LMFDB - Embedded newform 2736.2.cg.a.1151.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1151,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))

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

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

chi := DirichletCharacter("2736.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.cg (of order $6$, degree $2$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1151.9
Character	$\chi$	$=$	2736.1151
Dual form			2736.2.cg.a.2591.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.62081 + 0.935772i) q^{5} -2.07786i q^{7} +O(q^{10})$	$q+(1.62081 + 0.935772i) q^{5} -2.07786i q^{7} -5.79064 q^{11} +(3.25108 + 5.63103i) q^{13} +(3.56803 + 2.06000i) q^{17} +(-2.66462 + 3.44961i) q^{19} +(-2.02412 - 3.50588i) q^{23} +(-0.748660 - 1.29672i) q^{25} +(-2.90040 + 1.67455i) q^{29} +5.15471i q^{31} +(1.94441 - 3.36781i) q^{35} -6.99947 q^{37} +(-9.12968 - 5.27102i) q^{41} +(0.638308 + 0.368528i) q^{43} +(4.95759 + 8.58679i) q^{47} +2.68248 q^{49} +(-4.83526 + 2.79164i) q^{53} +(-9.38550 - 5.41872i) q^{55} +(-1.37628 + 2.38379i) q^{59} +(7.20914 + 12.4866i) q^{61} +12.1691i q^{65} +(-10.3881 + 5.99756i) q^{67} +(6.81563 - 11.8050i) q^{71} +(4.73667 - 8.20415i) q^{73} +12.0322i q^{77} +(9.40973 + 5.43271i) q^{79} -7.44365 q^{83} +(3.85539 + 6.67773i) q^{85} +(-1.89525 + 1.09422i) q^{89} +(11.7005 - 6.75529i) q^{91} +(-7.54688 + 3.09766i) q^{95} +(-0.380159 + 0.658455i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) $ 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	1.62081	+	0.935772i	0.724846	+	0.418490i	0.816534	−	0.577298i	$-0.195893\pi$
$5$	1.62081	+	0.935772i	0.724846	+	0.418490i	−0.0916876	+	0.995788i	$0.529226\pi$
$6$		0			0
$7$		−	2.07786i		−	0.785359i	−0.919675	−	0.392679i	$-0.871548\pi$
$7$		−	2.07786i		−	0.785359i	0.919675	−	0.392679i	$-0.128452\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.79064			−1.74594			−0.872971	−	0.487771i	$-0.837810\pi$
$11$	−5.79064			−1.74594			−0.872971	+	0.487771i	$0.837810\pi$
$12$		0			0
$13$	3.25108	+	5.63103i	0.901686	+	1.56177i	0.825305	+	0.564688i	$0.191003\pi$
$13$	3.25108	+	5.63103i	0.901686	+	1.56177i	0.0763815	+	0.997079i	$0.475663\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.56803	+	2.06000i	0.865374	+	0.499624i	0.865808	−	0.500376i	$-0.166805\pi$
$17$	3.56803	+	2.06000i	0.865374	+	0.499624i	−0.000434048		1.00000i	$0.500138\pi$
$18$		0			0
$19$	−2.66462	+	3.44961i	−0.611307	+	0.791394i
$19$	−2.66462	+	3.44961i	−0.611307	+	0.791394i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.02412	−	3.50588i	−0.422059	−	0.731027i	0.574082	−	0.818798i	$-0.305359\pi$
$23$	−2.02412	−	3.50588i	−0.422059	−	0.731027i	−0.996141	+	0.0877707i	$0.972026\pi$
$24$		0			0
$25$	−0.748660	−	1.29672i	−0.149732	−	0.259343i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.90040	+	1.67455i	−0.538590	+	0.310955i	−0.744507	−	0.667614i	$-0.767316\pi$
$29$	−2.90040	+	1.67455i	−0.538590	+	0.310955i	0.205917	+	0.978569i	$0.433982\pi$
$30$		0			0
$31$			5.15471i			0.925813i	0.886407	+	0.462906i	$0.153193\pi$
$31$			5.15471i			0.925813i	−0.886407	+	0.462906i	$0.846807\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.94441	−	3.36781i	0.328665	−	0.569264i
$36$		0			0
$37$	−6.99947			−1.15071			−0.575353	−	0.817905i	$-0.695135\pi$
$37$	−6.99947			−1.15071			−0.575353	+	0.817905i	$0.695135\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.12968	−	5.27102i	−1.42582	−	0.823196i	−0.429029	−	0.903291i	$-0.641144\pi$
$41$	−9.12968	−	5.27102i	−1.42582	−	0.823196i	−0.996787	+	0.0800951i	$0.974478\pi$
$42$		0			0
$43$	0.638308	+	0.368528i	0.0973411	+	0.0561999i	0.547880	−	0.836557i	$-0.315435\pi$
$43$	0.638308	+	0.368528i	0.0973411	+	0.0561999i	−0.450539	+	0.892757i	$0.648768\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.95759	+	8.58679i	0.723138	+	1.25251i	0.959736	+	0.280904i	$0.0906344\pi$
$47$	4.95759	+	8.58679i	0.723138	+	1.25251i	−0.236598	+	0.971608i	$0.576032\pi$
$48$		0			0
$49$	2.68248			0.383212
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.83526	+	2.79164i	−0.664173	+	0.383461i	−0.793865	−	0.608094i	$-0.791935\pi$
$53$	−4.83526	+	2.79164i	−0.664173	+	0.383461i	0.129692	+	0.991554i	$0.458601\pi$
$54$		0			0
$55$	−9.38550	−	5.41872i	−1.26554	−	0.730660i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.37628	+	2.38379i	−0.179176	+	0.310342i	−0.941599	−	0.336737i	$-0.890677\pi$
$59$	−1.37628	+	2.38379i	−0.179176	+	0.310342i	0.762422	+	0.647080i	$0.224010\pi$
$60$		0			0
$61$	7.20914	+	12.4866i	0.923036	+	1.59875i	0.794690	+	0.607016i	$0.207634\pi$
$61$	7.20914	+	12.4866i	0.923036	+	1.59875i	0.128347	+	0.991729i	$0.459033\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			12.1691i			1.50939i
$66$		0			0
$67$	−10.3881	+	5.99756i	−1.26911	+	0.732719i	−0.974819	−	0.222998i	$-0.928416\pi$
$67$	−10.3881	+	5.99756i	−1.26911	+	0.732719i	−0.294288	+	0.955717i	$0.595082\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.81563	−	11.8050i	0.808866	−	1.40100i	−0.104783	−	0.994495i	$-0.533415\pi$
$71$	6.81563	−	11.8050i	0.808866	−	1.40100i	0.913650	−	0.406503i	$-0.133252\pi$
$72$		0			0
$73$	4.73667	−	8.20415i	0.554385	−	0.960223i	−0.443566	−	0.896242i	$-0.646287\pi$
$73$	4.73667	−	8.20415i	0.554385	−	0.960223i	0.997951	−	0.0639811i	$-0.0203797\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			12.0322i			1.37119i
$78$		0			0
$79$	9.40973	+	5.43271i	1.05868	+	0.611228i	0.925066	−	0.379806i	$-0.124009\pi$
$79$	9.40973	+	5.43271i	1.05868	+	0.611228i	0.133611	+	0.991034i	$0.457343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.44365			−0.817047			−0.408523	−	0.912748i	$-0.633956\pi$
$83$	−7.44365			−0.817047			−0.408523	+	0.912748i	$0.633956\pi$
$84$		0			0
$85$	3.85539	+	6.67773i	0.418175	+	0.724301i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.89525	+	1.09422i	−0.200896	+	0.115987i	−0.597073	−	0.802187i	$-0.703670\pi$
$89$	−1.89525	+	1.09422i	−0.200896	+	0.115987i	0.396177	+	0.918174i	$0.370337\pi$
$90$		0			0
$91$	11.7005	−	6.75529i	1.22655	−	0.708147i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.54688	+	3.09766i	−0.774294	+	0.317813i
$96$		0			0
$97$	−0.380159	+	0.658455i	−0.0385993	+	0.0668559i	−0.884680	−	0.466199i	$-0.845623\pi$
$97$	−0.380159	+	0.658455i	−0.0385993	+	0.0668559i	0.846080	+	0.533055i	$0.178956\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.1779	+	6.45358i	−1.11225	+	0.642155i	−0.939410	−	0.342796i	$-0.888626\pi$
$101$	−11.1779	+	6.45358i	−1.11225	+	0.642155i	−0.172835	+	0.984951i	$0.555293\pi$
$102$		0			0
$103$			9.35178i			0.921458i	0.887541	+	0.460729i	$0.152412\pi$
$103$			9.35178i			0.921458i	−0.887541	+	0.460729i	$0.847588\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.89471			−0.666536			−0.333268	−	0.942832i	$-0.608151\pi$
$107$	−6.89471			−0.666536			−0.333268	+	0.942832i	$0.608151\pi$
$108$		0			0
$109$	5.06513	−	8.77306i	0.485152	−	0.840307i	−0.514703	−	0.857369i	$-0.672098\pi$
$109$	5.06513	−	8.77306i	0.485152	−	0.840307i	0.999854	+	0.0170615i	$0.00543112\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			18.9806i			1.78554i	0.450509	+	0.892772i	$0.351242\pi$
$113$			18.9806i			1.78554i	−0.450509	+	0.892772i	$0.648758\pi$
$114$		0			0
$115$		−	7.57647i		−	0.706510i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.28040	−	7.41388i	0.392384	−	0.679629i
$120$		0			0
$121$	22.5315			2.04832
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1600i		−	1.08763i
$126$		0			0
$127$	−5.07997	+	2.93292i	−0.450775	+	0.260255i	−0.708157	−	0.706055i	$-0.750473\pi$
$127$	−5.07997	+	2.93292i	−0.450775	+	0.260255i	0.257383	+	0.966310i	$0.417140\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.09510	+	12.2891i	−0.619902	+	1.07370i	0.369602	+	0.929190i	$0.379494\pi$
$131$	−7.09510	+	12.2891i	−0.619902	+	1.07370i	−0.989503	+	0.144511i	$0.953839\pi$
$132$		0			0
$133$	7.16781	+	5.53672i	0.621528	+	0.480095i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.96943	+	4.02380i	−0.595438	+	0.343776i	−0.767245	−	0.641354i	$-0.778373\pi$
$137$	−6.96943	+	4.02380i	−0.595438	+	0.343776i	0.171807	+	0.985131i	$0.445040\pi$
$138$		0			0
$139$	17.9930	−	10.3883i	1.52615	−	0.881121i	0.526628	−	0.850096i	$-0.323456\pi$
$139$	17.9930	−	10.3883i	1.52615	−	0.881121i	0.999519	−	0.0310252i	$-0.00987722\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.8258	−	32.6072i	−1.57429	−	2.72675i
$144$		0			0
$145$	−6.26797			−0.520527
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.96267	+	3.44255i	0.488481	+	0.282025i	0.723944	−	0.689859i	$-0.242327\pi$
$149$	5.96267	+	3.44255i	0.488481	+	0.282025i	−0.235463	+	0.971883i	$0.575661\pi$
$150$		0			0
$151$			14.6762i			1.19434i	0.802116	+	0.597168i	$0.203707\pi$
$151$			14.6762i			1.19434i	−0.802116	+	0.597168i	$0.796293\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.82363	+	8.35478i	−0.387443	+	0.671072i
$156$		0			0
$157$	10.8380	−	18.7720i	0.864970	−	1.49817i	−0.00210638	−	0.999998i	$-0.500670\pi$
$157$	10.8380	−	18.7720i	0.864970	−	1.49817i	0.867077	−	0.498175i	$-0.165996\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.28474	+	4.20585i	−0.574118	+	0.331467i
$162$		0			0
$163$			5.16217i			0.404332i	0.979351	+	0.202166i	$0.0647981\pi$
$163$			5.16217i			0.404332i	−0.979351	+	0.202166i	$0.935202\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.52114	−	14.7591i	−0.659386	−	1.14209i	−0.980775	−	0.195143i	$-0.937483\pi$
$167$	−8.52114	−	14.7591i	−0.659386	−	1.14209i	0.321389	−	0.946947i	$-0.395850\pi$
$168$		0			0
$169$	−14.6390	+	25.3555i	−1.12608	+	1.95042i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−17.9915	−	10.3874i	−1.36787	−	0.789738i	−0.377212	−	0.926127i	$-0.623117\pi$
$173$	−17.9915	−	10.3874i	−1.36787	−	0.789738i	−0.990655	+	0.136389i	$0.956450\pi$
$174$		0			0
$175$	−2.69440	+	1.55561i	−0.203678	+	0.117593i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.3106			1.06962			0.534812	−	0.844971i	$-0.320383\pi$
$179$	14.3106			1.06962			0.534812	+	0.844971i	$0.320383\pi$
$180$		0			0
$181$	0.0269230	+	0.0466320i	0.00200117	+	0.00346613i	0.867024	−	0.498266i	$-0.166030\pi$
$181$	0.0269230	+	0.0466320i	0.00200117	+	0.00346613i	−0.865023	+	0.501732i	$0.832696\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.3448	−	6.54991i	−0.834085	−	0.481559i
$186$		0			0
$187$	−20.6612	−	11.9287i	−1.51089	−	0.872315i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.76311			0.489361			0.244681	−	0.969604i	$-0.421317\pi$
$191$	6.76311			0.489361			0.244681	+	0.969604i	$0.421317\pi$
$192$		0			0
$193$	−3.75772	+	6.50857i	−0.270487	+	0.468497i	−0.968987	−	0.247113i	$-0.920518\pi$
$193$	−3.75772	+	6.50857i	−0.270487	+	0.468497i	0.698500	+	0.715610i	$0.253851\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.28995i			0.376893i	0.982083	+	0.188447i	$0.0603453\pi$
$197$			5.28995i			0.376893i	−0.982083	+	0.188447i	$0.939655\pi$
$198$		0			0
$199$	−10.2864	+	5.93888i	−0.729187	+	0.420996i	−0.818125	−	0.575041i	$-0.804986\pi$
$199$	−10.2864	+	5.93888i	−0.729187	+	0.420996i	0.0889379	+	0.996037i	$0.471653\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.47948	+	6.02663i	0.244211	+	0.422987i
$204$		0			0
$205$	−9.86496	−	17.0866i	−0.688998	−	1.19338i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.4299	−	19.9754i	1.06731	−	1.38173i
$210$		0			0
$211$	5.87255	+	3.39052i	0.404283	+	0.233413i	0.688330	−	0.725397i	$-0.258344\pi$
$211$	5.87255	+	3.39052i	0.404283	+	0.233413i	−0.284047	+	0.958810i	$0.591677\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.689716	+	1.19462i	0.0470382	+	0.0814726i
$216$		0			0
$217$	10.7108			0.727095
$218$		0			0
$219$		0			0
$220$		0			0
$221$			26.7889i			1.80202i
$222$		0			0
$223$	−6.92614	−	3.99881i	−0.463808	−	0.267780i	0.249836	−	0.968288i	$-0.419623\pi$
$223$	−6.92614	−	3.99881i	−0.463808	−	0.267780i	−0.713644	+	0.700508i	$0.752957\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0997			−0.670339			−0.335169	−	0.942158i	$-0.608794\pi$
$227$	−10.0997			−0.670339			−0.335169	+	0.942158i	$0.608794\pi$
$228$		0			0
$229$	−28.3277			−1.87194			−0.935972	−	0.352074i	$-0.885477\pi$
$229$	−28.3277			−1.87194			−0.935972	+	0.352074i	$0.885477\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.1589	+	6.44260i	0.731045	+	0.422069i	0.818804	−	0.574073i	$-0.194637\pi$
$233$	11.1589	+	6.44260i	0.731045	+	0.422069i	−0.0877596	+	0.996142i	$0.527971\pi$
$234$		0			0
$235$			18.5567i			1.21050i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.33973			0.410083			0.205041	−	0.978753i	$-0.434267\pi$
$239$	6.33973			0.410083			0.205041	+	0.978753i	$0.434267\pi$
$240$		0			0
$241$	10.0080	+	17.3344i	0.644673	+	1.11661i	0.984377	+	0.176074i	$0.0563398\pi$
$241$	10.0080	+	17.3344i	0.644673	+	1.11661i	−0.339704	+	0.940532i	$0.610327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.34778	+	2.51019i	0.277770	+	0.160370i
$246$		0			0
$247$	−28.0877	−	3.78964i	−1.78718	−	0.241129i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.1098	+	17.5107i	0.638127	+	1.10527i	0.985844	+	0.167668i	$0.0536237\pi$
$251$	10.1098	+	17.5107i	0.638127	+	1.10527i	−0.347717	+	0.937600i	$0.613043\pi$
$252$		0			0
$253$	11.7210	+	20.3013i	0.736890	+	1.27633i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.76931	−	2.75356i	0.297501	−	0.171762i	−0.343819	−	0.939036i	$-0.611721\pi$
$257$	4.76931	−	2.75356i	0.297501	−	0.171762i	0.641320	+	0.767274i	$0.278387\pi$
$258$		0			0
$259$			14.5439i			0.903717i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.79206	−	6.56805i	0.233829	−	0.405003i	−0.725103	−	0.688640i	$-0.758208\pi$
$263$	3.79206	−	6.56805i	0.233829	−	0.405003i	0.958932	+	0.283637i	$0.0915412\pi$
$264$		0			0
$265$	−10.4493			−0.641898
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.48406	−	4.32092i	−0.456311	−	0.263451i	0.254181	−	0.967157i	$-0.418194\pi$
$269$	−7.48406	−	4.32092i	−0.456311	−	0.263451i	−0.710492	+	0.703705i	$0.751528\pi$
$270$		0			0
$271$	−7.65160	−	4.41765i	−0.464801	−	0.268353i	0.249260	−	0.968437i	$-0.419813\pi$
$271$	−7.65160	−	4.41765i	−0.464801	−	0.268353i	−0.714061	+	0.700084i	$0.753146\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.33522	+	7.50882i	0.261424	+	0.452799i
$276$		0			0
$277$	7.77272			0.467018			0.233509	−	0.972355i	$-0.424979\pi$
$277$	7.77272			0.467018			0.233509	+	0.972355i	$0.424979\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.08861	−	1.78321i	0.184251	−	0.106377i	−0.405037	−	0.914300i	$-0.632741\pi$
$281$	3.08861	−	1.78321i	0.184251	−	0.106377i	0.589288	+	0.807923i	$0.299408\pi$
$282$		0			0
$283$	20.9714	+	12.1079i	1.24662	+	0.719738i	0.970434	−	0.241366i	$-0.0775954\pi$
$283$	20.9714	+	12.1079i	1.24662	+	0.719738i	0.276188	+	0.961104i	$0.410929\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.9525	+	18.9702i	−0.646504	+	1.11978i
$288$		0			0
$289$	−0.0127773	−	0.0221309i	−0.000751606	−	0.00130182i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			15.7164i			0.918164i	0.888394	+	0.459082i	$0.151822\pi$
$293$			15.7164i			0.918164i	−0.888394	+	0.459082i	$0.848178\pi$
$294$		0			0
$295$	−4.46136	+	2.57577i	−0.259750	+	0.149967i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.1611	−	22.7958i	0.761129	−	1.31831i
$300$		0			0
$301$	0.765750	−	1.32632i	0.0441371	−	0.0764477i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			26.9845i			1.54513i
$306$		0			0
$307$	5.75955	+	3.32528i	0.328715	+	0.189784i	0.655270	−	0.755394i	$-0.272555\pi$
$307$	5.75955	+	3.32528i	0.328715	+	0.189784i	−0.326556	+	0.945178i	$0.605888\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.7439			−0.609233			−0.304616	−	0.952475i	$-0.598528\pi$
$311$	−10.7439			−0.609233			−0.304616	+	0.952475i	$0.598528\pi$
$312$		0			0
$313$	4.08059	+	7.06779i	0.230648	+	0.399495i	0.957999	−	0.286771i	$-0.0925819\pi$
$313$	4.08059	+	7.06779i	0.230648	+	0.399495i	−0.727351	+	0.686266i	$0.759249\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.05344	−	4.07230i	0.396160	−	0.228723i	−0.288666	−	0.957430i	$-0.593212\pi$
$317$	7.05344	−	4.07230i	0.396160	−	0.228723i	0.684826	+	0.728707i	$0.259878\pi$
$318$		0			0
$319$	16.7952	−	9.69669i	0.940348	−	0.542910i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.6137	+	6.81916i	−0.924408	+	0.379428i
$324$		0			0
$325$	4.86790	−	8.43145i	0.270023	−	0.467693i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.8422	−	10.3012i	0.983671	−	0.567923i
$330$		0			0
$331$		−	0.852824i		−	0.0468755i	−0.999725	−	0.0234377i	$-0.992539\pi$
$331$		−	0.852824i		−	0.0468755i	0.999725	−	0.0234377i	$-0.00746114\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.4494			−1.22654
$336$		0			0
$337$	8.59680	−	14.8901i	0.468298	−	0.811115i	−0.531046	−	0.847343i	$-0.678201\pi$
$337$	8.59680	−	14.8901i	0.468298	−	0.811115i	0.999344	+	0.0362279i	$0.0115342\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	29.8490i		−	1.61642i
$342$		0			0
$343$		−	20.1189i		−	1.08632i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.51473	+	16.4800i	−0.510777	+	0.884692i	0.489145	+	0.872203i	$0.337309\pi$
$347$	−9.51473	+	16.4800i	−0.510777	+	0.884692i	−0.999922	+	0.0124895i	$0.996024\pi$
$348$		0			0
$349$	12.9061			0.690848			0.345424	−	0.938447i	$-0.387735\pi$
$349$	12.9061			0.690848			0.345424	+	0.938447i	$0.387735\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7.38465i			0.393045i	0.980499	+	0.196523i	$0.0629649\pi$
$353$			7.38465i			0.393045i	−0.980499	+	0.196523i	$0.937035\pi$
$354$		0			0
$355$	22.0936	−	12.7558i	1.17261	−	0.677005i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.42743	−	9.40058i	0.286449	−	0.496143i	−0.686511	−	0.727120i	$-0.740859\pi$
$359$	5.42743	−	9.40058i	0.286449	−	0.496143i	0.972959	+	0.230976i	$0.0741919\pi$
$360$		0			0
$361$	−4.79956	−	18.3838i	−0.252608	−	0.967569i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.3544	−	8.86488i	0.803687	−	0.464009i
$366$		0			0
$367$	10.1793	−	5.87701i	0.531354	−	0.306778i	−0.210213	−	0.977656i	$-0.567416\pi$
$367$	10.1793	−	5.87701i	0.531354	−	0.306778i	0.741568	+	0.670878i	$0.234083\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.80064	+	10.0470i	0.301154	+	0.521614i
$372$		0			0
$373$	−6.16324			−0.319121			−0.159560	−	0.987188i	$-0.551008\pi$
$373$	−6.16324			−0.319121			−0.159560	+	0.987188i	$0.551008\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.8588	−	10.8881i	−0.971279	−	0.560768i
$378$		0			0
$379$		−	33.7070i		−	1.73141i	−0.500554	−	0.865705i	$-0.666870\pi$
$379$		−	33.7070i		−	1.73141i	0.500554	−	0.865705i	$-0.333130\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.3544	−	30.0587i	0.886769	−	1.53593i	0.0430970	−	0.999071i	$-0.486278\pi$
$383$	17.3544	−	30.0587i	0.886769	−	1.53593i	0.843672	−	0.536859i	$-0.180389\pi$
$384$		0			0
$385$	−11.2594	+	19.5018i	−0.573830	+	0.993903i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.88292	−	5.12856i	0.450382	−	0.260028i	−0.257609	−	0.966249i	$-0.582935\pi$
$389$	8.88292	−	5.12856i	0.450382	−	0.260028i	0.707992	+	0.706221i	$0.249601\pi$
$390$		0			0
$391$		−	16.6788i		−	0.843483i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.1676	+	17.6107i	0.511586	+	0.886092i
$396$		0			0
$397$	1.74168	−	3.01667i	0.0874122	−	0.151402i	−0.819004	−	0.573787i	$-0.805474\pi$
$397$	1.74168	−	3.01667i	0.0874122	−	0.151402i	0.906417	+	0.422385i	$0.138807\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.2948	−	9.40781i	−0.813724	−	0.469804i	0.0345233	−	0.999404i	$-0.489009\pi$
$401$	−16.2948	−	9.40781i	−0.813724	−	0.469804i	−0.848247	+	0.529600i	$0.822342\pi$
$402$		0			0
$403$	−29.0263	+	16.7583i	−1.44590	+	0.834792i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	40.5314			2.00907
$408$		0			0
$409$	−4.38801	−	7.60025i	−0.216973	−	0.375808i	0.736908	−	0.675993i	$-0.236285\pi$
$409$	−4.38801	−	7.60025i	−0.216973	−	0.375808i	−0.953881	+	0.300185i	$0.902952\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.95318	+	2.85972i	0.243730	+	0.140718i
$414$		0			0
$415$	−12.0647	−	6.96556i	−0.592233	−	0.341926i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−21.6605			−1.05819			−0.529093	−	0.848564i	$-0.677468\pi$
$419$	−21.6605			−1.05819			−0.529093	+	0.848564i	$0.677468\pi$
$420$		0			0
$421$	−8.98499	+	15.5625i	−0.437902	+	0.758468i	−0.997527	−	0.0702775i	$-0.977612\pi$
$421$	−8.98499	+	15.5625i	−0.437902	+	0.758468i	0.559626	+	0.828745i	$0.310945\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.16897i		−	0.299239i
$426$		0			0
$427$	25.9455	−	14.9796i	1.25559	−	0.724914i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.76138	+	15.1751i	0.422021	+	0.730961i	0.996137	−	0.0878125i	$-0.0279876\pi$
$431$	8.76138	+	15.1751i	0.422021	+	0.730961i	−0.574116	+	0.818774i	$0.694654\pi$
$432$		0			0
$433$	−2.34289	−	4.05800i	−0.112592	−	0.195015i	0.804223	−	0.594328i	$-0.202582\pi$
$433$	−2.34289	−	4.05800i	−0.112592	−	0.195015i	−0.916815	+	0.399313i	$0.869249\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	17.4874	+	2.35943i	0.836538	+	0.112867i
$438$		0			0
$439$	−13.2560	−	7.65336i	−0.632674	−	0.365275i	0.149113	−	0.988820i	$-0.452358\pi$
$439$	−13.2560	−	7.65336i	−0.632674	−	0.365275i	−0.781787	+	0.623545i	$0.785692\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.07822	+	7.06369i	0.193762	+	0.335606i	0.946494	−	0.322721i	$-0.104598\pi$
$443$	4.07822	+	7.06369i	0.193762	+	0.335606i	−0.752732	+	0.658327i	$0.771264\pi$
$444$		0			0
$445$	−4.09577			−0.194158
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	23.8387i		−	1.12502i	−0.826792	−	0.562508i	$-0.809836\pi$
$449$		−	23.8387i		−	1.12502i	0.826792	−	0.562508i	$-0.190164\pi$
$450$		0			0
$451$	52.8667	+	30.5226i	2.48939	+	1.43725i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	25.2857			1.18541
$456$		0			0
$457$	−3.94548			−0.184562			−0.0922809	−	0.995733i	$-0.529416\pi$
$457$	−3.94548			−0.184562			−0.0922809	+	0.995733i	$0.529416\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.97618	+	4.02770i	0.324913	+	0.187589i	0.653580	−	0.756857i	$-0.273266\pi$
$461$	6.97618	+	4.02770i	0.324913	+	0.187589i	−0.328667	+	0.944446i	$0.606599\pi$
$462$		0			0
$463$		−	14.1392i		−	0.657106i	−0.944485	−	0.328553i	$-0.893439\pi$
$463$		−	14.1392i		−	0.657106i	0.944485	−	0.328553i	$-0.106561\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	39.2314			1.81541			0.907706	−	0.419606i	$-0.137832\pi$
$467$	39.2314			1.81541			0.907706	+	0.419606i	$0.137832\pi$
$468$		0			0
$469$	12.4621	+	21.5850i	0.575447	+	0.996704i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.69621	−	2.13401i	−0.169952	−	0.0981219i
$474$		0			0
$475$	6.46806	+	0.872682i	0.296775	+	0.0400414i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.5431	−	25.1894i	−0.664491	−	1.15093i	−0.979423	−	0.201818i	$-0.935315\pi$
$479$	−14.5431	−	25.1894i	−0.664491	−	1.15093i	0.314932	−	0.949114i	$-0.398018\pi$
$480$		0			0
$481$	−22.7558	−	39.4142i	−1.03758	−	1.79713i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.23233	+	0.711484i	−0.0559571	+	0.0323068i
$486$		0			0
$487$			18.9319i			0.857886i	0.903332	+	0.428943i	$0.141114\pi$
$487$			18.9319i			0.857886i	−0.903332	+	0.428943i	$0.858886\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.01606	−	12.1522i	0.316630	−	0.548420i	−0.663152	−	0.748484i	$-0.730782\pi$
$491$	7.01606	−	12.1522i	0.316630	−	0.548420i	0.979783	+	0.200065i	$0.0641152\pi$
$492$		0			0
$493$	−13.7983			−0.621443
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.5292	−	14.1619i	−1.10029	−	0.635250i
$498$		0			0
$499$	1.74316	+	1.00641i	0.0780346	+	0.0450533i	0.538510	−	0.842619i	$-0.318988\pi$
$499$	1.74316	+	1.00641i	0.0780346	+	0.0450533i	−0.460475	+	0.887673i	$0.652321\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.09091	+	7.08567i	0.182405	+	0.315934i	0.942699	−	0.333645i	$-0.108279\pi$
$503$	4.09091	+	7.08567i	0.182405	+	0.315934i	−0.760294	+	0.649579i	$0.774945\pi$
$504$		0			0
$505$	−24.1563			−1.07494
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.95195	−	2.28166i	0.175167	−	0.101133i	−0.409853	−	0.912152i	$-0.634420\pi$
$509$	3.95195	−	2.28166i	0.175167	−	0.101133i	0.585020	+	0.811019i	$0.301087\pi$
$510$		0			0
$511$	−17.0471	−	9.84214i	−0.754119	−	0.435391i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.75113	+	15.1574i	−0.385621	+	0.667915i
$516$		0			0
$517$	−28.7076	−	49.7230i	−1.26256	−	2.18681i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	23.2941i		−	1.02053i	−0.860016	−	0.510267i	$-0.829547\pi$
$521$		−	23.2941i		−	1.02053i	0.860016	−	0.510267i	$-0.170453\pi$
$522$		0			0
$523$	−23.9928	+	13.8523i	−1.04913	+	0.605717i	−0.922406	−	0.386221i	$-0.873780\pi$
$523$	−23.9928	+	13.8523i	−1.04913	+	0.605717i	−0.126726	+	0.991938i	$0.540447\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.6187	+	18.3921i	−0.462558	+	0.801174i
$528$		0			0
$529$	3.30586	−	5.72591i	0.143733	−	0.248953i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	68.5460i		−	2.96906i
$534$		0			0
$535$	−11.1750	−	6.45188i	−0.483136	−	0.278939i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.5333			−0.669066
$540$		0			0
$541$	−17.2110	−	29.8104i	−0.739961	−	1.28165i	−0.952512	−	0.304499i	$-0.901511\pi$
$541$	−17.2110	−	29.8104i	−0.739961	−	1.28165i	0.212552	−	0.977150i	$-0.431823\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.4192	−	9.47962i	0.703320	−	0.406062i
$546$		0			0
$547$	−19.8899	+	11.4835i	−0.850433	+	0.490998i	−0.860797	−	0.508949i	$-0.830034\pi$
$547$	−19.8899	+	11.4835i	−0.850433	+	0.490998i	0.0103641	+	0.999946i	$0.496701\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.95195	−	14.4673i	0.0831558	−	0.616326i
$552$		0			0
$553$	11.2884	−	19.5521i	0.480033	−	0.831441i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.6366	−	14.2239i	1.04389	−	0.602688i	0.122955	−	0.992412i	$-0.460763\pi$
$557$	24.6366	−	14.2239i	1.04389	−	0.602688i	0.920932	+	0.389724i	$0.127430\pi$
$558$		0			0
$559$			4.79244i			0.202699i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.7849			0.833835			0.416918	−	0.908944i	$-0.363110\pi$
$563$	19.7849			0.833835			0.416918	+	0.908944i	$0.363110\pi$
$564$		0			0
$565$	−17.7615	+	30.7638i	−0.747232	+	1.29424i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			38.6476i			1.62019i	0.586296	+	0.810097i	$0.300585\pi$
$569$			38.6476i			1.62019i	−0.586296	+	0.810097i	$0.699415\pi$
$570$		0			0
$571$			35.1985i			1.47301i	0.676431	+	0.736506i	$0.263526\pi$
$571$			35.1985i			1.47301i	−0.676431	+	0.736506i	$0.736474\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.03076	+	5.24943i	−0.126391	+	0.218916i
$576$		0			0
$577$	−0.994107			−0.0413852			−0.0206926	−	0.999786i	$-0.506587\pi$
$577$	−0.994107			−0.0413852			−0.0206926	+	0.999786i	$0.506587\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			15.4669i			0.641675i
$582$		0			0
$583$	27.9992	−	16.1654i	1.15961	−	0.669500i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.70043	−	16.8016i	0.400380	−	0.693478i	−0.593392	−	0.804914i	$-0.702212\pi$
$587$	9.70043	−	16.8016i	0.400380	−	0.693478i	0.993772	+	0.111436i	$0.0355449\pi$
$588$		0			0
$589$	−17.7817	−	13.7354i	−0.732682	−	0.565955i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20.5834	+	11.8838i	−0.845259	+	0.488011i	−0.859048	−	0.511894i	$-0.828944\pi$
$593$	−20.5834	+	11.8838i	−0.845259	+	0.488011i	0.0137892	+	0.999905i	$0.495611\pi$
$594$		0			0
$595$	13.8754	−	8.01097i	0.568836	−	0.328418i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.33561	+	12.7056i	0.299725	+	0.519138i	0.976073	−	0.217444i	$-0.0697718\pi$
$599$	7.33561	+	12.7056i	0.299725	+	0.519138i	−0.676348	+	0.736582i	$0.736439\pi$
$600$		0			0
$601$	4.43432			0.180880			0.0904398	−	0.995902i	$-0.471173\pi$
$601$	4.43432			0.180880			0.0904398	+	0.995902i	$0.471173\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	36.5191	+	21.0843i	1.48471	+	0.857200i
$606$		0			0
$607$			13.6113i			0.552466i	0.961091	+	0.276233i	$0.0890862\pi$
$607$			13.6113i			0.552466i	−0.961091	+	0.276233i	$0.910914\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.2350	+	55.8326i	−1.30409	+	2.25875i
$612$		0			0
$613$	16.4115	−	28.4255i	0.662853	−	1.14810i	−0.317010	−	0.948422i	$-0.602679\pi$
$613$	16.4115	−	28.4255i	0.662853	−	1.14810i	0.979863	−	0.199673i	$-0.0639879\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.4281	−	16.9903i	1.18473	−	0.684005i	0.227627	−	0.973748i	$-0.426903\pi$
$617$	29.4281	−	16.9903i	1.18473	−	0.684005i	0.957104	+	0.289744i	$0.0935701\pi$
$618$		0			0
$619$		−	22.0106i		−	0.884679i	−0.896848	−	0.442340i	$-0.854149\pi$
$619$		−	22.0106i		−	0.884679i	0.896848	−	0.442340i	$-0.145851\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.27364	+	3.93806i	0.0910916	+	0.157775i
$624$		0			0
$625$	7.63572	−	13.2254i	0.305429	−	0.529018i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.9743	−	14.4189i	−0.995791	−	0.574920i
$630$		0			0
$631$	4.38722	−	2.53296i	0.174652	−	0.100836i	−0.410125	−	0.912029i	$-0.634515\pi$
$631$	4.38722	−	2.53296i	0.174652	−	0.100836i	0.584778	+	0.811194i	$0.301182\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.9782			−0.435657
$636$		0			0
$637$	8.72096	+	15.1051i	0.345537	+	0.598488i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.0410	+	8.68392i	0.594084	+	0.342994i	0.766711	−	0.641993i	$-0.221892\pi$
$641$	15.0410	+	8.68392i	0.594084	+	0.342994i	−0.172627	+	0.984987i	$0.555226\pi$
$642$		0			0
$643$	34.9119	+	20.1564i	1.37679	+	0.794890i	0.991772	−	0.128020i	$-0.0408620\pi$
$643$	34.9119	+	20.1564i	1.37679	+	0.794890i	0.385018	+	0.922909i	$0.374195\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.7322			−0.657812			−0.328906	−	0.944363i	$-0.606680\pi$
$647$	−16.7322			−0.657812			−0.328906	+	0.944363i	$0.606680\pi$
$648$		0			0
$649$	7.96953	−	13.8036i	0.312832	−	0.541840i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	24.4774i		−	0.957873i	−0.877849	−	0.478937i	$-0.841022\pi$
$653$		−	24.4774i		−	0.957873i	0.877849	−	0.478937i	$-0.158978\pi$
$654$		0			0
$655$	−22.9995	+	13.2788i	−0.898667	+	0.518845i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.89814	+	5.01973i	0.112896	+	0.195541i	0.916937	−	0.399033i	$-0.130654\pi$
$659$	2.89814	+	5.01973i	0.112896	+	0.195541i	−0.804041	+	0.594574i	$0.797321\pi$
$660$		0			0
$661$	9.60042	+	16.6284i	0.373413	+	0.646770i	0.990088	−	0.140447i	$-0.0448541\pi$
$661$	9.60042	+	16.6284i	0.373413	+	0.646770i	−0.616675	+	0.787218i	$0.711521\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.43651	+	15.6814i	0.249597	+	0.608098i
$666$		0			0
$667$	11.7415	+	6.77897i	0.454633	+	0.262483i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−41.7455	−	72.3054i	−1.61157	−	2.79132i
$672$		0			0
$673$	−7.63501			−0.294308			−0.147154	−	0.989114i	$-0.547011\pi$
$673$	−7.63501			−0.294308			−0.147154	+	0.989114i	$0.547011\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			7.82123i			0.300594i	0.988641	+	0.150297i	$0.0480230\pi$
$677$			7.82123i			0.300594i	−0.988641	+	0.150297i	$0.951977\pi$
$678$		0			0
$679$	1.36818	+	0.789918i	0.0525059	+	0.0303143i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	46.6386			1.78458			0.892288	−	0.451467i	$-0.149099\pi$
$683$	46.6386			1.78458			0.892288	+	0.451467i	$0.149099\pi$
$684$		0			0
$685$	−15.0614			−0.575468
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−31.4396	−	18.1516i	−1.19775	−	0.691522i
$690$		0			0
$691$		−	42.0932i		−	1.60130i	−0.599131	−	0.800651i	$-0.704487\pi$
$691$		−	42.0932i		−	1.60130i	0.599131	−	0.800651i	$-0.295513\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	38.8842			1.47496
$696$		0			0
$697$	−21.7166	−	37.6143i	−0.822577	−	1.42474i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.85988	+	2.80585i	0.183555	+	0.105976i	0.588962	−	0.808161i	$-0.299537\pi$
$701$	4.85988	+	2.80585i	0.183555	+	0.105976i	−0.405407	+	0.914136i	$0.632870\pi$
$702$		0			0
$703$	18.6510	−	24.1454i	0.703434	−	0.910662i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.4097	+	23.2262i	0.504322	+	0.873511i
$708$		0			0
$709$	9.69188	+	16.7868i	0.363986	+	0.630443i	0.988613	−	0.150481i	$-0.0480821\pi$
$709$	9.69188	+	16.7868i	0.363986	+	0.630443i	−0.624627	+	0.780924i	$0.714749\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.0718	−	10.4338i	0.676794	−	0.390747i
$714$		0			0
$715$		−	70.4667i		−	2.63530i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.8754	−	22.3009i	0.480172	−	0.831682i	−0.519569	−	0.854428i	$-0.673908\pi$
$719$	12.8754	−	22.3009i	0.480172	−	0.831682i	0.999741	+	0.0227459i	$0.00724088\pi$
$720$		0			0
$721$	19.4317			0.723675
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.34282	+	2.50733i	0.161288	+	0.0931199i
$726$		0			0
$727$	−3.12435	−	1.80384i	−0.115876	−	0.0669008i	0.440942	−	0.897536i	$-0.354644\pi$
$727$	−3.12435	−	1.80384i	−0.115876	−	0.0669008i	−0.556818	+	0.830635i	$0.687978\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.51834	+	2.62983i	0.0561577	+	0.0972679i
$732$		0			0
$733$	18.6985			0.690646			0.345323	−	0.938484i	$-0.387769\pi$
$733$	18.6985			0.690646			0.345323	+	0.938484i	$0.387769\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	60.1536	−	34.7297i	2.21579	−	1.27929i
$738$		0			0
$739$	20.7648	+	11.9886i	0.763845	+	0.441006i	0.830675	−	0.556758i	$-0.187955\pi$
$739$	20.7648	+	11.9886i	0.763845	+	0.441006i	−0.0668296	+	0.997764i	$0.521288\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.3236	+	17.8810i	−0.378736	+	0.655991i	−0.990879	−	0.134757i	$-0.956975\pi$
$743$	−10.3236	+	17.8810i	−0.378736	+	0.655991i	0.612142	+	0.790748i	$0.290308\pi$
$744$		0			0
$745$	6.44289	+	11.1594i	0.236049	+	0.408849i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			14.3263i			0.523470i
$750$		0			0
$751$	−15.2520	+	8.80576i	−0.556554	+	0.321327i	−0.751761	−	0.659435i	$-0.770795\pi$
$751$	−15.2520	+	8.80576i	−0.556554	+	0.321327i	0.195207	+	0.980762i	$0.437462\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13.7336	+	23.7873i	−0.499818	+	0.865710i
$756$		0			0
$757$	8.78966	−	15.2241i	0.319466	−	0.553330i	−0.660911	−	0.750464i	$-0.729830\pi$
$757$	8.78966	−	15.2241i	0.319466	−	0.553330i	0.980377	+	0.197134i	$0.0631633\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	44.3559i		−	1.60790i	−0.594698	−	0.803949i	$-0.702728\pi$
$761$		−	44.3559i		−	1.60790i	0.594698	−	0.803949i	$-0.297272\pi$
$762$		0			0
$763$	−18.2292	−	10.5246i	−0.659942	−	0.381018i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−17.8975			−0.646243
$768$		0			0
$769$	9.60836	+	16.6422i	0.346486	+	0.600132i	0.985623	−	0.168961i	$-0.0540413\pi$
$769$	9.60836	+	16.6422i	0.346486	+	0.600132i	−0.639136	+	0.769094i	$0.720708\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.1877	+	10.5007i	−0.654167	+	0.377683i	−0.790051	−	0.613041i	$-0.789946\pi$
$773$	−18.1877	+	10.5007i	−0.654167	+	0.377683i	0.135884	+	0.990725i	$0.456613\pi$
$774$		0			0
$775$	6.68420	−	3.85912i	0.240103	−	0.138624i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	42.5101	−	17.4485i	1.52308	−	0.625157i
$780$		0			0
$781$	−39.4668	+	68.3586i	−1.41223	+	2.44606i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	35.1327	−	20.2839i	1.25394	−	0.723963i
$786$		0			0
$787$		−	4.79513i		−	0.170928i	−0.996341	−	0.0854640i	$-0.972763\pi$
$787$		−	4.79513i		−	0.170928i	0.996341	−	0.0854640i	$-0.0272373\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	39.4391			1.40229
$792$		0			0
$793$	−46.8749	+	81.1898i	−1.66458	+	2.88313i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	27.5076i		−	0.974369i	−0.873299	−	0.487184i	$-0.838024\pi$
$797$		−	27.5076i		−	0.974369i	0.873299	−	0.487184i	$-0.161976\pi$
$798$		0			0
$799$			40.8506i			1.44519i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.4283	+	47.5072i	−0.967924	+	1.67649i
$804$		0			0
$805$	−15.7429			−0.554863
$806$		0			0
$807$		0			0
$808$		0			0
$809$			12.1107i			0.425791i	0.977075	+	0.212896i	$0.0682894\pi$
$809$			12.1107i			0.425791i	−0.977075	+	0.212896i	$0.931711\pi$
$810$		0			0
$811$	38.2648	−	22.0922i	1.34366	−	0.775763i	0.356318	−	0.934365i	$-0.384032\pi$
$811$	38.2648	−	22.0922i	1.34366	−	0.775763i	0.987343	+	0.158602i	$0.0506987\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.83062	+	8.36687i	−0.169209	+	0.293079i
$816$		0			0
$817$	−2.97213	+	1.21993i	−0.103982	+	0.0426798i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.2241	−	15.7178i	0.950128	−	0.548556i	0.0570070	−	0.998374i	$-0.481844\pi$
$821$	27.2241	−	15.7178i	0.950128	−	0.548556i	0.893121	+	0.449817i	$0.148511\pi$
$822$		0			0
$823$	−35.9358	+	20.7475i	−1.25264	+	0.723213i	−0.971633	−	0.236492i	$-0.924002\pi$
$823$	−35.9358	+	20.7475i	−1.25264	+	0.723213i	−0.281008	+	0.959705i	$0.590669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.74298	−	16.8753i	−0.338797	−	0.586813i	0.645410	−	0.763836i	$-0.276686\pi$
$827$	−9.74298	−	16.8753i	−0.338797	−	0.586813i	−0.984207	+	0.177023i	$0.943353\pi$
$828$		0			0
$829$	−13.8041			−0.479438			−0.239719	−	0.970842i	$-0.577055\pi$
$829$	−13.8041			−0.479438			−0.239719	+	0.970842i	$0.577055\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.57118	+	5.52592i	0.331622	+	0.191462i
$834$		0			0
$835$		−	31.8954i		−	1.10379i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.60406	−	9.70652i	0.193474	−	0.335106i	−0.752925	−	0.658106i	$-0.771358\pi$
$839$	5.60406	−	9.70652i	0.193474	−	0.335106i	0.946399	+	0.323000i	$0.104691\pi$
$840$		0			0
$841$	−8.89180	+	15.4010i	−0.306614	+	0.531070i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−47.4539	+	27.3975i	−1.63246	+	0.942503i
$846$		0			0
$847$		−	46.8173i		−	1.60866i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14.1678	+	24.5393i	0.485665	+	0.841197i
$852$		0			0
$853$	12.9219	−	22.3814i	0.442437	−	0.766324i	−0.555433	−	0.831562i	$-0.687447\pi$
$853$	12.9219	−	22.3814i	0.442437	−	0.766324i	0.997870	+	0.0652380i	$0.0207807\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.8588	+	20.1257i	1.19075	+	0.687482i	0.958478	−	0.285168i	$-0.0920494\pi$
$857$	34.8588	+	20.1257i	1.19075	+	0.687482i	0.232276	+	0.972650i	$0.425383\pi$
$858$		0			0
$859$	−47.3942	+	27.3630i	−1.61707	+	0.933615i	−0.629395	+	0.777086i	$0.716697\pi$
$859$	−47.3942	+	27.3630i	−1.61707	+	0.933615i	−0.987673	+	0.156529i	$0.949970\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.6235			−0.361627			−0.180814	−	0.983517i	$-0.557873\pi$
$863$	−10.6235			−0.361627			−0.180814	+	0.983517i	$0.557873\pi$
$864$		0			0
$865$	−19.4405	−	33.6719i	−0.660996	−	1.14488i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−54.4883	−	31.4589i	−1.84839	−	1.06717i
$870$		0			0
$871$	−67.5449	−	38.9971i	−2.28867	−	1.32137i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.2669			−0.854176
$876$		0			0
$877$	11.7825	−	20.4079i	0.397867	−	0.689125i	−0.595596	−	0.803284i	$-0.703084\pi$
$877$	11.7825	−	20.4079i	0.397867	−	0.689125i	0.993462	+	0.114159i	$0.0364174\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			4.26456i			0.143677i	0.997416	+	0.0718384i	$0.0228866\pi$
$881$			4.26456i			0.143677i	−0.997416	+	0.0718384i	$0.977113\pi$
$882$		0			0
$883$	15.1967	−	8.77382i	0.511410	−	0.295262i	−0.222003	−	0.975046i	$-0.571260\pi$
$883$	15.1967	−	8.77382i	0.511410	−	0.295262i	0.733413	+	0.679783i	$0.237926\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.2307	+	17.7200i	0.343512	+	0.594981i	0.985082	−	0.172084i	$-0.0550500\pi$
$887$	10.2307	+	17.7200i	0.343512	+	0.594981i	−0.641570	+	0.767064i	$0.721717\pi$
$888$		0			0
$889$	6.09422	+	10.5555i	0.204393	+	0.354020i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−42.8311	−	5.77885i	−1.43329	−	0.193382i
$894$		0			0
$895$	23.1947	+	13.3915i	0.775312	+	0.447627i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.63179	−	14.9507i	−0.287886	−	0.498634i
$900$		0			0
$901$	−23.0031			−0.766345
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.100775i			0.00334988i
$906$		0			0
$907$	30.2084	+	17.4408i	1.00305	+	0.579114i	0.909151	−	0.416468i	$-0.136732\pi$
$907$	30.2084	+	17.4408i	1.00305	+	0.579114i	0.0939037	+	0.995581i	$0.470065\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.2960			0.506780			0.253390	−	0.967364i	$-0.418454\pi$
$911$	15.2960			0.506780			0.253390	+	0.967364i	$0.418454\pi$
$912$		0			0
$913$	43.1035			1.42652
$914$		0			0
$915$		0			0
$916$		0			0
$917$	25.5350	+	14.7426i	0.843240	+	0.486845i
$918$		0			0
$919$			11.0399i			0.364171i	0.983283	+	0.182086i	$0.0582848\pi$
$919$			11.0399i			0.364171i	−0.983283	+	0.182086i	$0.941715\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	88.6325			2.91737
$924$		0			0
$925$	5.24022	+	9.07633i	0.172298	+	0.298428i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−24.2544	−	14.0033i	−0.795760	−	0.459432i	0.0462264	−	0.998931i	$-0.485280\pi$
$929$	−24.2544	−	14.0033i	−0.795760	−	0.459432i	−0.841986	+	0.539499i	$0.818614\pi$
$930$		0			0
$931$	−7.14781	+	9.25351i	−0.234260	+	0.303272i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−22.3252	−	38.6683i	−0.730111	−	1.26459i
$936$		0			0
$937$	−3.59430	−	6.22551i	−0.117421	−	0.203378i	0.801324	−	0.598230i	$-0.204129\pi$
$937$	−3.59430	−	6.22551i	−0.117421	−	0.203378i	−0.918745	+	0.394852i	$0.870796\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.83887	−	2.21637i	0.125143	−	0.0722516i	−0.436121	−	0.899888i	$-0.643648\pi$
$941$	3.83887	−	2.21637i	0.125143	−	0.0722516i	0.561265	+	0.827636i	$0.310315\pi$
$942$		0			0
$943$			42.6768i			1.38975i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.19901	+	12.4691i	−0.233937	+	0.405190i	−0.958963	−	0.283531i	$-0.908494\pi$
$947$	−7.19901	+	12.4691i	−0.233937	+	0.405190i	0.725027	+	0.688721i	$0.241827\pi$
$948$		0			0
$949$	61.5970			1.99952
$950$		0			0
$951$		0			0
$952$		0			0
$953$	47.1305	+	27.2108i	1.52671	+	0.881445i	0.999497	+	0.0317103i	$0.0100954\pi$
$953$	47.1305	+	27.2108i	1.52671	+	0.881445i	0.527210	+	0.849735i	$0.323238\pi$
$954$		0			0
$955$	10.9617	+	6.32873i	0.354712	+	0.204793i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.36091	+	14.4815i	0.269988	+	0.467633i
$960$		0			0
$961$	4.42901			0.142871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.1811	+	7.03275i	−0.392123	+	0.226392i
$966$		0			0
$967$	22.6993	+	13.1054i	0.729960	+	0.421442i	0.818407	−	0.574638i	$-0.194857\pi$
$967$	22.6993	+	13.1054i	0.729960	+	0.421442i	−0.0884478	+	0.996081i	$0.528191\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.2189	+	22.8957i	−0.424213	+	0.734759i	−0.996347	−	0.0854015i	$-0.972783\pi$
$971$	−13.2189	+	22.8957i	−0.424213	+	0.734759i	0.572133	+	0.820161i	$0.306116\pi$
$972$		0			0
$973$	−21.5854	−	37.3870i	−0.691996	−	1.19857i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			23.4049i			0.748790i	0.927269	+	0.374395i	$0.122150\pi$
$977$			23.4049i			0.748790i	−0.927269	+	0.374395i	$0.877850\pi$
$978$		0			0
$979$	10.9747	−	6.33624i	0.350753	−	0.202507i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.1709	+	41.8652i	−0.770932	+	1.33529i	0.166120	+	0.986106i	$0.446876\pi$
$983$	−24.1709	+	41.8652i	−0.770932	+	1.33529i	−0.937053	+	0.349189i	$0.886457\pi$
$984$		0			0
$985$	−4.95019	+	8.57398i	−0.157726	+	0.273190i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	2.98378i		−	0.0948787i
$990$		0			0
$991$	−2.96161	−	1.70988i	−0.0940786	−	0.0543163i	0.452223	−	0.891905i	$-0.350631\pi$
$991$	−2.96161	−	1.70988i	−0.0940786	−	0.0543163i	−0.546301	+	0.837589i	$0.683965\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.2298			−0.704731
$996$		0			0
$997$	−9.88247	−	17.1169i	−0.312981	−	0.542099i	0.666025	−	0.745929i	$-0.267994\pi$
$997$	−9.88247	−	17.1169i	−0.312981	−	0.542099i	−0.979006	+	0.203830i	$0.934661\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.a.1151.9	yes	24
3.2	odd	2	inner	2736.2.cg.a.1151.4	✓	24
4.3	odd	2		2736.2.cg.b.1151.9	yes	24
12.11	even	2		2736.2.cg.b.1151.4	yes	24
19.7	even	3		2736.2.cg.b.2591.4	yes	24
57.26	odd	6		2736.2.cg.b.2591.9	yes	24
76.7	odd	6	inner	2736.2.cg.a.2591.4	yes	24
228.83	even	6	inner	2736.2.cg.a.2591.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.4	✓	24	3.2	odd	2	inner
2736.2.cg.a.1151.9	yes	24	1.1	even	1	trivial
2736.2.cg.a.2591.4	yes	24	76.7	odd	6	inner
2736.2.cg.a.2591.9	yes	24	228.83	even	6	inner
2736.2.cg.b.1151.4	yes	24	12.11	even	2
2736.2.cg.b.1151.9	yes	24	4.3	odd	2
2736.2.cg.b.2591.4	yes	24	19.7	even	3
2736.2.cg.b.2591.9	yes	24	57.26	odd	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.62081 + 0.935772i) q^{5} -2.07786i q^{7} +O(q^{10})\)	\(q+(1.62081 + 0.935772i) q^{5} -2.07786i q^{7} -5.79064 q^{11} +(3.25108 + 5.63103i) q^{13} +(3.56803 + 2.06000i) q^{17} +(-2.66462 + 3.44961i) q^{19} +(-2.02412 - 3.50588i) q^{23} +(-0.748660 - 1.29672i) q^{25} +(-2.90040 + 1.67455i) q^{29} +5.15471i q^{31} +(1.94441 - 3.36781i) q^{35} -6.99947 q^{37} +(-9.12968 - 5.27102i) q^{41} +(0.638308 + 0.368528i) q^{43} +(4.95759 + 8.58679i) q^{47} +2.68248 q^{49} +(-4.83526 + 2.79164i) q^{53} +(-9.38550 - 5.41872i) q^{55} +(-1.37628 + 2.38379i) q^{59} +(7.20914 + 12.4866i) q^{61} +12.1691i q^{65} +(-10.3881 + 5.99756i) q^{67} +(6.81563 - 11.8050i) q^{71} +(4.73667 - 8.20415i) q^{73} +12.0322i q^{77} +(9.40973 + 5.43271i) q^{79} -7.44365 q^{83} +(3.85539 + 6.67773i) q^{85} +(-1.89525 + 1.09422i) q^{89} +(11.7005 - 6.75529i) q^{91} +(-7.54688 + 3.09766i) q^{95} +(-0.380159 + 0.658455i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) \) 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Introduction

L-functions

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