LMFDB - Embedded newform 3900.2.j.h.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3900,2,Mod(649,3900)] 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("3900.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$31.1416567883$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3900.649
Dual form			3900.2.j.h.649.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{9} -3.46410i q^{11} +(3.46410 - 1.00000i) q^{13} -6.00000i q^{17} -6.92820i q^{19} +1.00000i q^{27} +6.00000 q^{29} +6.92820i q^{31} -3.46410 q^{33} +(-1.00000 - 3.46410i) q^{39} +3.46410i q^{41} +8.00000i q^{43} -3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} +6.00000i q^{53} -6.92820 q^{57} -3.46410i q^{59} +10.0000 q^{61} -13.8564 q^{67} -10.3923i q^{71} -6.92820 q^{73} +8.00000 q^{79} +1.00000 q^{81} -3.46410 q^{83} -6.00000i q^{87} -17.3205i q^{89} +6.92820 q^{93} -6.92820 q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 24 q^{29} - 4 q^{39} - 28 q^{49} - 24 q^{51} + 40 q^{61} + 32 q^{79} + 4 q^{81}+O(q^{100}) $ 4 * q - 4 * q^9 + 24 * q^29 - 4 * q^39 - 28 * q^49 - 24 * q^51 + 40 * q^61 + 32 * q^79 + 4 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$		−	6.92820i		−	1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$		−	6.92820i		−	1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			1.00000i			0.192450i
$28$		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
$31$			6.92820i			1.24434i	0.782881	+	0.622171i	$0.213749\pi$
$31$			6.92820i			1.24434i	−0.782881	+	0.622171i	$0.786251\pi$
$32$		0			0
$33$	−3.46410			−0.603023
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$	−1.00000	−	3.46410i	−0.160128	−	0.554700i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−6.00000			−0.840168
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.92820			−0.917663
$58$		0			0
$59$		−	3.46410i		−	0.450988i	−0.974245	−	0.225494i	$-0.927600\pi$
$59$		−	3.46410i		−	0.450988i	0.974245	−	0.225494i	$-0.0723995\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−13.8564			−1.69283			−0.846415	−	0.532524i	$-0.821244\pi$
$67$	−13.8564			−1.69283			−0.846415	+	0.532524i	$0.821244\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	−6.92820			−0.810885			−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820			−0.810885			−0.405442	+	0.914121i	$0.632883\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$		−	17.3205i		−	1.83597i	−0.396615	−	0.917985i	$-0.629815\pi$
$89$		−	17.3205i		−	1.83597i	0.396615	−	0.917985i	$-0.370185\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.92820			0.718421
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.92820			−0.703452			−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820			−0.703452			−0.351726	+	0.936103i	$0.614405\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.46410	+	1.00000i	−0.320256	+	0.0924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	3.46410			0.312348
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		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
$137$	10.3923			0.887875			0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923			0.887875			0.443937	+	0.896058i	$0.353581\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$			3.46410i			0.291730i
$142$		0			0
$143$	−3.46410	−	12.0000i	−0.289683	−	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$		−	17.3205i		−	1.41895i	−0.704730	−	0.709476i	$-0.748932\pi$
$149$		−	17.3205i		−	1.41895i	0.704730	−	0.709476i	$-0.251068\pi$
$150$		0			0
$151$		−	6.92820i		−	0.563809i	−0.959442	−	0.281905i	$-0.909034\pi$
$151$		−	6.92820i		−	0.563809i	0.959442	−	0.281905i	$-0.0909662\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−20.7846			−1.62798			−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846			−1.62798			−0.813988	+	0.580881i	$0.802708\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$		0			0
$169$	11.0000	−	6.92820i	0.846154	−	0.532939i
$170$		0			0
$171$			6.92820i			0.529813i
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.46410			−0.260378
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−20.7846			−1.51992
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	13.8564			0.997406			0.498703	−	0.866773i	$-0.333810\pi$
$193$	13.8564			0.997406			0.498703	+	0.866773i	$0.333810\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.3205			−1.23404			−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205			−1.23404			−0.617018	+	0.786949i	$0.711659\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$			13.8564i			0.977356i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−10.3923			−0.712069
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$			6.92820i			0.468165i
$220$		0			0
$221$	−6.00000	−	20.7846i	−0.403604	−	1.39812i
$222$		0			0
$223$	−6.92820			−0.463947			−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820			−0.463947			−0.231973	+	0.972722i	$0.574518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.3923			0.689761			0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923			0.689761			0.344881	+	0.938647i	$0.387919\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	8.00000i		−	0.519656i
$238$		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
$241$			20.7846i			1.33885i	0.742878	+	0.669427i	$0.233460\pi$
$241$			20.7846i			1.33885i	−0.742878	+	0.669427i	$0.766540\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.92820	−	24.0000i	−0.440831	−	1.52708i
$248$		0			0
$249$			3.46410i			0.219529i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−17.3205			−1.06000
$268$		0			0
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$			27.7128i			1.68343i	0.539919	+	0.841717i	$0.318455\pi$
$271$			27.7128i			1.68343i	−0.539919	+	0.841717i	$0.681545\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		0			0
$279$		−	6.92820i		−	0.414781i
$280$		0			0
$281$		−	17.3205i		−	1.03325i	−0.856210	−	0.516627i	$-0.827187\pi$
$281$		−	17.3205i		−	1.03325i	0.856210	−	0.516627i	$-0.172813\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$			6.92820i			0.406138i
$292$		0			0
$293$	3.46410			0.202375			0.101187	−	0.994867i	$-0.467736\pi$
$293$	3.46410			0.202375			0.101187	+	0.994867i	$0.467736\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.46410			0.201008
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	6.00000i		−	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		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
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.2487			1.36194			0.680972	−	0.732310i	$-0.261558\pi$
$317$	24.2487			1.36194			0.680972	+	0.732310i	$0.261558\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	−41.5692			−2.31297
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.92820			−0.383131
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	13.8564i		−	0.761617i	−0.924654	−	0.380808i	$-0.875646\pi$
$331$		−	13.8564i		−	0.761617i	0.924654	−	0.380808i	$-0.124354\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	10.0000i		−	0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$		−	10.0000i		−	0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$		0			0
$339$	−18.0000			−0.977626
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	1.00000	+	3.46410i	0.0533761	+	0.184900i
$352$		0			0
$353$	10.3923			0.553127			0.276563	−	0.960996i	$-0.410804\pi$
$353$	10.3923			0.553127			0.276563	+	0.960996i	$0.410804\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	24.2487i		−	1.27980i	−0.768459	−	0.639899i	$-0.778976\pi$
$359$		−	24.2487i		−	1.27980i	0.768459	−	0.639899i	$-0.221024\pi$
$360$		0			0
$361$	−29.0000			−1.52632
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			28.0000i			1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$			28.0000i			1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$		−	3.46410i		−	0.180334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	2.00000i		−	0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$		−	2.00000i		−	0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.7846	−	6.00000i	1.07046	−	0.309016i
$378$		0			0
$379$			20.7846i			1.06763i	0.845600	+	0.533817i	$0.179243\pi$
$379$			20.7846i			1.06763i	−0.845600	+	0.533817i	$0.820757\pi$
$380$		0			0
$381$	−16.0000			−0.819705
$382$		0			0
$383$	−24.2487			−1.23905			−0.619526	−	0.784976i	$-0.712675\pi$
$383$	−24.2487			−1.23905			−0.619526	+	0.784976i	$0.712675\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13.8564			−0.695433			−0.347717	−	0.937600i	$-0.613043\pi$
$397$	−13.8564			−0.695433			−0.347717	+	0.937600i	$0.613043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.3923i		−	0.518967i	−0.965748	−	0.259483i	$-0.916448\pi$
$401$		−	10.3923i		−	0.518967i	0.965748	−	0.259483i	$-0.0835523\pi$
$402$		0			0
$403$	6.92820	+	24.0000i	0.345118	+	1.19553i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			20.7846i			1.02773i	0.857870	+	0.513866i	$0.171787\pi$
$409$			20.7846i			1.02773i	−0.857870	+	0.513866i	$0.828213\pi$
$410$		0			0
$411$		−	10.3923i		−	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$			20.7846i			1.01298i	0.862246	+	0.506490i	$0.169057\pi$
$421$			20.7846i			1.01298i	−0.862246	+	0.506490i	$0.830943\pi$
$422$		0			0
$423$	3.46410			0.168430
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−12.0000	+	3.46410i	−0.579365	+	0.167248i
$430$		0			0
$431$		−	31.1769i		−	1.50174i	−0.660451	−	0.750870i	$-0.729635\pi$
$431$		−	31.1769i		−	1.50174i	0.660451	−	0.750870i	$-0.270365\pi$
$432$		0			0
$433$		−	2.00000i		−	0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$		−	2.00000i		−	0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−17.3205			−0.819232
$448$		0			0
$449$		−	3.46410i		−	0.163481i	−0.996654	−	0.0817405i	$-0.973952\pi$
$449$		−	3.46410i		−	0.163481i	0.996654	−	0.0817405i	$-0.0260479\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$	−6.92820			−0.325515
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.92820			−0.324088			−0.162044	−	0.986784i	$-0.551809\pi$
$457$	−6.92820			−0.324088			−0.162044	+	0.986784i	$0.551809\pi$
$458$		0			0
$459$	6.00000			0.280056
$460$		0			0
$461$			10.3923i			0.484018i	0.970274	+	0.242009i	$0.0778063\pi$
$461$			10.3923i			0.484018i	−0.970274	+	0.242009i	$0.922194\pi$
$462$		0			0
$463$	13.8564			0.643962			0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564			0.643962			0.321981	+	0.946746i	$0.395651\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$	27.7128			1.27424
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$		−	3.46410i		−	0.158279i	−0.996864	−	0.0791394i	$-0.974783\pi$
$479$		−	3.46410i		−	0.158279i	0.996864	−	0.0791394i	$-0.0252172\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−6.92820			−0.313947			−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820			−0.313947			−0.156973	+	0.987603i	$0.550174\pi$
$488$		0			0
$489$			20.7846i			0.939913i
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$		−	36.0000i		−	1.62136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$			17.3205i			0.773823i
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6.92820	−	11.0000i	−0.307692	−	0.488527i
$508$		0			0
$509$		−	24.2487i		−	1.07481i	−0.843326	−	0.537403i	$-0.819406\pi$
$509$		−	24.2487i		−	1.07481i	0.843326	−	0.537403i	$-0.180594\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.92820			0.305888
$514$		0			0
$515$		0			0
$516$		0			0
$517$			12.0000i			0.527759i
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			8.00000i			0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$			8.00000i			0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	41.5692			1.81078
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$			3.46410i			0.150329i
$532$		0			0
$533$	3.46410	+	12.0000i	0.150047	+	0.519778i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	12.0000i		−	0.517838i
$538$		0			0
$539$			24.2487i			1.04447i
$540$		0			0
$541$			20.7846i			0.893600i	0.894634	+	0.446800i	$0.147436\pi$
$541$			20.7846i			0.893600i	−0.894634	+	0.446800i	$0.852564\pi$
$542$		0			0
$543$			22.0000i			0.944110i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$		−	41.5692i		−	1.77091i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.2487			1.02745			0.513725	−	0.857955i	$-0.328265\pi$
$557$	24.2487			1.02745			0.513725	+	0.857955i	$0.328265\pi$
$558$		0			0
$559$	8.00000	+	27.7128i	0.338364	+	1.17213i
$560$		0			0
$561$			20.7846i			0.877527i
$562$		0			0
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		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
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	13.8564			0.576850			0.288425	−	0.957503i	$-0.406868\pi$
$577$	13.8564			0.576850			0.288425	+	0.957503i	$0.406868\pi$
$578$		0			0
$579$		−	13.8564i		−	0.575853i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20.7846			0.860811
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.1769			1.28681			0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769			1.28681			0.643404	+	0.765526i	$0.277521\pi$
$588$		0			0
$589$	48.0000			1.97781
$590$		0			0
$591$			17.3205i			0.712470i
$592$		0			0
$593$	24.2487			0.995775			0.497888	−	0.867242i	$-0.334109\pi$
$593$	24.2487			0.995775			0.497888	+	0.867242i	$0.334109\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	8.00000i		−	0.327418i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	13.8564			0.564276
$604$		0			0
$605$		0			0
$606$		0			0
$607$			28.0000i			1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$			28.0000i			1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	+	3.46410i	−0.485468	+	0.140143i
$612$		0			0
$613$	13.8564			0.559655			0.279827	−	0.960050i	$-0.409723\pi$
$613$	13.8564			0.559655			0.279827	+	0.960050i	$0.409723\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.1769			1.25514			0.627568	−	0.778562i	$-0.284051\pi$
$617$	31.1769			1.25514			0.627568	+	0.778562i	$0.284051\pi$
$618$		0			0
$619$		−	27.7128i		−	1.11387i	−0.830555	−	0.556936i	$-0.811977\pi$
$619$		−	27.7128i		−	1.11387i	0.830555	−	0.556936i	$-0.188023\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$			24.0000i			0.958468i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	20.7846i		−	0.827422i	−0.910408	−	0.413711i	$-0.864232\pi$
$631$		−	20.7846i		−	0.827422i	0.910408	−	0.413711i	$-0.135768\pi$
$632$		0			0
$633$		−	4.00000i		−	0.158986i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−24.2487	+	7.00000i	−0.960769	+	0.277350i
$638$		0			0
$639$			10.3923i			0.411113i
$640$		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
$643$	−41.5692			−1.63933			−0.819665	−	0.572843i	$-0.805840\pi$
$643$	−41.5692			−1.63933			−0.819665	+	0.572843i	$0.805840\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	6.00000i		−	0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$		−	6.00000i		−	0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	6.92820			0.270295
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$			13.8564i			0.538952i	0.963007	+	0.269476i	$0.0868504\pi$
$661$			13.8564i			0.538952i	−0.963007	+	0.269476i	$0.913150\pi$
$662$		0			0
$663$	−20.7846	+	6.00000i	−0.807207	+	0.233021i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			6.92820i			0.267860i
$670$		0			0
$671$		−	34.6410i		−	1.33730i
$672$		0			0
$673$			14.0000i			0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$			14.0000i			0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	10.3923i		−	0.398234i
$682$		0			0
$683$	3.46410			0.132550			0.0662751	−	0.997801i	$-0.478889\pi$
$683$	3.46410			0.132550			0.0662751	+	0.997801i	$0.478889\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6.92820			0.264327
$688$		0			0
$689$	6.00000	+	20.7846i	0.228582	+	0.791831i
$690$		0			0
$691$		−	27.7128i		−	1.05425i	−0.849789	−	0.527123i	$-0.823271\pi$
$691$		−	27.7128i		−	1.05425i	0.849789	−	0.527123i	$-0.176729\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	20.7846			0.787273
$698$		0			0
$699$	18.0000			0.680823
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	6.92820i		−	0.260194i	−0.991501	−	0.130097i	$-0.958471\pi$
$709$		−	6.92820i		−	0.260194i	0.991501	−	0.130097i	$-0.0415289\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.3923			0.388108
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	20.7846			0.772988
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	48.0000			1.77534
$732$		0			0
$733$	6.92820			0.255899			0.127950	−	0.991781i	$-0.459160\pi$
$733$	6.92820			0.255899			0.127950	+	0.991781i	$0.459160\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.0000i			1.76810i
$738$		0			0
$739$			27.7128i			1.01943i	0.860343	+	0.509716i	$0.170250\pi$
$739$			27.7128i			1.01943i	−0.860343	+	0.509716i	$0.829750\pi$
$740$		0			0
$741$	−24.0000	+	6.92820i	−0.881662	+	0.254514i
$742$		0			0
$743$	10.3923			0.381257			0.190628	−	0.981662i	$-0.438947\pi$
$743$	10.3923			0.381257			0.190628	+	0.981662i	$0.438947\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.46410			0.126745
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.0000			−1.02173			−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000			−1.02173			−0.510867	+	0.859660i	$0.670676\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			38.0000i			1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$			38.0000i			1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.1769i		−	1.13016i	−0.825035	−	0.565081i	$-0.808845\pi$
$761$		−	31.1769i		−	1.13016i	0.825035	−	0.565081i	$-0.191155\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.46410	−	12.0000i	−0.125081	−	0.433295i
$768$		0			0
$769$			27.7128i			0.999350i	0.866213	+	0.499675i	$0.166547\pi$
$769$			27.7128i			0.999350i	−0.866213	+	0.499675i	$0.833453\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$	51.9615			1.86893			0.934463	−	0.356060i	$-0.115880\pi$
$773$	51.9615			1.86893			0.934463	+	0.356060i	$0.115880\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	27.7128			0.987855			0.493928	−	0.869503i	$-0.335561\pi$
$787$	27.7128			0.987855			0.493928	+	0.869503i	$0.335561\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	34.6410	−	10.0000i	1.23014	−	0.355110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$			20.7846i			0.735307i
$800$		0			0
$801$			17.3205i			0.611990i
$802$		0			0
$803$			24.0000i			0.846942i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	30.0000i		−	1.05605i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$		−	6.92820i		−	0.243282i	−0.992574	−	0.121641i	$-0.961184\pi$
$811$		−	6.92820i		−	0.243282i	0.992574	−	0.121641i	$-0.0388157\pi$
$812$		0			0
$813$	27.7128			0.971931
$814$		0			0
$815$		0			0
$816$		0			0
$817$	55.4256			1.93910
$818$		0			0
$819$		0			0
$820$		0			0
$821$			3.46410i			0.120898i	0.998171	+	0.0604490i	$0.0192532\pi$
$821$			3.46410i			0.120898i	−0.998171	+	0.0604490i	$0.980747\pi$
$822$		0			0
$823$		−	4.00000i		−	0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$		−	4.00000i		−	0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−17.3205			−0.602293			−0.301147	−	0.953578i	$-0.597369\pi$
$827$	−17.3205			−0.602293			−0.301147	+	0.953578i	$0.597369\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$			42.0000i			1.45521i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−6.92820			−0.239474
$838$		0			0
$839$			10.3923i			0.358782i	0.983778	+	0.179391i	$0.0574128\pi$
$839$			10.3923i			0.358782i	−0.983778	+	0.179391i	$0.942587\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−17.3205			−0.596550
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$		0			0
$852$		0			0
$853$	13.8564			0.474434			0.237217	−	0.971457i	$-0.423765\pi$
$853$	13.8564			0.474434			0.237217	+	0.971457i	$0.423765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−38.1051			−1.29711			−0.648557	−	0.761166i	$-0.724627\pi$
$863$	−38.1051			−1.29711			−0.648557	+	0.761166i	$0.724627\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			19.0000i			0.645274i
$868$		0			0
$869$		−	27.7128i		−	0.940093i
$870$		0			0
$871$	−48.0000	+	13.8564i	−1.62642	+	0.469506i
$872$		0			0
$873$	6.92820			0.234484
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−41.5692			−1.40369			−0.701846	−	0.712328i	$-0.747641\pi$
$877$	−41.5692			−1.40369			−0.701846	+	0.712328i	$0.747641\pi$
$878$		0			0
$879$		−	3.46410i		−	0.116841i
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$		−	20.0000i		−	0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$		−	20.0000i		−	0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	3.46410i		−	0.116052i
$892$		0			0
$893$			24.0000i			0.803129i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			41.5692i			1.38641i
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$			12.0000i			0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3923	−	36.0000i	−0.342067	−	1.18495i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			4.00000i			0.131377i
$928$		0			0
$929$		−	31.1769i		−	1.02288i	−0.859319	−	0.511441i	$-0.829112\pi$
$929$		−	31.1769i		−	1.02288i	0.859319	−	0.511441i	$-0.170888\pi$
$930$		0			0
$931$			48.4974i			1.58944i
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$		0			0
$939$	10.0000			0.326338
$940$		0			0
$941$		−	10.3923i		−	0.338779i	−0.985549	−	0.169390i	$-0.945820\pi$
$941$		−	10.3923i		−	0.338779i	0.985549	−	0.169390i	$-0.0541797\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	45.0333			1.46339			0.731693	−	0.681634i	$-0.238730\pi$
$947$	45.0333			1.46339			0.731693	+	0.681634i	$0.238730\pi$
$948$		0			0
$949$	−24.0000	+	6.92820i	−0.779073	+	0.224899i
$950$		0			0
$951$		−	24.2487i		−	0.786318i
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−20.7846			−0.671871
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	34.6410			1.11398			0.556990	−	0.830519i	$-0.311956\pi$
$967$	34.6410			1.11398			0.556990	+	0.830519i	$0.311956\pi$
$968$		0			0
$969$			41.5692i			1.33540i
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.2487			0.775785			0.387893	−	0.921705i	$-0.373203\pi$
$977$	24.2487			0.775785			0.387893	+	0.921705i	$0.373203\pi$
$978$		0			0
$979$	−60.0000			−1.91761
$980$		0			0
$981$			6.92820i			0.221201i
$982$		0			0
$983$	−31.1769			−0.994389			−0.497195	−	0.867639i	$-0.665636\pi$
$983$	−31.1769			−0.994389			−0.497195	+	0.867639i	$0.665636\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$	−13.8564			−0.439720
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	38.0000i		−	1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$		−	38.0000i		−	1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.j.h.649.1		4
5.2	odd	4		156.2.b.a.25.1	✓	2
5.3	odd	4		3900.2.c.c.3301.1		2
5.4	even	2	inner	3900.2.j.h.649.3		4
13.12	even	2	inner	3900.2.j.h.649.2		4
15.2	even	4		468.2.b.a.181.2		2
20.7	even	4		624.2.c.f.337.1		2
35.27	even	4		7644.2.e.g.4705.2		2
40.27	even	4		2496.2.c.e.961.2		2
40.37	odd	4		2496.2.c.l.961.2		2
60.47	odd	4		1872.2.c.c.1585.2		2
65.2	even	12		2028.2.i.i.529.1		4
65.7	even	12		2028.2.i.i.2005.2		4
65.12	odd	4		156.2.b.a.25.2	yes	2
65.17	odd	12		2028.2.q.c.361.1		2
65.22	odd	12		2028.2.q.b.361.1		2
65.32	even	12		2028.2.i.i.2005.1		4
65.37	even	12		2028.2.i.i.529.2		4
65.38	odd	4		3900.2.c.c.3301.2		2
65.42	odd	12		2028.2.q.c.1837.1		2
65.47	even	4		2028.2.a.g.1.2		2
65.57	even	4		2028.2.a.g.1.1		2
65.62	odd	12		2028.2.q.b.1837.1		2
65.64	even	2	inner	3900.2.j.h.649.4		4
195.47	odd	4		6084.2.a.v.1.1		2
195.77	even	4		468.2.b.a.181.1		2
195.122	odd	4		6084.2.a.v.1.2		2
260.47	odd	4		8112.2.a.bs.1.2		2
260.187	odd	4		8112.2.a.bs.1.1		2
260.207	even	4		624.2.c.f.337.2		2
455.272	even	4		7644.2.e.g.4705.1		2
520.77	odd	4		2496.2.c.l.961.1		2
520.467	even	4		2496.2.c.e.961.1		2
780.467	odd	4		1872.2.c.c.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.a.25.1	✓	2	5.2	odd	4
156.2.b.a.25.2	yes	2	65.12	odd	4
468.2.b.a.181.1		2	195.77	even	4
468.2.b.a.181.2		2	15.2	even	4
624.2.c.f.337.1		2	20.7	even	4
624.2.c.f.337.2		2	260.207	even	4
1872.2.c.c.1585.1		2	780.467	odd	4
1872.2.c.c.1585.2		2	60.47	odd	4
2028.2.a.g.1.1		2	65.57	even	4
2028.2.a.g.1.2		2	65.47	even	4
2028.2.i.i.529.1		4	65.2	even	12
2028.2.i.i.529.2		4	65.37	even	12
2028.2.i.i.2005.1		4	65.32	even	12
2028.2.i.i.2005.2		4	65.7	even	12
2028.2.q.b.361.1		2	65.22	odd	12
2028.2.q.b.1837.1		2	65.62	odd	12
2028.2.q.c.361.1		2	65.17	odd	12
2028.2.q.c.1837.1		2	65.42	odd	12
2496.2.c.e.961.1		2	520.467	even	4
2496.2.c.e.961.2		2	40.27	even	4
2496.2.c.l.961.1		2	520.77	odd	4
2496.2.c.l.961.2		2	40.37	odd	4
3900.2.c.c.3301.1		2	5.3	odd	4
3900.2.c.c.3301.2		2	65.38	odd	4
3900.2.j.h.649.1		4	1.1	even	1	trivial
3900.2.j.h.649.2		4	13.12	even	2	inner
3900.2.j.h.649.3		4	5.4	even	2	inner
3900.2.j.h.649.4		4	65.64	even	2	inner
6084.2.a.v.1.1		2	195.47	odd	4
6084.2.a.v.1.2		2	195.122	odd	4
7644.2.e.g.4705.1		2	455.272	even	4
7644.2.e.g.4705.2		2	35.27	even	4
8112.2.a.bs.1.1		2	260.187	odd	4
8112.2.a.bs.1.2		2	260.47	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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{9} -3.46410i q^{11} +(3.46410 - 1.00000i) q^{13} -6.00000i q^{17} -6.92820i q^{19} +1.00000i q^{27} +6.00000 q^{29} +6.92820i q^{31} -3.46410 q^{33} +(-1.00000 - 3.46410i) q^{39} +3.46410i q^{41} +8.00000i q^{43} -3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} +6.00000i q^{53} -6.92820 q^{57} -3.46410i q^{59} +10.0000 q^{61} -13.8564 q^{67} -10.3923i q^{71} -6.92820 q^{73} +8.00000 q^{79} +1.00000 q^{81} -3.46410 q^{83} -6.00000i q^{87} -17.3205i q^{89} +6.92820 q^{93} -6.92820 q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 24 q^{29} - 4 q^{39} - 28 q^{49} - 24 q^{51} + 40 q^{61} + 32 q^{79} + 4 q^{81}+O(q^{100}) \) 4 * q - 4 * q^9 + 24 * q^29 - 4 * q^39 - 28 * q^49 - 24 * q^51 + 40 * q^61 + 32 * q^79 + 4 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more