LMFDB - Embedded newform 112.2.j.b.27.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,2,Mod(27,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.27"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	112.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.894324502638$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			27.1
Root			$-1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	112.27
Dual form			112.2.j.b.83.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.32288 - 0.500000i) q^{2} +(1.50000 + 1.32288i) q^{4} +2.64575 q^{7} +(-1.32288 - 2.50000i) q^{8} +3.00000i q^{9} +(4.64575 - 4.64575i) q^{11} +(-3.50000 - 1.32288i) q^{14} +(0.500000 + 3.96863i) q^{16} +(1.50000 - 3.96863i) q^{18} +(-8.46863 + 3.82288i) q^{22} -8.00000 q^{23} +5.00000i q^{25} +(3.96863 + 3.50000i) q^{28} +(-4.29150 + 4.29150i) q^{29} +(1.32288 - 5.50000i) q^{32} +(-3.96863 + 4.50000i) q^{36} +(-8.29150 - 8.29150i) q^{37} +(3.35425 - 3.35425i) q^{43} +(13.1144 - 0.822876i) q^{44} +(10.5830 + 4.00000i) q^{46} +7.00000 q^{49} +(2.50000 - 6.61438i) q^{50} +(0.291503 + 0.291503i) q^{53} +(-3.50000 - 6.61438i) q^{56} +(7.82288 - 3.53137i) q^{58} +7.93725i q^{63} +(-4.50000 + 6.61438i) q^{64} +(-5.93725 - 5.93725i) q^{67} -5.29150 q^{71} +(7.50000 - 3.96863i) q^{72} +(6.82288 + 15.1144i) q^{74} +(12.2915 - 12.2915i) q^{77} +8.00000i q^{79} -9.00000 q^{81} +(-6.11438 + 2.76013i) q^{86} +(-17.7601 - 5.46863i) q^{88} +(-12.0000 - 10.5830i) q^{92} +(-9.26013 - 3.50000i) q^{98} +(13.9373 + 13.9373i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 8 q^{11} - 14 q^{14} + 2 q^{16} + 6 q^{18} - 18 q^{22} - 32 q^{23} + 4 q^{29} - 12 q^{37} + 24 q^{43} + 26 q^{44} + 28 q^{49} + 10 q^{50} - 20 q^{53} - 14 q^{56} + 26 q^{58} - 18 q^{64}+ \cdots + 24 q^{99}+O(q^{100}) $ 4 * q + 6 * q^4 + 8 * q^11 - 14 * q^14 + 2 * q^16 + 6 * q^18 - 18 * q^22 - 32 * q^23 + 4 * q^29 - 12 * q^37 + 24 * q^43 + 26 * q^44 + 28 * q^49 + 10 * q^50 - 20 * q^53 - 14 * q^56 + 26 * q^58 - 18 * q^64 + 8 * q^67 + 30 * q^72 + 22 * q^74 + 28 * q^77 - 36 * q^81 + 2 * q^86 - 34 * q^88 - 48 * q^92 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.32288	−	0.500000i	−0.935414	−	0.353553i
$2$	−1.32288	−	0.500000i	−0.935414	−	0.353553i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$	1.50000	+	1.32288i	0.750000	+	0.661438i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$	−1.32288	−	2.50000i	−0.467707	−	0.883883i
$9$			3.00000i			1.00000i
$10$		0			0
$11$	4.64575	−	4.64575i	1.40075	−	1.40075i	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.64575	−	4.64575i	1.40075	−	1.40075i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$	−3.50000	−	1.32288i	−0.935414	−	0.353553i
$15$		0			0
$16$	0.500000	+	3.96863i	0.125000	+	0.992157i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	1.50000	−	3.96863i	0.353553	−	0.935414i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$	−8.46863	+	3.82288i	−1.80552	+	0.815040i
$23$	−8.00000			−1.66812			−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000			−1.66812			−0.834058	+	0.551677i	$0.813988\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$	3.96863	+	3.50000i	0.750000	+	0.661438i
$29$	−4.29150	+	4.29150i	−0.796912	+	0.796912i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	−4.29150	+	4.29150i	−0.796912	+	0.796912i	0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	1.32288	−	5.50000i	0.233854	−	0.972272i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.96863	+	4.50000i	−0.661438	+	0.750000i
$37$	−8.29150	−	8.29150i	−1.36311	−	1.36311i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	−8.29150	−	8.29150i	−1.36311	−	1.36311i	−0.493197	−	0.869918i	$-0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	3.35425	−	3.35425i	0.511518	−	0.511518i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	3.35425	−	3.35425i	0.511518	−	0.511518i	0.914991	+	0.403473i	$0.132197\pi$
$44$	13.1144	−	0.822876i	1.97707	−	0.124053i
$45$		0			0
$46$	10.5830	+	4.00000i	1.56038	+	0.589768i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	2.50000	−	6.61438i	0.353553	−	0.935414i
$51$		0			0
$52$		0			0
$53$	0.291503	+	0.291503i	0.0400410	+	0.0400410i	0.726844	−	0.686803i	$-0.240986\pi$
$53$	0.291503	+	0.291503i	0.0400410	+	0.0400410i	−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	−	6.61438i	−0.467707	−	0.883883i
$57$		0			0
$58$	7.82288	−	3.53137i	1.02719	−	0.463692i
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$			7.93725i			1.00000i
$64$	−4.50000	+	6.61438i	−0.562500	+	0.826797i
$65$		0			0
$66$		0			0
$67$	−5.93725	−	5.93725i	−0.725351	−	0.725351i	0.244339	−	0.969690i	$-0.421429\pi$
$67$	−5.93725	−	5.93725i	−0.725351	−	0.725351i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.29150			−0.627986			−0.313993	−	0.949425i	$-0.601667\pi$
$71$	−5.29150			−0.627986			−0.313993	+	0.949425i	$0.601667\pi$
$72$	7.50000	−	3.96863i	0.883883	−	0.467707i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	6.82288	+	15.1144i	0.793143	+	1.75701i
$75$		0			0
$76$		0			0
$77$	12.2915	−	12.2915i	1.40075	−	1.40075i
$78$		0			0
$79$			8.00000i			0.900070i	0.893011	+	0.450035i	$0.148589\pi$
$79$			8.00000i			0.900070i	−0.893011	+	0.450035i	$0.851411\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$	−6.11438	+	2.76013i	−0.659330	+	0.297632i
$87$		0			0
$88$	−17.7601	−	5.46863i	−1.89324	−	0.582958i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−12.0000	−	10.5830i	−1.25109	−	1.10335i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−9.26013	−	3.50000i	−0.935414	−	0.353553i
$99$	13.9373	+	13.9373i	1.40075	+	1.40075i
$100$	−6.61438	+	7.50000i	−0.661438	+	0.750000i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−0.239870	−	0.531373i	−0.0232983	−	0.0516115i
$107$	−12.6458	+	12.6458i	−1.22251	+	1.22251i	−0.255774	+	0.966736i	$0.582330\pi$
$107$	−12.6458	+	12.6458i	−1.22251	+	1.22251i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	−3.70850	+	3.70850i	−0.355210	+	0.355210i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−3.70850	+	3.70850i	−0.355210	+	0.355210i	0.506834	+	0.862044i	$0.330816\pi$
$110$		0			0
$111$		0			0
$112$	1.32288	+	10.5000i	0.125000	+	0.992157i
$113$	21.1660			1.99113			0.995565	−	0.0940721i	$-0.0299884\pi$
$113$	21.1660			1.99113			0.995565	+	0.0940721i	$0.0299884\pi$
$114$		0			0
$115$		0			0
$116$	−12.1144	+	0.760130i	−1.12479	+	0.0705763i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	32.1660i		−	2.92418i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	3.96863	−	10.5000i	0.353553	−	0.935414i
$127$		−	15.8745i		−	1.40863i	−0.709885	−	0.704317i	$-0.751253\pi$
$127$		−	15.8745i		−	1.40863i	0.709885	−	0.704317i	$-0.248747\pi$
$128$	9.26013	−	6.50000i	0.818488	−	0.574524i
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$	4.88562	+	10.8229i	0.422053	+	0.934954i
$135$		0			0
$136$		0			0
$137$			21.1660i			1.80833i	0.427179	+	0.904167i	$0.359507\pi$
$137$			21.1660i			1.80833i	−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$	7.00000	+	2.64575i	0.587427	+	0.222027i
$143$		0			0
$144$	−11.9059	+	1.50000i	−0.992157	+	0.125000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.46863	−	23.4059i	−0.120720	−	1.92395i
$149$	16.2915	+	16.2915i	1.33465	+	1.33465i	0.901155	+	0.433497i	$0.142720\pi$
$149$	16.2915	+	16.2915i	1.33465	+	1.33465i	0.433497	+	0.901155i	$0.357280\pi$
$150$		0			0
$151$	24.0000			1.95309			0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000			1.95309			0.976546	+	0.215308i	$0.0690756\pi$
$152$		0			0
$153$		0			0
$154$	−22.4059	+	10.1144i	−1.80552	+	0.815040i
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	4.00000	−	10.5830i	0.318223	−	0.841939i
$159$		0			0
$160$		0			0
$161$	−21.1660			−1.66812
$162$	11.9059	+	4.50000i	0.935414	+	0.353553i
$163$	−2.06275	−	2.06275i	−0.161567	−	0.161567i	0.621694	−	0.783260i	$-0.286445\pi$
$163$	−2.06275	−	2.06275i	−0.161567	−	0.161567i	−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$	9.46863	−	0.594119i	0.721976	−	0.0453012i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$	20.7601	+	16.1144i	1.56485	+	1.21467i
$177$		0			0
$178$		0			0
$179$	−15.2288	−	15.2288i	−1.13825	−	1.13825i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	−15.2288	−	15.2288i	−1.13825	−	1.13825i	−0.149487	−	0.988764i	$-0.547762\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	10.5830	+	20.0000i	0.780189	+	1.47442i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.00000i		−	0.578860i	−0.957199	−	0.289430i	$-0.906534\pi$
$191$		−	8.00000i		−	0.578860i	0.957199	−	0.289430i	$-0.0934657\pi$
$192$		0			0
$193$	−21.1660			−1.52356			−0.761781	−	0.647834i	$-0.775675\pi$
$193$	−21.1660			−1.52356			−0.761781	+	0.647834i	$0.775675\pi$
$194$		0			0
$195$		0			0
$196$	10.5000	+	9.26013i	0.750000	+	0.661438i
$197$	7.70850	+	7.70850i	0.549208	+	0.549208i	0.926212	−	0.377004i	$-0.123046\pi$
$197$	7.70850	+	7.70850i	0.549208	+	0.549208i	−0.377004	+	0.926212i	$0.623046\pi$
$198$	−11.4686	−	25.4059i	−0.815040	−	1.80552i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	12.5000	−	6.61438i	0.883883	−	0.467707i
$201$		0			0
$202$		0			0
$203$	−11.3542	+	11.3542i	−0.796912	+	0.796912i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	24.0000i		−	1.66812i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	7.22876	+	7.22876i	0.497648	+	0.497648i	0.910705	−	0.413057i	$-0.135539\pi$
$211$	7.22876	+	7.22876i	0.497648	+	0.497648i	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0.0516322	+	0.822876i	0.00354611	+	0.0565153i
$213$		0			0
$214$	23.0516	−	10.4059i	1.57578	−	0.711331i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	6.76013	−	3.05163i	0.457854	−	0.206683i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	3.50000	−	14.5516i	0.233854	−	0.972272i
$225$	−15.0000			−1.00000
$226$	−28.0000	−	10.5830i	−1.86253	−	0.703971i
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$	16.4059	+	5.05163i	1.07710	+	0.331656i
$233$			21.1660i			1.38663i	0.720634	+	0.693316i	$0.243851\pi$
$233$			21.1660i			1.38663i	−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	26.4575i		−	1.71139i	−0.517477	−	0.855697i	$-0.673129\pi$
$239$		−	26.4575i		−	1.71139i	0.517477	−	0.855697i	$-0.326871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−16.0830	+	42.5516i	−1.03385	+	2.73532i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$	−10.5000	+	11.9059i	−0.661438	+	0.750000i
$253$	−37.1660	+	37.1660i	−2.33661	+	2.33661i
$254$	−7.93725	+	21.0000i	−0.498028	+	1.31766i
$255$		0			0
$256$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−21.9373	−	21.9373i	−1.36311	−	1.36311i
$260$		0			0
$261$	−12.8745	−	12.8745i	−0.796912	−	0.796912i
$262$		0			0
$263$	5.29150			0.326288			0.163144	−	0.986602i	$-0.447836\pi$
$263$	5.29150			0.326288			0.163144	+	0.986602i	$0.447836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.05163	−	16.7601i	−0.0642387	−	1.02379i
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	10.5830	−	28.0000i	0.639343	−	1.69154i
$275$	23.2288	+	23.2288i	1.40075	+	1.40075i
$276$		0			0
$277$	20.8745	+	20.8745i	1.25423	+	1.25423i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	20.8745	+	20.8745i	1.25423	+	1.25423i	0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	21.1660i		−	1.26266i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$		−	21.1660i		−	1.26266i	0.775515	−	0.631329i	$-0.217490\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$	−7.93725	−	7.00000i	−0.470989	−	0.415374i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.5000	+	3.96863i	0.972272	+	0.233854i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$	−9.76013	+	31.6974i	−0.567296	+	1.84237i
$297$		0			0
$298$	−13.4059	−	29.6974i	−0.776582	−	1.72032i
$299$		0			0
$300$		0			0
$301$	8.87451	−	8.87451i	0.511518	−	0.511518i
$302$	−31.7490	−	12.0000i	−1.82695	−	0.690522i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$	34.6974	−	2.17712i	1.97707	−	0.124053i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−10.5830	+	12.0000i	−0.595341	+	0.675053i
$317$	11.7085	−	11.7085i	0.657615	−	0.657615i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	11.7085	−	11.7085i	0.657615	−	0.657615i	0.954815	+	0.297200i	$0.0960529\pi$
$318$		0			0
$319$			39.8745i			2.23254i
$320$		0			0
$321$		0			0
$322$	28.0000	+	10.5830i	1.56038	+	0.589768i
$323$		0			0
$324$	−13.5000	−	11.9059i	−0.750000	−	0.661438i
$325$		0			0
$326$	1.69738	+	3.76013i	0.0940094	+	0.208254i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.6458	−	20.6458i	1.13479	−	1.13479i	0.145424	−	0.989369i	$-0.453545\pi$
$331$	20.6458	−	20.6458i	1.13479	−	1.13479i	0.989369	−	0.145424i	$-0.0464545\pi$
$332$		0			0
$333$	24.8745	−	24.8745i	1.36311	−	1.36311i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	21.1660			1.15299			0.576493	−	0.817102i	$-0.304421\pi$
$337$	21.1660			1.15299			0.576493	+	0.817102i	$0.304421\pi$
$338$	−6.50000	+	17.1974i	−0.353553	+	0.935414i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$	−12.8229	−	3.94837i	−0.691363	−	0.212882i
$345$		0			0
$346$		0			0
$347$	16.5203	−	16.5203i	0.886854	−	0.886854i	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	16.5203	−	16.5203i	0.886854	−	0.886854i	0.994220	+	0.107366i	$0.0342415\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	6.61438	−	17.5000i	0.353553	−	0.935414i
$351$		0			0
$352$	−19.4059	−	31.6974i	−1.03434	−	1.68948i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	12.5314	+	27.7601i	0.662304	+	1.46717i
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−4.00000	−	31.7490i	−0.208514	−	1.65503i
$369$		0			0
$370$		0			0
$371$	0.771243	+	0.771243i	0.0400410	+	0.0400410i
$372$		0			0
$373$	4.87451	+	4.87451i	0.252392	+	0.252392i	0.821951	−	0.569558i	$-0.192886\pi$
$373$	4.87451	+	4.87451i	0.252392	+	0.252392i	−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−24.5203	+	24.5203i	−1.25952	+	1.25952i	−0.308199	+	0.951322i	$0.599726\pi$
$379$	−24.5203	+	24.5203i	−1.25952	+	1.25952i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	−4.00000	+	10.5830i	−0.204658	+	0.541474i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	28.0000	+	10.5830i	1.42516	+	0.538661i
$387$	10.0627	+	10.0627i	0.511518	+	0.511518i
$388$		0			0
$389$	−24.2915	−	24.2915i	−1.23163	−	1.23163i	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−24.2915	−	24.2915i	−1.23163	−	1.23163i	−0.268290	−	0.963338i	$-0.586458\pi$
$390$		0			0
$391$		0			0
$392$	−9.26013	−	17.5000i	−0.467707	−	0.883883i
$393$		0			0
$394$	−6.34313	−	14.0516i	−0.319563	−	0.707911i
$395$		0			0
$396$	2.46863	+	39.3431i	0.124053	+	1.97707i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$	−19.8431	+	2.50000i	−0.992157	+	0.125000i
$401$	21.1660			1.05698			0.528490	−	0.848939i	$-0.322758\pi$
$401$	21.1660			1.05698			0.528490	+	0.848939i	$0.322758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	20.6974	−	9.34313i	1.02719	−	0.463692i
$407$	−77.0405			−3.81876
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−12.0000	+	31.7490i	−0.589768	+	1.56038i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	−28.8745	−	28.8745i	−1.40726	−	1.40726i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	−28.8745	−	28.8745i	−1.40726	−	1.40726i	−0.633581	−	0.773676i	$-0.718416\pi$
$422$	−5.94837	−	13.1771i	−0.289562	−	0.641452i
$423$		0			0
$424$	0.343135	−	1.11438i	0.0166641	−	0.0541190i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−35.6974	+	2.23987i	−1.72550	+	0.108268i
$429$		0			0
$430$		0			0
$431$			26.4575i			1.27441i	0.770693	+	0.637207i	$0.219910\pi$
$431$			26.4575i			1.27441i	−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−10.4686	+	0.656865i	−0.501356	+	0.0314581i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$		0			0
$443$	−8.52026	+	8.52026i	−0.404810	+	0.404810i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	−8.52026	+	8.52026i	−0.404810	+	0.404810i	0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−11.9059	+	17.5000i	−0.562500	+	0.826797i
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$	19.8431	+	7.50000i	0.935414	+	0.353553i
$451$		0			0
$452$	31.7490	+	28.0000i	1.49335	+	1.31701i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	6.00000i		−	0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$		−	6.00000i		−	0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$			40.0000i			1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$			40.0000i			1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	−19.1771	−	14.8856i	−0.890276	−	0.691048i
$465$		0			0
$466$	10.5830	−	28.0000i	0.490248	−	1.29707i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$	−15.7085	−	15.7085i	−0.725351	−	0.725351i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	31.1660i		−	1.43301i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.874508	+	0.874508i	−0.0400410	+	0.0400410i
$478$	−13.2288	+	35.0000i	−0.605069	+	1.60086i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	42.5516	−	48.2490i	1.93417	−	2.19314i
$485$		0			0
$486$		0			0
$487$	−24.0000			−1.08754			−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000			−1.08754			−0.543772	+	0.839233i	$0.683004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.3542	−	19.3542i	0.873445	−	0.873445i	−0.119401	−	0.992846i	$-0.538097\pi$
$491$	19.3542	−	19.3542i	0.873445	−	0.873445i	0.992846	+	0.119401i	$0.0380974\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.0000			−0.627986
$498$		0			0
$499$	−31.2288	−	31.2288i	−1.39799	−	1.39799i	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−31.2288	−	31.2288i	−1.39799	−	1.39799i	−0.592200	−	0.805791i	$-0.701741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	19.8431	−	10.5000i	0.883883	−	0.467707i
$505$		0			0
$506$	67.7490	−	30.5830i	3.01181	−	1.35958i
$507$		0			0
$508$	21.0000	−	23.8118i	0.931724	−	1.05648i
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	22.4889	+	2.50000i	0.993878	+	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	18.0516	+	39.9889i	0.793143	+	1.75701i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	10.5941	+	23.4686i	0.463692	+	1.02719i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	−7.00000	−	2.64575i	−0.305215	−	0.115360i
$527$		0			0
$528$		0			0
$529$	41.0000			1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−6.98889	+	22.6974i	−0.301874	+	0.980378i
$537$		0			0
$538$		0			0
$539$	32.5203	−	32.5203i	1.40075	−	1.40075i
$540$		0			0
$541$	−32.8745	+	32.8745i	−1.41339	+	1.41339i	−0.682498	+	0.730887i	$0.739107\pi$
$541$	−32.8745	+	32.8745i	−1.41339	+	1.41339i	−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	29.9373	+	29.9373i	1.28002	+	1.28002i	0.940652	+	0.339372i	$0.110215\pi$
$547$	29.9373	+	29.9373i	1.28002	+	1.28002i	0.339372	+	0.940652i	$0.389785\pi$
$548$	−28.0000	+	31.7490i	−1.19610	+	1.35625i
$549$		0			0
$550$	−19.1144	−	42.3431i	−0.815040	−	1.80552i
$551$		0			0
$552$		0			0
$553$			21.1660i			0.900070i
$554$	−17.1771	−	38.0516i	−0.729786	−	1.61666i
$555$		0			0
$556$		0			0
$557$	28.2915	−	28.2915i	1.19875	−	1.19875i	0.224208	−	0.974541i	$-0.428020\pi$
$557$	28.2915	−	28.2915i	1.19875	−	1.19875i	0.974541	−	0.224208i	$-0.0719796\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−10.5830	+	28.0000i	−0.446417	+	1.18111i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.8118			−1.00000
$568$	7.00000	+	13.2288i	0.293713	+	0.555066i
$569$			22.0000i			0.922288i	0.887325	+	0.461144i	$0.152561\pi$
$569$			22.0000i			0.922288i	−0.887325	+	0.461144i	$0.847439\pi$
$570$		0			0
$571$	−25.8118	+	25.8118i	−1.08019	+	1.08019i	−0.0836974	+	0.996491i	$0.526673\pi$
$571$	−25.8118	+	25.8118i	−1.08019	+	1.08019i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	40.0000i		−	1.66812i
$576$	−19.8431	−	13.5000i	−0.826797	−	0.562500i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−22.4889	−	8.50000i	−0.935414	−	0.353553i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.70850			0.112174
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	28.7601	−	37.0516i	1.18203	−	1.52281i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	2.88562	+	45.9889i	0.118200	+	1.88378i
$597$		0			0
$598$		0			0
$599$	37.0405			1.51343			0.756717	−	0.653742i	$-0.226802\pi$
$599$	37.0405			1.51343			0.756717	+	0.653742i	$0.226802\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$	−16.1771	+	7.30262i	−0.659330	+	0.297632i
$603$	17.8118	−	17.8118i	0.725351	−	0.725351i
$604$	36.0000	+	31.7490i	1.46482	+	1.29185i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.12549	+	3.12549i	0.126237	+	0.126237i	0.767403	−	0.641165i	$-0.221549\pi$
$613$	3.12549	+	3.12549i	0.126237	+	0.126237i	−0.641165	+	0.767403i	$0.721549\pi$
$614$		0			0
$615$		0			0
$616$	−46.9889	−	14.4686i	−1.89324	−	0.582958i
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	47.6235			1.89586			0.947931	−	0.318475i	$-0.103171\pi$
$631$	47.6235			1.89586			0.947931	+	0.318475i	$0.103171\pi$
$632$	20.0000	−	10.5830i	0.795557	−	0.420969i
$633$		0			0
$634$	−21.3431	+	9.63464i	−0.847644	+	0.382640i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	19.9373	−	52.7490i	0.789323	−	2.08835i
$639$		−	15.8745i		−	0.627986i
$640$		0			0
$641$	−21.1660			−0.836007			−0.418004	−	0.908445i	$-0.637270\pi$
$641$	−21.1660			−0.836007			−0.418004	+	0.908445i	$0.637270\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$	−31.7490	−	28.0000i	−1.25109	−	1.10335i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	11.9059	+	22.5000i	0.467707	+	0.883883i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.365363	−	5.82288i	−0.0143087	−	0.228041i
$653$	−19.7085	+	19.7085i	−0.771253	+	0.771253i	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−19.7085	+	19.7085i	−0.771253	+	0.771253i	0.207072	+	0.978326i	$0.433606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.77124	−	8.77124i	−0.341679	−	0.341679i	0.515319	−	0.856998i	$-0.327673\pi$
$659$	−8.77124	−	8.77124i	−0.341679	−	0.341679i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$	−37.6346	+	16.9889i	−1.46271	+	0.660292i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−45.3431	+	20.4686i	−1.75701	+	0.793143i
$667$	34.3320	−	34.3320i	1.32934	−	1.32934i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$	−28.0000	−	10.5830i	−1.07852	−	0.407642i
$675$		0			0
$676$	17.1974	−	19.5000i	0.661438	−	0.750000i
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28.6458	+	28.6458i	−1.09610	+	1.09610i	−0.101237	+	0.994862i	$0.532280\pi$
$683$	−28.6458	+	28.6458i	−1.09610	+	1.09610i	−0.994862	+	0.101237i	$0.967720\pi$
$684$		0			0
$685$		0			0
$686$	−24.5000	−	9.26013i	−0.935414	−	0.353553i
$687$		0			0
$688$	14.9889	+	11.6346i	0.571446	+	0.443566i
$689$		0			0
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$	36.8745	+	36.8745i	1.40075	+	1.40075i
$694$	−30.1144	+	13.5941i	−1.14313	+	0.516026i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	+	19.8431i	−0.661438	+	0.750000i
$701$	25.4575	−	25.4575i	0.961517	−	0.961517i	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	25.4575	−	25.4575i	0.961517	−	0.961517i	0.999286	+	0.0377695i	$0.0120253\pi$
$702$		0			0
$703$		0			0
$704$	9.82288	+	51.6346i	0.370214	+	1.94605i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	29.4575	+	29.4575i	1.10630	+	1.10630i	0.993633	+	0.112667i	$0.0359394\pi$
$709$	29.4575	+	29.4575i	1.10630	+	1.10630i	0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$	−24.0000			−0.900070
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.69738	−	42.9889i	−0.100806	−	1.60657i
$717$		0			0
$718$	−10.5830	−	4.00000i	−0.394954	−	0.149279i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	9.50000	−	25.1346i	0.353553	−	0.935414i
$723$		0			0
$724$		0			0
$725$	−21.4575	−	21.4575i	−0.796912	−	0.796912i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	−10.5830	+	44.0000i	−0.390095	+	1.62186i
$737$	−55.1660			−2.03207
$738$		0			0
$739$	−18.0627	−	18.0627i	−0.664449	−	0.664449i	0.291977	−	0.956425i	$-0.405687\pi$
$739$	−18.0627	−	18.0627i	−0.664449	−	0.664449i	−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		0			0
$742$	−0.634637	−	1.40588i	−0.0232983	−	0.0516115i
$743$	40.0000			1.46746			0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000			1.46746			0.733729	+	0.679442i	$0.237778\pi$
$744$		0			0
$745$		0			0
$746$	−4.01111	−	8.88562i	−0.146857	−	0.325326i
$747$		0			0
$748$		0			0
$749$	−33.4575	+	33.4575i	−1.22251	+	1.22251i
$750$		0			0
$751$		−	26.4575i		−	0.965448i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$		−	26.4575i		−	0.965448i	0.875772	−	0.482724i	$-0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	32.2915	+	32.2915i	1.17365	+	1.17365i	0.981332	+	0.192323i	$0.0616021\pi$
$757$	32.2915	+	32.2915i	1.17365	+	1.17365i	0.192323	+	0.981332i	$0.438398\pi$
$758$	44.6974	−	20.1771i	1.62348	−	0.732866i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−9.81176	+	9.81176i	−0.355210	+	0.355210i
$764$	10.5830	−	12.0000i	0.382880	−	0.434145i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−31.7490	−	28.0000i	−1.14267	−	1.00774i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$	−8.28039	−	18.3431i	−0.297632	−	0.659330i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	19.9889	+	44.2804i	0.716636	+	1.58753i
$779$		0			0
$780$		0			0
$781$	−24.5830	+	24.5830i	−0.879649	+	0.879649i
$782$		0			0
$783$		0			0
$784$	3.50000	+	27.7804i	0.125000	+	0.992157i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	1.36536	+	21.7601i	0.0486390	+	0.775173i
$789$		0			0
$790$		0			0
$791$	56.0000			1.99113
$792$	16.4059	−	53.2804i	0.582958	−	1.89324i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	27.5000	+	6.61438i	0.972272	+	0.233854i
$801$		0			0
$802$	−28.0000	−	10.5830i	−0.988714	−	0.373699i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	38.0000i		−	1.33601i	−0.744157	−	0.668004i	$-0.767149\pi$
$809$		−	38.0000i		−	1.33601i	0.744157	−	0.668004i	$-0.232851\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$	−32.0516	+	2.01111i	−1.12479	+	0.0705763i
$813$		0			0
$814$	101.915	+	38.5203i	3.57212	+	1.35013i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37.4575	−	37.4575i	−1.30728	−	1.30728i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	−37.4575	−	37.4575i	−1.30728	−	1.30728i	−0.383903	−	0.923374i	$-0.625420\pi$
$822$		0			0
$823$	−47.6235			−1.66005			−0.830026	−	0.557725i	$-0.811674\pi$
$823$	−47.6235			−1.66005			−0.830026	+	0.557725i	$0.811674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−40.5203	+	40.5203i	−1.40903	+	1.40903i	−0.644013	+	0.765015i	$0.722732\pi$
$827$	−40.5203	+	40.5203i	−1.40903	+	1.40903i	−0.765015	+	0.644013i	$0.777268\pi$
$828$	31.7490	−	36.0000i	1.10335	−	1.25109i
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	7.83399i		−	0.270138i
$842$	23.7601	+	52.6346i	0.818828	+	1.81391i
$843$		0			0
$844$	1.28039	+	20.4059i	0.0440728	+	0.702399i
$845$		0			0
$846$		0			0
$847$		−	85.1033i		−	2.92418i
$848$	−1.01111	+	1.30262i	−0.0347218	+	0.0447320i
$849$		0			0
$850$		0			0
$851$	66.3320	+	66.3320i	2.27383	+	2.27383i
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	48.3431	+	14.8856i	1.65233	+	0.508780i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	13.2288	−	35.0000i	0.450573	−	1.19210i
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	37.1660	+	37.1660i	1.26077	+	1.26077i
$870$		0			0
$871$		0			0
$872$	14.1771	+	4.36536i	0.480098	+	0.147830i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	40.8745	−	40.8745i	1.38023	−	1.38023i	0.536044	−	0.844190i	$-0.319918\pi$
$877$	40.8745	−	40.8745i	1.38023	−	1.38023i	0.844190	−	0.536044i	$-0.180082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	10.5000	−	27.7804i	0.353553	−	0.935414i
$883$	−35.1033	−	35.1033i	−1.18132	−	1.18132i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	−35.1033	−	35.1033i	−1.18132	−	1.18132i	−0.201916	−	0.979403i	$-0.564717\pi$
$884$		0			0
$885$		0			0
$886$	15.5314	−	7.01111i	0.521787	−	0.235543i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		−	42.0000i		−	1.40863i
$890$		0			0
$891$	−41.8118	+	41.8118i	−1.40075	+	1.40075i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	24.5000	−	17.1974i	0.818488	−	0.574524i
$897$		0			0
$898$	2.64575	+	1.00000i	0.0882899	+	0.0333704i
$899$		0			0
$900$	−22.5000	−	19.8431i	−0.750000	−	0.661438i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−28.0000	−	52.9150i	−0.931266	−	1.75993i
$905$		0			0
$906$		0			0
$907$	−27.3542	+	27.3542i	−0.908283	+	0.908283i	−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−27.3542	+	27.3542i	−0.908283	+	0.908283i	0.0878507	+	0.996134i	$0.472000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			58.2065i			1.92847i	0.265052	+	0.964234i	$0.414611\pi$
$911$			58.2065i			1.92847i	−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		0			0
$914$	−3.00000	+	7.93725i	−0.0992312	+	0.262541i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−37.0405			−1.22185			−0.610927	−	0.791687i	$-0.709203\pi$
$919$	−37.0405			−1.22185			−0.610927	+	0.791687i	$0.709203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	41.4575	−	41.4575i	1.36311	−	1.36311i
$926$	20.0000	−	52.9150i	0.657241	−	1.73890i
$927$		0			0
$928$	17.9261	+	29.2804i	0.588454	+	0.961176i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−28.0000	+	31.7490i	−0.917170	+	1.03997i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	12.9261	+	28.6346i	0.422053	+	0.934954i
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−15.5830	+	41.2288i	−0.506647	+	1.34046i
$947$	−19.1033	−	19.1033i	−0.620773	−	0.620773i	0.324956	−	0.945729i	$-0.394650\pi$
$947$	−19.1033	−	19.1033i	−0.620773	−	0.620773i	−0.945729	+	0.324956i	$0.894650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.1660i		−	0.685634i	−0.939402	−	0.342817i	$-0.888619\pi$
$953$		−	21.1660i		−	0.685634i	0.939402	−	0.342817i	$-0.111381\pi$
$954$	1.59412	−	0.719611i	0.0516115	−	0.0232983i
$955$		0			0
$956$	35.0000	−	39.6863i	1.13198	−	1.28355i
$957$		0			0
$958$		0			0
$959$			56.0000i			1.80833i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−37.9373	−	37.9373i	−1.22251	−	1.22251i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	40.0000			1.28631			0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000			1.28631			0.643157	+	0.765735i	$0.277624\pi$
$968$	−80.4150	+	42.5516i	−2.58464	+	1.36766i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$	31.7490	+	12.0000i	1.01730	+	0.384505i
$975$		0			0
$976$		0			0
$977$	−46.0000			−1.47167			−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000			−1.47167			−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−11.1255	−	11.1255i	−0.355210	−	0.355210i
$982$	−35.2804	+	15.9261i	−1.12584	+	0.508224i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.8340	+	26.8340i	−0.853271	+	0.853271i
$990$		0			0
$991$			24.0000i			0.762385i	0.924496	+	0.381193i	$0.124487\pi$
$991$			24.0000i			0.762385i	−0.924496	+	0.381193i	$0.875513\pi$
$992$		0			0
$993$		0			0
$994$	18.5203	+	7.00000i	0.587427	+	0.222027i
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	25.6974	+	56.9261i	0.813436	+	1.80197i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.j.b.27.1	✓	4
4.3	odd	2		448.2.j.a.335.1		4
7.2	even	3		784.2.w.c.619.2		8
7.3	odd	6		784.2.w.c.411.2		8
7.4	even	3		784.2.w.c.411.2		8
7.5	odd	6		784.2.w.c.619.2		8
7.6	odd	2	CM	112.2.j.b.27.1	✓	4
8.3	odd	2		896.2.j.d.671.1		4
8.5	even	2		896.2.j.c.671.2		4
16.3	odd	4	inner	112.2.j.b.83.1	yes	4
16.5	even	4		896.2.j.d.223.1		4
16.11	odd	4		896.2.j.c.223.2		4
16.13	even	4		448.2.j.a.111.1		4
28.27	even	2		448.2.j.a.335.1		4
56.13	odd	2		896.2.j.c.671.2		4
56.27	even	2		896.2.j.d.671.1		4
112.3	even	12		784.2.w.c.19.2		8
112.13	odd	4		448.2.j.a.111.1		4
112.19	even	12		784.2.w.c.227.2		8
112.27	even	4		896.2.j.c.223.2		4
112.51	odd	12		784.2.w.c.227.2		8
112.67	odd	12		784.2.w.c.19.2		8
112.69	odd	4		896.2.j.d.223.1		4
112.83	even	4	inner	112.2.j.b.83.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.b.27.1	✓	4	1.1	even	1	trivial
112.2.j.b.27.1	✓	4	7.6	odd	2	CM
112.2.j.b.83.1	yes	4	16.3	odd	4	inner
112.2.j.b.83.1	yes	4	112.83	even	4	inner
448.2.j.a.111.1		4	16.13	even	4
448.2.j.a.111.1		4	112.13	odd	4
448.2.j.a.335.1		4	4.3	odd	2
448.2.j.a.335.1		4	28.27	even	2
784.2.w.c.19.2		8	112.3	even	12
784.2.w.c.19.2		8	112.67	odd	12
784.2.w.c.227.2		8	112.19	even	12
784.2.w.c.227.2		8	112.51	odd	12
784.2.w.c.411.2		8	7.3	odd	6
784.2.w.c.411.2		8	7.4	even	3
784.2.w.c.619.2		8	7.2	even	3
784.2.w.c.619.2		8	7.5	odd	6
896.2.j.c.223.2		4	16.11	odd	4
896.2.j.c.223.2		4	112.27	even	4
896.2.j.c.671.2		4	8.5	even	2
896.2.j.c.671.2		4	56.13	odd	2
896.2.j.d.223.1		4	16.5	even	4
896.2.j.d.223.1		4	112.69	odd	4
896.2.j.d.671.1		4	8.3	odd	2
896.2.j.d.671.1		4	56.27	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32288 - 0.500000i) q^{2} +(1.50000 + 1.32288i) q^{4} +2.64575 q^{7} +(-1.32288 - 2.50000i) q^{8} +3.00000i q^{9} +(4.64575 - 4.64575i) q^{11} +(-3.50000 - 1.32288i) q^{14} +(0.500000 + 3.96863i) q^{16} +(1.50000 - 3.96863i) q^{18} +(-8.46863 + 3.82288i) q^{22} -8.00000 q^{23} +5.00000i q^{25} +(3.96863 + 3.50000i) q^{28} +(-4.29150 + 4.29150i) q^{29} +(1.32288 - 5.50000i) q^{32} +(-3.96863 + 4.50000i) q^{36} +(-8.29150 - 8.29150i) q^{37} +(3.35425 - 3.35425i) q^{43} +(13.1144 - 0.822876i) q^{44} +(10.5830 + 4.00000i) q^{46} +7.00000 q^{49} +(2.50000 - 6.61438i) q^{50} +(0.291503 + 0.291503i) q^{53} +(-3.50000 - 6.61438i) q^{56} +(7.82288 - 3.53137i) q^{58} +7.93725i q^{63} +(-4.50000 + 6.61438i) q^{64} +(-5.93725 - 5.93725i) q^{67} -5.29150 q^{71} +(7.50000 - 3.96863i) q^{72} +(6.82288 + 15.1144i) q^{74} +(12.2915 - 12.2915i) q^{77} +8.00000i q^{79} -9.00000 q^{81} +(-6.11438 + 2.76013i) q^{86} +(-17.7601 - 5.46863i) q^{88} +(-12.0000 - 10.5830i) q^{92} +(-9.26013 - 3.50000i) q^{98} +(13.9373 + 13.9373i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 8 q^{11} - 14 q^{14} + 2 q^{16} + 6 q^{18} - 18 q^{22} - 32 q^{23} + 4 q^{29} - 12 q^{37} + 24 q^{43} + 26 q^{44} + 28 q^{49} + 10 q^{50} - 20 q^{53} - 14 q^{56} + 26 q^{58} - 18 q^{64}+ \cdots + 24 q^{99}+O(q^{100}) \) 4 * q + 6 * q^4 + 8 * q^11 - 14 * q^14 + 2 * q^16 + 6 * q^18 - 18 * q^22 - 32 * q^23 + 4 * q^29 - 12 * q^37 + 24 * q^43 + 26 * q^44 + 28 * q^49 + 10 * q^50 - 20 * q^53 - 14 * q^56 + 26 * q^58 - 18 * q^64 + 8 * q^67 + 30 * q^72 + 22 * q^74 + 28 * q^77 - 36 * q^81 + 2 * q^86 - 34 * q^88 - 48 * q^92 + 24 * q^99

Introduction

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