LMFDB - Embedded newform 112.2.f.a.111.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,2,Mod(111,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("112.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	112.f (of order $2$, degree $1$, 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:	$0.894324502638$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	112.111
Dual form			112.2.f.a.111.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +3.46410i q^{5} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +3.46410i q^{5} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +3.46410i q^{11} -3.46410i q^{13} -6.92820i q^{15} -2.00000 q^{19} +(-4.00000 - 3.46410i) q^{21} -3.46410i q^{23} -7.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} -6.92820i q^{33} +(-6.00000 + 6.92820i) q^{35} -2.00000 q^{37} +6.92820i q^{39} +6.92820i q^{41} -10.3923i q^{43} +3.46410i q^{45} +(1.00000 + 6.92820i) q^{49} +6.00000 q^{53} -12.0000 q^{55} +4.00000 q^{57} -6.00000 q^{59} -3.46410i q^{61} +(2.00000 + 1.73205i) q^{63} +12.0000 q^{65} +3.46410i q^{67} +6.92820i q^{69} +3.46410i q^{71} -6.92820i q^{73} +14.0000 q^{75} +(-6.00000 + 6.92820i) q^{77} +3.46410i q^{79} -11.0000 q^{81} +6.00000 q^{83} -12.0000 q^{87} +6.92820i q^{89} +(6.00000 - 6.92820i) q^{91} -16.0000 q^{93} -6.92820i q^{95} -13.8564i q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9} - 4 q^{19} - 8 q^{21} - 14 q^{25} + 8 q^{27} + 12 q^{29} + 16 q^{31} - 12 q^{35} - 4 q^{37} + 2 q^{49} + 12 q^{53} - 24 q^{55} + 8 q^{57} - 12 q^{59} + 4 q^{63} + 24 q^{65} + 28 q^{75} - 12 q^{77} - 22 q^{81} + 12 q^{83} - 24 q^{87} + 12 q^{91} - 32 q^{93}+O(q^{100}) $ 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9 - 4 * q^19 - 8 * q^21 - 14 * q^25 + 8 * q^27 + 12 * q^29 + 16 * q^31 - 12 * q^35 - 4 * q^37 + 2 * q^49 + 12 * q^53 - 24 * q^55 + 8 * q^57 - 12 * q^59 + 4 * q^63 + 24 * q^65 + 28 * q^75 - 12 * q^77 - 22 * q^81 + 12 * q^83 - 24 * q^87 + 12 * q^91 - 32 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−2.00000			−1.15470			−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000			−1.15470			−0.577350	+	0.816497i	$0.695913\pi$
$4$		0			0
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		−	6.92820i		−	1.78885i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$	−4.00000	−	3.46410i	−0.872872	−	0.755929i
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$	4.00000			0.769800
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		−	6.92820i		−	1.20605i
$34$		0			0
$35$	−6.00000	+	6.92820i	−1.01419	+	1.17108i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$			6.92820i			1.10940i
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$		−	10.3923i		−	1.58481i	−0.609994	−	0.792406i	$-0.708828\pi$
$43$		−	10.3923i		−	1.58481i	0.609994	−	0.792406i	$-0.291172\pi$
$44$		0			0
$45$			3.46410i			0.516398i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		−	3.46410i		−	0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$		−	3.46410i		−	0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$		0			0
$63$	2.00000	+	1.73205i	0.251976	+	0.218218i
$64$		0			0
$65$	12.0000			1.48842
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$			6.92820i			0.834058i
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	14.0000			1.61658
$76$		0			0
$77$	−6.00000	+	6.92820i	−0.683763	+	0.789542i
$78$		0			0
$79$			3.46410i			0.389742i	0.980829	+	0.194871i	$0.0624288\pi$
$79$			3.46410i			0.389742i	−0.980829	+	0.194871i	$0.937571\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−12.0000			−1.28654
$88$		0			0
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	6.00000	−	6.92820i	0.628971	−	0.726273i
$92$		0			0
$93$	−16.0000			−1.65912
$94$		0			0
$95$		−	6.92820i		−	0.710819i
$96$		0			0
$97$		−	13.8564i		−	1.40690i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$		−	13.8564i		−	1.40690i	0.710742	−	0.703452i	$-0.248359\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$		−	10.3923i		−	1.03407i	−0.855963	−	0.517036i	$-0.827035\pi$
$101$		−	10.3923i		−	1.03407i	0.855963	−	0.517036i	$-0.172965\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	12.0000	−	13.8564i	1.17108	−	1.35225i
$106$		0			0
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	12.0000			1.11901
$116$		0			0
$117$		−	3.46410i		−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	13.8564i		−	1.24939i
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$			20.7846i			1.82998i
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$	−4.00000	−	3.46410i	−0.346844	−	0.300376i
$134$		0			0
$135$			13.8564i			1.19257i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$			20.7846i			1.72607i
$146$		0			0
$147$	−2.00000	−	13.8564i	−0.164957	−	1.14286i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			27.7128i			2.22595i
$156$		0			0
$157$			10.3923i			0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$			10.3923i			0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$		0			0
$163$			17.3205i			1.35665i	0.734763	+	0.678323i	$0.237293\pi$
$163$			17.3205i			1.35665i	−0.734763	+	0.678323i	$0.762707\pi$
$164$		0			0
$165$	24.0000			1.86840
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$		−	3.46410i		−	0.263371i	−0.991292	−	0.131685i	$-0.957961\pi$
$173$		−	3.46410i		−	0.263371i	0.991292	−	0.131685i	$-0.0420389\pi$
$174$		0			0
$175$	−14.0000	−	12.1244i	−1.05830	−	0.916515i
$176$		0			0
$177$	12.0000			0.901975
$178$		0			0
$179$		−	10.3923i		−	0.776757i	−0.921500	−	0.388379i	$-0.873035\pi$
$179$		−	10.3923i		−	0.776757i	0.921500	−	0.388379i	$-0.126965\pi$
$180$		0			0
$181$			17.3205i			1.28742i	0.765268	+	0.643712i	$0.222606\pi$
$181$			17.3205i			1.28742i	−0.765268	+	0.643712i	$0.777394\pi$
$182$		0			0
$183$			6.92820i			0.512148i
$184$		0			0
$185$		−	6.92820i		−	0.509372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.00000	+	6.92820i	0.581914	+	0.503953i
$190$		0			0
$191$		−	24.2487i		−	1.75458i	−0.479965	−	0.877288i	$-0.659351\pi$
$191$		−	24.2487i		−	1.75458i	0.479965	−	0.877288i	$-0.340649\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	−24.0000			−1.71868
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		−	6.92820i		−	0.488678i
$202$		0			0
$203$	12.0000	+	10.3923i	0.842235	+	0.729397i
$204$		0			0
$205$	−24.0000			−1.67623
$206$		0			0
$207$		−	3.46410i		−	0.240772i
$208$		0			0
$209$		−	6.92820i		−	0.479234i
$210$		0			0
$211$			3.46410i			0.238479i	0.992866	+	0.119239i	$0.0380456\pi$
$211$			3.46410i			0.238479i	−0.992866	+	0.119239i	$0.961954\pi$
$212$		0			0
$213$		−	6.92820i		−	0.474713i
$214$		0			0
$215$	36.0000			2.45518
$216$		0			0
$217$	16.0000	+	13.8564i	1.08615	+	0.940634i
$218$		0			0
$219$			13.8564i			0.936329i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−7.00000			−0.466667
$226$		0			0
$227$	6.00000			0.398234			0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000			0.398234			0.199117	+	0.979976i	$0.436193\pi$
$228$		0			0
$229$		−	10.3923i		−	0.686743i	−0.939200	−	0.343371i	$-0.888431\pi$
$229$		−	10.3923i		−	0.686743i	0.939200	−	0.343371i	$-0.111569\pi$
$230$		0			0
$231$	12.0000	−	13.8564i	0.789542	−	0.911685i
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	6.92820i		−	0.450035i
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$		−	27.7128i		−	1.78514i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$		−	27.7128i		−	1.78514i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$	10.0000			0.641500
$244$		0			0
$245$	−24.0000	+	3.46410i	−1.53330	+	0.221313i
$246$		0			0
$247$			6.92820i			0.440831i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$	−30.0000			−1.89358			−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000			−1.89358			−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	27.7128i		−	1.72868i	−0.502910	−	0.864339i	$-0.667737\pi$
$257$		−	27.7128i		−	1.72868i	0.502910	−	0.864339i	$-0.332263\pi$
$258$		0			0
$259$	−4.00000	−	3.46410i	−0.248548	−	0.215249i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	24.2487i		−	1.49524i	−0.664127	−	0.747620i	$-0.731197\pi$
$263$		−	24.2487i		−	1.49524i	0.664127	−	0.747620i	$-0.268803\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$		0			0
$267$		−	13.8564i		−	0.847998i
$268$		0			0
$269$		−	3.46410i		−	0.211210i	−0.994408	−	0.105605i	$-0.966322\pi$
$269$		−	3.46410i		−	0.211210i	0.994408	−	0.105605i	$-0.0336779\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$	−12.0000	+	13.8564i	−0.726273	+	0.838628i
$274$		0			0
$275$		−	24.2487i		−	1.46225i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$			13.8564i			0.820783i
$286$		0			0
$287$	−12.0000	+	13.8564i	−0.708338	+	0.817918i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$			27.7128i			1.62455i
$292$		0			0
$293$		−	10.3923i		−	0.607125i	−0.952812	−	0.303562i	$-0.901824\pi$
$293$		−	10.3923i		−	0.607125i	0.952812	−	0.303562i	$-0.0981761\pi$
$294$		0			0
$295$		−	20.7846i		−	1.21013i
$296$		0			0
$297$			13.8564i			0.804030i
$298$		0			0
$299$	−12.0000			−0.693978
$300$		0			0
$301$	18.0000	−	20.7846i	1.03750	−	1.19800i
$302$		0			0
$303$			20.7846i			1.19404i
$304$		0			0
$305$	12.0000			0.687118
$306$		0			0
$307$	22.0000			1.25561			0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000			1.25561			0.627803	+	0.778372i	$0.283954\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	6.92820i		−	0.391605i	−0.980643	−	0.195803i	$-0.937269\pi$
$313$		−	6.92820i		−	0.391605i	0.980643	−	0.195803i	$-0.0627312\pi$
$314$		0			0
$315$	−6.00000	+	6.92820i	−0.338062	+	0.390360i
$316$		0			0
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$			20.7846i			1.16371i
$320$		0			0
$321$			20.7846i			1.16008i
$322$		0			0
$323$		0			0
$324$		0			0
$325$			24.2487i			1.34508i
$326$		0			0
$327$	−28.0000			−1.54840
$328$		0			0
$329$		0			0
$330$		0			0
$331$			3.46410i			0.190404i	0.995458	+	0.0952021i	$0.0303497\pi$
$331$			3.46410i			0.190404i	−0.995458	+	0.0952021i	$0.969650\pi$
$332$		0			0
$333$	−2.00000			−0.109599
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	36.0000			1.95525
$340$		0			0
$341$			27.7128i			1.50073i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$	−24.0000			−1.29212
$346$		0			0
$347$			17.3205i			0.929814i	0.885360	+	0.464907i	$0.153912\pi$
$347$			17.3205i			0.929814i	−0.885360	+	0.464907i	$0.846088\pi$
$348$		0			0
$349$		−	3.46410i		−	0.185429i	−0.995693	−	0.0927146i	$-0.970446\pi$
$349$		−	3.46410i		−	0.185429i	0.995693	−	0.0927146i	$-0.0295544\pi$
$350$		0			0
$351$		−	13.8564i		−	0.739600i
$352$		0			0
$353$			27.7128i			1.47500i	0.675345	+	0.737502i	$0.263995\pi$
$353$			27.7128i			1.47500i	−0.675345	+	0.737502i	$0.736005\pi$
$354$		0			0
$355$	−12.0000			−0.636894
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	31.1769i		−	1.64545i	−0.568436	−	0.822727i	$-0.692451\pi$
$359$		−	31.1769i		−	1.64545i	0.568436	−	0.822727i	$-0.307549\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	2.00000			0.104973
$364$		0			0
$365$	24.0000			1.25622
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$			6.92820i			0.360668i
$370$		0			0
$371$	12.0000	+	10.3923i	0.623009	+	0.539542i
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$			13.8564i			0.715542i
$376$		0			0
$377$		−	20.7846i		−	1.07046i
$378$		0			0
$379$		−	24.2487i		−	1.24557i	−0.782392	−	0.622786i	$-0.786001\pi$
$379$		−	24.2487i		−	1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$		0			0
$381$		−	20.7846i		−	1.06483i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$	−24.0000	−	20.7846i	−1.22315	−	1.05928i
$386$		0			0
$387$		−	10.3923i		−	0.528271i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	36.0000			1.81596
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$			10.3923i			0.521575i	0.965396	+	0.260787i	$0.0839822\pi$
$397$			10.3923i			0.521575i	−0.965396	+	0.260787i	$0.916018\pi$
$398$		0			0
$399$	8.00000	+	6.92820i	0.400501	+	0.346844i
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		−	27.7128i		−	1.38047i
$404$		0			0
$405$		−	38.1051i		−	1.89346i
$406$		0			0
$407$		−	6.92820i		−	0.343418i
$408$		0			0
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$	−12.0000	−	10.3923i	−0.590481	−	0.511372i
$414$		0			0
$415$			20.7846i			1.02028i
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00000	−	6.92820i	0.290360	−	0.335279i
$428$		0			0
$429$	−24.0000			−1.15873
$430$		0			0
$431$			38.1051i			1.83546i	0.397206	+	0.917729i	$0.369980\pi$
$431$			38.1051i			1.83546i	−0.397206	+	0.917729i	$0.630020\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		−	41.5692i		−	1.99309i
$436$		0			0
$437$			6.92820i			0.331421i
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	1.00000	+	6.92820i	0.0476190	+	0.329914i
$442$		0			0
$443$			3.46410i			0.164584i	0.996608	+	0.0822922i	$0.0262241\pi$
$443$			3.46410i			0.164584i	−0.996608	+	0.0822922i	$0.973776\pi$
$444$		0			0
$445$	−24.0000			−1.13771
$446$		0			0
$447$	−12.0000			−0.567581
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		0			0
$453$			6.92820i			0.325515i
$454$		0			0
$455$	24.0000	+	20.7846i	1.12514	+	0.974398i
$456$		0			0
$457$	−34.0000			−1.59045			−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000			−1.59045			−0.795226	+	0.606313i	$0.792648\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			38.1051i			1.77473i	0.461065	+	0.887366i	$0.347467\pi$
$461$			38.1051i			1.77473i	−0.461065	+	0.887366i	$0.652533\pi$
$462$		0			0
$463$		−	24.2487i		−	1.12693i	−0.826139	−	0.563467i	$-0.809467\pi$
$463$		−	24.2487i		−	1.12693i	0.826139	−	0.563467i	$-0.190533\pi$
$464$		0			0
$465$		−	55.4256i		−	2.57030i
$466$		0			0
$467$	30.0000			1.38823			0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000			1.38823			0.694117	+	0.719862i	$0.255795\pi$
$468$		0			0
$469$	−6.00000	+	6.92820i	−0.277054	+	0.319915i
$470$		0			0
$471$		−	20.7846i		−	0.957704i
$472$		0			0
$473$	36.0000			1.65528
$474$		0			0
$475$	14.0000			0.642364
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$			6.92820i			0.315899i
$482$		0			0
$483$	−12.0000	+	13.8564i	−0.546019	+	0.630488i
$484$		0			0
$485$	48.0000			2.17957
$486$		0			0
$487$			24.2487i			1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$			24.2487i			1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$		0			0
$489$		−	34.6410i		−	1.56652i
$490$		0			0
$491$			17.3205i			0.781664i	0.920462	+	0.390832i	$0.127813\pi$
$491$			17.3205i			0.781664i	−0.920462	+	0.390832i	$0.872187\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$	−6.00000	+	6.92820i	−0.269137	+	0.310772i
$498$		0			0
$499$		−	10.3923i		−	0.465223i	−0.972570	−	0.232612i	$-0.925273\pi$
$499$		−	10.3923i		−	0.465223i	0.972570	−	0.232612i	$-0.0747271\pi$
$500$		0			0
$501$	−24.0000			−1.07224
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$	−2.00000			−0.0888231
$508$		0			0
$509$		−	31.1769i		−	1.38189i	−0.722906	−	0.690946i	$-0.757194\pi$
$509$		−	31.1769i		−	1.38189i	0.722906	−	0.690946i	$-0.242806\pi$
$510$		0			0
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$		0			0
$513$	−8.00000			−0.353209
$514$		0			0
$515$		−	13.8564i		−	0.610586i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			6.92820i			0.304114i
$520$		0			0
$521$			34.6410i			1.51765i	0.651294	+	0.758825i	$0.274226\pi$
$521$			34.6410i			1.51765i	−0.651294	+	0.758825i	$0.725774\pi$
$522$		0			0
$523$	2.00000			0.0874539			0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000			0.0874539			0.0437269	+	0.999044i	$0.486077\pi$
$524$		0			0
$525$	28.0000	+	24.2487i	1.22202	+	1.05830i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$	−6.00000			−0.260378
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$	36.0000			1.55642
$536$		0			0
$537$			20.7846i			0.896922i
$538$		0			0
$539$	−24.0000	+	3.46410i	−1.03375	+	0.149209i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		−	34.6410i		−	1.48659i
$544$		0			0
$545$			48.4974i			2.07740i
$546$		0			0
$547$			17.3205i			0.740571i	0.928918	+	0.370286i	$0.120740\pi$
$547$			17.3205i			0.740571i	−0.928918	+	0.370286i	$0.879260\pi$
$548$		0			0
$549$		−	3.46410i		−	0.147844i
$550$		0			0
$551$	−12.0000			−0.511217
$552$		0			0
$553$	−6.00000	+	6.92820i	−0.255146	+	0.294617i
$554$		0			0
$555$			13.8564i			0.588172i
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		−	62.3538i		−	2.62325i
$566$		0			0
$567$	−22.0000	−	19.0526i	−0.923913	−	0.800132i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$			45.0333i			1.88459i	0.334790	+	0.942293i	$0.391335\pi$
$571$			45.0333i			1.88459i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$			48.4974i			2.02601i
$574$		0			0
$575$			24.2487i			1.01124i
$576$		0			0
$577$			27.7128i			1.15370i	0.816850	+	0.576850i	$0.195718\pi$
$577$			27.7128i			1.15370i	−0.816850	+	0.576850i	$0.804282\pi$
$578$		0			0
$579$	4.00000			0.166234
$580$		0			0
$581$	12.0000	+	10.3923i	0.497844	+	0.431145i
$582$		0			0
$583$			20.7846i			0.860811i
$584$		0			0
$585$	12.0000			0.496139
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	36.0000			1.48084
$592$		0			0
$593$			13.8564i			0.569014i	0.958674	+	0.284507i	$0.0918300\pi$
$593$			13.8564i			0.569014i	−0.958674	+	0.284507i	$0.908170\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	40.0000			1.63709
$598$		0			0
$599$		−	24.2487i		−	0.990775i	−0.868672	−	0.495388i	$-0.835026\pi$
$599$		−	24.2487i		−	0.990775i	0.868672	−	0.495388i	$-0.164974\pi$
$600$		0			0
$601$		−	20.7846i		−	0.847822i	−0.905704	−	0.423911i	$-0.860657\pi$
$601$		−	20.7846i		−	0.847822i	0.905704	−	0.423911i	$-0.139343\pi$
$602$		0			0
$603$			3.46410i			0.141069i
$604$		0			0
$605$		−	3.46410i		−	0.140836i
$606$		0			0
$607$	16.0000			0.649420			0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000			0.649420			0.324710	+	0.945814i	$0.394733\pi$
$608$		0			0
$609$	−24.0000	−	20.7846i	−0.972529	−	0.842235i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$	48.0000			1.93555
$616$		0			0
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		−	13.8564i		−	0.556038i
$622$		0			0
$623$	−12.0000	+	13.8564i	−0.480770	+	0.555145i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$			13.8564i			0.553372i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			3.46410i			0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$			3.46410i			0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$		0			0
$633$		−	6.92820i		−	0.275371i
$634$		0			0
$635$	−36.0000			−1.42862
$636$		0			0
$637$	24.0000	−	3.46410i	0.950915	−	0.137253i
$638$		0			0
$639$			3.46410i			0.137038i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	−26.0000			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$644$		0			0
$645$	−72.0000			−2.83500
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	−32.0000	−	27.7128i	−1.25418	−	1.08615i
$652$		0			0
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$		0			0
$655$		−	62.3538i		−	2.43637i
$656$		0			0
$657$		−	6.92820i		−	0.270295i
$658$		0			0
$659$			3.46410i			0.134942i	0.997721	+	0.0674711i	$0.0214931\pi$
$659$			3.46410i			0.134942i	−0.997721	+	0.0674711i	$0.978507\pi$
$660$		0			0
$661$			3.46410i			0.134738i	0.997728	+	0.0673690i	$0.0214605\pi$
$661$			3.46410i			0.134738i	−0.997728	+	0.0673690i	$0.978540\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	−	13.8564i	0.465340	−	0.537328i
$666$		0			0
$667$		−	20.7846i		−	0.804783i
$668$		0			0
$669$	−32.0000			−1.23719
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$	−28.0000			−1.07772
$676$		0			0
$677$			3.46410i			0.133136i	0.997782	+	0.0665681i	$0.0212050\pi$
$677$			3.46410i			0.133136i	−0.997782	+	0.0665681i	$0.978795\pi$
$678$		0			0
$679$	24.0000	−	27.7128i	0.921035	−	1.06352i
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$		−	38.1051i		−	1.45805i	−0.684486	−	0.729026i	$-0.739973\pi$
$683$		−	38.1051i		−	1.45805i	0.684486	−	0.729026i	$-0.260027\pi$
$684$		0			0
$685$		−	20.7846i		−	0.794139i
$686$		0			0
$687$			20.7846i			0.792982i
$688$		0			0
$689$		−	20.7846i		−	0.791831i
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$		0			0
$693$	−6.00000	+	6.92820i	−0.227921	+	0.263181i
$694$		0			0
$695$			6.92820i			0.262802i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	12.0000			0.453882
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.0000	−	20.7846i	0.676960	−	0.781686i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$			3.46410i			0.129914i
$712$		0			0
$713$		−	27.7128i		−	1.03785i
$714$		0			0
$715$			41.5692i			1.55460i
$716$		0			0
$717$		−	20.7846i		−	0.776215i
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	−8.00000	−	6.92820i	−0.297936	−	0.258020i
$722$		0			0
$723$			55.4256i			2.06130i
$724$		0			0
$725$	−42.0000			−1.55984
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$			51.9615i			1.91924i	0.281295	+	0.959621i	$0.409236\pi$
$733$			51.9615i			1.91924i	−0.281295	+	0.959621i	$0.590764\pi$
$734$		0			0
$735$	48.0000	−	6.92820i	1.77051	−	0.255551i
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$		−	10.3923i		−	0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$		−	10.3923i		−	0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$		0			0
$741$		−	13.8564i		−	0.509028i
$742$		0			0
$743$			24.2487i			0.889599i	0.895630	+	0.444799i	$0.146725\pi$
$743$			24.2487i			0.889599i	−0.895630	+	0.444799i	$0.853275\pi$
$744$		0			0
$745$			20.7846i			0.761489i
$746$		0			0
$747$	6.00000			0.219529
$748$		0			0
$749$	18.0000	−	20.7846i	0.657706	−	0.759453i
$750$		0			0
$751$		−	45.0333i		−	1.64329i	−0.570000	−	0.821645i	$-0.693057\pi$
$751$		−	45.0333i		−	1.64329i	0.570000	−	0.821645i	$-0.306943\pi$
$752$		0			0
$753$	60.0000			2.18652
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$	−24.0000			−0.871145
$760$		0			0
$761$			20.7846i			0.753442i	0.926327	+	0.376721i	$0.122948\pi$
$761$			20.7846i			0.753442i	−0.926327	+	0.376721i	$0.877052\pi$
$762$		0			0
$763$	28.0000	+	24.2487i	1.01367	+	0.877862i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20.7846i			0.750489i
$768$		0			0
$769$		−	13.8564i		−	0.499675i	−0.968288	−	0.249837i	$-0.919623\pi$
$769$		−	13.8564i		−	0.499675i	0.968288	−	0.249837i	$-0.0803772\pi$
$770$		0			0
$771$			55.4256i			1.99611i
$772$		0			0
$773$			3.46410i			0.124595i	0.998058	+	0.0622975i	$0.0198428\pi$
$773$			3.46410i			0.124595i	−0.998058	+	0.0622975i	$0.980157\pi$
$774$		0			0
$775$	−56.0000			−2.01158
$776$		0			0
$777$	8.00000	+	6.92820i	0.286998	+	0.248548i
$778$		0			0
$779$		−	13.8564i		−	0.496457i
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$	24.0000			0.857690
$784$		0			0
$785$	−36.0000			−1.28490
$786$		0			0
$787$	−34.0000			−1.21197			−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000			−1.21197			−0.605985	+	0.795476i	$0.707221\pi$
$788$		0			0
$789$			48.4974i			1.72655i
$790$		0			0
$791$	−36.0000	−	31.1769i	−1.28001	−	1.10852i
$792$		0			0
$793$	−12.0000			−0.426132
$794$		0			0
$795$		−	41.5692i		−	1.47431i
$796$		0			0
$797$		−	17.3205i		−	0.613524i	−0.951786	−	0.306762i	$-0.900754\pi$
$797$		−	17.3205i		−	0.613524i	0.951786	−	0.306762i	$-0.0992455\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			6.92820i			0.244796i
$802$		0			0
$803$	24.0000			0.846942
$804$		0			0
$805$	24.0000	+	20.7846i	0.845889	+	0.732561i
$806$		0			0
$807$			6.92820i			0.243884i
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	−60.0000			−2.10171
$816$		0			0
$817$			20.7846i			0.727161i
$818$		0			0
$819$	6.00000	−	6.92820i	0.209657	−	0.242091i
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$			3.46410i			0.120751i	0.998176	+	0.0603755i	$0.0192298\pi$
$823$			3.46410i			0.120751i	−0.998176	+	0.0603755i	$0.980770\pi$
$824$		0			0
$825$			48.4974i			1.68846i
$826$		0			0
$827$		−	24.2487i		−	0.843210i	−0.906780	−	0.421605i	$-0.861467\pi$
$827$		−	24.2487i		−	0.843210i	0.906780	−	0.421605i	$-0.138533\pi$
$828$		0			0
$829$		−	3.46410i		−	0.120313i	−0.998189	−	0.0601566i	$-0.980840\pi$
$829$		−	3.46410i		−	0.120313i	0.998189	−	0.0601566i	$-0.0191600\pi$
$830$		0			0
$831$	4.00000			0.138758
$832$		0			0
$833$		0			0
$834$		0			0
$835$			41.5692i			1.43856i
$836$		0			0
$837$	32.0000			1.10608
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	12.0000			0.413302
$844$		0			0
$845$			3.46410i			0.119169i
$846$		0			0
$847$	−2.00000	−	1.73205i	−0.0687208	−	0.0595140i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$			6.92820i			0.237496i
$852$		0			0
$853$			3.46410i			0.118609i	0.998240	+	0.0593043i	$0.0188882\pi$
$853$			3.46410i			0.118609i	−0.998240	+	0.0593043i	$0.981112\pi$
$854$		0			0
$855$		−	6.92820i		−	0.236940i
$856$		0			0
$857$		−	34.6410i		−	1.18331i	−0.806190	−	0.591657i	$-0.798474\pi$
$857$		−	34.6410i		−	1.18331i	0.806190	−	0.591657i	$-0.201526\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$	24.0000	−	27.7128i	0.817918	−	0.944450i
$862$		0			0
$863$			31.1769i			1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$			31.1769i			1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$	−34.0000			−1.15470
$868$		0			0
$869$	−12.0000			−0.407072
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$	12.0000	−	13.8564i	0.405674	−	0.468432i
$876$		0			0
$877$	38.0000			1.28317			0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000			1.28317			0.641584	+	0.767052i	$0.278277\pi$
$878$		0			0
$879$			20.7846i			0.701047i
$880$		0			0
$881$		−	13.8564i		−	0.466834i	−0.972377	−	0.233417i	$-0.925009\pi$
$881$		−	13.8564i		−	0.466834i	0.972377	−	0.233417i	$-0.0749907\pi$
$882$		0			0
$883$			3.46410i			0.116576i	0.998300	+	0.0582882i	$0.0185642\pi$
$883$			3.46410i			0.116576i	−0.998300	+	0.0582882i	$0.981436\pi$
$884$		0			0
$885$			41.5692i			1.39733i
$886$		0			0
$887$	−12.0000			−0.402921			−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000			−0.402921			−0.201460	+	0.979497i	$0.564569\pi$
$888$		0			0
$889$	−18.0000	+	20.7846i	−0.603701	+	0.697093i
$890$		0			0
$891$		−	38.1051i		−	1.27657i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	36.0000			1.20335
$896$		0			0
$897$	24.0000			0.801337
$898$		0			0
$899$	48.0000			1.60089
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−36.0000	+	41.5692i	−1.19800	+	1.38334i
$904$		0			0
$905$	−60.0000			−1.99447
$906$		0			0
$907$		−	24.2487i		−	0.805165i	−0.915384	−	0.402583i	$-0.868113\pi$
$907$		−	24.2487i		−	0.805165i	0.915384	−	0.402583i	$-0.131887\pi$
$908$		0			0
$909$		−	10.3923i		−	0.344691i
$910$		0			0
$911$			10.3923i			0.344312i	0.985070	+	0.172156i	$0.0550734\pi$
$911$			10.3923i			0.344312i	−0.985070	+	0.172156i	$0.944927\pi$
$912$		0			0
$913$			20.7846i			0.687870i
$914$		0			0
$915$	−24.0000			−0.793416
$916$		0			0
$917$	−36.0000	−	31.1769i	−1.18882	−	1.02955i
$918$		0			0
$919$			31.1769i			1.02843i	0.857661	+	0.514216i	$0.171917\pi$
$919$			31.1769i			1.02843i	−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$	−44.0000			−1.44985
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	14.0000			0.460317
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$		−	41.5692i		−	1.36384i	−0.731426	−	0.681921i	$-0.761145\pi$
$929$		−	41.5692i		−	1.36384i	0.731426	−	0.681921i	$-0.238855\pi$
$930$		0			0
$931$	−2.00000	−	13.8564i	−0.0655474	−	0.454125i
$932$		0			0
$933$	−24.0000			−0.785725
$934$		0			0
$935$		0			0
$936$		0			0
$937$			48.4974i			1.58434i	0.610299	+	0.792171i	$0.291049\pi$
$937$			48.4974i			1.58434i	−0.610299	+	0.792171i	$0.708951\pi$
$938$		0			0
$939$			13.8564i			0.452187i
$940$		0			0
$941$			24.2487i			0.790485i	0.918577	+	0.395243i	$0.129340\pi$
$941$			24.2487i			0.790485i	−0.918577	+	0.395243i	$0.870660\pi$
$942$		0			0
$943$	24.0000			0.781548
$944$		0			0
$945$	−24.0000	+	27.7128i	−0.780720	+	0.901498i
$946$		0			0
$947$		−	24.2487i		−	0.787977i	−0.919115	−	0.393989i	$-0.871095\pi$
$947$		−	24.2487i		−	0.787977i	0.919115	−	0.393989i	$-0.128905\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	−12.0000			−0.389127
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$	84.0000			2.71818
$956$		0			0
$957$		−	41.5692i		−	1.34374i
$958$		0			0
$959$	−12.0000	−	10.3923i	−0.387500	−	0.335585i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	10.3923i		−	0.334887i
$964$		0			0
$965$		−	6.92820i		−	0.223027i
$966$		0			0
$967$			24.2487i			0.779786i	0.920860	+	0.389893i	$0.127488\pi$
$967$			24.2487i			0.779786i	−0.920860	+	0.389893i	$0.872512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.0000			0.577647			0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000			0.577647			0.288824	+	0.957382i	$0.406736\pi$
$972$		0			0
$973$	4.00000	+	3.46410i	0.128234	+	0.111054i
$974$		0			0
$975$		−	48.4974i		−	1.55316i
$976$		0			0
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	14.0000			0.446986
$982$		0			0
$983$	36.0000			1.14822			0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000			1.14822			0.574111	+	0.818778i	$0.305348\pi$
$984$		0			0
$985$		−	62.3538i		−	1.98676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−36.0000			−1.14473
$990$		0			0
$991$			3.46410i			0.110041i	0.998485	+	0.0550204i	$0.0175224\pi$
$991$			3.46410i			0.110041i	−0.998485	+	0.0550204i	$0.982478\pi$
$992$		0			0
$993$		−	6.92820i		−	0.219860i
$994$		0			0
$995$		−	69.2820i		−	2.19639i
$996$		0			0
$997$		−	51.9615i		−	1.64564i	−0.568304	−	0.822819i	$-0.692400\pi$
$997$		−	51.9615i		−	1.64564i	0.568304	−	0.822819i	$-0.307600\pi$
$998$		0			0
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.f.a.111.2	yes	2
3.2	odd	2		1008.2.b.g.559.1		2
4.3	odd	2		112.2.f.b.111.2	yes	2
5.2	odd	4		2800.2.e.c.2799.3		4
5.3	odd	4		2800.2.e.c.2799.2		4
5.4	even	2		2800.2.k.e.2351.1		2
7.2	even	3		784.2.p.f.31.1		2
7.3	odd	6		784.2.p.b.607.1		2
7.4	even	3		784.2.p.e.607.1		2
7.5	odd	6		784.2.p.a.31.1		2
7.6	odd	2		112.2.f.b.111.1	yes	2
8.3	odd	2		448.2.f.a.447.1		2
8.5	even	2		448.2.f.c.447.1		2
12.11	even	2		1008.2.b.b.559.1		2
16.3	odd	4		1792.2.e.c.895.4		4
16.5	even	4		1792.2.e.a.895.3		4
16.11	odd	4		1792.2.e.c.895.1		4
16.13	even	4		1792.2.e.a.895.2		4
20.3	even	4		2800.2.e.b.2799.3		4
20.7	even	4		2800.2.e.b.2799.2		4
20.19	odd	2		2800.2.k.b.2351.2		2
21.20	even	2		1008.2.b.b.559.2		2
24.5	odd	2		4032.2.b.h.3583.2		2
24.11	even	2		4032.2.b.b.3583.2		2
28.3	even	6		784.2.p.f.607.1		2
28.11	odd	6		784.2.p.a.607.1		2
28.19	even	6		784.2.p.e.31.1		2
28.23	odd	6		784.2.p.b.31.1		2
28.27	even	2	inner	112.2.f.a.111.1	✓	2
35.13	even	4		2800.2.e.b.2799.4		4
35.27	even	4		2800.2.e.b.2799.1		4
35.34	odd	2		2800.2.k.b.2351.1		2
56.13	odd	2		448.2.f.a.447.2		2
56.27	even	2		448.2.f.c.447.2		2
84.83	odd	2		1008.2.b.g.559.2		2
112.13	odd	4		1792.2.e.c.895.3		4
112.27	even	4		1792.2.e.a.895.4		4
112.69	odd	4		1792.2.e.c.895.2		4
112.83	even	4		1792.2.e.a.895.1		4
140.27	odd	4		2800.2.e.c.2799.4		4
140.83	odd	4		2800.2.e.c.2799.1		4
140.139	even	2		2800.2.k.e.2351.2		2
168.83	odd	2		4032.2.b.h.3583.1		2
168.125	even	2		4032.2.b.b.3583.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.f.a.111.1	✓	2	28.27	even	2	inner
112.2.f.a.111.2	yes	2	1.1	even	1	trivial
112.2.f.b.111.1	yes	2	7.6	odd	2
112.2.f.b.111.2	yes	2	4.3	odd	2
448.2.f.a.447.1		2	8.3	odd	2
448.2.f.a.447.2		2	56.13	odd	2
448.2.f.c.447.1		2	8.5	even	2
448.2.f.c.447.2		2	56.27	even	2
784.2.p.a.31.1		2	7.5	odd	6
784.2.p.a.607.1		2	28.11	odd	6
784.2.p.b.31.1		2	28.23	odd	6
784.2.p.b.607.1		2	7.3	odd	6
784.2.p.e.31.1		2	28.19	even	6
784.2.p.e.607.1		2	7.4	even	3
784.2.p.f.31.1		2	7.2	even	3
784.2.p.f.607.1		2	28.3	even	6
1008.2.b.b.559.1		2	12.11	even	2
1008.2.b.b.559.2		2	21.20	even	2
1008.2.b.g.559.1		2	3.2	odd	2
1008.2.b.g.559.2		2	84.83	odd	2
1792.2.e.a.895.1		4	112.83	even	4
1792.2.e.a.895.2		4	16.13	even	4
1792.2.e.a.895.3		4	16.5	even	4
1792.2.e.a.895.4		4	112.27	even	4
1792.2.e.c.895.1		4	16.11	odd	4
1792.2.e.c.895.2		4	112.69	odd	4
1792.2.e.c.895.3		4	112.13	odd	4
1792.2.e.c.895.4		4	16.3	odd	4
2800.2.e.b.2799.1		4	35.27	even	4
2800.2.e.b.2799.2		4	20.7	even	4
2800.2.e.b.2799.3		4	20.3	even	4
2800.2.e.b.2799.4		4	35.13	even	4
2800.2.e.c.2799.1		4	140.83	odd	4
2800.2.e.c.2799.2		4	5.3	odd	4
2800.2.e.c.2799.3		4	5.2	odd	4
2800.2.e.c.2799.4		4	140.27	odd	4
2800.2.k.b.2351.1		2	35.34	odd	2
2800.2.k.b.2351.2		2	20.19	odd	2
2800.2.k.e.2351.1		2	5.4	even	2
2800.2.k.e.2351.2		2	140.139	even	2
4032.2.b.b.3583.1		2	168.125	even	2
4032.2.b.b.3583.2		2	24.11	even	2
4032.2.b.h.3583.1		2	168.83	odd	2
4032.2.b.h.3583.2		2	24.5	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +3.46410i q^{5} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +3.46410i q^{5} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +3.46410i q^{11} -3.46410i q^{13} -6.92820i q^{15} -2.00000 q^{19} +(-4.00000 - 3.46410i) q^{21} -3.46410i q^{23} -7.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} -6.92820i q^{33} +(-6.00000 + 6.92820i) q^{35} -2.00000 q^{37} +6.92820i q^{39} +6.92820i q^{41} -10.3923i q^{43} +3.46410i q^{45} +(1.00000 + 6.92820i) q^{49} +6.00000 q^{53} -12.0000 q^{55} +4.00000 q^{57} -6.00000 q^{59} -3.46410i q^{61} +(2.00000 + 1.73205i) q^{63} +12.0000 q^{65} +3.46410i q^{67} +6.92820i q^{69} +3.46410i q^{71} -6.92820i q^{73} +14.0000 q^{75} +(-6.00000 + 6.92820i) q^{77} +3.46410i q^{79} -11.0000 q^{81} +6.00000 q^{83} -12.0000 q^{87} +6.92820i q^{89} +(6.00000 - 6.92820i) q^{91} -16.0000 q^{93} -6.92820i q^{95} -13.8564i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9} - 4 q^{19} - 8 q^{21} - 14 q^{25} + 8 q^{27} + 12 q^{29} + 16 q^{31} - 12 q^{35} - 4 q^{37} + 2 q^{49} + 12 q^{53} - 24 q^{55} + 8 q^{57} - 12 q^{59} + 4 q^{63} + 24 q^{65} + 28 q^{75} - 12 q^{77} - 22 q^{81} + 12 q^{83} - 24 q^{87} + 12 q^{91} - 32 q^{93}+O(q^{100}) \) 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9 - 4 * q^19 - 8 * q^21 - 14 * q^25 + 8 * q^27 + 12 * q^29 + 16 * q^31 - 12 * q^35 - 4 * q^37 + 2 * q^49 + 12 * q^53 - 24 * q^55 + 8 * q^57 - 12 * q^59 + 4 * q^63 + 24 * q^65 + 28 * q^75 - 12 * q^77 - 22 * q^81 + 12 * q^83 - 24 * q^87 + 12 * q^91 - 32 * q^93

Introduction

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