LMFDB - Embedded newform 6160.2.a.bi.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.91729$ 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)

$f(q)$	$=$	$q+0.406728 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83457 q^{9} +O(q^{10})$	$q+0.406728 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83457 q^{9} -1.00000 q^{11} -0.813457 q^{13} -0.406728 q^{15} +3.83457 q^{17} -5.42784 q^{19} +0.406728 q^{21} +5.42784 q^{23} +1.00000 q^{25} -2.37309 q^{27} +6.61439 q^{29} -6.00000 q^{31} -0.406728 q^{33} -1.00000 q^{35} +8.24130 q^{37} -0.330856 q^{39} -5.59327 q^{41} +3.02112 q^{43} +2.83457 q^{45} +5.02112 q^{47} +1.00000 q^{49} +1.55963 q^{51} +6.61439 q^{53} +1.00000 q^{55} -2.20766 q^{57} -10.6480 q^{59} +0.978885 q^{61} -2.83457 q^{63} +0.813457 q^{65} -8.00000 q^{67} +2.20766 q^{69} -14.8557 q^{71} -6.20766 q^{73} +0.406728 q^{75} -1.00000 q^{77} -17.0970 q^{79} +7.53851 q^{81} +8.81346 q^{83} -3.83457 q^{85} +2.69026 q^{87} -1.18654 q^{89} -0.813457 q^{91} -2.44037 q^{93} +5.42784 q^{95} -3.42784 q^{97} +2.83457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9	$ 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9} - 3 q^{11} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{35} + 4 q^{37} - 40 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - 2 q^{47} + 3 q^{49} + 12 q^{51} + 4 q^{53} + 3 q^{55} + 8 q^{57} - 10 q^{59} + 20 q^{61} + 11 q^{63} - 24 q^{67} - 8 q^{69} - 8 q^{71} - 4 q^{73} - 3 q^{77} + 6 q^{79} + 47 q^{81} + 24 q^{83} + 8 q^{85} - 48 q^{87} - 6 q^{89} - 2 q^{95} + 8 q^{97} - 11 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9 - 3 * q^11 - 8 * q^17 + 2 * q^19 - 2 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^35 + 4 * q^37 - 40 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - 2 * q^47 + 3 * q^49 + 12 * q^51 + 4 * q^53 + 3 * q^55 + 8 * q^57 - 10 * q^59 + 20 * q^61 + 11 * q^63 - 24 * q^67 - 8 * q^69 - 8 * q^71 - 4 * q^73 - 3 * q^77 + 6 * q^79 + 47 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 - 6 * q^89 - 2 * q^95 + 8 * q^97 - 11 * 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
$3$	0.406728	0.234825	0.117412	−	0.993083i	$-0.462540\pi$
$3$	0.406728	0.234825	0.117412	+	0.993083i	$0.462540\pi$
$4$	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
$9$	−2.83457	−0.944857
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.813457	−0.225612	−0.112806	−	0.993617i	$-0.535984\pi$
$13$	−0.813457	−0.225612	−0.112806	+	0.993617i	$0.535984\pi$
$14$	0	0
$15$	−0.406728	−0.105017
$16$	0	0
$17$	3.83457	0.930020	0.465010	−	0.885305i	$-0.346051\pi$
$17$	3.83457	0.930020	0.465010	+	0.885305i	$0.346051\pi$
$18$	0	0
$19$	−5.42784	−1.24523	−0.622616	−	0.782527i	$-0.713930\pi$
$19$	−5.42784	−1.24523	−0.622616	+	0.782527i	$0.713930\pi$
$20$	0	0
$21$	0.406728	0.0887554
$22$	0	0
$23$	5.42784	1.13178	0.565892	−	0.824480i	$-0.308532\pi$
$23$	5.42784	1.13178	0.565892	+	0.824480i	$0.308532\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.37309	−0.456701
$28$	0	0
$29$	6.61439	1.22826	0.614130	−	0.789205i	$-0.289507\pi$
$29$	6.61439	1.22826	0.614130	+	0.789205i	$0.289507\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−0.406728	−0.0708023
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.24130	1.35486	0.677431	−	0.735587i	$-0.263093\pi$
$37$	8.24130	1.35486	0.677431	+	0.735587i	$0.263093\pi$
$38$	0	0
$39$	−0.330856	−0.0529794
$40$	0	0
$41$	−5.59327	−0.873522	−0.436761	−	0.899578i	$-0.643875\pi$
$41$	−5.59327	−0.873522	−0.436761	+	0.899578i	$0.643875\pi$
$42$	0	0
$43$	3.02112	0.460716	0.230358	−	0.973106i	$-0.426010\pi$
$43$	3.02112	0.460716	0.230358	+	0.973106i	$0.426010\pi$
$44$	0	0
$45$	2.83457	0.422553
$46$	0	0
$47$	5.02112	0.732405	0.366202	−	0.930535i	$-0.380658\pi$
$47$	5.02112	0.732405	0.366202	+	0.930535i	$0.380658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.55963	0.218392
$52$	0	0
$53$	6.61439	0.908556	0.454278	−	0.890860i	$-0.349897\pi$
$53$	6.61439	0.908556	0.454278	+	0.890860i	$0.349897\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.20766	−0.292411
$58$	0	0
$59$	−10.6480	−1.38626	−0.693128	−	0.720815i	$-0.743768\pi$
$59$	−10.6480	−1.38626	−0.693128	+	0.720815i	$0.743768\pi$
$60$	0	0
$61$	0.978885	0.125333	0.0626667	−	0.998035i	$-0.480039\pi$
$61$	0.978885	0.125333	0.0626667	+	0.998035i	$0.480039\pi$
$62$	0	0
$63$	−2.83457	−0.357123
$64$	0	0
$65$	0.813457	0.100897
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	2.20766	0.265771
$70$	0	0
$71$	−14.8557	−1.76305	−0.881523	−	0.472141i	$-0.843481\pi$
$71$	−14.8557	−1.76305	−0.881523	+	0.472141i	$0.843481\pi$
$72$	0	0
$73$	−6.20766	−0.726551	−0.363276	−	0.931682i	$-0.618342\pi$
$73$	−6.20766	−0.726551	−0.363276	+	0.931682i	$0.618342\pi$
$74$	0	0
$75$	0.406728	0.0469650
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−17.0970	−1.92356	−0.961781	−	0.273821i	$-0.911712\pi$
$79$	−17.0970	−1.92356	−0.961781	+	0.273821i	$0.911712\pi$
$80$	0	0
$81$	7.53851	0.837613
$82$	0	0
$83$	8.81346	0.967403	0.483701	−	0.875233i	$-0.339292\pi$
$83$	8.81346	0.967403	0.483701	+	0.875233i	$0.339292\pi$
$84$	0	0
$85$	−3.83457	−0.415918
$86$	0	0
$87$	2.69026	0.288426
$88$	0	0
$89$	−1.18654	−0.125773	−0.0628867	−	0.998021i	$-0.520031\pi$
$89$	−1.18654	−0.125773	−0.0628867	+	0.998021i	$0.520031\pi$
$90$	0	0
$91$	−0.813457	−0.0852734
$92$	0	0
$93$	−2.44037	−0.253055
$94$	0	0
$95$	5.42784	0.556885
$96$	0	0
$97$	−3.42784	−0.348045	−0.174022	−	0.984742i	$-0.555677\pi$
$97$	−3.42784	−0.348045	−0.174022	+	0.984742i	$0.555677\pi$
$98$	0	0
$99$	2.83457	0.284885
$100$	0	0
$101$	−18.6903	−1.85975	−0.929875	−	0.367875i	$-0.880085\pi$
$101$	−18.6903	−1.85975	−0.929875	+	0.367875i	$0.880085\pi$
$102$	0	0
$103$	−15.4615	−1.52347	−0.761733	−	0.647891i	$-0.775651\pi$
$103$	−15.4615	−1.52347	−0.761733	+	0.647891i	$0.775651\pi$
$104$	0	0
$105$	−0.406728	−0.0396926
$106$	0	0
$107$	11.0211	1.06545	0.532726	−	0.846288i	$-0.321168\pi$
$107$	11.0211	1.06545	0.532726	+	0.846288i	$0.321168\pi$
$108$	0	0
$109$	16.2413	1.55563	0.777817	−	0.628491i	$-0.216327\pi$
$109$	16.2413	1.55563	0.777817	+	0.628491i	$0.216327\pi$
$110$	0	0
$111$	3.35197	0.318155
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−5.42784	−0.506149
$116$	0	0
$117$	2.30580	0.213171
$118$	0	0
$119$	3.83457	0.351515
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.27494	−0.205125
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.2961	−1.53478	−0.767388	−	0.641182i	$-0.778444\pi$
$127$	−17.2961	−1.53478	−0.767388	+	0.641182i	$0.778444\pi$
$128$	0	0
$129$	1.22877	0.108187
$130$	0	0
$131$	−9.42784	−0.823715	−0.411857	−	0.911248i	$-0.635120\pi$
$131$	−9.42784	−0.823715	−0.411857	+	0.911248i	$0.635120\pi$
$132$	0	0
$133$	−5.42784	−0.470654
$134$	0	0
$135$	2.37309	0.204243
$136$	0	0
$137$	13.6691	1.16783	0.583917	−	0.811813i	$-0.301519\pi$
$137$	13.6691	1.16783	0.583917	+	0.811813i	$0.301519\pi$
$138$	0	0
$139$	11.8682	1.00665	0.503324	−	0.864098i	$-0.332110\pi$
$139$	11.8682	1.00665	0.503324	+	0.864098i	$0.332110\pi$
$140$	0	0
$141$	2.04223	0.171987
$142$	0	0
$143$	0.813457	0.0680247
$144$	0	0
$145$	−6.61439	−0.549295
$146$	0	0
$147$	0.406728	0.0335464
$148$	0	0
$149$	−22.2835	−1.82554	−0.912769	−	0.408476i	$-0.866060\pi$
$149$	−22.2835	−1.82554	−0.912769	+	0.408476i	$0.866060\pi$
$150$	0	0
$151$	3.05476	0.248593	0.124296	−	0.992245i	$-0.460333\pi$
$151$	3.05476	0.248593	0.124296	+	0.992245i	$0.460333\pi$
$152$	0	0
$153$	−10.8694	−0.878737
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	4.85569	0.387526	0.193763	−	0.981048i	$-0.437931\pi$
$157$	4.85569	0.387526	0.193763	+	0.981048i	$0.437931\pi$
$158$	0	0
$159$	2.69026	0.213351
$160$	0	0
$161$	5.42784	0.427774
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	0.406728	0.0316638
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.3383	−0.949099
$170$	0	0
$171$	15.3856	1.17657
$172$	0	0
$173$	−19.0211	−1.44615	−0.723074	−	0.690770i	$-0.757272\pi$
$173$	−19.0211	−1.44615	−0.723074	+	0.690770i	$0.757272\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.33086	−0.325527
$178$	0	0
$179$	−4.64803	−0.347410	−0.173705	−	0.984798i	$-0.555574\pi$
$179$	−4.64803	−0.347410	−0.173705	+	0.984798i	$0.555574\pi$
$180$	0	0
$181$	−25.3383	−1.88338	−0.941690	−	0.336482i	$-0.890763\pi$
$181$	−25.3383	−1.88338	−0.941690	+	0.336482i	$0.890763\pi$
$182$	0	0
$183$	0.398140	0.0294314
$184$	0	0
$185$	−8.24130	−0.605912
$186$	0	0
$187$	−3.83457	−0.280412
$188$	0	0
$189$	−2.37309	−0.172617
$190$	0	0
$191$	−15.6691	−1.13378	−0.566890	−	0.823794i	$-0.691853\pi$
$191$	−15.6691	−1.13378	−0.566890	+	0.823794i	$0.691853\pi$
$192$	0	0
$193$	−2.16543	−0.155871	−0.0779355	−	0.996958i	$-0.524833\pi$
$193$	−2.16543	−0.155871	−0.0779355	+	0.996958i	$0.524833\pi$
$194$	0	0
$195$	0.330856	0.0236931
$196$	0	0
$197$	2.81346	0.200451	0.100225	−	0.994965i	$-0.468044\pi$
$197$	2.81346	0.200451	0.100225	+	0.994965i	$0.468044\pi$
$198$	0	0
$199$	13.0211	0.923042	0.461521	−	0.887129i	$-0.347304\pi$
$199$	13.0211	0.923042	0.461521	+	0.887129i	$0.347304\pi$
$200$	0	0
$201$	−3.25383	−0.229507
$202$	0	0
$203$	6.61439	0.464239
$204$	0	0
$205$	5.59327	0.390651
$206$	0	0
$207$	−15.3856	−1.06937
$208$	0	0
$209$	5.42784	0.375452
$210$	0	0
$211$	−19.6691	−1.35408	−0.677040	−	0.735946i	$-0.736738\pi$
$211$	−19.6691	−1.35408	−0.677040	+	0.735946i	$0.736738\pi$
$212$	0	0
$213$	−6.04223	−0.414007
$214$	0	0
$215$	−3.02112	−0.206038
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	−2.52483	−0.170612
$220$	0	0
$221$	−3.11926	−0.209824
$222$	0	0
$223$	3.79234	0.253954	0.126977	−	0.991906i	$-0.459473\pi$
$223$	3.79234	0.253954	0.126977	+	0.991906i	$0.459473\pi$
$224$	0	0
$225$	−2.83457	−0.188971
$226$	0	0
$227$	14.0422	0.932016	0.466008	−	0.884781i	$-0.345692\pi$
$227$	14.0422	0.932016	0.466008	+	0.884781i	$0.345692\pi$
$228$	0	0
$229$	1.83457	0.121232	0.0606160	−	0.998161i	$-0.480694\pi$
$229$	1.83457	0.121232	0.0606160	+	0.998161i	$0.480694\pi$
$230$	0	0
$231$	−0.406728	−0.0267608
$232$	0	0
$233$	−8.04223	−0.526864	−0.263432	−	0.964678i	$-0.584854\pi$
$233$	−8.04223	−0.526864	−0.263432	+	0.964678i	$0.584854\pi$
$234$	0	0
$235$	−5.02112	−0.327541
$236$	0	0
$237$	−6.95383	−0.451700
$238$	0	0
$239$	15.4701	1.00068	0.500338	−	0.865830i	$-0.333209\pi$
$239$	15.4701	1.00068	0.500338	+	0.865830i	$0.333209\pi$
$240$	0	0
$241$	11.6355	0.749509	0.374754	−	0.927124i	$-0.377727\pi$
$241$	11.6355	0.749509	0.374754	+	0.927124i	$0.377727\pi$
$242$	0	0
$243$	10.1854	0.653393
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.41532	0.280940
$248$	0	0
$249$	3.58468	0.227170
$250$	0	0
$251$	−5.35197	−0.337813	−0.168907	−	0.985632i	$-0.554024\pi$
$251$	−5.35197	−0.337813	−0.168907	+	0.985632i	$0.554024\pi$
$252$	0	0
$253$	−5.42784	−0.341246
$254$	0	0
$255$	−1.55963	−0.0976678
$256$	0	0
$257$	12.7239	0.793695	0.396848	−	0.917885i	$-0.370104\pi$
$257$	12.7239	0.793695	0.396848	+	0.917885i	$0.370104\pi$
$258$	0	0
$259$	8.24130	0.512089
$260$	0	0
$261$	−18.7490	−1.16053
$262$	0	0
$263$	1.62691	0.100320	0.0501599	−	0.998741i	$-0.484027\pi$
$263$	1.62691	0.100320	0.0501599	+	0.998741i	$0.484027\pi$
$264$	0	0
$265$	−6.61439	−0.406319
$266$	0	0
$267$	−0.482601	−0.0295347
$268$	0	0
$269$	23.8768	1.45579	0.727897	−	0.685686i	$-0.240498\pi$
$269$	23.8768	1.45579	0.727897	+	0.685686i	$0.240498\pi$
$270$	0	0
$271$	−2.44037	−0.148242	−0.0741210	−	0.997249i	$-0.523615\pi$
$271$	−2.44037	−0.148242	−0.0741210	+	0.997249i	$0.523615\pi$
$272$	0	0
$273$	−0.330856	−0.0200243
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	17.0074	1.01821
$280$	0	0
$281$	−20.8557	−1.24415	−0.622073	−	0.782959i	$-0.713709\pi$
$281$	−20.8557	−1.24415	−0.622073	+	0.782959i	$0.713709\pi$
$282$	0	0
$283$	−0.330856	−0.0196673	−0.00983367	−	0.999952i	$-0.503130\pi$
$283$	−0.330856	−0.0196673	−0.00983367	+	0.999952i	$0.503130\pi$
$284$	0	0
$285$	2.20766	0.130770
$286$	0	0
$287$	−5.59327	−0.330160
$288$	0	0
$289$	−2.29606	−0.135062
$290$	0	0
$291$	−1.39420	−0.0817295
$292$	0	0
$293$	−3.18654	−0.186160	−0.0930799	−	0.995659i	$-0.529671\pi$
$293$	−3.18654	−0.186160	−0.0930799	+	0.995659i	$0.529671\pi$
$294$	0	0
$295$	10.6480	0.619952
$296$	0	0
$297$	2.37309	0.137700
$298$	0	0
$299$	−4.41532	−0.255344
$300$	0	0
$301$	3.02112	0.174134
$302$	0	0
$303$	−7.60186	−0.436715
$304$	0	0
$305$	−0.978885	−0.0560508
$306$	0	0
$307$	−20.4826	−1.16900	−0.584502	−	0.811392i	$-0.698710\pi$
$307$	−20.4826	−1.16900	−0.584502	+	0.811392i	$0.698710\pi$
$308$	0	0
$309$	−6.28863	−0.357747
$310$	0	0
$311$	−3.62691	−0.205663	−0.102832	−	0.994699i	$-0.532790\pi$
$311$	−3.62691	−0.205663	−0.102832	+	0.994699i	$0.532790\pi$
$312$	0	0
$313$	20.7239	1.17138	0.585692	−	0.810534i	$-0.300823\pi$
$313$	20.7239	1.17138	0.585692	+	0.810534i	$0.300823\pi$
$314$	0	0
$315$	2.83457	0.159710
$316$	0	0
$317$	−1.80093	−0.101150	−0.0505751	−	0.998720i	$-0.516105\pi$
$317$	−1.80093	−0.101150	−0.0505751	+	0.998720i	$0.516105\pi$
$318$	0	0
$319$	−6.61439	−0.370335
$320$	0	0
$321$	4.48260	0.250194
$322$	0	0
$323$	−20.8135	−1.15809
$324$	0	0
$325$	−0.813457	−0.0451225
$326$	0	0
$327$	6.60580	0.365301
$328$	0	0
$329$	5.02112	0.276823
$330$	0	0
$331$	3.02112	0.166056	0.0830278	−	0.996547i	$-0.473541\pi$
$331$	3.02112	0.166056	0.0830278	+	0.996547i	$0.473541\pi$
$332$	0	0
$333$	−23.3606	−1.28015
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−9.66914	−0.526712	−0.263356	−	0.964699i	$-0.584829\pi$
$337$	−9.66914	−0.526712	−0.263356	+	0.964699i	$0.584829\pi$
$338$	0	0
$339$	−5.69420	−0.309266
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.20766	−0.118856
$346$	0	0
$347$	−9.39420	−0.504307	−0.252154	−	0.967687i	$-0.581139\pi$
$347$	−9.39420	−0.504307	−0.252154	+	0.967687i	$0.581139\pi$
$348$	0	0
$349$	−18.2077	−0.974634	−0.487317	−	0.873225i	$-0.662024\pi$
$349$	−18.2077	−0.974634	−0.487317	+	0.873225i	$0.662024\pi$
$350$	0	0
$351$	1.93040	0.103037
$352$	0	0
$353$	12.7239	0.677225	0.338612	−	0.940926i	$-0.390042\pi$
$353$	12.7239	0.677225	0.338612	+	0.940926i	$0.390042\pi$
$354$	0	0
$355$	14.8557	0.788458
$356$	0	0
$357$	1.55963	0.0825443
$358$	0	0
$359$	−29.8432	−1.57506	−0.787531	−	0.616275i	$-0.788641\pi$
$359$	−29.8432	−1.57506	−0.787531	+	0.616275i	$0.788641\pi$
$360$	0	0
$361$	10.4615	0.550605
$362$	0	0
$363$	0.406728	0.0213477
$364$	0	0
$365$	6.20766	0.324924
$366$	0	0
$367$	−15.4615	−0.807083	−0.403541	−	0.914961i	$-0.632221\pi$
$367$	−15.4615	−0.807083	−0.403541	+	0.914961i	$0.632221\pi$
$368$	0	0
$369$	15.8545	0.825354
$370$	0	0
$371$	6.61439	0.343402
$372$	0	0
$373$	−8.10951	−0.419895	−0.209947	−	0.977713i	$-0.567329\pi$
$373$	−8.10951	−0.419895	−0.209947	+	0.977713i	$0.567329\pi$
$374$	0	0
$375$	−0.406728	−0.0210034
$376$	0	0
$377$	−5.38052	−0.277111
$378$	0	0
$379$	−22.0422	−1.13223	−0.566117	−	0.824325i	$-0.691555\pi$
$379$	−22.0422	−1.13223	−0.566117	+	0.824325i	$0.691555\pi$
$380$	0	0
$381$	−7.03480	−0.360404
$382$	0	0
$383$	−38.2499	−1.95448	−0.977239	−	0.212141i	$-0.931956\pi$
$383$	−38.2499	−1.95448	−0.977239	+	0.212141i	$0.931956\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−8.56357	−0.435311
$388$	0	0
$389$	16.8557	0.854617	0.427309	−	0.904106i	$-0.359462\pi$
$389$	16.8557	0.854617	0.427309	+	0.904106i	$0.359462\pi$
$390$	0	0
$391$	20.8135	1.05258
$392$	0	0
$393$	−3.83457	−0.193429
$394$	0	0
$395$	17.0970	0.860243
$396$	0	0
$397$	−4.44037	−0.222856	−0.111428	−	0.993773i	$-0.535542\pi$
$397$	−4.44037	−0.222856	−0.111428	+	0.993773i	$0.535542\pi$
$398$	0	0
$399$	−2.20766	−0.110521
$400$	0	0
$401$	−3.46149	−0.172858	−0.0864292	−	0.996258i	$-0.527546\pi$
$401$	−3.46149	−0.172858	−0.0864292	+	0.996258i	$0.527546\pi$
$402$	0	0
$403$	4.88074	0.243127
$404$	0	0
$405$	−7.53851	−0.374592
$406$	0	0
$407$	−8.24130	−0.408506
$408$	0	0
$409$	−17.2624	−0.853572	−0.426786	−	0.904353i	$-0.640354\pi$
$409$	−17.2624	−0.853572	−0.426786	+	0.904353i	$0.640354\pi$
$410$	0	0
$411$	5.55963	0.274236
$412$	0	0
$413$	−10.6480	−0.523955
$414$	0	0
$415$	−8.81346	−0.432636
$416$	0	0
$417$	4.82714	0.236386
$418$	0	0
$419$	−9.02112	−0.440710	−0.220355	−	0.975420i	$-0.570722\pi$
$419$	−9.02112	−0.440710	−0.220355	+	0.975420i	$0.570722\pi$
$420$	0	0
$421$	31.7114	1.54552	0.772759	−	0.634700i	$-0.218876\pi$
$421$	31.7114	1.54552	0.772759	+	0.634700i	$0.218876\pi$
$422$	0	0
$423$	−14.2327	−0.692018
$424$	0	0
$425$	3.83457	0.186004
$426$	0	0
$427$	0.978885	0.0473716
$428$	0	0
$429$	0.330856	0.0159739
$430$	0	0
$431$	3.05476	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$431$	3.05476	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$432$	0	0
$433$	27.9104	1.34129	0.670645	−	0.741778i	$-0.266017\pi$
$433$	27.9104	1.34129	0.670645	+	0.741778i	$0.266017\pi$
$434$	0	0
$435$	−2.69026	−0.128988
$436$	0	0
$437$	−29.4615	−1.40933
$438$	0	0
$439$	14.8557	0.709023	0.354512	−	0.935052i	$-0.384647\pi$
$439$	14.8557	0.709023	0.354512	+	0.935052i	$0.384647\pi$
$440$	0	0
$441$	−2.83457	−0.134980
$442$	0	0
$443$	−0.746173	−0.0354517	−0.0177259	−	0.999843i	$-0.505643\pi$
$443$	−0.746173	−0.0354517	−0.0177259	+	0.999843i	$0.505643\pi$
$444$	0	0
$445$	1.18654	0.0562475
$446$	0	0
$447$	−9.06335	−0.428682
$448$	0	0
$449$	9.83457	0.464122	0.232061	−	0.972701i	$-0.425453\pi$
$449$	9.83457	0.464122	0.232061	+	0.972701i	$0.425453\pi$
$450$	0	0
$451$	5.59327	0.263377
$452$	0	0
$453$	1.24246	0.0583757
$454$	0	0
$455$	0.813457	0.0381354
$456$	0	0
$457$	11.2961	0.528407	0.264204	−	0.964467i	$-0.414891\pi$
$457$	11.2961	0.528407	0.264204	+	0.964467i	$0.414891\pi$
$458$	0	0
$459$	−9.09977	−0.424741
$460$	0	0
$461$	−7.41926	−0.345549	−0.172775	−	0.984961i	$-0.555273\pi$
$461$	−7.41926	−0.345549	−0.172775	+	0.984961i	$0.555273\pi$
$462$	0	0
$463$	10.1740	0.472827	0.236413	−	0.971653i	$-0.424028\pi$
$463$	10.1740	0.472827	0.236413	+	0.971653i	$0.424028\pi$
$464$	0	0
$465$	2.44037	0.113169
$466$	0	0
$467$	32.4912	1.50351	0.751756	−	0.659441i	$-0.229207\pi$
$467$	32.4912	1.50351	0.751756	+	0.659441i	$0.229207\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	1.97495	0.0910007
$472$	0	0
$473$	−3.02112	−0.138911
$474$	0	0
$475$	−5.42784	−0.249047
$476$	0	0
$477$	−18.7490	−0.858456
$478$	0	0
$479$	11.6691	0.533177	0.266588	−	0.963810i	$-0.414104\pi$
$479$	11.6691	0.533177	0.266588	+	0.963810i	$0.414104\pi$
$480$	0	0
$481$	−6.70394	−0.305673
$482$	0	0
$483$	2.20766	0.100452
$484$	0	0
$485$	3.42784	0.155650
$486$	0	0
$487$	5.49513	0.249008	0.124504	−	0.992219i	$-0.460266\pi$
$487$	5.49513	0.249008	0.124504	+	0.992219i	$0.460266\pi$
$488$	0	0
$489$	3.25383	0.147143
$490$	0	0
$491$	13.2961	0.600043	0.300021	−	0.953932i	$-0.403006\pi$
$491$	13.2961	0.600043	0.300021	+	0.953932i	$0.403006\pi$
$492$	0	0
$493$	25.3633	1.14231
$494$	0	0
$495$	−2.83457	−0.127405
$496$	0	0
$497$	−14.8557	−0.666369
$498$	0	0
$499$	24.6480	1.10340	0.551699	−	0.834044i	$-0.313980\pi$
$499$	24.6480	1.10340	0.551699	+	0.834044i	$0.313980\pi$
$500$	0	0
$501$	−3.25383	−0.145370
$502$	0	0
$503$	−38.9230	−1.73549	−0.867745	−	0.497010i	$-0.834431\pi$
$503$	−38.9230	−1.73549	−0.867745	+	0.497010i	$0.834431\pi$
$504$	0	0
$505$	18.6903	0.831706
$506$	0	0
$507$	−5.01833	−0.222872
$508$	0	0
$509$	−39.0497	−1.73085	−0.865423	−	0.501042i	$-0.832950\pi$
$509$	−39.0497	−1.73085	−0.865423	+	0.501042i	$0.832950\pi$
$510$	0	0
$511$	−6.20766	−0.274611
$512$	0	0
$513$	12.8807	0.568699
$514$	0	0
$515$	15.4615	0.681314
$516$	0	0
$517$	−5.02112	−0.220828
$518$	0	0
$519$	−7.73643	−0.339592
$520$	0	0
$521$	−33.2710	−1.45763	−0.728815	−	0.684711i	$-0.759928\pi$
$521$	−33.2710	−1.45763	−0.728815	+	0.684711i	$0.759928\pi$
$522$	0	0
$523$	40.2362	1.75941	0.879703	−	0.475523i	$-0.157741\pi$
$523$	40.2362	1.75941	0.879703	+	0.475523i	$0.157741\pi$
$524$	0	0
$525$	0.406728	0.0177511
$526$	0	0
$527$	−23.0074	−1.00222
$528$	0	0
$529$	6.46149	0.280934
$530$	0	0
$531$	30.1826	1.30981
$532$	0	0
$533$	4.54989	0.197077
$534$	0	0
$535$	−11.0211	−0.476484
$536$	0	0
$537$	−1.89049	−0.0815805
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	26.2835	1.13002	0.565009	−	0.825085i	$-0.308873\pi$
$541$	26.2835	1.13002	0.565009	+	0.825085i	$0.308873\pi$
$542$	0	0
$543$	−10.3058	−0.442264
$544$	0	0
$545$	−16.2413	−0.695701
$546$	0	0
$547$	−13.6269	−0.582645	−0.291322	−	0.956625i	$-0.594095\pi$
$547$	−13.6269	−0.582645	−0.291322	+	0.956625i	$0.594095\pi$
$548$	0	0
$549$	−2.77472	−0.118422
$550$	0	0
$551$	−35.9019	−1.52947
$552$	0	0
$553$	−17.0970	−0.727038
$554$	0	0
$555$	−3.35197	−0.142283
$556$	0	0
$557$	14.8979	0.631245	0.315623	−	0.948885i	$-0.397787\pi$
$557$	14.8979	0.631245	0.315623	+	0.948885i	$0.397787\pi$
$558$	0	0
$559$	−2.45755	−0.103943
$560$	0	0
$561$	−1.55963	−0.0658476
$562$	0	0
$563$	−39.7536	−1.67541	−0.837707	−	0.546119i	$-0.816104\pi$
$563$	−39.7536	−1.67541	−0.837707	+	0.546119i	$0.816104\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	7.53851	0.316588
$568$	0	0
$569$	8.78840	0.368429	0.184215	−	0.982886i	$-0.441026\pi$
$569$	8.78840	0.368429	0.184215	+	0.982886i	$0.441026\pi$
$570$	0	0
$571$	5.22877	0.218817	0.109409	−	0.993997i	$-0.465104\pi$
$571$	5.22877	0.218817	0.109409	+	0.993997i	$0.465104\pi$
$572$	0	0
$573$	−6.37309	−0.266239
$574$	0	0
$575$	5.42784	0.226357
$576$	0	0
$577$	5.80093	0.241496	0.120748	−	0.992683i	$-0.461471\pi$
$577$	5.80093	0.241496	0.120748	+	0.992683i	$0.461471\pi$
$578$	0	0
$579$	−0.880741	−0.0366024
$580$	0	0
$581$	8.81346	0.365644
$582$	0	0
$583$	−6.61439	−0.273940
$584$	0	0
$585$	−2.30580	−0.0953332
$586$	0	0
$587$	−25.3046	−1.04443	−0.522217	−	0.852812i	$-0.674895\pi$
$587$	−25.3046	−1.04443	−0.522217	+	0.852812i	$0.674895\pi$
$588$	0	0
$589$	32.5671	1.34190
$590$	0	0
$591$	1.14431	0.0470707
$592$	0	0
$593$	−23.1729	−0.951595	−0.475798	−	0.879555i	$-0.657841\pi$
$593$	−23.1729	−0.951595	−0.475798	+	0.879555i	$0.657841\pi$
$594$	0	0
$595$	−3.83457	−0.157202
$596$	0	0
$597$	5.29606	0.216753
$598$	0	0
$599$	8.48260	0.346590	0.173295	−	0.984870i	$-0.444559\pi$
$599$	8.48260	0.346590	0.173295	+	0.984870i	$0.444559\pi$
$600$	0	0
$601$	−32.4490	−1.32362	−0.661810	−	0.749671i	$-0.730212\pi$
$601$	−32.4490	−1.32362	−0.661810	+	0.749671i	$0.730212\pi$
$602$	0	0
$603$	22.6766	0.923462
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	33.9441	1.37775	0.688874	−	0.724881i	$-0.258105\pi$
$607$	33.9441	1.37775	0.688874	+	0.724881i	$0.258105\pi$
$608$	0	0
$609$	2.69026	0.109015
$610$	0	0
$611$	−4.08446	−0.165240
$612$	0	0
$613$	8.77123	0.354267	0.177133	−	0.984187i	$-0.443318\pi$
$613$	8.77123	0.354267	0.177133	+	0.984187i	$0.443318\pi$
$614$	0	0
$615$	2.27494	0.0917345
$616$	0	0
$617$	−32.1095	−1.29268	−0.646340	−	0.763049i	$-0.723701\pi$
$617$	−32.1095	−1.29268	−0.646340	+	0.763049i	$0.723701\pi$
$618$	0	0
$619$	28.2749	1.13647	0.568233	−	0.822868i	$-0.307627\pi$
$619$	28.2749	1.13647	0.568233	+	0.822868i	$0.307627\pi$
$620$	0	0
$621$	−12.8807	−0.516886
$622$	0	0
$623$	−1.18654	−0.0475378
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.20766	0.0881654
$628$	0	0
$629$	31.6019	1.26005
$630$	0	0
$631$	−34.9230	−1.39026	−0.695131	−	0.718883i	$-0.744654\pi$
$631$	−34.9230	−1.39026	−0.695131	+	0.718883i	$0.744654\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	17.2961	0.686373
$636$	0	0
$637$	−0.813457	−0.0322303
$638$	0	0
$639$	42.1095	1.66583
$640$	0	0
$641$	−5.25383	−0.207514	−0.103757	−	0.994603i	$-0.533086\pi$
$641$	−5.25383	−0.207514	−0.103757	+	0.994603i	$0.533086\pi$
$642$	0	0
$643$	3.17796	0.125326	0.0626632	−	0.998035i	$-0.480041\pi$
$643$	3.17796	0.125326	0.0626632	+	0.998035i	$0.480041\pi$
$644$	0	0
$645$	−1.22877	−0.0483829
$646$	0	0
$647$	−37.9190	−1.49075	−0.745375	−	0.666645i	$-0.767730\pi$
$647$	−37.9190	−1.49075	−0.745375	+	0.666645i	$0.767730\pi$
$648$	0	0
$649$	10.6480	0.417972
$650$	0	0
$651$	−2.44037	−0.0956457
$652$	0	0
$653$	29.5374	1.15589	0.577943	−	0.816077i	$-0.303856\pi$
$653$	29.5374	1.15589	0.577943	+	0.816077i	$0.303856\pi$
$654$	0	0
$655$	9.42784	0.368376
$656$	0	0
$657$	17.5961	0.686487
$658$	0	0
$659$	−15.1865	−0.591584	−0.295792	−	0.955252i	$-0.595583\pi$
$659$	−15.1865	−0.591584	−0.295792	+	0.955252i	$0.595583\pi$
$660$	0	0
$661$	−44.6766	−1.73772	−0.868859	−	0.495060i	$-0.835146\pi$
$661$	−44.6766	−1.73772	−0.868859	+	0.495060i	$0.835146\pi$
$662$	0	0
$663$	−1.26869	−0.0492719
$664$	0	0
$665$	5.42784	0.210483
$666$	0	0
$667$	35.9019	1.39013
$668$	0	0
$669$	1.54245	0.0596347
$670$	0	0
$671$	−0.978885	−0.0377894
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	0	0
$675$	−2.37309	−0.0913401
$676$	0	0
$677$	−48.7325	−1.87294	−0.936471	−	0.350745i	$-0.885928\pi$
$677$	−48.7325	−1.87294	−0.936471	+	0.350745i	$0.885928\pi$
$678$	0	0
$679$	−3.42784	−0.131549
$680$	0	0
$681$	5.71137	0.218860
$682$	0	0
$683$	0.482601	0.0184662	0.00923310	−	0.999957i	$-0.497061\pi$
$683$	0.482601	0.0184662	0.00923310	+	0.999957i	$0.497061\pi$
$684$	0	0
$685$	−13.6691	−0.522271
$686$	0	0
$687$	0.746173	0.0284683
$688$	0	0
$689$	−5.38052	−0.204981
$690$	0	0
$691$	25.5037	0.970207	0.485104	−	0.874457i	$-0.338782\pi$
$691$	25.5037	0.970207	0.485104	+	0.874457i	$0.338782\pi$
$692$	0	0
$693$	2.83457	0.107676
$694$	0	0
$695$	−11.8682	−0.450187
$696$	0	0
$697$	−21.4478	−0.812393
$698$	0	0
$699$	−3.27100	−0.123721
$700$	0	0
$701$	33.8682	1.27918	0.639592	−	0.768714i	$-0.279103\pi$
$701$	33.8682	1.27918	0.639592	+	0.768714i	$0.279103\pi$
$702$	0	0
$703$	−44.7325	−1.68712
$704$	0	0
$705$	−2.04223	−0.0769148
$706$	0	0
$707$	−18.6903	−0.702920
$708$	0	0
$709$	−15.9749	−0.599952	−0.299976	−	0.953947i	$-0.596979\pi$
$709$	−15.9749	−0.599952	−0.299976	+	0.953947i	$0.596979\pi$
$710$	0	0
$711$	48.4626	1.81749
$712$	0	0
$713$	−32.5671	−1.21965
$714$	0	0
$715$	−0.813457	−0.0304216
$716$	0	0
$717$	6.29212	0.234983
$718$	0	0
$719$	13.4364	0.501094	0.250547	−	0.968104i	$-0.419389\pi$
$719$	13.4364	0.501094	0.250547	+	0.968104i	$0.419389\pi$
$720$	0	0
$721$	−15.4615	−0.575816
$722$	0	0
$723$	4.73249	0.176003
$724$	0	0
$725$	6.61439	0.245652
$726$	0	0
$727$	−6.64803	−0.246562	−0.123281	−	0.992372i	$-0.539342\pi$
$727$	−6.64803	−0.246562	−0.123281	+	0.992372i	$0.539342\pi$
$728$	0	0
$729$	−18.4729	−0.684180
$730$	0	0
$731$	11.5847	0.428475
$732$	0	0
$733$	38.0285	1.40462	0.702308	−	0.711873i	$-0.252153\pi$
$733$	38.0285	1.40462	0.702308	+	0.711873i	$0.252153\pi$
$734$	0	0
$735$	−0.406728	−0.0150024
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	52.6343	1.93619	0.968093	−	0.250592i	$-0.0806252\pi$
$739$	52.6343	1.93619	0.968093	+	0.250592i	$0.0806252\pi$
$740$	0	0
$741$	1.79583	0.0659716
$742$	0	0
$743$	−28.0845	−1.03032	−0.515159	−	0.857094i	$-0.672267\pi$
$743$	−28.0845	−1.03032	−0.515159	+	0.857094i	$0.672267\pi$
$744$	0	0
$745$	22.2835	0.816405
$746$	0	0
$747$	−24.9824	−0.914057
$748$	0	0
$749$	11.0211	0.402703
$750$	0	0
$751$	−48.4826	−1.76916	−0.884578	−	0.466393i	$-0.845553\pi$
$751$	−48.4826	−1.76916	−0.884578	+	0.466393i	$0.845553\pi$
$752$	0	0
$753$	−2.17680	−0.0793270
$754$	0	0
$755$	−3.05476	−0.111174
$756$	0	0
$757$	−11.8260	−0.429823	−0.214911	−	0.976634i	$-0.568946\pi$
$757$	−11.8260	−0.429823	−0.214911	+	0.976634i	$0.568946\pi$
$758$	0	0
$759$	−2.20766	−0.0801329
$760$	0	0
$761$	−19.5510	−0.708725	−0.354362	−	0.935108i	$-0.615302\pi$
$761$	−19.5510	−0.708725	−0.354362	+	0.935108i	$0.615302\pi$
$762$	0	0
$763$	16.2413	0.587975
$764$	0	0
$765$	10.8694	0.392983
$766$	0	0
$767$	8.66171	0.312756
$768$	0	0
$769$	6.07587	0.219102	0.109551	−	0.993981i	$-0.465059\pi$
$769$	6.07587	0.219102	0.109551	+	0.993981i	$0.465059\pi$
$770$	0	0
$771$	5.17517	0.186379
$772$	0	0
$773$	−2.33086	−0.0838351	−0.0419175	−	0.999121i	$-0.513347\pi$
$773$	−2.33086	−0.0838351	−0.0419175	+	0.999121i	$0.513347\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	3.35197	0.120251
$778$	0	0
$779$	30.3594	1.08774
$780$	0	0
$781$	14.8557	0.531578
$782$	0	0
$783$	−15.6965	−0.560948
$784$	0	0
$785$	−4.85569	−0.173307
$786$	0	0
$787$	17.3805	0.619549	0.309774	−	0.950810i	$-0.399747\pi$
$787$	17.3805	0.619549	0.309774	+	0.950810i	$0.399747\pi$
$788$	0	0
$789$	0.661712	0.0235576
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−0.796281	−0.0282768
$794$	0	0
$795$	−2.69026	−0.0954137
$796$	0	0
$797$	−15.6942	−0.555917	−0.277959	−	0.960593i	$-0.589658\pi$
$797$	−15.6942	−0.555917	−0.277959	+	0.960593i	$0.589658\pi$
$798$	0	0
$799$	19.2538	0.681151
$800$	0	0
$801$	3.36334	0.118838
$802$	0	0
$803$	6.20766	0.219064
$804$	0	0
$805$	−5.42784	−0.191306
$806$	0	0
$807$	9.71137	0.341857
$808$	0	0
$809$	23.1443	0.813711	0.406855	−	0.913493i	$-0.366625\pi$
$809$	23.1443	0.813711	0.406855	+	0.913493i	$0.366625\pi$
$810$	0	0
$811$	27.6218	0.969933	0.484967	−	0.874533i	$-0.338832\pi$
$811$	27.6218	0.969933	0.484967	+	0.874533i	$0.338832\pi$
$812$	0	0
$813$	−0.992568	−0.0348109
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−16.3981	−0.573698
$818$	0	0
$819$	2.30580	0.0805712
$820$	0	0
$821$	−39.5123	−1.37899	−0.689494	−	0.724291i	$-0.742167\pi$
$821$	−39.5123	−1.37899	−0.689494	+	0.724291i	$0.742167\pi$
$822$	0	0
$823$	−13.5123	−0.471009	−0.235505	−	0.971873i	$-0.575674\pi$
$823$	−13.5123	−0.471009	−0.235505	+	0.971873i	$0.575674\pi$
$824$	0	0
$825$	−0.406728	−0.0141605
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	13.8346	0.480495	0.240247	−	0.970712i	$-0.422771\pi$
$829$	13.8346	0.480495	0.240247	+	0.970712i	$0.422771\pi$
$830$	0	0
$831$	8.94803	0.310404
$832$	0	0
$833$	3.83457	0.132860
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	14.2385	0.492155
$838$	0	0
$839$	−44.0422	−1.52051	−0.760253	−	0.649627i	$-0.774925\pi$
$839$	−44.0422	−1.52051	−0.760253	+	0.649627i	$0.774925\pi$
$840$	0	0
$841$	14.7501	0.508625
$842$	0	0
$843$	−8.48260	−0.292156
$844$	0	0
$845$	12.3383	0.424450
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−0.134569	−0.00461838
$850$	0	0
$851$	44.7325	1.53341
$852$	0	0
$853$	−29.8095	−1.02066	−0.510329	−	0.859979i	$-0.670476\pi$
$853$	−29.8095	−1.02066	−0.510329	+	0.859979i	$0.670476\pi$
$854$	0	0
$855$	−15.3856	−0.526177
$856$	0	0
$857$	42.7575	1.46057	0.730285	−	0.683143i	$-0.239387\pi$
$857$	42.7575	1.46057	0.730285	+	0.683143i	$0.239387\pi$
$858$	0	0
$859$	−6.31717	−0.215539	−0.107770	−	0.994176i	$-0.534371\pi$
$859$	−6.31717	−0.215539	−0.107770	+	0.994176i	$0.534371\pi$
$860$	0	0
$861$	−2.27494	−0.0775298
$862$	0	0
$863$	48.3680	1.64647	0.823233	−	0.567704i	$-0.192168\pi$
$863$	48.3680	1.64647	0.823233	+	0.567704i	$0.192168\pi$
$864$	0	0
$865$	19.0211	0.646737
$866$	0	0
$867$	−0.933872	−0.0317160
$868$	0	0
$869$	17.0970	0.579976
$870$	0	0
$871$	6.50765	0.220503
$872$	0	0
$873$	9.71647	0.328853
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−24.1940	−0.816972	−0.408486	−	0.912764i	$-0.633943\pi$
$877$	−24.1940	−0.816972	−0.408486	+	0.912764i	$0.633943\pi$
$878$	0	0
$879$	−1.29606	−0.0437149
$880$	0	0
$881$	12.3731	0.416860	0.208430	−	0.978037i	$-0.433165\pi$
$881$	12.3731	0.416860	0.208430	+	0.978037i	$0.433165\pi$
$882$	0	0
$883$	14.5248	0.488799	0.244400	−	0.969675i	$-0.421409\pi$
$883$	14.5248	0.488799	0.244400	+	0.969675i	$0.421409\pi$
$884$	0	0
$885$	4.33086	0.145580
$886$	0	0
$887$	25.3805	0.852194	0.426097	−	0.904677i	$-0.359888\pi$
$887$	25.3805	0.852194	0.426097	+	0.904677i	$0.359888\pi$
$888$	0	0
$889$	−17.2961	−0.580091
$890$	0	0
$891$	−7.53851	−0.252550
$892$	0	0
$893$	−27.2538	−0.912015
$894$	0	0
$895$	4.64803	0.155366
$896$	0	0
$897$	−1.79583	−0.0599612
$898$	0	0
$899$	−39.6863	−1.32361
$900$	0	0
$901$	25.3633	0.844975
$902$	0	0
$903$	1.22877	0.0408910
$904$	0	0
$905$	25.3383	0.842273
$906$	0	0
$907$	7.93272	0.263402	0.131701	−	0.991290i	$-0.457956\pi$
$907$	7.93272	0.263402	0.131701	+	0.991290i	$0.457956\pi$
$908$	0	0
$909$	52.9789	1.75720
$910$	0	0
$911$	−43.7364	−1.44905	−0.724526	−	0.689247i	$-0.757941\pi$
$911$	−43.7364	−1.44905	−0.724526	+	0.689247i	$0.757941\pi$
$912$	0	0
$913$	−8.81346	−0.291683
$914$	0	0
$915$	−0.398140	−0.0131621
$916$	0	0
$917$	−9.42784	−0.311335
$918$	0	0
$919$	32.7661	1.08085	0.540427	−	0.841391i	$-0.318263\pi$
$919$	32.7661	1.08085	0.540427	+	0.841391i	$0.318263\pi$
$920$	0	0
$921$	−8.33086	−0.274511
$922$	0	0
$923$	12.0845	0.397765
$924$	0	0
$925$	8.24130	0.270972
$926$	0	0
$927$	43.8267	1.43946
$928$	0	0
$929$	59.4478	1.95042	0.975210	−	0.221283i	$-0.0710246\pi$
$929$	59.4478	1.95042	0.975210	+	0.221283i	$0.0710246\pi$
$930$	0	0
$931$	−5.42784	−0.177890
$932$	0	0
$933$	−1.47517	−0.0482949
$934$	0	0
$935$	3.83457	0.125404
$936$	0	0
$937$	53.2824	1.74066	0.870330	−	0.492470i	$-0.163906\pi$
$937$	53.2824	1.74066	0.870330	+	0.492470i	$0.163906\pi$
$938$	0	0
$939$	8.42900	0.275070
$940$	0	0
$941$	−5.72506	−0.186632	−0.0933158	−	0.995637i	$-0.529747\pi$
$941$	−5.72506	−0.186632	−0.0933158	+	0.995637i	$0.529747\pi$
$942$	0	0
$943$	−30.3594	−0.988638
$944$	0	0
$945$	2.37309	0.0771965
$946$	0	0
$947$	27.4056	0.890561	0.445281	−	0.895391i	$-0.353104\pi$
$947$	27.4056	0.890561	0.445281	+	0.895391i	$0.353104\pi$
$948$	0	0
$949$	5.04966	0.163919
$950$	0	0
$951$	−0.732489	−0.0237526
$952$	0	0
$953$	−10.7998	−0.349839	−0.174919	−	0.984583i	$-0.555967\pi$
$953$	−10.7998	−0.349839	−0.174919	+	0.984583i	$0.555967\pi$
$954$	0	0
$955$	15.6691	0.507042
$956$	0	0
$957$	−2.69026	−0.0869637
$958$	0	0
$959$	13.6691	0.441400
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−31.2401	−1.00670
$964$	0	0
$965$	2.16543	0.0697076
$966$	0	0
$967$	−11.2538	−0.361899	−0.180949	−	0.983492i	$-0.557917\pi$
$967$	−11.2538	−0.361899	−0.180949	+	0.983492i	$0.557917\pi$
$968$	0	0
$969$	−8.46542	−0.271949
$970$	0	0
$971$	−33.2401	−1.06673	−0.533363	−	0.845886i	$-0.679072\pi$
$971$	−33.2401	−1.06673	−0.533363	+	0.845886i	$0.679072\pi$
$972$	0	0
$973$	11.8682	0.380477
$974$	0	0
$975$	−0.330856	−0.0105959
$976$	0	0
$977$	−2.74617	−0.0878578	−0.0439289	−	0.999035i	$-0.513988\pi$
$977$	−2.74617	−0.0878578	−0.0439289	+	0.999035i	$0.513988\pi$
$978$	0	0
$979$	1.18654	0.0379221
$980$	0	0
$981$	−46.0371	−1.46985
$982$	0	0
$983$	−39.4615	−1.25863	−0.629313	−	0.777152i	$-0.716664\pi$
$983$	−39.4615	−1.25863	−0.629313	+	0.777152i	$0.716664\pi$
$984$	0	0
$985$	−2.81346	−0.0896442
$986$	0	0
$987$	2.04223	0.0650049
$988$	0	0
$989$	16.3981	0.521431
$990$	0	0
$991$	−20.9652	−0.665982	−0.332991	−	0.942930i	$-0.608058\pi$
$991$	−20.9652	−0.665982	−0.332991	+	0.942930i	$0.608058\pi$
$992$	0	0
$993$	1.22877	0.0389939
$994$	0	0
$995$	−13.0211	−0.412797
$996$	0	0
$997$	21.9441	0.694976	0.347488	−	0.937684i	$-0.387035\pi$
$997$	21.9441	0.694976	0.347488	+	0.937684i	$0.387035\pi$
$998$	0	0
$999$	−19.5573	−0.618766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bi.1.2		3
4.3	odd	2		770.2.a.l.1.2	✓	3
12.11	even	2		6930.2.a.cl.1.2		3
20.3	even	4		3850.2.c.z.1849.5		6
20.7	even	4		3850.2.c.z.1849.2		6
20.19	odd	2		3850.2.a.bu.1.2		3
28.27	even	2		5390.2.a.bz.1.2		3
44.43	even	2		8470.2.a.cl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.2	✓	3	4.3	odd	2
3850.2.a.bu.1.2		3	20.19	odd	2
3850.2.c.z.1849.2		6	20.7	even	4
3850.2.c.z.1849.5		6	20.3	even	4
5390.2.a.bz.1.2		3	28.27	even	2
6160.2.a.bi.1.2		3	1.1	even	1	trivial
6930.2.a.cl.1.2		3	12.11	even	2
8470.2.a.cl.1.2		3	44.43	even	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.406728 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83457 q^{9} +O(q^{10})\)	\(q+0.406728 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83457 q^{9} -1.00000 q^{11} -0.813457 q^{13} -0.406728 q^{15} +3.83457 q^{17} -5.42784 q^{19} +0.406728 q^{21} +5.42784 q^{23} +1.00000 q^{25} -2.37309 q^{27} +6.61439 q^{29} -6.00000 q^{31} -0.406728 q^{33} -1.00000 q^{35} +8.24130 q^{37} -0.330856 q^{39} -5.59327 q^{41} +3.02112 q^{43} +2.83457 q^{45} +5.02112 q^{47} +1.00000 q^{49} +1.55963 q^{51} +6.61439 q^{53} +1.00000 q^{55} -2.20766 q^{57} -10.6480 q^{59} +0.978885 q^{61} -2.83457 q^{63} +0.813457 q^{65} -8.00000 q^{67} +2.20766 q^{69} -14.8557 q^{71} -6.20766 q^{73} +0.406728 q^{75} -1.00000 q^{77} -17.0970 q^{79} +7.53851 q^{81} +8.81346 q^{83} -3.83457 q^{85} +2.69026 q^{87} -1.18654 q^{89} -0.813457 q^{91} -2.44037 q^{93} +5.42784 q^{95} -3.42784 q^{97} +2.83457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9	\( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9} - 3 q^{11} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{35} + 4 q^{37} - 40 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - 2 q^{47} + 3 q^{49} + 12 q^{51} + 4 q^{53} + 3 q^{55} + 8 q^{57} - 10 q^{59} + 20 q^{61} + 11 q^{63} - 24 q^{67} - 8 q^{69} - 8 q^{71} - 4 q^{73} - 3 q^{77} + 6 q^{79} + 47 q^{81} + 24 q^{83} + 8 q^{85} - 48 q^{87} - 6 q^{89} - 2 q^{95} + 8 q^{97} - 11 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 11 * q^9 - 3 * q^11 - 8 * q^17 + 2 * q^19 - 2 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^35 + 4 * q^37 - 40 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - 2 * q^47 + 3 * q^49 + 12 * q^51 + 4 * q^53 + 3 * q^55 + 8 * q^57 - 10 * q^59 + 20 * q^61 + 11 * q^63 - 24 * q^67 - 8 * q^69 - 8 * q^71 - 4 * q^73 - 3 * q^77 + 6 * q^79 + 47 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 - 6 * q^89 - 2 * q^95 + 8 * q^97 - 11 * q^99

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