LMFDB - Embedded newform 1040.4.a.h.1.2

Error: no document with id 190582215 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1040,4,Mod(1,1040)] 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("1040.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-10,0,-16,0,50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$61.3619864060$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{51}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 51 $ x^2 - 51
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.14143$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.14143 q^{3} -5.00000 q^{5} -22.2829 q^{7} +10.7171 q^{9} -43.8586 q^{11} +13.0000 q^{13} -30.7071 q^{15} +99.9800 q^{17} -93.2729 q^{19} -136.849 q^{21} +154.990 q^{23} +25.0000 q^{25} -100.000 q^{27} +191.677 q^{29} +213.839 q^{31} -269.354 q^{33} +111.414 q^{35} -401.697 q^{37} +79.8386 q^{39} +458.809 q^{41} +251.799 q^{43} -53.5857 q^{45} +261.071 q^{47} +153.526 q^{49} +614.020 q^{51} +151.374 q^{53} +219.293 q^{55} -572.829 q^{57} -263.899 q^{59} +644.203 q^{61} -238.809 q^{63} -65.0000 q^{65} -387.577 q^{67} +951.860 q^{69} +544.990 q^{71} +301.737 q^{73} +153.536 q^{75} +977.294 q^{77} +562.991 q^{79} -903.506 q^{81} +938.689 q^{83} -499.900 q^{85} +1177.17 q^{87} +261.514 q^{89} -289.677 q^{91} +1313.27 q^{93} +466.364 q^{95} -1371.50 q^{97} -470.039 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 10 q^{5} - 16 q^{7} + 50 q^{9} - 102 q^{11} + 26 q^{13} + 10 q^{15} - 58 q^{19} - 188 q^{21} + 210 q^{23} + 50 q^{25} - 200 q^{27} + 12 q^{29} + 242 q^{31} + 204 q^{33} + 80 q^{35} - 632 q^{37}+ \cdots - 2754 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 10 * q^5 - 16 * q^7 + 50 * q^9 - 102 * q^11 + 26 * q^13 + 10 * q^15 - 58 * q^19 - 188 * q^21 + 210 * q^23 + 50 * q^25 - 200 * q^27 + 12 * q^29 + 242 * q^31 + 204 * q^33 + 80 * q^35 - 632 * q^37 - 26 * q^39 + 432 * q^41 - 82 * q^43 - 250 * q^45 - 192 * q^47 - 150 * q^49 + 1428 * q^51 - 240 * q^53 + 510 * q^55 - 860 * q^57 - 942 * q^59 + 460 * q^61 + 8 * q^63 - 130 * q^65 + 596 * q^67 + 504 * q^69 + 990 * q^71 + 832 * q^73 - 50 * q^75 + 612 * q^77 - 388 * q^79 - 1150 * q^81 + 192 * q^83 + 2640 * q^87 + 1380 * q^89 - 208 * q^91 + 1084 * q^93 + 290 * q^95 - 572 * q^97 - 2754 * 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
$3$	6.14143	1.18192	0.590959	−	0.806701i	$-0.298749\pi$
$3$	6.14143	1.18192	0.590959	+	0.806701i	$0.298749\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−22.2829	−1.20316	−0.601581	−	0.798812i	$-0.705462\pi$
$7$	−22.2829	−1.20316	−0.601581	+	0.798812i	$0.705462\pi$
$8$	0	0
$9$	10.7171	0.396931
$10$	0	0
$11$	−43.8586	−1.20217	−0.601084	−	0.799186i	$-0.705264\pi$
$11$	−43.8586	−1.20217	−0.601084	+	0.799186i	$0.705264\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−30.7071	−0.528570
$16$	0	0
$17$	99.9800	1.42639	0.713197	−	0.700963i	$-0.247246\pi$
$17$	99.9800	1.42639	0.713197	+	0.700963i	$0.247246\pi$
$18$	0	0
$19$	−93.2729	−1.12622	−0.563112	−	0.826381i	$-0.690396\pi$
$19$	−93.2729	−1.12622	−0.563112	+	0.826381i	$0.690396\pi$
$20$	0	0
$21$	−136.849	−1.42204
$22$	0	0
$23$	154.990	1.40512	0.702558	−	0.711627i	$-0.252041\pi$
$23$	154.990	1.40512	0.702558	+	0.711627i	$0.252041\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	191.677	1.22736	0.613682	−	0.789553i	$-0.289688\pi$
$29$	191.677	1.22736	0.613682	+	0.789553i	$0.289688\pi$
$30$	0	0
$31$	213.839	1.23892	0.619460	−	0.785028i	$-0.287351\pi$
$31$	213.839	1.23892	0.619460	+	0.785028i	$0.287351\pi$
$32$	0	0
$33$	−269.354	−1.42087
$34$	0	0
$35$	111.414	0.538070
$36$	0	0
$37$	−401.697	−1.78483	−0.892414	−	0.451218i	$-0.850990\pi$
$37$	−401.697	−1.78483	−0.892414	+	0.451218i	$0.850990\pi$
$38$	0	0
$39$	79.8386	0.327805
$40$	0	0
$41$	458.809	1.74766	0.873828	−	0.486236i	$-0.161630\pi$
$41$	458.809	1.74766	0.873828	+	0.486236i	$0.161630\pi$
$42$	0	0
$43$	251.799	0.892998	0.446499	−	0.894784i	$-0.352671\pi$
$43$	251.799	0.892998	0.446499	+	0.894784i	$0.352671\pi$
$44$	0	0
$45$	−53.5857	−0.177513
$46$	0	0
$47$	261.071	0.810238	0.405119	−	0.914264i	$-0.367230\pi$
$47$	261.071	0.810238	0.405119	+	0.914264i	$0.367230\pi$
$48$	0	0
$49$	153.526	0.447597
$50$	0	0
$51$	614.020	1.68588
$52$	0	0
$53$	151.374	0.392318	0.196159	−	0.980572i	$-0.437153\pi$
$53$	151.374	0.392318	0.196159	+	0.980572i	$0.437153\pi$
$54$	0	0
$55$	219.293	0.537626
$56$	0	0
$57$	−572.829	−1.33111
$58$	0	0
$59$	−263.899	−0.582316	−0.291158	−	0.956675i	$-0.594041\pi$
$59$	−263.899	−0.582316	−0.291158	+	0.956675i	$0.594041\pi$
$60$	0	0
$61$	644.203	1.35216	0.676080	−	0.736829i	$-0.263677\pi$
$61$	644.203	1.35216	0.676080	+	0.736829i	$0.263677\pi$
$62$	0	0
$63$	−238.809	−0.477572
$64$	0	0
$65$	−65.0000	−0.124035
$66$	0	0
$67$	−387.577	−0.706718	−0.353359	−	0.935488i	$-0.614961\pi$
$67$	−387.577	−0.706718	−0.353359	+	0.935488i	$0.614961\pi$
$68$	0	0
$69$	951.860	1.66073
$70$	0	0
$71$	544.990	0.910964	0.455482	−	0.890245i	$-0.349467\pi$
$71$	544.990	0.910964	0.455482	+	0.890245i	$0.349467\pi$
$72$	0	0
$73$	301.737	0.483776	0.241888	−	0.970304i	$-0.422233\pi$
$73$	301.737	0.483776	0.241888	+	0.970304i	$0.422233\pi$
$74$	0	0
$75$	153.536	0.236384
$76$	0	0
$77$	977.294	1.44640
$78$	0	0
$79$	562.991	0.801791	0.400895	−	0.916124i	$-0.368699\pi$
$79$	562.991	0.801791	0.400895	+	0.916124i	$0.368699\pi$
$80$	0	0
$81$	−903.506	−1.23938
$82$	0	0
$83$	938.689	1.24138	0.620689	−	0.784056i	$-0.286853\pi$
$83$	938.689	1.24138	0.620689	+	0.784056i	$0.286853\pi$
$84$	0	0
$85$	−499.900	−0.637903
$86$	0	0
$87$	1177.17	1.45064
$88$	0	0
$89$	261.514	0.311466	0.155733	−	0.987799i	$-0.450226\pi$
$89$	261.514	0.311466	0.155733	+	0.987799i	$0.450226\pi$
$90$	0	0
$91$	−289.677	−0.333697
$92$	0	0
$93$	1313.27	1.46430
$94$	0	0
$95$	466.364	0.503663
$96$	0	0
$97$	−1371.50	−1.43561	−0.717806	−	0.696243i	$-0.754854\pi$
$97$	−1371.50	−1.43561	−0.717806	+	0.696243i	$0.754854\pi$
$98$	0	0
$99$	−470.039	−0.477178
$100$	0	0
$101$	259.697	0.255850	0.127925	−	0.991784i	$-0.459168\pi$
$101$	259.697	0.255850	0.127925	+	0.991784i	$0.459168\pi$
$102$	0	0
$103$	−749.053	−0.716567	−0.358283	−	0.933613i	$-0.616638\pi$
$103$	−749.053	−0.716567	−0.358283	+	0.933613i	$0.616638\pi$
$104$	0	0
$105$	684.243	0.635955
$106$	0	0
$107$	−797.133	−0.720203	−0.360102	−	0.932913i	$-0.617258\pi$
$107$	−797.133	−0.720203	−0.360102	+	0.932913i	$0.617258\pi$
$108$	0	0
$109$	648.949	0.570257	0.285128	−	0.958489i	$-0.407964\pi$
$109$	648.949	0.570257	0.285128	+	0.958489i	$0.407964\pi$
$110$	0	0
$111$	−2466.99	−2.10952
$112$	0	0
$113$	600.060	0.499548	0.249774	−	0.968304i	$-0.419644\pi$
$113$	600.060	0.499548	0.249774	+	0.968304i	$0.419644\pi$
$114$	0	0
$115$	−774.950	−0.628387
$116$	0	0
$117$	139.323	0.110089
$118$	0	0
$119$	−2227.84	−1.71618
$120$	0	0
$121$	592.574	0.445210
$122$	0	0
$123$	2817.74	2.06559
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1918.42	1.34041	0.670207	−	0.742174i	$-0.266205\pi$
$127$	1918.42	1.34041	0.670207	+	0.742174i	$0.266205\pi$
$128$	0	0
$129$	1546.40	1.05545
$130$	0	0
$131$	−1428.12	−0.952484	−0.476242	−	0.879314i	$-0.658001\pi$
$131$	−1428.12	−0.952484	−0.476242	+	0.879314i	$0.658001\pi$
$132$	0	0
$133$	2078.39	1.35503
$134$	0	0
$135$	500.000	0.318764
$136$	0	0
$137$	−1129.19	−0.704187	−0.352093	−	0.935965i	$-0.614530\pi$
$137$	−1129.19	−0.704187	−0.352093	+	0.935965i	$0.614530\pi$
$138$	0	0
$139$	−180.889	−0.110380	−0.0551898	−	0.998476i	$-0.517576\pi$
$139$	−180.889	−0.110380	−0.0551898	+	0.998476i	$0.517576\pi$
$140$	0	0
$141$	1603.35	0.957635
$142$	0	0
$143$	−570.161	−0.333422
$144$	0	0
$145$	−958.386	−0.548894
$146$	0	0
$147$	942.867	0.529023
$148$	0	0
$149$	−1515.13	−0.833050	−0.416525	−	0.909124i	$-0.636752\pi$
$149$	−1515.13	−0.833050	−0.416525	+	0.909124i	$0.636752\pi$
$150$	0	0
$151$	258.970	0.139567	0.0697837	−	0.997562i	$-0.477769\pi$
$151$	258.970	0.139567	0.0697837	+	0.997562i	$0.477769\pi$
$152$	0	0
$153$	1071.50	0.566181
$154$	0	0
$155$	−1069.19	−0.554062
$156$	0	0
$157$	−512.949	−0.260750	−0.130375	−	0.991465i	$-0.541618\pi$
$157$	−512.949	−0.260750	−0.130375	+	0.991465i	$0.541618\pi$
$158$	0	0
$159$	929.654	0.463688
$160$	0	0
$161$	−3453.62	−1.69058
$162$	0	0
$163$	3299.58	1.58554	0.792769	−	0.609522i	$-0.208638\pi$
$163$	3299.58	1.58554	0.792769	+	0.609522i	$0.208638\pi$
$164$	0	0
$165$	1346.77	0.635430
$166$	0	0
$167$	−81.4315	−0.0377327	−0.0188663	−	0.999822i	$-0.506006\pi$
$167$	−81.4315	−0.0377327	−0.0188663	+	0.999822i	$0.506006\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−999.619	−0.447034
$172$	0	0
$173$	−1038.49	−0.456386	−0.228193	−	0.973616i	$-0.573282\pi$
$173$	−1038.49	−0.456386	−0.228193	+	0.973616i	$0.573282\pi$
$174$	0	0
$175$	−557.071	−0.240632
$176$	0	0
$177$	−1620.71	−0.688251
$178$	0	0
$179$	−2001.70	−0.835834	−0.417917	−	0.908485i	$-0.637240\pi$
$179$	−2001.70	−0.835834	−0.417917	+	0.908485i	$0.637240\pi$
$180$	0	0
$181$	−3417.70	−1.40351	−0.701756	−	0.712417i	$-0.747600\pi$
$181$	−3417.70	−1.40351	−0.701756	+	0.712417i	$0.747600\pi$
$182$	0	0
$183$	3956.33	1.59814
$184$	0	0
$185$	2008.49	0.798199
$186$	0	0
$187$	−4384.98	−1.71477
$188$	0	0
$189$	2228.29	0.857587
$190$	0	0
$191$	2415.03	0.914899	0.457450	−	0.889236i	$-0.348763\pi$
$191$	2415.03	0.914899	0.457450	+	0.889236i	$0.348763\pi$
$192$	0	0
$193$	−3899.05	−1.45420	−0.727098	−	0.686534i	$-0.759131\pi$
$193$	−3899.05	−1.45420	−0.727098	+	0.686534i	$0.759131\pi$
$194$	0	0
$195$	−399.193	−0.146599
$196$	0	0
$197$	1664.75	0.602073	0.301037	−	0.953613i	$-0.402667\pi$
$197$	1664.75	0.602073	0.301037	+	0.953613i	$0.402667\pi$
$198$	0	0
$199$	4671.64	1.66414	0.832070	−	0.554671i	$-0.187156\pi$
$199$	4671.64	1.66414	0.832070	+	0.554671i	$0.187156\pi$
$200$	0	0
$201$	−2380.28	−0.835283
$202$	0	0
$203$	−4271.11	−1.47672
$204$	0	0
$205$	−2294.04	−0.781575
$206$	0	0
$207$	1661.05	0.557734
$208$	0	0
$209$	4090.81	1.35391
$210$	0	0
$211$	1405.82	0.458677	0.229338	−	0.973347i	$-0.426344\pi$
$211$	1405.82	0.458677	0.229338	+	0.973347i	$0.426344\pi$
$212$	0	0
$213$	3347.02	1.07668
$214$	0	0
$215$	−1258.99	−0.399361
$216$	0	0
$217$	−4764.93	−1.49062
$218$	0	0
$219$	1853.10	0.571784
$220$	0	0
$221$	1299.74	0.395611
$222$	0	0
$223$	2423.95	0.727890	0.363945	−	0.931420i	$-0.381430\pi$
$223$	2423.95	0.727890	0.363945	+	0.931420i	$0.381430\pi$
$224$	0	0
$225$	267.929	0.0793862
$226$	0	0
$227$	−185.926	−0.0543626	−0.0271813	−	0.999631i	$-0.508653\pi$
$227$	−185.926	−0.0543626	−0.0271813	+	0.999631i	$0.508653\pi$
$228$	0	0
$229$	6353.04	1.83328	0.916639	−	0.399715i	$-0.130891\pi$
$229$	6353.04	1.83328	0.916639	+	0.399715i	$0.130891\pi$
$230$	0	0
$231$	6001.98	1.70953
$232$	0	0
$233$	−1088.91	−0.306168	−0.153084	−	0.988213i	$-0.548921\pi$
$233$	−1088.91	−0.306168	−0.153084	+	0.988213i	$0.548921\pi$
$234$	0	0
$235$	−1305.36	−0.362349
$236$	0	0
$237$	3457.57	0.947651
$238$	0	0
$239$	1555.66	0.421036	0.210518	−	0.977590i	$-0.432485\pi$
$239$	1555.66	0.421036	0.210518	+	0.977590i	$0.432485\pi$
$240$	0	0
$241$	520.786	0.139198	0.0695991	−	0.997575i	$-0.477828\pi$
$241$	520.786	0.139198	0.0695991	+	0.997575i	$0.477828\pi$
$242$	0	0
$243$	−2848.82	−0.752064
$244$	0	0
$245$	−767.629	−0.200171
$246$	0	0
$247$	−1212.55	−0.312358
$248$	0	0
$249$	5764.89	1.46721
$250$	0	0
$251$	7079.53	1.78030	0.890151	−	0.455665i	$-0.150599\pi$
$251$	7079.53	1.78030	0.890151	+	0.455665i	$0.150599\pi$
$252$	0	0
$253$	−6797.64	−1.68919
$254$	0	0
$255$	−3070.10	−0.753949
$256$	0	0
$257$	−437.806	−0.106263	−0.0531315	−	0.998588i	$-0.516920\pi$
$257$	−437.806	−0.106263	−0.0531315	+	0.998588i	$0.516920\pi$
$258$	0	0
$259$	8950.96	2.14743
$260$	0	0
$261$	2054.23	0.487179
$262$	0	0
$263$	−1237.18	−0.290067	−0.145033	−	0.989427i	$-0.546329\pi$
$263$	−1237.18	−0.290067	−0.145033	+	0.989427i	$0.546329\pi$
$264$	0	0
$265$	−756.871	−0.175450
$266$	0	0
$267$	1606.07	0.368127
$268$	0	0
$269$	−6507.75	−1.47504	−0.737518	−	0.675328i	$-0.764002\pi$
$269$	−6507.75	−1.47504	−0.737518	+	0.675328i	$0.764002\pi$
$270$	0	0
$271$	−2547.66	−0.571069	−0.285534	−	0.958368i	$-0.592171\pi$
$271$	−2547.66	−0.571069	−0.285534	+	0.958368i	$0.592171\pi$
$272$	0	0
$273$	−1779.03	−0.394402
$274$	0	0
$275$	−1096.46	−0.240434
$276$	0	0
$277$	5214.12	1.13100	0.565499	−	0.824749i	$-0.308684\pi$
$277$	5214.12	1.13100	0.565499	+	0.824749i	$0.308684\pi$
$278$	0	0
$279$	2291.74	0.491766
$280$	0	0
$281$	7359.26	1.56234	0.781168	−	0.624320i	$-0.214624\pi$
$281$	7359.26	1.56234	0.781168	+	0.624320i	$0.214624\pi$
$282$	0	0
$283$	2985.29	0.627057	0.313528	−	0.949579i	$-0.398489\pi$
$283$	2985.29	0.627057	0.313528	+	0.949579i	$0.398489\pi$
$284$	0	0
$285$	2864.14	0.595288
$286$	0	0
$287$	−10223.6	−2.10271
$288$	0	0
$289$	5083.00	1.03460
$290$	0	0
$291$	−8422.95	−1.69678
$292$	0	0
$293$	9607.65	1.91565	0.957824	−	0.287357i	$-0.0927765\pi$
$293$	9607.65	1.91565	0.957824	+	0.287357i	$0.0927765\pi$
$294$	0	0
$295$	1319.49	0.260420
$296$	0	0
$297$	4385.86	0.856880
$298$	0	0
$299$	2014.87	0.389709
$300$	0	0
$301$	−5610.79	−1.07442
$302$	0	0
$303$	1594.91	0.302394
$304$	0	0
$305$	−3221.01	−0.604704
$306$	0	0
$307$	8851.79	1.64560	0.822798	−	0.568334i	$-0.192412\pi$
$307$	8851.79	1.64560	0.822798	+	0.568334i	$0.192412\pi$
$308$	0	0
$309$	−4600.25	−0.846923
$310$	0	0
$311$	−2645.58	−0.482370	−0.241185	−	0.970479i	$-0.577536\pi$
$311$	−2645.58	−0.482370	−0.241185	+	0.970479i	$0.577536\pi$
$312$	0	0
$313$	−6088.30	−1.09946	−0.549730	−	0.835342i	$-0.685269\pi$
$313$	−6088.30	−1.09946	−0.549730	+	0.835342i	$0.685269\pi$
$314$	0	0
$315$	1194.04	0.213577
$316$	0	0
$317$	9945.91	1.76220	0.881101	−	0.472928i	$-0.156803\pi$
$317$	9945.91	1.76220	0.881101	+	0.472928i	$0.156803\pi$
$318$	0	0
$319$	−8406.69	−1.47550
$320$	0	0
$321$	−4895.53	−0.851221
$322$	0	0
$323$	−9325.42	−1.60644
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	3985.47	0.673997
$328$	0	0
$329$	−5817.42	−0.974847
$330$	0	0
$331$	−2735.14	−0.454189	−0.227095	−	0.973873i	$-0.572923\pi$
$331$	−2735.14	−0.454189	−0.227095	+	0.973873i	$0.572923\pi$
$332$	0	0
$333$	−4305.05	−0.708454
$334$	0	0
$335$	1937.89	0.316054
$336$	0	0
$337$	7468.44	1.20722	0.603608	−	0.797281i	$-0.293729\pi$
$337$	7468.44	1.20722	0.603608	+	0.797281i	$0.293729\pi$
$338$	0	0
$339$	3685.23	0.590425
$340$	0	0
$341$	−9378.65	−1.48939
$342$	0	0
$343$	4222.03	0.664630
$344$	0	0
$345$	−4759.30	−0.742702
$346$	0	0
$347$	−9931.82	−1.53651	−0.768253	−	0.640146i	$-0.778874\pi$
$347$	−9931.82	−1.53651	−0.768253	+	0.640146i	$0.778874\pi$
$348$	0	0
$349$	1573.92	0.241404	0.120702	−	0.992689i	$-0.461486\pi$
$349$	1573.92	0.241404	0.120702	+	0.992689i	$0.461486\pi$
$350$	0	0
$351$	−1300.00	−0.197689
$352$	0	0
$353$	10869.2	1.63884	0.819420	−	0.573194i	$-0.194296\pi$
$353$	10869.2	1.63884	0.819420	+	0.573194i	$0.194296\pi$
$354$	0	0
$355$	−2724.95	−0.407395
$356$	0	0
$357$	−13682.1	−2.02839
$358$	0	0
$359$	7001.85	1.02937	0.514684	−	0.857380i	$-0.327909\pi$
$359$	7001.85	1.02937	0.514684	+	0.857380i	$0.327909\pi$
$360$	0	0
$361$	1840.83	0.268381
$362$	0	0
$363$	3639.25	0.526202
$364$	0	0
$365$	−1508.69	−0.216351
$366$	0	0
$367$	−11254.8	−1.60081	−0.800403	−	0.599462i	$-0.795381\pi$
$367$	−11254.8	−1.60081	−0.800403	+	0.599462i	$0.795381\pi$
$368$	0	0
$369$	4917.12	0.693699
$370$	0	0
$371$	−3373.05	−0.472022
$372$	0	0
$373$	−9551.67	−1.32592	−0.662958	−	0.748657i	$-0.730699\pi$
$373$	−9551.67	−1.32592	−0.662958	+	0.748657i	$0.730699\pi$
$374$	0	0
$375$	−767.679	−0.105714
$376$	0	0
$377$	2491.80	0.340409
$378$	0	0
$379$	−6124.55	−0.830071	−0.415036	−	0.909805i	$-0.636231\pi$
$379$	−6124.55	−0.830071	−0.415036	+	0.909805i	$0.636231\pi$
$380$	0	0
$381$	11781.9	1.58426
$382$	0	0
$383$	−2206.62	−0.294394	−0.147197	−	0.989107i	$-0.547025\pi$
$383$	−2206.62	−0.294394	−0.147197	+	0.989107i	$0.547025\pi$
$384$	0	0
$385$	−4886.47	−0.646851
$386$	0	0
$387$	2698.56	0.354459
$388$	0	0
$389$	−8458.16	−1.10243	−0.551216	−	0.834363i	$-0.685836\pi$
$389$	−8458.16	−1.10243	−0.551216	+	0.834363i	$0.685836\pi$
$390$	0	0
$391$	15495.9	2.00425
$392$	0	0
$393$	−8770.70	−1.12576
$394$	0	0
$395$	−2814.96	−0.358572
$396$	0	0
$397$	10377.1	1.31186	0.655931	−	0.754821i	$-0.272276\pi$
$397$	10377.1	1.31186	0.655931	+	0.754821i	$0.272276\pi$
$398$	0	0
$399$	12764.3	1.60153
$400$	0	0
$401$	8962.85	1.11617	0.558084	−	0.829784i	$-0.311537\pi$
$401$	8962.85	1.11617	0.558084	+	0.829784i	$0.311537\pi$
$402$	0	0
$403$	2779.90	0.343615
$404$	0	0
$405$	4517.53	0.554266
$406$	0	0
$407$	17617.9	2.14566
$408$	0	0
$409$	−1133.52	−0.137039	−0.0685193	−	0.997650i	$-0.521827\pi$
$409$	−1133.52	−0.137039	−0.0685193	+	0.997650i	$0.521827\pi$
$410$	0	0
$411$	−6934.87	−0.832291
$412$	0	0
$413$	5880.41	0.700620
$414$	0	0
$415$	−4693.44	−0.555162
$416$	0	0
$417$	−1110.91	−0.130460
$418$	0	0
$419$	−10588.0	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$419$	−10588.0	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$420$	0	0
$421$	−8296.85	−0.960484	−0.480242	−	0.877136i	$-0.659451\pi$
$421$	−8296.85	−0.960484	−0.480242	+	0.877136i	$0.659451\pi$
$422$	0	0
$423$	2797.94	0.321609
$424$	0	0
$425$	2499.50	0.285279
$426$	0	0
$427$	−14354.7	−1.62687
$428$	0	0
$429$	−3501.61	−0.394077
$430$	0	0
$431$	−3010.89	−0.336496	−0.168248	−	0.985745i	$-0.553811\pi$
$431$	−3010.89	−0.336496	−0.168248	+	0.985745i	$0.553811\pi$
$432$	0	0
$433$	−3310.31	−0.367399	−0.183699	−	0.982982i	$-0.558807\pi$
$433$	−3310.31	−0.367399	−0.183699	+	0.982982i	$0.558807\pi$
$434$	0	0
$435$	−5885.86	−0.648748
$436$	0	0
$437$	−14456.4	−1.58247
$438$	0	0
$439$	−2597.90	−0.282439	−0.141220	−	0.989978i	$-0.545102\pi$
$439$	−2597.90	−0.282439	−0.141220	+	0.989978i	$0.545102\pi$
$440$	0	0
$441$	1645.36	0.177665
$442$	0	0
$443$	16570.8	1.77721	0.888606	−	0.458671i	$-0.151675\pi$
$443$	16570.8	1.77721	0.888606	+	0.458671i	$0.151675\pi$
$444$	0	0
$445$	−1307.57	−0.139292
$446$	0	0
$447$	−9305.07	−0.984597
$448$	0	0
$449$	−6153.27	−0.646750	−0.323375	−	0.946271i	$-0.604817\pi$
$449$	−6153.27	−0.646750	−0.323375	+	0.946271i	$0.604817\pi$
$450$	0	0
$451$	−20122.7	−2.10098
$452$	0	0
$453$	1590.45	0.164957
$454$	0	0
$455$	1448.39	0.149234
$456$	0	0
$457$	−12968.9	−1.32748	−0.663742	−	0.747961i	$-0.731033\pi$
$457$	−12968.9	−1.32748	−0.663742	+	0.747961i	$0.731033\pi$
$458$	0	0
$459$	−9998.00	−1.01670
$460$	0	0
$461$	−12449.4	−1.25776	−0.628878	−	0.777504i	$-0.716485\pi$
$461$	−12449.4	−1.25776	−0.628878	+	0.777504i	$0.716485\pi$
$462$	0	0
$463$	−2012.71	−0.202028	−0.101014	−	0.994885i	$-0.532209\pi$
$463$	−2012.71	−0.202028	−0.101014	+	0.994885i	$0.532209\pi$
$464$	0	0
$465$	−6566.37	−0.654856
$466$	0	0
$467$	4541.22	0.449984	0.224992	−	0.974361i	$-0.427764\pi$
$467$	4541.22	0.449984	0.224992	+	0.974361i	$0.427764\pi$
$468$	0	0
$469$	8636.33	0.850295
$470$	0	0
$471$	−3150.24	−0.308185
$472$	0	0
$473$	−11043.5	−1.07353
$474$	0	0
$475$	−2331.82	−0.225245
$476$	0	0
$477$	1622.30	0.155723
$478$	0	0
$479$	−5671.15	−0.540963	−0.270482	−	0.962725i	$-0.587183\pi$
$479$	−5671.15	−0.540963	−0.270482	+	0.962725i	$0.587183\pi$
$480$	0	0
$481$	−5222.06	−0.495022
$482$	0	0
$483$	−21210.2	−1.99813
$484$	0	0
$485$	6857.49	0.642026
$486$	0	0
$487$	−5379.59	−0.500560	−0.250280	−	0.968174i	$-0.580523\pi$
$487$	−5379.59	−0.500560	−0.250280	+	0.968174i	$0.580523\pi$
$488$	0	0
$489$	20264.1	1.87398
$490$	0	0
$491$	883.306	0.0811874	0.0405937	−	0.999176i	$-0.487075\pi$
$491$	883.306	0.0811874	0.0405937	+	0.999176i	$0.487075\pi$
$492$	0	0
$493$	19163.9	1.75071
$494$	0	0
$495$	2350.19	0.213401
$496$	0	0
$497$	−12143.9	−1.09604
$498$	0	0
$499$	15251.5	1.36824	0.684119	−	0.729371i	$-0.260187\pi$
$499$	15251.5	1.36824	0.684119	+	0.729371i	$0.260187\pi$
$500$	0	0
$501$	−500.105	−0.0445969
$502$	0	0
$503$	9087.97	0.805592	0.402796	−	0.915290i	$-0.368038\pi$
$503$	9087.97	0.805592	0.402796	+	0.915290i	$0.368038\pi$
$504$	0	0
$505$	−1298.49	−0.114420
$506$	0	0
$507$	1037.90	0.0909168
$508$	0	0
$509$	−19916.8	−1.73438	−0.867189	−	0.497979i	$-0.834076\pi$
$509$	−19916.8	−1.73438	−0.867189	+	0.497979i	$0.834076\pi$
$510$	0	0
$511$	−6723.57	−0.582061
$512$	0	0
$513$	9327.29	0.802748
$514$	0	0
$515$	3745.26	0.320458
$516$	0	0
$517$	−11450.2	−0.974043
$518$	0	0
$519$	−6377.80	−0.539411
$520$	0	0
$521$	6154.94	0.517568	0.258784	−	0.965935i	$-0.416678\pi$
$521$	6154.94	0.517568	0.258784	+	0.965935i	$0.416678\pi$
$522$	0	0
$523$	−6032.73	−0.504384	−0.252192	−	0.967677i	$-0.581152\pi$
$523$	−6032.73	−0.504384	−0.252192	+	0.967677i	$0.581152\pi$
$524$	0	0
$525$	−3421.21	−0.284408
$526$	0	0
$527$	21379.6	1.76719
$528$	0	0
$529$	11854.9	0.974349
$530$	0	0
$531$	−2828.24	−0.231140
$532$	0	0
$533$	5964.51	0.484712
$534$	0	0
$535$	3985.66	0.322085
$536$	0	0
$537$	−12293.3	−0.987888
$538$	0	0
$539$	−6733.42	−0.538087
$540$	0	0
$541$	9116.03	0.724452	0.362226	−	0.932090i	$-0.382017\pi$
$541$	9116.03	0.724452	0.362226	+	0.932090i	$0.382017\pi$
$542$	0	0
$543$	−20989.6	−1.65884
$544$	0	0
$545$	−3244.74	−0.255027
$546$	0	0
$547$	12356.8	0.965885	0.482942	−	0.875652i	$-0.339568\pi$
$547$	12356.8	0.965885	0.482942	+	0.875652i	$0.339568\pi$
$548$	0	0
$549$	6904.01	0.536714
$550$	0	0
$551$	−17878.3	−1.38229
$552$	0	0
$553$	−12545.1	−0.964684
$554$	0	0
$555$	12335.0	0.943406
$556$	0	0
$557$	−10292.1	−0.782928	−0.391464	−	0.920194i	$-0.628031\pi$
$557$	−10292.1	−0.782928	−0.391464	+	0.920194i	$0.628031\pi$
$558$	0	0
$559$	3273.38	0.247673
$560$	0	0
$561$	−26930.0	−2.02672
$562$	0	0
$563$	−21015.4	−1.57317	−0.786585	−	0.617482i	$-0.788153\pi$
$563$	−21015.4	−1.57317	−0.786585	+	0.617482i	$0.788153\pi$
$564$	0	0
$565$	−3000.30	−0.223405
$566$	0	0
$567$	20132.7	1.49117
$568$	0	0
$569$	5207.27	0.383656	0.191828	−	0.981429i	$-0.438558\pi$
$569$	5207.27	0.383656	0.191828	+	0.981429i	$0.438558\pi$
$570$	0	0
$571$	−16827.8	−1.23331	−0.616655	−	0.787234i	$-0.711513\pi$
$571$	−16827.8	−1.23331	−0.616655	+	0.787234i	$0.711513\pi$
$572$	0	0
$573$	14831.8	1.08134
$574$	0	0
$575$	3874.75	0.281023
$576$	0	0
$577$	7624.71	0.550123	0.275061	−	0.961427i	$-0.411302\pi$
$577$	7624.71	0.550123	0.275061	+	0.961427i	$0.411302\pi$
$578$	0	0
$579$	−23945.7	−1.71874
$580$	0	0
$581$	−20916.7	−1.49358
$582$	0	0
$583$	−6639.06	−0.471633
$584$	0	0
$585$	−696.614	−0.0492333
$586$	0	0
$587$	−18164.5	−1.27722	−0.638612	−	0.769529i	$-0.720491\pi$
$587$	−18164.5	−1.27722	−0.638612	+	0.769529i	$0.720491\pi$
$588$	0	0
$589$	−19945.3	−1.39530
$590$	0	0
$591$	10223.9	0.711601
$592$	0	0
$593$	−13329.0	−0.923029	−0.461514	−	0.887133i	$-0.652694\pi$
$593$	−13329.0	−0.923029	−0.461514	+	0.887133i	$0.652694\pi$
$594$	0	0
$595$	11139.2	0.767500
$596$	0	0
$597$	28690.5	1.96688
$598$	0	0
$599$	10495.5	0.715920	0.357960	−	0.933737i	$-0.383472\pi$
$599$	10495.5	0.715920	0.357960	+	0.933737i	$0.383472\pi$
$600$	0	0
$601$	−12575.5	−0.853518	−0.426759	−	0.904365i	$-0.640345\pi$
$601$	−12575.5	−0.853518	−0.426759	+	0.904365i	$0.640345\pi$
$602$	0	0
$603$	−4153.72	−0.280518
$604$	0	0
$605$	−2962.87	−0.199104
$606$	0	0
$607$	−4144.35	−0.277124	−0.138562	−	0.990354i	$-0.544248\pi$
$607$	−4144.35	−0.277124	−0.138562	+	0.990354i	$0.544248\pi$
$608$	0	0
$609$	−26230.7	−1.74536
$610$	0	0
$611$	3393.93	0.224720
$612$	0	0
$613$	9975.52	0.657272	0.328636	−	0.944457i	$-0.393411\pi$
$613$	9975.52	0.657272	0.328636	+	0.944457i	$0.393411\pi$
$614$	0	0
$615$	−14088.7	−0.923758
$616$	0	0
$617$	19244.9	1.25571	0.627853	−	0.778332i	$-0.283934\pi$
$617$	19244.9	1.25571	0.627853	+	0.778332i	$0.283934\pi$
$618$	0	0
$619$	9885.84	0.641915	0.320958	−	0.947094i	$-0.395995\pi$
$619$	9885.84	0.641915	0.320958	+	0.947094i	$0.395995\pi$
$620$	0	0
$621$	−15499.0	−1.00154
$622$	0	0
$623$	−5827.29	−0.374744
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	25123.4	1.60021
$628$	0	0
$629$	−40161.7	−2.54587
$630$	0	0
$631$	5213.18	0.328896	0.164448	−	0.986386i	$-0.447416\pi$
$631$	5213.18	0.328896	0.164448	+	0.986386i	$0.447416\pi$
$632$	0	0
$633$	8633.76	0.542119
$634$	0	0
$635$	−9592.12	−0.599452
$636$	0	0
$637$	1995.83	0.124141
$638$	0	0
$639$	5840.74	0.361590
$640$	0	0
$641$	−1754.92	−0.108136	−0.0540680	−	0.998537i	$-0.517219\pi$
$641$	−1754.92	−0.108136	−0.0540680	+	0.998537i	$0.517219\pi$
$642$	0	0
$643$	8477.04	0.519909	0.259955	−	0.965621i	$-0.416292\pi$
$643$	8477.04	0.519909	0.259955	+	0.965621i	$0.416292\pi$
$644$	0	0
$645$	−7732.01	−0.472012
$646$	0	0
$647$	23346.4	1.41861	0.709306	−	0.704901i	$-0.249008\pi$
$647$	23346.4	1.41861	0.709306	+	0.704901i	$0.249008\pi$
$648$	0	0
$649$	11574.2	0.700043
$650$	0	0
$651$	−29263.5	−1.76179
$652$	0	0
$653$	−18810.5	−1.12727	−0.563637	−	0.826022i	$-0.690598\pi$
$653$	−18810.5	−1.12727	−0.563637	+	0.826022i	$0.690598\pi$
$654$	0	0
$655$	7140.60	0.425964
$656$	0	0
$657$	3233.76	0.192026
$658$	0	0
$659$	−2823.49	−0.166901	−0.0834503	−	0.996512i	$-0.526594\pi$
$659$	−2823.49	−0.166901	−0.0834503	+	0.996512i	$0.526594\pi$
$660$	0	0
$661$	14353.1	0.844586	0.422293	−	0.906459i	$-0.361225\pi$
$661$	14353.1	0.844586	0.422293	+	0.906459i	$0.361225\pi$
$662$	0	0
$663$	7982.26	0.467580
$664$	0	0
$665$	−10391.9	−0.605988
$666$	0	0
$667$	29708.0	1.72459
$668$	0	0
$669$	14886.5	0.860306
$670$	0	0
$671$	−28253.8	−1.62552
$672$	0	0
$673$	2021.69	0.115795	0.0578977	−	0.998323i	$-0.481560\pi$
$673$	2021.69	0.115795	0.0578977	+	0.998323i	$0.481560\pi$
$674$	0	0
$675$	−2500.00	−0.142556
$676$	0	0
$677$	16670.5	0.946380	0.473190	−	0.880960i	$-0.343102\pi$
$677$	16670.5	0.946380	0.473190	+	0.880960i	$0.343102\pi$
$678$	0	0
$679$	30560.9	1.72727
$680$	0	0
$681$	−1141.85	−0.0642522
$682$	0	0
$683$	−19539.7	−1.09468	−0.547341	−	0.836910i	$-0.684360\pi$
$683$	−19539.7	−1.09468	−0.547341	+	0.836910i	$0.684360\pi$
$684$	0	0
$685$	5645.97	0.314922
$686$	0	0
$687$	39016.8	2.16679
$688$	0	0
$689$	1967.87	0.108809
$690$	0	0
$691$	26847.1	1.47802	0.739011	−	0.673694i	$-0.235293\pi$
$691$	26847.1	1.47802	0.739011	+	0.673694i	$0.235293\pi$
$692$	0	0
$693$	10473.8	0.574122
$694$	0	0
$695$	904.443	0.0493633
$696$	0	0
$697$	45871.7	2.49285
$698$	0	0
$699$	−6687.49	−0.361866
$700$	0	0
$701$	−27878.5	−1.50208	−0.751039	−	0.660258i	$-0.770447\pi$
$701$	−27878.5	−1.50208	−0.751039	+	0.660258i	$0.770447\pi$
$702$	0	0
$703$	37467.4	2.01012
$704$	0	0
$705$	−8016.76	−0.428267
$706$	0	0
$707$	−5786.79	−0.307829
$708$	0	0
$709$	27805.6	1.47286	0.736431	−	0.676512i	$-0.236509\pi$
$709$	27805.6	1.47286	0.736431	+	0.676512i	$0.236509\pi$
$710$	0	0
$711$	6033.66	0.318256
$712$	0	0
$713$	33142.8	1.74083
$714$	0	0
$715$	2850.81	0.149111
$716$	0	0
$717$	9554.00	0.497630
$718$	0	0
$719$	33653.6	1.74557	0.872787	−	0.488101i	$-0.162310\pi$
$719$	33653.6	1.74557	0.872787	+	0.488101i	$0.162310\pi$
$720$	0	0
$721$	16691.0	0.862145
$722$	0	0
$723$	3198.37	0.164521
$724$	0	0
$725$	4791.93	0.245473
$726$	0	0
$727$	−1384.32	−0.0706210	−0.0353105	−	0.999376i	$-0.511242\pi$
$727$	−1384.32	−0.0706210	−0.0353105	+	0.999376i	$0.511242\pi$
$728$	0	0
$729$	6898.86	0.350498
$730$	0	0
$731$	25174.8	1.27377
$732$	0	0
$733$	−1212.64	−0.0611049	−0.0305524	−	0.999533i	$-0.509727\pi$
$733$	−1212.64	−0.0611049	−0.0305524	+	0.999533i	$0.509727\pi$
$734$	0	0
$735$	−4714.34	−0.236586
$736$	0	0
$737$	16998.6	0.849594
$738$	0	0
$739$	8168.18	0.406592	0.203296	−	0.979117i	$-0.434835\pi$
$739$	8168.18	0.406592	0.203296	+	0.979117i	$0.434835\pi$
$740$	0	0
$741$	−7446.77	−0.369182
$742$	0	0
$743$	11712.7	0.578328	0.289164	−	0.957280i	$-0.406623\pi$
$743$	11712.7	0.578328	0.289164	+	0.957280i	$0.406623\pi$
$744$	0	0
$745$	7575.66	0.372551
$746$	0	0
$747$	10060.1	0.492742
$748$	0	0
$749$	17762.4	0.866521
$750$	0	0
$751$	−29405.0	−1.42877	−0.714383	−	0.699754i	$-0.753293\pi$
$751$	−29405.0	−1.42877	−0.714383	+	0.699754i	$0.753293\pi$
$752$	0	0
$753$	43478.4	2.10417
$754$	0	0
$755$	−1294.85	−0.0624164
$756$	0	0
$757$	13612.9	0.653593	0.326797	−	0.945095i	$-0.394031\pi$
$757$	13612.9	0.653593	0.326797	+	0.945095i	$0.394031\pi$
$758$	0	0
$759$	−41747.2	−1.99648
$760$	0	0
$761$	4598.55	0.219050	0.109525	−	0.993984i	$-0.465067\pi$
$761$	4598.55	0.219050	0.109525	+	0.993984i	$0.465067\pi$
$762$	0	0
$763$	−14460.4	−0.686111
$764$	0	0
$765$	−5357.50	−0.253204
$766$	0	0
$767$	−3430.68	−0.161506
$768$	0	0
$769$	−9640.19	−0.452060	−0.226030	−	0.974120i	$-0.572575\pi$
$769$	−9640.19	−0.452060	−0.226030	+	0.974120i	$0.572575\pi$
$770$	0	0
$771$	−2688.75	−0.125594
$772$	0	0
$773$	−4369.66	−0.203319	−0.101660	−	0.994819i	$-0.532415\pi$
$773$	−4369.66	−0.203319	−0.101660	+	0.994819i	$0.532415\pi$
$774$	0	0
$775$	5345.96	0.247784
$776$	0	0
$777$	54971.7	2.53809
$778$	0	0
$779$	−42794.4	−1.96825
$780$	0	0
$781$	−23902.5	−1.09513
$782$	0	0
$783$	−19167.7	−0.874838
$784$	0	0
$785$	2564.74	0.116611
$786$	0	0
$787$	−36670.3	−1.66094	−0.830468	−	0.557066i	$-0.811927\pi$
$787$	−36670.3	−1.66094	−0.830468	+	0.557066i	$0.811927\pi$
$788$	0	0
$789$	−7598.03	−0.342835
$790$	0	0
$791$	−13371.1	−0.601036
$792$	0	0
$793$	8374.64	0.375022
$794$	0	0
$795$	−4648.27	−0.207368
$796$	0	0
$797$	−2533.75	−0.112610	−0.0563050	−	0.998414i	$-0.517932\pi$
$797$	−2533.75	−0.112610	−0.0563050	+	0.998414i	$0.517932\pi$
$798$	0	0
$799$	26101.9	1.15572
$800$	0	0
$801$	2802.69	0.123631
$802$	0	0
$803$	−13233.8	−0.581581
$804$	0	0
$805$	17268.1	0.756050
$806$	0	0
$807$	−39966.9	−1.74337
$808$	0	0
$809$	20002.0	0.869261	0.434630	−	0.900609i	$-0.356879\pi$
$809$	20002.0	0.869261	0.434630	+	0.900609i	$0.356879\pi$
$810$	0	0
$811$	−45063.8	−1.95118	−0.975589	−	0.219605i	$-0.929523\pi$
$811$	−45063.8	−1.95118	−0.975589	+	0.219605i	$0.929523\pi$
$812$	0	0
$813$	−15646.3	−0.674956
$814$	0	0
$815$	−16497.9	−0.709075
$816$	0	0
$817$	−23486.0	−1.00572
$818$	0	0
$819$	−3104.51	−0.132455
$820$	0	0
$821$	−17316.1	−0.736097	−0.368049	−	0.929807i	$-0.619974\pi$
$821$	−17316.1	−0.736097	−0.368049	+	0.929807i	$0.619974\pi$
$822$	0	0
$823$	32263.9	1.36653	0.683263	−	0.730173i	$-0.260560\pi$
$823$	32263.9	1.36653	0.683263	+	0.730173i	$0.260560\pi$
$824$	0	0
$825$	−6733.86	−0.284173
$826$	0	0
$827$	33244.3	1.39784	0.698921	−	0.715198i	$-0.253664\pi$
$827$	33244.3	1.39784	0.698921	+	0.715198i	$0.253664\pi$
$828$	0	0
$829$	30822.9	1.29134	0.645672	−	0.763614i	$-0.276577\pi$
$829$	30822.9	1.29134	0.645672	+	0.763614i	$0.276577\pi$
$830$	0	0
$831$	32022.2	1.33675
$832$	0	0
$833$	15349.5	0.638450
$834$	0	0
$835$	407.157	0.0168746
$836$	0	0
$837$	−21383.9	−0.883076
$838$	0	0
$839$	−21723.5	−0.893895	−0.446947	−	0.894560i	$-0.647489\pi$
$839$	−21723.5	−0.893895	−0.446947	+	0.894560i	$0.647489\pi$
$840$	0	0
$841$	12351.1	0.506422
$842$	0	0
$843$	45196.4	1.84655
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	−13204.2	−0.535659
$848$	0	0
$849$	18333.9	0.741130
$850$	0	0
$851$	−62259.0	−2.50789
$852$	0	0
$853$	−15600.6	−0.626206	−0.313103	−	0.949719i	$-0.601369\pi$
$853$	−15600.6	−0.626206	−0.313103	+	0.949719i	$0.601369\pi$
$854$	0	0
$855$	4998.09	0.199919
$856$	0	0
$857$	−34393.9	−1.37091	−0.685457	−	0.728113i	$-0.740398\pi$
$857$	−34393.9	−1.37091	−0.685457	+	0.728113i	$0.740398\pi$
$858$	0	0
$859$	−8181.69	−0.324978	−0.162489	−	0.986710i	$-0.551952\pi$
$859$	−8181.69	−0.324978	−0.162489	+	0.986710i	$0.551952\pi$
$860$	0	0
$861$	−62787.3	−2.48523
$862$	0	0
$863$	3270.88	0.129017	0.0645087	−	0.997917i	$-0.479452\pi$
$863$	3270.88	0.129017	0.0645087	+	0.997917i	$0.479452\pi$
$864$	0	0
$865$	5192.44	0.204102
$866$	0	0
$867$	31216.9	1.22282
$868$	0	0
$869$	−24692.0	−0.963888
$870$	0	0
$871$	−5038.50	−0.196008
$872$	0	0
$873$	−14698.5	−0.569840
$874$	0	0
$875$	2785.36	0.107614
$876$	0	0
$877$	15309.4	0.589466	0.294733	−	0.955580i	$-0.404769\pi$
$877$	15309.4	0.589466	0.294733	+	0.955580i	$0.404769\pi$
$878$	0	0
$879$	59004.7	2.26414
$880$	0	0
$881$	10407.6	0.398003	0.199001	−	0.979999i	$-0.436230\pi$
$881$	10407.6	0.398003	0.199001	+	0.979999i	$0.436230\pi$
$882$	0	0
$883$	39646.5	1.51100	0.755499	−	0.655150i	$-0.227395\pi$
$883$	39646.5	1.51100	0.755499	+	0.655150i	$0.227395\pi$
$884$	0	0
$885$	8103.57	0.307795
$886$	0	0
$887$	33937.4	1.28467	0.642337	−	0.766422i	$-0.277965\pi$
$887$	33937.4	1.28467	0.642337	+	0.766422i	$0.277965\pi$
$888$	0	0
$889$	−42748.0	−1.61273
$890$	0	0
$891$	39626.5	1.48994
$892$	0	0
$893$	−24350.9	−0.912510
$894$	0	0
$895$	10008.5	0.373796
$896$	0	0
$897$	12374.2	0.460604
$898$	0	0
$899$	40988.0	1.52061
$900$	0	0
$901$	15134.4	0.559600
$902$	0	0
$903$	−34458.3	−1.26988
$904$	0	0
$905$	17088.5	0.627670
$906$	0	0
$907$	−31969.4	−1.17037	−0.585185	−	0.810900i	$-0.698978\pi$
$907$	−31969.4	−1.17037	−0.585185	+	0.810900i	$0.698978\pi$
$908$	0	0
$909$	2783.21	0.101555
$910$	0	0
$911$	−160.760	−0.00584656	−0.00292328	−	0.999996i	$-0.500931\pi$
$911$	−160.760	−0.00584656	−0.00292328	+	0.999996i	$0.500931\pi$
$912$	0	0
$913$	−41169.5	−1.49235
$914$	0	0
$915$	−19781.6	−0.714711
$916$	0	0
$917$	31822.6	1.14599
$918$	0	0
$919$	−7482.72	−0.268588	−0.134294	−	0.990942i	$-0.542877\pi$
$919$	−7482.72	−0.268588	−0.134294	+	0.990942i	$0.542877\pi$
$920$	0	0
$921$	54362.6	1.94496
$922$	0	0
$923$	7084.87	0.252656
$924$	0	0
$925$	−10042.4	−0.356965
$926$	0	0
$927$	−8027.71	−0.284428
$928$	0	0
$929$	−34614.1	−1.22245	−0.611224	−	0.791458i	$-0.709322\pi$
$929$	−34614.1	−1.22245	−0.611224	+	0.791458i	$0.709322\pi$
$930$	0	0
$931$	−14319.8	−0.504094
$932$	0	0
$933$	−16247.6	−0.570122
$934$	0	0
$935$	21924.9	0.766867
$936$	0	0
$937$	43467.7	1.51551	0.757753	−	0.652542i	$-0.226297\pi$
$937$	43467.7	1.51551	0.757753	+	0.652542i	$0.226297\pi$
$938$	0	0
$939$	−37390.9	−1.29947
$940$	0	0
$941$	4890.69	0.169428	0.0847142	−	0.996405i	$-0.473002\pi$
$941$	4890.69	0.169428	0.0847142	+	0.996405i	$0.473002\pi$
$942$	0	0
$943$	71110.7	2.45566
$944$	0	0
$945$	−11141.4	−0.383525
$946$	0	0
$947$	20043.4	0.687776	0.343888	−	0.939011i	$-0.388256\pi$
$947$	20043.4	0.687776	0.343888	+	0.939011i	$0.388256\pi$
$948$	0	0
$949$	3922.58	0.134175
$950$	0	0
$951$	61082.1	2.08278
$952$	0	0
$953$	−33417.6	−1.13589	−0.567944	−	0.823067i	$-0.692261\pi$
$953$	−33417.6	−1.13589	−0.567944	+	0.823067i	$0.692261\pi$
$954$	0	0
$955$	−12075.2	−0.409155
$956$	0	0
$957$	−51629.1	−1.74392
$958$	0	0
$959$	25161.7	0.847250
$960$	0	0
$961$	15935.9	0.534924
$962$	0	0
$963$	−8542.99	−0.285871
$964$	0	0
$965$	19495.3	0.650336
$966$	0	0
$967$	48557.9	1.61481	0.807403	−	0.590000i	$-0.200872\pi$
$967$	48557.9	1.61481	0.807403	+	0.590000i	$0.200872\pi$
$968$	0	0
$969$	−57271.4	−1.89868
$970$	0	0
$971$	−2384.30	−0.0788009	−0.0394005	−	0.999223i	$-0.512545\pi$
$971$	−2384.30	−0.0788009	−0.0394005	+	0.999223i	$0.512545\pi$
$972$	0	0
$973$	4030.71	0.132804
$974$	0	0
$975$	1995.96	0.0655610
$976$	0	0
$977$	37231.4	1.21918	0.609590	−	0.792717i	$-0.291334\pi$
$977$	37231.4	1.21918	0.609590	+	0.792717i	$0.291334\pi$
$978$	0	0
$979$	−11469.6	−0.374435
$980$	0	0
$981$	6954.87	0.226353
$982$	0	0
$983$	−27677.9	−0.898055	−0.449027	−	0.893518i	$-0.648229\pi$
$983$	−27677.9	−0.898055	−0.449027	+	0.893518i	$0.648229\pi$
$984$	0	0
$985$	−8323.74	−0.269255
$986$	0	0
$987$	−35727.3	−1.15219
$988$	0	0
$989$	39026.3	1.25477
$990$	0	0
$991$	−22073.3	−0.707548	−0.353774	−	0.935331i	$-0.615102\pi$
$991$	−22073.3	−0.707548	−0.353774	+	0.935331i	$0.615102\pi$
$992$	0	0
$993$	−16797.6	−0.536815
$994$	0	0
$995$	−23358.2	−0.744226
$996$	0	0
$997$	−11349.0	−0.360507	−0.180253	−	0.983620i	$-0.557692\pi$
$997$	−11349.0	−0.360507	−0.180253	+	0.983620i	$0.557692\pi$
$998$	0	0
$999$	40169.7	1.27219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.4.a.h.1.2		2
4.3	odd	2		130.4.a.f.1.1	✓	2
12.11	even	2		1170.4.a.t.1.2		2
20.3	even	4		650.4.b.m.599.1		4
20.7	even	4		650.4.b.m.599.4		4
20.19	odd	2		650.4.a.k.1.2		2
52.51	odd	2		1690.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.4.a.f.1.1	✓	2	4.3	odd	2
650.4.a.k.1.2		2	20.19	odd	2
650.4.b.m.599.1		4	20.3	even	4
650.4.b.m.599.4		4	20.7	even	4
1040.4.a.h.1.2		2	1.1	even	1	trivial
1170.4.a.t.1.2		2	12.11	even	2
1690.4.a.o.1.1		2	52.51	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+6.14143 q^{3} -5.00000 q^{5} -22.2829 q^{7} +10.7171 q^{9} -43.8586 q^{11} +13.0000 q^{13} -30.7071 q^{15} +99.9800 q^{17} -93.2729 q^{19} -136.849 q^{21} +154.990 q^{23} +25.0000 q^{25} -100.000 q^{27} +191.677 q^{29} +213.839 q^{31} -269.354 q^{33} +111.414 q^{35} -401.697 q^{37} +79.8386 q^{39} +458.809 q^{41} +251.799 q^{43} -53.5857 q^{45} +261.071 q^{47} +153.526 q^{49} +614.020 q^{51} +151.374 q^{53} +219.293 q^{55} -572.829 q^{57} -263.899 q^{59} +644.203 q^{61} -238.809 q^{63} -65.0000 q^{65} -387.577 q^{67} +951.860 q^{69} +544.990 q^{71} +301.737 q^{73} +153.536 q^{75} +977.294 q^{77} +562.991 q^{79} -903.506 q^{81} +938.689 q^{83} -499.900 q^{85} +1177.17 q^{87} +261.514 q^{89} -289.677 q^{91} +1313.27 q^{93} +466.364 q^{95} -1371.50 q^{97} -470.039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 10 q^{5} - 16 q^{7} + 50 q^{9} - 102 q^{11} + 26 q^{13} + 10 q^{15} - 58 q^{19} - 188 q^{21} + 210 q^{23} + 50 q^{25} - 200 q^{27} + 12 q^{29} + 242 q^{31} + 204 q^{33} + 80 q^{35} - 632 q^{37}+ \cdots - 2754 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 10 * q^5 - 16 * q^7 + 50 * q^9 - 102 * q^11 + 26 * q^13 + 10 * q^15 - 58 * q^19 - 188 * q^21 + 210 * q^23 + 50 * q^25 - 200 * q^27 + 12 * q^29 + 242 * q^31 + 204 * q^33 + 80 * q^35 - 632 * q^37 - 26 * q^39 + 432 * q^41 - 82 * q^43 - 250 * q^45 - 192 * q^47 - 150 * q^49 + 1428 * q^51 - 240 * q^53 + 510 * q^55 - 860 * q^57 - 942 * q^59 + 460 * q^61 + 8 * q^63 - 130 * q^65 + 596 * q^67 + 504 * q^69 + 990 * q^71 + 832 * q^73 - 50 * q^75 + 612 * q^77 - 388 * q^79 - 1150 * q^81 + 192 * q^83 + 2640 * q^87 + 1380 * q^89 - 208 * q^91 + 1084 * q^93 + 290 * q^95 - 572 * q^97 - 2754 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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