LMFDB - Embedded newform 560.6.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,6,Mod(1,560)] 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("560.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-5,0,50,0,-98,0,91] 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:	$89.8149390953$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$17.3003$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-19.3003 q^{3} +25.0000 q^{5} -49.0000 q^{7} +129.501 q^{9} +10.9039 q^{11} +29.6967 q^{13} -482.507 q^{15} -432.519 q^{17} +956.234 q^{19} +945.715 q^{21} -979.429 q^{23} +625.000 q^{25} +2190.56 q^{27} +996.928 q^{29} -4790.53 q^{31} -210.448 q^{33} -1225.00 q^{35} -1889.95 q^{37} -573.156 q^{39} -1928.56 q^{41} +18079.7 q^{43} +3237.54 q^{45} +28563.5 q^{47} +2401.00 q^{49} +8347.75 q^{51} -287.102 q^{53} +272.597 q^{55} -18455.6 q^{57} -11271.3 q^{59} -32884.4 q^{61} -6345.57 q^{63} +742.418 q^{65} +37022.2 q^{67} +18903.3 q^{69} +63930.2 q^{71} +49142.9 q^{73} -12062.7 q^{75} -534.290 q^{77} -71237.6 q^{79} -73747.2 q^{81} -94396.8 q^{83} -10813.0 q^{85} -19241.0 q^{87} +78631.8 q^{89} -1455.14 q^{91} +92458.7 q^{93} +23905.8 q^{95} +93414.6 q^{97} +1412.07 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} + 50 q^{5} - 98 q^{7} + 91 q^{9} - 415 q^{11} + 429 q^{13} - 125 q^{15} + 1319 q^{17} - 1918 q^{19} + 245 q^{21} + 1334 q^{23} + 1250 q^{25} - 1835 q^{27} - 1131 q^{29} + 5472 q^{31} - 6301 q^{33}+ \cdots + 17810 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 + 50 * q^5 - 98 * q^7 + 91 * q^9 - 415 * q^11 + 429 * q^13 - 125 * q^15 + 1319 * q^17 - 1918 * q^19 + 245 * q^21 + 1334 * q^23 + 1250 * q^25 - 1835 * q^27 - 1131 * q^29 + 5472 * q^31 - 6301 * q^33 - 2450 * q^35 - 9156 * q^37 + 5137 * q^39 - 7822 * q^41 - 2078 * q^43 + 2275 * q^45 + 8507 * q^47 + 4802 * q^49 + 33395 * q^51 - 34242 * q^53 - 10375 * q^55 - 59558 * q^57 - 6952 * q^59 - 45138 * q^61 - 4459 * q^63 + 10725 * q^65 + 54556 * q^67 + 51986 * q^69 + 62272 * q^71 + 40896 * q^73 - 3125 * q^75 + 20335 * q^77 - 62069 * q^79 - 121958 * q^81 - 189600 * q^83 + 32975 * q^85 - 49671 * q^87 + 64190 * q^89 - 21021 * q^91 + 239216 * q^93 - 47950 * q^95 + 155547 * q^97 + 17810 * 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
$3$	−19.3003	−1.23811	−0.619057	−	0.785346i	$-0.712485\pi$
$3$	−19.3003	−1.23811	−0.619057	+	0.785346i	$0.712485\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	129.501	0.532928
$10$	0	0
$11$	10.9039	0.0271706	0.0135853	−	0.999908i	$-0.495676\pi$
$11$	10.9039	0.0271706	0.0135853	+	0.999908i	$0.495676\pi$
$12$	0	0
$13$	29.6967	0.0487360	0.0243680	−	0.999703i	$-0.492243\pi$
$13$	29.6967	0.0487360	0.0243680	+	0.999703i	$0.492243\pi$
$14$	0	0
$15$	−482.507	−0.553702
$16$	0	0
$17$	−432.519	−0.362980	−0.181490	−	0.983393i	$-0.558092\pi$
$17$	−432.519	−0.362980	−0.181490	+	0.983393i	$0.558092\pi$
$18$	0	0
$19$	956.234	0.607687	0.303844	−	0.952722i	$-0.401730\pi$
$19$	956.234	0.607687	0.303844	+	0.952722i	$0.401730\pi$
$20$	0	0
$21$	945.715	0.467963
$22$	0	0
$23$	−979.429	−0.386059	−0.193029	−	0.981193i	$-0.561831\pi$
$23$	−979.429	−0.386059	−0.193029	+	0.981193i	$0.561831\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2190.56	0.578289
$28$	0	0
$29$	996.928	0.220125	0.110062	−	0.993925i	$-0.464895\pi$
$29$	996.928	0.220125	0.110062	+	0.993925i	$0.464895\pi$
$30$	0	0
$31$	−4790.53	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$31$	−4790.53	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$32$	0	0
$33$	−210.448	−0.0336403
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−1889.95	−0.226959	−0.113479	−	0.993540i	$-0.536200\pi$
$37$	−1889.95	−0.226959	−0.113479	+	0.993540i	$0.536200\pi$
$38$	0	0
$39$	−573.156	−0.0603408
$40$	0	0
$41$	−1928.56	−0.179174	−0.0895869	−	0.995979i	$-0.528555\pi$
$41$	−1928.56	−0.179174	−0.0895869	+	0.995979i	$0.528555\pi$
$42$	0	0
$43$	18079.7	1.49115	0.745574	−	0.666422i	$-0.232175\pi$
$43$	18079.7	1.49115	0.745574	+	0.666422i	$0.232175\pi$
$44$	0	0
$45$	3237.54	0.238333
$46$	0	0
$47$	28563.5	1.88611	0.943055	−	0.332635i	$-0.107938\pi$
$47$	28563.5	1.88611	0.943055	+	0.332635i	$0.107938\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	8347.75	0.449411
$52$	0	0
$53$	−287.102	−0.0140393	−0.00701966	−	0.999975i	$-0.502234\pi$
$53$	−287.102	−0.0140393	−0.00701966	+	0.999975i	$0.502234\pi$
$54$	0	0
$55$	272.597	0.0121511
$56$	0	0
$57$	−18455.6	−0.752387
$58$	0	0
$59$	−11271.3	−0.421546	−0.210773	−	0.977535i	$-0.567598\pi$
$59$	−11271.3	−0.421546	−0.210773	+	0.977535i	$0.567598\pi$
$60$	0	0
$61$	−32884.4	−1.13153	−0.565764	−	0.824567i	$-0.691419\pi$
$61$	−32884.4	−1.13153	−0.565764	+	0.824567i	$0.691419\pi$
$62$	0	0
$63$	−6345.57	−0.201428
$64$	0	0
$65$	742.418	0.0217954
$66$	0	0
$67$	37022.2	1.00757	0.503785	−	0.863829i	$-0.331941\pi$
$67$	37022.2	1.00757	0.503785	+	0.863829i	$0.331941\pi$
$68$	0	0
$69$	18903.3	0.477985
$70$	0	0
$71$	63930.2	1.50508	0.752541	−	0.658546i	$-0.228828\pi$
$71$	63930.2	1.50508	0.752541	+	0.658546i	$0.228828\pi$
$72$	0	0
$73$	49142.9	1.07933	0.539664	−	0.841880i	$-0.318551\pi$
$73$	49142.9	1.07933	0.539664	+	0.841880i	$0.318551\pi$
$74$	0	0
$75$	−12062.7	−0.247623
$76$	0	0
$77$	−534.290	−0.0102695
$78$	0	0
$79$	−71237.6	−1.28423	−0.642113	−	0.766610i	$-0.721942\pi$
$79$	−71237.6	−1.28423	−0.642113	+	0.766610i	$0.721942\pi$
$80$	0	0
$81$	−73747.2	−1.24892
$82$	0	0
$83$	−94396.8	−1.50405	−0.752025	−	0.659135i	$-0.770923\pi$
$83$	−94396.8	−1.50405	−0.752025	+	0.659135i	$0.770923\pi$
$84$	0	0
$85$	−10813.0	−0.162330
$86$	0	0
$87$	−19241.0	−0.272540
$88$	0	0
$89$	78631.8	1.05226	0.526130	−	0.850404i	$-0.323642\pi$
$89$	78631.8	1.05226	0.526130	+	0.850404i	$0.323642\pi$
$90$	0	0
$91$	−1455.14	−0.0184205
$92$	0	0
$93$	92458.7	1.10851
$94$	0	0
$95$	23905.8	0.271766
$96$	0	0
$97$	93414.6	1.00806	0.504029	−	0.863687i	$-0.331851\pi$
$97$	93414.6	1.00806	0.504029	+	0.863687i	$0.331851\pi$
$98$	0	0
$99$	1412.07	0.0144800
$100$	0	0
$101$	−14190.4	−0.138417	−0.0692086	−	0.997602i	$-0.522047\pi$
$101$	−14190.4	−0.138417	−0.0692086	+	0.997602i	$0.522047\pi$
$102$	0	0
$103$	−197607.	−1.83531	−0.917657	−	0.397374i	$-0.869921\pi$
$103$	−197607.	−1.83531	−0.917657	+	0.397374i	$0.869921\pi$
$104$	0	0
$105$	23642.9	0.209280
$106$	0	0
$107$	−163104.	−1.37723	−0.688615	−	0.725128i	$-0.741781\pi$
$107$	−163104.	−1.37723	−0.688615	+	0.725128i	$0.741781\pi$
$108$	0	0
$109$	−67208.1	−0.541820	−0.270910	−	0.962605i	$-0.587325\pi$
$109$	−67208.1	−0.541820	−0.270910	+	0.962605i	$0.587325\pi$
$110$	0	0
$111$	36476.6	0.281001
$112$	0	0
$113$	55975.9	0.412387	0.206193	−	0.978511i	$-0.433892\pi$
$113$	55975.9	0.412387	0.206193	+	0.978511i	$0.433892\pi$
$114$	0	0
$115$	−24485.7	−0.172651
$116$	0	0
$117$	3845.77	0.0259728
$118$	0	0
$119$	21193.4	0.137194
$120$	0	0
$121$	−160932.	−0.999262
$122$	0	0
$123$	37221.9	0.221838
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	86091.4	0.473642	0.236821	−	0.971553i	$-0.423894\pi$
$127$	86091.4	0.473642	0.236821	+	0.971553i	$0.423894\pi$
$128$	0	0
$129$	−348944.	−1.84621
$130$	0	0
$131$	−221094.	−1.12564	−0.562820	−	0.826580i	$-0.690283\pi$
$131$	−221094.	−1.12564	−0.562820	+	0.826580i	$0.690283\pi$
$132$	0	0
$133$	−46855.5	−0.229684
$134$	0	0
$135$	54763.9	0.258619
$136$	0	0
$137$	−425173.	−1.93537	−0.967686	−	0.252158i	$-0.918860\pi$
$137$	−425173.	−1.93537	−0.967686	+	0.252158i	$0.918860\pi$
$138$	0	0
$139$	14492.4	0.0636216	0.0318108	−	0.999494i	$-0.489873\pi$
$139$	14492.4	0.0636216	0.0318108	+	0.999494i	$0.489873\pi$
$140$	0	0
$141$	−551285.	−2.33522
$142$	0	0
$143$	323.809	0.00132419
$144$	0	0
$145$	24923.2	0.0984427
$146$	0	0
$147$	−46340.0	−0.176874
$148$	0	0
$149$	−36977.8	−0.136451	−0.0682253	−	0.997670i	$-0.521734\pi$
$149$	−36977.8	−0.136451	−0.0682253	+	0.997670i	$0.521734\pi$
$150$	0	0
$151$	−81428.5	−0.290626	−0.145313	−	0.989386i	$-0.546419\pi$
$151$	−81428.5	−0.290626	−0.145313	+	0.989386i	$0.546419\pi$
$152$	0	0
$153$	−56011.9	−0.193442
$154$	0	0
$155$	−119763.	−0.400401
$156$	0	0
$157$	−113780.	−0.368397	−0.184199	−	0.982889i	$-0.558969\pi$
$157$	−113780.	−0.368397	−0.184199	+	0.982889i	$0.558969\pi$
$158$	0	0
$159$	5541.15	0.0173823
$160$	0	0
$161$	47992.0	0.145917
$162$	0	0
$163$	−440567.	−1.29880	−0.649401	−	0.760446i	$-0.724980\pi$
$163$	−440567.	−1.29880	−0.649401	+	0.760446i	$0.724980\pi$
$164$	0	0
$165$	−5261.20	−0.0150444
$166$	0	0
$167$	−621094.	−1.72332	−0.861661	−	0.507484i	$-0.830576\pi$
$167$	−621094.	−1.72332	−0.861661	+	0.507484i	$0.830576\pi$
$168$	0	0
$169$	−370411.	−0.997625
$170$	0	0
$171$	123834.	0.323854
$172$	0	0
$173$	506925.	1.28774	0.643871	−	0.765134i	$-0.277327\pi$
$173$	506925.	1.28774	0.643871	+	0.765134i	$0.277327\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	217540.	0.521923
$178$	0	0
$179$	−800840.	−1.86816	−0.934078	−	0.357070i	$-0.883776\pi$
$179$	−800840.	−1.86816	−0.934078	+	0.357070i	$0.883776\pi$
$180$	0	0
$181$	−91559.5	−0.207734	−0.103867	−	0.994591i	$-0.533122\pi$
$181$	−91559.5	−0.207734	−0.103867	+	0.994591i	$0.533122\pi$
$182$	0	0
$183$	634678.	1.40096
$184$	0	0
$185$	−47248.8	−0.101499
$186$	0	0
$187$	−4716.13	−0.00986239
$188$	0	0
$189$	−107337.	−0.218573
$190$	0	0
$191$	−409991.	−0.813187	−0.406594	−	0.913609i	$-0.633283\pi$
$191$	−409991.	−0.813187	−0.406594	+	0.913609i	$0.633283\pi$
$192$	0	0
$193$	−65112.4	−0.125826	−0.0629130	−	0.998019i	$-0.520039\pi$
$193$	−65112.4	−0.125826	−0.0629130	+	0.998019i	$0.520039\pi$
$194$	0	0
$195$	−14328.9	−0.0269852
$196$	0	0
$197$	183218.	0.336360	0.168180	−	0.985756i	$-0.446211\pi$
$197$	183218.	0.336360	0.168180	+	0.985756i	$0.446211\pi$
$198$	0	0
$199$	563040.	1.00787	0.503937	−	0.863740i	$-0.331884\pi$
$199$	563040.	1.00787	0.503937	+	0.863740i	$0.331884\pi$
$200$	0	0
$201$	−714539.	−1.24749
$202$	0	0
$203$	−48849.5	−0.0831993
$204$	0	0
$205$	−48214.1	−0.0801290
$206$	0	0
$207$	−126838.	−0.205742
$208$	0	0
$209$	10426.6	0.0165112
$210$	0	0
$211$	564811.	0.873367	0.436684	−	0.899615i	$-0.356153\pi$
$211$	564811.	0.873367	0.436684	+	0.899615i	$0.356153\pi$
$212$	0	0
$213$	−1.23387e6	−1.86346
$214$	0	0
$215$	451993.	0.666862
$216$	0	0
$217$	234736.	0.338400
$218$	0	0
$219$	−948473.	−1.33633
$220$	0	0
$221$	−12844.4	−0.0176902
$222$	0	0
$223$	1.17657e6	1.58437	0.792186	−	0.610280i	$-0.208943\pi$
$223$	1.17657e6	1.58437	0.792186	+	0.610280i	$0.208943\pi$
$224$	0	0
$225$	80938.4	0.106586
$226$	0	0
$227$	−984829.	−1.26852	−0.634258	−	0.773121i	$-0.718694\pi$
$227$	−984829.	−1.26852	−0.634258	+	0.773121i	$0.718694\pi$
$228$	0	0
$229$	174251.	0.219577	0.109789	−	0.993955i	$-0.464983\pi$
$229$	174251.	0.219577	0.109789	+	0.993955i	$0.464983\pi$
$230$	0	0
$231$	10311.9	0.0127148
$232$	0	0
$233$	895160.	1.08022	0.540108	−	0.841596i	$-0.318383\pi$
$233$	895160.	1.08022	0.540108	+	0.841596i	$0.318383\pi$
$234$	0	0
$235$	714088.	0.843494
$236$	0	0
$237$	1.37491e6	1.59002
$238$	0	0
$239$	−250128.	−0.283249	−0.141624	−	0.989920i	$-0.545232\pi$
$239$	−250128.	−0.283249	−0.141624	+	0.989920i	$0.545232\pi$
$240$	0	0
$241$	−824888.	−0.914855	−0.457427	−	0.889247i	$-0.651229\pi$
$241$	−824888.	−0.914855	−0.457427	+	0.889247i	$0.651229\pi$
$242$	0	0
$243$	891039.	0.968012
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	28397.0	0.0296163
$248$	0	0
$249$	1.82189e6	1.86219
$250$	0	0
$251$	−729516.	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$251$	−729516.	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$252$	0	0
$253$	−10679.6	−0.0104894
$254$	0	0
$255$	208694.	0.200983
$256$	0	0
$257$	118644.	0.112050	0.0560252	−	0.998429i	$-0.482157\pi$
$257$	118644.	0.112050	0.0560252	+	0.998429i	$0.482157\pi$
$258$	0	0
$259$	92607.7	0.0857823
$260$	0	0
$261$	129104.	0.117311
$262$	0	0
$263$	766194.	0.683045	0.341523	−	0.939874i	$-0.389057\pi$
$263$	766194.	0.683045	0.341523	+	0.939874i	$0.389057\pi$
$264$	0	0
$265$	−7177.54	−0.00627858
$266$	0	0
$267$	−1.51762e6	−1.30282
$268$	0	0
$269$	−774789.	−0.652834	−0.326417	−	0.945226i	$-0.605841\pi$
$269$	−774789.	−0.652834	−0.326417	+	0.945226i	$0.605841\pi$
$270$	0	0
$271$	1.63371e6	1.35130	0.675648	−	0.737224i	$-0.263864\pi$
$271$	1.63371e6	1.35130	0.675648	+	0.737224i	$0.263864\pi$
$272$	0	0
$273$	28084.6	0.0228067
$274$	0	0
$275$	6814.92	0.00543412
$276$	0	0
$277$	−2.14606e6	−1.68051	−0.840257	−	0.542189i	$-0.817596\pi$
$277$	−2.14606e6	−1.68051	−0.840257	+	0.542189i	$0.817596\pi$
$278$	0	0
$279$	−620381.	−0.477143
$280$	0	0
$281$	913760.	0.690345	0.345173	−	0.938539i	$-0.387820\pi$
$281$	913760.	0.690345	0.345173	+	0.938539i	$0.387820\pi$
$282$	0	0
$283$	1.11656e6	0.828738	0.414369	−	0.910109i	$-0.364002\pi$
$283$	1.11656e6	0.828738	0.414369	+	0.910109i	$0.364002\pi$
$284$	0	0
$285$	−461390.	−0.336477
$286$	0	0
$287$	94499.7	0.0677213
$288$	0	0
$289$	−1.23278e6	−0.868245
$290$	0	0
$291$	−1.80293e6	−1.24809
$292$	0	0
$293$	396556.	0.269858	0.134929	−	0.990855i	$-0.456919\pi$
$293$	396556.	0.269858	0.134929	+	0.990855i	$0.456919\pi$
$294$	0	0
$295$	−281783.	−0.188521
$296$	0	0
$297$	23885.5	0.0157124
$298$	0	0
$299$	−29085.8	−0.0188150
$300$	0	0
$301$	−885907.	−0.563601
$302$	0	0
$303$	273878.	0.171376
$304$	0	0
$305$	−822110.	−0.506034
$306$	0	0
$307$	−1.43537e6	−0.869197	−0.434598	−	0.900624i	$-0.643110\pi$
$307$	−1.43537e6	−0.869197	−0.434598	+	0.900624i	$0.643110\pi$
$308$	0	0
$309$	3.81388e6	2.27233
$310$	0	0
$311$	−1.61306e6	−0.945691	−0.472846	−	0.881145i	$-0.656773\pi$
$311$	−1.61306e6	−0.945691	−0.472846	+	0.881145i	$0.656773\pi$
$312$	0	0
$313$	2.21419e6	1.27748	0.638740	−	0.769423i	$-0.279456\pi$
$313$	2.21419e6	1.27748	0.638740	+	0.769423i	$0.279456\pi$
$314$	0	0
$315$	−158639.	−0.0900813
$316$	0	0
$317$	−2.33004e6	−1.30231	−0.651155	−	0.758944i	$-0.725715\pi$
$317$	−2.33004e6	−1.30231	−0.651155	+	0.758944i	$0.725715\pi$
$318$	0	0
$319$	10870.4	0.00598091
$320$	0	0
$321$	3.14796e6	1.70517
$322$	0	0
$323$	−413590.	−0.220579
$324$	0	0
$325$	18560.5	0.00974721
$326$	0	0
$327$	1.29714e6	0.670836
$328$	0	0
$329$	−1.39961e6	−0.712883
$330$	0	0
$331$	−2.63604e6	−1.32246	−0.661230	−	0.750184i	$-0.729965\pi$
$331$	−2.63604e6	−1.32246	−0.661230	+	0.750184i	$0.729965\pi$
$332$	0	0
$333$	−244752.	−0.120953
$334$	0	0
$335$	925554.	0.450599
$336$	0	0
$337$	1.13473e6	0.544275	0.272138	−	0.962258i	$-0.412269\pi$
$337$	1.13473e6	0.544275	0.272138	+	0.962258i	$0.412269\pi$
$338$	0	0
$339$	−1.08035e6	−0.510582
$340$	0	0
$341$	−52235.3	−0.0243264
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	472582.	0.213761
$346$	0	0
$347$	2.62759e6	1.17148	0.585740	−	0.810499i	$-0.300804\pi$
$347$	2.62759e6	1.17148	0.585740	+	0.810499i	$0.300804\pi$
$348$	0	0
$349$	−3.82656e6	−1.68169	−0.840844	−	0.541278i	$-0.817941\pi$
$349$	−3.82656e6	−1.68169	−0.840844	+	0.541278i	$0.817941\pi$
$350$	0	0
$351$	65052.3	0.0281835
$352$	0	0
$353$	1.49143e6	0.637040	0.318520	−	0.947916i	$-0.396814\pi$
$353$	1.49143e6	0.637040	0.318520	+	0.947916i	$0.396814\pi$
$354$	0	0
$355$	1.59825e6	0.673093
$356$	0	0
$357$	−409040.	−0.169862
$358$	0	0
$359$	406372.	0.166413	0.0832066	−	0.996532i	$-0.473484\pi$
$359$	406372.	0.166413	0.0832066	+	0.996532i	$0.473484\pi$
$360$	0	0
$361$	−1.56172e6	−0.630716
$362$	0	0
$363$	3.10604e6	1.23720
$364$	0	0
$365$	1.22857e6	0.482690
$366$	0	0
$367$	−2.09427e6	−0.811647	−0.405824	−	0.913951i	$-0.633015\pi$
$367$	−2.09427e6	−0.811647	−0.405824	+	0.913951i	$0.633015\pi$
$368$	0	0
$369$	−249752.	−0.0954867
$370$	0	0
$371$	14068.0	0.00530637
$372$	0	0
$373$	2.11130e6	0.785736	0.392868	−	0.919595i	$-0.371483\pi$
$373$	2.11130e6	0.785736	0.392868	+	0.919595i	$0.371483\pi$
$374$	0	0
$375$	−301567.	−0.110740
$376$	0	0
$377$	29605.5	0.0107280
$378$	0	0
$379$	−1.21058e6	−0.432907	−0.216453	−	0.976293i	$-0.569449\pi$
$379$	−1.21058e6	−0.432907	−0.216453	+	0.976293i	$0.569449\pi$
$380$	0	0
$381$	−1.66159e6	−0.586424
$382$	0	0
$383$	994517.	0.346430	0.173215	−	0.984884i	$-0.444584\pi$
$383$	994517.	0.346430	0.173215	+	0.984884i	$0.444584\pi$
$384$	0	0
$385$	−13357.2	−0.00459267
$386$	0	0
$387$	2.34135e6	0.794675
$388$	0	0
$389$	−2.49121e6	−0.834711	−0.417355	−	0.908743i	$-0.637043\pi$
$389$	−2.49121e6	−0.834711	−0.417355	+	0.908743i	$0.637043\pi$
$390$	0	0
$391$	423622.	0.140132
$392$	0	0
$393$	4.26719e6	1.39367
$394$	0	0
$395$	−1.78094e6	−0.574324
$396$	0	0
$397$	2.30834e6	0.735062	0.367531	−	0.930011i	$-0.380203\pi$
$397$	2.30834e6	0.735062	0.367531	+	0.930011i	$0.380203\pi$
$398$	0	0
$399$	904324.	0.284375
$400$	0	0
$401$	−633811.	−0.196833	−0.0984167	−	0.995145i	$-0.531378\pi$
$401$	−633811.	−0.196833	−0.0984167	+	0.995145i	$0.531378\pi$
$402$	0	0
$403$	−142263.	−0.0436345
$404$	0	0
$405$	−1.84368e6	−0.558532
$406$	0	0
$407$	−20607.8	−0.00616660
$408$	0	0
$409$	942669.	0.278645	0.139322	−	0.990247i	$-0.455508\pi$
$409$	942669.	0.278645	0.139322	+	0.990247i	$0.455508\pi$
$410$	0	0
$411$	8.20597e6	2.39621
$412$	0	0
$413$	552296.	0.159330
$414$	0	0
$415$	−2.35992e6	−0.672631
$416$	0	0
$417$	−279709.	−0.0787709
$418$	0	0
$419$	3.93604e6	1.09528	0.547639	−	0.836715i	$-0.315527\pi$
$419$	3.93604e6	1.09528	0.547639	+	0.836715i	$0.315527\pi$
$420$	0	0
$421$	−5.76142e6	−1.58425	−0.792125	−	0.610358i	$-0.791025\pi$
$421$	−5.76142e6	−1.58425	−0.792125	+	0.610358i	$0.791025\pi$
$422$	0	0
$423$	3.69902e6	1.00516
$424$	0	0
$425$	−270325.	−0.0725961
$426$	0	0
$427$	1.61133e6	0.427677
$428$	0	0
$429$	−6249.61	−0.00163949
$430$	0	0
$431$	−1.93127e6	−0.500785	−0.250392	−	0.968144i	$-0.580560\pi$
$431$	−1.93127e6	−0.500785	−0.250392	+	0.968144i	$0.580560\pi$
$432$	0	0
$433$	1.40458e6	0.360021	0.180010	−	0.983665i	$-0.442387\pi$
$433$	1.40458e6	0.360021	0.180010	+	0.983665i	$0.442387\pi$
$434$	0	0
$435$	−481025.	−0.121883
$436$	0	0
$437$	−936563.	−0.234603
$438$	0	0
$439$	3.66469e6	0.907562	0.453781	−	0.891113i	$-0.350075\pi$
$439$	3.66469e6	0.907562	0.453781	+	0.891113i	$0.350075\pi$
$440$	0	0
$441$	310933.	0.0761326
$442$	0	0
$443$	−4.68414e6	−1.13402	−0.567009	−	0.823711i	$-0.691900\pi$
$443$	−4.68414e6	−1.13402	−0.567009	+	0.823711i	$0.691900\pi$
$444$	0	0
$445$	1.96580e6	0.470585
$446$	0	0
$447$	713683.	0.168942
$448$	0	0
$449$	−873525.	−0.204484	−0.102242	−	0.994760i	$-0.532602\pi$
$449$	−873525.	−0.204484	−0.102242	+	0.994760i	$0.532602\pi$
$450$	0	0
$451$	−21028.8	−0.00486826
$452$	0	0
$453$	1.57160e6	0.359828
$454$	0	0
$455$	−36378.5	−0.00823789
$456$	0	0
$457$	5.20487e6	1.16579	0.582894	−	0.812548i	$-0.301920\pi$
$457$	5.20487e6	1.16579	0.582894	+	0.812548i	$0.301920\pi$
$458$	0	0
$459$	−947457.	−0.209908
$460$	0	0
$461$	−1.93474e6	−0.424004	−0.212002	−	0.977269i	$-0.567998\pi$
$461$	−1.93474e6	−0.424004	−0.212002	+	0.977269i	$0.567998\pi$
$462$	0	0
$463$	2.35881e6	0.511375	0.255688	−	0.966759i	$-0.417698\pi$
$463$	2.35881e6	0.511375	0.255688	+	0.966759i	$0.417698\pi$
$464$	0	0
$465$	2.31147e6	0.495742
$466$	0	0
$467$	3.37402e6	0.715905	0.357953	−	0.933740i	$-0.383475\pi$
$467$	3.37402e6	0.715905	0.357953	+	0.933740i	$0.383475\pi$
$468$	0	0
$469$	−1.81409e6	−0.380825
$470$	0	0
$471$	2.19599e6	0.456118
$472$	0	0
$473$	197139.	0.0405154
$474$	0	0
$475$	597646.	0.121537
$476$	0	0
$477$	−37180.1	−0.00748195
$478$	0	0
$479$	−4.74768e6	−0.945459	−0.472729	−	0.881208i	$-0.656731\pi$
$479$	−4.74768e6	−0.945459	−0.472729	+	0.881208i	$0.656731\pi$
$480$	0	0
$481$	−56125.4	−0.0110611
$482$	0	0
$483$	−926260.	−0.180661
$484$	0	0
$485$	2.33536e6	0.450817
$486$	0	0
$487$	−5.24868e6	−1.00283	−0.501416	−	0.865207i	$-0.667187\pi$
$487$	−5.24868e6	−1.00283	−0.501416	+	0.865207i	$0.667187\pi$
$488$	0	0
$489$	8.50307e6	1.60807
$490$	0	0
$491$	4.42605e6	0.828537	0.414269	−	0.910155i	$-0.364037\pi$
$491$	4.42605e6	0.828537	0.414269	+	0.910155i	$0.364037\pi$
$492$	0	0
$493$	−431191.	−0.0799009
$494$	0	0
$495$	35301.7	0.00647564
$496$	0	0
$497$	−3.13258e6	−0.568867
$498$	0	0
$499$	45902.0	0.00825240	0.00412620	−	0.999991i	$-0.498687\pi$
$499$	45902.0	0.00825240	0.00412620	+	0.999991i	$0.498687\pi$
$500$	0	0
$501$	1.19873e7	2.13367
$502$	0	0
$503$	2.04449e6	0.360301	0.180150	−	0.983639i	$-0.442342\pi$
$503$	2.04449e6	0.360301	0.180150	+	0.983639i	$0.442342\pi$
$504$	0	0
$505$	−354759.	−0.0619021
$506$	0	0
$507$	7.14904e6	1.23517
$508$	0	0
$509$	398969.	0.0682566	0.0341283	−	0.999417i	$-0.489135\pi$
$509$	398969.	0.0682566	0.0341283	+	0.999417i	$0.489135\pi$
$510$	0	0
$511$	−2.40800e6	−0.407948
$512$	0	0
$513$	2.09468e6	0.351419
$514$	0	0
$515$	−4.94019e6	−0.820777
$516$	0	0
$517$	311453.	0.0512467
$518$	0	0
$519$	−9.78381e6	−1.59437
$520$	0	0
$521$	1.02501e7	1.65437	0.827187	−	0.561927i	$-0.189940\pi$
$521$	1.02501e7	1.65437	0.827187	+	0.561927i	$0.189940\pi$
$522$	0	0
$523$	−1.51839e6	−0.242733	−0.121367	−	0.992608i	$-0.538728\pi$
$523$	−1.51839e6	−0.242733	−0.121367	+	0.992608i	$0.538728\pi$
$524$	0	0
$525$	591072.	0.0935927
$526$	0	0
$527$	2.07200e6	0.324985
$528$	0	0
$529$	−5.47706e6	−0.850959
$530$	0	0
$531$	−1.45966e6	−0.224654
$532$	0	0
$533$	−57272.1	−0.00873222
$534$	0	0
$535$	−4.07761e6	−0.615916
$536$	0	0
$537$	1.54564e7	2.31299
$538$	0	0
$539$	26180.2	0.00388151
$540$	0	0
$541$	3.59686e6	0.528360	0.264180	−	0.964473i	$-0.414899\pi$
$541$	3.59686e6	0.528360	0.264180	+	0.964473i	$0.414899\pi$
$542$	0	0
$543$	1.76713e6	0.257198
$544$	0	0
$545$	−1.68020e6	−0.242309
$546$	0	0
$547$	7.64729e6	1.09280	0.546398	−	0.837526i	$-0.315999\pi$
$547$	7.64729e6	1.09280	0.546398	+	0.837526i	$0.315999\pi$
$548$	0	0
$549$	−4.25858e6	−0.603023
$550$	0	0
$551$	953296.	0.133767
$552$	0	0
$553$	3.49064e6	0.485392
$554$	0	0
$555$	911916.	0.125667
$556$	0	0
$557$	−8.89209e6	−1.21441	−0.607206	−	0.794544i	$-0.707710\pi$
$557$	−8.89209e6	−1.21441	−0.607206	+	0.794544i	$0.707710\pi$
$558$	0	0
$559$	536909.	0.0726727
$560$	0	0
$561$	91022.8	0.0122108
$562$	0	0
$563$	3.33731e6	0.443737	0.221869	−	0.975077i	$-0.428784\pi$
$563$	3.33731e6	0.443737	0.221869	+	0.975077i	$0.428784\pi$
$564$	0	0
$565$	1.39940e6	0.184425
$566$	0	0
$567$	3.61361e6	0.472046
$568$	0	0
$569$	−1.17027e7	−1.51532	−0.757661	−	0.652648i	$-0.773658\pi$
$569$	−1.17027e7	−1.51532	−0.757661	+	0.652648i	$0.773658\pi$
$570$	0	0
$571$	−685949.	−0.0880444	−0.0440222	−	0.999031i	$-0.514017\pi$
$571$	−685949.	−0.0880444	−0.0440222	+	0.999031i	$0.514017\pi$
$572$	0	0
$573$	7.91294e6	1.00682
$574$	0	0
$575$	−612143.	−0.0772118
$576$	0	0
$577$	−1.10785e7	−1.38530	−0.692648	−	0.721276i	$-0.743556\pi$
$577$	−1.10785e7	−1.38530	−0.692648	+	0.721276i	$0.743556\pi$
$578$	0	0
$579$	1.25669e6	0.155787
$580$	0	0
$581$	4.62544e6	0.568477
$582$	0	0
$583$	−3130.52	−0.000381457 0
$584$	0	0
$585$	96144.3	0.0116154
$586$	0	0
$587$	8.07328e6	0.967063	0.483531	−	0.875327i	$-0.339354\pi$
$587$	8.07328e6	0.967063	0.483531	+	0.875327i	$0.339354\pi$
$588$	0	0
$589$	−4.58087e6	−0.544076
$590$	0	0
$591$	−3.53617e6	−0.416452
$592$	0	0
$593$	1.33847e7	1.56304	0.781521	−	0.623879i	$-0.214444\pi$
$593$	1.33847e7	1.56304	0.781521	+	0.623879i	$0.214444\pi$
$594$	0	0
$595$	529836.	0.0613549
$596$	0	0
$597$	−1.08668e7	−1.24786
$598$	0	0
$599$	2.30111e6	0.262042	0.131021	−	0.991380i	$-0.458174\pi$
$599$	2.30111e6	0.262042	0.131021	+	0.991380i	$0.458174\pi$
$600$	0	0
$601$	5.21404e6	0.588828	0.294414	−	0.955678i	$-0.404876\pi$
$601$	5.21404e6	0.588828	0.294414	+	0.955678i	$0.404876\pi$
$602$	0	0
$603$	4.79443e6	0.536962
$604$	0	0
$605$	−4.02330e6	−0.446883
$606$	0	0
$607$	−1.75294e7	−1.93106	−0.965529	−	0.260296i	$-0.916180\pi$
$607$	−1.75294e7	−1.93106	−0.965529	+	0.260296i	$0.916180\pi$
$608$	0	0
$609$	942809.	0.103010
$610$	0	0
$611$	848243.	0.0919216
$612$	0	0
$613$	−1.70871e7	−1.83661	−0.918304	−	0.395876i	$-0.870441\pi$
$613$	−1.70871e7	−1.83661	−0.918304	+	0.395876i	$0.870441\pi$
$614$	0	0
$615$	930547.	0.0992089
$616$	0	0
$617$	−4.74820e6	−0.502130	−0.251065	−	0.967970i	$-0.580781\pi$
$617$	−4.74820e6	−0.502130	−0.251065	+	0.967970i	$0.580781\pi$
$618$	0	0
$619$	−1.68570e6	−0.176829	−0.0884144	−	0.996084i	$-0.528180\pi$
$619$	−1.68570e6	−0.176829	−0.0884144	+	0.996084i	$0.528180\pi$
$620$	0	0
$621$	−2.14549e6	−0.223253
$622$	0	0
$623$	−3.85296e6	−0.397717
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−201237.	−0.0204428
$628$	0	0
$629$	817441.	0.0823815
$630$	0	0
$631$	8.18052e6	0.817913	0.408957	−	0.912554i	$-0.365893\pi$
$631$	8.18052e6	0.817913	0.408957	+	0.912554i	$0.365893\pi$
$632$	0	0
$633$	−1.09010e7	−1.08133
$634$	0	0
$635$	2.15229e6	0.211819
$636$	0	0
$637$	71301.8	0.00696229
$638$	0	0
$639$	8.27905e6	0.802100
$640$	0	0
$641$	7.88556e6	0.758032	0.379016	−	0.925390i	$-0.376263\pi$
$641$	7.88556e6	0.758032	0.379016	+	0.925390i	$0.376263\pi$
$642$	0	0
$643$	−7.75540e6	−0.739736	−0.369868	−	0.929084i	$-0.620597\pi$
$643$	−7.75540e6	−0.739736	−0.369868	+	0.929084i	$0.620597\pi$
$644$	0	0
$645$	−8.72361e6	−0.825652
$646$	0	0
$647$	7.60702e6	0.714420	0.357210	−	0.934024i	$-0.383728\pi$
$647$	7.60702e6	0.714420	0.357210	+	0.934024i	$0.383728\pi$
$648$	0	0
$649$	−122901.	−0.0114537
$650$	0	0
$651$	−4.53048e6	−0.418978
$652$	0	0
$653$	−1.23713e7	−1.13535	−0.567677	−	0.823252i	$-0.692158\pi$
$653$	−1.23713e7	−1.13535	−0.567677	+	0.823252i	$0.692158\pi$
$654$	0	0
$655$	−5.52736e6	−0.503401
$656$	0	0
$657$	6.36408e6	0.575204
$658$	0	0
$659$	−1.94708e7	−1.74650	−0.873252	−	0.487269i	$-0.837993\pi$
$659$	−1.94708e7	−1.74650	−0.873252	+	0.487269i	$0.837993\pi$
$660$	0	0
$661$	−3.17932e6	−0.283029	−0.141515	−	0.989936i	$-0.545197\pi$
$661$	−3.17932e6	−0.283029	−0.141515	+	0.989936i	$0.545197\pi$
$662$	0	0
$663$	247901.	0.0219025
$664$	0	0
$665$	−1.17139e6	−0.102718
$666$	0	0
$667$	−976420.	−0.0849811
$668$	0	0
$669$	−2.27082e7	−1.96163
$670$	0	0
$671$	−358567.	−0.0307443
$672$	0	0
$673$	−1.79934e7	−1.53136	−0.765678	−	0.643224i	$-0.777597\pi$
$673$	−1.79934e7	−1.53136	−0.765678	+	0.643224i	$0.777597\pi$
$674$	0	0
$675$	1.36910e6	0.115658
$676$	0	0
$677$	1.47781e7	1.23921	0.619606	−	0.784913i	$-0.287292\pi$
$677$	1.47781e7	1.23921	0.619606	+	0.784913i	$0.287292\pi$
$678$	0	0
$679$	−4.57731e6	−0.381010
$680$	0	0
$681$	1.90075e7	1.57057
$682$	0	0
$683$	882971.	0.0724260	0.0362130	−	0.999344i	$-0.488471\pi$
$683$	882971.	0.0724260	0.0362130	+	0.999344i	$0.488471\pi$
$684$	0	0
$685$	−1.06293e7	−0.865525
$686$	0	0
$687$	−3.36310e6	−0.271862
$688$	0	0
$689$	−8525.98	−0.000684221 0
$690$	0	0
$691$	1.56459e7	1.24654	0.623270	−	0.782007i	$-0.285804\pi$
$691$	1.56459e7	1.24654	0.623270	+	0.782007i	$0.285804\pi$
$692$	0	0
$693$	−69191.3	−0.00547291
$694$	0	0
$695$	362311.	0.0284525
$696$	0	0
$697$	834142.	0.0650366
$698$	0	0
$699$	−1.72769e7	−1.33743
$700$	0	0
$701$	−1.39490e7	−1.07213	−0.536067	−	0.844175i	$-0.680091\pi$
$701$	−1.39490e7	−1.07213	−0.536067	+	0.844175i	$0.680091\pi$
$702$	0	0
$703$	−1.80724e6	−0.137920
$704$	0	0
$705$	−1.37821e7	−1.04434
$706$	0	0
$707$	695328.	0.0523168
$708$	0	0
$709$	−7.66584e6	−0.572722	−0.286361	−	0.958122i	$-0.592446\pi$
$709$	−7.66584e6	−0.572722	−0.286361	+	0.958122i	$0.592446\pi$
$710$	0	0
$711$	−9.22538e6	−0.684400
$712$	0	0
$713$	4.69199e6	0.345647
$714$	0	0
$715$	8095.23	0.000592194 0
$716$	0	0
$717$	4.82755e6	0.350694
$718$	0	0
$719$	−1.46181e7	−1.05456	−0.527278	−	0.849693i	$-0.676787\pi$
$719$	−1.46181e7	−1.05456	−0.527278	+	0.849693i	$0.676787\pi$
$720$	0	0
$721$	9.68276e6	0.693683
$722$	0	0
$723$	1.59206e7	1.13270
$724$	0	0
$725$	623080.	0.0440249
$726$	0	0
$727$	−1.17581e7	−0.825089	−0.412544	−	0.910938i	$-0.635360\pi$
$727$	−1.17581e7	−0.825089	−0.412544	+	0.910938i	$0.635360\pi$
$728$	0	0
$729$	723267.	0.0504057
$730$	0	0
$731$	−7.81984e6	−0.541258
$732$	0	0
$733$	−9.42125e6	−0.647662	−0.323831	−	0.946115i	$-0.604971\pi$
$733$	−9.42125e6	−0.647662	−0.323831	+	0.946115i	$0.604971\pi$
$734$	0	0
$735$	−1.15850e6	−0.0791002
$736$	0	0
$737$	403685.	0.0273762
$738$	0	0
$739$	−8.69064e6	−0.585384	−0.292692	−	0.956207i	$-0.594551\pi$
$739$	−8.69064e6	−0.585384	−0.292692	+	0.956207i	$0.594551\pi$
$740$	0	0
$741$	−548071.	−0.0366683
$742$	0	0
$743$	−1.42528e7	−0.947172	−0.473586	−	0.880748i	$-0.657041\pi$
$743$	−1.42528e7	−0.947172	−0.473586	+	0.880748i	$0.657041\pi$
$744$	0	0
$745$	−924445.	−0.0610226
$746$	0	0
$747$	−1.22245e7	−0.801550
$748$	0	0
$749$	7.99212e6	0.520544
$750$	0	0
$751$	−1.37921e7	−0.892343	−0.446172	−	0.894947i	$-0.647213\pi$
$751$	−1.37921e7	−0.892343	−0.446172	+	0.894947i	$0.647213\pi$
$752$	0	0
$753$	1.40799e7	0.904922
$754$	0	0
$755$	−2.03571e6	−0.129972
$756$	0	0
$757$	2.62790e6	0.166674	0.0833371	−	0.996521i	$-0.473442\pi$
$757$	2.62790e6	0.166674	0.0833371	+	0.996521i	$0.473442\pi$
$758$	0	0
$759$	206119.	0.0129871
$760$	0	0
$761$	−7.93120e6	−0.496452	−0.248226	−	0.968702i	$-0.579848\pi$
$761$	−7.93120e6	−0.496452	−0.248226	+	0.968702i	$0.579848\pi$
$762$	0	0
$763$	3.29320e6	0.204789
$764$	0	0
$765$	−1.40030e6	−0.0865101
$766$	0	0
$767$	−334722.	−0.0205445
$768$	0	0
$769$	3.06343e6	0.186806	0.0934032	−	0.995628i	$-0.470225\pi$
$769$	3.06343e6	0.186806	0.0934032	+	0.995628i	$0.470225\pi$
$770$	0	0
$771$	−2.28986e6	−0.138731
$772$	0	0
$773$	1.17714e7	0.708562	0.354281	−	0.935139i	$-0.384726\pi$
$773$	1.17714e7	0.708562	0.354281	+	0.935139i	$0.384726\pi$
$774$	0	0
$775$	−2.99408e6	−0.179065
$776$	0	0
$777$	−1.78736e6	−0.106208
$778$	0	0
$779$	−1.84416e6	−0.108882
$780$	0	0
$781$	697086.	0.0408939
$782$	0	0
$783$	2.18382e6	0.127296
$784$	0	0
$785$	−2.84450e6	−0.164752
$786$	0	0
$787$	−1.28179e7	−0.737701	−0.368851	−	0.929489i	$-0.620249\pi$
$787$	−1.28179e7	−0.737701	−0.368851	+	0.929489i	$0.620249\pi$
$788$	0	0
$789$	−1.47878e7	−0.845689
$790$	0	0
$791$	−2.74282e6	−0.155868
$792$	0	0
$793$	−976559.	−0.0551462
$794$	0	0
$795$	138529.	0.00777360
$796$	0	0
$797$	−3.33470e7	−1.85957	−0.929783	−	0.368108i	$-0.880006\pi$
$797$	−3.33470e7	−1.85957	−0.929783	+	0.368108i	$0.880006\pi$
$798$	0	0
$799$	−1.23543e7	−0.684621
$800$	0	0
$801$	1.01829e7	0.560779
$802$	0	0
$803$	535848.	0.0293260
$804$	0	0
$805$	1.19980e6	0.0652558
$806$	0	0
$807$	1.49537e7	0.808284
$808$	0	0
$809$	1.95295e7	1.04911	0.524554	−	0.851377i	$-0.324232\pi$
$809$	1.95295e7	1.04911	0.524554	+	0.851377i	$0.324232\pi$
$810$	0	0
$811$	2.21959e7	1.18501	0.592504	−	0.805568i	$-0.298139\pi$
$811$	2.21959e7	1.18501	0.592504	+	0.805568i	$0.298139\pi$
$812$	0	0
$813$	−3.15310e7	−1.67306
$814$	0	0
$815$	−1.10142e7	−0.580842
$816$	0	0
$817$	1.72885e7	0.906152
$818$	0	0
$819$	−188443.	−0.00981679
$820$	0	0
$821$	2.37914e7	1.23186	0.615930	−	0.787801i	$-0.288780\pi$
$821$	2.37914e7	1.23186	0.615930	+	0.787801i	$0.288780\pi$
$822$	0	0
$823$	1.75911e7	0.905300	0.452650	−	0.891688i	$-0.350479\pi$
$823$	1.75911e7	0.905300	0.452650	+	0.891688i	$0.350479\pi$
$824$	0	0
$825$	−131530.	−0.00672806
$826$	0	0
$827$	1.50096e6	0.0763145	0.0381572	−	0.999272i	$-0.487851\pi$
$827$	1.50096e6	0.0763145	0.0381572	+	0.999272i	$0.487851\pi$
$828$	0	0
$829$	−3.12166e7	−1.57761	−0.788805	−	0.614644i	$-0.789300\pi$
$829$	−3.12166e7	−1.57761	−0.788805	+	0.614644i	$0.789300\pi$
$830$	0	0
$831$	4.14195e7	2.08067
$832$	0	0
$833$	−1.03848e6	−0.0518543
$834$	0	0
$835$	−1.55274e7	−0.770693
$836$	0	0
$837$	−1.04939e7	−0.517755
$838$	0	0
$839$	2.72358e7	1.33578	0.667890	−	0.744260i	$-0.267198\pi$
$839$	2.72358e7	1.33578	0.667890	+	0.744260i	$0.267198\pi$
$840$	0	0
$841$	−1.95173e7	−0.951545
$842$	0	0
$843$	−1.76358e7	−0.854727
$844$	0	0
$845$	−9.26028e6	−0.446151
$846$	0	0
$847$	7.88567e6	0.377685
$848$	0	0
$849$	−2.15500e7	−1.02607
$850$	0	0
$851$	1.85107e6	0.0876193
$852$	0	0
$853$	2.58892e6	0.121828	0.0609140	−	0.998143i	$-0.480598\pi$
$853$	2.58892e6	0.121828	0.0609140	+	0.998143i	$0.480598\pi$
$854$	0	0
$855$	3.09584e6	0.144832
$856$	0	0
$857$	3.19020e7	1.48377	0.741883	−	0.670529i	$-0.233933\pi$
$857$	3.19020e7	1.48377	0.741883	+	0.670529i	$0.233933\pi$
$858$	0	0
$859$	−2.89610e7	−1.33915	−0.669577	−	0.742742i	$-0.733525\pi$
$859$	−2.89610e7	−1.33915	−0.669577	+	0.742742i	$0.733525\pi$
$860$	0	0
$861$	−1.82387e6	−0.0838468
$862$	0	0
$863$	3.43636e7	1.57062	0.785311	−	0.619101i	$-0.212503\pi$
$863$	3.43636e7	1.57062	0.785311	+	0.619101i	$0.212503\pi$
$864$	0	0
$865$	1.26731e7	0.575896
$866$	0	0
$867$	2.37931e7	1.07499
$868$	0	0
$869$	−776766.	−0.0348932
$870$	0	0
$871$	1.09944e6	0.0491049
$872$	0	0
$873$	1.20973e7	0.537222
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	1.55161e7	0.681213	0.340607	−	0.940206i	$-0.389368\pi$
$877$	1.55161e7	0.681213	0.340607	+	0.940206i	$0.389368\pi$
$878$	0	0
$879$	−7.65365e6	−0.334115
$880$	0	0
$881$	−2.13678e7	−0.927515	−0.463757	−	0.885962i	$-0.653499\pi$
$881$	−2.13678e7	−0.927515	−0.463757	+	0.885962i	$0.653499\pi$
$882$	0	0
$883$	−1.70291e7	−0.735003	−0.367502	−	0.930023i	$-0.619787\pi$
$883$	−1.70291e7	−0.735003	−0.367502	+	0.930023i	$0.619787\pi$
$884$	0	0
$885$	5.43850e6	0.233411
$886$	0	0
$887$	−3.88954e7	−1.65993	−0.829964	−	0.557817i	$-0.811639\pi$
$887$	−3.88954e7	−1.65993	−0.829964	+	0.557817i	$0.811639\pi$
$888$	0	0
$889$	−4.21848e6	−0.179020
$890$	0	0
$891$	−804130.	−0.0339338
$892$	0	0
$893$	2.73134e7	1.14617
$894$	0	0
$895$	−2.00210e7	−0.835465
$896$	0	0
$897$	561365.	0.0232951
$898$	0	0
$899$	−4.77582e6	−0.197083
$900$	0	0
$901$	124177.	0.00509600
$902$	0	0
$903$	1.70983e7	0.697803
$904$	0	0
$905$	−2.28899e6	−0.0929014
$906$	0	0
$907$	9.89966e6	0.399578	0.199789	−	0.979839i	$-0.435974\pi$
$907$	9.89966e6	0.399578	0.199789	+	0.979839i	$0.435974\pi$
$908$	0	0
$909$	−1.83767e6	−0.0737664
$910$	0	0
$911$	−8.31180e6	−0.331817	−0.165909	−	0.986141i	$-0.553056\pi$
$911$	−8.31180e6	−0.331817	−0.165909	+	0.986141i	$0.553056\pi$
$912$	0	0
$913$	−1.02929e6	−0.0408659
$914$	0	0
$915$	1.58670e7	0.626529
$916$	0	0
$917$	1.08336e7	0.425452
$918$	0	0
$919$	1.40148e7	0.547393	0.273697	−	0.961816i	$-0.411754\pi$
$919$	1.40148e7	0.547393	0.273697	+	0.961816i	$0.411754\pi$
$920$	0	0
$921$	2.77031e7	1.07617
$922$	0	0
$923$	1.89852e6	0.0733517
$924$	0	0
$925$	−1.18122e6	−0.0453917
$926$	0	0
$927$	−2.55905e7	−0.978090
$928$	0	0
$929$	2.55744e7	0.972225	0.486112	−	0.873896i	$-0.338415\pi$
$929$	2.55744e7	0.972225	0.486112	+	0.873896i	$0.338415\pi$
$930$	0	0
$931$	2.29592e6	0.0868125
$932$	0	0
$933$	3.11325e7	1.17087
$934$	0	0
$935$	−117903.	−0.00441059
$936$	0	0
$937$	4.55745e7	1.69579	0.847896	−	0.530162i	$-0.177869\pi$
$937$	4.55745e7	1.69579	0.847896	+	0.530162i	$0.177869\pi$
$938$	0	0
$939$	−4.27345e7	−1.58167
$940$	0	0
$941$	−1.44078e7	−0.530425	−0.265213	−	0.964190i	$-0.585442\pi$
$941$	−1.44078e7	−0.530425	−0.265213	+	0.964190i	$0.585442\pi$
$942$	0	0
$943$	1.88889e6	0.0691716
$944$	0	0
$945$	−2.68343e6	−0.0977486
$946$	0	0
$947$	4.58169e7	1.66016	0.830082	−	0.557641i	$-0.188293\pi$
$947$	4.58169e7	1.66016	0.830082	+	0.557641i	$0.188293\pi$
$948$	0	0
$949$	1.45938e6	0.0526022
$950$	0	0
$951$	4.49704e7	1.61241
$952$	0	0
$953$	2.52608e7	0.900980	0.450490	−	0.892781i	$-0.351249\pi$
$953$	2.52608e7	0.900980	0.450490	+	0.892781i	$0.351249\pi$
$954$	0	0
$955$	−1.02498e7	−0.363668
$956$	0	0
$957$	−209801.	−0.00740506
$958$	0	0
$959$	2.08335e7	0.731502
$960$	0	0
$961$	−5.67994e6	−0.198397
$962$	0	0
$963$	−2.11223e7	−0.733964
$964$	0	0
$965$	−1.62781e6	−0.0562711
$966$	0	0
$967$	−5.68247e7	−1.95421	−0.977105	−	0.212758i	$-0.931755\pi$
$967$	−5.68247e7	−1.95421	−0.977105	+	0.212758i	$0.931755\pi$
$968$	0	0
$969$	7.98240e6	0.273102
$970$	0	0
$971$	2.46332e7	0.838441	0.419221	−	0.907884i	$-0.362303\pi$
$971$	2.46332e7	0.838441	0.419221	+	0.907884i	$0.362303\pi$
$972$	0	0
$973$	−710130.	−0.0240467
$974$	0	0
$975$	−358222.	−0.0120682
$976$	0	0
$977$	−4.16806e7	−1.39700	−0.698502	−	0.715608i	$-0.746150\pi$
$977$	−4.16806e7	−1.39700	−0.698502	+	0.715608i	$0.746150\pi$
$978$	0	0
$979$	857391.	0.0285905
$980$	0	0
$981$	−8.70355e6	−0.288751
$982$	0	0
$983$	−3.85482e6	−0.127239	−0.0636196	−	0.997974i	$-0.520264\pi$
$983$	−3.85482e6	−0.127239	−0.0636196	+	0.997974i	$0.520264\pi$
$984$	0	0
$985$	4.58046e6	0.150425
$986$	0	0
$987$	2.70129e7	0.882631
$988$	0	0
$989$	−1.77078e7	−0.575671
$990$	0	0
$991$	3.85510e7	1.24696	0.623478	−	0.781841i	$-0.285719\pi$
$991$	3.85510e7	1.24696	0.623478	+	0.781841i	$0.285719\pi$
$992$	0	0
$993$	5.08764e7	1.63736
$994$	0	0
$995$	1.40760e7	0.450735
$996$	0	0
$997$	3.80257e6	0.121155	0.0605773	−	0.998164i	$-0.480706\pi$
$997$	3.80257e6	0.121155	0.0605773	+	0.998164i	$0.480706\pi$
$998$	0	0
$999$	−4.14004e6	−0.131248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.k.1.1		2
4.3	odd	2		70.6.a.h.1.2	✓	2
12.11	even	2		630.6.a.s.1.2		2
20.3	even	4		350.6.c.k.99.2		4
20.7	even	4		350.6.c.k.99.3		4
20.19	odd	2		350.6.a.p.1.1		2
28.27	even	2		490.6.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.h.1.2	✓	2	4.3	odd	2
350.6.a.p.1.1		2	20.19	odd	2
350.6.c.k.99.2		4	20.3	even	4
350.6.c.k.99.3		4	20.7	even	4
490.6.a.u.1.1		2	28.27	even	2
560.6.a.k.1.1		2	1.1	even	1	trivial
630.6.a.s.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-19.3003 q^{3} +25.0000 q^{5} -49.0000 q^{7} +129.501 q^{9} +10.9039 q^{11} +29.6967 q^{13} -482.507 q^{15} -432.519 q^{17} +956.234 q^{19} +945.715 q^{21} -979.429 q^{23} +625.000 q^{25} +2190.56 q^{27} +996.928 q^{29} -4790.53 q^{31} -210.448 q^{33} -1225.00 q^{35} -1889.95 q^{37} -573.156 q^{39} -1928.56 q^{41} +18079.7 q^{43} +3237.54 q^{45} +28563.5 q^{47} +2401.00 q^{49} +8347.75 q^{51} -287.102 q^{53} +272.597 q^{55} -18455.6 q^{57} -11271.3 q^{59} -32884.4 q^{61} -6345.57 q^{63} +742.418 q^{65} +37022.2 q^{67} +18903.3 q^{69} +63930.2 q^{71} +49142.9 q^{73} -12062.7 q^{75} -534.290 q^{77} -71237.6 q^{79} -73747.2 q^{81} -94396.8 q^{83} -10813.0 q^{85} -19241.0 q^{87} +78631.8 q^{89} -1455.14 q^{91} +92458.7 q^{93} +23905.8 q^{95} +93414.6 q^{97} +1412.07 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} + 50 q^{5} - 98 q^{7} + 91 q^{9} - 415 q^{11} + 429 q^{13} - 125 q^{15} + 1319 q^{17} - 1918 q^{19} + 245 q^{21} + 1334 q^{23} + 1250 q^{25} - 1835 q^{27} - 1131 q^{29} + 5472 q^{31} - 6301 q^{33}+ \cdots + 17810 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 + 50 * q^5 - 98 * q^7 + 91 * q^9 - 415 * q^11 + 429 * q^13 - 125 * q^15 + 1319 * q^17 - 1918 * q^19 + 245 * q^21 + 1334 * q^23 + 1250 * q^25 - 1835 * q^27 - 1131 * q^29 + 5472 * q^31 - 6301 * q^33 - 2450 * q^35 - 9156 * q^37 + 5137 * q^39 - 7822 * q^41 - 2078 * q^43 + 2275 * q^45 + 8507 * q^47 + 4802 * q^49 + 33395 * q^51 - 34242 * q^53 - 10375 * q^55 - 59558 * q^57 - 6952 * q^59 - 45138 * q^61 - 4459 * q^63 + 10725 * q^65 + 54556 * q^67 + 51986 * q^69 + 62272 * q^71 + 40896 * q^73 - 3125 * q^75 + 20335 * q^77 - 62069 * q^79 - 121958 * q^81 - 189600 * q^83 + 32975 * q^85 - 49671 * q^87 + 64190 * q^89 - 21021 * q^91 + 239216 * q^93 - 47950 * q^95 + 155547 * q^97 + 17810 * q^99

Introduction

L-functions

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Database

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