LMFDB - Embedded newform 1840.2.e.d.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.527896576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 2x^5 + 7x^4 - 10x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$1.47984 + 1.47984i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.d.369.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95969i q^{3} +(0.479844 - 2.18398i) q^{5} -2.28394i q^{7} -0.840379 q^{9} +O(q^{10})$	$q-1.95969i q^{3} +(0.479844 - 2.18398i) q^{5} -2.28394i q^{7} -0.840379 q^{9} -1.12432 q^{11} -5.95969i q^{13} +(-4.27991 - 0.940345i) q^{15} +5.80007i q^{17} -4.08401 q^{19} -4.47581 q^{21} -1.00000i q^{23} +(-4.53950 - 2.09594i) q^{25} -4.23218i q^{27} +0.408263 q^{29} +3.19187 q^{31} +2.20332i q^{33} +(-4.98807 - 1.09594i) q^{35} -9.80345i q^{37} -11.6791 q^{39} +6.27087 q^{41} +7.75474i q^{43} +(-0.403251 + 1.83537i) q^{45} +6.40020i q^{47} +1.78361 q^{49} +11.3663 q^{51} +6.73590i q^{53} +(-0.539499 + 2.45549i) q^{55} +8.00339i q^{57} -4.75976 q^{59} -6.33265 q^{61} +1.91938i q^{63} +(-13.0158 - 2.85972i) q^{65} -0.283942i q^{67} -1.95969 q^{69} +13.9516 q^{71} -9.61659i q^{73} +(-4.10738 + 8.89600i) q^{75} +2.56788i q^{77} +4.48387 q^{79} -10.8149 q^{81} -10.8223i q^{83} +(12.6672 + 2.78313i) q^{85} -0.800068i q^{87} -5.68414 q^{89} -13.6116 q^{91} -6.25508i q^{93} +(-1.95969 + 8.91938i) q^{95} +11.0676i q^{97} +0.944856 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 6 * q^5 - 8 * q^9	$ 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.95969i		−	1.13143i	−0.824602	−	0.565713i	$-0.808601\pi$
$3$		−	1.95969i		−	1.13143i	0.824602	−	0.565713i	$-0.191399\pi$
$4$		0			0
$5$	0.479844	−	2.18398i	0.214593	−	0.976704i
$5$	0.479844	−	2.18398i	0.214593	−	0.976704i
$6$		0			0
$7$		−	2.28394i		−	0.863249i	−0.902053	−	0.431624i	$-0.857941\pi$
$7$		−	2.28394i		−	0.863249i	0.902053	−	0.431624i	$-0.142059\pi$
$8$		0			0
$9$	−0.840379			−0.280126
$10$		0			0
$11$	−1.12432			−0.338996			−0.169498	−	0.985531i	$-0.554215\pi$
$11$	−1.12432			−0.338996			−0.169498	+	0.985531i	$0.554215\pi$
$12$		0			0
$13$		−	5.95969i		−	1.65292i	−0.562995	−	0.826460i	$-0.690351\pi$
$13$		−	5.95969i		−	1.65292i	0.562995	−	0.826460i	$-0.309649\pi$
$14$		0			0
$15$	−4.27991	−	0.940345i	−1.10507	−	0.242796i
$16$		0			0
$17$			5.80007i			1.40672i	0.710832	+	0.703362i	$0.248318\pi$
$17$			5.80007i			1.40672i	−0.710832	+	0.703362i	$0.751682\pi$
$18$		0			0
$19$	−4.08401			−0.936936			−0.468468	−	0.883480i	$-0.655194\pi$
$19$	−4.08401			−0.936936			−0.468468	+	0.883480i	$0.655194\pi$
$20$		0			0
$21$	−4.47581			−0.976703
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.53950	−	2.09594i	−0.907900	−	0.419187i
$26$		0			0
$27$		−	4.23218i		−	0.814484i
$28$		0			0
$29$	0.408263			0.0758125			0.0379062	−	0.999281i	$-0.487931\pi$
$29$	0.408263			0.0758125			0.0379062	+	0.999281i	$0.487931\pi$
$30$		0			0
$31$	3.19187			0.573277			0.286639	−	0.958039i	$-0.407462\pi$
$31$	3.19187			0.573277			0.286639	+	0.958039i	$0.407462\pi$
$32$		0			0
$33$			2.20332i			0.383549i
$34$		0			0
$35$	−4.98807	−	1.09594i	−0.843138	−	0.185247i
$36$		0			0
$37$		−	9.80345i		−	1.61168i	−0.592135	−	0.805839i	$-0.701715\pi$
$37$		−	9.80345i		−	1.61168i	0.592135	−	0.805839i	$-0.298285\pi$
$38$		0			0
$39$	−11.6791			−1.87016
$40$		0			0
$41$	6.27087			0.979345			0.489673	−	0.871906i	$-0.337116\pi$
$41$	6.27087			0.979345			0.489673	+	0.871906i	$0.337116\pi$
$42$		0			0
$43$			7.75474i			1.18259i	0.806456	+	0.591294i	$0.201383\pi$
$43$			7.75474i			1.18259i	−0.806456	+	0.591294i	$0.798617\pi$
$44$		0			0
$45$	−0.403251	+	1.83537i	−0.0601131	+	0.273600i
$46$		0			0
$47$			6.40020i			0.933566i	0.884372	+	0.466783i	$0.154587\pi$
$47$			6.40020i			0.933566i	−0.884372	+	0.466783i	$0.845413\pi$
$48$		0			0
$49$	1.78361			0.254801
$50$		0			0
$51$	11.3663			1.59160
$52$		0			0
$53$			6.73590i			0.925247i	0.886555	+	0.462624i	$0.153092\pi$
$53$			6.73590i			0.925247i	−0.886555	+	0.462624i	$0.846908\pi$
$54$		0			0
$55$	−0.539499	+	2.45549i	−0.0727460	+	0.331098i
$56$		0			0
$57$			8.00339i			1.06007i
$58$		0			0
$59$	−4.75976			−0.619667			−0.309834	−	0.950791i	$-0.600273\pi$
$59$	−4.75976			−0.619667			−0.309834	+	0.950791i	$0.600273\pi$
$60$		0			0
$61$	−6.33265			−0.810813			−0.405406	−	0.914137i	$-0.632870\pi$
$61$	−6.33265			−0.810813			−0.405406	+	0.914137i	$0.632870\pi$
$62$		0			0
$63$			1.91938i			0.241819i
$64$		0			0
$65$	−13.0158	−	2.85972i	−1.61441	−	0.354705i
$66$		0			0
$67$		−	0.283942i		−	0.0346890i	−0.999850	−	0.0173445i	$-0.994479\pi$
$67$		−	0.283942i		−	0.0346890i	0.999850	−	0.0173445i	$-0.00552120\pi$
$68$		0			0
$69$	−1.95969			−0.235919
$70$		0			0
$71$	13.9516			1.65575			0.827877	−	0.560910i	$-0.189549\pi$
$71$	13.9516			1.65575			0.827877	+	0.560910i	$0.189549\pi$
$72$		0			0
$73$		−	9.61659i		−	1.12554i	−0.826615	−	0.562769i	$-0.809736\pi$
$73$		−	9.61659i		−	1.12554i	0.826615	−	0.562769i	$-0.190264\pi$
$74$		0			0
$75$	−4.10738	+	8.89600i	−0.474280	+	1.02722i
$76$		0			0
$77$			2.56788i			0.292637i
$78$		0			0
$79$	4.48387			0.504475			0.252238	−	0.967665i	$-0.418834\pi$
$79$	4.48387			0.504475			0.252238	+	0.967665i	$0.418834\pi$
$80$		0			0
$81$	−10.8149			−1.20166
$82$		0			0
$83$		−	10.8223i		−	1.18790i	−0.804501	−	0.593951i	$-0.797567\pi$
$83$		−	10.8223i		−	1.18790i	0.804501	−	0.593951i	$-0.202433\pi$
$84$		0			0
$85$	12.6672	+	2.78313i	1.37395	+	0.301873i
$86$		0			0
$87$		−	0.800068i		−	0.0857763i
$88$		0			0
$89$	−5.68414			−0.602518			−0.301259	−	0.953542i	$-0.597407\pi$
$89$	−5.68414			−0.602518			−0.301259	+	0.953542i	$0.597407\pi$
$90$		0			0
$91$	−13.6116			−1.42688
$92$		0			0
$93$		−	6.25508i		−	0.648621i
$94$		0			0
$95$	−1.95969	+	8.91938i	−0.201060	+	0.915109i
$96$		0			0
$97$			11.0676i			1.12374i	0.827226	+	0.561870i	$0.189918\pi$
$97$			11.0676i			1.12374i	−0.827226	+	0.561870i	$0.810082\pi$
$98$		0			0
$99$	0.944856			0.0949616
$100$		0			0
$101$	1.60014			0.159219			0.0796097	−	0.996826i	$-0.474633\pi$
$101$	1.60014			0.159219			0.0796097	+	0.996826i	$0.474633\pi$
$102$		0			0
$103$			1.23218i			0.121411i	0.998156	+	0.0607054i	$0.0193350\pi$
$103$			1.23218i			0.121411i	−0.998156	+	0.0607054i	$0.980665\pi$
$104$		0			0
$105$	−2.14769	+	9.77507i	−0.209593	+	0.953949i
$106$		0			0
$107$			0.235232i			0.0227408i	0.999935	+	0.0113704i	$0.00361938\pi$
$107$			0.235232i			0.0227408i	−0.999935	+	0.0113704i	$0.996381\pi$
$108$		0			0
$109$	−7.43550			−0.712192			−0.356096	−	0.934449i	$-0.615892\pi$
$109$	−7.43550			−0.712192			−0.356096	+	0.934449i	$0.615892\pi$
$110$		0			0
$111$	−19.2117			−1.82350
$112$		0			0
$113$			2.28394i			0.214855i	0.994213	+	0.107428i	$0.0342614\pi$
$113$			2.28394i			0.214855i	−0.994213	+	0.107428i	$0.965739\pi$
$114$		0			0
$115$	−2.18398	−	0.479844i	−0.203657	−	0.0447457i
$116$		0			0
$117$			5.00840i			0.463027i
$118$		0			0
$119$	13.2470			1.21435
$120$		0			0
$121$	−9.73590			−0.885082
$122$		0			0
$123$		−	12.2890i		−	1.10806i
$124$		0			0
$125$	−6.75573	+	8.90843i	−0.604251	+	0.796794i
$126$		0			0
$127$			10.2151i			0.906444i	0.891398	+	0.453222i	$0.149725\pi$
$127$			10.2151i			0.906444i	−0.891398	+	0.453222i	$0.850275\pi$
$128$		0			0
$129$	15.1969			1.33801
$130$		0			0
$131$	16.2232			1.41742			0.708712	−	0.705498i	$-0.249276\pi$
$131$	16.2232			1.41742			0.708712	+	0.705498i	$0.249276\pi$
$132$		0			0
$133$			9.32764i			0.808809i
$134$		0			0
$135$	−9.24299	−	2.03079i	−0.795510	−	0.174783i
$136$		0			0
$137$		−	4.89715i		−	0.418392i	−0.977874	−	0.209196i	$-0.932915\pi$
$137$		−	4.89715i		−	0.418392i	0.977874	−	0.209196i	$-0.0670846\pi$
$138$		0			0
$139$	4.43889			0.376502			0.188251	−	0.982121i	$-0.439718\pi$
$139$	4.43889			0.376502			0.188251	+	0.982121i	$0.439718\pi$
$140$		0			0
$141$	12.5424			1.05626
$142$		0			0
$143$			6.70060i			0.560333i
$144$		0			0
$145$	0.195903	−	0.891636i	0.0162688	−	0.0740463i
$146$		0			0
$147$		−	3.49532i		−	0.288289i
$148$		0			0
$149$	2.78361			0.228042			0.114021	−	0.993478i	$-0.463627\pi$
$149$	2.78361			0.228042			0.114021	+	0.993478i	$0.463627\pi$
$150$		0			0
$151$	−11.9278			−0.970669			−0.485334	−	0.874329i	$-0.661302\pi$
$151$	−11.9278			−0.970669			−0.485334	+	0.874329i	$0.661302\pi$
$152$		0			0
$153$		−	4.87426i		−	0.394060i
$154$		0			0
$155$	1.53160	−	6.97097i	0.123021	−	0.559922i
$156$		0			0
$157$			22.9550i			1.83201i	0.401167	+	0.916005i	$0.368605\pi$
$157$			22.9550i			1.83201i	−0.401167	+	0.916005i	$0.631395\pi$
$158$		0			0
$159$	13.2003			1.04685
$160$		0			0
$161$	−2.28394			−0.180000
$162$		0			0
$163$		−	10.2083i		−	0.799578i	−0.916607	−	0.399789i	$-0.869083\pi$
$163$		−	10.2083i		−	0.799578i	0.916607	−	0.399789i	$-0.130917\pi$
$164$		0			0
$165$	4.81199	+	1.05725i	0.374613	+	0.0823068i
$166$		0			0
$167$		−	6.84715i		−	0.529849i	−0.964269	−	0.264924i	$-0.914653\pi$
$167$		−	6.84715i		−	0.529849i	0.964269	−	0.264924i	$-0.0853470\pi$
$168$		0			0
$169$	−22.5179			−1.73215
$170$		0			0
$171$	3.43212			0.262461
$172$		0			0
$173$		−	4.29539i		−	0.326572i	−0.986579	−	0.163286i	$-0.947791\pi$
$173$		−	4.29539i		−	0.326572i	0.986579	−	0.163286i	$-0.0522094\pi$
$174$		0			0
$175$	−4.78700	+	10.3680i	−0.361863	+	0.783743i
$176$		0			0
$177$			9.32764i			0.701108i
$178$		0			0
$179$	−14.8626			−1.11088			−0.555442	−	0.831555i	$-0.687451\pi$
$179$	−14.8626			−1.11088			−0.555442	+	0.831555i	$0.687451\pi$
$180$		0			0
$181$	13.1311			0.976027			0.488013	−	0.872836i	$-0.337722\pi$
$181$	13.1311			0.976027			0.488013	+	0.872836i	$0.337722\pi$
$182$		0			0
$183$			12.4100i			0.917375i
$184$		0			0
$185$	−21.4105	−	4.70413i	−1.57413	−	0.345855i
$186$		0			0
$187$		−	6.52114i		−	0.476873i
$188$		0			0
$189$	−9.66606			−0.703103
$190$		0			0
$191$	−13.9819			−1.01170			−0.505848	−	0.862623i	$-0.668820\pi$
$191$	−13.9819			−1.01170			−0.505848	+	0.862623i	$0.668820\pi$
$192$		0			0
$193$		−	8.71267i		−	0.627152i	−0.949563	−	0.313576i	$-0.898473\pi$
$193$		−	8.71267i		−	0.627152i	0.949563	−	0.313576i	$-0.101527\pi$
$194$		0			0
$195$	−5.60417	+	25.5069i	−0.401323	+	1.82659i
$196$		0			0
$197$		−	22.4876i		−	1.60218i	−0.598547	−	0.801088i	$-0.704255\pi$
$197$		−	22.4876i		−	1.60218i	0.598547	−	0.801088i	$-0.295745\pi$
$198$		0			0
$199$	9.01078			0.638757			0.319379	−	0.947627i	$-0.396526\pi$
$199$	9.01078			0.638757			0.319379	+	0.947627i	$0.396526\pi$
$200$		0			0
$201$	−0.556437			−0.0392481
$202$		0			0
$203$		−	0.932448i		−	0.0654450i
$204$		0			0
$205$	3.00904	−	13.6954i	0.210161	−	0.956530i
$206$		0			0
$207$			0.840379i			0.0584104i
$208$		0			0
$209$	4.59174			0.317617
$210$		0			0
$211$	−5.60014			−0.385529			−0.192765	−	0.981245i	$-0.561745\pi$
$211$	−5.60014			−0.385529			−0.192765	+	0.981245i	$0.561745\pi$
$212$		0			0
$213$		−	27.3408i		−	1.87336i
$214$		0			0
$215$	16.9362	+	3.72107i	1.15504	+	0.253775i
$216$		0			0
$217$		−	7.29005i		−	0.494881i
$218$		0			0
$219$	−18.8455			−1.27346
$220$		0			0
$221$	34.5666			2.32520
$222$		0			0
$223$		−	7.06517i		−	0.473119i	−0.971617	−	0.236559i	$-0.923980\pi$
$223$		−	7.06517i		−	0.473119i	0.971617	−	0.236559i	$-0.0760198\pi$
$224$		0			0
$225$	3.81490	+	1.76138i	0.254327	+	0.117425i
$226$		0			0
$227$		−	25.1072i		−	1.66643i	−0.552952	−	0.833213i	$-0.686499\pi$
$227$		−	25.1072i		−	1.66643i	0.552952	−	0.833213i	$-0.313501\pi$
$228$		0			0
$229$	3.20366			0.211704			0.105852	−	0.994382i	$-0.466243\pi$
$229$	3.20366			0.211704			0.105852	+	0.994382i	$0.466243\pi$
$230$		0			0
$231$	5.03225			0.331098
$232$		0			0
$233$		−	18.6388i		−	1.22107i	−0.791989	−	0.610535i	$-0.790954\pi$
$233$		−	18.6388i		−	1.22107i	0.791989	−	0.610535i	$-0.209046\pi$
$234$		0			0
$235$	13.9779	+	3.07110i	0.911817	+	0.200337i
$236$		0			0
$237$		−	8.78700i		−	0.570777i
$238$		0			0
$239$	3.18185			0.205817			0.102908	−	0.994691i	$-0.467185\pi$
$239$	3.18185			0.205817			0.102908	+	0.994691i	$0.467185\pi$
$240$		0			0
$241$	4.79844			0.309095			0.154547	−	0.987985i	$-0.450608\pi$
$241$	4.79844			0.309095			0.154547	+	0.987985i	$0.450608\pi$
$242$		0			0
$243$			8.49728i			0.545101i
$244$		0			0
$245$	0.855855	−	3.89536i	0.0546786	−	0.248865i
$246$		0			0
$247$			24.3394i			1.54868i
$248$		0			0
$249$	−21.2083			−1.34402
$250$		0			0
$251$	2.24668			0.141809			0.0709045	−	0.997483i	$-0.477411\pi$
$251$	2.24668			0.141809			0.0709045	+	0.997483i	$0.477411\pi$
$252$		0			0
$253$			1.12432i			0.0706855i
$254$		0			0
$255$	5.45407	−	24.8238i	0.341547	−	1.55453i
$256$		0			0
$257$			11.0148i			0.687086i	0.939137	+	0.343543i	$0.111627\pi$
$257$			11.0148i			0.687086i	−0.939137	+	0.343543i	$0.888373\pi$
$258$		0			0
$259$	−22.3905			−1.39128
$260$		0			0
$261$	−0.343095			−0.0212371
$262$		0			0
$263$		−	11.0585i		−	0.681898i	−0.940082	−	0.340949i	$-0.889252\pi$
$263$		−	11.0585i		−	0.681898i	0.940082	−	0.340949i	$-0.110748\pi$
$264$		0			0
$265$	14.7110	+	3.23218i	0.903692	+	0.198551i
$266$		0			0
$267$			11.1392i			0.681705i
$268$		0			0
$269$	9.90392			0.603853			0.301926	−	0.953331i	$-0.402370\pi$
$269$	9.90392			0.603853			0.301926	+	0.953331i	$0.402370\pi$
$270$		0			0
$271$	−4.82655			−0.293192			−0.146596	−	0.989196i	$-0.546832\pi$
$271$	−4.82655			−0.293192			−0.146596	+	0.989196i	$0.546832\pi$
$272$		0			0
$273$			26.6745i			1.61441i
$274$		0			0
$275$	5.10385	+	2.35651i	0.307774	+	0.142103i
$276$		0			0
$277$			7.54303i			0.453217i	0.973986	+	0.226608i	$0.0727637\pi$
$277$			7.54303i			0.453217i	−0.973986	+	0.226608i	$0.927236\pi$
$278$		0			0
$279$	−2.68238			−0.160590
$280$		0			0
$281$	6.01145			0.358613			0.179306	−	0.983793i	$-0.442615\pi$
$281$	6.01145			0.358613			0.179306	+	0.983793i	$0.442615\pi$
$282$		0			0
$283$		−	15.6388i		−	0.929631i	−0.885407	−	0.464816i	$-0.846121\pi$
$283$		−	15.6388i		−	0.929631i	0.885407	−	0.464816i	$-0.153879\pi$
$284$		0			0
$285$	17.4792	+	3.84038i	1.03538	+	0.227484i
$286$		0			0
$287$		−	14.3223i		−	0.845419i
$288$		0			0
$289$	−16.6408			−0.978870
$290$		0			0
$291$	21.6890			1.27143
$292$		0			0
$293$			3.21639i			0.187904i	0.995577	+	0.0939518i	$0.0299499\pi$
$293$			3.21639i			0.187904i	−0.995577	+	0.0939518i	$0.970050\pi$
$294$		0			0
$295$	−2.28394	+	10.3952i	−0.132976	+	0.605231i
$296$		0			0
$297$			4.75833i			0.276106i
$298$		0			0
$299$	−5.95969			−0.344658
$300$		0			0
$301$	17.7114			1.02087
$302$		0			0
$303$		−	3.13577i		−	0.180145i
$304$		0			0
$305$	−3.03869	+	13.8304i	−0.173995	+	0.791924i
$306$		0			0
$307$			34.4702i			1.96732i	0.180044	+	0.983659i	$0.442376\pi$
$307$			34.4702i			1.96732i	−0.180044	+	0.983659i	$0.557624\pi$
$308$		0			0
$309$	2.41470			0.137367
$310$		0			0
$311$	18.7443			1.06289			0.531446	−	0.847092i	$-0.321649\pi$
$311$	18.7443			1.06289			0.531446	+	0.847092i	$0.321649\pi$
$312$		0			0
$313$		−	9.91260i		−	0.560294i	−0.959957	−	0.280147i	$-0.909617\pi$
$313$		−	9.91260i		−	0.560294i	0.959957	−	0.280147i	$-0.0903831\pi$
$314$		0			0
$315$	4.19187	+	0.921002i	0.236185	+	0.0518926i
$316$		0			0
$317$		−	9.24024i		−	0.518984i	−0.965745	−	0.259492i	$-0.916445\pi$
$317$		−	9.24024i		−	0.518984i	0.965745	−	0.259492i	$-0.0835551\pi$
$318$		0			0
$319$	−0.459018			−0.0257001
$320$		0			0
$321$	0.460982			0.0257295
$322$		0			0
$323$		−	23.6875i		−	1.31801i
$324$		0			0
$325$	−12.4911	+	27.0540i	−0.692883	+	1.50069i
$326$		0			0
$327$			14.5713i			0.805793i
$328$		0			0
$329$	14.6177			0.805899
$330$		0			0
$331$	23.9684			1.31742			0.658712	−	0.752395i	$-0.271102\pi$
$331$	23.9684			1.31742			0.658712	+	0.752395i	$0.271102\pi$
$332$		0			0
$333$			8.23862i			0.451474i
$334$		0			0
$335$	−0.620122	−	0.136248i	−0.0338809	−	0.00744401i
$336$		0			0
$337$			9.76477i			0.531921i	0.963984	+	0.265960i	$0.0856890\pi$
$337$			9.76477i			0.531921i	−0.963984	+	0.265960i	$0.914311\pi$
$338$		0			0
$339$	4.47581			0.243093
$340$		0			0
$341$	−3.58869			−0.194338
$342$		0			0
$343$		−	20.0613i		−	1.08321i
$344$		0			0
$345$	−0.940345	+	4.27991i	−0.0506265	+	0.230423i
$346$		0			0
$347$		−	1.38441i		−	0.0743190i	−0.999309	−	0.0371595i	$-0.988169\pi$
$347$		−	1.38441i		−	0.0743190i	0.999309	−	0.0371595i	$-0.0118310\pi$
$348$		0			0
$349$	32.3106			1.72954			0.864772	−	0.502164i	$-0.167463\pi$
$349$	32.3106			1.72954			0.864772	+	0.502164i	$0.167463\pi$
$350$		0			0
$351$	−25.2225			−1.34628
$352$		0			0
$353$		−	11.4386i		−	0.608813i	−0.952542	−	0.304406i	$-0.901542\pi$
$353$		−	11.4386i		−	0.608813i	0.952542	−	0.304406i	$-0.0984581\pi$
$354$		0			0
$355$	6.69461	−	30.4700i	0.355313	−	1.61718i
$356$		0			0
$357$		−	25.9600i		−	1.37395i
$358$		0			0
$359$	9.95568			0.525441			0.262720	−	0.964872i	$-0.415380\pi$
$359$	9.95568			0.525441			0.262720	+	0.964872i	$0.415380\pi$
$360$		0			0
$361$	−2.32087			−0.122151
$362$		0			0
$363$			19.0793i			1.00141i
$364$		0			0
$365$	−21.0024	−	4.61447i	−1.09932	−	0.241532i
$366$		0			0
$367$		−	27.6785i		−	1.44481i	−0.691472	−	0.722403i	$-0.743037\pi$
$367$		−	27.6785i		−	1.44481i	0.691472	−	0.722403i	$-0.256963\pi$
$368$		0			0
$369$	−5.26991			−0.274341
$370$		0			0
$371$	15.3844			0.798719
$372$		0			0
$373$		−	28.7356i		−	1.48787i	−0.668251	−	0.743936i	$-0.732957\pi$
$373$		−	28.7356i		−	1.48787i	0.668251	−	0.743936i	$-0.267043\pi$
$374$		0			0
$375$	17.4578	+	13.2391i	0.901514	+	0.683665i
$376$		0			0
$377$		−	2.43312i		−	0.125312i
$378$		0			0
$379$	−18.2715			−0.938546			−0.469273	−	0.883053i	$-0.655484\pi$
$379$	−18.2715			−0.938546			−0.469273	+	0.883053i	$0.655484\pi$
$380$		0			0
$381$	20.0184			1.02557
$382$		0			0
$383$		−	29.0356i		−	1.48365i	−0.670593	−	0.741826i	$-0.733960\pi$
$383$		−	29.0356i		−	1.48365i	0.670593	−	0.741826i	$-0.266040\pi$
$384$		0			0
$385$	5.60819	+	1.23218i	0.285820	+	0.0627979i
$386$		0			0
$387$		−	6.51693i		−	0.331274i
$388$		0			0
$389$	−2.86423			−0.145222			−0.0726112	−	0.997360i	$-0.523133\pi$
$389$	−2.86423			−0.145222			−0.0726112	+	0.997360i	$0.523133\pi$
$390$		0			0
$391$	5.80007			0.293322
$392$		0			0
$393$		−	31.7923i		−	1.60371i
$394$		0			0
$395$	2.15156	−	9.79267i	0.108257	−	0.492723i
$396$		0			0
$397$			16.5592i			0.831080i	0.909575	+	0.415540i	$0.136407\pi$
$397$			16.5592i			0.831080i	−0.909575	+	0.415540i	$0.863593\pi$
$398$		0			0
$399$	18.2793			0.915108
$400$		0			0
$401$	−3.09479			−0.154547			−0.0772733	−	0.997010i	$-0.524621\pi$
$401$	−3.09479			−0.154547			−0.0772733	+	0.997010i	$0.524621\pi$
$402$		0			0
$403$		−	19.0226i		−	0.947582i
$404$		0			0
$405$	−5.18947	+	23.6195i	−0.257867	+	1.17366i
$406$		0			0
$407$			11.0222i			0.546352i
$408$		0			0
$409$	−8.93617			−0.441865			−0.220933	−	0.975289i	$-0.570910\pi$
$409$	−8.93617			−0.441865			−0.220933	+	0.975289i	$0.570910\pi$
$410$		0			0
$411$	−9.59689			−0.473379
$412$		0			0
$413$			10.8710i			0.534927i
$414$		0			0
$415$	−23.6356	−	5.19302i	−1.16023	−	0.254915i
$416$		0			0
$417$		−	8.69884i		−	0.425984i
$418$		0			0
$419$	24.1237			1.17852			0.589260	−	0.807944i	$-0.299419\pi$
$419$	24.1237			1.17852			0.589260	+	0.807944i	$0.299419\pi$
$420$		0			0
$421$	23.8602			1.16288			0.581438	−	0.813591i	$-0.302490\pi$
$421$	23.8602			1.16288			0.581438	+	0.813591i	$0.302490\pi$
$422$		0			0
$423$		−	5.37860i		−	0.261516i
$424$		0			0
$425$	12.1566	−	26.3294i	0.589680	−	1.27716i
$426$		0			0
$427$			14.4634i			0.699933i
$428$		0			0
$429$	13.1311			0.633975
$430$		0			0
$431$	−4.45096			−0.214395			−0.107198	−	0.994238i	$-0.534188\pi$
$431$	−4.45096			−0.214395			−0.107198	+	0.994238i	$0.534188\pi$
$432$		0			0
$433$		−	8.10929i		−	0.389707i	−0.980832	−	0.194854i	$-0.937577\pi$
$433$		−	8.10929i		−	0.389707i	0.980832	−	0.194854i	$-0.0624232\pi$
$434$		0			0
$435$	−1.74733	−	0.383908i	−0.0837780	−	0.0184070i
$436$		0			0
$437$			4.08401i			0.195365i
$438$		0			0
$439$	4.47180			0.213428			0.106714	−	0.994290i	$-0.465967\pi$
$439$	4.47180			0.213428			0.106714	+	0.994290i	$0.465967\pi$
$440$		0			0
$441$	−1.49891			−0.0713766
$442$		0			0
$443$			9.60047i			0.456132i	0.973646	+	0.228066i	$0.0732402\pi$
$443$			9.60047i			0.456132i	−0.973646	+	0.228066i	$0.926760\pi$
$444$		0			0
$445$	−2.72750	+	12.4140i	−0.129296	+	0.588482i
$446$		0			0
$447$		−	5.45501i		−	0.258013i
$448$		0			0
$449$	6.79944			0.320886			0.160443	−	0.987045i	$-0.448708\pi$
$449$	6.79944			0.320886			0.160443	+	0.987045i	$0.448708\pi$
$450$		0			0
$451$	−7.05047			−0.331994
$452$		0			0
$453$			23.3747i			1.09824i
$454$		0			0
$455$	−6.53144	+	29.7274i	−0.306199	+	1.39364i
$456$		0			0
$457$		−	19.5582i		−	0.914894i	−0.889237	−	0.457447i	$-0.848764\pi$
$457$		−	19.5582i		−	0.914894i	0.889237	−	0.457447i	$-0.151236\pi$
$458$		0			0
$459$	24.5470			1.14575
$460$		0			0
$461$	−42.7081			−1.98912			−0.994558	−	0.104184i	$-0.966777\pi$
$461$	−42.7081			−1.98912			−0.994558	+	0.104184i	$0.966777\pi$
$462$		0			0
$463$		−	7.42209i		−	0.344934i	−0.985015	−	0.172467i	$-0.944826\pi$
$463$		−	7.42209i		−	0.344934i	0.985015	−	0.172467i	$-0.0551738\pi$
$464$		0			0
$465$	−13.6609	−	3.00146i	−0.633511	−	0.139189i
$466$		0			0
$467$			22.5041i			1.04136i	0.853751	+	0.520682i	$0.174322\pi$
$467$			22.5041i			1.04136i	−0.853751	+	0.520682i	$0.825678\pi$
$468$		0			0
$469$	−0.648506			−0.0299452
$470$		0			0
$471$	44.9847			2.07278
$472$		0			0
$473$		−	8.71882i		−	0.400892i
$474$		0			0
$475$	18.5394	+	8.55982i	0.850644	+	0.392752i
$476$		0			0
$477$		−	5.66071i		−	0.259186i
$478$		0			0
$479$	−10.1667			−0.464530			−0.232265	−	0.972653i	$-0.574614\pi$
$479$	−10.1667			−0.464530			−0.232265	+	0.972653i	$0.574614\pi$
$480$		0			0
$481$	−58.4255			−2.66398
$482$		0			0
$483$			4.47581i			0.203657i
$484$		0			0
$485$	24.1713	+	5.31070i	1.09756	+	0.241147i
$486$		0			0
$487$		−	34.9917i		−	1.58562i	−0.609467	−	0.792812i	$-0.708616\pi$
$487$		−	34.9917i		−	1.58562i	0.609467	−	0.792812i	$-0.291384\pi$
$488$		0			0
$489$	−20.0051			−0.904664
$490$		0			0
$491$	2.27087			0.102483			0.0512415	−	0.998686i	$-0.483682\pi$
$491$	2.27087			0.102483			0.0512415	+	0.998686i	$0.483682\pi$
$492$		0			0
$493$			2.36795i			0.106647i
$494$		0			0
$495$	0.453384	−	2.06354i	0.0203781	−	0.0927493i
$496$		0			0
$497$		−	31.8647i		−	1.42933i
$498$		0			0
$499$	20.9929			0.939773			0.469887	−	0.882727i	$-0.344295\pi$
$499$	20.9929			0.939773			0.469887	+	0.882727i	$0.344295\pi$
$500$		0			0
$501$	−13.4183			−0.599485
$502$		0			0
$503$		−	18.5041i		−	0.825055i	−0.910945	−	0.412528i	$-0.864646\pi$
$503$		−	18.5041i		−	0.825055i	0.910945	−	0.412528i	$-0.135354\pi$
$504$		0			0
$505$	0.767816	−	3.49466i	0.0341674	−	0.155510i
$506$		0			0
$507$			44.1280i			1.95980i
$508$		0			0
$509$	41.8472			1.85484			0.927421	−	0.374019i	$-0.122020\pi$
$509$	41.8472			1.85484			0.927421	+	0.374019i	$0.122020\pi$
$510$		0			0
$511$	−21.9637			−0.971619
$512$		0			0
$513$			17.2843i			0.763120i
$514$		0			0
$515$	2.69106	+	0.591257i	0.118582	+	0.0260539i
$516$		0			0
$517$		−	7.19588i		−	0.316475i
$518$		0			0
$519$	−8.41762			−0.369493
$520$		0			0
$521$	34.1103			1.49440			0.747199	−	0.664600i	$-0.231398\pi$
$521$	34.1103			1.49440			0.747199	+	0.664600i	$0.231398\pi$
$522$		0			0
$523$			36.4755i			1.59496i	0.603343	+	0.797482i	$0.293835\pi$
$523$			36.4755i			1.59496i	−0.603343	+	0.797482i	$0.706165\pi$
$524$		0			0
$525$	20.3180	+	9.38102i	0.886748	+	0.409421i
$526$		0			0
$527$			18.5131i			0.806442i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	37.3724i		−	1.61878i
$534$		0			0
$535$	0.513741	+	0.112875i	0.0222110	+	0.00488000i
$536$		0			0
$537$			29.1261i			1.25688i
$538$		0			0
$539$	−2.00535			−0.0863765
$540$		0			0
$541$	28.2171			1.21315			0.606573	−	0.795028i	$-0.292544\pi$
$541$	28.2171			1.21315			0.606573	+	0.795028i	$0.292544\pi$
$542$		0			0
$543$		−	25.7329i		−	1.10430i
$544$		0			0
$545$	−3.56788	+	16.2390i	−0.152831	+	0.695601i
$546$		0			0
$547$		−	15.1638i		−	0.648356i	−0.945996	−	0.324178i	$-0.894912\pi$
$547$		−	15.1638i		−	0.648356i	0.945996	−	0.324178i	$-0.105088\pi$
$548$		0			0
$549$	5.32183			0.227130
$550$		0			0
$551$	−1.66735			−0.0710314
$552$		0			0
$553$		−	10.2409i		−	0.435488i
$554$		0			0
$555$	−9.21863	+	41.9579i	−0.391309	+	1.78101i
$556$		0			0
$557$		−	9.52222i		−	0.403469i	−0.979440	−	0.201735i	$-0.935342\pi$
$557$		−	9.52222i		−	0.403469i	0.979440	−	0.201735i	$-0.0646579\pi$
$558$		0			0
$559$	46.2159			1.95472
$560$		0			0
$561$	−12.7794			−0.539547
$562$		0			0
$563$			32.7494i			1.38022i	0.723702	+	0.690112i	$0.242439\pi$
$563$			32.7494i			1.38022i	−0.723702	+	0.690112i	$0.757561\pi$
$564$		0			0
$565$	4.98807	+	1.09594i	0.209850	+	0.0461064i
$566$		0			0
$567$			24.7006i			1.03733i
$568$		0			0
$569$	26.8926			1.12740			0.563698	−	0.825981i	$-0.309378\pi$
$569$	26.8926			1.12740			0.563698	+	0.825981i	$0.309378\pi$
$570$		0			0
$571$	−0.920000			−0.0385008			−0.0192504	−	0.999815i	$-0.506128\pi$
$571$	−0.920000			−0.0385008			−0.0192504	+	0.999815i	$0.506128\pi$
$572$		0			0
$573$			27.4002i			1.14466i
$574$		0			0
$575$	−2.09594	+	4.53950i	−0.0874066	+	0.189310i
$576$		0			0
$577$			42.5888i			1.77300i	0.462732	+	0.886498i	$0.346869\pi$
$577$			42.5888i			1.77300i	−0.462732	+	0.886498i	$0.653131\pi$
$578$		0			0
$579$	−17.0741			−0.709576
$580$		0			0
$581$	−24.7175			−1.02545
$582$		0			0
$583$		−	7.57332i		−	0.313655i
$584$		0			0
$585$	10.9382	+	2.40325i	0.452240	+	0.0993622i
$586$		0			0
$587$		−	9.51212i		−	0.392607i	−0.980543	−	0.196304i	$-0.937106\pi$
$587$		−	9.51212i		−	0.392607i	0.980543	−	0.196304i	$-0.0628938\pi$
$588$		0			0
$589$	−13.0356			−0.537124
$590$		0			0
$591$	−44.0687			−1.81274
$592$		0			0
$593$			31.0719i			1.27597i	0.770048	+	0.637986i	$0.220232\pi$
$593$			31.0719i			1.27597i	−0.770048	+	0.637986i	$0.779768\pi$
$594$		0			0
$595$	6.35651	−	28.9312i	0.260591	−	1.18606i
$596$		0			0
$597$		−	17.6583i		−	0.722707i
$598$		0			0
$599$	2.04065			0.0833787			0.0416893	−	0.999131i	$-0.486726\pi$
$599$	2.04065			0.0833787			0.0416893	+	0.999131i	$0.486726\pi$
$600$		0			0
$601$	−25.2377			−1.02947			−0.514733	−	0.857351i	$-0.672109\pi$
$601$	−25.2377			−1.02947			−0.514733	+	0.857351i	$0.672109\pi$
$602$		0			0
$603$			0.238619i			0.00971731i
$604$		0			0
$605$	−4.67172	+	21.2630i	−0.189932	+	0.864463i
$606$		0			0
$607$		−	10.4305i		−	0.423361i	−0.977339	−	0.211680i	$-0.932106\pi$
$607$		−	10.4305i		−	0.423361i	0.977339	−	0.211680i	$-0.0678936\pi$
$608$		0			0
$609$	−1.82731			−0.0740463
$610$		0			0
$611$	38.1432			1.54311
$612$		0			0
$613$			4.97745i			0.201037i	0.994935	+	0.100519i	$0.0320502\pi$
$613$			4.97745i			0.201037i	−0.994935	+	0.100519i	$0.967950\pi$
$614$		0			0
$615$	−26.8388	−	5.89678i	−1.08224	−	0.237781i
$616$		0			0
$617$			30.5881i			1.23143i	0.787969	+	0.615715i	$0.211133\pi$
$617$			30.5881i			1.23143i	−0.787969	+	0.615715i	$0.788867\pi$
$618$		0			0
$619$	−48.0404			−1.93091			−0.965453	−	0.260579i	$-0.916087\pi$
$619$	−48.0404			−1.93091			−0.965453	+	0.260579i	$0.916087\pi$
$620$		0			0
$621$	−4.23218			−0.169832
$622$		0			0
$623$			12.9823i			0.520123i
$624$		0			0
$625$	16.2141	+	19.0290i	0.648564	+	0.761160i
$626$		0			0
$627$		−	8.99837i		−	0.359360i
$628$		0			0
$629$	56.8607			2.26718
$630$		0			0
$631$	−10.6157			−0.422605			−0.211303	−	0.977421i	$-0.567771\pi$
$631$	−10.6157			−0.422605			−0.211303	+	0.977421i	$0.567771\pi$
$632$		0			0
$633$			10.9745i			0.436198i
$634$		0			0
$635$	22.3095	+	4.90166i	0.885327	+	0.194516i
$636$		0			0
$637$		−	10.6298i		−	0.421166i
$638$		0			0
$639$	−11.7247			−0.463820
$640$		0			0
$641$	3.47844			0.137390			0.0686951	−	0.997638i	$-0.478116\pi$
$641$	3.47844			0.137390			0.0686951	+	0.997638i	$0.478116\pi$
$642$		0			0
$643$			2.05348i			0.0809813i	0.999180	+	0.0404906i	$0.0128921\pi$
$643$			2.05348i			0.0809813i	−0.999180	+	0.0404906i	$0.987108\pi$
$644$		0			0
$645$	7.29214	−	33.1896i	0.287128	−	1.30684i
$646$		0			0
$647$			19.3408i			0.760367i	0.924911	+	0.380184i	$0.124139\pi$
$647$			19.3408i			0.760367i	−0.924911	+	0.380184i	$0.875861\pi$
$648$		0			0
$649$	5.35149			0.210064
$650$		0			0
$651$	−14.2862			−0.559922
$652$		0			0
$653$		−	21.9288i		−	0.858139i	−0.903271	−	0.429070i	$-0.858841\pi$
$653$		−	21.9288i		−	0.858139i	0.903271	−	0.429070i	$-0.141159\pi$
$654$		0			0
$655$	7.78459	−	35.4310i	0.304169	−	1.38440i
$656$		0			0
$657$			8.08158i			0.315293i
$658$		0			0
$659$	38.1351			1.48553			0.742767	−	0.669550i	$-0.233513\pi$
$659$	38.1351			1.48553			0.742767	+	0.669550i	$0.233513\pi$
$660$		0			0
$661$	−28.9007			−1.12411			−0.562053	−	0.827101i	$-0.689988\pi$
$661$	−28.9007			−1.12411			−0.562053	+	0.827101i	$0.689988\pi$
$662$		0			0
$663$		−	67.7398i		−	2.63079i
$664$		0			0
$665$	20.3713	+	4.47581i	0.789967	+	0.173565i
$666$		0			0
$667$		−	0.408263i		−	0.0158080i
$668$		0			0
$669$	−13.8455			−0.535299
$670$		0			0
$671$	7.11993			0.274862
$672$		0			0
$673$		−	49.5051i		−	1.90828i	−0.299361	−	0.954140i	$-0.596774\pi$
$673$		−	49.5051i		−	1.90828i	0.299361	−	0.954140i	$-0.403226\pi$
$674$		0			0
$675$	−8.87039	+	19.2120i	−0.341421	+	0.739470i
$676$		0			0
$677$			6.87426i			0.264199i	0.991236	+	0.132100i	$0.0421719\pi$
$677$			6.87426i			0.264199i	−0.991236	+	0.132100i	$0.957828\pi$
$678$		0			0
$679$	25.2776			0.970067
$680$		0			0
$681$	−49.2024			−1.88544
$682$		0			0
$683$		−	36.9887i		−	1.41533i	−0.706547	−	0.707666i	$-0.749748\pi$
$683$		−	36.9887i		−	1.41533i	0.706547	−	0.707666i	$-0.250252\pi$
$684$		0			0
$685$	−10.6953	−	2.34987i	−0.408645	−	0.0897839i
$686$		0			0
$687$		−	6.27817i		−	0.239527i
$688$		0			0
$689$	40.1439			1.52936
$690$		0			0
$691$	−38.5485			−1.46645			−0.733227	−	0.679984i	$-0.761987\pi$
$691$	−38.5485			−1.46645			−0.733227	+	0.679984i	$0.761987\pi$
$692$		0			0
$693$		−	2.15800i		−	0.0819755i
$694$		0			0
$695$	2.12998	−	9.69443i	0.0807946	−	0.367731i
$696$		0			0
$697$			36.3715i			1.37767i
$698$		0			0
$699$	−36.5263			−1.38155
$700$		0			0
$701$	−36.3146			−1.37158			−0.685791	−	0.727798i	$-0.740544\pi$
$701$	−36.3146			−1.37158			−0.685791	+	0.727798i	$0.740544\pi$
$702$		0			0
$703$			40.0374i			1.51004i
$704$		0			0
$705$	6.01840	−	27.3923i	0.226666	−	1.03165i
$706$		0			0
$707$		−	3.65462i		−	0.137446i
$708$		0			0
$709$	3.24463			0.121855			0.0609274	−	0.998142i	$-0.480594\pi$
$709$	3.24463			0.121855			0.0609274	+	0.998142i	$0.480594\pi$
$710$		0			0
$711$	−3.76815			−0.141317
$712$		0			0
$713$		−	3.19187i		−	0.119537i
$714$		0			0
$715$	14.6340	+	3.21525i	0.547279	+	0.120243i
$716$		0			0
$717$		−	6.23543i		−	0.232867i
$718$		0			0
$719$	10.4214			0.388654			0.194327	−	0.980937i	$-0.437748\pi$
$719$	10.4214			0.388654			0.194327	+	0.980937i	$0.437748\pi$
$720$		0			0
$721$	2.81424			0.104808
$722$		0			0
$723$		−	9.40345i		−	0.349718i
$724$		0			0
$725$	−1.85331	−	0.855693i	−0.0688301	−	0.0317796i
$726$		0			0
$727$			9.67651i			0.358882i	0.983769	+	0.179441i	$0.0574289\pi$
$727$			9.67651i			0.358882i	−0.983769	+	0.179441i	$0.942571\pi$
$728$		0			0
$729$	−15.7927			−0.584914
$730$		0			0
$731$	−44.9780			−1.66357
$732$		0			0
$733$		−	15.5782i		−	0.575396i	−0.957721	−	0.287698i	$-0.907110\pi$
$733$		−	15.5782i		−	0.575396i	0.957721	−	0.287698i	$-0.0928899\pi$
$734$		0			0
$735$	−7.63369	−	1.67721i	−0.281573	−	0.0618648i
$736$		0			0
$737$			0.319242i			0.0117594i
$738$		0			0
$739$	14.4328			0.530919			0.265459	−	0.964122i	$-0.414476\pi$
$739$	14.4328			0.530919			0.265459	+	0.964122i	$0.414476\pi$
$740$		0			0
$741$	47.6977			1.75222
$742$		0			0
$743$		−	4.50305i		−	0.165201i	−0.996583	−	0.0826005i	$-0.973677\pi$
$743$		−	4.50305i		−	0.165201i	0.996583	−	0.0826005i	$-0.0263226\pi$
$744$		0			0
$745$	1.33570	−	6.07934i	0.0489362	−	0.222730i
$746$		0			0
$747$			9.09483i			0.332763i
$748$		0			0
$749$	0.537257			0.0196309
$750$		0			0
$751$	−18.5451			−0.676721			−0.338361	−	0.941017i	$-0.609872\pi$
$751$	−18.5451			−0.676721			−0.338361	+	0.941017i	$0.609872\pi$
$752$		0			0
$753$		−	4.40279i		−	0.160447i
$754$		0			0
$755$	−5.72347	+	26.0500i	−0.208299	+	0.948055i
$756$		0			0
$757$		−	27.0366i		−	0.982663i	−0.870973	−	0.491332i	$-0.836510\pi$
$757$		−	27.0366i		−	0.982663i	0.870973	−	0.491332i	$-0.163490\pi$
$758$		0			0
$759$	2.20332			0.0799754
$760$		0			0
$761$	15.4066			0.558490			0.279245	−	0.960220i	$-0.409916\pi$
$761$	15.4066			0.558490			0.279245	+	0.960220i	$0.409916\pi$
$762$		0			0
$763$			16.9823i			0.614799i
$764$		0			0
$765$	−10.6453	−	2.33888i	−0.384880	−	0.0845625i
$766$		0			0
$767$			28.3667i			1.02426i
$768$		0			0
$769$	−15.1216			−0.545299			−0.272650	−	0.962113i	$-0.587900\pi$
$769$	−15.1216			−0.545299			−0.272650	+	0.962113i	$0.587900\pi$
$770$		0			0
$771$	21.5856			0.777388
$772$		0			0
$773$			28.7131i			1.03274i	0.856366	+	0.516370i	$0.172717\pi$
$773$			28.7131i			1.03274i	−0.856366	+	0.516370i	$0.827283\pi$
$774$		0			0
$775$	−14.4895	−	6.68996i	−0.520478	−	0.240311i
$776$		0			0
$777$			43.8784i			1.57413i
$778$		0			0
$779$	−25.6103			−0.917584
$780$		0			0
$781$	−15.6861			−0.561293
$782$		0			0
$783$		−	1.72784i		−	0.0617481i
$784$		0			0
$785$	50.1332	+	11.0148i	1.78933	+	0.393136i
$786$		0			0
$787$			16.3131i			0.581501i	0.956799	+	0.290750i	$0.0939049\pi$
$787$			16.3131i			0.581501i	−0.956799	+	0.290750i	$0.906095\pi$
$788$		0			0
$789$	−21.6713			−0.771518
$790$		0			0
$791$	5.21639			0.185473
$792$		0			0
$793$			37.7406i			1.34021i
$794$		0			0
$795$	6.33407	−	28.8291i	0.224646	−	1.02246i
$796$		0			0
$797$			2.88374i			0.102147i	0.998695	+	0.0510736i	$0.0162643\pi$
$797$			2.88374i			0.102147i	−0.998695	+	0.0510736i	$0.983736\pi$
$798$		0			0
$799$	−37.1216			−1.31327
$800$		0			0
$801$	4.77684			0.168781
$802$		0			0
$803$			10.8121i			0.381552i
$804$		0			0
$805$	−1.09594	+	4.98807i	−0.0386267	+	0.175806i
$806$		0			0
$807$		−	19.4086i		−	0.683215i
$808$		0			0
$809$	−39.7878			−1.39886			−0.699432	−	0.714699i	$-0.746564\pi$
$809$	−39.7878			−1.39886			−0.699432	+	0.714699i	$0.746564\pi$
$810$		0			0
$811$	−38.5454			−1.35351			−0.676756	−	0.736208i	$-0.736615\pi$
$811$	−38.5454			−1.35351			−0.676756	+	0.736208i	$0.736615\pi$
$812$		0			0
$813$			9.45853i			0.331725i
$814$		0			0
$815$	−22.2947	−	4.89841i	−0.780951	−	0.171584i
$816$		0			0
$817$		−	31.6705i		−	1.10801i
$818$		0			0
$819$	11.4389			0.399707
$820$		0			0
$821$	−16.3428			−0.570368			−0.285184	−	0.958473i	$-0.592055\pi$
$821$	−16.3428			−0.570368			−0.285184	+	0.958473i	$0.592055\pi$
$822$		0			0
$823$			38.6446i			1.34707i	0.739157	+	0.673534i	$0.235224\pi$
$823$			38.6446i			1.34707i	−0.739157	+	0.673534i	$0.764776\pi$
$824$		0			0
$825$	4.61802	−	10.0020i	0.160779	−	0.348224i
$826$		0			0
$827$		−	21.6969i		−	0.754474i	−0.926117	−	0.377237i	$-0.876874\pi$
$827$		−	21.6969i		−	0.754474i	0.926117	−	0.377237i	$-0.123126\pi$
$828$		0			0
$829$	−13.9429			−0.484259			−0.242129	−	0.970244i	$-0.577846\pi$
$829$	−13.9429			−0.484259			−0.242129	+	0.970244i	$0.577846\pi$
$830$		0			0
$831$	14.7820			0.512781
$832$		0			0
$833$			10.3451i			0.358435i
$834$		0			0
$835$	−14.9540	−	3.28557i	−0.517505	−	0.113702i
$836$		0			0
$837$		−	13.5086i		−	0.466925i
$838$		0			0
$839$	52.4484			1.81072			0.905360	−	0.424644i	$-0.139601\pi$
$839$	52.4484			1.81072			0.905360	+	0.424644i	$0.139601\pi$
$840$		0			0
$841$	−28.8333			−0.994252
$842$		0			0
$843$		−	11.7806i		−	0.405744i
$844$		0			0
$845$	−10.8051	+	49.1785i	−0.371706	+	1.69179i
$846$		0			0
$847$			22.2362i			0.764046i
$848$		0			0
$849$	−30.6472			−1.05181
$850$		0			0
$851$	−9.80345			−0.336058
$852$		0			0
$853$		−	30.4634i		−	1.04305i	−0.853237	−	0.521524i	$-0.825364\pi$
$853$		−	30.4634i		−	1.04305i	0.853237	−	0.521524i	$-0.174636\pi$
$854$		0			0
$855$	1.64688	−	7.49566i	0.0563222	−	0.256346i
$856$		0			0
$857$		−	42.4911i		−	1.45147i	−0.687975	−	0.725735i	$-0.741500\pi$
$857$		−	42.4911i		−	1.45147i	0.687975	−	0.725735i	$-0.258500\pi$
$858$		0			0
$859$	−15.5856			−0.531775			−0.265888	−	0.964004i	$-0.585665\pi$
$859$	−15.5856			−0.531775			−0.265888	+	0.964004i	$0.585665\pi$
$860$		0			0
$861$	−28.0673			−0.956529
$862$		0			0
$863$			50.8852i			1.73215i	0.499913	+	0.866076i	$0.333365\pi$
$863$			50.8852i			1.73215i	−0.499913	+	0.866076i	$0.666635\pi$
$864$		0			0
$865$	−9.38102	−	2.06112i	−0.318964	−	0.0700801i
$866$		0			0
$867$			32.6108i			1.10752i
$868$		0			0
$869$	−5.04131			−0.171015
$870$		0			0
$871$	−1.69220			−0.0573382
$872$		0			0
$873$		−	9.30094i		−	0.314789i
$874$		0			0
$875$	20.3463	+	15.4297i	0.687832	+	0.521619i
$876$		0			0
$877$		−	30.1080i		−	1.01667i	−0.861158	−	0.508337i	$-0.830260\pi$
$877$		−	30.1080i		−	1.01667i	0.861158	−	0.508337i	$-0.169740\pi$
$878$		0			0
$879$	6.30312			0.212599
$880$		0			0
$881$	20.2245			0.681379			0.340690	−	0.940176i	$-0.389339\pi$
$881$	20.2245			0.681379			0.340690	+	0.940176i	$0.389339\pi$
$882$		0			0
$883$		−	30.6482i		−	1.03139i	−0.856771	−	0.515697i	$-0.827533\pi$
$883$		−	30.6482i		−	1.03139i	0.856771	−	0.515697i	$-0.172467\pi$
$884$		0			0
$885$	20.3713	+	4.47581i	0.684775	+	0.150453i
$886$		0			0
$887$		−	14.0064i		−	0.470290i	−0.971960	−	0.235145i	$-0.924443\pi$
$887$		−	14.0064i		−	0.470290i	0.971960	−	0.235145i	$-0.0755565\pi$
$888$		0			0
$889$	23.3307			0.782487
$890$		0			0
$891$	12.1594			0.407356
$892$		0			0
$893$		−	26.1385i		−	0.874691i
$894$		0			0
$895$	−7.13174	+	32.4596i	−0.238388	+	1.08500i
$896$		0			0
$897$			11.6791i			0.389955i
$898$		0			0
$899$	1.30312			0.0434616
$900$		0			0
$901$	−39.0687			−1.30157
$902$		0			0
$903$		−	34.7088i		−	1.15504i
$904$		0			0
$905$	6.30088	−	28.6780i	0.209448	−	0.953289i
$906$		0			0
$907$			36.4739i			1.21109i	0.795809	+	0.605547i	$0.207046\pi$
$907$			36.4739i			1.21109i	−0.795809	+	0.605547i	$0.792954\pi$
$908$		0			0
$909$	−1.34472			−0.0446016
$910$		0			0
$911$	14.2327			0.471549			0.235775	−	0.971808i	$-0.424237\pi$
$911$	14.2327			0.471549			0.235775	+	0.971808i	$0.424237\pi$
$912$		0			0
$913$			12.1677i			0.402693i
$914$		0			0
$915$	27.1032	+	5.95488i	0.896004	+	0.196862i
$916$		0			0
$917$		−	37.0528i		−	1.22359i
$918$		0			0
$919$	34.3246			1.13226			0.566132	−	0.824315i	$-0.308439\pi$
$919$	34.3246			1.13226			0.566132	+	0.824315i	$0.308439\pi$
$920$		0			0
$921$	67.5508			2.22588
$922$		0			0
$923$		−	83.1474i		−	2.73683i
$924$		0			0
$925$	−20.5474	+	44.5028i	−0.675595	+	1.46324i
$926$		0			0
$927$		−	1.03550i		−	0.0340103i
$928$		0			0
$929$	−6.14579			−0.201637			−0.100818	−	0.994905i	$-0.532146\pi$
$929$	−6.14579			−0.201637			−0.100818	+	0.994905i	$0.532146\pi$
$930$		0			0
$931$	−7.28428			−0.238733
$932$		0			0
$933$		−	36.7330i		−	1.20258i
$934$		0			0
$935$	−14.2420	−	3.12913i	−0.465763	−	0.102334i
$936$		0			0
$937$			18.4581i			0.602998i	0.953466	+	0.301499i	$0.0974871\pi$
$937$			18.4581i			0.602998i	−0.953466	+	0.301499i	$0.902513\pi$
$938$		0			0
$939$	−19.4256			−0.633931
$940$		0			0
$941$	30.7623			1.00282			0.501411	−	0.865209i	$-0.332815\pi$
$941$	30.7623			1.00282			0.501411	+	0.865209i	$0.332815\pi$
$942$		0			0
$943$		−	6.27087i		−	0.204208i
$944$		0			0
$945$	−4.63820	+	21.1104i	−0.150881	+	0.686723i
$946$		0			0
$947$		−	11.4692i		−	0.372698i	−0.982484	−	0.186349i	$-0.940334\pi$
$947$		−	11.4692i		−	0.372698i	0.982484	−	0.186349i	$-0.0596655\pi$
$948$		0			0
$949$	−57.3119			−1.86042
$950$		0			0
$951$	−18.1080			−0.587192
$952$		0			0
$953$		−	2.40664i		−	0.0779586i	−0.999240	−	0.0389793i	$-0.987589\pi$
$953$		−	2.40664i		−	0.0779586i	0.999240	−	0.0389793i	$-0.0124106\pi$
$954$		0			0
$955$	−6.70914	+	30.5362i	−0.217103	+	0.988127i
$956$		0			0
$957$			0.899533i			0.0290778i
$958$		0			0
$959$	−11.1848			−0.361176
$960$		0			0
$961$	−20.8119			−0.671353
$962$		0			0
$963$		−	0.197684i		−	0.00637029i
$964$		0			0
$965$	−19.0283	−	4.18073i	−0.612541	−	0.134582i
$966$		0			0
$967$		−	57.4766i		−	1.84832i	−0.382001	−	0.924162i	$-0.624765\pi$
$967$		−	57.4766i		−	1.84832i	0.382001	−	0.924162i	$-0.375235\pi$
$968$		0			0
$969$	−46.4202			−1.49123
$970$		0			0
$971$	−53.1909			−1.70698			−0.853489	−	0.521111i	$-0.825518\pi$
$971$	−53.1909			−1.70698			−0.853489	+	0.521111i	$0.825518\pi$
$972$		0			0
$973$		−	10.1382i		−	0.325015i
$974$		0			0
$975$	53.0174	+	24.4787i	1.69792	+	0.783946i
$976$		0			0
$977$			54.4716i			1.74270i	0.490661	+	0.871350i	$0.336755\pi$
$977$			54.4716i			1.74270i	−0.490661	+	0.871350i	$0.663245\pi$
$978$		0			0
$979$	6.39080			0.204251
$980$		0			0
$981$	6.24864			0.199504
$982$		0			0
$983$			37.5908i			1.19896i	0.800390	+	0.599480i	$0.204626\pi$
$983$			37.5908i			1.19896i	−0.800390	+	0.599480i	$0.795374\pi$
$984$		0			0
$985$	−49.1124	−	10.7905i	−1.56485	−	0.343815i
$986$		0			0
$987$		−	28.6461i		−	0.911816i
$988$		0			0
$989$	7.75474			0.246587
$990$		0			0
$991$	−14.0545			−0.446455			−0.223228	−	0.974766i	$-0.571659\pi$
$991$	−14.0545			−0.446455			−0.223228	+	0.974766i	$0.571659\pi$
$992$		0			0
$993$		−	46.9706i		−	1.49057i
$994$		0			0
$995$	4.32377	−	19.6793i	0.137073	−	0.623877i
$996$		0			0
$997$			10.9378i			0.346404i	0.984886	+	0.173202i	$0.0554113\pi$
$997$			10.9378i			0.346404i	−0.984886	+	0.173202i	$0.944589\pi$
$998$		0			0
$999$	−41.4900			−1.31269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.d.369.2		8
4.3	odd	2		115.2.b.b.24.8	yes	8
5.2	odd	4		9200.2.a.cq.1.1		4
5.3	odd	4		9200.2.a.ck.1.4		4
5.4	even	2	inner	1840.2.e.d.369.7		8
12.11	even	2		1035.2.b.e.829.1		8
20.3	even	4		575.2.a.j.1.4		4
20.7	even	4		575.2.a.i.1.1		4
20.19	odd	2		115.2.b.b.24.1	✓	8
60.23	odd	4		5175.2.a.bw.1.1		4
60.47	odd	4		5175.2.a.bv.1.4		4
60.59	even	2		1035.2.b.e.829.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.1	✓	8	20.19	odd	2
115.2.b.b.24.8	yes	8	4.3	odd	2
575.2.a.i.1.1		4	20.7	even	4
575.2.a.j.1.4		4	20.3	even	4
1035.2.b.e.829.1		8	12.11	even	2
1035.2.b.e.829.8		8	60.59	even	2
1840.2.e.d.369.2		8	1.1	even	1	trivial
1840.2.e.d.369.7		8	5.4	even	2	inner
5175.2.a.bv.1.4		4	60.47	odd	4
5175.2.a.bw.1.1		4	60.23	odd	4
9200.2.a.ck.1.4		4	5.3	odd	4
9200.2.a.cq.1.1		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.95969i q^{3} +(0.479844 - 2.18398i) q^{5} -2.28394i q^{7} -0.840379 q^{9} +O(q^{10})\)	\(q-1.95969i q^{3} +(0.479844 - 2.18398i) q^{5} -2.28394i q^{7} -0.840379 q^{9} -1.12432 q^{11} -5.95969i q^{13} +(-4.27991 - 0.940345i) q^{15} +5.80007i q^{17} -4.08401 q^{19} -4.47581 q^{21} -1.00000i q^{23} +(-4.53950 - 2.09594i) q^{25} -4.23218i q^{27} +0.408263 q^{29} +3.19187 q^{31} +2.20332i q^{33} +(-4.98807 - 1.09594i) q^{35} -9.80345i q^{37} -11.6791 q^{39} +6.27087 q^{41} +7.75474i q^{43} +(-0.403251 + 1.83537i) q^{45} +6.40020i q^{47} +1.78361 q^{49} +11.3663 q^{51} +6.73590i q^{53} +(-0.539499 + 2.45549i) q^{55} +8.00339i q^{57} -4.75976 q^{59} -6.33265 q^{61} +1.91938i q^{63} +(-13.0158 - 2.85972i) q^{65} -0.283942i q^{67} -1.95969 q^{69} +13.9516 q^{71} -9.61659i q^{73} +(-4.10738 + 8.89600i) q^{75} +2.56788i q^{77} +4.48387 q^{79} -10.8149 q^{81} -10.8223i q^{83} +(12.6672 + 2.78313i) q^{85} -0.800068i q^{87} -5.68414 q^{89} -13.6116 q^{91} -6.25508i q^{93} +(-1.95969 + 8.91938i) q^{95} +11.0676i q^{97} +0.944856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 6 * q^5 - 8 * q^9	\( 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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