LMFDB - Embedded newform 832.4.a.bf.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,4,Mod(1,832)] 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("832.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	832.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-11,0,5,0,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.0895891248$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 66x^{3} - 139x^{2} + 283x + 676 $ x^5 - x^4 - 66x^3 - 139x^2 + 283*x + 676
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.30745$ of defining polynomial
Character	$\chi$	$=$	832.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.48781 q^{3} +10.6440 q^{5} -7.29306 q^{7} +63.0185 q^{9} -39.6958 q^{11} -13.0000 q^{13} -100.988 q^{15} +111.834 q^{17} +70.5902 q^{19} +69.1951 q^{21} +3.12876 q^{23} -11.7058 q^{25} -341.736 q^{27} -262.801 q^{29} +86.5729 q^{31} +376.626 q^{33} -77.6272 q^{35} -61.7501 q^{37} +123.341 q^{39} -198.635 q^{41} -393.717 q^{43} +670.767 q^{45} +422.349 q^{47} -289.811 q^{49} -1061.06 q^{51} -193.654 q^{53} -422.521 q^{55} -669.746 q^{57} +512.261 q^{59} +714.769 q^{61} -459.597 q^{63} -138.372 q^{65} -498.678 q^{67} -29.6850 q^{69} -852.467 q^{71} +28.6535 q^{73} +111.062 q^{75} +289.504 q^{77} +747.882 q^{79} +1540.83 q^{81} +1047.78 q^{83} +1190.36 q^{85} +2493.40 q^{87} +278.121 q^{89} +94.8098 q^{91} -821.387 q^{93} +751.360 q^{95} +1607.75 q^{97} -2501.57 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 11 q^{3} + 5 q^{5} + 21 q^{7} + 44 q^{9} + 18 q^{11} - 65 q^{13} + 39 q^{15} + 123 q^{17} - 142 q^{19} + 27 q^{21} + 92 q^{23} + 194 q^{25} - 449 q^{27} + 206 q^{29} + 436 q^{31} + 302 q^{33} - 817 q^{35}+ \cdots - 1404 q^{99}+O(q^{100}) $ 5 * q - 11 * q^3 + 5 * q^5 + 21 * q^7 + 44 * q^9 + 18 * q^11 - 65 * q^13 + 39 * q^15 + 123 * q^17 - 142 * q^19 + 27 * q^21 + 92 * q^23 + 194 * q^25 - 449 * q^27 + 206 * q^29 + 436 * q^31 + 302 * q^33 - 817 * q^35 + 273 * q^37 + 143 * q^39 - 248 * q^41 - 1093 * q^43 + 74 * q^45 + 905 * q^47 + 672 * q^49 - 1359 * q^51 + 656 * q^53 + 686 * q^55 + 550 * q^57 - 634 * q^59 + 1080 * q^61 + 1374 * q^63 - 65 * q^65 - 1322 * q^67 + 1708 * q^69 + 215 * q^71 + 726 * q^73 - 1428 * q^75 + 1038 * q^77 + 1592 * q^79 + 1373 * q^81 - 300 * q^83 + 2165 * q^85 + 1246 * q^87 + 1786 * q^89 - 273 * q^91 + 2508 * q^93 + 894 * q^95 + 2970 * q^97 - 1404 * 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
$3$	−9.48781	−1.82593	−0.912965	−	0.408039i	$-0.866213\pi$
$3$	−9.48781	−1.82593	−0.912965	+	0.408039i	$0.866213\pi$
$4$	0	0
$5$	10.6440	0.952026	0.476013	−	0.879438i	$-0.342081\pi$
$5$	10.6440	0.952026	0.476013	+	0.879438i	$0.342081\pi$
$6$	0	0
$7$	−7.29306	−0.393788	−0.196894	−	0.980425i	$-0.563086\pi$
$7$	−7.29306	−0.393788	−0.196894	+	0.980425i	$0.563086\pi$
$8$	0	0
$9$	63.0185	2.33402
$10$	0	0
$11$	−39.6958	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$11$	−39.6958	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−100.988	−1.73833
$16$	0	0
$17$	111.834	1.59552	0.797759	−	0.602976i	$-0.206018\pi$
$17$	111.834	1.59552	0.797759	+	0.602976i	$0.206018\pi$
$18$	0	0
$19$	70.5902	0.852342	0.426171	−	0.904643i	$-0.359862\pi$
$19$	70.5902	0.852342	0.426171	+	0.904643i	$0.359862\pi$
$20$	0	0
$21$	69.1951	0.719030
$22$	0	0
$23$	3.12876	0.0283648	0.0141824	−	0.999899i	$-0.495485\pi$
$23$	3.12876	0.0283648	0.0141824	+	0.999899i	$0.495485\pi$
$24$	0	0
$25$	−11.7058	−0.0936462
$26$	0	0
$27$	−341.736	−2.43582
$28$	0	0
$29$	−262.801	−1.68279	−0.841395	−	0.540421i	$-0.818265\pi$
$29$	−262.801	−1.68279	−0.841395	+	0.540421i	$0.818265\pi$
$30$	0	0
$31$	86.5729	0.501579	0.250789	−	0.968042i	$-0.419310\pi$
$31$	86.5729	0.501579	0.250789	+	0.968042i	$0.419310\pi$
$32$	0	0
$33$	376.626	1.98673
$34$	0	0
$35$	−77.6272	−0.374897
$36$	0	0
$37$	−61.7501	−0.274369	−0.137184	−	0.990546i	$-0.543805\pi$
$37$	−61.7501	−0.274369	−0.137184	+	0.990546i	$0.543805\pi$
$38$	0	0
$39$	123.341	0.506422
$40$	0	0
$41$	−198.635	−0.756622	−0.378311	−	0.925678i	$-0.623495\pi$
$41$	−198.635	−0.756622	−0.378311	+	0.925678i	$0.623495\pi$
$42$	0	0
$43$	−393.717	−1.39631	−0.698155	−	0.715947i	$-0.745995\pi$
$43$	−393.717	−1.39631	−0.698155	+	0.715947i	$0.745995\pi$
$44$	0	0
$45$	670.767	2.22205
$46$	0	0
$47$	422.349	1.31076	0.655382	−	0.755298i	$-0.272508\pi$
$47$	422.349	1.31076	0.655382	+	0.755298i	$0.272508\pi$
$48$	0	0
$49$	−289.811	−0.844931
$50$	0	0
$51$	−1061.06	−2.91330
$52$	0	0
$53$	−193.654	−0.501895	−0.250947	−	0.968001i	$-0.580742\pi$
$53$	−193.654	−0.501895	−0.250947	+	0.968001i	$0.580742\pi$
$54$	0	0
$55$	−422.521	−1.03587
$56$	0	0
$57$	−669.746	−1.55632
$58$	0	0
$59$	512.261	1.13035	0.565176	−	0.824970i	$-0.308808\pi$
$59$	512.261	1.13035	0.565176	+	0.824970i	$0.308808\pi$
$60$	0	0
$61$	714.769	1.50028	0.750138	−	0.661281i	$-0.229987\pi$
$61$	714.769	1.50028	0.750138	+	0.661281i	$0.229987\pi$
$62$	0	0
$63$	−459.597	−0.919109
$64$	0	0
$65$	−138.372	−0.264045
$66$	0	0
$67$	−498.678	−0.909303	−0.454651	−	0.890670i	$-0.650236\pi$
$67$	−498.678	−0.909303	−0.454651	+	0.890670i	$0.650236\pi$
$68$	0	0
$69$	−29.6850	−0.0517921
$70$	0	0
$71$	−852.467	−1.42492	−0.712460	−	0.701713i	$-0.752419\pi$
$71$	−852.467	−1.42492	−0.712460	+	0.701713i	$0.752419\pi$
$72$	0	0
$73$	28.6535	0.0459403	0.0229702	−	0.999736i	$-0.492688\pi$
$73$	28.6535	0.0459403	0.0229702	+	0.999736i	$0.492688\pi$
$74$	0	0
$75$	111.062	0.170991
$76$	0	0
$77$	289.504	0.428467
$78$	0	0
$79$	747.882	1.06510	0.532552	−	0.846397i	$-0.321233\pi$
$79$	747.882	1.06510	0.532552	+	0.846397i	$0.321233\pi$
$80$	0	0
$81$	1540.83	2.11362
$82$	0	0
$83$	1047.78	1.38565	0.692823	−	0.721108i	$-0.256367\pi$
$83$	1047.78	1.38565	0.692823	+	0.721108i	$0.256367\pi$
$84$	0	0
$85$	1190.36	1.51898
$86$	0	0
$87$	2493.40	3.07266
$88$	0	0
$89$	278.121	0.331245	0.165622	−	0.986189i	$-0.447037\pi$
$89$	278.121	0.331245	0.165622	+	0.986189i	$0.447037\pi$
$90$	0	0
$91$	94.8098	0.109217
$92$	0	0
$93$	−821.387	−0.915847
$94$	0	0
$95$	751.360	0.811452
$96$	0	0
$97$	1607.75	1.68291	0.841455	−	0.540326i	$-0.181699\pi$
$97$	1607.75	1.68291	0.841455	+	0.540326i	$0.181699\pi$
$98$	0	0
$99$	−2501.57	−2.53956
$100$	0	0
$101$	1266.29	1.24753	0.623763	−	0.781613i	$-0.285603\pi$
$101$	1266.29	1.24753	0.623763	+	0.781613i	$0.285603\pi$
$102$	0	0
$103$	117.218	0.112135	0.0560673	−	0.998427i	$-0.482144\pi$
$103$	117.218	0.112135	0.0560673	+	0.998427i	$0.482144\pi$
$104$	0	0
$105$	736.512	0.684535
$106$	0	0
$107$	1543.43	1.39447	0.697237	−	0.716840i	$-0.254412\pi$
$107$	1543.43	1.39447	0.697237	+	0.716840i	$0.254412\pi$
$108$	0	0
$109$	612.637	0.538349	0.269174	−	0.963091i	$-0.413249\pi$
$109$	612.637	0.538349	0.269174	+	0.963091i	$0.413249\pi$
$110$	0	0
$111$	585.873	0.500978
$112$	0	0
$113$	−1545.43	−1.28657	−0.643284	−	0.765627i	$-0.722429\pi$
$113$	−1545.43	−1.28657	−0.643284	+	0.765627i	$0.722429\pi$
$114$	0	0
$115$	33.3024	0.0270040
$116$	0	0
$117$	−819.240	−0.647340
$118$	0	0
$119$	−815.615	−0.628297
$120$	0	0
$121$	244.753	0.183886
$122$	0	0
$123$	1884.61	1.38154
$124$	0	0
$125$	−1455.09	−1.04118
$126$	0	0
$127$	1358.05	0.948879	0.474440	−	0.880288i	$-0.342651\pi$
$127$	1358.05	0.948879	0.474440	+	0.880288i	$0.342651\pi$
$128$	0	0
$129$	3735.51	2.54956
$130$	0	0
$131$	−1301.51	−0.868039	−0.434020	−	0.900903i	$-0.642905\pi$
$131$	−1301.51	−0.868039	−0.434020	+	0.900903i	$0.642905\pi$
$132$	0	0
$133$	−514.819	−0.335642
$134$	0	0
$135$	−3637.43	−2.31896
$136$	0	0
$137$	−670.632	−0.418219	−0.209109	−	0.977892i	$-0.567056\pi$
$137$	−670.632	−0.418219	−0.209109	+	0.977892i	$0.567056\pi$
$138$	0	0
$139$	2142.07	1.30711	0.653553	−	0.756881i	$-0.273278\pi$
$139$	2142.07	1.30711	0.653553	+	0.756881i	$0.273278\pi$
$140$	0	0
$141$	−4007.16	−2.39336
$142$	0	0
$143$	516.045	0.301775
$144$	0	0
$145$	−2797.25	−1.60206
$146$	0	0
$147$	2749.67	1.54278
$148$	0	0
$149$	3065.68	1.68557	0.842786	−	0.538248i	$-0.180914\pi$
$149$	3065.68	1.68557	0.842786	+	0.538248i	$0.180914\pi$
$150$	0	0
$151$	2589.07	1.39534	0.697668	−	0.716422i	$-0.254221\pi$
$151$	2589.07	1.39534	0.697668	+	0.716422i	$0.254221\pi$
$152$	0	0
$153$	7047.63	3.72397
$154$	0	0
$155$	921.480	0.477516
$156$	0	0
$157$	3712.44	1.88716	0.943582	−	0.331140i	$-0.107433\pi$
$157$	3712.44	1.88716	0.943582	+	0.331140i	$0.107433\pi$
$158$	0	0
$159$	1837.35	0.916425
$160$	0	0
$161$	−22.8182	−0.0111697
$162$	0	0
$163$	729.775	0.350677	0.175339	−	0.984508i	$-0.443898\pi$
$163$	729.775	0.350677	0.175339	+	0.984508i	$0.443898\pi$
$164$	0	0
$165$	4008.79	1.89142
$166$	0	0
$167$	−1609.67	−0.745870	−0.372935	−	0.927858i	$-0.621649\pi$
$167$	−1609.67	−0.745870	−0.372935	+	0.927858i	$0.621649\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	4448.49	1.98938
$172$	0	0
$173$	−398.815	−0.175268	−0.0876340	−	0.996153i	$-0.527931\pi$
$173$	−398.815	−0.175268	−0.0876340	+	0.996153i	$0.527931\pi$
$174$	0	0
$175$	85.3709	0.0368768
$176$	0	0
$177$	−4860.24	−2.06394
$178$	0	0
$179$	−1632.70	−0.681751	−0.340876	−	0.940108i	$-0.610724\pi$
$179$	−1632.70	−0.681751	−0.340876	+	0.940108i	$0.610724\pi$
$180$	0	0
$181$	−192.549	−0.0790722	−0.0395361	−	0.999218i	$-0.512588\pi$
$181$	−192.549	−0.0790722	−0.0395361	+	0.999218i	$0.512588\pi$
$182$	0	0
$183$	−6781.59	−2.73940
$184$	0	0
$185$	−657.266	−0.261206
$186$	0	0
$187$	−4439.35	−1.73603
$188$	0	0
$189$	2492.30	0.959197
$190$	0	0
$191$	1170.90	0.443577	0.221789	−	0.975095i	$-0.428811\pi$
$191$	1170.90	0.443577	0.221789	+	0.975095i	$0.428811\pi$
$192$	0	0
$193$	1029.07	0.383805	0.191902	−	0.981414i	$-0.438534\pi$
$193$	1029.07	0.383805	0.191902	+	0.981414i	$0.438534\pi$
$194$	0	0
$195$	1312.84	0.482127
$196$	0	0
$197$	4507.31	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$197$	4507.31	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$198$	0	0
$199$	−3526.54	−1.25623	−0.628116	−	0.778120i	$-0.716174\pi$
$199$	−3526.54	−1.25623	−0.628116	+	0.778120i	$0.716174\pi$
$200$	0	0
$201$	4731.36	1.66032
$202$	0	0
$203$	1916.62	0.662663
$204$	0	0
$205$	−2114.26	−0.720324
$206$	0	0
$207$	197.169	0.0662040
$208$	0	0
$209$	−2802.13	−0.927404
$210$	0	0
$211$	−3918.76	−1.27857	−0.639285	−	0.768970i	$-0.720770\pi$
$211$	−3918.76	−1.27857	−0.639285	+	0.768970i	$0.720770\pi$
$212$	0	0
$213$	8088.04	2.60180
$214$	0	0
$215$	−4190.72	−1.32932
$216$	0	0
$217$	−631.381	−0.197516
$218$	0	0
$219$	−271.859	−0.0838838
$220$	0	0
$221$	−1453.85	−0.442517
$222$	0	0
$223$	856.191	0.257107	0.128553	−	0.991703i	$-0.458967\pi$
$223$	856.191	0.257107	0.128553	+	0.991703i	$0.458967\pi$
$224$	0	0
$225$	−737.680	−0.218572
$226$	0	0
$227$	4305.21	1.25880	0.629399	−	0.777083i	$-0.283301\pi$
$227$	4305.21	1.25880	0.629399	+	0.777083i	$0.283301\pi$
$228$	0	0
$229$	−1850.42	−0.533970	−0.266985	−	0.963701i	$-0.586027\pi$
$229$	−1850.42	−0.533970	−0.266985	+	0.963701i	$0.586027\pi$
$230$	0	0
$231$	−2746.75	−0.782351
$232$	0	0
$233$	2840.65	0.798699	0.399350	−	0.916799i	$-0.369236\pi$
$233$	2840.65	0.798699	0.399350	+	0.916799i	$0.369236\pi$
$234$	0	0
$235$	4495.47	1.24788
$236$	0	0
$237$	−7095.76	−1.94481
$238$	0	0
$239$	−36.8268	−0.00996706	−0.00498353	−	0.999988i	$-0.501586\pi$
$239$	−36.8268	−0.00996706	−0.00498353	+	0.999988i	$0.501586\pi$
$240$	0	0
$241$	−5864.68	−1.56754	−0.783771	−	0.621051i	$-0.786706\pi$
$241$	−5864.68	−1.56754	−0.783771	+	0.621051i	$0.786706\pi$
$242$	0	0
$243$	−5392.20	−1.42350
$244$	0	0
$245$	−3084.74	−0.804396
$246$	0	0
$247$	−917.673	−0.236397
$248$	0	0
$249$	−9941.12	−2.53009
$250$	0	0
$251$	−7472.76	−1.87919	−0.939594	−	0.342291i	$-0.888797\pi$
$251$	−7472.76	−1.87919	−0.939594	+	0.342291i	$0.888797\pi$
$252$	0	0
$253$	−124.198	−0.0308628
$254$	0	0
$255$	−11293.9	−2.77354
$256$	0	0
$257$	1445.76	0.350910	0.175455	−	0.984487i	$-0.443860\pi$
$257$	1445.76	0.350910	0.175455	+	0.984487i	$0.443860\pi$
$258$	0	0
$259$	450.347	0.108043
$260$	0	0
$261$	−16561.3	−3.92766
$262$	0	0
$263$	5572.95	1.30663	0.653314	−	0.757087i	$-0.273378\pi$
$263$	5572.95	1.30663	0.653314	+	0.757087i	$0.273378\pi$
$264$	0	0
$265$	−2061.25	−0.477817
$266$	0	0
$267$	−2638.76	−0.604829
$268$	0	0
$269$	8181.17	1.85433	0.927164	−	0.374655i	$-0.122239\pi$
$269$	8181.17	1.85433	0.927164	+	0.374655i	$0.122239\pi$
$270$	0	0
$271$	5195.73	1.16464	0.582321	−	0.812959i	$-0.302145\pi$
$271$	5195.73	1.16464	0.582321	+	0.812959i	$0.302145\pi$
$272$	0	0
$273$	−899.537	−0.199423
$274$	0	0
$275$	464.669	0.101893
$276$	0	0
$277$	206.429	0.0447766	0.0223883	−	0.999749i	$-0.492873\pi$
$277$	206.429	0.0447766	0.0223883	+	0.999749i	$0.492873\pi$
$278$	0	0
$279$	5455.69	1.17069
$280$	0	0
$281$	−3588.58	−0.761839	−0.380919	−	0.924608i	$-0.624392\pi$
$281$	−3588.58	−0.761839	−0.380919	+	0.924608i	$0.624392\pi$
$282$	0	0
$283$	235.366	0.0494383	0.0247192	−	0.999694i	$-0.492131\pi$
$283$	235.366	0.0494383	0.0247192	+	0.999694i	$0.492131\pi$
$284$	0	0
$285$	−7128.76	−1.48165
$286$	0	0
$287$	1448.65	0.297949
$288$	0	0
$289$	7593.93	1.54568
$290$	0	0
$291$	−15254.0	−3.07288
$292$	0	0
$293$	−7136.82	−1.42299	−0.711497	−	0.702689i	$-0.751983\pi$
$293$	−7136.82	−1.42299	−0.711497	+	0.702689i	$0.751983\pi$
$294$	0	0
$295$	5452.50	1.07612
$296$	0	0
$297$	13565.5	2.65033
$298$	0	0
$299$	−40.6738	−0.00786698
$300$	0	0
$301$	2871.40	0.549851
$302$	0	0
$303$	−12014.3	−2.27789
$304$	0	0
$305$	7607.99	1.42830
$306$	0	0
$307$	5070.49	0.942633	0.471317	−	0.881964i	$-0.343779\pi$
$307$	5070.49	0.942633	0.471317	+	0.881964i	$0.343779\pi$
$308$	0	0
$309$	−1112.14	−0.204750
$310$	0	0
$311$	−4474.80	−0.815893	−0.407946	−	0.913006i	$-0.633755\pi$
$311$	−4474.80	−0.815893	−0.407946	+	0.913006i	$0.633755\pi$
$312$	0	0
$313$	2308.38	0.416860	0.208430	−	0.978037i	$-0.433165\pi$
$313$	2308.38	0.416860	0.208430	+	0.978037i	$0.433165\pi$
$314$	0	0
$315$	−4891.94	−0.875015
$316$	0	0
$317$	1698.40	0.300920	0.150460	−	0.988616i	$-0.451925\pi$
$317$	1698.40	0.300920	0.150460	+	0.988616i	$0.451925\pi$
$318$	0	0
$319$	10432.1	1.83099
$320$	0	0
$321$	−14643.7	−2.54621
$322$	0	0
$323$	7894.41	1.35993
$324$	0	0
$325$	152.175	0.0259728
$326$	0	0
$327$	−5812.58	−0.982986
$328$	0	0
$329$	−3080.21	−0.516163
$330$	0	0
$331$	1404.53	0.233233	0.116617	−	0.993177i	$-0.462795\pi$
$331$	1404.53	0.233233	0.116617	+	0.993177i	$0.462795\pi$
$332$	0	0
$333$	−3891.39	−0.640382
$334$	0	0
$335$	−5307.92	−0.865680
$336$	0	0
$337$	4425.16	0.715293	0.357646	−	0.933857i	$-0.383579\pi$
$337$	4425.16	0.715293	0.357646	+	0.933857i	$0.383579\pi$
$338$	0	0
$339$	14662.8	2.34918
$340$	0	0
$341$	−3436.58	−0.545751
$342$	0	0
$343$	4615.13	0.726512
$344$	0	0
$345$	−315.967	−0.0493075
$346$	0	0
$347$	2853.98	0.441527	0.220763	−	0.975327i	$-0.429145\pi$
$347$	2853.98	0.441527	0.220763	+	0.975327i	$0.429145\pi$
$348$	0	0
$349$	9100.76	1.39585	0.697926	−	0.716170i	$-0.254106\pi$
$349$	9100.76	1.39585	0.697926	+	0.716170i	$0.254106\pi$
$350$	0	0
$351$	4442.57	0.675575
$352$	0	0
$353$	−3778.54	−0.569720	−0.284860	−	0.958569i	$-0.591947\pi$
$353$	−3778.54	−0.569720	−0.284860	+	0.958569i	$0.591947\pi$
$354$	0	0
$355$	−9073.64	−1.35656
$356$	0	0
$357$	7738.40	1.14723
$358$	0	0
$359$	8836.36	1.29907	0.649534	−	0.760333i	$-0.274964\pi$
$359$	8836.36	1.29907	0.649534	+	0.760333i	$0.274964\pi$
$360$	0	0
$361$	−1876.02	−0.273513
$362$	0	0
$363$	−2322.17	−0.335764
$364$	0	0
$365$	304.988	0.0437364
$366$	0	0
$367$	3430.71	0.487961	0.243981	−	0.969780i	$-0.421547\pi$
$367$	3430.71	0.487961	0.243981	+	0.969780i	$0.421547\pi$
$368$	0	0
$369$	−12517.6	−1.76597
$370$	0	0
$371$	1412.33	0.197640
$372$	0	0
$373$	3465.99	0.481132	0.240566	−	0.970633i	$-0.422667\pi$
$373$	3465.99	0.481132	0.240566	+	0.970633i	$0.422667\pi$
$374$	0	0
$375$	13805.6	1.90112
$376$	0	0
$377$	3416.41	0.466722
$378$	0	0
$379$	4792.08	0.649479	0.324740	−	0.945803i	$-0.394723\pi$
$379$	4792.08	0.649479	0.324740	+	0.945803i	$0.394723\pi$
$380$	0	0
$381$	−12884.9	−1.73259
$382$	0	0
$383$	11095.4	1.48029	0.740144	−	0.672448i	$-0.234757\pi$
$383$	11095.4	1.48029	0.740144	+	0.672448i	$0.234757\pi$
$384$	0	0
$385$	3081.47	0.407912
$386$	0	0
$387$	−24811.5	−3.25901
$388$	0	0
$389$	6852.37	0.893134	0.446567	−	0.894750i	$-0.352646\pi$
$389$	6852.37	0.893134	0.446567	+	0.894750i	$0.352646\pi$
$390$	0	0
$391$	349.902	0.0452566
$392$	0	0
$393$	12348.4	1.58498
$394$	0	0
$395$	7960.44	1.01401
$396$	0	0
$397$	−686.266	−0.0867575	−0.0433787	−	0.999059i	$-0.513812\pi$
$397$	−686.266	−0.0867575	−0.0433787	+	0.999059i	$0.513812\pi$
$398$	0	0
$399$	4884.50	0.612859
$400$	0	0
$401$	−7624.65	−0.949518	−0.474759	−	0.880116i	$-0.657465\pi$
$401$	−7624.65	−0.949518	−0.474759	+	0.880116i	$0.657465\pi$
$402$	0	0
$403$	−1125.45	−0.139113
$404$	0	0
$405$	16400.5	2.01222
$406$	0	0
$407$	2451.22	0.298531
$408$	0	0
$409$	−14874.4	−1.79827	−0.899137	−	0.437668i	$-0.855804\pi$
$409$	−14874.4	−1.79827	−0.899137	+	0.437668i	$0.855804\pi$
$410$	0	0
$411$	6362.83	0.763638
$412$	0	0
$413$	−3735.95	−0.445119
$414$	0	0
$415$	11152.5	1.31917
$416$	0	0
$417$	−20323.5	−2.38668
$418$	0	0
$419$	1690.42	0.197094	0.0985471	−	0.995132i	$-0.468580\pi$
$419$	1690.42	0.197094	0.0985471	+	0.995132i	$0.468580\pi$
$420$	0	0
$421$	−3389.19	−0.392350	−0.196175	−	0.980569i	$-0.562852\pi$
$421$	−3389.19	−0.392350	−0.196175	+	0.980569i	$0.562852\pi$
$422$	0	0
$423$	26615.8	3.05934
$424$	0	0
$425$	−1309.11	−0.149414
$426$	0	0
$427$	−5212.86	−0.590791
$428$	0	0
$429$	−4896.13	−0.551020
$430$	0	0
$431$	6202.53	0.693191	0.346596	−	0.938015i	$-0.387338\pi$
$431$	6202.53	0.693191	0.346596	+	0.938015i	$0.387338\pi$
$432$	0	0
$433$	−6347.41	−0.704474	−0.352237	−	0.935911i	$-0.614579\pi$
$433$	−6347.41	−0.704474	−0.352237	+	0.935911i	$0.614579\pi$
$434$	0	0
$435$	26539.7	2.92525
$436$	0	0
$437$	220.860	0.0241765
$438$	0	0
$439$	2751.71	0.299161	0.149581	−	0.988750i	$-0.452208\pi$
$439$	2751.71	0.299161	0.149581	+	0.988750i	$0.452208\pi$
$440$	0	0
$441$	−18263.5	−1.97208
$442$	0	0
$443$	−8029.18	−0.861123	−0.430562	−	0.902561i	$-0.641685\pi$
$443$	−8029.18	−0.861123	−0.430562	+	0.902561i	$0.641685\pi$
$444$	0	0
$445$	2960.31	0.315353
$446$	0	0
$447$	−29086.6	−3.07774
$448$	0	0
$449$	1173.84	0.123379	0.0616893	−	0.998095i	$-0.480351\pi$
$449$	1173.84	0.123379	0.0616893	+	0.998095i	$0.480351\pi$
$450$	0	0
$451$	7884.95	0.823255
$452$	0	0
$453$	−24564.6	−2.54778
$454$	0	0
$455$	1009.15	0.103978
$456$	0	0
$457$	12798.4	1.31003	0.655014	−	0.755616i	$-0.272663\pi$
$457$	12798.4	1.31003	0.655014	+	0.755616i	$0.272663\pi$
$458$	0	0
$459$	−38217.8	−3.88640
$460$	0	0
$461$	−17150.9	−1.73275	−0.866374	−	0.499396i	$-0.833555\pi$
$461$	−17150.9	−1.73275	−0.866374	+	0.499396i	$0.833555\pi$
$462$	0	0
$463$	−801.962	−0.0804974	−0.0402487	−	0.999190i	$-0.512815\pi$
$463$	−801.962	−0.0804974	−0.0402487	+	0.999190i	$0.512815\pi$
$464$	0	0
$465$	−8742.82	−0.871911
$466$	0	0
$467$	17094.0	1.69382	0.846912	−	0.531733i	$-0.178459\pi$
$467$	17094.0	1.69382	0.846912	+	0.531733i	$0.178459\pi$
$468$	0	0
$469$	3636.89	0.358073
$470$	0	0
$471$	−35222.9	−3.44583
$472$	0	0
$473$	15628.9	1.51928
$474$	0	0
$475$	−826.313	−0.0798186
$476$	0	0
$477$	−12203.8	−1.17143
$478$	0	0
$479$	−11493.8	−1.09638	−0.548189	−	0.836355i	$-0.684683\pi$
$479$	−11493.8	−1.09638	−0.548189	+	0.836355i	$0.684683\pi$
$480$	0	0
$481$	802.751	0.0760962
$482$	0	0
$483$	216.495	0.0203951
$484$	0	0
$485$	17112.9	1.60218
$486$	0	0
$487$	−9687.92	−0.901441	−0.450721	−	0.892665i	$-0.648833\pi$
$487$	−9687.92	−0.901441	−0.450721	+	0.892665i	$0.648833\pi$
$488$	0	0
$489$	−6923.97	−0.640312
$490$	0	0
$491$	−3137.29	−0.288358	−0.144179	−	0.989552i	$-0.546054\pi$
$491$	−3137.29	−0.288358	−0.144179	+	0.989552i	$0.546054\pi$
$492$	0	0
$493$	−29390.2	−2.68492
$494$	0	0
$495$	−26626.6	−2.41773
$496$	0	0
$497$	6217.10	0.561116
$498$	0	0
$499$	−5204.78	−0.466930	−0.233465	−	0.972365i	$-0.575006\pi$
$499$	−5204.78	−0.466930	−0.233465	+	0.972365i	$0.575006\pi$
$500$	0	0
$501$	15272.3	1.36191
$502$	0	0
$503$	15769.6	1.39787	0.698936	−	0.715184i	$-0.253657\pi$
$503$	15769.6	1.39787	0.698936	+	0.715184i	$0.253657\pi$
$504$	0	0
$505$	13478.3	1.18768
$506$	0	0
$507$	−1603.44	−0.140456
$508$	0	0
$509$	636.740	0.0554480	0.0277240	−	0.999616i	$-0.491174\pi$
$509$	636.740	0.0554480	0.0277240	+	0.999616i	$0.491174\pi$
$510$	0	0
$511$	−208.972	−0.0180908
$512$	0	0
$513$	−24123.2	−2.07615
$514$	0	0
$515$	1247.67	0.106755
$516$	0	0
$517$	−16765.4	−1.42620
$518$	0	0
$519$	3783.88	0.320027
$520$	0	0
$521$	14422.6	1.21279	0.606395	−	0.795163i	$-0.292615\pi$
$521$	14422.6	1.21279	0.606395	+	0.795163i	$0.292615\pi$
$522$	0	0
$523$	6737.99	0.563349	0.281675	−	0.959510i	$-0.409110\pi$
$523$	6737.99	0.563349	0.281675	+	0.959510i	$0.409110\pi$
$524$	0	0
$525$	−809.983	−0.0673344
$526$	0	0
$527$	9681.82	0.800279
$528$	0	0
$529$	−12157.2	−0.999195
$530$	0	0
$531$	32281.9	2.63826
$532$	0	0
$533$	2582.25	0.209849
$534$	0	0
$535$	16428.2	1.32758
$536$	0	0
$537$	15490.7	1.24483
$538$	0	0
$539$	11504.3	0.919340
$540$	0	0
$541$	−13997.2	−1.11236	−0.556181	−	0.831061i	$-0.687734\pi$
$541$	−13997.2	−1.11236	−0.556181	+	0.831061i	$0.687734\pi$
$542$	0	0
$543$	1826.87	0.144380
$544$	0	0
$545$	6520.89	0.512522
$546$	0	0
$547$	−18288.6	−1.42955	−0.714773	−	0.699356i	$-0.753470\pi$
$547$	−18288.6	−1.42955	−0.714773	+	0.699356i	$0.753470\pi$
$548$	0	0
$549$	45043.7	3.50167
$550$	0	0
$551$	−18551.2	−1.43431
$552$	0	0
$553$	−5454.35	−0.419426
$554$	0	0
$555$	6236.01	0.476944
$556$	0	0
$557$	11342.2	0.862809	0.431405	−	0.902159i	$-0.358018\pi$
$557$	11342.2	0.862809	0.431405	+	0.902159i	$0.358018\pi$
$558$	0	0
$559$	5118.33	0.387267
$560$	0	0
$561$	42119.7	3.16987
$562$	0	0
$563$	−4057.73	−0.303753	−0.151877	−	0.988399i	$-0.548532\pi$
$563$	−4057.73	−0.303753	−0.151877	+	0.988399i	$0.548532\pi$
$564$	0	0
$565$	−16449.6	−1.22485
$566$	0	0
$567$	−11237.3	−0.832318
$568$	0	0
$569$	−4996.10	−0.368097	−0.184049	−	0.982917i	$-0.558920\pi$
$569$	−4996.10	−0.368097	−0.184049	+	0.982917i	$0.558920\pi$
$570$	0	0
$571$	−12178.1	−0.892533	−0.446266	−	0.894900i	$-0.647247\pi$
$571$	−12178.1	−0.892533	−0.446266	+	0.894900i	$0.647247\pi$
$572$	0	0
$573$	−11109.3	−0.809940
$574$	0	0
$575$	−36.6245	−0.00265626
$576$	0	0
$577$	7682.73	0.554309	0.277154	−	0.960825i	$-0.410609\pi$
$577$	7682.73	0.554309	0.277154	+	0.960825i	$0.410609\pi$
$578$	0	0
$579$	−9763.65	−0.700801
$580$	0	0
$581$	−7641.51	−0.545651
$582$	0	0
$583$	7687.24	0.546095
$584$	0	0
$585$	−8719.97	−0.616284
$586$	0	0
$587$	27945.2	1.96494	0.982472	−	0.186411i	$-0.0596857\pi$
$587$	27945.2	1.96494	0.982472	+	0.186411i	$0.0596857\pi$
$588$	0	0
$589$	6111.20	0.427517
$590$	0	0
$591$	−42764.4	−2.97647
$592$	0	0
$593$	19707.1	1.36471	0.682355	−	0.731021i	$-0.260956\pi$
$593$	19707.1	1.36471	0.682355	+	0.731021i	$0.260956\pi$
$594$	0	0
$595$	−8681.39	−0.598155
$596$	0	0
$597$	33459.2	2.29379
$598$	0	0
$599$	−25278.2	−1.72428	−0.862138	−	0.506674i	$-0.830875\pi$
$599$	−25278.2	−1.72428	−0.862138	+	0.506674i	$0.830875\pi$
$600$	0	0
$601$	20128.1	1.36613	0.683065	−	0.730358i	$-0.260647\pi$
$601$	20128.1	1.36613	0.683065	+	0.730358i	$0.260647\pi$
$602$	0	0
$603$	−31425.9	−2.12233
$604$	0	0
$605$	2605.14	0.175065
$606$	0	0
$607$	22806.4	1.52501	0.762507	−	0.646980i	$-0.223968\pi$
$607$	22806.4	1.52501	0.762507	+	0.646980i	$0.223968\pi$
$608$	0	0
$609$	−18184.6	−1.20998
$610$	0	0
$611$	−5490.53	−0.363540
$612$	0	0
$613$	−3949.78	−0.260245	−0.130122	−	0.991498i	$-0.541537\pi$
$613$	−3949.78	−0.260245	−0.130122	+	0.991498i	$0.541537\pi$
$614$	0	0
$615$	20059.7	1.31526
$616$	0	0
$617$	13587.0	0.886536	0.443268	−	0.896389i	$-0.353819\pi$
$617$	13587.0	0.886536	0.443268	+	0.896389i	$0.353819\pi$
$618$	0	0
$619$	−22277.0	−1.44651	−0.723254	−	0.690583i	$-0.757354\pi$
$619$	−22277.0	−1.44651	−0.723254	+	0.690583i	$0.757354\pi$
$620$	0	0
$621$	−1069.21	−0.0690916
$622$	0	0
$623$	−2028.35	−0.130440
$624$	0	0
$625$	−14024.8	−0.897584
$626$	0	0
$627$	26586.1	1.69337
$628$	0	0
$629$	−6905.78	−0.437761
$630$	0	0
$631$	7148.74	0.451009	0.225505	−	0.974242i	$-0.427597\pi$
$631$	7148.74	0.451009	0.225505	+	0.974242i	$0.427597\pi$
$632$	0	0
$633$	37180.4	2.33458
$634$	0	0
$635$	14455.1	0.903358
$636$	0	0
$637$	3767.55	0.234342
$638$	0	0
$639$	−53721.2	−3.32579
$640$	0	0
$641$	7861.34	0.484406	0.242203	−	0.970226i	$-0.422130\pi$
$641$	7861.34	0.484406	0.242203	+	0.970226i	$0.422130\pi$
$642$	0	0
$643$	−469.877	−0.0288183	−0.0144091	−	0.999896i	$-0.504587\pi$
$643$	−469.877	−0.0288183	−0.0144091	+	0.999896i	$0.504587\pi$
$644$	0	0
$645$	39760.7	2.42725
$646$	0	0
$647$	−27438.5	−1.66726	−0.833630	−	0.552323i	$-0.813742\pi$
$647$	−27438.5	−1.66726	−0.833630	+	0.552323i	$0.813742\pi$
$648$	0	0
$649$	−20334.6	−1.22990
$650$	0	0
$651$	5990.42	0.360650
$652$	0	0
$653$	16891.2	1.01225	0.506127	−	0.862459i	$-0.331077\pi$
$653$	16891.2	1.01225	0.506127	+	0.862459i	$0.331077\pi$
$654$	0	0
$655$	−13853.2	−0.826396
$656$	0	0
$657$	1805.70	0.107225
$658$	0	0
$659$	12065.5	0.713211	0.356605	−	0.934255i	$-0.383934\pi$
$659$	12065.5	0.713211	0.356605	+	0.934255i	$0.383934\pi$
$660$	0	0
$661$	22835.9	1.34374	0.671870	−	0.740669i	$-0.265491\pi$
$661$	22835.9	1.34374	0.671870	+	0.740669i	$0.265491\pi$
$662$	0	0
$663$	13793.8	0.808005
$664$	0	0
$665$	−5479.72	−0.319540
$666$	0	0
$667$	−822.240	−0.0477320
$668$	0	0
$669$	−8123.38	−0.469459
$670$	0	0
$671$	−28373.3	−1.63240
$672$	0	0
$673$	−28793.4	−1.64919	−0.824593	−	0.565726i	$-0.808596\pi$
$673$	−28793.4	−1.64919	−0.824593	+	0.565726i	$0.808596\pi$
$674$	0	0
$675$	4000.28	0.228105
$676$	0	0
$677$	1852.46	0.105163	0.0525817	−	0.998617i	$-0.483255\pi$
$677$	1852.46	0.105163	0.0525817	+	0.998617i	$0.483255\pi$
$678$	0	0
$679$	−11725.4	−0.662711
$680$	0	0
$681$	−40847.0	−2.29847
$682$	0	0
$683$	8830.51	0.494715	0.247357	−	0.968924i	$-0.420438\pi$
$683$	8830.51	0.494715	0.247357	+	0.968924i	$0.420438\pi$
$684$	0	0
$685$	−7138.19	−0.398155
$686$	0	0
$687$	17556.4	0.974992
$688$	0	0
$689$	2517.50	0.139201
$690$	0	0
$691$	−14371.7	−0.791209	−0.395605	−	0.918421i	$-0.629465\pi$
$691$	−14371.7	−0.791209	−0.395605	+	0.918421i	$0.629465\pi$
$692$	0	0
$693$	18244.1	1.00005
$694$	0	0
$695$	22800.1	1.24440
$696$	0	0
$697$	−22214.2	−1.20721
$698$	0	0
$699$	−26951.5	−1.45837
$700$	0	0
$701$	18349.7	0.988670	0.494335	−	0.869272i	$-0.335412\pi$
$701$	18349.7	0.988670	0.494335	+	0.869272i	$0.335412\pi$
$702$	0	0
$703$	−4358.95	−0.233856
$704$	0	0
$705$	−42652.1	−2.27854
$706$	0	0
$707$	−9235.10	−0.491261
$708$	0	0
$709$	−29030.6	−1.53775	−0.768877	−	0.639396i	$-0.779184\pi$
$709$	−29030.6	−1.53775	−0.768877	+	0.639396i	$0.779184\pi$
$710$	0	0
$711$	47130.4	2.48597
$712$	0	0
$713$	270.865	0.0142272
$714$	0	0
$715$	5492.77	0.287298
$716$	0	0
$717$	349.405	0.0181991
$718$	0	0
$719$	−9586.00	−0.497215	−0.248607	−	0.968604i	$-0.579973\pi$
$719$	−9586.00	−0.497215	−0.248607	+	0.968604i	$0.579973\pi$
$720$	0	0
$721$	−854.881	−0.0441573
$722$	0	0
$723$	55643.0	2.86222
$724$	0	0
$725$	3076.29	0.157587
$726$	0	0
$727$	2317.04	0.118204	0.0591019	−	0.998252i	$-0.481176\pi$
$727$	2317.04	0.118204	0.0591019	+	0.998252i	$0.481176\pi$
$728$	0	0
$729$	9557.77	0.485585
$730$	0	0
$731$	−44031.1	−2.22784
$732$	0	0
$733$	−31945.3	−1.60972	−0.804861	−	0.593463i	$-0.797760\pi$
$733$	−31945.3	−1.60972	−0.804861	+	0.593463i	$0.797760\pi$
$734$	0	0
$735$	29267.5	1.46877
$736$	0	0
$737$	19795.4	0.989381
$738$	0	0
$739$	−34494.8	−1.71707	−0.858533	−	0.512759i	$-0.828623\pi$
$739$	−34494.8	−1.71707	−0.858533	+	0.512759i	$0.828623\pi$
$740$	0	0
$741$	8706.70	0.431645
$742$	0	0
$743$	−12841.5	−0.634063	−0.317032	−	0.948415i	$-0.602686\pi$
$743$	−12841.5	−0.634063	−0.317032	+	0.948415i	$0.602686\pi$
$744$	0	0
$745$	32631.0	1.60471
$746$	0	0
$747$	66029.4	3.23412
$748$	0	0
$749$	−11256.3	−0.549128
$750$	0	0
$751$	7226.58	0.351134	0.175567	−	0.984467i	$-0.443824\pi$
$751$	7226.58	0.351134	0.175567	+	0.984467i	$0.443824\pi$
$752$	0	0
$753$	70900.1	3.43126
$754$	0	0
$755$	27558.0	1.32840
$756$	0	0
$757$	−8855.54	−0.425178	−0.212589	−	0.977142i	$-0.568190\pi$
$757$	−8855.54	−0.425178	−0.212589	+	0.977142i	$0.568190\pi$
$758$	0	0
$759$	1178.37	0.0563532
$760$	0	0
$761$	29732.4	1.41629	0.708147	−	0.706065i	$-0.249531\pi$
$761$	29732.4	1.41629	0.708147	+	0.706065i	$0.249531\pi$
$762$	0	0
$763$	−4468.00	−0.211995
$764$	0	0
$765$	75014.8	3.54531
$766$	0	0
$767$	−6659.40	−0.313503
$768$	0	0
$769$	6712.10	0.314752	0.157376	−	0.987539i	$-0.449697\pi$
$769$	6712.10	0.314752	0.157376	+	0.987539i	$0.449697\pi$
$770$	0	0
$771$	−13717.1	−0.640737
$772$	0	0
$773$	35131.6	1.63466	0.817332	−	0.576167i	$-0.195452\pi$
$773$	35131.6	1.63466	0.817332	+	0.576167i	$0.195452\pi$
$774$	0	0
$775$	−1013.40	−0.0469709
$776$	0	0
$777$	−4272.81	−0.197279
$778$	0	0
$779$	−14021.7	−0.644901
$780$	0	0
$781$	33839.3	1.55041
$782$	0	0
$783$	89808.6	4.09897
$784$	0	0
$785$	39515.1	1.79663
$786$	0	0
$787$	14152.2	0.641007	0.320504	−	0.947247i	$-0.396148\pi$
$787$	14152.2	0.641007	0.320504	+	0.947247i	$0.396148\pi$
$788$	0	0
$789$	−52875.1	−2.38581
$790$	0	0
$791$	11270.9	0.506636
$792$	0	0
$793$	−9292.00	−0.416102
$794$	0	0
$795$	19556.7	0.872460
$796$	0	0
$797$	3018.60	0.134158	0.0670792	−	0.997748i	$-0.478632\pi$
$797$	3018.60	0.134158	0.0670792	+	0.997748i	$0.478632\pi$
$798$	0	0
$799$	47233.1	2.09135
$800$	0	0
$801$	17526.8	0.773130
$802$	0	0
$803$	−1137.42	−0.0499861
$804$	0	0
$805$	−242.877	−0.0106339
$806$	0	0
$807$	−77621.3	−3.38587
$808$	0	0
$809$	−3800.90	−0.165182	−0.0825912	−	0.996584i	$-0.526320\pi$
$809$	−3800.90	−0.165182	−0.0825912	+	0.996584i	$0.526320\pi$
$810$	0	0
$811$	3140.54	0.135979	0.0679896	−	0.997686i	$-0.478342\pi$
$811$	3140.54	0.135979	0.0679896	+	0.997686i	$0.478342\pi$
$812$	0	0
$813$	−49296.1	−2.12655
$814$	0	0
$815$	7767.71	0.333854
$816$	0	0
$817$	−27792.6	−1.19013
$818$	0	0
$819$	5974.77	0.254915
$820$	0	0
$821$	9299.69	0.395325	0.197662	−	0.980270i	$-0.436665\pi$
$821$	9299.69	0.395325	0.197662	+	0.980270i	$0.436665\pi$
$822$	0	0
$823$	−23549.7	−0.997438	−0.498719	−	0.866764i	$-0.666196\pi$
$823$	−23549.7	−0.997438	−0.498719	+	0.866764i	$0.666196\pi$
$824$	0	0
$825$	−4408.69	−0.186050
$826$	0	0
$827$	10237.6	0.430468	0.215234	−	0.976563i	$-0.430949\pi$
$827$	10237.6	0.430468	0.215234	+	0.976563i	$0.430949\pi$
$828$	0	0
$829$	25983.8	1.08861	0.544304	−	0.838888i	$-0.316794\pi$
$829$	25983.8	1.08861	0.544304	+	0.838888i	$0.316794\pi$
$830$	0	0
$831$	−1958.56	−0.0817589
$832$	0	0
$833$	−32410.9	−1.34810
$834$	0	0
$835$	−17133.3	−0.710088
$836$	0	0
$837$	−29585.1	−1.22176
$838$	0	0
$839$	25109.7	1.03323	0.516616	−	0.856217i	$-0.327191\pi$
$839$	25109.7	1.03323	0.516616	+	0.856217i	$0.327191\pi$
$840$	0	0
$841$	44675.3	1.83178
$842$	0	0
$843$	34047.7	1.39106
$844$	0	0
$845$	1798.83	0.0732328
$846$	0	0
$847$	−1785.00	−0.0724123
$848$	0	0
$849$	−2233.10	−0.0902708
$850$	0	0
$851$	−193.201	−0.00778242
$852$	0	0
$853$	5855.00	0.235019	0.117510	−	0.993072i	$-0.462509\pi$
$853$	5855.00	0.235019	0.117510	+	0.993072i	$0.462509\pi$
$854$	0	0
$855$	47349.6	1.89394
$856$	0	0
$857$	−24632.4	−0.981827	−0.490913	−	0.871208i	$-0.663337\pi$
$857$	−24632.4	−0.981827	−0.490913	+	0.871208i	$0.663337\pi$
$858$	0	0
$859$	43384.2	1.72323	0.861613	−	0.507566i	$-0.169455\pi$
$859$	43384.2	1.72323	0.861613	+	0.507566i	$0.169455\pi$
$860$	0	0
$861$	−13744.6	−0.544034
$862$	0	0
$863$	37140.7	1.46499	0.732494	−	0.680774i	$-0.238356\pi$
$863$	37140.7	1.46499	0.732494	+	0.680774i	$0.238356\pi$
$864$	0	0
$865$	−4244.98	−0.166860
$866$	0	0
$867$	−72049.7	−2.82230
$868$	0	0
$869$	−29687.7	−1.15890
$870$	0	0
$871$	6482.82	0.252195
$872$	0	0
$873$	101318.	3.92794
$874$	0	0
$875$	10612.1	0.410004
$876$	0	0
$877$	−29986.8	−1.15460	−0.577298	−	0.816533i	$-0.695893\pi$
$877$	−29986.8	−1.15460	−0.577298	+	0.816533i	$0.695893\pi$
$878$	0	0
$879$	67712.7	2.59829
$880$	0	0
$881$	3632.08	0.138897	0.0694483	−	0.997586i	$-0.477876\pi$
$881$	3632.08	0.138897	0.0694483	+	0.997586i	$0.477876\pi$
$882$	0	0
$883$	−7147.35	−0.272398	−0.136199	−	0.990681i	$-0.543489\pi$
$883$	−7147.35	−0.272398	−0.136199	+	0.990681i	$0.543489\pi$
$884$	0	0
$885$	−51732.2	−1.96493
$886$	0	0
$887$	1703.10	0.0644696	0.0322348	−	0.999480i	$-0.489738\pi$
$887$	1703.10	0.0644696	0.0322348	+	0.999480i	$0.489738\pi$
$888$	0	0
$889$	−9904.36	−0.373658
$890$	0	0
$891$	−61164.3	−2.29975
$892$	0	0
$893$	29813.7	1.11722
$894$	0	0
$895$	−17378.4	−0.649045
$896$	0	0
$897$	385.905	0.0143646
$898$	0	0
$899$	−22751.4	−0.844052
$900$	0	0
$901$	−21657.2	−0.800783
$902$	0	0
$903$	−27243.3	−1.00399
$904$	0	0
$905$	−2049.49	−0.0752788
$906$	0	0
$907$	−20054.7	−0.734185	−0.367093	−	0.930184i	$-0.619647\pi$
$907$	−20054.7	−0.734185	−0.367093	+	0.930184i	$0.619647\pi$
$908$	0	0
$909$	79799.4	2.91175
$910$	0	0
$911$	13635.0	0.495883	0.247941	−	0.968775i	$-0.420246\pi$
$911$	13635.0	0.495883	0.247941	+	0.968775i	$0.420246\pi$
$912$	0	0
$913$	−41592.4	−1.50767
$914$	0	0
$915$	−72183.1	−2.60798
$916$	0	0
$917$	9491.97	0.341824
$918$	0	0
$919$	3829.52	0.137458	0.0687292	−	0.997635i	$-0.478106\pi$
$919$	3829.52	0.137458	0.0687292	+	0.997635i	$0.478106\pi$
$920$	0	0
$921$	−48107.9	−1.72118
$922$	0	0
$923$	11082.1	0.395201
$924$	0	0
$925$	722.832	0.0256936
$926$	0	0
$927$	7386.92	0.261724
$928$	0	0
$929$	23603.4	0.833587	0.416793	−	0.909001i	$-0.363154\pi$
$929$	23603.4	0.833587	0.416793	+	0.909001i	$0.363154\pi$
$930$	0	0
$931$	−20457.8	−0.720170
$932$	0	0
$933$	42456.0	1.48976
$934$	0	0
$935$	−47252.3	−1.65274
$936$	0	0
$937$	13355.3	0.465632	0.232816	−	0.972521i	$-0.425206\pi$
$937$	13355.3	0.465632	0.232816	+	0.972521i	$0.425206\pi$
$938$	0	0
$939$	−21901.5	−0.761158
$940$	0	0
$941$	−25394.2	−0.879731	−0.439866	−	0.898064i	$-0.644974\pi$
$941$	−25394.2	−0.879731	−0.439866	+	0.898064i	$0.644974\pi$
$942$	0	0
$943$	−621.479	−0.0214615
$944$	0	0
$945$	26528.0	0.913181
$946$	0	0
$947$	−14704.3	−0.504569	−0.252284	−	0.967653i	$-0.581182\pi$
$947$	−14704.3	−0.504569	−0.252284	+	0.967653i	$0.581182\pi$
$948$	0	0
$949$	−372.496	−0.0127416
$950$	0	0
$951$	−16114.1	−0.549458
$952$	0	0
$953$	−21157.8	−0.719171	−0.359585	−	0.933112i	$-0.617082\pi$
$953$	−21157.8	−0.719171	−0.359585	+	0.933112i	$0.617082\pi$
$954$	0	0
$955$	12463.0	0.422297
$956$	0	0
$957$	−98977.6	−3.34325
$958$	0	0
$959$	4890.96	0.164690
$960$	0	0
$961$	−22296.1	−0.748419
$962$	0	0
$963$	97264.4	3.25473
$964$	0	0
$965$	10953.4	0.365392
$966$	0	0
$967$	14786.2	0.491718	0.245859	−	0.969306i	$-0.420930\pi$
$967$	14786.2	0.491718	0.245859	+	0.969306i	$0.420930\pi$
$968$	0	0
$969$	−74900.6	−2.48313
$970$	0	0
$971$	−1424.45	−0.0470780	−0.0235390	−	0.999723i	$-0.507493\pi$
$971$	−1424.45	−0.0470780	−0.0235390	+	0.999723i	$0.507493\pi$
$972$	0	0
$973$	−15622.2	−0.514723
$974$	0	0
$975$	−1443.81	−0.0474244
$976$	0	0
$977$	11574.6	0.379021	0.189510	−	0.981879i	$-0.439310\pi$
$977$	11574.6	0.379021	0.189510	+	0.981879i	$0.439310\pi$
$978$	0	0
$979$	−11040.2	−0.360416
$980$	0	0
$981$	38607.4	1.25651
$982$	0	0
$983$	−35095.4	−1.13873	−0.569364	−	0.822086i	$-0.692810\pi$
$983$	−35095.4	−1.13873	−0.569364	+	0.822086i	$0.692810\pi$
$984$	0	0
$985$	47975.6	1.55191
$986$	0	0
$987$	29224.5	0.942478
$988$	0	0
$989$	−1231.85	−0.0396061
$990$	0	0
$991$	−12213.6	−0.391501	−0.195751	−	0.980654i	$-0.562714\pi$
$991$	−12213.6	−0.391501	−0.195751	+	0.980654i	$0.562714\pi$
$992$	0	0
$993$	−13325.9	−0.425867
$994$	0	0
$995$	−37536.5	−1.19597
$996$	0	0
$997$	27951.6	0.887898	0.443949	−	0.896052i	$-0.353577\pi$
$997$	27951.6	0.887898	0.443949	+	0.896052i	$0.353577\pi$
$998$	0	0
$999$	21102.2	0.668313

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.4.a.bf.1.1		5
4.3	odd	2		832.4.a.bg.1.5		5
8.3	odd	2		416.4.a.h.1.1	✓	5
8.5	even	2		416.4.a.i.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.4.a.h.1.1	✓	5	8.3	odd	2
416.4.a.i.1.5	yes	5	8.5	even	2
832.4.a.bf.1.1		5	1.1	even	1	trivial
832.4.a.bg.1.5		5	4.3	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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	832.a (trivial)

\(f(q)\)	\(=\)	\(q-9.48781 q^{3} +10.6440 q^{5} -7.29306 q^{7} +63.0185 q^{9} -39.6958 q^{11} -13.0000 q^{13} -100.988 q^{15} +111.834 q^{17} +70.5902 q^{19} +69.1951 q^{21} +3.12876 q^{23} -11.7058 q^{25} -341.736 q^{27} -262.801 q^{29} +86.5729 q^{31} +376.626 q^{33} -77.6272 q^{35} -61.7501 q^{37} +123.341 q^{39} -198.635 q^{41} -393.717 q^{43} +670.767 q^{45} +422.349 q^{47} -289.811 q^{49} -1061.06 q^{51} -193.654 q^{53} -422.521 q^{55} -669.746 q^{57} +512.261 q^{59} +714.769 q^{61} -459.597 q^{63} -138.372 q^{65} -498.678 q^{67} -29.6850 q^{69} -852.467 q^{71} +28.6535 q^{73} +111.062 q^{75} +289.504 q^{77} +747.882 q^{79} +1540.83 q^{81} +1047.78 q^{83} +1190.36 q^{85} +2493.40 q^{87} +278.121 q^{89} +94.8098 q^{91} -821.387 q^{93} +751.360 q^{95} +1607.75 q^{97} -2501.57 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 11 q^{3} + 5 q^{5} + 21 q^{7} + 44 q^{9} + 18 q^{11} - 65 q^{13} + 39 q^{15} + 123 q^{17} - 142 q^{19} + 27 q^{21} + 92 q^{23} + 194 q^{25} - 449 q^{27} + 206 q^{29} + 436 q^{31} + 302 q^{33} - 817 q^{35}+ \cdots - 1404 q^{99}+O(q^{100}) \) 5 * q - 11 * q^3 + 5 * q^5 + 21 * q^7 + 44 * q^9 + 18 * q^11 - 65 * q^13 + 39 * q^15 + 123 * q^17 - 142 * q^19 + 27 * q^21 + 92 * q^23 + 194 * q^25 - 449 * q^27 + 206 * q^29 + 436 * q^31 + 302 * q^33 - 817 * q^35 + 273 * q^37 + 143 * q^39 - 248 * q^41 - 1093 * q^43 + 74 * q^45 + 905 * q^47 + 672 * q^49 - 1359 * q^51 + 656 * q^53 + 686 * q^55 + 550 * q^57 - 634 * q^59 + 1080 * q^61 + 1374 * q^63 - 65 * q^65 - 1322 * q^67 + 1708 * q^69 + 215 * q^71 + 726 * q^73 - 1428 * q^75 + 1038 * q^77 + 1592 * q^79 + 1373 * q^81 - 300 * q^83 + 2165 * q^85 + 1246 * q^87 + 1786 * q^89 - 273 * q^91 + 2508 * q^93 + 894 * q^95 + 2970 * q^97 - 1404 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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