LMFDB - Embedded newform 9660.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9660,2,Mod(1,9660)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9660, 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("9660.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9660.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:	$77.1354883526$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 9x^{4} + 16x^{3} + 21x^{2} - 15x + 1 $ x^6 - 3x^5 - 9x^4 + 16x^3 + 21x^2 - 15*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91856$ of defining polynomial
Character	$\chi$	$=$	9660.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.15777 q^{11} -4.51802 q^{13} -1.00000 q^{15} +5.42723 q^{17} +6.08569 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.83713 q^{29} +5.40633 q^{31} -3.15777 q^{33} +1.00000 q^{35} -3.31911 q^{37} -4.51802 q^{39} +3.54036 q^{41} -5.71070 q^{43} -1.00000 q^{45} +3.35052 q^{47} +1.00000 q^{49} +5.42723 q^{51} -11.5638 q^{53} +3.15777 q^{55} +6.08569 q^{57} +10.2867 q^{59} +7.59114 q^{61} -1.00000 q^{63} +4.51802 q^{65} -13.7428 q^{67} -1.00000 q^{69} -0.701785 q^{71} -10.6868 q^{73} +1.00000 q^{75} +3.15777 q^{77} -4.20742 q^{79} +1.00000 q^{81} +4.46709 q^{83} -5.42723 q^{85} -3.83713 q^{87} -4.54532 q^{89} +4.51802 q^{91} +5.40633 q^{93} -6.08569 q^{95} +6.37604 q^{97} -3.15777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9} - 3 q^{13} - 6 q^{15} - 3 q^{17} - 6 q^{21} - 6 q^{23} + 6 q^{25} + 6 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{35} - 15 q^{37} - 3 q^{39} + 6 q^{41} - 3 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 3 q^{51} - 3 q^{53} + 6 q^{59} - 18 q^{61} - 6 q^{63} + 3 q^{65} - 9 q^{67} - 6 q^{69} - 9 q^{73} + 6 q^{75} - 18 q^{79} + 6 q^{81} - 3 q^{83} + 3 q^{85} + 6 q^{87} - 6 q^{89} + 3 q^{91} + 6 q^{93} - 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9 - 3 * q^13 - 6 * q^15 - 3 * q^17 - 6 * q^21 - 6 * q^23 + 6 * q^25 + 6 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^35 - 15 * q^37 - 3 * q^39 + 6 * q^41 - 3 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^49 - 3 * q^51 - 3 * q^53 + 6 * q^59 - 18 * q^61 - 6 * q^63 + 3 * q^65 - 9 * q^67 - 6 * q^69 - 9 * q^73 + 6 * q^75 - 18 * q^79 + 6 * q^81 - 3 * q^83 + 3 * q^85 + 6 * q^87 - 6 * q^89 + 3 * q^91 + 6 * q^93 - 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.15777	−0.952103	−0.476052	−	0.879417i	$-0.657932\pi$
$11$	−3.15777	−0.952103	−0.476052	+	0.879417i	$0.657932\pi$
$12$	0	0
$13$	−4.51802	−1.25307	−0.626536	−	0.779392i	$-0.715528\pi$
$13$	−4.51802	−1.25307	−0.626536	+	0.779392i	$0.715528\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.42723	1.31630	0.658148	−	0.752889i	$-0.271340\pi$
$17$	5.42723	1.31630	0.658148	+	0.752889i	$0.271340\pi$
$18$	0	0
$19$	6.08569	1.39615	0.698076	−	0.716023i	$-0.254040\pi$
$19$	6.08569	1.39615	0.698076	+	0.716023i	$0.254040\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.83713	−0.712537	−0.356268	−	0.934384i	$-0.615951\pi$
$29$	−3.83713	−0.712537	−0.356268	+	0.934384i	$0.615951\pi$
$30$	0	0
$31$	5.40633	0.971005	0.485503	−	0.874235i	$-0.338637\pi$
$31$	5.40633	0.971005	0.485503	+	0.874235i	$0.338637\pi$
$32$	0	0
$33$	−3.15777	−0.549697
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.31911	−0.545659	−0.272829	−	0.962062i	$-0.587959\pi$
$37$	−3.31911	−0.545659	−0.272829	+	0.962062i	$0.587959\pi$
$38$	0	0
$39$	−4.51802	−0.723462
$40$	0	0
$41$	3.54036	0.552912	0.276456	−	0.961027i	$-0.410840\pi$
$41$	3.54036	0.552912	0.276456	+	0.961027i	$0.410840\pi$
$42$	0	0
$43$	−5.71070	−0.870873	−0.435436	−	0.900219i	$-0.643406\pi$
$43$	−5.71070	−0.870873	−0.435436	+	0.900219i	$0.643406\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.35052	0.488724	0.244362	−	0.969684i	$-0.421421\pi$
$47$	3.35052	0.488724	0.244362	+	0.969684i	$0.421421\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.42723	0.759964
$52$	0	0
$53$	−11.5638	−1.58842	−0.794208	−	0.607647i	$-0.792114\pi$
$53$	−11.5638	−1.58842	−0.794208	+	0.607647i	$0.792114\pi$
$54$	0	0
$55$	3.15777	0.425793
$56$	0	0
$57$	6.08569	0.806069
$58$	0	0
$59$	10.2867	1.33921	0.669607	−	0.742715i	$-0.266462\pi$
$59$	10.2867	1.33921	0.669607	+	0.742715i	$0.266462\pi$
$60$	0	0
$61$	7.59114	0.971946	0.485973	−	0.873974i	$-0.338465\pi$
$61$	7.59114	0.971946	0.485973	+	0.873974i	$0.338465\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.51802	0.560391
$66$	0	0
$67$	−13.7428	−1.67895	−0.839473	−	0.543402i	$-0.817136\pi$
$67$	−13.7428	−1.67895	−0.839473	+	0.543402i	$0.817136\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.701785	−0.0832865	−0.0416432	−	0.999133i	$-0.513259\pi$
$71$	−0.701785	−0.0832865	−0.0416432	+	0.999133i	$0.513259\pi$
$72$	0	0
$73$	−10.6868	−1.25080	−0.625399	−	0.780305i	$-0.715064\pi$
$73$	−10.6868	−1.25080	−0.625399	+	0.780305i	$0.715064\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	3.15777	0.359861
$78$	0	0
$79$	−4.20742	−0.473372	−0.236686	−	0.971586i	$-0.576061\pi$
$79$	−4.20742	−0.473372	−0.236686	+	0.971586i	$0.576061\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.46709	0.490327	0.245164	−	0.969482i	$-0.421158\pi$
$83$	4.46709	0.490327	0.245164	+	0.969482i	$0.421158\pi$
$84$	0	0
$85$	−5.42723	−0.588665
$86$	0	0
$87$	−3.83713	−0.411383
$88$	0	0
$89$	−4.54532	−0.481803	−0.240902	−	0.970550i	$-0.577443\pi$
$89$	−4.54532	−0.481803	−0.240902	+	0.970550i	$0.577443\pi$
$90$	0	0
$91$	4.51802	0.473617
$92$	0	0
$93$	5.40633	0.560610
$94$	0	0
$95$	−6.08569	−0.624379
$96$	0	0
$97$	6.37604	0.647389	0.323694	−	0.946162i	$-0.395075\pi$
$97$	6.37604	0.647389	0.323694	+	0.946162i	$0.395075\pi$
$98$	0	0
$99$	−3.15777	−0.317368
$100$	0	0
$101$	0.561521	0.0558735	0.0279367	−	0.999610i	$-0.491106\pi$
$101$	0.561521	0.0558735	0.0279367	+	0.999610i	$0.491106\pi$
$102$	0	0
$103$	4.59107	0.452371	0.226186	−	0.974084i	$-0.427374\pi$
$103$	4.59107	0.452371	0.226186	+	0.974084i	$0.427374\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	5.03603	0.486852	0.243426	−	0.969919i	$-0.421729\pi$
$107$	5.03603	0.486852	0.243426	+	0.969919i	$0.421729\pi$
$108$	0	0
$109$	−5.18019	−0.496173	−0.248086	−	0.968738i	$-0.579802\pi$
$109$	−5.18019	−0.496173	−0.248086	+	0.968738i	$0.579802\pi$
$110$	0	0
$111$	−3.31911	−0.315036
$112$	0	0
$113$	−3.40633	−0.320441	−0.160220	−	0.987081i	$-0.551220\pi$
$113$	−3.40633	−0.320441	−0.160220	+	0.987081i	$0.551220\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−4.51802	−0.417691
$118$	0	0
$119$	−5.42723	−0.497513
$120$	0	0
$121$	−1.02850	−0.0934998
$122$	0	0
$123$	3.54036	0.319224
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.03499	−0.446783	−0.223391	−	0.974729i	$-0.571713\pi$
$127$	−5.03499	−0.446783	−0.223391	+	0.974729i	$0.571713\pi$
$128$	0	0
$129$	−5.71070	−0.502799
$130$	0	0
$131$	−4.82609	−0.421658	−0.210829	−	0.977523i	$-0.567616\pi$
$131$	−4.82609	−0.421658	−0.210829	+	0.977523i	$0.567616\pi$
$132$	0	0
$133$	−6.08569	−0.527696
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	1.05441	0.0900847	0.0450423	−	0.998985i	$-0.485658\pi$
$137$	1.05441	0.0900847	0.0450423	+	0.998985i	$0.485658\pi$
$138$	0	0
$139$	4.83502	0.410101	0.205051	−	0.978751i	$-0.434264\pi$
$139$	4.83502	0.410101	0.205051	+	0.978751i	$0.434264\pi$
$140$	0	0
$141$	3.35052	0.282165
$142$	0	0
$143$	14.2669	1.19305
$144$	0	0
$145$	3.83713	0.318656
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	5.70917	0.467713	0.233857	−	0.972271i	$-0.424865\pi$
$149$	5.70917	0.467713	0.233857	+	0.972271i	$0.424865\pi$
$150$	0	0
$151$	−9.38390	−0.763651	−0.381826	−	0.924234i	$-0.624705\pi$
$151$	−9.38390	−0.763651	−0.381826	+	0.924234i	$0.624705\pi$
$152$	0	0
$153$	5.42723	0.438765
$154$	0	0
$155$	−5.40633	−0.434247
$156$	0	0
$157$	−0.0176653	−0.00140984	−0.000704922	−	1.00000i	$-0.500224\pi$
$157$	−0.0176653	−0.00140984	−0.000704922		1.00000i	$0.500224\pi$
$158$	0	0
$159$	−11.5638	−0.917072
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−20.2871	−1.58901	−0.794504	−	0.607259i	$-0.792269\pi$
$163$	−20.2871	−1.58901	−0.794504	+	0.607259i	$0.792269\pi$
$164$	0	0
$165$	3.15777	0.245832
$166$	0	0
$167$	1.98875	0.153894	0.0769469	−	0.997035i	$-0.475483\pi$
$167$	1.98875	0.153894	0.0769469	+	0.997035i	$0.475483\pi$
$168$	0	0
$169$	7.41248	0.570191
$170$	0	0
$171$	6.08569	0.465384
$172$	0	0
$173$	−19.7824	−1.50403	−0.752013	−	0.659148i	$-0.770917\pi$
$173$	−19.7824	−1.50403	−0.752013	+	0.659148i	$0.770917\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	10.2867	0.773196
$178$	0	0
$179$	−17.1764	−1.28382	−0.641912	−	0.766778i	$-0.721859\pi$
$179$	−17.1764	−1.28382	−0.641912	+	0.766778i	$0.721859\pi$
$180$	0	0
$181$	1.28049	0.0951781	0.0475890	−	0.998867i	$-0.484846\pi$
$181$	1.28049	0.0951781	0.0475890	+	0.998867i	$0.484846\pi$
$182$	0	0
$183$	7.59114	0.561154
$184$	0	0
$185$	3.31911	0.244026
$186$	0	0
$187$	−17.1379	−1.25325
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.62719	−0.262454	−0.131227	−	0.991352i	$-0.541892\pi$
$191$	−3.62719	−0.262454	−0.131227	+	0.991352i	$0.541892\pi$
$192$	0	0
$193$	0.103415	0.00744395	0.00372198	−	0.999993i	$-0.498815\pi$
$193$	0.103415	0.00744395	0.00372198	+	0.999993i	$0.498815\pi$
$194$	0	0
$195$	4.51802	0.323542
$196$	0	0
$197$	−7.65118	−0.545124	−0.272562	−	0.962138i	$-0.587871\pi$
$197$	−7.65118	−0.545124	−0.272562	+	0.962138i	$0.587871\pi$
$198$	0	0
$199$	−17.6566	−1.25164	−0.625821	−	0.779966i	$-0.715236\pi$
$199$	−17.6566	−1.25164	−0.625821	+	0.779966i	$0.715236\pi$
$200$	0	0
$201$	−13.7428	−0.969340
$202$	0	0
$203$	3.83713	0.269314
$204$	0	0
$205$	−3.54036	−0.247270
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−19.2172	−1.32928
$210$	0	0
$211$	16.1389	1.11105	0.555524	−	0.831501i	$-0.312518\pi$
$211$	16.1389	1.11105	0.555524	+	0.831501i	$0.312518\pi$
$212$	0	0
$213$	−0.701785	−0.0480855
$214$	0	0
$215$	5.71070	0.389466
$216$	0	0
$217$	−5.40633	−0.367006
$218$	0	0
$219$	−10.6868	−0.722148
$220$	0	0
$221$	−24.5203	−1.64941
$222$	0	0
$223$	13.0720	0.875365	0.437683	−	0.899130i	$-0.355799\pi$
$223$	13.0720	0.875365	0.437683	+	0.899130i	$0.355799\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−11.2162	−0.744442	−0.372221	−	0.928144i	$-0.621404\pi$
$227$	−11.2162	−0.744442	−0.372221	+	0.928144i	$0.621404\pi$
$228$	0	0
$229$	−19.8869	−1.31416	−0.657082	−	0.753819i	$-0.728210\pi$
$229$	−19.8869	−1.31416	−0.657082	+	0.753819i	$0.728210\pi$
$230$	0	0
$231$	3.15777	0.207766
$232$	0	0
$233$	17.3116	1.13412	0.567061	−	0.823676i	$-0.308080\pi$
$233$	17.3116	1.13412	0.567061	+	0.823676i	$0.308080\pi$
$234$	0	0
$235$	−3.35052	−0.218564
$236$	0	0
$237$	−4.20742	−0.273302
$238$	0	0
$239$	−13.8967	−0.898905	−0.449453	−	0.893304i	$-0.648381\pi$
$239$	−13.8967	−0.898905	−0.449453	+	0.893304i	$0.648381\pi$
$240$	0	0
$241$	4.12417	0.265661	0.132830	−	0.991139i	$-0.457593\pi$
$241$	4.12417	0.265661	0.132830	+	0.991139i	$0.457593\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−27.4952	−1.74948
$248$	0	0
$249$	4.46709	0.283091
$250$	0	0
$251$	−7.89816	−0.498527	−0.249264	−	0.968436i	$-0.580189\pi$
$251$	−7.89816	−0.498527	−0.249264	+	0.968436i	$0.580189\pi$
$252$	0	0
$253$	3.15777	0.198527
$254$	0	0
$255$	−5.42723	−0.339866
$256$	0	0
$257$	12.0225	0.749944	0.374972	−	0.927036i	$-0.377652\pi$
$257$	12.0225	0.749944	0.374972	+	0.927036i	$0.377652\pi$
$258$	0	0
$259$	3.31911	0.206240
$260$	0	0
$261$	−3.83713	−0.237512
$262$	0	0
$263$	−15.3475	−0.946365	−0.473183	−	0.880964i	$-0.656895\pi$
$263$	−15.3475	−0.946365	−0.473183	+	0.880964i	$0.656895\pi$
$264$	0	0
$265$	11.5638	0.710361
$266$	0	0
$267$	−4.54532	−0.278169
$268$	0	0
$269$	−6.84604	−0.417410	−0.208705	−	0.977979i	$-0.566925\pi$
$269$	−6.84604	−0.417410	−0.208705	+	0.977979i	$0.566925\pi$
$270$	0	0
$271$	−25.3666	−1.54091	−0.770457	−	0.637492i	$-0.779972\pi$
$271$	−25.3666	−1.54091	−0.770457	+	0.637492i	$0.779972\pi$
$272$	0	0
$273$	4.51802	0.273443
$274$	0	0
$275$	−3.15777	−0.190421
$276$	0	0
$277$	−16.0305	−0.963182	−0.481591	−	0.876396i	$-0.659941\pi$
$277$	−16.0305	−0.963182	−0.481591	+	0.876396i	$0.659941\pi$
$278$	0	0
$279$	5.40633	0.323668
$280$	0	0
$281$	−4.82091	−0.287592	−0.143796	−	0.989607i	$-0.545931\pi$
$281$	−4.82091	−0.287592	−0.143796	+	0.989607i	$0.545931\pi$
$282$	0	0
$283$	−17.5532	−1.04343	−0.521714	−	0.853120i	$-0.674707\pi$
$283$	−17.5532	−1.04343	−0.521714	+	0.853120i	$0.674707\pi$
$284$	0	0
$285$	−6.08569	−0.360485
$286$	0	0
$287$	−3.54036	−0.208981
$288$	0	0
$289$	12.4548	0.732634
$290$	0	0
$291$	6.37604	0.373770
$292$	0	0
$293$	−18.2049	−1.06354	−0.531771	−	0.846888i	$-0.678473\pi$
$293$	−18.2049	−1.06354	−0.531771	+	0.846888i	$0.678473\pi$
$294$	0	0
$295$	−10.2867	−0.598915
$296$	0	0
$297$	−3.15777	−0.183232
$298$	0	0
$299$	4.51802	0.261284
$300$	0	0
$301$	5.71070	0.329159
$302$	0	0
$303$	0.561521	0.0322586
$304$	0	0
$305$	−7.59114	−0.434668
$306$	0	0
$307$	−18.3866	−1.04938	−0.524690	−	0.851293i	$-0.675819\pi$
$307$	−18.3866	−1.04938	−0.524690	+	0.851293i	$0.675819\pi$
$308$	0	0
$309$	4.59107	0.261177
$310$	0	0
$311$	−25.7986	−1.46291	−0.731453	−	0.681892i	$-0.761157\pi$
$311$	−25.7986	−1.46291	−0.731453	+	0.681892i	$0.761157\pi$
$312$	0	0
$313$	−0.0176653	−0.000998501 0	−0.000499251	−	1.00000i	$-0.500159\pi$
$313$	−0.0176653	−0.000998501 0	−0.000499251		1.00000i	$0.500159\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−3.15259	−0.177067	−0.0885335	−	0.996073i	$-0.528218\pi$
$317$	−3.15259	−0.177067	−0.0885335	+	0.996073i	$0.528218\pi$
$318$	0	0
$319$	12.1168	0.678409
$320$	0	0
$321$	5.03603	0.281084
$322$	0	0
$323$	33.0284	1.83775
$324$	0	0
$325$	−4.51802	−0.250615
$326$	0	0
$327$	−5.18019	−0.286465
$328$	0	0
$329$	−3.35052	−0.184720
$330$	0	0
$331$	7.49809	0.412132	0.206066	−	0.978538i	$-0.433934\pi$
$331$	7.49809	0.412132	0.206066	+	0.978538i	$0.433934\pi$
$332$	0	0
$333$	−3.31911	−0.181886
$334$	0	0
$335$	13.7428	0.750847
$336$	0	0
$337$	−5.24914	−0.285939	−0.142969	−	0.989727i	$-0.545665\pi$
$337$	−5.24914	−0.285939	−0.142969	+	0.989727i	$0.545665\pi$
$338$	0	0
$339$	−3.40633	−0.185006
$340$	0	0
$341$	−17.0719	−0.924497
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	3.74261	0.200914	0.100457	−	0.994941i	$-0.467970\pi$
$347$	3.74261	0.200914	0.100457	+	0.994941i	$0.467970\pi$
$348$	0	0
$349$	−11.3715	−0.608705	−0.304352	−	0.952560i	$-0.598440\pi$
$349$	−11.3715	−0.608705	−0.304352	+	0.952560i	$0.598440\pi$
$350$	0	0
$351$	−4.51802	−0.241154
$352$	0	0
$353$	6.34431	0.337674	0.168837	−	0.985644i	$-0.445999\pi$
$353$	6.34431	0.337674	0.168837	+	0.985644i	$0.445999\pi$
$354$	0	0
$355$	0.701785	0.0372469
$356$	0	0
$357$	−5.42723	−0.287239
$358$	0	0
$359$	14.9005	0.786417	0.393208	−	0.919449i	$-0.371365\pi$
$359$	14.9005	0.786417	0.393208	+	0.919449i	$0.371365\pi$
$360$	0	0
$361$	18.0356	0.949243
$362$	0	0
$363$	−1.02850	−0.0539821
$364$	0	0
$365$	10.6868	0.559374
$366$	0	0
$367$	22.7784	1.18902	0.594511	−	0.804088i	$-0.297346\pi$
$367$	22.7784	1.18902	0.594511	+	0.804088i	$0.297346\pi$
$368$	0	0
$369$	3.54036	0.184304
$370$	0	0
$371$	11.5638	0.600364
$372$	0	0
$373$	2.75916	0.142864	0.0714319	−	0.997445i	$-0.477243\pi$
$373$	2.75916	0.142864	0.0714319	+	0.997445i	$0.477243\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	17.3362	0.892860
$378$	0	0
$379$	−32.3263	−1.66049	−0.830245	−	0.557399i	$-0.811799\pi$
$379$	−32.3263	−1.66049	−0.830245	+	0.557399i	$0.811799\pi$
$380$	0	0
$381$	−5.03499	−0.257950
$382$	0	0
$383$	4.49916	0.229896	0.114948	−	0.993371i	$-0.463330\pi$
$383$	4.49916	0.229896	0.114948	+	0.993371i	$0.463330\pi$
$384$	0	0
$385$	−3.15777	−0.160935
$386$	0	0
$387$	−5.71070	−0.290291
$388$	0	0
$389$	31.8369	1.61419	0.807097	−	0.590418i	$-0.201037\pi$
$389$	31.8369	1.61419	0.807097	+	0.590418i	$0.201037\pi$
$390$	0	0
$391$	−5.42723	−0.274467
$392$	0	0
$393$	−4.82609	−0.243444
$394$	0	0
$395$	4.20742	0.211698
$396$	0	0
$397$	7.97147	0.400077	0.200038	−	0.979788i	$-0.435893\pi$
$397$	7.97147	0.400077	0.200038	+	0.979788i	$0.435893\pi$
$398$	0	0
$399$	−6.08569	−0.304666
$400$	0	0
$401$	32.7366	1.63479	0.817393	−	0.576080i	$-0.195418\pi$
$401$	32.7366	1.63479	0.817393	+	0.576080i	$0.195418\pi$
$402$	0	0
$403$	−24.4259	−1.21674
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	10.4810	0.519523
$408$	0	0
$409$	33.6346	1.66312	0.831561	−	0.555433i	$-0.187448\pi$
$409$	33.6346	1.66312	0.831561	+	0.555433i	$0.187448\pi$
$410$	0	0
$411$	1.05441	0.0520104
$412$	0	0
$413$	−10.2867	−0.506175
$414$	0	0
$415$	−4.46709	−0.219281
$416$	0	0
$417$	4.83502	0.236772
$418$	0	0
$419$	−8.39766	−0.410252	−0.205126	−	0.978736i	$-0.565760\pi$
$419$	−8.39766	−0.410252	−0.205126	+	0.978736i	$0.565760\pi$
$420$	0	0
$421$	−10.2038	−0.497301	−0.248651	−	0.968593i	$-0.579987\pi$
$421$	−10.2038	−0.497301	−0.248651	+	0.968593i	$0.579987\pi$
$422$	0	0
$423$	3.35052	0.162908
$424$	0	0
$425$	5.42723	0.263259
$426$	0	0
$427$	−7.59114	−0.367361
$428$	0	0
$429$	14.2669	0.688810
$430$	0	0
$431$	−8.33539	−0.401501	−0.200751	−	0.979642i	$-0.564338\pi$
$431$	−8.33539	−0.401501	−0.200751	+	0.979642i	$0.564338\pi$
$432$	0	0
$433$	−26.4887	−1.27297	−0.636483	−	0.771291i	$-0.719611\pi$
$433$	−26.4887	−1.27297	−0.636483	+	0.771291i	$0.719611\pi$
$434$	0	0
$435$	3.83713	0.183976
$436$	0	0
$437$	−6.08569	−0.291118
$438$	0	0
$439$	26.9042	1.28407	0.642033	−	0.766677i	$-0.278091\pi$
$439$	26.9042	1.28407	0.642033	+	0.766677i	$0.278091\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	37.7523	1.79367	0.896834	−	0.442368i	$-0.145861\pi$
$443$	37.7523	1.79367	0.896834	+	0.442368i	$0.145861\pi$
$444$	0	0
$445$	4.54532	0.215469
$446$	0	0
$447$	5.70917	0.270034
$448$	0	0
$449$	−4.89035	−0.230790	−0.115395	−	0.993320i	$-0.536813\pi$
$449$	−4.89035	−0.230790	−0.115395	+	0.993320i	$0.536813\pi$
$450$	0	0
$451$	−11.1797	−0.526429
$452$	0	0
$453$	−9.38390	−0.440894
$454$	0	0
$455$	−4.51802	−0.211808
$456$	0	0
$457$	−19.0742	−0.892252	−0.446126	−	0.894970i	$-0.647197\pi$
$457$	−19.0742	−0.892252	−0.446126	+	0.894970i	$0.647197\pi$
$458$	0	0
$459$	5.42723	0.253321
$460$	0	0
$461$	−16.0332	−0.746739	−0.373370	−	0.927683i	$-0.621798\pi$
$461$	−16.0332	−0.746739	−0.373370	+	0.927683i	$0.621798\pi$
$462$	0	0
$463$	−21.3172	−0.990696	−0.495348	−	0.868695i	$-0.664959\pi$
$463$	−21.3172	−0.990696	−0.495348	+	0.868695i	$0.664959\pi$
$464$	0	0
$465$	−5.40633	−0.250713
$466$	0	0
$467$	−30.8179	−1.42608	−0.713040	−	0.701123i	$-0.752682\pi$
$467$	−30.8179	−1.42608	−0.713040	+	0.701123i	$0.752682\pi$
$468$	0	0
$469$	13.7428	0.634582
$470$	0	0
$471$	−0.0176653	−0.000813974 0
$472$	0	0
$473$	18.0331	0.829161
$474$	0	0
$475$	6.08569	0.279231
$476$	0	0
$477$	−11.5638	−0.529472
$478$	0	0
$479$	22.3125	1.01948	0.509742	−	0.860328i	$-0.329741\pi$
$479$	22.3125	1.01948	0.509742	+	0.860328i	$0.329741\pi$
$480$	0	0
$481$	14.9958	0.683750
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−6.37604	−0.289521
$486$	0	0
$487$	−32.8855	−1.49018	−0.745091	−	0.666962i	$-0.767594\pi$
$487$	−32.8855	−1.49018	−0.745091	+	0.666962i	$0.767594\pi$
$488$	0	0
$489$	−20.2871	−0.917414
$490$	0	0
$491$	41.6756	1.88079	0.940397	−	0.340078i	$-0.110454\pi$
$491$	41.6756	1.88079	0.940397	+	0.340078i	$0.110454\pi$
$492$	0	0
$493$	−20.8250	−0.937909
$494$	0	0
$495$	3.15777	0.141931
$496$	0	0
$497$	0.701785	0.0314793
$498$	0	0
$499$	3.37991	0.151306	0.0756528	−	0.997134i	$-0.475896\pi$
$499$	3.37991	0.151306	0.0756528	+	0.997134i	$0.475896\pi$
$500$	0	0
$501$	1.98875	0.0888506
$502$	0	0
$503$	−14.4307	−0.643431	−0.321716	−	0.946836i	$-0.604259\pi$
$503$	−14.4307	−0.643431	−0.321716	+	0.946836i	$0.604259\pi$
$504$	0	0
$505$	−0.561521	−0.0249874
$506$	0	0
$507$	7.41248	0.329200
$508$	0	0
$509$	−9.65455	−0.427930	−0.213965	−	0.976841i	$-0.568638\pi$
$509$	−9.65455	−0.427930	−0.213965	+	0.976841i	$0.568638\pi$
$510$	0	0
$511$	10.6868	0.472757
$512$	0	0
$513$	6.08569	0.268690
$514$	0	0
$515$	−4.59107	−0.202307
$516$	0	0
$517$	−10.5802	−0.465316
$518$	0	0
$519$	−19.7824	−0.868350
$520$	0	0
$521$	5.72536	0.250833	0.125416	−	0.992104i	$-0.459973\pi$
$521$	5.72536	0.250833	0.125416	+	0.992104i	$0.459973\pi$
$522$	0	0
$523$	−42.9527	−1.87819	−0.939095	−	0.343658i	$-0.888334\pi$
$523$	−42.9527	−1.87819	−0.939095	+	0.343658i	$0.888334\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	29.3414	1.27813
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.2867	0.446405
$532$	0	0
$533$	−15.9954	−0.692839
$534$	0	0
$535$	−5.03603	−0.217727
$536$	0	0
$537$	−17.1764	−0.741216
$538$	0	0
$539$	−3.15777	−0.136015
$540$	0	0
$541$	−15.7132	−0.675561	−0.337781	−	0.941225i	$-0.609676\pi$
$541$	−15.7132	−0.675561	−0.337781	+	0.941225i	$0.609676\pi$
$542$	0	0
$543$	1.28049	0.0549511
$544$	0	0
$545$	5.18019	0.221895
$546$	0	0
$547$	10.0783	0.430918	0.215459	−	0.976513i	$-0.430875\pi$
$547$	10.0783	0.430918	0.215459	+	0.976513i	$0.430875\pi$
$548$	0	0
$549$	7.59114	0.323982
$550$	0	0
$551$	−23.3516	−0.994810
$552$	0	0
$553$	4.20742	0.178918
$554$	0	0
$555$	3.31911	0.140888
$556$	0	0
$557$	−18.9659	−0.803612	−0.401806	−	0.915725i	$-0.631617\pi$
$557$	−18.9659	−0.803612	−0.401806	+	0.915725i	$0.631617\pi$
$558$	0	0
$559$	25.8010	1.09127
$560$	0	0
$561$	−17.1379	−0.723564
$562$	0	0
$563$	38.7277	1.63218	0.816090	−	0.577925i	$-0.196137\pi$
$563$	38.7277	1.63218	0.816090	+	0.577925i	$0.196137\pi$
$564$	0	0
$565$	3.40633	0.143305
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−24.1287	−1.01153	−0.505764	−	0.862672i	$-0.668789\pi$
$569$	−24.1287	−1.01153	−0.505764	+	0.862672i	$0.668789\pi$
$570$	0	0
$571$	−3.28263	−0.137374	−0.0686869	−	0.997638i	$-0.521881\pi$
$571$	−3.28263	−0.137374	−0.0686869	+	0.997638i	$0.521881\pi$
$572$	0	0
$573$	−3.62719	−0.151528
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	22.8805	0.952529	0.476265	−	0.879302i	$-0.341990\pi$
$577$	22.8805	0.952529	0.476265	+	0.879302i	$0.341990\pi$
$578$	0	0
$579$	0.103415	0.00429777
$580$	0	0
$581$	−4.46709	−0.185326
$582$	0	0
$583$	36.5159	1.51233
$584$	0	0
$585$	4.51802	0.186797
$586$	0	0
$587$	−15.5585	−0.642170	−0.321085	−	0.947050i	$-0.604047\pi$
$587$	−15.5585	−0.642170	−0.321085	+	0.947050i	$0.604047\pi$
$588$	0	0
$589$	32.9012	1.35567
$590$	0	0
$591$	−7.65118	−0.314727
$592$	0	0
$593$	−7.11939	−0.292358	−0.146179	−	0.989258i	$-0.546698\pi$
$593$	−7.11939	−0.292358	−0.146179	+	0.989258i	$0.546698\pi$
$594$	0	0
$595$	5.42723	0.222495
$596$	0	0
$597$	−17.6566	−0.722636
$598$	0	0
$599$	48.6612	1.98824	0.994122	−	0.108263i	$-0.0345288\pi$
$599$	48.6612	1.98824	0.994122	+	0.108263i	$0.0345288\pi$
$600$	0	0
$601$	−38.1540	−1.55634	−0.778168	−	0.628057i	$-0.783851\pi$
$601$	−38.1540	−1.55634	−0.778168	+	0.628057i	$0.783851\pi$
$602$	0	0
$603$	−13.7428	−0.559649
$604$	0	0
$605$	1.02850	0.0418144
$606$	0	0
$607$	10.4531	0.424276	0.212138	−	0.977240i	$-0.431957\pi$
$607$	10.4531	0.424276	0.212138	+	0.977240i	$0.431957\pi$
$608$	0	0
$609$	3.83713	0.155488
$610$	0	0
$611$	−15.1377	−0.612407
$612$	0	0
$613$	25.8907	1.04572	0.522858	−	0.852420i	$-0.324866\pi$
$613$	25.8907	1.04572	0.522858	+	0.852420i	$0.324866\pi$
$614$	0	0
$615$	−3.54036	−0.142761
$616$	0	0
$617$	39.5200	1.59102	0.795508	−	0.605943i	$-0.207204\pi$
$617$	39.5200	1.59102	0.795508	+	0.605943i	$0.207204\pi$
$618$	0	0
$619$	24.7924	0.996489	0.498244	−	0.867037i	$-0.333978\pi$
$619$	24.7924	0.996489	0.498244	+	0.867037i	$0.333978\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	4.54532	0.182105
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−19.2172	−0.767461
$628$	0	0
$629$	−18.0136	−0.718248
$630$	0	0
$631$	−34.8012	−1.38541	−0.692707	−	0.721219i	$-0.743582\pi$
$631$	−34.8012	−1.38541	−0.692707	+	0.721219i	$0.743582\pi$
$632$	0	0
$633$	16.1389	0.641463
$634$	0	0
$635$	5.03499	0.199807
$636$	0	0
$637$	−4.51802	−0.179010
$638$	0	0
$639$	−0.701785	−0.0277622
$640$	0	0
$641$	−11.6766	−0.461199	−0.230599	−	0.973049i	$-0.574069\pi$
$641$	−11.6766	−0.461199	−0.230599	+	0.973049i	$0.574069\pi$
$642$	0	0
$643$	13.2841	0.523872	0.261936	−	0.965085i	$-0.415639\pi$
$643$	13.2841	0.523872	0.261936	+	0.965085i	$0.415639\pi$
$644$	0	0
$645$	5.71070	0.224858
$646$	0	0
$647$	23.6571	0.930055	0.465028	−	0.885296i	$-0.346044\pi$
$647$	23.6571	0.930055	0.465028	+	0.885296i	$0.346044\pi$
$648$	0	0
$649$	−32.4830	−1.27507
$650$	0	0
$651$	−5.40633	−0.211891
$652$	0	0
$653$	18.4517	0.722071	0.361035	−	0.932552i	$-0.382423\pi$
$653$	18.4517	0.722071	0.361035	+	0.932552i	$0.382423\pi$
$654$	0	0
$655$	4.82609	0.188571
$656$	0	0
$657$	−10.6868	−0.416933
$658$	0	0
$659$	12.6533	0.492902	0.246451	−	0.969155i	$-0.420736\pi$
$659$	12.6533	0.492902	0.246451	+	0.969155i	$0.420736\pi$
$660$	0	0
$661$	−28.7927	−1.11990	−0.559952	−	0.828525i	$-0.689181\pi$
$661$	−28.7927	−1.11990	−0.559952	+	0.828525i	$0.689181\pi$
$662$	0	0
$663$	−24.5203	−0.952289
$664$	0	0
$665$	6.08569	0.235993
$666$	0	0
$667$	3.83713	0.148574
$668$	0	0
$669$	13.0720	0.505392
$670$	0	0
$671$	−23.9711	−0.925393
$672$	0	0
$673$	−5.72052	−0.220510	−0.110255	−	0.993903i	$-0.535167\pi$
$673$	−5.72052	−0.220510	−0.110255	+	0.993903i	$0.535167\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	26.6222	1.02317	0.511587	−	0.859232i	$-0.329058\pi$
$677$	26.6222	1.02317	0.511587	+	0.859232i	$0.329058\pi$
$678$	0	0
$679$	−6.37604	−0.244690
$680$	0	0
$681$	−11.2162	−0.429804
$682$	0	0
$683$	−14.9697	−0.572800	−0.286400	−	0.958110i	$-0.592459\pi$
$683$	−14.9697	−0.572800	−0.286400	+	0.958110i	$0.592459\pi$
$684$	0	0
$685$	−1.05441	−0.0402871
$686$	0	0
$687$	−19.8869	−0.758733
$688$	0	0
$689$	52.2456	1.99040
$690$	0	0
$691$	−24.5804	−0.935083	−0.467541	−	0.883971i	$-0.654860\pi$
$691$	−24.5804	−0.935083	−0.467541	+	0.883971i	$0.654860\pi$
$692$	0	0
$693$	3.15777	0.119954
$694$	0	0
$695$	−4.83502	−0.183403
$696$	0	0
$697$	19.2144	0.727796
$698$	0	0
$699$	17.3116	0.654786
$700$	0	0
$701$	−3.13139	−0.118271	−0.0591355	−	0.998250i	$-0.518834\pi$
$701$	−3.13139	−0.118271	−0.0591355	+	0.998250i	$0.518834\pi$
$702$	0	0
$703$	−20.1991	−0.761823
$704$	0	0
$705$	−3.35052	−0.126188
$706$	0	0
$707$	−0.561521	−0.0211182
$708$	0	0
$709$	−4.91145	−0.184453	−0.0922266	−	0.995738i	$-0.529398\pi$
$709$	−4.91145	−0.184453	−0.0922266	+	0.995738i	$0.529398\pi$
$710$	0	0
$711$	−4.20742	−0.157791
$712$	0	0
$713$	−5.40633	−0.202469
$714$	0	0
$715$	−14.2669	−0.533550
$716$	0	0
$717$	−13.8967	−0.518983
$718$	0	0
$719$	5.28704	0.197173	0.0985867	−	0.995128i	$-0.468568\pi$
$719$	5.28704	0.197173	0.0985867	+	0.995128i	$0.468568\pi$
$720$	0	0
$721$	−4.59107	−0.170980
$722$	0	0
$723$	4.12417	0.153379
$724$	0	0
$725$	−3.83713	−0.142507
$726$	0	0
$727$	−38.5314	−1.42905	−0.714525	−	0.699610i	$-0.753357\pi$
$727$	−38.5314	−1.42905	−0.714525	+	0.699610i	$0.753357\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.9932	−1.14633
$732$	0	0
$733$	21.1942	0.782826	0.391413	−	0.920215i	$-0.371986\pi$
$733$	21.1942	0.782826	0.391413	+	0.920215i	$0.371986\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	43.3965	1.59853
$738$	0	0
$739$	−8.41107	−0.309406	−0.154703	−	0.987961i	$-0.549442\pi$
$739$	−8.41107	−0.309406	−0.154703	+	0.987961i	$0.549442\pi$
$740$	0	0
$741$	−27.4952	−1.01006
$742$	0	0
$743$	−25.2857	−0.927642	−0.463821	−	0.885929i	$-0.653522\pi$
$743$	−25.2857	−0.927642	−0.463821	+	0.885929i	$0.653522\pi$
$744$	0	0
$745$	−5.70917	−0.209168
$746$	0	0
$747$	4.46709	0.163442
$748$	0	0
$749$	−5.03603	−0.184013
$750$	0	0
$751$	−50.4688	−1.84163	−0.920817	−	0.389994i	$-0.872477\pi$
$751$	−50.4688	−1.84163	−0.920817	+	0.389994i	$0.872477\pi$
$752$	0	0
$753$	−7.89816	−0.287825
$754$	0	0
$755$	9.38390	0.341515
$756$	0	0
$757$	16.7026	0.607066	0.303533	−	0.952821i	$-0.401834\pi$
$757$	16.7026	0.607066	0.303533	+	0.952821i	$0.401834\pi$
$758$	0	0
$759$	3.15777	0.114620
$760$	0	0
$761$	−36.4462	−1.32117	−0.660586	−	0.750750i	$-0.729692\pi$
$761$	−36.4462	−1.32117	−0.660586	+	0.750750i	$0.729692\pi$
$762$	0	0
$763$	5.18019	0.187536
$764$	0	0
$765$	−5.42723	−0.196222
$766$	0	0
$767$	−46.4755	−1.67813
$768$	0	0
$769$	53.1063	1.91506	0.957532	−	0.288327i	$-0.0930991\pi$
$769$	53.1063	1.91506	0.957532	+	0.288327i	$0.0930991\pi$
$770$	0	0
$771$	12.0225	0.432981
$772$	0	0
$773$	29.7797	1.07110	0.535550	−	0.844503i	$-0.320104\pi$
$773$	29.7797	1.07110	0.535550	+	0.844503i	$0.320104\pi$
$774$	0	0
$775$	5.40633	0.194201
$776$	0	0
$777$	3.31911	0.119072
$778$	0	0
$779$	21.5456	0.771950
$780$	0	0
$781$	2.21607	0.0792973
$782$	0	0
$783$	−3.83713	−0.137128
$784$	0	0
$785$	0.0176653	0.000630501 0
$786$	0	0
$787$	−34.6032	−1.23347	−0.616735	−	0.787170i	$-0.711545\pi$
$787$	−34.6032	−1.23347	−0.616735	+	0.787170i	$0.711545\pi$
$788$	0	0
$789$	−15.3475	−0.546384
$790$	0	0
$791$	3.40633	0.121115
$792$	0	0
$793$	−34.2969	−1.21792
$794$	0	0
$795$	11.5638	0.410127
$796$	0	0
$797$	6.33677	0.224460	0.112230	−	0.993682i	$-0.464201\pi$
$797$	6.33677	0.224460	0.112230	+	0.993682i	$0.464201\pi$
$798$	0	0
$799$	18.1840	0.643305
$800$	0	0
$801$	−4.54532	−0.160601
$802$	0	0
$803$	33.7465	1.19089
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	−6.84604	−0.240992
$808$	0	0
$809$	−9.12331	−0.320758	−0.160379	−	0.987055i	$-0.551272\pi$
$809$	−9.12331	−0.320758	−0.160379	+	0.987055i	$0.551272\pi$
$810$	0	0
$811$	31.1402	1.09348	0.546740	−	0.837302i	$-0.315869\pi$
$811$	31.1402	1.09348	0.546740	+	0.837302i	$0.315869\pi$
$812$	0	0
$813$	−25.3666	−0.889647
$814$	0	0
$815$	20.2871	0.710626
$816$	0	0
$817$	−34.7535	−1.21587
$818$	0	0
$819$	4.51802	0.157872
$820$	0	0
$821$	5.00576	0.174702	0.0873511	−	0.996178i	$-0.472160\pi$
$821$	5.00576	0.174702	0.0873511	+	0.996178i	$0.472160\pi$
$822$	0	0
$823$	3.62516	0.126365	0.0631826	−	0.998002i	$-0.479875\pi$
$823$	3.62516	0.126365	0.0631826	+	0.998002i	$0.479875\pi$
$824$	0	0
$825$	−3.15777	−0.109939
$826$	0	0
$827$	−11.0653	−0.384777	−0.192388	−	0.981319i	$-0.561623\pi$
$827$	−11.0653	−0.384777	−0.192388	+	0.981319i	$0.561623\pi$
$828$	0	0
$829$	−15.5356	−0.539574	−0.269787	−	0.962920i	$-0.586953\pi$
$829$	−15.5356	−0.539574	−0.269787	+	0.962920i	$0.586953\pi$
$830$	0	0
$831$	−16.0305	−0.556093
$832$	0	0
$833$	5.42723	0.188042
$834$	0	0
$835$	−1.98875	−0.0688234
$836$	0	0
$837$	5.40633	0.186870
$838$	0	0
$839$	−22.0108	−0.759898	−0.379949	−	0.925007i	$-0.624058\pi$
$839$	−22.0108	−0.759898	−0.379949	+	0.925007i	$0.624058\pi$
$840$	0	0
$841$	−14.2764	−0.492291
$842$	0	0
$843$	−4.82091	−0.166041
$844$	0	0
$845$	−7.41248	−0.254997
$846$	0	0
$847$	1.02850	0.0353396
$848$	0	0
$849$	−17.5532	−0.602423
$850$	0	0
$851$	3.31911	0.113778
$852$	0	0
$853$	−44.3844	−1.51969	−0.759847	−	0.650102i	$-0.774726\pi$
$853$	−44.3844	−1.51969	−0.759847	+	0.650102i	$0.774726\pi$
$854$	0	0
$855$	−6.08569	−0.208126
$856$	0	0
$857$	6.11421	0.208858	0.104429	−	0.994532i	$-0.466699\pi$
$857$	6.11421	0.208858	0.104429	+	0.994532i	$0.466699\pi$
$858$	0	0
$859$	20.2632	0.691370	0.345685	−	0.938351i	$-0.387647\pi$
$859$	20.2632	0.691370	0.345685	+	0.938351i	$0.387647\pi$
$860$	0	0
$861$	−3.54036	−0.120655
$862$	0	0
$863$	−21.3057	−0.725254	−0.362627	−	0.931934i	$-0.618120\pi$
$863$	−21.3057	−0.725254	−0.362627	+	0.931934i	$0.618120\pi$
$864$	0	0
$865$	19.7824	0.672621
$866$	0	0
$867$	12.4548	0.422986
$868$	0	0
$869$	13.2861	0.450699
$870$	0	0
$871$	62.0900	2.10384
$872$	0	0
$873$	6.37604	0.215796
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−27.1566	−0.917014	−0.458507	−	0.888691i	$-0.651616\pi$
$877$	−27.1566	−0.917014	−0.458507	+	0.888691i	$0.651616\pi$
$878$	0	0
$879$	−18.2049	−0.614037
$880$	0	0
$881$	−16.0557	−0.540930	−0.270465	−	0.962730i	$-0.587177\pi$
$881$	−16.0557	−0.540930	−0.270465	+	0.962730i	$0.587177\pi$
$882$	0	0
$883$	27.7282	0.933130	0.466565	−	0.884487i	$-0.345491\pi$
$883$	27.7282	0.933130	0.466565	+	0.884487i	$0.345491\pi$
$884$	0	0
$885$	−10.2867	−0.345784
$886$	0	0
$887$	29.4551	0.989005	0.494502	−	0.869176i	$-0.335350\pi$
$887$	29.4551	0.989005	0.494502	+	0.869176i	$0.335350\pi$
$888$	0	0
$889$	5.03499	0.168868
$890$	0	0
$891$	−3.15777	−0.105789
$892$	0	0
$893$	20.3902	0.682334
$894$	0	0
$895$	17.1764	0.574144
$896$	0	0
$897$	4.51802	0.150852
$898$	0	0
$899$	−20.7448	−0.691877
$900$	0	0
$901$	−62.7595	−2.09082
$902$	0	0
$903$	5.71070	0.190040
$904$	0	0
$905$	−1.28049	−0.0425649
$906$	0	0
$907$	−34.0511	−1.13065	−0.565324	−	0.824869i	$-0.691249\pi$
$907$	−34.0511	−1.13065	−0.565324	+	0.824869i	$0.691249\pi$
$908$	0	0
$909$	0.561521	0.0186245
$910$	0	0
$911$	56.3757	1.86781	0.933906	−	0.357519i	$-0.116377\pi$
$911$	56.3757	1.86781	0.933906	+	0.357519i	$0.116377\pi$
$912$	0	0
$913$	−14.1061	−0.466842
$914$	0	0
$915$	−7.59114	−0.250955
$916$	0	0
$917$	4.82609	0.159372
$918$	0	0
$919$	−23.7752	−0.784272	−0.392136	−	0.919907i	$-0.628264\pi$
$919$	−23.7752	−0.784272	−0.392136	+	0.919907i	$0.628264\pi$
$920$	0	0
$921$	−18.3866	−0.605860
$922$	0	0
$923$	3.17067	0.104364
$924$	0	0
$925$	−3.31911	−0.109132
$926$	0	0
$927$	4.59107	0.150790
$928$	0	0
$929$	15.1141	0.495877	0.247938	−	0.968776i	$-0.420247\pi$
$929$	15.1141	0.495877	0.247938	+	0.968776i	$0.420247\pi$
$930$	0	0
$931$	6.08569	0.199450
$932$	0	0
$933$	−25.7986	−0.844609
$934$	0	0
$935$	17.1379	0.560470
$936$	0	0
$937$	−9.00848	−0.294294	−0.147147	−	0.989115i	$-0.547009\pi$
$937$	−9.00848	−0.294294	−0.147147	+	0.989115i	$0.547009\pi$
$938$	0	0
$939$	−0.0176653	−0.000576485 0
$940$	0	0
$941$	13.1291	0.427996	0.213998	−	0.976834i	$-0.431351\pi$
$941$	13.1291	0.427996	0.213998	+	0.976834i	$0.431351\pi$
$942$	0	0
$943$	−3.54036	−0.115290
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	15.1623	0.492709	0.246354	−	0.969180i	$-0.420767\pi$
$947$	15.1623	0.492709	0.246354	+	0.969180i	$0.420767\pi$
$948$	0	0
$949$	48.2832	1.56734
$950$	0	0
$951$	−3.15259	−0.102230
$952$	0	0
$953$	−38.9340	−1.26120	−0.630598	−	0.776109i	$-0.717190\pi$
$953$	−38.9340	−1.26120	−0.630598	+	0.776109i	$0.717190\pi$
$954$	0	0
$955$	3.62719	0.117373
$956$	0	0
$957$	12.1168	0.391679
$958$	0	0
$959$	−1.05441	−0.0340488
$960$	0	0
$961$	−1.77160	−0.0571485
$962$	0	0
$963$	5.03603	0.162284
$964$	0	0
$965$	−0.103415	−0.00332904
$966$	0	0
$967$	21.3960	0.688048	0.344024	−	0.938961i	$-0.388210\pi$
$967$	21.3960	0.688048	0.344024	+	0.938961i	$0.388210\pi$
$968$	0	0
$969$	33.0284	1.06103
$970$	0	0
$971$	−30.4022	−0.975652	−0.487826	−	0.872941i	$-0.662210\pi$
$971$	−30.4022	−0.975652	−0.487826	+	0.872941i	$0.662210\pi$
$972$	0	0
$973$	−4.83502	−0.155004
$974$	0	0
$975$	−4.51802	−0.144692
$976$	0	0
$977$	36.5176	1.16830	0.584151	−	0.811645i	$-0.301428\pi$
$977$	36.5176	1.16830	0.584151	+	0.811645i	$0.301428\pi$
$978$	0	0
$979$	14.3531	0.458727
$980$	0	0
$981$	−5.18019	−0.165391
$982$	0	0
$983$	−37.8543	−1.20736	−0.603682	−	0.797225i	$-0.706300\pi$
$983$	−37.8543	−1.20736	−0.603682	+	0.797225i	$0.706300\pi$
$984$	0	0
$985$	7.65118	0.243787
$986$	0	0
$987$	−3.35052	−0.106648
$988$	0	0
$989$	5.71070	0.181590
$990$	0	0
$991$	−3.53779	−0.112382	−0.0561908	−	0.998420i	$-0.517896\pi$
$991$	−3.53779	−0.112382	−0.0561908	+	0.998420i	$0.517896\pi$
$992$	0	0
$993$	7.49809	0.237945
$994$	0	0
$995$	17.6566	0.559752
$996$	0	0
$997$	−2.34239	−0.0741842	−0.0370921	−	0.999312i	$-0.511809\pi$
$997$	−2.34239	−0.0741842	−0.0370921	+	0.999312i	$0.511809\pi$
$998$	0	0
$999$	−3.31911	−0.105012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9660.2.a.w.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9660.2.a.w.1.2	✓	6	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9660.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.15777 q^{11} -4.51802 q^{13} -1.00000 q^{15} +5.42723 q^{17} +6.08569 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.83713 q^{29} +5.40633 q^{31} -3.15777 q^{33} +1.00000 q^{35} -3.31911 q^{37} -4.51802 q^{39} +3.54036 q^{41} -5.71070 q^{43} -1.00000 q^{45} +3.35052 q^{47} +1.00000 q^{49} +5.42723 q^{51} -11.5638 q^{53} +3.15777 q^{55} +6.08569 q^{57} +10.2867 q^{59} +7.59114 q^{61} -1.00000 q^{63} +4.51802 q^{65} -13.7428 q^{67} -1.00000 q^{69} -0.701785 q^{71} -10.6868 q^{73} +1.00000 q^{75} +3.15777 q^{77} -4.20742 q^{79} +1.00000 q^{81} +4.46709 q^{83} -5.42723 q^{85} -3.83713 q^{87} -4.54532 q^{89} +4.51802 q^{91} +5.40633 q^{93} -6.08569 q^{95} +6.37604 q^{97} -3.15777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9} - 3 q^{13} - 6 q^{15} - 3 q^{17} - 6 q^{21} - 6 q^{23} + 6 q^{25} + 6 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{35} - 15 q^{37} - 3 q^{39} + 6 q^{41} - 3 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 3 q^{51} - 3 q^{53} + 6 q^{59} - 18 q^{61} - 6 q^{63} + 3 q^{65} - 9 q^{67} - 6 q^{69} - 9 q^{73} + 6 q^{75} - 18 q^{79} + 6 q^{81} - 3 q^{83} + 3 q^{85} + 6 q^{87} - 6 q^{89} + 3 q^{91} + 6 q^{93} - 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9 - 3 * q^13 - 6 * q^15 - 3 * q^17 - 6 * q^21 - 6 * q^23 + 6 * q^25 + 6 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^35 - 15 * q^37 - 3 * q^39 + 6 * q^41 - 3 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^49 - 3 * q^51 - 3 * q^53 + 6 * q^59 - 18 * q^61 - 6 * q^63 + 3 * q^65 - 9 * q^67 - 6 * q^69 - 9 * q^73 + 6 * q^75 - 18 * q^79 + 6 * q^81 - 3 * q^83 + 3 * q^85 + 6 * q^87 - 6 * q^89 + 3 * q^91 + 6 * q^93 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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