LMFDB - Embedded newform 2736.2.dc.b.1889.1

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			1889.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.1889
Dual form			2736.2.dc.b.449.1

$q$-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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

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

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	$-0.435750\pi$
$5$	−1.22474	−	0.707107i	−0.547723	−	0.316228i	−0.748203	+	0.663470i	$0.769083\pi$
$6$		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	0.437573	−	0.899183i	$-0.355838\pi$
$13$	5.17423	−	2.98735i	1.43507	−	0.828541i	0.997502	+	0.0706424i	$0.0225049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.77526	+	1.02494i	0.430563	+	0.248585i	0.699586	−	0.714548i	$-0.253368\pi$
$17$	1.77526	+	1.02494i	0.430563	+	0.248585i	−0.269024	+	0.963134i	$0.586701\pi$
$18$		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	$-0.867447\pi$
$23$	−4.89898	+	2.82843i	−1.02151	+	0.589768i	−0.106967	+	0.994263i	$0.534114\pi$
$24$		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	$-0.0936346\pi$
$29$	1.22474	+	2.12132i	0.227429	+	0.393919i	−0.729616	+	0.683857i	$0.760301\pi$
$30$		0			0
$31$		−	0.953512i		−	0.171256i	−0.996327	−	0.0856279i	$-0.972710\pi$
$31$		−	0.953512i		−	0.171256i	0.996327	−	0.0856279i	$-0.0272896\pi$
$32$		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	$0.0661648\pi$
$37$			2.51059i			0.412738i	−0.978474	+	0.206369i	$0.933835\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	1.94949	−	3.37662i	0.297294	−	0.514929i	−0.678222	−	0.734857i	$-0.737249\pi$
$43$	1.94949	−	3.37662i	0.297294	−	0.514929i	0.975516	+	0.219928i	$0.0705824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.77526	−	1.02494i	0.258948	−	0.149503i	−0.364907	−	0.931044i	$-0.618899\pi$
$47$	1.77526	−	1.02494i	0.258948	−	0.149503i	0.623854	+	0.781541i	$0.285566\pi$
$48$		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	$-0.275897\pi$
$53$	−2.44949	−	4.24264i	−0.336463	−	0.582772i	−0.983765	+	0.179463i	$0.942564\pi$
$54$		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	−0.176217	+	0.984351i	$0.556386\pi$
$59$	−7.22474	+	12.5136i	−0.940582	+	1.62914i	−0.764365	+	0.644784i	$0.776947\pi$
$60$		0			0
$61$	1.72474	+	2.98735i	0.220831	+	0.382490i	0.955061	−	0.296411i	$-0.0957898\pi$
$61$	1.72474	+	2.98735i	0.220831	+	0.382490i	−0.734230	+	0.678901i	$0.762456\pi$
$62$		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	0.149843	−	0.988710i	$-0.452123\pi$
$67$	8.84847	−	5.10867i	1.08101	−	0.624123i	0.931169	+	0.364587i	$0.118790\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	−0.987294	−	0.158901i	$-0.949205\pi$
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	0.631260	+	0.775571i	$0.282538\pi$
$72$		0			0
$73$	1.05051	−	1.81954i	0.122953	−	0.212961i	−0.797978	−	0.602687i	$-0.794097\pi$
$73$	1.05051	−	1.81954i	0.122953	−	0.212961i	0.920931	+	0.389726i	$0.127430\pi$
$74$		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	0.452827	−	0.891598i	$-0.350415\pi$
$79$	−0.825765	−	0.476756i	−0.0929059	−	0.0536392i	−0.545733	+	0.837959i	$0.683749\pi$
$80$		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	−0.983335	−	0.181803i	$-0.941807\pi$
$89$	−6.12372	−	10.6066i	−0.649113	−	1.12430i	0.334221	−	0.942495i	$-0.391527\pi$
$90$		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	−0.972742	−	0.231890i	$-0.925509\pi$
$97$	−16.3485	−	9.43879i	−1.65994	−	0.958364i	−0.687193	−	0.726474i	$-0.741158\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.4495	+	6.61037i	−1.13927	+	0.657756i	−0.946249	−	0.323439i	$-0.895161\pi$
$101$	−11.4495	+	6.61037i	−1.13927	+	0.657756i	−0.193018	+	0.981195i	$0.561828\pi$
$102$		0			0
$103$		−	14.4600i		−	1.42478i	−0.701782	−	0.712392i	$-0.747612\pi$
$103$		−	14.4600i		−	1.42478i	0.701782	−	0.712392i	$-0.252388\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.7980			−1.52725			−0.763623	−	0.645662i	$-0.776581\pi$
$107$	−15.7980			−1.52725			−0.763623	+	0.645662i	$0.776581\pi$
$108$		0			0
$109$	−14.6969	−	8.48528i	−1.40771	−	0.812743i	−0.412544	−	0.910938i	$-0.635360\pi$
$109$	−14.6969	−	8.48528i	−1.40771	−	0.812743i	−0.995167	+	0.0981950i	$0.968693\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.1464			−1.61300			−0.806500	−	0.591234i	$-0.798641\pi$
$113$	−17.1464			−1.61300			−0.806500	+	0.591234i	$0.798641\pi$
$114$		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	−0.789249	−	0.614073i	$-0.789530\pi$
$127$	−7.34847	+	4.24264i	−0.652071	+	0.376473i	0.137178	+	0.990546i	$0.456197\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.7980	−	7.38891i	−1.11816	−	0.645572i	−0.177232	−	0.984169i	$-0.556714\pi$
$131$	−12.7980	−	7.38891i	−1.11816	−	0.645572i	−0.940931	+	0.338598i	$0.890047\pi$
$132$		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	$-0.753295\pi$
$137$	−5.44949	+	3.14626i	−0.465581	+	0.268804i	0.248807	+	0.968553i	$0.419962\pi$
$138$		0			0
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		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	$-0.0243210\pi$
$149$	17.4495	+	10.0745i	1.42952	+	0.825333i	0.432435	+	0.901665i	$0.357654\pi$
$150$		0			0
$151$		−	1.55708i		−	0.126713i	−0.997991	−	0.0633566i	$-0.979819\pi$
$151$		−	1.55708i		−	0.126713i	0.997991	−	0.0633566i	$-0.0201806\pi$
$152$		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	−0.999965	−	0.00834275i	$-0.997344\pi$
$157$	−6.17423	+	10.6941i	−0.492758	+	0.853481i	0.507208	+	0.861824i	$0.330678\pi$
$158$		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	$-0.272678\pi$
$167$	−4.22474	−	7.31747i	−0.326921	−	0.566243i	−0.981899	+	0.189404i	$0.939344\pi$
$168$		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	$-0.987487\pi$
$173$	−6.12372	+	10.6066i	−0.465578	+	0.806405i	0.533649	+	0.845706i	$0.320820\pi$
$174$		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	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−4.65153	+	2.68556i	−0.345746	+	0.199616i	−0.662810	−	0.748788i	$-0.730636\pi$
$181$	−4.65153	+	2.68556i	−0.345746	+	0.199616i	0.317064	+	0.948404i	$0.397303\pi$
$182$		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	0.927858	−	0.372934i	$-0.121648\pi$
$193$	14.8485	+	8.57277i	1.06882	+	0.617081i	0.140958	+	0.990016i	$0.454982\pi$
$194$		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	0.991018	+	0.133726i	$0.0426943\pi$
$199$	8.62372	+	14.9367i	0.611320	+	1.05884i	−0.379699	+	0.925110i	$0.623972\pi$
$200$		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	0.387582	−	0.921835i	$-0.373311\pi$
$211$	−3.15153	−	1.81954i	−0.216960	−	0.125262i	−0.604542	+	0.796573i	$0.706644\pi$
$212$		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	0.571029	−	0.820930i	$-0.306544\pi$
$223$	2.17423	+	1.25529i	0.145598	+	0.0840608i	−0.425432	+	0.904991i	$0.639878\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	23.1464			1.53628			0.768141	−	0.640280i	$-0.221182\pi$
$227$	23.1464			1.53628			0.768141	+	0.640280i	$0.221182\pi$
$228$		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	$-0.528998\pi$
$233$	−15.2474	−	8.80312i	−0.998894	−	0.576711i	−0.907921	+	0.419142i	$0.862331\pi$
$234$		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	−0.0546547	−	0.998505i	$-0.517406\pi$
$241$	12.1515	−	7.01569i	0.782749	−	0.451920i	0.837404	+	0.546585i	$0.184072\pi$
$242$		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	$-0.680878\pi$
$251$	−1.22474	+	0.707107i	−0.0773052	+	0.0446322i	0.460849	+	0.887478i	$0.347545\pi$
$252$		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	$-0.288773\pi$
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	−0.990217	+	0.139533i	$0.955440\pi$
$258$		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	$-0.342377\pi$
$263$	−0.797959	−	0.460702i	−0.0492043	−	0.0284081i	−0.524400	+	0.851472i	$0.675710\pi$
$264$		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	−0.365432	−	0.930838i	$-0.619079\pi$
$269$	10.2247	−	17.7098i	0.623414	−	1.07978i	0.988845	−	0.148946i	$-0.0475881\pi$
$270$		0			0
$271$	15.6969	−	27.1879i	0.953521	−	1.65155i	0.215804	−	0.976437i	$-0.430763\pi$
$271$	15.6969	−	27.1879i	0.953521	−	1.65155i	0.737717	−	0.675110i	$-0.235904\pi$
$272$		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	−0.971707	−	0.236190i	$-0.924101\pi$
$281$	−11.5732	−	20.0454i	−0.690400	−	1.19581i	0.281307	−	0.959618i	$-0.409232\pi$
$282$		0			0
$283$	−7.44949	+	12.9029i	−0.442826	+	0.766997i	−0.997898	−	0.0648050i	$-0.979357\pi$
$283$	−7.44949	+	12.9029i	−0.442826	+	0.766997i	0.555072	+	0.831802i	$0.312691\pi$
$284$		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	$-0.600709\pi$
$293$	−10.6515			−0.622269			−0.311135	+	0.950366i	$0.600709\pi$
$294$		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	0.829955	−	0.557830i	$-0.188366\pi$
$307$	13.3485	+	7.70674i	0.761837	+	0.439847i	−0.0681177	+	0.997677i	$0.521699\pi$
$308$		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	−0.985108	−	0.171934i	$-0.944998\pi$
$313$	−11.3485	−	19.6561i	−0.641453	−	1.11103i	0.343655	−	0.939096i	$-0.388335\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.67423	+	6.36396i	0.206366	+	0.357436i	0.950567	−	0.310520i	$-0.100503\pi$
$317$	3.67423	+	6.36396i	0.206366	+	0.357436i	−0.744201	+	0.667955i	$0.767170\pi$
$318$		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	$-0.828165\pi$
$331$		−	18.7026i		−	1.02799i	0.857794	−	0.513994i	$-0.171835\pi$
$332$		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	0.548305	−	0.836278i	$-0.315273\pi$
$337$	1.80306	+	1.04100i	0.0982190	+	0.0567068i	−0.450086	+	0.892985i	$0.648607\pi$
$338$		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	0.999841	+	0.0178073i	$0.00566852\pi$
$347$	28.2247	+	16.2956i	1.51518	+	0.874792i	0.515342	+	0.856984i	$0.327665\pi$
$348$		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	$-0.429839\pi$
$359$	−9.79796	−	5.65685i	−0.517116	−	0.298557i	−0.735754	+	0.677249i	$0.763172\pi$
$360$		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	0.615759	−	0.787935i	$-0.288850\pi$
$367$	−7.17423	−	12.4261i	−0.374492	−	0.648639i	−0.990251	+	0.139295i	$0.955516\pi$
$368$		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	$0.103151\pi$
$373$			12.2993i			0.636835i	−0.947951	+	0.318418i	$0.896849\pi$
$374$		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	$0.162759\pi$
$379$			19.0526i			0.978664i	−0.872098	+	0.489332i	$0.837241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.44949	−	9.43879i	0.278456	−	0.482300i	−0.692545	−	0.721374i	$-0.743511\pi$
$383$	5.44949	−	9.43879i	0.278456	−	0.482300i	0.971001	+	0.239075i	$0.0768441\pi$
$384$		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	$-0.432824\pi$
$389$	22.8990	−	13.2207i	1.16102	−	0.670318i	0.951549	+	0.307496i	$0.0994912\pi$
$390$		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	0.455380	+	0.890297i	$0.349503\pi$
$397$	−10.8258	+	18.7508i	−0.543330	+	0.941074i	−0.998710	+	0.0507775i	$0.983830\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.123724	−	0.214297i	0.00617850	−	0.0107015i	−0.862920	−	0.505341i	$-0.831367\pi$
$401$	0.123724	−	0.214297i	0.00617850	−	0.0107015i	0.869098	+	0.494640i	$0.164700\pi$
$402$		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	−0.968770	−	0.247960i	$-0.920240\pi$
$409$	−25.0454	+	14.4600i	−1.23842	+	0.715000i	−0.269645	+	0.962960i	$0.586906\pi$
$410$		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	0.532500	−	0.846430i	$-0.321253\pi$
$421$	1.34847	+	0.778539i	0.0657204	+	0.0379437i	−0.466780	+	0.884374i	$0.654586\pi$
$422$		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	$-0.0403713\pi$
$431$	8.02270	+	13.8957i	0.386440	+	0.669334i	−0.605528	+	0.795824i	$0.707038\pi$
$432$		0			0
$433$	25.1969	−	14.5475i	1.21089	−	0.699106i	0.247935	−	0.968777i	$-0.420248\pi$
$433$	25.1969	−	14.5475i	1.21089	−	0.699106i	0.962953	+	0.269670i	$0.0869148\pi$
$434$		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	0.768127	−	0.640298i	$-0.221189\pi$
$439$	12.5227	+	7.22999i	0.597676	+	0.345068i	−0.170451	+	0.985366i	$0.554522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.7753	−	14.8814i	1.22462	−	0.707034i	0.258720	−	0.965952i	$-0.416699\pi$
$443$	25.7753	−	14.8814i	1.22462	−	0.707034i	0.965899	+	0.258918i	$0.0833660\pi$
$444$		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	$-0.641689\pi$
$449$	−18.2474			−0.861150			−0.430575	+	0.902555i	$0.641689\pi$
$450$		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	$-0.254435\pi$
$461$	9.12372	+	5.26758i	0.424934	+	0.245336i	−0.272252	+	0.962226i	$0.587768\pi$
$462$		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	$-0.597724\pi$
$479$	8.14643	−	4.70334i	0.372220	−	0.214901i	0.674428	+	0.738341i	$0.264390\pi$
$480$		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	$0.355439\pi$
$487$			39.6622i			1.79727i	−0.438702	+	0.898633i	$0.644561\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−26.1464	−	15.0956i	−1.17997	−	0.681257i	−0.223964	−	0.974597i	$-0.571900\pi$
$491$	−26.1464	−	15.0956i	−1.17997	−	0.681257i	−0.956008	+	0.293340i	$0.905233\pi$
$492$		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	−0.288909	+	0.957357i	$0.593292\pi$
$499$	−21.7474	+	37.6677i	−0.973550	+	1.68624i	−0.684641	+	0.728881i	$0.740041\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	35.5176	−	20.5061i	1.58365	−	0.914322i	0.589331	−	0.807891i	$-0.299391\pi$
$503$	35.5176	−	20.5061i	1.58365	−	0.914322i	0.994320	−	0.106430i	$-0.0339421\pi$
$504$		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	$-0.262263\pi$
$509$	−6.67423	−	11.5601i	−0.295830	−	0.512393i	−0.975178	+	0.221424i	$0.928930\pi$
$510$		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	$-0.557503\pi$
$521$	−8.20204			−0.359338			−0.179669	+	0.983727i	$0.557503\pi$
$522$		0			0
$523$	−13.5000	+	7.79423i	−0.590314	+	0.340818i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−13.5000	+	7.79423i	−0.590314	+	0.340818i	0.174908	+	0.984585i	$0.444037\pi$
$524$		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	−0.982541	−	0.186046i	$-0.940433\pi$
$541$	−15.1742	−	26.2825i	−0.652391	−	1.12997i	0.330150	−	0.943929i	$-0.392901\pi$
$542$		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	−0.460958	−	0.887422i	$-0.652494\pi$
$547$	1.80306	−	1.04100i	0.0770933	−	0.0445099i	0.538051	+	0.842912i	$0.319161\pi$
$548$		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	$-0.480239\pi$
$557$	22.5959	−	13.0458i	0.957420	−	0.552767i	0.895378	+	0.445307i	$0.146905\pi$
$558$		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	$-0.318663\pi$
$563$	25.5959			1.07874			0.539370	+	0.842069i	$0.318663\pi$
$564$		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	$-0.273967\pi$
$569$	31.1010			1.30382			0.651911	+	0.758295i	$0.273967\pi$
$570$		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	$-0.528278\pi$
$587$	−24.1237	−	13.9278i	−0.995693	−	0.574863i	−0.906971	+	0.421193i	$0.861611\pi$
$588$		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	$-0.770152\pi$
$593$	−13.4722	+	7.77817i	−0.553237	+	0.319411i	0.197190	+	0.980365i	$0.436818\pi$
$594$		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	−0.938611	−	0.344976i	$-0.887887\pi$
$599$	−18.7980	−	32.5590i	−0.768064	−	1.33033i	0.170548	−	0.985349i	$-0.445446\pi$
$600$		0			0
$601$			10.2173i			0.416774i	0.978046	+	0.208387i	$0.0668213\pi$
$601$			10.2173i			0.416774i	−0.978046	+	0.208387i	$0.933179\pi$
$602$		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	$-0.986038\pi$
$607$		−	2.16064i		−	0.0876979i	0.999038	−	0.0438489i	$-0.0139620\pi$
$608$		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	−0.991300	−	0.131623i	$-0.957981\pi$
$613$	−9.44949	+	16.3670i	−0.381661	+	0.661057i	0.609639	+	0.792680i	$0.291315\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.1010	+	16.2241i	−1.13130	+	0.653159i	−0.944263	−	0.329193i	$-0.893223\pi$
$617$	−28.1010	+	16.2241i	−1.13130	+	0.653159i	−0.187042	+	0.982352i	$0.559890\pi$
$618$		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	0.973720	−	0.227749i	$-0.0731366\pi$
$631$	7.27526	+	12.6011i	0.289623	+	0.501642i	−0.684096	+	0.729392i	$0.739803\pi$
$632$		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	0.0943619	−	0.995538i	$-0.469919\pi$
$641$	23.0227	−	39.8765i	0.909342	−	1.57503i	0.814980	−	0.579489i	$-0.196748\pi$
$642$		0			0
$643$	21.2980	−	36.8891i	0.839910	−	1.45477i	−0.0500601	−	0.998746i	$-0.515941\pi$
$643$	21.2980	−	36.8891i	0.839910	−	1.45477i	0.889970	−	0.456020i	$-0.150725\pi$
$644$		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	$-0.237899\pi$
$659$	−5.69694	−	9.86739i	−0.221921	−	0.384379i	−0.955391	+	0.295343i	$0.904566\pi$
$660$		0			0
$661$	−23.6969	+	13.6814i	−0.921704	+	0.532146i	−0.884178	−	0.467150i	$-0.845281\pi$
$661$	−23.6969	+	13.6814i	−0.921704	+	0.532146i	−0.0375258	+	0.999296i	$0.511948\pi$
$662$		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	$-0.880313\pi$
$673$		−	19.0526i		−	0.734422i	0.930138	−	0.367211i	$-0.119687\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	37.3485			1.43542			0.717709	−	0.696343i	$-0.245191\pi$
$677$	37.3485			1.43542			0.717709	+	0.696343i	$0.245191\pi$
$678$		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	$-0.319722\pi$
$683$	28.0454			1.07313			0.536564	+	0.843860i	$0.319722\pi$
$684$		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	$-0.238421\pi$
$701$	13.4722	+	7.77817i	0.508838	+	0.293778i	−0.223518	+	0.974700i	$0.571754\pi$
$702$		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	0.800244	−	0.599674i	$-0.204703\pi$
$709$	−3.17423	−	5.49794i	−0.119211	−	0.206479i	−0.919455	+	0.393195i	$0.871370\pi$
$710$		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	$-0.453698\pi$
$719$	−17.1464	−	9.89949i	−0.639454	−	0.369189i	−0.784404	+	0.620250i	$0.787031\pi$
$720$		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	−0.762553	−	0.646926i	$-0.776054\pi$
$727$	4.82577	−	8.35847i	0.178978	−	0.309999i	0.941531	+	0.336927i	$0.109388\pi$
$728$		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	−0.0248879	−	0.999690i	$-0.507923\pi$
$739$	23.1969	−	40.1783i	0.853313	−	1.47798i	0.878201	−	0.478292i	$-0.158744\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	23.6969	−	41.0443i	0.869356	−	1.50577i	0.00670079	−	0.999978i	$-0.497867\pi$
$743$	23.6969	−	41.0443i	0.869356	−	1.50577i	0.862656	−	0.505792i	$-0.168800\pi$
$744$		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	−0.871905	−	0.489675i	$-0.837116\pi$
$751$	−24.2196	+	13.9832i	−0.883787	+	0.510255i	−0.0118820	+	0.999929i	$0.503782\pi$
$752$		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	−0.997873	−	0.0651902i	$-0.979235\pi$
$757$	−12.1742	+	21.0864i	−0.442480	+	0.766398i	0.555393	+	0.831588i	$0.312568\pi$
$758$		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	0.888479	−	0.458917i	$-0.151763\pi$
$769$	1.29796	+	2.24813i	0.0468056	+	0.0810697i	−0.841673	+	0.539987i	$0.818429\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.47219	+	7.74607i	0.160854	+	0.278607i	0.935175	−	0.354186i	$-0.115242\pi$
$773$	4.47219	+	7.74607i	0.160854	+	0.278607i	−0.774321	+	0.632792i	$0.781909\pi$
$774$		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	$0.141158\pi$
$787$			24.0737i			0.858136i	−0.903272	+	0.429068i	$0.858842\pi$
$788$		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	$-0.264306\pi$
$797$	38.0908			1.34925			0.674623	+	0.738162i	$0.264306\pi$
$798$		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	−0.777423	−	0.628978i	$-0.783474\pi$
$811$	−17.6969	+	10.2173i	−0.621424	+	0.358779i	0.155999	+	0.987757i	$0.450140\pi$
$812$		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	$-0.908396\pi$
$821$	−34.1691	+	19.7276i	−1.19251	+	0.688497i	−0.233636	+	0.972324i	$0.575062\pi$
$822$		0			0
$823$	17.3485	+	30.0484i	0.604730	+	1.04742i	0.992094	+	0.125496i	$0.0400522\pi$
$823$	17.3485	+	30.0484i	0.604730	+	1.04742i	−0.387365	+	0.921927i	$0.626615\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	23.8207	+	41.2586i	0.828326	+	1.43470i	0.899351	+	0.437228i	$0.144040\pi$
$827$	23.8207	+	41.2586i	0.828326	+	1.43470i	−0.0710253	+	0.997475i	$0.522627\pi$
$828$		0			0
$829$			10.9959i			0.381902i	0.981600	+	0.190951i	$0.0611572\pi$
$829$			10.9959i			0.381902i	−0.981600	+	0.190951i	$0.938843\pi$
$830$		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	$-0.777857\pi$
$839$	5.02270	−	8.69958i	0.173403	−	0.300343i	0.939607	+	0.342254i	$0.111190\pi$
$840$		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	−0.492337	−	0.870405i	$-0.663857\pi$
$853$	14.8258	−	25.6790i	0.507625	−	0.879231i	0.999961	−	0.00882655i	$-0.00280961\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.42679	−	5.93537i	0.117057	−	0.202748i	−0.801543	−	0.597937i	$-0.795987\pi$
$857$	3.42679	−	5.93537i	0.117057	−	0.202748i	0.918600	+	0.395188i	$0.129321\pi$
$858$		0			0
$859$	1.94949	+	3.37662i	0.0665157	+	0.115209i	0.897365	−	0.441288i	$-0.145478\pi$
$859$	1.94949	+	3.37662i	0.0665157	+	0.115209i	−0.830850	+	0.556497i	$0.812145\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37.3485			−1.27136			−0.635678	−	0.771954i	$-0.719280\pi$
$863$	−37.3485			−1.27136			−0.635678	+	0.771954i	$0.719280\pi$
$864$		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	0.853192	−	0.521596i	$-0.174663\pi$
$877$	24.5227	+	14.1582i	0.828073	+	0.478088i	−0.0251195	+	0.999684i	$0.507997\pi$
$878$		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	0.812120	−	0.583491i	$-0.198314\pi$
$883$	−2.94949	−	5.10867i	−0.0992582	−	0.171920i	−0.911378	+	0.411571i	$0.864980\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.3258	+	19.6168i	0.380282	+	0.658668i	0.991102	−	0.133101i	$-0.0424936\pi$
$887$	11.3258	+	19.6168i	0.380282	+	0.658668i	−0.610820	+	0.791769i	$0.709160\pi$
$888$		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	−0.228063	−	0.973646i	$-0.573239\pi$
$907$	−35.6969	−	20.6096i	−1.18530	−	0.684332i	−0.957234	+	0.289315i	$0.906573\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		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	$-0.568688\pi$
$929$	−35.5732	−	20.5382i	−1.16712	−	0.673837i	−0.953000	+	0.302971i	$0.902022\pi$
$930$		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	0.700507	−	0.713646i	$-0.252957\pi$
$937$	−8.19694	−	14.1975i	−0.267782	−	0.463813i	−0.968289	+	0.249833i	$0.919624\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.4495	+	35.4196i	0.666634	+	1.15464i	0.978839	+	0.204630i	$0.0655990\pi$
$941$	20.4495	+	35.4196i	0.666634	+	1.15464i	−0.312205	+	0.950015i	$0.601068\pi$
$942$		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	$-0.549627\pi$
$947$	−33.4949	−	19.3383i	−1.08844	−	0.628410i	−0.933160	+	0.359461i	$0.882961\pi$
$948$		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	$-0.839010\pi$
$953$	−0.550510	+	0.953512i	−0.0178328	+	0.0308873i	0.856971	+	0.515364i	$0.172343\pi$
$954$		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	−0.749508	−	0.661995i	$-0.769710\pi$
$967$	6.17423	−	10.6941i	0.198550	−	0.343899i	0.948058	+	0.318096i	$0.103043\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.8207	+	25.6701i	−0.475618	+	0.823794i	−0.999610	−	0.0279290i	$-0.991109\pi$
$971$	−14.8207	+	25.6701i	−0.475618	+	0.823794i	0.523992	+	0.851723i	$0.324442\pi$
$972$		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	$-0.501260\pi$
$977$	−0.247449			−0.00791659			−0.00395829	+	0.999992i	$0.501260\pi$
$978$		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	$-0.738031\pi$
$983$	9.24745	−	16.0171i	0.294948	−	0.510865i	0.974973	+	0.222324i	$0.0713644\pi$
$984$		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	−0.961974	−	0.273141i	$-0.911937\pi$
$991$	−52.8712	−	30.5252i	−1.67951	−	0.969664i	−0.717534	−	0.696523i	$-0.754729\pi$
$992$		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	0.983132	+	0.182898i	$0.0585479\pi$
$997$	20.5227	+	35.5464i	0.649961	+	1.12576i	−0.333171	+	0.942866i	$0.608119\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.s.a.107.1	✓	4	228.107	odd	6
342.2.s.a.179.1	yes	4	4.3	odd	2
342.2.s.b.107.2	yes	4	76.31	even	6
342.2.s.b.179.2	yes	4	12.11	even	2
2736.2.dc.a.449.2		4	19.12	odd	6
2736.2.dc.a.1889.2		4	3.2	odd	2
2736.2.dc.b.449.1		4	57.50	even	6	inner
2736.2.dc.b.1889.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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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