LMFDB - Embedded newform 1200.2.h.h.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(1151,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.h (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:	$9.58204824255$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1200.1151
Dual form			1200.2.h.h.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} -3.00000 q^{11} -2.00000 q^{13} -5.19615i q^{17} -5.19615i q^{19} +6.00000 q^{23} -5.19615i q^{27} -10.3923i q^{29} +3.46410i q^{31} +(-4.50000 + 2.59808i) q^{33} +8.00000 q^{37} +(-3.00000 + 1.73205i) q^{39} -5.19615i q^{41} +3.46410i q^{43} -6.00000 q^{47} +7.00000 q^{49} +(-4.50000 - 7.79423i) q^{51} +10.3923i q^{53} +(-4.50000 - 7.79423i) q^{57} -12.0000 q^{59} +8.00000 q^{61} +12.1244i q^{67} +(9.00000 - 5.19615i) q^{69} +6.00000 q^{71} +1.00000 q^{73} -6.92820i q^{79} +(-4.50000 - 7.79423i) q^{81} -9.00000 q^{83} +(-9.00000 - 15.5885i) q^{87} -5.19615i q^{89} +(3.00000 + 5.19615i) q^{93} +10.0000 q^{97} +(-4.50000 + 7.79423i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 3 * q^9	$ 2 q + 3 q^{3} + 3 q^{9} - 6 q^{11} - 4 q^{13} + 12 q^{23} - 9 q^{33} + 16 q^{37} - 6 q^{39} - 12 q^{47} + 14 q^{49} - 9 q^{51} - 9 q^{57} - 24 q^{59} + 16 q^{61} + 18 q^{69} + 12 q^{71} + 2 q^{73} - 9 q^{81} - 18 q^{83} - 18 q^{87} + 6 q^{93} + 20 q^{97} - 9 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 - 4 * q^13 + 12 * q^23 - 9 * q^33 + 16 * q^37 - 6 * q^39 - 12 * q^47 + 14 * q^49 - 9 * q^51 - 9 * q^57 - 24 * q^59 + 16 * q^61 + 18 * q^69 + 12 * q^71 + 2 * q^73 - 9 * q^81 - 18 * q^83 - 18 * q^87 + 6 * q^93 + 20 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.19615i		−	1.26025i	−0.776493	−	0.630126i	$-0.783003\pi$
$17$		−	5.19615i		−	1.26025i	0.776493	−	0.630126i	$-0.216997\pi$
$18$		0			0
$19$		−	5.19615i		−	1.19208i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$		−	5.19615i		−	1.19208i	0.802955	−	0.596040i	$-0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$		−	10.3923i		−	1.92980i	−0.262613	−	0.964901i	$-0.584584\pi$
$29$		−	10.3923i		−	1.92980i	0.262613	−	0.964901i	$-0.415416\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$		0			0
$33$	−4.50000	+	2.59808i	−0.783349	+	0.452267i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$	−3.00000	+	1.73205i	−0.480384	+	0.277350i
$40$		0			0
$41$		−	5.19615i		−	0.811503i	−0.913984	−	0.405751i	$-0.867010\pi$
$41$		−	5.19615i		−	0.811503i	0.913984	−	0.405751i	$-0.132990\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−4.50000	−	7.79423i	−0.630126	−	1.09141i
$52$		0			0
$53$			10.3923i			1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$			10.3923i			1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.50000	−	7.79423i	−0.596040	−	1.03237i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			12.1244i			1.48123i	0.671932	+	0.740613i	$0.265465\pi$
$67$			12.1244i			1.48123i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$	9.00000	−	5.19615i	1.08347	−	0.625543i
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	6.92820i		−	0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$		−	6.92820i		−	0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−9.00000	−	15.5885i	−0.964901	−	1.67126i
$88$		0			0
$89$		−	5.19615i		−	0.550791i	−0.961331	−	0.275396i	$-0.911191\pi$
$89$		−	5.19615i		−	0.550791i	0.961331	−	0.275396i	$-0.0888088\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.00000	+	5.19615i	0.311086	+	0.538816i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	−4.50000	+	7.79423i	−0.452267	+	0.783349i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$			17.3205i			1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$			17.3205i			1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$	−8.00000			−0.766261			−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000			−0.766261			−0.383131	+	0.923694i	$0.625154\pi$
$110$		0			0
$111$	12.0000	−	6.92820i	1.13899	−	0.657596i
$112$		0			0
$113$			5.19615i			0.488813i	0.969673	+	0.244406i	$0.0785931\pi$
$113$			5.19615i			0.488813i	−0.969673	+	0.244406i	$0.921407\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.00000	+	5.19615i	−0.277350	+	0.480384i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	−4.50000	−	7.79423i	−0.405751	−	0.702782i
$124$		0			0
$125$		0			0
$126$		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
$129$	3.00000	+	5.19615i	0.264135	+	0.457496i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.19615i		−	0.443937i	−0.975054	−	0.221969i	$-0.928752\pi$
$137$		−	5.19615i		−	0.443937i	0.975054	−	0.221969i	$-0.0712483\pi$
$138$		0			0
$139$		−	8.66025i		−	0.734553i	−0.930112	−	0.367277i	$-0.880290\pi$
$139$		−	8.66025i		−	0.734553i	0.930112	−	0.367277i	$-0.119710\pi$
$140$		0			0
$141$	−9.00000	+	5.19615i	−0.757937	+	0.437595i
$142$		0			0
$143$	6.00000			0.501745
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.5000	−	6.06218i	0.866025	−	0.500000i
$148$		0			0
$149$			10.3923i			0.851371i	0.904871	+	0.425685i	$0.139967\pi$
$149$			10.3923i			0.851371i	−0.904871	+	0.425685i	$0.860033\pi$
$150$		0			0
$151$			20.7846i			1.69143i	0.533637	+	0.845714i	$0.320825\pi$
$151$			20.7846i			1.69143i	−0.533637	+	0.845714i	$0.679175\pi$
$152$		0			0
$153$	−13.5000	−	7.79423i	−1.09141	−	0.630126i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	9.00000	+	15.5885i	0.713746	+	1.23625i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	12.1244i		−	0.949653i	−0.880079	−	0.474826i	$-0.842511\pi$
$163$		−	12.1244i		−	0.949653i	0.880079	−	0.474826i	$-0.157489\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−13.5000	−	7.79423i	−1.03237	−	0.596040i
$172$		0			0
$173$		−	20.7846i		−	1.58022i	−0.612962	−	0.790112i	$-0.710022\pi$
$173$		−	20.7846i		−	1.58022i	0.612962	−	0.790112i	$-0.289978\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−18.0000	+	10.3923i	−1.35296	+	0.781133i
$178$		0			0
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	12.0000	−	6.92820i	0.887066	−	0.512148i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			15.5885i			1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$	23.0000			1.65558			0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000			1.65558			0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.3923i			0.740421i	0.928948	+	0.370211i	$0.120714\pi$
$197$			10.3923i			0.740421i	−0.928948	+	0.370211i	$0.879286\pi$
$198$		0			0
$199$		−	6.92820i		−	0.491127i	−0.969380	−	0.245564i	$-0.921027\pi$
$199$		−	6.92820i		−	0.491127i	0.969380	−	0.245564i	$-0.0789730\pi$
$200$		0			0
$201$	10.5000	+	18.1865i	0.740613	+	1.28278i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	9.00000	−	15.5885i	0.625543	−	1.08347i
$208$		0			0
$209$			15.5885i			1.07828i
$210$		0			0
$211$		−	1.73205i		−	0.119239i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$		−	1.73205i		−	0.119239i	0.998221	−	0.0596196i	$-0.0189888\pi$
$212$		0			0
$213$	9.00000	−	5.19615i	0.616670	−	0.356034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.50000	−	0.866025i	0.101361	−	0.0585206i
$220$		0			0
$221$			10.3923i			0.699062i
$222$		0			0
$223$		−	6.92820i		−	0.463947i	−0.972722	−	0.231973i	$-0.925482\pi$
$223$		−	6.92820i		−	0.463947i	0.972722	−	0.231973i	$-0.0745182\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.7846i		−	1.36165i	−0.732448	−	0.680823i	$-0.761622\pi$
$233$		−	20.7846i		−	1.36165i	0.732448	−	0.680823i	$-0.238378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.00000	−	10.3923i	−0.389742	−	0.675053i
$238$		0			0
$239$	30.0000			1.94054			0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000			1.94054			0.970269	+	0.242028i	$0.0778125\pi$
$240$		0			0
$241$	−5.00000			−0.322078			−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000			−0.322078			−0.161039	+	0.986948i	$0.551485\pi$
$242$		0			0
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			10.3923i			0.661247i
$248$		0			0
$249$	−13.5000	+	7.79423i	−0.855528	+	0.493939i
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.7846i			1.29651i	0.761424	+	0.648254i	$0.224501\pi$
$257$			20.7846i			1.29651i	−0.761424	+	0.648254i	$0.775499\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−27.0000	−	15.5885i	−1.67126	−	0.964901i
$262$		0			0
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.50000	−	7.79423i	−0.275396	−	0.476999i
$268$		0			0
$269$			20.7846i			1.26726i	0.773636	+	0.633630i	$0.218436\pi$
$269$			20.7846i			1.26726i	−0.773636	+	0.633630i	$0.781564\pi$
$270$		0			0
$271$			24.2487i			1.47300i	0.676435	+	0.736502i	$0.263524\pi$
$271$			24.2487i			1.47300i	−0.676435	+	0.736502i	$0.736476\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.0000			−0.841178			−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000			−0.841178			−0.420589	+	0.907251i	$0.638177\pi$
$278$		0			0
$279$	9.00000	+	5.19615i	0.538816	+	0.311086i
$280$		0			0
$281$			20.7846i			1.23991i	0.784639	+	0.619953i	$0.212848\pi$
$281$			20.7846i			1.23991i	−0.784639	+	0.619953i	$0.787152\pi$
$282$		0			0
$283$			12.1244i			0.720718i	0.932814	+	0.360359i	$0.117346\pi$
$283$			12.1244i			0.720718i	−0.932814	+	0.360359i	$0.882654\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−10.0000			−0.588235
$290$		0			0
$291$	15.0000	−	8.66025i	0.879316	−	0.507673i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			15.5885i			0.904534i
$298$		0			0
$299$	−12.0000			−0.693978
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			12.1244i			0.691974i	0.938239	+	0.345987i	$0.112456\pi$
$307$			12.1244i			0.691974i	−0.938239	+	0.345987i	$0.887544\pi$
$308$		0			0
$309$	15.0000	+	25.9808i	0.853320	+	1.47799i
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$			31.1769i			1.74557i
$320$		0			0
$321$	4.50000	−	2.59808i	0.251166	−	0.145010i
$322$		0			0
$323$	−27.0000			−1.50232
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−12.0000	+	6.92820i	−0.663602	+	0.383131i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	8.66025i		−	0.476011i	−0.971264	−	0.238005i	$-0.923506\pi$
$331$		−	8.66025i		−	0.476011i	0.971264	−	0.238005i	$-0.0764936\pi$
$332$		0			0
$333$	12.0000	−	20.7846i	0.657596	−	1.13899i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	25.0000			1.36184			0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000			1.36184			0.680918	+	0.732359i	$0.261581\pi$
$338$		0			0
$339$	4.50000	+	7.79423i	0.244406	+	0.423324i
$340$		0			0
$341$		−	10.3923i		−	0.562775i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.00000			0.483145			0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000			0.483145			0.241573	+	0.970383i	$0.422337\pi$
$348$		0			0
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$		0			0
$351$			10.3923i			0.554700i
$352$		0			0
$353$			20.7846i			1.10625i	0.833097	+	0.553127i	$0.186565\pi$
$353$			20.7846i			1.10625i	−0.833097	+	0.553127i	$0.813435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−8.00000			−0.421053
$362$		0			0
$363$	−3.00000	+	1.73205i	−0.157459	+	0.0909091i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	6.92820i		−	0.361649i	−0.983515	−	0.180825i	$-0.942123\pi$
$367$		−	6.92820i		−	0.361649i	0.983515	−	0.180825i	$-0.0578766\pi$
$368$		0			0
$369$	−13.5000	−	7.79423i	−0.702782	−	0.405751i
$370$		0			0
$371$		0			0
$372$		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
$375$		0			0
$376$		0			0
$377$			20.7846i			1.07046i
$378$		0			0
$379$		−	19.0526i		−	0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$		−	19.0526i		−	0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$		0			0
$381$	9.00000	+	15.5885i	0.461084	+	0.798621i
$382$		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
$385$		0			0
$386$		0			0
$387$	9.00000	+	5.19615i	0.457496	+	0.264135i
$388$		0			0
$389$			10.3923i			0.526911i	0.964672	+	0.263455i	$0.0848622\pi$
$389$			10.3923i			0.526911i	−0.964672	+	0.263455i	$0.915138\pi$
$390$		0			0
$391$		−	31.1769i		−	1.57668i
$392$		0			0
$393$	−18.0000	+	10.3923i	−0.907980	+	0.524222i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−28.0000			−1.40528			−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000			−1.40528			−0.702640	+	0.711546i	$0.747995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	5.19615i		−	0.259483i	−0.991548	−	0.129742i	$-0.958585\pi$
$401$		−	5.19615i		−	0.259483i	0.991548	−	0.129742i	$-0.0414148\pi$
$402$		0			0
$403$		−	6.92820i		−	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	25.0000			1.23617			0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000			1.23617			0.618085	+	0.786111i	$0.287909\pi$
$410$		0			0
$411$	−4.50000	−	7.79423i	−0.221969	−	0.384461i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−7.50000	−	12.9904i	−0.367277	−	0.636142i
$418$		0			0
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$		0			0
$423$	−9.00000	+	15.5885i	−0.437595	+	0.757937i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	9.00000	−	5.19615i	0.434524	−	0.250873i
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$	17.0000			0.816968			0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000			0.816968			0.408484	+	0.912766i	$0.366058\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	31.1769i		−	1.49139i
$438$		0			0
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	10.5000	−	18.1865i	0.500000	−	0.866025i
$442$		0			0
$443$	15.0000			0.712672			0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000			0.712672			0.356336	+	0.934358i	$0.384026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	9.00000	+	15.5885i	0.425685	+	0.737309i
$448$		0			0
$449$		−	5.19615i		−	0.245222i	−0.992455	−	0.122611i	$-0.960873\pi$
$449$		−	5.19615i		−	0.245222i	0.992455	−	0.122611i	$-0.0391267\pi$
$450$		0			0
$451$			15.5885i			0.734032i
$452$		0			0
$453$	18.0000	+	31.1769i	0.845714	+	1.46482i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.00000			0.0467780			0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000			0.0467780			0.0233890	+	0.999726i	$0.492554\pi$
$458$		0			0
$459$	−27.0000			−1.26025
$460$		0			0
$461$		−	20.7846i		−	0.968036i	−0.875058	−	0.484018i	$-0.839177\pi$
$461$		−	20.7846i		−	0.968036i	0.875058	−	0.484018i	$-0.160823\pi$
$462$		0			0
$463$			27.7128i			1.28792i	0.765058	+	0.643962i	$0.222710\pi$
$463$			27.7128i			1.28792i	−0.765058	+	0.643962i	$0.777290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.00000	−	1.73205i	0.138233	−	0.0798087i
$472$		0			0
$473$		−	10.3923i		−	0.477839i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	27.0000	+	15.5885i	1.23625	+	0.713746i
$478$		0			0
$479$	18.0000			0.822441			0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000			0.822441			0.411220	+	0.911536i	$0.365103\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	38.1051i		−	1.72671i	−0.504599	−	0.863354i	$-0.668360\pi$
$487$		−	38.1051i		−	1.72671i	0.504599	−	0.863354i	$-0.331640\pi$
$488$		0			0
$489$	−10.5000	−	18.1865i	−0.474826	−	0.822423i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−54.0000			−2.43204
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			38.1051i			1.70582i	0.522059	+	0.852910i	$0.325164\pi$
$499$			38.1051i			1.70582i	−0.522059	+	0.852910i	$0.674836\pi$
$500$		0			0
$501$	−9.00000	+	5.19615i	−0.402090	+	0.232147i
$502$		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
$505$		0			0
$506$		0			0
$507$	−13.5000	+	7.79423i	−0.599556	+	0.346154i
$508$		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
$511$		0			0
$512$		0			0
$513$	−27.0000			−1.19208
$514$		0			0
$515$		0			0
$516$		0			0
$517$	18.0000			0.791639
$518$		0			0
$519$	−18.0000	−	31.1769i	−0.790112	−	1.36851i
$520$		0			0
$521$			5.19615i			0.227648i	0.993501	+	0.113824i	$0.0363099\pi$
$521$			5.19615i			0.227648i	−0.993501	+	0.113824i	$0.963690\pi$
$522$		0			0
$523$		−	25.9808i		−	1.13606i	−0.823008	−	0.568030i	$-0.807706\pi$
$523$		−	25.9808i		−	1.13606i	0.823008	−	0.568030i	$-0.192294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.0000			0.784092
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−18.0000	+	31.1769i	−0.781133	+	1.35296i
$532$		0			0
$533$			10.3923i			0.450141i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	22.5000	−	12.9904i	0.970947	−	0.560576i
$538$		0			0
$539$	−21.0000			−0.904534
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$	15.0000	−	8.66025i	0.643712	−	0.371647i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	19.0526i		−	0.814629i	−0.913288	−	0.407314i	$-0.866465\pi$
$547$		−	19.0526i		−	0.814629i	0.913288	−	0.407314i	$-0.133535\pi$
$548$		0			0
$549$	12.0000	−	20.7846i	0.512148	−	0.887066i
$550$		0			0
$551$	−54.0000			−2.30048
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.1769i		−	1.32101i	−0.750822	−	0.660504i	$-0.770343\pi$
$557$		−	31.1769i		−	1.32101i	0.750822	−	0.660504i	$-0.229657\pi$
$558$		0			0
$559$		−	6.92820i		−	0.293032i
$560$		0			0
$561$	13.5000	+	23.3827i	0.569970	+	0.987218i
$562$		0			0
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.9808i		−	1.08917i	−0.838706	−	0.544585i	$-0.816687\pi$
$569$		−	25.9808i		−	1.08917i	0.838706	−	0.544585i	$-0.183313\pi$
$570$		0			0
$571$		−	31.1769i		−	1.30471i	−0.757912	−	0.652357i	$-0.773780\pi$
$571$		−	31.1769i		−	1.30471i	0.757912	−	0.652357i	$-0.226220\pi$
$572$		0			0
$573$	27.0000	−	15.5885i	1.12794	−	0.651217i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.00000			−0.291414			−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000			−0.291414			−0.145707	+	0.989328i	$0.546546\pi$
$578$		0			0
$579$	34.5000	−	19.9186i	1.43377	−	0.827788i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	31.1769i		−	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−27.0000			−1.11441			−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000			−1.11441			−0.557205	+	0.830375i	$0.688126\pi$
$588$		0			0
$589$	18.0000			0.741677
$590$		0			0
$591$	9.00000	+	15.5885i	0.370211	+	0.641223i
$592$		0			0
$593$			15.5885i			0.640141i	0.947394	+	0.320071i	$0.103707\pi$
$593$			15.5885i			0.640141i	−0.947394	+	0.320071i	$0.896293\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−6.00000	−	10.3923i	−0.245564	−	0.425329i
$598$		0			0
$599$	−36.0000			−1.47092			−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000			−1.47092			−0.735460	+	0.677568i	$0.763034\pi$
$600$		0			0
$601$	5.00000			0.203954			0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000			0.203954			0.101977	+	0.994787i	$0.467483\pi$
$602$		0			0
$603$	31.5000	+	18.1865i	1.28278	+	0.740613i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			6.92820i			0.281207i	0.990066	+	0.140604i	$0.0449043\pi$
$607$			6.92820i			0.281207i	−0.990066	+	0.140604i	$0.955096\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0000			0.485468
$612$		0			0
$613$	−46.0000			−1.85792			−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000			−1.85792			−0.928961	+	0.370177i	$0.879297\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	41.5692i		−	1.67351i	−0.547575	−	0.836757i	$-0.684449\pi$
$617$		−	41.5692i		−	1.67351i	0.547575	−	0.836757i	$-0.315551\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$		−	31.1769i		−	1.25109i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	13.5000	+	23.3827i	0.539138	+	0.933815i
$628$		0			0
$629$		−	41.5692i		−	1.65747i
$630$		0			0
$631$		−	3.46410i		−	0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$		−	3.46410i		−	0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$		0			0
$633$	−1.50000	−	2.59808i	−0.0596196	−	0.103264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−14.0000			−0.554700
$638$		0			0
$639$	9.00000	−	15.5885i	0.356034	−	0.616670i
$640$		0			0
$641$		−	20.7846i		−	0.820943i	−0.911873	−	0.410471i	$-0.865364\pi$
$641$		−	20.7846i		−	0.820943i	0.911873	−	0.410471i	$-0.134636\pi$
$642$		0			0
$643$			31.1769i			1.22950i	0.788723	+	0.614749i	$0.210743\pi$
$643$			31.1769i			1.22950i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		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
$649$	36.0000			1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.7846i			0.813365i	0.913570	+	0.406682i	$0.133314\pi$
$653$			20.7846i			0.813365i	−0.913570	+	0.406682i	$0.866686\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.50000	−	2.59808i	0.0585206	−	0.101361i
$658$		0			0
$659$	9.00000			0.350590			0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000			0.350590			0.175295	+	0.984516i	$0.443912\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$		0			0
$663$	9.00000	+	15.5885i	0.349531	+	0.605406i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	62.3538i		−	2.41435i
$668$		0			0
$669$	−6.00000	−	10.3923i	−0.231973	−	0.401790i
$670$		0			0
$671$	−24.0000			−0.926510
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			41.5692i			1.59763i	0.601574	+	0.798817i	$0.294541\pi$
$677$			41.5692i			1.59763i	−0.601574	+	0.798817i	$0.705459\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	36.0000	−	20.7846i	1.37952	−	0.796468i
$682$		0			0
$683$	21.0000			0.803543			0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000			0.803543			0.401771	+	0.915740i	$0.368395\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−30.0000	+	17.3205i	−1.14457	+	0.660819i
$688$		0			0
$689$		−	20.7846i		−	0.791831i
$690$		0			0
$691$		−	5.19615i		−	0.197671i	−0.995104	−	0.0988355i	$-0.968488\pi$
$691$		−	5.19615i		−	0.197671i	0.995104	−	0.0988355i	$-0.0315118\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−27.0000			−1.02270
$698$		0			0
$699$	−18.0000	−	31.1769i	−0.680823	−	1.17922i
$700$		0			0
$701$			31.1769i			1.17754i	0.808302	+	0.588768i	$0.200387\pi$
$701$			31.1769i			1.17754i	−0.808302	+	0.588768i	$0.799613\pi$
$702$		0			0
$703$		−	41.5692i		−	1.56781i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.00000			−0.300446			−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000			−0.300446			−0.150223	+	0.988652i	$0.547999\pi$
$710$		0			0
$711$	−18.0000	−	10.3923i	−0.675053	−	0.389742i
$712$		0			0
$713$			20.7846i			0.778390i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	45.0000	−	25.9808i	1.68056	−	0.970269i
$718$		0			0
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−7.50000	+	4.33013i	−0.278928	+	0.161039i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	17.3205i		−	0.642382i	−0.947014	−	0.321191i	$-0.895917\pi$
$727$		−	17.3205i		−	0.642382i	0.947014	−	0.321191i	$-0.104083\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	18.0000			0.665754
$732$		0			0
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	36.3731i		−	1.33982i
$738$		0			0
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$		0			0
$741$	9.00000	+	15.5885i	0.330623	+	0.572656i
$742$		0			0
$743$	6.00000			0.220119			0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000			0.220119			0.110059	+	0.993925i	$0.464896\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−13.5000	+	23.3827i	−0.493939	+	0.855528i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	38.1051i		−	1.39048i	−0.718780	−	0.695238i	$-0.755299\pi$
$751$		−	38.1051i		−	1.39048i	0.718780	−	0.695238i	$-0.244701\pi$
$752$		0			0
$753$	−31.5000	+	18.1865i	−1.14792	+	0.662754i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$	−27.0000	+	15.5885i	−0.980038	+	0.565825i
$760$		0			0
$761$		−	15.5885i		−	0.565081i	−0.959255	−	0.282541i	$-0.908823\pi$
$761$		−	15.5885i		−	0.565081i	0.959255	−	0.282541i	$-0.0911772\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000			0.866590
$768$		0			0
$769$	47.0000			1.69486			0.847432	−	0.530904i	$-0.178148\pi$
$769$	47.0000			1.69486			0.847432	+	0.530904i	$0.178148\pi$
$770$		0			0
$771$	18.0000	+	31.1769i	0.648254	+	1.12281i
$772$		0			0
$773$		−	10.3923i		−	0.373785i	−0.982380	−	0.186893i	$-0.940158\pi$
$773$		−	10.3923i		−	0.373785i	0.982380	−	0.186893i	$-0.0598416\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.0000			−0.967375
$780$		0			0
$781$	−18.0000			−0.644091
$782$		0			0
$783$	−54.0000			−1.92980
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	10.3923i		−	0.370446i	−0.982697	−	0.185223i	$-0.940699\pi$
$787$		−	10.3923i		−	0.370446i	0.982697	−	0.185223i	$-0.0593007\pi$
$788$		0			0
$789$	9.00000	−	5.19615i	0.320408	−	0.184988i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−16.0000			−0.568177
$794$		0			0
$795$		0			0
$796$		0			0
$797$			51.9615i			1.84057i	0.391247	+	0.920286i	$0.372044\pi$
$797$			51.9615i			1.84057i	−0.391247	+	0.920286i	$0.627956\pi$
$798$		0			0
$799$			31.1769i			1.10296i
$800$		0			0
$801$	−13.5000	−	7.79423i	−0.476999	−	0.275396i
$802$		0			0
$803$	−3.00000			−0.105868
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18.0000	+	31.1769i	0.633630	+	1.09748i
$808$		0			0
$809$			20.7846i			0.730748i	0.930861	+	0.365374i	$0.119059\pi$
$809$			20.7846i			0.730748i	−0.930861	+	0.365374i	$0.880941\pi$
$810$		0			0
$811$			17.3205i			0.608205i	0.952639	+	0.304103i	$0.0983566\pi$
$811$			17.3205i			0.608205i	−0.952639	+	0.304103i	$0.901643\pi$
$812$		0			0
$813$	21.0000	+	36.3731i	0.736502	+	1.27566i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	18.0000			0.629740
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	10.3923i		−	0.362694i	−0.983419	−	0.181347i	$-0.941954\pi$
$821$		−	10.3923i		−	0.362694i	0.983419	−	0.181347i	$-0.0580457\pi$
$822$		0			0
$823$			48.4974i			1.69051i	0.534360	+	0.845257i	$0.320553\pi$
$823$			48.4974i			1.69051i	−0.534360	+	0.845257i	$0.679447\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.0000			−1.35616			−0.678081	−	0.734987i	$-0.737188\pi$
$827$	−39.0000			−1.35616			−0.678081	+	0.734987i	$0.737188\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$	−21.0000	+	12.1244i	−0.728482	+	0.420589i
$832$		0			0
$833$		−	36.3731i		−	1.26025i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	18.0000			0.622171
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−79.0000			−2.72414
$842$		0			0
$843$	18.0000	+	31.1769i	0.619953	+	1.07379i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	10.5000	+	18.1865i	0.360359	+	0.624160i
$850$		0			0
$851$	48.0000			1.64542
$852$		0			0
$853$	−22.0000			−0.753266			−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000			−0.753266			−0.376633	+	0.926363i	$0.622918\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	15.5885i		−	0.532492i	−0.963905	−	0.266246i	$-0.914217\pi$
$857$		−	15.5885i		−	0.532492i	0.963905	−	0.266246i	$-0.0857833\pi$
$858$		0			0
$859$			22.5167i			0.768259i	0.923279	+	0.384129i	$0.125498\pi$
$859$			22.5167i			0.768259i	−0.923279	+	0.384129i	$0.874502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−54.0000			−1.83818			−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000			−1.83818			−0.919091	+	0.394046i	$0.871075\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.0000	+	8.66025i	−0.509427	+	0.294118i
$868$		0			0
$869$			20.7846i			0.705070i
$870$		0			0
$871$		−	24.2487i		−	0.821636i
$872$		0			0
$873$	15.0000	−	25.9808i	0.507673	−	0.879316i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−40.0000			−1.35070			−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000			−1.35070			−0.675352	+	0.737496i	$0.736008\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20.7846i		−	0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$		−	20.7846i		−	0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$		0			0
$883$		−	25.9808i		−	0.874322i	−0.899383	−	0.437161i	$-0.855984\pi$
$883$		−	25.9808i		−	0.874322i	0.899383	−	0.437161i	$-0.144016\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−48.0000			−1.61168			−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000			−1.61168			−0.805841	+	0.592132i	$0.798286\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	13.5000	+	23.3827i	0.452267	+	0.783349i
$892$		0			0
$893$			31.1769i			1.04330i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−18.0000	+	10.3923i	−0.601003	+	0.346989i
$898$		0			0
$899$	36.0000			1.20067
$900$		0			0
$901$	54.0000			1.79900
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	38.1051i		−	1.26526i	−0.774454	−	0.632630i	$-0.781975\pi$
$907$		−	38.1051i		−	1.26526i	0.774454	−	0.632630i	$-0.218025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	27.0000			0.893570
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	3.46410i		−	0.114270i	−0.998366	−	0.0571351i	$-0.981803\pi$
$919$		−	3.46410i		−	0.114270i	0.998366	−	0.0571351i	$-0.0181966\pi$
$920$		0			0
$921$	10.5000	+	18.1865i	0.345987	+	0.599267i
$922$		0			0
$923$	−12.0000			−0.394985
$924$		0			0
$925$		0			0
$926$		0			0
$927$	45.0000	+	25.9808i	1.47799	+	0.853320i
$928$		0			0
$929$			20.7846i			0.681921i	0.940078	+	0.340960i	$0.110752\pi$
$929$			20.7846i			0.681921i	−0.940078	+	0.340960i	$0.889248\pi$
$930$		0			0
$931$		−	36.3731i		−	1.19208i
$932$		0			0
$933$	36.0000	−	20.7846i	1.17859	−	0.680458i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−25.0000			−0.816714			−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000			−0.816714			−0.408357	+	0.912822i	$0.633898\pi$
$938$		0			0
$939$	21.0000	−	12.1244i	0.685309	−	0.395663i
$940$		0			0
$941$			10.3923i			0.338779i	0.985549	+	0.169390i	$0.0541797\pi$
$941$			10.3923i			0.338779i	−0.985549	+	0.169390i	$0.945820\pi$
$942$		0			0
$943$		−	31.1769i		−	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	36.0000			1.16984			0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000			1.16984			0.584921	+	0.811090i	$0.301125\pi$
$948$		0			0
$949$	−2.00000			−0.0649227
$950$		0			0
$951$		0			0
$952$		0			0
$953$			36.3731i			1.17824i	0.808046	+	0.589120i	$0.200525\pi$
$953$			36.3731i			1.17824i	−0.808046	+	0.589120i	$0.799475\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	27.0000	+	46.7654i	0.872786	+	1.51171i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	4.50000	−	7.79423i	0.145010	−	0.251166i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			13.8564i			0.445592i	0.974865	+	0.222796i	$0.0715184\pi$
$967$			13.8564i			0.445592i	−0.974865	+	0.222796i	$0.928482\pi$
$968$		0			0
$969$	−40.5000	+	23.3827i	−1.30105	+	0.751160i
$970$		0			0
$971$	−3.00000			−0.0962746			−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000			−0.0962746			−0.0481373	+	0.998841i	$0.515328\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.9808i			0.831198i	0.909548	+	0.415599i	$0.136428\pi$
$977$			25.9808i			0.831198i	−0.909548	+	0.415599i	$0.863572\pi$
$978$		0			0
$979$			15.5885i			0.498209i
$980$		0			0
$981$	−12.0000	+	20.7846i	−0.383131	+	0.663602i
$982$		0			0
$983$	−24.0000			−0.765481			−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000			−0.765481			−0.382741	+	0.923856i	$0.625020\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			20.7846i			0.660912i
$990$		0			0
$991$		−	20.7846i		−	0.660245i	−0.943938	−	0.330122i	$-0.892910\pi$
$991$		−	20.7846i		−	0.660245i	0.943938	−	0.330122i	$-0.107090\pi$
$992$		0			0
$993$	−7.50000	−	12.9904i	−0.238005	−	0.412237i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28.0000			−0.886769			−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000			−0.886769			−0.443384	+	0.896332i	$0.646222\pi$
$998$		0			0
$999$		−	41.5692i		−	1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.h.h.1151.1	yes	2
3.2	odd	2		1200.2.h.b.1151.1	yes	2
4.3	odd	2		1200.2.h.b.1151.2	yes	2
5.2	odd	4		1200.2.o.c.1199.1		4
5.3	odd	4		1200.2.o.c.1199.4		4
5.4	even	2		1200.2.h.a.1151.2	yes	2
12.11	even	2	inner	1200.2.h.h.1151.2	yes	2
15.2	even	4		1200.2.o.d.1199.2		4
15.8	even	4		1200.2.o.d.1199.3		4
15.14	odd	2		1200.2.h.i.1151.2	yes	2
20.3	even	4		1200.2.o.d.1199.1		4
20.7	even	4		1200.2.o.d.1199.4		4
20.19	odd	2		1200.2.h.i.1151.1	yes	2
60.23	odd	4		1200.2.o.c.1199.2		4
60.47	odd	4		1200.2.o.c.1199.3		4
60.59	even	2		1200.2.h.a.1151.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1200.2.h.a.1151.1	✓	2	60.59	even	2
1200.2.h.a.1151.2	yes	2	5.4	even	2
1200.2.h.b.1151.1	yes	2	3.2	odd	2
1200.2.h.b.1151.2	yes	2	4.3	odd	2
1200.2.h.h.1151.1	yes	2	1.1	even	1	trivial
1200.2.h.h.1151.2	yes	2	12.11	even	2	inner
1200.2.h.i.1151.1	yes	2	20.19	odd	2
1200.2.h.i.1151.2	yes	2	15.14	odd	2
1200.2.o.c.1199.1		4	5.2	odd	4
1200.2.o.c.1199.2		4	60.23	odd	4
1200.2.o.c.1199.3		4	60.47	odd	4
1200.2.o.c.1199.4		4	5.3	odd	4
1200.2.o.d.1199.1		4	20.3	even	4
1200.2.o.d.1199.2		4	15.2	even	4
1200.2.o.d.1199.3		4	15.8	even	4
1200.2.o.d.1199.4		4	20.7	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(1.50000 - 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{9} -3.00000 q^{11} -2.00000 q^{13} -5.19615i q^{17} -5.19615i q^{19} +6.00000 q^{23} -5.19615i q^{27} -10.3923i q^{29} +3.46410i q^{31} +(-4.50000 + 2.59808i) q^{33} +8.00000 q^{37} +(-3.00000 + 1.73205i) q^{39} -5.19615i q^{41} +3.46410i q^{43} -6.00000 q^{47} +7.00000 q^{49} +(-4.50000 - 7.79423i) q^{51} +10.3923i q^{53} +(-4.50000 - 7.79423i) q^{57} -12.0000 q^{59} +8.00000 q^{61} +12.1244i q^{67} +(9.00000 - 5.19615i) q^{69} +6.00000 q^{71} +1.00000 q^{73} -6.92820i q^{79} +(-4.50000 - 7.79423i) q^{81} -9.00000 q^{83} +(-9.00000 - 15.5885i) q^{87} -5.19615i q^{89} +(3.00000 + 5.19615i) q^{93} +10.0000 q^{97} +(-4.50000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 3 * q^9	\( 2 q + 3 q^{3} + 3 q^{9} - 6 q^{11} - 4 q^{13} + 12 q^{23} - 9 q^{33} + 16 q^{37} - 6 q^{39} - 12 q^{47} + 14 q^{49} - 9 q^{51} - 9 q^{57} - 24 q^{59} + 16 q^{61} + 18 q^{69} + 12 q^{71} + 2 q^{73} - 9 q^{81} - 18 q^{83} - 18 q^{87} + 6 q^{93} + 20 q^{97} - 9 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 - 4 * q^13 + 12 * q^23 - 9 * q^33 + 16 * q^37 - 6 * q^39 - 12 * q^47 + 14 * q^49 - 9 * q^51 - 9 * q^57 - 24 * q^59 + 16 * q^61 + 18 * q^69 + 12 * q^71 + 2 * q^73 - 9 * q^81 - 18 * q^83 - 18 * q^87 + 6 * q^93 + 20 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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