966$		0			0
$967$	10.3182			0.331812			0.165906	−	0.986142i	$-0.446945\pi$
$967$	10.3182			0.331812			0.165906	+	0.986142i	$0.446945\pi$
$968$		0			0
$969$	7.41838			0.238313
$970$		0			0
$971$	−60.2995			−1.93510			−0.967551	−	0.252675i	$-0.918690\pi$
$971$	−60.2995			−1.93510			−0.967551	+	0.252675i	$0.918690\pi$
$972$		0			0
$973$	24.9181			0.798838
$974$		0			0
$975$	−80.8439	+	4.03747i	−2.58908	+	0.129302i
$976$		0			0
$977$	6.82697			0.218414			0.109207	−	0.994019i	$-0.465169\pi$
$977$	6.82697			0.218414			0.109207	+	0.994019i	$0.465169\pi$
$978$		0			0
$979$			27.7856i			0.888033i
$980$		0			0
$981$			110.149i			3.51679i
$982$		0			0
$983$			42.5123i			1.35593i	0.735093	+	0.677966i	$0.237138\pi$
$983$			42.5123i			1.35593i	−0.735093	+	0.677966i	$0.762862\pi$
$984$		0			0
$985$	−25.6231	−	24.3753i	−0.816419	−	0.776663i
$986$		0			0
$987$			109.290i			3.47875i
$988$		0			0
$989$	−1.71123	−	0.209354i	−0.0544139	−	0.00665706i
$990$		0			0
$991$			14.2777i			0.453545i	0.973948	+	0.226772i	$0.0728173\pi$
$991$			14.2777i			0.453545i	−0.973948	+	0.226772i	$0.927183\pi$
$992$		0			0
$993$			31.2081i			0.990359i
$994$		0			0
$995$	−30.7547	+	32.3290i	−0.974991	+	1.02490i
$996$		0			0
$997$		−	19.4129i		−	0.614814i	−0.951578	−	0.307407i	$-0.900539\pi$
$997$		−	19.4129i		−	0.614814i	0.951578	−	0.307407i	$-0.0994612\pi$
$998$		0			0
$999$	114.135			3.61106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.m.g.1839.4	yes	40
4.3	odd	2	inner	1840.2.m.g.1839.37	yes	40
5.4	even	2	inner	1840.2.m.g.1839.38	yes	40
20.19	odd	2	inner	1840.2.m.g.1839.3	yes	40
23.22	odd	2	inner	1840.2.m.g.1839.2	yes	40
92.91	even	2	inner	1840.2.m.g.1839.39	yes	40
115.114	odd	2	inner	1840.2.m.g.1839.40	yes	40
460.459	even	2	inner	1840.2.m.g.1839.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.m.g.1839.1	✓	40	460.459	even	2	inner
1840.2.m.g.1839.2	yes	40	23.22	odd	2	inner
1840.2.m.g.1839.3	yes	40	20.19	odd	2	inner
1840.2.m.g.1839.4	yes	40	1.1	even	1	trivial
1840.2.m.g.1839.37	yes	40	4.3	odd	2	inner
1840.2.m.g.1839.38	yes	40	5.4	even	2	inner
1840.2.m.g.1839.39	yes	40	92.91	even	2	inner
1840.2.m.g.1839.40	yes	40	115.114	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} +O(q^{10})\)	\(q-3.28652 q^{3} +(1.54120 - 1.62009i) q^{5} -2.73174i q^{7} +7.80121 q^{9} -2.28639 q^{11} +4.92586i q^{13} +(-5.06519 + 5.32446i) q^{15} -5.97341 q^{17} +0.377877 q^{19} +8.97791i q^{21} +(-0.582385 + 4.76034i) q^{23} +(-0.249397 - 4.99378i) q^{25} -15.7793 q^{27} +3.70400 q^{29} -2.89324i q^{31} +7.51426 q^{33} +(-4.42567 - 4.21016i) q^{35} -7.23320 q^{37} -16.1889i q^{39} +0.451624 q^{41} +0.359476i q^{43} +(12.0232 - 12.6387i) q^{45} +12.1733 q^{47} -0.462395 q^{49} +19.6317 q^{51} -9.69298 q^{53} +(-3.52378 + 3.70416i) q^{55} -1.24190 q^{57} +10.7427i q^{59} +6.73249i q^{61} -21.3109i q^{63} +(7.98034 + 7.59174i) q^{65} -7.06480i q^{67} +(1.91402 - 15.6449i) q^{69} +7.34667i q^{71} -1.24543i q^{73} +(0.819647 + 16.4121i) q^{75} +6.24581i q^{77} -13.6417 q^{79} +28.4553 q^{81} +15.1841i q^{83} +(-9.20623 + 9.67748i) q^{85} -12.1733 q^{87} -12.1526i q^{89} +13.4562 q^{91} +9.50868i q^{93} +(0.582385 - 0.612196i) q^{95} +4.34220 q^{97} -17.8366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 80 q^{9}+O(q^{10}) \) 40 * q + 80 * q^9	\( 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) \) 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more