LMFDB - Embedded newform 1008.3.cg.q.145.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(145,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1008.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cg (of order $6$, degree $2$, 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + \cdots + 1148023744 $ x^16 - 2x^15 + 81x^14 - 118x^13 + 1960x^12 - 366x^11 + 37625x^10 - 83714x^9 + 623931x^8 + 289492x^7 + 241286x^6 - 4777608x^5 + 98362220x^4 - 239083824x^3 + 523977616x^2 - 403737152*x + 1148023744
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.8
Root			$-0.338813 + 7.55243i$ of defining polynomial
Character	$\chi$	$=$	1008.145
Dual form			1008.3.cg.q.577.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.85541 + 4.53532i) q^{5} +(-1.17763 + 6.90023i) q^{7} +O(q^{10})$	$q+(7.85541 + 4.53532i) q^{5} +(-1.17763 + 6.90023i) q^{7} +(8.81372 + 15.2658i) q^{11} -19.0631i q^{13} +(14.7807 - 8.53366i) q^{17} +(7.74221 + 4.46997i) q^{19} +(-4.72676 + 8.18700i) q^{23} +(28.6383 + 49.6030i) q^{25} +10.0174 q^{29} +(39.5511 - 22.8348i) q^{31} +(-40.5455 + 48.8632i) q^{35} +(-26.9278 + 46.6403i) q^{37} -58.9153i q^{41} -69.3894 q^{43} +(-41.7123 - 24.0826i) q^{47} +(-46.2264 - 16.2518i) q^{49} +(21.6496 + 37.4982i) q^{53} +159.892i q^{55} +(-0.930469 + 0.537207i) q^{59} +(45.3875 + 26.2045i) q^{61} +(86.4572 - 149.748i) q^{65} +(24.0006 + 41.5702i) q^{67} -53.5026 q^{71} +(-2.16558 + 1.25030i) q^{73} +(-115.717 + 42.8393i) q^{77} +(11.2927 - 19.5596i) q^{79} -51.2015i q^{83} +154.812 q^{85} +(-43.9536 - 25.3766i) q^{89} +(131.540 + 22.4492i) q^{91} +(40.5455 + 70.2269i) q^{95} +126.711i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	7.85541	+	4.53532i	1.57108	+	0.907064i	0.996037	+	0.0889425i	$0.0283487\pi$
$5$	7.85541	+	4.53532i	1.57108	+	0.907064i	0.575045	+	0.818122i	$0.304985\pi$
$6$		0			0
$7$	−1.17763	+	6.90023i	−0.168232	+	0.985747i
$7$	−1.17763	+	6.90023i	−0.168232	+	0.985747i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.81372	+	15.2658i	0.801247	+	1.38780i	0.918796	+	0.394733i	$0.129163\pi$
$11$	8.81372	+	15.2658i	0.801247	+	1.38780i	−0.117549	+	0.993067i	$0.537504\pi$
$12$		0			0
$13$		−	19.0631i		−	1.46639i	−0.680018	−	0.733195i	$-0.738028\pi$
$13$		−	19.0631i		−	1.46639i	0.680018	−	0.733195i	$-0.261972\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.7807	−	8.53366i	0.869454	−	0.501980i	0.00228757	−	0.999997i	$-0.499272\pi$
$17$	14.7807	−	8.53366i	0.869454	−	0.501980i	0.867167	+	0.498018i	$0.165939\pi$
$18$		0			0
$19$	7.74221	+	4.46997i	0.407485	+	0.235262i	0.689709	−	0.724087i	$-0.257739\pi$
$19$	7.74221	+	4.46997i	0.407485	+	0.235262i	−0.282224	+	0.959349i	$0.591072\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.72676	+	8.18700i	−0.205512	+	0.355956i	−0.950296	−	0.311349i	$-0.899219\pi$
$23$	−4.72676	+	8.18700i	−0.205512	+	0.355956i	0.744784	+	0.667306i	$0.232552\pi$
$24$		0			0
$25$	28.6383	+	49.6030i	1.14553	+	1.98412i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	10.0174			0.345428			0.172714	−	0.984972i	$-0.444746\pi$
$29$	10.0174			0.345428			0.172714	+	0.984972i	$0.444746\pi$
$30$		0			0
$31$	39.5511	−	22.8348i	1.27584	−	0.736608i	0.299761	−	0.954014i	$-0.403093\pi$
$31$	39.5511	−	22.8348i	1.27584	−	0.736608i	0.976081	+	0.217407i	$0.0697598\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−40.5455	+	48.8632i	−1.15844	+	1.39609i
$36$		0			0
$37$	−26.9278	+	46.6403i	−0.727778	+	1.26055i	0.230042	+	0.973181i	$0.426114\pi$
$37$	−26.9278	+	46.6403i	−0.727778	+	1.26055i	−0.957820	+	0.287368i	$0.907220\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	58.9153i		−	1.43696i	−0.695549	−	0.718479i	$-0.744839\pi$
$41$		−	58.9153i		−	1.43696i	0.695549	−	0.718479i	$-0.255161\pi$
$42$		0			0
$43$	−69.3894			−1.61371			−0.806853	−	0.590752i	$-0.798831\pi$
$43$	−69.3894			−1.61371			−0.806853	+	0.590752i	$0.798831\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−41.7123	−	24.0826i	−0.887495	−	0.512395i	−0.0143726	−	0.999897i	$-0.504575\pi$
$47$	−41.7123	−	24.0826i	−0.887495	−	0.512395i	−0.873122	+	0.487501i	$0.837908\pi$
$48$		0			0
$49$	−46.2264	−	16.2518i	−0.943396	−	0.331669i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	21.6496	+	37.4982i	0.408483	+	0.707514i	0.994720	−	0.102626i	$-0.0327245\pi$
$53$	21.6496	+	37.4982i	0.408483	+	0.707514i	−0.586237	+	0.810140i	$0.699391\pi$
$54$		0			0
$55$			159.892i			2.90713i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.930469	+	0.537207i	−0.0157707	+	0.00910520i	−0.507865	−	0.861437i	$-0.669565\pi$
$59$	−0.930469	+	0.537207i	−0.0157707	+	0.00910520i	0.492094	+	0.870542i	$0.336232\pi$
$60$		0			0
$61$	45.3875	+	26.2045i	0.744058	+	0.429582i	0.823543	−	0.567254i	$-0.191994\pi$
$61$	45.3875	+	26.2045i	0.744058	+	0.429582i	−0.0794850	+	0.996836i	$0.525328\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	86.4572	−	149.748i	1.33011	−	2.30382i
$66$		0			0
$67$	24.0006	+	41.5702i	0.358217	+	0.620451i	0.987663	−	0.156594i	$-0.0500513\pi$
$67$	24.0006	+	41.5702i	0.358217	+	0.620451i	−0.629446	+	0.777045i	$0.716718\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−53.5026			−0.753558			−0.376779	−	0.926303i	$-0.622968\pi$
$71$	−53.5026			−0.753558			−0.376779	+	0.926303i	$0.622968\pi$
$72$		0			0
$73$	−2.16558	+	1.25030i	−0.0296655	+	0.0171274i	−0.514759	−	0.857335i	$-0.672119\pi$
$73$	−2.16558	+	1.25030i	−0.0296655	+	0.0171274i	0.485094	+	0.874462i	$0.338785\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−115.717	+	42.8393i	−1.50282	+	0.556354i
$78$		0			0
$79$	11.2927	−	19.5596i	0.142946	−	0.247590i	−0.785659	−	0.618660i	$-0.787676\pi$
$79$	11.2927	−	19.5596i	0.142946	−	0.247590i	0.928605	+	0.371070i	$0.121009\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	51.2015i		−	0.616886i	−0.951243	−	0.308443i	$-0.900192\pi$
$83$		−	51.2015i		−	0.616886i	0.951243	−	0.308443i	$-0.0998078\pi$
$84$		0			0
$85$	154.812			1.82131
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−43.9536	−	25.3766i	−0.493861	−	0.285131i	0.232314	−	0.972641i	$-0.425370\pi$
$89$	−43.9536	−	25.3766i	−0.493861	−	0.285131i	−0.726175	+	0.687510i	$0.758704\pi$
$90$		0			0
$91$	131.540	+	22.4492i	1.44549	+	0.246694i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	40.5455	+	70.2269i	0.426795	+	0.739230i
$96$		0			0
$97$			126.711i			1.30630i	0.757230	+	0.653149i	$0.226552\pi$
$97$			126.711i			1.30630i	−0.757230	+	0.653149i	$0.773448\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.57104	−	1.48439i	0.0254558	−	0.0146969i	−0.487218	−	0.873280i	$-0.661988\pi$
$101$	2.57104	−	1.48439i	0.0254558	−	0.0146969i	0.512674	+	0.858583i	$0.328655\pi$
$102$		0			0
$103$	6.15906	+	3.55593i	0.0597967	+	0.0345236i	0.529600	−	0.848247i	$-0.322342\pi$
$103$	6.15906	+	3.55593i	0.0597967	+	0.0345236i	−0.469804	+	0.882771i	$0.655675\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	27.3521	−	47.3753i	0.255627	−	0.442759i	−0.709438	−	0.704767i	$-0.751051\pi$
$107$	27.3521	−	47.3753i	0.255627	−	0.442759i	0.965066	+	0.262008i	$0.0843847\pi$
$108$		0			0
$109$	−6.54259	−	11.3321i	−0.0600237	−	0.103964i	0.834452	−	0.551081i	$-0.185784\pi$
$109$	−6.54259	−	11.3321i	−0.0600237	−	0.103964i	−0.894476	+	0.447116i	$0.852451\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−39.6743			−0.351100			−0.175550	−	0.984471i	$-0.556170\pi$
$113$	−39.6743			−0.351100			−0.175550	+	0.984471i	$0.556170\pi$
$114$		0			0
$115$	−74.2613	+	42.8748i	−0.645751	+	0.372824i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	41.4780	+	112.040i	0.348555	+	0.941512i
$120$		0			0
$121$	−94.8632	+	164.308i	−0.783993	+	1.35792i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			292.769i			2.34215i
$126$		0			0
$127$	110.071			0.866701			0.433351	−	0.901225i	$-0.357331\pi$
$127$	110.071			0.866701			0.433351	+	0.901225i	$0.357331\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−200.578	−	115.804i	−1.53113	−	0.883998i	−0.999310	−	0.0371405i	$-0.988175\pi$
$131$	−200.578	−	115.804i	−1.53113	−	0.883998i	−0.531820	−	0.846858i	$-0.678492\pi$
$132$		0			0
$133$	−39.9613	+	48.1591i	−0.300461	+	0.362099i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	126.475	+	219.062i	0.923178	+	1.59899i	0.794465	+	0.607309i	$0.207751\pi$
$137$	126.475	+	219.062i	0.923178	+	1.59899i	0.128713	+	0.991682i	$0.458916\pi$
$138$		0			0
$139$		−	40.9631i		−	0.294699i	−0.989085	−	0.147349i	$-0.952926\pi$
$139$		−	40.9631i		−	0.294699i	0.989085	−	0.147349i	$-0.0470742\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	291.013	−	168.017i	2.03506	−	1.17494i
$144$		0			0
$145$	78.6908	+	45.4322i	0.542695	+	0.313325i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	79.7034	−	138.050i	0.534922	−	0.926512i	−0.464245	−	0.885707i	$-0.653674\pi$
$149$	79.7034	−	138.050i	0.534922	−	0.926512i	0.999167	−	0.0408055i	$-0.0129924\pi$
$150$		0			0
$151$	−27.4354	−	47.5195i	−0.181691	−	0.314699i	0.760765	−	0.649027i	$-0.224824\pi$
$151$	−27.4354	−	47.5195i	−0.181691	−	0.314699i	−0.942457	+	0.334328i	$0.891491\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	414.253			2.67260
$156$		0			0
$157$	95.5760	−	55.1808i	0.608764	−	0.351470i	−0.163717	−	0.986507i	$-0.552349\pi$
$157$	95.5760	−	55.1808i	0.608764	−	0.351470i	0.772482	+	0.635037i	$0.219015\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−50.9258	−	42.2570i	−0.316309	−	0.262466i
$162$		0			0
$163$	−113.229	+	196.118i	−0.694657	+	1.20318i	0.275640	+	0.961261i	$0.411110\pi$
$163$	−113.229	+	196.118i	−0.694657	+	1.20318i	−0.970296	+	0.241920i	$0.922223\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			175.214i			1.04918i	0.851354	+	0.524592i	$0.175782\pi$
$167$			175.214i			1.04918i	−0.851354	+	0.524592i	$0.824218\pi$
$168$		0			0
$169$	−194.401			−1.15030
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.3775	+	7.14614i	0.0715461	+	0.0413072i	0.535346	−	0.844633i	$-0.320181\pi$
$173$	12.3775	+	7.14614i	0.0715461	+	0.0413072i	−0.463800	+	0.885940i	$0.653514\pi$
$174$		0			0
$175$	−375.997	+	139.197i	−2.14856	+	0.795412i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.121347	+	0.210178i	0.000677914	+	0.00117418i	0.866364	−	0.499413i	$-0.166451\pi$
$179$	0.121347	+	0.210178i	0.000677914	+	0.00117418i	−0.865686	+	0.500587i	$0.833118\pi$
$180$		0			0
$181$			1.08434i			0.00599080i	0.999996	+	0.00299540i	$0.000953467\pi$
$181$			1.08434i			0.00599080i	−0.999996	+	0.00299540i	$0.999047\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−423.057	+	244.252i	−2.28680	+	1.32028i
$186$		0			0
$187$	260.546	+	150.426i	1.39330	+	0.804419i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	101.580	−	175.942i	0.531833	−	0.921162i	−0.467476	−	0.884006i	$-0.654837\pi$
$191$	101.580	−	175.942i	0.531833	−	0.921162i	0.999309	−	0.0371565i	$-0.0118300\pi$
$192$		0			0
$193$	−118.113	−	204.578i	−0.611985	−	1.05999i	−0.990905	−	0.134560i	$-0.957038\pi$
$193$	−118.113	−	204.578i	−0.611985	−	1.05999i	0.378920	−	0.925429i	$-0.376295\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	50.1145			0.254388			0.127194	−	0.991878i	$-0.459403\pi$
$197$	50.1145			0.254388			0.127194	+	0.991878i	$0.459403\pi$
$198$		0			0
$199$	106.338	−	61.3944i	0.534363	−	0.308514i	−0.208429	−	0.978038i	$-0.566835\pi$
$199$	106.338	−	61.3944i	0.534363	−	0.308514i	0.742791	+	0.669523i	$0.233502\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.7968	+	69.1224i	−0.0581121	+	0.340505i
$204$		0			0
$205$	267.200	−	462.803i	1.30341	−	2.25758i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			157.588i			0.754010i
$210$		0			0
$211$	247.973			1.17523			0.587614	−	0.809141i	$-0.300067\pi$
$211$	247.973			1.17523			0.587614	+	0.809141i	$0.300067\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−545.082	−	314.703i	−2.53526	−	1.46374i
$216$		0			0
$217$	110.989	+	299.803i	0.511471	+	1.38158i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−162.678	−	281.766i	−0.736099	−	1.27496i
$222$		0			0
$223$		−	205.854i		−	0.923114i	−0.887111	−	0.461557i	$-0.847291\pi$
$223$		−	205.854i		−	0.923114i	0.887111	−	0.461557i	$-0.152709\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.36322	+	0.787054i	−0.00600536	+	0.00346720i	−0.503000	−	0.864287i	$-0.667770\pi$
$227$	−1.36322	+	0.787054i	−0.00600536	+	0.00346720i	0.496994	+	0.867754i	$0.334437\pi$
$228$		0			0
$229$	19.4017	+	11.2016i	0.0847238	+	0.0489153i	0.541763	−	0.840531i	$-0.317757\pi$
$229$	19.4017	+	11.2016i	0.0847238	+	0.0489153i	−0.457040	+	0.889446i	$0.651090\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	39.2335	−	67.9543i	0.168384	−	0.291650i	−0.769468	−	0.638685i	$-0.779479\pi$
$233$	39.2335	−	67.9543i	0.168384	−	0.291650i	0.937852	+	0.347036i	$0.112812\pi$
$234$		0			0
$235$	−218.445	−	378.357i	−0.929551	−	1.61003i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−47.4323			−0.198462			−0.0992308	−	0.995064i	$-0.531638\pi$
$239$	−47.4323			−0.198462			−0.0992308	+	0.995064i	$0.531638\pi$
$240$		0			0
$241$	−146.065	+	84.3309i	−0.606080	+	0.349921i	−0.771430	−	0.636314i	$-0.780458\pi$
$241$	−146.065	+	84.3309i	−0.606080	+	0.349921i	0.165349	+	0.986235i	$0.447125\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−289.420	−	337.316i	−1.18131	−	1.37680i
$246$		0			0
$247$	85.2114	−	147.590i	0.344985	−	0.597532i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	482.804i		−	1.92352i	−0.273893	−	0.961760i	$-0.588311\pi$
$251$		−	482.804i		−	1.92352i	0.273893	−	0.961760i	$-0.411689\pi$
$252$		0			0
$253$	−166.641			−0.658662
$254$		0			0
$255$		0			0
$256$		0			0
$257$	81.5627	+	47.0903i	0.317365	+	0.183231i	0.650217	−	0.759748i	$-0.274678\pi$
$257$	81.5627	+	47.0903i	0.317365	+	0.183231i	−0.332853	+	0.942979i	$0.608011\pi$
$258$		0			0
$259$	−290.118	−	240.733i	−1.12015	−	0.929470i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−102.365	−	177.301i	−0.389220	−	0.674149i	0.603125	−	0.797647i	$-0.293922\pi$
$263$	−102.365	−	177.301i	−0.389220	−	0.674149i	−0.992345	+	0.123498i	$0.960589\pi$
$264$		0			0
$265$			392.752i			1.48208i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	155.173	−	89.5890i	0.576850	−	0.333045i	−0.183030	−	0.983107i	$-0.558591\pi$
$269$	155.173	−	89.5890i	0.576850	−	0.333045i	0.759881	+	0.650063i	$0.225257\pi$
$270$		0			0
$271$	−172.307	−	99.4817i	−0.635821	−	0.367091i	0.147182	−	0.989109i	$-0.452980\pi$
$271$	−172.307	−	99.4817i	−0.635821	−	0.367091i	−0.783003	+	0.622018i	$0.786313\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−504.819	+	874.373i	−1.83571	+	3.17954i
$276$		0			0
$277$	−212.444	−	367.963i	−0.766945	−	1.32839i	−0.939212	−	0.343338i	$-0.888442\pi$
$277$	−212.444	−	367.963i	−0.766945	−	1.32839i	0.172267	−	0.985050i	$-0.444891\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	38.2443			0.136101			0.0680504	−	0.997682i	$-0.478322\pi$
$281$	38.2443			0.136101			0.0680504	+	0.997682i	$0.478322\pi$
$282$		0			0
$283$	353.144	−	203.888i	1.24786	−	0.720452i	0.277178	−	0.960819i	$-0.410601\pi$
$283$	353.144	−	203.888i	1.24786	−	0.720452i	0.970682	+	0.240366i	$0.0772675\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	406.529	+	69.3801i	1.41648	+	0.241743i
$288$		0			0
$289$	1.14658	−	1.98594i	0.00396741	−	0.00687175i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	94.4090i		−	0.322215i	−0.986937	−	0.161107i	$-0.948493\pi$
$293$		−	94.4090i		−	0.322215i	0.986937	−	0.161107i	$-0.0515066\pi$
$294$		0			0
$295$	−9.74562			−0.0330360
$296$		0			0
$297$		0			0
$298$		0			0
$299$	156.069	+	90.1067i	0.521971	+	0.301360i
$300$		0			0
$301$	81.7147	−	478.803i	0.271477	−	1.59071i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	237.692	+	411.694i	0.779317	+	1.34982i
$306$		0			0
$307$			95.2741i			0.310339i	0.987888	+	0.155170i	$0.0495924\pi$
$307$			95.2741i			0.310339i	−0.987888	+	0.155170i	$0.950408\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.40688	−	3.69902i	0.0206009	−	0.0118939i	−0.489664	−	0.871911i	$-0.662881\pi$
$311$	6.40688	−	3.69902i	0.0206009	−	0.0118939i	0.510265	+	0.860017i	$0.329547\pi$
$312$		0			0
$313$	−445.196	−	257.034i	−1.42235	−	0.821196i	−0.425853	−	0.904792i	$-0.640026\pi$
$313$	−445.196	−	257.034i	−1.42235	−	0.821196i	−0.996500	+	0.0835965i	$0.973359\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	284.352	−	492.512i	0.897009	−	1.55367i	0.0657111	−	0.997839i	$-0.479068\pi$
$317$	284.352	−	492.512i	0.897009	−	1.55367i	0.831298	−	0.555827i	$-0.187598\pi$
$318$		0			0
$319$	88.2906	+	152.924i	0.276773	+	0.479385i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	152.581			0.472386
$324$		0			0
$325$	945.585	−	545.934i	2.90949	−	1.67980i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	215.297	−	259.464i	0.654398	−	0.788644i
$330$		0			0
$331$	−188.061	+	325.731i	−0.568159	+	0.984080i	0.428589	+	0.903499i	$0.359011\pi$
$331$	−188.061	+	325.731i	−0.568159	+	0.984080i	−0.996748	+	0.0805806i	$0.974323\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			435.401i			1.29971i
$336$		0			0
$337$	−48.6640			−0.144403			−0.0722017	−	0.997390i	$-0.523003\pi$
$337$	−48.6640			−0.144403			−0.0722017	+	0.997390i	$0.523003\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	697.184	+	402.520i	2.04453	+	1.18041i
$342$		0			0
$343$	166.578	−	299.834i	0.485652	−	0.874153i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	61.0477	+	105.738i	0.175930	+	0.304720i	0.940483	−	0.339841i	$-0.110373\pi$
$347$	61.0477	+	105.738i	0.175930	+	0.304720i	−0.764553	+	0.644561i	$0.777040\pi$
$348$		0			0
$349$		−	42.6904i		−	0.122322i	−0.998128	−	0.0611611i	$-0.980520\pi$
$349$		−	42.6904i		−	0.122322i	0.998128	−	0.0611611i	$-0.0194803\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−259.656	+	149.913i	−0.735571	+	0.424682i	−0.820457	−	0.571709i	$-0.806281\pi$
$353$	−259.656	+	149.913i	−0.735571	+	0.424682i	0.0848860	+	0.996391i	$0.472947\pi$
$354$		0			0
$355$	−420.285	−	242.652i	−1.18390	−	0.683525i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−288.227	+	499.224i	−0.802861	+	1.39060i	0.114864	+	0.993381i	$0.463357\pi$
$359$	−288.227	+	499.224i	−0.802861	+	1.39060i	−0.917725	+	0.397215i	$0.869977\pi$
$360$		0			0
$361$	−140.539	−	243.420i	−0.389304	−	0.674294i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−22.6821			−0.0621426
$366$		0			0
$367$	−129.450	+	74.7379i	−0.352724	+	0.203646i	−0.665885	−	0.746055i	$-0.731946\pi$
$367$	−129.450	+	74.7379i	−0.352724	+	0.203646i	0.313160	+	0.949700i	$0.398612\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−284.242	+	105.228i	−0.766150	+	0.283635i
$372$		0			0
$373$	−237.012	+	410.517i	−0.635421	+	1.10058i	0.351005	+	0.936374i	$0.385840\pi$
$373$	−237.012	+	410.517i	−0.635421	+	1.10058i	−0.986426	+	0.164208i	$0.947493\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	190.963i		−	0.506532i
$378$		0			0
$379$	−128.926			−0.340174			−0.170087	−	0.985429i	$-0.554405\pi$
$379$	−128.926			−0.340174			−0.170087	+	0.985429i	$0.554405\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	442.382	+	255.410i	1.15505	+	0.666866i	0.950112	−	0.311910i	$-0.100969\pi$
$383$	442.382	+	255.410i	1.15505	+	0.666866i	0.204933	+	0.978776i	$0.434302\pi$
$384$		0			0
$385$	−1103.29	−	188.293i	−2.86570	−	0.489073i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−99.7955	−	172.851i	−0.256544	−	0.444347i	0.708770	−	0.705440i	$-0.249250\pi$
$389$	−99.7955	−	172.851i	−0.256544	−	0.444347i	−0.965314	+	0.261093i	$0.915917\pi$
$390$		0			0
$391$			161.346i			0.412650i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	177.418	−	102.432i	0.449160	−	0.259323i
$396$		0			0
$397$	272.159	+	157.131i	0.685538	+	0.395795i	0.801938	−	0.597407i	$-0.203802\pi$
$397$	272.159	+	157.131i	0.685538	+	0.395795i	−0.116400	+	0.993202i	$0.537136\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	165.788	−	287.154i	0.413437	−	0.716094i	−0.581826	−	0.813314i	$-0.697661\pi$
$401$	165.788	−	287.154i	0.413437	−	0.716094i	0.995263	+	0.0972191i	$0.0309947\pi$
$402$		0			0
$403$	−435.302	−	753.966i	−1.08015	−	1.87088i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−949.335			−2.33252
$408$		0			0
$409$	44.7996	−	25.8650i	0.109534	−	0.0632397i	−0.444232	−	0.895912i	$-0.646523\pi$
$409$	44.7996	−	25.8650i	0.109534	−	0.0632397i	0.553766	+	0.832672i	$0.313190\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.61111	−	7.05308i	−0.00632229	−	0.0170777i
$414$		0			0
$415$	232.215	−	402.209i	0.559555	−	0.969178i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	779.507i		−	1.86040i	−0.367055	−	0.930199i	$-0.619634\pi$
$419$		−	779.507i		−	1.86040i	0.367055	−	0.930199i	$-0.380366\pi$
$420$		0			0
$421$	610.247			1.44952			0.724759	−	0.689003i	$-0.241951\pi$
$421$	610.247			1.44952			0.724759	+	0.689003i	$0.241951\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	846.589	+	488.779i	1.99198	+	1.15007i
$426$		0			0
$427$	−234.267	+	282.325i	−0.548634	+	0.661184i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−309.626	−	536.287i	−0.718389	−	1.24429i	−0.961638	−	0.274322i	$-0.911547\pi$
$431$	−309.626	−	536.287i	−0.718389	−	1.24429i	0.243249	−	0.969964i	$-0.421787\pi$
$432$		0			0
$433$			6.27297i			0.0144872i	0.999974	+	0.00724361i	$0.00230573\pi$
$433$			6.27297i			0.0144872i	−0.999974	+	0.00724361i	$0.997694\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−73.1913	+	42.2570i	−0.167486	+	0.0966979i
$438$		0			0
$439$	438.913	+	253.406i	0.999802	+	0.577236i	0.908190	−	0.418559i	$-0.137465\pi$
$439$	438.913	+	253.406i	0.999802	+	0.577236i	0.0916121	+	0.995795i	$0.470798\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−231.039	+	400.171i	−0.521533	+	0.903321i	0.478154	+	0.878276i	$0.341306\pi$
$443$	−231.039	+	400.171i	−0.521533	+	0.903321i	−0.999686	+	0.0250447i	$0.992027\pi$
$444$		0			0
$445$	−230.183	−	398.688i	−0.517264	−	0.895928i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−773.455			−1.72262			−0.861308	−	0.508083i	$-0.830354\pi$
$449$	−773.455			−1.72262			−0.861308	+	0.508083i	$0.830354\pi$
$450$		0			0
$451$	899.389	−	519.262i	1.99421	−	1.15136i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	931.484	+	772.922i	2.04722	+	1.69873i
$456$		0			0
$457$	−340.636	+	589.999i	−0.745374	+	1.29103i	0.204645	+	0.978836i	$0.434396\pi$
$457$	−340.636	+	589.999i	−0.745374	+	1.29103i	−0.950020	+	0.312190i	$0.898937\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	141.028i		−	0.305918i	−0.988233	−	0.152959i	$-0.951120\pi$
$461$		−	141.028i		−	0.305918i	0.988233	−	0.152959i	$-0.0488802\pi$
$462$		0			0
$463$	−488.209			−1.05445			−0.527224	−	0.849727i	$-0.676767\pi$
$463$	−488.209			−1.05445			−0.527224	+	0.849727i	$0.676767\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−403.532	−	232.979i	−0.864095	−	0.498885i	0.00128656	−	0.999999i	$-0.499590\pi$
$467$	−403.532	−	232.979i	−0.864095	−	0.498885i	−0.865381	+	0.501114i	$0.832924\pi$
$468$		0			0
$469$	−315.108	+	116.655i	−0.671872	+	0.248732i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−611.578	−	1059.28i	−1.29298	−	2.23950i
$474$		0			0
$475$			512.049i			1.07800i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	186.185	−	107.494i	0.388696	−	0.224414i	−0.292899	−	0.956143i	$-0.594620\pi$
$479$	186.185	−	107.494i	0.388696	−	0.224414i	0.681595	+	0.731730i	$0.261287\pi$
$480$		0			0
$481$	889.108	+	513.327i	1.84846	+	1.06721i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−574.674	+	995.365i	−1.18490	+	2.05230i
$486$		0			0
$487$	−177.999	−	308.303i	−0.365500	−	0.633065i	0.623356	−	0.781938i	$-0.285769\pi$
$487$	−177.999	−	308.303i	−0.365500	−	0.633065i	−0.988856	+	0.148873i	$0.952435\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	386.760			0.787698			0.393849	−	0.919175i	$-0.371143\pi$
$491$	386.760			0.787698			0.393849	+	0.919175i	$0.371143\pi$
$492$		0			0
$493$	148.065	−	85.4851i	0.300334	−	0.173398i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	63.0061	−	369.180i	0.126773	−	0.742818i
$498$		0			0
$499$	200.322	−	346.968i	0.401447	−	0.695327i	−0.592454	−	0.805604i	$-0.701841\pi$
$499$	200.322	−	346.968i	0.401447	−	0.695327i	0.993901	+	0.110278i	$0.0351741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	240.052i		−	0.477241i	−0.971113	−	0.238621i	$-0.923305\pi$
$503$		−	240.052i		−	0.477241i	0.971113	−	0.238621i	$-0.0766952\pi$
$504$		0			0
$505$	26.9287			0.0533242
$506$		0			0
$507$		0			0
$508$		0			0
$509$	441.516	+	254.909i	0.867419	+	0.500804i	0.866490	−	0.499195i	$-0.166371\pi$
$509$	441.516	+	254.909i	0.867419	+	0.500804i	0.000929124		1.00000i	$0.499704\pi$
$510$		0			0
$511$	−6.07712	−	16.4154i	−0.0118926	−	0.0321241i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	32.2546	+	55.8666i	0.0626303	+	0.108479i
$516$		0			0
$517$		−	849.028i		−	1.64222i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	183.183	−	105.761i	0.351600	−	0.202996i	−0.313790	−	0.949492i	$-0.601599\pi$
$521$	183.183	−	105.761i	0.351600	−	0.202996i	0.665390	+	0.746496i	$0.268265\pi$
$522$		0			0
$523$	500.693	+	289.075i	0.957347	+	0.552725i	0.895356	−	0.445352i	$-0.146921\pi$
$523$	500.693	+	289.075i	0.957347	+	0.552725i	0.0619917	+	0.998077i	$0.480255\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	389.729	−	675.031i	0.739524	−	1.28089i
$528$		0			0
$529$	219.815	+	380.731i	0.415530	+	0.719719i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1123.11			−2.10714
$534$		0			0
$535$	429.724	−	248.101i	0.803223	−	0.463741i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−159.330	−	848.922i	−0.295603	−	1.57499i
$540$		0			0
$541$	403.423	−	698.750i	0.745699	−	1.29159i	−0.204168	−	0.978936i	$-0.565449\pi$
$541$	403.423	−	698.750i	0.745699	−	1.29159i	0.949867	−	0.312653i	$-0.101218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	118.691i		−	0.217782i
$546$		0			0
$547$	−841.193			−1.53783			−0.768915	−	0.639351i	$-0.779203\pi$
$547$	−841.193			−1.53783			−0.768915	+	0.639351i	$0.779203\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	77.5569	+	44.7775i	0.140757	+	0.0812659i
$552$		0			0
$553$	121.667	+	100.956i	0.220013	+	0.182561i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	311.159	+	538.943i	0.558633	+	0.967582i	0.997611	+	0.0690834i	$0.0220075\pi$
$557$	311.159	+	538.943i	0.558633	+	0.967582i	−0.438977	+	0.898498i	$0.644659\pi$
$558$		0			0
$559$			1322.78i			2.36632i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	272.546	−	157.355i	0.484096	−	0.279493i	−0.238026	−	0.971259i	$-0.576500\pi$
$563$	272.546	−	157.355i	0.484096	−	0.279493i	0.722122	+	0.691766i	$0.243167\pi$
$564$		0			0
$565$	−311.658	−	179.936i	−0.551607	−	0.318470i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.6505	+	21.9114i	−0.0222329	+	0.0385085i	−0.876928	−	0.480622i	$-0.840411\pi$
$569$	−12.6505	+	21.9114i	−0.0222329	+	0.0385085i	0.854695	+	0.519131i	$0.173744\pi$
$570$		0			0
$571$	96.6400	+	167.385i	0.169247	+	0.293144i	0.938155	−	0.346215i	$-0.112533\pi$
$571$	96.6400	+	167.385i	0.169247	+	0.293144i	−0.768908	+	0.639359i	$0.779200\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−541.466			−0.941680
$576$		0			0
$577$	−245.268	+	141.605i	−0.425074	+	0.245417i	−0.697246	−	0.716832i	$-0.745591\pi$
$577$	−245.268	+	141.605i	−0.425074	+	0.245417i	0.272172	+	0.962249i	$0.412258\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	353.302	+	60.2962i	0.608093	+	0.103780i
$582$		0			0
$583$	−381.627	+	660.997i	−0.654592	+	1.13379i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	450.692i		−	0.767789i	−0.923377	−	0.383895i	$-0.874583\pi$
$587$		−	450.692i		−	0.767789i	0.923377	−	0.383895i	$-0.125417\pi$
$588$		0			0
$589$	408.284			0.693182
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−428.895	−	247.623i	−0.723264	−	0.417577i	0.0926890	−	0.995695i	$-0.470454\pi$
$593$	−428.895	−	247.623i	−0.723264	−	0.417577i	−0.815953	+	0.578119i	$0.803787\pi$
$594$		0			0
$595$	−182.310	+	1068.24i	−0.306403	+	1.79535i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−488.101	−	845.415i	−0.814859	−	1.41138i	−0.909429	−	0.415859i	$-0.863481\pi$
$599$	−488.101	−	845.415i	−0.814859	−	1.41138i	0.0945697	−	0.995518i	$-0.469852\pi$
$600$		0			0
$601$			609.471i			1.01410i	0.861918	+	0.507048i	$0.169263\pi$
$601$			609.471i			1.01410i	−0.861918	+	0.507048i	$0.830737\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1490.38	+	860.470i	−2.46343	+	1.42226i
$606$		0			0
$607$	347.737	+	200.766i	0.572877	+	0.330751i	0.758298	−	0.651908i	$-0.226031\pi$
$607$	347.737	+	200.766i	0.572877	+	0.330751i	−0.185420	+	0.982659i	$0.559365\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−459.088	+	795.164i	−0.751372	+	1.30141i
$612$		0			0
$613$	−21.8505	−	37.8461i	−0.0356451	−	0.0617391i	0.847652	−	0.530552i	$-0.178015\pi$
$613$	−21.8505	−	37.8461i	−0.0356451	−	0.0617391i	−0.883298	+	0.468813i	$0.844682\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	632.523			1.02516			0.512580	−	0.858640i	$-0.328690\pi$
$617$	632.523			1.02516			0.512580	+	0.858640i	$0.328690\pi$
$618$		0			0
$619$	206.161	−	119.027i	0.333055	−	0.192289i	−0.324142	−	0.946009i	$-0.605075\pi$
$619$	206.161	−	119.027i	0.333055	−	0.192289i	0.657196	+	0.753719i	$0.271742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	226.866	−	273.406i	0.364150	−	0.438854i
$624$		0			0
$625$	−611.846	+	1059.75i	−0.978954	+	1.69560i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			919.170i			1.46132i
$630$		0			0
$631$	861.236			1.36487			0.682437	−	0.730944i	$-0.260920\pi$
$631$	861.236			1.36487			0.682437	+	0.730944i	$0.260920\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	864.653	+	499.208i	1.36166	+	0.786154i
$636$		0			0
$637$	−309.809	+	881.217i	−0.486356	+	1.38339i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	206.326	+	357.367i	0.321881	+	0.557514i	0.980876	−	0.194633i	$-0.0623516\pi$
$641$	206.326	+	357.367i	0.321881	+	0.557514i	−0.658995	+	0.752147i	$0.729018\pi$
$642$		0			0
$643$			643.619i			1.00096i	0.865747	+	0.500482i	$0.166844\pi$
$643$			643.619i			1.00096i	−0.865747	+	0.500482i	$0.833156\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	673.471	−	388.829i	1.04091	−	0.600972i	0.120821	−	0.992674i	$-0.461447\pi$
$647$	673.471	−	388.829i	1.04091	−	0.600972i	0.920092	+	0.391703i	$0.128114\pi$
$648$		0			0
$649$	−16.4018	−	9.46957i	−0.0252724	−	0.0145910i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	117.781	−	204.003i	0.180370	−	0.312410i	−0.761637	−	0.648004i	$-0.775604\pi$
$653$	117.781	−	204.003i	0.180370	−	0.312410i	0.942007	+	0.335595i	$0.108937\pi$
$654$		0			0
$655$	−1050.41	−	1819.37i	−1.60369	−	2.77767i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1119.92			1.69943			0.849713	−	0.527246i	$-0.176775\pi$
$659$	1119.92			1.69943			0.849713	+	0.527246i	$0.176775\pi$
$660$		0			0
$661$	−174.861	+	100.956i	−0.264541	+	0.152733i	−0.626404	−	0.779499i	$-0.715474\pi$
$661$	−174.861	+	100.956i	−0.264541	+	0.152733i	0.361863	+	0.932231i	$0.382141\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−532.329	+	197.072i	−0.800495	+	0.296349i
$666$		0			0
$667$	−47.3499	+	82.0125i	−0.0709894	+	0.122957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			923.836i			1.37681i
$672$		0			0
$673$	167.885			0.249457			0.124729	−	0.992191i	$-0.460194\pi$
$673$	167.885			0.249457			0.124729	+	0.992191i	$0.460194\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−630.551	−	364.049i	−0.931391	−	0.537739i	−0.0441395	−	0.999025i	$-0.514055\pi$
$677$	−630.551	−	364.049i	−0.931391	−	0.537739i	−0.887251	+	0.461287i	$0.847388\pi$
$678$		0			0
$679$	−874.334	−	149.218i	−1.28768	−	0.219761i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	552.381	+	956.752i	0.808757	+	1.40081i	0.913725	+	0.406332i	$0.133192\pi$
$683$	552.381	+	956.752i	0.808757	+	1.40081i	−0.104969	+	0.994476i	$0.533474\pi$
$684$		0			0
$685$			2294.43i			3.34953i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	714.832	−	412.708i	1.03749	−	0.598996i
$690$		0			0
$691$	−846.017	−	488.448i	−1.22434	−	0.706872i	−0.258498	−	0.966012i	$-0.583228\pi$
$691$	−846.017	−	488.448i	−1.22434	−	0.706872i	−0.965840	+	0.259140i	$0.916561\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	185.781	−	321.782i	0.267311	−	0.462996i
$696$		0			0
$697$	−502.763	−	870.810i	−0.721324	−	1.24937i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1099.72			1.56878			0.784392	−	0.620266i	$-0.212975\pi$
$701$	1099.72			1.56878			0.784392	+	0.620266i	$0.212975\pi$
$702$		0			0
$703$	−416.961	+	240.733i	−0.593117	+	0.342436i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.21491	+	19.4888i	0.0102050	+	0.0275655i
$708$		0			0
$709$	−44.5147	+	77.1017i	−0.0627852	+	0.108747i	−0.895709	−	0.444640i	$-0.853332\pi$
$709$	−44.5147	+	77.1017i	−0.0627852	+	0.108747i	0.832924	+	0.553387i	$0.186665\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			431.740i			0.605525i
$714$		0			0
$715$	3048.04			4.26299
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−412.423	−	238.113i	−0.573607	−	0.331172i	0.184982	−	0.982742i	$-0.440777\pi$
$719$	−412.423	−	238.113i	−0.573607	−	0.331172i	−0.758589	+	0.651570i	$0.774111\pi$
$720$		0			0
$721$	−31.7898	+	38.3114i	−0.0440913	+	0.0531364i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	286.881	+	496.893i	0.395698	+	0.685370i
$726$		0			0
$727$			14.5314i			0.0199881i	0.999950	+	0.00999406i	$0.00318126\pi$
$727$			14.5314i			0.0199881i	−0.999950	+	0.00999406i	$0.996819\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1025.63	+	592.145i	−1.40304	+	0.810048i
$732$		0			0
$733$	1103.82	+	637.293i	1.50590	+	0.869431i	0.999977	+	0.00685249i	$0.00218123\pi$
$733$	1103.82	+	637.293i	1.50590	+	0.869431i	0.505923	+	0.862579i	$0.331152\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−423.068	+	732.776i	−0.574041	+	0.994269i
$738$		0			0
$739$	631.506	+	1093.80i	0.854541	+	1.48011i	0.877070	+	0.480362i	$0.159495\pi$
$739$	631.506	+	1093.80i	0.854541	+	1.48011i	−0.0225294	+	0.999746i	$0.507172\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1143.15			1.53856			0.769282	−	0.638909i	$-0.220614\pi$
$743$	1143.15			1.53856			0.769282	+	0.638909i	$0.220614\pi$
$744$		0			0
$745$	1252.21	−	722.961i	1.68081	−	0.970418i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	294.690	+	244.526i	0.393444	+	0.326470i
$750$		0			0
$751$	−315.689	+	546.790i	−0.420359	+	0.728082i	−0.995974	−	0.0896378i	$-0.971429\pi$
$751$	−315.689	+	546.790i	−0.420359	+	0.728082i	0.575616	+	0.817720i	$0.304762\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	497.714i		−	0.659223i
$756$		0			0
$757$	705.180			0.931546			0.465773	−	0.884904i	$-0.345776\pi$
$757$	705.180			0.931546			0.465773	+	0.884904i	$0.345776\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−43.2391	−	24.9641i	−0.0568188	−	0.0328043i	0.471322	−	0.881961i	$-0.343777\pi$
$761$	−43.2391	−	24.9641i	−0.0568188	−	0.0328043i	−0.528140	+	0.849157i	$0.677110\pi$
$762$		0			0
$763$	85.8988	−	31.8004i	0.112580	−	0.0416781i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.2408	+	17.7376i	0.0133518	+	0.0231260i
$768$		0			0
$769$			909.426i			1.18261i	0.806448	+	0.591304i	$0.201387\pi$
$769$			909.426i			1.18261i	−0.806448	+	0.591304i	$0.798613\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1079.76	−	623.402i	1.39685	−	0.806471i	0.402787	−	0.915294i	$-0.368041\pi$
$773$	1079.76	−	623.402i	1.39685	−	0.806471i	0.994061	+	0.108823i	$0.0347081\pi$
$774$		0			0
$775$	2265.35	+	1307.90i	2.92303	+	1.68761i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	263.349	−	456.135i	0.338061	−	0.585539i
$780$		0			0
$781$	−471.557	−	816.760i	−0.603786	−	1.04579i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1001.05			1.27522
$786$		0			0
$787$	−686.254	+	396.209i	−0.871988	+	0.503442i	−0.868008	−	0.496550i	$-0.834600\pi$
$787$	−686.254	+	396.209i	−0.871988	+	0.503442i	−0.00397939	+	0.999992i	$0.501267\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	46.7215	−	273.762i	0.0590663	−	0.346096i
$792$		0			0
$793$	499.539	−	865.226i	0.629935	−	1.09108i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1067.43i		−	1.33932i	−0.742670	−	0.669658i	$-0.766441\pi$
$797$		−	1067.43i		−	1.33932i	0.742670	−	0.669658i	$-0.233559\pi$
$798$		0			0
$799$	−822.050			−1.02885
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−38.1737	−	22.0396i	−0.0475388	−	0.0274466i
$804$		0			0
$805$	−208.394	−	562.911i	−0.258875	−	0.699268i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−492.247	−	852.596i	−0.608463	−	1.05389i	−0.991494	−	0.130154i	$-0.958453\pi$
$809$	−492.247	−	852.596i	−0.608463	−	1.05389i	0.383031	−	0.923736i	$-0.374880\pi$
$810$		0			0
$811$		−	1036.21i		−	1.27769i	−0.769336	−	0.638845i	$-0.779413\pi$
$811$		−	1036.21i		−	1.27769i	0.769336	−	0.638845i	$-0.220587\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1778.92	+	1027.06i	−2.18272	+	1.26020i
$816$		0			0
$817$	−537.227	−	310.168i	−0.657561	−	0.379643i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	495.184	−	857.685i	0.603148	−	1.04468i	−0.389193	−	0.921156i	$-0.627246\pi$
$821$	495.184	−	857.685i	0.603148	−	1.04468i	0.992341	−	0.123527i	$-0.0394205\pi$
$822$		0			0
$823$	535.120	+	926.855i	0.650206	+	1.12619i	0.983073	+	0.183217i	$0.0586509\pi$
$823$	535.120	+	926.855i	0.650206	+	1.12619i	−0.332866	+	0.942974i	$0.608016\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1035.41			−1.25201			−0.626005	−	0.779819i	$-0.715311\pi$
$827$	−1035.41			−1.25201			−0.626005	+	0.779819i	$0.715311\pi$
$828$		0			0
$829$	672.235	−	388.115i	0.810899	−	0.468173i	−0.0363691	−	0.999338i	$-0.511579\pi$
$829$	672.235	−	388.115i	0.810899	−	0.468173i	0.847268	+	0.531166i	$0.178246\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−821.947	+	154.267i	−0.986731	+	0.185194i
$834$		0			0
$835$	−794.651	+	1376.38i	−0.951677	+	1.64835i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1582.78i			1.88651i	0.332076	+	0.943253i	$0.392251\pi$
$839$			1582.78i			1.88651i	−0.332076	+	0.943253i	$0.607749\pi$
$840$		0			0
$841$	−740.652			−0.880680
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1527.10	−	881.671i	−1.80722	−	1.04340i
$846$		0			0
$847$	−1022.05	−	848.071i	−1.20667	−	1.00126i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−254.563	−	440.915i	−0.299134	−	0.518114i
$852$		0			0
$853$			499.247i			0.585283i	0.956222	+	0.292642i	$0.0945343\pi$
$853$			499.247i			0.585283i	−0.956222	+	0.292642i	$0.905466\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−200.112	+	115.535i	−0.233503	+	0.134813i	−0.612187	−	0.790713i	$-0.709710\pi$
$857$	−200.112	+	115.535i	−0.233503	+	0.134813i	0.378684	+	0.925526i	$0.376377\pi$
$858$		0			0
$859$	−1104.81	−	637.860i	−1.28615	−	0.742562i	−0.308188	−	0.951325i	$-0.599723\pi$
$859$	−1104.81	−	637.860i	−1.28615	−	0.742562i	−0.977966	+	0.208764i	$0.933056\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	554.074	−	959.684i	0.642032	−	1.11203i	−0.342946	−	0.939355i	$-0.611425\pi$
$863$	554.074	−	959.684i	0.642032	−	1.11203i	0.984978	−	0.172677i	$-0.0552418\pi$
$864$		0			0
$865$	64.8201	+	112.272i	0.0749365	+	0.129794i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	398.124			0.458141
$870$		0			0
$871$	792.456	−	457.525i	0.909823	−	0.525287i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2020.18	−	344.773i	−2.30877	−	0.394026i
$876$		0			0
$877$	−371.376	+	643.242i	−0.423462	+	0.733458i	−0.996275	−	0.0862280i	$-0.972519\pi$
$877$	−371.376	+	643.242i	−0.423462	+	0.733458i	0.572813	+	0.819686i	$0.305852\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			544.827i			0.618419i	0.950994	+	0.309209i	$0.100064\pi$
$881$			544.827i			0.618419i	−0.950994	+	0.309209i	$0.899936\pi$
$882$		0			0
$883$	−915.864			−1.03722			−0.518609	−	0.855011i	$-0.673550\pi$
$883$	−915.864			−1.03722			−0.518609	+	0.855011i	$0.673550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1344.46	−	776.222i	−1.51573	−	0.875109i	−0.999830	−	0.0184631i	$-0.994123\pi$
$887$	−1344.46	−	776.222i	−1.51573	−	0.875109i	−0.515904	−	0.856646i	$-0.672544\pi$
$888$		0			0
$889$	−129.623	+	759.516i	−0.145807	+	0.854349i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−215.297	−	372.905i	−0.241094	−	0.417587i
$894$		0			0
$895$			2.20138i			0.00245964i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	396.199	−	228.746i	0.440711	−	0.254445i
$900$		0			0
$901$	639.994	+	369.501i	0.710315	+	0.410101i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.91781	+	8.51790i	−0.00543404	+	0.00941204i
$906$		0			0
$907$	354.727	+	614.406i	0.391099	+	0.677404i	0.992595	−	0.121472i	$-0.0387615\pi$
$907$	354.727	+	614.406i	0.391099	+	0.677404i	−0.601495	+	0.798876i	$0.705428\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1178.08			−1.29317			−0.646584	−	0.762842i	$-0.723803\pi$
$911$	−1178.08			−1.29317			−0.646584	+	0.762842i	$0.723803\pi$
$912$		0			0
$913$	781.632	−	451.275i	0.856114	−	0.494278i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1035.28	−	1247.66i	1.12898	−	1.36059i
$918$		0			0
$919$	298.271	−	516.621i	0.324560	−	0.562155i	−0.656863	−	0.754010i	$-0.728117\pi$
$919$	298.271	−	516.621i	0.324560	−	0.562155i	0.981423	+	0.191855i	$0.0614503\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1019.92i			1.10501i
$924$		0			0
$925$	−3084.66			−3.33477
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−612.089	−	353.390i	−0.658868	−	0.380398i	0.132977	−	0.991119i	$-0.457546\pi$
$929$	−612.089	−	353.390i	−0.658868	−	0.380398i	−0.791846	+	0.610721i	$0.790880\pi$
$930$		0			0
$931$	−285.250	−	332.455i	−0.306391	−	0.357095i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1364.46	+	2363.32i	1.45932	+	2.52762i
$936$		0			0
$937$			732.454i			0.781701i	0.920454	+	0.390851i	$0.127819\pi$
$937$			732.454i			0.781701i	−0.920454	+	0.390851i	$0.872181\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.5804	−	21.6970i	0.0399366	−	0.0230574i	−0.479899	−	0.877324i	$-0.659327\pi$
$941$	37.5804	−	21.6970i	0.0399366	−	0.0230574i	0.519835	+	0.854266i	$0.325993\pi$
$942$		0			0
$943$	482.339	+	278.479i	0.511494	+	0.295311i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	463.757	−	803.251i	0.489712	−	0.848206i	−0.510218	−	0.860045i	$-0.670435\pi$
$947$	463.757	−	803.251i	0.489712	−	0.848206i	0.999930	+	0.0118393i	$0.00376865\pi$
$948$		0			0
$949$	23.8346	+	41.2827i	0.0251155	+	0.0435013i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−692.901			−0.727073			−0.363537	−	0.931580i	$-0.618431\pi$
$953$	−692.901			−0.727073			−0.363537	+	0.931580i	$0.618431\pi$
$954$		0			0
$955$	1595.91	−	921.397i	1.67111	−	0.964814i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1660.52	+	614.737i	−1.73151	+	0.641018i
$960$		0			0
$961$	562.360	−	974.036i	0.585182	−	1.01356i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2142.72i		−	2.22044i
$966$		0			0
$967$	−202.072			−0.208968			−0.104484	−	0.994527i	$-0.533319\pi$
$967$	−202.072			−0.208968			−0.104484	+	0.994527i	$0.533319\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−530.638	−	306.364i	−0.546486	−	0.315514i	0.201217	−	0.979547i	$-0.435510\pi$
$971$	−530.638	−	306.364i	−0.546486	−	0.315514i	−0.747704	+	0.664033i	$0.768844\pi$
$972$		0			0
$973$	282.655	+	48.2392i	0.290499	+	0.0495778i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	405.699	+	702.692i	0.415250	+	0.719234i	0.995455	−	0.0952366i	$-0.0303608\pi$
$977$	405.699	+	702.692i	0.415250	+	0.719234i	−0.580205	+	0.814471i	$0.697027\pi$
$978$		0			0
$979$		−	894.650i		−	0.913841i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1250.68	−	722.082i	1.27231	−	0.734570i	0.296889	−	0.954912i	$-0.404051\pi$
$983$	1250.68	−	722.082i	1.27231	−	0.734570i	0.975423	+	0.220342i	$0.0707174\pi$
$984$		0			0
$985$	393.670	+	227.285i	0.399664	+	0.230746i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	327.987	−	568.091i	0.331635	−	0.574409i
$990$		0			0
$991$	−413.395	−	716.021i	−0.417149	−	0.722523i	0.578502	−	0.815681i	$-0.303637\pi$
$991$	−413.395	−	716.021i	−0.417149	−	0.722523i	−0.995651	+	0.0931574i	$0.970304\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1113.77			1.11937
$996$		0			0
$997$	−437.754	+	252.737i	−0.439071	+	0.253498i	−0.703204	−	0.710989i	$-0.748248\pi$
$997$	−437.754	+	252.737i	−0.439071	+	0.253498i	0.264132	+	0.964486i	$0.414914\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cg.q.145.8		16
3.2	odd	2	inner	1008.3.cg.q.145.1		16
4.3	odd	2		504.3.by.d.145.8	yes	16
7.3	odd	6	inner	1008.3.cg.q.577.8		16
12.11	even	2		504.3.by.d.145.1	yes	16
21.17	even	6	inner	1008.3.cg.q.577.1		16
28.3	even	6		504.3.by.d.73.8	yes	16
28.19	even	6		3528.3.f.i.2449.16		16
28.23	odd	6		3528.3.f.i.2449.2		16
84.23	even	6		3528.3.f.i.2449.15		16
84.47	odd	6		3528.3.f.i.2449.1		16
84.59	odd	6		504.3.by.d.73.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.by.d.73.1	✓	16	84.59	odd	6
504.3.by.d.73.8	yes	16	28.3	even	6
504.3.by.d.145.1	yes	16	12.11	even	2
504.3.by.d.145.8	yes	16	4.3	odd	2
1008.3.cg.q.145.1		16	3.2	odd	2	inner
1008.3.cg.q.145.8		16	1.1	even	1	trivial
1008.3.cg.q.577.1		16	21.17	even	6	inner
1008.3.cg.q.577.8		16	7.3	odd	6	inner
3528.3.f.i.2449.1		16	84.47	odd	6
3528.3.f.i.2449.2		16	28.23	odd	6
3528.3.f.i.2449.15		16	84.23	even	6
3528.3.f.i.2449.16		16	28.19	even	6

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.85541 + 4.53532i) q^{5} +(-1.17763 + 6.90023i) q^{7} +O(q^{10})\)	\(q+(7.85541 + 4.53532i) q^{5} +(-1.17763 + 6.90023i) q^{7} +(8.81372 + 15.2658i) q^{11} -19.0631i q^{13} +(14.7807 - 8.53366i) q^{17} +(7.74221 + 4.46997i) q^{19} +(-4.72676 + 8.18700i) q^{23} +(28.6383 + 49.6030i) q^{25} +10.0174 q^{29} +(39.5511 - 22.8348i) q^{31} +(-40.5455 + 48.8632i) q^{35} +(-26.9278 + 46.6403i) q^{37} -58.9153i q^{41} -69.3894 q^{43} +(-41.7123 - 24.0826i) q^{47} +(-46.2264 - 16.2518i) q^{49} +(21.6496 + 37.4982i) q^{53} +159.892i q^{55} +(-0.930469 + 0.537207i) q^{59} +(45.3875 + 26.2045i) q^{61} +(86.4572 - 149.748i) q^{65} +(24.0006 + 41.5702i) q^{67} -53.5026 q^{71} +(-2.16558 + 1.25030i) q^{73} +(-115.717 + 42.8393i) q^{77} +(11.2927 - 19.5596i) q^{79} -51.2015i q^{83} +154.812 q^{85} +(-43.9536 - 25.3766i) q^{89} +(131.540 + 22.4492i) q^{91} +(40.5455 + 70.2269i) q^{95} +126.711i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

Introduction

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