LMFDB - Embedded newform 2736.2.dc.a.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(449,2736)]

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([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.a.1889.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 0.707107i) q^{5} -1.44949 q^{7} +O(q^{10})$	$q+(1.22474 - 0.707107i) q^{5} -1.44949 q^{7} -0.635674i q^{11} +(5.17423 + 2.98735i) q^{13} +(-1.77526 + 1.02494i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(4.89898 + 2.82843i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(-1.22474 + 2.12132i) q^{29} +0.953512i q^{31} +(-1.77526 + 1.02494i) q^{35} -2.51059i q^{37} +(1.94949 + 3.37662i) q^{43} +(-1.77526 - 1.02494i) q^{47} -4.89898 q^{49} +(2.44949 - 4.24264i) q^{53} +(-0.449490 - 0.778539i) q^{55} +(7.22474 + 12.5136i) q^{59} +(1.72474 - 2.98735i) q^{61} +8.44949 q^{65} +(8.84847 + 5.10867i) q^{67} +(3.00000 + 5.19615i) q^{71} +(1.05051 + 1.81954i) q^{73} +0.921404i q^{77} +(-0.825765 + 0.476756i) q^{79} -11.8065i q^{83} +(-1.44949 + 2.51059i) q^{85} +(6.12372 - 10.6066i) q^{89} +(-7.50000 - 4.33013i) q^{91} +(-3.67423 + 4.94975i) q^{95} +(-16.3485 + 9.43879i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 16 q^{19} - 6 q^{25} - 12 q^{35} - 2 q^{43} - 12 q^{47} + 8 q^{55} + 24 q^{59} + 2 q^{61} + 24 q^{65} + 6 q^{67} + 12 q^{71} + 14 q^{73} - 18 q^{79} + 4 q^{85} - 30 q^{91} - 36 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 16 * q^19 - 6 * q^25 - 12 * q^35 - 2 * q^43 - 12 * q^47 + 8 * q^55 + 24 * q^59 + 2 * q^61 + 24 * q^65 + 6 * q^67 + 12 * q^71 + 14 * q^73 - 18 * q^79 + 4 * q^85 - 30 * q^91 - 36 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$1$

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
$5$	1.22474	−	0.707107i	0.547723	−	0.316228i	−0.200480	−	0.979698i	$-0.564250\pi$
$5$	1.22474	−	0.707107i	0.547723	−	0.316228i	0.748203	+	0.663470i	$0.230917\pi$
$6$		0			0
$7$	−1.44949			−0.547856			−0.273928	−	0.961750i	$-0.588323\pi$
$7$	−1.44949			−0.547856			−0.273928	+	0.961750i	$0.588323\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.635674i		−	0.191663i	−0.995398	−	0.0958315i	$-0.969449\pi$
$11$		−	0.635674i		−	0.191663i	0.995398	−	0.0958315i	$-0.0305510\pi$
$12$		0			0
$13$	5.17423	+	2.98735i	1.43507	+	0.828541i	0.997502	−	0.0706424i	$-0.0225049\pi$
$13$	5.17423	+	2.98735i	1.43507	+	0.828541i	0.437573	+	0.899183i	$0.355838\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.77526	+	1.02494i	−0.430563	+	0.248585i	−0.699586	−	0.714548i	$-0.746632\pi$
$17$	−1.77526	+	1.02494i	−0.430563	+	0.248585i	0.269024	+	0.963134i	$0.413299\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.89898	+	2.82843i	1.02151	+	0.589768i	0.914540	−	0.404495i	$-0.132553\pi$
$23$	4.89898	+	2.82843i	1.02151	+	0.589768i	0.106967	+	0.994263i	$0.465886\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.22474	+	2.12132i	−0.227429	+	0.393919i	−0.957046	−	0.289938i	$-0.906365\pi$
$29$	−1.22474	+	2.12132i	−0.227429	+	0.393919i	0.729616	+	0.683857i	$0.239699\pi$
$30$		0			0
$31$			0.953512i			0.171256i	0.996327	+	0.0856279i	$0.0272896\pi$
$31$			0.953512i			0.171256i	−0.996327	+	0.0856279i	$0.972710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.77526	+	1.02494i	−0.300073	+	0.173247i
$36$		0			0
$37$		−	2.51059i		−	0.412738i	−0.978474	−	0.206369i	$-0.933835\pi$
$37$		−	2.51059i		−	0.412738i	0.978474	−	0.206369i	$-0.0661648\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	1.94949	+	3.37662i	0.297294	+	0.514929i	0.975516	−	0.219928i	$-0.0705824\pi$
$43$	1.94949	+	3.37662i	0.297294	+	0.514929i	−0.678222	+	0.734857i	$0.737249\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.77526	−	1.02494i	−0.258948	−	0.149503i	0.364907	−	0.931044i	$-0.381101\pi$
$47$	−1.77526	−	1.02494i	−0.258948	−	0.149503i	−0.623854	+	0.781541i	$0.714434\pi$
$48$		0			0
$49$	−4.89898			−0.699854
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.44949	−	4.24264i	0.336463	−	0.582772i	−0.647302	−	0.762234i	$-0.724103\pi$
$53$	2.44949	−	4.24264i	0.336463	−	0.582772i	0.983765	+	0.179463i	$0.0574359\pi$
$54$		0			0
$55$	−0.449490	−	0.778539i	−0.0606092	−	0.104978i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.22474	+	12.5136i	0.940582	+	1.62914i	0.764365	+	0.644784i	$0.223053\pi$
$59$	7.22474	+	12.5136i	0.940582	+	1.62914i	0.176217	+	0.984351i	$0.443614\pi$
$60$		0			0
$61$	1.72474	−	2.98735i	0.220831	−	0.382490i	−0.734230	−	0.678901i	$-0.762456\pi$
$61$	1.72474	−	2.98735i	0.220831	−	0.382490i	0.955061	+	0.296411i	$0.0957898\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.44949			1.04803
$66$		0			0
$67$	8.84847	+	5.10867i	1.08101	+	0.624123i	0.931169	−	0.364587i	$-0.118790\pi$
$67$	8.84847	+	5.10867i	1.08101	+	0.624123i	0.149843	+	0.988710i	$0.452123\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	0.987294	−	0.158901i	$-0.0507952\pi$
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	−0.631260	+	0.775571i	$0.717462\pi$
$72$		0			0
$73$	1.05051	+	1.81954i	0.122953	+	0.212961i	0.920931	−	0.389726i	$-0.127430\pi$
$73$	1.05051	+	1.81954i	0.122953	+	0.212961i	−0.797978	+	0.602687i	$0.794097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.921404i			0.105004i
$78$		0			0
$79$	−0.825765	+	0.476756i	−0.0929059	+	0.0536392i	−0.545733	−	0.837959i	$-0.683749\pi$
$79$	−0.825765	+	0.476756i	−0.0929059	+	0.0536392i	0.452827	+	0.891598i	$0.350415\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.8065i		−	1.29593i	−0.761669	−	0.647967i	$-0.775620\pi$
$83$		−	11.8065i		−	1.29593i	0.761669	−	0.647967i	$-0.224380\pi$
$84$		0			0
$85$	−1.44949	+	2.51059i	−0.157219	+	0.272312i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.12372	−	10.6066i	0.649113	−	1.12430i	−0.334221	−	0.942495i	$-0.608473\pi$
$89$	6.12372	−	10.6066i	0.649113	−	1.12430i	0.983335	−	0.181803i	$-0.0581933\pi$
$90$		0			0
$91$	−7.50000	−	4.33013i	−0.786214	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.67423	+	4.94975i	−0.376969	+	0.507833i
$96$		0			0
$97$	−16.3485	+	9.43879i	−1.65994	+	0.958364i	−0.687193	+	0.726474i	$0.741158\pi$
$97$	−16.3485	+	9.43879i	−1.65994	+	0.958364i	−0.972742	+	0.231890i	$0.925509\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.4495	+	6.61037i	1.13927	+	0.657756i	0.946249	−	0.323439i	$-0.104839\pi$
$101$	11.4495	+	6.61037i	1.13927	+	0.657756i	0.193018	+	0.981195i	$0.438172\pi$
$102$		0			0
$103$			14.4600i			1.42478i	0.701782	+	0.712392i	$0.252388\pi$
$103$			14.4600i			1.42478i	−0.701782	+	0.712392i	$0.747612\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.7980			1.52725			0.763623	−	0.645662i	$-0.223419\pi$
$107$	15.7980			1.52725			0.763623	+	0.645662i	$0.223419\pi$
$108$		0			0
$109$	−14.6969	+	8.48528i	−1.40771	+	0.812743i	−0.995167	−	0.0981950i	$-0.968693\pi$
$109$	−14.6969	+	8.48528i	−1.40771	+	0.812743i	−0.412544	+	0.910938i	$0.635360\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	17.1464			1.61300			0.806500	−	0.591234i	$-0.201359\pi$
$113$	17.1464			1.61300			0.806500	+	0.591234i	$0.201359\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.57321	−	1.48565i	0.235886	−	0.136189i
$120$		0			0
$121$	10.5959			0.963265
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$	−7.34847	−	4.24264i	−0.652071	−	0.376473i	0.137178	−	0.990546i	$-0.456197\pi$
$127$	−7.34847	−	4.24264i	−0.652071	−	0.376473i	−0.789249	+	0.614073i	$0.789530\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.7980	−	7.38891i	1.11816	−	0.645572i	0.177232	−	0.984169i	$-0.443286\pi$
$131$	12.7980	−	7.38891i	1.11816	−	0.645572i	0.940931	+	0.338598i	$0.109953\pi$
$132$		0			0
$133$	5.79796	−	2.51059i	0.502747	−	0.217696i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.44949	+	3.14626i	0.465581	+	0.268804i	0.714388	−	0.699750i	$-0.246705\pi$
$137$	5.44949	+	3.14626i	0.465581	+	0.268804i	−0.248807	+	0.968553i	$0.580038\pi$
$138$		0			0
$139$	−3.50000	+	6.06218i	−0.296866	+	0.514187i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−3.50000	+	6.06218i	−0.296866	+	0.514187i	0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.89898	−	3.28913i	0.158801	−	0.275051i
$144$		0			0
$145$			3.46410i			0.287678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.4495	+	10.0745i	−1.42952	+	0.825333i	−0.997082	−	0.0763323i	$-0.975679\pi$
$149$	−17.4495	+	10.0745i	−1.42952	+	0.825333i	−0.432435	+	0.901665i	$0.642346\pi$
$150$		0			0
$151$			1.55708i			0.126713i	0.997991	+	0.0633566i	$0.0201806\pi$
$151$			1.55708i			0.126713i	−0.997991	+	0.0633566i	$0.979819\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.674235	+	1.16781i	0.0541558	+	0.0938006i
$156$		0			0
$157$	−6.17423	−	10.6941i	−0.492758	−	0.853481i	0.507208	−	0.861824i	$-0.330678\pi$
$157$	−6.17423	−	10.6941i	−0.492758	−	0.853481i	−0.999965	+	0.00834275i	$0.997344\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.10102	−	4.09978i	−0.559639	−	0.323108i
$162$		0			0
$163$	−7.69694			−0.602871			−0.301435	−	0.953487i	$-0.597466\pi$
$163$	−7.69694			−0.602871			−0.301435	+	0.953487i	$0.597466\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.22474	−	7.31747i	0.326921	−	0.566243i	−0.654979	−	0.755647i	$-0.727322\pi$
$167$	4.22474	−	7.31747i	0.326921	−	0.566243i	0.981899	+	0.189404i	$0.0606557\pi$
$168$		0			0
$169$	11.3485	+	19.6561i	0.872959	+	1.51201i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.12372	+	10.6066i	0.465578	+	0.806405i	0.999227	−	0.0393009i	$-0.0125131\pi$
$173$	6.12372	+	10.6066i	0.465578	+	0.806405i	−0.533649	+	0.845706i	$0.679180\pi$
$174$		0			0
$175$	2.17423	−	3.76588i	0.164357	−	0.284674i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−4.65153	−	2.68556i	−0.345746	−	0.199616i	0.317064	−	0.948404i	$-0.397303\pi$
$181$	−4.65153	−	2.68556i	−0.345746	−	0.199616i	−0.662810	+	0.748788i	$0.730636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.77526	−	3.07483i	−0.130519	−	0.226066i
$186$		0			0
$187$	0.651531	+	1.12848i	0.0476446	+	0.0825230i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.2706i		−	1.10494i	−0.833532	−	0.552472i	$-0.813685\pi$
$191$		−	15.2706i		−	1.10494i	0.833532	−	0.552472i	$-0.186315\pi$
$192$		0			0
$193$	14.8485	−	8.57277i	1.06882	−	0.617081i	0.140958	−	0.990016i	$-0.454982\pi$
$193$	14.8485	−	8.57277i	1.06882	−	0.617081i	0.927858	+	0.372934i	$0.121648\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.2918i			1.44573i	0.690989	+	0.722865i	$0.257175\pi$
$197$			20.2918i			1.44573i	−0.690989	+	0.722865i	$0.742825\pi$
$198$		0			0
$199$	8.62372	−	14.9367i	0.611320	−	1.05884i	−0.379699	−	0.925110i	$-0.623972\pi$
$199$	8.62372	−	14.9367i	0.611320	−	1.05884i	0.991018	−	0.133726i	$-0.0426943\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.77526	−	3.07483i	0.124598	−	0.215811i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.10102	+	2.54270i	0.0761592	+	0.175882i
$210$		0			0
$211$	−3.15153	+	1.81954i	−0.216960	+	0.125262i	−0.604542	−	0.796573i	$-0.706644\pi$
$211$	−3.15153	+	1.81954i	−0.216960	+	0.125262i	0.387582	+	0.921835i	$0.373311\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.77526	+	2.75699i	0.325670	+	0.188025i
$216$		0			0
$217$		−	1.38211i		−	0.0938234i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.2474			−0.823853
$222$		0			0
$223$	2.17423	−	1.25529i	0.145598	−	0.0840608i	−0.425432	−	0.904991i	$-0.639878\pi$
$223$	2.17423	−	1.25529i	0.145598	−	0.0840608i	0.571029	+	0.820930i	$0.306544\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.1464			−1.53628			−0.768141	−	0.640280i	$-0.778818\pi$
$227$	−23.1464			−1.53628			−0.768141	+	0.640280i	$0.778818\pi$
$228$		0			0
$229$	−19.2474			−1.27191			−0.635954	−	0.771727i	$-0.719393\pi$
$229$	−19.2474			−1.27191			−0.635954	+	0.771727i	$0.719393\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.2474	−	8.80312i	0.998894	−	0.576711i	0.0909728	−	0.995853i	$-0.471002\pi$
$233$	15.2474	−	8.80312i	0.998894	−	0.576711i	0.907921	+	0.419142i	$0.137669\pi$
$234$		0			0
$235$	−2.89898			−0.189109
$236$		0			0
$237$		0			0
$238$		0			0
$239$			22.4846i			1.45440i	0.686423	+	0.727202i	$0.259180\pi$
$239$			22.4846i			1.45440i	−0.686423	+	0.727202i	$0.740820\pi$
$240$		0			0
$241$	12.1515	+	7.01569i	0.782749	+	0.451920i	0.837404	−	0.546585i	$-0.184072\pi$
$241$	12.1515	+	7.01569i	0.782749	+	0.451920i	−0.0546547	+	0.998505i	$0.517406\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	+	3.46410i	−0.383326	+	0.221313i
$246$		0			0
$247$	−25.8712	−	2.98735i	−1.64614	−	0.190080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.22474	+	0.707107i	0.0773052	+	0.0446322i	0.538154	−	0.842846i	$-0.319122\pi$
$251$	1.22474	+	0.707107i	0.0773052	+	0.0446322i	−0.460849	+	0.887478i	$0.652455\pi$
$252$		0			0
$253$	1.79796	−	3.11416i	0.113037	−	0.195785i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	−0.615948	−	0.787787i	$-0.711227\pi$
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	0.990217	+	0.139533i	$0.0445601\pi$
$258$		0			0
$259$			3.63907i			0.226121i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.797959	−	0.460702i	0.0492043	−	0.0284081i	−0.475196	−	0.879880i	$-0.657623\pi$
$263$	0.797959	−	0.460702i	0.0492043	−	0.0284081i	0.524400	+	0.851472i	$0.324290\pi$
$264$		0			0
$265$		−	6.92820i		−	0.425596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.2247	−	17.7098i	−0.623414	−	1.07978i	−0.988845	−	0.148946i	$-0.952412\pi$
$269$	−10.2247	−	17.7098i	−0.623414	−	1.07978i	0.365432	−	0.930838i	$-0.380921\pi$
$270$		0			0
$271$	15.6969	+	27.1879i	0.953521	+	1.65155i	0.737717	+	0.675110i	$0.235904\pi$
$271$	15.6969	+	27.1879i	0.953521	+	1.65155i	0.215804	+	0.976437i	$0.430763\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.65153	+	0.953512i	0.0995911	+	0.0574989i
$276$		0			0
$277$	9.10102			0.546827			0.273414	−	0.961897i	$-0.411847\pi$
$277$	9.10102			0.546827			0.273414	+	0.961897i	$0.411847\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.5732	−	20.0454i	0.690400	−	1.19581i	−0.281307	−	0.959618i	$-0.590768\pi$
$281$	11.5732	−	20.0454i	0.690400	−	1.19581i	0.971707	−	0.236190i	$-0.0758988\pi$
$282$		0			0
$283$	−7.44949	−	12.9029i	−0.442826	−	0.766997i	0.555072	−	0.831802i	$-0.312691\pi$
$283$	−7.44949	−	12.9029i	−0.442826	−	0.766997i	−0.997898	+	0.0648050i	$0.979357\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−6.39898	+	11.0834i	−0.376411	+	0.651962i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.6515			0.622269			0.311135	−	0.950366i	$-0.399291\pi$
$293$	10.6515			0.622269			0.311135	+	0.950366i	$0.399291\pi$
$294$		0			0
$295$	17.6969	+	10.2173i	1.03036	+	0.594876i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.8990	+	29.2699i	0.977293	+	1.69272i
$300$		0			0
$301$	−2.82577	−	4.89437i	−0.162874	−	0.282107i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4.87832i		−	0.279332i
$306$		0			0
$307$	13.3485	−	7.70674i	0.761837	−	0.439847i	−0.0681177	−	0.997677i	$-0.521699\pi$
$307$	13.3485	−	7.70674i	0.761837	−	0.439847i	0.829955	+	0.557830i	$0.188366\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5.16404i			0.292826i	0.989224	+	0.146413i	$0.0467729\pi$
$311$			5.16404i			0.292826i	−0.989224	+	0.146413i	$0.953227\pi$
$312$		0			0
$313$	−11.3485	+	19.6561i	−0.641453	+	1.11103i	0.343655	+	0.939096i	$0.388335\pi$
$313$	−11.3485	+	19.6561i	−0.641453	+	1.11103i	−0.985108	+	0.171934i	$0.944998\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.67423	+	6.36396i	−0.206366	+	0.357436i	−0.950567	−	0.310520i	$-0.899497\pi$
$317$	−3.67423	+	6.36396i	−0.206366	+	0.357436i	0.744201	+	0.667955i	$0.232830\pi$
$318$		0			0
$319$	1.34847	+	0.778539i	0.0754998	+	0.0435898i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.32577	−	7.17461i	0.296334	−	0.399206i
$324$		0			0
$325$	−15.5227	+	8.96204i	−0.861045	+	0.497124i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.57321	+	1.48565i	0.141866	+	0.0819063i
$330$		0			0
$331$			18.7026i			1.02799i	0.857794	+	0.513994i	$0.171835\pi$
$331$			18.7026i			1.02799i	−0.857794	+	0.513994i	$0.828165\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14.4495			0.789460
$336$		0			0
$337$	1.80306	−	1.04100i	0.0982190	−	0.0567068i	−0.450086	−	0.892985i	$-0.648607\pi$
$337$	1.80306	−	1.04100i	0.0982190	−	0.0567068i	0.548305	+	0.836278i	$0.315273\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.606123			0.0328234
$342$		0			0
$343$	17.2474			0.931275
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−28.2247	+	16.2956i	−1.51518	+	0.874792i	−0.515342	+	0.856984i	$0.672335\pi$
$347$	−28.2247	+	16.2956i	−1.51518	+	0.874792i	−0.999841	+	0.0178073i	$0.994331\pi$
$348$		0			0
$349$	23.2474			1.24441			0.622204	−	0.782855i	$-0.286238\pi$
$349$	23.2474			1.24441			0.622204	+	0.782855i	$0.286238\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3.32124i			0.176772i	0.996086	+	0.0883858i	$0.0281708\pi$
$353$			3.32124i			0.176772i	−0.996086	+	0.0883858i	$0.971829\pi$
$354$		0			0
$355$	7.34847	+	4.24264i	0.390016	+	0.225176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.79796	−	5.65685i	0.517116	−	0.298557i	−0.218638	−	0.975806i	$-0.570161\pi$
$359$	9.79796	−	5.65685i	0.517116	−	0.298557i	0.735754	+	0.677249i	$0.236828\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.57321	+	1.48565i	0.134688	+	0.0777623i
$366$		0			0
$367$	−7.17423	+	12.4261i	−0.374492	+	0.648639i	−0.990251	−	0.139295i	$-0.955516\pi$
$367$	−7.17423	+	12.4261i	−0.374492	+	0.648639i	0.615759	+	0.787935i	$0.288850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.55051	+	6.14966i	−0.184333	+	0.319275i
$372$		0			0
$373$		−	12.2993i		−	0.636835i	−0.947951	−	0.318418i	$-0.896849\pi$
$373$		−	12.2993i		−	0.636835i	0.947951	−	0.318418i	$-0.103151\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.6742	+	7.31747i	−0.652756	+	0.376869i
$378$		0			0
$379$		−	19.0526i		−	0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$		−	19.0526i		−	0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.44949	−	9.43879i	−0.278456	−	0.482300i	0.692545	−	0.721374i	$-0.256489\pi$
$383$	−5.44949	−	9.43879i	−0.278456	−	0.482300i	−0.971001	+	0.239075i	$0.923156\pi$
$384$		0			0
$385$	0.651531	+	1.12848i	0.0332051	+	0.0575129i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.8990	−	13.2207i	−1.16102	−	0.670318i	−0.209475	−	0.977814i	$-0.567176\pi$
$389$	−22.8990	−	13.2207i	−1.16102	−	0.670318i	−0.951549	+	0.307496i	$0.900509\pi$
$390$		0			0
$391$	−11.5959			−0.586431
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.674235	+	1.16781i	−0.0339244	+	0.0587588i
$396$		0			0
$397$	−10.8258	−	18.7508i	−0.543330	−	0.941074i	−0.998710	−	0.0507775i	$-0.983830\pi$
$397$	−10.8258	−	18.7508i	−0.543330	−	0.941074i	0.455380	−	0.890297i	$-0.349503\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.123724	−	0.214297i	−0.00617850	−	0.0107015i	0.862920	−	0.505341i	$-0.168633\pi$
$401$	−0.123724	−	0.214297i	−0.00617850	−	0.0107015i	−0.869098	+	0.494640i	$0.835300\pi$
$402$		0			0
$403$	−2.84847	+	4.93369i	−0.141892	+	0.245765i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.59592			−0.0791067
$408$		0			0
$409$	−25.0454	−	14.4600i	−1.23842	−	0.715000i	−0.269645	−	0.962960i	$-0.586906\pi$
$409$	−25.0454	−	14.4600i	−1.23842	−	0.715000i	−0.968770	+	0.247960i	$0.920240\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.4722	−	18.1384i	−0.515303	−	0.892531i
$414$		0			0
$415$	−8.34847	−	14.4600i	−0.409810	−	0.709812i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	3.74983i		−	0.183191i	−0.995796	−	0.0915956i	$-0.970803\pi$
$419$		−	3.74983i		−	0.183191i	0.995796	−	0.0915956i	$-0.0291967\pi$
$420$		0			0
$421$	1.34847	−	0.778539i	0.0657204	−	0.0379437i	−0.466780	−	0.884374i	$-0.654586\pi$
$421$	1.34847	−	0.778539i	0.0657204	−	0.0379437i	0.532500	+	0.846430i	$0.321253\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.14966i		−	0.298303i
$426$		0			0
$427$	−2.50000	+	4.33013i	−0.120983	+	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.02270	+	13.8957i	−0.386440	+	0.669334i	−0.991968	−	0.126490i	$-0.959629\pi$
$431$	−8.02270	+	13.8957i	−0.386440	+	0.669334i	0.605528	+	0.795824i	$0.292962\pi$
$432$		0			0
$433$	25.1969	+	14.5475i	1.21089	+	0.699106i	0.962953	−	0.269670i	$-0.0869148\pi$
$433$	25.1969	+	14.5475i	1.21089	+	0.699106i	0.247935	+	0.968777i	$0.420248\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.4949	−	2.82843i	−1.17175	−	0.135302i
$438$		0			0
$439$	12.5227	−	7.22999i	0.597676	−	0.345068i	−0.170451	−	0.985366i	$-0.554522\pi$
$439$	12.5227	−	7.22999i	0.597676	−	0.345068i	0.768127	+	0.640298i	$0.221189\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−25.7753	−	14.8814i	−1.22462	−	0.707034i	−0.258720	−	0.965952i	$-0.583301\pi$
$443$	−25.7753	−	14.8814i	−1.22462	−	0.707034i	−0.965899	+	0.258918i	$0.916634\pi$
$444$		0			0
$445$		−	17.3205i		−	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.2474			0.861150			0.430575	−	0.902555i	$-0.358311\pi$
$449$	18.2474			0.861150			0.430575	+	0.902555i	$0.358311\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.2474			−0.574169
$456$		0			0
$457$	19.6969			0.921384			0.460692	−	0.887560i	$-0.347601\pi$
$457$	19.6969			0.921384			0.460692	+	0.887560i	$0.347601\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.12372	+	5.26758i	−0.424934	+	0.245336i	−0.697186	−	0.716890i	$-0.745565\pi$
$461$	−9.12372	+	5.26758i	−0.424934	+	0.245336i	0.272252	+	0.962226i	$0.412232\pi$
$462$		0			0
$463$	−40.1464			−1.86576			−0.932881	−	0.360184i	$-0.882714\pi$
$463$	−40.1464			−1.86576			−0.932881	+	0.360184i	$0.882714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	21.0703i		−	0.975019i	−0.873118	−	0.487510i	$-0.837905\pi$
$467$		−	21.0703i		−	0.975019i	0.873118	−	0.487510i	$-0.162095\pi$
$468$		0			0
$469$	−12.8258	−	7.40496i	−0.592239	−	0.341929i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.14643	−	1.23924i	0.0986929	−	0.0569804i
$474$		0			0
$475$	1.50000	−	12.9904i	0.0688247	−	0.596040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.14643	−	4.70334i	−0.372220	−	0.214901i	0.302208	−	0.953242i	$-0.402276\pi$
$479$	−8.14643	−	4.70334i	−0.372220	−	0.214901i	−0.674428	+	0.738341i	$0.735610\pi$
$480$		0			0
$481$	7.50000	−	12.9904i	0.341971	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13.3485	+	23.1202i	−0.606123	+	1.04984i
$486$		0			0
$487$		−	39.6622i		−	1.79727i	−0.438702	−	0.898633i	$-0.644561\pi$
$487$		−	39.6622i		−	1.79727i	0.438702	−	0.898633i	$-0.355439\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.1464	−	15.0956i	1.17997	−	0.681257i	0.223964	−	0.974597i	$-0.428100\pi$
$491$	26.1464	−	15.0956i	1.17997	−	0.681257i	0.956008	+	0.293340i	$0.0947669\pi$
$492$		0			0
$493$		−	5.02118i		−	0.226143i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.34847	−	7.53177i	−0.195056	−	0.337846i
$498$		0			0
$499$	−21.7474	−	37.6677i	−0.973550	−	1.68624i	−0.684641	−	0.728881i	$-0.740041\pi$
$499$	−21.7474	−	37.6677i	−0.973550	−	1.68624i	−0.288909	−	0.957357i	$-0.593292\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.5176	−	20.5061i	−1.58365	−	0.914322i	−0.994320	−	0.106430i	$-0.966058\pi$
$503$	−35.5176	−	20.5061i	−1.58365	−	0.914322i	−0.589331	−	0.807891i	$-0.700609\pi$
$504$		0			0
$505$	18.6969			0.832003
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.67423	−	11.5601i	0.295830	−	0.512393i	−0.679347	−	0.733817i	$-0.737737\pi$
$509$	6.67423	−	11.5601i	0.295830	−	0.512393i	0.975178	+	0.221424i	$0.0710704\pi$
$510$		0			0
$511$	−1.52270	−	2.63740i	−0.0673605	−	0.116672i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.2247	+	17.7098i	0.450556	+	0.780386i
$516$		0			0
$517$	−0.651531	+	1.12848i	−0.0286543	+	0.0496307i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.20204			0.359338			0.179669	−	0.983727i	$-0.442497\pi$
$521$	8.20204			0.359338			0.179669	+	0.983727i	$0.442497\pi$
$522$		0			0
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.977296	−	1.69273i	−0.0425717	−	0.0737363i
$528$		0			0
$529$	4.50000	+	7.79423i	0.195652	+	0.338880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	19.3485	−	11.1708i	0.836507	−	0.482958i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			3.11416i			0.134136i
$540$		0			0
$541$	−15.1742	+	26.2825i	−0.652391	+	1.12997i	0.330150	+	0.943929i	$0.392901\pi$
$541$	−15.1742	+	26.2825i	−0.652391	+	1.12997i	−0.982541	+	0.186046i	$0.940433\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	+	20.7846i	−0.514024	+	0.890315i
$546$		0			0
$547$	1.80306	+	1.04100i	0.0770933	+	0.0445099i	0.538051	−	0.842912i	$-0.319161\pi$
$547$	1.80306	+	1.04100i	0.0770933	+	0.0445099i	−0.460958	+	0.887422i	$0.652494\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.22474	−	10.6066i	0.0521759	−	0.451856i
$552$		0			0
$553$	1.19694	−	0.691053i	0.0508990	−	0.0293866i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.5959	−	13.0458i	−0.957420	−	0.552767i	−0.0620418	−	0.998074i	$-0.519761\pi$
$557$	−22.5959	−	13.0458i	−0.957420	−	0.552767i	−0.895378	+	0.445307i	$0.853095\pi$
$558$		0			0
$559$			23.2952i			0.985282i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.5959			−1.07874			−0.539370	−	0.842069i	$-0.681337\pi$
$563$	−25.5959			−1.07874			−0.539370	+	0.842069i	$0.681337\pi$
$564$		0			0
$565$	21.0000	−	12.1244i	0.883477	−	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−31.1010			−1.30382			−0.651911	−	0.758295i	$-0.726033\pi$
$569$	−31.1010			−1.30382			−0.651911	+	0.758295i	$0.726033\pi$
$570$		0			0
$571$	3.69694			0.154712			0.0773560	−	0.997004i	$-0.475352\pi$
$571$	3.69694			0.154712			0.0773560	+	0.997004i	$0.475352\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14.6969	+	8.48528i	−0.612905	+	0.353861i
$576$		0			0
$577$	5.79796			0.241372			0.120686	−	0.992691i	$-0.461491\pi$
$577$	5.79796			0.241372			0.120686	+	0.992691i	$0.461491\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			17.1134i			0.709985i
$582$		0			0
$583$	−2.69694	−	1.55708i	−0.111696	−	0.0644876i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.1237	−	13.9278i	0.995693	−	0.574863i	0.0887216	−	0.996056i	$-0.471722\pi$
$587$	24.1237	−	13.9278i	0.995693	−	0.574863i	0.906971	+	0.421193i	$0.138389\pi$
$588$		0			0
$589$	−1.65153	−	3.81405i	−0.0680501	−	0.157155i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.4722	+	7.77817i	0.553237	+	0.319411i	0.750426	−	0.660954i	$-0.229848\pi$
$593$	13.4722	+	7.77817i	0.553237	+	0.319411i	−0.197190	+	0.980365i	$0.563182\pi$
$594$		0			0
$595$	2.10102	−	3.63907i	0.0861334	−	0.149188i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.7980	−	32.5590i	0.768064	−	1.33033i	−0.170548	−	0.985349i	$-0.554554\pi$
$599$	18.7980	−	32.5590i	0.768064	−	1.33033i	0.938611	−	0.344976i	$-0.112113\pi$
$600$		0			0
$601$		−	10.2173i		−	0.416774i	−0.978046	−	0.208387i	$-0.933179\pi$
$601$		−	10.2173i		−	0.416774i	0.978046	−	0.208387i	$-0.0668213\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.9773	−	7.49245i	0.527602	−	0.304611i
$606$		0			0
$607$			2.16064i			0.0876979i	0.999038	+	0.0438489i	$0.0139620\pi$
$607$			2.16064i			0.0876979i	−0.999038	+	0.0438489i	$0.986038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.12372	−	10.6066i	−0.247739	−	0.429097i
$612$		0			0
$613$	−9.44949	−	16.3670i	−0.381661	−	0.661057i	0.609639	−	0.792680i	$-0.291315\pi$
$613$	−9.44949	−	16.3670i	−0.381661	−	0.661057i	−0.991300	+	0.131623i	$0.957981\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	28.1010	+	16.2241i	1.13130	+	0.653159i	0.944263	−	0.329193i	$-0.106777\pi$
$617$	28.1010	+	16.2241i	1.13130	+	0.653159i	0.187042	+	0.982352i	$0.440110\pi$
$618$		0			0
$619$	36.3939			1.46279			0.731397	−	0.681952i	$-0.238869\pi$
$619$	36.3939			1.46279			0.731397	+	0.681952i	$0.238869\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.87628	+	15.3742i	−0.355620	+	0.615953i
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.57321	+	4.45694i	0.102601	+	0.177710i
$630$		0			0
$631$	7.27526	−	12.6011i	0.289623	−	0.501642i	−0.684096	−	0.729392i	$-0.739803\pi$
$631$	7.27526	−	12.6011i	0.289623	−	0.501642i	0.973720	+	0.227749i	$0.0731366\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000			−0.476205
$636$		0			0
$637$	−25.3485	−	14.6349i	−1.00434	−	0.579858i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.0227	−	39.8765i	−0.909342	−	1.57503i	−0.814980	−	0.579489i	$-0.803252\pi$
$641$	−23.0227	−	39.8765i	−0.909342	−	1.57503i	−0.0943619	−	0.995538i	$-0.530081\pi$
$642$		0			0
$643$	21.2980	+	36.8891i	0.839910	+	1.45477i	0.889970	+	0.456020i	$0.150725\pi$
$643$	21.2980	+	36.8891i	0.839910	+	1.45477i	−0.0500601	+	0.998746i	$0.515941\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.97129i		−	0.116814i	−0.998293	−	0.0584068i	$-0.981398\pi$
$647$		−	2.97129i		−	0.116814i	0.998293	−	0.0584068i	$-0.0186020\pi$
$648$		0			0
$649$	7.95459	−	4.59259i	0.312245	−	0.180275i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			38.3908i			1.50235i	0.660103	+	0.751175i	$0.270513\pi$
$653$			38.3908i			1.50235i	−0.660103	+	0.751175i	$0.729487\pi$
$654$		0			0
$655$	10.4495	−	18.0990i	0.408295	−	0.707188i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.69694	−	9.86739i	0.221921	−	0.384379i	−0.733470	−	0.679722i	$-0.762101\pi$
$659$	5.69694	−	9.86739i	0.221921	−	0.384379i	0.955391	+	0.295343i	$0.0954339\pi$
$660$		0			0
$661$	−23.6969	−	13.6814i	−0.921704	−	0.532146i	−0.0375258	−	0.999296i	$-0.511948\pi$
$661$	−23.6969	−	13.6814i	−0.921704	−	0.532146i	−0.884178	+	0.467150i	$0.845281\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.32577	−	7.17461i	0.206524	−	0.278219i
$666$		0			0
$667$	−12.0000	+	6.92820i	−0.464642	+	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.89898	−	1.09638i	−0.0733093	−	0.0423251i
$672$		0			0
$673$			19.0526i			0.734422i	0.930138	+	0.367211i	$0.119687\pi$
$673$			19.0526i			0.734422i	−0.930138	+	0.367211i	$0.880313\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.3485			−1.43542			−0.717709	−	0.696343i	$-0.754809\pi$
$677$	−37.3485			−1.43542			−0.717709	+	0.696343i	$0.754809\pi$
$678$		0			0
$679$	23.6969	−	13.6814i	0.909405	−	0.525045i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28.0454			−1.07313			−0.536564	−	0.843860i	$-0.680278\pi$
$683$	−28.0454			−1.07313			−0.536564	+	0.843860i	$0.680278\pi$
$684$		0			0
$685$	8.89898			0.340013
$686$		0			0
$687$		0			0
$688$		0			0
$689$	25.3485	−	14.6349i	0.965700	−	0.557547i
$690$		0			0
$691$	0.696938			0.0265128			0.0132564	−	0.999912i	$-0.495780\pi$
$691$	0.696938			0.0265128			0.0132564	+	0.999912i	$0.495780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			9.89949i			0.375509i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.4722	+	7.77817i	−0.508838	+	0.293778i	−0.732356	−	0.680922i	$-0.761579\pi$
$701$	−13.4722	+	7.77817i	−0.508838	+	0.293778i	0.223518	+	0.974700i	$0.428246\pi$
$702$		0			0
$703$	4.34847	+	10.0424i	0.164006	+	0.378755i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.5959	−	9.58166i	−0.624154	−	0.360355i
$708$		0			0
$709$	−3.17423	+	5.49794i	−0.119211	+	0.206479i	−0.919455	−	0.393195i	$-0.871370\pi$
$709$	−3.17423	+	5.49794i	−0.119211	+	0.206479i	0.800244	+	0.599674i	$0.204703\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.69694	+	4.67123i	−0.101001	+	0.174939i
$714$		0			0
$715$		−	5.37113i		−	0.200869i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.1464	−	9.89949i	0.639454	−	0.369189i	−0.144950	−	0.989439i	$-0.546302\pi$
$719$	17.1464	−	9.89949i	0.639454	−	0.369189i	0.784404	+	0.620250i	$0.212969\pi$
$720$		0			0
$721$		−	20.9596i		−	0.780576i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.67423	−	6.36396i	−0.136458	−	0.236352i
$726$		0			0
$727$	4.82577	+	8.35847i	0.178978	+	0.309999i	0.941531	−	0.336927i	$-0.109388\pi$
$727$	4.82577	+	8.35847i	0.178978	+	0.309999i	−0.762553	+	0.646926i	$0.776054\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.92168	−	3.99624i	−0.256008	−	0.147806i
$732$		0			0
$733$	−30.6969			−1.13382			−0.566909	−	0.823781i	$-0.691861\pi$
$733$	−30.6969			−1.13382			−0.566909	+	0.823781i	$0.691861\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.24745	−	5.62475i	0.119621	−	0.207190i
$738$		0			0
$739$	23.1969	+	40.1783i	0.853313	+	1.47798i	0.878201	+	0.478292i	$0.158744\pi$
$739$	23.1969	+	40.1783i	0.853313	+	1.47798i	−0.0248879	+	0.999690i	$0.507923\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−23.6969	−	41.0443i	−0.869356	−	1.50577i	−0.862656	−	0.505792i	$-0.831200\pi$
$743$	−23.6969	−	41.0443i	−0.869356	−	1.50577i	−0.00670079	−	0.999978i	$-0.502133\pi$
$744$		0			0
$745$	−14.2474	+	24.6773i	−0.521986	+	0.904107i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.8990			−0.836711
$750$		0			0
$751$	−24.2196	−	13.9832i	−0.883787	−	0.510255i	−0.0118820	−	0.999929i	$-0.503782\pi$
$751$	−24.2196	−	13.9832i	−0.883787	−	0.510255i	−0.871905	+	0.489675i	$0.837116\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.10102	+	1.90702i	0.0400702	+	0.0694037i
$756$		0			0
$757$	−12.1742	−	21.0864i	−0.442480	−	0.766398i	0.555393	−	0.831588i	$-0.312568\pi$
$757$	−12.1742	−	21.0864i	−0.442480	−	0.766398i	−0.997873	+	0.0651902i	$0.979235\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	44.8262i		−	1.62495i	−0.582996	−	0.812475i	$-0.698120\pi$
$761$		−	44.8262i		−	1.62495i	0.582996	−	0.812475i	$-0.301880\pi$
$762$		0			0
$763$	21.3031	−	12.2993i	0.771223	−	0.445266i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			86.3312i			3.11724i
$768$		0			0
$769$	1.29796	−	2.24813i	0.0468056	−	0.0810697i	−0.841673	−	0.539987i	$-0.818429\pi$
$769$	1.29796	−	2.24813i	0.0468056	−	0.0810697i	0.888479	+	0.458917i	$0.151763\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.47219	+	7.74607i	−0.160854	+	0.278607i	−0.935175	−	0.354186i	$-0.884758\pi$
$773$	−4.47219	+	7.74607i	−0.160854	+	0.278607i	0.774321	+	0.632792i	$0.218091\pi$
$774$		0			0
$775$	−2.47730	−	1.43027i	−0.0889871	−	0.0513767i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	3.30306	−	1.90702i	0.118193	−	0.0682387i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.1237	−	8.73169i	−0.539789	−	0.311647i
$786$		0			0
$787$		−	24.0737i		−	0.858136i	−0.903272	−	0.429068i	$-0.858842\pi$
$787$		−	24.0737i		−	0.858136i	0.903272	−	0.429068i	$-0.141158\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.8536			−0.883691
$792$		0			0
$793$	17.8485	−	10.3048i	0.633818	−	0.365935i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.0908			−1.34925			−0.674623	−	0.738162i	$-0.735694\pi$
$797$	−38.0908			−1.34925			−0.674623	+	0.738162i	$0.735694\pi$
$798$		0			0
$799$	4.20204			0.148658
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.15663	−	0.667783i	0.0408167	−	0.0235655i
$804$		0			0
$805$	−11.5959			−0.408702
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	5.16404i		−	0.181558i	−0.995871	−	0.0907791i	$-0.971064\pi$
$809$		−	5.16404i		−	0.181558i	0.995871	−	0.0907791i	$-0.0289357\pi$
$810$		0			0
$811$	−17.6969	−	10.2173i	−0.621424	−	0.358779i	0.155999	−	0.987757i	$-0.450140\pi$
$811$	−17.6969	−	10.2173i	−0.621424	−	0.358779i	−0.777423	+	0.628978i	$0.783474\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.42679	+	5.44256i	−0.330206	+	0.190644i
$816$		0			0
$817$	−13.6464	−	10.1298i	−0.477428	−	0.354398i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34.1691	+	19.7276i	1.19251	+	0.688497i	0.958875	−	0.283828i	$-0.0916044\pi$
$821$	34.1691	+	19.7276i	1.19251	+	0.688497i	0.233636	+	0.972324i	$0.424938\pi$
$822$		0			0
$823$	17.3485	−	30.0484i	0.604730	−	1.04742i	−0.387365	−	0.921927i	$-0.626615\pi$
$823$	17.3485	−	30.0484i	0.604730	−	1.04742i	0.992094	−	0.125496i	$-0.0400522\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−23.8207	+	41.2586i	−0.828326	+	1.43470i	0.0710253	+	0.997475i	$0.477373\pi$
$827$	−23.8207	+	41.2586i	−0.828326	+	1.43470i	−0.899351	+	0.437228i	$0.855960\pi$
$828$		0			0
$829$		−	10.9959i		−	0.381902i	−0.981600	−	0.190951i	$-0.938843\pi$
$829$		−	10.9959i		−	0.381902i	0.981600	−	0.190951i	$-0.0611572\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.69694	−	5.02118i	0.301331	−	0.173974i
$834$		0			0
$835$		−	11.9494i		−	0.413525i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.02270	−	8.69958i	−0.173403	−	0.300343i	0.766204	−	0.642597i	$-0.222143\pi$
$839$	−5.02270	−	8.69958i	−0.173403	−	0.300343i	−0.939607	+	0.342254i	$0.888810\pi$
$840$		0			0
$841$	11.5000	+	19.9186i	0.396552	+	0.686848i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	27.7980	+	16.0492i	0.956279	+	0.552108i
$846$		0			0
$847$	−15.3587			−0.527730
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.10102	−	12.2993i	0.243420	−	0.421616i
$852$		0			0
$853$	14.8258	+	25.6790i	0.507625	+	0.879231i	0.999961	+	0.00882655i	$0.00280961\pi$
$853$	14.8258	+	25.6790i	0.507625	+	0.879231i	−0.492337	+	0.870405i	$0.663857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.42679	−	5.93537i	−0.117057	−	0.202748i	0.801543	−	0.597937i	$-0.204013\pi$
$857$	−3.42679	−	5.93537i	−0.117057	−	0.202748i	−0.918600	+	0.395188i	$0.870679\pi$
$858$		0			0
$859$	1.94949	−	3.37662i	0.0665157	−	0.115209i	−0.830850	−	0.556497i	$-0.812145\pi$
$859$	1.94949	−	3.37662i	0.0665157	−	0.115209i	0.897365	+	0.441288i	$0.145478\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37.3485			1.27136			0.635678	−	0.771954i	$-0.280720\pi$
$863$	37.3485			1.27136			0.635678	+	0.771954i	$0.280720\pi$
$864$		0			0
$865$	15.0000	+	8.66025i	0.510015	+	0.294457i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.303062	+	0.524918i	0.0102807	+	0.0178066i
$870$		0			0
$871$	30.5227	+	52.8669i	1.03422	+	1.79133i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	16.3991i		−	0.554391i
$876$		0			0
$877$	24.5227	−	14.1582i	0.828073	−	0.478088i	−0.0251195	−	0.999684i	$-0.507997\pi$
$877$	24.5227	−	14.1582i	0.828073	−	0.478088i	0.853192	+	0.521596i	$0.174663\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	2.04989i		−	0.0690625i	−0.999404	−	0.0345312i	$-0.989006\pi$
$881$		−	2.04989i		−	0.0690625i	0.999404	−	0.0345312i	$-0.0109938\pi$
$882$		0			0
$883$	−2.94949	+	5.10867i	−0.0992582	+	0.171920i	−0.911378	−	0.411571i	$-0.864980\pi$
$883$	−2.94949	+	5.10867i	−0.0992582	+	0.171920i	0.812120	+	0.583491i	$0.198314\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.3258	+	19.6168i	−0.380282	+	0.658668i	−0.991102	−	0.133101i	$-0.957506\pi$
$887$	−11.3258	+	19.6168i	−0.380282	+	0.658668i	0.610820	+	0.791769i	$0.290840\pi$
$888$		0			0
$889$	10.6515	+	6.14966i	0.357241	+	0.206253i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.87628	+	1.02494i	0.297033	+	0.0342984i
$894$		0			0
$895$	−14.6969	+	8.48528i	−0.491264	+	0.283632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.02270	−	1.16781i	−0.0674610	−	0.0389486i
$900$		0			0
$901$			10.0424i			0.334560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.59592			−0.252497
$906$		0			0
$907$	−35.6969	+	20.6096i	−1.18530	+	0.684332i	−0.957234	−	0.289315i	$-0.906573\pi$
$907$	−35.6969	+	20.6096i	−1.18530	+	0.684332i	−0.228063	+	0.973646i	$0.573239\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	−7.50510			−0.248383
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.5505	+	10.7101i	−0.612592	+	0.353680i
$918$		0			0
$919$	−9.04541			−0.298380			−0.149190	−	0.988809i	$-0.547667\pi$
$919$	−9.04541			−0.298380			−0.149190	+	0.988809i	$0.547667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			35.8481i			1.17996i
$924$		0			0
$925$	6.52270	+	3.76588i	0.214465	+	0.123822i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.5732	−	20.5382i	1.16712	−	0.673837i	0.214119	−	0.976807i	$-0.431312\pi$
$929$	35.5732	−	20.5382i	1.16712	−	0.673837i	0.953000	+	0.302971i	$0.0979785\pi$
$930$		0			0
$931$	19.5959	−	8.48528i	0.642230	−	0.278094i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.59592	+	0.921404i	0.0521921	+	0.0301331i
$936$		0			0
$937$	−8.19694	+	14.1975i	−0.267782	+	0.463813i	−0.968289	−	0.249833i	$-0.919624\pi$
$937$	−8.19694	+	14.1975i	−0.267782	+	0.463813i	0.700507	+	0.713646i	$0.252957\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.4495	+	35.4196i	−0.666634	+	1.15464i	0.312205	+	0.950015i	$0.398932\pi$
$941$	−20.4495	+	35.4196i	−0.666634	+	1.15464i	−0.978839	+	0.204630i	$0.934401\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.4949	−	19.3383i	1.08844	−	0.628410i	0.155278	−	0.987871i	$-0.450373\pi$
$947$	33.4949	−	19.3383i	1.08844	−	0.628410i	0.933160	+	0.359461i	$0.117039\pi$
$948$		0			0
$949$			12.5529i			0.407486i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.550510	+	0.953512i	0.0178328	+	0.0308873i	0.874804	−	0.484477i	$-0.160990\pi$
$953$	0.550510	+	0.953512i	0.0178328	+	0.0308873i	−0.856971	+	0.515364i	$0.827657\pi$
$954$		0			0
$955$	−10.7980	−	18.7026i	−0.349414	−	0.605202i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.89898	−	4.56048i	−0.255071	−	0.147266i
$960$		0			0
$961$	30.0908			0.970671
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.1237	−	20.9989i	0.390276	−	0.675979i
$966$		0			0
$967$	6.17423	+	10.6941i	0.198550	+	0.343899i	0.948058	−	0.318096i	$-0.103043\pi$
$967$	6.17423	+	10.6941i	0.198550	+	0.343899i	−0.749508	+	0.661995i	$0.769710\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.8207	+	25.6701i	0.475618	+	0.823794i	0.999610	−	0.0279290i	$-0.00889124\pi$
$971$	14.8207	+	25.6701i	0.475618	+	0.823794i	−0.523992	+	0.851723i	$0.675558\pi$
$972$		0			0
$973$	5.07321	−	8.78706i	0.162640	−	0.281700i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.247449			0.00791659			0.00395829	−	0.999992i	$-0.498740\pi$
$977$	0.247449			0.00791659			0.00395829	+	0.999992i	$0.498740\pi$
$978$		0			0
$979$	−6.74235	−	3.89270i	−0.215486	−	0.124411i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.24745	−	16.0171i	−0.294948	−	0.510865i	0.680025	−	0.733189i	$-0.261969\pi$
$983$	−9.24745	−	16.0171i	−0.294948	−	0.510865i	−0.974973	+	0.222324i	$0.928636\pi$
$984$		0			0
$985$	14.3485	+	24.8523i	0.457180	+	0.791859i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			22.0560i			0.701339i
$990$		0			0
$991$	−52.8712	+	30.5252i	−1.67951	+	0.969664i	−0.717534	+	0.696523i	$0.754729\pi$
$991$	−52.8712	+	30.5252i	−1.67951	+	0.969664i	−0.961974	+	0.273141i	$0.911937\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	24.3916i		−	0.773265i
$996$		0			0
$997$	20.5227	−	35.5464i	0.649961	−	1.12576i	−0.333171	−	0.942866i	$-0.608119\pi$
$997$	20.5227	−	35.5464i	0.649961	−	1.12576i	0.983132	−	0.182898i	$-0.0585479\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.a.449.2		4
3.2	odd	2		2736.2.dc.b.449.1		4
4.3	odd	2		342.2.s.b.107.2	yes	4
12.11	even	2		342.2.s.a.107.1	✓	4
19.8	odd	6		2736.2.dc.b.1889.1		4
57.8	even	6	inner	2736.2.dc.a.1889.2		4
76.27	even	6		342.2.s.a.179.1	yes	4
228.179	odd	6		342.2.s.b.179.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.s.a.107.1	✓	4	12.11	even	2
342.2.s.a.179.1	yes	4	76.27	even	6
342.2.s.b.107.2	yes	4	4.3	odd	2
342.2.s.b.179.2	yes	4	228.179	odd	6
2736.2.dc.a.449.2		4	1.1	even	1	trivial
2736.2.dc.a.1889.2		4	57.8	even	6	inner
2736.2.dc.b.449.1		4	3.2	odd	2
2736.2.dc.b.1889.1		4	19.8	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.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{5} -1.44949 q^{7} +O(q^{10})\)	\(q+(1.22474 - 0.707107i) q^{5} -1.44949 q^{7} -0.635674i q^{11} +(5.17423 + 2.98735i) q^{13} +(-1.77526 + 1.02494i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(4.89898 + 2.82843i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(-1.22474 + 2.12132i) q^{29} +0.953512i q^{31} +(-1.77526 + 1.02494i) q^{35} -2.51059i q^{37} +(1.94949 + 3.37662i) q^{43} +(-1.77526 - 1.02494i) q^{47} -4.89898 q^{49} +(2.44949 - 4.24264i) q^{53} +(-0.449490 - 0.778539i) q^{55} +(7.22474 + 12.5136i) q^{59} +(1.72474 - 2.98735i) q^{61} +8.44949 q^{65} +(8.84847 + 5.10867i) q^{67} +(3.00000 + 5.19615i) q^{71} +(1.05051 + 1.81954i) q^{73} +0.921404i q^{77} +(-0.825765 + 0.476756i) q^{79} -11.8065i q^{83} +(-1.44949 + 2.51059i) q^{85} +(6.12372 - 10.6066i) q^{89} +(-7.50000 - 4.33013i) q^{91} +(-3.67423 + 4.94975i) q^{95} +(-16.3485 + 9.43879i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 16 q^{19} - 6 q^{25} - 12 q^{35} - 2 q^{43} - 12 q^{47} + 8 q^{55} + 24 q^{59} + 2 q^{61} + 24 q^{65} + 6 q^{67} + 12 q^{71} + 14 q^{73} - 18 q^{79} + 4 q^{85} - 30 q^{91} - 36 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 16 * q^19 - 6 * q^25 - 12 * q^35 - 2 * q^43 - 12 * q^47 + 8 * q^55 + 24 * q^59 + 2 * q^61 + 24 * q^65 + 6 * q^67 + 12 * q^71 + 14 * q^73 - 18 * q^79 + 4 * q^85 - 30 * q^91 - 36 * q^97

Introduction

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