LMFDB - Embedded newform 4140.2.f.a.829.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
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:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.5
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.a.829.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67513 - 1.48119i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(1.67513 - 1.48119i) q^{5} +1.00000i q^{7} +1.67513 q^{11} +0.869067i q^{13} -1.86907i q^{17} -0.869067 q^{19} -1.00000i q^{23} +(0.612127 - 4.96239i) q^{25} +2.44358 q^{29} -4.19394 q^{31} +(1.48119 + 1.67513i) q^{35} -2.76845i q^{37} +10.5999 q^{41} -7.11871i q^{43} -2.71274i q^{47} +6.00000 q^{49} +3.63752i q^{53} +(2.80606 - 2.48119i) q^{55} +3.48119 q^{59} -1.78067 q^{61} +(1.28726 + 1.45580i) q^{65} -11.0811i q^{67} +0.100615 q^{71} +2.16854i q^{73} +1.67513i q^{77} -5.08840 q^{79} +9.09332i q^{83} +(-2.76845 - 3.13093i) q^{85} +7.92478 q^{89} -0.869067 q^{91} +(-1.45580 + 1.28726i) q^{95} +4.99271i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 4 q^{19} + 2 q^{25} - 18 q^{29} - 26 q^{31} - 2 q^{35} + 10 q^{41} + 36 q^{49} + 16 q^{55} + 10 q^{59} - 40 q^{61} - 4 q^{65} + 14 q^{71} + 8 q^{79} + 6 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) $ 6 * q + 4 * q^19 + 2 * q^25 - 18 * q^29 - 26 * q^31 - 2 * q^35 + 10 * q^41 + 36 * q^49 + 16 * q^55 + 10 * q^59 - 40 * q^61 - 4 * q^65 + 14 * q^71 + 8 * q^79 + 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 $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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
$9$		0			0
$10$		0			0
$11$	1.67513			0.505071			0.252535	−	0.967588i	$-0.418736\pi$
$11$	1.67513			0.505071			0.252535	+	0.967588i	$0.418736\pi$
$12$		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
$15$		0			0
$16$		0			0
$17$		−	1.86907i		−	0.453315i	−0.973974	−	0.226658i	$-0.927220\pi$
$17$		−	1.86907i		−	0.453315i	0.973974	−	0.226658i	$-0.0727799\pi$
$18$		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
$21$		0			0
$22$		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
$27$		0			0
$28$		0			0
$29$	2.44358			0.453762			0.226881	−	0.973922i	$-0.427147\pi$
$29$	2.44358			0.453762			0.226881	+	0.973922i	$0.427147\pi$
$30$		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
$33$		0			0
$34$		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			0
$40$		0			0
$41$	10.5999			1.65543			0.827714	−	0.561151i	$-0.189641\pi$
$41$	10.5999			1.65543			0.827714	+	0.561151i	$0.189641\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$		−	2.71274i		−	0.395694i	−0.980233	−	0.197847i	$-0.936605\pi$
$47$		−	2.71274i		−	0.395694i	0.980233	−	0.197847i	$-0.0633949\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.63752i			0.499652i	0.968291	+	0.249826i	$0.0803734\pi$
$53$			3.63752i			0.499652i	−0.968291	+	0.249826i	$0.919627\pi$
$54$		0			0
$55$	2.80606	−	2.48119i	0.378370	−	0.334564i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.48119			0.453213			0.226606	−	0.973986i	$-0.427237\pi$
$59$	3.48119			0.453213			0.226606	+	0.973986i	$0.427237\pi$
$60$		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
$63$		0			0
$64$		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
$69$		0			0
$70$		0			0
$71$	0.100615			0.0119409			0.00597043	−	0.999982i	$-0.498100\pi$
$71$	0.100615			0.0119409			0.00597043	+	0.999982i	$0.498100\pi$
$72$		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
$75$		0			0
$76$		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
$81$		0			0
$82$		0			0
$83$			9.09332i			0.998122i	0.866567	+	0.499061i	$0.166322\pi$
$83$			9.09332i			0.998122i	−0.866567	+	0.499061i	$0.833678\pi$
$84$		0			0
$85$	−2.76845	−	3.13093i	−0.300281	−	0.339597i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.92478			0.840025			0.420012	−	0.907518i	$-0.362026\pi$
$89$	7.92478			0.840025			0.420012	+	0.907518i	$0.362026\pi$
$90$		0			0
$91$	−0.869067			−0.0911030
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$	8.28726			0.824613			0.412306	−	0.911045i	$-0.364723\pi$
$101$	8.28726			0.824613			0.412306	+	0.911045i	$0.364723\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$		−	3.45088i		−	0.333609i	−0.985990	−	0.166804i	$-0.946655\pi$
$107$		−	3.45088i		−	0.333609i	0.985990	−	0.166804i	$-0.0533449\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$		−	8.98778i		−	0.845499i	−0.906246	−	0.422750i	$-0.861065\pi$
$113$		−	8.98778i		−	0.845499i	0.906246	−	0.422750i	$-0.138935\pi$
$114$		0			0
$115$	−1.48119	−	1.67513i	−0.138122	−	0.156207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.86907			0.171337
$120$		0			0
$121$	−8.19394			−0.744903
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	20.9380			1.82936			0.914679	−	0.404182i	$-0.132444\pi$
$131$	20.9380			1.82936			0.914679	+	0.404182i	$0.132444\pi$
$132$		0			0
$133$		−	0.869067i		−	0.0753577i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11.8945i		−	1.01621i	−0.861294	−	0.508106i	$-0.830346\pi$
$137$		−	11.8945i		−	1.01621i	0.861294	−	0.508106i	$-0.169654\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$			1.45580i			0.121740i
$144$		0			0
$145$	4.09332	−	3.61942i	0.339932	−	0.300577i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.67513			−0.792618			−0.396309	−	0.918117i	$-0.629709\pi$
$149$	−9.67513			−0.792618			−0.396309	+	0.918117i	$0.629709\pi$
$150$		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
$153$		0			0
$154$		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
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$		−	9.08603i		−	0.703098i	−0.936170	−	0.351549i	$-0.885655\pi$
$167$		−	9.08603i		−	0.703098i	0.936170	−	0.351549i	$-0.114345\pi$
$168$		0			0
$169$	12.2447			0.941902
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	6.49929i		−	0.494132i	−0.968999	−	0.247066i	$-0.920534\pi$
$173$		−	6.49929i		−	0.494132i	0.968999	−	0.247066i	$-0.0794665\pi$
$174$		0			0
$175$	4.96239	+	0.612127i	0.375121	+	0.0462724i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.54420			0.190162			0.0950812	−	0.995470i	$-0.469689\pi$
$179$	2.54420			0.190162			0.0950812	+	0.995470i	$0.469689\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	−4.10062	−	4.63752i	−0.301483	−	0.340957i
$186$		0			0
$187$		−	3.13093i		−	0.228956i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.28726			0.527287			0.263644	−	0.964620i	$-0.415076\pi$
$191$	7.28726			0.527287			0.263644	+	0.964620i	$0.415076\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$		−	16.9380i		−	1.20678i	−0.797447	−	0.603390i	$-0.793816\pi$
$197$		−	16.9380i		−	1.20678i	0.797447	−	0.603390i	$-0.206184\pi$
$198$		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
$201$		0			0
$202$		0			0
$203$			2.44358i			0.171506i
$204$		0			0
$205$	17.7562	−	15.7005i	1.24015	−	1.09657i
$206$		0			0
$207$		0			0
$208$		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			0
$214$		0			0
$215$	−10.5442	−	11.9248i	−0.719108	−	0.813263i
$216$		0			0
$217$		−	4.19394i		−	0.284703i
$218$		0			0
$219$		0			0
$220$		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			0
$226$		0			0
$227$			5.14903i			0.341753i	0.985292	+	0.170877i	$0.0546599\pi$
$227$			5.14903i			0.341753i	−0.985292	+	0.170877i	$0.945340\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$		−	14.1768i		−	0.928753i	−0.885638	−	0.464376i	$-0.846279\pi$
$233$		−	14.1768i		−	0.928753i	0.885638	−	0.464376i	$-0.153721\pi$
$234$		0			0
$235$	−4.01810	−	4.54420i	−0.262112	−	0.296431i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.26423			−0.0817766			−0.0408883	−	0.999164i	$-0.513019\pi$
$239$	−1.26423			−0.0817766			−0.0408883	+	0.999164i	$0.513019\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	10.0508	−	8.88717i	0.642121	−	0.567780i
$246$		0			0
$247$		−	0.755278i		−	0.0480572i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.66291			−0.483679			−0.241839	−	0.970316i	$-0.577751\pi$
$251$	−7.66291			−0.483679			−0.241839	+	0.970316i	$0.577751\pi$
$252$		0			0
$253$		−	1.67513i		−	0.105315i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	9.65069i		−	0.601994i	−0.953625	−	0.300997i	$-0.902681\pi$
$257$		−	9.65069i		−	0.601994i	0.953625	−	0.300997i	$-0.0973194\pi$
$258$		0			0
$259$	2.76845			0.172023
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.8994i			0.980398i	0.871611	+	0.490199i	$0.163076\pi$
$263$			15.8994i			0.980398i	−0.871611	+	0.490199i	$0.836924\pi$
$264$		0			0
$265$	5.38787	+	6.09332i	0.330974	+	0.374310i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.85448			0.478896			0.239448	−	0.970909i	$-0.423034\pi$
$269$	7.85448			0.478896			0.239448	+	0.970909i	$0.423034\pi$
$270$		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			0
$274$		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
$279$		0			0
$280$		0			0
$281$	6.96002			0.415200			0.207600	−	0.978214i	$-0.433435\pi$
$281$	6.96002			0.415200			0.207600	+	0.978214i	$0.433435\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$			10.5999i			0.625693i
$288$		0			0
$289$	13.5066			0.794505
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.18172i		−	0.127457i	−0.997967	−	0.0637287i	$-0.979701\pi$
$293$		−	2.18172i		−	0.127457i	0.997967	−	0.0637287i	$-0.0202992\pi$
$294$		0			0
$295$	5.83146	−	5.15633i	0.339520	−	0.300213i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.869067			0.0502595
$300$		0			0
$301$	7.11871			0.410316
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	14.2619			0.808716			0.404358	−	0.914601i	$-0.367495\pi$
$311$	14.2619			0.808716			0.404358	+	0.914601i	$0.367495\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$		−	31.0616i		−	1.74459i	−0.488978	−	0.872296i	$-0.662630\pi$
$317$		−	31.0616i		−	1.74459i	0.488978	−	0.872296i	$-0.337370\pi$
$318$		0			0
$319$	4.09332			0.229182
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.62435i			0.0903810i
$324$		0			0
$325$	4.31265	+	0.531980i	0.239223	+	0.0295089i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$	−7.02539			−0.380446
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			15.9756i			0.857613i	0.903396	+	0.428807i	$0.141066\pi$
$347$			15.9756i			0.857613i	−0.903396	+	0.428807i	$0.858934\pi$
$348$		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			0
$352$		0			0
$353$		−	9.47390i		−	0.504245i	−0.967695	−	0.252122i	$-0.918871\pi$
$353$		−	9.47390i		−	0.504245i	0.967695	−	0.252122i	$-0.0811286\pi$
$354$		0			0
$355$	0.168544	−	0.149031i	0.00894538	−	0.00790974i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.5853			−1.45590			−0.727949	−	0.685632i	$-0.759526\pi$
$359$	−27.5853			−1.45590			−0.727949	+	0.685632i	$0.759526\pi$
$360$		0			0
$361$	−18.2447			−0.960249
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		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
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$		−	27.8300i		−	1.42205i	−0.703167	−	0.711024i	$-0.748232\pi$
$383$		−	27.8300i		−	1.42205i	0.703167	−	0.711024i	$-0.251768\pi$
$384$		0			0
$385$	2.48119	+	2.80606i	0.126453	+	0.143010i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.3733			−0.830158			−0.415079	−	0.909785i	$-0.636246\pi$
$389$	−16.3733			−0.830158			−0.415079	+	0.909785i	$0.636246\pi$
$390$		0			0
$391$	−1.86907			−0.0945228
$392$		0			0
$393$		0			0
$394$		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			0
$400$		0			0
$401$	12.2619			0.612328			0.306164	−	0.951979i	$-0.400954\pi$
$401$	12.2619			0.612328			0.306164	+	0.951979i	$0.400954\pi$
$402$		0			0
$403$		−	3.64481i		−	0.181561i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			3.48119i			0.171298i
$414$		0			0
$415$	13.4690	+	15.2325i	0.661166	+	0.747734i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.4363			1.38920			0.694602	−	0.719394i	$-0.255581\pi$
$419$	28.4363			1.38920			0.694602	+	0.719394i	$0.255581\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$	−9.27504	−	1.14411i	−0.449905	−	0.0554973i
$426$		0			0
$427$		−	1.78067i		−	0.0861727i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.62530			−0.319130			−0.159565	−	0.987187i	$-0.551009\pi$
$431$	−6.62530			−0.319130			−0.159565	+	0.987187i	$0.551009\pi$
$432$		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
$435$		0			0
$436$		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
$441$		0			0
$442$		0			0
$443$			12.2398i			0.581530i	0.956794	+	0.290765i	$0.0939098\pi$
$443$			12.2398i			0.581530i	−0.956794	+	0.290765i	$0.906090\pi$
$444$		0			0
$445$	13.2750	−	11.7381i	0.629297	−	0.556441i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−26.8315			−1.26625			−0.633127	−	0.774048i	$-0.718229\pi$
$449$	−26.8315			−1.26625			−0.633127	+	0.774048i	$0.718229\pi$
$450$		0			0
$451$	17.7562			0.836108
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	18.3127			0.852905			0.426453	−	0.904510i	$-0.359763\pi$
$461$	18.3127			0.852905			0.426453	+	0.904510i	$0.359763\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$			16.7767i			0.776333i	0.921589	+	0.388167i	$0.126891\pi$
$467$			16.7767i			0.776333i	−0.921589	+	0.388167i	$0.873109\pi$
$468$		0			0
$469$	11.0811			0.511678
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.9248i		−	0.548302i
$474$		0			0
$475$	−0.531980	+	4.31265i	−0.0244089	+	0.197878i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.6751			0.533450			0.266725	−	0.963773i	$-0.414058\pi$
$479$	11.6751			0.533450			0.266725	+	0.963773i	$0.414058\pi$
$480$		0			0
$481$	2.40597			0.109703
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	12.0155			0.542254			0.271127	−	0.962544i	$-0.412604\pi$
$491$	12.0155			0.542254			0.271127	+	0.962544i	$0.412604\pi$
$492$		0			0
$493$		−	4.56722i		−	0.205697i
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		0			0
$503$			5.35519i			0.238776i	0.992848	+	0.119388i	$0.0380932\pi$
$503$			5.35519i			0.238776i	−0.992848	+	0.119388i	$0.961907\pi$
$504$		0			0
$505$	13.8822	−	12.2750i	0.617752	−	0.546232i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.74543			−0.298986			−0.149493	−	0.988763i	$-0.547764\pi$
$509$	−6.74543			−0.298986			−0.149493	+	0.988763i	$0.547764\pi$
$510$		0			0
$511$	−2.16854			−0.0959307
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.15633	+	4.70052i	0.183150	+	0.207130i
$516$		0			0
$517$		−	4.54420i		−	0.199854i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.2252			0.535596			0.267798	−	0.963475i	$-0.413704\pi$
$521$	12.2252			0.535596			0.267798	+	0.963475i	$0.413704\pi$
$522$		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			0
$526$		0			0
$527$			7.83875i			0.341461i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9.21203i			0.399018i
$534$		0			0
$535$	−5.11142	−	5.78067i	−0.220986	−	0.249920i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$	−2.12364			−0.0904700
$552$		0			0
$553$		−	5.08840i		−	0.216381i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			35.6883i			1.51216i	0.654478	+	0.756081i	$0.272888\pi$
$557$			35.6883i			1.51216i	−0.654478	+	0.756081i	$0.727112\pi$
$558$		0			0
$559$	6.18664			0.261667
$560$		0			0
$561$		0			0
$562$		0			0
$563$			38.5510i			1.62473i	0.583148	+	0.812366i	$0.301821\pi$
$563$			38.5510i			1.62473i	−0.583148	+	0.812366i	$0.698179\pi$
$564$		0			0
$565$	−13.3127	−	15.0557i	−0.560067	−	0.633398i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.27645			−0.263123			−0.131561	−	0.991308i	$-0.541999\pi$
$569$	−6.27645			−0.263123			−0.131561	+	0.991308i	$0.541999\pi$
$570$		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
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	−9.09332			−0.377255
$582$		0			0
$583$			6.09332i			0.252360i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.68735i		−	0.0696444i	−0.999394	−	0.0348222i	$-0.988914\pi$
$587$		−	1.68735i		−	0.0696444i	0.999394	−	0.0348222i	$-0.0110865\pi$
$588$		0			0
$589$	3.64481			0.150182
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	37.7523i		−	1.55030i	−0.631777	−	0.775150i	$-0.717674\pi$
$593$		−	37.7523i		−	1.55030i	0.631777	−	0.775150i	$-0.282326\pi$
$594$		0			0
$595$	3.13093	−	2.76845i	0.128356	−	0.113495i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.5271			−0.552700			−0.276350	−	0.961057i	$-0.589125\pi$
$599$	−13.5271			−0.552700			−0.276350	+	0.961057i	$0.589125\pi$
$600$		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
$603$		0			0
$604$		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
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$			24.0411i			0.967859i	0.875107	+	0.483930i	$0.160791\pi$
$617$			24.0411i			0.967859i	−0.875107	+	0.483930i	$0.839209\pi$
$618$		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
$621$		0			0
$622$		0			0
$623$			7.92478i			0.317499i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−4.43629	−	5.01714i	−0.176049	−	0.199099i
$636$		0			0
$637$			5.21440i			0.206602i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	30.9257			1.22149			0.610746	−	0.791826i	$-0.290869\pi$
$641$	30.9257			1.22149			0.610746	+	0.791826i	$0.290869\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$		−	5.67513i		−	0.223112i	−0.993758	−	0.111556i	$-0.964416\pi$
$647$		−	5.67513i		−	0.223112i	0.993758	−	0.111556i	$-0.0355835\pi$
$648$		0			0
$649$	5.83146			0.228905
$650$		0			0
$651$		0			0
$652$		0			0
$653$			50.7245i			1.98500i	0.122232	+	0.992502i	$0.460995\pi$
$653$			50.7245i			1.98500i	−0.122232	+	0.992502i	$0.539005\pi$
$654$		0			0
$655$	35.0738	−	31.0132i	1.37045	−	1.21178i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.4641			0.914030			0.457015	−	0.889459i	$-0.348919\pi$
$659$	23.4641			0.914030			0.457015	+	0.889459i	$0.348919\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	−1.28726	−	1.45580i	−0.0499177	−	0.0564536i
$666$		0			0
$667$		−	2.44358i		−	0.0946159i
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$			2.52469i			0.0970315i	0.998822	+	0.0485158i	$0.0154491\pi$
$677$			2.52469i			0.0970315i	−0.998822	+	0.0485158i	$0.984551\pi$
$678$		0			0
$679$	−4.99271			−0.191602
$680$		0			0
$681$		0			0
$682$		0			0
$683$			24.5477i			0.939292i	0.882855	+	0.469646i	$0.155618\pi$
$683$			24.5477i			0.939292i	−0.882855	+	0.469646i	$0.844382\pi$
$684$		0			0
$685$	−17.6180	−	19.9248i	−0.673149	−	0.761287i
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	−10.6375	+	9.40597i	−0.403504	+	0.356789i
$696$		0			0
$697$		−	19.8119i		−	0.750431i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.9633			−0.451849			−0.225925	−	0.974145i	$-0.572540\pi$
$701$	−11.9633			−0.451849			−0.225925	+	0.974145i	$0.572540\pi$
$702$		0			0
$703$			2.40597i			0.0907429i
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$			4.19394i			0.157064i
$714$		0			0
$715$	2.15633	+	2.43866i	0.0806420	+	0.0912007i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−25.6190			−0.955426			−0.477713	−	0.878516i	$-0.658534\pi$
$719$	−25.6190			−0.955426			−0.477713	+	0.878516i	$0.658534\pi$
$720$		0			0
$721$	−2.80606			−0.104503
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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
$735$		0			0
$736$		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			0
$742$		0			0
$743$			45.9657i			1.68632i	0.537664	+	0.843159i	$0.319307\pi$
$743$			45.9657i			1.68632i	−0.537664	+	0.843159i	$0.680693\pi$
$744$		0			0
$745$	−16.2071	+	14.3307i	−0.593783	+	0.525038i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−11.0787			−0.401604			−0.200802	−	0.979632i	$-0.564355\pi$
$761$	−11.0787			−0.401604			−0.200802	+	0.979632i	$0.564355\pi$
$762$		0			0
$763$			13.3684i			0.483967i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$			33.7645i			1.21442i	0.794540	+	0.607212i	$0.207712\pi$
$773$			33.7645i			1.21442i	−0.794540	+	0.607212i	$0.792288\pi$
$774$		0			0
$775$	−2.56722	+	20.8119i	−0.0922173	+	0.747587i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.21203			−0.330055
$780$		0			0
$781$	0.168544			0.00603098
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	8.98778			0.319569
$792$		0			0
$793$		−	1.54752i		−	0.0549542i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	32.9633i		−	1.16762i	−0.811890	−	0.583811i	$-0.801561\pi$
$797$		−	32.9633i		−	1.16762i	0.811890	−	0.583811i	$-0.198439\pi$
$798$		0			0
$799$	−5.07030			−0.179374
$800$		0			0
$801$		0			0
$802$		0			0
$803$			3.63259i			0.128191i
$804$		0			0
$805$	1.67513	−	1.48119i	0.0590406	−	0.0522052i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.6639			0.972610			0.486305	−	0.873789i	$-0.338344\pi$
$809$	27.6639			0.972610			0.486305	+	0.873789i	$0.338344\pi$
$810$		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
$813$		0			0
$814$		0			0
$815$	−6.38787	−	7.22425i	−0.223757	−	0.253055i
$816$		0			0
$817$			6.18664i			0.216443i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.1319			0.353605			0.176803	−	0.984246i	$-0.443425\pi$
$821$	10.1319			0.353605			0.176803	+	0.984246i	$0.443425\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$			5.18901i			0.180440i	0.995922	+	0.0902198i	$0.0287570\pi$
$827$			5.18901i			0.180440i	−0.995922	+	0.0902198i	$0.971243\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$		−	11.2144i		−	0.388556i
$834$		0			0
$835$	−13.4582	−	15.2203i	−0.465739	−	0.526720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.2882			−0.838522			−0.419261	−	0.907866i	$-0.637711\pi$
$839$	−24.2882			−0.838522			−0.419261	+	0.907866i	$0.637711\pi$
$840$		0			0
$841$	−23.0289			−0.794100
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.5115	−	18.1368i	0.705617	−	0.623925i
$846$		0			0
$847$		−	8.19394i		−	0.281547i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$			27.2605i			0.931199i	0.884995	+	0.465600i	$0.154161\pi$
$857$			27.2605i			0.931199i	−0.884995	+	0.465600i	$0.845839\pi$
$858$		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
$861$		0			0
$862$		0			0
$863$			45.5026i			1.54893i	0.632619	+	0.774464i	$0.281980\pi$
$863$			45.5026i			1.54893i	−0.632619	+	0.774464i	$0.718020\pi$
$864$		0			0
$865$	−9.62672	−	10.8872i	−0.327318	−	0.370175i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.52373			−0.289148
$870$		0			0
$871$	9.63023			0.326308
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$	−26.2351			−0.883882			−0.441941	−	0.897044i	$-0.645710\pi$
$881$	−26.2351			−0.883882			−0.441941	+	0.897044i	$0.645710\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$		−	22.1866i		−	0.744955i	−0.928041	−	0.372477i	$-0.878508\pi$
$887$		−	22.1866i		−	0.744955i	0.928041	−	0.372477i	$-0.121492\pi$
$888$		0			0
$889$	2.99508			0.100452
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.35756i			0.0788926i
$894$		0			0
$895$	4.26187	−	3.76845i	0.142458	−	0.125965i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−10.2482			−0.341798
$900$		0			0
$901$	6.79877			0.226500
$902$		0			0
$903$		0			0
$904$		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
$909$		0			0
$910$		0			0
$911$	−54.0381			−1.79036			−0.895181	−	0.445702i	$-0.852954\pi$
$911$	−54.0381			−1.79036			−0.895181	+	0.445702i	$0.852954\pi$
$912$		0			0
$913$			15.2325i			0.504122i
$914$		0			0
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$			0.0874416i			0.00287817i
$924$		0			0
$925$	−13.7381	−	1.69464i	−0.451707	−	0.0557195i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.8183			−0.387745			−0.193873	−	0.981027i	$-0.562105\pi$
$929$	−11.8183			−0.387745			−0.193873	+	0.981027i	$0.562105\pi$
$930$		0			0
$931$	−5.21440			−0.170895
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	−25.0616			−0.816984			−0.408492	−	0.912762i	$-0.633945\pi$
$941$	−25.0616			−0.816984			−0.408492	+	0.912762i	$0.633945\pi$
$942$		0			0
$943$		−	10.5999i		−	0.345180i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	15.7626i		−	0.512215i	−0.966648	−	0.256107i	$-0.917560\pi$
$947$		−	15.7626i		−	0.512215i	0.966648	−	0.256107i	$-0.0824401\pi$
$948$		0			0
$949$	−1.88461			−0.0611771
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10.2520i			0.332095i	0.986118	+	0.166048i	$0.0531006\pi$
$953$			10.2520i			0.332095i	−0.986118	+	0.166048i	$0.946899\pi$
$954$		0			0
$955$	12.2071	−	10.7938i	0.395013	−	0.349281i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.8945			0.384092
$960$		0			0
$961$	−13.4109			−0.432610
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	−53.2144			−1.70773			−0.853866	−	0.520493i	$-0.825748\pi$
$971$	−53.2144			−1.70773			−0.853866	+	0.520493i	$0.825748\pi$
$972$		0			0
$973$		−	6.35026i		−	0.203580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			23.0590i			0.737724i	0.929484	+	0.368862i	$0.120252\pi$
$977$			23.0590i			0.737724i	−0.929484	+	0.368862i	$0.879748\pi$
$978$		0			0
$979$	13.2750			0.424272
$980$		0			0
$981$		0			0
$982$		0			0
$983$			14.7708i			0.471116i	0.971860	+	0.235558i	$0.0756918\pi$
$983$			14.7708i			0.471116i	−0.971860	+	0.235558i	$0.924308\pi$
$984$		0			0
$985$	−25.0884	−	28.3733i	−0.799383	−	0.904048i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		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
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.1	✓	6	15.14	odd	2
1380.2.f.a.829.4	yes	6	3.2	odd	2
4140.2.f.a.829.5		6	1.1	even	1	trivial
4140.2.f.a.829.6		6	5.4	even	2	inner
6900.2.a.y.1.1		3	15.8	even	4
6900.2.a.z.1.1		3	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

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