LMFDB - Embedded newform 6080.2.a.bn.1.2

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			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	$q-1.31111 q^{3} +1.00000 q^{5} +3.52543 q^{7} -1.28100 q^{9} +O(q^{10})$	$q-1.31111 q^{3} +1.00000 q^{5} +3.52543 q^{7} -1.28100 q^{9} -1.52543 q^{11} -6.02074 q^{13} -1.31111 q^{15} +5.05086 q^{17} -1.00000 q^{19} -4.62222 q^{21} +2.28100 q^{23} +1.00000 q^{25} +5.61285 q^{27} -4.62222 q^{29} -7.80642 q^{31} +2.00000 q^{33} +3.52543 q^{35} +9.39853 q^{37} +7.89384 q^{39} +2.70964 q^{43} -1.28100 q^{45} -5.33185 q^{47} +5.42864 q^{49} -6.62222 q^{51} +9.82717 q^{53} -1.52543 q^{55} +1.31111 q^{57} -13.1842 q^{59} -5.52543 q^{61} -4.51606 q^{63} -6.02074 q^{65} -6.06668 q^{67} -2.99063 q^{69} -0.193576 q^{71} -9.47949 q^{73} -1.31111 q^{75} -5.37778 q^{77} +10.0415 q^{79} -3.51606 q^{81} -13.7605 q^{83} +5.05086 q^{85} +6.06022 q^{87} +8.62222 q^{89} -21.2257 q^{91} +10.2351 q^{93} -1.00000 q^{95} +7.20495 q^{97} +1.95407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.31111	−0.756968	−0.378484	−	0.925608i	$-0.623555\pi$
$3$	−1.31111	−0.756968	−0.378484	+	0.925608i	$0.623555\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.52543	1.33249	0.666243	−	0.745735i	$-0.267901\pi$
$7$	3.52543	1.33249	0.666243	+	0.745735i	$0.267901\pi$
$8$	0	0
$9$	−1.28100	−0.426999
$10$	0	0
$11$	−1.52543	−0.459934	−0.229967	−	0.973198i	$-0.573862\pi$
$11$	−1.52543	−0.459934	−0.229967	+	0.973198i	$0.573862\pi$
$12$	0	0
$13$	−6.02074	−1.66985	−0.834927	−	0.550361i	$-0.814490\pi$
$13$	−6.02074	−1.66985	−0.834927	+	0.550361i	$0.814490\pi$
$14$	0	0
$15$	−1.31111	−0.338527
$16$	0	0
$17$	5.05086	1.22501	0.612506	−	0.790466i	$-0.290161\pi$
$17$	5.05086	1.22501	0.612506	+	0.790466i	$0.290161\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.62222	−1.00865
$22$	0	0
$23$	2.28100	0.475621	0.237810	−	0.971312i	$-0.423570\pi$
$23$	2.28100	0.475621	0.237810	+	0.971312i	$0.423570\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.61285	1.08019
$28$	0	0
$29$	−4.62222	−0.858324	−0.429162	−	0.903228i	$-0.641191\pi$
$29$	−4.62222	−0.858324	−0.429162	+	0.903228i	$0.641191\pi$
$30$	0	0
$31$	−7.80642	−1.40208	−0.701038	−	0.713124i	$-0.747279\pi$
$31$	−7.80642	−1.40208	−0.701038	+	0.713124i	$0.747279\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.52543	0.595906
$36$	0	0
$37$	9.39853	1.54511	0.772554	−	0.634949i	$-0.218979\pi$
$37$	9.39853	1.54511	0.772554	+	0.634949i	$0.218979\pi$
$38$	0	0
$39$	7.89384	1.26403
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.70964	0.413216	0.206608	−	0.978424i	$-0.433758\pi$
$43$	2.70964	0.413216	0.206608	+	0.978424i	$0.433758\pi$
$44$	0	0
$45$	−1.28100	−0.190960
$46$	0	0
$47$	−5.33185	−0.777730	−0.388865	−	0.921295i	$-0.627133\pi$
$47$	−5.33185	−0.777730	−0.388865	+	0.921295i	$0.627133\pi$
$48$	0	0
$49$	5.42864	0.775520
$50$	0	0
$51$	−6.62222	−0.927296
$52$	0	0
$53$	9.82717	1.34987	0.674933	−	0.737879i	$-0.264173\pi$
$53$	9.82717	1.34987	0.674933	+	0.737879i	$0.264173\pi$
$54$	0	0
$55$	−1.52543	−0.205689
$56$	0	0
$57$	1.31111	0.173660
$58$	0	0
$59$	−13.1842	−1.71644	−0.858219	−	0.513284i	$-0.828429\pi$
$59$	−13.1842	−1.71644	−0.858219	+	0.513284i	$0.828429\pi$
$60$	0	0
$61$	−5.52543	−0.707459	−0.353729	−	0.935348i	$-0.615087\pi$
$61$	−5.52543	−0.707459	−0.353729	+	0.935348i	$0.615087\pi$
$62$	0	0
$63$	−4.51606	−0.568970
$64$	0	0
$65$	−6.02074	−0.746781
$66$	0	0
$67$	−6.06668	−0.741163	−0.370581	−	0.928800i	$-0.620841\pi$
$67$	−6.06668	−0.741163	−0.370581	+	0.928800i	$0.620841\pi$
$68$	0	0
$69$	−2.99063	−0.360030
$70$	0	0
$71$	−0.193576	−0.0229733	−0.0114866	−	0.999934i	$-0.503656\pi$
$71$	−0.193576	−0.0229733	−0.0114866	+	0.999934i	$0.503656\pi$
$72$	0	0
$73$	−9.47949	−1.10949	−0.554745	−	0.832020i	$-0.687184\pi$
$73$	−9.47949	−1.10949	−0.554745	+	0.832020i	$0.687184\pi$
$74$	0	0
$75$	−1.31111	−0.151394
$76$	0	0
$77$	−5.37778	−0.612855
$78$	0	0
$79$	10.0415	1.12976	0.564878	−	0.825175i	$-0.308923\pi$
$79$	10.0415	1.12976	0.564878	+	0.825175i	$0.308923\pi$
$80$	0	0
$81$	−3.51606	−0.390673
$82$	0	0
$83$	−13.7605	−1.51041	−0.755205	−	0.655489i	$-0.772463\pi$
$83$	−13.7605	−1.51041	−0.755205	+	0.655489i	$0.772463\pi$
$84$	0	0
$85$	5.05086	0.547842
$86$	0	0
$87$	6.06022	0.649724
$88$	0	0
$89$	8.62222	0.913953	0.456977	−	0.889479i	$-0.348932\pi$
$89$	8.62222	0.913953	0.456977	+	0.889479i	$0.348932\pi$
$90$	0	0
$91$	−21.2257	−2.22506
$92$	0	0
$93$	10.2351	1.06133
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	7.20495	0.731552	0.365776	−	0.930703i	$-0.380804\pi$
$97$	7.20495	0.731552	0.365776	+	0.930703i	$0.380804\pi$
$98$	0	0
$99$	1.95407	0.196391
$100$	0	0
$101$	−9.56691	−0.951944	−0.475972	−	0.879461i	$-0.657904\pi$
$101$	−9.56691	−0.951944	−0.475972	+	0.879461i	$0.657904\pi$
$102$	0	0
$103$	−8.68889	−0.856142	−0.428071	−	0.903745i	$-0.640807\pi$
$103$	−8.68889	−0.856142	−0.428071	+	0.903745i	$0.640807\pi$
$104$	0	0
$105$	−4.62222	−0.451082
$106$	0	0
$107$	−19.0573	−1.84234	−0.921170	−	0.389161i	$-0.872765\pi$
$107$	−19.0573	−1.84234	−0.921170	+	0.389161i	$0.872765\pi$
$108$	0	0
$109$	−1.67307	−0.160251	−0.0801256	−	0.996785i	$-0.525532\pi$
$109$	−1.67307	−0.160251	−0.0801256	+	0.996785i	$0.525532\pi$
$110$	0	0
$111$	−12.3225	−1.16960
$112$	0	0
$113$	0.642959	0.0604845	0.0302423	−	0.999543i	$-0.490372\pi$
$113$	0.642959	0.0604845	0.0302423	+	0.999543i	$0.490372\pi$
$114$	0	0
$115$	2.28100	0.212704
$116$	0	0
$117$	7.71255	0.713026
$118$	0	0
$119$	17.8064	1.63231
$120$	0	0
$121$	−8.67307	−0.788461
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.97481	0.707650	0.353825	−	0.935312i	$-0.384881\pi$
$127$	7.97481	0.707650	0.353825	+	0.935312i	$0.384881\pi$
$128$	0	0
$129$	−3.55262	−0.312791
$130$	0	0
$131$	2.62222	0.229104	0.114552	−	0.993417i	$-0.463457\pi$
$131$	2.62222	0.229104	0.114552	+	0.993417i	$0.463457\pi$
$132$	0	0
$133$	−3.52543	−0.305693
$134$	0	0
$135$	5.61285	0.483077
$136$	0	0
$137$	2.85728	0.244114	0.122057	−	0.992523i	$-0.461051\pi$
$137$	2.85728	0.244114	0.122057	+	0.992523i	$0.461051\pi$
$138$	0	0
$139$	9.13828	0.775098	0.387549	−	0.921849i	$-0.373322\pi$
$139$	9.13828	0.775098	0.387549	+	0.921849i	$0.373322\pi$
$140$	0	0
$141$	6.99063	0.588717
$142$	0	0
$143$	9.18421	0.768022
$144$	0	0
$145$	−4.62222	−0.383854
$146$	0	0
$147$	−7.11753	−0.587044
$148$	0	0
$149$	13.8207	1.13224	0.566119	−	0.824324i	$-0.308444\pi$
$149$	13.8207	1.13224	0.566119	+	0.824324i	$0.308444\pi$
$150$	0	0
$151$	0.815792	0.0663882	0.0331941	−	0.999449i	$-0.489432\pi$
$151$	0.815792	0.0663882	0.0331941	+	0.999449i	$0.489432\pi$
$152$	0	0
$153$	−6.47013	−0.523079
$154$	0	0
$155$	−7.80642	−0.627027
$156$	0	0
$157$	14.8573	1.18574	0.592870	−	0.805298i	$-0.297995\pi$
$157$	14.8573	1.18574	0.592870	+	0.805298i	$0.297995\pi$
$158$	0	0
$159$	−12.8845	−1.02181
$160$	0	0
$161$	8.04149	0.633758
$162$	0	0
$163$	−12.3412	−0.966639	−0.483319	−	0.875444i	$-0.660569\pi$
$163$	−12.3412	−0.966639	−0.483319	+	0.875444i	$0.660569\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−15.5877	−1.20621	−0.603105	−	0.797662i	$-0.706070\pi$
$167$	−15.5877	−1.20621	−0.603105	+	0.797662i	$0.706070\pi$
$168$	0	0
$169$	23.2494	1.78841
$170$	0	0
$171$	1.28100	0.0979602
$172$	0	0
$173$	−5.03011	−0.382432	−0.191216	−	0.981548i	$-0.561243\pi$
$173$	−5.03011	−0.382432	−0.191216	+	0.981548i	$0.561243\pi$
$174$	0	0
$175$	3.52543	0.266497
$176$	0	0
$177$	17.2859	1.29929
$178$	0	0
$179$	−22.0415	−1.64746	−0.823729	−	0.566984i	$-0.808110\pi$
$179$	−22.0415	−1.64746	−0.823729	+	0.566984i	$0.808110\pi$
$180$	0	0
$181$	−2.85728	−0.212380	−0.106190	−	0.994346i	$-0.533865\pi$
$181$	−2.85728	−0.212380	−0.106190	+	0.994346i	$0.533865\pi$
$182$	0	0
$183$	7.24443	0.535524
$184$	0	0
$185$	9.39853	0.690994
$186$	0	0
$187$	−7.70471	−0.563424
$188$	0	0
$189$	19.7877	1.43934
$190$	0	0
$191$	−14.6637	−1.06103	−0.530514	−	0.847676i	$-0.678001\pi$
$191$	−14.6637	−1.06103	−0.530514	+	0.847676i	$0.678001\pi$
$192$	0	0
$193$	−8.40790	−0.605214	−0.302607	−	0.953115i	$-0.597857\pi$
$193$	−8.40790	−0.605214	−0.302607	+	0.953115i	$0.597857\pi$
$194$	0	0
$195$	7.89384	0.565290
$196$	0	0
$197$	2.16193	0.154031	0.0770157	−	0.997030i	$-0.475461\pi$
$197$	2.16193	0.154031	0.0770157	+	0.997030i	$0.475461\pi$
$198$	0	0
$199$	−11.3461	−0.804307	−0.402153	−	0.915572i	$-0.631738\pi$
$199$	−11.3461	−0.804307	−0.402153	+	0.915572i	$0.631738\pi$
$200$	0	0
$201$	7.95407	0.561037
$202$	0	0
$203$	−16.2953	−1.14370
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.92195	−0.203089
$208$	0	0
$209$	1.52543	0.105516
$210$	0	0
$211$	24.0415	1.65508	0.827542	−	0.561404i	$-0.189738\pi$
$211$	24.0415	1.65508	0.827542	+	0.561404i	$0.189738\pi$
$212$	0	0
$213$	0.253799	0.0173900
$214$	0	0
$215$	2.70964	0.184796
$216$	0	0
$217$	−27.5210	−1.86825
$218$	0	0
$219$	12.4286	0.839850
$220$	0	0
$221$	−30.4099	−2.04559
$222$	0	0
$223$	−12.9240	−0.865452	−0.432726	−	0.901525i	$-0.642448\pi$
$223$	−12.9240	−0.865452	−0.432726	+	0.901525i	$0.642448\pi$
$224$	0	0
$225$	−1.28100	−0.0853998
$226$	0	0
$227$	−24.9842	−1.65826	−0.829129	−	0.559057i	$-0.811163\pi$
$227$	−24.9842	−1.65826	−0.829129	+	0.559057i	$0.811163\pi$
$228$	0	0
$229$	−5.65878	−0.373943	−0.186971	−	0.982365i	$-0.559867\pi$
$229$	−5.65878	−0.373943	−0.186971	+	0.982365i	$0.559867\pi$
$230$	0	0
$231$	7.05086	0.463912
$232$	0	0
$233$	18.0415	1.18194	0.590969	−	0.806695i	$-0.298746\pi$
$233$	18.0415	1.18194	0.590969	+	0.806695i	$0.298746\pi$
$234$	0	0
$235$	−5.33185	−0.347812
$236$	0	0
$237$	−13.1655	−0.855189
$238$	0	0
$239$	8.16193	0.527952	0.263976	−	0.964529i	$-0.414966\pi$
$239$	8.16193	0.527952	0.263976	+	0.964529i	$0.414966\pi$
$240$	0	0
$241$	−16.2953	−1.04967	−0.524836	−	0.851203i	$-0.675873\pi$
$241$	−16.2953	−1.04967	−0.524836	+	0.851203i	$0.675873\pi$
$242$	0	0
$243$	−12.2286	−0.784466
$244$	0	0
$245$	5.42864	0.346823
$246$	0	0
$247$	6.02074	0.383091
$248$	0	0
$249$	18.0415	1.14333
$250$	0	0
$251$	16.7654	1.05822	0.529112	−	0.848552i	$-0.322525\pi$
$251$	16.7654	1.05822	0.529112	+	0.848552i	$0.322525\pi$
$252$	0	0
$253$	−3.47949	−0.218754
$254$	0	0
$255$	−6.62222	−0.414699
$256$	0	0
$257$	−18.2558	−1.13877	−0.569383	−	0.822072i	$-0.692818\pi$
$257$	−18.2558	−1.13877	−0.569383	+	0.822072i	$0.692818\pi$
$258$	0	0
$259$	33.1338	2.05884
$260$	0	0
$261$	5.92104	0.366503
$262$	0	0
$263$	14.4558	0.891385	0.445693	−	0.895186i	$-0.352957\pi$
$263$	14.4558	0.891385	0.445693	+	0.895186i	$0.352957\pi$
$264$	0	0
$265$	9.82717	0.603678
$266$	0	0
$267$	−11.3047	−0.691834
$268$	0	0
$269$	−2.62222	−0.159879	−0.0799397	−	0.996800i	$-0.525473\pi$
$269$	−2.62222	−0.159879	−0.0799397	+	0.996800i	$0.525473\pi$
$270$	0	0
$271$	−9.61729	−0.584209	−0.292104	−	0.956386i	$-0.594356\pi$
$271$	−9.61729	−0.584209	−0.292104	+	0.956386i	$0.594356\pi$
$272$	0	0
$273$	27.8292	1.68430
$274$	0	0
$275$	−1.52543	−0.0919867
$276$	0	0
$277$	23.7146	1.42487	0.712435	−	0.701738i	$-0.247592\pi$
$277$	23.7146	1.42487	0.712435	+	0.701738i	$0.247592\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−19.2257	−1.14691	−0.573454	−	0.819237i	$-0.694397\pi$
$281$	−19.2257	−1.14691	−0.573454	+	0.819237i	$0.694397\pi$
$282$	0	0
$283$	21.1066	1.25466	0.627330	−	0.778754i	$-0.284148\pi$
$283$	21.1066	1.25466	0.627330	+	0.778754i	$0.284148\pi$
$284$	0	0
$285$	1.31111	0.0776633
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.51114	0.500655
$290$	0	0
$291$	−9.44647	−0.553762
$292$	0	0
$293$	10.7763	0.629559	0.314779	−	0.949165i	$-0.398069\pi$
$293$	10.7763	0.629559	0.314779	+	0.949165i	$0.398069\pi$
$294$	0	0
$295$	−13.1842	−0.767614
$296$	0	0
$297$	−8.56199	−0.496817
$298$	0	0
$299$	−13.7333	−0.794217
$300$	0	0
$301$	9.55262	0.550604
$302$	0	0
$303$	12.5433	0.720591
$304$	0	0
$305$	−5.52543	−0.316385
$306$	0	0
$307$	−23.5274	−1.34278	−0.671391	−	0.741103i	$-0.734303\pi$
$307$	−23.5274	−1.34278	−0.671391	+	0.741103i	$0.734303\pi$
$308$	0	0
$309$	11.3921	0.648072
$310$	0	0
$311$	17.9956	1.02043	0.510217	−	0.860046i	$-0.329565\pi$
$311$	17.9956	1.02043	0.510217	+	0.860046i	$0.329565\pi$
$312$	0	0
$313$	−30.8988	−1.74650	−0.873251	−	0.487271i	$-0.837992\pi$
$313$	−30.8988	−1.74650	−0.873251	+	0.487271i	$0.837992\pi$
$314$	0	0
$315$	−4.51606	−0.254451
$316$	0	0
$317$	31.0114	1.74177	0.870886	−	0.491485i	$-0.163546\pi$
$317$	31.0114	1.74177	0.870886	+	0.491485i	$0.163546\pi$
$318$	0	0
$319$	7.05086	0.394772
$320$	0	0
$321$	24.9862	1.39459
$322$	0	0
$323$	−5.05086	−0.281037
$324$	0	0
$325$	−6.02074	−0.333971
$326$	0	0
$327$	2.19358	0.121305
$328$	0	0
$329$	−18.7971	−1.03632
$330$	0	0
$331$	7.00937	0.385270	0.192635	−	0.981271i	$-0.438297\pi$
$331$	7.00937	0.385270	0.192635	+	0.981271i	$0.438297\pi$
$332$	0	0
$333$	−12.0395	−0.659759
$334$	0	0
$335$	−6.06668	−0.331458
$336$	0	0
$337$	−32.7130	−1.78199	−0.890996	−	0.454011i	$-0.849993\pi$
$337$	−32.7130	−1.78199	−0.890996	+	0.454011i	$0.849993\pi$
$338$	0	0
$339$	−0.842989	−0.0457849
$340$	0	0
$341$	11.9081	0.644862
$342$	0	0
$343$	−5.53972	−0.299117
$344$	0	0
$345$	−2.99063	−0.161010
$346$	0	0
$347$	−23.4050	−1.25645	−0.628223	−	0.778034i	$-0.716217\pi$
$347$	−23.4050	−1.25645	−0.628223	+	0.778034i	$0.716217\pi$
$348$	0	0
$349$	−26.0415	−1.39397	−0.696984	−	0.717086i	$-0.745475\pi$
$349$	−26.0415	−1.39397	−0.696984	+	0.717086i	$0.745475\pi$
$350$	0	0
$351$	−33.7935	−1.80376
$352$	0	0
$353$	3.92687	0.209006	0.104503	−	0.994525i	$-0.466675\pi$
$353$	3.92687	0.209006	0.104503	+	0.994525i	$0.466675\pi$
$354$	0	0
$355$	−0.193576	−0.0102740
$356$	0	0
$357$	−23.3461	−1.23561
$358$	0	0
$359$	−22.8528	−1.20613	−0.603063	−	0.797693i	$-0.706053\pi$
$359$	−22.8528	−1.20613	−0.603063	+	0.797693i	$0.706053\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.3713	0.596840
$364$	0	0
$365$	−9.47949	−0.496179
$366$	0	0
$367$	25.9353	1.35381	0.676907	−	0.736069i	$-0.263320\pi$
$367$	25.9353	1.35381	0.676907	+	0.736069i	$0.263320\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	34.6450	1.79868
$372$	0	0
$373$	−31.5921	−1.63578	−0.817889	−	0.575377i	$-0.804855\pi$
$373$	−31.5921	−1.63578	−0.817889	+	0.575377i	$0.804855\pi$
$374$	0	0
$375$	−1.31111	−0.0677053
$376$	0	0
$377$	27.8292	1.43328
$378$	0	0
$379$	−29.0607	−1.49275	−0.746374	−	0.665527i	$-0.768207\pi$
$379$	−29.0607	−1.49275	−0.746374	+	0.665527i	$0.768207\pi$
$380$	0	0
$381$	−10.4558	−0.535669
$382$	0	0
$383$	−4.82225	−0.246405	−0.123203	−	0.992382i	$-0.539317\pi$
$383$	−4.82225	−0.246405	−0.123203	+	0.992382i	$0.539317\pi$
$384$	0	0
$385$	−5.37778	−0.274077
$386$	0	0
$387$	−3.47103	−0.176443
$388$	0	0
$389$	27.6731	1.40308	0.701540	−	0.712630i	$-0.252496\pi$
$389$	27.6731	1.40308	0.701540	+	0.712630i	$0.252496\pi$
$390$	0	0
$391$	11.5210	0.582641
$392$	0	0
$393$	−3.43801	−0.173425
$394$	0	0
$395$	10.0415	0.505242
$396$	0	0
$397$	−0.326929	−0.0164081	−0.00820405	−	0.999966i	$-0.502611\pi$
$397$	−0.326929	−0.0164081	−0.00820405	+	0.999966i	$0.502611\pi$
$398$	0	0
$399$	4.62222	0.231400
$400$	0	0
$401$	−21.6414	−1.08072	−0.540361	−	0.841434i	$-0.681712\pi$
$401$	−21.6414	−1.08072	−0.540361	+	0.841434i	$0.681712\pi$
$402$	0	0
$403$	47.0005	2.34126
$404$	0	0
$405$	−3.51606	−0.174714
$406$	0	0
$407$	−14.3368	−0.710647
$408$	0	0
$409$	−12.1334	−0.599956	−0.299978	−	0.953946i	$-0.596979\pi$
$409$	−12.1334	−0.599956	−0.299978	+	0.953946i	$0.596979\pi$
$410$	0	0
$411$	−3.74620	−0.184786
$412$	0	0
$413$	−46.4800	−2.28713
$414$	0	0
$415$	−13.7605	−0.675476
$416$	0	0
$417$	−11.9813	−0.586725
$418$	0	0
$419$	−4.04149	−0.197440	−0.0987198	−	0.995115i	$-0.531475\pi$
$419$	−4.04149	−0.197440	−0.0987198	+	0.995115i	$0.531475\pi$
$420$	0	0
$421$	−20.7654	−1.01204	−0.506022	−	0.862520i	$-0.668885\pi$
$421$	−20.7654	−1.01204	−0.506022	+	0.862520i	$0.668885\pi$
$422$	0	0
$423$	6.83008	0.332090
$424$	0	0
$425$	5.05086	0.245002
$426$	0	0
$427$	−19.4795	−0.942679
$428$	0	0
$429$	−12.0415	−0.581368
$430$	0	0
$431$	−9.64449	−0.464559	−0.232279	−	0.972649i	$-0.574618\pi$
$431$	−9.64449	−0.464559	−0.232279	+	0.972649i	$0.574618\pi$
$432$	0	0
$433$	−27.4400	−1.31868	−0.659341	−	0.751844i	$-0.729165\pi$
$433$	−27.4400	−1.31868	−0.659341	+	0.751844i	$0.729165\pi$
$434$	0	0
$435$	6.06022	0.290565
$436$	0	0
$437$	−2.28100	−0.109115
$438$	0	0
$439$	−29.5625	−1.41094	−0.705470	−	0.708740i	$-0.749264\pi$
$439$	−29.5625	−1.41094	−0.705470	+	0.708740i	$0.749264\pi$
$440$	0	0
$441$	−6.95407	−0.331146
$442$	0	0
$443$	3.82071	0.181528	0.0907638	−	0.995872i	$-0.471069\pi$
$443$	3.82071	0.181528	0.0907638	+	0.995872i	$0.471069\pi$
$444$	0	0
$445$	8.62222	0.408732
$446$	0	0
$447$	−18.1204	−0.857068
$448$	0	0
$449$	9.08250	0.428630	0.214315	−	0.976765i	$-0.431248\pi$
$449$	9.08250	0.428630	0.214315	+	0.976765i	$0.431248\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−1.06959	−0.0502538
$454$	0	0
$455$	−21.2257	−0.995076
$456$	0	0
$457$	13.8796	0.649258	0.324629	−	0.945841i	$-0.394761\pi$
$457$	13.8796	0.649258	0.324629	+	0.945841i	$0.394761\pi$
$458$	0	0
$459$	28.3497	1.32325
$460$	0	0
$461$	30.3497	1.41353	0.706763	−	0.707451i	$-0.250155\pi$
$461$	30.3497	1.41353	0.706763	+	0.707451i	$0.250155\pi$
$462$	0	0
$463$	7.52543	0.349736	0.174868	−	0.984592i	$-0.444050\pi$
$463$	7.52543	0.349736	0.174868	+	0.984592i	$0.444050\pi$
$464$	0	0
$465$	10.2351	0.474640
$466$	0	0
$467$	−15.5254	−0.718431	−0.359216	−	0.933255i	$-0.616956\pi$
$467$	−15.5254	−0.718431	−0.359216	+	0.933255i	$0.616956\pi$
$468$	0	0
$469$	−21.3876	−0.987589
$470$	0	0
$471$	−19.4795	−0.897568
$472$	0	0
$473$	−4.13335	−0.190052
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−12.5886	−0.576391
$478$	0	0
$479$	23.0651	1.05387	0.526937	−	0.849905i	$-0.323340\pi$
$479$	23.0651	1.05387	0.526937	+	0.849905i	$0.323340\pi$
$480$	0	0
$481$	−56.5861	−2.58011
$482$	0	0
$483$	−10.5433	−0.479735
$484$	0	0
$485$	7.20495	0.327160
$486$	0	0
$487$	17.3210	0.784887	0.392444	−	0.919776i	$-0.371630\pi$
$487$	17.3210	0.784887	0.392444	+	0.919776i	$0.371630\pi$
$488$	0	0
$489$	16.1807	0.731715
$490$	0	0
$491$	5.21585	0.235388	0.117694	−	0.993050i	$-0.462450\pi$
$491$	5.21585	0.235388	0.117694	+	0.993050i	$0.462450\pi$
$492$	0	0
$493$	−23.3461	−1.05146
$494$	0	0
$495$	1.95407	0.0878288
$496$	0	0
$497$	−0.682439	−0.0306116
$498$	0	0
$499$	−35.7101	−1.59860	−0.799302	−	0.600929i	$-0.794797\pi$
$499$	−35.7101	−1.59860	−0.799302	+	0.600929i	$0.794797\pi$
$500$	0	0
$501$	20.4371	0.913062
$502$	0	0
$503$	−34.6593	−1.54538	−0.772690	−	0.634784i	$-0.781089\pi$
$503$	−34.6593	−1.54538	−0.772690	+	0.634784i	$0.781089\pi$
$504$	0	0
$505$	−9.56691	−0.425722
$506$	0	0
$507$	−30.4824	−1.35377
$508$	0	0
$509$	−15.0825	−0.668520	−0.334260	−	0.942481i	$-0.608486\pi$
$509$	−15.0825	−0.668520	−0.334260	+	0.942481i	$0.608486\pi$
$510$	0	0
$511$	−33.4193	−1.47838
$512$	0	0
$513$	−5.61285	−0.247813
$514$	0	0
$515$	−8.68889	−0.382878
$516$	0	0
$517$	8.13335	0.357704
$518$	0	0
$519$	6.59502	0.289489
$520$	0	0
$521$	1.90813	0.0835969	0.0417984	−	0.999126i	$-0.486691\pi$
$521$	1.90813	0.0835969	0.0417984	+	0.999126i	$0.486691\pi$
$522$	0	0
$523$	−19.5462	−0.854694	−0.427347	−	0.904088i	$-0.640552\pi$
$523$	−19.5462	−0.854694	−0.427347	+	0.904088i	$0.640552\pi$
$524$	0	0
$525$	−4.62222	−0.201730
$526$	0	0
$527$	−39.4291	−1.71756
$528$	0	0
$529$	−17.7971	−0.773785
$530$	0	0
$531$	16.8889	0.732917
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−19.0573	−0.823919
$536$	0	0
$537$	28.8988	1.24707
$538$	0	0
$539$	−8.28100	−0.356688
$540$	0	0
$541$	0.830082	0.0356880	0.0178440	−	0.999841i	$-0.494320\pi$
$541$	0.830082	0.0356880	0.0178440	+	0.999841i	$0.494320\pi$
$542$	0	0
$543$	3.74620	0.160765
$544$	0	0
$545$	−1.67307	−0.0716665
$546$	0	0
$547$	34.3847	1.47018	0.735092	−	0.677967i	$-0.237139\pi$
$547$	34.3847	1.47018	0.735092	+	0.677967i	$0.237139\pi$
$548$	0	0
$549$	7.07805	0.302084
$550$	0	0
$551$	4.62222	0.196913
$552$	0	0
$553$	35.4005	1.50538
$554$	0	0
$555$	−12.3225	−0.523060
$556$	0	0
$557$	0.164996	0.00699110	0.00349555	−	0.999994i	$-0.498887\pi$
$557$	0.164996	0.00699110	0.00349555	+	0.999994i	$0.498887\pi$
$558$	0	0
$559$	−16.3140	−0.690010
$560$	0	0
$561$	10.1017	0.426495
$562$	0	0
$563$	18.5555	0.782023	0.391011	−	0.920386i	$-0.372125\pi$
$563$	18.5555	0.782023	0.391011	+	0.920386i	$0.372125\pi$
$564$	0	0
$565$	0.642959	0.0270495
$566$	0	0
$567$	−12.3956	−0.520567
$568$	0	0
$569$	12.2163	0.512135	0.256068	−	0.966659i	$-0.417573\pi$
$569$	12.2163	0.512135	0.256068	+	0.966659i	$0.417573\pi$
$570$	0	0
$571$	−21.6084	−0.904283	−0.452142	−	0.891946i	$-0.649340\pi$
$571$	−21.6084	−0.904283	−0.452142	+	0.891946i	$0.649340\pi$
$572$	0	0
$573$	19.2257	0.803165
$574$	0	0
$575$	2.28100	0.0951241
$576$	0	0
$577$	0.206483	0.00859601	0.00429800	−	0.999991i	$-0.498632\pi$
$577$	0.206483	0.00859601	0.00429800	+	0.999991i	$0.498632\pi$
$578$	0	0
$579$	11.0237	0.458128
$580$	0	0
$581$	−48.5116	−2.01260
$582$	0	0
$583$	−14.9906	−0.620849
$584$	0	0
$585$	7.71255	0.318875
$586$	0	0
$587$	20.7195	0.855184	0.427592	−	0.903972i	$-0.359362\pi$
$587$	20.7195	0.855184	0.427592	+	0.903972i	$0.359362\pi$
$588$	0	0
$589$	7.80642	0.321658
$590$	0	0
$591$	−2.83453	−0.116597
$592$	0	0
$593$	11.1240	0.456807	0.228404	−	0.973567i	$-0.426649\pi$
$593$	11.1240	0.456807	0.228404	+	0.973567i	$0.426649\pi$
$594$	0	0
$595$	17.8064	0.729992
$596$	0	0
$597$	14.8760	0.608835
$598$	0	0
$599$	23.5812	0.963502	0.481751	−	0.876308i	$-0.340001\pi$
$599$	23.5812	0.963502	0.481751	+	0.876308i	$0.340001\pi$
$600$	0	0
$601$	18.1936	0.742131	0.371066	−	0.928607i	$-0.378992\pi$
$601$	18.1936	0.742131	0.371066	+	0.928607i	$0.378992\pi$
$602$	0	0
$603$	7.77139	0.316475
$604$	0	0
$605$	−8.67307	−0.352610
$606$	0	0
$607$	−45.6291	−1.85203	−0.926015	−	0.377487i	$-0.876788\pi$
$607$	−45.6291	−1.85203	−0.926015	+	0.377487i	$0.876788\pi$
$608$	0	0
$609$	21.3649	0.865749
$610$	0	0
$611$	32.1017	1.29870
$612$	0	0
$613$	−41.6829	−1.68356	−0.841779	−	0.539823i	$-0.818491\pi$
$613$	−41.6829	−1.68356	−0.841779	+	0.539823i	$0.818491\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.0736	−1.25098	−0.625488	−	0.780234i	$-0.715100\pi$
$617$	−31.0736	−1.25098	−0.625488	+	0.780234i	$0.715100\pi$
$618$	0	0
$619$	−20.5763	−0.827031	−0.413515	−	0.910497i	$-0.635699\pi$
$619$	−20.5763	−0.827031	−0.413515	+	0.910497i	$0.635699\pi$
$620$	0	0
$621$	12.8029	0.513762
$622$	0	0
$623$	30.3970	1.21783
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.00000	−0.0798723
$628$	0	0
$629$	47.4706	1.89278
$630$	0	0
$631$	−5.10970	−0.203414	−0.101707	−	0.994814i	$-0.532430\pi$
$631$	−5.10970	−0.203414	−0.101707	+	0.994814i	$0.532430\pi$
$632$	0	0
$633$	−31.5210	−1.25285
$634$	0	0
$635$	7.97481	0.316471
$636$	0	0
$637$	−32.6844	−1.29500
$638$	0	0
$639$	0.247970	0.00980955
$640$	0	0
$641$	9.21585	0.364004	0.182002	−	0.983298i	$-0.441742\pi$
$641$	9.21585	0.364004	0.182002	+	0.983298i	$0.441742\pi$
$642$	0	0
$643$	11.9126	0.469786	0.234893	−	0.972021i	$-0.424526\pi$
$643$	11.9126	0.469786	0.234893	+	0.972021i	$0.424526\pi$
$644$	0	0
$645$	−3.55262	−0.139884
$646$	0	0
$647$	−10.8301	−0.425774	−0.212887	−	0.977077i	$-0.568287\pi$
$647$	−10.8301	−0.425774	−0.212887	+	0.977077i	$0.568287\pi$
$648$	0	0
$649$	20.1116	0.789448
$650$	0	0
$651$	36.0830	1.41420
$652$	0	0
$653$	9.27607	0.363001	0.181500	−	0.983391i	$-0.441905\pi$
$653$	9.27607	0.363001	0.181500	+	0.983391i	$0.441905\pi$
$654$	0	0
$655$	2.62222	0.102458
$656$	0	0
$657$	12.1432	0.473751
$658$	0	0
$659$	19.0192	0.740883	0.370442	−	0.928856i	$-0.379206\pi$
$659$	19.0192	0.740883	0.370442	+	0.928856i	$0.379206\pi$
$660$	0	0
$661$	−45.5022	−1.76983	−0.884916	−	0.465750i	$-0.845784\pi$
$661$	−45.5022	−1.76983	−0.884916	+	0.465750i	$0.845784\pi$
$662$	0	0
$663$	39.8707	1.54845
$664$	0	0
$665$	−3.52543	−0.136710
$666$	0	0
$667$	−10.5433	−0.408237
$668$	0	0
$669$	16.9447	0.655120
$670$	0	0
$671$	8.42864	0.325384
$672$	0	0
$673$	42.2242	1.62762	0.813811	−	0.581130i	$-0.197389\pi$
$673$	42.2242	1.62762	0.813811	+	0.581130i	$0.197389\pi$
$674$	0	0
$675$	5.61285	0.216039
$676$	0	0
$677$	−24.4494	−0.939666	−0.469833	−	0.882755i	$-0.655686\pi$
$677$	−24.4494	−0.939666	−0.469833	+	0.882755i	$0.655686\pi$
$678$	0	0
$679$	25.4005	0.974783
$680$	0	0
$681$	32.7570	1.25525
$682$	0	0
$683$	−35.9333	−1.37495	−0.687475	−	0.726208i	$-0.741281\pi$
$683$	−35.9333	−1.37495	−0.687475	+	0.726208i	$0.741281\pi$
$684$	0	0
$685$	2.85728	0.109171
$686$	0	0
$687$	7.41927	0.283063
$688$	0	0
$689$	−59.1669	−2.25408
$690$	0	0
$691$	6.08742	0.231576	0.115788	−	0.993274i	$-0.463061\pi$
$691$	6.08742	0.231576	0.115788	+	0.993274i	$0.463061\pi$
$692$	0	0
$693$	6.88892	0.261689
$694$	0	0
$695$	9.13828	0.346635
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−23.6543	−0.894689
$700$	0	0
$701$	36.0973	1.36337	0.681687	−	0.731644i	$-0.261246\pi$
$701$	36.0973	1.36337	0.681687	+	0.731644i	$0.261246\pi$
$702$	0	0
$703$	−9.39853	−0.354472
$704$	0	0
$705$	6.99063	0.263282
$706$	0	0
$707$	−33.7275	−1.26845
$708$	0	0
$709$	40.7338	1.52979	0.764894	−	0.644156i	$-0.222791\pi$
$709$	40.7338	1.52979	0.764894	+	0.644156i	$0.222791\pi$
$710$	0	0
$711$	−12.8631	−0.482404
$712$	0	0
$713$	−17.8064	−0.666856
$714$	0	0
$715$	9.18421	0.343470
$716$	0	0
$717$	−10.7012	−0.399643
$718$	0	0
$719$	25.9126	0.966376	0.483188	−	0.875517i	$-0.339479\pi$
$719$	25.9126	0.966376	0.483188	+	0.875517i	$0.339479\pi$
$720$	0	0
$721$	−30.6321	−1.14080
$722$	0	0
$723$	21.3649	0.794568
$724$	0	0
$725$	−4.62222	−0.171665
$726$	0	0
$727$	−25.6400	−0.950937	−0.475468	−	0.879733i	$-0.657721\pi$
$727$	−25.6400	−0.950937	−0.475468	+	0.879733i	$0.657721\pi$
$728$	0	0
$729$	26.5812	0.984489
$730$	0	0
$731$	13.6860	0.506194
$732$	0	0
$733$	49.7659	1.83815	0.919073	−	0.394088i	$-0.128940\pi$
$733$	49.7659	1.83815	0.919073	+	0.394088i	$0.128940\pi$
$734$	0	0
$735$	−7.11753	−0.262534
$736$	0	0
$737$	9.25428	0.340886
$738$	0	0
$739$	18.8760	0.694365	0.347183	−	0.937798i	$-0.387138\pi$
$739$	18.8760	0.694365	0.347183	+	0.937798i	$0.387138\pi$
$740$	0	0
$741$	−7.89384	−0.289988
$742$	0	0
$743$	43.4039	1.59234	0.796168	−	0.605076i	$-0.206857\pi$
$743$	43.4039	1.59234	0.796168	+	0.605076i	$0.206857\pi$
$744$	0	0
$745$	13.8207	0.506352
$746$	0	0
$747$	17.6271	0.644943
$748$	0	0
$749$	−67.1852	−2.45489
$750$	0	0
$751$	−34.8256	−1.27081	−0.635403	−	0.772181i	$-0.719166\pi$
$751$	−34.8256	−1.27081	−0.635403	+	0.772181i	$0.719166\pi$
$752$	0	0
$753$	−21.9813	−0.801042
$754$	0	0
$755$	0.815792	0.0296897
$756$	0	0
$757$	−31.3689	−1.14012	−0.570061	−	0.821602i	$-0.693080\pi$
$757$	−31.3689	−1.14012	−0.570061	+	0.821602i	$0.693080\pi$
$758$	0	0
$759$	4.56199	0.165590
$760$	0	0
$761$	25.1798	0.912766	0.456383	−	0.889784i	$-0.349145\pi$
$761$	25.1798	0.912766	0.456383	+	0.889784i	$0.349145\pi$
$762$	0	0
$763$	−5.89829	−0.213532
$764$	0	0
$765$	−6.47013	−0.233928
$766$	0	0
$767$	79.3787	2.86620
$768$	0	0
$769$	8.91306	0.321413	0.160707	−	0.987002i	$-0.448623\pi$
$769$	8.91306	0.321413	0.160707	+	0.987002i	$0.448623\pi$
$770$	0	0
$771$	23.9353	0.862010
$772$	0	0
$773$	19.9891	0.718958	0.359479	−	0.933153i	$-0.382954\pi$
$773$	19.9891	0.718958	0.359479	+	0.933153i	$0.382954\pi$
$774$	0	0
$775$	−7.80642	−0.280415
$776$	0	0
$777$	−43.4420	−1.55847
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.295286	0.0105662
$782$	0	0
$783$	−25.9438	−0.927156
$784$	0	0
$785$	14.8573	0.530279
$786$	0	0
$787$	28.8736	1.02923	0.514616	−	0.857421i	$-0.327935\pi$
$787$	28.8736	1.02923	0.514616	+	0.857421i	$0.327935\pi$
$788$	0	0
$789$	−18.9532	−0.674750
$790$	0	0
$791$	2.26671	0.0805948
$792$	0	0
$793$	33.2672	1.18135
$794$	0	0
$795$	−12.8845	−0.456965
$796$	0	0
$797$	17.4400	0.617757	0.308879	−	0.951101i	$-0.400046\pi$
$797$	17.4400	0.617757	0.308879	+	0.951101i	$0.400046\pi$
$798$	0	0
$799$	−26.9304	−0.952729
$800$	0	0
$801$	−11.0450	−0.390257
$802$	0	0
$803$	14.4603	0.510292
$804$	0	0
$805$	8.04149	0.283425
$806$	0	0
$807$	3.43801	0.121024
$808$	0	0
$809$	−0.530350	−0.0186461	−0.00932305	−	0.999957i	$-0.502968\pi$
$809$	−0.530350	−0.0186461	−0.00932305	+	0.999957i	$0.502968\pi$
$810$	0	0
$811$	7.21585	0.253383	0.126691	−	0.991942i	$-0.459564\pi$
$811$	7.21585	0.253383	0.126691	+	0.991942i	$0.459564\pi$
$812$	0	0
$813$	12.6093	0.442228
$814$	0	0
$815$	−12.3412	−0.432294
$816$	0	0
$817$	−2.70964	−0.0947982
$818$	0	0
$819$	27.1900	0.950097
$820$	0	0
$821$	50.2449	1.75356	0.876780	−	0.480892i	$-0.159687\pi$
$821$	50.2449	1.75356	0.876780	+	0.480892i	$0.159687\pi$
$822$	0	0
$823$	0.369800	0.0128904	0.00644520	−	0.999979i	$-0.497948\pi$
$823$	0.369800	0.0128904	0.00644520	+	0.999979i	$0.497948\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	8.27010	0.287579	0.143790	−	0.989608i	$-0.454071\pi$
$827$	8.27010	0.287579	0.143790	+	0.989608i	$0.454071\pi$
$828$	0	0
$829$	22.4099	0.778328	0.389164	−	0.921168i	$-0.372764\pi$
$829$	22.4099	0.778328	0.389164	+	0.921168i	$0.372764\pi$
$830$	0	0
$831$	−31.0923	−1.07858
$832$	0	0
$833$	27.4193	0.950021
$834$	0	0
$835$	−15.5877	−0.539433
$836$	0	0
$837$	−43.8163	−1.51451
$838$	0	0
$839$	26.0129	0.898065	0.449033	−	0.893515i	$-0.351769\pi$
$839$	26.0129	0.898065	0.449033	+	0.893515i	$0.351769\pi$
$840$	0	0
$841$	−7.63512	−0.263280
$842$	0	0
$843$	25.2070	0.868174
$844$	0	0
$845$	23.2494	0.799802
$846$	0	0
$847$	−30.5763	−1.05061
$848$	0	0
$849$	−27.6731	−0.949737
$850$	0	0
$851$	21.4380	0.734885
$852$	0	0
$853$	11.8064	0.404244	0.202122	−	0.979360i	$-0.435216\pi$
$853$	11.8064	0.404244	0.202122	+	0.979360i	$0.435216\pi$
$854$	0	0
$855$	1.28100	0.0438091
$856$	0	0
$857$	15.8459	0.541286	0.270643	−	0.962680i	$-0.412764\pi$
$857$	15.8459	0.541286	0.270643	+	0.962680i	$0.412764\pi$
$858$	0	0
$859$	0.225219	0.00768437	0.00384219	−	0.999993i	$-0.498777\pi$
$859$	0.225219	0.00768437	0.00384219	+	0.999993i	$0.498777\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.7719	−1.59213	−0.796066	−	0.605209i	$-0.793089\pi$
$863$	−46.7719	−1.59213	−0.796066	+	0.605209i	$0.793089\pi$
$864$	0	0
$865$	−5.03011	−0.171029
$866$	0	0
$867$	−11.1590	−0.378980
$868$	0	0
$869$	−15.3176	−0.519613
$870$	0	0
$871$	36.5259	1.23763
$872$	0	0
$873$	−9.22952	−0.312372
$874$	0	0
$875$	3.52543	0.119181
$876$	0	0
$877$	30.9797	1.04611	0.523056	−	0.852299i	$-0.324792\pi$
$877$	30.9797	1.04611	0.523056	+	0.852299i	$0.324792\pi$
$878$	0	0
$879$	−14.1289	−0.476556
$880$	0	0
$881$	−28.5477	−0.961797	−0.480898	−	0.876776i	$-0.659689\pi$
$881$	−28.5477	−0.961797	−0.480898	+	0.876776i	$0.659689\pi$
$882$	0	0
$883$	−5.06515	−0.170456	−0.0852279	−	0.996361i	$-0.527162\pi$
$883$	−5.06515	−0.170456	−0.0852279	+	0.996361i	$0.527162\pi$
$884$	0	0
$885$	17.2859	0.581060
$886$	0	0
$887$	40.3303	1.35416	0.677080	−	0.735910i	$-0.263245\pi$
$887$	40.3303	1.35416	0.677080	+	0.735910i	$0.263245\pi$
$888$	0	0
$889$	28.1146	0.942934
$890$	0	0
$891$	5.36349	0.179684
$892$	0	0
$893$	5.33185	0.178424
$894$	0	0
$895$	−22.0415	−0.736766
$896$	0	0
$897$	18.0058	0.601197
$898$	0	0
$899$	36.0830	1.20343
$900$	0	0
$901$	49.6356	1.65360
$902$	0	0
$903$	−12.5245	−0.416790
$904$	0	0
$905$	−2.85728	−0.0949792
$906$	0	0
$907$	−28.3017	−0.939744	−0.469872	−	0.882735i	$-0.655700\pi$
$907$	−28.3017	−0.939744	−0.469872	+	0.882735i	$0.655700\pi$
$908$	0	0
$909$	12.2552	0.406479
$910$	0	0
$911$	24.3368	0.806313	0.403157	−	0.915131i	$-0.367913\pi$
$911$	24.3368	0.806313	0.403157	+	0.915131i	$0.367913\pi$
$912$	0	0
$913$	20.9906	0.694689
$914$	0	0
$915$	7.24443	0.239494
$916$	0	0
$917$	9.24443	0.305278
$918$	0	0
$919$	−32.3654	−1.06763	−0.533817	−	0.845600i	$-0.679243\pi$
$919$	−32.3654	−1.06763	−0.533817	+	0.845600i	$0.679243\pi$
$920$	0	0
$921$	30.8470	1.01644
$922$	0	0
$923$	1.16547	0.0383620
$924$	0	0
$925$	9.39853	0.309022
$926$	0	0
$927$	11.1304	0.365572
$928$	0	0
$929$	−32.6923	−1.07260	−0.536300	−	0.844028i	$-0.680178\pi$
$929$	−32.6923	−1.07260	−0.536300	+	0.844028i	$0.680178\pi$
$930$	0	0
$931$	−5.42864	−0.177916
$932$	0	0
$933$	−23.5941	−0.772437
$934$	0	0
$935$	−7.70471	−0.251971
$936$	0	0
$937$	−42.9906	−1.40444	−0.702221	−	0.711959i	$-0.747808\pi$
$937$	−42.9906	−1.40444	−0.702221	+	0.711959i	$0.747808\pi$
$938$	0	0
$939$	40.5116	1.32205
$940$	0	0
$941$	36.9501	1.20454	0.602269	−	0.798293i	$-0.294263\pi$
$941$	36.9501	1.20454	0.602269	+	0.798293i	$0.294263\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	19.7877	0.643694
$946$	0	0
$947$	−6.30958	−0.205034	−0.102517	−	0.994731i	$-0.532690\pi$
$947$	−6.30958	−0.205034	−0.102517	+	0.994731i	$0.532690\pi$
$948$	0	0
$949$	57.0736	1.85269
$950$	0	0
$951$	−40.6593	−1.31847
$952$	0	0
$953$	−45.0400	−1.45899	−0.729494	−	0.683988i	$-0.760244\pi$
$953$	−45.0400	−1.45899	−0.729494	+	0.683988i	$0.760244\pi$
$954$	0	0
$955$	−14.6637	−0.474506
$956$	0	0
$957$	−9.24443	−0.298830
$958$	0	0
$959$	10.0731	0.325278
$960$	0	0
$961$	29.9403	0.965815
$962$	0	0
$963$	24.4123	0.786677
$964$	0	0
$965$	−8.40790	−0.270660
$966$	0	0
$967$	−14.3511	−0.461499	−0.230750	−	0.973013i	$-0.574118\pi$
$967$	−14.3511	−0.461499	−0.230750	+	0.973013i	$0.574118\pi$
$968$	0	0
$969$	6.62222	0.212736
$970$	0	0
$971$	34.8573	1.11862	0.559312	−	0.828957i	$-0.311066\pi$
$971$	34.8573	1.11862	0.559312	+	0.828957i	$0.311066\pi$
$972$	0	0
$973$	32.2163	1.03281
$974$	0	0
$975$	7.89384	0.252805
$976$	0	0
$977$	7.91903	0.253352	0.126676	−	0.991944i	$-0.459569\pi$
$977$	7.91903	0.253352	0.126676	+	0.991944i	$0.459569\pi$
$978$	0	0
$979$	−13.1526	−0.420358
$980$	0	0
$981$	2.14320	0.0684270
$982$	0	0
$983$	56.4766	1.80132	0.900662	−	0.434521i	$-0.143082\pi$
$983$	56.4766	1.80132	0.900662	+	0.434521i	$0.143082\pi$
$984$	0	0
$985$	2.16193	0.0688849
$986$	0	0
$987$	24.6450	0.784458
$988$	0	0
$989$	6.18067	0.196534
$990$	0	0
$991$	−41.0192	−1.30302	−0.651509	−	0.758641i	$-0.725864\pi$
$991$	−41.0192	−1.30302	−0.651509	+	0.758641i	$0.725864\pi$
$992$	0	0
$993$	−9.19004	−0.291637
$994$	0	0
$995$	−11.3461	−0.359697
$996$	0	0
$997$	11.6030	0.367471	0.183735	−	0.982976i	$-0.441181\pi$
$997$	11.6030	0.367471	0.183735	+	0.982976i	$0.441181\pi$
$998$	0	0
$999$	52.7525	1.66902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bn.1.2		3
4.3	odd	2		6080.2.a.cb.1.2		3
8.3	odd	2		3040.2.a.i.1.2	✓	3
8.5	even	2		3040.2.a.o.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.i.1.2	✓	3	8.3	odd	2
3040.2.a.o.1.2	yes	3	8.5	even	2
6080.2.a.bn.1.2		3	1.1	even	1	trivial
6080.2.a.cb.1.2		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)

\(f(q)\)	\(=\)	\(q-1.31111 q^{3} +1.00000 q^{5} +3.52543 q^{7} -1.28100 q^{9} +O(q^{10})\)	\(q-1.31111 q^{3} +1.00000 q^{5} +3.52543 q^{7} -1.28100 q^{9} -1.52543 q^{11} -6.02074 q^{13} -1.31111 q^{15} +5.05086 q^{17} -1.00000 q^{19} -4.62222 q^{21} +2.28100 q^{23} +1.00000 q^{25} +5.61285 q^{27} -4.62222 q^{29} -7.80642 q^{31} +2.00000 q^{33} +3.52543 q^{35} +9.39853 q^{37} +7.89384 q^{39} +2.70964 q^{43} -1.28100 q^{45} -5.33185 q^{47} +5.42864 q^{49} -6.62222 q^{51} +9.82717 q^{53} -1.52543 q^{55} +1.31111 q^{57} -13.1842 q^{59} -5.52543 q^{61} -4.51606 q^{63} -6.02074 q^{65} -6.06668 q^{67} -2.99063 q^{69} -0.193576 q^{71} -9.47949 q^{73} -1.31111 q^{75} -5.37778 q^{77} +10.0415 q^{79} -3.51606 q^{81} -13.7605 q^{83} +5.05086 q^{85} +6.06022 q^{87} +8.62222 q^{89} -21.2257 q^{91} +10.2351 q^{93} -1.00000 q^{95} +7.20495 q^{97} +1.95407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 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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