LMFDB - Embedded newform 1400.2.bh.c.849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(249,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))

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

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

chi := DirichletCharacter("1400.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.bh (of order $6$, degree $2$, not 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			849.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1400.849
Dual form			1400.2.bh.c.249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{3} +(2.59808 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})$	$q+(0.866025 - 0.500000i) q^{3} +(2.59808 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(1.00000 + 1.73205i) q^{11} -4.00000i q^{13} +(3.00000 - 5.19615i) q^{19} +(2.50000 - 0.866025i) q^{21} +(2.59808 + 1.50000i) q^{23} +5.00000i q^{27} +3.00000 q^{29} +(1.73205 + 1.00000i) q^{33} +(10.3923 + 6.00000i) q^{37} +(-2.00000 - 3.46410i) q^{39} -7.00000 q^{41} +9.00000i q^{43} +(6.50000 + 2.59808i) q^{49} +(5.19615 - 3.00000i) q^{53} -6.00000i q^{57} +(-5.00000 - 8.66025i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-3.46410 + 4.00000i) q^{63} +(9.52628 - 5.50000i) q^{67} +3.00000 q^{69} -10.0000 q^{71} +(6.92820 - 4.00000i) q^{73} +(1.73205 + 5.00000i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-0.500000 - 0.866025i) q^{81} +3.00000i q^{83} +(2.59808 - 1.50000i) q^{87} +(8.50000 - 14.7224i) q^{89} +(2.00000 - 10.3923i) q^{91} -2.00000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 4 q^{11} + 12 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{39} - 28 q^{41} + 26 q^{49} - 20 q^{59} - 10 q^{61} + 12 q^{69} - 40 q^{71} + 12 q^{79} - 2 q^{81} + 34 q^{89} + 8 q^{91} - 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^11 + 12 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^39 - 28 * q^41 + 26 * q^49 - 20 * q^59 - 10 * q^61 + 12 * q^69 - 40 * q^71 + 12 * q^79 - 2 * q^81 + 34 * q^89 + 8 * q^91 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

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
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	0.728714	+	0.684819i	$0.240119\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$8$		0			0
$9$	−1.00000	+	1.73205i	−0.333333	+	0.577350i
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	3.00000	−	5.19615i	0.688247	−	1.19208i	−0.284157	−	0.958778i	$-0.591714\pi$
$19$	3.00000	−	5.19615i	0.688247	−	1.19208i	0.972404	−	0.233301i	$-0.0749529\pi$
$20$		0			0
$21$	2.50000	−	0.866025i	0.545545	−	0.188982i
$22$		0			0
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	0.745782	−	0.666190i	$-0.232076\pi$
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	−0.204046	+	0.978961i	$0.565409\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$	1.73205	+	1.00000i	0.301511	+	0.174078i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.3923	+	6.00000i	1.70848	+	0.986394i	0.936442	+	0.350823i	$0.114098\pi$
$37$	10.3923	+	6.00000i	1.70848	+	0.986394i	0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$	−2.00000	−	3.46410i	−0.320256	−	0.554700i
$40$		0			0
$41$	−7.00000			−1.09322			−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000			−1.09322			−0.546608	+	0.837389i	$0.684081\pi$
$42$		0			0
$43$			9.00000i			1.37249i	0.727372	+	0.686244i	$0.240742\pi$
$43$			9.00000i			1.37249i	−0.727372	+	0.686244i	$0.759258\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.19615	−	3.00000i	0.713746	−	0.412082i	−0.0987002	−	0.995117i	$-0.531468\pi$
$53$	5.19615	−	3.00000i	0.713746	−	0.412082i	0.812447	+	0.583036i	$0.198135\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$		0			0
$63$	−3.46410	+	4.00000i	−0.436436	+	0.503953i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	9.52628	−	5.50000i	1.16382	−	0.671932i	0.211604	−	0.977356i	$-0.432131\pi$
$67$	9.52628	−	5.50000i	1.16382	−	0.671932i	0.952217	+	0.305424i	$0.0987981\pi$
$68$		0			0
$69$	3.00000			0.361158
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$	6.92820	−	4.00000i	0.810885	−	0.468165i	−0.0363782	−	0.999338i	$-0.511582\pi$
$73$	6.92820	−	4.00000i	0.810885	−	0.468165i	0.847263	+	0.531174i	$0.178249\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.73205	+	5.00000i	0.197386	+	0.569803i
$78$		0			0
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	−0.646440	−	0.762964i	$-0.723743\pi$
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	0.983967	+	0.178352i	$0.0570765\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			3.00000i			0.329293i	0.986353	+	0.164646i	$0.0526483\pi$
$83$			3.00000i			0.329293i	−0.986353	+	0.164646i	$0.947352\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.59808	−	1.50000i	0.278543	−	0.160817i
$88$		0			0
$89$	8.50000	−	14.7224i	0.900998	−	1.56057i	0.0747975	−	0.997199i	$-0.476169\pi$
$89$	8.50000	−	14.7224i	0.900998	−	1.56057i	0.826201	−	0.563376i	$-0.190498\pi$
$90$		0			0
$91$	2.00000	−	10.3923i	0.209657	−	1.08941i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	0.884941	+	0.465704i	$0.154199\pi$
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	−0.0391591	+	0.999233i	$0.512468\pi$
$102$		0			0
$103$	−12.9904	−	7.50000i	−1.27998	−	0.738997i	−0.303136	−	0.952947i	$-0.598034\pi$
$103$	−12.9904	−	7.50000i	−1.27998	−	0.738997i	−0.976845	+	0.213950i	$0.931367\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.866025	+	0.500000i	0.0837218	+	0.0483368i	0.541276	−	0.840845i	$-0.317941\pi$
$107$	0.866025	+	0.500000i	0.0837218	+	0.0483368i	−0.457555	+	0.889182i	$0.651275\pi$
$108$		0			0
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	0.721102	−	0.692829i	$-0.243636\pi$
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	−0.960558	+	0.278078i	$0.910303\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.92820	+	4.00000i	0.640513	+	0.369800i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	−6.06218	+	3.50000i	−0.546608	+	0.315584i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$	4.50000	+	7.79423i	0.396203	+	0.686244i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	10.3923	−	12.0000i	0.901127	−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.46410	+	2.00000i	−0.295958	+	0.170872i	−0.640626	−	0.767853i	$-0.721325\pi$
$137$	−3.46410	+	2.00000i	−0.295958	+	0.170872i	0.344668	+	0.938725i	$0.387992\pi$
$138$		0			0
$139$	−18.0000			−1.52674			−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000			−1.52674			−0.763370	+	0.645961i	$0.776457\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.92820	−	4.00000i	0.579365	−	0.334497i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	6.92820	−	1.00000i	0.571429	−	0.0824786i
$148$		0			0
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	0.273377	+	0.961907i	$0.411859\pi$
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	−0.969724	+	0.244202i	$0.921474\pi$
$150$		0			0
$151$	−10.0000	−	17.3205i	−0.813788	−	1.40952i	−0.910195	−	0.414181i	$-0.864068\pi$
$151$	−10.0000	−	17.3205i	−0.813788	−	1.40952i	0.0964061	−	0.995342i	$-0.469265\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.1244	+	7.00000i	−0.967629	+	0.558661i	−0.898513	−	0.438948i	$-0.855351\pi$
$157$	−12.1244	+	7.00000i	−0.967629	+	0.558661i	−0.0691164	+	0.997609i	$0.522018\pi$
$158$		0			0
$159$	3.00000	−	5.19615i	0.237915	−	0.412082i
$160$		0			0
$161$	6.00000	+	5.19615i	0.472866	+	0.409514i
$162$		0			0
$163$	−10.3923	−	6.00000i	−0.813988	−	0.469956i	0.0343508	−	0.999410i	$-0.489064\pi$
$163$	−10.3923	−	6.00000i	−0.813988	−	0.469956i	−0.848339	+	0.529454i	$0.822397\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	6.00000	+	10.3923i	0.458831	+	0.794719i
$172$		0			0
$173$	−6.92820	−	4.00000i	−0.526742	−	0.304114i	0.212947	−	0.977064i	$-0.431694\pi$
$173$	−6.92820	−	4.00000i	−0.526742	−	0.304114i	−0.739689	+	0.672949i	$0.765027\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.66025	−	5.00000i	−0.650945	−	0.375823i
$178$		0			0
$179$	−4.00000	−	6.92820i	−0.298974	−	0.517838i	0.676927	−	0.736050i	$-0.263311\pi$
$179$	−4.00000	−	6.92820i	−0.298974	−	0.517838i	−0.975901	+	0.218212i	$0.929978\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$			5.00000i			0.369611i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.50000	+	12.9904i	−0.181848	+	0.944911i
$190$		0			0
$191$	−7.00000	+	12.1244i	−0.506502	+	0.877288i	0.493469	+	0.869763i	$0.335728\pi$
$191$	−7.00000	+	12.1244i	−0.506502	+	0.877288i	−0.999972	+	0.00752447i	$0.997605\pi$
$192$		0			0
$193$	19.0526	−	11.0000i	1.37143	−	0.791797i	0.380325	−	0.924853i	$-0.375812\pi$
$193$	19.0526	−	11.0000i	1.37143	−	0.791797i	0.991109	+	0.133056i	$0.0424789\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	6.00000	+	10.3923i	0.425329	+	0.736691i	0.996451	−	0.0841740i	$-0.0268252\pi$
$199$	6.00000	+	10.3923i	0.425329	+	0.736691i	−0.571122	+	0.820865i	$0.693492\pi$
$200$		0			0
$201$	5.50000	−	9.52628i	0.387940	−	0.671932i
$202$		0			0
$203$	7.79423	+	1.50000i	0.547048	+	0.105279i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.19615	+	3.00000i	−0.361158	+	0.208514i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$	−8.66025	+	5.00000i	−0.593391	+	0.342594i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.00000	−	6.92820i	0.270295	−	0.468165i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.46410	+	2.00000i	−0.229920	+	0.132745i	−0.610535	−	0.791989i	$-0.709046\pi$
$227$	−3.46410	+	2.00000i	−0.229920	+	0.132745i	0.380615	+	0.924734i	$0.375712\pi$
$228$		0			0
$229$	−1.00000	+	1.73205i	−0.0660819	+	0.114457i	−0.897173	−	0.441679i	$-0.854383\pi$
$229$	−1.00000	+	1.73205i	−0.0660819	+	0.114457i	0.831092	+	0.556136i	$0.187717\pi$
$230$		0			0
$231$	4.00000	+	3.46410i	0.263181	+	0.227921i
$232$		0			0
$233$	−3.46410	−	2.00000i	−0.226941	−	0.131024i	0.382219	−	0.924072i	$-0.375160\pi$
$233$	−3.46410	−	2.00000i	−0.226941	−	0.131024i	−0.609160	+	0.793047i	$0.708493\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	6.00000i		−	0.389742i
$238$		0			0
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$	−9.00000	−	15.5885i	−0.579741	−	1.00414i	−0.995509	−	0.0946700i	$-0.969820\pi$
$241$	−9.00000	−	15.5885i	−0.579741	−	1.00414i	0.415768	−	0.909471i	$-0.363513\pi$
$242$		0			0
$243$	−13.8564	−	8.00000i	−0.888889	−	0.513200i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−20.7846	−	12.0000i	−1.32249	−	0.763542i
$248$		0			0
$249$	1.50000	+	2.59808i	0.0950586	+	0.164646i
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$			6.00000i			0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.7846	−	12.0000i	−1.29651	−	0.748539i	−0.316709	−	0.948523i	$-0.602578\pi$
$257$	−20.7846	−	12.0000i	−1.29651	−	0.748539i	−0.979799	+	0.199983i	$0.935911\pi$
$258$		0			0
$259$	24.0000	+	20.7846i	1.49129	+	1.29149i
$260$		0			0
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$	−18.1865	+	10.5000i	−1.12143	+	0.647458i	−0.941766	−	0.336270i	$-0.890834\pi$
$263$	−18.1865	+	10.5000i	−1.12143	+	0.647458i	−0.179664	+	0.983728i	$0.557501\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	17.0000i		−	1.04038i
$268$		0			0
$269$	15.5000	+	26.8468i	0.945052	+	1.63688i	0.755648	+	0.654978i	$0.227322\pi$
$269$	15.5000	+	26.8468i	0.945052	+	1.63688i	0.189404	+	0.981899i	$0.439344\pi$
$270$		0			0
$271$	1.00000	−	1.73205i	0.0607457	−	0.105215i	−0.834053	−	0.551684i	$-0.813985\pi$
$271$	1.00000	−	1.73205i	0.0607457	−	0.105215i	0.894799	+	0.446469i	$0.147319\pi$
$272$		0			0
$273$	−3.46410	−	10.0000i	−0.209657	−	0.605228i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.92820	−	4.00000i	0.416275	−	0.240337i	−0.277207	−	0.960810i	$-0.589409\pi$
$277$	6.92820	−	4.00000i	0.416275	−	0.240337i	0.693482	+	0.720473i	$0.256075\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$	24.2487	−	14.0000i	1.44144	−	0.832214i	0.443491	−	0.896279i	$-0.353740\pi$
$283$	24.2487	−	14.0000i	1.44144	−	0.832214i	0.997946	+	0.0640654i	$0.0204066\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−18.1865	−	3.50000i	−1.07352	−	0.206598i
$288$		0			0
$289$	−8.50000	+	14.7224i	−0.500000	+	0.866025i
$290$		0			0
$291$	−1.00000	−	1.73205i	−0.0586210	−	0.101535i
$292$		0			0
$293$			4.00000i			0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$			4.00000i			0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−8.66025	+	5.00000i	−0.502519	+	0.290129i
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$	−4.50000	+	23.3827i	−0.259376	+	1.34776i
$302$		0			0
$303$	14.7224	+	8.50000i	0.845782	+	0.488312i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	21.0000i		−	1.19853i	−0.800549	−	0.599267i	$-0.795459\pi$
$307$		−	21.0000i		−	1.19853i	0.800549	−	0.599267i	$-0.204541\pi$
$308$		0			0
$309$	−15.0000			−0.853320
$310$		0			0
$311$	−5.00000	−	8.66025i	−0.283524	−	0.491078i	0.688726	−	0.725022i	$-0.258170\pi$
$311$	−5.00000	−	8.66025i	−0.283524	−	0.491078i	−0.972250	+	0.233944i	$0.924837\pi$
$312$		0			0
$313$	−13.8564	−	8.00000i	−0.783210	−	0.452187i	0.0543564	−	0.998522i	$-0.482689\pi$
$313$	−13.8564	−	8.00000i	−0.783210	−	0.452187i	−0.837567	+	0.546335i	$0.816023\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.3923	+	6.00000i	0.583690	+	0.336994i	0.762598	−	0.646872i	$-0.223923\pi$
$317$	10.3923	+	6.00000i	0.583690	+	0.336994i	−0.178908	+	0.983866i	$0.557257\pi$
$318$		0			0
$319$	3.00000	+	5.19615i	0.167968	+	0.290929i
$320$		0			0
$321$	1.00000			0.0558146
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.33013	−	2.50000i	−0.239457	−	0.138250i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$	−20.7846	+	12.0000i	−1.13899	+	0.657596i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			10.0000i			0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$			10.0000i			0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$		0			0
$339$	9.00000	+	15.5885i	0.488813	+	0.846649i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.6506	+	12.5000i	−1.16227	+	0.671035i	−0.951846	−	0.306576i	$-0.900817\pi$
$347$	−21.6506	+	12.5000i	−1.16227	+	0.671035i	−0.210421	+	0.977611i	$0.567483\pi$
$348$		0			0
$349$	−17.0000			−0.909989			−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000			−0.909989			−0.454995	+	0.890494i	$0.650359\pi$
$350$		0			0
$351$	20.0000			1.06752
$352$		0			0
$353$	−15.5885	+	9.00000i	−0.829690	+	0.479022i	−0.853746	−	0.520689i	$-0.825675\pi$
$353$	−15.5885	+	9.00000i	−0.829690	+	0.479022i	0.0240566	+	0.999711i	$0.492342\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	0.353529	+	0.935423i	$0.384981\pi$
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	−0.986865	+	0.161546i	$0.948352\pi$
$360$		0			0
$361$	−8.50000	−	14.7224i	−0.447368	−	0.774865i
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.2583	−	6.50000i	0.587680	−	0.339297i	−0.176500	−	0.984301i	$-0.556477\pi$
$367$	11.2583	−	6.50000i	0.587680	−	0.339297i	0.764180	+	0.645003i	$0.223144\pi$
$368$		0			0
$369$	7.00000	−	12.1244i	0.364405	−	0.631169i
$370$		0			0
$371$	15.0000	−	5.19615i	0.778761	−	0.269771i
$372$		0			0
$373$	31.1769	+	18.0000i	1.61428	+	0.932005i	0.988363	+	0.152115i	$0.0486083\pi$
$373$	31.1769	+	18.0000i	1.61428	+	0.932005i	0.625917	+	0.779890i	$0.284725\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	−34.0000			−1.74646			−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000			−1.74646			−0.873231	+	0.487306i	$0.837980\pi$
$380$		0			0
$381$	4.00000	+	6.92820i	0.204926	+	0.354943i
$382$		0			0
$383$	−28.5788	−	16.5000i	−1.46031	−	0.843111i	−0.461285	−	0.887252i	$-0.652611\pi$
$383$	−28.5788	−	16.5000i	−1.46031	−	0.843111i	−0.999025	+	0.0441413i	$0.985945\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−15.5885	−	9.00000i	−0.792406	−	0.457496i
$388$		0			0
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	0.632189	−	0.774814i	$-0.282157\pi$
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	−0.987103	+	0.160085i	$0.948823\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.73205	+	1.00000i	0.0869291	+	0.0501886i	0.542834	−	0.839840i	$-0.317351\pi$
$397$	1.73205	+	1.00000i	0.0869291	+	0.0501886i	−0.455905	+	0.890028i	$0.650684\pi$
$398$		0			0
$399$	3.00000	−	15.5885i	0.150188	−	0.780399i
$400$		0			0
$401$	16.5000	−	28.5788i	0.823971	−	1.42716i	−0.0787327	−	0.996896i	$-0.525087\pi$
$401$	16.5000	−	28.5788i	0.823971	−	1.42716i	0.902703	−	0.430263i	$-0.141579\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.0000i			1.18964i
$408$		0			0
$409$	−6.50000	−	11.2583i	−0.321404	−	0.556689i	0.659374	−	0.751815i	$-0.270822\pi$
$409$	−6.50000	−	11.2583i	−0.321404	−	0.556689i	−0.980778	+	0.195127i	$0.937488\pi$
$410$		0			0
$411$	−2.00000	+	3.46410i	−0.0986527	+	0.170872i
$412$		0			0
$413$	−8.66025	−	25.0000i	−0.426143	−	1.23017i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−15.5885	+	9.00000i	−0.763370	+	0.440732i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−31.0000			−1.51085			−0.755424	−	0.655237i	$-0.772569\pi$
$421$	−31.0000			−1.51085			−0.755424	+	0.655237i	$0.772569\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.66025	+	10.0000i	−0.419099	+	0.483934i
$428$		0			0
$429$	4.00000	−	6.92820i	0.193122	−	0.334497i
$430$		0			0
$431$	11.0000	+	19.0526i	0.529851	+	0.917729i	0.999394	+	0.0348195i	$0.0110856\pi$
$431$	11.0000	+	19.0526i	0.529851	+	0.917729i	−0.469542	+	0.882910i	$0.655581\pi$
$432$		0			0
$433$		−	18.0000i		−	0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$		−	18.0000i		−	0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.5885	−	9.00000i	0.745697	−	0.430528i
$438$		0			0
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	−0.909798	−	0.415051i	$-0.863764\pi$
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	0.814344	+	0.580383i	$0.197097\pi$
$440$		0			0
$441$	−11.0000	+	8.66025i	−0.523810	+	0.412393i
$442$		0			0
$443$	−18.1865	−	10.5000i	−0.864068	−	0.498870i	0.00130426	−	0.999999i	$-0.499585\pi$
$443$	−18.1865	−	10.5000i	−0.864068	−	0.498870i	−0.865373	+	0.501129i	$0.832918\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			17.0000i			0.804072i
$448$		0			0
$449$	−39.0000			−1.84052			−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000			−1.84052			−0.920262	+	0.391303i	$0.872024\pi$
$450$		0			0
$451$	−7.00000	−	12.1244i	−0.329617	−	0.570914i
$452$		0			0
$453$	−17.3205	−	10.0000i	−0.813788	−	0.469841i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.92820	−	4.00000i	−0.324088	−	0.187112i	0.329125	−	0.944286i	$-0.393246\pi$
$457$	−6.92820	−	4.00000i	−0.324088	−	0.187112i	−0.653213	+	0.757174i	$0.726579\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$		−	29.0000i		−	1.34774i	−0.738848	−	0.673872i	$-0.764630\pi$
$463$		−	29.0000i		−	1.34774i	0.738848	−	0.673872i	$-0.235370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.5788	+	16.5000i	1.32247	+	0.763529i	0.984122	−	0.177492i	$-0.0567983\pi$
$467$	28.5788	+	16.5000i	1.32247	+	0.763529i	0.338349	+	0.941021i	$0.390132\pi$
$468$		0			0
$469$	27.5000	−	9.52628i	1.26983	−	0.439883i
$470$		0			0
$471$	−7.00000	+	12.1244i	−0.322543	+	0.558661i
$472$		0			0
$473$	−15.5885	+	9.00000i	−0.716758	+	0.413820i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$	18.0000	+	31.1769i	0.822441	+	1.42451i	0.903859	+	0.427830i	$0.140722\pi$
$479$	18.0000	+	31.1769i	0.822441	+	1.42451i	−0.0814184	+	0.996680i	$0.525945\pi$
$480$		0			0
$481$	24.0000	−	41.5692i	1.09431	−	1.89539i
$482$		0			0
$483$	7.79423	+	1.50000i	0.354650	+	0.0682524i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	27.7128	−	16.0000i	1.25579	−	0.725029i	0.283535	−	0.958962i	$-0.408493\pi$
$487$	27.7128	−	16.0000i	1.25579	−	0.725029i	0.972253	+	0.233933i	$0.0751596\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−25.9808	−	5.00000i	−1.16540	−	0.224281i
$498$		0			0
$499$	−5.00000	+	8.66025i	−0.223831	+	0.387686i	−0.955968	−	0.293471i	$-0.905190\pi$
$499$	−5.00000	+	8.66025i	−0.223831	+	0.387686i	0.732137	+	0.681157i	$0.238523\pi$
$500$		0			0
$501$	1.50000	+	2.59808i	0.0670151	+	0.116073i
$502$		0			0
$503$			37.0000i			1.64975i	0.565316	+	0.824874i	$0.308754\pi$
$503$			37.0000i			1.64975i	−0.565316	+	0.824874i	$0.691246\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.59808	+	1.50000i	−0.115385	+	0.0666173i
$508$		0			0
$509$	10.5000	−	18.1865i	0.465404	−	0.806104i	−0.533815	−	0.845601i	$-0.679242\pi$
$509$	10.5000	−	18.1865i	0.465404	−	0.806104i	0.999220	+	0.0394971i	$0.0125756\pi$
$510$		0			0
$511$	20.0000	−	6.92820i	0.884748	−	0.306486i
$512$		0			0
$513$	25.9808	+	15.0000i	1.14708	+	0.662266i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−8.00000			−0.351161
$520$		0			0
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	0.993011	−	0.118020i	$-0.0376547\pi$
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	−0.598714	+	0.800963i	$0.704321\pi$
$522$		0			0
$523$	24.2487	+	14.0000i	1.06032	+	0.612177i	0.925521	−	0.378695i	$-0.123627\pi$
$523$	24.2487	+	14.0000i	1.06032	+	0.612177i	0.134801	+	0.990873i	$0.456961\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.00000	−	12.1244i	−0.304348	−	0.527146i
$530$		0			0
$531$	20.0000			0.867926
$532$		0			0
$533$			28.0000i			1.21281i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.92820	−	4.00000i	−0.298974	−	0.172613i
$538$		0			0
$539$	2.00000	+	13.8564i	0.0861461	+	0.596838i
$540$		0			0
$541$	−19.5000	+	33.7750i	−0.838370	+	1.45210i	0.0528859	+	0.998601i	$0.483158\pi$
$541$	−19.5000	+	33.7750i	−0.838370	+	1.45210i	−0.891256	+	0.453500i	$0.850175\pi$
$542$		0			0
$543$	−4.33013	+	2.50000i	−0.185824	+	0.107285i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	13.0000i		−	0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$		−	13.0000i		−	0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$		0			0
$549$	−5.00000	−	8.66025i	−0.213395	−	0.369611i
$550$		0			0
$551$	9.00000	−	15.5885i	0.383413	−	0.664091i
$552$		0			0
$553$	10.3923	−	12.0000i	0.441926	−	0.510292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.9808	−	15.0000i	1.10084	−	0.635570i	0.164399	−	0.986394i	$-0.447432\pi$
$557$	25.9808	−	15.0000i	1.10084	−	0.635570i	0.936442	+	0.350824i	$0.114098\pi$
$558$		0			0
$559$	36.0000			1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.52628	−	5.50000i	0.401485	−	0.231797i	−0.285640	−	0.958337i	$-0.592206\pi$
$563$	9.52628	−	5.50000i	0.401485	−	0.231797i	0.687124	+	0.726540i	$0.258873\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.866025	−	2.50000i	−0.0363696	−	0.104990i
$568$		0			0
$569$	11.0000	−	19.0526i	0.461144	−	0.798725i	−0.537874	−	0.843025i	$-0.680772\pi$
$569$	11.0000	−	19.0526i	0.461144	−	0.798725i	0.999018	+	0.0443003i	$0.0141058\pi$
$570$		0			0
$571$	9.00000	+	15.5885i	0.376638	+	0.652357i	0.990571	−	0.137002i	$-0.0437466\pi$
$571$	9.00000	+	15.5885i	0.376638	+	0.652357i	−0.613933	+	0.789359i	$0.710413\pi$
$572$		0			0
$573$			14.0000i			0.584858i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.73205	−	1.00000i	0.0721062	−	0.0416305i	−0.463513	−	0.886090i	$-0.653411\pi$
$577$	1.73205	−	1.00000i	0.0721062	−	0.0416305i	0.535620	+	0.844459i	$0.320078\pi$
$578$		0			0
$579$	11.0000	−	19.0526i	0.457144	−	0.791797i
$580$		0			0
$581$	−1.50000	+	7.79423i	−0.0622305	+	0.323359i
$582$		0			0
$583$	10.3923	+	6.00000i	0.430405	+	0.248495i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			20.0000i			0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$			20.0000i			0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−3.00000	−	5.19615i	−0.123404	−	0.213741i
$592$		0			0
$593$	−8.66025	−	5.00000i	−0.355634	−	0.205325i	0.311530	−	0.950236i	$-0.399159\pi$
$593$	−8.66025	−	5.00000i	−0.355634	−	0.205325i	−0.667164	+	0.744911i	$0.732492\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10.3923	+	6.00000i	0.425329	+	0.245564i
$598$		0			0
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$			22.0000i			0.895909i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.06218	+	3.50000i	0.246056	+	0.142061i	0.617957	−	0.786212i	$-0.287961\pi$
$607$	6.06218	+	3.50000i	0.246056	+	0.142061i	−0.371901	+	0.928272i	$0.621294\pi$
$608$		0			0
$609$	7.50000	−	2.59808i	0.303915	−	0.105279i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.92820	+	4.00000i	−0.279827	+	0.161558i	−0.633345	−	0.773869i	$-0.718319\pi$
$613$	−6.92820	+	4.00000i	−0.279827	+	0.161558i	0.353518	+	0.935428i	$0.384985\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.0000i		−	0.805170i	−0.915383	−	0.402585i	$-0.868112\pi$
$617$		−	20.0000i		−	0.805170i	0.915383	−	0.402585i	$-0.131888\pi$
$618$		0			0
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	0.999657	+	0.0261952i	$0.00833914\pi$
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	−0.477143	+	0.878826i	$0.658328\pi$
$620$		0			0
$621$	−7.50000	+	12.9904i	−0.300965	+	0.521286i
$622$		0			0
$623$	29.4449	−	34.0000i	1.17968	−	1.36218i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	10.3923	−	6.00000i	0.415029	−	0.239617i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−6.00000			−0.238856			−0.119428	−	0.992843i	$-0.538106\pi$
$631$	−6.00000			−0.238856			−0.119428	+	0.992843i	$0.538106\pi$
$632$		0			0
$633$	−12.1244	+	7.00000i	−0.481900	+	0.278225i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10.3923	−	26.0000i	0.411758	−	1.03016i
$638$		0			0
$639$	10.0000	−	17.3205i	0.395594	−	0.685189i
$640$		0			0
$641$	−6.50000	−	11.2583i	−0.256735	−	0.444677i	0.708631	−	0.705580i	$-0.249313\pi$
$641$	−6.50000	−	11.2583i	−0.256735	−	0.444677i	−0.965365	+	0.260902i	$0.915980\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	38.9711	−	22.5000i	1.53211	−	0.884566i	0.532850	−	0.846210i	$-0.321121\pi$
$647$	38.9711	−	22.5000i	1.53211	−	0.884566i	0.999264	−	0.0383563i	$-0.0122122\pi$
$648$		0			0
$649$	10.0000	−	17.3205i	0.392534	−	0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			16.0000i			0.624219i
$658$		0			0
$659$	46.0000			1.79191			0.895953	−	0.444149i	$-0.146494\pi$
$659$	46.0000			1.79191			0.895953	+	0.444149i	$0.146494\pi$
$660$		0			0
$661$	3.50000	+	6.06218i	0.136134	+	0.235791i	0.926030	−	0.377450i	$-0.123199\pi$
$661$	3.50000	+	6.06218i	0.136134	+	0.235791i	−0.789896	+	0.613241i	$0.789865\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.79423	+	4.50000i	0.301794	+	0.174241i
$668$		0			0
$669$	2.00000	+	3.46410i	0.0773245	+	0.133930i
$670$		0			0
$671$	−10.0000			−0.386046
$672$		0			0
$673$			16.0000i			0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$			16.0000i			0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.1769	+	18.0000i	1.19823	+	0.691796i	0.960159	−	0.279453i	$-0.0901530\pi$
$677$	31.1769	+	18.0000i	1.19823	+	0.691796i	0.238067	+	0.971249i	$0.423486\pi$
$678$		0			0
$679$	1.00000	−	5.19615i	0.0383765	−	0.199410i
$680$		0			0
$681$	−2.00000	+	3.46410i	−0.0766402	+	0.132745i
$682$		0			0
$683$	−6.06218	+	3.50000i	−0.231963	+	0.133924i	−0.611477	−	0.791262i	$-0.709424\pi$
$683$	−6.06218	+	3.50000i	−0.231963	+	0.133924i	0.379514	+	0.925186i	$0.376091\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			2.00000i			0.0763048i
$688$		0			0
$689$	−12.0000	−	20.7846i	−0.457164	−	0.791831i
$690$		0			0
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	−0.755110	−	0.655598i	$-0.772417\pi$
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	0.945319	+	0.326146i	$0.105750\pi$
$692$		0			0
$693$	−10.3923	−	2.00000i	−0.394771	−	0.0759737i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−4.00000			−0.151294
$700$		0			0
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$	62.3538	−	36.0000i	2.35172	−	1.35777i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.7224	+	42.5000i	0.553694	+	1.59838i
$708$		0			0
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	−0.856484	−	0.516174i	$-0.827356\pi$
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$	6.00000	+	10.3923i	0.225018	+	0.389742i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−17.3205	+	10.0000i	−0.646846	+	0.373457i
$718$		0			0
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	−0.515030	−	0.857172i	$-0.672219\pi$
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	0.999848	+	0.0174426i	$0.00555244\pi$
$720$		0			0
$721$	−30.0000	−	25.9808i	−1.11726	−	0.967574i
$722$		0			0
$723$	−15.5885	−	9.00000i	−0.579741	−	0.334714i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	13.0000i		−	0.482143i	−0.970507	−	0.241072i	$-0.922501\pi$
$727$		−	13.0000i		−	0.482143i	0.970507	−	0.241072i	$-0.0774989\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$	12.1244	+	7.00000i	0.447823	+	0.258551i	0.706910	−	0.707303i	$-0.250088\pi$
$733$	12.1244	+	7.00000i	0.447823	+	0.258551i	−0.259087	+	0.965854i	$0.583422\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19.0526	+	11.0000i	0.701810	+	0.405190i
$738$		0			0
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	0.759287	−	0.650756i	$-0.225548\pi$
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	−0.943215	+	0.332184i	$0.892215\pi$
$740$		0			0
$741$	−24.0000			−0.881662
$742$		0			0
$743$		−	9.00000i		−	0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$		−	9.00000i		−	0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.19615	−	3.00000i	−0.190117	−	0.109764i
$748$		0			0
$749$	2.00000	+	1.73205i	0.0730784	+	0.0632878i
$750$		0			0
$751$	−14.0000	+	24.2487i	−0.510867	+	0.884848i	0.489053	+	0.872254i	$0.337342\pi$
$751$	−14.0000	+	24.2487i	−0.510867	+	0.884848i	−0.999921	+	0.0125942i	$0.995991\pi$
$752$		0			0
$753$	−3.46410	+	2.00000i	−0.126239	+	0.0728841i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			20.0000i			0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$			20.0000i			0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$		0			0
$759$	3.00000	+	5.19615i	0.108893	+	0.188608i
$760$		0			0
$761$	11.0000	−	19.0526i	0.398750	−	0.690655i	−0.594822	−	0.803857i	$-0.702778\pi$
$761$	11.0000	−	19.0526i	0.398750	−	0.690655i	0.993572	+	0.113203i	$0.0361109\pi$
$762$		0			0
$763$	−4.33013	−	12.5000i	−0.156761	−	0.452530i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−34.6410	+	20.0000i	−1.25081	+	0.722158i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$	−24.0000			−0.864339
$772$		0			0
$773$	−3.46410	+	2.00000i	−0.124595	+	0.0719350i	−0.561002	−	0.827814i	$-0.689584\pi$
$773$	−3.46410	+	2.00000i	−0.124595	+	0.0719350i	0.436407	+	0.899749i	$0.356251\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	31.1769	+	6.00000i	1.11847	+	0.215249i
$778$		0			0
$779$	−21.0000	+	36.3731i	−0.752403	+	1.30320i
$780$		0			0
$781$	−10.0000	−	17.3205i	−0.357828	−	0.619777i
$782$		0			0
$783$			15.0000i			0.536056i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−32.0429	+	18.5000i	−1.14221	+	0.659454i	−0.946976	−	0.321303i	$-0.895879\pi$
$787$	−32.0429	+	18.5000i	−1.14221	+	0.659454i	−0.195231	+	0.980757i	$0.562546\pi$
$788$		0			0
$789$	−10.5000	+	18.1865i	−0.373810	+	0.647458i
$790$		0			0
$791$	−9.00000	+	46.7654i	−0.320003	+	1.66279i
$792$		0			0
$793$	17.3205	+	10.0000i	0.615069	+	0.355110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	17.0000	+	29.4449i	0.600665	+	1.04038i
$802$		0			0
$803$	13.8564	+	8.00000i	0.488982	+	0.282314i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	26.8468	+	15.5000i	0.945052	+	0.545626i
$808$		0			0
$809$	−5.50000	−	9.52628i	−0.193370	−	0.334926i	0.752995	−	0.658026i	$-0.228608\pi$
$809$	−5.50000	−	9.52628i	−0.193370	−	0.334926i	−0.946365	+	0.323100i	$0.895275\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$		0			0
$813$		−	2.00000i		−	0.0701431i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	46.7654	+	27.0000i	1.63611	+	0.944610i
$818$		0			0
$819$	16.0000	+	13.8564i	0.559085	+	0.484182i
$820$		0			0
$821$	17.0000	−	29.4449i	0.593304	−	1.02763i	−0.400480	−	0.916306i	$-0.631157\pi$
$821$	17.0000	−	29.4449i	0.593304	−	1.02763i	0.993784	−	0.111327i	$-0.0355102\pi$
$822$		0			0
$823$	11.2583	−	6.50000i	0.392441	−	0.226576i	−0.290776	−	0.956791i	$-0.593914\pi$
$823$	11.2583	−	6.50000i	0.392441	−	0.226576i	0.683217	+	0.730215i	$0.260580\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	31.0000i		−	1.07798i	−0.842314	−	0.538988i	$-0.818807\pi$
$827$		−	31.0000i		−	1.07798i	0.842314	−	0.538988i	$-0.181193\pi$
$828$		0			0
$829$	−7.00000	−	12.1244i	−0.243120	−	0.421096i	0.718481	−	0.695546i	$-0.244838\pi$
$829$	−7.00000	−	12.1244i	−0.243120	−	0.421096i	−0.961601	+	0.274450i	$0.911504\pi$
$830$		0			0
$831$	4.00000	−	6.92820i	0.138758	−	0.240337i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−46.0000			−1.58810			−0.794048	−	0.607855i	$-0.792030\pi$
$839$	−46.0000			−1.58810			−0.794048	+	0.607855i	$0.792030\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$	22.5167	−	13.0000i	0.775515	−	0.447744i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	12.1244	−	14.0000i	0.416598	−	0.481046i
$848$		0			0
$849$	14.0000	−	24.2487i	0.480479	−	0.832214i
$850$		0			0
$851$	18.0000	+	31.1769i	0.617032	+	1.06873i
$852$		0			0
$853$		−	34.0000i		−	1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$		−	34.0000i		−	1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.5167	+	13.0000i	−0.769154	+	0.444072i	−0.832573	−	0.553915i	$-0.813133\pi$
$857$	−22.5167	+	13.0000i	−0.769154	+	0.444072i	0.0634184	+	0.997987i	$0.479800\pi$
$858$		0			0
$859$	28.0000	−	48.4974i	0.955348	−	1.65471i	0.221777	−	0.975097i	$-0.428814\pi$
$859$	28.0000	−	48.4974i	0.955348	−	1.65471i	0.733571	−	0.679613i	$-0.237852\pi$
$860$		0			0
$861$	−17.5000	+	6.06218i	−0.596398	+	0.206598i
$862$		0			0
$863$	−37.2391	−	21.5000i	−1.26763	−	0.731869i	−0.293094	−	0.956084i	$-0.594685\pi$
$863$	−37.2391	−	21.5000i	−1.26763	−	0.731869i	−0.974540	+	0.224215i	$0.928018\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			17.0000i			0.577350i
$868$		0			0
$869$	12.0000			0.407072
$870$		0			0
$871$	−22.0000	−	38.1051i	−0.745442	−	1.29114i
$872$		0			0
$873$	3.46410	+	2.00000i	0.117242	+	0.0676897i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.0526	+	11.0000i	0.643359	+	0.371444i	0.785907	−	0.618344i	$-0.212196\pi$
$877$	19.0526	+	11.0000i	0.643359	+	0.371444i	−0.142548	+	0.989788i	$0.545530\pi$
$878$		0			0
$879$	2.00000	+	3.46410i	0.0674583	+	0.116841i
$880$		0			0
$881$	41.0000			1.38133			0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000			1.38133			0.690663	+	0.723177i	$0.257319\pi$
$882$		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
$887$	−32.0429	−	18.5000i	−1.07590	−	0.621169i	−0.146110	−	0.989268i	$-0.546675\pi$
$887$	−32.0429	−	18.5000i	−1.07590	−	0.621169i	−0.929787	+	0.368099i	$0.880009\pi$
$888$		0			0
$889$	−4.00000	+	20.7846i	−0.134156	+	0.697093i
$890$		0			0
$891$	1.00000	−	1.73205i	0.0335013	−	0.0580259i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	12.0000i		−	0.400668i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	7.79423	+	22.5000i	0.259376	+	0.748753i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7.79423	−	4.50000i	0.258803	−	0.149420i	−0.364985	−	0.931013i	$-0.618926\pi$
$907$	7.79423	−	4.50000i	0.258803	−	0.149420i	0.623788	+	0.781593i	$0.285593\pi$
$908$		0			0
$909$	−34.0000			−1.12771
$910$		0			0
$911$	−34.0000			−1.12647			−0.563235	−	0.826297i	$-0.690443\pi$
$911$	−34.0000			−1.12647			−0.563235	+	0.826297i	$0.690443\pi$
$912$		0			0
$913$	−5.19615	+	3.00000i	−0.171968	+	0.0992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	17.0000	−	29.4449i	0.560778	−	0.971296i	−0.436650	−	0.899631i	$-0.643835\pi$
$919$	17.0000	−	29.4449i	0.560778	−	0.971296i	0.997429	−	0.0716652i	$-0.0228313\pi$
$920$		0			0
$921$	−10.5000	−	18.1865i	−0.345987	−	0.599267i
$922$		0			0
$923$			40.0000i			1.31662i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	25.9808	−	15.0000i	0.853320	−	0.492665i
$928$		0			0
$929$	13.5000	−	23.3827i	0.442921	−	0.767161i	−0.554984	−	0.831861i	$-0.687276\pi$
$929$	13.5000	−	23.3827i	0.442921	−	0.767161i	0.997905	+	0.0646999i	$0.0206090\pi$
$930$		0			0
$931$	33.0000	−	25.9808i	1.08153	−	0.851485i
$932$		0			0
$933$	−8.66025	−	5.00000i	−0.283524	−	0.163693i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			4.00000i			0.130674i	0.997863	+	0.0653372i	$0.0208123\pi$
$937$			4.00000i			0.130674i	−0.997863	+	0.0653372i	$0.979188\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	0.989599	+	0.143856i	$0.0459502\pi$
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	−0.370216	+	0.928946i	$0.620716\pi$
$942$		0			0
$943$	−18.1865	−	10.5000i	−0.592235	−	0.341927i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.2583	+	6.50000i	0.365847	+	0.211222i	0.671642	−	0.740875i	$-0.265589\pi$
$947$	11.2583	+	6.50000i	0.365847	+	0.211222i	−0.305796	+	0.952097i	$0.598922\pi$
$948$		0			0
$949$	−16.0000	−	27.7128i	−0.519382	−	0.899596i
$950$		0			0
$951$	12.0000			0.389127
$952$		0			0
$953$			28.0000i			0.907009i	0.891254	+	0.453504i	$0.149826\pi$
$953$			28.0000i			0.907009i	−0.891254	+	0.453504i	$0.850174\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	5.19615	+	3.00000i	0.167968	+	0.0969762i
$958$		0			0
$959$	−10.0000	+	3.46410i	−0.322917	+	0.111862i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$	−1.73205	+	1.00000i	−0.0558146	+	0.0322245i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			13.0000i			0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$			13.0000i			0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.0000	+	34.6410i	−0.641831	+	1.11168i	0.343193	+	0.939265i	$0.388491\pi$
$971$	−20.0000	+	34.6410i	−0.641831	+	1.11168i	−0.985024	+	0.172418i	$0.944842\pi$
$972$		0			0
$973$	−46.7654	−	9.00000i	−1.49923	−	0.288527i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.5885	−	9.00000i	0.498719	−	0.287936i	−0.229465	−	0.973317i	$-0.573698\pi$
$977$	15.5885	−	9.00000i	0.498719	−	0.287936i	0.728184	+	0.685381i	$0.240364\pi$
$978$		0			0
$979$	34.0000			1.08664
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−7.79423	+	4.50000i	−0.248597	+	0.143528i	−0.619122	−	0.785295i	$-0.712511\pi$
$983$	−7.79423	+	4.50000i	−0.248597	+	0.143528i	0.370525	+	0.928823i	$0.379178\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.5000	+	23.3827i	−0.429275	+	0.743526i
$990$		0			0
$991$	−10.0000	−	17.3205i	−0.317660	−	0.550204i	0.662339	−	0.749204i	$-0.269564\pi$
$991$	−10.0000	−	17.3205i	−0.317660	−	0.550204i	−0.979999	+	0.199000i	$0.936231\pi$
$992$		0			0
$993$			4.00000i			0.126936i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	12.1244	−	7.00000i	0.383982	−	0.221692i	−0.295567	−	0.955322i	$-0.595509\pi$
$997$	12.1244	−	7.00000i	0.383982	−	0.221692i	0.679549	+	0.733630i	$0.262175\pi$
$998$		0			0
$999$	−30.0000	+	51.9615i	−0.949158	+	1.64399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.bh.c.849.2		4
5.2	odd	4		1400.2.q.c.401.1		2
5.3	odd	4		280.2.q.b.121.1	yes	2
5.4	even	2	inner	1400.2.bh.c.849.1		4
7.4	even	3	inner	1400.2.bh.c.249.1		4
15.8	even	4		2520.2.bi.d.1801.1		2
20.3	even	4		560.2.q.e.401.1		2
35.2	odd	12		9800.2.a.z.1.1		1
35.3	even	12		1960.2.q.d.361.1		2
35.4	even	6	inner	1400.2.bh.c.249.2		4
35.12	even	12		9800.2.a.o.1.1		1
35.13	even	4		1960.2.q.d.961.1		2
35.18	odd	12		280.2.q.b.81.1	✓	2
35.23	odd	12		1960.2.a.c.1.1		1
35.32	odd	12		1400.2.q.c.1201.1		2
35.33	even	12		1960.2.a.l.1.1		1
105.53	even	12		2520.2.bi.d.361.1		2
140.23	even	12		3920.2.a.v.1.1		1
140.103	odd	12		3920.2.a.q.1.1		1
140.123	even	12		560.2.q.e.81.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.b.81.1	✓	2	35.18	odd	12
280.2.q.b.121.1	yes	2	5.3	odd	4
560.2.q.e.81.1		2	140.123	even	12
560.2.q.e.401.1		2	20.3	even	4
1400.2.q.c.401.1		2	5.2	odd	4
1400.2.q.c.1201.1		2	35.32	odd	12
1400.2.bh.c.249.1		4	7.4	even	3	inner
1400.2.bh.c.249.2		4	35.4	even	6	inner
1400.2.bh.c.849.1		4	5.4	even	2	inner
1400.2.bh.c.849.2		4	1.1	even	1	trivial
1960.2.a.c.1.1		1	35.23	odd	12
1960.2.a.l.1.1		1	35.33	even	12
1960.2.q.d.361.1		2	35.3	even	12
1960.2.q.d.961.1		2	35.13	even	4
2520.2.bi.d.361.1		2	105.53	even	12
2520.2.bi.d.1801.1		2	15.8	even	4
3920.2.a.q.1.1		1	140.103	odd	12
3920.2.a.v.1.1		1	140.23	even	12
9800.2.a.o.1.1		1	35.12	even	12
9800.2.a.z.1.1		1	35.2	odd	12

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(2.59808 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{3} +(2.59808 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(1.00000 + 1.73205i) q^{11} -4.00000i q^{13} +(3.00000 - 5.19615i) q^{19} +(2.50000 - 0.866025i) q^{21} +(2.59808 + 1.50000i) q^{23} +5.00000i q^{27} +3.00000 q^{29} +(1.73205 + 1.00000i) q^{33} +(10.3923 + 6.00000i) q^{37} +(-2.00000 - 3.46410i) q^{39} -7.00000 q^{41} +9.00000i q^{43} +(6.50000 + 2.59808i) q^{49} +(5.19615 - 3.00000i) q^{53} -6.00000i q^{57} +(-5.00000 - 8.66025i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-3.46410 + 4.00000i) q^{63} +(9.52628 - 5.50000i) q^{67} +3.00000 q^{69} -10.0000 q^{71} +(6.92820 - 4.00000i) q^{73} +(1.73205 + 5.00000i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-0.500000 - 0.866025i) q^{81} +3.00000i q^{83} +(2.59808 - 1.50000i) q^{87} +(8.50000 - 14.7224i) q^{89} +(2.00000 - 10.3923i) q^{91} -2.00000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 4 q^{11} + 12 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{39} - 28 q^{41} + 26 q^{49} - 20 q^{59} - 10 q^{61} + 12 q^{69} - 40 q^{71} + 12 q^{79} - 2 q^{81} + 34 q^{89} + 8 q^{91} - 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^11 + 12 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^39 - 28 * q^41 + 26 * q^49 - 20 * q^59 - 10 * q^61 + 12 * q^69 - 40 * q^71 + 12 * q^79 - 2 * q^81 + 34 * q^89 + 8 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more