LMFDB - Embedded newform 1380.2.f.a.829.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(829,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1380.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.4
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1380.829
Dual form			1380.2.f.a.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.67513 + 1.48119i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-1.67513 + 1.48119i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -1.67513 q^{11} +0.869067i q^{13} +(-1.48119 - 1.67513i) q^{15} +1.86907i q^{17} -0.869067 q^{19} -1.00000 q^{21} +1.00000i q^{23} +(0.612127 - 4.96239i) q^{25} -1.00000i q^{27} -2.44358 q^{29} -4.19394 q^{31} -1.67513i q^{33} +(-1.48119 - 1.67513i) q^{35} -2.76845i q^{37} -0.869067 q^{39} -10.5999 q^{41} -7.11871i q^{43} +(1.67513 - 1.48119i) q^{45} +2.71274i q^{47} +6.00000 q^{49} -1.86907 q^{51} -3.63752i q^{53} +(2.80606 - 2.48119i) q^{55} -0.869067i q^{57} -3.48119 q^{59} -1.78067 q^{61} -1.00000i q^{63} +(-1.28726 - 1.45580i) q^{65} -11.0811i q^{67} -1.00000 q^{69} -0.100615 q^{71} +2.16854i q^{73} +(4.96239 + 0.612127i) q^{75} -1.67513i q^{77} -5.08840 q^{79} +1.00000 q^{81} -9.09332i q^{83} +(-2.76845 - 3.13093i) q^{85} -2.44358i q^{87} -7.92478 q^{89} -0.869067 q^{91} -4.19394i q^{93} +(1.45580 - 1.28726i) q^{95} +4.99271i q^{97} +1.67513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 2 q^{15} + 4 q^{19} - 6 q^{21} + 2 q^{25} + 18 q^{29} - 26 q^{31} + 2 q^{35} + 4 q^{39} - 10 q^{41} + 36 q^{49} - 2 q^{51} + 16 q^{55} - 10 q^{59} - 40 q^{61} + 4 q^{65} - 6 q^{69} - 14 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 6 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95}+O(q^{100}) $ 6 * q - 6 * q^9 + 2 * q^15 + 4 * q^19 - 6 * q^21 + 2 * q^25 + 18 * q^29 - 26 * q^31 + 2 * q^35 + 4 * q^39 - 10 * q^41 + 36 * q^49 - 2 * q^51 + 16 * q^55 - 10 * q^59 - 40 * q^61 + 4 * q^65 - 6 * q^69 - 14 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 6 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−1.67513	+	1.48119i	−0.749141	+	0.662410i
$5$	−1.67513	+	1.48119i	−0.749141	+	0.662410i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.67513			−0.505071			−0.252535	−	0.967588i	$-0.581264\pi$
$11$	−1.67513			−0.505071			−0.252535	+	0.967588i	$0.581264\pi$
$12$		0			0
$13$			0.869067i			0.241036i	0.992711	+	0.120518i	$0.0384555\pi$
$13$			0.869067i			0.241036i	−0.992711	+	0.120518i	$0.961544\pi$
$14$		0			0
$15$	−1.48119	−	1.67513i	−0.382443	−	0.432517i
$16$		0			0
$17$			1.86907i			0.453315i	0.973974	+	0.226658i	$0.0727799\pi$
$17$			1.86907i			0.453315i	−0.973974	+	0.226658i	$0.927220\pi$
$18$		0			0
$19$	−0.869067			−0.199378			−0.0996889	−	0.995019i	$-0.531785\pi$
$19$	−0.869067			−0.199378			−0.0996889	+	0.995019i	$0.531785\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	0.612127	−	4.96239i	0.122425	−	0.992478i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.44358			−0.453762			−0.226881	−	0.973922i	$-0.572853\pi$
$29$	−2.44358			−0.453762			−0.226881	+	0.973922i	$0.572853\pi$
$30$		0			0
$31$	−4.19394			−0.753253			−0.376627	−	0.926365i	$-0.622916\pi$
$31$	−4.19394			−0.753253			−0.376627	+	0.926365i	$0.622916\pi$
$32$		0			0
$33$		−	1.67513i		−	0.291603i
$34$		0			0
$35$	−1.48119	−	1.67513i	−0.250368	−	0.283149i
$36$		0			0
$37$		−	2.76845i		−	0.455131i	−0.973763	−	0.227565i	$-0.926923\pi$
$37$		−	2.76845i		−	0.455131i	0.973763	−	0.227565i	$-0.0730765\pi$
$38$		0			0
$39$	−0.869067			−0.139162
$40$		0			0
$41$	−10.5999			−1.65543			−0.827714	−	0.561151i	$-0.810359\pi$
$41$	−10.5999			−1.65543			−0.827714	+	0.561151i	$0.810359\pi$
$42$		0			0
$43$		−	7.11871i		−	1.08559i	−0.839864	−	0.542797i	$-0.817365\pi$
$43$		−	7.11871i		−	1.08559i	0.839864	−	0.542797i	$-0.182635\pi$
$44$		0			0
$45$	1.67513	−	1.48119i	0.249714	−	0.220803i
$46$		0			0
$47$			2.71274i			0.395694i	0.980233	+	0.197847i	$0.0633949\pi$
$47$			2.71274i			0.395694i	−0.980233	+	0.197847i	$0.936605\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	−1.86907			−0.261722
$52$		0			0
$53$		−	3.63752i		−	0.499652i	−0.968291	−	0.249826i	$-0.919627\pi$
$53$		−	3.63752i		−	0.499652i	0.968291	−	0.249826i	$-0.0803734\pi$
$54$		0			0
$55$	2.80606	−	2.48119i	0.378370	−	0.334564i
$56$		0			0
$57$		−	0.869067i		−	0.115111i
$58$		0			0
$59$	−3.48119			−0.453213			−0.226606	−	0.973986i	$-0.572763\pi$
$59$	−3.48119			−0.453213			−0.226606	+	0.973986i	$0.572763\pi$
$60$		0			0
$61$	−1.78067			−0.227992			−0.113996	−	0.993481i	$-0.536365\pi$
$61$	−1.78067			−0.227992			−0.113996	+	0.993481i	$0.536365\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−1.28726	−	1.45580i	−0.159665	−	0.180570i
$66$		0			0
$67$		−	11.0811i		−	1.35377i	−0.736088	−	0.676886i	$-0.763329\pi$
$67$		−	11.0811i		−	1.35377i	0.736088	−	0.676886i	$-0.236671\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$	−0.100615			−0.0119409			−0.00597043	−	0.999982i	$-0.501900\pi$
$71$	−0.100615			−0.0119409			−0.00597043	+	0.999982i	$0.501900\pi$
$72$		0			0
$73$			2.16854i			0.253809i	0.991915	+	0.126904i	$0.0405042\pi$
$73$			2.16854i			0.253809i	−0.991915	+	0.126904i	$0.959496\pi$
$74$		0			0
$75$	4.96239	+	0.612127i	0.573007	+	0.0706823i
$76$		0			0
$77$		−	1.67513i		−	0.190899i
$78$		0			0
$79$	−5.08840			−0.572489			−0.286245	−	0.958157i	$-0.592407\pi$
$79$	−5.08840			−0.572489			−0.286245	+	0.958157i	$0.592407\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.09332i		−	0.998122i	−0.866567	−	0.499061i	$-0.833678\pi$
$83$		−	9.09332i		−	0.998122i	0.866567	−	0.499061i	$-0.166322\pi$
$84$		0			0
$85$	−2.76845	−	3.13093i	−0.300281	−	0.339597i
$86$		0			0
$87$		−	2.44358i		−	0.261980i
$88$		0			0
$89$	−7.92478			−0.840025			−0.420012	−	0.907518i	$-0.637974\pi$
$89$	−7.92478			−0.840025			−0.420012	+	0.907518i	$0.637974\pi$
$90$		0			0
$91$	−0.869067			−0.0911030
$92$		0			0
$93$		−	4.19394i		−	0.434891i
$94$		0			0
$95$	1.45580	−	1.28726i	0.149362	−	0.132070i
$96$		0			0
$97$			4.99271i			0.506932i	0.967344	+	0.253466i	$0.0815707\pi$
$97$			4.99271i			0.506932i	−0.967344	+	0.253466i	$0.918429\pi$
$98$		0			0
$99$	1.67513			0.168357
$100$		0			0
$101$	−8.28726			−0.824613			−0.412306	−	0.911045i	$-0.635277\pi$
$101$	−8.28726			−0.824613			−0.412306	+	0.911045i	$0.635277\pi$
$102$		0			0
$103$			2.80606i			0.276490i	0.990398	+	0.138245i	$0.0441461\pi$
$103$			2.80606i			0.276490i	−0.990398	+	0.138245i	$0.955854\pi$
$104$		0			0
$105$	1.67513	−	1.48119i	0.163476	−	0.144550i
$106$		0			0
$107$			3.45088i			0.333609i	0.985990	+	0.166804i	$0.0533449\pi$
$107$			3.45088i			0.333609i	−0.985990	+	0.166804i	$0.946655\pi$
$108$		0			0
$109$	13.3684			1.28046			0.640228	−	0.768185i	$-0.278840\pi$
$109$	13.3684			1.28046			0.640228	+	0.768185i	$0.278840\pi$
$110$		0			0
$111$	2.76845			0.262770
$112$		0			0
$113$			8.98778i			0.845499i	0.906246	+	0.422750i	$0.138935\pi$
$113$			8.98778i			0.845499i	−0.906246	+	0.422750i	$0.861065\pi$
$114$		0			0
$115$	−1.48119	−	1.67513i	−0.138122	−	0.156207i
$116$		0			0
$117$		−	0.869067i		−	0.0803453i
$118$		0			0
$119$	−1.86907			−0.171337
$120$		0			0
$121$	−8.19394			−0.744903
$122$		0			0
$123$		−	10.5999i		−	0.955762i
$124$		0			0
$125$	6.32487	+	9.21933i	0.565713	+	0.824602i
$126$		0			0
$127$		−	2.99508i		−	0.265770i	−0.991131	−	0.132885i	$-0.957576\pi$
$127$		−	2.99508i		−	0.265770i	0.991131	−	0.132885i	$-0.0424241\pi$
$128$		0			0
$129$	7.11871			0.626768
$130$		0			0
$131$	−20.9380			−1.82936			−0.914679	−	0.404182i	$-0.867556\pi$
$131$	−20.9380			−1.82936			−0.914679	+	0.404182i	$0.867556\pi$
$132$		0			0
$133$		−	0.869067i		−	0.0753577i
$134$		0			0
$135$	1.48119	+	1.67513i	0.127481	+	0.144172i
$136$		0			0
$137$			11.8945i			1.01621i	0.861294	+	0.508106i	$0.169654\pi$
$137$			11.8945i			1.01621i	−0.861294	+	0.508106i	$0.830346\pi$
$138$		0			0
$139$	−6.35026			−0.538622			−0.269311	−	0.963053i	$-0.586796\pi$
$139$	−6.35026			−0.538622			−0.269311	+	0.963053i	$0.586796\pi$
$140$		0			0
$141$	−2.71274			−0.228454
$142$		0			0
$143$		−	1.45580i		−	0.121740i
$144$		0			0
$145$	4.09332	−	3.61942i	0.339932	−	0.300577i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	9.67513			0.792618			0.396309	−	0.918117i	$-0.370291\pi$
$149$	9.67513			0.792618			0.396309	+	0.918117i	$0.370291\pi$
$150$		0			0
$151$	2.15633			0.175479			0.0877396	−	0.996143i	$-0.472036\pi$
$151$	2.15633			0.175479			0.0877396	+	0.996143i	$0.472036\pi$
$152$		0			0
$153$		−	1.86907i		−	0.151105i
$154$		0			0
$155$	7.02539	−	6.21203i	0.564293	−	0.498963i
$156$		0			0
$157$			4.42548i			0.353192i	0.984283	+	0.176596i	$0.0565086\pi$
$157$			4.42548i			0.353192i	−0.984283	+	0.176596i	$0.943491\pi$
$158$		0			0
$159$	3.63752			0.288474
$160$		0			0
$161$	−1.00000			−0.0788110
$162$		0			0
$163$		−	4.31265i		−	0.337793i	−0.985634	−	0.168896i	$-0.945980\pi$
$163$		−	4.31265i		−	0.337793i	0.985634	−	0.168896i	$-0.0540203\pi$
$164$		0			0
$165$	2.48119	+	2.80606i	0.193161	+	0.218452i
$166$		0			0
$167$			9.08603i			0.703098i	0.936170	+	0.351549i	$0.114345\pi$
$167$			9.08603i			0.703098i	−0.936170	+	0.351549i	$0.885655\pi$
$168$		0			0
$169$	12.2447			0.941902
$170$		0			0
$171$	0.869067			0.0664592
$172$		0			0
$173$			6.49929i			0.494132i	0.968999	+	0.247066i	$0.0794665\pi$
$173$			6.49929i			0.494132i	−0.968999	+	0.247066i	$0.920534\pi$
$174$		0			0
$175$	4.96239	+	0.612127i	0.375121	+	0.0462724i
$176$		0			0
$177$		−	3.48119i		−	0.261663i
$178$		0			0
$179$	−2.54420			−0.190162			−0.0950812	−	0.995470i	$-0.530311\pi$
$179$	−2.54420			−0.190162			−0.0950812	+	0.995470i	$0.530311\pi$
$180$		0			0
$181$	−8.12601			−0.604001			−0.302001	−	0.953308i	$-0.597655\pi$
$181$	−8.12601			−0.604001			−0.302001	+	0.953308i	$0.597655\pi$
$182$		0			0
$183$		−	1.78067i		−	0.131631i
$184$		0			0
$185$	4.10062	+	4.63752i	0.301483	+	0.340957i
$186$		0			0
$187$		−	3.13093i		−	0.228956i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−7.28726			−0.527287			−0.263644	−	0.964620i	$-0.584924\pi$
$191$	−7.28726			−0.527287			−0.263644	+	0.964620i	$0.584924\pi$
$192$		0			0
$193$			13.5877i			0.978063i	0.872266	+	0.489032i	$0.162650\pi$
$193$			13.5877i			0.978063i	−0.872266	+	0.489032i	$0.837350\pi$
$194$		0			0
$195$	1.45580	−	1.28726i	0.104252	−	0.0921824i
$196$		0			0
$197$			16.9380i			1.20678i	0.797447	+	0.603390i	$0.206184\pi$
$197$			16.9380i			1.20678i	−0.797447	+	0.603390i	$0.793816\pi$
$198$		0			0
$199$	−16.4241			−1.16427			−0.582136	−	0.813092i	$-0.697783\pi$
$199$	−16.4241			−1.16427			−0.582136	+	0.813092i	$0.697783\pi$
$200$		0			0
$201$	11.0811			0.781601
$202$		0			0
$203$		−	2.44358i		−	0.171506i
$204$		0			0
$205$	17.7562	−	15.7005i	1.24015	−	1.09657i
$206$		0			0
$207$		−	1.00000i		−	0.0695048i
$208$		0			0
$209$	1.45580			0.100700
$210$		0			0
$211$	−20.1490			−1.38712			−0.693558	−	0.720401i	$-0.743958\pi$
$211$	−20.1490			−1.38712			−0.693558	+	0.720401i	$0.743958\pi$
$212$		0			0
$213$		−	0.100615i		−	0.00689405i
$214$		0			0
$215$	10.5442	+	11.9248i	0.719108	+	0.813263i
$216$		0			0
$217$		−	4.19394i		−	0.284703i
$218$		0			0
$219$	−2.16854			−0.146537
$220$		0			0
$221$	−1.62435			−0.109265
$222$		0			0
$223$			17.9248i			1.20033i	0.799876	+	0.600166i	$0.204899\pi$
$223$			17.9248i			1.20033i	−0.799876	+	0.600166i	$0.795101\pi$
$224$		0			0
$225$	−0.612127	+	4.96239i	−0.0408085	+	0.330826i
$226$		0			0
$227$		−	5.14903i		−	0.341753i	−0.985292	−	0.170877i	$-0.945340\pi$
$227$		−	5.14903i		−	0.341753i	0.985292	−	0.170877i	$-0.0546599\pi$
$228$		0			0
$229$	20.7513			1.37129			0.685643	−	0.727938i	$-0.259521\pi$
$229$	20.7513			1.37129			0.685643	+	0.727938i	$0.259521\pi$
$230$		0			0
$231$	1.67513			0.110216
$232$		0			0
$233$			14.1768i			0.928753i	0.885638	+	0.464376i	$0.153721\pi$
$233$			14.1768i			0.928753i	−0.885638	+	0.464376i	$0.846279\pi$
$234$		0			0
$235$	−4.01810	−	4.54420i	−0.262112	−	0.296431i
$236$		0			0
$237$		−	5.08840i		−	0.330527i
$238$		0			0
$239$	1.26423			0.0817766			0.0408883	−	0.999164i	$-0.486981\pi$
$239$	1.26423			0.0817766			0.0408883	+	0.999164i	$0.486981\pi$
$240$		0			0
$241$	6.45676			0.415916			0.207958	−	0.978138i	$-0.433318\pi$
$241$	6.45676			0.415916			0.207958	+	0.978138i	$0.433318\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−10.0508	+	8.88717i	−0.642121	+	0.567780i
$246$		0			0
$247$		−	0.755278i		−	0.0480572i
$248$		0			0
$249$	9.09332			0.576266
$250$		0			0
$251$	7.66291			0.483679			0.241839	−	0.970316i	$-0.422249\pi$
$251$	7.66291			0.483679			0.241839	+	0.970316i	$0.422249\pi$
$252$		0			0
$253$		−	1.67513i		−	0.105315i
$254$		0			0
$255$	3.13093	−	2.76845i	0.196067	−	0.173367i
$256$		0			0
$257$			9.65069i			0.601994i	0.953625	+	0.300997i	$0.0973194\pi$
$257$			9.65069i			0.601994i	−0.953625	+	0.300997i	$0.902681\pi$
$258$		0			0
$259$	2.76845			0.172023
$260$		0			0
$261$	2.44358			0.151254
$262$		0			0
$263$		−	15.8994i		−	0.980398i	−0.871611	−	0.490199i	$-0.836924\pi$
$263$		−	15.8994i		−	0.980398i	0.871611	−	0.490199i	$-0.163076\pi$
$264$		0			0
$265$	5.38787	+	6.09332i	0.330974	+	0.374310i
$266$		0			0
$267$		−	7.92478i		−	0.484988i
$268$		0			0
$269$	−7.85448			−0.478896			−0.239448	−	0.970909i	$-0.576966\pi$
$269$	−7.85448			−0.478896			−0.239448	+	0.970909i	$0.576966\pi$
$270$		0			0
$271$	−9.96239			−0.605172			−0.302586	−	0.953122i	$-0.597850\pi$
$271$	−9.96239			−0.605172			−0.302586	+	0.953122i	$0.597850\pi$
$272$		0			0
$273$		−	0.869067i		−	0.0525984i
$274$		0			0
$275$	−1.02539	+	8.31265i	−0.0618335	+	0.501272i
$276$		0			0
$277$			5.61213i			0.337200i	0.985685	+	0.168600i	$0.0539246\pi$
$277$			5.61213i			0.337200i	−0.985685	+	0.168600i	$0.946075\pi$
$278$		0			0
$279$	4.19394			0.251084
$280$		0			0
$281$	−6.96002			−0.415200			−0.207600	−	0.978214i	$-0.566565\pi$
$281$	−6.96002			−0.415200			−0.207600	+	0.978214i	$0.566565\pi$
$282$		0			0
$283$		−	9.05079i		−	0.538013i	−0.963138	−	0.269007i	$-0.913305\pi$
$283$		−	9.05079i		−	0.538013i	0.963138	−	0.269007i	$-0.0866954\pi$
$284$		0			0
$285$	1.28726	+	1.45580i	0.0762506	+	0.0862342i
$286$		0			0
$287$		−	10.5999i		−	0.625693i
$288$		0			0
$289$	13.5066			0.794505
$290$		0			0
$291$	−4.99271			−0.292678
$292$		0			0
$293$			2.18172i			0.127457i	0.997967	+	0.0637287i	$0.0202992\pi$
$293$			2.18172i			0.127457i	−0.997967	+	0.0637287i	$0.979701\pi$
$294$		0			0
$295$	5.83146	−	5.15633i	0.339520	−	0.300213i
$296$		0			0
$297$			1.67513i			0.0972010i
$298$		0			0
$299$	−0.869067			−0.0502595
$300$		0			0
$301$	7.11871			0.410316
$302$		0			0
$303$		−	8.28726i		−	0.476091i
$304$		0			0
$305$	2.98286	−	2.63752i	0.170798	−	0.151024i
$306$		0			0
$307$			4.91985i			0.280791i	0.990096	+	0.140395i	$0.0448374\pi$
$307$			4.91985i			0.280791i	−0.990096	+	0.140395i	$0.955163\pi$
$308$		0			0
$309$	−2.80606			−0.159631
$310$		0			0
$311$	−14.2619			−0.808716			−0.404358	−	0.914601i	$-0.632505\pi$
$311$	−14.2619			−0.808716			−0.404358	+	0.914601i	$0.632505\pi$
$312$		0			0
$313$			4.32979i			0.244734i	0.992485	+	0.122367i	$0.0390486\pi$
$313$			4.32979i			0.244734i	−0.992485	+	0.122367i	$0.960951\pi$
$314$		0			0
$315$	1.48119	+	1.67513i	0.0834558	+	0.0943829i
$316$		0			0
$317$			31.0616i			1.74459i	0.488978	+	0.872296i	$0.337370\pi$
$317$			31.0616i			1.74459i	−0.488978	+	0.872296i	$0.662630\pi$
$318$		0			0
$319$	4.09332			0.229182
$320$		0			0
$321$	−3.45088			−0.192609
$322$		0			0
$323$		−	1.62435i		−	0.0903810i
$324$		0			0
$325$	4.31265	+	0.531980i	0.239223	+	0.0295089i
$326$		0			0
$327$			13.3684i			0.739272i
$328$		0			0
$329$	−2.71274			−0.149558
$330$		0			0
$331$	3.80606			0.209200			0.104600	−	0.994514i	$-0.466644\pi$
$331$	3.80606			0.209200			0.104600	+	0.994514i	$0.466644\pi$
$332$		0			0
$333$			2.76845i			0.151710i
$334$		0			0
$335$	16.4133	+	18.5623i	0.896753	+	1.01417i
$336$		0			0
$337$			6.62530i			0.360903i	0.983584	+	0.180452i	$0.0577559\pi$
$337$			6.62530i			0.360903i	−0.983584	+	0.180452i	$0.942244\pi$
$338$		0			0
$339$	−8.98778			−0.488149
$340$		0			0
$341$	7.02539			0.380446
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$	1.67513	−	1.48119i	0.0901860	−	0.0797448i
$346$		0			0
$347$		−	15.9756i		−	0.857613i	−0.903396	−	0.428807i	$-0.858934\pi$
$347$		−	15.9756i		−	0.857613i	0.903396	−	0.428807i	$-0.141066\pi$
$348$		0			0
$349$	22.2750			1.19236			0.596178	−	0.802852i	$-0.296685\pi$
$349$	22.2750			1.19236			0.596178	+	0.802852i	$0.296685\pi$
$350$		0			0
$351$	0.869067			0.0463874
$352$		0			0
$353$			9.47390i			0.504245i	0.967695	+	0.252122i	$0.0811286\pi$
$353$			9.47390i			0.504245i	−0.967695	+	0.252122i	$0.918871\pi$
$354$		0			0
$355$	0.168544	−	0.149031i	0.00894538	−	0.00790974i
$356$		0			0
$357$		−	1.86907i		−	0.0989215i
$358$		0			0
$359$	27.5853			1.45590			0.727949	−	0.685632i	$-0.240474\pi$
$359$	27.5853			1.45590			0.727949	+	0.685632i	$0.240474\pi$
$360$		0			0
$361$	−18.2447			−0.960249
$362$		0			0
$363$		−	8.19394i		−	0.430070i
$364$		0			0
$365$	−3.21203	−	3.63259i	−0.168126	−	0.190139i
$366$		0			0
$367$		−	10.3199i		−	0.538697i	−0.963043	−	0.269348i	$-0.913192\pi$
$367$		−	10.3199i		−	0.538697i	0.963043	−	0.269348i	$-0.0868083\pi$
$368$		0			0
$369$	10.5999			0.551809
$370$		0			0
$371$	3.63752			0.188851
$372$		0			0
$373$			25.9551i			1.34390i	0.740595	+	0.671952i	$0.234544\pi$
$373$			25.9551i			1.34390i	−0.740595	+	0.671952i	$0.765456\pi$
$374$		0			0
$375$	−9.21933	+	6.32487i	−0.476084	+	0.326615i
$376$		0			0
$377$		−	2.12364i		−	0.109373i
$378$		0			0
$379$	7.84955			0.403205			0.201602	−	0.979467i	$-0.435385\pi$
$379$	7.84955			0.403205			0.201602	+	0.979467i	$0.435385\pi$
$380$		0			0
$381$	2.99508			0.153442
$382$		0			0
$383$			27.8300i			1.42205i	0.703167	+	0.711024i	$0.251768\pi$
$383$			27.8300i			1.42205i	−0.703167	+	0.711024i	$0.748232\pi$
$384$		0			0
$385$	2.48119	+	2.80606i	0.126453	+	0.143010i
$386$		0			0
$387$			7.11871i			0.361865i
$388$		0			0
$389$	16.3733			0.830158			0.415079	−	0.909785i	$-0.363754\pi$
$389$	16.3733			0.830158			0.415079	+	0.909785i	$0.363754\pi$
$390$		0			0
$391$	−1.86907			−0.0945228
$392$		0			0
$393$		−	20.9380i		−	1.05618i
$394$		0			0
$395$	8.52373	−	7.53690i	0.428875	−	0.379223i
$396$		0			0
$397$		−	19.0884i		−	0.958019i	−0.877810	−	0.479010i	$-0.840996\pi$
$397$		−	19.0884i		−	0.958019i	0.877810	−	0.479010i	$-0.159004\pi$
$398$		0			0
$399$	0.869067			0.0435078
$400$		0			0
$401$	−12.2619			−0.612328			−0.306164	−	0.951979i	$-0.599046\pi$
$401$	−12.2619			−0.612328			−0.306164	+	0.951979i	$0.599046\pi$
$402$		0			0
$403$		−	3.64481i		−	0.181561i
$404$		0			0
$405$	−1.67513	+	1.48119i	−0.0832379	+	0.0736011i
$406$		0			0
$407$			4.63752i			0.229873i
$408$		0			0
$409$	−20.8251			−1.02974			−0.514868	−	0.857270i	$-0.672159\pi$
$409$	−20.8251			−1.02974			−0.514868	+	0.857270i	$0.672159\pi$
$410$		0			0
$411$	−11.8945			−0.586710
$412$		0			0
$413$		−	3.48119i		−	0.171298i
$414$		0			0
$415$	13.4690	+	15.2325i	0.661166	+	0.747734i
$416$		0			0
$417$		−	6.35026i		−	0.310974i
$418$		0			0
$419$	−28.4363			−1.38920			−0.694602	−	0.719394i	$-0.744419\pi$
$419$	−28.4363			−1.38920			−0.694602	+	0.719394i	$0.744419\pi$
$420$		0			0
$421$	18.2555			0.889720			0.444860	−	0.895600i	$-0.353253\pi$
$421$	18.2555			0.889720			0.444860	+	0.895600i	$0.353253\pi$
$422$		0			0
$423$		−	2.71274i		−	0.131898i
$424$		0			0
$425$	9.27504	+	1.14411i	0.449905	+	0.0554973i
$426$		0			0
$427$		−	1.78067i		−	0.0861727i
$428$		0			0
$429$	1.45580			0.0702868
$430$		0			0
$431$	6.62530			0.319130			0.159565	−	0.987187i	$-0.448991\pi$
$431$	6.62530			0.319130			0.159565	+	0.987187i	$0.448991\pi$
$432$		0			0
$433$			2.93937i			0.141257i	0.997503	+	0.0706284i	$0.0225005\pi$
$433$			2.93937i			0.141257i	−0.997503	+	0.0706284i	$0.977500\pi$
$434$		0			0
$435$	3.61942	+	4.09332i	0.173538	+	0.196260i
$436$		0			0
$437$		−	0.869067i		−	0.0415731i
$438$		0			0
$439$	26.7572			1.27705			0.638525	−	0.769601i	$-0.279545\pi$
$439$	26.7572			1.27705			0.638525	+	0.769601i	$0.279545\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$		−	12.2398i		−	0.581530i	−0.956794	−	0.290765i	$-0.906090\pi$
$443$		−	12.2398i		−	0.581530i	0.956794	−	0.290765i	$-0.0939098\pi$
$444$		0			0
$445$	13.2750	−	11.7381i	0.629297	−	0.556441i
$446$		0			0
$447$			9.67513i			0.457618i
$448$		0			0
$449$	26.8315			1.26625			0.633127	−	0.774048i	$-0.281771\pi$
$449$	26.8315			1.26625			0.633127	+	0.774048i	$0.281771\pi$
$450$		0			0
$451$	17.7562			0.836108
$452$		0			0
$453$			2.15633i			0.101313i
$454$		0			0
$455$	1.45580	−	1.28726i	0.0682490	−	0.0603476i
$456$		0			0
$457$		−	17.8568i		−	0.835308i	−0.908606	−	0.417654i	$-0.862852\pi$
$457$		−	17.8568i		−	0.835308i	0.908606	−	0.417654i	$-0.137148\pi$
$458$		0			0
$459$	1.86907			0.0872406
$460$		0			0
$461$	−18.3127			−0.852905			−0.426453	−	0.904510i	$-0.640237\pi$
$461$	−18.3127			−0.852905			−0.426453	+	0.904510i	$0.640237\pi$
$462$		0			0
$463$			28.7840i			1.33771i	0.743395	+	0.668853i	$0.233214\pi$
$463$			28.7840i			1.33771i	−0.743395	+	0.668853i	$0.766786\pi$
$464$		0			0
$465$	6.21203	+	7.02539i	0.288076	+	0.325795i
$466$		0			0
$467$		−	16.7767i		−	0.776333i	−0.921589	−	0.388167i	$-0.873109\pi$
$467$		−	16.7767i		−	0.776333i	0.921589	−	0.388167i	$-0.126891\pi$
$468$		0			0
$469$	11.0811			0.511678
$470$		0			0
$471$	−4.42548			−0.203916
$472$		0			0
$473$			11.9248i			0.548302i
$474$		0			0
$475$	−0.531980	+	4.31265i	−0.0244089	+	0.197878i
$476$		0			0
$477$			3.63752i			0.166551i
$478$		0			0
$479$	−11.6751			−0.533450			−0.266725	−	0.963773i	$-0.585942\pi$
$479$	−11.6751			−0.533450			−0.266725	+	0.963773i	$0.585942\pi$
$480$		0			0
$481$	2.40597			0.109703
$482$		0			0
$483$		−	1.00000i		−	0.0455016i
$484$		0			0
$485$	−7.39517	−	8.36344i	−0.335797	−	0.379764i
$486$		0			0
$487$			2.16854i			0.0982661i	0.998792	+	0.0491331i	$0.0156458\pi$
$487$			2.16854i			0.0982661i	−0.998792	+	0.0491331i	$0.984354\pi$
$488$		0			0
$489$	4.31265			0.195025
$490$		0			0
$491$	−12.0155			−0.542254			−0.271127	−	0.962544i	$-0.587396\pi$
$491$	−12.0155			−0.542254			−0.271127	+	0.962544i	$0.587396\pi$
$492$		0			0
$493$		−	4.56722i		−	0.205697i
$494$		0			0
$495$	−2.80606	+	2.48119i	−0.126123	+	0.111521i
$496$		0			0
$497$		−	0.100615i		−	0.00451322i
$498$		0			0
$499$	−34.5936			−1.54862			−0.774310	−	0.632806i	$-0.781903\pi$
$499$	−34.5936			−1.54862			−0.774310	+	0.632806i	$0.781903\pi$
$500$		0			0
$501$	−9.08603			−0.405934
$502$		0			0
$503$		−	5.35519i		−	0.238776i	−0.992848	−	0.119388i	$-0.961907\pi$
$503$		−	5.35519i		−	0.238776i	0.992848	−	0.119388i	$-0.0380932\pi$
$504$		0			0
$505$	13.8822	−	12.2750i	0.617752	−	0.546232i
$506$		0			0
$507$			12.2447i			0.543807i
$508$		0			0
$509$	6.74543			0.298986			0.149493	−	0.988763i	$-0.452236\pi$
$509$	6.74543			0.298986			0.149493	+	0.988763i	$0.452236\pi$
$510$		0			0
$511$	−2.16854			−0.0959307
$512$		0			0
$513$			0.869067i			0.0383703i
$514$		0			0
$515$	−4.15633	−	4.70052i	−0.183150	−	0.207130i
$516$		0			0
$517$		−	4.54420i		−	0.199854i
$518$		0			0
$519$	−6.49929			−0.285287
$520$		0			0
$521$	−12.2252			−0.535596			−0.267798	−	0.963475i	$-0.586296\pi$
$521$	−12.2252			−0.535596			−0.267798	+	0.963475i	$0.586296\pi$
$522$		0			0
$523$		−	3.51247i		−	0.153589i	−0.997047	−	0.0767947i	$-0.975531\pi$
$523$		−	3.51247i		−	0.153589i	0.997047	−	0.0767947i	$-0.0244686\pi$
$524$		0			0
$525$	−0.612127	+	4.96239i	−0.0267154	+	0.216576i
$526$		0			0
$527$		−	7.83875i		−	0.341461i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	3.48119			0.151071
$532$		0			0
$533$		−	9.21203i		−	0.399018i
$534$		0			0
$535$	−5.11142	−	5.78067i	−0.220986	−	0.249920i
$536$		0			0
$537$		−	2.54420i		−	0.109790i
$538$		0			0
$539$	−10.0508			−0.432918
$540$		0			0
$541$	−6.56864			−0.282408			−0.141204	−	0.989981i	$-0.545097\pi$
$541$	−6.56864			−0.282408			−0.141204	+	0.989981i	$0.545097\pi$
$542$		0			0
$543$		−	8.12601i		−	0.348720i
$544$		0			0
$545$	−22.3938	+	19.8011i	−0.959243	+	0.848188i
$546$		0			0
$547$		−	21.9102i		−	0.936812i	−0.883513	−	0.468406i	$-0.844828\pi$
$547$		−	21.9102i		−	0.936812i	0.883513	−	0.468406i	$-0.155172\pi$
$548$		0			0
$549$	1.78067			0.0759972
$550$		0			0
$551$	2.12364			0.0904700
$552$		0			0
$553$		−	5.08840i		−	0.216381i
$554$		0			0
$555$	−4.63752	+	4.10062i	−0.196852	+	0.174061i
$556$		0			0
$557$		−	35.6883i		−	1.51216i	−0.654478	−	0.756081i	$-0.727112\pi$
$557$		−	35.6883i		−	1.51216i	0.654478	−	0.756081i	$-0.272888\pi$
$558$		0			0
$559$	6.18664			0.261667
$560$		0			0
$561$	3.13093			0.132188
$562$		0			0
$563$		−	38.5510i		−	1.62473i	−0.583148	−	0.812366i	$-0.698179\pi$
$563$		−	38.5510i		−	1.62473i	0.583148	−	0.812366i	$-0.301821\pi$
$564$		0			0
$565$	−13.3127	−	15.0557i	−0.560067	−	0.633398i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	6.27645			0.263123			0.131561	−	0.991308i	$-0.458001\pi$
$569$	6.27645			0.263123			0.131561	+	0.991308i	$0.458001\pi$
$570$		0			0
$571$	−22.4323			−0.938763			−0.469382	−	0.882995i	$-0.655523\pi$
$571$	−22.4323			−0.938763			−0.469382	+	0.882995i	$0.655523\pi$
$572$		0			0
$573$		−	7.28726i		−	0.304430i
$574$		0			0
$575$	4.96239	+	0.612127i	0.206946	+	0.0255275i
$576$		0			0
$577$		−	24.9525i		−	1.03879i	−0.854535	−	0.519394i	$-0.826158\pi$
$577$		−	24.9525i		−	1.03879i	0.854535	−	0.519394i	$-0.173842\pi$
$578$		0			0
$579$	−13.5877			−0.564685
$580$		0			0
$581$	9.09332			0.377255
$582$		0			0
$583$			6.09332i			0.252360i
$584$		0			0
$585$	1.28726	+	1.45580i	0.0532216	+	0.0601900i
$586$		0			0
$587$			1.68735i			0.0696444i	0.999394	+	0.0348222i	$0.0110865\pi$
$587$			1.68735i			0.0696444i	−0.999394	+	0.0348222i	$0.988914\pi$
$588$		0			0
$589$	3.64481			0.150182
$590$		0			0
$591$	−16.9380			−0.696734
$592$		0			0
$593$			37.7523i			1.55030i	0.631777	+	0.775150i	$0.282326\pi$
$593$			37.7523i			1.55030i	−0.631777	+	0.775150i	$0.717674\pi$
$594$		0			0
$595$	3.13093	−	2.76845i	0.128356	−	0.113495i
$596$		0			0
$597$		−	16.4241i		−	0.672192i
$598$		0			0
$599$	13.5271			0.552700			0.276350	−	0.961057i	$-0.410875\pi$
$599$	13.5271			0.552700			0.276350	+	0.961057i	$0.410875\pi$
$600$		0			0
$601$	−29.4544			−1.20147			−0.600735	−	0.799448i	$-0.705125\pi$
$601$	−29.4544			−1.20147			−0.600735	+	0.799448i	$0.705125\pi$
$602$		0			0
$603$			11.0811i			0.451257i
$604$		0			0
$605$	13.7259	−	12.1368i	0.558038	−	0.493432i
$606$		0			0
$607$			28.6942i			1.16466i	0.812952	+	0.582331i	$0.197859\pi$
$607$			28.6942i			1.16466i	−0.812952	+	0.582331i	$0.802141\pi$
$608$		0			0
$609$	2.44358			0.0990190
$610$		0			0
$611$	−2.35756			−0.0953765
$612$		0			0
$613$			22.9018i			0.924993i	0.886621	+	0.462497i	$0.153046\pi$
$613$			22.9018i			0.924993i	−0.886621	+	0.462497i	$0.846954\pi$
$614$		0			0
$615$	15.7005	+	17.7562i	0.633106	+	0.716000i
$616$		0			0
$617$		−	24.0411i		−	0.967859i	−0.875107	−	0.483930i	$-0.839209\pi$
$617$		−	24.0411i		−	0.967859i	0.875107	−	0.483930i	$-0.160791\pi$
$618$		0			0
$619$	−34.5647			−1.38927			−0.694636	−	0.719362i	$-0.744434\pi$
$619$	−34.5647			−1.38927			−0.694636	+	0.719362i	$0.744434\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$		−	7.92478i		−	0.317499i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$			1.45580i			0.0581391i
$628$		0			0
$629$	5.17442			0.206318
$630$		0			0
$631$	−29.8677			−1.18901			−0.594506	−	0.804091i	$-0.702652\pi$
$631$	−29.8677			−1.18901			−0.594506	+	0.804091i	$0.702652\pi$
$632$		0			0
$633$		−	20.1490i		−	0.800852i
$634$		0			0
$635$	4.43629	+	5.01714i	0.176049	+	0.199099i
$636$		0			0
$637$			5.21440i			0.206602i
$638$		0			0
$639$	0.100615			0.00398028
$640$		0			0
$641$	−30.9257			−1.22149			−0.610746	−	0.791826i	$-0.709131\pi$
$641$	−30.9257			−1.22149			−0.610746	+	0.791826i	$0.709131\pi$
$642$		0			0
$643$			36.2955i			1.43136i	0.698431	+	0.715678i	$0.253882\pi$
$643$			36.2955i			1.43136i	−0.698431	+	0.715678i	$0.746118\pi$
$644$		0			0
$645$	−11.9248	+	10.5442i	−0.469538	+	0.415177i
$646$		0			0
$647$			5.67513i			0.223112i	0.993758	+	0.111556i	$0.0355835\pi$
$647$			5.67513i			0.223112i	−0.993758	+	0.111556i	$0.964416\pi$
$648$		0			0
$649$	5.83146			0.228905
$650$		0			0
$651$	4.19394			0.164373
$652$		0			0
$653$		−	50.7245i		−	1.98500i	−0.122232	−	0.992502i	$-0.539005\pi$
$653$		−	50.7245i		−	1.98500i	0.122232	−	0.992502i	$-0.460995\pi$
$654$		0			0
$655$	35.0738	−	31.0132i	1.37045	−	1.21178i
$656$		0			0
$657$		−	2.16854i		−	0.0846030i
$658$		0			0
$659$	−23.4641			−0.914030			−0.457015	−	0.889459i	$-0.651081\pi$
$659$	−23.4641			−0.914030			−0.457015	+	0.889459i	$0.651081\pi$
$660$		0			0
$661$	−18.9135			−0.735650			−0.367825	−	0.929895i	$-0.619898\pi$
$661$	−18.9135			−0.735650			−0.367825	+	0.929895i	$0.619898\pi$
$662$		0			0
$663$		−	1.62435i		−	0.0630844i
$664$		0			0
$665$	1.28726	+	1.45580i	0.0499177	+	0.0564536i
$666$		0			0
$667$		−	2.44358i		−	0.0946159i
$668$		0			0
$669$	−17.9248			−0.693012
$670$		0			0
$671$	2.98286			0.115152
$672$		0			0
$673$			15.8559i			0.611200i	0.952160	+	0.305600i	$0.0988570\pi$
$673$			15.8559i			0.611200i	−0.952160	+	0.305600i	$0.901143\pi$
$674$		0			0
$675$	−4.96239	−	0.612127i	−0.191002	−	0.0235608i
$676$		0			0
$677$		−	2.52469i		−	0.0970315i	−0.998822	−	0.0485158i	$-0.984551\pi$
$677$		−	2.52469i		−	0.0970315i	0.998822	−	0.0485158i	$-0.0154491\pi$
$678$		0			0
$679$	−4.99271			−0.191602
$680$		0			0
$681$	5.14903			0.197311
$682$		0			0
$683$		−	24.5477i		−	0.939292i	−0.882855	−	0.469646i	$-0.844382\pi$
$683$		−	24.5477i		−	0.939292i	0.882855	−	0.469646i	$-0.155618\pi$
$684$		0			0
$685$	−17.6180	−	19.9248i	−0.673149	−	0.761287i
$686$		0			0
$687$			20.7513i			0.791712i
$688$		0			0
$689$	3.16125			0.120434
$690$		0			0
$691$	23.7685			0.904195			0.452097	−	0.891969i	$-0.350676\pi$
$691$	23.7685			0.904195			0.452097	+	0.891969i	$0.350676\pi$
$692$		0			0
$693$			1.67513i			0.0636330i
$694$		0			0
$695$	10.6375	−	9.40597i	0.403504	−	0.356789i
$696$		0			0
$697$		−	19.8119i		−	0.750431i
$698$		0			0
$699$	−14.1768			−0.536216
$700$		0			0
$701$	11.9633			0.451849			0.225925	−	0.974145i	$-0.427460\pi$
$701$	11.9633			0.451849			0.225925	+	0.974145i	$0.427460\pi$
$702$		0			0
$703$			2.40597i			0.0907429i
$704$		0			0
$705$	4.54420	−	4.01810i	0.171144	−	0.151330i
$706$		0			0
$707$		−	8.28726i		−	0.311674i
$708$		0			0
$709$	−29.4093			−1.10449			−0.552245	−	0.833682i	$-0.686229\pi$
$709$	−29.4093			−1.10449			−0.552245	+	0.833682i	$0.686229\pi$
$710$		0			0
$711$	5.08840			0.190830
$712$		0			0
$713$		−	4.19394i		−	0.157064i
$714$		0			0
$715$	2.15633	+	2.43866i	0.0806420	+	0.0912007i
$716$		0			0
$717$			1.26423i			0.0472137i
$718$		0			0
$719$	25.6190			0.955426			0.477713	−	0.878516i	$-0.341466\pi$
$719$	25.6190			0.955426			0.477713	+	0.878516i	$0.341466\pi$
$720$		0			0
$721$	−2.80606			−0.104503
$722$		0			0
$723$			6.45676i			0.240129i
$724$		0			0
$725$	−1.49578	+	12.1260i	−0.0555520	+	0.450349i
$726$		0			0
$727$		−	31.4894i		−	1.16788i	−0.811797	−	0.583939i	$-0.801511\pi$
$727$		−	31.4894i		−	1.16788i	0.811797	−	0.583939i	$-0.198489\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	13.3054			0.492116
$732$		0			0
$733$			37.0059i			1.36684i	0.730024	+	0.683422i	$0.239509\pi$
$733$			37.0059i			1.36684i	−0.730024	+	0.683422i	$0.760491\pi$
$734$		0			0
$735$	−8.88717	−	10.0508i	−0.327808	−	0.370729i
$736$		0			0
$737$			18.5623i			0.683751i
$738$		0			0
$739$	−15.7816			−0.580536			−0.290268	−	0.956945i	$-0.593745\pi$
$739$	−15.7816			−0.580536			−0.290268	+	0.956945i	$0.593745\pi$
$740$		0			0
$741$	0.755278			0.0277458
$742$		0			0
$743$		−	45.9657i		−	1.68632i	−0.537664	−	0.843159i	$-0.680693\pi$
$743$		−	45.9657i		−	1.68632i	0.537664	−	0.843159i	$-0.319307\pi$
$744$		0			0
$745$	−16.2071	+	14.3307i	−0.593783	+	0.525038i
$746$		0			0
$747$			9.09332i			0.332707i
$748$		0			0
$749$	−3.45088			−0.126092
$750$		0			0
$751$	−15.3538			−0.560267			−0.280134	−	0.959961i	$-0.590379\pi$
$751$	−15.3538			−0.560267			−0.280134	+	0.959961i	$0.590379\pi$
$752$		0			0
$753$			7.66291i			0.279252i
$754$		0			0
$755$	−3.61213	+	3.19394i	−0.131459	+	0.116239i
$756$		0			0
$757$		−	24.6385i		−	0.895501i	−0.894159	−	0.447750i	$-0.852225\pi$
$757$		−	24.6385i		−	0.895501i	0.894159	−	0.447750i	$-0.147775\pi$
$758$		0			0
$759$	1.67513			0.0608034
$760$		0			0
$761$	11.0787			0.401604			0.200802	−	0.979632i	$-0.435645\pi$
$761$	11.0787			0.401604			0.200802	+	0.979632i	$0.435645\pi$
$762$		0			0
$763$			13.3684i			0.483967i
$764$		0			0
$765$	2.76845	+	3.13093i	0.100094	+	0.113199i
$766$		0			0
$767$		−	3.02539i		−	0.109241i
$768$		0			0
$769$	−22.3961			−0.807625			−0.403812	−	0.914842i	$-0.632315\pi$
$769$	−22.3961			−0.807625			−0.403812	+	0.914842i	$0.632315\pi$
$770$		0			0
$771$	−9.65069			−0.347561
$772$		0			0
$773$		−	33.7645i		−	1.21442i	−0.794540	−	0.607212i	$-0.792288\pi$
$773$		−	33.7645i		−	1.21442i	0.794540	−	0.607212i	$-0.207712\pi$
$774$		0			0
$775$	−2.56722	+	20.8119i	−0.0922173	+	0.747587i
$776$		0			0
$777$			2.76845i			0.0993177i
$778$		0			0
$779$	9.21203			0.330055
$780$		0			0
$781$	0.168544			0.00603098
$782$		0			0
$783$			2.44358i			0.0873265i
$784$		0			0
$785$	−6.55500	−	7.41327i	−0.233958	−	0.264591i
$786$		0			0
$787$			25.7718i			0.918665i	0.888265	+	0.459332i	$0.151911\pi$
$787$			25.7718i			0.918665i	−0.888265	+	0.459332i	$0.848089\pi$
$788$		0			0
$789$	15.8994			0.566033
$790$		0			0
$791$	−8.98778			−0.319569
$792$		0			0
$793$		−	1.54752i		−	0.0549542i
$794$		0			0
$795$	−6.09332	+	5.38787i	−0.216108	+	0.191088i
$796$		0			0
$797$			32.9633i			1.16762i	0.811890	+	0.583811i	$0.198439\pi$
$797$			32.9633i			1.16762i	−0.811890	+	0.583811i	$0.801561\pi$
$798$		0			0
$799$	−5.07030			−0.179374
$800$		0			0
$801$	7.92478			0.280008
$802$		0			0
$803$		−	3.63259i		−	0.128191i
$804$		0			0
$805$	1.67513	−	1.48119i	0.0590406	−	0.0522052i
$806$		0			0
$807$		−	7.85448i		−	0.276491i
$808$		0			0
$809$	−27.6639			−0.972610			−0.486305	−	0.873789i	$-0.661656\pi$
$809$	−27.6639			−0.972610			−0.486305	+	0.873789i	$0.661656\pi$
$810$		0			0
$811$	13.2228			0.464317			0.232158	−	0.972678i	$-0.425421\pi$
$811$	13.2228			0.464317			0.232158	+	0.972678i	$0.425421\pi$
$812$		0			0
$813$		−	9.96239i		−	0.349396i
$814$		0			0
$815$	6.38787	+	7.22425i	0.223757	+	0.253055i
$816$		0			0
$817$			6.18664i			0.216443i
$818$		0			0
$819$	0.869067			0.0303677
$820$		0			0
$821$	−10.1319			−0.353605			−0.176803	−	0.984246i	$-0.556575\pi$
$821$	−10.1319			−0.353605			−0.176803	+	0.984246i	$0.556575\pi$
$822$		0			0
$823$			13.9394i			0.485896i	0.970039	+	0.242948i	$0.0781144\pi$
$823$			13.9394i			0.485896i	−0.970039	+	0.242948i	$0.921886\pi$
$824$		0			0
$825$	−8.31265	−	1.02539i	−0.289409	−	0.0356996i
$826$		0			0
$827$		−	5.18901i		−	0.180440i	−0.995922	−	0.0902198i	$-0.971243\pi$
$827$		−	5.18901i		−	0.180440i	0.995922	−	0.0902198i	$-0.0287570\pi$
$828$		0			0
$829$	−29.1173			−1.01129			−0.505643	−	0.862743i	$-0.668745\pi$
$829$	−29.1173			−1.01129			−0.505643	+	0.862743i	$0.668745\pi$
$830$		0			0
$831$	−5.61213			−0.194683
$832$		0			0
$833$			11.2144i			0.388556i
$834$		0			0
$835$	−13.4582	−	15.2203i	−0.465739	−	0.526720i
$836$		0			0
$837$			4.19394i			0.144964i
$838$		0			0
$839$	24.2882			0.838522			0.419261	−	0.907866i	$-0.362289\pi$
$839$	24.2882			0.838522			0.419261	+	0.907866i	$0.362289\pi$
$840$		0			0
$841$	−23.0289			−0.794100
$842$		0			0
$843$		−	6.96002i		−	0.239716i
$844$		0			0
$845$	−20.5115	+	18.1368i	−0.705617	+	0.623925i
$846$		0			0
$847$		−	8.19394i		−	0.281547i
$848$		0			0
$849$	9.05079			0.310622
$850$		0			0
$851$	2.76845			0.0949013
$852$		0			0
$853$		−	15.8397i		−	0.542341i	−0.962531	−	0.271171i	$-0.912589\pi$
$853$		−	15.8397i		−	0.542341i	0.962531	−	0.271171i	$-0.0874108\pi$
$854$		0			0
$855$	−1.45580	+	1.28726i	−0.0497874	+	0.0440233i
$856$		0			0
$857$		−	27.2605i		−	0.931199i	−0.884995	−	0.465600i	$-0.845839\pi$
$857$		−	27.2605i		−	0.931199i	0.884995	−	0.465600i	$-0.154161\pi$
$858$		0			0
$859$	−22.5223			−0.768451			−0.384226	−	0.923239i	$-0.625532\pi$
$859$	−22.5223			−0.768451			−0.384226	+	0.923239i	$0.625532\pi$
$860$		0			0
$861$	10.5999			0.361244
$862$		0			0
$863$		−	45.5026i		−	1.54893i	−0.632619	−	0.774464i	$-0.718020\pi$
$863$		−	45.5026i		−	1.54893i	0.632619	−	0.774464i	$-0.281980\pi$
$864$		0			0
$865$	−9.62672	−	10.8872i	−0.327318	−	0.370175i
$866$		0			0
$867$			13.5066i			0.458708i
$868$		0			0
$869$	8.52373			0.289148
$870$		0			0
$871$	9.63023			0.326308
$872$		0			0
$873$		−	4.99271i		−	0.168977i
$874$		0			0
$875$	−9.21933	+	6.32487i	−0.311670	+	0.213820i
$876$		0			0
$877$			58.0019i			1.95859i	0.202450	+	0.979293i	$0.435110\pi$
$877$			58.0019i			1.95859i	−0.202450	+	0.979293i	$0.564890\pi$
$878$		0			0
$879$	−2.18172			−0.0735875
$880$		0			0
$881$	26.2351			0.883882			0.441941	−	0.897044i	$-0.354290\pi$
$881$	26.2351			0.883882			0.441941	+	0.897044i	$0.354290\pi$
$882$		0			0
$883$			27.8578i			0.937490i	0.883334	+	0.468745i	$0.155294\pi$
$883$			27.8578i			0.937490i	−0.883334	+	0.468745i	$0.844706\pi$
$884$		0			0
$885$	5.15633	+	5.83146i	0.173328	+	0.196022i
$886$		0			0
$887$			22.1866i			0.744955i	0.928041	+	0.372477i	$0.121492\pi$
$887$			22.1866i			0.744955i	−0.928041	+	0.372477i	$0.878508\pi$
$888$		0			0
$889$	2.99508			0.100452
$890$		0			0
$891$	−1.67513			−0.0561190
$892$		0			0
$893$		−	2.35756i		−	0.0788926i
$894$		0			0
$895$	4.26187	−	3.76845i	0.142458	−	0.125965i
$896$		0			0
$897$		−	0.869067i		−	0.0290173i
$898$		0			0
$899$	10.2482			0.341798
$900$		0			0
$901$	6.79877			0.226500
$902$		0			0
$903$			7.11871i			0.236896i
$904$		0			0
$905$	13.6121	−	12.0362i	0.452482	−	0.400097i
$906$		0			0
$907$		−	22.0943i		−	0.733628i	−0.930294	−	0.366814i	$-0.880448\pi$
$907$		−	22.0943i		−	0.733628i	0.930294	−	0.366814i	$-0.119552\pi$
$908$		0			0
$909$	8.28726			0.274871
$910$		0			0
$911$	54.0381			1.79036			0.895181	−	0.445702i	$-0.147046\pi$
$911$	54.0381			1.79036			0.895181	+	0.445702i	$0.147046\pi$
$912$		0			0
$913$			15.2325i			0.504122i
$914$		0			0
$915$	2.63752	+	2.98286i	0.0871937	+	0.0986102i
$916$		0			0
$917$		−	20.9380i		−	0.691432i
$918$		0			0
$919$	−23.9854			−0.791206			−0.395603	−	0.918422i	$-0.629464\pi$
$919$	−23.9854			−0.791206			−0.395603	+	0.918422i	$0.629464\pi$
$920$		0			0
$921$	−4.91985			−0.162115
$922$		0			0
$923$		−	0.0874416i		−	0.00287817i
$924$		0			0
$925$	−13.7381	−	1.69464i	−0.451707	−	0.0557195i
$926$		0			0
$927$		−	2.80606i		−	0.0921632i
$928$		0			0
$929$	11.8183			0.387745			0.193873	−	0.981027i	$-0.437895\pi$
$929$	11.8183			0.387745			0.193873	+	0.981027i	$0.437895\pi$
$930$		0			0
$931$	−5.21440			−0.170895
$932$		0			0
$933$		−	14.2619i		−	0.466913i
$934$		0			0
$935$	4.63752	+	5.24472i	0.151663	+	0.171521i
$936$		0			0
$937$			5.70782i			0.186466i	0.995644	+	0.0932331i	$0.0297202\pi$
$937$			5.70782i			0.186466i	−0.995644	+	0.0932331i	$0.970280\pi$
$938$		0			0
$939$	−4.32979			−0.141297
$940$		0			0
$941$	25.0616			0.816984			0.408492	−	0.912762i	$-0.366055\pi$
$941$	25.0616			0.816984			0.408492	+	0.912762i	$0.366055\pi$
$942$		0			0
$943$		−	10.5999i		−	0.345180i
$944$		0			0
$945$	−1.67513	+	1.48119i	−0.0544920	+	0.0481833i
$946$		0			0
$947$			15.7626i			0.512215i	0.966648	+	0.256107i	$0.0824401\pi$
$947$			15.7626i			0.512215i	−0.966648	+	0.256107i	$0.917560\pi$
$948$		0			0
$949$	−1.88461			−0.0611771
$950$		0			0
$951$	−31.0616			−1.00724
$952$		0			0
$953$		−	10.2520i		−	0.332095i	−0.986118	−	0.166048i	$-0.946899\pi$
$953$		−	10.2520i		−	0.332095i	0.986118	−	0.166048i	$-0.0531006\pi$
$954$		0			0
$955$	12.2071	−	10.7938i	0.395013	−	0.349281i
$956$		0			0
$957$			4.09332i			0.132318i
$958$		0			0
$959$	−11.8945			−0.384092
$960$		0			0
$961$	−13.4109			−0.432610
$962$		0			0
$963$		−	3.45088i		−	0.111203i
$964$		0			0
$965$	−20.1260	−	22.7612i	−0.647879	−	0.732708i
$966$		0			0
$967$			14.5075i			0.466531i	0.972413	+	0.233266i	$0.0749411\pi$
$967$			14.5075i			0.466531i	−0.972413	+	0.233266i	$0.925059\pi$
$968$		0			0
$969$	1.62435			0.0521815
$970$		0			0
$971$	53.2144			1.70773			0.853866	−	0.520493i	$-0.174252\pi$
$971$	53.2144			1.70773			0.853866	+	0.520493i	$0.174252\pi$
$972$		0			0
$973$		−	6.35026i		−	0.203580i
$974$		0			0
$975$	−0.531980	+	4.31265i	−0.0170370	+	0.138115i
$976$		0			0
$977$		−	23.0590i		−	0.737724i	−0.929484	−	0.368862i	$-0.879748\pi$
$977$		−	23.0590i		−	0.737724i	0.929484	−	0.368862i	$-0.120252\pi$
$978$		0			0
$979$	13.2750			0.424272
$980$		0			0
$981$	−13.3684			−0.426819
$982$		0			0
$983$		−	14.7708i		−	0.471116i	−0.971860	−	0.235558i	$-0.924308\pi$
$983$		−	14.7708i		−	0.471116i	0.971860	−	0.235558i	$-0.0756918\pi$
$984$		0			0
$985$	−25.0884	−	28.3733i	−0.799383	−	0.904048i
$986$		0			0
$987$		−	2.71274i		−	0.0863475i
$988$		0			0
$989$	7.11871			0.226362
$990$		0			0
$991$	52.8905			1.68012			0.840061	−	0.542492i	$-0.182519\pi$
$991$	52.8905			1.68012			0.840061	+	0.542492i	$0.182519\pi$
$992$		0			0
$993$			3.80606i			0.120782i
$994$		0			0
$995$	27.5125	−	24.3272i	0.872204	−	0.771225i
$996$		0			0
$997$			5.87399i			0.186031i	0.995665	+	0.0930156i	$0.0296506\pi$
$997$			5.87399i			0.186031i	−0.995665	+	0.0930156i	$0.970349\pi$
$998$		0			0
$999$	−2.76845			−0.0875899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.f.a.829.4	yes	6
3.2	odd	2		4140.2.f.a.829.5		6
5.2	odd	4		6900.2.a.z.1.1		3
5.3	odd	4		6900.2.a.y.1.1		3
5.4	even	2	inner	1380.2.f.a.829.1	✓	6
15.14	odd	2		4140.2.f.a.829.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.1	✓	6	5.4	even	2	inner
1380.2.f.a.829.4	yes	6	1.1	even	1	trivial
4140.2.f.a.829.5		6	3.2	odd	2
4140.2.f.a.829.6		6	15.14	odd	2
6900.2.a.y.1.1		3	5.3	odd	4
6900.2.a.z.1.1		3	5.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.67513 + 1.48119i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.67513 + 1.48119i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -1.67513 q^{11} +0.869067i q^{13} +(-1.48119 - 1.67513i) q^{15} +1.86907i q^{17} -0.869067 q^{19} -1.00000 q^{21} +1.00000i q^{23} +(0.612127 - 4.96239i) q^{25} -1.00000i q^{27} -2.44358 q^{29} -4.19394 q^{31} -1.67513i q^{33} +(-1.48119 - 1.67513i) q^{35} -2.76845i q^{37} -0.869067 q^{39} -10.5999 q^{41} -7.11871i q^{43} +(1.67513 - 1.48119i) q^{45} +2.71274i q^{47} +6.00000 q^{49} -1.86907 q^{51} -3.63752i q^{53} +(2.80606 - 2.48119i) q^{55} -0.869067i q^{57} -3.48119 q^{59} -1.78067 q^{61} -1.00000i q^{63} +(-1.28726 - 1.45580i) q^{65} -11.0811i q^{67} -1.00000 q^{69} -0.100615 q^{71} +2.16854i q^{73} +(4.96239 + 0.612127i) q^{75} -1.67513i q^{77} -5.08840 q^{79} +1.00000 q^{81} -9.09332i q^{83} +(-2.76845 - 3.13093i) q^{85} -2.44358i q^{87} -7.92478 q^{89} -0.869067 q^{91} -4.19394i q^{93} +(1.45580 - 1.28726i) q^{95} +4.99271i q^{97} +1.67513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 2 q^{15} + 4 q^{19} - 6 q^{21} + 2 q^{25} + 18 q^{29} - 26 q^{31} + 2 q^{35} + 4 q^{39} - 10 q^{41} + 36 q^{49} - 2 q^{51} + 16 q^{55} - 10 q^{59} - 40 q^{61} + 4 q^{65} - 6 q^{69} - 14 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 6 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95}+O(q^{100}) \) 6 * q - 6 * q^9 + 2 * q^15 + 4 * q^19 - 6 * q^21 + 2 * q^25 + 18 * q^29 - 26 * q^31 + 2 * q^35 + 4 * q^39 - 10 * q^41 + 36 * q^49 - 2 * q^51 + 16 * q^55 - 10 * q^59 - 40 * q^61 + 4 * q^65 - 6 * q^69 - 14 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 6 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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