LMFDB - Embedded newform 4160.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.a (trivial)

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:	yes
Analytic conductor:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25088.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 2 $ x^4 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.37608$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.841723 q^{3} +1.00000 q^{5} -4.75216 q^{7} -2.29150 q^{9} +O(q^{10})$	$q-0.841723 q^{3} +1.00000 q^{5} -4.75216 q^{7} -2.29150 q^{9} -3.91044 q^{11} +1.00000 q^{13} -0.841723 q^{15} +6.00000 q^{17} -3.91044 q^{19} +4.00000 q^{21} -6.97915 q^{23} +1.00000 q^{25} +4.45398 q^{27} -5.29150 q^{29} +5.59388 q^{31} +3.29150 q^{33} -4.75216 q^{35} +1.29150 q^{37} -0.841723 q^{39} -8.58301 q^{41} -2.52517 q^{43} -2.29150 q^{45} -4.75216 q^{47} +15.5830 q^{49} -5.05034 q^{51} -12.5830 q^{53} -3.91044 q^{55} +3.29150 q^{57} +7.27733 q^{59} -13.2915 q^{61} +10.8896 q^{63} +1.00000 q^{65} -3.06871 q^{67} +5.87451 q^{69} +11.7313 q^{71} +9.29150 q^{73} -0.841723 q^{75} +18.5830 q^{77} +7.82087 q^{79} +3.12549 q^{81} -4.75216 q^{83} +6.00000 q^{85} +4.45398 q^{87} +16.5830 q^{89} -4.75216 q^{91} -4.70850 q^{93} -3.91044 q^{95} -2.00000 q^{97} +8.96077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 12 * q^9	$ 4 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{17} + 16 q^{21} + 4 q^{25} - 8 q^{33} - 16 q^{37} + 8 q^{41} + 12 q^{45} + 20 q^{49} - 8 q^{53} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 40 q^{69} + 16 q^{73} + 32 q^{77} + 76 q^{81} + 24 q^{85} + 24 q^{89} - 40 q^{93} - 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^17 + 16 * q^21 + 4 * q^25 - 8 * q^33 - 16 * q^37 + 8 * q^41 + 12 * q^45 + 20 * q^49 - 8 * q^53 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 40 * q^69 + 16 * q^73 + 32 * q^77 + 76 * q^81 + 24 * q^85 + 24 * q^89 - 40 * q^93 - 8 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−0.841723	−0.485969	−0.242984	−	0.970030i	$-0.578126\pi$
$3$	−0.841723	−0.485969	−0.242984	+	0.970030i	$0.578126\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.75216	−1.79615	−0.898073	−	0.439846i	$-0.855033\pi$
$7$	−4.75216	−1.79615	−0.898073	+	0.439846i	$0.855033\pi$
$8$	0	0
$9$	−2.29150	−0.763834
$10$	0	0
$11$	−3.91044	−1.17904	−0.589520	−	0.807754i	$-0.700683\pi$
$11$	−3.91044	−1.17904	−0.589520	+	0.807754i	$0.700683\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.841723	−0.217332
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−3.91044	−0.897115	−0.448558	−	0.893754i	$-0.648062\pi$
$19$	−3.91044	−0.897115	−0.448558	+	0.893754i	$0.648062\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−6.97915	−1.45525	−0.727626	−	0.685974i	$-0.759377\pi$
$23$	−6.97915	−1.45525	−0.727626	+	0.685974i	$0.759377\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.45398	0.857169
$28$	0	0
$29$	−5.29150	−0.982607	−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150	−0.982607	−0.491304	+	0.870988i	$0.663479\pi$
$30$	0	0
$31$	5.59388	1.00469	0.502345	−	0.864667i	$-0.332471\pi$
$31$	5.59388	1.00469	0.502345	+	0.864667i	$0.332471\pi$
$32$	0	0
$33$	3.29150	0.572977
$34$	0	0
$35$	−4.75216	−0.803261
$36$	0	0
$37$	1.29150	0.212322	0.106161	−	0.994349i	$-0.466144\pi$
$37$	1.29150	0.212322	0.106161	+	0.994349i	$0.466144\pi$
$38$	0	0
$39$	−0.841723	−0.134784
$40$	0	0
$41$	−8.58301	−1.34044	−0.670220	−	0.742162i	$-0.733800\pi$
$41$	−8.58301	−1.34044	−0.670220	+	0.742162i	$0.733800\pi$
$42$	0	0
$43$	−2.52517	−0.385085	−0.192542	−	0.981289i	$-0.561673\pi$
$43$	−2.52517	−0.385085	−0.192542	+	0.981289i	$0.561673\pi$
$44$	0	0
$45$	−2.29150	−0.341597
$46$	0	0
$47$	−4.75216	−0.693173	−0.346587	−	0.938018i	$-0.612659\pi$
$47$	−4.75216	−0.693173	−0.346587	+	0.938018i	$0.612659\pi$
$48$	0	0
$49$	15.5830	2.22614
$50$	0	0
$51$	−5.05034	−0.707189
$52$	0	0
$53$	−12.5830	−1.72841	−0.864204	−	0.503141i	$-0.832178\pi$
$53$	−12.5830	−1.72841	−0.864204	+	0.503141i	$0.832178\pi$
$54$	0	0
$55$	−3.91044	−0.527283
$56$	0	0
$57$	3.29150	0.435970
$58$	0	0
$59$	7.27733	0.947427	0.473714	−	0.880679i	$-0.342913\pi$
$59$	7.27733	0.947427	0.473714	+	0.880679i	$0.342913\pi$
$60$	0	0
$61$	−13.2915	−1.70180	−0.850901	−	0.525326i	$-0.823944\pi$
$61$	−13.2915	−1.70180	−0.850901	+	0.525326i	$0.823944\pi$
$62$	0	0
$63$	10.8896	1.37196
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.06871	−0.374903	−0.187451	−	0.982274i	$-0.560023\pi$
$67$	−3.06871	−0.374903	−0.187451	+	0.982274i	$0.560023\pi$
$68$	0	0
$69$	5.87451	0.707208
$70$	0	0
$71$	11.7313	1.39225	0.696125	−	0.717921i	$-0.254906\pi$
$71$	11.7313	1.39225	0.696125	+	0.717921i	$0.254906\pi$
$72$	0	0
$73$	9.29150	1.08749	0.543744	−	0.839251i	$-0.317006\pi$
$73$	9.29150	1.08749	0.543744	+	0.839251i	$0.317006\pi$
$74$	0	0
$75$	−0.841723	−0.0971938
$76$	0	0
$77$	18.5830	2.11773
$78$	0	0
$79$	7.82087	0.879917	0.439958	−	0.898018i	$-0.354993\pi$
$79$	7.82087	0.879917	0.439958	+	0.898018i	$0.354993\pi$
$80$	0	0
$81$	3.12549	0.347277
$82$	0	0
$83$	−4.75216	−0.521617	−0.260809	−	0.965391i	$-0.583989\pi$
$83$	−4.75216	−0.521617	−0.260809	+	0.965391i	$0.583989\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	4.45398	0.477517
$88$	0	0
$89$	16.5830	1.75780	0.878898	−	0.477011i	$-0.158280\pi$
$89$	16.5830	1.75780	0.878898	+	0.477011i	$0.158280\pi$
$90$	0	0
$91$	−4.75216	−0.498162
$92$	0	0
$93$	−4.70850	−0.488248
$94$	0	0
$95$	−3.91044	−0.401202
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	8.96077	0.900591
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	14.8000	1.45829	0.729145	−	0.684360i	$-0.239918\pi$
$103$	14.8000	1.45829	0.729145	+	0.684360i	$0.239918\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	10.3460	1.00019	0.500095	−	0.865971i	$-0.333299\pi$
$107$	10.3460	1.00019	0.500095	+	0.865971i	$0.333299\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−1.08709	−0.103182
$112$	0	0
$113$	12.5830	1.18371	0.591855	−	0.806045i	$-0.298396\pi$
$113$	12.5830	1.18371	0.591855	+	0.806045i	$0.298396\pi$
$114$	0	0
$115$	−6.97915	−0.650809
$116$	0	0
$117$	−2.29150	−0.211849
$118$	0	0
$119$	−28.5129	−2.61378
$120$	0	0
$121$	4.29150	0.390137
$122$	0	0
$123$	7.22451	0.651412
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.8000	−1.31329	−0.656645	−	0.754200i	$-0.728025\pi$
$127$	−14.8000	−1.31329	−0.656645	+	0.754200i	$0.728025\pi$
$128$	0	0
$129$	2.12549	0.187139
$130$	0	0
$131$	12.8712	1.12456	0.562281	−	0.826946i	$-0.309924\pi$
$131$	12.8712	1.12456	0.562281	+	0.826946i	$0.309924\pi$
$132$	0	0
$133$	18.5830	1.61135
$134$	0	0
$135$	4.45398	0.383337
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	7.82087	0.663358	0.331679	−	0.943392i	$-0.392385\pi$
$139$	7.82087	0.663358	0.331679	+	0.943392i	$0.392385\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−3.91044	−0.327007
$144$	0	0
$145$	−5.29150	−0.439435
$146$	0	0
$147$	−13.1166	−1.08184
$148$	0	0
$149$	12.5830	1.03084	0.515420	−	0.856938i	$-0.327636\pi$
$149$	12.5830	1.03084	0.515420	+	0.856938i	$0.327636\pi$
$150$	0	0
$151$	0.543544	0.0442330	0.0221165	−	0.999755i	$-0.492960\pi$
$151$	0.543544	0.0442330	0.0221165	+	0.999755i	$0.492960\pi$
$152$	0	0
$153$	−13.7490	−1.11154
$154$	0	0
$155$	5.59388	0.449311
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	10.5914	0.839953
$160$	0	0
$161$	33.1660	2.61385
$162$	0	0
$163$	9.80250	0.767791	0.383895	−	0.923377i	$-0.374582\pi$
$163$	9.80250	0.767791	0.383895	+	0.923377i	$0.374582\pi$
$164$	0	0
$165$	3.29150	0.256243
$166$	0	0
$167$	−23.7608	−1.83867	−0.919333	−	0.393481i	$-0.871271\pi$
$167$	−23.7608	−1.83867	−0.919333	+	0.393481i	$0.871271\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.96077	0.685247
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−4.75216	−0.359229
$176$	0	0
$177$	−6.12549	−0.460420
$178$	0	0
$179$	19.0086	1.42077	0.710386	−	0.703812i	$-0.248520\pi$
$179$	19.0086	1.42077	0.710386	+	0.703812i	$0.248520\pi$
$180$	0	0
$181$	−5.29150	−0.393314	−0.196657	−	0.980472i	$-0.563009\pi$
$181$	−5.29150	−0.393314	−0.196657	+	0.980472i	$0.563009\pi$
$182$	0	0
$183$	11.1878	0.827023
$184$	0	0
$185$	1.29150	0.0949532
$186$	0	0
$187$	−23.4626	−1.71576
$188$	0	0
$189$	−21.1660	−1.53960
$190$	0	0
$191$	12.8712	0.931328	0.465664	−	0.884962i	$-0.345816\pi$
$191$	12.8712	0.931328	0.465664	+	0.884962i	$0.345816\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	−0.841723	−0.0602770
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	12.2748	0.870141	0.435070	−	0.900396i	$-0.356723\pi$
$199$	12.2748	0.870141	0.435070	+	0.900396i	$0.356723\pi$
$200$	0	0
$201$	2.58301	0.182191
$202$	0	0
$203$	25.1461	1.76491
$204$	0	0
$205$	−8.58301	−0.599463
$206$	0	0
$207$	15.9927	1.11157
$208$	0	0
$209$	15.2915	1.05774
$210$	0	0
$211$	12.8712	0.886090	0.443045	−	0.896499i	$-0.353898\pi$
$211$	12.8712	0.886090	0.443045	+	0.896499i	$0.353898\pi$
$212$	0	0
$213$	−9.87451	−0.676590
$214$	0	0
$215$	−2.52517	−0.172215
$216$	0	0
$217$	−26.5830	−1.80457
$218$	0	0
$219$	−7.82087	−0.528485
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	−17.6234	−1.18015	−0.590074	−	0.807349i	$-0.700901\pi$
$223$	−17.6234	−1.18015	−0.590074	+	0.807349i	$0.700901\pi$
$224$	0	0
$225$	−2.29150	−0.152767
$226$	0	0
$227$	−6.43560	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$227$	−6.43560	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$228$	0	0
$229$	−4.58301	−0.302854	−0.151427	−	0.988468i	$-0.548387\pi$
$229$	−4.58301	−0.302854	−0.151427	+	0.988468i	$0.548387\pi$
$230$	0	0
$231$	−15.6417	−1.02915
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−4.75216	−0.309997
$236$	0	0
$237$	−6.58301	−0.427612
$238$	0	0
$239$	7.27733	0.470731	0.235366	−	0.971907i	$-0.424371\pi$
$239$	7.27733	0.470731	0.235366	+	0.971907i	$0.424371\pi$
$240$	0	0
$241$	−16.5830	−1.06821	−0.534103	−	0.845420i	$-0.679350\pi$
$241$	−16.5830	−1.06821	−0.534103	+	0.845420i	$0.679350\pi$
$242$	0	0
$243$	−15.9927	−1.02593
$244$	0	0
$245$	15.5830	0.995562
$246$	0	0
$247$	−3.91044	−0.248815
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−10.5914	−0.668523	−0.334262	−	0.942480i	$-0.608487\pi$
$251$	−10.5914	−0.668523	−0.334262	+	0.942480i	$0.608487\pi$
$252$	0	0
$253$	27.2915	1.71580
$254$	0	0
$255$	−5.05034	−0.316264
$256$	0	0
$257$	−4.58301	−0.285880	−0.142940	−	0.989731i	$-0.545656\pi$
$257$	−4.58301	−0.285880	−0.142940	+	0.989731i	$0.545656\pi$
$258$	0	0
$259$	−6.13742	−0.381361
$260$	0	0
$261$	12.1255	0.750549
$262$	0	0
$263$	6.97915	0.430353	0.215176	−	0.976575i	$-0.430967\pi$
$263$	6.97915	0.430353	0.215176	+	0.976575i	$0.430967\pi$
$264$	0	0
$265$	−12.5830	−0.772968
$266$	0	0
$267$	−13.9583	−0.854234
$268$	0	0
$269$	24.5830	1.49885	0.749426	−	0.662088i	$-0.230329\pi$
$269$	24.5830	1.49885	0.749426	+	0.662088i	$0.230329\pi$
$270$	0	0
$271$	−24.0062	−1.45827	−0.729135	−	0.684370i	$-0.760077\pi$
$271$	−24.0062	−1.45827	−0.729135	+	0.684370i	$0.760077\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	−3.91044	−0.235808
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−12.8184	−0.767417
$280$	0	0
$281$	−0.583005	−0.0347792	−0.0173896	−	0.999849i	$-0.505536\pi$
$281$	−0.583005	−0.0347792	−0.0173896	+	0.999849i	$0.505536\pi$
$282$	0	0
$283$	−21.5338	−1.28005	−0.640026	−	0.768353i	$-0.721076\pi$
$283$	−21.5338	−1.28005	−0.640026	+	0.768353i	$0.721076\pi$
$284$	0	0
$285$	3.29150	0.194972
$286$	0	0
$287$	40.7878	2.40763
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	1.68345	0.0986853
$292$	0	0
$293$	−0.125492	−0.00733133	−0.00366566	−	0.999993i	$-0.501167\pi$
$293$	−0.125492	−0.00733133	−0.00366566	+	0.999993i	$0.501167\pi$
$294$	0	0
$295$	7.27733	0.423702
$296$	0	0
$297$	−17.4170	−1.01064
$298$	0	0
$299$	−6.97915	−0.403615
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	5.05034	0.290134
$304$	0	0
$305$	−13.2915	−0.761069
$306$	0	0
$307$	−7.52269	−0.429343	−0.214671	−	0.976686i	$-0.568868\pi$
$307$	−7.52269	−0.429343	−0.214671	+	0.976686i	$0.568868\pi$
$308$	0	0
$309$	−12.4575	−0.708683
$310$	0	0
$311$	24.0590	1.36426	0.682129	−	0.731231i	$-0.261054\pi$
$311$	24.0590	1.36426	0.682129	+	0.731231i	$0.261054\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	10.8896	0.613558
$316$	0	0
$317$	−13.2915	−0.746525	−0.373263	−	0.927726i	$-0.621761\pi$
$317$	−13.2915	−0.746525	−0.373263	+	0.927726i	$0.621761\pi$
$318$	0	0
$319$	20.6921	1.15853
$320$	0	0
$321$	−8.70850	−0.486061
$322$	0	0
$323$	−23.4626	−1.30549
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	1.68345	0.0930948
$328$	0	0
$329$	22.5830	1.24504
$330$	0	0
$331$	−21.8320	−1.19999	−0.599997	−	0.800002i	$-0.704832\pi$
$331$	−21.8320	−1.19999	−0.599997	+	0.800002i	$0.704832\pi$
$332$	0	0
$333$	−2.95948	−0.162179
$334$	0	0
$335$	−3.06871	−0.167662
$336$	0	0
$337$	20.5830	1.12123	0.560614	−	0.828077i	$-0.310565\pi$
$337$	20.5830	1.12123	0.560614	+	0.828077i	$0.310565\pi$
$338$	0	0
$339$	−10.5914	−0.575246
$340$	0	0
$341$	−21.8745	−1.18457
$342$	0	0
$343$	−40.7878	−2.20233
$344$	0	0
$345$	5.87451	0.316273
$346$	0	0
$347$	−18.1669	−0.975251	−0.487625	−	0.873053i	$-0.662137\pi$
$347$	−18.1669	−0.975251	−0.487625	+	0.873053i	$0.662137\pi$
$348$	0	0
$349$	31.1660	1.66828	0.834139	−	0.551554i	$-0.185965\pi$
$349$	31.1660	1.66828	0.834139	+	0.551554i	$0.185965\pi$
$350$	0	0
$351$	4.45398	0.237736
$352$	0	0
$353$	−26.4575	−1.40819	−0.704096	−	0.710105i	$-0.748647\pi$
$353$	−26.4575	−1.40819	−0.704096	+	0.710105i	$0.748647\pi$
$354$	0	0
$355$	11.7313	0.622633
$356$	0	0
$357$	24.0000	1.27021
$358$	0	0
$359$	8.96077	0.472931	0.236466	−	0.971640i	$-0.424011\pi$
$359$	8.96077	0.472931	0.236466	+	0.971640i	$0.424011\pi$
$360$	0	0
$361$	−3.70850	−0.195184
$362$	0	0
$363$	−3.61226	−0.189594
$364$	0	0
$365$	9.29150	0.486339
$366$	0	0
$367$	24.3043	1.26868	0.634338	−	0.773056i	$-0.281273\pi$
$367$	24.3043	1.26868	0.634338	+	0.773056i	$0.281273\pi$
$368$	0	0
$369$	19.6680	1.02387
$370$	0	0
$371$	59.7964	3.10448
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	−0.841723	−0.0434664
$376$	0	0
$377$	−5.29150	−0.272526
$378$	0	0
$379$	5.59388	0.287338	0.143669	−	0.989626i	$-0.454110\pi$
$379$	5.59388	0.287338	0.143669	+	0.989626i	$0.454110\pi$
$380$	0	0
$381$	12.4575	0.638218
$382$	0	0
$383$	−10.8896	−0.556432	−0.278216	−	0.960519i	$-0.589743\pi$
$383$	−10.8896	−0.556432	−0.278216	+	0.960519i	$0.589743\pi$
$384$	0	0
$385$	18.5830	0.947078
$386$	0	0
$387$	5.78643	0.294141
$388$	0	0
$389$	0.583005	0.0295595	0.0147798	−	0.999891i	$-0.495295\pi$
$389$	0.583005	0.0295595	0.0147798	+	0.999891i	$0.495295\pi$
$390$	0	0
$391$	−41.8749	−2.11770
$392$	0	0
$393$	−10.8340	−0.546502
$394$	0	0
$395$	7.82087	0.393511
$396$	0	0
$397$	17.2915	0.867836	0.433918	−	0.900952i	$-0.357131\pi$
$397$	17.2915	0.867836	0.433918	+	0.900952i	$0.357131\pi$
$398$	0	0
$399$	−15.6417	−0.783066
$400$	0	0
$401$	0.583005	0.0291139	0.0145569	−	0.999894i	$-0.495366\pi$
$401$	0.583005	0.0291139	0.0145569	+	0.999894i	$0.495366\pi$
$402$	0	0
$403$	5.59388	0.278651
$404$	0	0
$405$	3.12549	0.155307
$406$	0	0
$407$	−5.05034	−0.250336
$408$	0	0
$409$	12.5830	0.622190	0.311095	−	0.950379i	$-0.399304\pi$
$409$	12.5830	0.622190	0.311095	+	0.950379i	$0.399304\pi$
$410$	0	0
$411$	−5.05034	−0.249115
$412$	0	0
$413$	−34.5830	−1.70172
$414$	0	0
$415$	−4.75216	−0.233274
$416$	0	0
$417$	−6.58301	−0.322371
$418$	0	0
$419$	17.9215	0.875525	0.437762	−	0.899091i	$-0.355771\pi$
$419$	17.9215	0.875525	0.437762	+	0.899091i	$0.355771\pi$
$420$	0	0
$421$	11.1660	0.544198	0.272099	−	0.962269i	$-0.412282\pi$
$421$	11.1660	0.544198	0.272099	+	0.962269i	$0.412282\pi$
$422$	0	0
$423$	10.8896	0.529470
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	63.1633	3.05669
$428$	0	0
$429$	3.29150	0.158915
$430$	0	0
$431$	36.8774	1.77632	0.888160	−	0.459534i	$-0.151984\pi$
$431$	36.8774	1.77632	0.888160	+	0.459534i	$0.151984\pi$
$432$	0	0
$433$	23.1660	1.11329	0.556644	−	0.830751i	$-0.312089\pi$
$433$	23.1660	1.11329	0.556644	+	0.830751i	$0.312089\pi$
$434$	0	0
$435$	4.45398	0.213552
$436$	0	0
$437$	27.2915	1.30553
$438$	0	0
$439$	31.2835	1.49308	0.746540	−	0.665341i	$-0.231714\pi$
$439$	31.2835	1.49308	0.746540	+	0.665341i	$0.231714\pi$
$440$	0	0
$441$	−35.7085	−1.70040
$442$	0	0
$443$	−38.8590	−1.84625	−0.923123	−	0.384505i	$-0.874372\pi$
$443$	−38.8590	−1.84625	−0.923123	+	0.384505i	$0.874372\pi$
$444$	0	0
$445$	16.5830	0.786110
$446$	0	0
$447$	−10.5914	−0.500956
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	33.5633	1.58043
$452$	0	0
$453$	−0.457513	−0.0214958
$454$	0	0
$455$	−4.75216	−0.222785
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	26.7239	1.24736
$460$	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
$463$	−32.1780	−1.49544	−0.747720	−	0.664015i	$-0.768851\pi$
$463$	−32.1780	−1.49544	−0.747720	+	0.664015i	$0.768851\pi$
$464$	0	0
$465$	−4.70850	−0.218351
$466$	0	0
$467$	18.7633	0.868260	0.434130	−	0.900850i	$-0.357056\pi$
$467$	18.7633	0.868260	0.434130	+	0.900850i	$0.357056\pi$
$468$	0	0
$469$	14.5830	0.673381
$470$	0	0
$471$	−11.7841	−0.542984
$472$	0	0
$473$	9.87451	0.454030
$474$	0	0
$475$	−3.91044	−0.179423
$476$	0	0
$477$	28.8340	1.32022
$478$	0	0
$479$	−16.7816	−0.766773	−0.383386	−	0.923588i	$-0.625242\pi$
$479$	−16.7816	−0.766773	−0.383386	+	0.923588i	$0.625242\pi$
$480$	0	0
$481$	1.29150	0.0588875
$482$	0	0
$483$	−27.9166	−1.27025
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	3.06871	0.139057	0.0695283	−	0.997580i	$-0.477851\pi$
$487$	3.06871	0.139057	0.0695283	+	0.997580i	$0.477851\pi$
$488$	0	0
$489$	−8.25098	−0.373122
$490$	0	0
$491$	−32.9669	−1.48778	−0.743888	−	0.668304i	$-0.767021\pi$
$491$	−32.9669	−1.48778	−0.743888	+	0.668304i	$0.767021\pi$
$492$	0	0
$493$	−31.7490	−1.42990
$494$	0	0
$495$	8.96077	0.402757
$496$	0	0
$497$	−55.7490	−2.50069
$498$	0	0
$499$	−7.27733	−0.325778	−0.162889	−	0.986644i	$-0.552081\pi$
$499$	−7.27733	−0.325778	−0.162889	+	0.986644i	$0.552081\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	25.3914	1.13215	0.566074	−	0.824355i	$-0.308462\pi$
$503$	25.3914	1.13215	0.566074	+	0.824355i	$0.308462\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−0.841723	−0.0373822
$508$	0	0
$509$	−40.3320	−1.78769	−0.893843	−	0.448381i	$-0.852001\pi$
$509$	−40.3320	−1.78769	−0.893843	+	0.448381i	$0.852001\pi$
$510$	0	0
$511$	−44.1547	−1.95329
$512$	0	0
$513$	−17.4170	−0.768979
$514$	0	0
$515$	14.8000	0.652167
$516$	0	0
$517$	18.5830	0.817280
$518$	0	0
$519$	18.5179	0.812846
$520$	0	0
$521$	−18.4575	−0.808638	−0.404319	−	0.914618i	$-0.632491\pi$
$521$	−18.4575	−0.808638	−0.404319	+	0.914618i	$0.632491\pi$
$522$	0	0
$523$	21.5338	0.941607	0.470804	−	0.882238i	$-0.343964\pi$
$523$	21.5338	0.941607	0.470804	+	0.882238i	$0.343964\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	33.5633	1.46204
$528$	0	0
$529$	25.7085	1.11776
$530$	0	0
$531$	−16.6760	−0.723677
$532$	0	0
$533$	−8.58301	−0.371771
$534$	0	0
$535$	10.3460	0.447298
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−60.9363	−2.62471
$540$	0	0
$541$	−15.4170	−0.662828	−0.331414	−	0.943485i	$-0.607526\pi$
$541$	−15.4170	−0.662828	−0.331414	+	0.943485i	$0.607526\pi$
$542$	0	0
$543$	4.45398	0.191139
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	41.6295	1.77995	0.889975	−	0.456010i	$-0.150722\pi$
$547$	41.6295	1.77995	0.889975	+	0.456010i	$0.150722\pi$
$548$	0	0
$549$	30.4575	1.29989
$550$	0	0
$551$	20.6921	0.881512
$552$	0	0
$553$	−37.1660	−1.58046
$554$	0	0
$555$	−1.08709	−0.0461443
$556$	0	0
$557$	−6.70850	−0.284248	−0.142124	−	0.989849i	$-0.545393\pi$
$557$	−6.70850	−0.284248	−0.142124	+	0.989849i	$0.545393\pi$
$558$	0	0
$559$	−2.52517	−0.106803
$560$	0	0
$561$	19.7490	0.833804
$562$	0	0
$563$	−5.89206	−0.248321	−0.124160	−	0.992262i	$-0.539624\pi$
$563$	−5.89206	−0.248321	−0.124160	+	0.992262i	$0.539624\pi$
$564$	0	0
$565$	12.5830	0.529371
$566$	0	0
$567$	−14.8528	−0.623760
$568$	0	0
$569$	30.4575	1.27685	0.638423	−	0.769686i	$-0.279587\pi$
$569$	30.4575	1.27685	0.638423	+	0.769686i	$0.279587\pi$
$570$	0	0
$571$	−20.6921	−0.865936	−0.432968	−	0.901409i	$-0.642534\pi$
$571$	−20.6921	−0.865936	−0.432968	+	0.901409i	$0.642534\pi$
$572$	0	0
$573$	−10.8340	−0.452596
$574$	0	0
$575$	−6.97915	−0.291051
$576$	0	0
$577$	33.2915	1.38594	0.692972	−	0.720965i	$-0.256301\pi$
$577$	33.2915	1.38594	0.692972	+	0.720965i	$0.256301\pi$
$578$	0	0
$579$	1.68345	0.0699616
$580$	0	0
$581$	22.5830	0.936901
$582$	0	0
$583$	49.2050	2.03786
$584$	0	0
$585$	−2.29150	−0.0947420
$586$	0	0
$587$	0.298179	0.0123072	0.00615359	−	0.999981i	$-0.498041\pi$
$587$	0.298179	0.0123072	0.00615359	+	0.999981i	$0.498041\pi$
$588$	0	0
$589$	−21.8745	−0.901323
$590$	0	0
$591$	1.68345	0.0692477
$592$	0	0
$593$	−23.1660	−0.951314	−0.475657	−	0.879631i	$-0.657790\pi$
$593$	−23.1660	−0.951314	−0.475657	+	0.879631i	$0.657790\pi$
$594$	0	0
$595$	−28.5129	−1.16892
$596$	0	0
$597$	−10.3320	−0.422861
$598$	0	0
$599$	−14.5547	−0.594687	−0.297344	−	0.954771i	$-0.596101\pi$
$599$	−14.5547	−0.594687	−0.297344	+	0.954771i	$0.596101\pi$
$600$	0	0
$601$	40.5830	1.65542	0.827708	−	0.561160i	$-0.189645\pi$
$601$	40.5830	1.65542	0.827708	+	0.561160i	$0.189645\pi$
$602$	0	0
$603$	7.03196	0.286364
$604$	0	0
$605$	4.29150	0.174474
$606$	0	0
$607$	−32.1252	−1.30392	−0.651961	−	0.758253i	$-0.726053\pi$
$607$	−32.1252	−1.30392	−0.651961	+	0.758253i	$0.726053\pi$
$608$	0	0
$609$	−21.1660	−0.857690
$610$	0	0
$611$	−4.75216	−0.192252
$612$	0	0
$613$	−7.16601	−0.289432	−0.144716	−	0.989473i	$-0.546227\pi$
$613$	−7.16601	−0.289432	−0.144716	+	0.989473i	$0.546227\pi$
$614$	0	0
$615$	7.22451	0.291320
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−34.1068	−1.37087	−0.685435	−	0.728134i	$-0.740388\pi$
$619$	−34.1068	−1.37087	−0.685435	+	0.728134i	$0.740388\pi$
$620$	0	0
$621$	−31.0850	−1.24740
$622$	0	0
$623$	−78.8051	−3.15726
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.8712	−0.514027
$628$	0	0
$629$	7.74902	0.308973
$630$	0	0
$631$	29.0565	1.15672	0.578360	−	0.815781i	$-0.303693\pi$
$631$	29.0565	1.15672	0.578360	+	0.815781i	$0.303693\pi$
$632$	0	0
$633$	−10.8340	−0.430612
$634$	0	0
$635$	−14.8000	−0.587321
$636$	0	0
$637$	15.5830	0.617421
$638$	0	0
$639$	−26.8823	−1.06345
$640$	0	0
$641$	37.7490	1.49100	0.745498	−	0.666508i	$-0.232212\pi$
$641$	37.7490	1.49100	0.745498	+	0.666508i	$0.232212\pi$
$642$	0	0
$643$	17.6234	0.694998	0.347499	−	0.937680i	$-0.387031\pi$
$643$	17.6234	0.694998	0.347499	+	0.937680i	$0.387031\pi$
$644$	0	0
$645$	2.12549	0.0836912
$646$	0	0
$647$	7.57551	0.297824	0.148912	−	0.988850i	$-0.452423\pi$
$647$	7.57551	0.297824	0.148912	+	0.988850i	$0.452423\pi$
$648$	0	0
$649$	−28.4575	−1.11706
$650$	0	0
$651$	22.3755	0.876966
$652$	0	0
$653$	−32.5830	−1.27507	−0.637536	−	0.770421i	$-0.720046\pi$
$653$	−32.5830	−1.27507	−0.637536	+	0.770421i	$0.720046\pi$
$654$	0	0
$655$	12.8712	0.502920
$656$	0	0
$657$	−21.2915	−0.830661
$658$	0	0
$659$	−17.9215	−0.698124	−0.349062	−	0.937100i	$-0.613500\pi$
$659$	−17.9215	−0.698124	−0.349062	+	0.937100i	$0.613500\pi$
$660$	0	0
$661$	27.1660	1.05664	0.528318	−	0.849047i	$-0.322823\pi$
$661$	27.1660	1.05664	0.528318	+	0.849047i	$0.322823\pi$
$662$	0	0
$663$	−5.05034	−0.196139
$664$	0	0
$665$	18.5830	0.720618
$666$	0	0
$667$	36.9302	1.42994
$668$	0	0
$669$	14.8340	0.573515
$670$	0	0
$671$	51.9756	2.00649
$672$	0	0
$673$	−15.1660	−0.584607	−0.292303	−	0.956326i	$-0.594422\pi$
$673$	−15.1660	−0.584607	−0.292303	+	0.956326i	$0.594422\pi$
$674$	0	0
$675$	4.45398	0.171434
$676$	0	0
$677$	3.41699	0.131326	0.0656629	−	0.997842i	$-0.479084\pi$
$677$	3.41699	0.131326	0.0656629	+	0.997842i	$0.479084\pi$
$678$	0	0
$679$	9.50432	0.364742
$680$	0	0
$681$	5.41699	0.207580
$682$	0	0
$683$	37.7191	1.44328	0.721640	−	0.692268i	$-0.243388\pi$
$683$	37.7191	1.44328	0.721640	+	0.692268i	$0.243388\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	3.85762	0.147177
$688$	0	0
$689$	−12.5830	−0.479374
$690$	0	0
$691$	−8.36441	−0.318197	−0.159099	−	0.987263i	$-0.550859\pi$
$691$	−8.36441	−0.318197	−0.159099	+	0.987263i	$0.550859\pi$
$692$	0	0
$693$	−42.5830	−1.61759
$694$	0	0
$695$	7.82087	0.296663
$696$	0	0
$697$	−51.4980	−1.95063
$698$	0	0
$699$	−8.41723	−0.318369
$700$	0	0
$701$	11.4170	0.431214	0.215607	−	0.976480i	$-0.430827\pi$
$701$	11.4170	0.431214	0.215607	+	0.976480i	$0.430827\pi$
$702$	0	0
$703$	−5.05034	−0.190477
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	28.5129	1.07234
$708$	0	0
$709$	19.4170	0.729221	0.364610	−	0.931160i	$-0.381202\pi$
$709$	19.4170	0.729221	0.364610	+	0.931160i	$0.381202\pi$
$710$	0	0
$711$	−17.9215	−0.672110
$712$	0	0
$713$	−39.0405	−1.46208
$714$	0	0
$715$	−3.91044	−0.146242
$716$	0	0
$717$	−6.12549	−0.228761
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−70.3320	−2.61930
$722$	0	0
$723$	13.9583	0.519115
$724$	0	0
$725$	−5.29150	−0.196521
$726$	0	0
$727$	29.3547	1.08870	0.544352	−	0.838857i	$-0.316776\pi$
$727$	29.3547	1.08870	0.544352	+	0.838857i	$0.316776\pi$
$728$	0	0
$729$	4.08497	0.151295
$730$	0	0
$731$	−15.1510	−0.560380
$732$	0	0
$733$	0.833990	0.0308041	0.0154021	−	0.999881i	$-0.495097\pi$
$733$	0.833990	0.0308041	0.0154021	+	0.999881i	$0.495097\pi$
$734$	0	0
$735$	−13.1166	−0.483812
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	6.68097	0.245763	0.122882	−	0.992421i	$-0.460786\pi$
$739$	6.68097	0.245763	0.122882	+	0.992421i	$0.460786\pi$
$740$	0	0
$741$	3.29150	0.120916
$742$	0	0
$743$	1.38527	0.0508205	0.0254102	−	0.999677i	$-0.491911\pi$
$743$	1.38527	0.0508205	0.0254102	+	0.999677i	$0.491911\pi$
$744$	0	0
$745$	12.5830	0.461006
$746$	0	0
$747$	10.8896	0.398429
$748$	0	0
$749$	−49.1660	−1.79649
$750$	0	0
$751$	29.6000	1.08012	0.540060	−	0.841626i	$-0.318401\pi$
$751$	29.6000	1.08012	0.540060	+	0.841626i	$0.318401\pi$
$752$	0	0
$753$	8.91503	0.324882
$754$	0	0
$755$	0.543544	0.0197816
$756$	0	0
$757$	−43.1660	−1.56890	−0.784448	−	0.620195i	$-0.787053\pi$
$757$	−43.1660	−1.56890	−0.784448	+	0.620195i	$0.787053\pi$
$758$	0	0
$759$	−22.9719	−0.833826
$760$	0	0
$761$	−35.1660	−1.27477	−0.637383	−	0.770547i	$-0.719983\pi$
$761$	−35.1660	−1.27477	−0.637383	+	0.770547i	$0.719983\pi$
$762$	0	0
$763$	9.50432	0.344079
$764$	0	0
$765$	−13.7490	−0.497097
$766$	0	0
$767$	7.27733	0.262769
$768$	0	0
$769$	−0.833990	−0.0300744	−0.0150372	−	0.999887i	$-0.504787\pi$
$769$	−0.833990	−0.0300744	−0.0150372	+	0.999887i	$0.504787\pi$
$770$	0	0
$771$	3.85762	0.138929
$772$	0	0
$773$	−22.7085	−0.816768	−0.408384	−	0.912810i	$-0.633908\pi$
$773$	−22.7085	−0.816768	−0.408384	+	0.912810i	$0.633908\pi$
$774$	0	0
$775$	5.59388	0.200938
$776$	0	0
$777$	5.16601	0.185330
$778$	0	0
$779$	33.5633	1.20253
$780$	0	0
$781$	−45.8745	−1.64152
$782$	0	0
$783$	−23.5682	−0.842260
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−38.8062	−1.38329	−0.691645	−	0.722237i	$-0.743114\pi$
$787$	−38.8062	−1.38329	−0.691645	+	0.722237i	$0.743114\pi$
$788$	0	0
$789$	−5.87451	−0.209138
$790$	0	0
$791$	−59.7964	−2.12612
$792$	0	0
$793$	−13.2915	−0.471995
$794$	0	0
$795$	10.5914	0.375638
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−28.5129	−1.00872
$800$	0	0
$801$	−38.0000	−1.34266
$802$	0	0
$803$	−36.3338	−1.28219
$804$	0	0
$805$	33.1660	1.16895
$806$	0	0
$807$	−20.6921	−0.728396
$808$	0	0
$809$	−14.7085	−0.517123	−0.258562	−	0.965995i	$-0.583249\pi$
$809$	−14.7085	−0.517123	−0.258562	+	0.965995i	$0.583249\pi$
$810$	0	0
$811$	−27.3730	−0.961198	−0.480599	−	0.876941i	$-0.659581\pi$
$811$	−27.3730	−0.961198	−0.480599	+	0.876941i	$0.659581\pi$
$812$	0	0
$813$	20.2065	0.708674
$814$	0	0
$815$	9.80250	0.343366
$816$	0	0
$817$	9.87451	0.345465
$818$	0	0
$819$	10.8896	0.380513
$820$	0	0
$821$	25.7490	0.898647	0.449323	−	0.893369i	$-0.351665\pi$
$821$	25.7490	0.898647	0.449323	+	0.893369i	$0.351665\pi$
$822$	0	0
$823$	11.4331	0.398534	0.199267	−	0.979945i	$-0.436144\pi$
$823$	11.4331	0.398534	0.199267	+	0.979945i	$0.436144\pi$
$824$	0	0
$825$	3.29150	0.114595
$826$	0	0
$827$	−0.788908	−0.0274330	−0.0137165	−	0.999906i	$-0.504366\pi$
$827$	−0.788908	−0.0274330	−0.0137165	+	0.999906i	$0.504366\pi$
$828$	0	0
$829$	−21.2915	−0.739484	−0.369742	−	0.929134i	$-0.620554\pi$
$829$	−21.2915	−0.739484	−0.369742	+	0.929134i	$0.620554\pi$
$830$	0	0
$831$	−8.41723	−0.291990
$832$	0	0
$833$	93.4980	3.23951
$834$	0	0
$835$	−23.7608	−0.822276
$836$	0	0
$837$	24.9150	0.861189
$838$	0	0
$839$	−0.543544	−0.0187652	−0.00938261	−	0.999956i	$-0.502987\pi$
$839$	−0.543544	−0.0187652	−0.00938261	+	0.999956i	$0.502987\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	0.490729	0.0169016
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−20.3939	−0.700743
$848$	0	0
$849$	18.1255	0.622065
$850$	0	0
$851$	−9.01359	−0.308982
$852$	0	0
$853$	−21.2915	−0.729007	−0.364504	−	0.931202i	$-0.618761\pi$
$853$	−21.2915	−0.729007	−0.364504	+	0.931202i	$0.618761\pi$
$854$	0	0
$855$	8.96077	0.306452
$856$	0	0
$857$	24.5830	0.839739	0.419870	−	0.907584i	$-0.362076\pi$
$857$	24.5830	0.839739	0.419870	+	0.907584i	$0.362076\pi$
$858$	0	0
$859$	16.7288	0.570780	0.285390	−	0.958411i	$-0.407877\pi$
$859$	16.7288	0.570780	0.285390	+	0.958411i	$0.407877\pi$
$860$	0	0
$861$	−34.3320	−1.17003
$862$	0	0
$863$	−20.9903	−0.714517	−0.357258	−	0.934006i	$-0.616288\pi$
$863$	−20.9903	−0.714517	−0.357258	+	0.934006i	$0.616288\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	0	0
$867$	−15.9927	−0.543142
$868$	0	0
$869$	−30.5830	−1.03746
$870$	0	0
$871$	−3.06871	−0.103979
$872$	0	0
$873$	4.58301	0.155111
$874$	0	0
$875$	−4.75216	−0.160652
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	0	0
$879$	0.105630	0.00356280
$880$	0	0
$881$	−34.4575	−1.16090	−0.580452	−	0.814295i	$-0.697124\pi$
$881$	−34.4575	−1.16090	−0.580452	+	0.814295i	$0.697124\pi$
$882$	0	0
$883$	51.1338	1.72079	0.860395	−	0.509627i	$-0.170217\pi$
$883$	51.1338	1.72079	0.860395	+	0.509627i	$0.170217\pi$
$884$	0	0
$885$	−6.12549	−0.205906
$886$	0	0
$887$	34.4050	1.15521	0.577604	−	0.816317i	$-0.303988\pi$
$887$	34.4050	1.15521	0.577604	+	0.816317i	$0.303988\pi$
$888$	0	0
$889$	70.3320	2.35886
$890$	0	0
$891$	−12.2220	−0.409454
$892$	0	0
$893$	18.5830	0.621857
$894$	0	0
$895$	19.0086	0.635388
$896$	0	0
$897$	5.87451	0.196144
$898$	0	0
$899$	−29.6000	−0.987216
$900$	0	0
$901$	−75.4980	−2.51520
$902$	0	0
$903$	−10.1007	−0.336129
$904$	0	0
$905$	−5.29150	−0.175895
$906$	0	0
$907$	−38.2626	−1.27049	−0.635245	−	0.772311i	$-0.719101\pi$
$907$	−38.2626	−1.27049	−0.635245	+	0.772311i	$0.719101\pi$
$908$	0	0
$909$	13.7490	0.456026
$910$	0	0
$911$	−9.50432	−0.314892	−0.157446	−	0.987528i	$-0.550326\pi$
$911$	−9.50432	−0.314892	−0.157446	+	0.987528i	$0.550326\pi$
$912$	0	0
$913$	18.5830	0.615008
$914$	0	0
$915$	11.1878	0.369856
$916$	0	0
$917$	−61.1660	−2.01988
$918$	0	0
$919$	−20.0957	−0.662897	−0.331448	−	0.943473i	$-0.607537\pi$
$919$	−20.0957	−0.662897	−0.331448	+	0.943473i	$0.607537\pi$
$920$	0	0
$921$	6.33202	0.208647
$922$	0	0
$923$	11.7313	0.386141
$924$	0	0
$925$	1.29150	0.0424643
$926$	0	0
$927$	−33.9143	−1.11389
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	−60.9363	−1.99711
$932$	0	0
$933$	−20.2510	−0.662987
$934$	0	0
$935$	−23.4626	−0.767309
$936$	0	0
$937$	31.1660	1.01815	0.509075	−	0.860722i	$-0.329988\pi$
$937$	31.1660	1.01815	0.509075	+	0.860722i	$0.329988\pi$
$938$	0	0
$939$	1.68345	0.0549372
$940$	0	0
$941$	−49.7490	−1.62177	−0.810886	−	0.585204i	$-0.801014\pi$
$941$	−49.7490	−1.62177	−0.810886	+	0.585204i	$0.801014\pi$
$942$	0	0
$943$	59.9021	1.95068
$944$	0	0
$945$	−21.1660	−0.688530
$946$	0	0
$947$	37.2284	1.20976	0.604880	−	0.796317i	$-0.293221\pi$
$947$	37.2284	1.20976	0.604880	+	0.796317i	$0.293221\pi$
$948$	0	0
$949$	9.29150	0.301615
$950$	0	0
$951$	11.1878	0.362788
$952$	0	0
$953$	−25.7490	−0.834092	−0.417046	−	0.908885i	$-0.636935\pi$
$953$	−25.7490	−0.834092	−0.417046	+	0.908885i	$0.636935\pi$
$954$	0	0
$955$	12.8712	0.416502
$956$	0	0
$957$	−17.4170	−0.563011
$958$	0	0
$959$	−28.5129	−0.920731
$960$	0	0
$961$	0.291503	0.00940331
$962$	0	0
$963$	−23.7080	−0.763979
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−38.3154	−1.23214	−0.616071	−	0.787691i	$-0.711276\pi$
$967$	−38.3154	−1.23214	−0.616071	+	0.787691i	$0.711276\pi$
$968$	0	0
$969$	19.7490	0.634430
$970$	0	0
$971$	49.6958	1.59481	0.797406	−	0.603443i	$-0.206205\pi$
$971$	49.6958	1.59481	0.797406	+	0.603443i	$0.206205\pi$
$972$	0	0
$973$	−37.1660	−1.19149
$974$	0	0
$975$	−0.841723	−0.0269567
$976$	0	0
$977$	−1.54249	−0.0493485	−0.0246743	−	0.999696i	$-0.507855\pi$
$977$	−1.54249	−0.0493485	−0.0246743	+	0.999696i	$0.507855\pi$
$978$	0	0
$979$	−64.8468	−2.07251
$980$	0	0
$981$	4.58301	0.146324
$982$	0	0
$983$	−34.3522	−1.09566	−0.547832	−	0.836588i	$-0.684547\pi$
$983$	−34.3522	−1.09566	−0.547832	+	0.836588i	$0.684547\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−19.0086	−0.605051
$988$	0	0
$989$	17.6235	0.560395
$990$	0	0
$991$	−59.7964	−1.89950	−0.949748	−	0.313015i	$-0.898661\pi$
$991$	−59.7964	−1.89950	−0.949748	+	0.313015i	$0.898661\pi$
$992$	0	0
$993$	18.3765	0.583160
$994$	0	0
$995$	12.2748	0.389139
$996$	0	0
$997$	40.5830	1.28528	0.642638	−	0.766170i	$-0.277840\pi$
$997$	40.5830	1.28528	0.642638	+	0.766170i	$0.277840\pi$
$998$	0	0
$999$	5.75233	0.181996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bv.1.2		4
4.3	odd	2	inner	4160.2.a.bv.1.3		4
8.3	odd	2		2080.2.a.r.1.2	✓	4
8.5	even	2		2080.2.a.r.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.r.1.2	✓	4	8.3	odd	2
2080.2.a.r.1.3	yes	4	8.5	even	2
4160.2.a.bv.1.2		4	1.1	even	1	trivial
4160.2.a.bv.1.3		4	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q-0.841723 q^{3} +1.00000 q^{5} -4.75216 q^{7} -2.29150 q^{9} +O(q^{10})\)	\(q-0.841723 q^{3} +1.00000 q^{5} -4.75216 q^{7} -2.29150 q^{9} -3.91044 q^{11} +1.00000 q^{13} -0.841723 q^{15} +6.00000 q^{17} -3.91044 q^{19} +4.00000 q^{21} -6.97915 q^{23} +1.00000 q^{25} +4.45398 q^{27} -5.29150 q^{29} +5.59388 q^{31} +3.29150 q^{33} -4.75216 q^{35} +1.29150 q^{37} -0.841723 q^{39} -8.58301 q^{41} -2.52517 q^{43} -2.29150 q^{45} -4.75216 q^{47} +15.5830 q^{49} -5.05034 q^{51} -12.5830 q^{53} -3.91044 q^{55} +3.29150 q^{57} +7.27733 q^{59} -13.2915 q^{61} +10.8896 q^{63} +1.00000 q^{65} -3.06871 q^{67} +5.87451 q^{69} +11.7313 q^{71} +9.29150 q^{73} -0.841723 q^{75} +18.5830 q^{77} +7.82087 q^{79} +3.12549 q^{81} -4.75216 q^{83} +6.00000 q^{85} +4.45398 q^{87} +16.5830 q^{89} -4.75216 q^{91} -4.70850 q^{93} -3.91044 q^{95} -2.00000 q^{97} +8.96077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 12 * q^9	\( 4 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{17} + 16 q^{21} + 4 q^{25} - 8 q^{33} - 16 q^{37} + 8 q^{41} + 12 q^{45} + 20 q^{49} - 8 q^{53} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 40 q^{69} + 16 q^{73} + 32 q^{77} + 76 q^{81} + 24 q^{85} + 24 q^{89} - 40 q^{93} - 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^17 + 16 * q^21 + 4 * q^25 - 8 * q^33 - 16 * q^37 + 8 * q^41 + 12 * q^45 + 20 * q^49 - 8 * q^53 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 40 * q^69 + 16 * q^73 + 32 * q^77 + 76 * q^81 + 24 * q^85 + 24 * q^89 - 40 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more