LMFDB - Embedded newform 1840.2.e.e.369.4

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:	8.0.11574317056.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{4} + 4 $ x^8 + 45*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.4
Root			$-1.83051 - 1.83051i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.e.369.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.706585i q^{3} +(-1.83051 + 1.28422i) q^{5} -2.40815i q^{7} +2.50074 q^{9} +O(q^{10})$	$q-0.706585i q^{3} +(-1.83051 + 1.28422i) q^{5} -2.40815i q^{7} +2.50074 q^{9} -4.95444 q^{11} +4.40815i q^{13} +(0.907411 + 1.29341i) q^{15} +0.200825i q^{17} +6.65600 q^{19} -1.70156 q^{21} -1.00000i q^{23} +(1.70156 - 4.70156i) q^{25} -3.88674i q^{27} -1.67942 q^{29} +1.82132 q^{31} +3.50074i q^{33} +(3.09259 + 4.40815i) q^{35} -8.47732i q^{37} +3.11473 q^{39} +11.0171 q^{41} +2.06268i q^{43} +(-4.57763 + 3.21149i) q^{45} -0.505760i q^{47} +1.20083 q^{49} +0.141900 q^{51} -5.84621i q^{53} +(9.06918 - 6.36259i) q^{55} -4.70304i q^{57} -4.41464 q^{59} -6.67815 q^{61} -6.02214i q^{63} +(-5.66103 - 8.06918i) q^{65} -12.6625i q^{67} -0.706585 q^{69} -3.61400 q^{71} -14.4589i q^{73} +(-3.32206 - 1.20230i) q^{75} +11.9310i q^{77} +9.95317 q^{79} +4.75590 q^{81} +13.3037i q^{83} +(-0.257904 - 0.367613i) q^{85} +1.18665i q^{87} +5.04576 q^{89} +10.6155 q^{91} -1.28692i q^{93} +(-12.1839 + 8.54777i) q^{95} +3.79917i q^{97} -12.3898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * 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
$3$		−	0.706585i		−	0.407947i	−0.978976	−	0.203974i	$-0.934614\pi$
$3$		−	0.706585i		−	0.407947i	0.978976	−	0.203974i	$-0.0653857\pi$
$4$		0			0
$5$	−1.83051	+	1.28422i	−0.818631	+	0.574320i
$5$	−1.83051	+	1.28422i	−0.818631	+	0.574320i
$6$		0			0
$7$		−	2.40815i		−	0.910194i	−0.890442	−	0.455097i	$-0.849605\pi$
$7$		−	2.40815i		−	0.910194i	0.890442	−	0.455097i	$-0.150395\pi$
$8$		0			0
$9$	2.50074			0.833579
$10$		0			0
$11$	−4.95444			−1.49382			−0.746910	−	0.664925i	$-0.768464\pi$
$11$	−4.95444			−1.49382			−0.746910	+	0.664925i	$0.768464\pi$
$12$		0			0
$13$			4.40815i			1.22260i	0.791399	+	0.611300i	$0.209353\pi$
$13$			4.40815i			1.22260i	−0.791399	+	0.611300i	$0.790647\pi$
$14$		0			0
$15$	0.907411	+	1.29341i	0.234292	+	0.333958i
$16$		0			0
$17$			0.200825i			0.0487073i	0.999703	+	0.0243536i	$0.00775277\pi$
$17$			0.200825i			0.0487073i	−0.999703	+	0.0243536i	$0.992247\pi$
$18$		0			0
$19$	6.65600			1.52699			0.763496	−	0.645812i	$-0.223481\pi$
$19$	6.65600			1.52699			0.763496	+	0.645812i	$0.223481\pi$
$20$		0			0
$21$	−1.70156			−0.371311
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	1.70156	−	4.70156i	0.340312	−	0.940312i
$26$		0			0
$27$		−	3.88674i		−	0.748004i
$28$		0			0
$29$	−1.67942			−0.311860			−0.155930	−	0.987768i	$-0.549837\pi$
$29$	−1.67942			−0.311860			−0.155930	+	0.987768i	$0.549837\pi$
$30$		0			0
$31$	1.82132			0.327118			0.163559	−	0.986534i	$-0.447703\pi$
$31$	1.82132			0.327118			0.163559	+	0.986534i	$0.447703\pi$
$32$		0			0
$33$			3.50074i			0.609400i
$34$		0			0
$35$	3.09259	+	4.40815i	0.522743	+	0.745113i
$36$		0			0
$37$		−	8.47732i		−	1.39366i	−0.717235	−	0.696832i	$-0.754592\pi$
$37$		−	8.47732i		−	1.39366i	0.717235	−	0.696832i	$-0.245408\pi$
$38$		0			0
$39$	3.11473			0.498756
$40$		0			0
$41$	11.0171			1.72059			0.860293	−	0.509801i	$-0.170281\pi$
$41$	11.0171			1.72059			0.860293	+	0.509801i	$0.170281\pi$
$42$		0			0
$43$			2.06268i			0.314555i	0.987554	+	0.157278i	$0.0502718\pi$
$43$			2.06268i			0.314555i	−0.987554	+	0.157278i	$0.949728\pi$
$44$		0			0
$45$	−4.57763	+	3.21149i	−0.682393	+	0.478741i
$46$		0			0
$47$		−	0.505760i		−	0.0737727i	−0.999319	−	0.0368864i	$-0.988256\pi$
$47$		−	0.505760i		−	0.0737727i	0.999319	−	0.0368864i	$-0.0117440\pi$
$48$		0			0
$49$	1.20083			0.171546
$50$		0			0
$51$	0.141900			0.0198700
$52$		0			0
$53$		−	5.84621i		−	0.803038i	−0.915851	−	0.401519i	$-0.868482\pi$
$53$		−	5.84621i		−	0.803038i	0.915851	−	0.401519i	$-0.131518\pi$
$54$		0			0
$55$	9.06918	−	6.36259i	1.22289	−	0.857931i
$56$		0			0
$57$		−	4.70304i		−	0.622932i
$58$		0			0
$59$	−4.41464			−0.574738			−0.287369	−	0.957820i	$-0.592781\pi$
$59$	−4.41464			−0.574738			−0.287369	+	0.957820i	$0.592781\pi$
$60$		0			0
$61$	−6.67815			−0.855049			−0.427525	−	0.904004i	$-0.640614\pi$
$61$	−6.67815			−0.855049			−0.427525	+	0.904004i	$0.640614\pi$
$62$		0			0
$63$		−	6.02214i		−	0.758719i
$64$		0			0
$65$	−5.66103	−	8.06918i	−0.702164	−	1.00086i
$66$		0			0
$67$		−	12.6625i		−	1.54697i	−0.633814	−	0.773485i	$-0.718512\pi$
$67$		−	12.6625i		−	1.54697i	0.633814	−	0.773485i	$-0.281488\pi$
$68$		0			0
$69$	−0.706585			−0.0850629
$70$		0			0
$71$	−3.61400			−0.428902			−0.214451	−	0.976735i	$-0.568796\pi$
$71$	−3.61400			−0.428902			−0.214451	+	0.976735i	$0.568796\pi$
$72$		0			0
$73$		−	14.4589i		−	1.69229i	−0.532953	−	0.846145i	$-0.678918\pi$
$73$		−	14.4589i		−	1.69229i	0.532953	−	0.846145i	$-0.321082\pi$
$74$		0			0
$75$	−3.32206	−	1.20230i	−0.383598	−	0.138830i
$76$		0			0
$77$			11.9310i			1.35967i
$78$		0			0
$79$	9.95317			1.11982			0.559910	−	0.828554i	$-0.310836\pi$
$79$	9.95317			1.11982			0.559910	+	0.828554i	$0.310836\pi$
$80$		0			0
$81$	4.75590			0.528433
$82$		0			0
$83$			13.3037i			1.46027i	0.683305	+	0.730133i	$0.260542\pi$
$83$			13.3037i			1.46027i	−0.683305	+	0.730133i	$0.739458\pi$
$84$		0			0
$85$	−0.257904	−	0.367613i	−0.0279736	−	0.0398733i
$86$		0			0
$87$			1.18665i			0.127223i
$88$		0			0
$89$	5.04576			0.534850			0.267425	−	0.963579i	$-0.413827\pi$
$89$	5.04576			0.534850			0.267425	+	0.963579i	$0.413827\pi$
$90$		0			0
$91$	10.6155			1.11280
$92$		0			0
$93$		−	1.28692i		−	0.133447i
$94$		0			0
$95$	−12.1839	+	8.54777i	−1.25004	+	0.876983i
$96$		0			0
$97$			3.79917i			0.385748i	0.981224	+	0.192874i	$0.0617808\pi$
$97$			3.79917i			0.385748i	−0.981224	+	0.192874i	$0.938219\pi$
$98$		0			0
$99$	−12.3898			−1.24522
$100$		0			0
$101$	13.2410			1.31753			0.658764	−	0.752350i	$-0.271080\pi$
$101$	13.2410			1.31753			0.658764	+	0.752350i	$0.271080\pi$
$102$		0			0
$103$		−	10.4202i		−	1.02674i	−0.858168	−	0.513369i	$-0.828397\pi$
$103$		−	10.4202i		−	1.02674i	0.858168	−	0.513369i	$-0.171603\pi$
$104$		0			0
$105$	3.11473	−	2.18518i	0.303967	−	0.213252i
$106$		0			0
$107$		−	9.98161i		−	0.964959i	−0.875907	−	0.482479i	$-0.839736\pi$
$107$		−	9.98161i		−	0.964959i	0.875907	−	0.482479i	$-0.160264\pi$
$108$		0			0
$109$	13.8739			1.32888			0.664442	−	0.747340i	$-0.268669\pi$
$109$	13.8739			1.32888			0.664442	+	0.747340i	$0.268669\pi$
$110$		0			0
$111$	−5.98995			−0.568541
$112$		0			0
$113$		−	18.4957i		−	1.73993i	−0.493113	−	0.869965i	$-0.664141\pi$
$113$		−	18.4957i		−	1.73993i	0.493113	−	0.869965i	$-0.335859\pi$
$114$		0			0
$115$	1.28422	+	1.83051i	0.119754	+	0.170696i
$116$		0			0
$117$			11.0236i			1.01913i
$118$		0			0
$119$	0.483617			0.0443331
$120$		0			0
$121$	13.5465			1.23150
$122$		0			0
$123$		−	7.78454i		−	0.701908i
$124$		0			0
$125$	2.92310	+	10.7915i	0.261450	+	0.965217i
$126$		0			0
$127$			18.1384i			1.60952i	0.593601	+	0.804759i	$0.297706\pi$
$127$			18.1384i			1.60952i	−0.593601	+	0.804759i	$0.702294\pi$
$128$		0			0
$129$	1.45746			0.128322
$130$		0			0
$131$	1.98995			0.173863			0.0869315	−	0.996214i	$-0.472294\pi$
$131$	1.98995			0.173863			0.0869315	+	0.996214i	$0.472294\pi$
$132$		0			0
$133$		−	16.0286i		−	1.38986i
$134$		0			0
$135$	4.99143	+	7.11473i	0.429594	+	0.612339i
$136$		0			0
$137$		−	1.14337i		−	0.0976850i	−0.998806	−	0.0488425i	$-0.984447\pi$
$137$		−	1.14337i		−	0.0976850i	0.998806	−	0.0488425i	$-0.0155532\pi$
$138$		0			0
$139$	22.4489			1.90409			0.952045	−	0.305959i	$-0.0989769\pi$
$139$	22.4489			1.90409			0.952045	+	0.305959i	$0.0989769\pi$
$140$		0			0
$141$	−0.357363			−0.0300954
$142$		0			0
$143$		−	21.8399i		−	1.82635i
$144$		0			0
$145$	3.07420	−	2.15674i	0.255298	−	0.179108i
$146$		0			0
$147$		−	0.848486i		−	0.0699819i
$148$		0			0
$149$	−8.20458			−0.672145			−0.336073	−	0.941836i	$-0.609099\pi$
$149$	−8.20458			−0.672145			−0.336073	+	0.941836i	$0.609099\pi$
$150$		0			0
$151$	−10.5244			−0.856460			−0.428230	−	0.903670i	$-0.640863\pi$
$151$	−10.5244			−0.856460			−0.428230	+	0.903670i	$0.640863\pi$
$152$		0			0
$153$			0.502211i			0.0406013i
$154$		0			0
$155$	−3.33395	+	2.33897i	−0.267789	+	0.187871i
$156$		0			0
$157$			1.02991i			0.0821959i	0.999155	+	0.0410979i	$0.0130856\pi$
$157$			1.02991i			0.0821959i	−0.999155	+	0.0410979i	$0.986914\pi$
$158$		0			0
$159$	−4.13084			−0.327597
$160$		0			0
$161$	−2.40815			−0.189789
$162$		0			0
$163$		−	11.1802i		−	0.875697i	−0.899049	−	0.437849i	$-0.855741\pi$
$163$		−	11.1802i		−	0.875697i	0.899049	−	0.437849i	$-0.144259\pi$
$164$		0			0
$165$	−4.49571	−	6.40815i	−0.349991	−	0.498874i
$166$		0			0
$167$		−	3.49424i		−	0.270392i	−0.990819	−	0.135196i	$-0.956834\pi$
$167$		−	3.49424i		−	0.270392i	0.990819	−	0.135196i	$-0.0431665\pi$
$168$		0			0
$169$	−6.43177			−0.494751
$170$		0			0
$171$	16.6449			1.27287
$172$		0			0
$173$		−	8.61547i		−	0.655022i	−0.944847	−	0.327511i	$-0.893790\pi$
$173$		−	8.61547i		−	0.655022i	0.944847	−	0.327511i	$-0.106210\pi$
$174$		0			0
$175$	−11.3221	−	4.09761i	−0.855867	−	0.309750i
$176$		0			0
$177$			3.11932i			0.234463i
$178$		0			0
$179$	−3.36634			−0.251612			−0.125806	−	0.992055i	$-0.540152\pi$
$179$	−3.36634			−0.251612			−0.125806	+	0.992055i	$0.540152\pi$
$180$		0			0
$181$	−15.8970			−1.18161			−0.590807	−	0.806813i	$-0.701191\pi$
$181$	−15.8970			−1.18161			−0.590807	+	0.806813i	$0.701191\pi$
$182$		0			0
$183$			4.71868i			0.348815i
$184$		0			0
$185$	10.8867	+	15.5179i	0.800409	+	1.14090i
$186$		0			0
$187$		−	0.994977i		−	0.0727599i
$188$		0			0
$189$	−9.35985			−0.680829
$190$		0			0
$191$	6.74527			0.488071			0.244035	−	0.969766i	$-0.421529\pi$
$191$	6.74527			0.488071			0.244035	+	0.969766i	$0.421529\pi$
$192$		0			0
$193$			4.81630i			0.346685i	0.984862	+	0.173342i	$0.0554567\pi$
$193$			4.81630i			0.346685i	−0.984862	+	0.173342i	$0.944543\pi$
$194$		0			0
$195$	−5.70156	+	4.00000i	−0.408297	+	0.286446i
$196$		0			0
$197$			12.8349i			0.914448i	0.889352	+	0.457224i	$0.151156\pi$
$197$			12.8349i			0.914448i	−0.889352	+	0.457224i	$0.848844\pi$
$198$		0			0
$199$	−12.4957			−0.885798			−0.442899	−	0.896572i	$-0.646050\pi$
$199$	−12.4957			−0.885798			−0.442899	+	0.896572i	$0.646050\pi$
$200$		0			0
$201$	−8.94714			−0.631083
$202$		0			0
$203$			4.04429i			0.283853i
$204$		0			0
$205$	−20.1670	+	14.1484i	−1.40852	+	0.988167i
$206$		0			0
$207$		−	2.50074i		−	0.173813i
$208$		0			0
$209$	−32.9768			−2.28105
$210$		0			0
$211$	16.0130			1.10238			0.551190	−	0.834380i	$-0.314174\pi$
$211$	16.0130			1.10238			0.551190	+	0.834380i	$0.314174\pi$
$212$		0			0
$213$			2.55360i			0.174970i
$214$		0			0
$215$	−2.64893	−	3.77576i	−0.180656	−	0.257505i
$216$		0			0
$217$		−	4.38600i		−	0.297741i
$218$		0			0
$219$	−10.2165			−0.690365
$220$		0			0
$221$	−0.885267			−0.0595495
$222$		0			0
$223$			18.2094i			1.21939i	0.792636	+	0.609695i	$0.208708\pi$
$223$			18.2094i			1.21939i	−0.792636	+	0.609695i	$0.791292\pi$
$224$		0			0
$225$	4.25516	−	11.7574i	0.283677	−	0.783825i
$226$		0			0
$227$			19.5331i			1.29646i	0.761445	+	0.648230i	$0.224490\pi$
$227$			19.5331i			1.29646i	−0.761445	+	0.648230i	$0.775510\pi$
$228$		0			0
$229$	22.3419			1.47640			0.738198	−	0.674584i	$-0.235677\pi$
$229$	22.3419			1.47640			0.738198	+	0.674584i	$0.235677\pi$
$230$		0			0
$231$	8.43029			0.554672
$232$		0			0
$233$		−	6.42469i		−	0.420896i	−0.977605	−	0.210448i	$-0.932508\pi$
$233$		−	6.42469i		−	0.420896i	0.977605	−	0.210448i	$-0.0674922\pi$
$234$		0			0
$235$	0.649507	+	0.925801i	0.0423692	+	0.0603926i
$236$		0			0
$237$		−	7.03277i		−	0.456827i
$238$		0			0
$239$	−9.52848			−0.616346			−0.308173	−	0.951330i	$-0.599718\pi$
$239$	−9.52848			−0.616346			−0.308173	+	0.951330i	$0.599718\pi$
$240$		0			0
$241$	−13.4842			−0.868593			−0.434297	−	0.900770i	$-0.643003\pi$
$241$	−13.4842			−0.868593			−0.434297	+	0.900770i	$0.643003\pi$
$242$		0			0
$243$		−	15.0207i		−	0.963576i
$244$		0			0
$245$	−2.19813	+	1.54212i	−0.140433	+	0.0985226i
$246$		0			0
$247$			29.3406i			1.86690i
$248$		0			0
$249$	9.40018			0.595712
$250$		0			0
$251$	10.1402			0.640044			0.320022	−	0.947410i	$-0.396310\pi$
$251$	10.1402			0.640044			0.320022	+	0.947410i	$0.396310\pi$
$252$		0			0
$253$			4.95444i			0.311483i
$254$		0			0
$255$	−0.259750	+	0.182231i	−0.0162662	+	0.0114117i
$256$		0			0
$257$		−	8.76196i		−	0.546556i	−0.961935	−	0.273278i	$-0.911892\pi$
$257$		−	8.76196i		−	0.546556i	0.961935	−	0.273278i	$-0.0881079\pi$
$258$		0			0
$259$	−20.4146			−1.26850
$260$		0			0
$261$	−4.19978			−0.259960
$262$		0			0
$263$		−	15.7448i		−	0.970868i	−0.874273	−	0.485434i	$-0.838662\pi$
$263$		−	15.7448i		−	0.970868i	0.874273	−	0.485434i	$-0.161338\pi$
$264$		0			0
$265$	7.50781	+	10.7016i	0.461201	+	0.657392i
$266$		0			0
$267$		−	3.56526i		−	0.218190i
$268$		0			0
$269$	−23.5515			−1.43596			−0.717981	−	0.696063i	$-0.754933\pi$
$269$	−23.5515			−1.43596			−0.717981	+	0.696063i	$0.754933\pi$
$270$		0			0
$271$	1.62863			0.0989325			0.0494662	−	0.998776i	$-0.484248\pi$
$271$	1.62863			0.0989325			0.0494662	+	0.998776i	$0.484248\pi$
$272$		0			0
$273$		−	7.50074i		−	0.453965i
$274$		0			0
$275$	−8.43029	+	23.2936i	−0.508366	+	1.40466i
$276$		0			0
$277$			21.9063i			1.31622i	0.752920	+	0.658112i	$0.228645\pi$
$277$			21.9063i			1.31622i	−0.752920	+	0.658112i	$0.771355\pi$
$278$		0			0
$279$	4.55464			0.272679
$280$		0			0
$281$	−5.64264			−0.336612			−0.168306	−	0.985735i	$-0.553830\pi$
$281$	−5.64264			−0.336612			−0.168306	+	0.985735i	$0.553830\pi$
$282$		0			0
$283$		−	6.15674i		−	0.365980i	−0.983115	−	0.182990i	$-0.941422\pi$
$283$		−	6.15674i		−	0.365980i	0.983115	−	0.182990i	$-0.0585776\pi$
$284$		0			0
$285$	6.03973	+	8.60897i	0.357763	+	0.509952i
$286$		0			0
$287$		−	26.5309i		−	1.56607i
$288$		0			0
$289$	16.9597			0.997628
$290$		0			0
$291$	2.68444			0.157365
$292$		0			0
$293$			21.4888i			1.25539i	0.778459	+	0.627696i	$0.216002\pi$
$293$			21.4888i			1.25539i	−0.778459	+	0.627696i	$0.783998\pi$
$294$		0			0
$295$	8.08107	−	5.66937i	0.470498	−	0.330084i
$296$		0			0
$297$			19.2566i			1.11738i
$298$		0			0
$299$	4.40815			0.254930
$300$		0			0
$301$	4.96723			0.286307
$302$		0			0
$303$		−	9.35589i		−	0.537482i
$304$		0			0
$305$	12.2244	−	8.57621i	0.699970	−	0.491072i
$306$		0			0
$307$			4.09672i			0.233812i	0.993143	+	0.116906i	$0.0372976\pi$
$307$			4.09672i			0.233812i	−0.993143	+	0.116906i	$0.962702\pi$
$308$		0			0
$309$	−7.36279			−0.418855
$310$		0			0
$311$	−24.0105			−1.36151			−0.680754	−	0.732512i	$-0.738348\pi$
$311$	−24.0105			−1.36151			−0.680754	+	0.732512i	$0.738348\pi$
$312$		0			0
$313$		−	10.8696i		−	0.614387i	−0.951647	−	0.307194i	$-0.900610\pi$
$313$		−	10.8696i		−	0.614387i	0.951647	−	0.307194i	$-0.0993899\pi$
$314$		0			0
$315$	7.73375	+	11.0236i	0.435748	+	0.621110i
$316$		0			0
$317$		−	22.1661i		−	1.24497i	−0.782631	−	0.622486i	$-0.786123\pi$
$317$		−	22.1661i		−	1.24497i	0.782631	−	0.622486i	$-0.213877\pi$
$318$		0			0
$319$	8.32058			0.465863
$320$		0			0
$321$	−7.05286			−0.393652
$322$		0			0
$323$			1.33669i			0.0743756i
$324$		0			0
$325$	20.7252	+	7.50074i	1.14963	+	0.416066i
$326$		0			0
$327$		−	9.80313i		−	0.542114i
$328$		0			0
$329$	−1.21795			−0.0671475
$330$		0			0
$331$	9.39308			0.516290			0.258145	−	0.966106i	$-0.416889\pi$
$331$	9.39308			0.516290			0.258145	+	0.966106i	$0.416889\pi$
$332$		0			0
$333$		−	21.1996i		−	1.16173i
$334$		0			0
$335$	16.2614	+	23.1789i	0.888457	+	1.26640i
$336$		0			0
$337$			19.8113i			1.07919i	0.841925	+	0.539594i	$0.181422\pi$
$337$			19.8113i			1.07919i	−0.841925	+	0.539594i	$0.818578\pi$
$338$		0			0
$339$	−13.0688			−0.709800
$340$		0			0
$341$	−9.02362			−0.488656
$342$		0			0
$343$		−	19.7488i		−	1.06633i
$344$		0			0
$345$	1.29341	−	0.907411i	0.0696351	−	0.0488533i
$346$		0			0
$347$			19.5128i			1.04750i	0.851871	+	0.523752i	$0.175468\pi$
$347$			19.5128i			1.04750i	−0.851871	+	0.523752i	$0.824532\pi$
$348$		0			0
$349$	22.7352			1.21699			0.608494	−	0.793558i	$-0.291774\pi$
$349$	22.7352			1.21699			0.608494	+	0.793558i	$0.291774\pi$
$350$		0			0
$351$	17.1333			0.914509
$352$		0			0
$353$			24.6471i			1.31183i	0.754835	+	0.655915i	$0.227717\pi$
$353$			24.6471i			1.31183i	−0.754835	+	0.655915i	$0.772283\pi$
$354$		0			0
$355$	6.61547	−	4.64116i	0.351113	−	0.246327i
$356$		0			0
$357$		−	0.341716i		−	0.0180856i
$358$		0			0
$359$	15.2778			0.806330			0.403165	−	0.915127i	$-0.367910\pi$
$359$	15.2778			0.806330			0.403165	+	0.915127i	$0.367910\pi$
$360$		0			0
$361$	25.3024			1.33170
$362$		0			0
$363$		−	9.57176i		−	0.502387i
$364$		0			0
$365$	18.5684	+	26.4673i	0.971916	+	1.38536i
$366$		0			0
$367$		−	9.59581i		−	0.500897i	−0.968130	−	0.250449i	$-0.919422\pi$
$367$		−	9.59581i		−	0.500897i	0.968130	−	0.250449i	$-0.0805781\pi$
$368$		0			0
$369$	27.5509			1.43424
$370$		0			0
$371$	−14.0785			−0.730921
$372$		0			0
$373$			17.2125i			0.891232i	0.895224	+	0.445616i	$0.147015\pi$
$373$			17.2125i			0.891232i	−0.895224	+	0.445616i	$0.852985\pi$
$374$		0			0
$375$	7.62508	−	2.06542i	0.393758	−	0.106658i
$376$		0			0
$377$		−	7.40312i		−	0.381280i
$378$		0			0
$379$	−20.7354			−1.06511			−0.532554	−	0.846396i	$-0.678768\pi$
$379$	−20.7354			−1.06511			−0.532554	+	0.846396i	$0.678768\pi$
$380$		0			0
$381$	12.8163			0.656599
$382$		0			0
$383$			13.0115i			0.664858i	0.943128	+	0.332429i	$0.107868\pi$
$383$			13.0115i			0.664858i	−0.943128	+	0.332429i	$0.892132\pi$
$384$		0			0
$385$	−15.3221	−	21.8399i	−0.780884	−	1.11307i
$386$		0			0
$387$			5.15822i			0.262207i
$388$		0			0
$389$	−6.55279			−0.332240			−0.166120	−	0.986106i	$-0.553124\pi$
$389$	−6.55279			−0.332240			−0.166120	+	0.986106i	$0.553124\pi$
$390$		0			0
$391$	0.200825			0.0101562
$392$		0			0
$393$		−	1.40607i		−	0.0709270i
$394$		0			0
$395$	−18.2194	+	12.7821i	−0.916718	+	0.643135i
$396$		0			0
$397$			6.79516i			0.341039i	0.985354	+	0.170520i	$0.0545446\pi$
$397$			6.79516i			0.341039i	−0.985354	+	0.170520i	$0.945455\pi$
$398$		0			0
$399$	−11.3256			−0.566989
$400$		0			0
$401$	−30.5660			−1.52639			−0.763196	−	0.646167i	$-0.776371\pi$
$401$	−30.5660			−1.52639			−0.763196	+	0.646167i	$0.776371\pi$
$402$		0			0
$403$			8.02864i			0.399935i
$404$		0			0
$405$	−8.70573	+	6.10761i	−0.432591	+	0.303490i
$406$		0			0
$407$			42.0004i			2.08188i
$408$		0			0
$409$	−19.8610			−0.982066			−0.491033	−	0.871141i	$-0.663380\pi$
$409$	−19.8610			−0.982066			−0.491033	+	0.871141i	$0.663380\pi$
$410$		0			0
$411$	−0.807892			−0.0398503
$412$		0			0
$413$			10.6311i			0.523123i
$414$		0			0
$415$	−17.0848	−	24.3525i	−0.838661	−	1.19542i
$416$		0			0
$417$		−	15.8621i		−	0.776768i
$418$		0			0
$419$	4.19057			0.204723			0.102361	−	0.994747i	$-0.467360\pi$
$419$	4.19057			0.204723			0.102361	+	0.994747i	$0.467360\pi$
$420$		0			0
$421$	−32.2306			−1.57082			−0.785411	−	0.618975i	$-0.787548\pi$
$421$	−32.2306			−1.57082			−0.785411	+	0.618975i	$0.787548\pi$
$422$		0			0
$423$		−	1.26477i		−	0.0614954i
$424$		0			0
$425$	0.944192	+	0.341716i	0.0458000	+	0.0165757i
$426$		0			0
$427$			16.0820i			0.778261i
$428$		0			0
$429$	−15.4318			−0.745053
$430$		0			0
$431$	−18.7583			−0.903554			−0.451777	−	0.892131i	$-0.649210\pi$
$431$	−18.7583			−0.903554			−0.451777	+	0.892131i	$0.649210\pi$
$432$		0			0
$433$		−	28.3271i		−	1.36131i	−0.732603	−	0.680656i	$-0.761695\pi$
$433$		−	28.3271i		−	1.36131i	0.732603	−	0.680656i	$-0.238305\pi$
$434$		0			0
$435$	−1.52392	−	2.17218i	−0.0730665	−	0.104148i
$436$		0			0
$437$		−	6.65600i		−	0.318400i
$438$		0			0
$439$	23.7532			1.13368			0.566840	−	0.823828i	$-0.308166\pi$
$439$	23.7532			1.13368			0.566840	+	0.823828i	$0.308166\pi$
$440$		0			0
$441$	3.00295			0.142998
$442$		0			0
$443$			9.91284i			0.470973i	0.971878	+	0.235487i	$0.0756684\pi$
$443$			9.91284i			0.470973i	−0.971878	+	0.235487i	$0.924332\pi$
$444$		0			0
$445$	−9.23634	+	6.47986i	−0.437844	+	0.307175i
$446$		0			0
$447$			5.79724i			0.274200i
$448$		0			0
$449$	39.3758			1.85826			0.929129	−	0.369755i	$-0.120558\pi$
$449$	39.3758			1.85826			0.929129	+	0.369755i	$0.120558\pi$
$450$		0			0
$451$	−54.5837			−2.57025
$452$		0			0
$453$			7.43636i			0.349390i
$454$		0			0
$455$	−19.4318	+	13.6326i	−0.910975	+	0.639106i
$456$		0			0
$457$			15.1737i			0.709794i	0.934905	+	0.354897i	$0.115484\pi$
$457$			15.1737i			0.709794i	−0.934905	+	0.354897i	$0.884516\pi$
$458$		0			0
$459$	0.780555			0.0364332
$460$		0			0
$461$	8.44848			0.393485			0.196742	−	0.980455i	$-0.436964\pi$
$461$	8.44848			0.393485			0.196742	+	0.980455i	$0.436964\pi$
$462$		0			0
$463$			19.8888i			0.924311i	0.886799	+	0.462155i	$0.152924\pi$
$463$			19.8888i			0.924311i	−0.886799	+	0.462155i	$0.847076\pi$
$464$		0			0
$465$	1.65268	+	2.35572i	0.0766414	+	0.109244i
$466$		0			0
$467$			27.3066i			1.26360i	0.775132	+	0.631800i	$0.217683\pi$
$467$			27.3066i			1.26360i	−0.775132	+	0.631800i	$0.782317\pi$
$468$		0			0
$469$	−30.4932			−1.40804
$470$		0			0
$471$	0.727720			0.0335316
$472$		0			0
$473$		−	10.2194i		−	0.469889i
$474$		0			0
$475$	11.3256	−	31.2936i	0.519654	−	1.43585i
$476$		0			0
$477$		−	14.6198i		−	0.669396i
$478$		0			0
$479$	21.0558			0.962064			0.481032	−	0.876703i	$-0.340262\pi$
$479$	21.0558			0.962064			0.481032	+	0.876703i	$0.340262\pi$
$480$		0			0
$481$	37.3693			1.70389
$482$		0			0
$483$			1.70156i			0.0774238i
$484$		0			0
$485$	−4.87897	−	6.95444i	−0.221543	−	0.315785i
$486$		0			0
$487$			22.5530i			1.02197i	0.859589	+	0.510987i	$0.170720\pi$
$487$			22.5530i			1.02197i	−0.859589	+	0.510987i	$0.829280\pi$
$488$		0			0
$489$	−7.89974			−0.357238
$490$		0			0
$491$	17.1369			0.773376			0.386688	−	0.922211i	$-0.373619\pi$
$491$	17.1369			0.773376			0.386688	+	0.922211i	$0.373619\pi$
$492$		0			0
$493$		−	0.337269i		−	0.0151899i
$494$		0			0
$495$	22.6796	−	15.9112i	1.01937	−	0.715154i
$496$		0			0
$497$			8.70304i			0.390385i
$498$		0			0
$499$	−12.1254			−0.542805			−0.271403	−	0.962466i	$-0.587488\pi$
$499$	−12.1254			−0.542805			−0.271403	+	0.962466i	$0.587488\pi$
$500$		0			0
$501$	−2.46898			−0.110306
$502$		0			0
$503$			38.5026i			1.71675i	0.513025	+	0.858373i	$0.328525\pi$
$503$			38.5026i			1.71675i	−0.513025	+	0.858373i	$0.671475\pi$
$504$		0			0
$505$	−24.2378	+	17.0043i	−1.07857	+	0.756683i
$506$		0			0
$507$			4.54459i			0.201832i
$508$		0			0
$509$	−11.8646			−0.525889			−0.262945	−	0.964811i	$-0.584694\pi$
$509$	−11.8646			−0.525889			−0.262945	+	0.964811i	$0.584694\pi$
$510$		0			0
$511$	−34.8192			−1.54031
$512$		0			0
$513$		−	25.8702i		−	1.14220i
$514$		0			0
$515$	13.3819	+	19.0744i	0.589676	+	0.840519i
$516$		0			0
$517$			2.50576i			0.110203i
$518$		0			0
$519$	−6.08757			−0.267214
$520$		0			0
$521$	0.876118			0.0383834			0.0191917	−	0.999816i	$-0.493891\pi$
$521$	0.876118			0.0383834			0.0191917	+	0.999816i	$0.493891\pi$
$522$		0			0
$523$		−	35.9716i		−	1.57293i	−0.617637	−	0.786463i	$-0.711910\pi$
$523$		−	35.9716i		−	1.57293i	0.617637	−	0.786463i	$-0.288090\pi$
$524$		0			0
$525$	−2.89531	+	8.00000i	−0.126362	+	0.349149i
$526$		0			0
$527$			0.365767i			0.0159330i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−11.0399			−0.479089
$532$		0			0
$533$			48.5651i			2.10359i
$534$		0			0
$535$	12.8186	+	18.2715i	0.554195	+	0.789945i
$536$		0			0
$537$			2.37861i			0.102645i
$538$		0			0
$539$	−5.94942			−0.256260
$540$		0			0
$541$	−32.7051			−1.40610			−0.703051	−	0.711140i	$-0.748179\pi$
$541$	−32.7051			−1.40610			−0.703051	+	0.711140i	$0.748179\pi$
$542$		0			0
$543$			11.2326i			0.482036i
$544$		0			0
$545$	−25.3965	+	17.8172i	−1.08786	+	0.763205i
$546$		0			0
$547$		−	17.1463i		−	0.733124i	−0.930394	−	0.366562i	$-0.880535\pi$
$547$		−	17.1463i		−	0.733124i	0.930394	−	0.366562i	$-0.119465\pi$
$548$		0			0
$549$	−16.7003			−0.712751
$550$		0			0
$551$	−11.1782			−0.476208
$552$		0			0
$553$		−	23.9687i		−	1.01925i
$554$		0			0
$555$	10.9647	−	7.69241i	0.465425	−	0.326525i
$556$		0			0
$557$		−	5.35049i		−	0.226708i	−0.993555	−	0.113354i	$-0.963841\pi$
$557$		−	5.35049i		−	0.226708i	0.993555	−	0.113354i	$-0.0361594\pi$
$558$		0			0
$559$	−9.09259			−0.384576
$560$		0			0
$561$	−0.703036			−0.0296822
$562$		0			0
$563$		−	12.7511i		−	0.537394i	−0.963225	−	0.268697i	$-0.913407\pi$
$563$		−	12.7511i		−	0.537394i	0.963225	−	0.268697i	$-0.0865930\pi$
$564$		0			0
$565$	23.7526	+	33.8567i	0.999277	+	1.42436i
$566$		0			0
$567$		−	11.4529i		−	0.480977i
$568$		0			0
$569$	−38.8368			−1.62812			−0.814062	−	0.580779i	$-0.802748\pi$
$569$	−38.8368			−1.62812			−0.814062	+	0.580779i	$0.802748\pi$
$570$		0			0
$571$	8.80350			0.368415			0.184208	−	0.982887i	$-0.441028\pi$
$571$	8.80350			0.368415			0.184208	+	0.982887i	$0.441028\pi$
$572$		0			0
$573$		−	4.76611i		−	0.199107i
$574$		0			0
$575$	−4.70156	−	1.70156i	−0.196069	−	0.0709600i
$576$		0			0
$577$		−	16.5133i		−	0.687456i	−0.939069	−	0.343728i	$-0.888310\pi$
$577$		−	16.5133i		−	0.687456i	0.939069	−	0.343728i	$-0.111690\pi$
$578$		0			0
$579$	3.40312			0.141429
$580$		0			0
$581$	32.0372			1.32913
$582$		0			0
$583$			28.9647i			1.19959i
$584$		0			0
$585$	−14.1567	−	20.1789i	−0.585309	−	0.834294i
$586$		0			0
$587$		−	31.0543i		−	1.28175i	−0.767646	−	0.640874i	$-0.778572\pi$
$587$		−	31.0543i		−	1.28175i	0.767646	−	0.640874i	$-0.221428\pi$
$588$		0			0
$589$	12.1227			0.499507
$590$		0			0
$591$	9.06895			0.373047
$592$		0			0
$593$		−	10.6211i		−	0.436155i	−0.975931	−	0.218078i	$-0.930021\pi$
$593$		−	10.6211i		−	0.436155i	0.975931	−	0.218078i	$-0.0699786\pi$
$594$		0			0
$595$	−0.885267	+	0.621070i	−0.0362924	+	0.0254614i
$596$		0			0
$597$			8.82929i			0.361359i
$598$		0			0
$599$	5.31351			0.217104			0.108552	−	0.994091i	$-0.465379\pi$
$599$	5.31351			0.217104			0.108552	+	0.994091i	$0.465379\pi$
$600$		0			0
$601$	−15.6336			−0.637708			−0.318854	−	0.947804i	$-0.603298\pi$
$601$	−15.6336			−0.637708			−0.318854	+	0.947804i	$0.603298\pi$
$602$		0			0
$603$		−	31.6656i		−	1.28952i
$604$		0			0
$605$	−24.7971	+	17.3967i	−1.00814	+	0.707275i
$606$		0			0
$607$		−	15.0588i		−	0.611216i	−0.952157	−	0.305608i	$-0.901140\pi$
$607$		−	15.0588i		−	0.611216i	0.952157	−	0.305608i	$-0.0988597\pi$
$608$		0			0
$609$	2.85763			0.115797
$610$		0			0
$611$	2.22947			0.0901945
$612$		0			0
$613$		−	44.1496i		−	1.78319i	−0.452836	−	0.891594i	$-0.649588\pi$
$613$		−	44.1496i		−	1.78319i	0.452836	−	0.891594i	$-0.350412\pi$
$614$		0			0
$615$	9.99705	+	14.2497i	0.403120	+	0.574604i
$616$		0			0
$617$			10.1871i			0.410117i	0.978750	+	0.205058i	$0.0657384\pi$
$617$			10.1871i			0.410117i	−0.978750	+	0.205058i	$0.934262\pi$
$618$		0			0
$619$	2.48087			0.0997147			0.0498573	−	0.998756i	$-0.484123\pi$
$619$	2.48087			0.0997147			0.0498573	+	0.998756i	$0.484123\pi$
$620$		0			0
$621$	−3.88674			−0.155970
$622$		0			0
$623$		−	12.1509i		−	0.486817i
$624$		0			0
$625$	−19.2094	−	16.0000i	−0.768375	−	0.640000i
$626$		0			0
$627$			23.3009i			0.930549i
$628$		0			0
$629$	1.70246			0.0678815
$630$		0			0
$631$	−14.0029			−0.557449			−0.278724	−	0.960371i	$-0.589912\pi$
$631$	−14.0029			−0.557449			−0.278724	+	0.960371i	$0.589912\pi$
$632$		0			0
$633$		−	11.3145i		−	0.449713i
$634$		0			0
$635$	−23.2936	−	33.2025i	−0.924379	−	1.31760i
$636$		0			0
$637$			5.29341i			0.209733i
$638$		0			0
$639$	−9.03765			−0.357524
$640$		0			0
$641$	21.3592			0.843639			0.421820	−	0.906680i	$-0.361392\pi$
$641$	21.3592			0.843639			0.421820	+	0.906680i	$0.361392\pi$
$642$		0			0
$643$		−	7.65098i		−	0.301725i	−0.988555	−	0.150863i	$-0.951795\pi$
$643$		−	7.65098i		−	0.301725i	0.988555	−	0.150863i	$-0.0482051\pi$
$644$		0			0
$645$	−2.66790	+	1.87170i	−0.105048	+	0.0736980i
$646$		0			0
$647$		−	9.66388i		−	0.379926i	−0.981791	−	0.189963i	$-0.939163\pi$
$647$		−	9.66388i		−	0.379926i	0.981791	−	0.189963i	$-0.0608369\pi$
$648$		0			0
$649$	21.8721			0.858555
$650$		0			0
$651$	−3.09909			−0.121463
$652$		0			0
$653$			7.76698i			0.303946i	0.988385	+	0.151973i	$0.0485626\pi$
$653$			7.76698i			0.303946i	−0.988385	+	0.151973i	$0.951437\pi$
$654$		0			0
$655$	−3.64264	+	2.55554i	−0.142330	+	0.0998531i
$656$		0			0
$657$		−	36.1580i		−	1.41066i
$658$		0			0
$659$	10.6157			0.413528			0.206764	−	0.978391i	$-0.433707\pi$
$659$	10.6157			0.413528			0.206764	+	0.978391i	$0.433707\pi$
$660$		0			0
$661$	24.7693			0.963413			0.481706	−	0.876333i	$-0.340017\pi$
$661$	24.7693			0.963413			0.481706	+	0.876333i	$0.340017\pi$
$662$		0			0
$663$			0.625517i			0.0242931i
$664$		0			0
$665$	20.5843	+	29.3406i	0.798224	+	1.13778i
$666$		0			0
$667$			1.67942i			0.0650273i
$668$		0			0
$669$	12.8665			0.497447
$670$		0			0
$671$	33.0865			1.27729
$672$		0			0
$673$			42.4961i			1.63811i	0.573719	+	0.819053i	$0.305500\pi$
$673$			42.4961i			1.63811i	−0.573719	+	0.819053i	$0.694500\pi$
$674$		0			0
$675$	−18.2738	−	6.61353i	−0.703357	−	0.254555i
$676$		0			0
$677$		−	42.9463i		−	1.65056i	−0.564723	−	0.825280i	$-0.691017\pi$
$677$		−	42.9463i		−	1.65056i	0.564723	−	0.825280i	$-0.308983\pi$
$678$		0			0
$679$	9.14897			0.351105
$680$		0			0
$681$	13.8018			0.528887
$682$		0			0
$683$		−	42.2263i		−	1.61575i	−0.589357	−	0.807873i	$-0.700619\pi$
$683$		−	42.2263i		−	1.61575i	0.589357	−	0.807873i	$-0.299381\pi$
$684$		0			0
$685$	1.46834	+	2.09296i	0.0561025	+	0.0799680i
$686$		0			0
$687$		−	15.7865i		−	0.602292i
$688$		0			0
$689$	25.7709			0.981795
$690$		0			0
$691$	−47.6928			−1.81432			−0.907160	−	0.420785i	$-0.861755\pi$
$691$	−47.6928			−1.81432			−0.907160	+	0.420785i	$0.861755\pi$
$692$		0			0
$693$			29.8364i			1.13339i
$694$		0			0
$695$	−41.0930	+	28.8293i	−1.55875	+	1.09356i
$696$		0			0
$697$			2.21251i			0.0838050i
$698$		0			0
$699$	−4.53959			−0.171703
$700$		0			0
$701$	−4.52516			−0.170913			−0.0854565	−	0.996342i	$-0.527235\pi$
$701$	−4.52516			−0.170913			−0.0854565	+	0.996342i	$0.527235\pi$
$702$		0			0
$703$		−	56.4251i		−	2.12811i
$704$		0			0
$705$	0.654158	−	0.458932i	0.0246370	−	0.0172844i
$706$		0			0
$707$		−	31.8863i		−	1.19921i
$708$		0			0
$709$	21.5986			0.811151			0.405575	−	0.914062i	$-0.367071\pi$
$709$	21.5986			0.811151			0.405575	+	0.914062i	$0.367071\pi$
$710$		0			0
$711$	24.8903			0.933458
$712$		0			0
$713$		−	1.82132i		−	0.0682089i
$714$		0			0
$715$	28.0472	+	39.9783i	1.04891	+	1.49510i
$716$		0			0
$717$			6.73269i			0.251437i
$718$		0			0
$719$	16.8098			0.626900			0.313450	−	0.949605i	$-0.398515\pi$
$719$	16.8098			0.626900			0.313450	+	0.949605i	$0.398515\pi$
$720$		0			0
$721$	−25.0935			−0.934530
$722$		0			0
$723$			9.52773i			0.354340i
$724$		0			0
$725$	−2.85763	+	7.89589i	−0.106130	+	0.293246i
$726$		0			0
$727$			5.65373i			0.209685i	0.994489	+	0.104843i	$0.0334339\pi$
$727$			5.65373i			0.209685i	−0.994489	+	0.104843i	$0.966566\pi$
$728$		0			0
$729$	3.65430			0.135345
$730$		0			0
$731$	−0.414238			−0.0153211
$732$		0			0
$733$		−	40.2821i		−	1.48785i	−0.668261	−	0.743927i	$-0.732961\pi$
$733$		−	40.2821i		−	1.48785i	0.668261	−	0.743927i	$-0.267039\pi$
$734$		0			0
$735$	1.08964	+	1.55316i	0.0401920	+	0.0572893i
$736$		0			0
$737$			62.7356i			2.31090i
$738$		0			0
$739$	1.97327			0.0725877			0.0362939	−	0.999341i	$-0.488445\pi$
$739$	1.97327			0.0725877			0.0362939	+	0.999341i	$0.488445\pi$
$740$		0			0
$741$	20.7317			0.761597
$742$		0			0
$743$		−	24.8403i		−	0.911304i	−0.890158	−	0.455652i	$-0.849406\pi$
$743$		−	24.8403i		−	0.911304i	0.890158	−	0.455652i	$-0.150594\pi$
$744$		0			0
$745$	15.0186	−	10.5365i	0.550239	−	0.386027i
$746$		0			0
$747$			33.2690i			1.21725i
$748$		0			0
$749$	−24.0372			−0.878300
$750$		0			0
$751$	−3.61665			−0.131973			−0.0659867	−	0.997821i	$-0.521019\pi$
$751$	−3.61665			−0.131973			−0.0659867	+	0.997821i	$0.521019\pi$
$752$		0			0
$753$		−	7.16492i		−	0.261104i
$754$		0			0
$755$	19.2650	−	13.5156i	0.701124	−	0.491882i
$756$		0			0
$757$		−	22.3992i		−	0.814113i	−0.913403	−	0.407056i	$-0.866555\pi$
$757$		−	22.3992i		−	0.814113i	0.913403	−	0.407056i	$-0.133445\pi$
$758$		0			0
$759$	3.50074			0.127069
$760$		0			0
$761$	25.5137			0.924872			0.462436	−	0.886653i	$-0.346975\pi$
$761$	25.5137			0.924872			0.462436	+	0.886653i	$0.346975\pi$
$762$		0			0
$763$		−	33.4105i		−	1.20954i
$764$		0			0
$765$	−0.644949	−	0.919304i	−0.0233182	−	0.0332375i
$766$		0			0
$767$		−	19.4604i		−	0.702675i
$768$		0			0
$769$	13.8123			0.498084			0.249042	−	0.968493i	$-0.419884\pi$
$769$	13.8123			0.498084			0.249042	+	0.968493i	$0.419884\pi$
$770$		0			0
$771$	−6.19107			−0.222966
$772$		0			0
$773$		−	3.36423i		−	0.121003i	−0.998168	−	0.0605015i	$-0.980730\pi$
$773$		−	3.36423i		−	0.121003i	0.998168	−	0.0605015i	$-0.0192700\pi$
$774$		0			0
$775$	3.09909	−	8.56304i	0.111322	−	0.307594i
$776$		0			0
$777$			14.4247i			0.517483i
$778$		0			0
$779$	73.3300			2.62732
$780$		0			0
$781$	17.9053			0.640703
$782$		0			0
$783$			6.52746i			0.233273i
$784$		0			0
$785$	−1.32263	−	1.88527i	−0.0472068	−	0.0672881i
$786$		0			0
$787$		−	1.29067i		−	0.0460074i	−0.999735	−	0.0230037i	$-0.992677\pi$
$787$		−	1.29067i		−	0.0460074i	0.999735	−	0.0230037i	$-0.00732296\pi$
$788$		0			0
$789$	−11.1251			−0.396063
$790$		0			0
$791$	−44.5404			−1.58367
$792$		0			0
$793$		−	29.4383i		−	1.04538i
$794$		0			0
$795$	7.56157	−	5.30491i	0.268181	−	0.188146i
$796$		0			0
$797$		−	22.4673i		−	0.795832i	−0.917422	−	0.397916i	$-0.869734\pi$
$797$		−	22.4673i		−	0.795832i	0.917422	−	0.397916i	$-0.130266\pi$
$798$		0			0
$799$	0.101569			0.00359327
$800$		0			0
$801$	12.6181			0.445839
$802$		0			0
$803$			71.6359i			2.52798i
$804$		0			0
$805$	4.40815	−	3.09259i	0.155367	−	0.108999i
$806$		0			0
$807$			16.6412i			0.585797i
$808$		0			0
$809$	22.3547			0.785950			0.392975	−	0.919549i	$-0.371446\pi$
$809$	22.3547			0.785950			0.392975	+	0.919549i	$0.371446\pi$
$810$		0			0
$811$	49.6698			1.74414			0.872071	−	0.489380i	$-0.162777\pi$
$811$	49.6698			1.74414			0.872071	+	0.489380i	$0.162777\pi$
$812$		0			0
$813$		−	1.15077i		−	0.0403592i
$814$		0			0
$815$	14.3578	+	20.4654i	0.502931	+	0.716873i
$816$		0			0
$817$			13.7292i			0.480324i
$818$		0			0
$819$	26.5465			0.927610
$820$		0			0
$821$	−50.2031			−1.75210			−0.876051	−	0.482219i	$-0.839831\pi$
$821$	−50.2031			−1.75210			−0.876051	+	0.482219i	$0.839831\pi$
$822$		0			0
$823$		−	34.2979i		−	1.19555i	−0.801663	−	0.597776i	$-0.796051\pi$
$823$		−	34.2979i		−	1.19555i	0.801663	−	0.597776i	$-0.203949\pi$
$824$		0			0
$825$	16.4589	+	5.95672i	0.573026	+	0.207386i
$826$		0			0
$827$			44.4915i			1.54712i	0.633723	+	0.773560i	$0.281526\pi$
$827$			44.4915i			1.54712i	−0.633723	+	0.773560i	$0.718474\pi$
$828$		0			0
$829$	−18.0055			−0.625356			−0.312678	−	0.949859i	$-0.601226\pi$
$829$	−18.0055			−0.625356			−0.312678	+	0.949859i	$0.601226\pi$
$830$		0			0
$831$	15.4787			0.536950
$832$		0			0
$833$			0.241156i			0.00835556i
$834$		0			0
$835$	4.48737	+	6.39625i	0.155292	+	0.221352i
$836$		0			0
$837$		−	7.07899i		−	0.244686i
$838$		0			0
$839$	−6.47816			−0.223651			−0.111825	−	0.993728i	$-0.535670\pi$
$839$	−6.47816			−0.223651			−0.111825	+	0.993728i	$0.535670\pi$
$840$		0			0
$841$	−26.1796			−0.902743
$842$		0			0
$843$			3.98701i			0.137320i
$844$		0			0
$845$	11.7734	−	8.25980i	0.405018	−	0.284146i
$846$		0			0
$847$		−	32.6220i		−	1.12090i
$848$		0			0
$849$	−4.35026			−0.149301
$850$		0			0
$851$	−8.47732			−0.290599
$852$		0			0
$853$			45.4861i			1.55742i	0.627387	+	0.778708i	$0.284124\pi$
$853$			45.4861i			1.55742i	−0.627387	+	0.778708i	$0.715876\pi$
$854$		0			0
$855$	−30.4688	+	21.3757i	−1.04201	+	0.731034i
$856$		0			0
$857$			19.4734i			0.665199i	0.943068	+	0.332599i	$0.107926\pi$
$857$			19.4734i			0.665199i	−0.943068	+	0.332599i	$0.892074\pi$
$858$		0			0
$859$	−32.7724			−1.11818			−0.559090	−	0.829107i	$-0.688849\pi$
$859$	−32.7724			−1.11818			−0.559090	+	0.829107i	$0.688849\pi$
$860$		0			0
$861$	−18.7463			−0.638873
$862$		0			0
$863$		−	9.47045i		−	0.322378i	−0.986924	−	0.161189i	$-0.948467\pi$
$863$		−	9.47045i		−	0.322378i	0.986924	−	0.161189i	$-0.0515329\pi$
$864$		0			0
$865$	11.0642	+	15.7707i	0.376192	+	0.536221i
$866$		0			0
$867$		−	11.9835i		−	0.406980i
$868$		0			0
$869$	−49.3124			−1.67281
$870$		0			0
$871$	55.8182			1.89133
$872$		0			0
$873$			9.50074i			0.321551i
$874$		0			0
$875$	25.9874	−	7.03926i	0.878535	−	0.237971i
$876$		0			0
$877$		−	8.59538i		−	0.290245i	−0.989414	−	0.145123i	$-0.953642\pi$
$877$		−	8.59538i		−	0.290245i	0.989414	−	0.145123i	$-0.0463577\pi$
$878$		0			0
$879$	15.1837			0.512134
$880$		0			0
$881$	18.6633			0.628784			0.314392	−	0.949293i	$-0.398199\pi$
$881$	18.6633			0.628784			0.314392	+	0.949293i	$0.398199\pi$
$882$		0			0
$883$			48.3855i			1.62830i	0.580653	+	0.814151i	$0.302797\pi$
$883$			48.3855i			1.62830i	−0.580653	+	0.814151i	$0.697203\pi$
$884$		0			0
$885$	−4.00590	−	5.70997i	−0.134657	−	0.191938i
$886$		0			0
$887$			21.8123i			0.732385i	0.930539	+	0.366192i	$0.119339\pi$
$887$			21.8123i			0.732385i	−0.930539	+	0.366192i	$0.880661\pi$
$888$		0			0
$889$	43.6798			1.46497
$890$		0			0
$891$	−23.5628			−0.789384
$892$		0			0
$893$		−	3.36634i		−	0.112650i
$894$		0			0
$895$	6.16214	−	4.32312i	0.205978	−	0.144506i
$896$		0			0
$897$		−	3.11473i		−	0.103998i
$898$		0			0
$899$	−3.05876			−0.102015
$900$		0			0
$901$	1.17407			0.0391138
$902$		0			0
$903$		−	3.50977i		−	0.116798i
$904$		0			0
$905$	29.0997	−	20.4152i	0.967306	−	0.678625i
$906$		0			0
$907$		−	20.7436i		−	0.688779i	−0.938827	−	0.344389i	$-0.888086\pi$
$907$		−	20.7436i		−	0.688779i	0.938827	−	0.344389i	$-0.111914\pi$
$908$		0			0
$909$	33.1122			1.09826
$910$		0			0
$911$	52.4263			1.73696			0.868480	−	0.495725i	$-0.165098\pi$
$911$	52.4263			1.73696			0.868480	+	0.495725i	$0.165098\pi$
$912$		0			0
$913$		−	65.9122i		−	2.18138i
$914$		0			0
$915$	−6.05982	−	8.63761i	−0.200332	−	0.285551i
$916$		0			0
$917$		−	4.79210i		−	0.158249i
$918$		0			0
$919$	10.9334			0.360660			0.180330	−	0.983606i	$-0.442284\pi$
$919$	10.9334			0.360660			0.180330	+	0.983606i	$0.442284\pi$
$920$		0			0
$921$	2.89468			0.0953829
$922$		0			0
$923$		−	15.9310i		−	0.524376i
$924$		0			0
$925$	−39.8567	−	14.4247i	−1.31048	−	0.474281i
$926$		0			0
$927$		−	26.0583i		−	0.855867i
$928$		0			0
$929$	−30.7620			−1.00927			−0.504634	−	0.863334i	$-0.668372\pi$
$929$	−30.7620			−1.00927			−0.504634	+	0.863334i	$0.668372\pi$
$930$		0			0
$931$	7.99270			0.261950
$932$		0			0
$933$			16.9654i			0.555424i
$934$		0			0
$935$	1.27777	+	1.82132i	0.0417875	+	0.0595635i
$936$		0			0
$937$			11.3523i			0.370865i	0.982657	+	0.185432i	$0.0593686\pi$
$937$			11.3523i			0.370865i	−0.982657	+	0.185432i	$0.940631\pi$
$938$		0			0
$939$	−7.68032			−0.250638
$940$		0			0
$941$	11.5019			0.374953			0.187476	−	0.982269i	$-0.439969\pi$
$941$	11.5019			0.374953			0.187476	+	0.982269i	$0.439969\pi$
$942$		0			0
$943$		−	11.0171i		−	0.358767i
$944$		0			0
$945$	17.1333	−	12.0201i	0.557347	−	0.391014i
$946$		0			0
$947$			27.0735i			0.879771i	0.898054	+	0.439885i	$0.144981\pi$
$947$			27.0735i			0.879771i	−0.898054	+	0.439885i	$0.855019\pi$
$948$		0			0
$949$	63.7371			2.06899
$950$		0			0
$951$	−15.6622			−0.507883
$952$		0			0
$953$			8.05878i			0.261050i	0.991445	+	0.130525i	$0.0416662\pi$
$953$			8.05878i			0.261050i	−0.991445	+	0.130525i	$0.958334\pi$
$954$		0			0
$955$	−12.3473	+	8.66241i	−0.399550	+	0.280309i
$956$		0			0
$957$		−	5.87920i		−	0.190048i
$958$		0			0
$959$	−2.75341			−0.0889124
$960$		0			0
$961$	−27.6828			−0.892994
$962$		0			0
$963$		−	24.9614i		−	0.804369i
$964$		0			0
$965$	−6.18518	−	8.81630i	−0.199108	−	0.283807i
$966$		0			0
$967$			35.8257i			1.15208i	0.817423	+	0.576038i	$0.195402\pi$
$967$			35.8257i			1.15208i	−0.817423	+	0.576038i	$0.804598\pi$
$968$		0			0
$969$	0.944488			0.0303413
$970$		0			0
$971$	−18.5815			−0.596310			−0.298155	−	0.954517i	$-0.596371\pi$
$971$	−18.5815			−0.596310			−0.298155	+	0.954517i	$0.596371\pi$
$972$		0			0
$973$		−	54.0602i		−	1.73309i
$974$		0			0
$975$	5.29991	−	14.6441i	0.169733	−	0.468987i
$976$		0			0
$977$			56.7588i			1.81588i	0.419105	+	0.907938i	$0.362344\pi$
$977$			56.7588i			1.81588i	−0.419105	+	0.907938i	$0.637656\pi$
$978$		0			0
$979$	−24.9989			−0.798969
$980$		0			0
$981$	34.6951			1.10773
$982$		0			0
$983$		−	61.1897i		−	1.95165i	−0.218556	−	0.975824i	$-0.570135\pi$
$983$		−	61.1897i		−	1.95165i	0.218556	−	0.975824i	$-0.429865\pi$
$984$		0			0
$985$	−16.4828	−	23.4944i	−0.525186	−	0.748595i
$986$		0			0
$987$			0.860583i			0.0273926i
$988$		0			0
$989$	2.06268			0.0655893
$990$		0			0
$991$	34.9833			1.11128			0.555640	−	0.831423i	$-0.312473\pi$
$991$	34.9833			1.11128			0.555640	+	0.831423i	$0.312473\pi$
$992$		0			0
$993$		−	6.63701i		−	0.210619i
$994$		0			0
$995$	22.8736	−	16.0472i	0.725141	−	0.508732i
$996$		0			0
$997$		−	41.6506i		−	1.31909i	−0.751667	−	0.659543i	$-0.770750\pi$
$997$		−	41.6506i		−	1.31909i	0.751667	−	0.659543i	$-0.229250\pi$
$998$		0			0
$999$	−32.9492			−1.04247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.e.369.4		8
4.3	odd	2		230.2.b.b.139.2	✓	8
5.2	odd	4		9200.2.a.cj.1.3		4
5.3	odd	4		9200.2.a.cr.1.2		4
5.4	even	2	inner	1840.2.e.e.369.5		8
12.11	even	2		2070.2.d.f.829.8		8
20.3	even	4		1150.2.a.r.1.3		4
20.7	even	4		1150.2.a.s.1.2		4
20.19	odd	2		230.2.b.b.139.7	yes	8
60.59	even	2		2070.2.d.f.829.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.b.139.2	✓	8	4.3	odd	2
230.2.b.b.139.7	yes	8	20.19	odd	2
1150.2.a.r.1.3		4	20.3	even	4
1150.2.a.s.1.2		4	20.7	even	4
1840.2.e.e.369.4		8	1.1	even	1	trivial
1840.2.e.e.369.5		8	5.4	even	2	inner
2070.2.d.f.829.4		8	60.59	even	2
2070.2.d.f.829.8		8	12.11	even	2
9200.2.a.cj.1.3		4	5.2	odd	4
9200.2.a.cr.1.2		4	5.3	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)

\(f(q)\)	\(=\)	\(q-0.706585i q^{3} +(-1.83051 + 1.28422i) q^{5} -2.40815i q^{7} +2.50074 q^{9} +O(q^{10})\)	\(q-0.706585i q^{3} +(-1.83051 + 1.28422i) q^{5} -2.40815i q^{7} +2.50074 q^{9} -4.95444 q^{11} +4.40815i q^{13} +(0.907411 + 1.29341i) q^{15} +0.200825i q^{17} +6.65600 q^{19} -1.70156 q^{21} -1.00000i q^{23} +(1.70156 - 4.70156i) q^{25} -3.88674i q^{27} -1.67942 q^{29} +1.82132 q^{31} +3.50074i q^{33} +(3.09259 + 4.40815i) q^{35} -8.47732i q^{37} +3.11473 q^{39} +11.0171 q^{41} +2.06268i q^{43} +(-4.57763 + 3.21149i) q^{45} -0.505760i q^{47} +1.20083 q^{49} +0.141900 q^{51} -5.84621i q^{53} +(9.06918 - 6.36259i) q^{55} -4.70304i q^{57} -4.41464 q^{59} -6.67815 q^{61} -6.02214i q^{63} +(-5.66103 - 8.06918i) q^{65} -12.6625i q^{67} -0.706585 q^{69} -3.61400 q^{71} -14.4589i q^{73} +(-3.32206 - 1.20230i) q^{75} +11.9310i q^{77} +9.95317 q^{79} +4.75590 q^{81} +13.3037i q^{83} +(-0.257904 - 0.367613i) q^{85} +1.18665i q^{87} +5.04576 q^{89} +10.6155 q^{91} -1.28692i q^{93} +(-12.1839 + 8.54777i) q^{95} +3.79917i q^{97} -12.3898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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