LMFDB - Embedded newform 1040.4.a.i.1.2

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)

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")

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);

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,-10,0,0,0,-14] 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{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
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			$4.35890$ 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)

$f(q)$	$=$	$q+3.35890 q^{3} -5.00000 q^{5} -26.1534 q^{7} -15.7178 q^{9} -5.51229 q^{11} +13.0000 q^{13} -16.7945 q^{15} -105.025 q^{17} +112.383 q^{19} -87.8466 q^{21} +74.0767 q^{23} +25.0000 q^{25} -143.485 q^{27} -225.896 q^{29} +205.359 q^{31} -18.5152 q^{33} +130.767 q^{35} +299.485 q^{37} +43.6657 q^{39} -416.203 q^{41} +418.942 q^{43} +78.5890 q^{45} -419.589 q^{47} +341.000 q^{49} -352.767 q^{51} +655.786 q^{53} +27.5615 q^{55} +377.485 q^{57} +221.666 q^{59} +597.687 q^{61} +411.074 q^{63} -65.0000 q^{65} +310.503 q^{67} +248.816 q^{69} +310.334 q^{71} -486.859 q^{73} +83.9725 q^{75} +144.165 q^{77} -52.3501 q^{79} -57.5703 q^{81} +1125.85 q^{83} +525.123 q^{85} -758.761 q^{87} -112.466 q^{89} -339.994 q^{91} +689.780 q^{93} -561.917 q^{95} +159.963 q^{97} +86.6411 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 10 q^{5} - 14 q^{9} + 50 q^{11} + 26 q^{13} + 10 q^{15} - 88 q^{17} + 94 q^{19} - 228 q^{21} + 122 q^{23} + 50 q^{25} - 8 q^{27} - 260 q^{29} + 402 q^{31} - 316 q^{33} + 320 q^{37} - 26 q^{39}+ \cdots + 182 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 10 * q^5 - 14 * q^9 + 50 * q^11 + 26 * q^13 + 10 * q^15 - 88 * q^17 + 94 * q^19 - 228 * q^21 + 122 * q^23 + 50 * q^25 - 8 * q^27 - 260 * q^29 + 402 * q^31 - 316 * q^33 + 320 * q^37 - 26 * q^39 - 536 * q^41 + 62 * q^43 + 70 * q^45 - 752 * q^47 + 682 * q^49 - 444 * q^51 + 248 * q^53 - 250 * q^55 + 476 * q^57 + 330 * q^59 + 620 * q^61 + 456 * q^63 - 130 * q^65 - 460 * q^67 - 8 * q^69 + 734 * q^71 + 456 * q^73 - 50 * q^75 + 1596 * q^77 + 924 * q^79 - 830 * q^81 + 2304 * q^83 + 440 * q^85 - 576 * q^87 - 748 * q^89 - 364 * q^93 - 470 * q^95 + 1924 * q^97 + 182 * 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
$3$	3.35890	0.646420	0.323210	−	0.946327i	$-0.395238\pi$
$3$	3.35890	0.646420	0.323210	+	0.946327i	$0.395238\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−26.1534	−1.41215	−0.706075	−	0.708137i	$-0.749536\pi$
$7$	−26.1534	−1.41215	−0.706075	+	0.708137i	$0.749536\pi$
$8$	0	0
$9$	−15.7178	−0.582141
$10$	0	0
$11$	−5.51229	−0.151093	−0.0755463	−	0.997142i	$-0.524070\pi$
$11$	−5.51229	−0.151093	−0.0755463	+	0.997142i	$0.524070\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−16.7945	−0.289088
$16$	0	0
$17$	−105.025	−1.49836	−0.749182	−	0.662364i	$-0.769553\pi$
$17$	−105.025	−1.49836	−0.749182	+	0.662364i	$0.769553\pi$
$18$	0	0
$19$	112.383	1.35698	0.678488	−	0.734612i	$-0.262636\pi$
$19$	112.383	1.35698	0.678488	+	0.734612i	$0.262636\pi$
$20$	0	0
$21$	−87.8466	−0.912843
$22$	0	0
$23$	74.0767	0.671568	0.335784	−	0.941939i	$-0.390999\pi$
$23$	74.0767	0.671568	0.335784	+	0.941939i	$0.390999\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−143.485	−1.02273
$28$	0	0
$29$	−225.896	−1.44648	−0.723238	−	0.690599i	$-0.757347\pi$
$29$	−225.896	−1.44648	−0.723238	+	0.690599i	$0.757347\pi$
$30$	0	0
$31$	205.359	1.18979	0.594896	−	0.803803i	$-0.297193\pi$
$31$	205.359	1.18979	0.594896	+	0.803803i	$0.297193\pi$
$32$	0	0
$33$	−18.5152	−0.0976693
$34$	0	0
$35$	130.767	0.631533
$36$	0	0
$37$	299.485	1.33068	0.665338	−	0.746542i	$-0.268288\pi$
$37$	299.485	1.33068	0.665338	+	0.746542i	$0.268288\pi$
$38$	0	0
$39$	43.6657	0.179285
$40$	0	0
$41$	−416.203	−1.58536	−0.792682	−	0.609635i	$-0.791316\pi$
$41$	−416.203	−1.58536	−0.792682	+	0.609635i	$0.791316\pi$
$42$	0	0
$43$	418.942	1.48577	0.742884	−	0.669420i	$-0.233457\pi$
$43$	418.942	1.48577	0.742884	+	0.669420i	$0.233457\pi$
$44$	0	0
$45$	78.5890	0.260341
$46$	0	0
$47$	−419.589	−1.30220	−0.651099	−	0.758992i	$-0.725692\pi$
$47$	−419.589	−1.30220	−0.651099	+	0.758992i	$0.725692\pi$
$48$	0	0
$49$	341.000	0.994169
$50$	0	0
$51$	−352.767	−0.968574
$52$	0	0
$53$	655.786	1.69961	0.849803	−	0.527101i	$-0.176721\pi$
$53$	655.786	1.69961	0.849803	+	0.527101i	$0.176721\pi$
$54$	0	0
$55$	27.5615	0.0675707
$56$	0	0
$57$	377.485	0.877177
$58$	0	0
$59$	221.666	0.489126	0.244563	−	0.969633i	$-0.421356\pi$
$59$	221.666	0.489126	0.244563	+	0.969633i	$0.421356\pi$
$60$	0	0
$61$	597.687	1.25452	0.627262	−	0.778808i	$-0.284175\pi$
$61$	597.687	1.25452	0.627262	+	0.778808i	$0.284175\pi$
$62$	0	0
$63$	411.074	0.822070
$64$	0	0
$65$	−65.0000	−0.124035
$66$	0	0
$67$	310.503	0.566180	0.283090	−	0.959093i	$-0.408641\pi$
$67$	310.503	0.566180	0.283090	+	0.959093i	$0.408641\pi$
$68$	0	0
$69$	248.816	0.434115
$70$	0	0
$71$	310.334	0.518731	0.259366	−	0.965779i	$-0.416487\pi$
$71$	310.334	0.518731	0.259366	+	0.965779i	$0.416487\pi$
$72$	0	0
$73$	−486.859	−0.780583	−0.390292	−	0.920691i	$-0.627626\pi$
$73$	−486.859	−0.780583	−0.390292	+	0.920691i	$0.627626\pi$
$74$	0	0
$75$	83.9725	0.129284
$76$	0	0
$77$	144.165	0.213366
$78$	0	0
$79$	−52.3501	−0.0745550	−0.0372775	−	0.999305i	$-0.511869\pi$
$79$	−52.3501	−0.0745550	−0.0372775	+	0.999305i	$0.511869\pi$
$80$	0	0
$81$	−57.5703	−0.0789716
$82$	0	0
$83$	1125.85	1.48889	0.744444	−	0.667685i	$-0.232715\pi$
$83$	1125.85	1.48889	0.744444	+	0.667685i	$0.232715\pi$
$84$	0	0
$85$	525.123	0.670089
$86$	0	0
$87$	−758.761	−0.935031
$88$	0	0
$89$	−112.466	−0.133948	−0.0669740	−	0.997755i	$-0.521334\pi$
$89$	−112.466	−0.133948	−0.0669740	+	0.997755i	$0.521334\pi$
$90$	0	0
$91$	−339.994	−0.391660
$92$	0	0
$93$	689.780	0.769106
$94$	0	0
$95$	−561.917	−0.606858
$96$	0	0
$97$	159.963	0.167441	0.0837203	−	0.996489i	$-0.473320\pi$
$97$	159.963	0.167441	0.0837203	+	0.996489i	$0.473320\pi$
$98$	0	0
$99$	86.6411	0.0879572
$100$	0	0
$101$	408.663	0.402609	0.201304	−	0.979529i	$-0.435482\pi$
$101$	408.663	0.402609	0.201304	+	0.979529i	$0.435482\pi$
$102$	0	0
$103$	1534.30	1.46776	0.733881	−	0.679278i	$-0.237707\pi$
$103$	1534.30	1.46776	0.733881	+	0.679278i	$0.237707\pi$
$104$	0	0
$105$	439.233	0.408236
$106$	0	0
$107$	35.7876	0.0323338	0.0161669	−	0.999869i	$-0.494854\pi$
$107$	35.7876	0.0323338	0.0161669	+	0.999869i	$0.494854\pi$
$108$	0	0
$109$	717.265	0.630289	0.315144	−	0.949044i	$-0.397947\pi$
$109$	717.265	0.630289	0.315144	+	0.949044i	$0.397947\pi$
$110$	0	0
$111$	1005.94	0.860176
$112$	0	0
$113$	−901.835	−0.750774	−0.375387	−	0.926868i	$-0.622490\pi$
$113$	−901.835	−0.750774	−0.375387	+	0.926868i	$0.622490\pi$
$114$	0	0
$115$	−370.383	−0.300334
$116$	0	0
$117$	−204.331	−0.161457
$118$	0	0
$119$	2746.75	2.11592
$120$	0	0
$121$	−1300.61	−0.977171
$122$	0	0
$123$	−1397.98	−1.02481
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1763.63	−1.23226	−0.616129	−	0.787645i	$-0.711300\pi$
$127$	−1763.63	−1.23226	−0.616129	+	0.787645i	$0.711300\pi$
$128$	0	0
$129$	1407.18	0.960431
$130$	0	0
$131$	−415.497	−0.277115	−0.138558	−	0.990354i	$-0.544247\pi$
$131$	−415.497	−0.277115	−0.138558	+	0.990354i	$0.544247\pi$
$132$	0	0
$133$	−2939.21	−1.91625
$134$	0	0
$135$	717.424	0.457378
$136$	0	0
$137$	574.527	0.358286	0.179143	−	0.983823i	$-0.442668\pi$
$137$	574.527	0.358286	0.179143	+	0.983823i	$0.442668\pi$
$138$	0	0
$139$	−2136.85	−1.30392	−0.651962	−	0.758251i	$-0.726054\pi$
$139$	−2136.85	−1.30392	−0.651962	+	0.758251i	$0.726054\pi$
$140$	0	0
$141$	−1409.36	−0.841768
$142$	0	0
$143$	−71.6598	−0.0419056
$144$	0	0
$145$	1129.48	0.646884
$146$	0	0
$147$	1145.38	0.642651
$148$	0	0
$149$	−222.295	−0.122222	−0.0611111	−	0.998131i	$-0.519464\pi$
$149$	−222.295	−0.122222	−0.0611111	+	0.998131i	$0.519464\pi$
$150$	0	0
$151$	−1185.07	−0.638674	−0.319337	−	0.947641i	$-0.603460\pi$
$151$	−1185.07	−0.638674	−0.319337	+	0.947641i	$0.603460\pi$
$152$	0	0
$153$	1650.76	0.872259
$154$	0	0
$155$	−1026.79	−0.532091
$156$	0	0
$157$	2797.58	1.42211	0.711056	−	0.703136i	$-0.248217\pi$
$157$	2797.58	1.42211	0.711056	+	0.703136i	$0.248217\pi$
$158$	0	0
$159$	2202.72	1.09866
$160$	0	0
$161$	−1937.36	−0.948355
$162$	0	0
$163$	−1556.22	−0.747808	−0.373904	−	0.927467i	$-0.621981\pi$
$163$	−1556.22	−0.747808	−0.373904	+	0.927467i	$0.621981\pi$
$164$	0	0
$165$	92.5762	0.0436791
$166$	0	0
$167$	2743.80	1.27139	0.635693	−	0.771942i	$-0.280714\pi$
$167$	2743.80	1.27139	0.635693	+	0.771942i	$0.280714\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1766.42	−0.789951
$172$	0	0
$173$	3563.32	1.56598	0.782989	−	0.622035i	$-0.213694\pi$
$173$	3563.32	1.56598	0.782989	+	0.622035i	$0.213694\pi$
$174$	0	0
$175$	−653.835	−0.282430
$176$	0	0
$177$	744.553	0.316181
$178$	0	0
$179$	−481.054	−0.200870	−0.100435	−	0.994944i	$-0.532023\pi$
$179$	−481.054	−0.200870	−0.100435	+	0.994944i	$0.532023\pi$
$180$	0	0
$181$	4736.18	1.94496	0.972479	−	0.232989i	$-0.0748505\pi$
$181$	4736.18	1.94496	0.972479	+	0.232989i	$0.0748505\pi$
$182$	0	0
$183$	2007.57	0.810951
$184$	0	0
$185$	−1497.42	−0.595096
$186$	0	0
$187$	578.926	0.226392
$188$	0	0
$189$	3752.61	1.44425
$190$	0	0
$191$	−1700.58	−0.644238	−0.322119	−	0.946699i	$-0.604395\pi$
$191$	−1700.58	−0.644238	−0.322119	+	0.946699i	$0.604395\pi$
$192$	0	0
$193$	−1455.94	−0.543009	−0.271505	−	0.962437i	$-0.587521\pi$
$193$	−1455.94	−0.543009	−0.271505	+	0.962437i	$0.587521\pi$
$194$	0	0
$195$	−218.328	−0.0801786
$196$	0	0
$197$	−3617.28	−1.30822	−0.654112	−	0.756397i	$-0.726958\pi$
$197$	−3617.28	−1.30822	−0.654112	+	0.756397i	$0.726958\pi$
$198$	0	0
$199$	3445.88	1.22750	0.613749	−	0.789501i	$-0.289661\pi$
$199$	3445.88	1.22750	0.613749	+	0.789501i	$0.289661\pi$
$200$	0	0
$201$	1042.95	0.365990
$202$	0	0
$203$	5907.94	2.04264
$204$	0	0
$205$	2081.01	0.708996
$206$	0	0
$207$	−1164.32	−0.390947
$208$	0	0
$209$	−619.491	−0.205029
$210$	0	0
$211$	4173.44	1.36166	0.680832	−	0.732440i	$-0.261618\pi$
$211$	4173.44	1.36166	0.680832	+	0.732440i	$0.261618\pi$
$212$	0	0
$213$	1042.38	0.335318
$214$	0	0
$215$	−2094.71	−0.664456
$216$	0	0
$217$	−5370.83	−1.68017
$218$	0	0
$219$	−1635.31	−0.504585
$220$	0	0
$221$	−1365.32	−0.415572
$222$	0	0
$223$	−1103.24	−0.331292	−0.165646	−	0.986185i	$-0.552971\pi$
$223$	−1103.24	−0.331292	−0.165646	+	0.986185i	$0.552971\pi$
$224$	0	0
$225$	−392.945	−0.116428
$226$	0	0
$227$	−2494.39	−0.729333	−0.364667	−	0.931138i	$-0.618817\pi$
$227$	−2494.39	−0.729333	−0.364667	+	0.931138i	$0.618817\pi$
$228$	0	0
$229$	−2443.33	−0.705066	−0.352533	−	0.935799i	$-0.614680\pi$
$229$	−2443.33	−0.705066	−0.352533	+	0.935799i	$0.614680\pi$
$230$	0	0
$231$	484.236	0.137924
$232$	0	0
$233$	6343.70	1.78365	0.891823	−	0.452385i	$-0.149427\pi$
$233$	6343.70	1.78365	0.891823	+	0.452385i	$0.149427\pi$
$234$	0	0
$235$	2097.94	0.582361
$236$	0	0
$237$	−175.839	−0.0481939
$238$	0	0
$239$	−2939.07	−0.795449	−0.397725	−	0.917505i	$-0.630200\pi$
$239$	−2939.07	−0.795449	−0.397725	+	0.917505i	$0.630200\pi$
$240$	0	0
$241$	−4466.81	−1.19391	−0.596955	−	0.802275i	$-0.703623\pi$
$241$	−4466.81	−1.19391	−0.596955	+	0.802275i	$0.703623\pi$
$242$	0	0
$243$	3680.72	0.971679
$244$	0	0
$245$	−1705.00	−0.444606
$246$	0	0
$247$	1460.99	0.376357
$248$	0	0
$249$	3781.60	0.962448
$250$	0	0
$251$	5118.30	1.28711	0.643554	−	0.765400i	$-0.277459\pi$
$251$	5118.30	1.28711	0.643554	+	0.765400i	$0.277459\pi$
$252$	0	0
$253$	−408.332	−0.101469
$254$	0	0
$255$	1763.83	0.433159
$256$	0	0
$257$	747.695	0.181478	0.0907392	−	0.995875i	$-0.471077\pi$
$257$	747.695	0.181478	0.0907392	+	0.995875i	$0.471077\pi$
$258$	0	0
$259$	−7832.54	−1.87911
$260$	0	0
$261$	3550.58	0.842052
$262$	0	0
$263$	2135.10	0.500592	0.250296	−	0.968169i	$-0.419472\pi$
$263$	2135.10	0.500592	0.250296	+	0.968169i	$0.419472\pi$
$264$	0	0
$265$	−3278.93	−0.760087
$266$	0	0
$267$	−377.762	−0.0865868
$268$	0	0
$269$	−780.553	−0.176919	−0.0884594	−	0.996080i	$-0.528194\pi$
$269$	−780.553	−0.176919	−0.0884594	+	0.996080i	$0.528194\pi$
$270$	0	0
$271$	5412.95	1.21333	0.606667	−	0.794956i	$-0.292506\pi$
$271$	5412.95	1.21333	0.606667	+	0.794956i	$0.292506\pi$
$272$	0	0
$273$	−1142.01	−0.253177
$274$	0	0
$275$	−137.807	−0.0302185
$276$	0	0
$277$	−1738.26	−0.377047	−0.188524	−	0.982069i	$-0.560370\pi$
$277$	−1738.26	−0.377047	−0.188524	+	0.982069i	$0.560370\pi$
$278$	0	0
$279$	−3227.79	−0.692626
$280$	0	0
$281$	3923.08	0.832851	0.416426	−	0.909170i	$-0.363283\pi$
$281$	3923.08	0.832851	0.416426	+	0.909170i	$0.363283\pi$
$282$	0	0
$283$	−356.775	−0.0749401	−0.0374701	−	0.999298i	$-0.511930\pi$
$283$	−356.775	−0.0749401	−0.0374701	+	0.999298i	$0.511930\pi$
$284$	0	0
$285$	−1887.42	−0.392285
$286$	0	0
$287$	10885.1	2.23877
$288$	0	0
$289$	6117.16	1.24510
$290$	0	0
$291$	537.298	0.108237
$292$	0	0
$293$	−6936.33	−1.38302	−0.691510	−	0.722367i	$-0.743054\pi$
$293$	−6936.33	−1.38302	−0.691510	+	0.722367i	$0.743054\pi$
$294$	0	0
$295$	−1108.33	−0.218744
$296$	0	0
$297$	790.930	0.154527
$298$	0	0
$299$	962.997	0.186259
$300$	0	0
$301$	−10956.8	−2.09813
$302$	0	0
$303$	1372.66	0.260254
$304$	0	0
$305$	−2988.44	−0.561041
$306$	0	0
$307$	3919.02	0.728568	0.364284	−	0.931288i	$-0.381314\pi$
$307$	3919.02	0.728568	0.364284	+	0.931288i	$0.381314\pi$
$308$	0	0
$309$	5153.58	0.948792
$310$	0	0
$311$	−1817.13	−0.331319	−0.165659	−	0.986183i	$-0.552975\pi$
$311$	−1817.13	−0.331319	−0.165659	+	0.986183i	$0.552975\pi$
$312$	0	0
$313$	1464.09	0.264394	0.132197	−	0.991223i	$-0.457797\pi$
$313$	1464.09	0.264394	0.132197	+	0.991223i	$0.457797\pi$
$314$	0	0
$315$	−2055.37	−0.367641
$316$	0	0
$317$	4610.16	0.816821	0.408411	−	0.912798i	$-0.366083\pi$
$317$	4610.16	0.816821	0.408411	+	0.912798i	$0.366083\pi$
$318$	0	0
$319$	1245.20	0.218552
$320$	0	0
$321$	120.207	0.0209012
$322$	0	0
$323$	−11803.0	−2.03324
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	2409.22	0.407432
$328$	0	0
$329$	10973.7	1.83890
$330$	0	0
$331$	−2243.30	−0.372516	−0.186258	−	0.982501i	$-0.559636\pi$
$331$	−2243.30	−0.372516	−0.186258	+	0.982501i	$0.559636\pi$
$332$	0	0
$333$	−4707.24	−0.774640
$334$	0	0
$335$	−1552.52	−0.253203
$336$	0	0
$337$	−6004.40	−0.970566	−0.485283	−	0.874357i	$-0.661283\pi$
$337$	−6004.40	−0.970566	−0.485283	+	0.874357i	$0.661283\pi$
$338$	0	0
$339$	−3029.17	−0.485316
$340$	0	0
$341$	−1132.00	−0.179769
$342$	0	0
$343$	52.3068	0.00823411
$344$	0	0
$345$	−1244.08	−0.194142
$346$	0	0
$347$	−1757.95	−0.271964	−0.135982	−	0.990711i	$-0.543419\pi$
$347$	−1757.95	−0.271964	−0.135982	+	0.990711i	$0.543419\pi$
$348$	0	0
$349$	−5774.74	−0.885715	−0.442857	−	0.896592i	$-0.646035\pi$
$349$	−5774.74	−0.885715	−0.442857	+	0.896592i	$0.646035\pi$
$350$	0	0
$351$	−1865.30	−0.283654
$352$	0	0
$353$	286.265	0.0431625	0.0215812	−	0.999767i	$-0.493130\pi$
$353$	286.265	0.0431625	0.0215812	+	0.999767i	$0.493130\pi$
$354$	0	0
$355$	−1551.67	−0.231984
$356$	0	0
$357$	9226.05	1.36777
$358$	0	0
$359$	6004.87	0.882799	0.441399	−	0.897311i	$-0.354482\pi$
$359$	6004.87	0.882799	0.441399	+	0.897311i	$0.354482\pi$
$360$	0	0
$361$	5771.05	0.841383
$362$	0	0
$363$	−4368.63	−0.631663
$364$	0	0
$365$	2434.30	0.349087
$366$	0	0
$367$	3793.70	0.539590	0.269795	−	0.962918i	$-0.413044\pi$
$367$	3793.70	0.539590	0.269795	+	0.962918i	$0.413044\pi$
$368$	0	0
$369$	6541.79	0.922905
$370$	0	0
$371$	−17151.0	−2.40010
$372$	0	0
$373$	8923.03	1.23865	0.619326	−	0.785134i	$-0.287406\pi$
$373$	8923.03	1.23865	0.619326	+	0.785134i	$0.287406\pi$
$374$	0	0
$375$	−419.862	−0.0578176
$376$	0	0
$377$	−2936.65	−0.401180
$378$	0	0
$379$	10977.5	1.48781	0.743903	−	0.668287i	$-0.232972\pi$
$379$	10977.5	1.48781	0.743903	+	0.668287i	$0.232972\pi$
$380$	0	0
$381$	−5923.86	−0.796557
$382$	0	0
$383$	−10570.4	−1.41024	−0.705121	−	0.709087i	$-0.749107\pi$
$383$	−10570.4	−1.41024	−0.705121	+	0.709087i	$0.749107\pi$
$384$	0	0
$385$	−720.826	−0.0954200
$386$	0	0
$387$	−6584.85	−0.864926
$388$	0	0
$389$	517.213	0.0674132	0.0337066	−	0.999432i	$-0.489269\pi$
$389$	517.213	0.0674132	0.0337066	+	0.999432i	$0.489269\pi$
$390$	0	0
$391$	−7779.87	−1.00625
$392$	0	0
$393$	−1395.61	−0.179133
$394$	0	0
$395$	261.750	0.0333420
$396$	0	0
$397$	6979.33	0.882323	0.441162	−	0.897428i	$-0.354567\pi$
$397$	6979.33	0.882323	0.441162	+	0.897428i	$0.354567\pi$
$398$	0	0
$399$	−9872.51	−1.23871
$400$	0	0
$401$	3751.66	0.467204	0.233602	−	0.972332i	$-0.424949\pi$
$401$	3751.66	0.467204	0.233602	+	0.972332i	$0.424949\pi$
$402$	0	0
$403$	2669.67	0.329989
$404$	0	0
$405$	287.851	0.0353172
$406$	0	0
$407$	−1650.85	−0.201055
$408$	0	0
$409$	−5059.92	−0.611729	−0.305864	−	0.952075i	$-0.598945\pi$
$409$	−5059.92	−0.611729	−0.305864	+	0.952075i	$0.598945\pi$
$410$	0	0
$411$	1929.78	0.231603
$412$	0	0
$413$	−5797.31	−0.690719
$414$	0	0
$415$	−5629.23	−0.665851
$416$	0	0
$417$	−7177.48	−0.842884
$418$	0	0
$419$	−1758.72	−0.205057	−0.102529	−	0.994730i	$-0.532693\pi$
$419$	−1758.72	−0.205057	−0.102529	+	0.994730i	$0.532693\pi$
$420$	0	0
$421$	3013.57	0.348866	0.174433	−	0.984669i	$-0.444191\pi$
$421$	3013.57	0.348866	0.174433	+	0.984669i	$0.444191\pi$
$422$	0	0
$423$	6595.01	0.758063
$424$	0	0
$425$	−2625.61	−0.299673
$426$	0	0
$427$	−15631.6	−1.77158
$428$	0	0
$429$	−240.698	−0.0270886
$430$	0	0
$431$	−9218.86	−1.03029	−0.515147	−	0.857102i	$-0.672263\pi$
$431$	−9218.86	−1.03029	−0.515147	+	0.857102i	$0.672263\pi$
$432$	0	0
$433$	−14908.9	−1.65468	−0.827340	−	0.561701i	$-0.810147\pi$
$433$	−14908.9	−1.65468	−0.827340	+	0.561701i	$0.810147\pi$
$434$	0	0
$435$	3793.81	0.418159
$436$	0	0
$437$	8325.00	0.911301
$438$	0	0
$439$	14586.5	1.58582	0.792910	−	0.609339i	$-0.208565\pi$
$439$	14586.5	1.58582	0.792910	+	0.609339i	$0.208565\pi$
$440$	0	0
$441$	−5359.77	−0.578746
$442$	0	0
$443$	−1461.45	−0.156740	−0.0783698	−	0.996924i	$-0.524971\pi$
$443$	−1461.45	−0.156740	−0.0783698	+	0.996924i	$0.524971\pi$
$444$	0	0
$445$	562.330	0.0599034
$446$	0	0
$447$	−746.667	−0.0790070
$448$	0	0
$449$	−580.289	−0.0609923	−0.0304961	−	0.999535i	$-0.509709\pi$
$449$	−580.289	−0.0609923	−0.0304961	+	0.999535i	$0.509709\pi$
$450$	0	0
$451$	2294.23	0.239537
$452$	0	0
$453$	−3980.54	−0.412852
$454$	0	0
$455$	1699.97	0.175156
$456$	0	0
$457$	−21.7638	−0.00222772	−0.00111386	−	0.999999i	$-0.500355\pi$
$457$	−21.7638	−0.00222772	−0.00111386	+	0.999999i	$0.500355\pi$
$458$	0	0
$459$	15069.4	1.53242
$460$	0	0
$461$	−14548.5	−1.46983	−0.734917	−	0.678157i	$-0.762779\pi$
$461$	−14548.5	−1.46983	−0.734917	+	0.678157i	$0.762779\pi$
$462$	0	0
$463$	−5988.97	−0.601147	−0.300574	−	0.953759i	$-0.597178\pi$
$463$	−5988.97	−0.601147	−0.300574	+	0.953759i	$0.597178\pi$
$464$	0	0
$465$	−3448.90	−0.343955
$466$	0	0
$467$	6294.85	0.623749	0.311875	−	0.950123i	$-0.399043\pi$
$467$	6294.85	0.623749	0.311875	+	0.950123i	$0.399043\pi$
$468$	0	0
$469$	−8120.72	−0.799531
$470$	0	0
$471$	9396.80	0.919282
$472$	0	0
$473$	−2309.33	−0.224489
$474$	0	0
$475$	2809.59	0.271395
$476$	0	0
$477$	−10307.5	−0.989409
$478$	0	0
$479$	−5353.36	−0.510649	−0.255325	−	0.966855i	$-0.582182\pi$
$479$	−5353.36	−0.510649	−0.255325	+	0.966855i	$0.582182\pi$
$480$	0	0
$481$	3893.30	0.369063
$482$	0	0
$483$	−6507.39	−0.613036
$484$	0	0
$485$	−799.813	−0.0748817
$486$	0	0
$487$	15339.9	1.42735	0.713673	−	0.700479i	$-0.247030\pi$
$487$	15339.9	1.42735	0.713673	+	0.700479i	$0.247030\pi$
$488$	0	0
$489$	−5227.19	−0.483398
$490$	0	0
$491$	−16288.5	−1.49713	−0.748563	−	0.663063i	$-0.769256\pi$
$491$	−16288.5	−1.49713	−0.748563	+	0.663063i	$0.769256\pi$
$492$	0	0
$493$	23724.6	2.16735
$494$	0	0
$495$	−433.206	−0.0393356
$496$	0	0
$497$	−8116.30	−0.732526
$498$	0	0
$499$	−5857.06	−0.525447	−0.262723	−	0.964871i	$-0.584621\pi$
$499$	−5857.06	−0.525447	−0.262723	+	0.964871i	$0.584621\pi$
$500$	0	0
$501$	9216.14	0.821850
$502$	0	0
$503$	4444.45	0.393972	0.196986	−	0.980406i	$-0.436885\pi$
$503$	4444.45	0.393972	0.196986	+	0.980406i	$0.436885\pi$
$504$	0	0
$505$	−2043.31	−0.180052
$506$	0	0
$507$	567.654	0.0497246
$508$	0	0
$509$	−19485.0	−1.69677	−0.848385	−	0.529379i	$-0.822425\pi$
$509$	−19485.0	−1.69677	−0.848385	+	0.529379i	$0.822425\pi$
$510$	0	0
$511$	12733.0	1.10230
$512$	0	0
$513$	−16125.3	−1.38782
$514$	0	0
$515$	−7671.52	−0.656403
$516$	0	0
$517$	2312.90	0.196753
$518$	0	0
$519$	11968.8	1.01228
$520$	0	0
$521$	12935.7	1.08776	0.543879	−	0.839164i	$-0.316955\pi$
$521$	12935.7	1.08776	0.543879	+	0.839164i	$0.316955\pi$
$522$	0	0
$523$	−10562.9	−0.883142	−0.441571	−	0.897226i	$-0.645579\pi$
$523$	−10562.9	−0.883142	−0.441571	+	0.897226i	$0.645579\pi$
$524$	0	0
$525$	−2196.17	−0.182569
$526$	0	0
$527$	−21567.7	−1.78274
$528$	0	0
$529$	−6679.64	−0.548997
$530$	0	0
$531$	−3484.10	−0.284740
$532$	0	0
$533$	−5410.63	−0.439701
$534$	0	0
$535$	−178.938	−0.0144601
$536$	0	0
$537$	−1615.81	−0.129846
$538$	0	0
$539$	−1879.69	−0.150212
$540$	0	0
$541$	−13076.9	−1.03922	−0.519610	−	0.854403i	$-0.673923\pi$
$541$	−13076.9	−1.03922	−0.519610	+	0.854403i	$0.673923\pi$
$542$	0	0
$543$	15908.4	1.25726
$544$	0	0
$545$	−3586.32	−0.281874
$546$	0	0
$547$	12729.9	0.995049	0.497524	−	0.867450i	$-0.334243\pi$
$547$	12729.9	0.995049	0.497524	+	0.867450i	$0.334243\pi$
$548$	0	0
$549$	−9394.33	−0.730310
$550$	0	0
$551$	−25387.0	−1.96283
$552$	0	0
$553$	1369.13	0.105283
$554$	0	0
$555$	−5029.70	−0.384682
$556$	0	0
$557$	14621.7	1.11228	0.556142	−	0.831087i	$-0.312281\pi$
$557$	14621.7	1.11228	0.556142	+	0.831087i	$0.312281\pi$
$558$	0	0
$559$	5446.25	0.412078
$560$	0	0
$561$	1944.55	0.146344
$562$	0	0
$563$	−11104.7	−0.831271	−0.415636	−	0.909531i	$-0.636441\pi$
$563$	−11104.7	−0.831271	−0.415636	+	0.909531i	$0.636441\pi$
$564$	0	0
$565$	4509.17	0.335756
$566$	0	0
$567$	1505.66	0.111520
$568$	0	0
$569$	−26150.1	−1.92666	−0.963331	−	0.268316i	$-0.913533\pi$
$569$	−26150.1	−1.92666	−0.963331	+	0.268316i	$0.913533\pi$
$570$	0	0
$571$	6762.85	0.495651	0.247825	−	0.968805i	$-0.420284\pi$
$571$	6762.85	0.495651	0.247825	+	0.968805i	$0.420284\pi$
$572$	0	0
$573$	−5712.07	−0.416449
$574$	0	0
$575$	1851.92	0.134314
$576$	0	0
$577$	−58.4468	−0.00421693	−0.00210847	−	0.999998i	$-0.500671\pi$
$577$	−58.4468	−0.00421693	−0.00210847	+	0.999998i	$0.500671\pi$
$578$	0	0
$579$	−4890.35	−0.351012
$580$	0	0
$581$	−29444.7	−2.10253
$582$	0	0
$583$	−3614.88	−0.256798
$584$	0	0
$585$	1021.66	0.0722057
$586$	0	0
$587$	−11211.0	−0.788290	−0.394145	−	0.919048i	$-0.628959\pi$
$587$	−11211.0	−0.788290	−0.394145	+	0.919048i	$0.628959\pi$
$588$	0	0
$589$	23078.9	1.61452
$590$	0	0
$591$	−12150.1	−0.845663
$592$	0	0
$593$	5124.35	0.354860	0.177430	−	0.984133i	$-0.443222\pi$
$593$	5124.35	0.354860	0.177430	+	0.984133i	$0.443222\pi$
$594$	0	0
$595$	−13733.7	−0.946267
$596$	0	0
$597$	11574.4	0.793479
$598$	0	0
$599$	9235.37	0.629961	0.314981	−	0.949098i	$-0.398002\pi$
$599$	9235.37	0.629961	0.314981	+	0.949098i	$0.398002\pi$
$600$	0	0
$601$	−26519.4	−1.79992	−0.899958	−	0.435977i	$-0.856403\pi$
$601$	−26519.4	−1.79992	−0.899958	+	0.435977i	$0.856403\pi$
$602$	0	0
$603$	−4880.43	−0.329596
$604$	0	0
$605$	6503.07	0.437004
$606$	0	0
$607$	8337.23	0.557492	0.278746	−	0.960365i	$-0.410081\pi$
$607$	8337.23	0.557492	0.278746	+	0.960365i	$0.410081\pi$
$608$	0	0
$609$	19844.2	1.32041
$610$	0	0
$611$	−5454.66	−0.361165
$612$	0	0
$613$	7475.93	0.492578	0.246289	−	0.969196i	$-0.420789\pi$
$613$	7475.93	0.492578	0.246289	+	0.969196i	$0.420789\pi$
$614$	0	0
$615$	6989.91	0.458310
$616$	0	0
$617$	21101.5	1.37684	0.688422	−	0.725310i	$-0.258304\pi$
$617$	21101.5	1.37684	0.688422	+	0.725310i	$0.258304\pi$
$618$	0	0
$619$	−19468.7	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$619$	−19468.7	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$620$	0	0
$621$	−10628.9	−0.686831
$622$	0	0
$623$	2941.37	0.189155
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2080.81	−0.132535
$628$	0	0
$629$	−31453.3	−1.99384
$630$	0	0
$631$	6405.28	0.404105	0.202052	−	0.979375i	$-0.435239\pi$
$631$	6405.28	0.404105	0.202052	+	0.979375i	$0.435239\pi$
$632$	0	0
$633$	14018.1	0.880208
$634$	0	0
$635$	8818.15	0.551083
$636$	0	0
$637$	4433.00	0.275733
$638$	0	0
$639$	−4877.77	−0.301974
$640$	0	0
$641$	−3794.20	−0.233794	−0.116897	−	0.993144i	$-0.537295\pi$
$641$	−3794.20	−0.233794	−0.116897	+	0.993144i	$0.537295\pi$
$642$	0	0
$643$	22637.5	1.38839	0.694196	−	0.719786i	$-0.255760\pi$
$643$	22637.5	1.38839	0.694196	+	0.719786i	$0.255760\pi$
$644$	0	0
$645$	−7035.92	−0.429518
$646$	0	0
$647$	26719.3	1.62356	0.811780	−	0.583964i	$-0.198499\pi$
$647$	26719.3	1.62356	0.811780	+	0.583964i	$0.198499\pi$
$648$	0	0
$649$	−1221.89	−0.0739033
$650$	0	0
$651$	−18040.1	−1.08609
$652$	0	0
$653$	26866.7	1.61007	0.805036	−	0.593226i	$-0.202146\pi$
$653$	26866.7	1.61007	0.805036	+	0.593226i	$0.202146\pi$
$654$	0	0
$655$	2077.48	0.123930
$656$	0	0
$657$	7652.36	0.454409
$658$	0	0
$659$	14165.9	0.837368	0.418684	−	0.908132i	$-0.362491\pi$
$659$	14165.9	0.837368	0.418684	+	0.908132i	$0.362491\pi$
$660$	0	0
$661$	−14186.8	−0.834799	−0.417399	−	0.908723i	$-0.637058\pi$
$661$	−14186.8	−0.834799	−0.417399	+	0.908723i	$0.637058\pi$
$662$	0	0
$663$	−4585.97	−0.268634
$664$	0	0
$665$	14696.0	0.856975
$666$	0	0
$667$	−16733.6	−0.971406
$668$	0	0
$669$	−3705.66	−0.214154
$670$	0	0
$671$	−3294.63	−0.189549
$672$	0	0
$673$	23199.0	1.32876	0.664380	−	0.747395i	$-0.268696\pi$
$673$	23199.0	1.32876	0.664380	+	0.747395i	$0.268696\pi$
$674$	0	0
$675$	−3587.12	−0.204546
$676$	0	0
$677$	28477.3	1.61665	0.808324	−	0.588738i	$-0.200375\pi$
$677$	28477.3	1.61665	0.808324	+	0.588738i	$0.200375\pi$
$678$	0	0
$679$	−4183.56	−0.236451
$680$	0	0
$681$	−8378.42	−0.471456
$682$	0	0
$683$	29441.0	1.64938	0.824691	−	0.565583i	$-0.191349\pi$
$683$	29441.0	1.64938	0.824691	+	0.565583i	$0.191349\pi$
$684$	0	0
$685$	−2872.63	−0.160230
$686$	0	0
$687$	−8206.91	−0.455769
$688$	0	0
$689$	8525.21	0.471386
$690$	0	0
$691$	−10587.3	−0.582863	−0.291431	−	0.956592i	$-0.594132\pi$
$691$	−10587.3	−0.582863	−0.291431	+	0.956592i	$0.594132\pi$
$692$	0	0
$693$	−2265.96	−0.124209
$694$	0	0
$695$	10684.3	0.583133
$696$	0	0
$697$	43711.5	2.37545
$698$	0	0
$699$	21307.8	1.15298
$700$	0	0
$701$	−2837.65	−0.152891	−0.0764456	−	0.997074i	$-0.524357\pi$
$701$	−2837.65	−0.152891	−0.0764456	+	0.997074i	$0.524357\pi$
$702$	0	0
$703$	33657.1	1.80569
$704$	0	0
$705$	7046.79	0.376450
$706$	0	0
$707$	−10687.9	−0.568544
$708$	0	0
$709$	20574.7	1.08984	0.544921	−	0.838487i	$-0.316560\pi$
$709$	20574.7	1.08984	0.544921	+	0.838487i	$0.316560\pi$
$710$	0	0
$711$	822.828	0.0434015
$712$	0	0
$713$	15212.3	0.799026
$714$	0	0
$715$	358.299	0.0187407
$716$	0	0
$717$	−9872.03	−0.514195
$718$	0	0
$719$	23117.7	1.19909	0.599544	−	0.800342i	$-0.295349\pi$
$719$	23117.7	1.19909	0.599544	+	0.800342i	$0.295349\pi$
$720$	0	0
$721$	−40127.3	−2.07270
$722$	0	0
$723$	−15003.6	−0.771768
$724$	0	0
$725$	−5647.39	−0.289295
$726$	0	0
$727$	34214.3	1.74545	0.872723	−	0.488216i	$-0.162352\pi$
$727$	34214.3	1.74545	0.872723	+	0.488216i	$0.162352\pi$
$728$	0	0
$729$	13917.6	0.707085
$730$	0	0
$731$	−43999.2	−2.22622
$732$	0	0
$733$	−9083.99	−0.457742	−0.228871	−	0.973457i	$-0.573503\pi$
$733$	−9083.99	−0.457742	−0.228871	+	0.973457i	$0.573503\pi$
$734$	0	0
$735$	−5726.92	−0.287402
$736$	0	0
$737$	−1711.59	−0.0855456
$738$	0	0
$739$	13479.4	0.670971	0.335485	−	0.942045i	$-0.391100\pi$
$739$	13479.4	0.670971	0.335485	+	0.942045i	$0.391100\pi$
$740$	0	0
$741$	4907.30	0.243285
$742$	0	0
$743$	−27595.0	−1.36253	−0.681266	−	0.732036i	$-0.738570\pi$
$743$	−27595.0	−1.36253	−0.681266	+	0.732036i	$0.738570\pi$
$744$	0	0
$745$	1111.48	0.0546595
$746$	0	0
$747$	−17695.8	−0.866742
$748$	0	0
$749$	−935.966	−0.0456602
$750$	0	0
$751$	−1384.29	−0.0672618	−0.0336309	−	0.999434i	$-0.510707\pi$
$751$	−1384.29	−0.0672618	−0.0336309	+	0.999434i	$0.510707\pi$
$752$	0	0
$753$	17191.9	0.832013
$754$	0	0
$755$	5925.36	0.285624
$756$	0	0
$757$	−16314.4	−0.783301	−0.391650	−	0.920114i	$-0.628096\pi$
$757$	−16314.4	−0.783301	−0.391650	+	0.920114i	$0.628096\pi$
$758$	0	0
$759$	−1371.55	−0.0655916
$760$	0	0
$761$	1939.46	0.0923856	0.0461928	−	0.998933i	$-0.485291\pi$
$761$	1939.46	0.0923856	0.0461928	+	0.998933i	$0.485291\pi$
$762$	0	0
$763$	−18758.9	−0.890063
$764$	0	0
$765$	−8253.78	−0.390086
$766$	0	0
$767$	2881.65	0.135659
$768$	0	0
$769$	−11834.6	−0.554961	−0.277480	−	0.960731i	$-0.589499\pi$
$769$	−11834.6	−0.554961	−0.277480	+	0.960731i	$0.589499\pi$
$770$	0	0
$771$	2511.43	0.117311
$772$	0	0
$773$	−33715.5	−1.56878	−0.784388	−	0.620271i	$-0.787023\pi$
$773$	−33715.5	−1.56878	−0.784388	+	0.620271i	$0.787023\pi$
$774$	0	0
$775$	5133.97	0.237958
$776$	0	0
$777$	−26308.7	−1.21470
$778$	0	0
$779$	−46774.3	−2.15130
$780$	0	0
$781$	−1710.65	−0.0783764
$782$	0	0
$783$	32412.6	1.47935
$784$	0	0
$785$	−13987.9	−0.635987
$786$	0	0
$787$	25730.6	1.16543	0.582716	−	0.812676i	$-0.301990\pi$
$787$	25730.6	1.16543	0.582716	+	0.812676i	$0.301990\pi$
$788$	0	0
$789$	7171.57	0.323593
$790$	0	0
$791$	23586.0	1.06021
$792$	0	0
$793$	7769.94	0.347943
$794$	0	0
$795$	−11013.6	−0.491336
$796$	0	0
$797$	19638.7	0.872823	0.436412	−	0.899747i	$-0.356249\pi$
$797$	19638.7	0.872823	0.436412	+	0.899747i	$0.356249\pi$
$798$	0	0
$799$	44067.2	1.95117
$800$	0	0
$801$	1767.72	0.0779766
$802$	0	0
$803$	2683.71	0.117940
$804$	0	0
$805$	9686.79	0.424117
$806$	0	0
$807$	−2621.80	−0.114364
$808$	0	0
$809$	6908.35	0.300228	0.150114	−	0.988669i	$-0.452036\pi$
$809$	6908.35	0.300228	0.150114	+	0.988669i	$0.452036\pi$
$810$	0	0
$811$	−11961.9	−0.517929	−0.258964	−	0.965887i	$-0.583381\pi$
$811$	−11961.9	−0.517929	−0.258964	+	0.965887i	$0.583381\pi$
$812$	0	0
$813$	18181.6	0.784324
$814$	0	0
$815$	7781.11	0.334430
$816$	0	0
$817$	47082.2	2.01615
$818$	0	0
$819$	5343.96	0.228001
$820$	0	0
$821$	44644.7	1.89782	0.948909	−	0.315549i	$-0.102189\pi$
$821$	44644.7	1.89782	0.948909	+	0.315549i	$0.102189\pi$
$822$	0	0
$823$	−46052.2	−1.95052	−0.975260	−	0.221063i	$-0.929047\pi$
$823$	−46052.2	−1.95052	−0.975260	+	0.221063i	$0.929047\pi$
$824$	0	0
$825$	−462.881	−0.0195339
$826$	0	0
$827$	24966.6	1.04979	0.524894	−	0.851168i	$-0.324105\pi$
$827$	24966.6	1.04979	0.524894	+	0.851168i	$0.324105\pi$
$828$	0	0
$829$	1562.61	0.0654665	0.0327333	−	0.999464i	$-0.489579\pi$
$829$	1562.61	0.0654665	0.0327333	+	0.999464i	$0.489579\pi$
$830$	0	0
$831$	−5838.65	−0.243731
$832$	0	0
$833$	−35813.4	−1.48963
$834$	0	0
$835$	−13719.0	−0.568581
$836$	0	0
$837$	−29465.9	−1.21683
$838$	0	0
$839$	13249.3	0.545191	0.272596	−	0.962129i	$-0.412118\pi$
$839$	13249.3	0.545191	0.272596	+	0.962129i	$0.412118\pi$
$840$	0	0
$841$	26639.9	1.09229
$842$	0	0
$843$	13177.2	0.538372
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	34015.5	1.37991
$848$	0	0
$849$	−1198.37	−0.0484428
$850$	0	0
$851$	22184.8	0.893639
$852$	0	0
$853$	39445.1	1.58332	0.791662	−	0.610959i	$-0.209216\pi$
$853$	39445.1	1.58332	0.791662	+	0.610959i	$0.209216\pi$
$854$	0	0
$855$	8832.10	0.353277
$856$	0	0
$857$	14494.7	0.577749	0.288874	−	0.957367i	$-0.406719\pi$
$857$	14494.7	0.577749	0.288874	+	0.957367i	$0.406719\pi$
$858$	0	0
$859$	−22349.5	−0.887724	−0.443862	−	0.896095i	$-0.646392\pi$
$859$	−22349.5	−0.887724	−0.443862	+	0.896095i	$0.646392\pi$
$860$	0	0
$861$	36562.0	1.44719
$862$	0	0
$863$	12406.4	0.489360	0.244680	−	0.969604i	$-0.421317\pi$
$863$	12406.4	0.489360	0.244680	+	0.969604i	$0.421317\pi$
$864$	0	0
$865$	−17816.6	−0.700327
$866$	0	0
$867$	20546.9	0.804856
$868$	0	0
$869$	288.569	0.0112647
$870$	0	0
$871$	4036.55	0.157030
$872$	0	0
$873$	−2514.26	−0.0974740
$874$	0	0
$875$	3269.17	0.126307
$876$	0	0
$877$	−28964.4	−1.11523	−0.557616	−	0.830099i	$-0.688284\pi$
$877$	−28964.4	−1.11523	−0.557616	+	0.830099i	$0.688284\pi$
$878$	0	0
$879$	−23298.4	−0.894013
$880$	0	0
$881$	33358.6	1.27569	0.637844	−	0.770166i	$-0.279827\pi$
$881$	33358.6	1.27569	0.637844	+	0.770166i	$0.279827\pi$
$882$	0	0
$883$	17482.9	0.666303	0.333151	−	0.942873i	$-0.391888\pi$
$883$	17482.9	0.666303	0.333151	+	0.942873i	$0.391888\pi$
$884$	0	0
$885$	−3722.76	−0.141400
$886$	0	0
$887$	39666.1	1.50153	0.750766	−	0.660568i	$-0.229685\pi$
$887$	39666.1	1.50153	0.750766	+	0.660568i	$0.229685\pi$
$888$	0	0
$889$	46124.9	1.74014
$890$	0	0
$891$	317.344	0.0119320
$892$	0	0
$893$	−47154.9	−1.76705
$894$	0	0
$895$	2405.27	0.0898316
$896$	0	0
$897$	3234.61	0.120402
$898$	0	0
$899$	−46389.7	−1.72100
$900$	0	0
$901$	−68873.6	−2.54663
$902$	0	0
$903$	−36802.6	−1.35627
$904$	0	0
$905$	−23680.9	−0.869812
$906$	0	0
$907$	16181.4	0.592388	0.296194	−	0.955128i	$-0.404283\pi$
$907$	16181.4	0.592388	0.296194	+	0.955128i	$0.404283\pi$
$908$	0	0
$909$	−6423.28	−0.234375
$910$	0	0
$911$	−22957.0	−0.834906	−0.417453	−	0.908699i	$-0.637077\pi$
$911$	−22957.0	−0.834906	−0.417453	+	0.908699i	$0.637077\pi$
$912$	0	0
$913$	−6206.00	−0.224960
$914$	0	0
$915$	−10037.9	−0.362668
$916$	0	0
$917$	10866.6	0.391328
$918$	0	0
$919$	7028.37	0.252279	0.126140	−	0.992012i	$-0.459741\pi$
$919$	7028.37	0.252279	0.126140	+	0.992012i	$0.459741\pi$
$920$	0	0
$921$	13163.6	0.470961
$922$	0	0
$923$	4034.35	0.143870
$924$	0	0
$925$	7487.12	0.266135
$926$	0	0
$927$	−24115.9	−0.854444
$928$	0	0
$929$	6014.14	0.212398	0.106199	−	0.994345i	$-0.466132\pi$
$929$	6014.14	0.212398	0.106199	+	0.994345i	$0.466132\pi$
$930$	0	0
$931$	38322.8	1.34906
$932$	0	0
$933$	−6103.56	−0.214171
$934$	0	0
$935$	−2894.63	−0.101246
$936$	0	0
$937$	−35125.0	−1.22464	−0.612318	−	0.790612i	$-0.709763\pi$
$937$	−35125.0	−1.22464	−0.612318	+	0.790612i	$0.709763\pi$
$938$	0	0
$939$	4917.74	0.170910
$940$	0	0
$941$	−22190.2	−0.768736	−0.384368	−	0.923180i	$-0.625581\pi$
$941$	−22190.2	−0.768736	−0.384368	+	0.923180i	$0.625581\pi$
$942$	0	0
$943$	−30830.9	−1.06468
$944$	0	0
$945$	−18763.1	−0.645886
$946$	0	0
$947$	43661.7	1.49822	0.749111	−	0.662445i	$-0.230481\pi$
$947$	43661.7	1.49822	0.749111	+	0.662445i	$0.230481\pi$
$948$	0	0
$949$	−6329.17	−0.216495
$950$	0	0
$951$	15485.1	0.528010
$952$	0	0
$953$	−11102.5	−0.377381	−0.188691	−	0.982037i	$-0.560424\pi$
$953$	−11102.5	−0.377381	−0.188691	+	0.982037i	$0.560424\pi$
$954$	0	0
$955$	8502.89	0.288112
$956$	0	0
$957$	4182.51	0.141276
$958$	0	0
$959$	−15025.8	−0.505953
$960$	0	0
$961$	12381.3	0.415605
$962$	0	0
$963$	−562.502	−0.0188228
$964$	0	0
$965$	7279.70	0.242841
$966$	0	0
$967$	−36593.6	−1.21693	−0.608465	−	0.793581i	$-0.708214\pi$
$967$	−36593.6	−1.21693	−0.608465	+	0.793581i	$0.708214\pi$
$968$	0	0
$969$	−39645.2	−1.31433
$970$	0	0
$971$	−34303.3	−1.13372	−0.566861	−	0.823813i	$-0.691842\pi$
$971$	−34303.3	−1.13372	−0.566861	+	0.823813i	$0.691842\pi$
$972$	0	0
$973$	55886.0	1.84134
$974$	0	0
$975$	1091.64	0.0358570
$976$	0	0
$977$	14168.1	0.463948	0.231974	−	0.972722i	$-0.425482\pi$
$977$	14168.1	0.463948	0.231974	+	0.972722i	$0.425482\pi$
$978$	0	0
$979$	619.946	0.0202386
$980$	0	0
$981$	−11273.8	−0.366917
$982$	0	0
$983$	49509.4	1.60641	0.803207	−	0.595700i	$-0.203125\pi$
$983$	49509.4	1.60641	0.803207	+	0.595700i	$0.203125\pi$
$984$	0	0
$985$	18086.4	0.585056
$986$	0	0
$987$	36859.5	1.18870
$988$	0	0
$989$	31033.8	0.997794
$990$	0	0
$991$	−45392.1	−1.45502	−0.727511	−	0.686096i	$-0.759323\pi$
$991$	−45392.1	−1.45502	−0.727511	+	0.686096i	$0.759323\pi$
$992$	0	0
$993$	−7535.01	−0.240802
$994$	0	0
$995$	−17229.4	−0.548954
$996$	0	0
$997$	40408.9	1.28361	0.641807	−	0.766866i	$-0.278185\pi$
$997$	40408.9	1.28361	0.641807	+	0.766866i	$0.278185\pi$
$998$	0	0
$999$	−42971.5	−1.36092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.4.a.i.1.2		2
4.3	odd	2		130.4.a.d.1.1	✓	2
12.11	even	2		1170.4.a.ba.1.2		2
20.3	even	4		650.4.b.l.599.3		4
20.7	even	4		650.4.b.l.599.2		4
20.19	odd	2		650.4.a.r.1.2		2
52.51	odd	2		1690.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.4.a.d.1.1	✓	2	4.3	odd	2
650.4.a.r.1.2		2	20.19	odd	2
650.4.b.l.599.2		4	20.7	even	4
650.4.b.l.599.3		4	20.3	even	4
1040.4.a.i.1.2		2	1.1	even	1	trivial
1170.4.a.ba.1.2		2	12.11	even	2
1690.4.a.s.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)

\(f(q)\)	\(=\)	\(q+3.35890 q^{3} -5.00000 q^{5} -26.1534 q^{7} -15.7178 q^{9} -5.51229 q^{11} +13.0000 q^{13} -16.7945 q^{15} -105.025 q^{17} +112.383 q^{19} -87.8466 q^{21} +74.0767 q^{23} +25.0000 q^{25} -143.485 q^{27} -225.896 q^{29} +205.359 q^{31} -18.5152 q^{33} +130.767 q^{35} +299.485 q^{37} +43.6657 q^{39} -416.203 q^{41} +418.942 q^{43} +78.5890 q^{45} -419.589 q^{47} +341.000 q^{49} -352.767 q^{51} +655.786 q^{53} +27.5615 q^{55} +377.485 q^{57} +221.666 q^{59} +597.687 q^{61} +411.074 q^{63} -65.0000 q^{65} +310.503 q^{67} +248.816 q^{69} +310.334 q^{71} -486.859 q^{73} +83.9725 q^{75} +144.165 q^{77} -52.3501 q^{79} -57.5703 q^{81} +1125.85 q^{83} +525.123 q^{85} -758.761 q^{87} -112.466 q^{89} -339.994 q^{91} +689.780 q^{93} -561.917 q^{95} +159.963 q^{97} +86.6411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 10 q^{5} - 14 q^{9} + 50 q^{11} + 26 q^{13} + 10 q^{15} - 88 q^{17} + 94 q^{19} - 228 q^{21} + 122 q^{23} + 50 q^{25} - 8 q^{27} - 260 q^{29} + 402 q^{31} - 316 q^{33} + 320 q^{37} - 26 q^{39}+ \cdots + 182 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 10 * q^5 - 14 * q^9 + 50 * q^11 + 26 * q^13 + 10 * q^15 - 88 * q^17 + 94 * q^19 - 228 * q^21 + 122 * q^23 + 50 * q^25 - 8 * q^27 - 260 * q^29 + 402 * q^31 - 316 * q^33 + 320 * q^37 - 26 * q^39 - 536 * q^41 + 62 * q^43 + 70 * q^45 - 752 * q^47 + 682 * q^49 - 444 * q^51 + 248 * q^53 - 250 * q^55 + 476 * q^57 + 330 * q^59 + 620 * q^61 + 456 * q^63 - 130 * q^65 - 460 * q^67 - 8 * q^69 + 734 * q^71 + 456 * q^73 - 50 * q^75 + 1596 * q^77 + 924 * q^79 - 830 * q^81 + 2304 * q^83 + 440 * q^85 - 576 * q^87 - 748 * q^89 - 364 * q^93 - 470 * q^95 + 1924 * q^97 + 182 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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