LMFDB - Embedded newform 6160.2.a.w.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{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			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +1.00000 q^{11} -4.47214 q^{13} -4.47214 q^{17} -6.47214 q^{19} +2.47214 q^{23} +1.00000 q^{25} +4.47214 q^{29} -2.47214 q^{31} +1.00000 q^{35} +4.47214 q^{37} -2.00000 q^{41} -4.00000 q^{43} +3.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} -0.472136 q^{53} -1.00000 q^{55} -10.4721 q^{59} +4.47214 q^{61} +3.00000 q^{63} +4.47214 q^{65} +4.94427 q^{67} +16.4721 q^{73} -1.00000 q^{77} -1.52786 q^{79} +9.00000 q^{81} +4.47214 q^{85} -2.94427 q^{89} +4.47214 q^{91} +6.47214 q^{95} -4.47214 q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^9	$ 2 q - 2 q^{5} - 2 q^{7} - 6 q^{9} + 2 q^{11} - 4 q^{19} - 4 q^{23} + 2 q^{25} + 4 q^{31} + 2 q^{35} - 4 q^{41} - 8 q^{43} + 6 q^{45} - 4 q^{47} + 2 q^{49} + 8 q^{53} - 2 q^{55} - 12 q^{59} + 6 q^{63} - 8 q^{67} + 24 q^{73} - 2 q^{77} - 12 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{95} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^9 + 2 * q^11 - 4 * q^19 - 4 * q^23 + 2 * q^25 + 4 * q^31 + 2 * q^35 - 4 * q^41 - 8 * q^43 + 6 * q^45 - 4 * q^47 + 2 * q^49 + 8 * q^53 - 2 * q^55 - 12 * q^59 + 6 * q^63 - 8 * q^67 + 24 * q^73 - 2 * q^77 - 12 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^95 - 6 * 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	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	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
$9$	−3.00000	−1.00000
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4721	−1.36336	−0.681678	−	0.731652i	$-0.738749\pi$
$59$	−10.4721	−1.36336	−0.681678	+	0.731652i	$0.738749\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	16.4721	1.92792	0.963959	−	0.266051i	$-0.0857191\pi$
$73$	16.4721	1.92792	0.963959	+	0.266051i	$0.0857191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.52786	−0.171898	−0.0859491	−	0.996300i	$-0.527392\pi$
$79$	−1.52786	−0.171898	−0.0859491	+	0.996300i	$0.527392\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.94427	−0.312092	−0.156046	−	0.987750i	$-0.549875\pi$
$89$	−2.94427	−0.312092	−0.156046	+	0.987750i	$0.549875\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.47214	0.664027
$96$	0	0
$97$	−4.47214	−0.454077	−0.227038	−	0.973886i	$-0.572904\pi$
$97$	−4.47214	−0.454077	−0.227038	+	0.973886i	$0.572904\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−0.472136	−0.0469793	−0.0234896	−	0.999724i	$-0.507478\pi$
$101$	−0.472136	−0.0469793	−0.0234896	+	0.999724i	$0.507478\pi$
$102$	0	0
$103$	−1.52786	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$103$	−1.52786	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.9443	1.63806	0.819032	−	0.573747i	$-0.194511\pi$
$107$	16.9443	1.63806	0.819032	+	0.573747i	$0.194511\pi$
$108$	0	0
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−2.47214	−0.230528
$116$	0	0
$117$	13.4164	1.24035
$118$	0	0
$119$	4.47214	0.409960
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.52786	0.832453	0.416227	−	0.909261i	$-0.363352\pi$
$131$	9.52786	0.832453	0.416227	+	0.909261i	$0.363352\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−1.52786	−0.129592	−0.0647959	−	0.997899i	$-0.520640\pi$
$139$	−1.52786	−0.129592	−0.0647959	+	0.997899i	$0.520640\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.47214	−0.373979
$144$	0	0
$145$	−4.47214	−0.371391
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.47214	−0.694064	−0.347032	−	0.937853i	$-0.612811\pi$
$149$	−8.47214	−0.694064	−0.347032	+	0.937853i	$0.612811\pi$
$150$	0	0
$151$	6.47214	0.526695	0.263347	−	0.964701i	$-0.415173\pi$
$151$	6.47214	0.526695	0.263347	+	0.964701i	$0.415173\pi$
$152$	0	0
$153$	13.4164	1.08465
$154$	0	0
$155$	2.47214	0.198567
$156$	0	0
$157$	−2.94427	−0.234978	−0.117489	−	0.993074i	$-0.537485\pi$
$157$	−2.94427	−0.234978	−0.117489	+	0.993074i	$0.537485\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	3.05573	0.239343	0.119672	−	0.992814i	$-0.461816\pi$
$163$	3.05573	0.239343	0.119672	+	0.992814i	$0.461816\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.8885	−1.38426	−0.692129	−	0.721774i	$-0.743327\pi$
$167$	−17.8885	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	19.4164	1.48481
$172$	0	0
$173$	3.52786	0.268219	0.134109	−	0.990967i	$-0.457183\pi$
$173$	3.52786	0.268219	0.134109	+	0.990967i	$0.457183\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−2.94427	−0.218846	−0.109423	−	0.993995i	$-0.534900\pi$
$181$	−2.94427	−0.218846	−0.109423	+	0.993995i	$0.534900\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.47214	−0.328798
$186$	0	0
$187$	−4.47214	−0.327035
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.8885	−1.29437	−0.647185	−	0.762333i	$-0.724054\pi$
$191$	−17.8885	−1.29437	−0.647185	+	0.762333i	$0.724054\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.9443	−0.779747	−0.389874	−	0.920868i	$-0.627481\pi$
$197$	−10.9443	−0.779747	−0.389874	+	0.920868i	$0.627481\pi$
$198$	0	0
$199$	−18.4721	−1.30945	−0.654727	−	0.755865i	$-0.727217\pi$
$199$	−18.4721	−1.30945	−0.654727	+	0.755865i	$0.727217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.47214	−0.313882
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−7.41641	−0.515476
$208$	0	0
$209$	−6.47214	−0.447687
$210$	0	0
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	2.47214	0.167820
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	−9.52786	−0.638033	−0.319016	−	0.947749i	$-0.603353\pi$
$223$	−9.52786	−0.638033	−0.319016	+	0.947749i	$0.603353\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−2.94427	−0.194563	−0.0972815	−	0.995257i	$-0.531015\pi$
$229$	−2.94427	−0.194563	−0.0972815	+	0.995257i	$0.531015\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.05573	−0.331212	−0.165606	−	0.986192i	$-0.552958\pi$
$233$	−5.05573	−0.331212	−0.165606	+	0.986192i	$0.552958\pi$
$234$	0	0
$235$	6.47214	0.422196
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.4164	−0.738466	−0.369233	−	0.929337i	$-0.620380\pi$
$239$	−11.4164	−0.738466	−0.369233	+	0.929337i	$0.620380\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.583592	0.0368360	0.0184180	−	0.999830i	$-0.494137\pi$
$251$	0.583592	0.0368360	0.0184180	+	0.999830i	$0.494137\pi$
$252$	0	0
$253$	2.47214	0.155422
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.4721	1.02750	0.513752	−	0.857939i	$-0.328255\pi$
$257$	16.4721	1.02750	0.513752	+	0.857939i	$0.328255\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	−13.4164	−0.830455
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.05573	0.308253	0.154127	−	0.988051i	$-0.450744\pi$
$269$	5.05573	0.308253	0.154127	+	0.988051i	$0.450744\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	0	0
$279$	7.41641	0.444009
$280$	0	0
$281$	19.8885	1.18645	0.593226	−	0.805036i	$-0.297854\pi$
$281$	19.8885	1.18645	0.593226	+	0.805036i	$0.297854\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.3607	1.54001	0.770004	−	0.638039i	$-0.220254\pi$
$293$	26.3607	1.54001	0.770004	+	0.638039i	$0.220254\pi$
$294$	0	0
$295$	10.4721	0.609711
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.0557	−0.639369
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.47214	−0.256074
$306$	0	0
$307$	12.9443	0.738769	0.369384	−	0.929277i	$-0.379569\pi$
$307$	12.9443	0.738769	0.369384	+	0.929277i	$0.379569\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.52786	−0.313456	−0.156728	−	0.987642i	$-0.550095\pi$
$311$	−5.52786	−0.313456	−0.156728	+	0.987642i	$0.550095\pi$
$312$	0	0
$313$	−9.41641	−0.532247	−0.266123	−	0.963939i	$-0.585743\pi$
$313$	−9.41641	−0.532247	−0.266123	+	0.963939i	$0.585743\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	−29.4164	−1.65219	−0.826095	−	0.563531i	$-0.809443\pi$
$317$	−29.4164	−1.65219	−0.826095	+	0.563531i	$0.809443\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	0	0
$321$	0	0
$322$	0	0
$323$	28.9443	1.61050
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	−26.8328	−1.47486	−0.737432	−	0.675421i	$-0.763962\pi$
$331$	−26.8328	−1.47486	−0.737432	+	0.675421i	$0.763962\pi$
$332$	0	0
$333$	−13.4164	−0.735215
$334$	0	0
$335$	−4.94427	−0.270134
$336$	0	0
$337$	−24.8328	−1.35273	−0.676365	−	0.736567i	$-0.736446\pi$
$337$	−24.8328	−1.35273	−0.676365	+	0.736567i	$0.736446\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.47214	−0.133874
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	9.41641	0.504049	0.252024	−	0.967721i	$-0.418904\pi$
$349$	9.41641	0.504049	0.252024	+	0.967721i	$0.418904\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.3607	1.40304	0.701519	−	0.712651i	$-0.252506\pi$
$353$	26.3607	1.40304	0.701519	+	0.712651i	$0.252506\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.3607	1.28571	0.642854	−	0.765989i	$-0.277750\pi$
$359$	24.3607	1.28571	0.642854	+	0.765989i	$0.277750\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.4721	−0.862191
$366$	0	0
$367$	−1.52786	−0.0797539	−0.0398769	−	0.999205i	$-0.512697\pi$
$367$	−1.52786	−0.0797539	−0.0398769	+	0.999205i	$0.512697\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0.472136	0.0245121
$372$	0	0
$373$	−20.8328	−1.07868	−0.539341	−	0.842087i	$-0.681327\pi$
$373$	−20.8328	−1.07868	−0.539341	+	0.842087i	$0.681327\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−8.94427	−0.459436	−0.229718	−	0.973257i	$-0.573780\pi$
$379$	−8.94427	−0.459436	−0.229718	+	0.973257i	$0.573780\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	37.3050	1.90619	0.953097	−	0.302665i	$-0.0978763\pi$
$383$	37.3050	1.90619	0.953097	+	0.302665i	$0.0978763\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	−11.0557	−0.559112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.52786	0.0768752
$396$	0	0
$397$	22.9443	1.15154	0.575770	−	0.817612i	$-0.304702\pi$
$397$	22.9443	1.15154	0.575770	+	0.817612i	$0.304702\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	11.0557	0.550725
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	4.47214	0.221676
$408$	0	0
$409$	20.8328	1.03012	0.515058	−	0.857155i	$-0.327770\pi$
$409$	20.8328	1.03012	0.515058	+	0.857155i	$0.327770\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.4721	0.515300
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.4164	1.14397	0.571983	−	0.820265i	$-0.306174\pi$
$419$	23.4164	1.14397	0.571983	+	0.820265i	$0.306174\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	19.4164	0.944058
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	−4.47214	−0.216422
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.4164	−1.70595	−0.852974	−	0.521953i	$-0.825204\pi$
$431$	−35.4164	−1.70595	−0.852974	+	0.521953i	$0.825204\pi$
$432$	0	0
$433$	−27.3050	−1.31219	−0.656096	−	0.754677i	$-0.727793\pi$
$433$	−27.3050	−1.31219	−0.656096	+	0.754677i	$0.727793\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	2.94427	0.139572
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.47214	−0.209657
$456$	0	0
$457$	−0.111456	−0.00521370	−0.00260685	−	0.999997i	$-0.500830\pi$
$457$	−0.111456	−0.00521370	−0.00260685	+	0.999997i	$0.500830\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.4164	1.55636	0.778179	−	0.628043i	$-0.216144\pi$
$461$	33.4164	1.55636	0.778179	+	0.628043i	$0.216144\pi$
$462$	0	0
$463$	−23.4164	−1.08825	−0.544126	−	0.839003i	$-0.683139\pi$
$463$	−23.4164	−1.08825	−0.544126	+	0.839003i	$0.683139\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.8885	1.93837	0.969185	−	0.246333i	$-0.0792256\pi$
$467$	41.8885	1.93837	0.969185	+	0.246333i	$0.0792256\pi$
$468$	0	0
$469$	−4.94427	−0.228305
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−6.47214	−0.296962
$476$	0	0
$477$	1.41641	0.0648529
$478$	0	0
$479$	−11.0557	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$479$	−11.0557	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.47214	0.203069
$486$	0	0
$487$	0.583592	0.0264451	0.0132225	−	0.999913i	$-0.495791\pi$
$487$	0.583592	0.0264451	0.0132225	+	0.999913i	$0.495791\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0557	0.679455	0.339728	−	0.940524i	$-0.389665\pi$
$491$	15.0557	0.679455	0.339728	+	0.940524i	$0.389665\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.8885	−0.979866	−0.489933	−	0.871760i	$-0.662979\pi$
$499$	−21.8885	−0.979866	−0.489933	+	0.871760i	$0.662979\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	0.472136	0.0210098
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.8328	1.80988	0.904941	−	0.425536i	$-0.139915\pi$
$509$	40.8328	1.80988	0.904941	+	0.425536i	$0.139915\pi$
$510$	0	0
$511$	−16.4721	−0.728684
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.52786	0.0673257
$516$	0	0
$517$	−6.47214	−0.284644
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	0	0
$523$	−30.8328	−1.34822	−0.674112	−	0.738629i	$-0.735474\pi$
$523$	−30.8328	−1.34822	−0.674112	+	0.738629i	$0.735474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.0557	0.481595
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	31.4164	1.36336
$532$	0	0
$533$	8.94427	0.387419
$534$	0	0
$535$	−16.9443	−0.732565
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	12.4721	0.536219	0.268110	−	0.963388i	$-0.413601\pi$
$541$	12.4721	0.536219	0.268110	+	0.963388i	$0.413601\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.4721	−0.534248
$546$	0	0
$547$	2.11146	0.0902793	0.0451397	−	0.998981i	$-0.485627\pi$
$547$	2.11146	0.0902793	0.0451397	+	0.998981i	$0.485627\pi$
$548$	0	0
$549$	−13.4164	−0.572598
$550$	0	0
$551$	−28.9443	−1.23307
$552$	0	0
$553$	1.52786	0.0649714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.8328	−1.22169	−0.610843	−	0.791752i	$-0.709169\pi$
$557$	−28.8328	−1.22169	−0.610843	+	0.791752i	$0.709169\pi$
$558$	0	0
$559$	17.8885	0.756605
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.05573	0.128784	0.0643918	−	0.997925i	$-0.479489\pi$
$563$	3.05573	0.128784	0.0643918	+	0.997925i	$0.479489\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−17.0557	−0.715013	−0.357507	−	0.933911i	$-0.616373\pi$
$569$	−17.0557	−0.715013	−0.357507	+	0.933911i	$0.616373\pi$
$570$	0	0
$571$	34.8328	1.45771	0.728854	−	0.684669i	$-0.240053\pi$
$571$	34.8328	1.45771	0.728854	+	0.684669i	$0.240053\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.47214	0.103095
$576$	0	0
$577$	39.3050	1.63629	0.818143	−	0.575014i	$-0.195004\pi$
$577$	39.3050	1.63629	0.818143	+	0.575014i	$0.195004\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.472136	−0.0195539
$584$	0	0
$585$	−13.4164	−0.554700
$586$	0	0
$587$	25.8885	1.06853	0.534267	−	0.845316i	$-0.320588\pi$
$587$	25.8885	1.06853	0.534267	+	0.845316i	$0.320588\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.5836	0.598876	0.299438	−	0.954116i	$-0.403201\pi$
$593$	14.5836	0.598876	0.299438	+	0.954116i	$0.403201\pi$
$594$	0	0
$595$	−4.47214	−0.183340
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.11146	0.249707	0.124854	−	0.992175i	$-0.460154\pi$
$599$	6.11146	0.249707	0.124854	+	0.992175i	$0.460154\pi$
$600$	0	0
$601$	−32.8328	−1.33928	−0.669639	−	0.742687i	$-0.733551\pi$
$601$	−32.8328	−1.33928	−0.669639	+	0.742687i	$0.733551\pi$
$602$	0	0
$603$	−14.8328	−0.604039
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	14.8328	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$607$	14.8328	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.9443	1.17096
$612$	0	0
$613$	8.11146	0.327619	0.163809	−	0.986492i	$-0.447622\pi$
$613$	8.11146	0.327619	0.163809	+	0.986492i	$0.447622\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.8328	−1.48283	−0.741417	−	0.671045i	$-0.765846\pi$
$617$	−36.8328	−1.48283	−0.741417	+	0.671045i	$0.765846\pi$
$618$	0	0
$619$	−39.4164	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$619$	−39.4164	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.94427	0.117960
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	17.8885	0.712132	0.356066	−	0.934461i	$-0.384118\pi$
$631$	17.8885	0.712132	0.356066	+	0.934461i	$0.384118\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.8885	1.10153	0.550766	−	0.834660i	$-0.314336\pi$
$641$	27.8885	1.10153	0.550766	+	0.834660i	$0.314336\pi$
$642$	0	0
$643$	11.0557	0.435995	0.217998	−	0.975949i	$-0.430047\pi$
$643$	11.0557	0.435995	0.217998	+	0.975949i	$0.430047\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.5279	0.689091	0.344546	−	0.938770i	$-0.388033\pi$
$647$	17.5279	0.689091	0.344546	+	0.938770i	$0.388033\pi$
$648$	0	0
$649$	−10.4721	−0.411067
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.472136	−0.0184761	−0.00923805	−	0.999957i	$-0.502941\pi$
$653$	−0.472136	−0.0184761	−0.00923805	+	0.999957i	$0.502941\pi$
$654$	0	0
$655$	−9.52786	−0.372284
$656$	0	0
$657$	−49.4164	−1.92792
$658$	0	0
$659$	−2.11146	−0.0822507	−0.0411253	−	0.999154i	$-0.513094\pi$
$659$	−2.11146	−0.0822507	−0.0411253	+	0.999154i	$0.513094\pi$
$660$	0	0
$661$	3.88854	0.151247	0.0756234	−	0.997136i	$-0.475905\pi$
$661$	3.88854	0.151247	0.0756234	+	0.997136i	$0.475905\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.47214	−0.250979
$666$	0	0
$667$	11.0557	0.428080
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.47214	0.172645
$672$	0	0
$673$	−8.83282	−0.340480	−0.170240	−	0.985403i	$-0.554454\pi$
$673$	−8.83282	−0.340480	−0.170240	+	0.985403i	$0.554454\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.4164	1.13056	0.565282	−	0.824898i	$-0.308767\pi$
$677$	29.4164	1.13056	0.565282	+	0.824898i	$0.308767\pi$
$678$	0	0
$679$	4.47214	0.171625
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.7214	1.86427	0.932136	−	0.362110i	$-0.117943\pi$
$683$	48.7214	1.86427	0.932136	+	0.362110i	$0.117943\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.11146	0.0804401
$690$	0	0
$691$	−7.41641	−0.282133	−0.141067	−	0.990000i	$-0.545053\pi$
$691$	−7.41641	−0.282133	−0.141067	+	0.990000i	$0.545053\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	1.52786	0.0579552
$696$	0	0
$697$	8.94427	0.338788
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.4164	0.657809	0.328904	−	0.944363i	$-0.393321\pi$
$701$	17.4164	0.657809	0.328904	+	0.944363i	$0.393321\pi$
$702$	0	0
$703$	−28.9443	−1.09165
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.472136	0.0177565
$708$	0	0
$709$	36.8328	1.38329	0.691643	−	0.722240i	$-0.256887\pi$
$709$	36.8328	1.38329	0.691643	+	0.722240i	$0.256887\pi$
$710$	0	0
$711$	4.58359	0.171898
$712$	0	0
$713$	−6.11146	−0.228876
$714$	0	0
$715$	4.47214	0.167248
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−39.4164	−1.46998	−0.734992	−	0.678076i	$-0.762814\pi$
$719$	−39.4164	−1.46998	−0.734992	+	0.678076i	$0.762814\pi$
$720$	0	0
$721$	1.52786	0.0569006
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.47214	0.166091
$726$	0	0
$727$	−11.4164	−0.423411	−0.211706	−	0.977333i	$-0.567902\pi$
$727$	−11.4164	−0.423411	−0.211706	+	0.977333i	$0.567902\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	26.3607	0.973654	0.486827	−	0.873498i	$-0.338154\pi$
$733$	26.3607	0.973654	0.486827	+	0.873498i	$0.338154\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.94427	0.182125
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.8885	−1.24325	−0.621625	−	0.783315i	$-0.713527\pi$
$743$	−33.8885	−1.24325	−0.621625	+	0.783315i	$0.713527\pi$
$744$	0	0
$745$	8.47214	0.310395
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−16.9443	−0.619130
$750$	0	0
$751$	25.8885	0.944686	0.472343	−	0.881415i	$-0.343408\pi$
$751$	25.8885	0.944686	0.472343	+	0.881415i	$0.343408\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.47214	−0.235545
$756$	0	0
$757$	−48.4721	−1.76175	−0.880875	−	0.473349i	$-0.843045\pi$
$757$	−48.4721	−1.76175	−0.880875	+	0.473349i	$0.843045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.05573	0.0382701	0.0191351	−	0.999817i	$-0.493909\pi$
$761$	1.05573	0.0382701	0.0191351	+	0.999817i	$0.493909\pi$
$762$	0	0
$763$	−12.4721	−0.451522
$764$	0	0
$765$	−13.4164	−0.485071
$766$	0	0
$767$	46.8328	1.69103
$768$	0	0
$769$	49.7771	1.79501	0.897504	−	0.441007i	$-0.145378\pi$
$769$	49.7771	1.79501	0.897504	+	0.441007i	$0.145378\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.94427	0.249768	0.124884	−	0.992171i	$-0.460144\pi$
$773$	6.94427	0.249768	0.124884	+	0.992171i	$0.460144\pi$
$774$	0	0
$775$	−2.47214	−0.0888017
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.9443	0.463777
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.94427	0.105086
$786$	0	0
$787$	20.9443	0.746583	0.373291	−	0.927714i	$-0.378229\pi$
$787$	20.9443	0.746583	0.373291	+	0.927714i	$0.378229\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.0557	0.462458	0.231229	−	0.972899i	$-0.425725\pi$
$797$	13.0557	0.462458	0.231229	+	0.972899i	$0.425725\pi$
$798$	0	0
$799$	28.9443	1.02397
$800$	0	0
$801$	8.83282	0.312092
$802$	0	0
$803$	16.4721	0.581289
$804$	0	0
$805$	2.47214	0.0871313
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.83282	−0.169913	−0.0849564	−	0.996385i	$-0.527075\pi$
$809$	−4.83282	−0.169913	−0.0849564	+	0.996385i	$0.527075\pi$
$810$	0	0
$811$	25.5279	0.896405	0.448202	−	0.893932i	$-0.352064\pi$
$811$	25.5279	0.896405	0.448202	+	0.893932i	$0.352064\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.05573	−0.107037
$816$	0	0
$817$	25.8885	0.905725
$818$	0	0
$819$	−13.4164	−0.468807
$820$	0	0
$821$	−19.5279	−0.681527	−0.340764	−	0.940149i	$-0.610686\pi$
$821$	−19.5279	−0.681527	−0.340764	+	0.940149i	$0.610686\pi$
$822$	0	0
$823$	−8.58359	−0.299205	−0.149603	−	0.988746i	$-0.547799\pi$
$823$	−8.58359	−0.299205	−0.149603	+	0.988746i	$0.547799\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.9443	−0.589210	−0.294605	−	0.955619i	$-0.595188\pi$
$827$	−16.9443	−0.589210	−0.294605	+	0.955619i	$0.595188\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.47214	−0.154950
$834$	0	0
$835$	17.8885	0.619059
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.2492	−1.04432	−0.522160	−	0.852848i	$-0.674873\pi$
$839$	−30.2492	−1.04432	−0.522160	+	0.852848i	$0.674873\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.00000	−0.240807
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.0557	0.378985
$852$	0	0
$853$	1.63932	0.0561293	0.0280646	−	0.999606i	$-0.491066\pi$
$853$	1.63932	0.0561293	0.0280646	+	0.999606i	$0.491066\pi$
$854$	0	0
$855$	−19.4164	−0.664027
$856$	0	0
$857$	−43.3050	−1.47927	−0.739634	−	0.673009i	$-0.765002\pi$
$857$	−43.3050	−1.47927	−0.739634	+	0.673009i	$0.765002\pi$
$858$	0	0
$859$	−41.3050	−1.40931	−0.704653	−	0.709552i	$-0.748897\pi$
$859$	−41.3050	−1.40931	−0.704653	+	0.709552i	$0.748897\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.5279	1.27746	0.638732	−	0.769430i	$-0.279459\pi$
$863$	37.5279	1.27746	0.638732	+	0.769430i	$0.279459\pi$
$864$	0	0
$865$	−3.52786	−0.119951
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.52786	−0.0518292
$870$	0	0
$871$	−22.1115	−0.749218
$872$	0	0
$873$	13.4164	0.454077
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−12.1115	−0.408975	−0.204487	−	0.978869i	$-0.565553\pi$
$877$	−12.1115	−0.408975	−0.204487	+	0.978869i	$0.565553\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−50.9443	−1.71636	−0.858178	−	0.513352i	$-0.828403\pi$
$881$	−50.9443	−1.71636	−0.858178	+	0.513352i	$0.828403\pi$
$882$	0	0
$883$	12.9443	0.435609	0.217805	−	0.975992i	$-0.430110\pi$
$883$	12.9443	0.435609	0.217805	+	0.975992i	$0.430110\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.8328	−0.766651	−0.383325	−	0.923613i	$-0.625221\pi$
$887$	−22.8328	−0.766651	−0.383325	+	0.923613i	$0.625221\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	41.8885	1.40175
$894$	0	0
$895$	8.94427	0.298974
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.0557	−0.368729
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.94427	0.0978709
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	1.41641	0.0469793
$910$	0	0
$911$	27.7771	0.920296	0.460148	−	0.887842i	$-0.347796\pi$
$911$	27.7771	0.920296	0.460148	+	0.887842i	$0.347796\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.52786	−0.314638
$918$	0	0
$919$	−22.4721	−0.741287	−0.370644	−	0.928775i	$-0.620863\pi$
$919$	−22.4721	−0.741287	−0.370644	+	0.928775i	$0.620863\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.47214	0.147043
$926$	0	0
$927$	4.58359	0.150545
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	−6.47214	−0.212116
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.47214	0.146254
$936$	0	0
$937$	−26.5836	−0.868448	−0.434224	−	0.900805i	$-0.642977\pi$
$937$	−26.5836	−0.868448	−0.434224	+	0.900805i	$0.642977\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.4164	−0.437362	−0.218681	−	0.975796i	$-0.570175\pi$
$941$	−13.4164	−0.437362	−0.218681	+	0.975796i	$0.570175\pi$
$942$	0	0
$943$	−4.94427	−0.161008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.9443	0.680597	0.340299	−	0.940317i	$-0.389472\pi$
$947$	20.9443	0.680597	0.340299	+	0.940317i	$0.389472\pi$
$948$	0	0
$949$	−73.6656	−2.39129
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	17.8885	0.578860
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	−50.8328	−1.63806
$964$	0	0
$965$	−22.0000	−0.708205
$966$	0	0
$967$	−44.9443	−1.44531	−0.722655	−	0.691209i	$-0.757079\pi$
$967$	−44.9443	−1.44531	−0.722655	+	0.691209i	$0.757079\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	52.3607	1.68033	0.840167	−	0.542328i	$-0.182457\pi$
$971$	52.3607	1.68033	0.840167	+	0.542328i	$0.182457\pi$
$972$	0	0
$973$	1.52786	0.0489811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.8885	1.14818	0.574088	−	0.818794i	$-0.305357\pi$
$977$	35.8885	1.14818	0.574088	+	0.818794i	$0.305357\pi$
$978$	0	0
$979$	−2.94427	−0.0940993
$980$	0	0
$981$	−37.4164	−1.19461
$982$	0	0
$983$	−1.52786	−0.0487313	−0.0243656	−	0.999703i	$-0.507757\pi$
$983$	−1.52786	−0.0487313	−0.0243656	+	0.999703i	$0.507757\pi$
$984$	0	0
$985$	10.9443	0.348713
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.88854	−0.314437
$990$	0	0
$991$	−44.9443	−1.42770	−0.713851	−	0.700298i	$-0.753051\pi$
$991$	−44.9443	−1.42770	−0.713851	+	0.700298i	$0.753051\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.4721	0.585606
$996$	0	0
$997$	29.4164	0.931627	0.465813	−	0.884883i	$-0.345762\pi$
$997$	29.4164	0.931627	0.465813	+	0.884883i	$0.345762\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.w.1.1		2
4.3	odd	2		3080.2.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.f.1.1	✓	2	4.3	odd	2
6160.2.a.w.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +1.00000 q^{11} -4.47214 q^{13} -4.47214 q^{17} -6.47214 q^{19} +2.47214 q^{23} +1.00000 q^{25} +4.47214 q^{29} -2.47214 q^{31} +1.00000 q^{35} +4.47214 q^{37} -2.00000 q^{41} -4.00000 q^{43} +3.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} -0.472136 q^{53} -1.00000 q^{55} -10.4721 q^{59} +4.47214 q^{61} +3.00000 q^{63} +4.47214 q^{65} +4.94427 q^{67} +16.4721 q^{73} -1.00000 q^{77} -1.52786 q^{79} +9.00000 q^{81} +4.47214 q^{85} -2.94427 q^{89} +4.47214 q^{91} +6.47214 q^{95} -4.47214 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^9	\( 2 q - 2 q^{5} - 2 q^{7} - 6 q^{9} + 2 q^{11} - 4 q^{19} - 4 q^{23} + 2 q^{25} + 4 q^{31} + 2 q^{35} - 4 q^{41} - 8 q^{43} + 6 q^{45} - 4 q^{47} + 2 q^{49} + 8 q^{53} - 2 q^{55} - 12 q^{59} + 6 q^{63} - 8 q^{67} + 24 q^{73} - 2 q^{77} - 12 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{95} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^9 + 2 * q^11 - 4 * q^19 - 4 * q^23 + 2 * q^25 + 4 * q^31 + 2 * q^35 - 4 * q^41 - 8 * q^43 + 6 * q^45 - 4 * q^47 + 2 * q^49 + 8 * q^53 - 2 * q^55 - 12 * q^59 + 6 * q^63 - 8 * q^67 + 24 * q^73 - 2 * q^77 - 12 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^95 - 6 * q^99

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