LMFDB - Embedded newform 350.6.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(1,350)] 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("350.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.1343369345$
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	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -19.3003 q^{3} +16.0000 q^{4} +77.2012 q^{6} -49.0000 q^{7} -64.0000 q^{8} +129.501 q^{9} -10.9039 q^{11} -308.805 q^{12} -29.6967 q^{13} +196.000 q^{14} +256.000 q^{16} +432.519 q^{17} -518.006 q^{18} -956.234 q^{19} +945.715 q^{21} +43.6155 q^{22} -979.429 q^{23} +1235.22 q^{24} +118.787 q^{26} +2190.56 q^{27} -784.000 q^{28} +996.928 q^{29} +4790.53 q^{31} -1024.00 q^{32} +210.448 q^{33} -1730.08 q^{34} +2072.02 q^{36} +1889.95 q^{37} +3824.94 q^{38} +573.156 q^{39} -1928.56 q^{41} -3782.86 q^{42} +18079.7 q^{43} -174.462 q^{44} +3917.72 q^{46} +28563.5 q^{47} -4940.88 q^{48} +2401.00 q^{49} -8347.75 q^{51} -475.148 q^{52} +287.102 q^{53} -8762.22 q^{54} +3136.00 q^{56} +18455.6 q^{57} -3987.71 q^{58} +11271.3 q^{59} -32884.4 q^{61} -19162.1 q^{62} -6345.57 q^{63} +4096.00 q^{64} -841.792 q^{66} +37022.2 q^{67} +6920.31 q^{68} +18903.3 q^{69} -63930.2 q^{71} -8288.10 q^{72} -49142.9 q^{73} -7559.81 q^{74} -15299.7 q^{76} +534.290 q^{77} -2292.62 q^{78} +71237.6 q^{79} -73747.2 q^{81} +7714.26 q^{82} -94396.8 q^{83} +15131.4 q^{84} -72319.0 q^{86} -19241.0 q^{87} +697.848 q^{88} +78631.8 q^{89} +1455.14 q^{91} -15670.9 q^{92} -92458.7 q^{93} -114254. q^{94} +19763.5 q^{96} -93414.6 q^{97} -9604.00 q^{98} -1412.07 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 5 q^{3} + 32 q^{4} + 20 q^{6} - 98 q^{7} - 128 q^{8} + 91 q^{9} + 415 q^{11} - 80 q^{12} - 429 q^{13} + 392 q^{14} + 512 q^{16} - 1319 q^{17} - 364 q^{18} + 1918 q^{19} + 245 q^{21} - 1660 q^{22}+ \cdots - 17810 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 - 5 * q^3 + 32 * q^4 + 20 * q^6 - 98 * q^7 - 128 * q^8 + 91 * q^9 + 415 * q^11 - 80 * q^12 - 429 * q^13 + 392 * q^14 + 512 * q^16 - 1319 * q^17 - 364 * q^18 + 1918 * q^19 + 245 * q^21 - 1660 * q^22 + 1334 * q^23 + 320 * q^24 + 1716 * q^26 - 1835 * q^27 - 1568 * q^28 - 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 6301 * q^33 + 5276 * q^34 + 1456 * q^36 + 9156 * q^37 - 7672 * q^38 - 5137 * q^39 - 7822 * q^41 - 980 * q^42 - 2078 * q^43 + 6640 * q^44 - 5336 * q^46 + 8507 * q^47 - 1280 * q^48 + 4802 * q^49 - 33395 * q^51 - 6864 * q^52 + 34242 * q^53 + 7340 * q^54 + 6272 * q^56 + 59558 * q^57 + 4524 * q^58 + 6952 * q^59 - 45138 * q^61 + 21888 * q^62 - 4459 * q^63 + 8192 * q^64 - 25204 * q^66 + 54556 * q^67 - 21104 * q^68 + 51986 * q^69 - 62272 * q^71 - 5824 * q^72 - 40896 * q^73 - 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 20548 * q^78 + 62069 * q^79 - 121958 * q^81 + 31288 * q^82 - 189600 * q^83 + 3920 * q^84 + 8312 * q^86 - 49671 * q^87 - 26560 * q^88 + 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 239216 * q^93 - 34028 * q^94 + 5120 * q^96 - 155547 * q^97 - 19208 * q^98 - 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−19.3003	−1.23811	−0.619057	−	0.785346i	$-0.712485\pi$
$3$	−19.3003	−1.23811	−0.619057	+	0.785346i	$0.712485\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	77.2012	0.875479
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	129.501	0.532928
$10$	0	0
$11$	−10.9039	−0.0271706	−0.0135853	−	0.999908i	$-0.504324\pi$
$11$	−10.9039	−0.0271706	−0.0135853	+	0.999908i	$0.504324\pi$
$12$	−308.805	−0.619057
$13$	−29.6967	−0.0487360	−0.0243680	−	0.999703i	$-0.507757\pi$
$13$	−29.6967	−0.0487360	−0.0243680	+	0.999703i	$0.507757\pi$
$14$	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	$0.441908\pi$
$18$	−518.006	−0.376837
$19$	−956.234	−0.607687	−0.303844	−	0.952722i	$-0.598270\pi$
$19$	−956.234	−0.607687	−0.303844	+	0.952722i	$0.598270\pi$
$20$	0	0
$21$	945.715	0.467963
$22$	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	$0.561831\pi$
$24$	1235.22	0.437740
$25$	0	0
$26$	118.787	0.0344616
$27$	2190.56	0.578289
$28$	−784.000	−0.188982
$29$	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	$-0.352257\pi$
$31$	4790.53	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$32$	−1024.00	−0.176777
$33$	210.448	0.0336403
$34$	−1730.08	−0.256666
$35$	0	0
$36$	2072.02	0.266464
$37$	1889.95	0.226959	0.113479	−	0.993540i	$-0.463800\pi$
$37$	1889.95	0.226959	0.113479	+	0.993540i	$0.463800\pi$
$38$	3824.94	0.429700
$39$	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	$0.528555\pi$
$42$	−3782.86	−0.330900
$43$	18079.7	1.49115	0.745574	−	0.666422i	$-0.232175\pi$
$43$	18079.7	1.49115	0.745574	+	0.666422i	$0.232175\pi$
$44$	−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	$0.107938\pi$
$48$	−4940.88	−0.309529
$49$	2401.00	0.142857
$50$	0	0
$51$	−8347.75	−0.449411
$52$	−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	$0.497766\pi$
$54$	−8762.22	−0.408912
$55$	0	0
$56$	3136.00	0.133631
$57$	18455.6	0.752387
$58$	−3987.71	−0.155652
$59$	11271.3	0.421546	0.210773	−	0.977535i	$-0.432402\pi$
$59$	11271.3	0.421546	0.210773	+	0.977535i	$0.432402\pi$
$60$	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
$63$	−6345.57	−0.201428
$64$	4096.00	0.125000
$65$	0	0
$66$	−841.792	−0.0237873
$67$	37022.2	1.00757	0.503785	−	0.863829i	$-0.331941\pi$
$67$	37022.2	1.00757	0.503785	+	0.863829i	$0.331941\pi$
$68$	6920.31	0.181490
$69$	18903.3	0.477985
$70$	0	0
$71$	−63930.2	−1.50508	−0.752541	−	0.658546i	$-0.771172\pi$
$71$	−63930.2	−1.50508	−0.752541	+	0.658546i	$0.771172\pi$
$72$	−8288.10	−0.188418
$73$	−49142.9	−1.07933	−0.539664	−	0.841880i	$-0.681449\pi$
$73$	−49142.9	−1.07933	−0.539664	+	0.841880i	$0.681449\pi$
$74$	−7559.81	−0.160484
$75$	0	0
$76$	−15299.7	−0.303844
$77$	534.290	0.0102695
$78$	−2292.62	−0.0426674
$79$	71237.6	1.28423	0.642113	−	0.766610i	$-0.278058\pi$
$79$	71237.6	1.28423	0.642113	+	0.766610i	$0.278058\pi$
$80$	0	0
$81$	−73747.2	−1.24892
$82$	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	$0.770923\pi$
$84$	15131.4	0.233982
$85$	0	0
$86$	−72319.0	−1.05440
$87$	−19241.0	−0.272540
$88$	697.848	0.00960625
$89$	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
$92$	−15670.9	−0.193029
$93$	−92458.7	−1.10851
$94$	−114254.	−1.33368
$95$	0	0
$96$	19763.5	0.218870
$97$	−93414.6	−1.00806	−0.504029	−	0.863687i	$-0.668149\pi$
$97$	−93414.6	−1.00806	−0.504029	+	0.863687i	$0.668149\pi$
$98$	−9604.00	−0.101015
$99$	−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	$0.522047\pi$
$102$	33391.0	0.317782
$103$	−197607.	−1.83531	−0.917657	−	0.397374i	$-0.869921\pi$
$103$	−197607.	−1.83531	−0.917657	+	0.397374i	$0.869921\pi$
$104$	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	$0.741781\pi$
$108$	35048.9	0.289144
$109$	−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
$112$	−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	$0.566108\pi$
$114$	−73822.4	−0.532018
$115$	0	0
$116$	15950.8	0.110062
$117$	−3845.77	−0.0259728
$118$	−45085.4	−0.298078
$119$	−21193.4	−0.137194
$120$	0	0
$121$	−160932.	−0.999262
$122$	131538.	0.800111
$123$	37221.9	0.221838
$124$	76648.5	0.447661
$125$	0	0
$126$	25382.3	0.142431
$127$	86091.4	0.473642	0.236821	−	0.971553i	$-0.423894\pi$
$127$	86091.4	0.473642	0.236821	+	0.971553i	$0.423894\pi$
$128$	−16384.0	−0.0883883
$129$	−348944.	−1.84621
$130$	0	0
$131$	221094.	1.12564	0.562820	−	0.826580i	$-0.309717\pi$
$131$	221094.	1.12564	0.562820	+	0.826580i	$0.309717\pi$
$132$	3367.17	0.0168201
$133$	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	$0.0811404\pi$
$138$	−75613.1	−0.337986
$139$	−14492.4	−0.0636216	−0.0318108	−	0.999494i	$-0.510127\pi$
$139$	−14492.4	−0.0636216	−0.0318108	+	0.999494i	$0.510127\pi$
$140$	0	0
$141$	−551285.	−2.33522
$142$	255721.	1.06425
$143$	323.809	0.00132419
$144$	33152.4	0.133232
$145$	0	0
$146$	196572.	0.763201
$147$	−46340.0	−0.176874
$148$	30239.2	0.113479
$149$	−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	$-0.453581\pi$
$151$	81428.5	0.290626	0.145313	+	0.989386i	$0.453581\pi$
$152$	61199.0	0.214850
$153$	56011.9	0.193442
$154$	−2137.16	−0.00726164
$155$	0	0
$156$	9170.49	0.0301704
$157$	113780.	0.368397	0.184199	−	0.982889i	$-0.441031\pi$
$157$	113780.	0.368397	0.184199	+	0.982889i	$0.441031\pi$
$158$	−284950.	−0.908085
$159$	−5541.15	−0.0173823
$160$	0	0
$161$	47992.0	0.145917
$162$	294989.	0.883117
$163$	−440567.	−1.29880	−0.649401	−	0.760446i	$-0.724980\pi$
$163$	−440567.	−1.29880	−0.649401	+	0.760446i	$0.724980\pi$
$164$	−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	$0.830576\pi$
$168$	−60525.7	−0.165450
$169$	−370411.	−0.997625
$170$	0	0
$171$	−123834.	−0.323854
$172$	289276.	0.745574
$173$	−506925.	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$173$	−506925.	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$174$	76964.0	0.192715
$175$	0	0
$176$	−2791.39	−0.00679265
$177$	−217540.	−0.521923
$178$	−314527.	−0.744061
$179$	800840.	1.86816	0.934078	−	0.357070i	$-0.116224\pi$
$179$	800840.	1.86816	0.934078	+	0.357070i	$0.116224\pi$
$180$	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
$183$	634678.	1.40096
$184$	62683.5	0.136492
$185$	0	0
$186$	369835.	0.783837
$187$	−4716.13	−0.00986239
$188$	457016.	0.943055
$189$	−107337.	−0.218573
$190$	0	0
$191$	409991.	0.813187	0.406594	−	0.913609i	$-0.366717\pi$
$191$	409991.	0.813187	0.406594	+	0.913609i	$0.366717\pi$
$192$	−79054.0	−0.154764
$193$	65112.4	0.125826	0.0629130	−	0.998019i	$-0.479961\pi$
$193$	65112.4	0.125826	0.0629130	+	0.998019i	$0.479961\pi$
$194$	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	$0.553789\pi$
$198$	5648.27	0.0102389
$199$	−563040.	−1.00787	−0.503937	−	0.863740i	$-0.668116\pi$
$199$	−563040.	−1.00787	−0.503937	+	0.863740i	$0.668116\pi$
$200$	0	0
$201$	−714539.	−1.24749
$202$	56761.5	0.0978758
$203$	−48849.5	−0.0831993
$204$	−133564.	−0.224706
$205$	0	0
$206$	790430.	1.29776
$207$	−126838.	−0.205742
$208$	−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
$213$	1.23387e6	1.86346
$214$	652418.	0.973848
$215$	0	0
$216$	−140196.	−0.204456
$217$	−234736.	−0.338400
$218$	268832.	0.383125
$219$	948473.	1.33633
$220$	0	0
$221$	−12844.4	−0.0176902
$222$	145907.	0.198698
$223$	1.17657e6	1.58437	0.792186	−	0.610280i	$-0.208943\pi$
$223$	1.17657e6	1.58437	0.792186	+	0.610280i	$0.208943\pi$
$224$	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	$0.718694\pi$
$228$	295290.	0.376193
$229$	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
$232$	−63803.4	−0.0778258
$233$	−895160.	−1.08022	−0.540108	−	0.841596i	$-0.681617\pi$
$233$	−895160.	−1.08022	−0.540108	+	0.841596i	$0.681617\pi$
$234$	15383.1	0.0183655
$235$	0	0
$236$	180341.	0.210773
$237$	−1.37491e6	−1.59002
$238$	84773.8	0.0970106
$239$	250128.	0.283249	0.141624	−	0.989920i	$-0.454768\pi$
$239$	250128.	0.283249	0.141624	+	0.989920i	$0.454768\pi$
$240$	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
$243$	891039.	0.968012
$244$	−526150.	−0.565764
$245$	0	0
$246$	−148888.	−0.156863
$247$	28397.0	0.0296163
$248$	−306594.	−0.316544
$249$	1.82189e6	1.86219
$250$	0	0
$251$	729516.	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$251$	729516.	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$252$	−101529.	−0.100714
$253$	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	$0.517843\pi$
$258$	1.39578e6	1.30547
$259$	−92607.7	−0.0857823
$260$	0	0
$261$	129104.	0.117311
$262$	−884377.	−0.795947
$263$	766194.	0.683045	0.341523	−	0.939874i	$-0.389057\pi$
$263$	766194.	0.683045	0.341523	+	0.939874i	$0.389057\pi$
$264$	−13468.7	−0.0118936
$265$	0	0
$266$	−187422.	−0.162411
$267$	−1.51762e6	−1.30282
$268$	592355.	0.503785
$269$	−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	$-0.736136\pi$
$271$	−1.63371e6	−1.35130	−0.675648	+	0.737224i	$0.736136\pi$
$272$	110725.	0.0907451
$273$	−28084.6	−0.0228067
$274$	−1.70069e6	−1.36851
$275$	0	0
$276$	302452.	0.238993
$277$	2.14606e6	1.68051	0.840257	−	0.542189i	$-0.182404\pi$
$277$	2.14606e6	1.68051	0.840257	+	0.542189i	$0.182404\pi$
$278$	57969.8	0.0449873
$279$	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	$0.387820\pi$
$282$	2.20514e6	1.65125
$283$	1.11656e6	0.828738	0.414369	−	0.910109i	$-0.364002\pi$
$283$	1.11656e6	0.828738	0.414369	+	0.910109i	$0.364002\pi$
$284$	−1.02288e6	−0.752541
$285$	0	0
$286$	−1295.24	−0.000936341 0
$287$	94499.7	0.0677213
$288$	−132610.	−0.0942092
$289$	−1.23278e6	−0.868245
$290$	0	0
$291$	1.80293e6	1.24809
$292$	−786287.	−0.539664
$293$	−396556.	−0.269858	−0.134929	−	0.990855i	$-0.543081\pi$
$293$	−396556.	−0.269858	−0.134929	+	0.990855i	$0.543081\pi$
$294$	185360.	0.125068
$295$	0	0
$296$	−120957.	−0.0802420
$297$	−23885.5	−0.0157124
$298$	147911.	0.0964852
$299$	29085.8	0.0188150
$300$	0	0
$301$	−885907.	−0.563601
$302$	−325714.	−0.205504
$303$	273878.	0.171376
$304$	−244796.	−0.151922
$305$	0	0
$306$	−224048.	−0.136784
$307$	−1.43537e6	−0.869197	−0.434598	−	0.900624i	$-0.643110\pi$
$307$	−1.43537e6	−0.869197	−0.434598	+	0.900624i	$0.643110\pi$
$308$	8548.63	0.00513476
$309$	3.81388e6	2.27233
$310$	0	0
$311$	1.61306e6	0.945691	0.472846	−	0.881145i	$-0.343227\pi$
$311$	1.61306e6	0.945691	0.472846	+	0.881145i	$0.343227\pi$
$312$	−36682.0	−0.0213337
$313$	−2.21419e6	−1.27748	−0.638740	−	0.769423i	$-0.720544\pi$
$313$	−2.21419e6	−1.27748	−0.638740	+	0.769423i	$0.720544\pi$
$314$	−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	$0.274285\pi$
$318$	22164.6	0.0122911
$319$	−10870.4	−0.00598091
$320$	0	0
$321$	3.14796e6	1.70517
$322$	−191968.	−0.103179
$323$	−413590.	−0.220579
$324$	−1.17996e6	−0.624458
$325$	0	0
$326$	1.76227e6	0.918391
$327$	1.29714e6	0.670836
$328$	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
$333$	244752.	0.120953
$334$	2.48438e6	1.21857
$335$	0	0
$336$	242103.	0.116991
$337$	−1.13473e6	−0.544275	−0.272138	−	0.962258i	$-0.587731\pi$
$337$	−1.13473e6	−0.544275	−0.272138	+	0.962258i	$0.587731\pi$
$338$	1.48164e6	0.705427
$339$	1.08035e6	0.510582
$340$	0	0
$341$	−52235.3	−0.0243264
$342$	495335.	0.228999
$343$	−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	$0.300804\pi$
$348$	−307856.	−0.136270
$349$	−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
$352$	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	$0.603186\pi$
$354$	870161.	0.369055
$355$	0	0
$356$	1.25811e6	0.526130
$357$	409040.	0.169862
$358$	−3.20336e6	−1.32099
$359$	−406372.	−0.166413	−0.0832066	−	0.996532i	$-0.526516\pi$
$359$	−406372.	−0.166413	−0.0832066	+	0.996532i	$0.526516\pi$
$360$	0	0
$361$	−1.56172e6	−0.630716
$362$	366238.	0.146890
$363$	3.10604e6	1.23720
$364$	23282.2	0.00921024
$365$	0	0
$366$	−2.53871e6	−0.990629
$367$	−2.09427e6	−0.811647	−0.405824	−	0.913951i	$-0.633015\pi$
$367$	−2.09427e6	−0.811647	−0.405824	+	0.913951i	$0.633015\pi$
$368$	−250734.	−0.0965147
$369$	−249752.	−0.0954867
$370$	0	0
$371$	−14068.0	−0.00530637
$372$	−1.47934e6	−0.554256
$373$	−2.11130e6	−0.785736	−0.392868	−	0.919595i	$-0.628517\pi$
$373$	−2.11130e6	−0.785736	−0.392868	+	0.919595i	$0.628517\pi$
$374$	18864.5	0.00697376
$375$	0	0
$376$	−1.82807e6	−0.666841
$377$	−29605.5	−0.0107280
$378$	429349.	0.154554
$379$	1.21058e6	0.432907	0.216453	−	0.976293i	$-0.430551\pi$
$379$	1.21058e6	0.432907	0.216453	+	0.976293i	$0.430551\pi$
$380$	0	0
$381$	−1.66159e6	−0.586424
$382$	−1.63996e6	−0.575010
$383$	994517.	0.346430	0.173215	−	0.984884i	$-0.444584\pi$
$383$	994517.	0.346430	0.173215	+	0.984884i	$0.444584\pi$
$384$	316216.	0.109435
$385$	0	0
$386$	−260449.	−0.0889724
$387$	2.34135e6	0.794675
$388$	−1.49463e6	−0.504029
$389$	−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
$392$	−153664.	−0.0505076
$393$	−4.26719e6	−1.39367
$394$	732874.	0.237842
$395$	0	0
$396$	−22593.1	−0.00723998
$397$	−2.30834e6	−0.735062	−0.367531	−	0.930011i	$-0.619797\pi$
$397$	−2.30834e6	−0.735062	−0.367531	+	0.930011i	$0.619797\pi$
$398$	2.25216e6	0.712675
$399$	−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	$0.531378\pi$
$402$	2.85816e6	0.882106
$403$	−142263.	−0.0436345
$404$	−227046.	−0.0692086
$405$	0	0
$406$	195398.	0.0588308
$407$	−20607.8	−0.00616660
$408$	534256.	0.158891
$409$	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
$412$	−3.16172e6	−0.917657
$413$	−552296.	−0.159330
$414$	507350.	0.145481
$415$	0	0
$416$	30409.4	0.00861540
$417$	279709.	0.0787709
$418$	−41706.6	−0.0116752
$419$	−3.93604e6	−1.09528	−0.547639	−	0.836715i	$-0.684473\pi$
$419$	−3.93604e6	−1.09528	−0.547639	+	0.836715i	$0.684473\pi$
$420$	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
$423$	3.69902e6	1.00516
$424$	−18374.5	−0.00496365
$425$	0	0
$426$	−4.93549e6	−1.31767
$427$	1.61133e6	0.427677
$428$	−2.60967e6	−0.688615
$429$	−6249.61	−0.00163949
$430$	0	0
$431$	1.93127e6	0.500785	0.250392	−	0.968144i	$-0.419440\pi$
$431$	1.93127e6	0.500785	0.250392	+	0.968144i	$0.419440\pi$
$432$	560782.	0.144572
$433$	−1.40458e6	−0.360021	−0.180010	−	0.983665i	$-0.557613\pi$
$433$	−1.40458e6	−0.360021	−0.180010	+	0.983665i	$0.557613\pi$
$434$	938945.	0.239285
$435$	0	0
$436$	−1.07533e6	−0.270910
$437$	936563.	0.234603
$438$	−3.79389e6	−0.944930
$439$	−3.66469e6	−0.907562	−0.453781	−	0.891113i	$-0.649925\pi$
$439$	−3.66469e6	−0.907562	−0.453781	+	0.891113i	$0.649925\pi$
$440$	0	0
$441$	310933.	0.0761326
$442$	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	$0.691900\pi$
$444$	−583626.	−0.140500
$445$	0	0
$446$	−4.70630e6	−1.12032
$447$	713683.	0.168942
$448$	−200704.	−0.0472456
$449$	−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
$452$	−895614.	−0.206193
$453$	−1.57160e6	−0.359828
$454$	3.93932e6	0.896977
$455$	0	0
$456$	−1.18116e6	−0.266009
$457$	−5.20487e6	−1.16579	−0.582894	−	0.812548i	$-0.698080\pi$
$457$	−5.20487e6	−1.16579	−0.582894	+	0.812548i	$0.698080\pi$
$458$	−697005.	−0.155264
$459$	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	$0.567998\pi$
$462$	41247.8	0.00899075
$463$	2.35881e6	0.511375	0.255688	−	0.966759i	$-0.417698\pi$
$463$	2.35881e6	0.511375	0.255688	+	0.966759i	$0.417698\pi$
$464$	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	$0.383475\pi$
$468$	−61532.3	−0.0129864
$469$	−1.81409e6	−0.380825
$470$	0	0
$471$	−2.19599e6	−0.456118
$472$	−721366.	−0.149039
$473$	−197139.	−0.0405154
$474$	5.49963e6	1.12431
$475$	0	0
$476$	−339095.	−0.0685969
$477$	37180.1	0.00748195
$478$	−1.00051e6	−0.200287
$479$	4.74768e6	0.945459	0.472729	−	0.881208i	$-0.343269\pi$
$479$	4.74768e6	0.945459	0.472729	+	0.881208i	$0.343269\pi$
$480$	0	0
$481$	−56125.4	−0.0110611
$482$	3.29955e6	0.646900
$483$	−926260.	−0.180661
$484$	−2.57491e6	−0.499631
$485$	0	0
$486$	−3.56415e6	−0.684488
$487$	−5.24868e6	−1.00283	−0.501416	−	0.865207i	$-0.667187\pi$
$487$	−5.24868e6	−1.00283	−0.501416	+	0.865207i	$0.667187\pi$
$488$	2.10460e6	0.400055
$489$	8.50307e6	1.60807
$490$	0	0
$491$	−4.42605e6	−0.828537	−0.414269	−	0.910155i	$-0.635963\pi$
$491$	−4.42605e6	−0.828537	−0.414269	+	0.910155i	$0.635963\pi$
$492$	595550.	0.110919
$493$	431191.	0.0799009
$494$	−113588.	−0.0209419
$495$	0	0
$496$	1.22638e6	0.223831
$497$	3.13258e6	0.568867
$498$	−7.28754e6	−1.31676
$499$	−45902.0	−0.00825240	−0.00412620	−	0.999991i	$-0.501313\pi$
$499$	−45902.0	−0.00825240	−0.00412620	+	0.999991i	$0.501313\pi$
$500$	0	0
$501$	1.19873e7	2.13367
$502$	−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	$0.442342\pi$
$504$	406117.	0.0712155
$505$	0	0
$506$	−42718.3	−0.00741716
$507$	7.14904e6	1.23517
$508$	1.37746e6	0.236821
$509$	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
$512$	−262144.	−0.0441942
$513$	−2.09468e6	−0.351419
$514$	474576.	0.0792315
$515$	0	0
$516$	−5.58311e6	−0.923107
$517$	−311453.	−0.0512467
$518$	370431.	0.0606572
$519$	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	$0.189940\pi$
$522$	−516414.	−0.0829511
$523$	−1.51839e6	−0.242733	−0.121367	−	0.992608i	$-0.538728\pi$
$523$	−1.51839e6	−0.242733	−0.121367	+	0.992608i	$0.538728\pi$
$524$	3.53751e6	0.562820
$525$	0	0
$526$	−3.06478e6	−0.482986
$527$	2.07200e6	0.324985
$528$	53874.7	0.00841007
$529$	−5.47706e6	−0.850959
$530$	0	0
$531$	1.45966e6	0.224654
$532$	749687.	0.114842
$533$	57272.1	0.00873222
$534$	6.07047e6	0.921232
$535$	0	0
$536$	−2.36942e6	−0.356230
$537$	−1.54564e7	−2.31299
$538$	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
$543$	1.76713e6	0.257198
$544$	−442900.	−0.0641665
$545$	0	0
$546$	112339.	0.0161268
$547$	7.64729e6	1.09280	0.546398	−	0.837526i	$-0.315999\pi$
$547$	7.64729e6	1.09280	0.546398	+	0.837526i	$0.315999\pi$
$548$	6.80277e6	0.967686
$549$	−4.25858e6	−0.603023
$550$	0	0
$551$	−953296.	−0.133767
$552$	−1.20981e6	−0.168993
$553$	−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	$0.292290\pi$
$558$	−2.48152e6	−0.337391
$559$	−536909.	−0.0726727
$560$	0	0
$561$	91022.8	0.0122108
$562$	−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	$0.428784\pi$
$564$	−8.82055e6	−1.16761
$565$	0	0
$566$	−4.46625e6	−0.586006
$567$	3.61361e6	0.472046
$568$	4.09153e6	0.532127
$569$	−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	$-0.485983\pi$
$571$	685949.	0.0880444	0.0440222	+	0.999031i	$0.485983\pi$
$572$	5180.95	0.000662093 0
$573$	−7.91294e6	−1.00682
$574$	−377999.	−0.0478862
$575$	0	0
$576$	530438.	0.0666160
$577$	1.10785e7	1.38530	0.692648	−	0.721276i	$-0.256444\pi$
$577$	1.10785e7	1.38530	0.692648	+	0.721276i	$0.256444\pi$
$578$	4.93114e6	0.613942
$579$	−1.25669e6	−0.155787
$580$	0	0
$581$	4.62544e6	0.568477
$582$	−7.21172e6	−0.882533
$583$	−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	$0.339354\pi$
$588$	−741440.	−0.0884368
$589$	−4.58087e6	−0.544076
$590$	0	0
$591$	3.53617e6	0.416452
$592$	483828.	0.0567396
$593$	−1.33847e7	−1.56304	−0.781521	−	0.623879i	$-0.785556\pi$
$593$	−1.33847e7	−1.56304	−0.781521	+	0.623879i	$0.785556\pi$
$594$	95542.1	0.0111104
$595$	0	0
$596$	−591645.	−0.0682253
$597$	1.08668e7	1.24786
$598$	−116343.	−0.0133042
$599$	−2.30111e6	−0.262042	−0.131021	−	0.991380i	$-0.541826\pi$
$599$	−2.30111e6	−0.262042	−0.131021	+	0.991380i	$0.541826\pi$
$600$	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
$603$	4.79443e6	0.536962
$604$	1.30286e6	0.145313
$605$	0	0
$606$	−1.09551e6	−0.121181
$607$	−1.75294e7	−1.93106	−0.965529	−	0.260296i	$-0.916180\pi$
$607$	−1.75294e7	−1.93106	−0.965529	+	0.260296i	$0.916180\pi$
$608$	979184.	0.107425
$609$	942809.	0.103010
$610$	0	0
$611$	−848243.	−0.0919216
$612$	896190.	0.0967212
$613$	1.70871e7	1.83661	0.918304	−	0.395876i	$-0.129559\pi$
$613$	1.70871e7	1.83661	0.918304	+	0.395876i	$0.129559\pi$
$614$	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	$0.419219\pi$
$618$	−1.52555e7	−1.60678
$619$	1.68570e6	0.176829	0.0884144	−	0.996084i	$-0.471820\pi$
$619$	1.68570e6	0.176829	0.0884144	+	0.996084i	$0.471820\pi$
$620$	0	0
$621$	−2.14549e6	−0.223253
$622$	−6.45224e6	−0.668705
$623$	−3.85296e6	−0.397717
$624$	146728.	0.0150852
$625$	0	0
$626$	8.85676e6	0.903315
$627$	−201237.	−0.0204428
$628$	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
$633$	1.09010e7	1.08133
$634$	−9.32015e6	−0.920873
$635$	0	0
$636$	−88658.4	−0.00869115
$637$	−71301.8	−0.00696229
$638$	43481.5	0.00422914
$639$	−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	$0.376263\pi$
$642$	−1.25919e7	−1.20574
$643$	−7.75540e6	−0.739736	−0.369868	−	0.929084i	$-0.620597\pi$
$643$	−7.75540e6	−0.739736	−0.369868	+	0.929084i	$0.620597\pi$
$644$	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	$0.383728\pi$
$648$	4.71982e6	0.441558
$649$	−122901.	−0.0114537
$650$	0	0
$651$	4.53048e6	0.418978
$652$	−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	$0.307842\pi$
$654$	−5.18854e6	−0.474352
$655$	0	0
$656$	−493713.	−0.0447935
$657$	−6.36408e6	−0.575204
$658$	5.59845e6	0.504084
$659$	1.94708e7	1.74650	0.873252	−	0.487269i	$-0.162007\pi$
$659$	1.94708e7	1.74650	0.873252	+	0.487269i	$0.162007\pi$
$660$	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
$663$	247901.	0.0219025
$664$	6.04139e6	0.531762
$665$	0	0
$666$	−979007.	−0.0855264
$667$	−976420.	−0.0849811
$668$	−9.93751e6	−0.861661
$669$	−2.27082e7	−1.96163
$670$	0	0
$671$	358567.	0.0307443
$672$	−968412.	−0.0827250
$673$	1.79934e7	1.53136	0.765678	−	0.643224i	$-0.222403\pi$
$673$	1.79934e7	1.53136	0.765678	+	0.643224i	$0.222403\pi$
$674$	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	$0.712708\pi$
$678$	−4.32140e6	−0.361036
$679$	4.57731e6	0.381010
$680$	0	0
$681$	1.90075e7	1.57057
$682$	208941.	0.0172014
$683$	882971.	0.0724260	0.0362130	−	0.999344i	$-0.488471\pi$
$683$	882971.	0.0724260	0.0362130	+	0.999344i	$0.488471\pi$
$684$	−1.98134e6	−0.161927
$685$	0	0
$686$	470596.	0.0381802
$687$	−3.36310e6	−0.271862
$688$	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
$693$	69191.3	0.00547291
$694$	−1.05104e7	−0.828361
$695$	0	0
$696$	1.23142e6	0.0963573
$697$	−834142.	−0.0650366
$698$	1.53063e7	1.18913
$699$	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	$0.680091\pi$
$702$	260209.	0.0199287
$703$	−1.80724e6	−0.137920
$704$	−44662.2	−0.00339632
$705$	0	0
$706$	5.96573e6	0.450455
$707$	695328.	0.0523168
$708$	−3.48064e6	−0.260961
$709$	−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
$712$	−5.03244e6	−0.372030
$713$	−4.69199e6	−0.345647
$714$	−1.63616e6	−0.120110
$715$	0	0
$716$	1.28134e7	0.934078
$717$	−4.82755e6	−0.350694
$718$	1.62549e6	0.117672
$719$	1.46181e7	1.05456	0.527278	−	0.849693i	$-0.323213\pi$
$719$	1.46181e7	1.05456	0.527278	+	0.849693i	$0.323213\pi$
$720$	0	0
$721$	9.68276e6	0.693683
$722$	6.24686e6	0.445984
$723$	1.59206e7	1.13270
$724$	−1.46495e6	−0.103867
$725$	0	0
$726$	−1.24242e7	−0.874833
$727$	−1.17581e7	−0.825089	−0.412544	−	0.910938i	$-0.635360\pi$
$727$	−1.17581e7	−0.825089	−0.412544	+	0.910938i	$0.635360\pi$
$728$	−93128.9	−0.00651263
$729$	723267.	0.0504057
$730$	0	0
$731$	7.81984e6	0.541258
$732$	1.01549e7	0.700480
$733$	9.42125e6	0.647662	0.323831	−	0.946115i	$-0.395029\pi$
$733$	9.42125e6	0.647662	0.323831	+	0.946115i	$0.395029\pi$
$734$	8.37707e6	0.573921
$735$	0	0
$736$	1.00294e6	0.0682462
$737$	−403685.	−0.0273762
$738$	999008.	0.0675193
$739$	8.69064e6	0.585384	0.292692	−	0.956207i	$-0.405449\pi$
$739$	8.69064e6	0.585384	0.292692	+	0.956207i	$0.405449\pi$
$740$	0	0
$741$	−548071.	−0.0366683
$742$	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	$0.657041\pi$
$744$	5.91736e6	0.391918
$745$	0	0
$746$	8.44518e6	0.555600
$747$	−1.22245e7	−0.801550
$748$	−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
$753$	−1.40799e7	−0.904922
$754$	118422.	0.00758584
$755$	0	0
$756$	−1.71740e6	−0.109286
$757$	−2.62790e6	−0.166674	−0.0833371	−	0.996521i	$-0.526558\pi$
$757$	−2.62790e6	−0.166674	−0.0833371	+	0.996521i	$0.526558\pi$
$758$	−4.84231e6	−0.306111
$759$	−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	$0.579848\pi$
$762$	6.64636e6	0.414664
$763$	3.29320e6	0.204789
$764$	6.55985e6	0.406594
$765$	0	0
$766$	−3.97807e6	−0.244963
$767$	−334722.	−0.0205445
$768$	−1.26486e6	−0.0773822
$769$	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
$772$	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	$0.615274\pi$
$774$	−9.36541e6	−0.561920
$775$	0	0
$776$	5.97853e6	0.356402
$777$	1.78736e6	0.106208
$778$	9.96483e6	0.590230
$779$	1.84416e6	0.108882
$780$	0	0
$781$	697086.	0.0408939
$782$	1.69449e6	0.0990881
$783$	2.18382e6	0.127296
$784$	614656.	0.0357143
$785$	0	0
$786$	1.70687e7	0.985474
$787$	−1.28179e7	−0.737701	−0.368851	−	0.929489i	$-0.620249\pi$
$787$	−1.28179e7	−0.737701	−0.368851	+	0.929489i	$0.620249\pi$
$788$	−2.93150e6	−0.168180
$789$	−1.47878e7	−0.845689
$790$	0	0
$791$	2.74282e6	0.155868
$792$	90372.3	0.00511944
$793$	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	$0.119994\pi$
$798$	3.61730e6	0.201084
$799$	1.23543e7	0.684621
$800$	0	0
$801$	1.01829e7	0.560779
$802$	2.53524e6	0.139182
$803$	535848.	0.0293260
$804$	−1.14326e7	−0.623743
$805$	0	0
$806$	569053.	0.0308542
$807$	1.49537e7	0.808284
$808$	908184.	0.0489379
$809$	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	$-0.701861\pi$
$811$	−2.21959e7	−1.18501	−0.592504	+	0.805568i	$0.701861\pi$
$812$	−781591.	−0.0415996
$813$	3.15310e7	1.67306
$814$	82431.2	0.00436044
$815$	0	0
$816$	−2.13702e6	−0.112353
$817$	−1.72885e7	−0.906152
$818$	−3.77068e6	−0.197032
$819$	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	$0.288780\pi$
$822$	3.28239e7	1.69438
$823$	1.75911e7	0.905300	0.452650	−	0.891688i	$-0.350479\pi$
$823$	1.75911e7	0.905300	0.452650	+	0.891688i	$0.350479\pi$
$824$	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	$0.487851\pi$
$828$	−2.02940e6	−0.102871
$829$	−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
$832$	−121638.	−0.00609200
$833$	1.03848e6	0.0518543
$834$	−1.11883e6	−0.0556994
$835$	0	0
$836$	166826.	0.00825561
$837$	1.04939e7	0.517755
$838$	1.57442e7	0.774478
$839$	−2.72358e7	−1.33578	−0.667890	−	0.744260i	$-0.732802\pi$
$839$	−2.72358e7	−1.33578	−0.667890	+	0.744260i	$0.732802\pi$
$840$	0	0
$841$	−1.95173e7	−0.951545
$842$	2.30457e7	1.12023
$843$	−1.76358e7	−0.854727
$844$	−9.03697e6	−0.436684
$845$	0	0
$846$	−1.47961e7	−0.710756
$847$	7.88567e6	0.377685
$848$	73498.1	0.00350983
$849$	−2.15500e7	−1.02607
$850$	0	0
$851$	−1.85107e6	−0.0876193
$852$	1.97419e7	0.931732
$853$	−2.58892e6	−0.121828	−0.0609140	−	0.998143i	$-0.519402\pi$
$853$	−2.58892e6	−0.121828	−0.0609140	+	0.998143i	$0.519402\pi$
$854$	−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	$0.766067\pi$
$858$	24998.5	0.00115930
$859$	2.89610e7	1.33915	0.669577	−	0.742742i	$-0.266475\pi$
$859$	2.89610e7	1.33915	0.669577	+	0.742742i	$0.266475\pi$
$860$	0	0
$861$	−1.82387e6	−0.0838468
$862$	−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	$0.212503\pi$
$864$	−2.24313e6	−0.102228
$865$	0	0
$866$	5.61833e6	0.254573
$867$	2.37931e7	1.07499
$868$	−3.75578e6	−0.169200
$869$	−776766.	−0.0348932
$870$	0	0
$871$	−1.09944e6	−0.0491049
$872$	4.30132e6	0.191562
$873$	−1.20973e7	−0.537222
$874$	−3.74625e6	−0.165889
$875$	0	0
$876$	1.51756e7	0.668166
$877$	−1.55161e7	−0.681213	−0.340607	−	0.940206i	$-0.610632\pi$
$877$	−1.55161e7	−0.681213	−0.340607	+	0.940206i	$0.610632\pi$
$878$	1.46588e7	0.641743
$879$	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	$0.653499\pi$
$882$	−1.24373e6	−0.0538339
$883$	−1.70291e7	−0.735003	−0.367502	−	0.930023i	$-0.619787\pi$
$883$	−1.70291e7	−0.735003	−0.367502	+	0.930023i	$0.619787\pi$
$884$	−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	$0.811639\pi$
$888$	2.33451e6	0.0993488
$889$	−4.21848e6	−0.179020
$890$	0	0
$891$	804130.	0.0339338
$892$	1.88252e7	0.792186
$893$	−2.73134e7	−1.14617
$894$	−2.85473e6	−0.119460
$895$	0	0
$896$	802816.	0.0334077
$897$	−561365.	−0.0232951
$898$	3.49410e6	0.144592
$899$	4.77582e6	0.197083
$900$	0	0
$901$	124177.	0.00509600
$902$	−84115.3	−0.00344238
$903$	1.70983e7	0.697803
$904$	3.58245e6	0.145801
$905$	0	0
$906$	6.28638e6	0.254437
$907$	9.89966e6	0.399578	0.199789	−	0.979839i	$-0.435974\pi$
$907$	9.89966e6	0.399578	0.199789	+	0.979839i	$0.435974\pi$
$908$	−1.57573e7	−0.634258
$909$	−1.83767e6	−0.0737664
$910$	0	0
$911$	8.31180e6	0.331817	0.165909	−	0.986141i	$-0.446944\pi$
$911$	8.31180e6	0.331817	0.165909	+	0.986141i	$0.446944\pi$
$912$	4.72463e6	0.188097
$913$	1.02929e6	0.0408659
$914$	2.08195e7	0.824337
$915$	0	0
$916$	2.78802e6	0.109789
$917$	−1.08336e7	−0.425452
$918$	−3.78983e6	−0.148427
$919$	−1.40148e7	−0.547393	−0.273697	−	0.961816i	$-0.588246\pi$
$919$	−1.40148e7	−0.547393	−0.273697	+	0.961816i	$0.588246\pi$
$920$	0	0
$921$	2.77031e7	1.07617
$922$	7.73896e6	0.299816
$923$	1.89852e6	0.0733517
$924$	−164991.	−0.00635742
$925$	0	0
$926$	−9.43522e6	−0.361597
$927$	−2.55905e7	−0.978090
$928$	−1.02085e6	−0.0389129
$929$	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
$932$	−1.43226e7	−0.540108
$933$	−3.11325e7	−1.17087
$934$	−1.34961e7	−0.506221
$935$	0	0
$936$	246129.	0.00918277
$937$	−4.55745e7	−1.69579	−0.847896	−	0.530162i	$-0.822131\pi$
$937$	−4.55745e7	−1.69579	−0.847896	+	0.530162i	$0.822131\pi$
$938$	7.25635e6	0.269284
$939$	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	$0.585442\pi$
$942$	8.78395e6	0.322524
$943$	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	$0.188293\pi$
$948$	−2.19985e7	−0.795010
$949$	1.45938e6	0.0526022
$950$	0	0
$951$	−4.49704e7	−1.61241
$952$	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	$0.648751\pi$
$954$	−148720.	−0.00529054
$955$	0	0
$956$	4.00205e6	0.141624
$957$	209801.	0.00740506
$958$	−1.89907e7	−0.668540
$959$	−2.08335e7	−0.731502
$960$	0	0
$961$	−5.67994e6	−0.198397
$962$	224502.	0.00782135
$963$	−2.11223e7	−0.733964
$964$	−1.31982e7	−0.457427
$965$	0	0
$966$	3.70504e6	0.127747
$967$	−5.68247e7	−1.95421	−0.977105	−	0.212758i	$-0.931755\pi$
$967$	−5.68247e7	−1.95421	−0.977105	+	0.212758i	$0.931755\pi$
$968$	1.02997e7	0.353292
$969$	7.98240e6	0.273102
$970$	0	0
$971$	−2.46332e7	−0.838441	−0.419221	−	0.907884i	$-0.637697\pi$
$971$	−2.46332e7	−0.838441	−0.419221	+	0.907884i	$0.637697\pi$
$972$	1.42566e7	0.484006
$973$	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	$0.253850\pi$
$978$	−3.40123e7	−1.13707
$979$	−857391.	−0.0285905
$980$	0	0
$981$	−8.70355e6	−0.288751
$982$	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	$0.520264\pi$
$984$	−2.38220e6	−0.0784315
$985$	0	0
$986$	−1.72476e6	−0.0564985
$987$	2.70129e7	0.882631
$988$	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
$993$	−5.08764e7	−1.63736
$994$	−1.25303e7	−0.402250
$995$	0	0
$996$	2.91502e7	0.931093
$997$	−3.80257e6	−0.121155	−0.0605773	−	0.998164i	$-0.519294\pi$
$997$	−3.80257e6	−0.121155	−0.0605773	+	0.998164i	$0.519294\pi$
$998$	183608.	0.00583533
$999$	4.14004e6	0.131248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.p.1.1		2
5.2	odd	4		350.6.c.k.99.2		4
5.3	odd	4		350.6.c.k.99.3		4
5.4	even	2		70.6.a.h.1.2	✓	2
15.14	odd	2		630.6.a.s.1.2		2
20.19	odd	2		560.6.a.k.1.1		2
35.34	odd	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	5.4	even	2
350.6.a.p.1.1		2	1.1	even	1	trivial
350.6.c.k.99.2		4	5.2	odd	4
350.6.c.k.99.3		4	5.3	odd	4
490.6.a.u.1.1		2	35.34	odd	2
560.6.a.k.1.1		2	20.19	odd	2
630.6.a.s.1.2		2	15.14	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -19.3003 q^{3} +16.0000 q^{4} +77.2012 q^{6} -49.0000 q^{7} -64.0000 q^{8} +129.501 q^{9} -10.9039 q^{11} -308.805 q^{12} -29.6967 q^{13} +196.000 q^{14} +256.000 q^{16} +432.519 q^{17} -518.006 q^{18} -956.234 q^{19} +945.715 q^{21} +43.6155 q^{22} -979.429 q^{23} +1235.22 q^{24} +118.787 q^{26} +2190.56 q^{27} -784.000 q^{28} +996.928 q^{29} +4790.53 q^{31} -1024.00 q^{32} +210.448 q^{33} -1730.08 q^{34} +2072.02 q^{36} +1889.95 q^{37} +3824.94 q^{38} +573.156 q^{39} -1928.56 q^{41} -3782.86 q^{42} +18079.7 q^{43} -174.462 q^{44} +3917.72 q^{46} +28563.5 q^{47} -4940.88 q^{48} +2401.00 q^{49} -8347.75 q^{51} -475.148 q^{52} +287.102 q^{53} -8762.22 q^{54} +3136.00 q^{56} +18455.6 q^{57} -3987.71 q^{58} +11271.3 q^{59} -32884.4 q^{61} -19162.1 q^{62} -6345.57 q^{63} +4096.00 q^{64} -841.792 q^{66} +37022.2 q^{67} +6920.31 q^{68} +18903.3 q^{69} -63930.2 q^{71} -8288.10 q^{72} -49142.9 q^{73} -7559.81 q^{74} -15299.7 q^{76} +534.290 q^{77} -2292.62 q^{78} +71237.6 q^{79} -73747.2 q^{81} +7714.26 q^{82} -94396.8 q^{83} +15131.4 q^{84} -72319.0 q^{86} -19241.0 q^{87} +697.848 q^{88} +78631.8 q^{89} +1455.14 q^{91} -15670.9 q^{92} -92458.7 q^{93} -114254. q^{94} +19763.5 q^{96} -93414.6 q^{97} -9604.00 q^{98} -1412.07 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 5 q^{3} + 32 q^{4} + 20 q^{6} - 98 q^{7} - 128 q^{8} + 91 q^{9} + 415 q^{11} - 80 q^{12} - 429 q^{13} + 392 q^{14} + 512 q^{16} - 1319 q^{17} - 364 q^{18} + 1918 q^{19} + 245 q^{21} - 1660 q^{22}+ \cdots - 17810 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 - 5 * q^3 + 32 * q^4 + 20 * q^6 - 98 * q^7 - 128 * q^8 + 91 * q^9 + 415 * q^11 - 80 * q^12 - 429 * q^13 + 392 * q^14 + 512 * q^16 - 1319 * q^17 - 364 * q^18 + 1918 * q^19 + 245 * q^21 - 1660 * q^22 + 1334 * q^23 + 320 * q^24 + 1716 * q^26 - 1835 * q^27 - 1568 * q^28 - 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 6301 * q^33 + 5276 * q^34 + 1456 * q^36 + 9156 * q^37 - 7672 * q^38 - 5137 * q^39 - 7822 * q^41 - 980 * q^42 - 2078 * q^43 + 6640 * q^44 - 5336 * q^46 + 8507 * q^47 - 1280 * q^48 + 4802 * q^49 - 33395 * q^51 - 6864 * q^52 + 34242 * q^53 + 7340 * q^54 + 6272 * q^56 + 59558 * q^57 + 4524 * q^58 + 6952 * q^59 - 45138 * q^61 + 21888 * q^62 - 4459 * q^63 + 8192 * q^64 - 25204 * q^66 + 54556 * q^67 - 21104 * q^68 + 51986 * q^69 - 62272 * q^71 - 5824 * q^72 - 40896 * q^73 - 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 20548 * q^78 + 62069 * q^79 - 121958 * q^81 + 31288 * q^82 - 189600 * q^83 + 3920 * q^84 + 8312 * q^86 - 49671 * q^87 - 26560 * q^88 + 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 239216 * q^93 - 34028 * q^94 + 5120 * q^96 - 155547 * q^97 - 19208 * q^98 - 17810 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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