LMFDB - Embedded newform 2380.2.m.d.169.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2380,2,Mod(169,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2380.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [26,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$19.0043956811$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.12
Character	$\chi$	$=$	2380.169
Dual form			2380.2.m.d.169.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0401658 q^{3} +(1.78015 - 1.35317i) q^{5} +1.00000 q^{7} -2.99839 q^{9} +4.41190i q^{11} +0.124430i q^{13} +(-0.0715012 + 0.0543509i) q^{15} +(-3.87296 - 1.41427i) q^{17} -5.48398 q^{19} -0.0401658 q^{21} -5.88539 q^{23} +(1.33789 - 4.81768i) q^{25} +0.240930 q^{27} +1.55880i q^{29} -7.94462i q^{31} -0.177207i q^{33} +(1.78015 - 1.35317i) q^{35} -8.67968 q^{37} -0.00499783i q^{39} -12.2935i q^{41} +3.08398i q^{43} +(-5.33759 + 4.05731i) q^{45} -4.44458i q^{47} +1.00000 q^{49} +(0.155560 + 0.0568053i) q^{51} -3.96580i q^{53} +(5.97003 + 7.85385i) q^{55} +0.220268 q^{57} -5.88784 q^{59} -5.75018i q^{61} -2.99839 q^{63} +(0.168374 + 0.221504i) q^{65} +11.9079i q^{67} +0.236391 q^{69} -13.3221i q^{71} +1.87848 q^{73} +(-0.0537372 + 0.193506i) q^{75} +4.41190i q^{77} +5.59040i q^{79} +8.98548 q^{81} +8.93663i q^{83} +(-8.80821 + 2.72314i) q^{85} -0.0626105i q^{87} -0.343504 q^{89} +0.124430i q^{91} +0.319102i q^{93} +(-9.76233 + 7.42074i) q^{95} -9.97802 q^{97} -13.2286i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) $ 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2380\mathbb{Z}\right)^\times$.

$n$	$477$	$1191$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−0.0401658			−0.0231897			−0.0115949	−	0.999933i	$-0.503691\pi$
$3$	−0.0401658			−0.0231897			−0.0115949	+	0.999933i	$0.503691\pi$
$4$		0			0
$5$	1.78015	−	1.35317i	0.796108	−	0.605154i
$5$	1.78015	−	1.35317i	0.796108	−	0.605154i
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$	−2.99839			−0.999462
$10$		0			0
$11$			4.41190i			1.33024i	0.746738	+	0.665119i	$0.231619\pi$
$11$			4.41190i			1.33024i	−0.746738	+	0.665119i	$0.768381\pi$
$12$		0			0
$13$			0.124430i			0.0345107i	0.999851	+	0.0172553i	$0.00549282\pi$
$13$			0.124430i			0.0345107i	−0.999851	+	0.0172553i	$0.994507\pi$
$14$		0			0
$15$	−0.0715012	+	0.0543509i	−0.0184615	+	0.0140333i
$16$		0			0
$17$	−3.87296	−	1.41427i	−0.939331	−	0.343011i
$17$	−3.87296	−	1.41427i	−0.939331	−	0.343011i
$18$		0			0
$19$	−5.48398			−1.25811			−0.629056	−	0.777360i	$-0.716558\pi$
$19$	−5.48398			−1.25811			−0.629056	+	0.777360i	$0.716558\pi$
$20$		0			0
$21$	−0.0401658			−0.00876489
$22$		0			0
$23$	−5.88539			−1.22719			−0.613594	−	0.789622i	$-0.710277\pi$
$23$	−5.88539			−1.22719			−0.613594	+	0.789622i	$0.710277\pi$
$24$		0			0
$25$	1.33789	−	4.81768i	0.267577	−	0.963536i
$26$		0			0
$27$	0.240930			0.0463669
$28$		0			0
$29$			1.55880i			0.289462i	0.989471	+	0.144731i	$0.0462317\pi$
$29$			1.55880i			0.289462i	−0.989471	+	0.144731i	$0.953768\pi$
$30$		0			0
$31$		−	7.94462i		−	1.42690i	−0.700708	−	0.713448i	$-0.747133\pi$
$31$		−	7.94462i		−	1.42690i	0.700708	−	0.713448i	$-0.252867\pi$
$32$		0			0
$33$		−	0.177207i		−	0.0308478i
$34$		0			0
$35$	1.78015	−	1.35317i	0.300901	−	0.228727i
$36$		0			0
$37$	−8.67968			−1.42693			−0.713465	−	0.700691i	$-0.752875\pi$
$37$	−8.67968			−1.42693			−0.713465	+	0.700691i	$0.752875\pi$
$38$		0			0
$39$		−	0.00499783i		−	0.000800293i
$40$		0			0
$41$		−	12.2935i		−	1.91993i	−0.280122	−	0.959964i	$-0.590375\pi$
$41$		−	12.2935i		−	1.91993i	0.280122	−	0.959964i	$-0.409625\pi$
$42$		0			0
$43$			3.08398i			0.470303i	0.971959	+	0.235152i	$0.0755586\pi$
$43$			3.08398i			0.470303i	−0.971959	+	0.235152i	$0.924441\pi$
$44$		0			0
$45$	−5.33759	+	4.05731i	−0.795680	+	0.604829i
$46$		0			0
$47$		−	4.44458i		−	0.648309i	−0.946004	−	0.324155i	$-0.894920\pi$
$47$		−	4.44458i		−	0.648309i	0.946004	−	0.324155i	$-0.105080\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	0.155560	+	0.0568053i	0.0217828	+	0.00795433i
$52$		0			0
$53$		−	3.96580i		−	0.544744i	−0.962192	−	0.272372i	$-0.912192\pi$
$53$		−	3.96580i		−	0.544744i	0.962192	−	0.272372i	$-0.0878082\pi$
$54$		0			0
$55$	5.97003	+	7.85385i	0.804998	+	1.05901i
$56$		0			0
$57$	0.220268			0.0291752
$58$		0			0
$59$	−5.88784			−0.766531			−0.383265	−	0.923638i	$-0.625201\pi$
$59$	−5.88784			−0.766531			−0.383265	+	0.923638i	$0.625201\pi$
$60$		0			0
$61$		−	5.75018i		−	0.736235i	−0.929779	−	0.368117i	$-0.880002\pi$
$61$		−	5.75018i		−	0.736235i	0.929779	−	0.368117i	$-0.119998\pi$
$62$		0			0
$63$	−2.99839			−0.377761
$64$		0			0
$65$	0.168374	+	0.221504i	0.0208843	+	0.0274743i
$66$		0			0
$67$			11.9079i			1.45478i	0.686225	+	0.727389i	$0.259266\pi$
$67$			11.9079i			1.45478i	−0.686225	+	0.727389i	$0.740734\pi$
$68$		0			0
$69$	0.236391			0.0284581
$70$		0			0
$71$		−	13.3221i		−	1.58105i	−0.612433	−	0.790523i	$-0.709809\pi$
$71$		−	13.3221i		−	1.58105i	0.612433	−	0.790523i	$-0.290191\pi$
$72$		0			0
$73$	1.87848			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$73$	1.87848			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$74$		0			0
$75$	−0.0537372	+	0.193506i	−0.00620504	+	0.0223441i
$76$		0			0
$77$			4.41190i			0.502782i
$78$		0			0
$79$			5.59040i			0.628969i	0.949263	+	0.314485i	$0.101832\pi$
$79$			5.59040i			0.628969i	−0.949263	+	0.314485i	$0.898168\pi$
$80$		0			0
$81$	8.98548			0.998387
$82$		0			0
$83$			8.93663i			0.980922i	0.871463	+	0.490461i	$0.163172\pi$
$83$			8.93663i			0.980922i	−0.871463	+	0.490461i	$0.836828\pi$
$84$		0			0
$85$	−8.80821	+	2.72314i	−0.955384	+	0.295366i
$86$		0			0
$87$		−	0.0626105i		−	0.00671255i
$88$		0			0
$89$	−0.343504			−0.0364114			−0.0182057	−	0.999834i	$-0.505795\pi$
$89$	−0.343504			−0.0364114			−0.0182057	+	0.999834i	$0.505795\pi$
$90$		0			0
$91$			0.124430i			0.0130438i
$92$		0			0
$93$			0.319102i			0.0330893i
$94$		0			0
$95$	−9.76233	+	7.42074i	−1.00159	+	0.761351i
$96$		0			0
$97$	−9.97802			−1.01311			−0.506557	−	0.862206i	$-0.669082\pi$
$97$	−9.97802			−1.01311			−0.506557	+	0.862206i	$0.669082\pi$
$98$		0			0
$99$		−	13.2286i		−	1.32952i
$100$		0			0
$101$	3.40363			0.338673			0.169337	−	0.985558i	$-0.445837\pi$
$101$	3.40363			0.338673			0.169337	+	0.985558i	$0.445837\pi$
$102$		0			0
$103$			0.471544i			0.0464626i	0.999730	+	0.0232313i	$0.00739542\pi$
$103$			0.471544i			0.0464626i	−0.999730	+	0.0232313i	$0.992605\pi$
$104$		0			0
$105$	−0.0715012	+	0.0543509i	−0.00697780	+	0.00530411i
$106$		0			0
$107$	7.94370			0.767947			0.383973	−	0.923344i	$-0.374555\pi$
$107$	7.94370			0.767947			0.383973	+	0.923344i	$0.374555\pi$
$108$		0			0
$109$			3.72199i			0.356502i	0.983985	+	0.178251i	$0.0570439\pi$
$109$			3.72199i			0.356502i	−0.983985	+	0.178251i	$0.942956\pi$
$110$		0			0
$111$	0.348626			0.0330901
$112$		0			0
$113$	−1.84147			−0.173231			−0.0866154	−	0.996242i	$-0.527605\pi$
$113$	−1.84147			−0.173231			−0.0866154	+	0.996242i	$0.527605\pi$
$114$		0			0
$115$	−10.4769	+	7.96390i	−0.976974	+	0.742638i
$116$		0			0
$117$		−	0.373089i		−	0.0344921i
$118$		0			0
$119$	−3.87296	−	1.41427i	−0.355034	−	0.129646i
$120$		0			0
$121$	−8.46484			−0.769531
$122$		0			0
$123$			0.493779i			0.0445226i
$124$		0			0
$125$	−4.13748	−	10.3866i	−0.370067	−	0.929005i
$126$		0			0
$127$		−	1.91181i		−	0.169646i	−0.996396	−	0.0848230i	$-0.972968\pi$
$127$		−	1.91181i		−	0.169646i	0.996396	−	0.0848230i	$-0.0270325\pi$
$128$		0			0
$129$		−	0.123871i		−	0.0109062i
$130$		0			0
$131$			8.21039i			0.717345i	0.933464	+	0.358672i	$0.116770\pi$
$131$			8.21039i			0.717345i	−0.933464	+	0.358672i	$0.883230\pi$
$132$		0			0
$133$	−5.48398			−0.475522
$134$		0			0
$135$	0.428892	−	0.326018i	0.0369131	−	0.0280591i
$136$		0			0
$137$			23.0153i			1.96633i	0.182730	+	0.983163i	$0.441507\pi$
$137$			23.0153i			1.96633i	−0.182730	+	0.983163i	$0.558493\pi$
$138$		0			0
$139$		−	5.59882i		−	0.474885i	−0.971402	−	0.237443i	$-0.923691\pi$
$139$		−	5.59882i		−	0.474885i	0.971402	−	0.237443i	$-0.0763092\pi$
$140$		0			0
$141$			0.178520i			0.0150341i
$142$		0			0
$143$	−0.548973			−0.0459074
$144$		0			0
$145$	2.10932	+	2.77491i	0.175169	+	0.230443i
$146$		0			0
$147$	−0.0401658			−0.00331282
$148$		0			0
$149$	−14.1850			−1.16208			−0.581039	−	0.813875i	$-0.697354\pi$
$149$	−14.1850			−1.16208			−0.581039	+	0.813875i	$0.697354\pi$
$150$		0			0
$151$	4.66075			0.379287			0.189643	−	0.981853i	$-0.439267\pi$
$151$	4.66075			0.379287			0.189643	+	0.981853i	$0.439267\pi$
$152$		0			0
$153$	11.6126	+	4.24053i	0.938826	+	0.342827i
$154$		0			0
$155$	−10.7504	−	14.1426i	−0.863492	−	1.13596i
$156$		0			0
$157$		−	15.7578i		−	1.25761i	−0.777564	−	0.628804i	$-0.783545\pi$
$157$		−	15.7578i		−	1.25761i	0.777564	−	0.628804i	$-0.216455\pi$
$158$		0			0
$159$			0.159289i			0.0126325i
$160$		0			0
$161$	−5.88539			−0.463833
$162$		0			0
$163$	3.78015			0.296084			0.148042	−	0.988981i	$-0.452703\pi$
$163$	3.78015			0.296084			0.148042	+	0.988981i	$0.452703\pi$
$164$		0			0
$165$	−0.239791	−	0.315456i	−0.0186677	−	0.0245582i
$166$		0			0
$167$	−2.19371			−0.169754			−0.0848770	−	0.996391i	$-0.527050\pi$
$167$	−2.19371			−0.169754			−0.0848770	+	0.996391i	$0.527050\pi$
$168$		0			0
$169$	12.9845			0.998809
$170$		0			0
$171$	16.4431			1.25744
$172$		0			0
$173$	−16.7430			−1.27294			−0.636472	−	0.771300i	$-0.719607\pi$
$173$	−16.7430			−1.27294			−0.636472	+	0.771300i	$0.719607\pi$
$174$		0			0
$175$	1.33789	−	4.81768i	0.101135	−	0.364183i
$176$		0			0
$177$	0.236489			0.0177756
$178$		0			0
$179$	19.3684			1.44766			0.723830	−	0.689978i	$-0.242380\pi$
$179$	19.3684			1.44766			0.723830	+	0.689978i	$0.242380\pi$
$180$		0			0
$181$			2.45892i			0.182770i	0.995816	+	0.0913851i	$0.0291294\pi$
$181$			2.45892i			0.182770i	−0.995816	+	0.0913851i	$0.970871\pi$
$182$		0			0
$183$			0.230960i			0.0170731i
$184$		0			0
$185$	−15.4512	+	11.7450i	−1.13599	+	0.863513i
$186$		0			0
$187$	6.23962	−	17.0871i	0.456286	−	1.24953i
$188$		0			0
$189$	0.240930			0.0175251
$190$		0			0
$191$	23.3129			1.68687			0.843433	−	0.537235i	$-0.180531\pi$
$191$	23.3129			1.68687			0.843433	+	0.537235i	$0.180531\pi$
$192$		0			0
$193$	2.47857			0.178411			0.0892057	−	0.996013i	$-0.471567\pi$
$193$	2.47857			0.178411			0.0892057	+	0.996013i	$0.471567\pi$
$194$		0			0
$195$	−0.00676289	−	0.00889689i	−0.000484300	−	0.000637120i
$196$		0			0
$197$	−25.7420			−1.83404			−0.917021	−	0.398838i	$-0.869413\pi$
$197$	−25.7420			−1.83404			−0.917021	+	0.398838i	$0.869413\pi$
$198$		0			0
$199$		−	0.857328i		−	0.0607744i	−0.999538	−	0.0303872i	$-0.990326\pi$
$199$		−	0.857328i		−	0.0607744i	0.999538	−	0.0303872i	$-0.00967403\pi$
$200$		0			0
$201$		−	0.478289i		−	0.0337359i
$202$		0			0
$203$			1.55880i			0.109406i
$204$		0			0
$205$	−16.6352	−	21.8844i	−1.16185	−	1.52847i
$206$		0			0
$207$	17.6467			1.22653
$208$		0			0
$209$		−	24.1948i		−	1.67359i
$210$		0			0
$211$			25.0841i			1.72686i	0.504465	+	0.863432i	$0.331690\pi$
$211$			25.0841i			1.72686i	−0.504465	+	0.863432i	$0.668310\pi$
$212$		0			0
$213$			0.535093i			0.0366640i
$214$		0			0
$215$	4.17314	+	5.48996i	0.284606	+	0.374412i
$216$		0			0
$217$		−	7.94462i		−	0.539316i
$218$		0			0
$219$	−0.0754507			−0.00509849
$220$		0			0
$221$	0.175978	−	0.481913i	0.0118376	−	0.0324170i
$222$		0			0
$223$		−	18.3312i		−	1.22755i	−0.789481	−	0.613775i	$-0.789650\pi$
$223$		−	18.3312i		−	1.22755i	0.789481	−	0.613775i	$-0.210350\pi$
$224$		0			0
$225$	−4.01150	+	14.4453i	−0.267433	+	0.963018i
$226$		0			0
$227$	−23.4124			−1.55394			−0.776968	−	0.629540i	$-0.783244\pi$
$227$	−23.4124			−1.55394			−0.776968	+	0.629540i	$0.783244\pi$
$228$		0			0
$229$	−12.1193			−0.800867			−0.400434	−	0.916326i	$-0.631140\pi$
$229$	−12.1193			−0.800867			−0.400434	+	0.916326i	$0.631140\pi$
$230$		0			0
$231$		−	0.177207i		−	0.0116594i
$232$		0			0
$233$	−19.9972			−1.31006			−0.655032	−	0.755601i	$-0.727345\pi$
$233$	−19.9972			−1.31006			−0.655032	+	0.755601i	$0.727345\pi$
$234$		0			0
$235$	−6.01426	−	7.91204i	−0.392327	−	0.516124i
$236$		0			0
$237$		−	0.224543i		−	0.0145856i
$238$		0			0
$239$	−21.3028			−1.37796			−0.688982	−	0.724778i	$-0.741942\pi$
$239$	−21.3028			−1.37796			−0.688982	+	0.724778i	$0.741942\pi$
$240$		0			0
$241$		−	11.8586i		−	0.763882i	−0.924187	−	0.381941i	$-0.875256\pi$
$241$		−	11.8586i		−	0.763882i	0.924187	−	0.381941i	$-0.124744\pi$
$242$		0			0
$243$	−1.08370			−0.0695192
$244$		0			0
$245$	1.78015	−	1.35317i	0.113730	−	0.0864506i
$246$		0			0
$247$		−	0.682372i		−	0.0434183i
$248$		0			0
$249$		−	0.358946i		−	0.0227473i
$250$		0			0
$251$	24.9516			1.57493			0.787464	−	0.616360i	$-0.211393\pi$
$251$	24.9516			1.57493			0.787464	+	0.616360i	$0.211393\pi$
$252$		0			0
$253$		−	25.9657i		−	1.63245i
$254$		0			0
$255$	0.353788	−	0.109377i	0.0221551	−	0.00684945i
$256$		0			0
$257$			24.6722i			1.53901i	0.638643	+	0.769503i	$0.279496\pi$
$257$			24.6722i			1.53901i	−0.638643	+	0.769503i	$0.720504\pi$
$258$		0			0
$259$	−8.67968			−0.539329
$260$		0			0
$261$		−	4.67389i		−	0.289307i
$262$		0			0
$263$		−	10.4482i		−	0.644266i	−0.946694	−	0.322133i	$-0.895600\pi$
$263$		−	10.4482i		−	0.644266i	0.946694	−	0.322133i	$-0.104400\pi$
$264$		0			0
$265$	−5.36638	−	7.05973i	−0.329654	−	0.433676i
$266$		0			0
$267$	0.0137971			0.000844369
$268$		0			0
$269$		−	16.8467i		−	1.02716i	−0.858040	−	0.513582i	$-0.828318\pi$
$269$		−	16.8467i		−	1.02716i	0.858040	−	0.513582i	$-0.171682\pi$
$270$		0			0
$271$	−2.70894			−0.164557			−0.0822783	−	0.996609i	$-0.526220\pi$
$271$	−2.70894			−0.164557			−0.0822783	+	0.996609i	$0.526220\pi$
$272$		0			0
$273$		−	0.00499783i		−	0.000302482i
$274$		0			0
$275$	21.2551	+	5.90262i	1.28173	+	0.355941i
$276$		0			0
$277$	−6.60716			−0.396986			−0.198493	−	0.980102i	$-0.563605\pi$
$277$	−6.60716			−0.396986			−0.198493	+	0.980102i	$0.563605\pi$
$278$		0			0
$279$			23.8210i			1.42613i
$280$		0			0
$281$	−6.96201			−0.415318			−0.207659	−	0.978201i	$-0.566585\pi$
$281$	−6.96201			−0.415318			−0.207659	+	0.978201i	$0.566585\pi$
$282$		0			0
$283$	14.2220			0.845413			0.422706	−	0.906267i	$-0.361080\pi$
$283$	14.2220			0.845413			0.422706	+	0.906267i	$0.361080\pi$
$284$		0			0
$285$	0.392111	−	0.298059i	0.0232267	−	0.0176555i
$286$		0			0
$287$		−	12.2935i		−	0.725665i
$288$		0			0
$289$	12.9997	+	10.9548i	0.764687	+	0.644402i
$290$		0			0
$291$	0.400775			0.0234938
$292$		0			0
$293$			27.9514i			1.63294i	0.577388	+	0.816470i	$0.304072\pi$
$293$			27.9514i			1.63294i	−0.577388	+	0.816470i	$0.695928\pi$
$294$		0			0
$295$	−10.4812	+	7.96722i	−0.610242	+	0.463869i
$296$		0			0
$297$			1.06296i			0.0616790i
$298$		0			0
$299$		−	0.732319i		−	0.0423511i
$300$		0			0
$301$			3.08398i			0.177758i
$302$		0			0
$303$	−0.136709			−0.00785374
$304$		0			0
$305$	−7.78094	−	10.2362i	−0.445535	−	0.586123i
$306$		0			0
$307$			18.5924i			1.06113i	0.847645	+	0.530564i	$0.178020\pi$
$307$			18.5924i			1.06113i	−0.847645	+	0.530564i	$0.821980\pi$
$308$		0			0
$309$		−	0.0189399i		−	0.00107745i
$310$		0			0
$311$		−	14.4894i		−	0.821620i	−0.911721	−	0.410810i	$-0.865246\pi$
$311$		−	14.4894i		−	0.821620i	0.911721	−	0.410810i	$-0.134754\pi$
$312$		0			0
$313$	−23.8817			−1.34987			−0.674937	−	0.737875i	$-0.735829\pi$
$313$	−23.8817			−1.34987			−0.674937	+	0.737875i	$0.735829\pi$
$314$		0			0
$315$	−5.33759	+	4.05731i	−0.300739	+	0.228604i
$316$		0			0
$317$	−8.43782			−0.473915			−0.236958	−	0.971520i	$-0.576150\pi$
$317$	−8.43782			−0.473915			−0.236958	+	0.971520i	$0.576150\pi$
$318$		0			0
$319$	−6.87728			−0.385054
$320$		0			0
$321$	−0.319065			−0.0178085
$322$		0			0
$323$	21.2393	+	7.75584i	1.18178	+	0.431547i
$324$		0			0
$325$	0.599464	+	0.166473i	0.0332523	+	0.00923427i
$326$		0			0
$327$		−	0.149497i		−	0.00826718i
$328$		0			0
$329$		−	4.44458i		−	0.245038i
$330$		0			0
$331$	−4.80202			−0.263943			−0.131972	−	0.991254i	$-0.542131\pi$
$331$	−4.80202			−0.263943			−0.131972	+	0.991254i	$0.542131\pi$
$332$		0			0
$333$	26.0250			1.42616
$334$		0			0
$335$	16.1133	+	21.1978i	0.880365	+	1.15816i
$336$		0			0
$337$	13.9612			0.760516			0.380258	−	0.924880i	$-0.375835\pi$
$337$	13.9612			0.760516			0.380258	+	0.924880i	$0.375835\pi$
$338$		0			0
$339$	0.0739640			0.00401717
$340$		0			0
$341$	35.0509			1.89811
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$	0.420812	−	0.319876i	0.0226558	−	0.0172215i
$346$		0			0
$347$	12.6764			0.680505			0.340253	−	0.940334i	$-0.389487\pi$
$347$	12.6764			0.680505			0.340253	+	0.940334i	$0.389487\pi$
$348$		0			0
$349$	18.0226			0.964728			0.482364	−	0.875971i	$-0.339778\pi$
$349$	18.0226			0.964728			0.482364	+	0.875971i	$0.339778\pi$
$350$		0			0
$351$			0.0299789i			0.00160016i
$352$		0			0
$353$			15.6792i			0.834518i	0.908788	+	0.417259i	$0.137009\pi$
$353$			15.6792i			0.834518i	−0.908788	+	0.417259i	$0.862991\pi$
$354$		0			0
$355$	−18.0270	−	23.7154i	−0.956776	−	1.25868i
$356$		0			0
$357$	0.155560	+	0.0568053i	0.00823313	+	0.00300645i
$358$		0			0
$359$	−25.3814			−1.33958			−0.669789	−	0.742551i	$-0.733616\pi$
$359$	−25.3814			−1.33958			−0.669789	+	0.742551i	$0.733616\pi$
$360$		0			0
$361$	11.0741			0.582846
$362$		0			0
$363$	0.339997			0.0178452
$364$		0			0
$365$	3.34399	−	2.54190i	0.175032	−	0.133049i
$366$		0			0
$367$	26.2380			1.36961			0.684805	−	0.728726i	$-0.259887\pi$
$367$	26.2380			1.36961			0.684805	+	0.728726i	$0.259887\pi$
$368$		0			0
$369$			36.8608i			1.91890i
$370$		0			0
$371$		−	3.96580i		−	0.205894i
$372$		0			0
$373$			5.61849i			0.290914i	0.989365	+	0.145457i	$0.0464653\pi$
$373$			5.61849i			0.290914i	−0.989365	+	0.145457i	$0.953535\pi$
$374$		0			0
$375$	0.166185	+	0.417185i	0.00858176	+	0.0215434i
$376$		0			0
$377$	−0.193962			−0.00998954
$378$		0			0
$379$			34.9673i			1.79615i	0.439840	+	0.898076i	$0.355035\pi$
$379$			34.9673i			1.79615i	−0.439840	+	0.898076i	$0.644965\pi$
$380$		0			0
$381$			0.0767895i			0.00393404i
$382$		0			0
$383$			8.41466i			0.429969i	0.976617	+	0.214984i	$0.0689701\pi$
$383$			8.41466i			0.429969i	−0.976617	+	0.214984i	$0.931030\pi$
$384$		0			0
$385$	5.97003	+	7.85385i	0.304261	+	0.400269i
$386$		0			0
$387$		−	9.24698i		−	0.470050i
$388$		0			0
$389$	−4.29965			−0.218001			−0.109000	−	0.994042i	$-0.534765\pi$
$389$	−4.29965			−0.218001			−0.109000	+	0.994042i	$0.534765\pi$
$390$		0			0
$391$	22.7939	+	8.32353i	1.15274	+	0.420939i
$392$		0			0
$393$		−	0.329776i		−	0.0166350i
$394$		0			0
$395$	7.56474	+	9.95177i	0.380623	+	0.500728i
$396$		0			0
$397$	20.3257			1.02012			0.510059	−	0.860140i	$-0.329624\pi$
$397$	20.3257			1.02012			0.510059	+	0.860140i	$0.329624\pi$
$398$		0			0
$399$	0.220268			0.0110272
$400$		0			0
$401$		−	5.19389i		−	0.259371i	−0.991555	−	0.129685i	$-0.958603\pi$
$401$		−	5.19389i		−	0.259371i	0.991555	−	0.129685i	$-0.0413967\pi$
$402$		0			0
$403$	0.988550			0.0492432
$404$		0			0
$405$	15.9955	−	12.1588i	0.794824	−	0.604178i
$406$		0			0
$407$		−	38.2939i		−	1.89816i
$408$		0			0
$409$	22.9619			1.13540			0.567698	−	0.823237i	$-0.307834\pi$
$409$	22.9619			1.13540			0.567698	+	0.823237i	$0.307834\pi$
$410$		0			0
$411$		−	0.924425i		−	0.0455985i
$412$		0			0
$413$	−5.88784			−0.289721
$414$		0			0
$415$	12.0927	+	15.9086i	0.593609	+	0.780920i
$416$		0			0
$417$			0.224881i			0.0110125i
$418$		0			0
$419$		−	10.7578i		−	0.525551i	−0.964857	−	0.262776i	$-0.915362\pi$
$419$		−	10.7578i		−	0.525551i	0.964857	−	0.262776i	$-0.0846379\pi$
$420$		0			0
$421$	−17.2909			−0.842705			−0.421353	−	0.906897i	$-0.638445\pi$
$421$	−17.2909			−0.842705			−0.421353	+	0.906897i	$0.638445\pi$
$422$		0			0
$423$			13.3266i			0.647961i
$424$		0			0
$425$	−11.9951	+	16.7666i	−0.581847	+	0.813298i
$426$		0			0
$427$		−	5.75018i		−	0.278271i
$428$		0			0
$429$	0.0220499			0.00106458
$430$		0			0
$431$		−	10.6368i		−	0.512355i	−0.966630	−	0.256178i	$-0.917537\pi$
$431$		−	10.6368i		−	0.512355i	0.966630	−	0.256178i	$-0.0824632\pi$
$432$		0			0
$433$			13.3302i			0.640607i	0.947315	+	0.320303i	$0.103785\pi$
$433$			13.3302i			0.640607i	−0.947315	+	0.320303i	$0.896215\pi$
$434$		0			0
$435$	−0.0847223	−	0.111456i	−0.00406212	−	0.00534392i
$436$		0			0
$437$	32.2754			1.54394
$438$		0			0
$439$			9.24191i			0.441092i	0.975377	+	0.220546i	$0.0707840\pi$
$439$			9.24191i			0.441092i	−0.975377	+	0.220546i	$0.929216\pi$
$440$		0			0
$441$	−2.99839			−0.142780
$442$		0			0
$443$		−	33.5874i		−	1.59579i	−0.602799	−	0.797893i	$-0.705948\pi$
$443$		−	33.5874i		−	1.59579i	0.602799	−	0.797893i	$-0.294052\pi$
$444$		0			0
$445$	−0.611490	+	0.464818i	−0.0289874	+	0.0220345i
$446$		0			0
$447$	0.569750			0.0269483
$448$		0			0
$449$		−	29.3113i		−	1.38328i	−0.722241	−	0.691642i	$-0.756888\pi$
$449$		−	29.3113i		−	1.38328i	0.722241	−	0.691642i	$-0.243112\pi$
$450$		0			0
$451$	54.2379			2.55396
$452$		0			0
$453$	−0.187203			−0.00879555
$454$		0			0
$455$	0.168374	+	0.221504i	0.00789352	+	0.0103843i
$456$		0			0
$457$		−	14.5134i		−	0.678909i	−0.940622	−	0.339454i	$-0.889758\pi$
$457$		−	14.5134i		−	0.678909i	0.940622	−	0.339454i	$-0.110242\pi$
$458$		0			0
$459$	−0.933112	−	0.340740i	−0.0435539	−	0.0159044i
$460$		0			0
$461$	26.8490			1.25048			0.625241	−	0.780431i	$-0.285001\pi$
$461$	26.8490			1.25048			0.625241	+	0.780431i	$0.285001\pi$
$462$		0			0
$463$			3.70245i			0.172067i	0.996292	+	0.0860336i	$0.0274193\pi$
$463$			3.70245i			0.172067i	−0.996292	+	0.0860336i	$0.972581\pi$
$464$		0			0
$465$	0.431797	+	0.568050i	0.0200241	+	0.0263427i
$466$		0			0
$467$			23.9050i			1.10619i	0.833117	+	0.553097i	$0.186554\pi$
$467$			23.9050i			1.10619i	−0.833117	+	0.553097i	$0.813446\pi$
$468$		0			0
$469$			11.9079i			0.549855i
$470$		0			0
$471$			0.632923i			0.0291636i
$472$		0			0
$473$	−13.6062			−0.625615
$474$		0			0
$475$	−7.33694	+	26.4201i	−0.336642	+	1.21224i
$476$		0			0
$477$			11.8910i			0.544451i
$478$		0			0
$479$			8.74666i			0.399645i	0.979832	+	0.199822i	$0.0640365\pi$
$479$			8.74666i			0.399645i	−0.979832	+	0.199822i	$0.935963\pi$
$480$		0			0
$481$		−	1.08001i		−	0.0492444i
$482$		0			0
$483$	0.236391			0.0107562
$484$		0			0
$485$	−17.7624	+	13.5019i	−0.806549	+	0.613090i
$486$		0			0
$487$	8.58421			0.388988			0.194494	−	0.980904i	$-0.437694\pi$
$487$	8.58421			0.388988			0.194494	+	0.980904i	$0.437694\pi$
$488$		0			0
$489$	−0.151833			−0.00686611
$490$		0			0
$491$	−12.2745			−0.553942			−0.276971	−	0.960878i	$-0.589331\pi$
$491$	−12.2745			−0.553942			−0.276971	+	0.960878i	$0.589331\pi$
$492$		0			0
$493$	2.20457	−	6.03718i	0.0992888	−	0.271901i
$494$		0			0
$495$	−17.9005	−	23.5489i	−0.804566	−	1.05844i
$496$		0			0
$497$		−	13.3221i		−	0.597579i
$498$		0			0
$499$		−	28.6807i		−	1.28393i	−0.766736	−	0.641963i	$-0.778120\pi$
$499$		−	28.6807i		−	1.28393i	0.766736	−	0.641963i	$-0.221880\pi$
$500$		0			0
$501$	0.0881119			0.00393655
$502$		0			0
$503$	7.27389			0.324327			0.162163	−	0.986764i	$-0.448153\pi$
$503$	7.27389			0.324327			0.162163	+	0.986764i	$0.448153\pi$
$504$		0			0
$505$	6.05897	−	4.60567i	0.269621	−	0.204950i
$506$		0			0
$507$	−0.521533			−0.0231621
$508$		0			0
$509$	−9.37318			−0.415459			−0.207729	−	0.978186i	$-0.566607\pi$
$509$	−9.37318			−0.415459			−0.207729	+	0.978186i	$0.566607\pi$
$510$		0			0
$511$	1.87848			0.0830992
$512$		0			0
$513$	−1.32125			−0.0583348
$514$		0			0
$515$	0.638077	+	0.839421i	0.0281171	+	0.0369893i
$516$		0			0
$517$	19.6091			0.862405
$518$		0			0
$519$	0.672493			0.0295192
$520$		0			0
$521$			28.5357i			1.25017i	0.780555	+	0.625087i	$0.214936\pi$
$521$			28.5357i			1.25017i	−0.780555	+	0.625087i	$0.785064\pi$
$522$		0			0
$523$		−	3.43057i		−	0.150008i	−0.997183	−	0.0750042i	$-0.976103\pi$
$523$		−	3.43057i		−	0.150008i	0.997183	−	0.0750042i	$-0.0238970\pi$
$524$		0			0
$525$	−0.0537372	+	0.193506i	−0.00234528	+	0.00844529i
$526$		0			0
$527$	−11.2359	+	30.7692i	−0.489441	+	1.34033i
$528$		0			0
$529$	11.6378			0.505989
$530$		0			0
$531$	17.6540			0.766119
$532$		0			0
$533$	1.52969			0.0662581
$534$		0			0
$535$	14.1410	−	10.7491i	0.611369	−	0.464726i
$536$		0			0
$537$	−0.777946			−0.0335708
$538$		0			0
$539$			4.41190i			0.190034i
$540$		0			0
$541$		−	16.7869i		−	0.721725i	−0.932619	−	0.360862i	$-0.882482\pi$
$541$		−	16.7869i		−	0.721725i	0.932619	−	0.360862i	$-0.117518\pi$
$542$		0			0
$543$		−	0.0987645i		−	0.00423839i
$544$		0			0
$545$	5.03647	+	6.62571i	0.215739	+	0.283814i
$546$		0			0
$547$	−6.39587			−0.273468			−0.136734	−	0.990608i	$-0.543660\pi$
$547$	−6.39587			−0.273468			−0.136734	+	0.990608i	$0.543660\pi$
$548$		0			0
$549$			17.2413i			0.735839i
$550$		0			0
$551$		−	8.54845i		−	0.364176i
$552$		0			0
$553$			5.59040i			0.237728i
$554$		0			0
$555$	0.620607	−	0.471748i	0.0263433	−	0.0200246i
$556$		0			0
$557$		−	22.4422i		−	0.950908i	−0.879741	−	0.475454i	$-0.842284\pi$
$557$		−	22.4422i		−	0.950908i	0.879741	−	0.475454i	$-0.157716\pi$
$558$		0			0
$559$	−0.383740			−0.0162305
$560$		0			0
$561$	−0.250619	+	0.686317i	−0.0105811	+	0.0289763i
$562$		0			0
$563$			1.15013i			0.0484722i	0.999706	+	0.0242361i	$0.00771535\pi$
$563$			1.15013i			0.0484722i	−0.999706	+	0.0242361i	$0.992285\pi$
$564$		0			0
$565$	−3.27809	+	2.49181i	−0.137910	+	0.104831i
$566$		0			0
$567$	8.98548			0.377355
$568$		0			0
$569$	−30.5170			−1.27934			−0.639670	−	0.768650i	$-0.720929\pi$
$569$	−30.5170			−1.27934			−0.639670	+	0.768650i	$0.720929\pi$
$570$		0			0
$571$			16.2445i			0.679809i	0.940460	+	0.339905i	$0.110395\pi$
$571$			16.2445i			0.679809i	−0.940460	+	0.339905i	$0.889605\pi$
$572$		0			0
$573$	−0.936382			−0.0391179
$574$		0			0
$575$	−7.87398	+	28.3539i	−0.328367	+	1.18244i
$576$		0			0
$577$		−	9.90217i		−	0.412233i	−0.978527	−	0.206116i	$-0.933917\pi$
$577$		−	9.90217i		−	0.412233i	0.978527	−	0.206116i	$-0.0660825\pi$
$578$		0			0
$579$	−0.0995536			−0.00413731
$580$		0			0
$581$			8.93663i			0.370754i
$582$		0			0
$583$	17.4967			0.724639
$584$		0			0
$585$	−0.504852	−	0.664156i	−0.0208731	−	0.0274595i
$586$		0			0
$587$		−	9.43130i		−	0.389271i	−0.980876	−	0.194636i	$-0.937648\pi$
$587$		−	9.43130i		−	0.389271i	0.980876	−	0.194636i	$-0.0623524\pi$
$588$		0			0
$589$			43.5682i			1.79519i
$590$		0			0
$591$	1.03395			0.0425309
$592$		0			0
$593$		−	45.0280i		−	1.84908i	−0.381089	−	0.924539i	$-0.624451\pi$
$593$		−	45.0280i		−	1.84908i	0.381089	−	0.924539i	$-0.375549\pi$
$594$		0			0
$595$	−8.80821	+	2.72314i	−0.361101	+	0.111638i
$596$		0			0
$597$			0.0344352i			0.00140934i
$598$		0			0
$599$	−17.6020			−0.719200			−0.359600	−	0.933107i	$-0.617087\pi$
$599$	−17.6020			−0.719200			−0.359600	+	0.933107i	$0.617087\pi$
$600$		0			0
$601$		−	40.7666i		−	1.66291i	−0.555595	−	0.831453i	$-0.687510\pi$
$601$		−	40.7666i		−	1.66291i	0.555595	−	0.831453i	$-0.312490\pi$
$602$		0			0
$603$		−	35.7044i		−	1.45400i
$604$		0			0
$605$	−15.0687	+	11.4543i	−0.612630	+	0.465685i
$606$		0			0
$607$	36.6008			1.48558			0.742791	−	0.669523i	$-0.233502\pi$
$607$	36.6008			1.48558			0.742791	+	0.669523i	$0.233502\pi$
$608$		0			0
$609$		−	0.0626105i		−	0.00253710i
$610$		0			0
$611$	0.553040			0.0223736
$612$		0			0
$613$			25.6102i			1.03438i	0.855869	+	0.517192i	$0.173023\pi$
$613$			25.6102i			1.03438i	−0.855869	+	0.517192i	$0.826977\pi$
$614$		0			0
$615$	0.668165	+	0.879003i	0.0269430	+	0.0354448i
$616$		0			0
$617$	−31.1511			−1.25410			−0.627048	−	0.778981i	$-0.715737\pi$
$617$	−31.1511			−1.25410			−0.627048	+	0.778981i	$0.715737\pi$
$618$		0			0
$619$			22.8119i			0.916888i	0.888723	+	0.458444i	$0.151593\pi$
$619$			22.8119i			0.916888i	−0.888723	+	0.458444i	$0.848407\pi$
$620$		0			0
$621$	−1.41796			−0.0569009
$622$		0			0
$623$	−0.343504			−0.0137622
$624$		0			0
$625$	−21.4201	−	12.8910i	−0.856805	−	0.515641i
$626$		0			0
$627$			0.971801i			0.0388100i
$628$		0			0
$629$	33.6161	+	12.2754i	1.34036	+	0.489453i
$630$		0			0
$631$	17.1990			0.684683			0.342341	−	0.939576i	$-0.388780\pi$
$631$	17.1990			0.684683			0.342341	+	0.939576i	$0.388780\pi$
$632$		0			0
$633$		−	1.00752i		−	0.0400455i
$634$		0			0
$635$	−2.58700	−	3.40332i	−0.102662	−	0.135057i
$636$		0			0
$637$			0.124430i			0.00493010i
$638$		0			0
$639$			39.9449i			1.58019i
$640$		0			0
$641$		−	29.4142i		−	1.16179i	−0.813978	−	0.580895i	$-0.802703\pi$
$641$		−	29.4142i		−	1.16179i	0.813978	−	0.580895i	$-0.197297\pi$
$642$		0			0
$643$	0.837979			0.0330467			0.0165233	−	0.999863i	$-0.494740\pi$
$643$	0.837979			0.0330467			0.0165233	+	0.999863i	$0.494740\pi$
$644$		0			0
$645$	−0.167617	−	0.220508i	−0.00659993	−	0.00868251i
$646$		0			0
$647$			33.4902i			1.31664i	0.752740	+	0.658318i	$0.228732\pi$
$647$			33.4902i			1.31664i	−0.752740	+	0.658318i	$0.771268\pi$
$648$		0			0
$649$		−	25.9765i		−	1.01967i
$650$		0			0
$651$			0.319102i			0.0125066i
$652$		0			0
$653$	8.99856			0.352141			0.176070	−	0.984378i	$-0.443661\pi$
$653$	8.99856			0.352141			0.176070	+	0.984378i	$0.443661\pi$
$654$		0			0
$655$	11.1100	+	14.6157i	0.434104	+	0.571084i
$656$		0			0
$657$	−5.63242			−0.219742
$658$		0			0
$659$	−43.3245			−1.68768			−0.843841	−	0.536594i	$-0.819711\pi$
$659$	−43.3245			−1.68768			−0.843841	+	0.536594i	$0.819711\pi$
$660$		0			0
$661$	−22.5144			−0.875709			−0.437854	−	0.899046i	$-0.644261\pi$
$661$	−22.5144			−0.875709			−0.437854	+	0.899046i	$0.644261\pi$
$662$		0			0
$663$	−0.00706828	+	0.0193564i	−0.000274509	+	0.000751740i
$664$		0			0
$665$	−9.76233	+	7.42074i	−0.378567	+	0.287764i
$666$		0			0
$667$		−	9.17415i		−	0.355225i
$668$		0			0
$669$			0.736287i			0.0284665i
$670$		0			0
$671$	25.3692			0.979367
$672$		0			0
$673$	42.0850			1.62226			0.811128	−	0.584868i	$-0.198854\pi$
$673$	42.0850			1.62226			0.811128	+	0.584868i	$0.198854\pi$
$674$		0			0
$675$	0.322337	−	1.16072i	0.0124067	−	0.0446762i
$676$		0			0
$677$	15.8497			0.609152			0.304576	−	0.952488i	$-0.401485\pi$
$677$	15.8497			0.609152			0.304576	+	0.952488i	$0.401485\pi$
$678$		0			0
$679$	−9.97802			−0.382921
$680$		0			0
$681$	0.940377			0.0360353
$682$		0			0
$683$	11.8248			0.452464			0.226232	−	0.974073i	$-0.427359\pi$
$683$	11.8248			0.452464			0.226232	+	0.974073i	$0.427359\pi$
$684$		0			0
$685$	31.1435	+	40.9707i	1.18993	+	1.56541i
$686$		0			0
$687$	0.486782			0.0185719
$688$		0			0
$689$	0.493465			0.0187995
$690$		0			0
$691$		−	20.2191i		−	0.769170i	−0.923089	−	0.384585i	$-0.874344\pi$
$691$		−	20.2191i		−	0.769170i	0.923089	−	0.384585i	$-0.125656\pi$
$692$		0			0
$693$		−	13.2286i		−	0.502512i
$694$		0			0
$695$	−7.57613	−	9.96675i	−0.287379	−	0.378060i
$696$		0			0
$697$	−17.3864	+	47.6124i	−0.658557	+	1.80345i
$698$		0			0
$699$	0.803204			0.0303800
$700$		0			0
$701$	16.8812			0.637594			0.318797	−	0.947823i	$-0.396721\pi$
$701$	16.8812			0.637594			0.318797	+	0.947823i	$0.396721\pi$
$702$		0			0
$703$	47.5992			1.79524
$704$		0			0
$705$	0.241567	+	0.317793i	0.00909795	+	0.0119688i
$706$		0			0
$707$	3.40363			0.128006
$708$		0			0
$709$		−	40.4270i		−	1.51827i	−0.650934	−	0.759135i	$-0.725622\pi$
$709$		−	40.4270i		−	1.51827i	0.650934	−	0.759135i	$-0.274378\pi$
$710$		0			0
$711$		−	16.7622i		−	0.628631i
$712$		0			0
$713$			46.7571i			1.75107i
$714$		0			0
$715$	−0.977255	+	0.742851i	−0.0365473	+	0.0277811i
$716$		0			0
$717$	0.855643			0.0319546
$718$		0			0
$719$		−	25.2148i		−	0.940354i	−0.882572	−	0.470177i	$-0.844190\pi$
$719$		−	25.2148i		−	0.940354i	0.882572	−	0.470177i	$-0.155810\pi$
$720$		0			0
$721$			0.471544i			0.0175612i
$722$		0			0
$723$			0.476311i			0.0177142i
$724$		0			0
$725$	7.50981	+	2.08550i	0.278908	+	0.0774535i
$726$		0			0
$727$		−	28.3494i		−	1.05142i	−0.850664	−	0.525710i	$-0.823800\pi$
$727$		−	28.3494i		−	1.05142i	0.850664	−	0.525710i	$-0.176200\pi$
$728$		0			0
$729$	−26.9129			−0.996775
$730$		0			0
$731$	4.36159	−	11.9442i	0.161319	−	0.441770i
$732$		0			0
$733$		−	22.1845i		−	0.819403i	−0.912220	−	0.409702i	$-0.865633\pi$
$733$		−	22.1845i		−	0.819403i	0.912220	−	0.409702i	$-0.134367\pi$
$734$		0			0
$735$	−0.0715012	+	0.0543509i	−0.00263736	+	0.00200476i
$736$		0			0
$737$	−52.5363			−1.93520
$738$		0			0
$739$	6.41874			0.236117			0.118059	−	0.993007i	$-0.462333\pi$
$739$	6.41874			0.236117			0.118059	+	0.993007i	$0.462333\pi$
$740$		0			0
$741$			0.0274080i			0.00100686i
$742$		0			0
$743$	12.7967			0.469467			0.234734	−	0.972060i	$-0.424578\pi$
$743$	12.7967			0.469467			0.234734	+	0.972060i	$0.424578\pi$
$744$		0			0
$745$	−25.2514	+	19.1946i	−0.925141	+	0.703237i
$746$		0			0
$747$		−	26.7955i		−	0.980395i
$748$		0			0
$749$	7.94370			0.290257
$750$		0			0
$751$			26.9892i			0.984851i	0.870355	+	0.492426i	$0.163890\pi$
$751$			26.9892i			0.984851i	−0.870355	+	0.492426i	$0.836110\pi$
$752$		0			0
$753$	−1.00220			−0.0365221
$754$		0			0
$755$	8.29685	−	6.30677i	0.301953	−	0.229527i
$756$		0			0
$757$		−	35.1468i		−	1.27743i	−0.769442	−	0.638716i	$-0.779466\pi$
$757$		−	35.1468i		−	1.27743i	0.769442	−	0.638716i	$-0.220534\pi$
$758$		0			0
$759$			1.04293i			0.0378561i
$760$		0			0
$761$	−16.5325			−0.599304			−0.299652	−	0.954049i	$-0.596870\pi$
$761$	−16.5325			−0.599304			−0.299652	+	0.954049i	$0.596870\pi$
$762$		0			0
$763$			3.72199i			0.134745i
$764$		0			0
$765$	26.4104	−	8.16503i	0.954870	−	0.295207i
$766$		0			0
$767$		−	0.732624i		−	0.0264535i
$768$		0			0
$769$	40.9878			1.47806			0.739029	−	0.673674i	$-0.235285\pi$
$769$	40.9878			1.47806			0.739029	+	0.673674i	$0.235285\pi$
$770$		0			0
$771$		−	0.990976i		−	0.0356891i
$772$		0			0
$773$		−	38.8349i		−	1.39680i	−0.715710	−	0.698398i	$-0.753897\pi$
$773$		−	38.8349i		−	1.39680i	0.715710	−	0.698398i	$-0.246103\pi$
$774$		0			0
$775$	−38.2747	−	10.6290i	−1.37487	−	0.381805i
$776$		0			0
$777$	0.348626			0.0125069
$778$		0			0
$779$			67.4176i			2.41549i
$780$		0			0
$781$	58.7758			2.10317
$782$		0			0
$783$			0.375562i			0.0134215i
$784$		0			0
$785$	−21.3229	−	28.0513i	−0.761046	−	1.00119i
$786$		0			0
$787$	−0.155779			−0.00555292			−0.00277646	−	0.999996i	$-0.500884\pi$
$787$	−0.155779			−0.00555292			−0.00277646	+	0.999996i	$0.500884\pi$
$788$		0			0
$789$			0.419662i			0.0149403i
$790$		0			0
$791$	−1.84147			−0.0654751
$792$		0			0
$793$	0.715495			0.0254080
$794$		0			0
$795$	0.215545	+	0.283559i	0.00764458	+	0.0100568i
$796$		0			0
$797$			19.0375i			0.674345i	0.941443	+	0.337172i	$0.109470\pi$
$797$			19.0375i			0.674345i	−0.941443	+	0.337172i	$0.890530\pi$
$798$		0			0
$799$	−6.28585	+	17.2137i	−0.222377	+	0.608977i
$800$		0			0
$801$	1.02996			0.0363918
$802$		0			0
$803$			8.28768i			0.292466i
$804$		0			0
$805$	−10.4769	+	7.96390i	−0.369262	+	0.280691i
$806$		0			0
$807$			0.676662i			0.0238196i
$808$		0			0
$809$		−	53.0643i		−	1.86564i	−0.360339	−	0.932821i	$-0.617339\pi$
$809$		−	53.0643i		−	1.86564i	0.360339	−	0.932821i	$-0.382661\pi$
$810$		0			0
$811$		−	7.08316i		−	0.248723i	−0.992237	−	0.124362i	$-0.960312\pi$
$811$		−	7.08316i		−	0.248723i	0.992237	−	0.124362i	$-0.0396883\pi$
$812$		0			0
$813$	0.108807			0.00381602
$814$		0			0
$815$	6.72925	−	5.11517i	0.235715	−	0.179177i
$816$		0			0
$817$		−	16.9125i		−	0.591694i
$818$		0			0
$819$		−	0.373089i		−	0.0130368i
$820$		0			0
$821$			22.6005i			0.788762i	0.918947	+	0.394381i	$0.129041\pi$
$821$			22.6005i			0.788762i	−0.918947	+	0.394381i	$0.870959\pi$
$822$		0			0
$823$	−37.3499			−1.30194			−0.650968	−	0.759105i	$-0.725637\pi$
$823$	−37.3499			−1.30194			−0.650968	+	0.759105i	$0.725637\pi$
$824$		0			0
$825$	−0.853728	−	0.237083i	−0.0297230	−	0.00825417i
$826$		0			0
$827$	−3.00350			−0.104442			−0.0522210	−	0.998636i	$-0.516630\pi$
$827$	−3.00350			−0.104442			−0.0522210	+	0.998636i	$0.516630\pi$
$828$		0			0
$829$	−44.1018			−1.53172			−0.765860	−	0.643007i	$-0.777686\pi$
$829$	−44.1018			−1.53172			−0.765860	+	0.643007i	$0.777686\pi$
$830$		0			0
$831$	0.265382			0.00920598
$832$		0			0
$833$	−3.87296	−	1.41427i	−0.134190	−	0.0490016i
$834$		0			0
$835$	−3.90513	+	2.96845i	−0.135143	+	0.102727i
$836$		0			0
$837$		−	1.91410i		−	0.0661608i
$838$		0			0
$839$			26.8617i			0.927369i	0.886000	+	0.463685i	$0.153473\pi$
$839$			26.8617i			0.927369i	−0.886000	+	0.463685i	$0.846527\pi$
$840$		0			0
$841$	26.5701			0.916212
$842$		0			0
$843$	0.279634			0.00963111
$844$		0			0
$845$	23.1144	−	17.5702i	0.795160	−	0.604433i
$846$		0			0
$847$	−8.46484			−0.290855
$848$		0			0
$849$	−0.571239			−0.0196049
$850$		0			0
$851$	51.0833			1.75111
$852$		0			0
$853$	−20.0239			−0.685605			−0.342803	−	0.939407i	$-0.611376\pi$
$853$	−20.0239			−0.685605			−0.342803	+	0.939407i	$0.611376\pi$
$854$		0			0
$855$	29.2712	−	22.2502i	1.00105	−	0.760942i
$856$		0			0
$857$	10.6832			0.364931			0.182466	−	0.983212i	$-0.441592\pi$
$857$	10.6832			0.364931			0.182466	+	0.983212i	$0.441592\pi$
$858$		0			0
$859$	−27.5694			−0.940656			−0.470328	−	0.882492i	$-0.655864\pi$
$859$	−27.5694			−0.940656			−0.470328	+	0.882492i	$0.655864\pi$
$860$		0			0
$861$			0.493779i			0.0168280i
$862$		0			0
$863$		−	52.4099i		−	1.78405i	−0.451982	−	0.892027i	$-0.649283\pi$
$863$		−	52.4099i		−	1.78405i	0.451982	−	0.892027i	$-0.350717\pi$
$864$		0			0
$865$	−29.8050	+	22.6560i	−1.01340	+	0.770327i
$866$		0			0
$867$	−0.522142	−	0.440009i	−0.0177329	−	0.0149435i
$868$		0			0
$869$	−24.6643			−0.836678
$870$		0			0
$871$	−1.48170			−0.0502054
$872$		0			0
$873$	29.9180			1.01257
$874$		0			0
$875$	−4.13748	−	10.3866i	−0.139872	−	0.351131i
$876$		0			0
$877$	−0.704352			−0.0237843			−0.0118921	−	0.999929i	$-0.503785\pi$
$877$	−0.704352			−0.0237843			−0.0118921	+	0.999929i	$0.503785\pi$
$878$		0			0
$879$		−	1.12269i		−	0.0378674i
$880$		0			0
$881$			18.8242i			0.634204i	0.948391	+	0.317102i	$0.102710\pi$
$881$			18.8242i			0.634204i	−0.948391	+	0.317102i	$0.897290\pi$
$882$		0			0
$883$		−	38.2228i		−	1.28630i	−0.765741	−	0.643150i	$-0.777627\pi$
$883$		−	38.2228i		−	1.28630i	0.765741	−	0.643150i	$-0.222373\pi$
$884$		0			0
$885$	0.420987	−	0.320009i	0.0141513	−	0.0107570i
$886$		0			0
$887$	−34.0956			−1.14482			−0.572408	−	0.819969i	$-0.693991\pi$
$887$	−34.0956			−1.14482			−0.572408	+	0.819969i	$0.693991\pi$
$888$		0			0
$889$		−	1.91181i		−	0.0641202i
$890$		0			0
$891$			39.6430i			1.32809i
$892$		0			0
$893$			24.3740i			0.815646i
$894$		0			0
$895$	34.4787	−	26.2086i	1.15250	−	0.876058i
$896$		0			0
$897$			0.0294141i			0.000982109i
$898$		0			0
$899$	12.3841			0.413033
$900$		0			0
$901$	−5.60872	+	15.3594i	−0.186853	+	0.511695i
$902$		0			0
$903$		−	0.123871i		−	0.00412215i
$904$		0			0
$905$	3.32733	+	4.37726i	0.110604	+	0.145505i
$906$		0			0
$907$	51.7651			1.71883			0.859416	−	0.511277i	$-0.170827\pi$
$907$	51.7651			1.71883			0.859416	+	0.511277i	$0.170827\pi$
$908$		0			0
$909$	−10.2054			−0.338491
$910$		0			0
$911$		−	37.9026i		−	1.25577i	−0.778306	−	0.627885i	$-0.783921\pi$
$911$		−	37.9026i		−	1.25577i	0.778306	−	0.627885i	$-0.216079\pi$
$912$		0			0
$913$	−39.4275			−1.30486
$914$		0			0
$915$	0.312527	+	0.411144i	0.0103318	+	0.0135920i
$916$		0			0
$917$			8.21039i			0.271131i
$918$		0			0
$919$	5.90232			0.194699			0.0973497	−	0.995250i	$-0.468963\pi$
$919$	5.90232			0.194699			0.0973497	+	0.995250i	$0.468963\pi$
$920$		0			0
$921$		−	0.746780i		−	0.0246072i
$922$		0			0
$923$	1.65767			0.0545630
$924$		0			0
$925$	−11.6124	+	41.8159i	−0.381814	+	1.37490i
$926$		0			0
$927$		−	1.41387i		−	0.0464377i
$928$		0			0
$929$		−	16.4841i		−	0.540825i	−0.962745	−	0.270412i	$-0.912840\pi$
$929$		−	16.4841i		−	0.540825i	0.962745	−	0.270412i	$-0.0871601\pi$
$930$		0			0
$931$	−5.48398			−0.179730
$932$		0			0
$933$			0.581978i			0.0190531i
$934$		0			0
$935$	−12.0142	−	38.8609i	−0.392907	−	1.27089i
$936$		0			0
$937$			35.5388i			1.16100i	0.814260	+	0.580500i	$0.197143\pi$
$937$			35.5388i			1.16100i	−0.814260	+	0.580500i	$0.802857\pi$
$938$		0			0
$939$	0.959227			0.0313032
$940$		0			0
$941$			8.02989i			0.261767i	0.991398	+	0.130883i	$0.0417814\pi$
$941$			8.02989i			0.261767i	−0.991398	+	0.130883i	$0.958219\pi$
$942$		0			0
$943$			72.3522i			2.35611i
$944$		0			0
$945$	0.428892	−	0.326018i	0.0139518	−	0.0106054i
$946$		0			0
$947$	−50.9056			−1.65421			−0.827105	−	0.562048i	$-0.810014\pi$
$947$	−50.9056			−1.65421			−0.827105	+	0.562048i	$0.810014\pi$
$948$		0			0
$949$			0.233740i			0.00758752i
$950$		0			0
$951$	0.338911			0.0109900
$952$		0			0
$953$			46.6306i			1.51051i	0.655428	+	0.755257i	$0.272488\pi$
$953$			46.6306i			1.51051i	−0.655428	+	0.755257i	$0.727512\pi$
$954$		0			0
$955$	41.5006	−	31.5463i	1.34293	−	1.02081i
$956$		0			0
$957$	0.276231			0.00892928
$958$		0			0
$959$			23.0153i			0.743202i
$960$		0			0
$961$	−32.1170			−1.03603
$962$		0			0
$963$	−23.8183			−0.767534
$964$		0			0
$965$	4.41223	−	3.35392i	0.142035	−	0.107966i
$966$		0			0
$967$			3.87224i			0.124523i	0.998060	+	0.0622614i	$0.0198312\pi$
$967$			3.87224i			0.124523i	−0.998060	+	0.0622614i	$0.980169\pi$
$968$		0			0
$969$	−0.853091	−	0.311519i	−0.0274052	−	0.0100074i
$970$		0			0
$971$	−22.0669			−0.708159			−0.354080	−	0.935215i	$-0.615206\pi$
$971$	−22.0669			−0.708159			−0.354080	+	0.935215i	$0.615206\pi$
$972$		0			0
$973$		−	5.59882i		−	0.179490i
$974$		0			0
$975$	−0.0240779	−	0.00668652i	−0.000771111	−	0.000214140i
$976$		0			0
$977$			5.95334i			0.190464i	0.995455	+	0.0952322i	$0.0303594\pi$
$977$			5.95334i			0.190464i	−0.995455	+	0.0952322i	$0.969641\pi$
$978$		0			0
$979$		−	1.51551i		−	0.0484358i
$980$		0			0
$981$		−	11.1600i		−	0.356310i
$982$		0			0
$983$	−53.6308			−1.71056			−0.855279	−	0.518168i	$-0.826614\pi$
$983$	−53.6308			−1.71056			−0.855279	+	0.518168i	$0.826614\pi$
$984$		0			0
$985$	−45.8247	+	34.8332i	−1.46010	+	1.10988i
$986$		0			0
$987$			0.178520i			0.00568236i
$988$		0			0
$989$		−	18.1504i		−	0.577150i
$990$		0			0
$991$		−	15.7975i		−	0.501825i	−0.968010	−	0.250913i	$-0.919269\pi$
$991$		−	15.7975i		−	0.501825i	0.968010	−	0.250913i	$-0.0807307\pi$
$992$		0			0
$993$	0.192877			0.00612076
$994$		0			0
$995$	−1.16011	−	1.52617i	−0.0367779	−	0.0483830i
$996$		0			0
$997$	34.2790			1.08563			0.542814	−	0.839853i	$-0.317359\pi$
$997$	34.2790			1.08563			0.542814	+	0.839853i	$0.317359\pi$
$998$		0			0
$999$	−2.09119			−0.0661624

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2380.2.m.d.169.12	yes	26
5.4	even	2		2380.2.m.c.169.16	yes	26
17.16	even	2		2380.2.m.c.169.15	✓	26
85.84	even	2	inner	2380.2.m.d.169.11	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2380.2.m.c.169.15	✓	26	17.16	even	2
2380.2.m.c.169.16	yes	26	5.4	even	2
2380.2.m.d.169.11	yes	26	85.84	even	2	inner
2380.2.m.d.169.12	yes	26	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2380.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.0401658 q^{3} +(1.78015 - 1.35317i) q^{5} +1.00000 q^{7} -2.99839 q^{9} +4.41190i q^{11} +0.124430i q^{13} +(-0.0715012 + 0.0543509i) q^{15} +(-3.87296 - 1.41427i) q^{17} -5.48398 q^{19} -0.0401658 q^{21} -5.88539 q^{23} +(1.33789 - 4.81768i) q^{25} +0.240930 q^{27} +1.55880i q^{29} -7.94462i q^{31} -0.177207i q^{33} +(1.78015 - 1.35317i) q^{35} -8.67968 q^{37} -0.00499783i q^{39} -12.2935i q^{41} +3.08398i q^{43} +(-5.33759 + 4.05731i) q^{45} -4.44458i q^{47} +1.00000 q^{49} +(0.155560 + 0.0568053i) q^{51} -3.96580i q^{53} +(5.97003 + 7.85385i) q^{55} +0.220268 q^{57} -5.88784 q^{59} -5.75018i q^{61} -2.99839 q^{63} +(0.168374 + 0.221504i) q^{65} +11.9079i q^{67} +0.236391 q^{69} -13.3221i q^{71} +1.87848 q^{73} +(-0.0537372 + 0.193506i) q^{75} +4.41190i q^{77} +5.59040i q^{79} +8.98548 q^{81} +8.93663i q^{83} +(-8.80821 + 2.72314i) q^{85} -0.0626105i q^{87} -0.343504 q^{89} +0.124430i q^{91} +0.319102i q^{93} +(-9.76233 + 7.42074i) q^{95} -9.97802 q^{97} -13.2286i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) \) 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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