LMFDB - Embedded newform 630.6.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [630,6,Mod(1,630)] 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("630.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(630, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	630.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,-50,0,98,-128,0,200,-415] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.041806482$
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:	$\Z[a_1, \ldots, a_{13}]$
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			1.2
Root			$-16.3003$ of defining polynomial
Character	$\chi$	$=$	630.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +100.000 q^{10} +10.9039 q^{11} +29.6967 q^{13} -196.000 q^{14} +256.000 q^{16} +432.519 q^{17} -956.234 q^{19} -400.000 q^{20} -43.6155 q^{22} -979.429 q^{23} +625.000 q^{25} -118.787 q^{26} +784.000 q^{28} -996.928 q^{29} +4790.53 q^{31} -1024.00 q^{32} -1730.08 q^{34} -1225.00 q^{35} -1889.95 q^{37} +3824.94 q^{38} +1600.00 q^{40} +1928.56 q^{41} -18079.7 q^{43} +174.462 q^{44} +3917.72 q^{46} +28563.5 q^{47} +2401.00 q^{49} -2500.00 q^{50} +475.148 q^{52} +287.102 q^{53} -272.597 q^{55} -3136.00 q^{56} +3987.71 q^{58} -11271.3 q^{59} -32884.4 q^{61} -19162.1 q^{62} +4096.00 q^{64} -742.418 q^{65} -37022.2 q^{67} +6920.31 q^{68} +4900.00 q^{70} +63930.2 q^{71} +49142.9 q^{73} +7559.81 q^{74} -15299.7 q^{76} +534.290 q^{77} +71237.6 q^{79} -6400.00 q^{80} -7714.26 q^{82} -94396.8 q^{83} -10813.0 q^{85} +72319.0 q^{86} -697.848 q^{88} -78631.8 q^{89} +1455.14 q^{91} -15670.9 q^{92} -114254. q^{94} +23905.8 q^{95} +93414.6 q^{97} -9604.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 50 q^{5} + 98 q^{7} - 128 q^{8} + 200 q^{10} - 415 q^{11} + 429 q^{13} - 392 q^{14} + 512 q^{16} - 1319 q^{17} + 1918 q^{19} - 800 q^{20} + 1660 q^{22} + 1334 q^{23} + 1250 q^{25}+ \cdots - 19208 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 50 * q^5 + 98 * q^7 - 128 * q^8 + 200 * q^10 - 415 * q^11 + 429 * q^13 - 392 * q^14 + 512 * q^16 - 1319 * q^17 + 1918 * q^19 - 800 * q^20 + 1660 * q^22 + 1334 * q^23 + 1250 * q^25 - 1716 * q^26 + 1568 * q^28 + 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 5276 * q^34 - 2450 * q^35 - 9156 * q^37 - 7672 * q^38 + 3200 * q^40 + 7822 * q^41 + 2078 * q^43 - 6640 * q^44 - 5336 * q^46 + 8507 * q^47 + 4802 * q^49 - 5000 * q^50 + 6864 * q^52 + 34242 * q^53 + 10375 * q^55 - 6272 * q^56 - 4524 * q^58 - 6952 * q^59 - 45138 * q^61 + 21888 * q^62 + 8192 * q^64 - 10725 * q^65 - 54556 * q^67 - 21104 * q^68 + 9800 * q^70 + 62272 * q^71 + 40896 * q^73 + 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 62069 * q^79 - 12800 * q^80 - 31288 * q^82 - 189600 * q^83 + 32975 * q^85 - 8312 * q^86 + 26560 * q^88 - 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 34028 * q^94 - 47950 * q^95 + 155547 * q^97 - 19208 * q^98

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	0	0
$10$	100.000	0.316228
$11$	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	$0.492243\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	432.519	0.362980	0.181490	−	0.983393i	$-0.441908\pi$
$17$	432.519	0.362980	0.181490	+	0.983393i	$0.441908\pi$
$18$	0	0
$19$	−956.234	−0.607687	−0.303844	−	0.952722i	$-0.598270\pi$
$19$	−956.234	−0.607687	−0.303844	+	0.952722i	$0.598270\pi$
$20$	−400.000	−0.223607
$21$	0	0
$22$	−43.6155	−0.0192125
$23$	−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
$26$	−118.787	−0.0344616
$27$	0	0
$28$	784.000	0.188982
$29$	−996.928	−0.220125	−0.110062	−	0.993925i	$-0.535105\pi$
$29$	−996.928	−0.220125	−0.110062	+	0.993925i	$0.535105\pi$
$30$	0	0
$31$	4790.53	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$31$	4790.53	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−1730.08	−0.256666
$35$	−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	$0.536200\pi$
$38$	3824.94	0.429700
$39$	0	0
$40$	1600.00	0.158114
$41$	1928.56	0.179174	0.0895869	−	0.995979i	$-0.471445\pi$
$41$	1928.56	0.179174	0.0895869	+	0.995979i	$0.471445\pi$
$42$	0	0
$43$	−18079.7	−1.49115	−0.745574	−	0.666422i	$-0.767825\pi$
$43$	−18079.7	−1.49115	−0.745574	+	0.666422i	$0.767825\pi$
$44$	174.462	0.0135853
$45$	0	0
$46$	3917.72	0.272985
$47$	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
$50$	−2500.00	−0.141421
$51$	0	0
$52$	475.148	0.0243680
$53$	287.102	0.0140393	0.00701966	−	0.999975i	$-0.497766\pi$
$53$	287.102	0.0140393	0.00701966	+	0.999975i	$0.497766\pi$
$54$	0	0
$55$	−272.597	−0.0121511
$56$	−3136.00	−0.133631
$57$	0	0
$58$	3987.71	0.155652
$59$	−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	$0.691419\pi$
$62$	−19162.1	−0.633089
$63$	0	0
$64$	4096.00	0.125000
$65$	−742.418	−0.0217954
$66$	0	0
$67$	−37022.2	−1.00757	−0.503785	−	0.863829i	$-0.668059\pi$
$67$	−37022.2	−1.00757	−0.503785	+	0.863829i	$0.668059\pi$
$68$	6920.31	0.181490
$69$	0	0
$70$	4900.00	0.119523
$71$	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	$0.318551\pi$
$74$	7559.81	0.160484
$75$	0	0
$76$	−15299.7	−0.303844
$77$	534.290	0.0102695
$78$	0	0
$79$	71237.6	1.28423	0.642113	−	0.766610i	$-0.278058\pi$
$79$	71237.6	1.28423	0.642113	+	0.766610i	$0.278058\pi$
$80$	−6400.00	−0.111803
$81$	0	0
$82$	−7714.26	−0.126695
$83$	−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
$86$	72319.0	1.05440
$87$	0	0
$88$	−697.848	−0.00960625
$89$	−78631.8	−1.05226	−0.526130	−	0.850404i	$-0.676358\pi$
$89$	−78631.8	−1.05226	−0.526130	+	0.850404i	$0.676358\pi$
$90$	0	0
$91$	1455.14	0.0184205
$92$	−15670.9	−0.193029
$93$	0	0
$94$	−114254.	−1.33368
$95$	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	$0.331851\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	10000.0	0.100000
$101$	14190.4	0.138417	0.0692086	−	0.997602i	$-0.477953\pi$
$101$	14190.4	0.138417	0.0692086	+	0.997602i	$0.477953\pi$
$102$	0	0
$103$	197607.	1.83531	0.917657	−	0.397374i	$-0.130079\pi$
$103$	197607.	1.83531	0.917657	+	0.397374i	$0.130079\pi$
$104$	−1900.59	−0.0172308
$105$	0	0
$106$	−1148.41	−0.00992730
$107$	−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	$0.587325\pi$
$110$	1090.39	0.00859209
$111$	0	0
$112$	12544.0	0.0944911
$113$	−55975.9	−0.412387	−0.206193	−	0.978511i	$-0.566108\pi$
$113$	−55975.9	−0.412387	−0.206193	+	0.978511i	$0.566108\pi$
$114$	0	0
$115$	24485.7	0.172651
$116$	−15950.8	−0.110062
$117$	0	0
$118$	45085.4	0.298078
$119$	21193.4	0.137194
$120$	0	0
$121$	−160932.	−0.999262
$122$	131538.	0.800111
$123$	0	0
$124$	76648.5	0.447661
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−86091.4	−0.473642	−0.236821	−	0.971553i	$-0.576106\pi$
$127$	−86091.4	−0.473642	−0.236821	+	0.971553i	$0.576106\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	2969.67	0.0154117
$131$	−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
$134$	148089.	0.712459
$135$	0	0
$136$	−27681.2	−0.128333
$137$	425173.	1.93537	0.967686	−	0.252158i	$-0.0811404\pi$
$137$	425173.	1.93537	0.967686	+	0.252158i	$0.0811404\pi$
$138$	0	0
$139$	−14492.4	−0.0636216	−0.0318108	−	0.999494i	$-0.510127\pi$
$139$	−14492.4	−0.0636216	−0.0318108	+	0.999494i	$0.510127\pi$
$140$	−19600.0	−0.0845154
$141$	0	0
$142$	−255721.	−1.06425
$143$	323.809	0.00132419
$144$	0	0
$145$	24923.2	0.0984427
$146$	−196572.	−0.763201
$147$	0	0
$148$	−30239.2	−0.113479
$149$	36977.8	0.136451	0.0682253	−	0.997670i	$-0.478266\pi$
$149$	36977.8	0.136451	0.0682253	+	0.997670i	$0.478266\pi$
$150$	0	0
$151$	81428.5	0.290626	0.145313	−	0.989386i	$-0.453581\pi$
$151$	81428.5	0.290626	0.145313	+	0.989386i	$0.453581\pi$
$152$	61199.0	0.214850
$153$	0	0
$154$	−2137.16	−0.00726164
$155$	−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	$0.558969\pi$
$158$	−284950.	−0.908085
$159$	0	0
$160$	25600.0	0.0790569
$161$	−47992.0	−0.145917
$162$	0	0
$163$	440567.	1.29880	0.649401	−	0.760446i	$-0.275020\pi$
$163$	440567.	1.29880	0.649401	+	0.760446i	$0.275020\pi$
$164$	30857.0	0.0895869
$165$	0	0
$166$	377587.	1.06352
$167$	−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
$170$	43251.9	0.114784
$171$	0	0
$172$	−289276.	−0.745574
$173$	−506925.	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$173$	−506925.	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	2791.39	0.00679265
$177$	0	0
$178$	314527.	0.744061
$179$	−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	$0.533122\pi$
$182$	−5820.56	−0.0130253
$183$	0	0
$184$	62683.5	0.136492
$185$	47248.8	0.101499
$186$	0	0
$187$	4716.13	0.00986239
$188$	457016.	0.943055
$189$	0	0
$190$	−95623.4	−0.192168
$191$	−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	$0.520039\pi$
$194$	−373658.	−0.712804
$195$	0	0
$196$	38416.0	0.0714286
$197$	−183218.	−0.336360	−0.168180	−	0.985756i	$-0.553789\pi$
$197$	−183218.	−0.336360	−0.168180	+	0.985756i	$0.553789\pi$
$198$	0	0
$199$	−563040.	−1.00787	−0.503937	−	0.863740i	$-0.668116\pi$
$199$	−563040.	−1.00787	−0.503937	+	0.863740i	$0.668116\pi$
$200$	−40000.0	−0.0707107
$201$	0	0
$202$	−56761.5	−0.0978758
$203$	−48849.5	−0.0831993
$204$	0	0
$205$	−48214.1	−0.0801290
$206$	−790430.	−1.29776
$207$	0	0
$208$	7602.36	0.0121840
$209$	−10426.6	−0.0165112
$210$	0	0
$211$	−564811.	−0.873367	−0.436684	−	0.899615i	$-0.643847\pi$
$211$	−564811.	−0.873367	−0.436684	+	0.899615i	$0.643847\pi$
$212$	4593.63	0.00701966
$213$	0	0
$214$	652418.	0.973848
$215$	451993.	0.666862
$216$	0	0
$217$	234736.	0.338400
$218$	268832.	0.383125
$219$	0	0
$220$	−4361.55	−0.00607553
$221$	12844.4	0.0176902
$222$	0	0
$223$	−1.17657e6	−1.58437	−0.792186	−	0.610280i	$-0.791057\pi$
$223$	−1.17657e6	−1.58437	−0.792186	+	0.610280i	$0.791057\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	223903.	0.291601
$227$	−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	$0.464983\pi$
$230$	−97942.9	−0.122083
$231$	0	0
$232$	63803.4	0.0778258
$233$	−895160.	−1.08022	−0.540108	−	0.841596i	$-0.681617\pi$
$233$	−895160.	−1.08022	−0.540108	+	0.841596i	$0.681617\pi$
$234$	0	0
$235$	−714088.	−0.843494
$236$	−180341.	−0.210773
$237$	0	0
$238$	−84773.8	−0.0970106
$239$	−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	$0.651229\pi$
$242$	643728.	0.706585
$243$	0	0
$244$	−526150.	−0.565764
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−28397.0	−0.0296163
$248$	−306594.	−0.316544
$249$	0	0
$250$	62500.0	0.0632456
$251$	−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
$254$	344366.	0.334916
$255$	0	0
$256$	65536.0	0.0625000
$257$	−118644.	−0.112050	−0.0560252	−	0.998429i	$-0.517843\pi$
$257$	−118644.	−0.112050	−0.0560252	+	0.998429i	$0.517843\pi$
$258$	0	0
$259$	−92607.7	−0.0857823
$260$	−11878.7	−0.0108977
$261$	0	0
$262$	884377.	0.795947
$263$	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
$266$	187422.	0.162411
$267$	0	0
$268$	−592355.	−0.503785
$269$	774789.	0.652834	0.326417	−	0.945226i	$-0.394159\pi$
$269$	774789.	0.652834	0.326417	+	0.945226i	$0.394159\pi$
$270$	0	0
$271$	−1.63371e6	−1.35130	−0.675648	−	0.737224i	$-0.736136\pi$
$271$	−1.63371e6	−1.35130	−0.675648	+	0.737224i	$0.736136\pi$
$272$	110725.	0.0907451
$273$	0	0
$274$	−1.70069e6	−1.36851
$275$	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	$0.817596\pi$
$278$	57969.8	0.0449873
$279$	0	0
$280$	78400.0	0.0597614
$281$	−913760.	−0.690345	−0.345173	−	0.938539i	$-0.612180\pi$
$281$	−913760.	−0.690345	−0.345173	+	0.938539i	$0.612180\pi$
$282$	0	0
$283$	−1.11656e6	−0.828738	−0.414369	−	0.910109i	$-0.635998\pi$
$283$	−1.11656e6	−0.828738	−0.414369	+	0.910109i	$0.635998\pi$
$284$	1.02288e6	0.752541
$285$	0	0
$286$	−1295.24	−0.000936341 0
$287$	94499.7	0.0677213
$288$	0	0
$289$	−1.23278e6	−0.868245
$290$	−99692.8	−0.0696095
$291$	0	0
$292$	786287.	0.539664
$293$	−396556.	−0.269858	−0.134929	−	0.990855i	$-0.543081\pi$
$293$	−396556.	−0.269858	−0.134929	+	0.990855i	$0.543081\pi$
$294$	0	0
$295$	281783.	0.188521
$296$	120957.	0.0802420
$297$	0	0
$298$	−147911.	−0.0964852
$299$	−29085.8	−0.0188150
$300$	0	0
$301$	−885907.	−0.563601
$302$	−325714.	−0.205504
$303$	0	0
$304$	−244796.	−0.151922
$305$	822110.	0.506034
$306$	0	0
$307$	1.43537e6	0.869197	0.434598	−	0.900624i	$-0.356890\pi$
$307$	1.43537e6	0.869197	0.434598	+	0.900624i	$0.356890\pi$
$308$	8548.63	0.00513476
$309$	0	0
$310$	479053.	0.283126
$311$	−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	$0.279456\pi$
$314$	455120.	0.260496
$315$	0	0
$316$	1.13980e6	0.642113
$317$	2.33004e6	1.30231	0.651155	−	0.758944i	$-0.274285\pi$
$317$	2.33004e6	1.30231	0.651155	+	0.758944i	$0.274285\pi$
$318$	0	0
$319$	−10870.4	−0.00598091
$320$	−102400.	−0.0559017
$321$	0	0
$322$	191968.	0.103179
$323$	−413590.	−0.220579
$324$	0	0
$325$	18560.5	0.00974721
$326$	−1.76227e6	−0.918391
$327$	0	0
$328$	−123428.	−0.0633475
$329$	1.39961e6	0.712883
$330$	0	0
$331$	2.63604e6	1.32246	0.661230	−	0.750184i	$-0.270035\pi$
$331$	2.63604e6	1.32246	0.661230	+	0.750184i	$0.270035\pi$
$332$	−1.51035e6	−0.752025
$333$	0	0
$334$	2.48438e6	1.21857
$335$	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	$0.412269\pi$
$338$	1.48164e6	0.705427
$339$	0	0
$340$	−173008.	−0.0811649
$341$	52235.3	0.0243264
$342$	0	0
$343$	117649.	0.0539949
$344$	1.15710e6	0.527201
$345$	0	0
$346$	2.02770e6	0.910571
$347$	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	$0.817941\pi$
$350$	−122500.	−0.0534522
$351$	0	0
$352$	−11165.6	−0.00480313
$353$	−1.49143e6	−0.637040	−0.318520	−	0.947916i	$-0.603186\pi$
$353$	−1.49143e6	−0.637040	−0.318520	+	0.947916i	$0.603186\pi$
$354$	0	0
$355$	−1.59825e6	−0.673093
$356$	−1.25811e6	−0.526130
$357$	0	0
$358$	3.20336e6	1.32099
$359$	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
$362$	366238.	0.146890
$363$	0	0
$364$	23282.2	0.00921024
$365$	−1.22857e6	−0.482690
$366$	0	0
$367$	2.09427e6	0.811647	0.405824	−	0.913951i	$-0.366985\pi$
$367$	2.09427e6	0.811647	0.405824	+	0.913951i	$0.366985\pi$
$368$	−250734.	−0.0965147
$369$	0	0
$370$	−188995.	−0.0717706
$371$	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	$0.371483\pi$
$374$	−18864.5	−0.00697376
$375$	0	0
$376$	−1.82807e6	−0.666841
$377$	−29605.5	−0.0107280
$378$	0	0
$379$	1.21058e6	0.432907	0.216453	−	0.976293i	$-0.430551\pi$
$379$	1.21058e6	0.432907	0.216453	+	0.976293i	$0.430551\pi$
$380$	382494.	0.135883
$381$	0	0
$382$	1.63996e6	0.575010
$383$	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
$386$	260449.	0.0889724
$387$	0	0
$388$	1.49463e6	0.504029
$389$	2.49121e6	0.834711	0.417355	−	0.908743i	$-0.362957\pi$
$389$	2.49121e6	0.834711	0.417355	+	0.908743i	$0.362957\pi$
$390$	0	0
$391$	−423622.	−0.140132
$392$	−153664.	−0.0505076
$393$	0	0
$394$	732874.	0.237842
$395$	−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	$0.380203\pi$
$398$	2.25216e6	0.712675
$399$	0	0
$400$	160000.	0.0500000
$401$	633811.	0.196833	0.0984167	−	0.995145i	$-0.468622\pi$
$401$	633811.	0.196833	0.0984167	+	0.995145i	$0.468622\pi$
$402$	0	0
$403$	142263.	0.0436345
$404$	227046.	0.0692086
$405$	0	0
$406$	195398.	0.0588308
$407$	−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	$0.455508\pi$
$410$	192856.	0.0566597
$411$	0	0
$412$	3.16172e6	0.917657
$413$	−552296.	−0.159330
$414$	0	0
$415$	2.35992e6	0.672631
$416$	−30409.4	−0.00861540
$417$	0	0
$418$	41706.6	0.0116752
$419$	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	$0.791025\pi$
$422$	2.25924e6	0.617564
$423$	0	0
$424$	−18374.5	−0.00496365
$425$	270325.	0.0725961
$426$	0	0
$427$	−1.61133e6	−0.427677
$428$	−2.60967e6	−0.688615
$429$	0	0
$430$	−1.80797e6	−0.471543
$431$	−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	$0.442387\pi$
$434$	−938945.	−0.239285
$435$	0	0
$436$	−1.07533e6	−0.270910
$437$	936563.	0.234603
$438$	0	0
$439$	−3.66469e6	−0.907562	−0.453781	−	0.891113i	$-0.649925\pi$
$439$	−3.66469e6	−0.907562	−0.453781	+	0.891113i	$0.649925\pi$
$440$	17446.2	0.00429605
$441$	0	0
$442$	−51377.6	−0.0125089
$443$	−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
$446$	4.70630e6	1.12032
$447$	0	0
$448$	200704.	0.0472456
$449$	873525.	0.204484	0.102242	−	0.994760i	$-0.467398\pi$
$449$	873525.	0.204484	0.102242	+	0.994760i	$0.467398\pi$
$450$	0	0
$451$	21028.8	0.00486826
$452$	−895614.	−0.206193
$453$	0	0
$454$	3.93932e6	0.896977
$455$	−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	$0.301920\pi$
$458$	−697005.	−0.155264
$459$	0	0
$460$	391772.	0.0863254
$461$	1.93474e6	0.424004	0.212002	−	0.977269i	$-0.432002\pi$
$461$	1.93474e6	0.424004	0.212002	+	0.977269i	$0.432002\pi$
$462$	0	0
$463$	−2.35881e6	−0.511375	−0.255688	−	0.966759i	$-0.582302\pi$
$463$	−2.35881e6	−0.511375	−0.255688	+	0.966759i	$0.582302\pi$
$464$	−255213.	−0.0550312
$465$	0	0
$466$	3.58064e6	0.763828
$467$	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
$470$	2.85635e6	0.596441
$471$	0	0
$472$	721366.	0.149039
$473$	−197139.	−0.0405154
$474$	0	0
$475$	−597646.	−0.121537
$476$	339095.	0.0685969
$477$	0	0
$478$	1.00051e6	0.200287
$479$	−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
$482$	3.29955e6	0.646900
$483$	0	0
$484$	−2.57491e6	−0.499631
$485$	−2.33536e6	−0.450817
$486$	0	0
$487$	5.24868e6	1.00283	0.501416	−	0.865207i	$-0.332813\pi$
$487$	5.24868e6	1.00283	0.501416	+	0.865207i	$0.332813\pi$
$488$	2.10460e6	0.400055
$489$	0	0
$490$	240100.	0.0451754
$491$	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
$494$	113588.	0.0209419
$495$	0	0
$496$	1.22638e6	0.223831
$497$	3.13258e6	0.568867
$498$	0	0
$499$	−45902.0	−0.00825240	−0.00412620	−	0.999991i	$-0.501313\pi$
$499$	−45902.0	−0.00825240	−0.00412620	+	0.999991i	$0.501313\pi$
$500$	−250000.	−0.0447214
$501$	0	0
$502$	2.91806e6	0.516815
$503$	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
$506$	42718.3	0.00741716
$507$	0	0
$508$	−1.37746e6	−0.236821
$509$	−398969.	−0.0682566	−0.0341283	−	0.999417i	$-0.510865\pi$
$509$	−398969.	−0.0682566	−0.0341283	+	0.999417i	$0.510865\pi$
$510$	0	0
$511$	2.40800e6	0.407948
$512$	−262144.	−0.0441942
$513$	0	0
$514$	474576.	0.0792315
$515$	−4.94019e6	−0.820777
$516$	0	0
$517$	311453.	0.0512467
$518$	370431.	0.0606572
$519$	0	0
$520$	47514.8	0.00770584
$521$	−1.02501e7	−1.65437	−0.827187	−	0.561927i	$-0.810060\pi$
$521$	−1.02501e7	−1.65437	−0.827187	+	0.561927i	$0.810060\pi$
$522$	0	0
$523$	1.51839e6	0.242733	0.121367	−	0.992608i	$-0.461272\pi$
$523$	1.51839e6	0.242733	0.121367	+	0.992608i	$0.461272\pi$
$524$	−3.53751e6	−0.562820
$525$	0	0
$526$	−3.06478e6	−0.482986
$527$	2.07200e6	0.324985
$528$	0	0
$529$	−5.47706e6	−0.850959
$530$	28710.2	0.00443962
$531$	0	0
$532$	−749687.	−0.114842
$533$	57272.1	0.00873222
$534$	0	0
$535$	4.07761e6	0.615916
$536$	2.36942e6	0.356230
$537$	0	0
$538$	−3.09916e6	−0.461624
$539$	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	$0.414899\pi$
$542$	6.53483e6	0.955511
$543$	0	0
$544$	−442900.	−0.0641665
$545$	1.68020e6	0.242309
$546$	0	0
$547$	−7.64729e6	−1.09280	−0.546398	−	0.837526i	$-0.684001\pi$
$547$	−7.64729e6	−1.09280	−0.546398	+	0.837526i	$0.684001\pi$
$548$	6.80277e6	0.967686
$549$	0	0
$550$	−27259.7	−0.00384250
$551$	953296.	0.133767
$552$	0	0
$553$	3.49064e6	0.485392
$554$	8.58423e6	1.18830
$555$	0	0
$556$	−231879.	−0.0318108
$557$	8.89209e6	1.21441	0.607206	−	0.794544i	$-0.292290\pi$
$557$	8.89209e6	1.21441	0.607206	+	0.794544i	$0.292290\pi$
$558$	0	0
$559$	−536909.	−0.0726727
$560$	−313600.	−0.0422577
$561$	0	0
$562$	3.65504e6	0.488148
$563$	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
$566$	4.46625e6	0.586006
$567$	0	0
$568$	−4.09153e6	−0.532127
$569$	1.17027e7	1.51532	0.757661	−	0.652648i	$-0.226342\pi$
$569$	1.17027e7	1.51532	0.757661	+	0.652648i	$0.226342\pi$
$570$	0	0
$571$	685949.	0.0880444	0.0440222	−	0.999031i	$-0.485983\pi$
$571$	685949.	0.0880444	0.0440222	+	0.999031i	$0.485983\pi$
$572$	5180.95	0.000662093 0
$573$	0	0
$574$	−377999.	−0.0478862
$575$	−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	$0.743556\pi$
$578$	4.93114e6	0.613942
$579$	0	0
$580$	398771.	0.0492214
$581$	−4.62544e6	−0.568477
$582$	0	0
$583$	3130.52	0.000381457 0
$584$	−3.14515e6	−0.381600
$585$	0	0
$586$	1.58622e6	0.190819
$587$	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
$590$	−1.12713e6	−0.133305
$591$	0	0
$592$	−483828.	−0.0567396
$593$	−1.33847e7	−1.56304	−0.781521	−	0.623879i	$-0.785556\pi$
$593$	−1.33847e7	−1.56304	−0.781521	+	0.623879i	$0.785556\pi$
$594$	0	0
$595$	−529836.	−0.0613549
$596$	591645.	0.0682253
$597$	0	0
$598$	116343.	0.0133042
$599$	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	$0.404876\pi$
$602$	3.54363e6	0.398526
$603$	0	0
$604$	1.30286e6	0.145313
$605$	4.02330e6	0.446883
$606$	0	0
$607$	1.75294e7	1.93106	0.965529	−	0.260296i	$-0.0838200\pi$
$607$	1.75294e7	1.93106	0.965529	+	0.260296i	$0.0838200\pi$
$608$	979184.	0.107425
$609$	0	0
$610$	−3.28844e6	−0.357820
$611$	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	$0.870441\pi$
$614$	−5.74148e6	−0.614615
$615$	0	0
$616$	−34194.5	−0.00363082
$617$	4.74820e6	0.502130	0.251065	−	0.967970i	$-0.419219\pi$
$617$	4.74820e6	0.502130	0.251065	+	0.967970i	$0.419219\pi$
$618$	0	0
$619$	1.68570e6	0.176829	0.0884144	−	0.996084i	$-0.471820\pi$
$619$	1.68570e6	0.176829	0.0884144	+	0.996084i	$0.471820\pi$
$620$	−1.91621e6	−0.200200
$621$	0	0
$622$	6.45224e6	0.668705
$623$	−3.85296e6	−0.397717
$624$	0	0
$625$	390625.	0.0400000
$626$	−8.85676e6	−0.903315
$627$	0	0
$628$	−1.82048e6	−0.184199
$629$	−817441.	−0.0823815
$630$	0	0
$631$	−8.18052e6	−0.817913	−0.408957	−	0.912554i	$-0.634107\pi$
$631$	−8.18052e6	−0.817913	−0.408957	+	0.912554i	$0.634107\pi$
$632$	−4.55921e6	−0.454043
$633$	0	0
$634$	−9.32015e6	−0.920873
$635$	2.15229e6	0.211819
$636$	0	0
$637$	71301.8	0.00696229
$638$	43481.5	0.00422914
$639$	0	0
$640$	409600.	0.0395285
$641$	−7.88556e6	−0.758032	−0.379016	−	0.925390i	$-0.623737\pi$
$641$	−7.88556e6	−0.758032	−0.379016	+	0.925390i	$0.623737\pi$
$642$	0	0
$643$	7.75540e6	0.739736	0.369868	−	0.929084i	$-0.379403\pi$
$643$	7.75540e6	0.739736	0.369868	+	0.929084i	$0.379403\pi$
$644$	−767872.	−0.0729583
$645$	0	0
$646$	1.65436e6	0.155973
$647$	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
$650$	−74241.8	−0.00689232
$651$	0	0
$652$	7.04907e6	0.649401
$653$	1.23713e7	1.13535	0.567677	−	0.823252i	$-0.307842\pi$
$653$	1.23713e7	1.13535	0.567677	+	0.823252i	$0.307842\pi$
$654$	0	0
$655$	5.52736e6	0.503401
$656$	493713.	0.0447935
$657$	0	0
$658$	−5.59845e6	−0.504084
$659$	−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	$0.545197\pi$
$662$	−1.05442e7	−0.935120
$663$	0	0
$664$	6.04139e6	0.531762
$665$	1.17139e6	0.102718
$666$	0	0
$667$	976420.	0.0849811
$668$	−9.93751e6	−0.861661
$669$	0	0
$670$	−3.70222e6	−0.318621
$671$	−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	$0.777597\pi$
$674$	−4.53893e6	−0.384861
$675$	0	0
$676$	−5.92658e6	−0.498812
$677$	−1.47781e7	−1.23921	−0.619606	−	0.784913i	$-0.712708\pi$
$677$	−1.47781e7	−1.23921	−0.619606	+	0.784913i	$0.712708\pi$
$678$	0	0
$679$	4.57731e6	0.381010
$680$	692031.	0.0573922
$681$	0	0
$682$	−208941.	−0.0172014
$683$	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
$686$	−470596.	−0.0381802
$687$	0	0
$688$	−4.62841e6	−0.372787
$689$	8525.98	0.000684221 0
$690$	0	0
$691$	−1.56459e7	−1.24654	−0.623270	−	0.782007i	$-0.714196\pi$
$691$	−1.56459e7	−1.24654	−0.623270	+	0.782007i	$0.714196\pi$
$692$	−8.11081e6	−0.643871
$693$	0	0
$694$	−1.05104e7	−0.828361
$695$	362311.	0.0284525
$696$	0	0
$697$	834142.	0.0650366
$698$	1.53063e7	1.18913
$699$	0	0
$700$	490000.	0.0377964
$701$	1.39490e7	1.07213	0.536067	−	0.844175i	$-0.319909\pi$
$701$	1.39490e7	1.07213	0.536067	+	0.844175i	$0.319909\pi$
$702$	0	0
$703$	1.80724e6	0.137920
$704$	44662.2	0.00339632
$705$	0	0
$706$	5.96573e6	0.450455
$707$	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	$0.592446\pi$
$710$	6.39302e6	0.475949
$711$	0	0
$712$	5.03244e6	0.372030
$713$	−4.69199e6	−0.345647
$714$	0	0
$715$	−8095.23	−0.000592194 0
$716$	−1.28134e7	−0.934078
$717$	0	0
$718$	−1.62549e6	−0.117672
$719$	−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
$722$	6.24686e6	0.445984
$723$	0	0
$724$	−1.46495e6	−0.103867
$725$	−623080.	−0.0440249
$726$	0	0
$727$	1.17581e7	0.825089	0.412544	−	0.910938i	$-0.364640\pi$
$727$	1.17581e7	0.825089	0.412544	+	0.910938i	$0.364640\pi$
$728$	−93128.9	−0.00651263
$729$	0	0
$730$	4.91429e6	0.341314
$731$	−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	$0.604971\pi$
$734$	−8.37707e6	−0.573921
$735$	0	0
$736$	1.00294e6	0.0682462
$737$	−403685.	−0.0273762
$738$	0	0
$739$	8.69064e6	0.585384	0.292692	−	0.956207i	$-0.405449\pi$
$739$	8.69064e6	0.585384	0.292692	+	0.956207i	$0.405449\pi$
$740$	755981.	0.0507495
$741$	0	0
$742$	−56272.0	−0.00375217
$743$	−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
$746$	−8.44518e6	−0.555600
$747$	0	0
$748$	75458.1	0.00493119
$749$	−7.99212e6	−0.520544
$750$	0	0
$751$	1.37921e7	0.892343	0.446172	−	0.894947i	$-0.352787\pi$
$751$	1.37921e7	0.892343	0.446172	+	0.894947i	$0.352787\pi$
$752$	7.31226e6	0.471528
$753$	0	0
$754$	118422.	0.00758584
$755$	−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	$0.473442\pi$
$758$	−4.84231e6	−0.306111
$759$	0	0
$760$	−1.52997e6	−0.0960838
$761$	7.93120e6	0.496452	0.248226	−	0.968702i	$-0.420152\pi$
$761$	7.93120e6	0.496452	0.248226	+	0.968702i	$0.420152\pi$
$762$	0	0
$763$	−3.29320e6	−0.204789
$764$	−6.55985e6	−0.406594
$765$	0	0
$766$	−3.97807e6	−0.244963
$767$	−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	$0.470225\pi$
$770$	53429.0	0.00324751
$771$	0	0
$772$	−1.04180e6	−0.0629130
$773$	−1.17714e7	−0.708562	−0.354281	−	0.935139i	$-0.615274\pi$
$773$	−1.17714e7	−0.708562	−0.354281	+	0.935139i	$0.615274\pi$
$774$	0	0
$775$	2.99408e6	0.179065
$776$	−5.97853e6	−0.356402
$777$	0	0
$778$	−9.96483e6	−0.590230
$779$	−1.84416e6	−0.108882
$780$	0	0
$781$	697086.	0.0408939
$782$	1.69449e6	0.0990881
$783$	0	0
$784$	614656.	0.0357143
$785$	2.84450e6	0.164752
$786$	0	0
$787$	1.28179e7	0.737701	0.368851	−	0.929489i	$-0.379751\pi$
$787$	1.28179e7	0.737701	0.368851	+	0.929489i	$0.379751\pi$
$788$	−2.93150e6	−0.168180
$789$	0	0
$790$	7.12376e6	0.406108
$791$	−2.74282e6	−0.155868
$792$	0	0
$793$	−976559.	−0.0551462
$794$	−9.23337e6	−0.519767
$795$	0	0
$796$	−9.00864e6	−0.503937
$797$	3.33470e7	1.85957	0.929783	−	0.368108i	$-0.119994\pi$
$797$	3.33470e7	1.85957	0.929783	+	0.368108i	$0.119994\pi$
$798$	0	0
$799$	1.23543e7	0.684621
$800$	−640000.	−0.0353553
$801$	0	0
$802$	−2.53524e6	−0.139182
$803$	535848.	0.0293260
$804$	0	0
$805$	1.19980e6	0.0652558
$806$	−569053.	−0.0308542
$807$	0	0
$808$	−908184.	−0.0489379
$809$	−1.95295e7	−1.04911	−0.524554	−	0.851377i	$-0.675768\pi$
$809$	−1.95295e7	−1.04911	−0.524554	+	0.851377i	$0.675768\pi$
$810$	0	0
$811$	−2.21959e7	−1.18501	−0.592504	−	0.805568i	$-0.701861\pi$
$811$	−2.21959e7	−1.18501	−0.592504	+	0.805568i	$0.701861\pi$
$812$	−781591.	−0.0415996
$813$	0	0
$814$	82431.2	0.00436044
$815$	−1.10142e7	−0.580842
$816$	0	0
$817$	1.72885e7	0.906152
$818$	−3.77068e6	−0.197032
$819$	0	0
$820$	−771426.	−0.0400645
$821$	−2.37914e7	−1.23186	−0.615930	−	0.787801i	$-0.711220\pi$
$821$	−2.37914e7	−1.23186	−0.615930	+	0.787801i	$0.711220\pi$
$822$	0	0
$823$	−1.75911e7	−0.905300	−0.452650	−	0.891688i	$-0.649521\pi$
$823$	−1.75911e7	−0.905300	−0.452650	+	0.891688i	$0.649521\pi$
$824$	−1.26469e7	−0.648881
$825$	0	0
$826$	2.20918e6	0.112663
$827$	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	$0.789300\pi$
$830$	−9.43968e6	−0.475622
$831$	0	0
$832$	121638.	0.00609200
$833$	1.03848e6	0.0518543
$834$	0	0
$835$	1.55274e7	0.770693
$836$	−166826.	−0.00825561
$837$	0	0
$838$	−1.57442e7	−0.774478
$839$	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
$842$	2.30457e7	1.12023
$843$	0	0
$844$	−9.03697e6	−0.436684
$845$	9.26028e6	0.446151
$846$	0	0
$847$	−7.88567e6	−0.377685
$848$	73498.1	0.00350983
$849$	0	0
$850$	−1.08130e6	−0.0513332
$851$	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	$0.480598\pi$
$854$	6.44534e6	0.302413
$855$	0	0
$856$	1.04387e7	0.486924
$857$	−3.19020e7	−1.48377	−0.741883	−	0.670529i	$-0.766067\pi$
$857$	−3.19020e7	−1.48377	−0.741883	+	0.670529i	$0.766067\pi$
$858$	0	0
$859$	2.89610e7	1.33915	0.669577	−	0.742742i	$-0.266475\pi$
$859$	2.89610e7	1.33915	0.669577	+	0.742742i	$0.266475\pi$
$860$	7.23190e6	0.333431
$861$	0	0
$862$	7.72510e6	0.354108
$863$	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
$866$	−5.61833e6	−0.254573
$867$	0	0
$868$	3.75578e6	0.169200
$869$	776766.	0.0348932
$870$	0	0
$871$	−1.09944e6	−0.0491049
$872$	4.30132e6	0.191562
$873$	0	0
$874$	−3.74625e6	−0.165889
$875$	−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	$0.389368\pi$
$878$	1.46588e7	0.641743
$879$	0	0
$880$	−69784.8	−0.00303776
$881$	2.13678e7	0.927515	0.463757	−	0.885962i	$-0.346501\pi$
$881$	2.13678e7	0.927515	0.463757	+	0.885962i	$0.346501\pi$
$882$	0	0
$883$	1.70291e7	0.735003	0.367502	−	0.930023i	$-0.380213\pi$
$883$	1.70291e7	0.735003	0.367502	+	0.930023i	$0.380213\pi$
$884$	205511.	0.00884511
$885$	0	0
$886$	1.87365e7	0.801872
$887$	−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
$890$	−7.86318e6	−0.332754
$891$	0	0
$892$	−1.88252e7	−0.792186
$893$	−2.73134e7	−1.14617
$894$	0	0
$895$	2.00210e7	0.835465
$896$	−802816.	−0.0334077
$897$	0	0
$898$	−3.49410e6	−0.144592
$899$	−4.77582e6	−0.197083
$900$	0	0
$901$	124177.	0.00509600
$902$	−84115.3	−0.00344238
$903$	0	0
$904$	3.58245e6	0.145801
$905$	2.28899e6	0.0929014
$906$	0	0
$907$	−9.89966e6	−0.399578	−0.199789	−	0.979839i	$-0.564026\pi$
$907$	−9.89966e6	−0.399578	−0.199789	+	0.979839i	$0.564026\pi$
$908$	−1.57573e7	−0.634258
$909$	0	0
$910$	145514.	0.00582507
$911$	−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
$914$	−2.08195e7	−0.824337
$915$	0	0
$916$	2.78802e6	0.109789
$917$	−1.08336e7	−0.425452
$918$	0	0
$919$	−1.40148e7	−0.547393	−0.273697	−	0.961816i	$-0.588246\pi$
$919$	−1.40148e7	−0.547393	−0.273697	+	0.961816i	$0.588246\pi$
$920$	−1.56709e6	−0.0610413
$921$	0	0
$922$	−7.73896e6	−0.299816
$923$	1.89852e6	0.0733517
$924$	0	0
$925$	−1.18122e6	−0.0453917
$926$	9.43522e6	0.361597
$927$	0	0
$928$	1.02085e6	0.0389129
$929$	−2.55744e7	−0.972225	−0.486112	−	0.873896i	$-0.661585\pi$
$929$	−2.55744e7	−0.972225	−0.486112	+	0.873896i	$0.661585\pi$
$930$	0	0
$931$	−2.29592e6	−0.0868125
$932$	−1.43226e7	−0.540108
$933$	0	0
$934$	−1.34961e7	−0.506221
$935$	−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	$0.177869\pi$
$938$	7.25635e6	0.269284
$939$	0	0
$940$	−1.14254e7	−0.421747
$941$	1.44078e7	0.530425	0.265213	−	0.964190i	$-0.414558\pi$
$941$	1.44078e7	0.530425	0.265213	+	0.964190i	$0.414558\pi$
$942$	0	0
$943$	−1.88889e6	−0.0691716
$944$	−2.88546e6	−0.105387
$945$	0	0
$946$	788556.	0.0286487
$947$	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
$950$	2.39058e6	0.0859400
$951$	0	0
$952$	−1.35638e6	−0.0485053
$953$	−2.52608e7	−0.900980	−0.450490	−	0.892781i	$-0.648751\pi$
$953$	−2.52608e7	−0.900980	−0.450490	+	0.892781i	$0.648751\pi$
$954$	0	0
$955$	1.02498e7	0.363668
$956$	−4.00205e6	−0.141624
$957$	0	0
$958$	1.89907e7	0.668540
$959$	2.08335e7	0.731502
$960$	0	0
$961$	−5.67994e6	−0.198397
$962$	224502.	0.00782135
$963$	0	0
$964$	−1.31982e7	−0.457427
$965$	1.62781e6	0.0562711
$966$	0	0
$967$	5.68247e7	1.95421	0.977105	−	0.212758i	$-0.0682447\pi$
$967$	5.68247e7	1.95421	0.977105	+	0.212758i	$0.0682447\pi$
$968$	1.02997e7	0.353292
$969$	0	0
$970$	9.34146e6	0.318776
$971$	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
$974$	−2.09947e7	−0.709109
$975$	0	0
$976$	−8.41840e6	−0.282882
$977$	4.16806e7	1.39700	0.698502	−	0.715608i	$-0.253850\pi$
$977$	4.16806e7	1.39700	0.698502	+	0.715608i	$0.253850\pi$
$978$	0	0
$979$	−857391.	−0.0285905
$980$	−960400.	−0.0319438
$981$	0	0
$982$	−1.77042e7	−0.585864
$983$	−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
$986$	1.72476e6	0.0564985
$987$	0	0
$988$	−454352.	−0.0148081
$989$	1.77078e7	0.575671
$990$	0	0
$991$	−3.85510e7	−1.24696	−0.623478	−	0.781841i	$-0.714281\pi$
$991$	−3.85510e7	−1.24696	−0.623478	+	0.781841i	$0.714281\pi$
$992$	−4.90551e6	−0.158272
$993$	0	0
$994$	−1.25303e7	−0.402250
$995$	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	$0.480706\pi$
$998$	183608.	0.00583533
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.6.a.s.1.2		2
3.2	odd	2		70.6.a.h.1.2	✓	2
12.11	even	2		560.6.a.k.1.1		2
15.2	even	4		350.6.c.k.99.3		4
15.8	even	4		350.6.c.k.99.2		4
15.14	odd	2		350.6.a.p.1.1		2
21.20	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	3.2	odd	2
350.6.a.p.1.1		2	15.14	odd	2
350.6.c.k.99.2		4	15.8	even	4
350.6.c.k.99.3		4	15.2	even	4
490.6.a.u.1.1		2	21.20	even	2
560.6.a.k.1.1		2	12.11	even	2
630.6.a.s.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +100.000 q^{10} +10.9039 q^{11} +29.6967 q^{13} -196.000 q^{14} +256.000 q^{16} +432.519 q^{17} -956.234 q^{19} -400.000 q^{20} -43.6155 q^{22} -979.429 q^{23} +625.000 q^{25} -118.787 q^{26} +784.000 q^{28} -996.928 q^{29} +4790.53 q^{31} -1024.00 q^{32} -1730.08 q^{34} -1225.00 q^{35} -1889.95 q^{37} +3824.94 q^{38} +1600.00 q^{40} +1928.56 q^{41} -18079.7 q^{43} +174.462 q^{44} +3917.72 q^{46} +28563.5 q^{47} +2401.00 q^{49} -2500.00 q^{50} +475.148 q^{52} +287.102 q^{53} -272.597 q^{55} -3136.00 q^{56} +3987.71 q^{58} -11271.3 q^{59} -32884.4 q^{61} -19162.1 q^{62} +4096.00 q^{64} -742.418 q^{65} -37022.2 q^{67} +6920.31 q^{68} +4900.00 q^{70} +63930.2 q^{71} +49142.9 q^{73} +7559.81 q^{74} -15299.7 q^{76} +534.290 q^{77} +71237.6 q^{79} -6400.00 q^{80} -7714.26 q^{82} -94396.8 q^{83} -10813.0 q^{85} +72319.0 q^{86} -697.848 q^{88} -78631.8 q^{89} +1455.14 q^{91} -15670.9 q^{92} -114254. q^{94} +23905.8 q^{95} +93414.6 q^{97} -9604.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 50 q^{5} + 98 q^{7} - 128 q^{8} + 200 q^{10} - 415 q^{11} + 429 q^{13} - 392 q^{14} + 512 q^{16} - 1319 q^{17} + 1918 q^{19} - 800 q^{20} + 1660 q^{22} + 1334 q^{23} + 1250 q^{25}+ \cdots - 19208 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 50 * q^5 + 98 * q^7 - 128 * q^8 + 200 * q^10 - 415 * q^11 + 429 * q^13 - 392 * q^14 + 512 * q^16 - 1319 * q^17 + 1918 * q^19 - 800 * q^20 + 1660 * q^22 + 1334 * q^23 + 1250 * q^25 - 1716 * q^26 + 1568 * q^28 + 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 5276 * q^34 - 2450 * q^35 - 9156 * q^37 - 7672 * q^38 + 3200 * q^40 + 7822 * q^41 + 2078 * q^43 - 6640 * q^44 - 5336 * q^46 + 8507 * q^47 + 4802 * q^49 - 5000 * q^50 + 6864 * q^52 + 34242 * q^53 + 10375 * q^55 - 6272 * q^56 - 4524 * q^58 - 6952 * q^59 - 45138 * q^61 + 21888 * q^62 + 8192 * q^64 - 10725 * q^65 - 54556 * q^67 - 21104 * q^68 + 9800 * q^70 + 62272 * q^71 + 40896 * q^73 + 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 62069 * q^79 - 12800 * q^80 - 31288 * q^82 - 189600 * q^83 + 32975 * q^85 - 8312 * q^86 + 26560 * q^88 - 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 34028 * q^94 - 47950 * q^95 + 155547 * q^97 - 19208 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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