LMFDB - Embedded newform 6160.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.116404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 2x + 12 $ x^4 - 10x^2 - 2x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.77987$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77987 q^{3} +1.00000 q^{5} +1.00000 q^{7} +4.72768 q^{9} +O(q^{10})$	$q-2.77987 q^{3} +1.00000 q^{5} +1.00000 q^{7} +4.72768 q^{9} +1.00000 q^{11} +2.77987 q^{13} -2.77987 q^{15} -2.02287 q^{17} -3.55974 q^{19} -2.77987 q^{21} -5.72768 q^{23} +1.00000 q^{25} -4.80274 q^{27} +8.80274 q^{29} +7.75055 q^{31} -2.77987 q^{33} +1.00000 q^{35} +0.272316 q^{37} -7.72768 q^{39} -0.507555 q^{41} -6.97068 q^{43} +4.72768 q^{45} +8.23524 q^{47} +1.00000 q^{49} +5.62331 q^{51} +3.72768 q^{53} +1.00000 q^{55} +9.89562 q^{57} +2.19081 q^{59} +8.06730 q^{61} +4.72768 q^{63} +2.77987 q^{65} +4.48469 q^{67} +15.9222 q^{69} -3.45537 q^{71} +6.02287 q^{73} -2.77987 q^{75} +1.00000 q^{77} -6.14881 q^{79} -0.832059 q^{81} +0.802738 q^{83} -2.02287 q^{85} -24.4705 q^{87} -17.7135 q^{89} +2.77987 q^{91} -21.5455 q^{93} -3.55974 q^{95} -10.9107 q^{97} +4.72768 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} + 4 q^{11} + 6 q^{17} + 8 q^{19} - 12 q^{23} + 4 q^{25} + 6 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{35} + 12 q^{37} - 20 q^{39} + 20 q^{41} - 14 q^{43} + 8 q^{45} + 4 q^{49} + 12 q^{51} + 4 q^{53} + 4 q^{55} + 40 q^{57} + 6 q^{59} - 12 q^{61} + 8 q^{63} + 10 q^{67} - 6 q^{69} + 8 q^{71} + 10 q^{73} + 4 q^{77} + 6 q^{79} + 8 q^{81} - 22 q^{83} + 6 q^{85} - 32 q^{87} - 2 q^{89} - 6 q^{93} + 8 q^{95} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 + 4 * q^11 + 6 * q^17 + 8 * q^19 - 12 * q^23 + 4 * q^25 + 6 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^35 + 12 * q^37 - 20 * q^39 + 20 * q^41 - 14 * q^43 + 8 * q^45 + 4 * q^49 + 12 * q^51 + 4 * q^53 + 4 * q^55 + 40 * q^57 + 6 * q^59 - 12 * q^61 + 8 * q^63 + 10 * q^67 - 6 * q^69 + 8 * q^71 + 10 * q^73 + 4 * q^77 + 6 * q^79 + 8 * q^81 - 22 * q^83 + 6 * q^85 - 32 * q^87 - 2 * q^89 - 6 * q^93 + 8 * q^95 + 8 * 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
$3$	−2.77987	−1.60496	−0.802480	−	0.596679i	$-0.796486\pi$
$3$	−2.77987	−1.60496	−0.802480	+	0.596679i	$0.796486\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.72768	1.57589
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.77987	0.770998	0.385499	−	0.922708i	$-0.374029\pi$
$13$	2.77987	0.770998	0.385499	+	0.922708i	$0.374029\pi$
$14$	0	0
$15$	−2.77987	−0.717760
$16$	0	0
$17$	−2.02287	−0.490617	−0.245309	−	0.969445i	$-0.578889\pi$
$17$	−2.02287	−0.490617	−0.245309	+	0.969445i	$0.578889\pi$
$18$	0	0
$19$	−3.55974	−0.816661	−0.408330	−	0.912834i	$-0.633889\pi$
$19$	−3.55974	−0.816661	−0.408330	+	0.912834i	$0.633889\pi$
$20$	0	0
$21$	−2.77987	−0.606618
$22$	0	0
$23$	−5.72768	−1.19430	−0.597152	−	0.802128i	$-0.703701\pi$
$23$	−5.72768	−1.19430	−0.597152	+	0.802128i	$0.703701\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.80274	−0.924287
$28$	0	0
$29$	8.80274	1.63463	0.817314	−	0.576193i	$-0.195462\pi$
$29$	8.80274	1.63463	0.817314	+	0.576193i	$0.195462\pi$
$30$	0	0
$31$	7.75055	1.39204	0.696020	−	0.718022i	$-0.254953\pi$
$31$	7.75055	1.39204	0.696020	+	0.718022i	$0.254953\pi$
$32$	0	0
$33$	−2.77987	−0.483913
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	0.272316	0.0447685	0.0223843	−	0.999749i	$-0.492874\pi$
$37$	0.272316	0.0447685	0.0223843	+	0.999749i	$0.492874\pi$
$38$	0	0
$39$	−7.72768	−1.23742
$40$	0	0
$41$	−0.507555	−0.0792667	−0.0396334	−	0.999214i	$-0.512619\pi$
$41$	−0.507555	−0.0792667	−0.0396334	+	0.999214i	$0.512619\pi$
$42$	0	0
$43$	−6.97068	−1.06302	−0.531509	−	0.847052i	$-0.678375\pi$
$43$	−6.97068	−1.06302	−0.531509	+	0.847052i	$0.678375\pi$
$44$	0	0
$45$	4.72768	0.704761
$46$	0	0
$47$	8.23524	1.20123	0.600616	−	0.799537i	$-0.294922\pi$
$47$	8.23524	1.20123	0.600616	+	0.799537i	$0.294922\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.62331	0.787421
$52$	0	0
$53$	3.72768	0.512037	0.256018	−	0.966672i	$-0.417589\pi$
$53$	3.72768	0.512037	0.256018	+	0.966672i	$0.417589\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	9.89562	1.31071
$58$	0	0
$59$	2.19081	0.285219	0.142609	−	0.989779i	$-0.454451\pi$
$59$	2.19081	0.285219	0.142609	+	0.989779i	$0.454451\pi$
$60$	0	0
$61$	8.06730	1.03291	0.516456	−	0.856314i	$-0.327251\pi$
$61$	8.06730	1.03291	0.516456	+	0.856314i	$0.327251\pi$
$62$	0	0
$63$	4.72768	0.595632
$64$	0	0
$65$	2.77987	0.344801
$66$	0	0
$67$	4.48469	0.547892	0.273946	−	0.961745i	$-0.411671\pi$
$67$	4.48469	0.547892	0.273946	+	0.961745i	$0.411671\pi$
$68$	0	0
$69$	15.9222	1.91681
$70$	0	0
$71$	−3.45537	−0.410077	−0.205038	−	0.978754i	$-0.565732\pi$
$71$	−3.45537	−0.410077	−0.205038	+	0.978754i	$0.565732\pi$
$72$	0	0
$73$	6.02287	0.704923	0.352462	−	0.935826i	$-0.385345\pi$
$73$	6.02287	0.704923	0.352462	+	0.935826i	$0.385345\pi$
$74$	0	0
$75$	−2.77987	−0.320992
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−6.14881	−0.691795	−0.345897	−	0.938272i	$-0.612425\pi$
$79$	−6.14881	−0.691795	−0.345897	+	0.938272i	$0.612425\pi$
$80$	0	0
$81$	−0.832059	−0.0924510
$82$	0	0
$83$	0.802738	0.0881119	0.0440560	−	0.999029i	$-0.485972\pi$
$83$	0.802738	0.0881119	0.0440560	+	0.999029i	$0.485972\pi$
$84$	0	0
$85$	−2.02287	−0.219411
$86$	0	0
$87$	−24.4705	−2.62351
$88$	0	0
$89$	−17.7135	−1.87762	−0.938812	−	0.344430i	$-0.888072\pi$
$89$	−17.7135	−1.87762	−0.938812	+	0.344430i	$0.888072\pi$
$90$	0	0
$91$	2.77987	0.291410
$92$	0	0
$93$	−21.5455	−2.23417
$94$	0	0
$95$	−3.55974	−0.365222
$96$	0	0
$97$	−10.9107	−1.10782	−0.553909	−	0.832577i	$-0.686864\pi$
$97$	−10.9107	−1.10782	−0.553909	+	0.832577i	$0.686864\pi$
$98$	0	0
$99$	4.72768	0.475150
$100$	0	0
$101$	12.1908	1.21303	0.606515	−	0.795072i	$-0.292567\pi$
$101$	12.1908	1.21303	0.606515	+	0.795072i	$0.292567\pi$
$102$	0	0
$103$	−3.43250	−0.338214	−0.169107	−	0.985598i	$-0.554088\pi$
$103$	−3.43250	−0.338214	−0.169107	+	0.985598i	$0.554088\pi$
$104$	0	0
$105$	−2.77987	−0.271288
$106$	0	0
$107$	−11.4554	−1.10743	−0.553716	−	0.832706i	$-0.686791\pi$
$107$	−11.4554	−1.10743	−0.553716	+	0.832706i	$0.686791\pi$
$108$	0	0
$109$	−9.81785	−0.940379	−0.470190	−	0.882565i	$-0.655814\pi$
$109$	−9.81785	−0.940379	−0.470190	+	0.882565i	$0.655814\pi$
$110$	0	0
$111$	−0.757005	−0.0718517
$112$	0	0
$113$	−0.104375	−0.00981879	−0.00490939	−	0.999988i	$-0.501563\pi$
$113$	−0.104375	−0.00981879	−0.00490939	+	0.999988i	$0.501563\pi$
$114$	0	0
$115$	−5.72768	−0.534109
$116$	0	0
$117$	13.1424	1.21501
$118$	0	0
$119$	−2.02287	−0.185436
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.41094	0.127220
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.55974	−0.848290	−0.424145	−	0.905594i	$-0.639425\pi$
$127$	−9.55974	−0.848290	−0.424145	+	0.905594i	$0.639425\pi$
$128$	0	0
$129$	19.3776	1.70610
$130$	0	0
$131$	11.7721	1.02853	0.514267	−	0.857630i	$-0.328064\pi$
$131$	11.7721	1.02853	0.514267	+	0.857630i	$0.328064\pi$
$132$	0	0
$133$	−3.55974	−0.308669
$134$	0	0
$135$	−4.80274	−0.413354
$136$	0	0
$137$	6.53042	0.557932	0.278966	−	0.960301i	$-0.410008\pi$
$137$	6.53042	0.557932	0.278966	+	0.960301i	$0.410008\pi$
$138$	0	0
$139$	−6.21237	−0.526926	−0.263463	−	0.964669i	$-0.584865\pi$
$139$	−6.21237	−0.526926	−0.263463	+	0.964669i	$0.584865\pi$
$140$	0	0
$141$	−22.8929	−1.92793
$142$	0	0
$143$	2.77987	0.232465
$144$	0	0
$145$	8.80274	0.731028
$146$	0	0
$147$	−2.77987	−0.229280
$148$	0	0
$149$	14.6984	1.20414	0.602068	−	0.798445i	$-0.294343\pi$
$149$	14.6984	1.20414	0.602068	+	0.798445i	$0.294343\pi$
$150$	0	0
$151$	19.1639	1.55954	0.779769	−	0.626068i	$-0.215337\pi$
$151$	19.1639	1.55954	0.779769	+	0.626068i	$0.215337\pi$
$152$	0	0
$153$	−9.56347	−0.773161
$154$	0	0
$155$	7.75055	0.622539
$156$	0	0
$157$	−4.69836	−0.374970	−0.187485	−	0.982267i	$-0.560034\pi$
$157$	−4.69836	−0.374970	−0.187485	+	0.982267i	$0.560034\pi$
$158$	0	0
$159$	−10.3625	−0.821798
$160$	0	0
$161$	−5.72768	−0.451405
$162$	0	0
$163$	−2.97068	−0.232682	−0.116341	−	0.993209i	$-0.537116\pi$
$163$	−2.97068	−0.232682	−0.116341	+	0.993209i	$0.537116\pi$
$164$	0	0
$165$	−2.77987	−0.216413
$166$	0	0
$167$	9.81785	0.759728	0.379864	−	0.925042i	$-0.375971\pi$
$167$	9.81785	0.759728	0.379864	+	0.925042i	$0.375971\pi$
$168$	0	0
$169$	−5.27232	−0.405563
$170$	0	0
$171$	−16.8293	−1.28697
$172$	0	0
$173$	−4.12724	−0.313788	−0.156894	−	0.987615i	$-0.550148\pi$
$173$	−4.12724	−0.313788	−0.156894	+	0.987615i	$0.550148\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−6.09016	−0.457765
$178$	0	0
$179$	15.6641	1.17079	0.585396	−	0.810748i	$-0.300939\pi$
$179$	15.6641	1.17079	0.585396	+	0.810748i	$0.300939\pi$
$180$	0	0
$181$	23.8178	1.77037	0.885183	−	0.465242i	$-0.154033\pi$
$181$	23.8178	1.77037	0.885183	+	0.465242i	$0.154033\pi$
$182$	0	0
$183$	−22.4260	−1.65778
$184$	0	0
$185$	0.272316	0.0200211
$186$	0	0
$187$	−2.02287	−0.147927
$188$	0	0
$189$	−4.80274	−0.349348
$190$	0	0
$191$	−9.01511	−0.652310	−0.326155	−	0.945316i	$-0.605753\pi$
$191$	−9.01511	−0.652310	−0.326155	+	0.945316i	$0.605753\pi$
$192$	0	0
$193$	−6.14881	−0.442601	−0.221300	−	0.975206i	$-0.571030\pi$
$193$	−6.14881	−0.442601	−0.221300	+	0.975206i	$0.571030\pi$
$194$	0	0
$195$	−7.72768	−0.553391
$196$	0	0
$197$	−15.5455	−1.10757	−0.553787	−	0.832659i	$-0.686818\pi$
$197$	−15.5455	−1.10757	−0.553787	+	0.832659i	$0.686818\pi$
$198$	0	0
$199$	−1.09792	−0.0778295	−0.0389148	−	0.999243i	$-0.512390\pi$
$199$	−1.09792	−0.0778295	−0.0389148	+	0.999243i	$0.512390\pi$
$200$	0	0
$201$	−12.4669	−0.879344
$202$	0	0
$203$	8.80274	0.617831
$204$	0	0
$205$	−0.507555	−0.0354492
$206$	0	0
$207$	−27.0787	−1.88210
$208$	0	0
$209$	−3.55974	−0.246233
$210$	0	0
$211$	−0.926249	−0.0637656	−0.0318828	−	0.999492i	$-0.510150\pi$
$211$	−0.926249	−0.0637656	−0.0318828	+	0.999492i	$0.510150\pi$
$212$	0	0
$213$	9.60548	0.658156
$214$	0	0
$215$	−6.97068	−0.475396
$216$	0	0
$217$	7.75055	0.526142
$218$	0	0
$219$	−16.7428	−1.13137
$220$	0	0
$221$	−5.62331	−0.378265
$222$	0	0
$223$	27.2312	1.82354	0.911768	−	0.410705i	$-0.134717\pi$
$223$	27.2312	1.82354	0.911768	+	0.410705i	$0.134717\pi$
$224$	0	0
$225$	4.72768	0.315179
$226$	0	0
$227$	10.4860	0.695980	0.347990	−	0.937498i	$-0.386864\pi$
$227$	10.4860	0.695980	0.347990	+	0.937498i	$0.386864\pi$
$228$	0	0
$229$	−6.59037	−0.435504	−0.217752	−	0.976004i	$-0.569872\pi$
$229$	−6.59037	−0.435504	−0.217752	+	0.976004i	$0.569872\pi$
$230$	0	0
$231$	−2.77987	−0.182902
$232$	0	0
$233$	−25.1053	−1.64470	−0.822351	−	0.568981i	$-0.807338\pi$
$233$	−25.1053	−1.64470	−0.822351	+	0.568981i	$0.807338\pi$
$234$	0	0
$235$	8.23524	0.537208
$236$	0	0
$237$	17.0929	1.11030
$238$	0	0
$239$	8.90711	0.576153	0.288077	−	0.957607i	$-0.406984\pi$
$239$	8.90711	0.576153	0.288077	+	0.957607i	$0.406984\pi$
$240$	0	0
$241$	−1.05219	−0.0677774	−0.0338887	−	0.999426i	$-0.510789\pi$
$241$	−1.05219	−0.0677774	−0.0338887	+	0.999426i	$0.510789\pi$
$242$	0	0
$243$	16.7212	1.07267
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−9.89562	−0.629644
$248$	0	0
$249$	−2.23151	−0.141416
$250$	0	0
$251$	15.4147	0.972965	0.486483	−	0.873690i	$-0.338280\pi$
$251$	15.4147	0.972965	0.486483	+	0.873690i	$0.338280\pi$
$252$	0	0
$253$	−5.72768	−0.360096
$254$	0	0
$255$	5.62331	0.352145
$256$	0	0
$257$	−1.89562	−0.118246	−0.0591229	−	0.998251i	$-0.518830\pi$
$257$	−1.89562	−0.118246	−0.0591229	+	0.998251i	$0.518830\pi$
$258$	0	0
$259$	0.272316	0.0169209
$260$	0	0
$261$	41.6166	2.57600
$262$	0	0
$263$	16.4705	1.01561	0.507807	−	0.861471i	$-0.330456\pi$
$263$	16.4705	1.01561	0.507807	+	0.861471i	$0.330456\pi$
$264$	0	0
$265$	3.72768	0.228990
$266$	0	0
$267$	49.2412	3.01351
$268$	0	0
$269$	13.8765	0.846064	0.423032	−	0.906115i	$-0.360966\pi$
$269$	13.8765	0.846064	0.423032	+	0.906115i	$0.360966\pi$
$270$	0	0
$271$	4.04573	0.245761	0.122880	−	0.992421i	$-0.460787\pi$
$271$	4.04573	0.245761	0.122880	+	0.992421i	$0.460787\pi$
$272$	0	0
$273$	−7.72768	−0.467701
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	30.9552	1.85992	0.929958	−	0.367666i	$-0.119843\pi$
$277$	30.9552	1.85992	0.929958	+	0.367666i	$0.119843\pi$
$278$	0	0
$279$	36.6421	2.19371
$280$	0	0
$281$	−6.63349	−0.395721	−0.197861	−	0.980230i	$-0.563399\pi$
$281$	−6.63349	−0.395721	−0.197861	+	0.980230i	$0.563399\pi$
$282$	0	0
$283$	−6.57485	−0.390834	−0.195417	−	0.980720i	$-0.562606\pi$
$283$	−6.57485	−0.390834	−0.195417	+	0.980720i	$0.562606\pi$
$284$	0	0
$285$	9.89562	0.586166
$286$	0	0
$287$	−0.507555	−0.0299600
$288$	0	0
$289$	−12.9080	−0.759295
$290$	0	0
$291$	30.3304	1.77800
$292$	0	0
$293$	−5.47823	−0.320042	−0.160021	−	0.987114i	$-0.551156\pi$
$293$	−5.47823	−0.320042	−0.160021	+	0.987114i	$0.551156\pi$
$294$	0	0
$295$	2.19081	0.127554
$296$	0	0
$297$	−4.80274	−0.278683
$298$	0	0
$299$	−15.9222	−0.920806
$300$	0	0
$301$	−6.97068	−0.401783
$302$	0	0
$303$	−33.8889	−1.94687
$304$	0	0
$305$	8.06730	0.461932
$306$	0	0
$307$	14.9960	0.855865	0.427933	−	0.903811i	$-0.359242\pi$
$307$	14.9960	0.855865	0.427933	+	0.903811i	$0.359242\pi$
$308$	0	0
$309$	9.54191	0.542820
$310$	0	0
$311$	29.1410	1.65244	0.826219	−	0.563350i	$-0.190488\pi$
$311$	29.1410	1.65244	0.826219	+	0.563350i	$0.190488\pi$
$312$	0	0
$313$	3.28873	0.185890	0.0929450	−	0.995671i	$-0.470372\pi$
$313$	3.28873	0.185890	0.0929450	+	0.995671i	$0.470372\pi$
$314$	0	0
$315$	4.72768	0.266375
$316$	0	0
$317$	28.3496	1.59227	0.796135	−	0.605119i	$-0.206874\pi$
$317$	28.3496	1.59227	0.796135	+	0.605119i	$0.206874\pi$
$318$	0	0
$319$	8.80274	0.492859
$320$	0	0
$321$	31.8444	1.77738
$322$	0	0
$323$	7.20088	0.400668
$324$	0	0
$325$	2.77987	0.154200
$326$	0	0
$327$	27.2924	1.50927
$328$	0	0
$329$	8.23524	0.454023
$330$	0	0
$331$	−8.92625	−0.490631	−0.245315	−	0.969443i	$-0.578892\pi$
$331$	−8.92625	−0.490631	−0.245315	+	0.969443i	$0.578892\pi$
$332$	0	0
$333$	1.28743	0.0705505
$334$	0	0
$335$	4.48469	0.245025
$336$	0	0
$337$	−7.54553	−0.411031	−0.205516	−	0.978654i	$-0.565887\pi$
$337$	−7.54553	−0.411031	−0.205516	+	0.978654i	$0.565887\pi$
$338$	0	0
$339$	0.290149	0.0157588
$340$	0	0
$341$	7.75055	0.419716
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.9222	0.857224
$346$	0	0
$347$	28.6650	1.53882	0.769409	−	0.638756i	$-0.220551\pi$
$347$	28.6650	1.53882	0.769409	+	0.638756i	$0.220551\pi$
$348$	0	0
$349$	−2.29880	−0.123052	−0.0615261	−	0.998105i	$-0.519597\pi$
$349$	−2.29880	−0.123052	−0.0615261	+	0.998105i	$0.519597\pi$
$350$	0	0
$351$	−13.3510	−0.712623
$352$	0	0
$353$	−4.80274	−0.255624	−0.127812	−	0.991798i	$-0.540795\pi$
$353$	−4.80274	−0.255624	−0.127812	+	0.991798i	$0.540795\pi$
$354$	0	0
$355$	−3.45537	−0.183392
$356$	0	0
$357$	5.62331	0.297617
$358$	0	0
$359$	−5.17943	−0.273360	−0.136680	−	0.990615i	$-0.543643\pi$
$359$	−5.17943	−0.273360	−0.136680	+	0.990615i	$0.543643\pi$
$360$	0	0
$361$	−6.32823	−0.333065
$362$	0	0
$363$	−2.77987	−0.145905
$364$	0	0
$365$	6.02287	0.315251
$366$	0	0
$367$	−14.3396	−0.748522	−0.374261	−	0.927323i	$-0.622104\pi$
$367$	−14.3396	−0.748522	−0.374261	+	0.927323i	$0.622104\pi$
$368$	0	0
$369$	−2.39956	−0.124916
$370$	0	0
$371$	3.72768	0.193532
$372$	0	0
$373$	−16.0902	−0.833117	−0.416559	−	0.909109i	$-0.636764\pi$
$373$	−16.0902	−0.833117	−0.416559	+	0.909109i	$0.636764\pi$
$374$	0	0
$375$	−2.77987	−0.143552
$376$	0	0
$377$	24.4705	1.26029
$378$	0	0
$379$	4.54463	0.233442	0.116721	−	0.993165i	$-0.462762\pi$
$379$	4.54463	0.233442	0.116721	+	0.993165i	$0.462762\pi$
$380$	0	0
$381$	26.5749	1.36147
$382$	0	0
$383$	−10.5089	−0.536978	−0.268489	−	0.963283i	$-0.586524\pi$
$383$	−10.5089	−0.536978	−0.268489	+	0.963283i	$0.586524\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−32.9552	−1.67521
$388$	0	0
$389$	35.5313	1.80151	0.900755	−	0.434328i	$-0.143014\pi$
$389$	35.5313	1.80151	0.900755	+	0.434328i	$0.143014\pi$
$390$	0	0
$391$	11.5863	0.585946
$392$	0	0
$393$	−32.7250	−1.65076
$394$	0	0
$395$	−6.14881	−0.309380
$396$	0	0
$397$	18.8485	0.945978	0.472989	−	0.881068i	$-0.343175\pi$
$397$	18.8485	0.945978	0.472989	+	0.881068i	$0.343175\pi$
$398$	0	0
$399$	9.89562	0.495401
$400$	0	0
$401$	−10.7632	−0.537490	−0.268745	−	0.963211i	$-0.586609\pi$
$401$	−10.7632	−0.537490	−0.268745	+	0.963211i	$0.586609\pi$
$402$	0	0
$403$	21.5455	1.07326
$404$	0	0
$405$	−0.832059	−0.0413453
$406$	0	0
$407$	0.272316	0.0134982
$408$	0	0
$409$	35.2490	1.74295	0.871477	−	0.490437i	$-0.163163\pi$
$409$	35.2490	1.74295	0.871477	+	0.490437i	$0.163163\pi$
$410$	0	0
$411$	−18.1537	−0.895457
$412$	0	0
$413$	2.19081	0.107803
$414$	0	0
$415$	0.802738	0.0394048
$416$	0	0
$417$	17.2696	0.845696
$418$	0	0
$419$	9.26094	0.452426	0.226213	−	0.974078i	$-0.427365\pi$
$419$	9.26094	0.452426	0.226213	+	0.974078i	$0.427365\pi$
$420$	0	0
$421$	−7.28743	−0.355167	−0.177584	−	0.984106i	$-0.556828\pi$
$421$	−7.28743	−0.355167	−0.177584	+	0.984106i	$0.556828\pi$
$422$	0	0
$423$	38.9336	1.89302
$424$	0	0
$425$	−2.02287	−0.0981234
$426$	0	0
$427$	8.06730	0.390404
$428$	0	0
$429$	−7.72768	−0.373096
$430$	0	0
$431$	9.60417	0.462617	0.231308	−	0.972880i	$-0.425699\pi$
$431$	9.60417	0.462617	0.231308	+	0.972880i	$0.425699\pi$
$432$	0	0
$433$	−29.6091	−1.42292	−0.711461	−	0.702725i	$-0.751966\pi$
$433$	−29.6091	−1.42292	−0.711461	+	0.702725i	$0.751966\pi$
$434$	0	0
$435$	−24.4705	−1.17327
$436$	0	0
$437$	20.3891	0.975342
$438$	0	0
$439$	27.5011	1.31256	0.656278	−	0.754519i	$-0.272130\pi$
$439$	27.5011	1.31256	0.656278	+	0.754519i	$0.272130\pi$
$440$	0	0
$441$	4.72768	0.225128
$442$	0	0
$443$	9.70855	0.461267	0.230634	−	0.973041i	$-0.425920\pi$
$443$	9.70855	0.461267	0.230634	+	0.973041i	$0.425920\pi$
$444$	0	0
$445$	−17.7135	−0.839699
$446$	0	0
$447$	−40.8596	−1.93259
$448$	0	0
$449$	−19.9209	−0.940126	−0.470063	−	0.882633i	$-0.655769\pi$
$449$	−19.9209	−0.940126	−0.470063	+	0.882633i	$0.655769\pi$
$450$	0	0
$451$	−0.507555	−0.0238998
$452$	0	0
$453$	−53.2732	−2.50299
$454$	0	0
$455$	2.77987	0.130322
$456$	0	0
$457$	5.17943	0.242283	0.121142	−	0.992635i	$-0.461344\pi$
$457$	5.17943	0.242283	0.121142	+	0.992635i	$0.461344\pi$
$458$	0	0
$459$	9.71530	0.453471
$460$	0	0
$461$	31.8981	1.48564	0.742820	−	0.669491i	$-0.233488\pi$
$461$	31.8981	1.48564	0.742820	+	0.669491i	$0.233488\pi$
$462$	0	0
$463$	−29.5264	−1.37221	−0.686104	−	0.727503i	$-0.740681\pi$
$463$	−29.5264	−1.37221	−0.686104	+	0.727503i	$0.740681\pi$
$464$	0	0
$465$	−21.5455	−0.999150
$466$	0	0
$467$	−23.3418	−1.08013	−0.540065	−	0.841623i	$-0.681600\pi$
$467$	−23.3418	−1.08013	−0.540065	+	0.841623i	$0.681600\pi$
$468$	0	0
$469$	4.48469	0.207084
$470$	0	0
$471$	13.0608	0.601812
$472$	0	0
$473$	−6.97068	−0.320512
$474$	0	0
$475$	−3.55974	−0.163332
$476$	0	0
$477$	17.6233	0.806916
$478$	0	0
$479$	37.4820	1.71259	0.856297	−	0.516483i	$-0.172759\pi$
$479$	37.4820	1.71259	0.856297	+	0.516483i	$0.172759\pi$
$480$	0	0
$481$	0.757005	0.0345164
$482$	0	0
$483$	15.9222	0.724486
$484$	0	0
$485$	−10.9107	−0.495431
$486$	0	0
$487$	10.0253	0.454289	0.227145	−	0.973861i	$-0.427061\pi$
$487$	10.0253	0.454289	0.227145	+	0.973861i	$0.427061\pi$
$488$	0	0
$489$	8.25810	0.373444
$490$	0	0
$491$	−0.742793	−0.0335218	−0.0167609	−	0.999860i	$-0.505335\pi$
$491$	−0.742793	−0.0335218	−0.0167609	+	0.999860i	$0.505335\pi$
$492$	0	0
$493$	−17.8068	−0.801976
$494$	0	0
$495$	4.72768	0.212494
$496$	0	0
$497$	−3.45537	−0.154994
$498$	0	0
$499$	−7.02802	−0.314617	−0.157309	−	0.987549i	$-0.550282\pi$
$499$	−7.02802	−0.314617	−0.157309	+	0.987549i	$0.550282\pi$
$500$	0	0
$501$	−27.2924	−1.21933
$502$	0	0
$503$	−30.6242	−1.36547	−0.682733	−	0.730668i	$-0.739209\pi$
$503$	−30.6242	−1.36547	−0.682733	+	0.730668i	$0.739209\pi$
$504$	0	0
$505$	12.1908	0.542484
$506$	0	0
$507$	14.6564	0.650912
$508$	0	0
$509$	36.7094	1.62712	0.813559	−	0.581483i	$-0.197527\pi$
$509$	36.7094	1.62712	0.813559	+	0.581483i	$0.197527\pi$
$510$	0	0
$511$	6.02287	0.266436
$512$	0	0
$513$	17.0965	0.754829
$514$	0	0
$515$	−3.43250	−0.151254
$516$	0	0
$517$	8.23524	0.362185
$518$	0	0
$519$	11.4732	0.503618
$520$	0	0
$521$	1.33186	0.0583497	0.0291748	−	0.999574i	$-0.490712\pi$
$521$	1.33186	0.0583497	0.0291748	+	0.999574i	$0.490712\pi$
$522$	0	0
$523$	−1.59925	−0.0699303	−0.0349651	−	0.999389i	$-0.511132\pi$
$523$	−1.59925	−0.0699303	−0.0349651	+	0.999389i	$0.511132\pi$
$524$	0	0
$525$	−2.77987	−0.121324
$526$	0	0
$527$	−15.6783	−0.682959
$528$	0	0
$529$	9.80636	0.426363
$530$	0	0
$531$	10.3574	0.449475
$532$	0	0
$533$	−1.41094	−0.0611145
$534$	0	0
$535$	−11.4554	−0.495259
$536$	0	0
$537$	−43.5442	−1.87907
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	36.3038	1.56082	0.780412	−	0.625266i	$-0.215009\pi$
$541$	36.3038	1.56082	0.780412	+	0.625266i	$0.215009\pi$
$542$	0	0
$543$	−66.2105	−2.84137
$544$	0	0
$545$	−9.81785	−0.420550
$546$	0	0
$547$	32.1346	1.37398	0.686988	−	0.726669i	$-0.258932\pi$
$547$	32.1346	1.37398	0.686988	+	0.726669i	$0.258932\pi$
$548$	0	0
$549$	38.1396	1.62776
$550$	0	0
$551$	−31.3355	−1.33494
$552$	0	0
$553$	−6.14881	−0.261474
$554$	0	0
$555$	−0.757005	−0.0321331
$556$	0	0
$557$	−16.1359	−0.683700	−0.341850	−	0.939755i	$-0.611053\pi$
$557$	−16.1359	−0.683700	−0.341850	+	0.939755i	$0.611053\pi$
$558$	0	0
$559$	−19.3776	−0.819585
$560$	0	0
$561$	5.62331	0.237416
$562$	0	0
$563$	−1.22386	−0.0515795	−0.0257898	−	0.999667i	$-0.508210\pi$
$563$	−1.22386	−0.0515795	−0.0257898	+	0.999667i	$0.508210\pi$
$564$	0	0
$565$	−0.104375	−0.00439109
$566$	0	0
$567$	−0.832059	−0.0349432
$568$	0	0
$569$	19.6512	0.823822	0.411911	−	0.911224i	$-0.364862\pi$
$569$	19.6512	0.823822	0.411911	+	0.911224i	$0.364862\pi$
$570$	0	0
$571$	9.55974	0.400063	0.200031	−	0.979789i	$-0.435896\pi$
$571$	9.55974	0.400063	0.200031	+	0.979789i	$0.435896\pi$
$572$	0	0
$573$	25.0608	1.04693
$574$	0	0
$575$	−5.72768	−0.238861
$576$	0	0
$577$	43.2083	1.79879	0.899393	−	0.437140i	$-0.144009\pi$
$577$	43.2083	1.79879	0.899393	+	0.437140i	$0.144009\pi$
$578$	0	0
$579$	17.0929	0.710356
$580$	0	0
$581$	0.802738	0.0333032
$582$	0	0
$583$	3.72768	0.154385
$584$	0	0
$585$	13.1424	0.543369
$586$	0	0
$587$	−27.6777	−1.14238	−0.571190	−	0.820818i	$-0.693518\pi$
$587$	−27.6777	−1.14238	−0.571190	+	0.820818i	$0.693518\pi$
$588$	0	0
$589$	−27.5900	−1.13682
$590$	0	0
$591$	43.2146	1.77761
$592$	0	0
$593$	−45.6981	−1.87659	−0.938297	−	0.345830i	$-0.887598\pi$
$593$	−45.6981	−1.87659	−0.938297	+	0.345830i	$0.887598\pi$
$594$	0	0
$595$	−2.02287	−0.0829294
$596$	0	0
$597$	3.05208	0.124913
$598$	0	0
$599$	41.3967	1.69142	0.845712	−	0.533639i	$-0.179176\pi$
$599$	41.3967	1.69142	0.845712	+	0.533639i	$0.179176\pi$
$600$	0	0
$601$	18.7008	0.762821	0.381411	−	0.924406i	$-0.375438\pi$
$601$	18.7008	0.762821	0.381411	+	0.924406i	$0.375438\pi$
$602$	0	0
$603$	21.2022	0.863420
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−20.4211	−0.828868	−0.414434	−	0.910079i	$-0.636020\pi$
$607$	−20.4211	−0.828868	−0.414434	+	0.910079i	$0.636020\pi$
$608$	0	0
$609$	−24.4705	−0.991594
$610$	0	0
$611$	22.8929	0.926148
$612$	0	0
$613$	12.1359	0.490164	0.245082	−	0.969502i	$-0.421185\pi$
$613$	12.1359	0.490164	0.245082	+	0.969502i	$0.421185\pi$
$614$	0	0
$615$	1.41094	0.0568945
$616$	0	0
$617$	36.5606	1.47188	0.735938	−	0.677049i	$-0.236741\pi$
$617$	36.5606	1.47188	0.735938	+	0.677049i	$0.236741\pi$
$618$	0	0
$619$	8.12594	0.326609	0.163305	−	0.986576i	$-0.447785\pi$
$619$	8.12594	0.326609	0.163305	+	0.986576i	$0.447785\pi$
$620$	0	0
$621$	27.5086	1.10388
$622$	0	0
$623$	−17.7135	−0.709675
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.89562	0.395193
$628$	0	0
$629$	−0.550860	−0.0219642
$630$	0	0
$631$	45.6659	1.81793	0.908966	−	0.416871i	$-0.136873\pi$
$631$	45.6659	1.81793	0.908966	+	0.416871i	$0.136873\pi$
$632$	0	0
$633$	2.57485	0.102341
$634$	0	0
$635$	−9.55974	−0.379367
$636$	0	0
$637$	2.77987	0.110143
$638$	0	0
$639$	−16.3359	−0.646238
$640$	0	0
$641$	−37.7708	−1.49186	−0.745929	−	0.666026i	$-0.767994\pi$
$641$	−37.7708	−1.49186	−0.745929	+	0.666026i	$0.767994\pi$
$642$	0	0
$643$	46.8367	1.84706	0.923529	−	0.383528i	$-0.125291\pi$
$643$	46.8367	1.84706	0.923529	+	0.383528i	$0.125291\pi$
$644$	0	0
$645$	19.3776	0.762992
$646$	0	0
$647$	18.5484	0.729211	0.364606	−	0.931162i	$-0.381204\pi$
$647$	18.5484	0.729211	0.364606	+	0.931162i	$0.381204\pi$
$648$	0	0
$649$	2.19081	0.0859967
$650$	0	0
$651$	−21.5455	−0.844436
$652$	0	0
$653$	49.0262	1.91854	0.959272	−	0.282485i	$-0.0911589\pi$
$653$	49.0262	1.91854	0.959272	+	0.282485i	$0.0911589\pi$
$654$	0	0
$655$	11.7721	0.459975
$656$	0	0
$657$	28.4742	1.11088
$658$	0	0
$659$	31.6894	1.23444	0.617222	−	0.786789i	$-0.288258\pi$
$659$	31.6894	1.23444	0.617222	+	0.786789i	$0.288258\pi$
$660$	0	0
$661$	−36.2883	−1.41145	−0.705726	−	0.708485i	$-0.749379\pi$
$661$	−36.2883	−1.41145	−0.705726	+	0.708485i	$0.749379\pi$
$662$	0	0
$663$	15.6321	0.607099
$664$	0	0
$665$	−3.55974	−0.138041
$666$	0	0
$667$	−50.4193	−1.95224
$668$	0	0
$669$	−75.6993	−2.92670
$670$	0	0
$671$	8.06730	0.311435
$672$	0	0
$673$	0.545936	0.0210443	0.0105221	−	0.999945i	$-0.496651\pi$
$673$	0.545936	0.0210443	0.0105221	+	0.999945i	$0.496651\pi$
$674$	0	0
$675$	−4.80274	−0.184857
$676$	0	0
$677$	−49.2832	−1.89411	−0.947053	−	0.321077i	$-0.895955\pi$
$677$	−49.2832	−1.89411	−0.947053	+	0.321077i	$0.895955\pi$
$678$	0	0
$679$	−10.9107	−0.418716
$680$	0	0
$681$	−29.1497	−1.11702
$682$	0	0
$683$	−33.0009	−1.26274	−0.631372	−	0.775480i	$-0.717508\pi$
$683$	−33.0009	−1.26274	−0.631372	+	0.775480i	$0.717508\pi$
$684$	0	0
$685$	6.53042	0.249515
$686$	0	0
$687$	18.3204	0.698966
$688$	0	0
$689$	10.3625	0.394779
$690$	0	0
$691$	−24.4334	−0.929490	−0.464745	−	0.885444i	$-0.653854\pi$
$691$	−24.4334	−0.929490	−0.464745	+	0.885444i	$0.653854\pi$
$692$	0	0
$693$	4.72768	0.179590
$694$	0	0
$695$	−6.21237	−0.235649
$696$	0	0
$697$	1.02672	0.0388896
$698$	0	0
$699$	69.7894	2.63968
$700$	0	0
$701$	−32.6397	−1.23279	−0.616393	−	0.787439i	$-0.711407\pi$
$701$	−32.6397	−1.23279	−0.616393	+	0.787439i	$0.711407\pi$
$702$	0	0
$703$	−0.969376	−0.0365607
$704$	0	0
$705$	−22.8929	−0.862197
$706$	0	0
$707$	12.1908	0.458483
$708$	0	0
$709$	4.03022	0.151358	0.0756790	−	0.997132i	$-0.475888\pi$
$709$	4.03022	0.151358	0.0756790	+	0.997132i	$0.475888\pi$
$710$	0	0
$711$	−29.0696	−1.09020
$712$	0	0
$713$	−44.3927	−1.66252
$714$	0	0
$715$	2.77987	0.103961
$716$	0	0
$717$	−24.7606	−0.924703
$718$	0	0
$719$	30.7008	1.14495	0.572473	−	0.819923i	$-0.305984\pi$
$719$	30.7008	1.14495	0.572473	+	0.819923i	$0.305984\pi$
$720$	0	0
$721$	−3.43250	−0.127833
$722$	0	0
$723$	2.92495	0.108780
$724$	0	0
$725$	8.80274	0.326925
$726$	0	0
$727$	33.7666	1.25233	0.626166	−	0.779690i	$-0.284623\pi$
$727$	33.7666	1.25233	0.626166	+	0.779690i	$0.284623\pi$
$728$	0	0
$729$	−43.9867	−1.62914
$730$	0	0
$731$	14.1008	0.521535
$732$	0	0
$733$	16.3458	0.603747	0.301874	−	0.953348i	$-0.402388\pi$
$733$	16.3458	0.603747	0.301874	+	0.953348i	$0.402388\pi$
$734$	0	0
$735$	−2.77987	−0.102537
$736$	0	0
$737$	4.48469	0.165196
$738$	0	0
$739$	−38.7935	−1.42704	−0.713520	−	0.700635i	$-0.752900\pi$
$739$	−38.7935	−1.42704	−0.713520	+	0.700635i	$0.752900\pi$
$740$	0	0
$741$	27.5086	1.01055
$742$	0	0
$743$	−41.9031	−1.53728	−0.768638	−	0.639685i	$-0.779065\pi$
$743$	−41.9031	−1.53728	−0.768638	+	0.639685i	$0.779065\pi$
$744$	0	0
$745$	14.6984	0.538506
$746$	0	0
$747$	3.79509	0.138855
$748$	0	0
$749$	−11.4554	−0.418570
$750$	0	0
$751$	−12.0457	−0.439555	−0.219778	−	0.975550i	$-0.570533\pi$
$751$	−12.0457	−0.439555	−0.219778	+	0.975550i	$0.570533\pi$
$752$	0	0
$753$	−42.8508	−1.56157
$754$	0	0
$755$	19.1639	0.697446
$756$	0	0
$757$	−26.9071	−0.977956	−0.488978	−	0.872296i	$-0.662630\pi$
$757$	−26.9071	−0.977956	−0.488978	+	0.872296i	$0.662630\pi$
$758$	0	0
$759$	15.9222	0.577940
$760$	0	0
$761$	49.7066	1.80186	0.900932	−	0.433961i	$-0.142884\pi$
$761$	49.7066	1.80186	0.900932	+	0.433961i	$0.142884\pi$
$762$	0	0
$763$	−9.81785	−0.355430
$764$	0	0
$765$	−9.56347	−0.345768
$766$	0	0
$767$	6.09016	0.219903
$768$	0	0
$769$	13.1437	0.473972	0.236986	−	0.971513i	$-0.423840\pi$
$769$	13.1437	0.473972	0.236986	+	0.971513i	$0.423840\pi$
$770$	0	0
$771$	5.26959	0.189780
$772$	0	0
$773$	3.13862	0.112888	0.0564442	−	0.998406i	$-0.482024\pi$
$773$	3.13862	0.112888	0.0564442	+	0.998406i	$0.482024\pi$
$774$	0	0
$775$	7.75055	0.278408
$776$	0	0
$777$	−0.757005	−0.0271574
$778$	0	0
$779$	1.80676	0.0647340
$780$	0	0
$781$	−3.45537	−0.123643
$782$	0	0
$783$	−42.2772	−1.51087
$784$	0	0
$785$	−4.69836	−0.167692
$786$	0	0
$787$	−30.6206	−1.09151	−0.545753	−	0.837946i	$-0.683756\pi$
$787$	−30.6206	−1.09151	−0.545753	+	0.837946i	$0.683756\pi$
$788$	0	0
$789$	−45.7858	−1.63002
$790$	0	0
$791$	−0.104375	−0.00371115
$792$	0	0
$793$	22.4260	0.796372
$794$	0	0
$795$	−10.3625	−0.367519
$796$	0	0
$797$	−2.93733	−0.104046	−0.0520228	−	0.998646i	$-0.516567\pi$
$797$	−2.93733	−0.104046	−0.0520228	+	0.998646i	$0.516567\pi$
$798$	0	0
$799$	−16.6588	−0.589346
$800$	0	0
$801$	−83.7437	−2.95894
$802$	0	0
$803$	6.02287	0.212542
$804$	0	0
$805$	−5.72768	−0.201874
$806$	0	0
$807$	−38.5749	−1.35790
$808$	0	0
$809$	−48.5366	−1.70646	−0.853228	−	0.521539i	$-0.825358\pi$
$809$	−48.5366	−1.70646	−0.853228	+	0.521539i	$0.825358\pi$
$810$	0	0
$811$	20.4834	0.719269	0.359634	−	0.933093i	$-0.382901\pi$
$811$	20.4834	0.719269	0.359634	+	0.933093i	$0.382901\pi$
$812$	0	0
$813$	−11.2466	−0.394436
$814$	0	0
$815$	−2.97068	−0.104058
$816$	0	0
$817$	24.8138	0.868126
$818$	0	0
$819$	13.1424	0.459231
$820$	0	0
$821$	−15.8765	−0.554093	−0.277047	−	0.960856i	$-0.589356\pi$
$821$	−15.8765	−0.554093	−0.277047	+	0.960856i	$0.589356\pi$
$822$	0	0
$823$	−15.2874	−0.532886	−0.266443	−	0.963851i	$-0.585848\pi$
$823$	−15.2874	−0.532886	−0.266443	+	0.963851i	$0.585848\pi$
$824$	0	0
$825$	−2.77987	−0.0967827
$826$	0	0
$827$	17.2479	0.599769	0.299884	−	0.953976i	$-0.403052\pi$
$827$	17.2479	0.599769	0.299884	+	0.953976i	$0.403052\pi$
$828$	0	0
$829$	1.57526	0.0547109	0.0273555	−	0.999626i	$-0.491291\pi$
$829$	1.57526	0.0547109	0.0273555	+	0.999626i	$0.491291\pi$
$830$	0	0
$831$	−86.0514	−2.98509
$832$	0	0
$833$	−2.02287	−0.0700882
$834$	0	0
$835$	9.81785	0.339761
$836$	0	0
$837$	−37.2239	−1.28664
$838$	0	0
$839$	53.2782	1.83937	0.919685	−	0.392658i	$-0.128444\pi$
$839$	53.2782	1.83937	0.919685	+	0.392658i	$0.128444\pi$
$840$	0	0
$841$	48.4882	1.67201
$842$	0	0
$843$	18.4403	0.635116
$844$	0	0
$845$	−5.27232	−0.181373
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	18.2772	0.627273
$850$	0	0
$851$	−1.55974	−0.0534673
$852$	0	0
$853$	20.6013	0.705376	0.352688	−	0.935741i	$-0.385268\pi$
$853$	20.6013	0.705376	0.352688	+	0.935741i	$0.385268\pi$
$854$	0	0
$855$	−16.8293	−0.575551
$856$	0	0
$857$	1.26224	0.0431173	0.0215587	−	0.999768i	$-0.493137\pi$
$857$	1.26224	0.0431173	0.0215587	+	0.999768i	$0.493137\pi$
$858$	0	0
$859$	2.66128	0.0908019	0.0454009	−	0.998969i	$-0.485543\pi$
$859$	2.66128	0.0908019	0.0454009	+	0.998969i	$0.485543\pi$
$860$	0	0
$861$	1.41094	0.0480846
$862$	0	0
$863$	2.15243	0.0732695	0.0366347	−	0.999329i	$-0.488336\pi$
$863$	2.15243	0.0732695	0.0366347	+	0.999329i	$0.488336\pi$
$864$	0	0
$865$	−4.12724	−0.140330
$866$	0	0
$867$	35.8826	1.21864
$868$	0	0
$869$	−6.14881	−0.208584
$870$	0	0
$871$	12.4669	0.422423
$872$	0	0
$873$	−51.5825	−1.74580
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−30.5020	−1.02998	−0.514990	−	0.857196i	$-0.672204\pi$
$877$	−30.5020	−1.02998	−0.514990	+	0.857196i	$0.672204\pi$
$878$	0	0
$879$	15.2288	0.513654
$880$	0	0
$881$	1.69616	0.0571451	0.0285726	−	0.999592i	$-0.490904\pi$
$881$	1.69616	0.0571451	0.0285726	+	0.999592i	$0.490904\pi$
$882$	0	0
$883$	−56.8810	−1.91420	−0.957099	−	0.289761i	$-0.906424\pi$
$883$	−56.8810	−1.91420	−0.957099	+	0.289761i	$0.906424\pi$
$884$	0	0
$885$	−6.09016	−0.204719
$886$	0	0
$887$	35.6814	1.19807	0.599033	−	0.800725i	$-0.295552\pi$
$887$	35.6814	1.19807	0.599033	+	0.800725i	$0.295552\pi$
$888$	0	0
$889$	−9.55974	−0.320623
$890$	0	0
$891$	−0.832059	−0.0278750
$892$	0	0
$893$	−29.3153	−0.981000
$894$	0	0
$895$	15.6641	0.523594
$896$	0	0
$897$	44.2617	1.47786
$898$	0	0
$899$	68.2261	2.27547
$900$	0	0
$901$	−7.54061	−0.251214
$902$	0	0
$903$	19.3776	0.644846
$904$	0	0
$905$	23.8178	0.791732
$906$	0	0
$907$	55.7103	1.84983	0.924916	−	0.380172i	$-0.124135\pi$
$907$	55.7103	1.84983	0.924916	+	0.380172i	$0.124135\pi$
$908$	0	0
$909$	57.6343	1.91161
$910$	0	0
$911$	3.45537	0.114481	0.0572407	−	0.998360i	$-0.481770\pi$
$911$	3.45537	0.114481	0.0572407	+	0.998360i	$0.481770\pi$
$912$	0	0
$913$	0.802738	0.0265667
$914$	0	0
$915$	−22.4260	−0.741382
$916$	0	0
$917$	11.7721	0.388749
$918$	0	0
$919$	5.33186	0.175882	0.0879409	−	0.996126i	$-0.471971\pi$
$919$	5.33186	0.175882	0.0879409	+	0.996126i	$0.471971\pi$
$920$	0	0
$921$	−41.6869	−1.37363
$922$	0	0
$923$	−9.60548	−0.316168
$924$	0	0
$925$	0.272316	0.00895371
$926$	0	0
$927$	−16.2278	−0.532990
$928$	0	0
$929$	−44.2426	−1.45155	−0.725776	−	0.687931i	$-0.758519\pi$
$929$	−44.2426	−1.45155	−0.725776	+	0.687931i	$0.758519\pi$
$930$	0	0
$931$	−3.55974	−0.116666
$932$	0	0
$933$	−81.0084	−2.65209
$934$	0	0
$935$	−2.02287	−0.0661548
$936$	0	0
$937$	28.3853	0.927309	0.463654	−	0.886016i	$-0.346538\pi$
$937$	28.3853	0.927309	0.463654	+	0.886016i	$0.346538\pi$
$938$	0	0
$939$	−9.14224	−0.298346
$940$	0	0
$941$	−49.2912	−1.60685	−0.803423	−	0.595409i	$-0.796990\pi$
$941$	−49.2912	−1.60685	−0.803423	+	0.595409i	$0.796990\pi$
$942$	0	0
$943$	2.90711	0.0946686
$944$	0	0
$945$	−4.80274	−0.156233
$946$	0	0
$947$	16.5304	0.537167	0.268583	−	0.963256i	$-0.413445\pi$
$947$	16.5304	0.537167	0.268583	+	0.963256i	$0.413445\pi$
$948$	0	0
$949$	16.7428	0.543494
$950$	0	0
$951$	−78.8082	−2.55553
$952$	0	0
$953$	59.5887	1.93027	0.965133	−	0.261760i	$-0.0843029\pi$
$953$	59.5887	1.93027	0.965133	+	0.261760i	$0.0843029\pi$
$954$	0	0
$955$	−9.01511	−0.291722
$956$	0	0
$957$	−24.4705	−0.791018
$958$	0	0
$959$	6.53042	0.210878
$960$	0	0
$961$	29.0710	0.937775
$962$	0	0
$963$	−54.1574	−1.74520
$964$	0	0
$965$	−6.14881	−0.197937
$966$	0	0
$967$	−23.8512	−0.767003	−0.383501	−	0.923540i	$-0.625282\pi$
$967$	−23.8512	−0.767003	−0.383501	+	0.923540i	$0.625282\pi$
$968$	0	0
$969$	−20.0175	−0.643056
$970$	0	0
$971$	37.7413	1.21117	0.605587	−	0.795779i	$-0.292938\pi$
$971$	37.7413	1.21117	0.605587	+	0.795779i	$0.292938\pi$
$972$	0	0
$973$	−6.21237	−0.199159
$974$	0	0
$975$	−7.72768	−0.247484
$976$	0	0
$977$	32.8366	1.05054	0.525268	−	0.850937i	$-0.323965\pi$
$977$	32.8366	1.05054	0.525268	+	0.850937i	$0.323965\pi$
$978$	0	0
$979$	−17.7135	−0.566125
$980$	0	0
$981$	−46.4157	−1.48194
$982$	0	0
$983$	35.2374	1.12390	0.561950	−	0.827171i	$-0.310051\pi$
$983$	35.2374	1.12390	0.561950	+	0.827171i	$0.310051\pi$
$984$	0	0
$985$	−15.5455	−0.495322
$986$	0	0
$987$	−22.8929	−0.728689
$988$	0	0
$989$	39.9258	1.26957
$990$	0	0
$991$	−49.0151	−1.55702	−0.778508	−	0.627635i	$-0.784023\pi$
$991$	−49.0151	−1.55702	−0.778508	+	0.627635i	$0.784023\pi$
$992$	0	0
$993$	24.8138	0.787443
$994$	0	0
$995$	−1.09792	−0.0348064
$996$	0	0
$997$	−5.93886	−0.188086	−0.0940428	−	0.995568i	$-0.529979\pi$
$997$	−5.93886	−0.188086	−0.0940428	+	0.995568i	$0.529979\pi$
$998$	0	0
$999$	−1.30786	−0.0413790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bt.1.1		4
4.3	odd	2		3080.2.a.p.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.p.1.4	✓	4	4.3	odd	2
6160.2.a.bt.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77987 q^{3} +1.00000 q^{5} +1.00000 q^{7} +4.72768 q^{9} +O(q^{10})\)	\(q-2.77987 q^{3} +1.00000 q^{5} +1.00000 q^{7} +4.72768 q^{9} +1.00000 q^{11} +2.77987 q^{13} -2.77987 q^{15} -2.02287 q^{17} -3.55974 q^{19} -2.77987 q^{21} -5.72768 q^{23} +1.00000 q^{25} -4.80274 q^{27} +8.80274 q^{29} +7.75055 q^{31} -2.77987 q^{33} +1.00000 q^{35} +0.272316 q^{37} -7.72768 q^{39} -0.507555 q^{41} -6.97068 q^{43} +4.72768 q^{45} +8.23524 q^{47} +1.00000 q^{49} +5.62331 q^{51} +3.72768 q^{53} +1.00000 q^{55} +9.89562 q^{57} +2.19081 q^{59} +8.06730 q^{61} +4.72768 q^{63} +2.77987 q^{65} +4.48469 q^{67} +15.9222 q^{69} -3.45537 q^{71} +6.02287 q^{73} -2.77987 q^{75} +1.00000 q^{77} -6.14881 q^{79} -0.832059 q^{81} +0.802738 q^{83} -2.02287 q^{85} -24.4705 q^{87} -17.7135 q^{89} +2.77987 q^{91} -21.5455 q^{93} -3.55974 q^{95} -10.9107 q^{97} +4.72768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} + 4 q^{11} + 6 q^{17} + 8 q^{19} - 12 q^{23} + 4 q^{25} + 6 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{35} + 12 q^{37} - 20 q^{39} + 20 q^{41} - 14 q^{43} + 8 q^{45} + 4 q^{49} + 12 q^{51} + 4 q^{53} + 4 q^{55} + 40 q^{57} + 6 q^{59} - 12 q^{61} + 8 q^{63} + 10 q^{67} - 6 q^{69} + 8 q^{71} + 10 q^{73} + 4 q^{77} + 6 q^{79} + 8 q^{81} - 22 q^{83} + 6 q^{85} - 32 q^{87} - 2 q^{89} - 6 q^{93} + 8 q^{95} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 + 4 * q^11 + 6 * q^17 + 8 * q^19 - 12 * q^23 + 4 * q^25 + 6 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^35 + 12 * q^37 - 20 * q^39 + 20 * q^41 - 14 * q^43 + 8 * q^45 + 4 * q^49 + 12 * q^51 + 4 * q^53 + 4 * q^55 + 40 * q^57 + 6 * q^59 - 12 * q^61 + 8 * q^63 + 10 * q^67 - 6 * q^69 + 8 * q^71 + 10 * q^73 + 4 * q^77 + 6 * q^79 + 8 * q^81 - 22 * q^83 + 6 * q^85 - 32 * q^87 - 2 * q^89 - 6 * q^93 + 8 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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