LMFDB - Embedded newform 1856.4.a.bh.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 164x^{8} + 8446x^{6} - 181180x^{4} + 1701497x^{2} - 5718568 $ x^10 - 164x^8 + 8446x^6 - 181180x^4 + 1701497x^2 - 5718568
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.85699$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85699 q^{3} +11.7382 q^{5} +6.79488 q^{7} -18.8376 q^{9} +O(q^{10})$	$q-2.85699 q^{3} +11.7382 q^{5} +6.79488 q^{7} -18.8376 q^{9} -30.6592 q^{11} +36.3320 q^{13} -33.5359 q^{15} -65.6005 q^{17} -12.4185 q^{19} -19.4129 q^{21} +161.593 q^{23} +12.7852 q^{25} +130.958 q^{27} -29.0000 q^{29} +215.733 q^{31} +87.5929 q^{33} +79.7596 q^{35} -28.1451 q^{37} -103.800 q^{39} -110.999 q^{41} -263.774 q^{43} -221.119 q^{45} +18.3700 q^{47} -296.830 q^{49} +187.420 q^{51} -353.900 q^{53} -359.883 q^{55} +35.4794 q^{57} -380.865 q^{59} -267.631 q^{61} -127.999 q^{63} +426.472 q^{65} -157.876 q^{67} -461.670 q^{69} -287.925 q^{71} -77.3816 q^{73} -36.5273 q^{75} -208.325 q^{77} +1188.88 q^{79} +134.471 q^{81} -145.846 q^{83} -770.031 q^{85} +82.8527 q^{87} +1225.09 q^{89} +246.871 q^{91} -616.346 q^{93} -145.770 q^{95} +36.3732 q^{97} +577.545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 40 q^{5} + 58 q^{9}+O(q^{10}) $ 10 * q - 40 * q^5 + 58 * q^9	$ 10 q - 40 q^{5} + 58 q^{9} - 136 q^{13} + 144 q^{17} - 92 q^{21} + 334 q^{25} - 290 q^{29} + 536 q^{33} - 212 q^{37} + 400 q^{41} - 748 q^{45} + 626 q^{49} - 1304 q^{53} + 408 q^{57} - 1600 q^{61} + 1808 q^{65} - 1916 q^{69} + 1332 q^{73} - 1724 q^{77} + 730 q^{81} - 3484 q^{85} + 2256 q^{89} - 3500 q^{93} + 560 q^{97}+O(q^{100}) $ 10 * q - 40 * q^5 + 58 * q^9 - 136 * q^13 + 144 * q^17 - 92 * q^21 + 334 * q^25 - 290 * q^29 + 536 * q^33 - 212 * q^37 + 400 * q^41 - 748 * q^45 + 626 * q^49 - 1304 * q^53 + 408 * q^57 - 1600 * q^61 + 1808 * q^65 - 1916 * q^69 + 1332 * q^73 - 1724 * q^77 + 730 * q^81 - 3484 * q^85 + 2256 * q^89 - 3500 * q^93 + 560 * 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
$3$	−2.85699	−0.549828	−0.274914	−	0.961469i	$-0.588649\pi$
$3$	−2.85699	−0.549828	−0.274914	+	0.961469i	$0.588649\pi$
$4$	0	0
$5$	11.7382	1.04990	0.524948	−	0.851134i	$-0.324085\pi$
$5$	11.7382	1.04990	0.524948	+	0.851134i	$0.324085\pi$
$6$	0	0
$7$	6.79488	0.366889	0.183444	−	0.983030i	$-0.441275\pi$
$7$	6.79488	0.366889	0.183444	+	0.983030i	$0.441275\pi$
$8$	0	0
$9$	−18.8376	−0.697689
$10$	0	0
$11$	−30.6592	−0.840371	−0.420186	−	0.907438i	$-0.638035\pi$
$11$	−30.6592	−0.840371	−0.420186	+	0.907438i	$0.638035\pi$
$12$	0	0
$13$	36.3320	0.775129	0.387564	−	0.921843i	$-0.373317\pi$
$13$	36.3320	0.775129	0.387564	+	0.921843i	$0.373317\pi$
$14$	0	0
$15$	−33.5359	−0.577262
$16$	0	0
$17$	−65.6005	−0.935909	−0.467954	−	0.883753i	$-0.655009\pi$
$17$	−65.6005	−0.935909	−0.467954	+	0.883753i	$0.655009\pi$
$18$	0	0
$19$	−12.4185	−0.149947	−0.0749734	−	0.997186i	$-0.523887\pi$
$19$	−12.4185	−0.149947	−0.0749734	+	0.997186i	$0.523887\pi$
$20$	0	0
$21$	−19.4129	−0.201726
$22$	0	0
$23$	161.593	1.46498	0.732489	−	0.680779i	$-0.238359\pi$
$23$	161.593	1.46498	0.732489	+	0.680779i	$0.238359\pi$
$24$	0	0
$25$	12.7852	0.102282
$26$	0	0
$27$	130.958	0.933437
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	215.733	1.24990	0.624948	−	0.780667i	$-0.285120\pi$
$31$	215.733	1.24990	0.624948	+	0.780667i	$0.285120\pi$
$32$	0	0
$33$	87.5929	0.462060
$34$	0	0
$35$	79.7596	0.385195
$36$	0	0
$37$	−28.1451	−0.125055	−0.0625273	−	0.998043i	$-0.519916\pi$
$37$	−28.1451	−0.125055	−0.0625273	+	0.998043i	$0.519916\pi$
$38$	0	0
$39$	−103.800	−0.426187
$40$	0	0
$41$	−110.999	−0.422809	−0.211404	−	0.977399i	$-0.567804\pi$
$41$	−110.999	−0.422809	−0.211404	+	0.977399i	$0.567804\pi$
$42$	0	0
$43$	−263.774	−0.935470	−0.467735	−	0.883869i	$-0.654930\pi$
$43$	−263.774	−0.935470	−0.467735	+	0.883869i	$0.654930\pi$
$44$	0	0
$45$	−221.119	−0.732501
$46$	0	0
$47$	18.3700	0.0570116	0.0285058	−	0.999594i	$-0.490925\pi$
$47$	18.3700	0.0570116	0.0285058	+	0.999594i	$0.490925\pi$
$48$	0	0
$49$	−296.830	−0.865393
$50$	0	0
$51$	187.420	0.514589
$52$	0	0
$53$	−353.900	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$53$	−353.900	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$54$	0	0
$55$	−359.883	−0.882302
$56$	0	0
$57$	35.4794	0.0824449
$58$	0	0
$59$	−380.865	−0.840414	−0.420207	−	0.907428i	$-0.638043\pi$
$59$	−380.865	−0.840414	−0.420207	+	0.907428i	$0.638043\pi$
$60$	0	0
$61$	−267.631	−0.561748	−0.280874	−	0.959745i	$-0.590624\pi$
$61$	−267.631	−0.561748	−0.280874	+	0.959745i	$0.590624\pi$
$62$	0	0
$63$	−127.999	−0.255974
$64$	0	0
$65$	426.472	0.813804
$66$	0	0
$67$	−157.876	−0.287874	−0.143937	−	0.989587i	$-0.545976\pi$
$67$	−157.876	−0.287874	−0.143937	+	0.989587i	$0.545976\pi$
$68$	0	0
$69$	−461.670	−0.805486
$70$	0	0
$71$	−287.925	−0.481273	−0.240636	−	0.970615i	$-0.577356\pi$
$71$	−287.925	−0.481273	−0.240636	+	0.970615i	$0.577356\pi$
$72$	0	0
$73$	−77.3816	−0.124066	−0.0620331	−	0.998074i	$-0.519758\pi$
$73$	−77.3816	−0.124066	−0.0620331	+	0.998074i	$0.519758\pi$
$74$	0	0
$75$	−36.5273	−0.0562374
$76$	0	0
$77$	−208.325	−0.308323
$78$	0	0
$79$	1188.88	1.69316	0.846582	−	0.532258i	$-0.178656\pi$
$79$	1188.88	1.69316	0.846582	+	0.532258i	$0.178656\pi$
$80$	0	0
$81$	134.471	0.184459
$82$	0	0
$83$	−145.846	−0.192876	−0.0964378	−	0.995339i	$-0.530745\pi$
$83$	−145.846	−0.192876	−0.0964378	+	0.995339i	$0.530745\pi$
$84$	0	0
$85$	−770.031	−0.982607
$86$	0	0
$87$	82.8527	0.102101
$88$	0	0
$89$	1225.09	1.45909	0.729547	−	0.683930i	$-0.239731\pi$
$89$	1225.09	1.45909	0.729547	+	0.683930i	$0.239731\pi$
$90$	0	0
$91$	246.871	0.284386
$92$	0	0
$93$	−616.346	−0.687227
$94$	0	0
$95$	−145.770	−0.157428
$96$	0	0
$97$	36.3732	0.0380736	0.0190368	−	0.999819i	$-0.493940\pi$
$97$	36.3732	0.0380736	0.0190368	+	0.999819i	$0.493940\pi$
$98$	0	0
$99$	577.545	0.586318
$100$	0	0
$101$	−218.909	−0.215666	−0.107833	−	0.994169i	$-0.534391\pi$
$101$	−218.909	−0.215666	−0.107833	+	0.994169i	$0.534391\pi$
$102$	0	0
$103$	1263.62	1.20882	0.604408	−	0.796675i	$-0.293410\pi$
$103$	1263.62	1.20882	0.604408	+	0.796675i	$0.293410\pi$
$104$	0	0
$105$	−227.872	−0.211791
$106$	0	0
$107$	394.527	0.356452	0.178226	−	0.983990i	$-0.442964\pi$
$107$	394.527	0.356452	0.178226	+	0.983990i	$0.442964\pi$
$108$	0	0
$109$	−1349.05	−1.18546	−0.592732	−	0.805400i	$-0.701951\pi$
$109$	−1349.05	−1.18546	−0.592732	+	0.805400i	$0.701951\pi$
$110$	0	0
$111$	80.4102	0.0687585
$112$	0	0
$113$	−344.781	−0.287029	−0.143515	−	0.989648i	$-0.545840\pi$
$113$	−344.781	−0.287029	−0.143515	+	0.989648i	$0.545840\pi$
$114$	0	0
$115$	1896.81	1.53807
$116$	0	0
$117$	−684.407	−0.540799
$118$	0	0
$119$	−445.747	−0.343374
$120$	0	0
$121$	−391.016	−0.293776
$122$	0	0
$123$	317.124	0.232472
$124$	0	0
$125$	−1317.20	−0.942511
$126$	0	0
$127$	−1219.86	−0.852324	−0.426162	−	0.904647i	$-0.640135\pi$
$127$	−1219.86	−0.852324	−0.426162	+	0.904647i	$0.640135\pi$
$128$	0	0
$129$	753.601	0.514348
$130$	0	0
$131$	1095.65	0.730745	0.365373	−	0.930861i	$-0.380942\pi$
$131$	1095.65	0.730745	0.365373	+	0.930861i	$0.380942\pi$
$132$	0	0
$133$	−84.3819	−0.0550138
$134$	0	0
$135$	1537.21	0.980012
$136$	0	0
$137$	−25.5768	−0.0159502	−0.00797509	−	0.999968i	$-0.502539\pi$
$137$	−25.5768	−0.0159502	−0.00797509	+	0.999968i	$0.502539\pi$
$138$	0	0
$139$	−2603.62	−1.58875	−0.794374	−	0.607429i	$-0.792201\pi$
$139$	−2603.62	−1.58875	−0.794374	+	0.607429i	$0.792201\pi$
$140$	0	0
$141$	−52.4830	−0.0313466
$142$	0	0
$143$	−1113.91	−0.651396
$144$	0	0
$145$	−340.408	−0.194961
$146$	0	0
$147$	848.039	0.475817
$148$	0	0
$149$	644.672	0.354453	0.177227	−	0.984170i	$-0.443287\pi$
$149$	644.672	0.354453	0.177227	+	0.984170i	$0.443287\pi$
$150$	0	0
$151$	−2995.94	−1.61461	−0.807306	−	0.590133i	$-0.799075\pi$
$151$	−2995.94	−1.61461	−0.807306	+	0.590133i	$0.799075\pi$
$152$	0	0
$153$	1235.76	0.652973
$154$	0	0
$155$	2532.31	1.31226
$156$	0	0
$157$	−1577.98	−0.802143	−0.401072	−	0.916047i	$-0.631362\pi$
$157$	−1577.98	−0.802143	−0.401072	+	0.916047i	$0.631362\pi$
$158$	0	0
$159$	1011.09	0.504306
$160$	0	0
$161$	1098.01	0.537484
$162$	0	0
$163$	−2243.56	−1.07809	−0.539047	−	0.842276i	$-0.681216\pi$
$163$	−2243.56	−1.07809	−0.539047	+	0.842276i	$0.681216\pi$
$164$	0	0
$165$	1028.18	0.485115
$166$	0	0
$167$	3393.65	1.57250	0.786252	−	0.617906i	$-0.212019\pi$
$167$	3393.65	1.57250	0.786252	+	0.617906i	$0.212019\pi$
$168$	0	0
$169$	−876.989	−0.399176
$170$	0	0
$171$	233.934	0.104616
$172$	0	0
$173$	−3446.70	−1.51473	−0.757363	−	0.652995i	$-0.773512\pi$
$173$	−3446.70	−1.51473	−0.757363	+	0.652995i	$0.773512\pi$
$174$	0	0
$175$	86.8741	0.0375261
$176$	0	0
$177$	1088.13	0.462083
$178$	0	0
$179$	−2499.65	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$179$	−2499.65	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$180$	0	0
$181$	879.619	0.361224	0.180612	−	0.983554i	$-0.442192\pi$
$181$	879.619	0.361224	0.180612	+	0.983554i	$0.442192\pi$
$182$	0	0
$183$	764.619	0.308865
$184$	0	0
$185$	−330.372	−0.131294
$186$	0	0
$187$	2011.25	0.786511
$188$	0	0
$189$	889.841	0.342468
$190$	0	0
$191$	1124.30	0.425923	0.212961	−	0.977061i	$-0.431689\pi$
$191$	1124.30	0.425923	0.212961	+	0.977061i	$0.431689\pi$
$192$	0	0
$193$	−875.528	−0.326538	−0.163269	−	0.986582i	$-0.552204\pi$
$193$	−875.528	−0.326538	−0.163269	+	0.986582i	$0.552204\pi$
$194$	0	0
$195$	−1218.43	−0.447453
$196$	0	0
$197$	538.701	0.194827	0.0974133	−	0.995244i	$-0.468943\pi$
$197$	538.701	0.194827	0.0974133	+	0.995244i	$0.468943\pi$
$198$	0	0
$199$	−2295.69	−0.817775	−0.408887	−	0.912585i	$-0.634083\pi$
$199$	−2295.69	−0.817775	−0.408887	+	0.912585i	$0.634083\pi$
$200$	0	0
$201$	451.049	0.158281
$202$	0	0
$203$	−197.051	−0.0681296
$204$	0	0
$205$	−1302.93	−0.443906
$206$	0	0
$207$	−3044.03	−1.02210
$208$	0	0
$209$	380.739	0.126011
$210$	0	0
$211$	2969.42	0.968831	0.484415	−	0.874838i	$-0.339032\pi$
$211$	2969.42	0.968831	0.484415	+	0.874838i	$0.339032\pi$
$212$	0	0
$213$	822.598	0.264617
$214$	0	0
$215$	−3096.24	−0.982147
$216$	0	0
$217$	1465.88	0.458573
$218$	0	0
$219$	221.078	0.0682151
$220$	0	0
$221$	−2383.39	−0.725450
$222$	0	0
$223$	−2840.08	−0.852851	−0.426426	−	0.904523i	$-0.640227\pi$
$223$	−2840.08	−0.852851	−0.426426	+	0.904523i	$0.640227\pi$
$224$	0	0
$225$	−240.843	−0.0713609
$226$	0	0
$227$	−4247.54	−1.24193	−0.620967	−	0.783836i	$-0.713260\pi$
$227$	−4247.54	−1.24193	−0.620967	+	0.783836i	$0.713260\pi$
$228$	0	0
$229$	3859.91	1.11384	0.556922	−	0.830565i	$-0.311982\pi$
$229$	3859.91	1.11384	0.556922	+	0.830565i	$0.311982\pi$
$230$	0	0
$231$	595.183	0.169525
$232$	0	0
$233$	−325.762	−0.0915939	−0.0457970	−	0.998951i	$-0.514583\pi$
$233$	−325.762	−0.0915939	−0.0457970	+	0.998951i	$0.514583\pi$
$234$	0	0
$235$	215.631	0.0598562
$236$	0	0
$237$	−3396.63	−0.930950
$238$	0	0
$239$	4495.69	1.21675	0.608373	−	0.793651i	$-0.291823\pi$
$239$	4495.69	1.21675	0.608373	+	0.793651i	$0.291823\pi$
$240$	0	0
$241$	−3949.64	−1.05568	−0.527839	−	0.849344i	$-0.676998\pi$
$241$	−3949.64	−1.05568	−0.527839	+	0.849344i	$0.676998\pi$
$242$	0	0
$243$	−3920.04	−1.03486
$244$	0	0
$245$	−3484.24	−0.908572
$246$	0	0
$247$	−451.187	−0.116228
$248$	0	0
$249$	416.681	0.106048
$250$	0	0
$251$	1479.36	0.372016	0.186008	−	0.982548i	$-0.440445\pi$
$251$	1479.36	0.372016	0.186008	+	0.982548i	$0.440445\pi$
$252$	0	0
$253$	−4954.31	−1.23113
$254$	0	0
$255$	2199.97	0.540265
$256$	0	0
$257$	4046.13	0.982064	0.491032	−	0.871141i	$-0.336620\pi$
$257$	4046.13	0.982064	0.491032	+	0.871141i	$0.336620\pi$
$258$	0	0
$259$	−191.242	−0.0458811
$260$	0	0
$261$	546.291	0.129558
$262$	0	0
$263$	−3003.05	−0.704092	−0.352046	−	0.935983i	$-0.614514\pi$
$263$	−3003.05	−0.704092	−0.352046	+	0.935983i	$0.614514\pi$
$264$	0	0
$265$	−4154.15	−0.962971
$266$	0	0
$267$	−3500.07	−0.802251
$268$	0	0
$269$	−5158.95	−1.16932	−0.584660	−	0.811279i	$-0.698772\pi$
$269$	−5158.95	−1.16932	−0.584660	+	0.811279i	$0.698772\pi$
$270$	0	0
$271$	7094.89	1.59035	0.795173	−	0.606382i	$-0.207380\pi$
$271$	7094.89	1.59035	0.795173	+	0.606382i	$0.207380\pi$
$272$	0	0
$273$	−705.309	−0.156363
$274$	0	0
$275$	−391.984	−0.0859547
$276$	0	0
$277$	−2798.04	−0.606924	−0.303462	−	0.952843i	$-0.598143\pi$
$277$	−2798.04	−0.606924	−0.303462	+	0.952843i	$0.598143\pi$
$278$	0	0
$279$	−4063.89	−0.872038
$280$	0	0
$281$	−6730.11	−1.42877	−0.714385	−	0.699753i	$-0.753294\pi$
$281$	−6730.11	−1.42877	−0.714385	+	0.699753i	$0.753294\pi$
$282$	0	0
$283$	−3076.66	−0.646249	−0.323125	−	0.946356i	$-0.604733\pi$
$283$	−3076.66	−0.646249	−0.323125	+	0.946356i	$0.604733\pi$
$284$	0	0
$285$	416.464	0.0865586
$286$	0	0
$287$	−754.226	−0.155124
$288$	0	0
$289$	−609.580	−0.124075
$290$	0	0
$291$	−103.918	−0.0209339
$292$	0	0
$293$	−5418.27	−1.08034	−0.540168	−	0.841557i	$-0.681639\pi$
$293$	−5418.27	−1.08034	−0.540168	+	0.841557i	$0.681639\pi$
$294$	0	0
$295$	−4470.67	−0.882347
$296$	0	0
$297$	−4015.05	−0.784434
$298$	0	0
$299$	5870.99	1.13555
$300$	0	0
$301$	−1792.32	−0.343214
$302$	0	0
$303$	625.421	0.118579
$304$	0	0
$305$	−3141.50	−0.589777
$306$	0	0
$307$	−2574.55	−0.478623	−0.239311	−	0.970943i	$-0.576922\pi$
$307$	−2574.55	−0.478623	−0.239311	+	0.970943i	$0.576922\pi$
$308$	0	0
$309$	−3610.14	−0.664641
$310$	0	0
$311$	2245.78	0.409475	0.204738	−	0.978817i	$-0.434366\pi$
$311$	2245.78	0.409475	0.204738	+	0.978817i	$0.434366\pi$
$312$	0	0
$313$	−8667.66	−1.56526	−0.782628	−	0.622490i	$-0.786121\pi$
$313$	−8667.66	−1.56526	−0.782628	+	0.622490i	$0.786121\pi$
$314$	0	0
$315$	−1502.48	−0.268747
$316$	0	0
$317$	−588.306	−0.104235	−0.0521176	−	0.998641i	$-0.516597\pi$
$317$	−588.306	−0.104235	−0.0521176	+	0.998641i	$0.516597\pi$
$318$	0	0
$319$	889.116	0.156053
$320$	0	0
$321$	−1127.16	−0.195987
$322$	0	0
$323$	814.656	0.140336
$324$	0	0
$325$	464.512	0.0792816
$326$	0	0
$327$	3854.23	0.651802
$328$	0	0
$329$	124.822	0.0209169
$330$	0	0
$331$	−7999.54	−1.32838	−0.664191	−	0.747563i	$-0.731224\pi$
$331$	−7999.54	−1.32838	−0.664191	+	0.747563i	$0.731224\pi$
$332$	0	0
$333$	530.186	0.0872492
$334$	0	0
$335$	−1853.18	−0.302238
$336$	0	0
$337$	−4123.38	−0.666513	−0.333256	−	0.942836i	$-0.608147\pi$
$337$	−4123.38	−0.666513	−0.333256	+	0.942836i	$0.608147\pi$
$338$	0	0
$339$	985.037	0.157817
$340$	0	0
$341$	−6614.18	−1.05038
$342$	0	0
$343$	−4347.56	−0.684392
$344$	0	0
$345$	−5419.17	−0.845677
$346$	0	0
$347$	−570.393	−0.0882429	−0.0441215	−	0.999026i	$-0.514049\pi$
$347$	−570.393	−0.0882429	−0.0441215	+	0.999026i	$0.514049\pi$
$348$	0	0
$349$	2565.75	0.393528	0.196764	−	0.980451i	$-0.436957\pi$
$349$	2565.75	0.393528	0.196764	+	0.980451i	$0.436957\pi$
$350$	0	0
$351$	4757.95	0.723534
$352$	0	0
$353$	−7876.07	−1.18754	−0.593769	−	0.804636i	$-0.702361\pi$
$353$	−7876.07	−1.18754	−0.593769	+	0.804636i	$0.702361\pi$
$354$	0	0
$355$	−3379.72	−0.505286
$356$	0	0
$357$	1273.50	0.188797
$358$	0	0
$359$	909.285	0.133677	0.0668387	−	0.997764i	$-0.478709\pi$
$359$	909.285	0.133677	0.0668387	+	0.997764i	$0.478709\pi$
$360$	0	0
$361$	−6704.78	−0.977516
$362$	0	0
$363$	1117.13	0.161526
$364$	0	0
$365$	−908.320	−0.130257
$366$	0	0
$367$	1159.74	0.164954	0.0824768	−	0.996593i	$-0.473717\pi$
$367$	1159.74	0.164954	0.0824768	+	0.996593i	$0.473717\pi$
$368$	0	0
$369$	2090.96	0.294989
$370$	0	0
$371$	−2404.71	−0.336513
$372$	0	0
$373$	10726.1	1.48894	0.744470	−	0.667656i	$-0.232702\pi$
$373$	10726.1	1.48894	0.744470	+	0.667656i	$0.232702\pi$
$374$	0	0
$375$	3763.22	0.518219
$376$	0	0
$377$	−1053.63	−0.143938
$378$	0	0
$379$	9963.98	1.35044	0.675218	−	0.737618i	$-0.264050\pi$
$379$	9963.98	1.35044	0.675218	+	0.737618i	$0.264050\pi$
$380$	0	0
$381$	3485.13	0.468632
$382$	0	0
$383$	467.860	0.0624191	0.0312096	−	0.999513i	$-0.490064\pi$
$383$	467.860	0.0624191	0.0312096	+	0.999513i	$0.490064\pi$
$384$	0	0
$385$	−2445.36	−0.323707
$386$	0	0
$387$	4968.88	0.652668
$388$	0	0
$389$	6058.21	0.789623	0.394812	−	0.918762i	$-0.370810\pi$
$389$	6058.21	0.789623	0.394812	+	0.918762i	$0.370810\pi$
$390$	0	0
$391$	−10600.6	−1.37109
$392$	0	0
$393$	−3130.27	−0.401784
$394$	0	0
$395$	13955.4	1.77765
$396$	0	0
$397$	4794.24	0.606086	0.303043	−	0.952977i	$-0.401997\pi$
$397$	4794.24	0.606086	0.303043	+	0.952977i	$0.401997\pi$
$398$	0	0
$399$	241.078	0.0302481
$400$	0	0
$401$	−11956.6	−1.48899	−0.744496	−	0.667627i	$-0.767310\pi$
$401$	−11956.6	−1.48899	−0.744496	+	0.667627i	$0.767310\pi$
$402$	0	0
$403$	7837.99	0.968829
$404$	0	0
$405$	1578.44	0.193663
$406$	0	0
$407$	862.904	0.105092
$408$	0	0
$409$	4230.52	0.511456	0.255728	−	0.966749i	$-0.417685\pi$
$409$	4230.52	0.511456	0.255728	+	0.966749i	$0.417685\pi$
$410$	0	0
$411$	73.0727	0.00876986
$412$	0	0
$413$	−2587.93	−0.308339
$414$	0	0
$415$	−1711.97	−0.202499
$416$	0	0
$417$	7438.51	0.873538
$418$	0	0
$419$	206.537	0.0240812	0.0120406	−	0.999928i	$-0.496167\pi$
$419$	206.537	0.0240812	0.0120406	+	0.999928i	$0.496167\pi$
$420$	0	0
$421$	10287.5	1.19094	0.595468	−	0.803379i	$-0.296967\pi$
$421$	10287.5	1.19094	0.595468	+	0.803379i	$0.296967\pi$
$422$	0	0
$423$	−346.047	−0.0397764
$424$	0	0
$425$	−838.717	−0.0957265
$426$	0	0
$427$	−1818.52	−0.206099
$428$	0	0
$429$	3182.42	0.358156
$430$	0	0
$431$	−11272.6	−1.25981	−0.629907	−	0.776671i	$-0.716907\pi$
$431$	−11272.6	−1.25981	−0.629907	+	0.776671i	$0.716907\pi$
$432$	0	0
$433$	−6976.32	−0.774274	−0.387137	−	0.922022i	$-0.626536\pi$
$433$	−6976.32	−0.774274	−0.387137	+	0.922022i	$0.626536\pi$
$434$	0	0
$435$	972.541	0.107195
$436$	0	0
$437$	−2006.74	−0.219669
$438$	0	0
$439$	7977.75	0.867328	0.433664	−	0.901075i	$-0.357220\pi$
$439$	7977.75	0.867328	0.433664	+	0.901075i	$0.357220\pi$
$440$	0	0
$441$	5591.56	0.603775
$442$	0	0
$443$	4387.79	0.470587	0.235294	−	0.971924i	$-0.424395\pi$
$443$	4387.79	0.470587	0.235294	+	0.971924i	$0.424395\pi$
$444$	0	0
$445$	14380.4	1.53190
$446$	0	0
$447$	−1841.82	−0.194888
$448$	0	0
$449$	−5369.61	−0.564382	−0.282191	−	0.959358i	$-0.591061\pi$
$449$	−5369.61	−0.564382	−0.282191	+	0.959358i	$0.591061\pi$
$450$	0	0
$451$	3403.14	0.355316
$452$	0	0
$453$	8559.38	0.887759
$454$	0	0
$455$	2897.82	0.298576
$456$	0	0
$457$	−7730.75	−0.791312	−0.395656	−	0.918399i	$-0.629483\pi$
$457$	−7730.75	−0.791312	−0.395656	+	0.918399i	$0.629483\pi$
$458$	0	0
$459$	−8590.88	−0.873612
$460$	0	0
$461$	−3737.00	−0.377547	−0.188774	−	0.982021i	$-0.560451\pi$
$461$	−3737.00	−0.377547	−0.188774	+	0.982021i	$0.560451\pi$
$462$	0	0
$463$	−7801.71	−0.783102	−0.391551	−	0.920156i	$-0.628061\pi$
$463$	−7801.71	−0.783102	−0.391551	+	0.920156i	$0.628061\pi$
$464$	0	0
$465$	−7234.80	−0.721517
$466$	0	0
$467$	−16998.7	−1.68438	−0.842190	−	0.539181i	$-0.818734\pi$
$467$	−16998.7	−1.68438	−0.842190	+	0.539181i	$0.818734\pi$
$468$	0	0
$469$	−1072.75	−0.105618
$470$	0	0
$471$	4508.27	0.441041
$472$	0	0
$473$	8087.10	0.786142
$474$	0	0
$475$	−158.773	−0.0153368
$476$	0	0
$477$	6666.63	0.639925
$478$	0	0
$479$	16768.5	1.59953	0.799763	−	0.600316i	$-0.204959\pi$
$479$	16768.5	1.59953	0.799763	+	0.600316i	$0.204959\pi$
$480$	0	0
$481$	−1022.57	−0.0969334
$482$	0	0
$483$	−3136.99	−0.295524
$484$	0	0
$485$	426.955	0.0399733
$486$	0	0
$487$	−14146.1	−1.31627	−0.658133	−	0.752901i	$-0.728654\pi$
$487$	−14146.1	−1.31627	−0.658133	+	0.752901i	$0.728654\pi$
$488$	0	0
$489$	6409.84	0.592767
$490$	0	0
$491$	−7062.57	−0.649143	−0.324572	−	0.945861i	$-0.605220\pi$
$491$	−7062.57	−0.649143	−0.324572	+	0.945861i	$0.605220\pi$
$492$	0	0
$493$	1902.41	0.173794
$494$	0	0
$495$	6779.34	0.615573
$496$	0	0
$497$	−1956.41	−0.176574
$498$	0	0
$499$	−9407.60	−0.843972	−0.421986	−	0.906602i	$-0.638667\pi$
$499$	−9407.60	−0.843972	−0.421986	+	0.906602i	$0.638667\pi$
$500$	0	0
$501$	−9695.62	−0.864607
$502$	0	0
$503$	1470.30	0.130333	0.0651667	−	0.997874i	$-0.479242\pi$
$503$	1470.30	0.130333	0.0651667	+	0.997874i	$0.479242\pi$
$504$	0	0
$505$	−2569.60	−0.226427
$506$	0	0
$507$	2505.55	0.219478
$508$	0	0
$509$	−11508.2	−1.00215	−0.501075	−	0.865404i	$-0.667062\pi$
$509$	−11508.2	−1.00215	−0.501075	+	0.865404i	$0.667062\pi$
$510$	0	0
$511$	−525.798	−0.0455185
$512$	0	0
$513$	−1626.29	−0.139966
$514$	0	0
$515$	14832.6	1.26913
$516$	0	0
$517$	−563.210	−0.0479109
$518$	0	0
$519$	9847.18	0.832838
$520$	0	0
$521$	16270.3	1.36817	0.684084	−	0.729403i	$-0.260202\pi$
$521$	16270.3	1.36817	0.684084	+	0.729403i	$0.260202\pi$
$522$	0	0
$523$	12281.6	1.02684	0.513421	−	0.858137i	$-0.328378\pi$
$523$	12281.6	1.02684	0.513421	+	0.858137i	$0.328378\pi$
$524$	0	0
$525$	−248.198	−0.0206329
$526$	0	0
$527$	−14152.2	−1.16979
$528$	0	0
$529$	13945.3	1.14616
$530$	0	0
$531$	7174.59	0.586348
$532$	0	0
$533$	−4032.82	−0.327731
$534$	0	0
$535$	4631.03	0.374237
$536$	0	0
$537$	7141.48	0.573887
$538$	0	0
$539$	9100.55	0.727251
$540$	0	0
$541$	−7760.81	−0.616753	−0.308377	−	0.951264i	$-0.599786\pi$
$541$	−7760.81	−0.616753	−0.308377	+	0.951264i	$0.599786\pi$
$542$	0	0
$543$	−2513.06	−0.198611
$544$	0	0
$545$	−15835.4	−1.24461
$546$	0	0
$547$	−16834.1	−1.31586	−0.657929	−	0.753080i	$-0.728567\pi$
$547$	−16834.1	−1.31586	−0.657929	+	0.753080i	$0.728567\pi$
$548$	0	0
$549$	5041.53	0.391925
$550$	0	0
$551$	360.135	0.0278444
$552$	0	0
$553$	8078.33	0.621203
$554$	0	0
$555$	943.870	0.0721893
$556$	0	0
$557$	11615.1	0.883571	0.441786	−	0.897121i	$-0.354345\pi$
$557$	11615.1	0.883571	0.441786	+	0.897121i	$0.354345\pi$
$558$	0	0
$559$	−9583.44	−0.725110
$560$	0	0
$561$	−5746.14	−0.432446
$562$	0	0
$563$	−4729.90	−0.354070	−0.177035	−	0.984205i	$-0.556651\pi$
$563$	−4729.90	−0.354070	−0.177035	+	0.984205i	$0.556651\pi$
$564$	0	0
$565$	−4047.11	−0.301351
$566$	0	0
$567$	913.712	0.0676760
$568$	0	0
$569$	2613.98	0.192590	0.0962950	−	0.995353i	$-0.469301\pi$
$569$	2613.98	0.192590	0.0962950	+	0.995353i	$0.469301\pi$
$570$	0	0
$571$	−24.3628	−0.00178556	−0.000892778	−	1.00000i	$-0.500284\pi$
$571$	−24.3628	−0.00178556	−0.000892778		1.00000i	$0.500284\pi$
$572$	0	0
$573$	−3212.10	−0.234184
$574$	0	0
$575$	2066.01	0.149841
$576$	0	0
$577$	12631.3	0.911345	0.455673	−	0.890147i	$-0.349399\pi$
$577$	12631.3	0.911345	0.455673	+	0.890147i	$0.349399\pi$
$578$	0	0
$579$	2501.38	0.179540
$580$	0	0
$581$	−991.006	−0.0707639
$582$	0	0
$583$	10850.3	0.770794
$584$	0	0
$585$	−8033.70	−0.567783
$586$	0	0
$587$	1344.30	0.0945231	0.0472616	−	0.998883i	$-0.484951\pi$
$587$	1344.30	0.0945231	0.0472616	+	0.998883i	$0.484951\pi$
$588$	0	0
$589$	−2679.07	−0.187418
$590$	0	0
$591$	−1539.06	−0.107121
$592$	0	0
$593$	−12920.2	−0.894722	−0.447361	−	0.894354i	$-0.647636\pi$
$593$	−12920.2	−0.894722	−0.447361	+	0.894354i	$0.647636\pi$
$594$	0	0
$595$	−5232.27	−0.360508
$596$	0	0
$597$	6558.77	0.449636
$598$	0	0
$599$	−12895.9	−0.879654	−0.439827	−	0.898082i	$-0.644960\pi$
$599$	−12895.9	−0.879654	−0.439827	+	0.898082i	$0.644960\pi$
$600$	0	0
$601$	19327.1	1.31176	0.655881	−	0.754864i	$-0.272297\pi$
$601$	19327.1	1.31176	0.655881	+	0.754864i	$0.272297\pi$
$602$	0	0
$603$	2974.00	0.200847
$604$	0	0
$605$	−4589.82	−0.308435
$606$	0	0
$607$	12597.5	0.842370	0.421185	−	0.906975i	$-0.361614\pi$
$607$	12597.5	0.842370	0.421185	+	0.906975i	$0.361614\pi$
$608$	0	0
$609$	562.974	0.0374595
$610$	0	0
$611$	667.419	0.0441913
$612$	0	0
$613$	7122.68	0.469302	0.234651	−	0.972080i	$-0.424605\pi$
$613$	7122.68	0.469302	0.234651	+	0.972080i	$0.424605\pi$
$614$	0	0
$615$	3722.46	0.244072
$616$	0	0
$617$	20442.7	1.33386	0.666929	−	0.745121i	$-0.267608\pi$
$617$	20442.7	1.33386	0.666929	+	0.745121i	$0.267608\pi$
$618$	0	0
$619$	3163.39	0.205408	0.102704	−	0.994712i	$-0.467251\pi$
$619$	3163.39	0.205408	0.102704	+	0.994712i	$0.467251\pi$
$620$	0	0
$621$	21161.8	1.36746
$622$	0	0
$623$	8324.35	0.535326
$624$	0	0
$625$	−17059.7	−1.09182
$626$	0	0
$627$	−1087.77	−0.0692843
$628$	0	0
$629$	1846.33	0.117040
$630$	0	0
$631$	18156.0	1.14545	0.572725	−	0.819748i	$-0.305886\pi$
$631$	18156.0	1.14545	0.572725	+	0.819748i	$0.305886\pi$
$632$	0	0
$633$	−8483.60	−0.532690
$634$	0	0
$635$	−14319.0	−0.894852
$636$	0	0
$637$	−10784.4	−0.670790
$638$	0	0
$639$	5423.81	0.335779
$640$	0	0
$641$	−7470.13	−0.460300	−0.230150	−	0.973155i	$-0.573922\pi$
$641$	−7470.13	−0.460300	−0.230150	+	0.973155i	$0.573922\pi$
$642$	0	0
$643$	21085.3	1.29319	0.646597	−	0.762832i	$-0.276192\pi$
$643$	21085.3	1.29319	0.646597	+	0.762832i	$0.276192\pi$
$644$	0	0
$645$	8845.92	0.540012
$646$	0	0
$647$	−15537.4	−0.944109	−0.472054	−	0.881569i	$-0.656487\pi$
$647$	−15537.4	−0.944109	−0.472054	+	0.881569i	$0.656487\pi$
$648$	0	0
$649$	11677.0	0.706260
$650$	0	0
$651$	−4188.00	−0.252136
$652$	0	0
$653$	−8849.63	−0.530341	−0.265171	−	0.964201i	$-0.585428\pi$
$653$	−8849.63	−0.530341	−0.265171	+	0.964201i	$0.585428\pi$
$654$	0	0
$655$	12861.0	0.767207
$656$	0	0
$657$	1457.68	0.0865596
$658$	0	0
$659$	10071.2	0.595323	0.297662	−	0.954671i	$-0.403793\pi$
$659$	10071.2	0.595323	0.297662	+	0.954671i	$0.403793\pi$
$660$	0	0
$661$	15736.3	0.925978	0.462989	−	0.886364i	$-0.346777\pi$
$661$	15736.3	0.925978	0.462989	+	0.886364i	$0.346777\pi$
$662$	0	0
$663$	6809.33	0.398873
$664$	0	0
$665$	−990.491	−0.0577588
$666$	0	0
$667$	−4686.20	−0.272040
$668$	0	0
$669$	8114.08	0.468922
$670$	0	0
$671$	8205.34	0.472077
$672$	0	0
$673$	850.654	0.0487226	0.0243613	−	0.999703i	$-0.492245\pi$
$673$	850.654	0.0487226	0.0243613	+	0.999703i	$0.492245\pi$
$674$	0	0
$675$	1674.32	0.0954737
$676$	0	0
$677$	23022.7	1.30699	0.653497	−	0.756929i	$-0.273301\pi$
$677$	23022.7	1.30699	0.653497	+	0.756929i	$0.273301\pi$
$678$	0	0
$679$	247.151	0.0139688
$680$	0	0
$681$	12135.2	0.682851
$682$	0	0
$683$	−26319.0	−1.47448	−0.737239	−	0.675632i	$-0.763871\pi$
$683$	−26319.0	−1.47448	−0.737239	+	0.675632i	$0.763871\pi$
$684$	0	0
$685$	−300.226	−0.0167460
$686$	0	0
$687$	−11027.7	−0.612423
$688$	0	0
$689$	−12857.9	−0.710953
$690$	0	0
$691$	29831.0	1.64230	0.821148	−	0.570715i	$-0.193334\pi$
$691$	29831.0	1.64230	0.821148	+	0.570715i	$0.193334\pi$
$692$	0	0
$693$	3924.35	0.215114
$694$	0	0
$695$	−30561.8	−1.66802
$696$	0	0
$697$	7281.60	0.395711
$698$	0	0
$699$	930.699	0.0503609
$700$	0	0
$701$	−23666.6	−1.27514	−0.637570	−	0.770392i	$-0.720060\pi$
$701$	−23666.6	−1.27514	−0.637570	+	0.770392i	$0.720060\pi$
$702$	0	0
$703$	349.518	0.0187515
$704$	0	0
$705$	−616.056	−0.0329106
$706$	0	0
$707$	−1487.46	−0.0791254
$708$	0	0
$709$	5838.89	0.309287	0.154643	−	0.987970i	$-0.450577\pi$
$709$	5838.89	0.309287	0.154643	+	0.987970i	$0.450577\pi$
$710$	0	0
$711$	−22395.7	−1.18130
$712$	0	0
$713$	34860.9	1.83107
$714$	0	0
$715$	−13075.3	−0.683898
$716$	0	0
$717$	−12844.2	−0.669001
$718$	0	0
$719$	585.151	0.0303511	0.0151755	−	0.999885i	$-0.495169\pi$
$719$	585.151	0.0303511	0.0151755	+	0.999885i	$0.495169\pi$
$720$	0	0
$721$	8586.13	0.443501
$722$	0	0
$723$	11284.1	0.580442
$724$	0	0
$725$	−370.772	−0.0189933
$726$	0	0
$727$	15779.0	0.804966	0.402483	−	0.915428i	$-0.368147\pi$
$727$	15779.0	0.804966	0.402483	+	0.915428i	$0.368147\pi$
$728$	0	0
$729$	7568.80	0.384535
$730$	0	0
$731$	17303.7	0.875515
$732$	0	0
$733$	−12985.8	−0.654353	−0.327176	−	0.944963i	$-0.606097\pi$
$733$	−12985.8	−0.654353	−0.327176	+	0.944963i	$0.606097\pi$
$734$	0	0
$735$	9954.45	0.499559
$736$	0	0
$737$	4840.33	0.241921
$738$	0	0
$739$	−25722.9	−1.28042	−0.640211	−	0.768199i	$-0.721153\pi$
$739$	−25722.9	−1.28042	−0.640211	+	0.768199i	$0.721153\pi$
$740$	0	0
$741$	1289.04	0.0639054
$742$	0	0
$743$	22473.0	1.10963	0.554815	−	0.831974i	$-0.312789\pi$
$743$	22473.0	1.10963	0.554815	+	0.831974i	$0.312789\pi$
$744$	0	0
$745$	7567.28	0.372139
$746$	0	0
$747$	2747.39	0.134567
$748$	0	0
$749$	2680.76	0.130778
$750$	0	0
$751$	−1083.85	−0.0526636	−0.0263318	−	0.999653i	$-0.508383\pi$
$751$	−1083.85	−0.0526636	−0.0263318	+	0.999653i	$0.508383\pi$
$752$	0	0
$753$	−4226.50	−0.204545
$754$	0	0
$755$	−35167.0	−1.69517
$756$	0	0
$757$	−31939.2	−1.53349	−0.766744	−	0.641953i	$-0.778124\pi$
$757$	−31939.2	−1.53349	−0.766744	+	0.641953i	$0.778124\pi$
$758$	0	0
$759$	14154.4	0.676907
$760$	0	0
$761$	37214.4	1.77270	0.886348	−	0.463020i	$-0.153234\pi$
$761$	37214.4	1.77270	0.886348	+	0.463020i	$0.153234\pi$
$762$	0	0
$763$	−9166.63	−0.434934
$764$	0	0
$765$	14505.5	0.685554
$766$	0	0
$767$	−13837.6	−0.651429
$768$	0	0
$769$	33666.3	1.57872	0.789361	−	0.613930i	$-0.210412\pi$
$769$	33666.3	1.57872	0.789361	+	0.613930i	$0.210412\pi$
$770$	0	0
$771$	−11559.7	−0.539967
$772$	0	0
$773$	−7806.37	−0.363229	−0.181614	−	0.983370i	$-0.558132\pi$
$773$	−7806.37	−0.363229	−0.181614	+	0.983370i	$0.558132\pi$
$774$	0	0
$775$	2758.19	0.127842
$776$	0	0
$777$	546.377	0.0252267
$778$	0	0
$779$	1378.44	0.0633988
$780$	0	0
$781$	8827.53	0.404448
$782$	0	0
$783$	−3797.77	−0.173335
$784$	0	0
$785$	−18522.6	−0.842167
$786$	0	0
$787$	16899.8	0.765455	0.382727	−	0.923861i	$-0.374985\pi$
$787$	16899.8	0.765455	0.382727	+	0.923861i	$0.374985\pi$
$788$	0	0
$789$	8579.69	0.387129
$790$	0	0
$791$	−2342.75	−0.105308
$792$	0	0
$793$	−9723.56	−0.435427
$794$	0	0
$795$	11868.4	0.529469
$796$	0	0
$797$	−2561.48	−0.113842	−0.0569212	−	0.998379i	$-0.518128\pi$
$797$	−2561.48	−0.113842	−0.0569212	+	0.998379i	$0.518128\pi$
$798$	0	0
$799$	−1205.08	−0.0533576
$800$	0	0
$801$	−23077.8	−1.01799
$802$	0	0
$803$	2372.45	0.104262
$804$	0	0
$805$	12888.6	0.564303
$806$	0	0
$807$	14739.1	0.642925
$808$	0	0
$809$	28309.8	1.23031	0.615154	−	0.788407i	$-0.289094\pi$
$809$	28309.8	1.23031	0.615154	+	0.788407i	$0.289094\pi$
$810$	0	0
$811$	−29448.8	−1.27508	−0.637538	−	0.770419i	$-0.720047\pi$
$811$	−29448.8	−1.27508	−0.637538	+	0.770419i	$0.720047\pi$
$812$	0	0
$813$	−20270.0	−0.874417
$814$	0	0
$815$	−26335.4	−1.13189
$816$	0	0
$817$	3275.67	0.140271
$818$	0	0
$819$	−4650.46	−0.198413
$820$	0	0
$821$	−6589.36	−0.280110	−0.140055	−	0.990144i	$-0.544728\pi$
$821$	−6589.36	−0.280110	−0.140055	+	0.990144i	$0.544728\pi$
$822$	0	0
$823$	14021.1	0.593857	0.296929	−	0.954900i	$-0.404038\pi$
$823$	14021.1	0.593857	0.296929	+	0.954900i	$0.404038\pi$
$824$	0	0
$825$	1119.90	0.0472603
$826$	0	0
$827$	29499.5	1.24038	0.620192	−	0.784450i	$-0.287055\pi$
$827$	29499.5	1.24038	0.620192	+	0.784450i	$0.287055\pi$
$828$	0	0
$829$	−21997.8	−0.921612	−0.460806	−	0.887501i	$-0.652440\pi$
$829$	−21997.8	−0.921612	−0.460806	+	0.887501i	$0.652440\pi$
$830$	0	0
$831$	7993.98	0.333704
$832$	0	0
$833$	19472.2	0.809928
$834$	0	0
$835$	39835.3	1.65097
$836$	0	0
$837$	28251.8	1.16670
$838$	0	0
$839$	−18823.8	−0.774576	−0.387288	−	0.921959i	$-0.626588\pi$
$839$	−18823.8	−0.774576	−0.387288	+	0.921959i	$0.626588\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	19227.9	0.785578
$844$	0	0
$845$	−10294.3	−0.419093
$846$	0	0
$847$	−2656.91	−0.107783
$848$	0	0
$849$	8789.99	0.355326
$850$	0	0
$851$	−4548.05	−0.183202
$852$	0	0
$853$	−7137.62	−0.286503	−0.143252	−	0.989686i	$-0.545756\pi$
$853$	−7137.62	−0.286503	−0.143252	+	0.989686i	$0.545756\pi$
$854$	0	0
$855$	2745.96	0.109836
$856$	0	0
$857$	33624.1	1.34023	0.670116	−	0.742257i	$-0.266245\pi$
$857$	33624.1	1.34023	0.670116	+	0.742257i	$0.266245\pi$
$858$	0	0
$859$	−30535.7	−1.21288	−0.606441	−	0.795128i	$-0.707403\pi$
$859$	−30535.7	−1.21288	−0.606441	+	0.795128i	$0.707403\pi$
$860$	0	0
$861$	2154.82	0.0852915
$862$	0	0
$863$	−32997.4	−1.30156	−0.650778	−	0.759268i	$-0.725557\pi$
$863$	−32997.4	−1.30156	−0.650778	+	0.759268i	$0.725557\pi$
$864$	0	0
$865$	−40458.0	−1.59030
$866$	0	0
$867$	1741.57	0.0682199
$868$	0	0
$869$	−36450.2	−1.42289
$870$	0	0
$871$	−5735.93	−0.223140
$872$	0	0
$873$	−685.183	−0.0265635
$874$	0	0
$875$	−8950.21	−0.345797
$876$	0	0
$877$	−23147.0	−0.891241	−0.445621	−	0.895222i	$-0.647017\pi$
$877$	−23147.0	−0.891241	−0.445621	+	0.895222i	$0.647017\pi$
$878$	0	0
$879$	15479.9	0.593999
$880$	0	0
$881$	33539.6	1.28261	0.641304	−	0.767287i	$-0.278394\pi$
$881$	33539.6	1.28261	0.641304	+	0.767287i	$0.278394\pi$
$882$	0	0
$883$	−33611.7	−1.28100	−0.640500	−	0.767959i	$-0.721273\pi$
$883$	−33611.7	−1.28100	−0.640500	+	0.767959i	$0.721273\pi$
$884$	0	0
$885$	12772.7	0.485139
$886$	0	0
$887$	−40888.0	−1.54778	−0.773892	−	0.633318i	$-0.781693\pi$
$887$	−40888.0	−1.54778	−0.773892	+	0.633318i	$0.781693\pi$
$888$	0	0
$889$	−8288.81	−0.312708
$890$	0	0
$891$	−4122.76	−0.155014
$892$	0	0
$893$	−228.127	−0.00854870
$894$	0	0
$895$	−29341.4	−1.09584
$896$	0	0
$897$	−16773.4	−0.624355
$898$	0	0
$899$	−6256.25	−0.232100
$900$	0	0
$901$	23216.0	0.858421
$902$	0	0
$903$	5120.63	0.188709
$904$	0	0
$905$	10325.1	0.379248
$906$	0	0
$907$	30206.2	1.10582	0.552910	−	0.833241i	$-0.313517\pi$
$907$	30206.2	1.10582	0.552910	+	0.833241i	$0.313517\pi$
$908$	0	0
$909$	4123.72	0.150468
$910$	0	0
$911$	43441.0	1.57987	0.789937	−	0.613188i	$-0.210113\pi$
$911$	43441.0	1.57987	0.789937	+	0.613188i	$0.210113\pi$
$912$	0	0
$913$	4471.52	0.162087
$914$	0	0
$915$	8975.25	0.324276
$916$	0	0
$917$	7444.83	0.268102
$918$	0	0
$919$	−17639.1	−0.633144	−0.316572	−	0.948568i	$-0.602532\pi$
$919$	−17639.1	−0.633144	−0.316572	+	0.948568i	$0.602532\pi$
$920$	0	0
$921$	7355.46	0.263160
$922$	0	0
$923$	−10460.9	−0.373048
$924$	0	0
$925$	−359.841	−0.0127908
$926$	0	0
$927$	−23803.5	−0.843377
$928$	0	0
$929$	22192.9	0.783773	0.391886	−	0.920014i	$-0.371823\pi$
$929$	22192.9	0.783773	0.391886	+	0.920014i	$0.371823\pi$
$930$	0	0
$931$	3686.16	0.129763
$932$	0	0
$933$	−6416.18	−0.225141
$934$	0	0
$935$	23608.5	0.825755
$936$	0	0
$937$	20250.5	0.706035	0.353018	−	0.935617i	$-0.385156\pi$
$937$	20250.5	0.706035	0.353018	+	0.935617i	$0.385156\pi$
$938$	0	0
$939$	24763.4	0.860621
$940$	0	0
$941$	29805.1	1.03254	0.516269	−	0.856426i	$-0.327320\pi$
$941$	29805.1	1.03254	0.516269	+	0.856426i	$0.327320\pi$
$942$	0	0
$943$	−17936.7	−0.619406
$944$	0	0
$945$	10445.1	0.359556
$946$	0	0
$947$	−2899.49	−0.0994941	−0.0497470	−	0.998762i	$-0.515842\pi$
$947$	−2899.49	−0.0994941	−0.0497470	+	0.998762i	$0.515842\pi$
$948$	0	0
$949$	−2811.42	−0.0961672
$950$	0	0
$951$	1680.79	0.0573115
$952$	0	0
$953$	33181.1	1.12785	0.563926	−	0.825826i	$-0.309290\pi$
$953$	33181.1	1.12785	0.563926	+	0.825826i	$0.309290\pi$
$954$	0	0
$955$	13197.2	0.447174
$956$	0	0
$957$	−2540.19	−0.0858023
$958$	0	0
$959$	−173.791	−0.00585194
$960$	0	0
$961$	16749.6	0.562238
$962$	0	0
$963$	−7431.94	−0.248692
$964$	0	0
$965$	−10277.1	−0.342831
$966$	0	0
$967$	−52522.2	−1.74664	−0.873320	−	0.487147i	$-0.838038\pi$
$967$	−52522.2	−1.74664	−0.873320	+	0.487147i	$0.838038\pi$
$968$	0	0
$969$	−2327.46	−0.0771609
$970$	0	0
$971$	−13112.8	−0.433377	−0.216689	−	0.976241i	$-0.569526\pi$
$971$	−13112.8	−0.433377	−0.216689	+	0.976241i	$0.569526\pi$
$972$	0	0
$973$	−17691.3	−0.582894
$974$	0	0
$975$	−1327.11	−0.0435912
$976$	0	0
$977$	37611.6	1.23163	0.615814	−	0.787892i	$-0.288827\pi$
$977$	37611.6	1.23163	0.615814	+	0.787892i	$0.288827\pi$
$978$	0	0
$979$	−37560.3	−1.22618
$980$	0	0
$981$	25412.9	0.827086
$982$	0	0
$983$	12941.5	0.419909	0.209955	−	0.977711i	$-0.432668\pi$
$983$	12941.5	0.419909	0.209955	+	0.977711i	$0.432668\pi$
$984$	0	0
$985$	6323.37	0.204548
$986$	0	0
$987$	−356.616	−0.0115007
$988$	0	0
$989$	−42624.1	−1.37044
$990$	0	0
$991$	3548.57	0.113748	0.0568738	−	0.998381i	$-0.481887\pi$
$991$	3548.57	0.113748	0.0568738	+	0.998381i	$0.481887\pi$
$992$	0	0
$993$	22854.6	0.730382
$994$	0	0
$995$	−26947.3	−0.858579
$996$	0	0
$997$	24567.8	0.780411	0.390206	−	0.920728i	$-0.372404\pi$
$997$	24567.8	0.780411	0.390206	+	0.920728i	$0.372404\pi$
$998$	0	0
$999$	−3685.81	−0.116731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bh.1.5		10
4.3	odd	2	inner	1856.4.a.bh.1.6		10
8.3	odd	2		928.4.a.g.1.5	✓	10
8.5	even	2		928.4.a.g.1.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.g.1.5	✓	10	8.3	odd	2
928.4.a.g.1.6	yes	10	8.5	even	2
1856.4.a.bh.1.5		10	1.1	even	1	trivial
1856.4.a.bh.1.6		10	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-2.85699 q^{3} +11.7382 q^{5} +6.79488 q^{7} -18.8376 q^{9} +O(q^{10})\)	\(q-2.85699 q^{3} +11.7382 q^{5} +6.79488 q^{7} -18.8376 q^{9} -30.6592 q^{11} +36.3320 q^{13} -33.5359 q^{15} -65.6005 q^{17} -12.4185 q^{19} -19.4129 q^{21} +161.593 q^{23} +12.7852 q^{25} +130.958 q^{27} -29.0000 q^{29} +215.733 q^{31} +87.5929 q^{33} +79.7596 q^{35} -28.1451 q^{37} -103.800 q^{39} -110.999 q^{41} -263.774 q^{43} -221.119 q^{45} +18.3700 q^{47} -296.830 q^{49} +187.420 q^{51} -353.900 q^{53} -359.883 q^{55} +35.4794 q^{57} -380.865 q^{59} -267.631 q^{61} -127.999 q^{63} +426.472 q^{65} -157.876 q^{67} -461.670 q^{69} -287.925 q^{71} -77.3816 q^{73} -36.5273 q^{75} -208.325 q^{77} +1188.88 q^{79} +134.471 q^{81} -145.846 q^{83} -770.031 q^{85} +82.8527 q^{87} +1225.09 q^{89} +246.871 q^{91} -616.346 q^{93} -145.770 q^{95} +36.3732 q^{97} +577.545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 40 q^{5} + 58 q^{9}+O(q^{10}) \) 10 * q - 40 * q^5 + 58 * q^9	\( 10 q - 40 q^{5} + 58 q^{9} - 136 q^{13} + 144 q^{17} - 92 q^{21} + 334 q^{25} - 290 q^{29} + 536 q^{33} - 212 q^{37} + 400 q^{41} - 748 q^{45} + 626 q^{49} - 1304 q^{53} + 408 q^{57} - 1600 q^{61} + 1808 q^{65} - 1916 q^{69} + 1332 q^{73} - 1724 q^{77} + 730 q^{81} - 3484 q^{85} + 2256 q^{89} - 3500 q^{93} + 560 q^{97}+O(q^{100}) \) 10 * q - 40 * q^5 + 58 * q^9 - 136 * q^13 + 144 * q^17 - 92 * q^21 + 334 * q^25 - 290 * q^29 + 536 * q^33 - 212 * q^37 + 400 * q^41 - 748 * q^45 + 626 * q^49 - 1304 * q^53 + 408 * q^57 - 1600 * q^61 + 1808 * q^65 - 1916 * q^69 + 1332 * q^73 - 1724 * q^77 + 730 * q^81 - 3484 * q^85 + 2256 * q^89 - 3500 * q^93 + 560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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