LMFDB - Embedded newform 1792.2.e.a.895.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1792,2,Mod(895,1792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1792.895"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-8,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.a.895.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -3.46410 q^{5} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{9} -3.46410 q^{11} -3.46410 q^{13} +6.92820i q^{15} -2.00000i q^{19} +(3.46410 + 4.00000i) q^{21} -3.46410i q^{23} +7.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} +6.92820i q^{33} +(6.92820 - 6.00000i) q^{35} +2.00000i q^{37} +6.92820i q^{39} +6.92820i q^{41} +10.3923 q^{43} +3.46410 q^{45} +(1.00000 - 6.92820i) q^{49} -6.00000i q^{53} +12.0000 q^{55} -4.00000 q^{57} +6.00000i q^{59} -3.46410 q^{61} +(2.00000 - 1.73205i) q^{63} +12.0000 q^{65} +3.46410 q^{67} -6.92820 q^{69} +3.46410i q^{71} -6.92820i q^{73} -14.0000i q^{75} +(6.92820 - 6.00000i) q^{77} -3.46410i q^{79} -11.0000 q^{81} +6.00000i q^{83} +12.0000 q^{87} +6.92820i q^{89} +(6.92820 - 6.00000i) q^{91} -16.0000i q^{93} +6.92820i q^{95} +13.8564i q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 4 q^{9} + 28 q^{25} + 32 q^{31} + 4 q^{49} + 48 q^{55} - 16 q^{57} + 8 q^{63} + 48 q^{65} - 44 q^{81} + 48 q^{87}+O(q^{100}) $ 4 * q - 8 * q^7 - 4 * q^9 + 28 * q^25 + 32 * q^31 + 4 * q^49 + 48 * q^55 - 16 * q^57 + 8 * q^63 + 48 * q^65 - 44 * q^81 + 48 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$		0			0
$5$	−3.46410			−1.54919			−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410			−1.54919			−0.774597	+	0.632456i	$0.782047\pi$
$6$		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
$11$	−3.46410			−1.04447			−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410			−1.04447			−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$	−3.46410			−0.960769			−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410			−0.960769			−0.480384	+	0.877058i	$0.659503\pi$
$14$		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
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$	3.46410	+	4.00000i	0.755929	+	0.872872i
$22$		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
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		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
$35$	6.92820	−	6.00000i	1.17108	−	1.01419i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		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
$43$	10.3923			1.58481			0.792406	−	0.609994i	$-0.208828\pi$
$43$	10.3923			1.58481			0.792406	+	0.609994i	$0.208828\pi$
$44$		0			0
$45$	3.46410			0.516398
$46$		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
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	−3.46410			−0.443533			−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410			−0.443533			−0.221766	+	0.975100i	$0.571182\pi$
$62$		0			0
$63$	2.00000	−	1.73205i	0.251976	−	0.218218i
$64$		0			0
$65$	12.0000			1.48842
$66$		0			0
$67$	3.46410			0.423207			0.211604	−	0.977356i	$-0.432131\pi$
$67$	3.46410			0.423207			0.211604	+	0.977356i	$0.432131\pi$
$68$		0			0
$69$	−6.92820			−0.834058
$70$		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
$75$		−	14.0000i		−	1.61658i
$76$		0			0
$77$	6.92820	−	6.00000i	0.789542	−	0.683763i
$78$		0			0
$79$		−	3.46410i		−	0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$		−	3.46410i		−	0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		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
$91$	6.92820	−	6.00000i	0.726273	−	0.628971i
$92$		0			0
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$			6.92820i			0.710819i
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	3.46410			0.348155
$100$		0			0
$101$	10.3923			1.03407			0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923			1.03407			0.517036	+	0.855963i	$0.327035\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	−12.0000	−	13.8564i	−1.17108	−	1.35225i
$106$		0			0
$107$	10.3923			1.00466			0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923			1.00466			0.502331	+	0.864675i	$0.332476\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		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
$115$			12.0000i			1.11901i
$116$		0			0
$117$	3.46410			0.320256
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	13.8564			1.24939
$124$		0			0
$125$	−6.92820			−0.619677
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		0			0
$129$		−	20.7846i		−	1.82998i
$130$		0			0
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$		0			0
$133$	3.46410	+	4.00000i	0.300376	+	0.346844i
$134$		0			0
$135$			13.8564i			1.19257i
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$		−	2.00000i		−	0.169638i	−0.996396	−	0.0848189i	$-0.972969\pi$
$139$		−	2.00000i		−	0.169638i	0.996396	−	0.0848189i	$-0.0270312\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$		−	20.7846i		−	1.72607i
$146$		0			0
$147$	−13.8564	−	2.00000i	−1.14286	−	0.164957i
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		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
$155$	−27.7128			−2.22595
$156$		0			0
$157$	10.3923			0.829396			0.414698	−	0.909959i	$-0.363887\pi$
$157$	10.3923			0.829396			0.414698	+	0.909959i	$0.363887\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$	6.00000	+	6.92820i	0.472866	+	0.546019i
$162$		0			0
$163$	17.3205			1.35665			0.678323	−	0.734763i	$-0.262707\pi$
$163$	17.3205			1.35665			0.678323	+	0.734763i	$0.262707\pi$
$164$		0			0
$165$		−	24.0000i		−	1.86840i
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$			2.00000i			0.152944i
$172$		0			0
$173$	−3.46410			−0.263371			−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410			−0.263371			−0.131685	+	0.991292i	$0.542039\pi$
$174$		0			0
$175$	−14.0000	+	12.1244i	−1.05830	+	0.916515i
$176$		0			0
$177$	12.0000			0.901975
$178$		0			0
$179$	−10.3923			−0.776757			−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923			−0.776757			−0.388379	+	0.921500i	$0.626965\pi$
$180$		0			0
$181$	−17.3205			−1.28742			−0.643712	−	0.765268i	$-0.722606\pi$
$181$	−17.3205			−1.28742			−0.643712	+	0.765268i	$0.722606\pi$
$182$		0			0
$183$			6.92820i			0.512148i
$184$		0			0
$185$		−	6.92820i		−	0.509372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	6.92820	+	8.00000i	0.503953	+	0.581914i
$190$		0			0
$191$			24.2487i			1.75458i	0.479965	+	0.877288i	$0.340649\pi$
$191$			24.2487i			1.75458i	−0.479965	+	0.877288i	$0.659351\pi$
$192$		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
$195$		−	24.0000i		−	1.71868i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		−	6.92820i		−	0.488678i
$202$		0			0
$203$	−10.3923	−	12.0000i	−0.729397	−	0.842235i
$204$		0			0
$205$		−	24.0000i		−	1.67623i
$206$		0			0
$207$			3.46410i			0.240772i
$208$		0			0
$209$			6.92820i			0.479234i
$210$		0			0
$211$	3.46410			0.238479			0.119239	−	0.992866i	$-0.461954\pi$
$211$	3.46410			0.238479			0.119239	+	0.992866i	$0.461954\pi$
$212$		0			0
$213$	6.92820			0.474713
$214$		0			0
$215$	−36.0000			−2.45518
$216$		0			0
$217$	−16.0000	+	13.8564i	−1.08615	+	0.940634i
$218$		0			0
$219$	−13.8564			−0.936329
$220$		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
$227$			6.00000i			0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$			6.00000i			0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$		0			0
$229$	10.3923			0.686743			0.343371	−	0.939200i	$-0.388431\pi$
$229$	10.3923			0.686743			0.343371	+	0.939200i	$0.388431\pi$
$230$		0			0
$231$	−12.0000	−	13.8564i	−0.789542	−	0.911685i
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.92820			−0.450035
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$			27.7128i			1.78514i	0.450910	+	0.892570i	$0.351100\pi$
$241$			27.7128i			1.78514i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	−3.46410	+	24.0000i	−0.221313	+	1.53330i
$246$		0			0
$247$			6.92820i			0.440831i
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$			30.0000i			1.89358i	0.321847	+	0.946792i	$0.395696\pi$
$251$			30.0000i			1.89358i	−0.321847	+	0.946792i	$0.604304\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			27.7128i			1.72868i	0.502910	+	0.864339i	$0.332263\pi$
$257$			27.7128i			1.72868i	−0.502910	+	0.864339i	$0.667737\pi$
$258$		0			0
$259$	−3.46410	−	4.00000i	−0.215249	−	0.248548i
$260$		0			0
$261$		−	6.00000i		−	0.371391i
$262$		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
$267$	13.8564			0.847998
$268$		0			0
$269$	−3.46410			−0.211210			−0.105605	−	0.994408i	$-0.533678\pi$
$269$	−3.46410			−0.211210			−0.105605	+	0.994408i	$0.533678\pi$
$270$		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
$275$	−24.2487			−1.46225
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$		0			0
$285$	13.8564			0.820783
$286$		0			0
$287$	−12.0000	−	13.8564i	−0.708338	−	0.817918i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	27.7128			1.62455
$292$		0			0
$293$	10.3923			0.607125			0.303562	−	0.952812i	$-0.401824\pi$
$293$	10.3923			0.607125			0.303562	+	0.952812i	$0.401824\pi$
$294$		0			0
$295$		−	20.7846i		−	1.21013i
$296$		0			0
$297$			13.8564i			0.804030i
$298$		0			0
$299$			12.0000i			0.693978i
$300$		0			0
$301$	−20.7846	+	18.0000i	−1.19800	+	1.03750i
$302$		0			0
$303$		−	20.7846i		−	1.19404i
$304$		0			0
$305$	12.0000			0.687118
$306$		0			0
$307$			22.0000i			1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$			22.0000i			1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$		0			0
$309$		−	8.00000i		−	0.455104i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		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
$315$	−6.92820	+	6.00000i	−0.390360	+	0.338062i
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$		−	20.7846i		−	1.16008i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.2487			−1.34508
$326$		0			0
$327$	28.0000			1.54840
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.46410			−0.190404			−0.0952021	−	0.995458i	$-0.530350\pi$
$331$	−3.46410			−0.190404			−0.0952021	+	0.995458i	$0.530350\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		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
$339$			36.0000i			1.95525i
$340$		0			0
$341$	−27.7128			−1.50073
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	24.0000			1.29212
$346$		0			0
$347$	−17.3205			−0.929814			−0.464907	−	0.885360i	$-0.653912\pi$
$347$	−17.3205			−0.929814			−0.464907	+	0.885360i	$0.653912\pi$
$348$		0			0
$349$	−3.46410			−0.185429			−0.0927146	−	0.995693i	$-0.529554\pi$
$349$	−3.46410			−0.185429			−0.0927146	+	0.995693i	$0.529554\pi$
$350$		0			0
$351$			13.8564i			0.739600i
$352$		0			0
$353$		−	27.7128i		−	1.47500i	−0.675345	−	0.737502i	$-0.736005\pi$
$353$		−	27.7128i		−	1.47500i	0.675345	−	0.737502i	$-0.263995\pi$
$354$		0			0
$355$		−	12.0000i		−	0.636894i
$356$		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
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$			24.0000i			1.25622i
$366$		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
$371$	10.3923	+	12.0000i	0.539542	+	0.623009i
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$			13.8564i			0.715542i
$376$		0			0
$377$		−	20.7846i		−	1.07046i
$378$		0			0
$379$	24.2487			1.24557			0.622786	−	0.782392i	$-0.286001\pi$
$379$	24.2487			1.24557			0.622786	+	0.782392i	$0.286001\pi$
$380$		0			0
$381$	−20.7846			−1.06483
$382$		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
$387$	−10.3923			−0.528271
$388$		0			0
$389$			18.0000i			0.912636i	0.889817	+	0.456318i	$0.150832\pi$
$389$			18.0000i			0.912636i	−0.889817	+	0.456318i	$0.849168\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−36.0000			−1.81596
$394$		0			0
$395$			12.0000i			0.603786i
$396$		0			0
$397$	10.3923			0.521575			0.260787	−	0.965396i	$-0.416018\pi$
$397$	10.3923			0.521575			0.260787	+	0.965396i	$0.416018\pi$
$398$		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
$403$	−27.7128			−1.38047
$404$		0			0
$405$	38.1051			1.89346
$406$		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
$411$		−	12.0000i		−	0.591916i
$412$		0			0
$413$	−10.3923	−	12.0000i	−0.511372	−	0.590481i
$414$		0			0
$415$		−	20.7846i		−	1.02028i
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$			30.0000i			1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$419$			30.0000i			1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$420$		0			0
$421$			26.0000i			1.26716i	0.773676	+	0.633581i	$0.218416\pi$
$421$			26.0000i			1.26716i	−0.773676	+	0.633581i	$0.781584\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.92820	−	6.00000i	0.335279	−	0.290360i
$428$		0			0
$429$		−	24.0000i		−	1.15873i
$430$		0			0
$431$		−	38.1051i		−	1.83546i	−0.397206	−	0.917729i	$-0.630020\pi$
$431$		−	38.1051i		−	1.83546i	0.397206	−	0.917729i	$-0.369980\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−41.5692			−1.99309
$436$		0			0
$437$	−6.92820			−0.331421
$438$		0			0
$439$	−28.0000			−1.33637			−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000			−1.33637			−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$	−1.00000	+	6.92820i	−0.0476190	+	0.329914i
$442$		0			0
$443$	−3.46410			−0.164584			−0.0822922	−	0.996608i	$-0.526224\pi$
$443$	−3.46410			−0.164584			−0.0822922	+	0.996608i	$0.526224\pi$
$444$		0			0
$445$		−	24.0000i		−	1.13771i
$446$		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
$451$		−	24.0000i		−	1.13012i
$452$		0			0
$453$	−6.92820			−0.325515
$454$		0			0
$455$	−24.0000	+	20.7846i	−1.12514	+	0.974398i
$456$		0			0
$457$	34.0000			1.59045			0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000			1.59045			0.795226	+	0.606313i	$0.207352\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	38.1051			1.77473			0.887366	−	0.461065i	$-0.152533\pi$
$461$	38.1051			1.77473			0.887366	+	0.461065i	$0.152533\pi$
$462$		0			0
$463$			24.2487i			1.12693i	0.826139	+	0.563467i	$0.190533\pi$
$463$			24.2487i			1.12693i	−0.826139	+	0.563467i	$0.809467\pi$
$464$		0			0
$465$			55.4256i			2.57030i
$466$		0			0
$467$			30.0000i			1.38823i	0.719862	+	0.694117i	$0.244205\pi$
$467$			30.0000i			1.38823i	−0.719862	+	0.694117i	$0.755795\pi$
$468$		0			0
$469$	−6.92820	+	6.00000i	−0.319915	+	0.277054i
$470$		0			0
$471$		−	20.7846i		−	0.957704i
$472$		0			0
$473$	−36.0000			−1.65528
$474$		0			0
$475$		−	14.0000i		−	0.642364i
$476$		0			0
$477$			6.00000i			0.274721i
$478$		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
$483$	13.8564	−	12.0000i	0.630488	−	0.546019i
$484$		0			0
$485$		−	48.0000i		−	2.17957i
$486$		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
$491$	−17.3205			−0.781664			−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205			−0.781664			−0.390832	+	0.920462i	$0.627813\pi$
$492$		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
$499$	−10.3923			−0.465223			−0.232612	−	0.972570i	$-0.574727\pi$
$499$	−10.3923			−0.465223			−0.232612	+	0.972570i	$0.574727\pi$
$500$		0			0
$501$			24.0000i			1.07224i
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$		0			0
$507$			2.00000i			0.0888231i
$508$		0			0
$509$	−31.1769			−1.38189			−0.690946	−	0.722906i	$-0.742806\pi$
$509$	−31.1769			−1.38189			−0.690946	+	0.722906i	$0.742806\pi$
$510$		0			0
$511$	12.0000	+	13.8564i	0.530849	+	0.612971i
$512$		0			0
$513$	−8.00000			−0.353209
$514$		0			0
$515$	−13.8564			−0.610586
$516$		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
$523$		−	2.00000i		−	0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$		−	2.00000i		−	0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$		0			0
$525$	24.2487	+	28.0000i	1.05830	+	1.22202i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$		−	24.0000i		−	1.03956i
$534$		0			0
$535$	−36.0000			−1.55642
$536$		0			0
$537$			20.7846i			0.896922i
$538$		0			0
$539$	−3.46410	+	24.0000i	−0.149209	+	1.03375i
$540$		0			0
$541$		−	2.00000i		−	0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$		−	2.00000i		−	0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$		0			0
$543$			34.6410i			1.48659i
$544$		0			0
$545$		−	48.4974i		−	2.07740i
$546$		0			0
$547$	17.3205			0.740571			0.370286	−	0.928918i	$-0.379260\pi$
$547$	17.3205			0.740571			0.370286	+	0.928918i	$0.379260\pi$
$548$		0			0
$549$	3.46410			0.147844
$550$		0			0
$551$	12.0000			0.511217
$552$		0			0
$553$	6.00000	+	6.92820i	0.255146	+	0.294617i
$554$		0			0
$555$	−13.8564			−0.588172
$556$		0			0
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	18.0000i		−	0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$		−	18.0000i		−	0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$		0			0
$565$	62.3538			2.62325
$566$		0			0
$567$	22.0000	−	19.0526i	0.923913	−	0.800132i
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	−45.0333			−1.88459			−0.942293	−	0.334790i	$-0.891335\pi$
$571$	−45.0333			−1.88459			−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$	48.4974			2.02601
$574$		0			0
$575$		−	24.2487i		−	1.01124i
$576$		0			0
$577$		−	27.7128i		−	1.15370i	−0.816850	−	0.576850i	$-0.804282\pi$
$577$		−	27.7128i		−	1.15370i	0.816850	−	0.576850i	$-0.195718\pi$
$578$		0			0
$579$			4.00000i			0.166234i
$580$		0			0
$581$	−10.3923	−	12.0000i	−0.431145	−	0.497844i
$582$		0			0
$583$			20.7846i			0.860811i
$584$		0			0
$585$	−12.0000			−0.496139
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$		0			0
$591$	36.0000			1.48084
$592$		0			0
$593$		−	13.8564i		−	0.569014i	−0.958674	−	0.284507i	$-0.908170\pi$
$593$		−	13.8564i		−	0.569014i	0.958674	−	0.284507i	$-0.0918300\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	40.0000i		−	1.63709i
$598$		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
$603$	−3.46410			−0.141069
$604$		0			0
$605$	−3.46410			−0.140836
$606$		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
$613$		−	38.0000i		−	1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$		−	38.0000i		−	1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$		0			0
$615$	−48.0000			−1.93555
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$		0			0
$621$	−13.8564			−0.556038
$622$		0			0
$623$	−12.0000	−	13.8564i	−0.480770	−	0.555145i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	13.8564			0.553372
$628$		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
$635$			36.0000i			1.42862i
$636$		0			0
$637$	−3.46410	+	24.0000i	−0.137253	+	0.950915i
$638$		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
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$			72.0000i			2.83500i
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	27.7128	+	32.0000i	1.08615	+	1.25418i
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$			62.3538i			2.43637i
$656$		0			0
$657$			6.92820i			0.270295i
$658$		0			0
$659$	3.46410			0.134942			0.0674711	−	0.997721i	$-0.478507\pi$
$659$	3.46410			0.134942			0.0674711	+	0.997721i	$0.478507\pi$
$660$		0			0
$661$	−3.46410			−0.134738			−0.0673690	−	0.997728i	$-0.521460\pi$
$661$	−3.46410			−0.134738			−0.0673690	+	0.997728i	$0.521460\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000	−	13.8564i	−0.465340	−	0.537328i
$666$		0			0
$667$	20.7846			0.804783
$668$		0			0
$669$		−	32.0000i		−	1.23719i
$670$		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
$675$		−	28.0000i		−	1.07772i
$676$		0			0
$677$	−3.46410			−0.133136			−0.0665681	−	0.997782i	$-0.521205\pi$
$677$	−3.46410			−0.133136			−0.0665681	+	0.997782i	$0.521205\pi$
$678$		0			0
$679$	−24.0000	−	27.7128i	−0.921035	−	1.06352i
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$	38.1051			1.45805			0.729026	−	0.684486i	$-0.239973\pi$
$683$	38.1051			1.45805			0.729026	+	0.684486i	$0.239973\pi$
$684$		0			0
$685$	−20.7846			−0.794139
$686$		0			0
$687$		−	20.7846i		−	0.792982i
$688$		0			0
$689$			20.7846i			0.791831i
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$		0			0
$693$	−6.92820	+	6.00000i	−0.263181	+	0.227921i
$694$		0			0
$695$			6.92820i			0.262802i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	12.0000i		−	0.453882i
$700$		0			0
$701$			30.0000i			1.13308i	0.824033	+	0.566542i	$0.191719\pi$
$701$			30.0000i			1.13308i	−0.824033	+	0.566542i	$0.808281\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.7846	+	18.0000i	−0.781686	+	0.676960i
$708$		0			0
$709$			10.0000i			0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$			10.0000i			0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$		0			0
$711$			3.46410i			0.129914i
$712$		0			0
$713$		−	27.7128i		−	1.03785i
$714$		0			0
$715$	−41.5692			−1.55460
$716$		0			0
$717$	−20.7846			−0.776215
$718$		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
$723$	55.4256			2.06130
$724$		0			0
$725$			42.0000i			1.55984i
$726$		0			0
$727$	28.0000			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$	51.9615			1.91924			0.959621	−	0.281295i	$-0.0907639\pi$
$733$	51.9615			1.91924			0.959621	+	0.281295i	$0.0907639\pi$
$734$		0			0
$735$	48.0000	+	6.92820i	1.77051	+	0.255551i
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$	−10.3923			−0.382287			−0.191144	−	0.981562i	$-0.561220\pi$
$739$	−10.3923			−0.382287			−0.191144	+	0.981562i	$0.561220\pi$
$740$		0			0
$741$	13.8564			0.509028
$742$		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
$747$		−	6.00000i		−	0.219529i
$748$		0			0
$749$	−20.7846	+	18.0000i	−0.759453	+	0.657706i
$750$		0			0
$751$			45.0333i			1.64329i	0.570000	+	0.821645i	$0.306943\pi$
$751$			45.0333i			1.64329i	−0.570000	+	0.821645i	$0.693057\pi$
$752$		0			0
$753$	60.0000			2.18652
$754$		0			0
$755$			12.0000i			0.436725i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$		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
$763$	−24.2487	−	28.0000i	−0.877862	−	1.01367i
$764$		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	$0.0803772\pi$
$769$			13.8564i			0.499675i	−0.968288	+	0.249837i	$0.919623\pi$
$770$		0			0
$771$	55.4256			1.99611
$772$		0			0
$773$	−3.46410			−0.124595			−0.0622975	−	0.998058i	$-0.519843\pi$
$773$	−3.46410			−0.124595			−0.0622975	+	0.998058i	$0.519843\pi$
$774$		0			0
$775$	56.0000			2.01158
$776$		0			0
$777$	−8.00000	+	6.92820i	−0.286998	+	0.248548i
$778$		0			0
$779$	13.8564			0.496457
$780$		0			0
$781$		−	12.0000i		−	0.429394i
$782$		0			0
$783$	24.0000			0.857690
$784$		0			0
$785$	−36.0000			−1.28490
$786$		0			0
$787$		−	34.0000i		−	1.21197i	−0.795476	−	0.605985i	$-0.792779\pi$
$787$		−	34.0000i		−	1.21197i	0.795476	−	0.605985i	$-0.207221\pi$
$788$		0			0
$789$	−48.4974			−1.72655
$790$		0			0
$791$	36.0000	−	31.1769i	1.28001	−	1.10852i
$792$		0			0
$793$	12.0000			0.426132
$794$		0			0
$795$	41.5692			1.47431
$796$		0			0
$797$	−17.3205			−0.613524			−0.306762	−	0.951786i	$-0.599246\pi$
$797$	−17.3205			−0.613524			−0.306762	+	0.951786i	$0.599246\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$			24.0000i			0.846942i
$804$		0			0
$805$	−20.7846	−	24.0000i	−0.732561	−	0.845889i
$806$		0			0
$807$			6.92820i			0.243884i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$			22.0000i			0.772524i	0.922389	+	0.386262i	$0.126234\pi$
$811$			22.0000i			0.772524i	−0.922389	+	0.386262i	$0.873766\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$	−60.0000			−2.10171
$816$		0			0
$817$		−	20.7846i		−	0.727161i
$818$		0			0
$819$	−6.92820	+	6.00000i	−0.242091	+	0.209657i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		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
$827$	24.2487			0.843210			0.421605	−	0.906780i	$-0.361467\pi$
$827$	24.2487			0.843210			0.421605	+	0.906780i	$0.361467\pi$
$828$		0			0
$829$	−3.46410			−0.120313			−0.0601566	−	0.998189i	$-0.519160\pi$
$829$	−3.46410			−0.120313			−0.0601566	+	0.998189i	$0.519160\pi$
$830$		0			0
$831$	4.00000			0.138758
$832$		0			0
$833$		0			0
$834$		0			0
$835$	41.5692			1.43856
$836$		0			0
$837$		−	32.0000i		−	1.10608i
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		−	12.0000i		−	0.413302i
$844$		0			0
$845$	3.46410			0.119169
$846$		0			0
$847$	−2.00000	+	1.73205i	−0.0687208	+	0.0595140i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$	6.92820			0.237496
$852$		0			0
$853$	−3.46410			−0.118609			−0.0593043	−	0.998240i	$-0.518888\pi$
$853$	−3.46410			−0.118609			−0.0593043	+	0.998240i	$0.518888\pi$
$854$		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
$859$			14.0000i			0.477674i	0.971060	+	0.238837i	$0.0767661\pi$
$859$			14.0000i			0.477674i	−0.971060	+	0.238837i	$0.923234\pi$
$860$		0			0
$861$	−27.7128	+	24.0000i	−0.944450	+	0.817918i
$862$		0			0
$863$		−	31.1769i		−	1.06127i	−0.847599	−	0.530637i	$-0.821953\pi$
$863$		−	31.1769i		−	1.06127i	0.847599	−	0.530637i	$-0.178047\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$		−	34.0000i		−	1.15470i
$868$		0			0
$869$			12.0000i			0.407072i
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$	13.8564	−	12.0000i	0.468432	−	0.405674i
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$		0			0
$879$		−	20.7846i		−	0.701047i
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$	3.46410			0.116576			0.0582882	−	0.998300i	$-0.481436\pi$
$883$	3.46410			0.116576			0.0582882	+	0.998300i	$0.481436\pi$
$884$		0			0
$885$	−41.5692			−1.39733
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$	18.0000	+	20.7846i	0.603701	+	0.697093i
$890$		0			0
$891$	38.1051			1.27657
$892$		0			0
$893$		0			0
$894$		0			0
$895$	36.0000			1.20335
$896$		0			0
$897$	24.0000			0.801337
$898$		0			0
$899$			48.0000i			1.60089i
$900$		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
$907$	24.2487			0.805165			0.402583	−	0.915384i	$-0.368113\pi$
$907$	24.2487			0.805165			0.402583	+	0.915384i	$0.368113\pi$
$908$		0			0
$909$	−10.3923			−0.344691
$910$		0			0
$911$		−	10.3923i		−	0.344312i	−0.985070	−	0.172156i	$-0.944927\pi$
$911$		−	10.3923i		−	0.344312i	0.985070	−	0.172156i	$-0.0550734\pi$
$912$		0			0
$913$		−	20.7846i		−	0.687870i
$914$		0			0
$915$		−	24.0000i		−	0.793416i
$916$		0			0
$917$	31.1769	+	36.0000i	1.02955	+	1.18882i
$918$		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
$923$		−	12.0000i		−	0.394985i
$924$		0			0
$925$			14.0000i			0.460317i
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$			41.5692i			1.36384i	0.731426	+	0.681921i	$0.238855\pi$
$929$			41.5692i			1.36384i	−0.731426	+	0.681921i	$0.761145\pi$
$930$		0			0
$931$	−13.8564	−	2.00000i	−0.454125	−	0.0655474i
$932$		0			0
$933$			24.0000i			0.785725i
$934$		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
$939$	−13.8564			−0.452187
$940$		0			0
$941$	24.2487			0.790485			0.395243	−	0.918577i	$-0.370660\pi$
$941$	24.2487			0.790485			0.395243	+	0.918577i	$0.370660\pi$
$942$		0			0
$943$	24.0000			0.781548
$944$		0			0
$945$	−24.0000	−	27.7128i	−0.780720	−	0.901498i
$946$		0			0
$947$	−24.2487			−0.787977			−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487			−0.787977			−0.393989	+	0.919115i	$0.628905\pi$
$948$		0			0
$949$			24.0000i			0.779073i
$950$		0			0
$951$	12.0000			0.389127
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		−	84.0000i		−	2.71818i
$956$		0			0
$957$	−41.5692			−1.34374
$958$		0			0
$959$	−12.0000	+	10.3923i	−0.387500	+	0.335585i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−10.3923			−0.334887
$964$		0			0
$965$	6.92820			0.223027
$966$		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
$971$		−	18.0000i		−	0.577647i	−0.957382	−	0.288824i	$-0.906736\pi$
$971$		−	18.0000i		−	0.577647i	0.957382	−	0.288824i	$-0.0932642\pi$
$972$		0			0
$973$	3.46410	+	4.00000i	0.111054	+	0.128234i
$974$		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
$979$		−	24.0000i		−	0.767043i
$980$		0			0
$981$		−	14.0000i		−	0.446986i
$982$		0			0
$983$	−36.0000			−1.14822			−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000			−1.14822			−0.574111	+	0.818778i	$0.694652\pi$
$984$		0			0
$985$		−	62.3538i		−	1.98676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	36.0000i		−	1.14473i
$990$		0			0
$991$		−	3.46410i		−	0.110041i	−0.998485	−	0.0550204i	$-0.982478\pi$
$991$		−	3.46410i		−	0.110041i	0.998485	−	0.0550204i	$-0.0175224\pi$
$992$		0			0
$993$			6.92820i			0.219860i
$994$		0			0
$995$	−69.2820			−2.19639
$996$		0			0
$997$	51.9615			1.64564			0.822819	−	0.568304i	$-0.192400\pi$
$997$	51.9615			1.64564			0.822819	+	0.568304i	$0.192400\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.a.895.1		4
4.3	odd	2		1792.2.e.c.895.3		4
7.6	odd	2		1792.2.e.c.895.4		4
8.3	odd	2		1792.2.e.c.895.2		4
8.5	even	2	inner	1792.2.e.a.895.4		4
16.3	odd	4		448.2.f.a.447.2		2
16.5	even	4		112.2.f.a.111.1	✓	2
16.11	odd	4		112.2.f.b.111.1	yes	2
16.13	even	4		448.2.f.c.447.2		2
28.27	even	2	inner	1792.2.e.a.895.2		4
48.5	odd	4		1008.2.b.g.559.2		2
48.11	even	4		1008.2.b.b.559.2		2
48.29	odd	4		4032.2.b.h.3583.1		2
48.35	even	4		4032.2.b.b.3583.1		2
56.13	odd	2		1792.2.e.c.895.1		4
56.27	even	2	inner	1792.2.e.a.895.3		4
80.27	even	4		2800.2.e.b.2799.1		4
80.37	odd	4		2800.2.e.c.2799.4		4
80.43	even	4		2800.2.e.b.2799.4		4
80.53	odd	4		2800.2.e.c.2799.1		4
80.59	odd	4		2800.2.k.b.2351.1		2
80.69	even	4		2800.2.k.e.2351.2		2
112.5	odd	12		784.2.p.b.31.1		2
112.11	odd	12		784.2.p.b.607.1		2
112.13	odd	4		448.2.f.a.447.1		2
112.27	even	4		112.2.f.a.111.2	yes	2
112.37	even	12		784.2.p.e.31.1		2
112.53	even	12		784.2.p.f.607.1		2
112.59	even	12		784.2.p.e.607.1		2
112.69	odd	4		112.2.f.b.111.2	yes	2
112.75	even	12		784.2.p.f.31.1		2
112.83	even	4		448.2.f.c.447.1		2
112.101	odd	12		784.2.p.a.607.1		2
112.107	odd	12		784.2.p.a.31.1		2
336.83	odd	4		4032.2.b.h.3583.2		2
336.125	even	4		4032.2.b.b.3583.2		2
336.251	odd	4		1008.2.b.g.559.1		2
336.293	even	4		1008.2.b.b.559.1		2
560.27	odd	4		2800.2.e.c.2799.3		4
560.69	odd	4		2800.2.k.b.2351.2		2
560.139	even	4		2800.2.k.e.2351.1		2
560.293	even	4		2800.2.e.b.2799.3		4
560.363	odd	4		2800.2.e.c.2799.2		4
560.517	even	4		2800.2.e.b.2799.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.f.a.111.1	✓	2	16.5	even	4
112.2.f.a.111.2	yes	2	112.27	even	4
112.2.f.b.111.1	yes	2	16.11	odd	4
112.2.f.b.111.2	yes	2	112.69	odd	4
448.2.f.a.447.1		2	112.13	odd	4
448.2.f.a.447.2		2	16.3	odd	4
448.2.f.c.447.1		2	112.83	even	4
448.2.f.c.447.2		2	16.13	even	4
784.2.p.a.31.1		2	112.107	odd	12
784.2.p.a.607.1		2	112.101	odd	12
784.2.p.b.31.1		2	112.5	odd	12
784.2.p.b.607.1		2	112.11	odd	12
784.2.p.e.31.1		2	112.37	even	12
784.2.p.e.607.1		2	112.59	even	12
784.2.p.f.31.1		2	112.75	even	12
784.2.p.f.607.1		2	112.53	even	12
1008.2.b.b.559.1		2	336.293	even	4
1008.2.b.b.559.2		2	48.11	even	4
1008.2.b.g.559.1		2	336.251	odd	4
1008.2.b.g.559.2		2	48.5	odd	4
1792.2.e.a.895.1		4	1.1	even	1	trivial
1792.2.e.a.895.2		4	28.27	even	2	inner
1792.2.e.a.895.3		4	56.27	even	2	inner
1792.2.e.a.895.4		4	8.5	even	2	inner
1792.2.e.c.895.1		4	56.13	odd	2
1792.2.e.c.895.2		4	8.3	odd	2
1792.2.e.c.895.3		4	4.3	odd	2
1792.2.e.c.895.4		4	7.6	odd	2
2800.2.e.b.2799.1		4	80.27	even	4
2800.2.e.b.2799.2		4	560.517	even	4
2800.2.e.b.2799.3		4	560.293	even	4
2800.2.e.b.2799.4		4	80.43	even	4
2800.2.e.c.2799.1		4	80.53	odd	4
2800.2.e.c.2799.2		4	560.363	odd	4
2800.2.e.c.2799.3		4	560.27	odd	4
2800.2.e.c.2799.4		4	80.37	odd	4
2800.2.k.b.2351.1		2	80.59	odd	4
2800.2.k.b.2351.2		2	560.69	odd	4
2800.2.k.e.2351.1		2	560.139	even	4
2800.2.k.e.2351.2		2	80.69	even	4
4032.2.b.b.3583.1		2	48.35	even	4
4032.2.b.b.3583.2		2	336.125	even	4
4032.2.b.h.3583.1		2	48.29	odd	4
4032.2.b.h.3583.2		2	336.83	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -3.46410 q^{5} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{9} -3.46410 q^{11} -3.46410 q^{13} +6.92820i q^{15} -2.00000i q^{19} +(3.46410 + 4.00000i) q^{21} -3.46410i q^{23} +7.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} +6.92820i q^{33} +(6.92820 - 6.00000i) q^{35} +2.00000i q^{37} +6.92820i q^{39} +6.92820i q^{41} +10.3923 q^{43} +3.46410 q^{45} +(1.00000 - 6.92820i) q^{49} -6.00000i q^{53} +12.0000 q^{55} -4.00000 q^{57} +6.00000i q^{59} -3.46410 q^{61} +(2.00000 - 1.73205i) q^{63} +12.0000 q^{65} +3.46410 q^{67} -6.92820 q^{69} +3.46410i q^{71} -6.92820i q^{73} -14.0000i q^{75} +(6.92820 - 6.00000i) q^{77} -3.46410i q^{79} -11.0000 q^{81} +6.00000i q^{83} +12.0000 q^{87} +6.92820i q^{89} +(6.92820 - 6.00000i) q^{91} -16.0000i q^{93} +6.92820i q^{95} +13.8564i q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 4 q^{9} + 28 q^{25} + 32 q^{31} + 4 q^{49} + 48 q^{55} - 16 q^{57} + 8 q^{63} + 48 q^{65} - 44 q^{81} + 48 q^{87}+O(q^{100}) \) 4 * q - 8 * q^7 - 4 * q^9 + 28 * q^25 + 32 * q^31 + 4 * q^49 + 48 * q^55 - 16 * q^57 + 8 * q^63 + 48 * q^65 - 44 * q^81 + 48 * q^87

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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