LMFDB - Embedded newform 6080.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +O(q^{10})$	$q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +4.15633 q^{11} +2.06300 q^{13} +0.481194 q^{15} -6.31265 q^{17} -1.00000 q^{19} -1.03761 q^{21} +3.76845 q^{23} +1.00000 q^{25} -2.77575 q^{27} -1.03761 q^{29} -3.61213 q^{31} +2.00000 q^{33} -2.15633 q^{35} +4.89938 q^{37} +0.992706 q^{39} -3.58181 q^{43} -2.76845 q^{45} +4.54420 q^{47} -2.35026 q^{49} -3.03761 q^{51} -2.45088 q^{53} +4.15633 q^{55} -0.481194 q^{57} -12.5745 q^{59} +0.156325 q^{61} +5.96968 q^{63} +2.06300 q^{65} -11.4436 q^{67} +1.81336 q^{69} -4.38787 q^{71} +9.66291 q^{73} +0.481194 q^{75} -8.96239 q^{77} -6.12601 q^{79} +6.96968 q^{81} +3.89446 q^{83} -6.31265 q^{85} -0.499293 q^{87} +5.03761 q^{89} -4.44851 q^{91} -1.73813 q^{93} -1.00000 q^{95} -1.48849 q^{97} -11.5066 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9	$ 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 3 q^{19} - 14 q^{21} + 3 q^{25} - 10 q^{27} - 14 q^{29} - 10 q^{31} + 6 q^{33} + 4 q^{35} + 8 q^{37} - 10 q^{39} - 12 q^{43} + 3 q^{45} + 4 q^{47} + 3 q^{49} - 20 q^{51} - 4 q^{53} + 2 q^{55} + 4 q^{57} - 26 q^{59} - 10 q^{61} + 20 q^{63} + 2 q^{65} - 18 q^{67} + 18 q^{69} - 14 q^{71} - 2 q^{73} - 4 q^{75} - 16 q^{77} - 10 q^{79} + 23 q^{81} - 8 q^{83} + 2 q^{85} + 32 q^{87} + 26 q^{89} - 10 q^{91} + 4 q^{93} - 3 q^{95} - 12 q^{97} - 14 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 - 3 * q^19 - 14 * q^21 + 3 * q^25 - 10 * q^27 - 14 * q^29 - 10 * q^31 + 6 * q^33 + 4 * q^35 + 8 * q^37 - 10 * q^39 - 12 * q^43 + 3 * q^45 + 4 * q^47 + 3 * q^49 - 20 * q^51 - 4 * q^53 + 2 * q^55 + 4 * q^57 - 26 * q^59 - 10 * q^61 + 20 * q^63 + 2 * q^65 - 18 * q^67 + 18 * q^69 - 14 * q^71 - 2 * q^73 - 4 * q^75 - 16 * q^77 - 10 * q^79 + 23 * q^81 - 8 * q^83 + 2 * q^85 + 32 * q^87 + 26 * q^89 - 10 * q^91 + 4 * q^93 - 3 * q^95 - 12 * q^97 - 14 * q^99

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
$3$	0.481194	0.277818	0.138909	−	0.990305i	$-0.455641\pi$
$3$	0.481194	0.277818	0.138909	+	0.990305i	$0.455641\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.15633	−0.815014	−0.407507	−	0.913202i	$-0.633602\pi$
$7$	−2.15633	−0.815014	−0.407507	+	0.913202i	$0.633602\pi$
$8$	0	0
$9$	−2.76845	−0.922817
$10$	0	0
$11$	4.15633	1.25318	0.626590	−	0.779349i	$-0.284450\pi$
$11$	4.15633	1.25318	0.626590	+	0.779349i	$0.284450\pi$
$12$	0	0
$13$	2.06300	0.572174	0.286087	−	0.958204i	$-0.407645\pi$
$13$	2.06300	0.572174	0.286087	+	0.958204i	$0.407645\pi$
$14$	0	0
$15$	0.481194	0.124244
$16$	0	0
$17$	−6.31265	−1.53104	−0.765521	−	0.643411i	$-0.777519\pi$
$17$	−6.31265	−1.53104	−0.765521	+	0.643411i	$0.777519\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.03761	−0.226425
$22$	0	0
$23$	3.76845	0.785777	0.392888	−	0.919586i	$-0.371476\pi$
$23$	3.76845	0.785777	0.392888	+	0.919586i	$0.371476\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.77575	−0.534193
$28$	0	0
$29$	−1.03761	−0.192680	−0.0963398	−	0.995349i	$-0.530714\pi$
$29$	−1.03761	−0.192680	−0.0963398	+	0.995349i	$0.530714\pi$
$30$	0	0
$31$	−3.61213	−0.648757	−0.324379	−	0.945927i	$-0.605155\pi$
$31$	−3.61213	−0.648757	−0.324379	+	0.945927i	$0.605155\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−2.15633	−0.364485
$36$	0	0
$37$	4.89938	0.805454	0.402727	−	0.915320i	$-0.368062\pi$
$37$	4.89938	0.805454	0.402727	+	0.915320i	$0.368062\pi$
$38$	0	0
$39$	0.992706	0.158960
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.58181	−0.546221	−0.273110	−	0.961983i	$-0.588052\pi$
$43$	−3.58181	−0.546221	−0.273110	+	0.961983i	$0.588052\pi$
$44$	0	0
$45$	−2.76845	−0.412696
$46$	0	0
$47$	4.54420	0.662839	0.331420	−	0.943483i	$-0.392472\pi$
$47$	4.54420	0.662839	0.331420	+	0.943483i	$0.392472\pi$
$48$	0	0
$49$	−2.35026	−0.335752
$50$	0	0
$51$	−3.03761	−0.425351
$52$	0	0
$53$	−2.45088	−0.336654	−0.168327	−	0.985731i	$-0.553836\pi$
$53$	−2.45088	−0.336654	−0.168327	+	0.985731i	$0.553836\pi$
$54$	0	0
$55$	4.15633	0.560439
$56$	0	0
$57$	−0.481194	−0.0637357
$58$	0	0
$59$	−12.5745	−1.63706	−0.818531	−	0.574462i	$-0.805211\pi$
$59$	−12.5745	−1.63706	−0.818531	+	0.574462i	$0.805211\pi$
$60$	0	0
$61$	0.156325	0.0200154	0.0100077	−	0.999950i	$-0.496814\pi$
$61$	0.156325	0.0200154	0.0100077	+	0.999950i	$0.496814\pi$
$62$	0	0
$63$	5.96968	0.752109
$64$	0	0
$65$	2.06300	0.255884
$66$	0	0
$67$	−11.4436	−1.39806	−0.699028	−	0.715094i	$-0.746384\pi$
$67$	−11.4436	−1.39806	−0.699028	+	0.715094i	$0.746384\pi$
$68$	0	0
$69$	1.81336	0.218303
$70$	0	0
$71$	−4.38787	−0.520745	−0.260372	−	0.965508i	$-0.583845\pi$
$71$	−4.38787	−0.520745	−0.260372	+	0.965508i	$0.583845\pi$
$72$	0	0
$73$	9.66291	1.13096	0.565479	−	0.824763i	$-0.308691\pi$
$73$	9.66291	1.13096	0.565479	+	0.824763i	$0.308691\pi$
$74$	0	0
$75$	0.481194	0.0555635
$76$	0	0
$77$	−8.96239	−1.02136
$78$	0	0
$79$	−6.12601	−0.689230	−0.344615	−	0.938744i	$-0.611990\pi$
$79$	−6.12601	−0.689230	−0.344615	+	0.938744i	$0.611990\pi$
$80$	0	0
$81$	6.96968	0.774409
$82$	0	0
$83$	3.89446	0.427473	0.213736	−	0.976891i	$-0.431437\pi$
$83$	3.89446	0.427473	0.213736	+	0.976891i	$0.431437\pi$
$84$	0	0
$85$	−6.31265	−0.684703
$86$	0	0
$87$	−0.499293	−0.0535298
$88$	0	0
$89$	5.03761	0.533986	0.266993	−	0.963699i	$-0.413970\pi$
$89$	5.03761	0.533986	0.266993	+	0.963699i	$0.413970\pi$
$90$	0	0
$91$	−4.44851	−0.466330
$92$	0	0
$93$	−1.73813	−0.180236
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−1.48849	−0.151133	−0.0755666	−	0.997141i	$-0.524077\pi$
$97$	−1.48849	−0.151133	−0.0755666	+	0.997141i	$0.524077\pi$
$98$	0	0
$99$	−11.5066	−1.15646
$100$	0	0
$101$	12.2823	1.22214	0.611069	−	0.791577i	$-0.290740\pi$
$101$	12.2823	1.22214	0.611069	+	0.791577i	$0.290740\pi$
$102$	0	0
$103$	−10.4812	−1.03274	−0.516371	−	0.856365i	$-0.672718\pi$
$103$	−10.4812	−1.03274	−0.516371	+	0.856365i	$0.672718\pi$
$104$	0	0
$105$	−1.03761	−0.101261
$106$	0	0
$107$	−19.6302	−1.89773	−0.948863	−	0.315689i	$-0.897764\pi$
$107$	−19.6302	−1.89773	−0.948863	+	0.315689i	$0.897764\pi$
$108$	0	0
$109$	13.2750	1.27152	0.635759	−	0.771888i	$-0.280687\pi$
$109$	13.2750	1.27152	0.635759	+	0.771888i	$0.280687\pi$
$110$	0	0
$111$	2.35756	0.223769
$112$	0	0
$113$	−11.0254	−1.03718	−0.518591	−	0.855023i	$-0.673543\pi$
$113$	−11.0254	−1.03718	−0.518591	+	0.855023i	$0.673543\pi$
$114$	0	0
$115$	3.76845	0.351410
$116$	0	0
$117$	−5.71133	−0.528012
$118$	0	0
$119$	13.6121	1.24782
$120$	0	0
$121$	6.27504	0.570458
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.5696	−1.20411	−0.602053	−	0.798456i	$-0.705651\pi$
$127$	−13.5696	−1.20411	−0.602053	+	0.798456i	$0.705651\pi$
$128$	0	0
$129$	−1.72355	−0.151750
$130$	0	0
$131$	−0.962389	−0.0840843	−0.0420421	−	0.999116i	$-0.513386\pi$
$131$	−0.962389	−0.0840843	−0.0420421	+	0.999116i	$0.513386\pi$
$132$	0	0
$133$	2.15633	0.186977
$134$	0	0
$135$	−2.77575	−0.238898
$136$	0	0
$137$	−12.7005	−1.08508	−0.542539	−	0.840030i	$-0.682537\pi$
$137$	−12.7005	−1.08508	−0.542539	+	0.840030i	$0.682537\pi$
$138$	0	0
$139$	−4.93207	−0.418333	−0.209166	−	0.977880i	$-0.567075\pi$
$139$	−4.93207	−0.418333	−0.209166	+	0.977880i	$0.567075\pi$
$140$	0	0
$141$	2.18664	0.184149
$142$	0	0
$143$	8.57452	0.717037
$144$	0	0
$145$	−1.03761	−0.0861689
$146$	0	0
$147$	−1.13093	−0.0932777
$148$	0	0
$149$	−10.3938	−0.851489	−0.425745	−	0.904843i	$-0.639988\pi$
$149$	−10.3938	−0.851489	−0.425745	+	0.904843i	$0.639988\pi$
$150$	0	0
$151$	1.42548	0.116004	0.0580021	−	0.998316i	$-0.481527\pi$
$151$	1.42548	0.116004	0.0580021	+	0.998316i	$0.481527\pi$
$152$	0	0
$153$	17.4763	1.41287
$154$	0	0
$155$	−3.61213	−0.290133
$156$	0	0
$157$	−0.700523	−0.0559079	−0.0279539	−	0.999609i	$-0.508899\pi$
$157$	−0.700523	−0.0559079	−0.0279539	+	0.999609i	$0.508899\pi$
$158$	0	0
$159$	−1.17935	−0.0935284
$160$	0	0
$161$	−8.12601	−0.640419
$162$	0	0
$163$	−7.26916	−0.569365	−0.284682	−	0.958622i	$-0.591888\pi$
$163$	−7.26916	−0.569365	−0.284682	+	0.958622i	$0.591888\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	14.3453	1.11008	0.555038	−	0.831825i	$-0.312704\pi$
$167$	14.3453	1.11008	0.555038	+	0.831825i	$0.312704\pi$
$168$	0	0
$169$	−8.74401	−0.672616
$170$	0	0
$171$	2.76845	0.211709
$172$	0	0
$173$	−1.75035	−0.133077	−0.0665385	−	0.997784i	$-0.521196\pi$
$173$	−1.75035	−0.133077	−0.0665385	+	0.997784i	$0.521196\pi$
$174$	0	0
$175$	−2.15633	−0.163003
$176$	0	0
$177$	−6.05079	−0.454805
$178$	0	0
$179$	−5.87399	−0.439043	−0.219521	−	0.975608i	$-0.570450\pi$
$179$	−5.87399	−0.439043	−0.219521	+	0.975608i	$0.570450\pi$
$180$	0	0
$181$	12.7005	0.944022	0.472011	−	0.881593i	$-0.343528\pi$
$181$	12.7005	0.944022	0.472011	+	0.881593i	$0.343528\pi$
$182$	0	0
$183$	0.0752228	0.00556063
$184$	0	0
$185$	4.89938	0.360210
$186$	0	0
$187$	−26.2374	−1.91867
$188$	0	0
$189$	5.98541	0.435375
$190$	0	0
$191$	5.08840	0.368183	0.184092	−	0.982909i	$-0.441066\pi$
$191$	5.08840	0.368183	0.184092	+	0.982909i	$0.441066\pi$
$192$	0	0
$193$	−8.71274	−0.627157	−0.313578	−	0.949562i	$-0.601528\pi$
$193$	−8.71274	−0.627157	−0.313578	+	0.949562i	$0.601528\pi$
$194$	0	0
$195$	0.992706	0.0710891
$196$	0	0
$197$	−27.1246	−1.93255	−0.966274	−	0.257518i	$-0.917095\pi$
$197$	−27.1246	−1.93255	−0.966274	+	0.257518i	$0.917095\pi$
$198$	0	0
$199$	18.5501	1.31498	0.657490	−	0.753463i	$-0.271618\pi$
$199$	18.5501	1.31498	0.657490	+	0.753463i	$0.271618\pi$
$200$	0	0
$201$	−5.50659	−0.388405
$202$	0	0
$203$	2.23743	0.157037
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.4328	−0.725128
$208$	0	0
$209$	−4.15633	−0.287499
$210$	0	0
$211$	7.87399	0.542068	0.271034	−	0.962570i	$-0.412634\pi$
$211$	7.87399	0.542068	0.271034	+	0.962570i	$0.412634\pi$
$212$	0	0
$213$	−2.11142	−0.144672
$214$	0	0
$215$	−3.58181	−0.244277
$216$	0	0
$217$	7.78892	0.528746
$218$	0	0
$219$	4.64974	0.314200
$220$	0	0
$221$	−13.0230	−0.876023
$222$	0	0
$223$	−2.74306	−0.183689	−0.0918444	−	0.995773i	$-0.529276\pi$
$223$	−2.74306	−0.183689	−0.0918444	+	0.995773i	$0.529276\pi$
$224$	0	0
$225$	−2.76845	−0.184563
$226$	0	0
$227$	−8.24377	−0.547158	−0.273579	−	0.961850i	$-0.588208\pi$
$227$	−8.24377	−0.547158	−0.273579	+	0.961850i	$0.588208\pi$
$228$	0	0
$229$	−10.7308	−0.709114	−0.354557	−	0.935034i	$-0.615368\pi$
$229$	−10.7308	−0.709114	−0.354557	+	0.935034i	$0.615368\pi$
$230$	0	0
$231$	−4.31265	−0.283752
$232$	0	0
$233$	1.87399	0.122769	0.0613846	−	0.998114i	$-0.480448\pi$
$233$	1.87399	0.122769	0.0613846	+	0.998114i	$0.480448\pi$
$234$	0	0
$235$	4.54420	0.296431
$236$	0	0
$237$	−2.94780	−0.191480
$238$	0	0
$239$	−21.1246	−1.36644	−0.683218	−	0.730214i	$-0.739420\pi$
$239$	−21.1246	−1.36644	−0.683218	+	0.730214i	$0.739420\pi$
$240$	0	0
$241$	2.23743	0.144125	0.0720627	−	0.997400i	$-0.477042\pi$
$241$	2.23743	0.144125	0.0720627	+	0.997400i	$0.477042\pi$
$242$	0	0
$243$	11.6810	0.749337
$244$	0	0
$245$	−2.35026	−0.150153
$246$	0	0
$247$	−2.06300	−0.131266
$248$	0	0
$249$	1.87399	0.118759
$250$	0	0
$251$	−25.7137	−1.62303	−0.811517	−	0.584329i	$-0.801358\pi$
$251$	−25.7137	−1.62303	−0.811517	+	0.584329i	$0.801358\pi$
$252$	0	0
$253$	15.6629	0.984719
$254$	0	0
$255$	−3.03761	−0.190223
$256$	0	0
$257$	1.80114	0.112352	0.0561760	−	0.998421i	$-0.482109\pi$
$257$	1.80114	0.112352	0.0561760	+	0.998421i	$0.482109\pi$
$258$	0	0
$259$	−10.5647	−0.656456
$260$	0	0
$261$	2.87258	0.177808
$262$	0	0
$263$	10.5296	0.649284	0.324642	−	0.945837i	$-0.394756\pi$
$263$	10.5296	0.649284	0.324642	+	0.945837i	$0.394756\pi$
$264$	0	0
$265$	−2.45088	−0.150556
$266$	0	0
$267$	2.42407	0.148351
$268$	0	0
$269$	0.962389	0.0586779	0.0293389	−	0.999570i	$-0.490660\pi$
$269$	0.962389	0.0586779	0.0293389	+	0.999570i	$0.490660\pi$
$270$	0	0
$271$	−30.8568	−1.87442	−0.937210	−	0.348765i	$-0.886601\pi$
$271$	−30.8568	−1.87442	−0.937210	+	0.348765i	$0.886601\pi$
$272$	0	0
$273$	−2.14060	−0.129555
$274$	0	0
$275$	4.15633	0.250636
$276$	0	0
$277$	−7.40105	−0.444686	−0.222343	−	0.974969i	$-0.571370\pi$
$277$	−7.40105	−0.444686	−0.222343	+	0.974969i	$0.571370\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−2.44851	−0.146066	−0.0730329	−	0.997330i	$-0.523268\pi$
$281$	−2.44851	−0.146066	−0.0730329	+	0.997330i	$0.523268\pi$
$282$	0	0
$283$	−26.4445	−1.57196	−0.785982	−	0.618249i	$-0.787842\pi$
$283$	−26.4445	−1.57196	−0.785982	+	0.618249i	$0.787842\pi$
$284$	0	0
$285$	−0.481194	−0.0285035
$286$	0	0
$287$	0	0
$288$	0	0
$289$	22.8496	1.34409
$290$	0	0
$291$	−0.716252	−0.0419874
$292$	0	0
$293$	9.86177	0.576131	0.288065	−	0.957611i	$-0.406988\pi$
$293$	9.86177	0.576131	0.288065	+	0.957611i	$0.406988\pi$
$294$	0	0
$295$	−12.5745	−0.732117
$296$	0	0
$297$	−11.5369	−0.669439
$298$	0	0
$299$	7.77433	0.449601
$300$	0	0
$301$	7.72355	0.445178
$302$	0	0
$303$	5.91019	0.339531
$304$	0	0
$305$	0.156325	0.00895115
$306$	0	0
$307$	−0.153956	−0.00878670	−0.00439335	−	0.999990i	$-0.501398\pi$
$307$	−0.153956	−0.00878670	−0.00439335	+	0.999990i	$0.501398\pi$
$308$	0	0
$309$	−5.04349	−0.286914
$310$	0	0
$311$	−11.6326	−0.659624	−0.329812	−	0.944047i	$-0.606985\pi$
$311$	−11.6326	−0.659624	−0.329812	+	0.944047i	$0.606985\pi$
$312$	0	0
$313$	0.826531	0.0467183	0.0233592	−	0.999727i	$-0.492564\pi$
$313$	0.826531	0.0467183	0.0233592	+	0.999727i	$0.492564\pi$
$314$	0	0
$315$	5.96968	0.336354
$316$	0	0
$317$	18.1236	1.01792	0.508962	−	0.860789i	$-0.330029\pi$
$317$	18.1236	1.01792	0.508962	+	0.860789i	$0.330029\pi$
$318$	0	0
$319$	−4.31265	−0.241462
$320$	0	0
$321$	−9.44595	−0.527222
$322$	0	0
$323$	6.31265	0.351245
$324$	0	0
$325$	2.06300	0.114435
$326$	0	0
$327$	6.38787	0.353250
$328$	0	0
$329$	−9.79877	−0.540224
$330$	0	0
$331$	11.8134	0.649321	0.324660	−	0.945831i	$-0.394750\pi$
$331$	11.8134	0.649321	0.324660	+	0.945831i	$0.394750\pi$
$332$	0	0
$333$	−13.5637	−0.743287
$334$	0	0
$335$	−11.4436	−0.625230
$336$	0	0
$337$	35.1632	1.91546	0.957730	−	0.287670i	$-0.0928805\pi$
$337$	35.1632	1.91546	0.957730	+	0.287670i	$0.0928805\pi$
$338$	0	0
$339$	−5.30536	−0.288147
$340$	0	0
$341$	−15.0132	−0.813009
$342$	0	0
$343$	20.1622	1.08866
$344$	0	0
$345$	1.81336	0.0976279
$346$	0	0
$347$	−30.8423	−1.65570	−0.827850	−	0.560950i	$-0.810436\pi$
$347$	−30.8423	−1.65570	−0.827850	+	0.560950i	$0.810436\pi$
$348$	0	0
$349$	−9.87399	−0.528543	−0.264271	−	0.964448i	$-0.585131\pi$
$349$	−9.87399	−0.528543	−0.264271	+	0.964448i	$0.585131\pi$
$350$	0	0
$351$	−5.72638	−0.305651
$352$	0	0
$353$	−13.3865	−0.712489	−0.356245	−	0.934393i	$-0.615943\pi$
$353$	−13.3865	−0.712489	−0.356245	+	0.934393i	$0.615943\pi$
$354$	0	0
$355$	−4.38787	−0.232884
$356$	0	0
$357$	6.55008	0.346667
$358$	0	0
$359$	22.3331	1.17870	0.589348	−	0.807879i	$-0.299385\pi$
$359$	22.3331	1.17870	0.589348	+	0.807879i	$0.299385\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.01951	0.158483
$364$	0	0
$365$	9.66291	0.505780
$366$	0	0
$367$	2.86670	0.149640	0.0748202	−	0.997197i	$-0.476162\pi$
$367$	2.86670	0.149640	0.0748202	+	0.997197i	$0.476162\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.28489	0.274378
$372$	0	0
$373$	−31.2873	−1.61999	−0.809996	−	0.586435i	$-0.800531\pi$
$373$	−31.2873	−1.61999	−0.809996	+	0.586435i	$0.800531\pi$
$374$	0	0
$375$	0.481194	0.0248488
$376$	0	0
$377$	−2.14060	−0.110246
$378$	0	0
$379$	31.9511	1.64122	0.820610	−	0.571489i	$-0.193634\pi$
$379$	31.9511	1.64122	0.820610	+	0.571489i	$0.193634\pi$
$380$	0	0
$381$	−6.52961	−0.334522
$382$	0	0
$383$	−17.3684	−0.887482	−0.443741	−	0.896155i	$-0.646349\pi$
$383$	−17.3684	−0.887482	−0.443741	+	0.896155i	$0.646349\pi$
$384$	0	0
$385$	−8.96239	−0.456766
$386$	0	0
$387$	9.91607	0.504062
$388$	0	0
$389$	12.7250	0.645181	0.322591	−	0.946539i	$-0.395446\pi$
$389$	12.7250	0.645181	0.322591	+	0.946539i	$0.395446\pi$
$390$	0	0
$391$	−23.7889	−1.20306
$392$	0	0
$393$	−0.463096	−0.0233601
$394$	0	0
$395$	−6.12601	−0.308233
$396$	0	0
$397$	−15.2750	−0.766632	−0.383316	−	0.923617i	$-0.625218\pi$
$397$	−15.2750	−0.766632	−0.383316	+	0.923617i	$0.625218\pi$
$398$	0	0
$399$	1.03761	0.0519455
$400$	0	0
$401$	26.7875	1.33770	0.668852	−	0.743396i	$-0.266786\pi$
$401$	26.7875	1.33770	0.668852	+	0.743396i	$0.266786\pi$
$402$	0	0
$403$	−7.45183	−0.371202
$404$	0	0
$405$	6.96968	0.346326
$406$	0	0
$407$	20.3634	1.00938
$408$	0	0
$409$	−22.8872	−1.13170	−0.565849	−	0.824509i	$-0.691451\pi$
$409$	−22.8872	−1.13170	−0.565849	+	0.824509i	$0.691451\pi$
$410$	0	0
$411$	−6.11142	−0.301454
$412$	0	0
$413$	27.1147	1.33423
$414$	0	0
$415$	3.89446	0.191172
$416$	0	0
$417$	−2.37328	−0.116220
$418$	0	0
$419$	12.1260	0.592394	0.296197	−	0.955127i	$-0.404281\pi$
$419$	12.1260	0.592394	0.296197	+	0.955127i	$0.404281\pi$
$420$	0	0
$421$	21.7137	1.05826	0.529130	−	0.848541i	$-0.322518\pi$
$421$	21.7137	1.05826	0.529130	+	0.848541i	$0.322518\pi$
$422$	0	0
$423$	−12.5804	−0.611680
$424$	0	0
$425$	−6.31265	−0.306209
$426$	0	0
$427$	−0.337088	−0.0163128
$428$	0	0
$429$	4.12601	0.199206
$430$	0	0
$431$	−34.7367	−1.67321	−0.836604	−	0.547807i	$-0.815463\pi$
$431$	−34.7367	−1.67321	−0.836604	+	0.547807i	$0.815463\pi$
$432$	0	0
$433$	−6.77338	−0.325508	−0.162754	−	0.986667i	$-0.552038\pi$
$433$	−6.77338	−0.325508	−0.162754	+	0.986667i	$0.552038\pi$
$434$	0	0
$435$	−0.499293	−0.0239393
$436$	0	0
$437$	−3.76845	−0.180270
$438$	0	0
$439$	21.9149	1.04594	0.522971	−	0.852350i	$-0.324824\pi$
$439$	21.9149	1.04594	0.522971	+	0.852350i	$0.324824\pi$
$440$	0	0
$441$	6.50659	0.309837
$442$	0	0
$443$	−20.3938	−0.968936	−0.484468	−	0.874809i	$-0.660987\pi$
$443$	−20.3938	−0.968936	−0.484468	+	0.874809i	$0.660987\pi$
$444$	0	0
$445$	5.03761	0.238806
$446$	0	0
$447$	−5.00141	−0.236559
$448$	0	0
$449$	31.1998	1.47241	0.736205	−	0.676758i	$-0.236616\pi$
$449$	31.1998	1.47241	0.736205	+	0.676758i	$0.236616\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0.685935	0.0322280
$454$	0	0
$455$	−4.44851	−0.208549
$456$	0	0
$457$	26.9986	1.26294	0.631470	−	0.775400i	$-0.282452\pi$
$457$	26.9986	1.26294	0.631470	+	0.775400i	$0.282452\pi$
$458$	0	0
$459$	17.5223	0.817872
$460$	0	0
$461$	19.5223	0.909245	0.454622	−	0.890684i	$-0.349774\pi$
$461$	19.5223	0.909245	0.454622	+	0.890684i	$0.349774\pi$
$462$	0	0
$463$	1.84367	0.0856828	0.0428414	−	0.999082i	$-0.486359\pi$
$463$	1.84367	0.0856828	0.0428414	+	0.999082i	$0.486359\pi$
$464$	0	0
$465$	−1.73813	−0.0806041
$466$	0	0
$467$	−9.84367	−0.455511	−0.227755	−	0.973718i	$-0.573139\pi$
$467$	−9.84367	−0.455511	−0.227755	+	0.973718i	$0.573139\pi$
$468$	0	0
$469$	24.6761	1.13944
$470$	0	0
$471$	−0.337088	−0.0155322
$472$	0	0
$473$	−14.8872	−0.684513
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	6.78514	0.310670
$478$	0	0
$479$	−8.31853	−0.380083	−0.190042	−	0.981776i	$-0.560862\pi$
$479$	−8.31853	−0.380083	−0.190042	+	0.981776i	$0.560862\pi$
$480$	0	0
$481$	10.1075	0.460860
$482$	0	0
$483$	−3.91019	−0.177920
$484$	0	0
$485$	−1.48849	−0.0675888
$486$	0	0
$487$	−34.1197	−1.54611	−0.773055	−	0.634339i	$-0.781272\pi$
$487$	−34.1197	−1.54611	−0.773055	+	0.634339i	$0.781272\pi$
$488$	0	0
$489$	−3.49788	−0.158180
$490$	0	0
$491$	38.0870	1.71884	0.859421	−	0.511269i	$-0.170824\pi$
$491$	38.0870	1.71884	0.859421	+	0.511269i	$0.170824\pi$
$492$	0	0
$493$	6.55008	0.295001
$494$	0	0
$495$	−11.5066	−0.517183
$496$	0	0
$497$	9.46168	0.424414
$498$	0	0
$499$	25.0336	1.12066	0.560330	−	0.828270i	$-0.310674\pi$
$499$	25.0336	1.12066	0.560330	+	0.828270i	$0.310674\pi$
$500$	0	0
$501$	6.90289	0.308399
$502$	0	0
$503$	14.7210	0.656377	0.328188	−	0.944612i	$-0.393562\pi$
$503$	14.7210	0.656377	0.328188	+	0.944612i	$0.393562\pi$
$504$	0	0
$505$	12.2823	0.546557
$506$	0	0
$507$	−4.20757	−0.186865
$508$	0	0
$509$	−37.1998	−1.64885	−0.824426	−	0.565969i	$-0.808502\pi$
$509$	−37.1998	−1.64885	−0.824426	+	0.565969i	$0.808502\pi$
$510$	0	0
$511$	−20.8364	−0.921747
$512$	0	0
$513$	2.77575	0.122552
$514$	0	0
$515$	−10.4812	−0.461857
$516$	0	0
$517$	18.8872	0.830657
$518$	0	0
$519$	−0.842260	−0.0369711
$520$	0	0
$521$	−25.0132	−1.09585	−0.547924	−	0.836528i	$-0.684582\pi$
$521$	−25.0132	−1.09585	−0.547924	+	0.836528i	$0.684582\pi$
$522$	0	0
$523$	−5.78067	−0.252771	−0.126386	−	0.991981i	$-0.540338\pi$
$523$	−5.78067	−0.252771	−0.126386	+	0.991981i	$0.540338\pi$
$524$	0	0
$525$	−1.03761	−0.0452851
$526$	0	0
$527$	22.8021	0.993275
$528$	0	0
$529$	−8.79877	−0.382555
$530$	0	0
$531$	34.8119	1.51071
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−19.6302	−0.848689
$536$	0	0
$537$	−2.82653	−0.121974
$538$	0	0
$539$	−9.76845	−0.420757
$540$	0	0
$541$	−18.5804	−0.798834	−0.399417	−	0.916769i	$-0.630787\pi$
$541$	−18.5804	−0.798834	−0.399417	+	0.916769i	$0.630787\pi$
$542$	0	0
$543$	6.11142	0.262266
$544$	0	0
$545$	13.2750	0.568640
$546$	0	0
$547$	−4.54657	−0.194397	−0.0971986	−	0.995265i	$-0.530988\pi$
$547$	−4.54657	−0.194397	−0.0971986	+	0.995265i	$0.530988\pi$
$548$	0	0
$549$	−0.432779	−0.0184705
$550$	0	0
$551$	1.03761	0.0442037
$552$	0	0
$553$	13.2097	0.561732
$554$	0	0
$555$	2.35756	0.100073
$556$	0	0
$557$	44.3996	1.88127	0.940636	−	0.339416i	$-0.110229\pi$
$557$	44.3996	1.88127	0.940636	+	0.339416i	$0.110229\pi$
$558$	0	0
$559$	−7.38929	−0.312534
$560$	0	0
$561$	−12.6253	−0.533041
$562$	0	0
$563$	9.59403	0.404340	0.202170	−	0.979350i	$-0.435201\pi$
$563$	9.59403	0.404340	0.202170	+	0.979350i	$0.435201\pi$
$564$	0	0
$565$	−11.0254	−0.463842
$566$	0	0
$567$	−15.0289	−0.631155
$568$	0	0
$569$	−9.36485	−0.392595	−0.196297	−	0.980544i	$-0.562892\pi$
$569$	−9.36485	−0.392595	−0.196297	+	0.980544i	$0.562892\pi$
$570$	0	0
$571$	16.4083	0.686668	0.343334	−	0.939213i	$-0.388444\pi$
$571$	16.4083	0.686668	0.343334	+	0.939213i	$0.388444\pi$
$572$	0	0
$573$	2.44851	0.102288
$574$	0	0
$575$	3.76845	0.157155
$576$	0	0
$577$	28.2736	1.17705	0.588523	−	0.808480i	$-0.299710\pi$
$577$	28.2736	1.17705	0.588523	+	0.808480i	$0.299710\pi$
$578$	0	0
$579$	−4.19252	−0.174235
$580$	0	0
$581$	−8.39772	−0.348396
$582$	0	0
$583$	−10.1866	−0.421888
$584$	0	0
$585$	−5.71133	−0.236134
$586$	0	0
$587$	−35.2203	−1.45370	−0.726848	−	0.686798i	$-0.759016\pi$
$587$	−35.2203	−1.45370	−0.726848	+	0.686798i	$0.759016\pi$
$588$	0	0
$589$	3.61213	0.148835
$590$	0	0
$591$	−13.0522	−0.536896
$592$	0	0
$593$	17.0738	0.701137	0.350569	−	0.936537i	$-0.385988\pi$
$593$	17.0738	0.701137	0.350569	+	0.936537i	$0.385988\pi$
$594$	0	0
$595$	13.6121	0.558043
$596$	0	0
$597$	8.92619	0.365325
$598$	0	0
$599$	−18.2882	−0.747236	−0.373618	−	0.927583i	$-0.621883\pi$
$599$	−18.2882	−0.747236	−0.373618	+	0.927583i	$0.621883\pi$
$600$	0	0
$601$	22.3879	0.913220	0.456610	−	0.889667i	$-0.349063\pi$
$601$	22.3879	0.913220	0.456610	+	0.889667i	$0.349063\pi$
$602$	0	0
$603$	31.6810	1.29015
$604$	0	0
$605$	6.27504	0.255117
$606$	0	0
$607$	0.471345	0.0191313	0.00956566	−	0.999954i	$-0.496955\pi$
$607$	0.471345	0.0191313	0.00956566	+	0.999954i	$0.496955\pi$
$608$	0	0
$609$	1.07664	0.0436275
$610$	0	0
$611$	9.37470	0.379260
$612$	0	0
$613$	22.9135	0.925468	0.462734	−	0.886497i	$-0.346868\pi$
$613$	22.9135	0.925468	0.462734	+	0.886497i	$0.346868\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.06537	0.244183	0.122091	−	0.992519i	$-0.461040\pi$
$617$	6.06537	0.244183	0.122091	+	0.992519i	$0.461040\pi$
$618$	0	0
$619$	−3.53102	−0.141924	−0.0709619	−	0.997479i	$-0.522607\pi$
$619$	−3.53102	−0.141924	−0.0709619	+	0.997479i	$0.522607\pi$
$620$	0	0
$621$	−10.4603	−0.419756
$622$	0	0
$623$	−10.8627	−0.435206
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.00000	−0.0798723
$628$	0	0
$629$	−30.9281	−1.23318
$630$	0	0
$631$	−31.0797	−1.23726	−0.618631	−	0.785681i	$-0.712313\pi$
$631$	−31.0797	−1.23726	−0.618631	+	0.785681i	$0.712313\pi$
$632$	0	0
$633$	3.78892	0.150596
$634$	0	0
$635$	−13.5696	−0.538493
$636$	0	0
$637$	−4.84860	−0.192109
$638$	0	0
$639$	12.1476	0.480552
$640$	0	0
$641$	42.0870	1.66234	0.831168	−	0.556021i	$-0.187673\pi$
$641$	42.0870	1.66234	0.831168	+	0.556021i	$0.187673\pi$
$642$	0	0
$643$	14.6194	0.576534	0.288267	−	0.957550i	$-0.406921\pi$
$643$	14.6194	0.576534	0.288267	+	0.957550i	$0.406921\pi$
$644$	0	0
$645$	−1.72355	−0.0678646
$646$	0	0
$647$	8.58040	0.337330	0.168665	−	0.985673i	$-0.446054\pi$
$647$	8.58040	0.337330	0.168665	+	0.985673i	$0.446054\pi$
$648$	0	0
$649$	−52.2638	−2.05153
$650$	0	0
$651$	3.74798	0.146895
$652$	0	0
$653$	35.5877	1.39265	0.696327	−	0.717725i	$-0.254816\pi$
$653$	35.5877	1.39265	0.696327	+	0.717725i	$0.254816\pi$
$654$	0	0
$655$	−0.962389	−0.0376036
$656$	0	0
$657$	−26.7513	−1.04367
$658$	0	0
$659$	−25.8251	−1.00600	−0.503002	−	0.864285i	$-0.667771\pi$
$659$	−25.8251	−1.00600	−0.503002	+	0.864285i	$0.667771\pi$
$660$	0	0
$661$	−0.584365	−0.0227291	−0.0113646	−	0.999935i	$-0.503618\pi$
$661$	−0.584365	−0.0227291	−0.0113646	+	0.999935i	$0.503618\pi$
$662$	0	0
$663$	−6.26660	−0.243375
$664$	0	0
$665$	2.15633	0.0836187
$666$	0	0
$667$	−3.91019	−0.151403
$668$	0	0
$669$	−1.31994	−0.0510320
$670$	0	0
$671$	0.649738	0.0250829
$672$	0	0
$673$	−11.3136	−0.436107	−0.218054	−	0.975937i	$-0.569971\pi$
$673$	−11.3136	−0.436107	−0.218054	+	0.975937i	$0.569971\pi$
$674$	0	0
$675$	−2.77575	−0.106839
$676$	0	0
$677$	−8.58673	−0.330015	−0.165008	−	0.986292i	$-0.552765\pi$
$677$	−8.58673	−0.330015	−0.165008	+	0.986292i	$0.552765\pi$
$678$	0	0
$679$	3.20967	0.123176
$680$	0	0
$681$	−3.96685	−0.152010
$682$	0	0
$683$	−30.5564	−1.16921	−0.584604	−	0.811318i	$-0.698750\pi$
$683$	−30.5564	−1.16921	−0.584604	+	0.811318i	$0.698750\pi$
$684$	0	0
$685$	−12.7005	−0.485262
$686$	0	0
$687$	−5.16362	−0.197004
$688$	0	0
$689$	−5.05617	−0.192625
$690$	0	0
$691$	3.38058	0.128603	0.0643016	−	0.997931i	$-0.479518\pi$
$691$	3.38058	0.128603	0.0643016	+	0.997931i	$0.479518\pi$
$692$	0	0
$693$	24.8119	0.942528
$694$	0	0
$695$	−4.93207	−0.187084
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0.901754	0.0341075
$700$	0	0
$701$	−16.2579	−0.614052	−0.307026	−	0.951701i	$-0.599334\pi$
$701$	−16.2579	−0.614052	−0.307026	+	0.951701i	$0.599334\pi$
$702$	0	0
$703$	−4.89938	−0.184784
$704$	0	0
$705$	2.18664	0.0823537
$706$	0	0
$707$	−26.4847	−0.996060
$708$	0	0
$709$	−35.2262	−1.32295	−0.661473	−	0.749969i	$-0.730068\pi$
$709$	−35.2262	−1.32295	−0.661473	+	0.749969i	$0.730068\pi$
$710$	0	0
$711$	16.9596	0.636033
$712$	0	0
$713$	−13.6121	−0.509778
$714$	0	0
$715$	8.57452	0.320669
$716$	0	0
$717$	−10.1650	−0.379620
$718$	0	0
$719$	28.6194	1.06732	0.533662	−	0.845698i	$-0.320815\pi$
$719$	28.6194	1.06732	0.533662	+	0.845698i	$0.320815\pi$
$720$	0	0
$721$	22.6009	0.841700
$722$	0	0
$723$	1.07664	0.0400406
$724$	0	0
$725$	−1.03761	−0.0385359
$726$	0	0
$727$	−21.1041	−0.782709	−0.391354	−	0.920240i	$-0.627993\pi$
$727$	−21.1041	−0.782709	−0.391354	+	0.920240i	$0.627993\pi$
$728$	0	0
$729$	−15.2882	−0.566230
$730$	0	0
$731$	22.6107	0.836287
$732$	0	0
$733$	−47.1655	−1.74210	−0.871049	−	0.491196i	$-0.836560\pi$
$733$	−47.1655	−1.74210	−0.871049	+	0.491196i	$0.836560\pi$
$734$	0	0
$735$	−1.13093	−0.0417151
$736$	0	0
$737$	−47.5633	−1.75201
$738$	0	0
$739$	12.9262	0.475498	0.237749	−	0.971327i	$-0.423590\pi$
$739$	12.9262	0.475498	0.237749	+	0.971327i	$0.423590\pi$
$740$	0	0
$741$	−0.992706	−0.0364680
$742$	0	0
$743$	−40.3717	−1.48109	−0.740547	−	0.672005i	$-0.765433\pi$
$743$	−40.3717	−1.48109	−0.740547	+	0.672005i	$0.765433\pi$
$744$	0	0
$745$	−10.3938	−0.380798
$746$	0	0
$747$	−10.7816	−0.394479
$748$	0	0
$749$	42.3291	1.54667
$750$	0	0
$751$	14.2130	0.518639	0.259320	−	0.965792i	$-0.416502\pi$
$751$	14.2130	0.518639	0.259320	+	0.965792i	$0.416502\pi$
$752$	0	0
$753$	−12.3733	−0.450908
$754$	0	0
$755$	1.42548	0.0518787
$756$	0	0
$757$	24.3028	0.883300	0.441650	−	0.897187i	$-0.354393\pi$
$757$	24.3028	0.883300	0.441650	+	0.897187i	$0.354393\pi$
$758$	0	0
$759$	7.53690	0.273572
$760$	0	0
$761$	−5.05808	−0.183355	−0.0916776	−	0.995789i	$-0.529223\pi$
$761$	−5.05808	−0.183355	−0.0916776	+	0.995789i	$0.529223\pi$
$762$	0	0
$763$	−28.6253	−1.03631
$764$	0	0
$765$	17.4763	0.631856
$766$	0	0
$767$	−25.9413	−0.936685
$768$	0	0
$769$	−42.8324	−1.54458	−0.772288	−	0.635272i	$-0.780888\pi$
$769$	−42.8324	−1.54458	−0.772288	+	0.635272i	$0.780888\pi$
$770$	0	0
$771$	0.866698	0.0312134
$772$	0	0
$773$	−21.5755	−0.776016	−0.388008	−	0.921656i	$-0.626837\pi$
$773$	−21.5755	−0.776016	−0.388008	+	0.921656i	$0.626837\pi$
$774$	0	0
$775$	−3.61213	−0.129751
$776$	0	0
$777$	−5.08366	−0.182375
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−18.2374	−0.652586
$782$	0	0
$783$	2.88015	0.102928
$784$	0	0
$785$	−0.700523	−0.0250028
$786$	0	0
$787$	−24.3961	−0.869628	−0.434814	−	0.900520i	$-0.643186\pi$
$787$	−24.3961	−0.869628	−0.434814	+	0.900520i	$0.643186\pi$
$788$	0	0
$789$	5.06679	0.180382
$790$	0	0
$791$	23.7743	0.845318
$792$	0	0
$793$	0.322499	0.0114523
$794$	0	0
$795$	−1.17935	−0.0418272
$796$	0	0
$797$	−3.22662	−0.114293	−0.0571464	−	0.998366i	$-0.518200\pi$
$797$	−3.22662	−0.114293	−0.0571464	+	0.998366i	$0.518200\pi$
$798$	0	0
$799$	−28.6859	−1.01484
$800$	0	0
$801$	−13.9464	−0.492771
$802$	0	0
$803$	40.1622	1.41729
$804$	0	0
$805$	−8.12601	−0.286404
$806$	0	0
$807$	0.463096	0.0163017
$808$	0	0
$809$	29.9756	1.05388	0.526942	−	0.849901i	$-0.323338\pi$
$809$	29.9756	1.05388	0.526942	+	0.849901i	$0.323338\pi$
$810$	0	0
$811$	40.0870	1.40764	0.703822	−	0.710376i	$-0.251475\pi$
$811$	40.0870	1.40764	0.703822	+	0.710376i	$0.251475\pi$
$812$	0	0
$813$	−14.8481	−0.520747
$814$	0	0
$815$	−7.26916	−0.254628
$816$	0	0
$817$	3.58181	0.125312
$818$	0	0
$819$	12.3155	0.430338
$820$	0	0
$821$	−11.3766	−0.397046	−0.198523	−	0.980096i	$-0.563615\pi$
$821$	−11.3766	−0.397046	−0.198523	+	0.980096i	$0.563615\pi$
$822$	0	0
$823$	−44.7426	−1.55963	−0.779814	−	0.626011i	$-0.784687\pi$
$823$	−44.7426	−1.55963	−0.779814	+	0.626011i	$0.784687\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−31.8070	−1.10604	−0.553019	−	0.833169i	$-0.686524\pi$
$827$	−31.8070	−1.10604	−0.553019	+	0.833169i	$0.686524\pi$
$828$	0	0
$829$	5.02302	0.174457	0.0872284	−	0.996188i	$-0.472199\pi$
$829$	5.02302	0.174457	0.0872284	+	0.996188i	$0.472199\pi$
$830$	0	0
$831$	−3.56134	−0.123542
$832$	0	0
$833$	14.8364	0.514050
$834$	0	0
$835$	14.3453	0.496441
$836$	0	0
$837$	10.0263	0.346561
$838$	0	0
$839$	49.8858	1.72225	0.861124	−	0.508395i	$-0.169761\pi$
$839$	49.8858	1.72225	0.861124	+	0.508395i	$0.169761\pi$
$840$	0	0
$841$	−27.9234	−0.962875
$842$	0	0
$843$	−1.17821	−0.0405796
$844$	0	0
$845$	−8.74401	−0.300803
$846$	0	0
$847$	−13.5310	−0.464932
$848$	0	0
$849$	−12.7250	−0.436720
$850$	0	0
$851$	18.4631	0.632907
$852$	0	0
$853$	7.61213	0.260634	0.130317	−	0.991472i	$-0.458400\pi$
$853$	7.61213	0.260634	0.130317	+	0.991472i	$0.458400\pi$
$854$	0	0
$855$	2.76845	0.0946791
$856$	0	0
$857$	13.1758	0.450078	0.225039	−	0.974350i	$-0.427749\pi$
$857$	13.1758	0.450078	0.225039	+	0.974350i	$0.427749\pi$
$858$	0	0
$859$	37.9003	1.29314	0.646571	−	0.762853i	$-0.276202\pi$
$859$	37.9003	1.29314	0.646571	+	0.762853i	$0.276202\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.2292	−0.552448	−0.276224	−	0.961093i	$-0.589083\pi$
$863$	−16.2292	−0.552448	−0.276224	+	0.961093i	$0.589083\pi$
$864$	0	0
$865$	−1.75035	−0.0595138
$866$	0	0
$867$	10.9951	0.373412
$868$	0	0
$869$	−25.4617	−0.863728
$870$	0	0
$871$	−23.6082	−0.799932
$872$	0	0
$873$	4.12081	0.139468
$874$	0	0
$875$	−2.15633	−0.0728971
$876$	0	0
$877$	−15.3888	−0.519644	−0.259822	−	0.965657i	$-0.583664\pi$
$877$	−15.3888	−0.519644	−0.259822	+	0.965657i	$0.583664\pi$
$878$	0	0
$879$	4.74543	0.160059
$880$	0	0
$881$	−51.5428	−1.73652	−0.868260	−	0.496109i	$-0.834762\pi$
$881$	−51.5428	−1.73652	−0.868260	+	0.496109i	$0.834762\pi$
$882$	0	0
$883$	26.3185	0.885689	0.442845	−	0.896598i	$-0.353969\pi$
$883$	26.3185	0.885689	0.442845	+	0.896598i	$0.353969\pi$
$884$	0	0
$885$	−6.05079	−0.203395
$886$	0	0
$887$	−6.30631	−0.211745	−0.105873	−	0.994380i	$-0.533764\pi$
$887$	−6.30631	−0.211745	−0.105873	+	0.994380i	$0.533764\pi$
$888$	0	0
$889$	29.2605	0.981364
$890$	0	0
$891$	28.9683	0.970473
$892$	0	0
$893$	−4.54420	−0.152066
$894$	0	0
$895$	−5.87399	−0.196346
$896$	0	0
$897$	3.74096	0.124907
$898$	0	0
$899$	3.74798	0.125002
$900$	0	0
$901$	15.4715	0.515431
$902$	0	0
$903$	3.71653	0.123678
$904$	0	0
$905$	12.7005	0.422180
$906$	0	0
$907$	−21.7054	−0.720718	−0.360359	−	0.932814i	$-0.617346\pi$
$907$	−21.7054	−0.720718	−0.360359	+	0.932814i	$0.617346\pi$
$908$	0	0
$909$	−34.0031	−1.12781
$910$	0	0
$911$	−10.3634	−0.343356	−0.171678	−	0.985153i	$-0.554919\pi$
$911$	−10.3634	−0.343356	−0.171678	+	0.985153i	$0.554919\pi$
$912$	0	0
$913$	16.1866	0.535700
$914$	0	0
$915$	0.0752228	0.00248679
$916$	0	0
$917$	2.07522	0.0685299
$918$	0	0
$919$	42.3752	1.39783	0.698914	−	0.715205i	$-0.253667\pi$
$919$	42.3752	1.39783	0.698914	+	0.715205i	$0.253667\pi$
$920$	0	0
$921$	−0.0740825	−0.00244110
$922$	0	0
$923$	−9.05220	−0.297957
$924$	0	0
$925$	4.89938	0.161091
$926$	0	0
$927$	29.0167	0.953033
$928$	0	0
$929$	27.1002	0.889127	0.444564	−	0.895747i	$-0.353359\pi$
$929$	27.1002	0.889127	0.444564	+	0.895747i	$0.353359\pi$
$930$	0	0
$931$	2.35026	0.0770267
$932$	0	0
$933$	−5.59754	−0.183255
$934$	0	0
$935$	−26.2374	−0.858056
$936$	0	0
$937$	−38.1866	−1.24750	−0.623752	−	0.781623i	$-0.714392\pi$
$937$	−38.1866	−1.24750	−0.623752	+	0.781623i	$0.714392\pi$
$938$	0	0
$939$	0.397722	0.0129792
$940$	0	0
$941$	−60.5910	−1.97521	−0.987605	−	0.156958i	$-0.949831\pi$
$941$	−60.5910	−1.97521	−0.987605	+	0.156958i	$0.949831\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.98541	0.194705
$946$	0	0
$947$	32.2433	1.04777	0.523883	−	0.851790i	$-0.324483\pi$
$947$	32.2433	1.04777	0.523883	+	0.851790i	$0.324483\pi$
$948$	0	0
$949$	19.9346	0.647105
$950$	0	0
$951$	8.72099	0.282798
$952$	0	0
$953$	7.88812	0.255521	0.127761	−	0.991805i	$-0.459221\pi$
$953$	7.88812	0.255521	0.127761	+	0.991805i	$0.459221\pi$
$954$	0	0
$955$	5.08840	0.164657
$956$	0	0
$957$	−2.07522	−0.0670824
$958$	0	0
$959$	27.3865	0.884355
$960$	0	0
$961$	−17.9525	−0.579114
$962$	0	0
$963$	54.3453	1.75125
$964$	0	0
$965$	−8.71274	−0.280473
$966$	0	0
$967$	40.3693	1.29819	0.649095	−	0.760708i	$-0.275148\pi$
$967$	40.3693	1.29819	0.649095	+	0.760708i	$0.275148\pi$
$968$	0	0
$969$	3.03761	0.0975821
$970$	0	0
$971$	19.2995	0.619350	0.309675	−	0.950843i	$-0.399780\pi$
$971$	19.2995	0.619350	0.309675	+	0.950843i	$0.399780\pi$
$972$	0	0
$973$	10.6351	0.340947
$974$	0	0
$975$	0.992706	0.0317920
$976$	0	0
$977$	22.5623	0.721832	0.360916	−	0.932598i	$-0.382464\pi$
$977$	22.5623	0.721832	0.360916	+	0.932598i	$0.382464\pi$
$978$	0	0
$979$	20.9380	0.669180
$980$	0	0
$981$	−36.7513	−1.17338
$982$	0	0
$983$	44.4666	1.41826	0.709132	−	0.705076i	$-0.249087\pi$
$983$	44.4666	1.41826	0.709132	+	0.705076i	$0.249087\pi$
$984$	0	0
$985$	−27.1246	−0.864261
$986$	0	0
$987$	−4.71511	−0.150084
$988$	0	0
$989$	−13.4979	−0.429208
$990$	0	0
$991$	3.82512	0.121509	0.0607544	−	0.998153i	$-0.480649\pi$
$991$	3.82512	0.121509	0.0607544	+	0.998153i	$0.480649\pi$
$992$	0	0
$993$	5.68452	0.180393
$994$	0	0
$995$	18.5501	0.588077
$996$	0	0
$997$	52.8627	1.67418	0.837090	−	0.547066i	$-0.184255\pi$
$997$	52.8627	1.67418	0.837090	+	0.547066i	$0.184255\pi$
$998$	0	0
$999$	−13.5994	−0.430268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bn.1.3		3
4.3	odd	2		6080.2.a.cb.1.1		3
8.3	odd	2		3040.2.a.i.1.3	✓	3
8.5	even	2		3040.2.a.o.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.i.1.3	✓	3	8.3	odd	2
3040.2.a.o.1.1	yes	3	8.5	even	2
6080.2.a.bn.1.3		3	1.1	even	1	trivial
6080.2.a.cb.1.1		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +O(q^{10})\)	\(q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +4.15633 q^{11} +2.06300 q^{13} +0.481194 q^{15} -6.31265 q^{17} -1.00000 q^{19} -1.03761 q^{21} +3.76845 q^{23} +1.00000 q^{25} -2.77575 q^{27} -1.03761 q^{29} -3.61213 q^{31} +2.00000 q^{33} -2.15633 q^{35} +4.89938 q^{37} +0.992706 q^{39} -3.58181 q^{43} -2.76845 q^{45} +4.54420 q^{47} -2.35026 q^{49} -3.03761 q^{51} -2.45088 q^{53} +4.15633 q^{55} -0.481194 q^{57} -12.5745 q^{59} +0.156325 q^{61} +5.96968 q^{63} +2.06300 q^{65} -11.4436 q^{67} +1.81336 q^{69} -4.38787 q^{71} +9.66291 q^{73} +0.481194 q^{75} -8.96239 q^{77} -6.12601 q^{79} +6.96968 q^{81} +3.89446 q^{83} -6.31265 q^{85} -0.499293 q^{87} +5.03761 q^{89} -4.44851 q^{91} -1.73813 q^{93} -1.00000 q^{95} -1.48849 q^{97} -11.5066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9	\( 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 3 q^{19} - 14 q^{21} + 3 q^{25} - 10 q^{27} - 14 q^{29} - 10 q^{31} + 6 q^{33} + 4 q^{35} + 8 q^{37} - 10 q^{39} - 12 q^{43} + 3 q^{45} + 4 q^{47} + 3 q^{49} - 20 q^{51} - 4 q^{53} + 2 q^{55} + 4 q^{57} - 26 q^{59} - 10 q^{61} + 20 q^{63} + 2 q^{65} - 18 q^{67} + 18 q^{69} - 14 q^{71} - 2 q^{73} - 4 q^{75} - 16 q^{77} - 10 q^{79} + 23 q^{81} - 8 q^{83} + 2 q^{85} + 32 q^{87} + 26 q^{89} - 10 q^{91} + 4 q^{93} - 3 q^{95} - 12 q^{97} - 14 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 - 3 * q^19 - 14 * q^21 + 3 * q^25 - 10 * q^27 - 14 * q^29 - 10 * q^31 + 6 * q^33 + 4 * q^35 + 8 * q^37 - 10 * q^39 - 12 * q^43 + 3 * q^45 + 4 * q^47 + 3 * q^49 - 20 * q^51 - 4 * q^53 + 2 * q^55 + 4 * q^57 - 26 * q^59 - 10 * q^61 + 20 * q^63 + 2 * q^65 - 18 * q^67 + 18 * q^69 - 14 * q^71 - 2 * q^73 - 4 * q^75 - 16 * q^77 - 10 * q^79 + 23 * q^81 - 8 * q^83 + 2 * q^85 + 32 * q^87 + 26 * q^89 - 10 * q^91 + 4 * q^93 - 3 * q^95 - 12 * q^97 - 14 * q^99

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