LMFDB - Embedded newform 8464.2.a.cg.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-1,0,0,0,-10,0,16,0,-23,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.12777$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.12777 q^{3} -0.858783 q^{5} -3.76707 q^{7} +1.52739 q^{9} -1.71756 q^{11} +5.36442 q^{13} -1.82729 q^{15} +3.04105 q^{17} +0.100133 q^{19} -8.01545 q^{21} -4.26249 q^{25} -3.13337 q^{27} +5.96694 q^{29} -7.34827 q^{31} -3.65457 q^{33} +3.23510 q^{35} -0.0146689 q^{37} +11.4142 q^{39} +6.41160 q^{41} -6.30968 q^{43} -1.31170 q^{45} +6.93875 q^{47} +7.19083 q^{49} +6.47064 q^{51} +4.55195 q^{53} +1.47501 q^{55} +0.213059 q^{57} -9.66697 q^{59} -13.2396 q^{61} -5.75380 q^{63} -4.60687 q^{65} +2.86759 q^{67} +13.6105 q^{71} -11.0999 q^{73} -9.06959 q^{75} +6.47017 q^{77} -9.75487 q^{79} -11.2492 q^{81} +11.4398 q^{83} -2.61160 q^{85} +12.6963 q^{87} -16.9454 q^{89} -20.2082 q^{91} -15.6354 q^{93} -0.0859921 q^{95} -13.3684 q^{97} -2.62339 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} + q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} + 18 q^{33} + 18 q^{35} + 24 q^{37} + 19 q^{39} + 9 q^{41} - 48 q^{43} + 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) $ 15 * q - q^3 - 10 * q^7 + 16 * q^9 - 23 * q^11 - 10 * q^15 - 29 * q^19 + q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 + 18 * q^33 + 18 * q^35 + 24 * q^37 + 19 * q^39 + 9 * q^41 - 48 * q^43 + 4 * q^45 + 36 * q^47 + 25 * q^49 - 35 * q^51 - 5 * q^53 + 10 * q^55 + 23 * q^57 + 22 * q^59 + 12 * q^61 - 35 * q^63 - 26 * q^65 - 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 - 26 * q^79 - 21 * q^81 - 68 * q^83 - 72 * q^85 - 19 * q^87 - 6 * q^89 - 71 * q^91 - 55 * q^93 + 12 * q^95 + 40 * q^97 - 63 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.12777	1.22847	0.614233	−	0.789124i	$-0.289465\pi$
$3$	2.12777	1.22847	0.614233	+	0.789124i	$0.289465\pi$
$4$	0	0
$5$	−0.858783	−0.384059	−0.192030	−	0.981389i	$-0.561507\pi$
$5$	−0.858783	−0.384059	−0.192030	+	0.981389i	$0.561507\pi$
$6$	0	0
$7$	−3.76707	−1.42382	−0.711910	−	0.702271i	$-0.752169\pi$
$7$	−3.76707	−1.42382	−0.711910	+	0.702271i	$0.752169\pi$
$8$	0	0
$9$	1.52739	0.509131
$10$	0	0
$11$	−1.71756	−0.517864	−0.258932	−	0.965896i	$-0.583371\pi$
$11$	−1.71756	−0.517864	−0.258932	+	0.965896i	$0.583371\pi$
$12$	0	0
$13$	5.36442	1.48782	0.743912	−	0.668278i	$-0.232968\pi$
$13$	5.36442	1.48782	0.743912	+	0.668278i	$0.232968\pi$
$14$	0	0
$15$	−1.82729	−0.471804
$16$	0	0
$17$	3.04105	0.737562	0.368781	−	0.929516i	$-0.379775\pi$
$17$	3.04105	0.737562	0.368781	+	0.929516i	$0.379775\pi$
$18$	0	0
$19$	0.100133	0.0229720	0.0114860	−	0.999934i	$-0.496344\pi$
$19$	0.100133	0.0229720	0.0114860	+	0.999934i	$0.496344\pi$
$20$	0	0
$21$	−8.01545	−1.74911
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.26249	−0.852498
$26$	0	0
$27$	−3.13337	−0.603017
$28$	0	0
$29$	5.96694	1.10803	0.554016	−	0.832506i	$-0.313094\pi$
$29$	5.96694	1.10803	0.554016	+	0.832506i	$0.313094\pi$
$30$	0	0
$31$	−7.34827	−1.31979	−0.659894	−	0.751359i	$-0.729399\pi$
$31$	−7.34827	−1.31979	−0.659894	+	0.751359i	$0.729399\pi$
$32$	0	0
$33$	−3.65457	−0.636179
$34$	0	0
$35$	3.23510	0.546831
$36$	0	0
$37$	−0.0146689	−0.00241155	−0.00120577	−	0.999999i	$-0.500384\pi$
$37$	−0.0146689	−0.00241155	−0.00120577	+	0.999999i	$0.500384\pi$
$38$	0	0
$39$	11.4142	1.82774
$40$	0	0
$41$	6.41160	1.00132	0.500662	−	0.865643i	$-0.333090\pi$
$41$	6.41160	1.00132	0.500662	+	0.865643i	$0.333090\pi$
$42$	0	0
$43$	−6.30968	−0.962217	−0.481108	−	0.876661i	$-0.659766\pi$
$43$	−6.30968	−0.962217	−0.481108	+	0.876661i	$0.659766\pi$
$44$	0	0
$45$	−1.31170	−0.195536
$46$	0	0
$47$	6.93875	1.01212	0.506060	−	0.862498i	$-0.331101\pi$
$47$	6.93875	1.01212	0.506060	+	0.862498i	$0.331101\pi$
$48$	0	0
$49$	7.19083	1.02726
$50$	0	0
$51$	6.47064	0.906071
$52$	0	0
$53$	4.55195	0.625258	0.312629	−	0.949875i	$-0.398790\pi$
$53$	4.55195	0.625258	0.312629	+	0.949875i	$0.398790\pi$
$54$	0	0
$55$	1.47501	0.198891
$56$	0	0
$57$	0.213059	0.0282203
$58$	0	0
$59$	−9.66697	−1.25853	−0.629266	−	0.777190i	$-0.716644\pi$
$59$	−9.66697	−1.25853	−0.629266	+	0.777190i	$0.716644\pi$
$60$	0	0
$61$	−13.2396	−1.69515	−0.847576	−	0.530675i	$-0.821939\pi$
$61$	−13.2396	−1.69515	−0.847576	+	0.530675i	$0.821939\pi$
$62$	0	0
$63$	−5.75380	−0.724910
$64$	0	0
$65$	−4.60687	−0.571412
$66$	0	0
$67$	2.86759	0.350332	0.175166	−	0.984539i	$-0.443954\pi$
$67$	2.86759	0.350332	0.175166	+	0.984539i	$0.443954\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6105	1.61527	0.807634	−	0.589685i	$-0.200748\pi$
$71$	13.6105	1.61527	0.807634	+	0.589685i	$0.200748\pi$
$72$	0	0
$73$	−11.0999	−1.29915	−0.649575	−	0.760298i	$-0.725053\pi$
$73$	−11.0999	−1.29915	−0.649575	+	0.760298i	$0.725053\pi$
$74$	0	0
$75$	−9.06959	−1.04727
$76$	0	0
$77$	6.47017	0.737345
$78$	0	0
$79$	−9.75487	−1.09751	−0.548755	−	0.835984i	$-0.684898\pi$
$79$	−9.75487	−1.09751	−0.548755	+	0.835984i	$0.684898\pi$
$80$	0	0
$81$	−11.2492	−1.24992
$82$	0	0
$83$	11.4398	1.25568	0.627840	−	0.778342i	$-0.283939\pi$
$83$	11.4398	1.25568	0.627840	+	0.778342i	$0.283939\pi$
$84$	0	0
$85$	−2.61160	−0.283268
$86$	0	0
$87$	12.6963	1.36118
$88$	0	0
$89$	−16.9454	−1.79621	−0.898107	−	0.439777i	$-0.855057\pi$
$89$	−16.9454	−1.79621	−0.898107	+	0.439777i	$0.855057\pi$
$90$	0	0
$91$	−20.2082	−2.11839
$92$	0	0
$93$	−15.6354	−1.62132
$94$	0	0
$95$	−0.0859921	−0.00882260
$96$	0	0
$97$	−13.3684	−1.35735	−0.678676	−	0.734438i	$-0.737446\pi$
$97$	−13.3684	−1.35735	−0.678676	+	0.734438i	$0.737446\pi$
$98$	0	0
$99$	−2.62339	−0.263661
$100$	0	0
$101$	0.605407	0.0602402	0.0301201	−	0.999546i	$-0.490411\pi$
$101$	0.605407	0.0602402	0.0301201	+	0.999546i	$0.490411\pi$
$102$	0	0
$103$	−10.0931	−0.994504	−0.497252	−	0.867606i	$-0.665658\pi$
$103$	−10.0931	−0.994504	−0.497252	+	0.867606i	$0.665658\pi$
$104$	0	0
$105$	6.88353	0.671764
$106$	0	0
$107$	−11.8653	−1.14706	−0.573530	−	0.819184i	$-0.694427\pi$
$107$	−11.8653	−1.14706	−0.573530	+	0.819184i	$0.694427\pi$
$108$	0	0
$109$	14.8230	1.41978	0.709892	−	0.704310i	$-0.248744\pi$
$109$	14.8230	1.41978	0.709892	+	0.704310i	$0.248744\pi$
$110$	0	0
$111$	−0.0312119	−0.00296250
$112$	0	0
$113$	−5.96573	−0.561209	−0.280604	−	0.959824i	$-0.590535\pi$
$113$	−5.96573	−0.561209	−0.280604	+	0.959824i	$0.590535\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.19358	0.757497
$118$	0	0
$119$	−11.4558	−1.05016
$120$	0	0
$121$	−8.04998	−0.731817
$122$	0	0
$123$	13.6424	1.23009
$124$	0	0
$125$	7.95447	0.711469
$126$	0	0
$127$	0.760254	0.0674616	0.0337308	−	0.999431i	$-0.489261\pi$
$127$	0.760254	0.0674616	0.0337308	+	0.999431i	$0.489261\pi$
$128$	0	0
$129$	−13.4255	−1.18205
$130$	0	0
$131$	7.43862	0.649915	0.324957	−	0.945729i	$-0.394650\pi$
$131$	7.43862	0.649915	0.324957	+	0.945729i	$0.394650\pi$
$132$	0	0
$133$	−0.377207	−0.0327080
$134$	0	0
$135$	2.69088	0.231594
$136$	0	0
$137$	7.22485	0.617261	0.308630	−	0.951182i	$-0.400129\pi$
$137$	7.22485	0.617261	0.308630	+	0.951182i	$0.400129\pi$
$138$	0	0
$139$	−4.87522	−0.413511	−0.206755	−	0.978393i	$-0.566290\pi$
$139$	−4.87522	−0.413511	−0.206755	+	0.978393i	$0.566290\pi$
$140$	0	0
$141$	14.7640	1.24336
$142$	0	0
$143$	−9.21372	−0.770490
$144$	0	0
$145$	−5.12430	−0.425550
$146$	0	0
$147$	15.3004	1.26196
$148$	0	0
$149$	1.62165	0.132851	0.0664253	−	0.997791i	$-0.478841\pi$
$149$	1.62165	0.132851	0.0664253	+	0.997791i	$0.478841\pi$
$150$	0	0
$151$	−5.97087	−0.485903	−0.242951	−	0.970038i	$-0.578116\pi$
$151$	−5.97087	−0.485903	−0.242951	+	0.970038i	$0.578116\pi$
$152$	0	0
$153$	4.64487	0.375516
$154$	0	0
$155$	6.31056	0.506877
$156$	0	0
$157$	18.2225	1.45432	0.727158	−	0.686470i	$-0.240841\pi$
$157$	18.2225	1.45432	0.727158	+	0.686470i	$0.240841\pi$
$158$	0	0
$159$	9.68549	0.768109
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.2964	−1.51141	−0.755704	−	0.654913i	$-0.772705\pi$
$163$	−19.2964	−1.51141	−0.755704	+	0.654913i	$0.772705\pi$
$164$	0	0
$165$	3.13848	0.244330
$166$	0	0
$167$	−13.6764	−1.05831	−0.529154	−	0.848526i	$-0.677491\pi$
$167$	−13.6764	−1.05831	−0.529154	+	0.848526i	$0.677491\pi$
$168$	0	0
$169$	15.7770	1.21362
$170$	0	0
$171$	0.152942	0.0116957
$172$	0	0
$173$	−15.6424	−1.18927	−0.594636	−	0.803995i	$-0.702704\pi$
$173$	−15.6424	−1.18927	−0.594636	+	0.803995i	$0.702704\pi$
$174$	0	0
$175$	16.0571	1.21380
$176$	0	0
$177$	−20.5691	−1.54606
$178$	0	0
$179$	−3.05412	−0.228276	−0.114138	−	0.993465i	$-0.536411\pi$
$179$	−3.05412	−0.228276	−0.114138	+	0.993465i	$0.536411\pi$
$180$	0	0
$181$	20.3332	1.51136	0.755678	−	0.654943i	$-0.227307\pi$
$181$	20.3332	1.51136	0.755678	+	0.654943i	$0.227307\pi$
$182$	0	0
$183$	−28.1707	−2.08244
$184$	0	0
$185$	0.0125974	0.000926176 0
$186$	0	0
$187$	−5.22318	−0.381957
$188$	0	0
$189$	11.8036	0.858587
$190$	0	0
$191$	−15.3487	−1.11059	−0.555297	−	0.831652i	$-0.687396\pi$
$191$	−15.3487	−1.11059	−0.555297	+	0.831652i	$0.687396\pi$
$192$	0	0
$193$	10.9489	0.788118	0.394059	−	0.919085i	$-0.371071\pi$
$193$	10.9489	0.788118	0.394059	+	0.919085i	$0.371071\pi$
$194$	0	0
$195$	−9.80235	−0.701961
$196$	0	0
$197$	−12.4714	−0.888553	−0.444277	−	0.895890i	$-0.646539\pi$
$197$	−12.4714	−0.888553	−0.444277	+	0.895890i	$0.646539\pi$
$198$	0	0
$199$	−18.1299	−1.28520	−0.642599	−	0.766203i	$-0.722144\pi$
$199$	−18.1299	−1.28520	−0.642599	+	0.766203i	$0.722144\pi$
$200$	0	0
$201$	6.10156	0.430371
$202$	0	0
$203$	−22.4779	−1.57764
$204$	0	0
$205$	−5.50617	−0.384567
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.171984	−0.0118964
$210$	0	0
$211$	12.3055	0.847146	0.423573	−	0.905862i	$-0.360776\pi$
$211$	12.3055	0.847146	0.423573	+	0.905862i	$0.360776\pi$
$212$	0	0
$213$	28.9599	1.98430
$214$	0	0
$215$	5.41864	0.369548
$216$	0	0
$217$	27.6814	1.87914
$218$	0	0
$219$	−23.6181	−1.59596
$220$	0	0
$221$	16.3135	1.09736
$222$	0	0
$223$	−3.31255	−0.221825	−0.110912	−	0.993830i	$-0.535377\pi$
$223$	−3.31255	−0.221825	−0.110912	+	0.993830i	$0.535377\pi$
$224$	0	0
$225$	−6.51050	−0.434033
$226$	0	0
$227$	0.326839	0.0216930	0.0108465	−	0.999941i	$-0.496547\pi$
$227$	0.326839	0.0216930	0.0108465	+	0.999941i	$0.496547\pi$
$228$	0	0
$229$	−24.0700	−1.59059	−0.795295	−	0.606223i	$-0.792684\pi$
$229$	−24.0700	−1.59059	−0.795295	+	0.606223i	$0.792684\pi$
$230$	0	0
$231$	13.7670	0.905804
$232$	0	0
$233$	−19.0426	−1.24752	−0.623762	−	0.781614i	$-0.714397\pi$
$233$	−19.0426	−1.24752	−0.623762	+	0.781614i	$0.714397\pi$
$234$	0	0
$235$	−5.95888	−0.388714
$236$	0	0
$237$	−20.7561	−1.34825
$238$	0	0
$239$	7.25321	0.469171	0.234586	−	0.972095i	$-0.424627\pi$
$239$	7.25321	0.469171	0.234586	+	0.972095i	$0.424627\pi$
$240$	0	0
$241$	24.8974	1.60379	0.801893	−	0.597468i	$-0.203827\pi$
$241$	24.8974	1.60379	0.801893	+	0.597468i	$0.203827\pi$
$242$	0	0
$243$	−14.5357	−0.932464
$244$	0	0
$245$	−6.17536	−0.394529
$246$	0	0
$247$	0.537153	0.0341783
$248$	0	0
$249$	24.3412	1.54256
$250$	0	0
$251$	−13.0011	−0.820620	−0.410310	−	0.911946i	$-0.634579\pi$
$251$	−13.0011	−0.820620	−0.410310	+	0.911946i	$0.634579\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.55687	−0.347985
$256$	0	0
$257$	−24.1388	−1.50574	−0.752870	−	0.658169i	$-0.771331\pi$
$257$	−24.1388	−1.50574	−0.752870	+	0.658169i	$0.771331\pi$
$258$	0	0
$259$	0.0552586	0.00343360
$260$	0	0
$261$	9.11386	0.564134
$262$	0	0
$263$	12.4450	0.767391	0.383695	−	0.923460i	$-0.374651\pi$
$263$	12.4450	0.767391	0.383695	+	0.923460i	$0.374651\pi$
$264$	0	0
$265$	−3.90914	−0.240136
$266$	0	0
$267$	−36.0560	−2.20659
$268$	0	0
$269$	4.78373	0.291669	0.145835	−	0.989309i	$-0.453413\pi$
$269$	4.78373	0.291669	0.145835	+	0.989309i	$0.453413\pi$
$270$	0	0
$271$	−15.8837	−0.964865	−0.482433	−	0.875933i	$-0.660247\pi$
$271$	−15.8837	−0.964865	−0.482433	+	0.875933i	$0.660247\pi$
$272$	0	0
$273$	−42.9983	−2.60237
$274$	0	0
$275$	7.32109	0.441478
$276$	0	0
$277$	−3.57213	−0.214629	−0.107314	−	0.994225i	$-0.534225\pi$
$277$	−3.57213	−0.214629	−0.107314	+	0.994225i	$0.534225\pi$
$278$	0	0
$279$	−11.2237	−0.671944
$280$	0	0
$281$	−24.2185	−1.44475	−0.722377	−	0.691499i	$-0.756950\pi$
$281$	−24.2185	−1.44475	−0.722377	+	0.691499i	$0.756950\pi$
$282$	0	0
$283$	−19.9904	−1.18830	−0.594152	−	0.804353i	$-0.702512\pi$
$283$	−19.9904	−1.18830	−0.594152	+	0.804353i	$0.702512\pi$
$284$	0	0
$285$	−0.182971	−0.0108383
$286$	0	0
$287$	−24.1529	−1.42570
$288$	0	0
$289$	−7.75203	−0.456002
$290$	0	0
$291$	−28.4448	−1.66746
$292$	0	0
$293$	−4.58063	−0.267603	−0.133802	−	0.991008i	$-0.542719\pi$
$293$	−4.58063	−0.267603	−0.133802	+	0.991008i	$0.542719\pi$
$294$	0	0
$295$	8.30182	0.483351
$296$	0	0
$297$	5.38175	0.312281
$298$	0	0
$299$	0	0
$300$	0	0
$301$	23.7690	1.37002
$302$	0	0
$303$	1.28816	0.0740031
$304$	0	0
$305$	11.3699	0.651039
$306$	0	0
$307$	−8.92424	−0.509333	−0.254667	−	0.967029i	$-0.581966\pi$
$307$	−8.92424	−0.509333	−0.254667	+	0.967029i	$0.581966\pi$
$308$	0	0
$309$	−21.4758	−1.22171
$310$	0	0
$311$	−6.79232	−0.385157	−0.192579	−	0.981282i	$-0.561685\pi$
$311$	−6.79232	−0.385157	−0.192579	+	0.981282i	$0.561685\pi$
$312$	0	0
$313$	0.351053	0.0198427	0.00992134	−	0.999951i	$-0.496842\pi$
$313$	0.351053	0.0198427	0.00992134	+	0.999951i	$0.496842\pi$
$314$	0	0
$315$	4.94126	0.278408
$316$	0	0
$317$	11.7658	0.660831	0.330415	−	0.943836i	$-0.392811\pi$
$317$	11.7658	0.660831	0.330415	+	0.943836i	$0.392811\pi$
$318$	0	0
$319$	−10.2486	−0.573810
$320$	0	0
$321$	−25.2466	−1.40913
$322$	0	0
$323$	0.304508	0.0169433
$324$	0	0
$325$	−22.8658	−1.26837
$326$	0	0
$327$	31.5399	1.74416
$328$	0	0
$329$	−26.1388	−1.44108
$330$	0	0
$331$	19.1570	1.05296	0.526481	−	0.850187i	$-0.323511\pi$
$331$	19.1570	1.05296	0.526481	+	0.850187i	$0.323511\pi$
$332$	0	0
$333$	−0.0224051	−0.00122779
$334$	0	0
$335$	−2.46263	−0.134548
$336$	0	0
$337$	11.7413	0.639590	0.319795	−	0.947487i	$-0.396386\pi$
$337$	11.7413	0.639590	0.319795	+	0.947487i	$0.396386\pi$
$338$	0	0
$339$	−12.6937	−0.689426
$340$	0	0
$341$	12.6211	0.683471
$342$	0	0
$343$	−0.718863	−0.0388149
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9732	1.17958	0.589792	−	0.807555i	$-0.299210\pi$
$347$	21.9732	1.17958	0.589792	+	0.807555i	$0.299210\pi$
$348$	0	0
$349$	−2.99272	−0.160197	−0.0800983	−	0.996787i	$-0.525523\pi$
$349$	−2.99272	−0.160197	−0.0800983	+	0.996787i	$0.525523\pi$
$350$	0	0
$351$	−16.8087	−0.897182
$352$	0	0
$353$	−35.8015	−1.90552	−0.952761	−	0.303720i	$-0.901771\pi$
$353$	−35.8015	−1.90552	−0.952761	+	0.303720i	$0.901771\pi$
$354$	0	0
$355$	−11.6884	−0.620358
$356$	0	0
$357$	−24.3754	−1.29008
$358$	0	0
$359$	−17.4895	−0.923058	−0.461529	−	0.887125i	$-0.652699\pi$
$359$	−17.4895	−0.923058	−0.461529	+	0.887125i	$0.652699\pi$
$360$	0	0
$361$	−18.9900	−0.999472
$362$	0	0
$363$	−17.1285	−0.899013
$364$	0	0
$365$	9.53244	0.498951
$366$	0	0
$367$	0.118951	0.00620922	0.00310461	−	0.999995i	$-0.499012\pi$
$367$	0.118951	0.00620922	0.00310461	+	0.999995i	$0.499012\pi$
$368$	0	0
$369$	9.79302	0.509804
$370$	0	0
$371$	−17.1475	−0.890255
$372$	0	0
$373$	27.8585	1.44246	0.721229	−	0.692697i	$-0.243578\pi$
$373$	27.8585	1.44246	0.721229	+	0.692697i	$0.243578\pi$
$374$	0	0
$375$	16.9253	0.874016
$376$	0	0
$377$	32.0092	1.64856
$378$	0	0
$379$	−16.7136	−0.858519	−0.429259	−	0.903181i	$-0.641225\pi$
$379$	−16.7136	−0.858519	−0.429259	+	0.903181i	$0.641225\pi$
$380$	0	0
$381$	1.61764	0.0828744
$382$	0	0
$383$	−6.13802	−0.313638	−0.156819	−	0.987627i	$-0.550124\pi$
$383$	−6.13802	−0.313638	−0.156819	+	0.987627i	$0.550124\pi$
$384$	0	0
$385$	−5.55647	−0.283184
$386$	0	0
$387$	−9.63735	−0.489894
$388$	0	0
$389$	−19.8366	−1.00575	−0.502877	−	0.864358i	$-0.667725\pi$
$389$	−19.8366	−1.00575	−0.502877	+	0.864358i	$0.667725\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	15.8276	0.798399
$394$	0	0
$395$	8.37732	0.421509
$396$	0	0
$397$	4.60642	0.231190	0.115595	−	0.993296i	$-0.463123\pi$
$397$	4.60642	0.231190	0.115595	+	0.993296i	$0.463123\pi$
$398$	0	0
$399$	−0.802608	−0.0401806
$400$	0	0
$401$	11.4131	0.569942	0.284971	−	0.958536i	$-0.408016\pi$
$401$	11.4131	0.569942	0.284971	+	0.958536i	$0.408016\pi$
$402$	0	0
$403$	−39.4192	−1.96361
$404$	0	0
$405$	9.66066	0.480042
$406$	0	0
$407$	0.0251947	0.00124885
$408$	0	0
$409$	26.0017	1.28570	0.642851	−	0.765991i	$-0.277751\pi$
$409$	26.0017	1.28570	0.642851	+	0.765991i	$0.277751\pi$
$410$	0	0
$411$	15.3728	0.758285
$412$	0	0
$413$	36.4162	1.79192
$414$	0	0
$415$	−9.82430	−0.482256
$416$	0	0
$417$	−10.3733	−0.507984
$418$	0	0
$419$	−33.4800	−1.63560	−0.817802	−	0.575500i	$-0.804807\pi$
$419$	−33.4800	−1.63560	−0.817802	+	0.575500i	$0.804807\pi$
$420$	0	0
$421$	29.7629	1.45056	0.725278	−	0.688457i	$-0.241711\pi$
$421$	29.7629	1.45056	0.725278	+	0.688457i	$0.241711\pi$
$422$	0	0
$423$	10.5982	0.515302
$424$	0	0
$425$	−12.9624	−0.628771
$426$	0	0
$427$	49.8744	2.41359
$428$	0	0
$429$	−19.6047	−0.946522
$430$	0	0
$431$	6.28557	0.302765	0.151383	−	0.988475i	$-0.451627\pi$
$431$	6.28557	0.302765	0.151383	+	0.988475i	$0.451627\pi$
$432$	0	0
$433$	−2.63739	−0.126745	−0.0633723	−	0.997990i	$-0.520186\pi$
$433$	−2.63739	−0.126745	−0.0633723	+	0.997990i	$0.520186\pi$
$434$	0	0
$435$	−10.9033	−0.522774
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.29021	0.300215	0.150108	−	0.988670i	$-0.452038\pi$
$439$	6.29021	0.300215	0.150108	+	0.988670i	$0.452038\pi$
$440$	0	0
$441$	10.9832	0.523010
$442$	0	0
$443$	−0.160116	−0.00760734	−0.00380367	−	0.999993i	$-0.501211\pi$
$443$	−0.160116	−0.00760734	−0.00380367	+	0.999993i	$0.501211\pi$
$444$	0	0
$445$	14.5525	0.689853
$446$	0	0
$447$	3.45049	0.163202
$448$	0	0
$449$	16.8646	0.795889	0.397945	−	0.917409i	$-0.369724\pi$
$449$	16.8646	0.795889	0.397945	+	0.917409i	$0.369724\pi$
$450$	0	0
$451$	−11.0123	−0.518549
$452$	0	0
$453$	−12.7046	−0.596916
$454$	0	0
$455$	17.3544	0.813588
$456$	0	0
$457$	12.3828	0.579245	0.289622	−	0.957141i	$-0.406470\pi$
$457$	12.3828	0.579245	0.289622	+	0.957141i	$0.406470\pi$
$458$	0	0
$459$	−9.52872	−0.444762
$460$	0	0
$461$	13.4581	0.626807	0.313404	−	0.949620i	$-0.398531\pi$
$461$	13.4581	0.626807	0.313404	+	0.949620i	$0.398531\pi$
$462$	0	0
$463$	−14.0460	−0.652775	−0.326388	−	0.945236i	$-0.605831\pi$
$463$	−14.0460	−0.652775	−0.326388	+	0.945236i	$0.605831\pi$
$464$	0	0
$465$	13.4274	0.622681
$466$	0	0
$467$	14.0230	0.648906	0.324453	−	0.945902i	$-0.394820\pi$
$467$	14.0230	0.648906	0.324453	+	0.945902i	$0.394820\pi$
$468$	0	0
$469$	−10.8024	−0.498809
$470$	0	0
$471$	38.7733	1.78658
$472$	0	0
$473$	10.8373	0.498298
$474$	0	0
$475$	−0.426814	−0.0195836
$476$	0	0
$477$	6.95261	0.318338
$478$	0	0
$479$	23.3842	1.06845	0.534225	−	0.845342i	$-0.320603\pi$
$479$	23.3842	1.06845	0.534225	+	0.845342i	$0.320603\pi$
$480$	0	0
$481$	−0.0786900	−0.00358795
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.4805	0.521303
$486$	0	0
$487$	−13.5311	−0.613154	−0.306577	−	0.951846i	$-0.599184\pi$
$487$	−13.5311	−0.613154	−0.306577	+	0.951846i	$0.599184\pi$
$488$	0	0
$489$	−41.0582	−1.85672
$490$	0	0
$491$	19.0139	0.858087	0.429044	−	0.903284i	$-0.358851\pi$
$491$	19.0139	0.858087	0.429044	+	0.903284i	$0.358851\pi$
$492$	0	0
$493$	18.1457	0.817243
$494$	0	0
$495$	2.25292	0.101261
$496$	0	0
$497$	−51.2717	−2.29985
$498$	0	0
$499$	−15.1055	−0.676216	−0.338108	−	0.941107i	$-0.609787\pi$
$499$	−15.1055	−0.676216	−0.338108	+	0.941107i	$0.609787\pi$
$500$	0	0
$501$	−29.1001	−1.30010
$502$	0	0
$503$	−12.5584	−0.559951	−0.279975	−	0.960007i	$-0.590326\pi$
$503$	−12.5584	−0.559951	−0.279975	+	0.960007i	$0.590326\pi$
$504$	0	0
$505$	−0.519913	−0.0231358
$506$	0	0
$507$	33.5699	1.49089
$508$	0	0
$509$	−25.3241	−1.12247	−0.561236	−	0.827656i	$-0.689674\pi$
$509$	−25.3241	−1.12247	−0.561236	+	0.827656i	$0.689674\pi$
$510$	0	0
$511$	41.8143	1.84975
$512$	0	0
$513$	−0.313752	−0.0138525
$514$	0	0
$515$	8.66779	0.381948
$516$	0	0
$517$	−11.9177	−0.524141
$518$	0	0
$519$	−33.2835	−1.46098
$520$	0	0
$521$	−4.54884	−0.199288	−0.0996442	−	0.995023i	$-0.531770\pi$
$521$	−4.54884	−0.199288	−0.0996442	+	0.995023i	$0.531770\pi$
$522$	0	0
$523$	14.3395	0.627022	0.313511	−	0.949585i	$-0.398495\pi$
$523$	14.3395	0.627022	0.313511	+	0.949585i	$0.398495\pi$
$524$	0	0
$525$	34.1658	1.49112
$526$	0	0
$527$	−22.3464	−0.973426
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−14.7652	−0.640757
$532$	0	0
$533$	34.3945	1.48979
$534$	0	0
$535$	10.1897	0.440539
$536$	0	0
$537$	−6.49846	−0.280429
$538$	0	0
$539$	−12.3507	−0.531982
$540$	0	0
$541$	−20.9132	−0.899128	−0.449564	−	0.893248i	$-0.648421\pi$
$541$	−20.9132	−0.899128	−0.449564	+	0.893248i	$0.648421\pi$
$542$	0	0
$543$	43.2644	1.85665
$544$	0	0
$545$	−12.7297	−0.545281
$546$	0	0
$547$	25.3324	1.08314	0.541568	−	0.840657i	$-0.317831\pi$
$547$	25.3324	1.08314	0.541568	+	0.840657i	$0.317831\pi$
$548$	0	0
$549$	−20.2220	−0.863054
$550$	0	0
$551$	0.597485	0.0254537
$552$	0	0
$553$	36.7473	1.56265
$554$	0	0
$555$	0.0268042	0.00113778
$556$	0	0
$557$	11.0736	0.469202	0.234601	−	0.972092i	$-0.424622\pi$
$557$	11.0736	0.469202	0.234601	+	0.972092i	$0.424622\pi$
$558$	0	0
$559$	−33.8478	−1.43161
$560$	0	0
$561$	−11.1137	−0.469222
$562$	0	0
$563$	−32.3640	−1.36398	−0.681989	−	0.731362i	$-0.738885\pi$
$563$	−32.3640	−1.36398	−0.681989	+	0.731362i	$0.738885\pi$
$564$	0	0
$565$	5.12327	0.215537
$566$	0	0
$567$	42.3767	1.77966
$568$	0	0
$569$	1.04907	0.0439792	0.0219896	−	0.999758i	$-0.493000\pi$
$569$	1.04907	0.0439792	0.0219896	+	0.999758i	$0.493000\pi$
$570$	0	0
$571$	35.3068	1.47754	0.738771	−	0.673956i	$-0.235406\pi$
$571$	35.3068	1.47754	0.738771	+	0.673956i	$0.235406\pi$
$572$	0	0
$573$	−32.6585	−1.36433
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13.5437	−0.563830	−0.281915	−	0.959439i	$-0.590970\pi$
$577$	−13.5437	−0.563830	−0.281915	+	0.959439i	$0.590970\pi$
$578$	0	0
$579$	23.2967	0.968176
$580$	0	0
$581$	−43.0945	−1.78786
$582$	0	0
$583$	−7.81825	−0.323799
$584$	0	0
$585$	−7.03650	−0.290924
$586$	0	0
$587$	28.8357	1.19018	0.595088	−	0.803661i	$-0.297117\pi$
$587$	28.8357	1.19018	0.595088	+	0.803661i	$0.297117\pi$
$588$	0	0
$589$	−0.735801	−0.0303181
$590$	0	0
$591$	−26.5363	−1.09156
$592$	0	0
$593$	−6.61536	−0.271660	−0.135830	−	0.990732i	$-0.543370\pi$
$593$	−6.61536	−0.271660	−0.135830	+	0.990732i	$0.543370\pi$
$594$	0	0
$595$	9.83808	0.403322
$596$	0	0
$597$	−38.5763	−1.57882
$598$	0	0
$599$	22.5893	0.922975	0.461488	−	0.887147i	$-0.347316\pi$
$599$	22.5893	0.922975	0.461488	+	0.887147i	$0.347316\pi$
$600$	0	0
$601$	19.9777	0.814908	0.407454	−	0.913226i	$-0.366417\pi$
$601$	19.9777	0.814908	0.407454	+	0.913226i	$0.366417\pi$
$602$	0	0
$603$	4.37993	0.178365
$604$	0	0
$605$	6.91319	0.281061
$606$	0	0
$607$	−3.32236	−0.134851	−0.0674253	−	0.997724i	$-0.521478\pi$
$607$	−3.32236	−0.134851	−0.0674253	+	0.997724i	$0.521478\pi$
$608$	0	0
$609$	−47.8277	−1.93808
$610$	0	0
$611$	37.2224	1.50586
$612$	0	0
$613$	15.3787	0.621142	0.310571	−	0.950550i	$-0.399480\pi$
$613$	15.3787	0.621142	0.310571	+	0.950550i	$0.399480\pi$
$614$	0	0
$615$	−11.7158	−0.472428
$616$	0	0
$617$	−44.0312	−1.77263	−0.886315	−	0.463083i	$-0.846743\pi$
$617$	−44.0312	−1.77263	−0.886315	+	0.463083i	$0.846743\pi$
$618$	0	0
$619$	−18.3764	−0.738612	−0.369306	−	0.929308i	$-0.620405\pi$
$619$	−18.3764	−0.738612	−0.369306	+	0.929308i	$0.620405\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	63.8347	2.55748
$624$	0	0
$625$	14.4813	0.579252
$626$	0	0
$627$	−0.365941	−0.0146143
$628$	0	0
$629$	−0.0446087	−0.00177867
$630$	0	0
$631$	31.3039	1.24619	0.623094	−	0.782147i	$-0.285875\pi$
$631$	31.3039	1.24619	0.623094	+	0.782147i	$0.285875\pi$
$632$	0	0
$633$	26.1833	1.04069
$634$	0	0
$635$	−0.652893	−0.0259093
$636$	0	0
$637$	38.5746	1.52838
$638$	0	0
$639$	20.7885	0.822382
$640$	0	0
$641$	5.64968	0.223149	0.111574	−	0.993756i	$-0.464411\pi$
$641$	5.64968	0.223149	0.111574	+	0.993756i	$0.464411\pi$
$642$	0	0
$643$	−1.03420	−0.0407849	−0.0203925	−	0.999792i	$-0.506492\pi$
$643$	−1.03420	−0.0407849	−0.0203925	+	0.999792i	$0.506492\pi$
$644$	0	0
$645$	11.5296	0.453978
$646$	0	0
$647$	−5.05484	−0.198726	−0.0993631	−	0.995051i	$-0.531681\pi$
$647$	−5.05484	−0.198726	−0.0993631	+	0.995051i	$0.531681\pi$
$648$	0	0
$649$	16.6036	0.651748
$650$	0	0
$651$	58.8997	2.30846
$652$	0	0
$653$	20.9767	0.820880	0.410440	−	0.911888i	$-0.365375\pi$
$653$	20.9767	0.820880	0.410440	+	0.911888i	$0.365375\pi$
$654$	0	0
$655$	−6.38815	−0.249606
$656$	0	0
$657$	−16.9540	−0.661437
$658$	0	0
$659$	32.3918	1.26180	0.630902	−	0.775863i	$-0.282685\pi$
$659$	32.3918	1.26180	0.630902	+	0.775863i	$0.282685\pi$
$660$	0	0
$661$	11.6338	0.452501	0.226250	−	0.974069i	$-0.427353\pi$
$661$	11.6338	0.452501	0.226250	+	0.974069i	$0.427353\pi$
$662$	0	0
$663$	34.7113	1.34807
$664$	0	0
$665$	0.323938	0.0125618
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−7.04833	−0.272504
$670$	0	0
$671$	22.7397	0.877858
$672$	0	0
$673$	−18.6674	−0.719574	−0.359787	−	0.933034i	$-0.617151\pi$
$673$	−18.6674	−0.719574	−0.359787	+	0.933034i	$0.617151\pi$
$674$	0	0
$675$	13.3559	0.514071
$676$	0	0
$677$	−1.25998	−0.0484250	−0.0242125	−	0.999707i	$-0.507708\pi$
$677$	−1.25998	−0.0484250	−0.0242125	+	0.999707i	$0.507708\pi$
$678$	0	0
$679$	50.3596	1.93262
$680$	0	0
$681$	0.695436	0.0266492
$682$	0	0
$683$	−42.1366	−1.61231	−0.806155	−	0.591704i	$-0.798456\pi$
$683$	−42.1366	−1.61231	−0.806155	+	0.591704i	$0.798456\pi$
$684$	0	0
$685$	−6.20458	−0.237065
$686$	0	0
$687$	−51.2153	−1.95399
$688$	0	0
$689$	24.4186	0.930274
$690$	0	0
$691$	0.374561	0.0142490	0.00712449	−	0.999975i	$-0.497732\pi$
$691$	0.374561	0.0142490	0.00712449	+	0.999975i	$0.497732\pi$
$692$	0	0
$693$	9.88249	0.375405
$694$	0	0
$695$	4.18676	0.158813
$696$	0	0
$697$	19.4980	0.738538
$698$	0	0
$699$	−40.5183	−1.53254
$700$	0	0
$701$	−11.3741	−0.429592	−0.214796	−	0.976659i	$-0.568909\pi$
$701$	−11.3741	−0.429592	−0.214796	+	0.976659i	$0.568909\pi$
$702$	0	0
$703$	−0.00146883	−5.53980e−5 0
$704$	0	0
$705$	−12.6791	−0.477523
$706$	0	0
$707$	−2.28061	−0.0857712
$708$	0	0
$709$	−13.6229	−0.511617	−0.255809	−	0.966727i	$-0.582342\pi$
$709$	−13.6229	−0.511617	−0.255809	+	0.966727i	$0.582342\pi$
$710$	0	0
$711$	−14.8995	−0.558776
$712$	0	0
$713$	0	0
$714$	0	0
$715$	7.91259	0.295914
$716$	0	0
$717$	15.4331	0.576361
$718$	0	0
$719$	−1.54491	−0.0576155	−0.0288078	−	0.999585i	$-0.509171\pi$
$719$	−1.54491	−0.0576155	−0.0288078	+	0.999585i	$0.509171\pi$
$720$	0	0
$721$	38.0215	1.41599
$722$	0	0
$723$	52.9760	1.97020
$724$	0	0
$725$	−25.4340	−0.944596
$726$	0	0
$727$	−16.7024	−0.619457	−0.309729	−	0.950825i	$-0.600238\pi$
$727$	−16.7024	−0.619457	−0.309729	+	0.950825i	$0.600238\pi$
$728$	0	0
$729$	2.81920	0.104415
$730$	0	0
$731$	−19.1880	−0.709695
$732$	0	0
$733$	21.9685	0.811425	0.405713	−	0.914001i	$-0.367023\pi$
$733$	21.9685	0.811425	0.405713	+	0.914001i	$0.367023\pi$
$734$	0	0
$735$	−13.1397	−0.484666
$736$	0	0
$737$	−4.92526	−0.181424
$738$	0	0
$739$	−23.5219	−0.865268	−0.432634	−	0.901570i	$-0.642416\pi$
$739$	−23.5219	−0.865268	−0.432634	+	0.901570i	$0.642416\pi$
$740$	0	0
$741$	1.14294	0.0419869
$742$	0	0
$743$	−4.82040	−0.176843	−0.0884217	−	0.996083i	$-0.528182\pi$
$743$	−4.82040	−0.176843	−0.0884217	+	0.996083i	$0.528182\pi$
$744$	0	0
$745$	−1.39264	−0.0510225
$746$	0	0
$747$	17.4731	0.639306
$748$	0	0
$749$	44.6974	1.63321
$750$	0	0
$751$	47.6674	1.73941	0.869703	−	0.493575i	$-0.164310\pi$
$751$	47.6674	1.73941	0.869703	+	0.493575i	$0.164310\pi$
$752$	0	0
$753$	−27.6632	−1.00810
$754$	0	0
$755$	5.12768	0.186616
$756$	0	0
$757$	28.8898	1.05002	0.525008	−	0.851097i	$-0.324062\pi$
$757$	28.8898	1.05002	0.525008	+	0.851097i	$0.324062\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.2511	0.371600	0.185800	−	0.982588i	$-0.440512\pi$
$761$	10.2511	0.371600	0.185800	+	0.982588i	$0.440512\pi$
$762$	0	0
$763$	−55.8392	−2.02152
$764$	0	0
$765$	−3.98894	−0.144220
$766$	0	0
$767$	−51.8577	−1.87247
$768$	0	0
$769$	3.73955	0.134852	0.0674258	−	0.997724i	$-0.478521\pi$
$769$	3.73955	0.134852	0.0674258	+	0.997724i	$0.478521\pi$
$770$	0	0
$771$	−51.3618	−1.84975
$772$	0	0
$773$	5.80430	0.208766	0.104383	−	0.994537i	$-0.466713\pi$
$773$	5.80430	0.208766	0.104383	+	0.994537i	$0.466713\pi$
$774$	0	0
$775$	31.3219	1.12512
$776$	0	0
$777$	0.117578	0.00421807
$778$	0	0
$779$	0.642010	0.0230024
$780$	0	0
$781$	−23.3768	−0.836489
$782$	0	0
$783$	−18.6966	−0.668162
$784$	0	0
$785$	−15.6492	−0.558543
$786$	0	0
$787$	43.8288	1.56233	0.781164	−	0.624326i	$-0.214626\pi$
$787$	43.8288	1.56233	0.781164	+	0.624326i	$0.214626\pi$
$788$	0	0
$789$	26.4800	0.942714
$790$	0	0
$791$	22.4733	0.799060
$792$	0	0
$793$	−71.0226	−2.52209
$794$	0	0
$795$	−8.31773	−0.294999
$796$	0	0
$797$	−22.4379	−0.794792	−0.397396	−	0.917647i	$-0.630086\pi$
$797$	−22.4379	−0.794792	−0.397396	+	0.917647i	$0.630086\pi$
$798$	0	0
$799$	21.1011	0.746502
$800$	0	0
$801$	−25.8823	−0.914508
$802$	0	0
$803$	19.0648	0.672783
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.1787	0.358306
$808$	0	0
$809$	−34.6629	−1.21868	−0.609342	−	0.792908i	$-0.708566\pi$
$809$	−34.6629	−1.21868	−0.609342	+	0.792908i	$0.708566\pi$
$810$	0	0
$811$	−6.49718	−0.228147	−0.114073	−	0.993472i	$-0.536390\pi$
$811$	−6.49718	−0.228147	−0.114073	+	0.993472i	$0.536390\pi$
$812$	0	0
$813$	−33.7968	−1.18531
$814$	0	0
$815$	16.5714	0.580470
$816$	0	0
$817$	−0.631804	−0.0221040
$818$	0	0
$819$	−30.8658	−1.07854
$820$	0	0
$821$	56.2273	1.96235	0.981174	−	0.193128i	$-0.0618632\pi$
$821$	56.2273	1.96235	0.981174	+	0.193128i	$0.0618632\pi$
$822$	0	0
$823$	−11.5045	−0.401021	−0.200510	−	0.979692i	$-0.564260\pi$
$823$	−11.5045	−0.401021	−0.200510	+	0.979692i	$0.564260\pi$
$824$	0	0
$825$	15.5776	0.542342
$826$	0	0
$827$	34.2674	1.19159	0.595797	−	0.803135i	$-0.296836\pi$
$827$	34.2674	1.19159	0.595797	+	0.803135i	$0.296836\pi$
$828$	0	0
$829$	−10.8712	−0.377571	−0.188786	−	0.982018i	$-0.560455\pi$
$829$	−10.8712	−0.377571	−0.188786	+	0.982018i	$0.560455\pi$
$830$	0	0
$831$	−7.60067	−0.263664
$832$	0	0
$833$	21.8676	0.757669
$834$	0	0
$835$	11.7450	0.406453
$836$	0	0
$837$	23.0248	0.795854
$838$	0	0
$839$	50.0919	1.72936	0.864682	−	0.502319i	$-0.167520\pi$
$839$	50.0919	1.72936	0.864682	+	0.502319i	$0.167520\pi$
$840$	0	0
$841$	6.60436	0.227737
$842$	0	0
$843$	−51.5313	−1.77483
$844$	0	0
$845$	−13.5490	−0.466101
$846$	0	0
$847$	30.3249	1.04197
$848$	0	0
$849$	−42.5349	−1.45979
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−33.7070	−1.15410	−0.577052	−	0.816707i	$-0.695797\pi$
$853$	−33.7070	−1.15410	−0.577052	+	0.816707i	$0.695797\pi$
$854$	0	0
$855$	−0.131344	−0.00449186
$856$	0	0
$857$	30.1722	1.03066	0.515331	−	0.856991i	$-0.327669\pi$
$857$	30.1722	1.03066	0.515331	+	0.856991i	$0.327669\pi$
$858$	0	0
$859$	−0.982353	−0.0335174	−0.0167587	−	0.999860i	$-0.505335\pi$
$859$	−0.982353	−0.0335174	−0.0167587	+	0.999860i	$0.505335\pi$
$860$	0	0
$861$	−51.3918	−1.75143
$862$	0	0
$863$	−32.5410	−1.10771	−0.553855	−	0.832613i	$-0.686844\pi$
$863$	−32.5410	−1.10771	−0.553855	+	0.832613i	$0.686844\pi$
$864$	0	0
$865$	13.4335	0.456751
$866$	0	0
$867$	−16.4945	−0.560183
$868$	0	0
$869$	16.7546	0.568361
$870$	0	0
$871$	15.3830	0.521232
$872$	0	0
$873$	−20.4187	−0.691069
$874$	0	0
$875$	−29.9650	−1.01300
$876$	0	0
$877$	−35.8660	−1.21111	−0.605554	−	0.795804i	$-0.707048\pi$
$877$	−35.8660	−1.21111	−0.605554	+	0.795804i	$0.707048\pi$
$878$	0	0
$879$	−9.74651	−0.328742
$880$	0	0
$881$	−17.4344	−0.587379	−0.293690	−	0.955901i	$-0.594883\pi$
$881$	−17.4344	−0.587379	−0.293690	+	0.955901i	$0.594883\pi$
$882$	0	0
$883$	26.3312	0.886114	0.443057	−	0.896493i	$-0.353894\pi$
$883$	26.3312	0.886114	0.443057	+	0.896493i	$0.353894\pi$
$884$	0	0
$885$	17.6643	0.593780
$886$	0	0
$887$	−43.7225	−1.46806	−0.734028	−	0.679119i	$-0.762362\pi$
$887$	−43.7225	−1.46806	−0.734028	+	0.679119i	$0.762362\pi$
$888$	0	0
$889$	−2.86393	−0.0960531
$890$	0	0
$891$	19.3213	0.647287
$892$	0	0
$893$	0.694795	0.0232504
$894$	0	0
$895$	2.62283	0.0876715
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−43.8467	−1.46237
$900$	0	0
$901$	13.8427	0.461167
$902$	0	0
$903$	50.5749	1.68303
$904$	0	0
$905$	−17.4618	−0.580451
$906$	0	0
$907$	−27.0613	−0.898556	−0.449278	−	0.893392i	$-0.648319\pi$
$907$	−27.0613	−0.898556	−0.449278	+	0.893392i	$0.648319\pi$
$908$	0	0
$909$	0.924693	0.0306701
$910$	0	0
$911$	18.7087	0.619846	0.309923	−	0.950762i	$-0.399697\pi$
$911$	18.7087	0.619846	0.309923	+	0.950762i	$0.399697\pi$
$912$	0	0
$913$	−19.6485	−0.650272
$914$	0	0
$915$	24.1925	0.799779
$916$	0	0
$917$	−28.0218	−0.925361
$918$	0	0
$919$	50.5427	1.66725	0.833626	−	0.552330i	$-0.186261\pi$
$919$	50.5427	1.66725	0.833626	+	0.552330i	$0.186261\pi$
$920$	0	0
$921$	−18.9887	−0.625699
$922$	0	0
$923$	73.0124	2.40323
$924$	0	0
$925$	0.0625259	0.00205584
$926$	0	0
$927$	−15.4161	−0.506332
$928$	0	0
$929$	60.0626	1.97059	0.985295	−	0.170863i	$-0.0546555\pi$
$929$	60.0626	1.97059	0.985295	+	0.170863i	$0.0546555\pi$
$930$	0	0
$931$	0.720036	0.0235982
$932$	0	0
$933$	−14.4525	−0.473153
$934$	0	0
$935$	4.48558	0.146694
$936$	0	0
$937$	−28.2242	−0.922043	−0.461022	−	0.887389i	$-0.652517\pi$
$937$	−28.2242	−0.922043	−0.461022	+	0.887389i	$0.652517\pi$
$938$	0	0
$939$	0.746959	0.0243761
$940$	0	0
$941$	32.2198	1.05034	0.525169	−	0.850998i	$-0.324002\pi$
$941$	32.2198	1.05034	0.525169	+	0.850998i	$0.324002\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−10.1367	−0.329748
$946$	0	0
$947$	0.685936	0.0222899	0.0111450	−	0.999938i	$-0.496452\pi$
$947$	0.685936	0.0222899	0.0111450	+	0.999938i	$0.496452\pi$
$948$	0	0
$949$	−59.5448	−1.93291
$950$	0	0
$951$	25.0348	0.811809
$952$	0	0
$953$	11.1663	0.361711	0.180856	−	0.983510i	$-0.442113\pi$
$953$	11.1663	0.361711	0.180856	+	0.983510i	$0.442113\pi$
$954$	0	0
$955$	13.1812	0.426534
$956$	0	0
$957$	−21.8066	−0.704907
$958$	0	0
$959$	−27.2165	−0.878868
$960$	0	0
$961$	22.9970	0.741839
$962$	0	0
$963$	−18.1229	−0.584004
$964$	0	0
$965$	−9.40271	−0.302684
$966$	0	0
$967$	12.0546	0.387650	0.193825	−	0.981036i	$-0.437911\pi$
$967$	12.0546	0.387650	0.193825	+	0.981036i	$0.437911\pi$
$968$	0	0
$969$	0.647922	0.0208142
$970$	0	0
$971$	−42.3430	−1.35885	−0.679426	−	0.733744i	$-0.737771\pi$
$971$	−42.3430	−1.35885	−0.679426	+	0.733744i	$0.737771\pi$
$972$	0	0
$973$	18.3653	0.588765
$974$	0	0
$975$	−48.6531	−1.55815
$976$	0	0
$977$	38.7026	1.23820	0.619102	−	0.785310i	$-0.287497\pi$
$977$	38.7026	1.23820	0.619102	+	0.785310i	$0.287497\pi$
$978$	0	0
$979$	29.1048	0.930195
$980$	0	0
$981$	22.6405	0.722856
$982$	0	0
$983$	2.94386	0.0938946	0.0469473	−	0.998897i	$-0.485051\pi$
$983$	2.94386	0.0938946	0.0469473	+	0.998897i	$0.485051\pi$
$984$	0	0
$985$	10.7103	0.341257
$986$	0	0
$987$	−55.6172	−1.77031
$988$	0	0
$989$	0	0
$990$	0	0
$991$	54.2855	1.72444	0.862218	−	0.506538i	$-0.169075\pi$
$991$	54.2855	1.72444	0.862218	+	0.506538i	$0.169075\pi$
$992$	0	0
$993$	40.7615	1.29353
$994$	0	0
$995$	15.5697	0.493592
$996$	0	0
$997$	14.7079	0.465803	0.232901	−	0.972500i	$-0.425178\pi$
$997$	14.7079	0.465803	0.232901	+	0.972500i	$0.425178\pi$
$998$	0	0
$999$	0.0459629	0.00145420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.cg.1.13		15
4.3	odd	2		4232.2.a.bb.1.3		15
23.4	even	11		368.2.m.e.177.3		30
23.6	even	11		368.2.m.e.289.3		30
23.22	odd	2		8464.2.a.ch.1.13		15
92.27	odd	22		184.2.i.b.177.1	yes	30
92.75	odd	22		184.2.i.b.105.1	✓	30
92.91	even	2		4232.2.a.ba.1.3		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.105.1	✓	30	92.75	odd	22
184.2.i.b.177.1	yes	30	92.27	odd	22
368.2.m.e.177.3		30	23.4	even	11
368.2.m.e.289.3		30	23.6	even	11
4232.2.a.ba.1.3		15	92.91	even	2
4232.2.a.bb.1.3		15	4.3	odd	2
8464.2.a.cg.1.13		15	1.1	even	1	trivial
8464.2.a.ch.1.13		15	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.12777 q^{3} -0.858783 q^{5} -3.76707 q^{7} +1.52739 q^{9} -1.71756 q^{11} +5.36442 q^{13} -1.82729 q^{15} +3.04105 q^{17} +0.100133 q^{19} -8.01545 q^{21} -4.26249 q^{25} -3.13337 q^{27} +5.96694 q^{29} -7.34827 q^{31} -3.65457 q^{33} +3.23510 q^{35} -0.0146689 q^{37} +11.4142 q^{39} +6.41160 q^{41} -6.30968 q^{43} -1.31170 q^{45} +6.93875 q^{47} +7.19083 q^{49} +6.47064 q^{51} +4.55195 q^{53} +1.47501 q^{55} +0.213059 q^{57} -9.66697 q^{59} -13.2396 q^{61} -5.75380 q^{63} -4.60687 q^{65} +2.86759 q^{67} +13.6105 q^{71} -11.0999 q^{73} -9.06959 q^{75} +6.47017 q^{77} -9.75487 q^{79} -11.2492 q^{81} +11.4398 q^{83} -2.61160 q^{85} +12.6963 q^{87} -16.9454 q^{89} -20.2082 q^{91} -15.6354 q^{93} -0.0859921 q^{95} -13.3684 q^{97} -2.62339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} + q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} + 18 q^{33} + 18 q^{35} + 24 q^{37} + 19 q^{39} + 9 q^{41} - 48 q^{43} + 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 15 * q - q^3 - 10 * q^7 + 16 * q^9 - 23 * q^11 - 10 * q^15 - 29 * q^19 + q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 + 18 * q^33 + 18 * q^35 + 24 * q^37 + 19 * q^39 + 9 * q^41 - 48 * q^43 + 4 * q^45 + 36 * q^47 + 25 * q^49 - 35 * q^51 - 5 * q^53 + 10 * q^55 + 23 * q^57 + 22 * q^59 + 12 * q^61 - 35 * q^63 - 26 * q^65 - 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 - 26 * q^79 - 21 * q^81 - 68 * q^83 - 72 * q^85 - 19 * q^87 - 6 * q^89 - 71 * q^91 - 55 * q^93 + 12 * q^95 + 40 * q^97 - 63 * q^99

Introduction

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