LMFDB - Embedded newform 2112.2.t.b.1585.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(529,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2112.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.t (of order $4$, 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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 528)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1585.5
Character	$\chi$	$=$	2112.1585
Dual form			2112.2.t.b.529.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(0.582740 + 0.582740i) q^{5} -4.89753i q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{3} +(0.582740 + 0.582740i) q^{5} -4.89753i q^{7} -1.00000i q^{9} +(0.707107 + 0.707107i) q^{11} +(1.72543 - 1.72543i) q^{13} -0.824119 q^{15} -2.20698 q^{17} +(4.64217 - 4.64217i) q^{19} +(3.46307 + 3.46307i) q^{21} +8.23649i q^{23} -4.32083i q^{25} +(0.707107 + 0.707107i) q^{27} +(-4.49302 + 4.49302i) q^{29} -3.82276 q^{31} -1.00000 q^{33} +(2.85399 - 2.85399i) q^{35} +(-4.65242 - 4.65242i) q^{37} +2.44013i q^{39} -10.2041i q^{41} +(1.22055 + 1.22055i) q^{43} +(0.582740 - 0.582740i) q^{45} -9.80364 q^{47} -16.9858 q^{49} +(1.56057 - 1.56057i) q^{51} +(-1.14015 - 1.14015i) q^{53} +0.824119i q^{55} +6.56503i q^{57} +(-2.17677 - 2.17677i) q^{59} +(2.35153 - 2.35153i) q^{61} -4.89753 q^{63} +2.01096 q^{65} +(8.69638 - 8.69638i) q^{67} +(-5.82408 - 5.82408i) q^{69} -9.44670i q^{71} +14.2278i q^{73} +(3.05529 + 3.05529i) q^{75} +(3.46307 - 3.46307i) q^{77} +1.80715 q^{79} -1.00000 q^{81} +(3.24879 - 3.24879i) q^{83} +(-1.28609 - 1.28609i) q^{85} -6.35409i q^{87} +5.30085i q^{89} +(-8.45035 - 8.45035i) q^{91} +(2.70310 - 2.70310i) q^{93} +5.41036 q^{95} +7.23833 q^{97} +(0.707107 - 0.707107i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 8 q^{15} - 8 q^{19} - 8 q^{31} - 40 q^{33} + 24 q^{35} + 64 q^{47} - 40 q^{49} + 8 q^{51} + 16 q^{59} - 64 q^{61} - 24 q^{63} + 16 q^{65} - 24 q^{67} - 16 q^{69} + 72 q^{79} - 40 q^{81} + 16 q^{85} - 16 q^{91} + 32 q^{93} - 128 q^{95} - 32 q^{97}+O(q^{100}) $ 40 * q + 8 * q^15 - 8 * q^19 - 8 * q^31 - 40 * q^33 + 24 * q^35 + 64 * q^47 - 40 * q^49 + 8 * q^51 + 16 * q^59 - 64 * q^61 - 24 * q^63 + 16 * q^65 - 24 * q^67 - 16 * q^69 + 72 * q^79 - 40 * q^81 + 16 * q^85 - 16 * q^91 + 32 * q^93 - 128 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$	0.582740	+	0.582740i	0.260609	+	0.260609i	0.825302	−	0.564692i	$-0.191005\pi$
$5$	0.582740	+	0.582740i	0.260609	+	0.260609i	−0.564692	+	0.825302i	$0.691005\pi$
$6$		0			0
$7$		−	4.89753i		−	1.85109i	−0.378637	−	0.925545i	$-0.623607\pi$
$7$		−	4.89753i		−	1.85109i	0.378637	−	0.925545i	$-0.376393\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	0.707107	+	0.707107i	0.213201	+	0.213201i
$11$	0.707107	+	0.707107i	0.213201	+	0.213201i
$12$		0			0
$13$	1.72543	−	1.72543i	0.478549	−	0.478549i	−0.426119	−	0.904667i	$-0.640119\pi$
$13$	1.72543	−	1.72543i	0.478549	−	0.478549i	0.904667	+	0.426119i	$0.140119\pi$
$14$		0			0
$15$	−0.824119			−0.212787
$16$		0			0
$17$	−2.20698			−0.535271			−0.267635	−	0.963520i	$-0.586242\pi$
$17$	−2.20698			−0.535271			−0.267635	+	0.963520i	$0.586242\pi$
$18$		0			0
$19$	4.64217	−	4.64217i	1.06499	−	1.06499i	0.0672519	−	0.997736i	$-0.478577\pi$
$19$	4.64217	−	4.64217i	1.06499	−	1.06499i	0.997736	−	0.0672519i	$-0.0214231\pi$
$20$		0			0
$21$	3.46307	+	3.46307i	0.755705	+	0.755705i
$22$		0			0
$23$			8.23649i			1.71743i	0.512456	+	0.858714i	$0.328736\pi$
$23$			8.23649i			1.71743i	−0.512456	+	0.858714i	$0.671264\pi$
$24$		0			0
$25$		−	4.32083i		−	0.864166i
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$	−4.49302	+	4.49302i	−0.834333	+	0.834333i	−0.988106	−	0.153773i	$-0.950858\pi$
$29$	−4.49302	+	4.49302i	−0.834333	+	0.834333i	0.153773	+	0.988106i	$0.450858\pi$
$30$		0			0
$31$	−3.82276			−0.686588			−0.343294	−	0.939228i	$-0.611543\pi$
$31$	−3.82276			−0.686588			−0.343294	+	0.939228i	$0.611543\pi$
$32$		0			0
$33$	−1.00000			−0.174078
$34$		0			0
$35$	2.85399	−	2.85399i	0.482412	−	0.482412i
$36$		0			0
$37$	−4.65242	−	4.65242i	−0.764853	−	0.764853i	0.212342	−	0.977195i	$-0.431891\pi$
$37$	−4.65242	−	4.65242i	−0.764853	−	0.764853i	−0.977195	+	0.212342i	$0.931891\pi$
$38$		0			0
$39$			2.44013i			0.390733i
$40$		0			0
$41$		−	10.2041i		−	1.59361i	−0.604234	−	0.796807i	$-0.706521\pi$
$41$		−	10.2041i		−	1.59361i	0.604234	−	0.796807i	$-0.293479\pi$
$42$		0			0
$43$	1.22055	+	1.22055i	0.186131	+	0.186131i	0.794021	−	0.607890i	$-0.207984\pi$
$43$	1.22055	+	1.22055i	0.186131	+	0.186131i	−0.607890	+	0.794021i	$0.707984\pi$
$44$		0			0
$45$	0.582740	−	0.582740i	0.0868698	−	0.0868698i
$46$		0			0
$47$	−9.80364			−1.43001			−0.715004	−	0.699120i	$-0.753575\pi$
$47$	−9.80364			−1.43001			−0.715004	+	0.699120i	$0.753575\pi$
$48$		0			0
$49$	−16.9858			−2.42654
$50$		0			0
$51$	1.56057	−	1.56057i	0.218523	−	0.218523i
$52$		0			0
$53$	−1.14015	−	1.14015i	−0.156611	−	0.156611i	0.624452	−	0.781063i	$-0.285322\pi$
$53$	−1.14015	−	1.14015i	−0.156611	−	0.156611i	−0.781063	+	0.624452i	$0.785322\pi$
$54$		0			0
$55$			0.824119i			0.111124i
$56$		0			0
$57$			6.56503i			0.869559i
$58$		0			0
$59$	−2.17677	−	2.17677i	−0.283392	−	0.283392i	0.551068	−	0.834460i	$-0.314220\pi$
$59$	−2.17677	−	2.17677i	−0.283392	−	0.283392i	−0.834460	+	0.551068i	$0.814220\pi$
$60$		0			0
$61$	2.35153	−	2.35153i	0.301082	−	0.301082i	−0.540355	−	0.841437i	$-0.681710\pi$
$61$	2.35153	−	2.35153i	0.301082	−	0.301082i	0.841437	+	0.540355i	$0.181710\pi$
$62$		0			0
$63$	−4.89753			−0.617030
$64$		0			0
$65$	2.01096			0.249429
$66$		0			0
$67$	8.69638	−	8.69638i	1.06243	−	1.06243i	0.0645155	−	0.997917i	$-0.479450\pi$
$67$	8.69638	−	8.69638i	1.06243	−	1.06243i	0.997917	−	0.0645155i	$-0.0205502\pi$
$68$		0			0
$69$	−5.82408	−	5.82408i	−0.701137	−	0.701137i
$70$		0			0
$71$		−	9.44670i		−	1.12112i	−0.828115	−	0.560559i	$-0.810586\pi$
$71$		−	9.44670i		−	1.12112i	0.828115	−	0.560559i	$-0.189414\pi$
$72$		0			0
$73$			14.2278i			1.66524i	0.553843	+	0.832621i	$0.313161\pi$
$73$			14.2278i			1.66524i	−0.553843	+	0.832621i	$0.686839\pi$
$74$		0			0
$75$	3.05529	+	3.05529i	0.352794	+	0.352794i
$76$		0			0
$77$	3.46307	−	3.46307i	0.394654	−	0.394654i
$78$		0			0
$79$	1.80715			0.203320			0.101660	−	0.994819i	$-0.467585\pi$
$79$	1.80715			0.203320			0.101660	+	0.994819i	$0.467585\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	3.24879	−	3.24879i	0.356601	−	0.356601i	−0.505958	−	0.862558i	$-0.668861\pi$
$83$	3.24879	−	3.24879i	0.356601	−	0.356601i	0.862558	+	0.505958i	$0.168861\pi$
$84$		0			0
$85$	−1.28609	−	1.28609i	−0.139497	−	0.139497i
$86$		0			0
$87$		−	6.35409i		−	0.681230i
$88$		0			0
$89$			5.30085i			0.561889i	0.959724	+	0.280945i	$0.0906477\pi$
$89$			5.30085i			0.561889i	−0.959724	+	0.280945i	$0.909352\pi$
$90$		0			0
$91$	−8.45035	−	8.45035i	−0.885837	−	0.885837i
$92$		0			0
$93$	2.70310	−	2.70310i	0.280298	−	0.280298i
$94$		0			0
$95$	5.41036			0.555092
$96$		0			0
$97$	7.23833			0.734942			0.367471	−	0.930035i	$-0.380224\pi$
$97$	7.23833			0.734942			0.367471	+	0.930035i	$0.380224\pi$
$98$		0			0
$99$	0.707107	−	0.707107i	0.0710669	−	0.0710669i
$100$		0			0
$101$	−7.36735	−	7.36735i	−0.733078	−	0.733078i	0.238150	−	0.971228i	$-0.423459\pi$
$101$	−7.36735	−	7.36735i	−0.733078	−	0.733078i	−0.971228	+	0.238150i	$0.923459\pi$
$102$		0			0
$103$		−	7.81962i		−	0.770490i	−0.922814	−	0.385245i	$-0.874117\pi$
$103$		−	7.81962i		−	0.770490i	0.922814	−	0.385245i	$-0.125883\pi$
$104$		0			0
$105$			4.03614i			0.393887i
$106$		0			0
$107$	−1.94604	−	1.94604i	−0.188131	−	0.188131i	0.606757	−	0.794887i	$-0.292470\pi$
$107$	−1.94604	−	1.94604i	−0.188131	−	0.188131i	−0.794887	+	0.606757i	$0.792470\pi$
$108$		0			0
$109$	−0.638617	+	0.638617i	−0.0611684	+	0.0611684i	−0.737029	−	0.675861i	$-0.763772\pi$
$109$	−0.638617	+	0.638617i	−0.0611684	+	0.0611684i	0.675861	+	0.737029i	$0.263772\pi$
$110$		0			0
$111$	6.57952			0.624500
$112$		0			0
$113$	0.698357			0.0656959			0.0328480	−	0.999460i	$-0.489542\pi$
$113$	0.698357			0.0656959			0.0328480	+	0.999460i	$0.489542\pi$
$114$		0			0
$115$	−4.79974	+	4.79974i	−0.447578	+	0.447578i
$116$		0			0
$117$	−1.72543	−	1.72543i	−0.159516	−	0.159516i
$118$		0			0
$119$			10.8087i			0.990835i
$120$		0			0
$121$			1.00000i			0.0909091i
$122$		0			0
$123$	7.21539	+	7.21539i	0.650590	+	0.650590i
$124$		0			0
$125$	5.43162	−	5.43162i	0.485819	−	0.485819i
$126$		0			0
$127$	10.8716			0.964696			0.482348	−	0.875980i	$-0.339784\pi$
$127$	10.8716			0.964696			0.482348	+	0.875980i	$0.339784\pi$
$128$		0			0
$129$	−1.72611			−0.151976
$130$		0			0
$131$	−10.4734	+	10.4734i	−0.915062	+	0.915062i	−0.996665	−	0.0816027i	$-0.973996\pi$
$131$	−10.4734	+	10.4734i	−0.915062	+	0.915062i	0.0816027	+	0.996665i	$0.473996\pi$
$132$		0			0
$133$	−22.7352	−	22.7352i	−1.97139	−	1.97139i
$134$		0			0
$135$			0.824119i			0.0709289i
$136$		0			0
$137$		−	18.9181i		−	1.61628i	−0.588992	−	0.808139i	$-0.700475\pi$
$137$		−	18.9181i		−	1.61628i	0.588992	−	0.808139i	$-0.299525\pi$
$138$		0			0
$139$	−3.03763	−	3.03763i	−0.257649	−	0.257649i	0.566449	−	0.824097i	$-0.308317\pi$
$139$	−3.03763	−	3.03763i	−0.257649	−	0.257649i	−0.824097	+	0.566449i	$0.808317\pi$
$140$		0			0
$141$	6.93222	−	6.93222i	0.583799	−	0.583799i
$142$		0			0
$143$	2.44013			0.204054
$144$		0			0
$145$	−5.23653			−0.434870
$146$		0			0
$147$	12.0107	−	12.0107i	0.990630	−	0.990630i
$148$		0			0
$149$	11.8476	+	11.8476i	0.970592	+	0.970592i	0.999580	−	0.0289874i	$-0.00922828\pi$
$149$	11.8476	+	11.8476i	0.970592	+	0.970592i	−0.0289874	+	0.999580i	$0.509228\pi$
$150$		0			0
$151$		−	14.6279i		−	1.19040i	−0.803578	−	0.595200i	$-0.797073\pi$
$151$		−	14.6279i		−	1.19040i	0.803578	−	0.595200i	$-0.202927\pi$
$152$		0			0
$153$			2.20698i			0.178424i
$154$		0			0
$155$	−2.22768	−	2.22768i	−0.178931	−	0.178931i
$156$		0			0
$157$	0.217412	−	0.217412i	0.0173514	−	0.0173514i	−0.698378	−	0.715729i	$-0.746094\pi$
$157$	0.217412	−	0.217412i	0.0173514	−	0.0173514i	0.715729	+	0.698378i	$0.246094\pi$
$158$		0			0
$159$	1.61241			0.127873
$160$		0			0
$161$	40.3384			3.17911
$162$		0			0
$163$	17.0683	−	17.0683i	1.33689	−	1.33689i	0.437841	−	0.899053i	$-0.355743\pi$
$163$	17.0683	−	17.0683i	1.33689	−	1.33689i	0.899053	−	0.437841i	$-0.144257\pi$
$164$		0			0
$165$	−0.582740	−	0.582740i	−0.0453663	−	0.0453663i
$166$		0			0
$167$			4.03060i			0.311897i	0.987765	+	0.155948i	$0.0498434\pi$
$167$			4.03060i			0.311897i	−0.987765	+	0.155948i	$0.950157\pi$
$168$		0			0
$169$			7.04577i			0.541982i
$170$		0			0
$171$	−4.64217	−	4.64217i	−0.354996	−	0.354996i
$172$		0			0
$173$	6.86259	−	6.86259i	0.521753	−	0.521753i	−0.396347	−	0.918101i	$-0.629722\pi$
$173$	6.86259	−	6.86259i	0.521753	−	0.521753i	0.918101	+	0.396347i	$0.129722\pi$
$174$		0			0
$175$	−21.1614			−1.59965
$176$		0			0
$177$	3.07842			0.231388
$178$		0			0
$179$	4.44665	−	4.44665i	0.332358	−	0.332358i	−0.521123	−	0.853481i	$-0.674487\pi$
$179$	4.44665	−	4.44665i	0.332358	−	0.332358i	0.853481	+	0.521123i	$0.174487\pi$
$180$		0			0
$181$	−12.2861	−	12.2861i	−0.913218	−	0.913218i	0.0833059	−	0.996524i	$-0.473452\pi$
$181$	−12.2861	−	12.2861i	−0.913218	−	0.913218i	−0.996524	+	0.0833059i	$0.973452\pi$
$182$		0			0
$183$			3.32556i			0.245833i
$184$		0			0
$185$		−	5.42231i		−	0.398656i
$186$		0			0
$187$	−1.56057	−	1.56057i	−0.114120	−	0.114120i
$188$		0			0
$189$	3.46307	−	3.46307i	0.251902	−	0.251902i
$190$		0			0
$191$	−21.9781			−1.59028			−0.795140	−	0.606426i	$-0.792603\pi$
$191$	−21.9781			−1.59028			−0.795140	+	0.606426i	$0.792603\pi$
$192$		0			0
$193$	7.33803			0.528203			0.264102	−	0.964495i	$-0.414925\pi$
$193$	7.33803			0.528203			0.264102	+	0.964495i	$0.414925\pi$
$194$		0			0
$195$	−1.42196	+	1.42196i	−0.101829	+	0.101829i
$196$		0			0
$197$	−5.33250	−	5.33250i	−0.379925	−	0.379925i	0.491150	−	0.871075i	$-0.336577\pi$
$197$	−5.33250	−	5.33250i	−0.379925	−	0.379925i	−0.871075	+	0.491150i	$0.836577\pi$
$198$		0			0
$199$			21.9645i			1.55702i	0.627630	+	0.778511i	$0.284025\pi$
$199$			21.9645i			1.55702i	−0.627630	+	0.778511i	$0.715975\pi$
$200$		0			0
$201$			12.2985i			0.867472i
$202$		0			0
$203$	22.0047	+	22.0047i	1.54443	+	1.54443i
$204$		0			0
$205$	5.94634	−	5.94634i	0.415311	−	0.415311i
$206$		0			0
$207$	8.23649			0.572476
$208$		0			0
$209$	6.56503			0.454112
$210$		0			0
$211$	−8.27179	+	8.27179i	−0.569453	+	0.569453i	−0.931975	−	0.362522i	$-0.881916\pi$
$211$	−8.27179	+	8.27179i	−0.569453	+	0.569453i	0.362522	+	0.931975i	$0.381916\pi$
$212$		0			0
$213$	6.67983	+	6.67983i	0.457694	+	0.457694i
$214$		0			0
$215$			1.42252i			0.0970152i
$216$		0			0
$217$			18.7221i			1.27094i
$218$		0			0
$219$	−10.0606	−	10.0606i	−0.679832	−	0.679832i
$220$		0			0
$221$	−3.80799	+	3.80799i	−0.256153	+	0.256153i
$222$		0			0
$223$	−24.9613			−1.67153			−0.835767	−	0.549084i	$-0.814977\pi$
$223$	−24.9613			−1.67153			−0.835767	+	0.549084i	$0.814977\pi$
$224$		0			0
$225$	−4.32083			−0.288055
$226$		0			0
$227$	15.9235	−	15.9235i	1.05688	−	1.05688i	0.0585975	−	0.998282i	$-0.481337\pi$
$227$	15.9235	−	15.9235i	1.05688	−	1.05688i	0.998282	−	0.0585975i	$-0.0186629\pi$
$228$		0			0
$229$	11.5279	+	11.5279i	0.761788	+	0.761788i	0.976645	−	0.214857i	$-0.0689286\pi$
$229$	11.5279	+	11.5279i	0.761788	+	0.761788i	−0.214857	+	0.976645i	$0.568929\pi$
$230$		0			0
$231$			4.89753i			0.322234i
$232$		0			0
$233$		−	5.87804i		−	0.385083i	−0.981289	−	0.192542i	$-0.938327\pi$
$233$		−	5.87804i		−	0.385083i	0.981289	−	0.192542i	$-0.0616731\pi$
$234$		0			0
$235$	−5.71298	−	5.71298i	−0.372674	−	0.372674i
$236$		0			0
$237$	−1.27785	+	1.27785i	−0.0830052	+	0.0830052i
$238$		0			0
$239$	0.0664988			0.00430145			0.00215072	−	0.999998i	$-0.499315\pi$
$239$	0.0664988			0.00430145			0.00215072	+	0.999998i	$0.499315\pi$
$240$		0			0
$241$	−23.1161			−1.48904			−0.744520	−	0.667600i	$-0.767322\pi$
$241$	−23.1161			−1.48904			−0.744520	+	0.667600i	$0.767322\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$	−9.89829	−	9.89829i	−0.632378	−	0.632378i
$246$		0			0
$247$		−	16.0195i		−	1.01930i
$248$		0			0
$249$			4.59448i			0.291163i
$250$		0			0
$251$	7.72118	+	7.72118i	0.487356	+	0.487356i	0.907471	−	0.420115i	$-0.138010\pi$
$251$	7.72118	+	7.72118i	0.487356	+	0.487356i	−0.420115	+	0.907471i	$0.638010\pi$
$252$		0			0
$253$	−5.82408	+	5.82408i	−0.366157	+	0.366157i
$254$		0			0
$255$	1.81881			0.113898
$256$		0			0
$257$	−1.47156			−0.0917932			−0.0458966	−	0.998946i	$-0.514614\pi$
$257$	−1.47156			−0.0917932			−0.0458966	+	0.998946i	$0.514614\pi$
$258$		0			0
$259$	−22.7854	+	22.7854i	−1.41581	+	1.41581i
$260$		0			0
$261$	4.49302	+	4.49302i	0.278111	+	0.278111i
$262$		0			0
$263$		−	19.5129i		−	1.20322i	−0.798790	−	0.601610i	$-0.794526\pi$
$263$		−	19.5129i		−	1.20322i	0.798790	−	0.601610i	$-0.205474\pi$
$264$		0			0
$265$		−	1.32882i		−	0.0816287i
$266$		0			0
$267$	−3.74827	−	3.74827i	−0.229390	−	0.229390i
$268$		0			0
$269$	5.88616	−	5.88616i	0.358886	−	0.358886i	−0.504517	−	0.863402i	$-0.668329\pi$
$269$	5.88616	−	5.88616i	0.358886	−	0.358886i	0.863402	+	0.504517i	$0.168329\pi$
$270$		0			0
$271$	−13.3108			−0.808571			−0.404286	−	0.914633i	$-0.632480\pi$
$271$	−13.3108			−0.808571			−0.404286	+	0.914633i	$0.632480\pi$
$272$		0			0
$273$	11.9506			0.723283
$274$		0			0
$275$	3.05529	−	3.05529i	0.184241	−	0.184241i
$276$		0			0
$277$	−0.343580	−	0.343580i	−0.0206438	−	0.0206438i	0.696710	−	0.717353i	$-0.254647\pi$
$277$	−0.343580	−	0.343580i	−0.0206438	−	0.0206438i	−0.717353	+	0.696710i	$0.754647\pi$
$278$		0			0
$279$			3.82276i			0.228863i
$280$		0			0
$281$			0.183598i			0.0109525i	0.999985	+	0.00547627i	$0.00174316\pi$
$281$			0.183598i			0.0109525i	−0.999985	+	0.00547627i	$0.998257\pi$
$282$		0			0
$283$	−12.3011	−	12.3011i	−0.731225	−	0.731225i	0.239637	−	0.970863i	$-0.422971\pi$
$283$	−12.3011	−	12.3011i	−0.731225	−	0.731225i	−0.970863	+	0.239637i	$0.922971\pi$
$284$		0			0
$285$	−3.82571	+	3.82571i	−0.226615	+	0.226615i
$286$		0			0
$287$	−49.9749			−2.94992
$288$		0			0
$289$	−12.1293			−0.713485
$290$		0			0
$291$	−5.11828	+	5.11828i	−0.300039	+	0.300039i
$292$		0			0
$293$	2.19204	+	2.19204i	0.128060	+	0.128060i	0.768232	−	0.640172i	$-0.221137\pi$
$293$	2.19204	+	2.19204i	0.128060	+	0.128060i	−0.640172	+	0.768232i	$0.721137\pi$
$294$		0			0
$295$		−	2.53699i		−	0.147709i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	14.2115	+	14.2115i	0.821873	+	0.821873i
$300$		0			0
$301$	5.97765	−	5.97765i	0.344546	−	0.344546i
$302$		0			0
$303$	10.4190			0.598556
$304$		0			0
$305$	2.74066			0.156930
$306$		0			0
$307$	−4.73919	+	4.73919i	−0.270480	+	0.270480i	−0.829293	−	0.558814i	$-0.811257\pi$
$307$	−4.73919	+	4.73919i	−0.270480	+	0.270480i	0.558814	+	0.829293i	$0.311257\pi$
$308$		0			0
$309$	5.52930	+	5.52930i	0.314551	+	0.314551i
$310$		0			0
$311$		−	1.86379i		−	0.105686i	−0.998603	−	0.0528431i	$-0.983172\pi$
$311$		−	1.86379i		−	0.105686i	0.998603	−	0.0528431i	$-0.0168283\pi$
$312$		0			0
$313$		−	2.88059i		−	0.162821i	−0.996681	−	0.0814103i	$-0.974058\pi$
$313$		−	2.88059i		−	0.162821i	0.996681	−	0.0814103i	$-0.0259424\pi$
$314$		0			0
$315$	−2.85399	−	2.85399i	−0.160804	−	0.160804i
$316$		0			0
$317$	−10.6057	+	10.6057i	−0.595673	+	0.595673i	−0.939158	−	0.343485i	$-0.888393\pi$
$317$	−10.6057	+	10.6057i	−0.595673	+	0.595673i	0.343485	+	0.939158i	$0.388393\pi$
$318$		0			0
$319$	−6.35409			−0.355761
$320$		0			0
$321$	2.75211			0.153608
$322$		0			0
$323$	−10.2452	+	10.2452i	−0.570057	+	0.570057i
$324$		0			0
$325$	−7.45530	−	7.45530i	−0.413545	−	0.413545i
$326$		0			0
$327$		−	0.903141i		−	0.0499438i
$328$		0			0
$329$			48.0136i			2.64708i
$330$		0			0
$331$	9.01783	+	9.01783i	0.495665	+	0.495665i	0.910085	−	0.414421i	$-0.136016\pi$
$331$	9.01783	+	9.01783i	0.495665	+	0.495665i	−0.414421	+	0.910085i	$0.636016\pi$
$332$		0			0
$333$	−4.65242	+	4.65242i	−0.254951	+	0.254951i
$334$		0			0
$335$	10.1355			0.553760
$336$		0			0
$337$	19.9937			1.08913			0.544563	−	0.838720i	$-0.316695\pi$
$337$	19.9937			1.08913			0.544563	+	0.838720i	$0.316695\pi$
$338$		0			0
$339$	−0.493813	+	0.493813i	−0.0268202	+	0.0268202i
$340$		0			0
$341$	−2.70310	−	2.70310i	−0.146381	−	0.146381i
$342$		0			0
$343$			48.9055i			2.64065i
$344$		0			0
$345$		−	6.78785i		−	0.365446i
$346$		0			0
$347$	14.4537	+	14.4537i	0.775916	+	0.775916i	0.979134	−	0.203217i	$-0.0651398\pi$
$347$	14.4537	+	14.4537i	0.775916	+	0.775916i	−0.203217	+	0.979134i	$0.565140\pi$
$348$		0			0
$349$	20.2751	−	20.2751i	1.08530	−	1.08530i	0.0892971	−	0.996005i	$-0.471538\pi$
$349$	20.2751	−	20.2751i	1.08530	−	1.08530i	0.996005	−	0.0892971i	$-0.0284621\pi$
$350$		0			0
$351$	2.44013			0.130244
$352$		0			0
$353$	10.2789			0.547090			0.273545	−	0.961859i	$-0.411804\pi$
$353$	10.2789			0.547090			0.273545	+	0.961859i	$0.411804\pi$
$354$		0			0
$355$	5.50497	−	5.50497i	0.292174	−	0.292174i
$356$		0			0
$357$	−7.64293	−	7.64293i	−0.404507	−	0.404507i
$358$		0			0
$359$			11.6208i			0.613324i	0.951818	+	0.306662i	$0.0992121\pi$
$359$			11.6208i			0.613324i	−0.951818	+	0.306662i	$0.900788\pi$
$360$		0			0
$361$		−	24.0996i		−	1.26840i
$362$		0			0
$363$	−0.707107	−	0.707107i	−0.0371135	−	0.0371135i
$364$		0			0
$365$	−8.29113	+	8.29113i	−0.433978	+	0.433978i
$366$		0			0
$367$	0.813135			0.0424453			0.0212226	−	0.999775i	$-0.493244\pi$
$367$	0.813135			0.0424453			0.0212226	+	0.999775i	$0.493244\pi$
$368$		0			0
$369$	−10.2041			−0.531205
$370$		0			0
$371$	−5.58390	+	5.58390i	−0.289902	+	0.289902i
$372$		0			0
$373$	22.8768	+	22.8768i	1.18451	+	1.18451i	0.978562	+	0.205953i	$0.0660294\pi$
$373$	22.8768	+	22.8768i	1.18451	+	1.18451i	0.205953	+	0.978562i	$0.433971\pi$
$374$		0			0
$375$			7.68147i			0.396670i
$376$		0			0
$377$			15.5048i			0.798538i
$378$		0			0
$379$	3.21310	+	3.21310i	0.165046	+	0.165046i	0.784798	−	0.619752i	$-0.212767\pi$
$379$	3.21310	+	3.21310i	0.165046	+	0.165046i	−0.619752	+	0.784798i	$0.712767\pi$
$380$		0			0
$381$	−7.68736	+	7.68736i	−0.393835	+	0.393835i
$382$		0			0
$383$	−9.08904			−0.464428			−0.232214	−	0.972665i	$-0.574597\pi$
$383$	−9.08904			−0.464428			−0.232214	+	0.972665i	$0.574597\pi$
$384$		0			0
$385$	4.03614			0.205701
$386$		0			0
$387$	1.22055	−	1.22055i	0.0620438	−	0.0620438i
$388$		0			0
$389$	8.32569	+	8.32569i	0.422129	+	0.422129i	0.885936	−	0.463807i	$-0.153517\pi$
$389$	8.32569	+	8.32569i	0.422129	+	0.422129i	−0.463807	+	0.885936i	$0.653517\pi$
$390$		0			0
$391$		−	18.1778i		−	0.919288i
$392$		0			0
$393$		−	14.8116i		−	0.747145i
$394$		0			0
$395$	1.05310	+	1.05310i	0.0529872	+	0.0529872i
$396$		0			0
$397$	−15.4784	+	15.4784i	−0.776840	+	0.776840i	−0.979292	−	0.202452i	$-0.935109\pi$
$397$	−15.4784	+	15.4784i	−0.776840	+	0.776840i	0.202452	+	0.979292i	$0.435109\pi$
$398$		0			0
$399$	32.1524			1.60963
$400$		0			0
$401$	8.52766			0.425851			0.212926	−	0.977068i	$-0.431701\pi$
$401$	8.52766			0.425851			0.212926	+	0.977068i	$0.431701\pi$
$402$		0			0
$403$	−6.59591	+	6.59591i	−0.328566	+	0.328566i
$404$		0			0
$405$	−0.582740	−	0.582740i	−0.0289566	−	0.0289566i
$406$		0			0
$407$		−	6.57952i		−	0.326135i
$408$		0			0
$409$			0.959341i			0.0474364i	0.999719	+	0.0237182i	$0.00755044\pi$
$409$			0.959341i			0.0474364i	−0.999719	+	0.0237182i	$0.992450\pi$
$410$		0			0
$411$	13.3771	+	13.3771i	0.659843	+	0.659843i
$412$		0			0
$413$	−10.6608	+	10.6608i	−0.524584	+	0.524584i
$414$		0			0
$415$	3.78640			0.185867
$416$		0			0
$417$	4.29586			0.210369
$418$		0			0
$419$	15.9502	−	15.9502i	0.779216	−	0.779216i	−0.200481	−	0.979697i	$-0.564251\pi$
$419$	15.9502	−	15.9502i	0.779216	−	0.779216i	0.979697	+	0.200481i	$0.0642506\pi$
$420$		0			0
$421$	6.78917	+	6.78917i	0.330884	+	0.330884i	0.852922	−	0.522038i	$-0.174828\pi$
$421$	6.78917	+	6.78917i	0.330884	+	0.330884i	−0.522038	+	0.852922i	$0.674828\pi$
$422$		0			0
$423$			9.80364i			0.476669i
$424$		0			0
$425$			9.53597i			0.462562i
$426$		0			0
$427$	−11.5167	−	11.5167i	−0.557331	−	0.557331i
$428$		0			0
$429$	−1.72543	+	1.72543i	−0.0833047	+	0.0833047i
$430$		0			0
$431$	5.46458			0.263219			0.131610	−	0.991302i	$-0.457985\pi$
$431$	5.46458			0.263219			0.131610	+	0.991302i	$0.457985\pi$
$432$		0			0
$433$	33.8187			1.62522			0.812611	−	0.582806i	$-0.198045\pi$
$433$	33.8187			1.62522			0.812611	+	0.582806i	$0.198045\pi$
$434$		0			0
$435$	3.70278	−	3.70278i	0.177535	−	0.177535i
$436$		0			0
$437$	38.2352	+	38.2352i	1.82904	+	1.82904i
$438$		0			0
$439$			20.8628i			0.995729i	0.867255	+	0.497864i	$0.165882\pi$
$439$			20.8628i			0.995729i	−0.867255	+	0.497864i	$0.834118\pi$
$440$		0			0
$441$			16.9858i			0.808846i
$442$		0			0
$443$	23.9878	+	23.9878i	1.13970	+	1.13970i	0.988505	+	0.151191i	$0.0483108\pi$
$443$	23.9878	+	23.9878i	1.13970	+	1.13970i	0.151191	+	0.988505i	$0.451689\pi$
$444$		0			0
$445$	−3.08902	+	3.08902i	−0.146434	+	0.146434i
$446$		0			0
$447$	−16.7550			−0.792485
$448$		0			0
$449$	−23.6534			−1.11627			−0.558135	−	0.829750i	$-0.688483\pi$
$449$	−23.6534			−1.11627			−0.558135	+	0.829750i	$0.688483\pi$
$450$		0			0
$451$	7.21539	−	7.21539i	0.339760	−	0.339760i
$452$		0			0
$453$	10.3435	+	10.3435i	0.485979	+	0.485979i
$454$		0			0
$455$		−	9.84872i		−	0.461715i
$456$		0			0
$457$			0.845965i			0.0395726i	0.999804	+	0.0197863i	$0.00629858\pi$
$457$			0.845965i			0.0395726i	−0.999804	+	0.0197863i	$0.993701\pi$
$458$		0			0
$459$	−1.56057	−	1.56057i	−0.0728411	−	0.0728411i
$460$		0			0
$461$	−7.70908	+	7.70908i	−0.359047	+	0.359047i	−0.863462	−	0.504414i	$-0.831708\pi$
$461$	−7.70908	+	7.70908i	−0.359047	+	0.359047i	0.504414	+	0.863462i	$0.331708\pi$
$462$		0			0
$463$	29.6331			1.37717			0.688584	−	0.725156i	$-0.258233\pi$
$463$	29.6331			1.37717			0.688584	+	0.725156i	$0.258233\pi$
$464$		0			0
$465$	3.15041			0.146097
$466$		0			0
$467$	22.1239	−	22.1239i	1.02377	−	1.02377i	0.0240610	−	0.999710i	$-0.492340\pi$
$467$	22.1239	−	22.1239i	1.02377	−	1.02377i	0.999710	−	0.0240610i	$-0.00765961\pi$
$468$		0			0
$469$	−42.5908	−	42.5908i	−1.96666	−	1.96666i
$470$		0			0
$471$			0.307467i			0.0141674i
$472$		0			0
$473$			1.72611i			0.0793667i
$474$		0			0
$475$	−20.0580	−	20.0580i	−0.920326	−	0.920326i
$476$		0			0
$477$	−1.14015	+	1.14015i	−0.0522037	+	0.0522037i
$478$		0			0
$479$	−10.8545			−0.495953			−0.247976	−	0.968766i	$-0.579766\pi$
$479$	−10.8545			−0.495953			−0.247976	+	0.968766i	$0.579766\pi$
$480$		0			0
$481$	−16.0549			−0.732039
$482$		0			0
$483$	−28.5236	+	28.5236i	−1.29787	+	1.29787i
$484$		0			0
$485$	4.21807	+	4.21807i	0.191533	+	0.191533i
$486$		0			0
$487$		−	6.22316i		−	0.281999i	−0.990010	−	0.140999i	$-0.954968\pi$
$487$		−	6.22316i		−	0.281999i	0.990010	−	0.140999i	$-0.0450315\pi$
$488$		0			0
$489$			24.1382i			1.09157i
$490$		0			0
$491$	26.7116	+	26.7116i	1.20548	+	1.20548i	0.972476	+	0.233002i	$0.0748550\pi$
$491$	26.7116	+	26.7116i	1.20548	+	1.20548i	0.233002	+	0.972476i	$0.425145\pi$
$492$		0			0
$493$	9.91600	−	9.91600i	0.446594	−	0.446594i
$494$		0			0
$495$	0.824119			0.0370414
$496$		0			0
$497$	−46.2655			−2.07529
$498$		0			0
$499$	−11.3391	+	11.3391i	−0.507607	+	0.507607i	−0.913791	−	0.406184i	$-0.866859\pi$
$499$	−11.3391	+	11.3391i	−0.507607	+	0.507607i	0.406184	+	0.913791i	$0.366859\pi$
$500$		0			0
$501$	−2.85006	−	2.85006i	−0.127331	−	0.127331i
$502$		0			0
$503$		−	28.6656i		−	1.27814i	−0.769151	−	0.639068i	$-0.779320\pi$
$503$		−	28.6656i		−	1.27814i	0.769151	−	0.639068i	$-0.220680\pi$
$504$		0			0
$505$		−	8.58650i		−	0.382094i
$506$		0			0
$507$	−4.98211	−	4.98211i	−0.221263	−	0.221263i
$508$		0			0
$509$	−5.69583	+	5.69583i	−0.252463	+	0.252463i	−0.821980	−	0.569517i	$-0.807130\pi$
$509$	−5.69583	+	5.69583i	−0.252463	+	0.252463i	0.569517	+	0.821980i	$0.307130\pi$
$510$		0			0
$511$	69.6812			3.08251
$512$		0			0
$513$	6.56503			0.289853
$514$		0			0
$515$	4.55681	−	4.55681i	0.200797	−	0.200797i
$516$		0			0
$517$	−6.93222	−	6.93222i	−0.304879	−	0.304879i
$518$		0			0
$519$			9.70517i			0.426010i
$520$		0			0
$521$		−	3.76148i		−	0.164794i	−0.996600	−	0.0823968i	$-0.973743\pi$
$521$		−	3.76148i		−	0.164794i	0.996600	−	0.0823968i	$-0.0262575\pi$
$522$		0			0
$523$	6.40138	+	6.40138i	0.279913	+	0.279913i	0.833074	−	0.553161i	$-0.186579\pi$
$523$	6.40138	+	6.40138i	0.279913	+	0.279913i	−0.553161	+	0.833074i	$0.686579\pi$
$524$		0			0
$525$	14.9633	−	14.9633i	0.653054	−	0.653054i
$526$		0			0
$527$	8.43675			0.367510
$528$		0			0
$529$	−44.8398			−1.94956
$530$		0			0
$531$	−2.17677	+	2.17677i	−0.0944639	+	0.0944639i
$532$		0			0
$533$	−17.6065	−	17.6065i	−0.762622	−	0.762622i
$534$		0			0
$535$		−	2.26807i		−	0.0980572i
$536$		0			0
$537$			6.28851i			0.271369i
$538$		0			0
$539$	−12.0107	−	12.0107i	−0.517339	−	0.517339i
$540$		0			0
$541$	7.83193	−	7.83193i	0.336721	−	0.336721i	−0.518411	−	0.855132i	$-0.673476\pi$
$541$	7.83193	−	7.83193i	0.336721	−	0.336721i	0.855132	+	0.518411i	$0.173476\pi$
$542$		0			0
$543$	17.3752			0.745639
$544$		0			0
$545$	−0.744296			−0.0318821
$546$		0			0
$547$	9.88804	−	9.88804i	0.422782	−	0.422782i	−0.463378	−	0.886161i	$-0.653363\pi$
$547$	9.88804	−	9.88804i	0.422782	−	0.422782i	0.886161	+	0.463378i	$0.153363\pi$
$548$		0			0
$549$	−2.35153	−	2.35153i	−0.100361	−	0.100361i
$550$		0			0
$551$			41.7148i			1.77711i
$552$		0			0
$553$		−	8.85057i		−	0.376365i
$554$		0			0
$555$	3.83415	+	3.83415i	0.162751	+	0.162751i
$556$		0			0
$557$	−11.7849	+	11.7849i	−0.499342	+	0.499342i	−0.911233	−	0.411891i	$-0.864868\pi$
$557$	−11.7849	+	11.7849i	−0.499342	+	0.499342i	0.411891	+	0.911233i	$0.364868\pi$
$558$		0			0
$559$	4.21194			0.178146
$560$		0			0
$561$	2.20698			0.0931787
$562$		0			0
$563$	6.86919	−	6.86919i	0.289502	−	0.289502i	−0.547381	−	0.836883i	$-0.684375\pi$
$563$	6.86919	−	6.86919i	0.289502	−	0.289502i	0.836883	+	0.547381i	$0.184375\pi$
$564$		0			0
$565$	0.406961	+	0.406961i	0.0171210	+	0.0171210i
$566$		0			0
$567$			4.89753i			0.205677i
$568$		0			0
$569$		−	2.65812i		−	0.111434i	−0.998447	−	0.0557170i	$-0.982256\pi$
$569$		−	2.65812i		−	0.111434i	0.998447	−	0.0557170i	$-0.0177445\pi$
$570$		0			0
$571$	3.47521	+	3.47521i	0.145433	+	0.145433i	0.776074	−	0.630641i	$-0.217208\pi$
$571$	3.47521	+	3.47521i	0.145433	+	0.145433i	−0.630641	+	0.776074i	$0.717208\pi$
$572$		0			0
$573$	15.5409	−	15.5409i	0.649229	−	0.649229i
$574$		0			0
$575$	35.5885			1.48414
$576$		0			0
$577$	20.5740			0.856506			0.428253	−	0.903659i	$-0.359129\pi$
$577$	20.5740			0.856506			0.428253	+	0.903659i	$0.359129\pi$
$578$		0			0
$579$	−5.18877	+	5.18877i	−0.215638	+	0.215638i
$580$		0			0
$581$	−15.9110	−	15.9110i	−0.660100	−	0.660100i
$582$		0			0
$583$		−	1.61241i		−	0.0667792i
$584$		0			0
$585$		−	2.01096i		−	0.0831429i
$586$		0			0
$587$	−3.77592	−	3.77592i	−0.155849	−	0.155849i	0.624875	−	0.780724i	$-0.285150\pi$
$587$	−3.77592	−	3.77592i	−0.155849	−	0.155849i	−0.780724	+	0.624875i	$0.785150\pi$
$588$		0			0
$589$	−17.7459	+	17.7459i	−0.731208	+	0.731208i
$590$		0			0
$591$	7.54129			0.310207
$592$		0			0
$593$	26.4648			1.08678			0.543390	−	0.839481i	$-0.317141\pi$
$593$	26.4648			1.08678			0.543390	+	0.839481i	$0.317141\pi$
$594$		0			0
$595$	−6.29868	+	6.29868i	−0.258221	+	0.258221i
$596$		0			0
$597$	−15.5313	−	15.5313i	−0.635652	−	0.635652i
$598$		0			0
$599$			6.53747i			0.267114i	0.991041	+	0.133557i	$0.0426399\pi$
$599$			6.53747i			0.267114i	−0.991041	+	0.133557i	$0.957360\pi$
$600$		0			0
$601$			13.0445i			0.532096i	0.963960	+	0.266048i	$0.0857179\pi$
$601$			13.0445i			0.532096i	−0.963960	+	0.266048i	$0.914282\pi$
$602$		0			0
$603$	−8.69638	−	8.69638i	−0.354144	−	0.354144i
$604$		0			0
$605$	−0.582740	+	0.582740i	−0.0236918	+	0.0236918i
$606$		0			0
$607$	10.2549			0.416235			0.208117	−	0.978104i	$-0.433266\pi$
$607$	10.2549			0.416235			0.208117	+	0.978104i	$0.433266\pi$
$608$		0			0
$609$	−31.1193			−1.26102
$610$		0			0
$611$	−16.9155	+	16.9155i	−0.684329	+	0.684329i
$612$		0			0
$613$	6.91080	+	6.91080i	0.279125	+	0.279125i	0.832759	−	0.553635i	$-0.186760\pi$
$613$	6.91080	+	6.91080i	0.279125	+	0.279125i	−0.553635	+	0.832759i	$0.686760\pi$
$614$		0			0
$615$			8.40940i			0.339100i
$616$		0			0
$617$		−	2.64616i		−	0.106531i	−0.998580	−	0.0532653i	$-0.983037\pi$
$617$		−	2.64616i		−	0.106531i	0.998580	−	0.0532653i	$-0.0169629\pi$
$618$		0			0
$619$	19.2581	+	19.2581i	0.774050	+	0.774050i	0.978812	−	0.204762i	$-0.0656419\pi$
$619$	19.2581	+	19.2581i	0.774050	+	0.774050i	−0.204762	+	0.978812i	$0.565642\pi$
$620$		0			0
$621$	−5.82408	+	5.82408i	−0.233712	+	0.233712i
$622$		0			0
$623$	25.9611			1.04011
$624$		0			0
$625$	−15.2737			−0.610948
$626$		0			0
$627$	−4.64217	+	4.64217i	−0.185391	+	0.185391i
$628$		0			0
$629$	10.2678	+	10.2678i	0.409404	+	0.409404i
$630$		0			0
$631$			14.1966i			0.565156i	0.959244	+	0.282578i	$0.0911896\pi$
$631$			14.1966i			0.565156i	−0.959244	+	0.282578i	$0.908810\pi$
$632$		0			0
$633$		−	11.6981i		−	0.464957i
$634$		0			0
$635$	6.33530	+	6.33530i	0.251409	+	0.251409i
$636$		0			0
$637$	−29.3078	+	29.3078i	−1.16122	+	1.16122i
$638$		0			0
$639$	−9.44670			−0.373706
$640$		0			0
$641$	−21.6912			−0.856753			−0.428376	−	0.903600i	$-0.640914\pi$
$641$	−21.6912			−0.856753			−0.428376	+	0.903600i	$0.640914\pi$
$642$		0			0
$643$	−32.7693	+	32.7693i	−1.29229	+	1.29229i	−0.358930	+	0.933365i	$0.616858\pi$
$643$	−32.7693	+	32.7693i	−1.29229	+	1.29229i	−0.933365	+	0.358930i	$0.883142\pi$
$644$		0			0
$645$	−1.00588	−	1.00588i	−0.0396063	−	0.0396063i
$646$		0			0
$647$		−	49.5258i		−	1.94706i	−0.228563	−	0.973529i	$-0.573403\pi$
$647$		−	49.5258i		−	1.94706i	0.228563	−	0.973529i	$-0.426597\pi$
$648$		0			0
$649$		−	3.07842i		−	0.120839i
$650$		0			0
$651$	−13.2385	−	13.2385i	−0.518858	−	0.518858i
$652$		0			0
$653$	−31.1015	+	31.1015i	−1.21710	+	1.21710i	−0.248450	+	0.968645i	$0.579921\pi$
$653$	−31.1015	+	31.1015i	−1.21710	+	1.21710i	−0.968645	+	0.248450i	$0.920079\pi$
$654$		0			0
$655$	−12.2065			−0.476948
$656$		0			0
$657$	14.2278			0.555081
$658$		0			0
$659$	−12.4434	+	12.4434i	−0.484727	+	0.484727i	−0.906637	−	0.421911i	$-0.861360\pi$
$659$	−12.4434	+	12.4434i	−0.484727	+	0.484727i	0.421911	+	0.906637i	$0.361360\pi$
$660$		0			0
$661$	22.2673	+	22.2673i	0.866098	+	0.866098i	0.992038	−	0.125940i	$-0.0401948\pi$
$661$	22.2673	+	22.2673i	0.866098	+	0.866098i	−0.125940	+	0.992038i	$0.540195\pi$
$662$		0			0
$663$		−	5.38531i		−	0.209148i
$664$		0			0
$665$		−	26.4974i		−	1.02753i
$666$		0			0
$667$	−37.0067	−	37.0067i	−1.43291	−	1.43291i
$668$		0			0
$669$	17.6503	−	17.6503i	0.682401	−	0.682401i
$670$		0			0
$671$	3.32556			0.128382
$672$		0			0
$673$	11.8771			0.457829			0.228914	−	0.973447i	$-0.426482\pi$
$673$	11.8771			0.457829			0.228914	+	0.973447i	$0.426482\pi$
$674$		0			0
$675$	3.05529	−	3.05529i	0.117598	−	0.117598i
$676$		0			0
$677$	−5.07654	−	5.07654i	−0.195107	−	0.195107i	0.602792	−	0.797899i	$-0.294055\pi$
$677$	−5.07654	−	5.07654i	−0.195107	−	0.195107i	−0.797899	+	0.602792i	$0.794055\pi$
$678$		0			0
$679$		−	35.4499i		−	1.36044i
$680$		0			0
$681$			22.5192i			0.862938i
$682$		0			0
$683$	9.91820	+	9.91820i	0.379510	+	0.379510i	0.870925	−	0.491416i	$-0.163520\pi$
$683$	9.91820	+	9.91820i	0.379510	+	0.379510i	−0.491416	+	0.870925i	$0.663520\pi$
$684$		0			0
$685$	11.0243	−	11.0243i	0.421217	−	0.421217i
$686$		0			0
$687$	−16.3030			−0.621998
$688$		0			0
$689$	−3.93449			−0.149892
$690$		0			0
$691$	20.2640	−	20.2640i	0.770880	−	0.770880i	−0.207380	−	0.978260i	$-0.566494\pi$
$691$	20.2640	−	20.2640i	0.770880	−	0.770880i	0.978260	+	0.207380i	$0.0664938\pi$
$692$		0			0
$693$	−3.46307	−	3.46307i	−0.131551	−	0.131551i
$694$		0			0
$695$		−	3.54030i		−	0.134291i
$696$		0			0
$697$			22.5202i			0.853015i
$698$		0			0
$699$	4.15640	+	4.15640i	0.157210	+	0.157210i
$700$		0			0
$701$	3.05442	−	3.05442i	0.115364	−	0.115364i	−0.647068	−	0.762432i	$-0.724005\pi$
$701$	3.05442	−	3.05442i	0.115364	−	0.115364i	0.762432	+	0.647068i	$0.224005\pi$
$702$		0			0
$703$	−43.1947			−1.62912
$704$		0			0
$705$	8.07937			0.304287
$706$		0			0
$707$	−36.0818	+	36.0818i	−1.35699	+	1.35699i
$708$		0			0
$709$	8.62217	+	8.62217i	0.323812	+	0.323812i	0.850228	−	0.526415i	$-0.176464\pi$
$709$	8.62217	+	8.62217i	0.323812	+	0.323812i	−0.526415	+	0.850228i	$0.676464\pi$
$710$		0			0
$711$		−	1.80715i		−	0.0677735i
$712$		0			0
$713$		−	31.4861i		−	1.17917i
$714$		0			0
$715$	1.42196	+	1.42196i	0.0531784	+	0.0531784i
$716$		0			0
$717$	−0.0470217	+	0.0470217i	−0.00175606	+	0.00175606i
$718$		0			0
$719$	−24.8479			−0.926670			−0.463335	−	0.886183i	$-0.653347\pi$
$719$	−24.8479			−0.926670			−0.463335	+	0.886183i	$0.653347\pi$
$720$		0			0
$721$	−38.2968			−1.42625
$722$		0			0
$723$	16.3456	−	16.3456i	0.607898	−	0.607898i
$724$		0			0
$725$	19.4136	+	19.4136i	0.721002	+	0.721002i
$726$		0			0
$727$		−	14.2040i		−	0.526796i	−0.964687	−	0.263398i	$-0.915157\pi$
$727$		−	14.2040i		−	0.526796i	0.964687	−	0.263398i	$-0.0848433\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	−2.69372	−	2.69372i	−0.0996307	−	0.0996307i
$732$		0			0
$733$	7.98976	−	7.98976i	0.295109	−	0.295109i	−0.543986	−	0.839094i	$-0.683085\pi$
$733$	7.98976	−	7.98976i	0.295109	−	0.295109i	0.839094	+	0.543986i	$0.183085\pi$
$734$		0			0
$735$	13.9983			0.516335
$736$		0			0
$737$	12.2985			0.453023
$738$		0			0
$739$	25.9202	−	25.9202i	0.953489	−	0.953489i	−0.0454768	−	0.998965i	$-0.514481\pi$
$739$	25.9202	−	25.9202i	0.953489	−	0.953489i	0.998965	+	0.0454768i	$0.0144807\pi$
$740$		0			0
$741$	11.3275	+	11.3275i	0.416126	+	0.416126i
$742$		0			0
$743$			23.5780i			0.864991i	0.901636	+	0.432496i	$0.142367\pi$
$743$			23.5780i			0.864991i	−0.901636	+	0.432496i	$0.857633\pi$
$744$		0			0
$745$			13.8081i			0.505891i
$746$		0			0
$747$	−3.24879	−	3.24879i	−0.118867	−	0.118867i
$748$		0			0
$749$	−9.53077	+	9.53077i	−0.348247	+	0.348247i
$750$		0			0
$751$	−48.9405			−1.78586			−0.892931	−	0.450193i	$-0.851355\pi$
$751$	−48.9405			−1.78586			−0.892931	+	0.450193i	$0.851355\pi$
$752$		0			0
$753$	−10.9194			−0.397925
$754$		0			0
$755$	8.52425	−	8.52425i	0.310229	−	0.310229i
$756$		0			0
$757$	14.8100	+	14.8100i	0.538277	+	0.538277i	0.923023	−	0.384745i	$-0.125711\pi$
$757$	14.8100	+	14.8100i	0.538277	+	0.538277i	−0.384745	+	0.923023i	$0.625711\pi$
$758$		0			0
$759$		−	8.23649i		−	0.298966i
$760$		0			0
$761$		−	13.6678i		−	0.495456i	−0.968830	−	0.247728i	$-0.920316\pi$
$761$		−	13.6678i		−	0.495456i	0.968830	−	0.247728i	$-0.0796840\pi$
$762$		0			0
$763$	3.12764	+	3.12764i	0.113228	+	0.113228i
$764$		0			0
$765$	−1.28609	+	1.28609i	−0.0464988	+	0.0464988i
$766$		0			0
$767$	−7.51175			−0.271234
$768$		0			0
$769$	7.88350			0.284286			0.142143	−	0.989846i	$-0.454601\pi$
$769$	7.88350			0.284286			0.142143	+	0.989846i	$0.454601\pi$
$770$		0			0
$771$	1.04055	−	1.04055i	0.0374744	−	0.0374744i
$772$		0			0
$773$	10.1154	+	10.1154i	0.363826	+	0.363826i	0.865220	−	0.501393i	$-0.167179\pi$
$773$	10.1154	+	10.1154i	0.363826	+	0.363826i	−0.501393	+	0.865220i	$0.667179\pi$
$774$		0			0
$775$			16.5175i			0.593326i
$776$		0			0
$777$		−	32.2234i		−	1.15601i
$778$		0			0
$779$	−47.3692	−	47.3692i	−1.69718	−	1.69718i
$780$		0			0
$781$	6.67983	−	6.67983i	0.239023	−	0.239023i
$782$		0			0
$783$	−6.35409			−0.227077
$784$		0			0
$785$	0.253390			0.00904387
$786$		0			0
$787$	15.5278	−	15.5278i	0.553508	−	0.553508i	−0.373943	−	0.927452i	$-0.621995\pi$
$787$	15.5278	−	15.5278i	0.553508	−	0.553508i	0.927452	+	0.373943i	$0.121995\pi$
$788$		0			0
$789$	13.7977	+	13.7977i	0.491212	+	0.491212i
$790$		0			0
$791$		−	3.42022i		−	0.121609i
$792$		0			0
$793$		−	8.11480i		−	0.288165i
$794$		0			0
$795$	0.939617	+	0.939617i	0.0333248	+	0.0333248i
$796$		0			0
$797$	31.0037	−	31.0037i	1.09821	−	1.09821i	0.103586	−	0.994620i	$-0.466968\pi$
$797$	31.0037	−	31.0037i	1.09821	−	1.09821i	0.994620	−	0.103586i	$-0.0330317\pi$
$798$		0			0
$799$	21.6364			0.765442
$800$		0			0
$801$	5.30085			0.187296
$802$		0			0
$803$	−10.0606	+	10.0606i	−0.355031	+	0.355031i
$804$		0			0
$805$	23.5068	+	23.5068i	0.828507	+	0.828507i
$806$		0			0
$807$			8.32429i			0.293029i
$808$		0			0
$809$		−	31.8240i		−	1.11887i	−0.828874	−	0.559436i	$-0.811018\pi$
$809$		−	31.8240i		−	1.11887i	0.828874	−	0.559436i	$-0.188982\pi$
$810$		0			0
$811$	27.7285	+	27.7285i	0.973678	+	0.973678i	0.999662	−	0.0259841i	$-0.00827192\pi$
$811$	27.7285	+	27.7285i	0.973678	+	0.973678i	−0.0259841	+	0.999662i	$0.508272\pi$
$812$		0			0
$813$	9.41213	−	9.41213i	0.330098	−	0.330098i
$814$		0			0
$815$	19.8928			0.696814
$816$		0			0
$817$	11.3320			0.396456
$818$		0			0
$819$	−8.45035	+	8.45035i	−0.295279	+	0.295279i
$820$		0			0
$821$	1.84278	+	1.84278i	0.0643133	+	0.0643133i	0.738532	−	0.674219i	$-0.235519\pi$
$821$	1.84278	+	1.84278i	0.0643133	+	0.0643133i	−0.674219	+	0.738532i	$0.735519\pi$
$822$		0			0
$823$			47.7183i			1.66336i	0.555259	+	0.831678i	$0.312619\pi$
$823$			47.7183i			1.66336i	−0.555259	+	0.831678i	$0.687381\pi$
$824$		0			0
$825$			4.32083i			0.150432i
$826$		0			0
$827$	−10.0174	−	10.0174i	−0.348341	−	0.348341i	0.511151	−	0.859491i	$-0.329219\pi$
$827$	−10.0174	−	10.0174i	−0.348341	−	0.348341i	−0.859491	+	0.511151i	$0.829219\pi$
$828$		0			0
$829$	22.1564	−	22.1564i	0.769524	−	0.769524i	−0.208498	−	0.978023i	$-0.566858\pi$
$829$	22.1564	−	22.1564i	0.769524	−	0.769524i	0.978023	+	0.208498i	$0.0668576\pi$
$830$		0			0
$831$	0.485896			0.0168556
$832$		0			0
$833$	37.4872			1.29885
$834$		0			0
$835$	−2.34879	+	2.34879i	−0.0812833	+	0.0812833i
$836$		0			0
$837$	−2.70310	−	2.70310i	−0.0934328	−	0.0934328i
$838$		0			0
$839$		−	1.68311i		−	0.0581073i	−0.999578	−	0.0290537i	$-0.990751\pi$
$839$		−	1.68311i		−	0.0581073i	0.999578	−	0.0290537i	$-0.00924937\pi$
$840$		0			0
$841$		−	11.3745i		−	0.392223i
$842$		0			0
$843$	−0.129823	−	0.129823i	−0.00447135	−	0.00447135i
$844$		0			0
$845$	−4.10585	+	4.10585i	−0.141246	+	0.141246i
$846$		0			0
$847$	4.89753			0.168281
$848$		0			0
$849$	17.3964			0.597043
$850$		0			0
$851$	38.3196	−	38.3196i	1.31358	−	1.31358i
$852$		0			0
$853$	−10.3815	−	10.3815i	−0.355455	−	0.355455i	0.506680	−	0.862134i	$-0.330873\pi$
$853$	−10.3815	−	10.3815i	−0.355455	−	0.355455i	−0.862134	+	0.506680i	$0.830873\pi$
$854$		0			0
$855$		−	5.41036i		−	0.185031i
$856$		0			0
$857$		−	42.3908i		−	1.44804i	−0.689777	−	0.724022i	$-0.742291\pi$
$857$		−	42.3908i		−	1.44804i	0.689777	−	0.724022i	$-0.257709\pi$
$858$		0			0
$859$	−27.7450	−	27.7450i	−0.946649	−	0.946649i	0.0519984	−	0.998647i	$-0.483441\pi$
$859$	−27.7450	−	27.7450i	−0.946649	−	0.946649i	−0.998647	+	0.0519984i	$0.983441\pi$
$860$		0			0
$861$	35.3376	−	35.3376i	1.20430	−	1.20430i
$862$		0			0
$863$	−0.888441			−0.0302429			−0.0151214	−	0.999886i	$-0.504813\pi$
$863$	−0.888441			−0.0302429			−0.0151214	+	0.999886i	$0.504813\pi$
$864$		0			0
$865$	7.99822			0.271947
$866$		0			0
$867$	8.57668	−	8.57668i	0.291279	−	0.291279i
$868$		0			0
$869$	1.27785	+	1.27785i	0.0433481	+	0.0433481i
$870$		0			0
$871$		−	30.0100i		−	1.01685i
$872$		0			0
$873$		−	7.23833i		−	0.244981i
$874$		0			0
$875$	−26.6015	−	26.6015i	−0.899295	−	0.899295i
$876$		0			0
$877$	21.6802	−	21.6802i	0.732089	−	0.732089i	−0.238944	−	0.971033i	$-0.576801\pi$
$877$	21.6802	−	21.6802i	0.732089	−	0.732089i	0.971033	+	0.238944i	$0.0768014\pi$
$878$		0			0
$879$	−3.10001			−0.104561
$880$		0			0
$881$	5.60419			0.188810			0.0944050	−	0.995534i	$-0.469905\pi$
$881$	5.60419			0.188810			0.0944050	+	0.995534i	$0.469905\pi$
$882$		0			0
$883$	13.8664	−	13.8664i	0.466641	−	0.466641i	−0.434183	−	0.900825i	$-0.642963\pi$
$883$	13.8664	−	13.8664i	0.466641	−	0.466641i	0.900825	+	0.434183i	$0.142963\pi$
$884$		0			0
$885$	1.79392	+	1.79392i	0.0603020	+	0.0603020i
$886$		0			0
$887$		−	55.7634i		−	1.87235i	−0.351533	−	0.936176i	$-0.614339\pi$
$887$		−	55.7634i		−	1.87235i	0.351533	−	0.936176i	$-0.385661\pi$
$888$		0			0
$889$		−	53.2438i		−	1.78574i
$890$		0			0
$891$	−0.707107	−	0.707107i	−0.0236890	−	0.0236890i
$892$		0			0
$893$	−45.5102	+	45.5102i	−1.52294	+	1.52294i
$894$		0			0
$895$	5.18248			0.173231
$896$		0			0
$897$	−20.0981			−0.671056
$898$		0			0
$899$	17.1757	−	17.1757i	0.572843	−	0.572843i
$900$		0			0
$901$	2.51628	+	2.51628i	0.0838294	+	0.0838294i
$902$		0			0
$903$			8.45368i			0.281321i
$904$		0			0
$905$		−	14.3192i		−	0.475986i
$906$		0			0
$907$	31.3840	+	31.3840i	1.04209	+	1.04209i	0.999075	+	0.0430131i	$0.0136957\pi$
$907$	31.3840	+	31.3840i	1.04209	+	1.04209i	0.0430131	+	0.999075i	$0.486304\pi$
$908$		0			0
$909$	−7.36735	+	7.36735i	−0.244359	+	0.244359i
$910$		0			0
$911$	24.5585			0.813661			0.406831	−	0.913504i	$-0.366634\pi$
$911$	24.5585			0.813661			0.406831	+	0.913504i	$0.366634\pi$
$912$		0			0
$913$	4.59448			0.152055
$914$		0			0
$915$	−1.93794	+	1.93794i	−0.0640663	+	0.0640663i
$916$		0			0
$917$	51.2936	+	51.2936i	1.69386	+	1.69386i
$918$		0			0
$919$			55.9358i			1.84515i	0.385815	+	0.922576i	$0.373920\pi$
$919$			55.9358i			1.84515i	−0.385815	+	0.922576i	$0.626080\pi$
$920$		0			0
$921$		−	6.70222i		−	0.220846i
$922$		0			0
$923$	−16.2996	−	16.2996i	−0.536509	−	0.536509i
$924$		0			0
$925$	−20.1023	+	20.1023i	−0.660960	+	0.660960i
$926$		0			0
$927$	−7.81962			−0.256830
$928$		0			0
$929$	26.9636			0.884647			0.442324	−	0.896856i	$-0.354154\pi$
$929$	26.9636			0.884647			0.442324	+	0.896856i	$0.354154\pi$
$930$		0			0
$931$	−78.8509	+	78.8509i	−2.58423	+	2.58423i
$932$		0			0
$933$	1.31790	+	1.31790i	0.0431462	+	0.0431462i
$934$		0			0
$935$		−	1.81881i		−	0.0594815i
$936$		0			0
$937$			46.8857i			1.53169i	0.643027	+	0.765844i	$0.277678\pi$
$937$			46.8857i			1.53169i	−0.643027	+	0.765844i	$0.722322\pi$
$938$		0			0
$939$	2.03688	+	2.03688i	0.0664712	+	0.0664712i
$940$		0			0
$941$	−26.1894	+	26.1894i	−0.853752	+	0.853752i	−0.990593	−	0.136841i	$-0.956305\pi$
$941$	−26.1894	+	26.1894i	−0.853752	+	0.853752i	0.136841	+	0.990593i	$0.456305\pi$
$942$		0			0
$943$	84.0460			2.73692
$944$		0			0
$945$	4.03614			0.131296
$946$		0			0
$947$	−30.1063	+	30.1063i	−0.978322	+	0.978322i	−0.999770	−	0.0214476i	$-0.993172\pi$
$947$	−30.1063	+	30.1063i	−0.978322	+	0.978322i	0.0214476	+	0.999770i	$0.493172\pi$
$948$		0			0
$949$	24.5492	+	24.5492i	0.796900	+	0.796900i
$950$		0			0
$951$		−	14.9987i		−	0.486365i
$952$		0			0
$953$		−	34.2473i		−	1.10938i	−0.832057	−	0.554690i	$-0.812837\pi$
$953$		−	34.2473i		−	1.10938i	0.832057	−	0.554690i	$-0.187163\pi$
$954$		0			0
$955$	−12.8075	−	12.8075i	−0.414442	−	0.414442i
$956$		0			0
$957$	4.49302	−	4.49302i	0.145239	−	0.145239i
$958$		0			0
$959$	−92.6517			−2.99188
$960$		0			0
$961$	−16.3865			−0.528597
$962$		0			0
$963$	−1.94604	+	1.94604i	−0.0627102	+	0.0627102i
$964$		0			0
$965$	4.27617	+	4.27617i	0.137655	+	0.137655i
$966$		0			0
$967$		−	26.2775i		−	0.845029i	−0.906356	−	0.422514i	$-0.861148\pi$
$967$		−	26.2775i		−	0.845029i	0.906356	−	0.422514i	$-0.138852\pi$
$968$		0			0
$969$		−	14.4889i		−	0.465449i
$970$		0			0
$971$	15.2593	+	15.2593i	0.489695	+	0.489695i	0.908210	−	0.418515i	$-0.137449\pi$
$971$	15.2593	+	15.2593i	0.489695	+	0.489695i	−0.418515	+	0.908210i	$0.637449\pi$
$972$		0			0
$973$	−14.8769	+	14.8769i	−0.476931	+	0.476931i
$974$		0			0
$975$	10.5434			0.337658
$976$		0			0
$977$	12.4229			0.397445			0.198723	−	0.980056i	$-0.436321\pi$
$977$	12.4229			0.397445			0.198723	+	0.980056i	$0.436321\pi$
$978$		0			0
$979$	−3.74827	+	3.74827i	−0.119795	+	0.119795i
$980$		0			0
$981$	0.638617	+	0.638617i	0.0203895	+	0.0203895i
$982$		0			0
$983$			21.4959i			0.685613i	0.939406	+	0.342807i	$0.111378\pi$
$983$			21.4959i			0.685613i	−0.939406	+	0.342807i	$0.888622\pi$
$984$		0			0
$985$		−	6.21492i		−	0.198024i
$986$		0			0
$987$	−33.9507	−	33.9507i	−1.08066	−	1.08066i
$988$		0			0
$989$	−10.0530	+	10.0530i	−0.319667	+	0.319667i
$990$		0			0
$991$	−17.3220			−0.550251			−0.275126	−	0.961408i	$-0.588719\pi$
$991$	−17.3220			−0.550251			−0.275126	+	0.961408i	$0.588719\pi$
$992$		0			0
$993$	−12.7531			−0.404708
$994$		0			0
$995$	−12.7996	+	12.7996i	−0.405775	+	0.405775i
$996$		0			0
$997$	−8.28691	−	8.28691i	−0.262449	−	0.262449i	0.563599	−	0.826048i	$-0.309416\pi$
$997$	−8.28691	−	8.28691i	−0.262449	−	0.262449i	−0.826048	+	0.563599i	$0.809416\pi$
$998$		0			0
$999$		−	6.57952i		−	0.208167i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.t.b.1585.5		40
4.3	odd	2		528.2.t.b.133.19	✓	40
16.3	odd	4		528.2.t.b.397.19	yes	40
16.13	even	4	inner	2112.2.t.b.529.5		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.2.t.b.133.19	✓	40	4.3	odd	2
528.2.t.b.397.19	yes	40	16.3	odd	4
2112.2.t.b.529.5		40	16.13	even	4	inner
2112.2.t.b.1585.5		40	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(0.582740 + 0.582740i) q^{5} -4.89753i q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} +(0.582740 + 0.582740i) q^{5} -4.89753i q^{7} -1.00000i q^{9} +(0.707107 + 0.707107i) q^{11} +(1.72543 - 1.72543i) q^{13} -0.824119 q^{15} -2.20698 q^{17} +(4.64217 - 4.64217i) q^{19} +(3.46307 + 3.46307i) q^{21} +8.23649i q^{23} -4.32083i q^{25} +(0.707107 + 0.707107i) q^{27} +(-4.49302 + 4.49302i) q^{29} -3.82276 q^{31} -1.00000 q^{33} +(2.85399 - 2.85399i) q^{35} +(-4.65242 - 4.65242i) q^{37} +2.44013i q^{39} -10.2041i q^{41} +(1.22055 + 1.22055i) q^{43} +(0.582740 - 0.582740i) q^{45} -9.80364 q^{47} -16.9858 q^{49} +(1.56057 - 1.56057i) q^{51} +(-1.14015 - 1.14015i) q^{53} +0.824119i q^{55} +6.56503i q^{57} +(-2.17677 - 2.17677i) q^{59} +(2.35153 - 2.35153i) q^{61} -4.89753 q^{63} +2.01096 q^{65} +(8.69638 - 8.69638i) q^{67} +(-5.82408 - 5.82408i) q^{69} -9.44670i q^{71} +14.2278i q^{73} +(3.05529 + 3.05529i) q^{75} +(3.46307 - 3.46307i) q^{77} +1.80715 q^{79} -1.00000 q^{81} +(3.24879 - 3.24879i) q^{83} +(-1.28609 - 1.28609i) q^{85} -6.35409i q^{87} +5.30085i q^{89} +(-8.45035 - 8.45035i) q^{91} +(2.70310 - 2.70310i) q^{93} +5.41036 q^{95} +7.23833 q^{97} +(0.707107 - 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 8 q^{15} - 8 q^{19} - 8 q^{31} - 40 q^{33} + 24 q^{35} + 64 q^{47} - 40 q^{49} + 8 q^{51} + 16 q^{59} - 64 q^{61} - 24 q^{63} + 16 q^{65} - 24 q^{67} - 16 q^{69} + 72 q^{79} - 40 q^{81} + 16 q^{85} - 16 q^{91} + 32 q^{93} - 128 q^{95} - 32 q^{97}+O(q^{100}) \) 40 * q + 8 * q^15 - 8 * q^19 - 8 * q^31 - 40 * q^33 + 24 * q^35 + 64 * q^47 - 40 * q^49 + 8 * q^51 + 16 * q^59 - 64 * q^61 - 24 * q^63 + 16 * q^65 - 24 * q^67 - 16 * q^69 + 72 * q^79 - 40 * q^81 + 16 * q^85 - 16 * q^91 + 32 * q^93 - 128 * q^95 - 32 * q^97

Introduction

L-functions

Modular forms

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Database

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