LMFDB - Embedded newform 350.6.a.s.1.2

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,-20] 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{79}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.88819$ 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)

$f(q)$	$=$	\(q+4.00000 q^{2} +7.77639 q^{3} +16.0000 q^{4} +31.1056 q^{6} +49.0000 q^{7} +64.0000 q^{8} -182.528 q^{9} -588.329 q^{11} +124.422 q^{12} +147.515 q^{13} +196.000 q^{14} +256.000 q^{16} +63.1806 q^{17} -730.111 q^{18} -1612.51 q^{19} +381.043 q^{21} -2353.32 q^{22} -1484.73 q^{23} +497.689 q^{24} +590.061 q^{26} -3309.07 q^{27} +784.000 q^{28} -1691.84 q^{29} -7446.52 q^{31} +1024.00 q^{32} -4575.08 q^{33} +252.722 q^{34} -2920.44 q^{36} +2439.79 q^{37} -6450.03 q^{38} +1147.14 q^{39} +334.413 q^{41} +1524.17 q^{42} +11933.5 q^{43} -9413.27 q^{44} -5938.91 q^{46} -5866.15 q^{47} +1990.76 q^{48} +2401.00 q^{49} +491.317 q^{51} +2360.24 q^{52} +25017.1 q^{53} -13236.3 q^{54} +3136.00 q^{56} -12539.5 q^{57} -6767.38 q^{58} -52348.0 q^{59} +16847.5 q^{61} -29786.1 q^{62} -8943.86 q^{63} +4096.00 q^{64} -18300.3 q^{66} -69080.7 q^{67} +1010.89 q^{68} -11545.8 q^{69} +30528.0 q^{71} -11681.8 q^{72} -46165.7 q^{73} +9759.17 q^{74} -25800.1 q^{76} -28828.1 q^{77} +4588.54 q^{78} -1869.00 q^{79} +18621.6 q^{81} +1337.65 q^{82} +106022. q^{83} +6096.69 q^{84} +47734.0 q^{86} -13156.4 q^{87} -37653.1 q^{88} -38290.7 q^{89} +7228.25 q^{91} -23755.6 q^{92} -57907.0 q^{93} -23464.6 q^{94} +7963.02 q^{96} -90755.9 q^{97} +9604.00 q^{98} +107386. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 20 q^{3} + 32 q^{4} - 80 q^{6} + 98 q^{7} + 128 q^{8} + 346 q^{9} - 1070 q^{11} - 320 q^{12} - 736 q^{13} + 392 q^{14} + 512 q^{16} + 1904 q^{17} + 1384 q^{18} + 828 q^{19} - 980 q^{21}+ \cdots - 147190 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 - 20 * q^3 + 32 * q^4 - 80 * q^6 + 98 * q^7 + 128 * q^8 + 346 * q^9 - 1070 * q^11 - 320 * q^12 - 736 * q^13 + 392 * q^14 + 512 * q^16 + 1904 * q^17 + 1384 * q^18 + 828 * q^19 - 980 * q^21 - 4280 * q^22 - 1654 * q^23 - 1280 * q^24 - 2944 * q^26 - 11240 * q^27 + 1568 * q^28 - 2246 * q^29 + 288 * q^31 + 2048 * q^32 + 8804 * q^33 + 7616 * q^34 + 5536 * q^36 - 8986 * q^37 + 3312 * q^38 + 25688 * q^39 + 11228 * q^41 - 3920 * q^42 - 3402 * q^43 - 17120 * q^44 - 6616 * q^46 - 15572 * q^47 - 5120 * q^48 + 4802 * q^49 - 50640 * q^51 - 11776 * q^52 + 16188 * q^53 - 44960 * q^54 + 6272 * q^56 - 80328 * q^57 - 8984 * q^58 - 92928 * q^59 + 15172 * q^61 + 1152 * q^62 + 16954 * q^63 + 8192 * q^64 + 35216 * q^66 - 78966 * q^67 + 30464 * q^68 - 6844 * q^69 - 20822 * q^71 + 22144 * q^72 - 72884 * q^73 - 35944 * q^74 + 13248 * q^76 - 52430 * q^77 + 102752 * q^78 - 72106 * q^79 + 110482 * q^81 + 44912 * q^82 + 47400 * q^83 - 15680 * q^84 - 13608 * q^86 + 2236 * q^87 - 68480 * q^88 - 9920 * q^89 - 36064 * q^91 - 26464 * q^92 - 272744 * q^93 - 62288 * q^94 - 20480 * q^96 + 26792 * q^97 + 19208 * q^98 - 147190 * 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
$3$	7.77639	0.498856	0.249428	−	0.968393i	$-0.419757\pi$
$3$	7.77639	0.498856	0.249428	+	0.968393i	$0.419757\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	31.1056	0.352744
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	−182.528	−0.751143
$10$	0	0
$11$	−588.329	−1.46602	−0.733008	−	0.680220i	$-0.761884\pi$
$11$	−588.329	−1.46602	−0.733008	+	0.680220i	$0.761884\pi$
$12$	124.422	0.249428
$13$	147.515	0.242091	0.121045	−	0.992647i	$-0.461375\pi$
$13$	147.515	0.242091	0.121045	+	0.992647i	$0.461375\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	63.1806	0.0530226	0.0265113	−	0.999649i	$-0.491560\pi$
$17$	63.1806	0.0530226	0.0265113	+	0.999649i	$0.491560\pi$
$18$	−730.111	−0.531138
$19$	−1612.51	−1.02475	−0.512375	−	0.858762i	$-0.671234\pi$
$19$	−1612.51	−1.02475	−0.512375	+	0.858762i	$0.671234\pi$
$20$	0	0
$21$	381.043	0.188550
$22$	−2353.32	−1.03663
$23$	−1484.73	−0.585230	−0.292615	−	0.956230i	$-0.594525\pi$
$23$	−1484.73	−0.585230	−0.292615	+	0.956230i	$0.594525\pi$
$24$	497.689	0.176372
$25$	0	0
$26$	590.061	0.171184
$27$	−3309.07	−0.873568
$28$	784.000	0.188982
$29$	−1691.84	−0.373564	−0.186782	−	0.982401i	$-0.559806\pi$
$29$	−1691.84	−0.373564	−0.186782	+	0.982401i	$0.559806\pi$
$30$	0	0
$31$	−7446.52	−1.39171	−0.695855	−	0.718182i	$-0.744975\pi$
$31$	−7446.52	−1.39171	−0.695855	+	0.718182i	$0.744975\pi$
$32$	1024.00	0.176777
$33$	−4575.08	−0.731330
$34$	252.722	0.0374927
$35$	0	0
$36$	−2920.44	−0.375572
$37$	2439.79	0.292987	0.146494	−	0.989212i	$-0.453201\pi$
$37$	2439.79	0.292987	0.146494	+	0.989212i	$0.453201\pi$
$38$	−6450.03	−0.724608
$39$	1147.14	0.120768
$40$	0	0
$41$	334.413	0.0310687	0.0155343	−	0.999879i	$-0.495055\pi$
$41$	334.413	0.0310687	0.0155343	+	0.999879i	$0.495055\pi$
$42$	1524.17	0.133325
$43$	11933.5	0.984229	0.492115	−	0.870530i	$-0.336224\pi$
$43$	11933.5	0.984229	0.492115	+	0.870530i	$0.336224\pi$
$44$	−9413.27	−0.733008
$45$	0	0
$46$	−5938.91	−0.413820
$47$	−5866.15	−0.387354	−0.193677	−	0.981065i	$-0.562041\pi$
$47$	−5866.15	−0.387354	−0.193677	+	0.981065i	$0.562041\pi$
$48$	1990.76	0.124714
$49$	2401.00	0.142857
$50$	0	0
$51$	491.317	0.0264506
$52$	2360.24	0.121045
$53$	25017.1	1.22334	0.611671	−	0.791112i	$-0.290498\pi$
$53$	25017.1	1.22334	0.611671	+	0.791112i	$0.290498\pi$
$54$	−13236.3	−0.617706
$55$	0	0
$56$	3136.00	0.133631
$57$	−12539.5	−0.511202
$58$	−6767.38	−0.264150
$59$	−52348.0	−1.95781	−0.978904	−	0.204322i	$-0.934501\pi$
$59$	−52348.0	−1.95781	−0.978904	+	0.204322i	$0.934501\pi$
$60$	0	0
$61$	16847.5	0.579710	0.289855	−	0.957071i	$-0.406393\pi$
$61$	16847.5	0.579710	0.289855	+	0.957071i	$0.406393\pi$
$62$	−29786.1	−0.984088
$63$	−8943.86	−0.283905
$64$	4096.00	0.125000
$65$	0	0
$66$	−18300.3	−0.517129
$67$	−69080.7	−1.88005	−0.940026	−	0.341104i	$-0.889199\pi$
$67$	−69080.7	−1.88005	−0.940026	+	0.341104i	$0.889199\pi$
$68$	1010.89	0.0265113
$69$	−11545.8	−0.291945
$70$	0	0
$71$	30528.0	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$71$	30528.0	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$72$	−11681.8	−0.265569
$73$	−46165.7	−1.01394	−0.506970	−	0.861964i	$-0.669234\pi$
$73$	−46165.7	−1.01394	−0.506970	+	0.861964i	$0.669234\pi$
$74$	9759.17	0.207173
$75$	0	0
$76$	−25800.1	−0.512375
$77$	−28828.1	−0.554102
$78$	4588.54	0.0853962
$79$	−1869.00	−0.0336932	−0.0168466	−	0.999858i	$-0.505363\pi$
$79$	−1869.00	−0.0336932	−0.0168466	+	0.999858i	$0.505363\pi$
$80$	0	0
$81$	18621.6	0.315359
$82$	1337.65	0.0219689
$83$	106022.	1.68928	0.844642	−	0.535332i	$-0.179813\pi$
$83$	106022.	1.68928	0.844642	+	0.535332i	$0.179813\pi$
$84$	6096.69	0.0942748
$85$	0	0
$86$	47734.0	0.695955
$87$	−13156.4	−0.186355
$88$	−37653.1	−0.518315
$89$	−38290.7	−0.512411	−0.256206	−	0.966622i	$-0.582472\pi$
$89$	−38290.7	−0.512411	−0.256206	+	0.966622i	$0.582472\pi$
$90$	0	0
$91$	7228.25	0.0915018
$92$	−23755.6	−0.292615
$93$	−57907.0	−0.694263
$94$	−23464.6	−0.273901
$95$	0	0
$96$	7963.02	0.0881860
$97$	−90755.9	−0.979367	−0.489683	−	0.871900i	$-0.662888\pi$
$97$	−90755.9	−0.979367	−0.489683	+	0.871900i	$0.662888\pi$
$98$	9604.00	0.101015
$99$	107386.	1.10119
$100$	0	0
$101$	115241.	1.12409	0.562046	−	0.827106i	$-0.310014\pi$
$101$	115241.	1.12409	0.562046	+	0.827106i	$0.310014\pi$
$102$	1965.27	0.0187034
$103$	8842.67	0.0821278	0.0410639	−	0.999157i	$-0.486925\pi$
$103$	8842.67	0.0821278	0.0410639	+	0.999157i	$0.486925\pi$
$104$	9440.98	0.0855921
$105$	0	0
$106$	100068.	0.865033
$107$	−131403.	−1.10955	−0.554775	−	0.832000i	$-0.687196\pi$
$107$	−131403.	−1.10955	−0.554775	+	0.832000i	$0.687196\pi$
$108$	−52945.1	−0.436784
$109$	−115386.	−0.930223	−0.465112	−	0.885252i	$-0.653986\pi$
$109$	−115386.	−0.930223	−0.465112	+	0.885252i	$0.653986\pi$
$110$	0	0
$111$	18972.8	0.146158
$112$	12544.0	0.0944911
$113$	−114989.	−0.847152	−0.423576	−	0.905861i	$-0.639225\pi$
$113$	−114989.	−0.847152	−0.423576	+	0.905861i	$0.639225\pi$
$114$	−50158.0	−0.361475
$115$	0	0
$116$	−27069.5	−0.186782
$117$	−26925.6	−0.181845
$118$	−209392.	−1.38438
$119$	3095.85	0.0200407
$120$	0	0
$121$	185080.	1.14920
$122$	67390.0	0.409917
$123$	2600.52	0.0154988
$124$	−119144.	−0.695855
$125$	0	0
$126$	−35775.4	−0.200751
$127$	−205537.	−1.13079	−0.565394	−	0.824821i	$-0.691276\pi$
$127$	−205537.	−1.13079	−0.565394	+	0.824821i	$0.691276\pi$
$128$	16384.0	0.0883883
$129$	92799.5	0.490988
$130$	0	0
$131$	188494.	0.959662	0.479831	−	0.877361i	$-0.340698\pi$
$131$	188494.	0.959662	0.479831	+	0.877361i	$0.340698\pi$
$132$	−73201.2	−0.365665
$133$	−79012.9	−0.387319
$134$	−276323.	−1.32940
$135$	0	0
$136$	4043.56	0.0187463
$137$	40844.0	0.185920	0.0929602	−	0.995670i	$-0.470367\pi$
$137$	40844.0	0.185920	0.0929602	+	0.995670i	$0.470367\pi$
$138$	−46183.2	−0.206437
$139$	−7349.31	−0.0322633	−0.0161317	−	0.999870i	$-0.505135\pi$
$139$	−7349.31	−0.0322633	−0.0161317	+	0.999870i	$0.505135\pi$
$140$	0	0
$141$	−45617.5	−0.193234
$142$	122112.	0.508204
$143$	−86787.5	−0.354909
$144$	−46727.1	−0.187786
$145$	0	0
$146$	−184663.	−0.716964
$147$	18671.1	0.0712651
$148$	39036.7	0.146494
$149$	277980.	1.02576	0.512882	−	0.858459i	$-0.328578\pi$
$149$	277980.	1.02576	0.512882	+	0.858459i	$0.328578\pi$
$150$	0	0
$151$	100921.	0.360197	0.180099	−	0.983649i	$-0.442358\pi$
$151$	100921.	0.360197	0.180099	+	0.983649i	$0.442358\pi$
$152$	−103201.	−0.362304
$153$	−11532.2	−0.0398276
$154$	−115313.	−0.391809
$155$	0	0
$156$	18354.2	0.0603842
$157$	−202146.	−0.654511	−0.327255	−	0.944936i	$-0.606124\pi$
$157$	−202146.	−0.654511	−0.327255	+	0.944936i	$0.606124\pi$
$158$	−7476.02	−0.0238247
$159$	194543.	0.610271
$160$	0	0
$161$	−72751.6	−0.221196
$162$	74486.6	0.222993
$163$	−120232.	−0.354448	−0.177224	−	0.984171i	$-0.556712\pi$
$163$	−120232.	−0.354448	−0.177224	+	0.984171i	$0.556712\pi$
$164$	5350.60	0.0155343
$165$	0	0
$166$	424090.	1.19450
$167$	368050.	1.02121	0.510605	−	0.859815i	$-0.329421\pi$
$167$	368050.	1.02121	0.510605	+	0.859815i	$0.329421\pi$
$168$	24386.8	0.0666624
$169$	−349532.	−0.941392
$170$	0	0
$171$	294328.	0.769734
$172$	190936.	0.492115
$173$	50079.2	0.127216	0.0636080	−	0.997975i	$-0.479739\pi$
$173$	50079.2	0.127216	0.0636080	+	0.997975i	$0.479739\pi$
$174$	−52625.8	−0.131773
$175$	0	0
$176$	−150612.	−0.366504
$177$	−407078.	−0.976663
$178$	−153163.	−0.362329
$179$	−366552.	−0.855073	−0.427536	−	0.903998i	$-0.640618\pi$
$179$	−366552.	−0.855073	−0.427536	+	0.903998i	$0.640618\pi$
$180$	0	0
$181$	237963.	0.539899	0.269949	−	0.962874i	$-0.412993\pi$
$181$	237963.	0.539899	0.269949	+	0.962874i	$0.412993\pi$
$182$	28913.0	0.0647015
$183$	131013.	0.289192
$184$	−95022.5	−0.206910
$185$	0	0
$186$	−231628.	−0.490918
$187$	−37171.0	−0.0777320
$188$	−93858.4	−0.193677
$189$	−162144.	−0.330177
$190$	0	0
$191$	180213.	0.357439	0.178720	−	0.983900i	$-0.442804\pi$
$191$	180213.	0.357439	0.178720	+	0.983900i	$0.442804\pi$
$192$	31852.1	0.0623569
$193$	671065.	1.29679	0.648397	−	0.761302i	$-0.275440\pi$
$193$	671065.	1.29679	0.648397	+	0.761302i	$0.275440\pi$
$194$	−363023.	−0.692517
$195$	0	0
$196$	38416.0	0.0714286
$197$	−432242.	−0.793527	−0.396763	−	0.917921i	$-0.629867\pi$
$197$	−432242.	−0.793527	−0.396763	+	0.917921i	$0.629867\pi$
$198$	429546.	0.778657
$199$	912639.	1.63368	0.816839	−	0.576866i	$-0.195724\pi$
$199$	912639.	1.63368	0.816839	+	0.576866i	$0.195724\pi$
$200$	0	0
$201$	−537198.	−0.937874
$202$	460962.	0.794853
$203$	−82900.4	−0.141194
$204$	7861.07	0.0132253
$205$	0	0
$206$	35370.7	0.0580731
$207$	271004.	0.439592
$208$	37763.9	0.0605227
$209$	948686.	1.50230
$210$	0	0
$211$	−984209.	−1.52188	−0.760941	−	0.648821i	$-0.775262\pi$
$211$	−984209.	−1.52188	−0.760941	+	0.648821i	$0.775262\pi$
$212$	400274.	0.611671
$213$	237398.	0.358532
$214$	−525614.	−0.784571
$215$	0	0
$216$	−211780.	−0.308853
$217$	−364879.	−0.526017
$218$	−461544.	−0.657767
$219$	−359002.	−0.505809
$220$	0	0
$221$	9320.10	0.0128363
$222$	75891.1	0.103349
$223$	363329.	0.489257	0.244629	−	0.969617i	$-0.421334\pi$
$223$	363329.	0.489257	0.244629	+	0.969617i	$0.421334\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−459957.	−0.599027
$227$	587124.	0.756250	0.378125	−	0.925755i	$-0.376569\pi$
$227$	587124.	0.756250	0.378125	+	0.925755i	$0.376569\pi$
$228$	−200632.	−0.255601
$229$	1.21082e6	1.52578	0.762891	−	0.646528i	$-0.223779\pi$
$229$	1.21082e6	1.52578	0.762891	+	0.646528i	$0.223779\pi$
$230$	0	0
$231$	−224179.	−0.276417
$232$	−108278.	−0.132075
$233$	163067.	0.196778	0.0983890	−	0.995148i	$-0.468631\pi$
$233$	163067.	0.196778	0.0983890	+	0.995148i	$0.468631\pi$
$234$	−107703.	−0.128584
$235$	0	0
$236$	−837568.	−0.978904
$237$	−14534.1	−0.0168081
$238$	12383.4	0.0141709
$239$	1.27658e6	1.44562	0.722809	−	0.691048i	$-0.242851\pi$
$239$	1.27658e6	1.44562	0.722809	+	0.691048i	$0.242851\pi$
$240$	0	0
$241$	−506946.	−0.562237	−0.281118	−	0.959673i	$-0.590705\pi$
$241$	−506946.	−0.562237	−0.281118	+	0.959673i	$0.590705\pi$
$242$	740321.	0.812609
$243$	948913.	1.03089
$244$	269560.	0.289855
$245$	0	0
$246$	10402.1	0.0109593
$247$	−237870.	−0.248083
$248$	−476577.	−0.492044
$249$	824472.	0.842709
$250$	0	0
$251$	−1.72797e6	−1.73122	−0.865611	−	0.500717i	$-0.833070\pi$
$251$	−1.72797e6	−1.73122	−0.865611	+	0.500717i	$0.833070\pi$
$252$	−143102.	−0.141953
$253$	873508.	0.857957
$254$	−822148.	−0.799587
$255$	0	0
$256$	65536.0	0.0625000
$257$	−104724.	−0.0989043	−0.0494522	−	0.998776i	$-0.515748\pi$
$257$	−104724.	−0.0989043	−0.0494522	+	0.998776i	$0.515748\pi$
$258$	371198.	0.347181
$259$	119550.	0.110739
$260$	0	0
$261$	308809.	0.280600
$262$	753974.	0.678584
$263$	1.36123e6	1.21351	0.606755	−	0.794889i	$-0.292471\pi$
$263$	1.36123e6	1.21351	0.606755	+	0.794889i	$0.292471\pi$
$264$	−292805.	−0.258564
$265$	0	0
$266$	−316052.	−0.273876
$267$	−297764.	−0.255619
$268$	−1.10529e6	−0.940026
$269$	−347865.	−0.293109	−0.146555	−	0.989203i	$-0.546818\pi$
$269$	−347865.	−0.293109	−0.146555	+	0.989203i	$0.546818\pi$
$270$	0	0
$271$	1.18265e6	0.978209	0.489104	−	0.872225i	$-0.337324\pi$
$271$	1.18265e6	0.978209	0.489104	+	0.872225i	$0.337324\pi$
$272$	16174.2	0.0132557
$273$	56209.7	0.0456462
$274$	163376.	0.131466
$275$	0	0
$276$	−184733.	−0.145973
$277$	628741.	0.492348	0.246174	−	0.969226i	$-0.420826\pi$
$277$	628741.	0.492348	0.246174	+	0.969226i	$0.420826\pi$
$278$	−29397.2	−0.0228136
$279$	1.35920e6	1.04537
$280$	0	0
$281$	1.51236e6	1.14258	0.571292	−	0.820747i	$-0.306442\pi$
$281$	1.51236e6	1.14258	0.571292	+	0.820747i	$0.306442\pi$
$282$	−182470.	−0.136637
$283$	−1.17843e6	−0.874654	−0.437327	−	0.899303i	$-0.644075\pi$
$283$	−1.17843e6	−0.874654	−0.437327	+	0.899303i	$0.644075\pi$
$284$	488448.	0.359354
$285$	0	0
$286$	−347150.	−0.250959
$287$	16386.2	0.0117429
$288$	−186908.	−0.132785
$289$	−1.41587e6	−0.997189
$290$	0	0
$291$	−705753.	−0.488563
$292$	−738651.	−0.506970
$293$	−490855.	−0.334029	−0.167014	−	0.985954i	$-0.553413\pi$
$293$	−490855.	−0.334029	−0.167014	+	0.985954i	$0.553413\pi$
$294$	74684.4	0.0503920
$295$	0	0
$296$	156147.	0.103587
$297$	1.94682e6	1.28066
$298$	1.11192e6	0.725325
$299$	−219020.	−0.141679
$300$	0	0
$301$	584741.	0.372004
$302$	403685.	0.254698
$303$	896155.	0.560759
$304$	−412802.	−0.256188
$305$	0	0
$306$	−46128.8	−0.0281623
$307$	2.29867e6	1.39197	0.695986	−	0.718055i	$-0.254967\pi$
$307$	2.29867e6	1.39197	0.695986	+	0.718055i	$0.254967\pi$
$308$	−461250.	−0.277051
$309$	68764.0	0.0409699
$310$	0	0
$311$	−1.61778e6	−0.948459	−0.474229	−	0.880401i	$-0.657273\pi$
$311$	−1.61778e6	−0.948459	−0.474229	+	0.880401i	$0.657273\pi$
$312$	73416.7	0.0426981
$313$	174399.	0.100620	0.0503099	−	0.998734i	$-0.483979\pi$
$313$	174399.	0.100620	0.0503099	+	0.998734i	$0.483979\pi$
$314$	−808586.	−0.462809
$315$	0	0
$316$	−29904.1	−0.0168466
$317$	3.28000e6	1.83327	0.916633	−	0.399730i	$-0.130896\pi$
$317$	3.28000e6	1.83327	0.916633	+	0.399730i	$0.130896\pi$
$318$	778171.	0.431527
$319$	995361.	0.547651
$320$	0	0
$321$	−1.02184e6	−0.553506
$322$	−291006.	−0.156409
$323$	−101879.	−0.0543349
$324$	297946.	0.157680
$325$	0	0
$326$	−480929.	−0.250632
$327$	−897287.	−0.464047
$328$	21402.4	0.0109844
$329$	−287441.	−0.146406
$330$	0	0
$331$	−82124.6	−0.0412006	−0.0206003	−	0.999788i	$-0.506558\pi$
$331$	−82124.6	−0.0412006	−0.0206003	+	0.999788i	$0.506558\pi$
$332$	1.69636e6	0.844642
$333$	−445330.	−0.220075
$334$	1.47220e6	0.722105
$335$	0	0
$336$	97547.0	0.0471374
$337$	−1.84526e6	−0.885079	−0.442540	−	0.896749i	$-0.645922\pi$
$337$	−1.84526e6	−0.885079	−0.442540	+	0.896749i	$0.645922\pi$
$338$	−1.39813e6	−0.665665
$339$	−894201.	−0.422606
$340$	0	0
$341$	4.38100e6	2.04027
$342$	1.17731e6	0.544284
$343$	117649.	0.0539949
$344$	763743.	0.347978
$345$	0	0
$346$	200317.	0.0899553
$347$	2.68081e6	1.19521	0.597603	−	0.801792i	$-0.296120\pi$
$347$	2.68081e6	1.19521	0.597603	+	0.801792i	$0.296120\pi$
$348$	−210503.	−0.0931773
$349$	3.46523e6	1.52289	0.761444	−	0.648231i	$-0.224491\pi$
$349$	3.46523e6	1.52289	0.761444	+	0.648231i	$0.224491\pi$
$350$	0	0
$351$	−488138.	−0.211483
$352$	−602449.	−0.259157
$353$	−237146.	−0.101293	−0.0506465	−	0.998717i	$-0.516128\pi$
$353$	−237146.	−0.101293	−0.0506465	+	0.998717i	$0.516128\pi$
$354$	−1.62831e6	−0.690605
$355$	0	0
$356$	−612652.	−0.256206
$357$	24074.5	0.00999740
$358$	−1.46621e6	−0.604628
$359$	−1.33358e6	−0.546115	−0.273057	−	0.961998i	$-0.588035\pi$
$359$	−1.33358e6	−0.546115	−0.273057	+	0.961998i	$0.588035\pi$
$360$	0	0
$361$	124084.	0.0501127
$362$	951851.	0.381766
$363$	1.43926e6	0.573286
$364$	115652.	0.0457509
$365$	0	0
$366$	524051.	0.204489
$367$	−4.51463e6	−1.74967	−0.874837	−	0.484418i	$-0.839031\pi$
$367$	−4.51463e6	−1.74967	−0.874837	+	0.484418i	$0.839031\pi$
$368$	−380090.	−0.146308
$369$	−61039.6	−0.0233370
$370$	0	0
$371$	1.22584e6	0.462380
$372$	−926512.	−0.347131
$373$	−3.81113e6	−1.41834	−0.709172	−	0.705035i	$-0.750931\pi$
$373$	−3.81113e6	−1.41834	−0.709172	+	0.705035i	$0.750931\pi$
$374$	−148684.	−0.0549648
$375$	0	0
$376$	−375434.	−0.136950
$377$	−249573.	−0.0904366
$378$	−648578.	−0.233471
$379$	−896148.	−0.320466	−0.160233	−	0.987079i	$-0.551225\pi$
$379$	−896148.	−0.320466	−0.160233	+	0.987079i	$0.551225\pi$
$380$	0	0
$381$	−1.59834e6	−0.564100
$382$	720852.	0.252748
$383$	1.51304e6	0.527051	0.263526	−	0.964652i	$-0.415115\pi$
$383$	1.51304e6	0.527051	0.263526	+	0.964652i	$0.415115\pi$
$384$	127408.	0.0440930
$385$	0	0
$386$	2.68426e6	0.916972
$387$	−2.17819e6	−0.739297
$388$	−1.45209e6	−0.489683
$389$	−5.36445e6	−1.79743	−0.898714	−	0.438536i	$-0.855497\pi$
$389$	−5.36445e6	−1.79743	−0.898714	+	0.438536i	$0.855497\pi$
$390$	0	0
$391$	−93805.8	−0.0310304
$392$	153664.	0.0505076
$393$	1.46580e6	0.478733
$394$	−1.72897e6	−0.561108
$395$	0	0
$396$	1.71818e6	0.550594
$397$	−2.84949e6	−0.907385	−0.453692	−	0.891158i	$-0.649893\pi$
$397$	−2.84949e6	−0.907385	−0.453692	+	0.891158i	$0.649893\pi$
$398$	3.65056e6	1.15518
$399$	−614435.	−0.193216
$400$	0	0
$401$	2.81190e6	0.873250	0.436625	−	0.899644i	$-0.356174\pi$
$401$	2.81190e6	0.873250	0.436625	+	0.899644i	$0.356174\pi$
$402$	−2.14879e6	−0.663177
$403$	−1.09848e6	−0.336921
$404$	1.84385e6	0.562046
$405$	0	0
$406$	−331602.	−0.0998393
$407$	−1.43540e6	−0.429524
$408$	31444.3	0.00935171
$409$	3.76521e6	1.11296	0.556482	−	0.830859i	$-0.312151\pi$
$409$	3.76521e6	1.11296	0.556482	+	0.830859i	$0.312151\pi$
$410$	0	0
$411$	317619.	0.0927474
$412$	141483.	0.0410639
$413$	−2.56505e6	−0.739982
$414$	1.08402e6	0.310838
$415$	0	0
$416$	151056.	0.0427960
$417$	−57151.1	−0.0160948
$418$	3.79474e6	1.06229
$419$	−1.22257e6	−0.340202	−0.170101	−	0.985427i	$-0.554409\pi$
$419$	−1.22257e6	−0.340202	−0.170101	+	0.985427i	$0.554409\pi$
$420$	0	0
$421$	5.01813e6	1.37987	0.689933	−	0.723874i	$-0.257640\pi$
$421$	5.01813e6	1.37987	0.689933	+	0.723874i	$0.257640\pi$
$422$	−3.93684e6	−1.07613
$423$	1.07074e6	0.290959
$424$	1.60110e6	0.432517
$425$	0	0
$426$	949591.	0.253520
$427$	825527.	0.219110
$428$	−2.10245e6	−0.554775
$429$	−674894.	−0.177048
$430$	0	0
$431$	−4.55140e6	−1.18019	−0.590095	−	0.807334i	$-0.700910\pi$
$431$	−4.55140e6	−1.18019	−0.590095	+	0.807334i	$0.700910\pi$
$432$	−847122.	−0.218392
$433$	−3.75433e6	−0.962306	−0.481153	−	0.876637i	$-0.659782\pi$
$433$	−3.75433e6	−0.962306	−0.481153	+	0.876637i	$0.659782\pi$
$434$	−1.45952e6	−0.371950
$435$	0	0
$436$	−1.84618e6	−0.465112
$437$	2.39413e6	0.599715
$438$	−1.43601e6	−0.357661
$439$	4.57813e6	1.13378	0.566888	−	0.823795i	$-0.308147\pi$
$439$	4.57813e6	1.13378	0.566888	+	0.823795i	$0.308147\pi$
$440$	0	0
$441$	−438249.	−0.107306
$442$	37280.4	0.00907663
$443$	−4.09528e6	−0.991459	−0.495729	−	0.868477i	$-0.665099\pi$
$443$	−4.09528e6	−0.991459	−0.495729	+	0.868477i	$0.665099\pi$
$444$	303564.	0.0730791
$445$	0	0
$446$	1.45331e6	0.345957
$447$	2.16168e6	0.511708
$448$	200704.	0.0472456
$449$	−5.87346e6	−1.37492	−0.687461	−	0.726221i	$-0.741275\pi$
$449$	−5.87346e6	−1.37492	−0.687461	+	0.726221i	$0.741275\pi$
$450$	0	0
$451$	−196745.	−0.0455472
$452$	−1.83983e6	−0.423576
$453$	784804.	0.179687
$454$	2.34850e6	0.534750
$455$	0	0
$456$	−802527.	−0.180737
$457$	−5.40551e6	−1.21073	−0.605364	−	0.795949i	$-0.706972\pi$
$457$	−5.40551e6	−1.21073	−0.605364	+	0.795949i	$0.706972\pi$
$458$	4.84330e6	1.07889
$459$	−209069.	−0.0463188
$460$	0	0
$461$	−2.02999e6	−0.444879	−0.222440	−	0.974946i	$-0.571402\pi$
$461$	−2.02999e6	−0.444879	−0.222440	+	0.974946i	$0.571402\pi$
$462$	−896715.	−0.195456
$463$	−2.20634e6	−0.478321	−0.239160	−	0.970980i	$-0.576872\pi$
$463$	−2.20634e6	−0.478321	−0.239160	+	0.970980i	$0.576872\pi$
$464$	−433112.	−0.0933911
$465$	0	0
$466$	652268.	0.139143
$467$	−9.24988e6	−1.96266	−0.981328	−	0.192342i	$-0.938392\pi$
$467$	−9.24988e6	−1.96266	−0.981328	+	0.192342i	$0.938392\pi$
$468$	−430810.	−0.0909225
$469$	−3.38495e6	−0.710593
$470$	0	0
$471$	−1.57197e6	−0.326506
$472$	−3.35027e6	−0.692189
$473$	−7.02082e6	−1.44290
$474$	−58136.4	−0.0118851
$475$	0	0
$476$	49533.6	0.0100203
$477$	−4.56632e6	−0.918904
$478$	5.10632e6	1.02221
$479$	−8.73710e6	−1.73992	−0.869959	−	0.493125i	$-0.835855\pi$
$479$	−8.73710e6	−1.73992	−0.869959	+	0.493125i	$0.835855\pi$
$480$	0	0
$481$	359907.	0.0709295
$482$	−2.02779e6	−0.397561
$483$	−565745.	−0.110345
$484$	2.96128e6	0.574601
$485$	0	0
$486$	3.79565e6	0.728947
$487$	5.75797e6	1.10014	0.550069	−	0.835119i	$-0.314602\pi$
$487$	5.75797e6	1.10014	0.550069	+	0.835119i	$0.314602\pi$
$488$	1.07824e6	0.204958
$489$	−934973.	−0.176818
$490$	0	0
$491$	21277.9	0.00398314	0.00199157	−	0.999998i	$-0.499366\pi$
$491$	21277.9	0.00398314	0.00199157	+	0.999998i	$0.499366\pi$
$492$	41608.3	0.00774939
$493$	−106892.	−0.0198074
$494$	−951478.	−0.175421
$495$	0	0
$496$	−1.90631e6	−0.347928
$497$	1.49587e6	0.271646
$498$	3.29789e6	0.595885
$499$	−8.04194e6	−1.44580	−0.722902	−	0.690950i	$-0.757192\pi$
$499$	−8.04194e6	−1.44580	−0.722902	+	0.690950i	$0.757192\pi$
$500$	0	0
$501$	2.86210e6	0.509437
$502$	−6.91189e6	−1.22416
$503$	5.82427e6	1.02641	0.513206	−	0.858266i	$-0.328458\pi$
$503$	5.82427e6	1.02641	0.513206	+	0.858266i	$0.328458\pi$
$504$	−572407.	−0.100376
$505$	0	0
$506$	3.49403e6	0.606667
$507$	−2.71810e6	−0.469619
$508$	−3.28859e6	−0.565394
$509$	1.43672e6	0.245798	0.122899	−	0.992419i	$-0.460781\pi$
$509$	1.43672e6	0.245798	0.122899	+	0.992419i	$0.460781\pi$
$510$	0	0
$511$	−2.26212e6	−0.383233
$512$	262144.	0.0441942
$513$	5.33590e6	0.895188
$514$	−418898.	−0.0699359
$515$	0	0
$516$	1.48479e6	0.245494
$517$	3.45123e6	0.567868
$518$	478199.	0.0783041
$519$	389435.	0.0634624
$520$	0	0
$521$	−8.10030e6	−1.30739	−0.653697	−	0.756756i	$-0.726783\pi$
$521$	−8.10030e6	−1.30739	−0.653697	+	0.756756i	$0.726783\pi$
$522$	1.23523e6	0.198414
$523$	375315.	0.0599987	0.0299993	−	0.999550i	$-0.490449\pi$
$523$	375315.	0.0599987	0.0299993	+	0.999550i	$0.490449\pi$
$524$	3.01590e6	0.479831
$525$	0	0
$526$	5.44493e6	0.858081
$527$	−470475.	−0.0737922
$528$	−1.17122e6	−0.182833
$529$	−4.23193e6	−0.657505
$530$	0	0
$531$	9.55496e6	1.47059
$532$	−1.26421e6	−0.193660
$533$	49331.0	0.00752145
$534$	−1.19105e6	−0.180750
$535$	0	0
$536$	−4.42116e6	−0.664698
$537$	−2.85045e6	−0.426558
$538$	−1.39146e6	−0.207260
$539$	−1.41258e6	−0.209431
$540$	0	0
$541$	−8.56631e6	−1.25835	−0.629174	−	0.777265i	$-0.716607\pi$
$541$	−8.56631e6	−1.25835	−0.629174	+	0.777265i	$0.716607\pi$
$542$	4.73058e6	0.691698
$543$	1.85049e6	0.269332
$544$	64696.9	0.00937316
$545$	0	0
$546$	224839.	0.0322767
$547$	−5.94955e6	−0.850190	−0.425095	−	0.905149i	$-0.639759\pi$
$547$	−5.94955e6	−0.850190	−0.425095	+	0.905149i	$0.639759\pi$
$548$	653504.	0.0929602
$549$	−3.07514e6	−0.435445
$550$	0	0
$551$	2.72811e6	0.382810
$552$	−738932.	−0.103218
$553$	−91581.2	−0.0127348
$554$	2.51497e6	0.348143
$555$	0	0
$556$	−117589.	−0.0161317
$557$	3.48878e6	0.476470	0.238235	−	0.971208i	$-0.423431\pi$
$557$	3.48878e6	0.476470	0.238235	+	0.971208i	$0.423431\pi$
$558$	5.43679e6	0.739191
$559$	1.76037e6	0.238273
$560$	0	0
$561$	−289056.	−0.0387770
$562$	6.04943e6	0.807930
$563$	−4.65749e6	−0.619271	−0.309635	−	0.950855i	$-0.600207\pi$
$563$	−4.65749e6	−0.619271	−0.309635	+	0.950855i	$0.600207\pi$
$564$	−729879.	−0.0966170
$565$	0	0
$566$	−4.71371e6	−0.618474
$567$	912460.	0.119195
$568$	1.95379e6	0.254102
$569$	−5.01530e6	−0.649406	−0.324703	−	0.945816i	$-0.605264\pi$
$569$	−5.01530e6	−0.649406	−0.324703	+	0.945816i	$0.605264\pi$
$570$	0	0
$571$	−8.50936e6	−1.09221	−0.546106	−	0.837716i	$-0.683890\pi$
$571$	−8.50936e6	−1.09221	−0.546106	+	0.837716i	$0.683890\pi$
$572$	−1.38860e6	−0.177455
$573$	1.40141e6	0.178311
$574$	65544.9	0.00830346
$575$	0	0
$576$	−747634.	−0.0938929
$577$	1.46073e7	1.82654	0.913270	−	0.407354i	$-0.133548\pi$
$577$	1.46073e7	1.82654	0.913270	+	0.407354i	$0.133548\pi$
$578$	−5.66346e6	−0.705119
$579$	5.21846e6	0.646913
$580$	0	0
$581$	5.19510e6	0.638489
$582$	−2.82301e6	−0.345466
$583$	−1.47183e7	−1.79344
$584$	−2.95460e6	−0.358482
$585$	0	0
$586$	−1.96342e6	−0.236194
$587$	3.92273e6	0.469887	0.234943	−	0.972009i	$-0.424509\pi$
$587$	3.92273e6	0.469887	0.234943	+	0.972009i	$0.424509\pi$
$588$	298738.	0.0356325
$589$	1.20076e7	1.42616
$590$	0	0
$591$	−3.36128e6	−0.395855
$592$	624587.	0.0732468
$593$	−1.98554e6	−0.231869	−0.115934	−	0.993257i	$-0.536986\pi$
$593$	−1.98554e6	−0.231869	−0.115934	+	0.993257i	$0.536986\pi$
$594$	7.78729e6	0.905566
$595$	0	0
$596$	4.44768e6	0.512882
$597$	7.09704e6	0.814969
$598$	−876079.	−0.100182
$599$	−1.21357e7	−1.38196	−0.690982	−	0.722871i	$-0.742822\pi$
$599$	−1.21357e7	−1.38196	−0.690982	+	0.722871i	$0.742822\pi$
$600$	0	0
$601$	1.45565e7	1.64388	0.821942	−	0.569570i	$-0.192890\pi$
$601$	1.45565e7	1.64388	0.821942	+	0.569570i	$0.192890\pi$
$602$	2.33896e6	0.263046
$603$	1.26091e7	1.41219
$604$	1.61474e6	0.180099
$605$	0	0
$606$	3.58462e6	0.396517
$607$	−4.81923e6	−0.530892	−0.265446	−	0.964126i	$-0.585519\pi$
$607$	−4.81923e6	−0.530892	−0.265446	+	0.964126i	$0.585519\pi$
$608$	−1.65121e6	−0.181152
$609$	−644666.	−0.0704354
$610$	0	0
$611$	−865347.	−0.0937750
$612$	−184515.	−0.0199138
$613$	−1.27574e7	−1.37123	−0.685616	−	0.727964i	$-0.740467\pi$
$613$	−1.27574e7	−1.37123	−0.685616	+	0.727964i	$0.740467\pi$
$614$	9.19468e6	0.984273
$615$	0	0
$616$	−1.84500e6	−0.195905
$617$	−845244.	−0.0893859	−0.0446930	−	0.999001i	$-0.514231\pi$
$617$	−845244.	−0.0893859	−0.0446930	+	0.999001i	$0.514231\pi$
$618$	275056.	0.0289701
$619$	−1.64444e7	−1.72501	−0.862505	−	0.506048i	$-0.831106\pi$
$619$	−1.64444e7	−1.72501	−0.862505	+	0.506048i	$0.831106\pi$
$620$	0	0
$621$	4.91306e6	0.511238
$622$	−6.47112e6	−0.670662
$623$	−1.87625e6	−0.193673
$624$	293667.	0.0301921
$625$	0	0
$626$	697596.	0.0711489
$627$	7.37735e6	0.749431
$628$	−3.23434e6	−0.327255
$629$	154147.	0.0155349
$630$	0	0
$631$	−1.36092e7	−1.36069	−0.680343	−	0.732894i	$-0.738169\pi$
$631$	−1.36092e7	−1.36069	−0.680343	+	0.732894i	$0.738169\pi$
$632$	−119616.	−0.0119124
$633$	−7.65359e6	−0.759200
$634$	1.31200e7	1.29631
$635$	0	0
$636$	3.11269e6	0.305135
$637$	354184.	0.0345844
$638$	3.98145e6	0.387248
$639$	−5.57221e6	−0.539853
$640$	0	0
$641$	1.20347e6	0.115688	0.0578440	−	0.998326i	$-0.481577\pi$
$641$	1.20347e6	0.115688	0.0578440	+	0.998326i	$0.481577\pi$
$642$	−4.08738e6	−0.391388
$643$	1.31941e7	1.25849	0.629247	−	0.777205i	$-0.283363\pi$
$643$	1.31941e7	1.25849	0.629247	+	0.777205i	$0.283363\pi$
$644$	−1.16403e6	−0.110598
$645$	0	0
$646$	−407517.	−0.0384206
$647$	−382498.	−0.0359227	−0.0179613	−	0.999839i	$-0.505718\pi$
$647$	−382498.	−0.0359227	−0.0179613	+	0.999839i	$0.505718\pi$
$648$	1.19178e6	0.111496
$649$	3.07978e7	2.87018
$650$	0	0
$651$	−2.83744e6	−0.262407
$652$	−1.92372e6	−0.177224
$653$	−1.34972e7	−1.23868	−0.619342	−	0.785121i	$-0.712601\pi$
$653$	−1.34972e7	−1.23868	−0.619342	+	0.785121i	$0.712601\pi$
$654$	−3.58915e6	−0.328131
$655$	0	0
$656$	85609.6	0.00776717
$657$	8.42652e6	0.761614
$658$	−1.14977e6	−0.103525
$659$	1.01824e7	0.913353	0.456676	−	0.889633i	$-0.349040\pi$
$659$	1.01824e7	0.913353	0.456676	+	0.889633i	$0.349040\pi$
$660$	0	0
$661$	140948.	0.0125475	0.00627373	−	0.999980i	$-0.498003\pi$
$661$	140948.	0.0125475	0.00627373	+	0.999980i	$0.498003\pi$
$662$	−328498.	−0.0291332
$663$	72476.7	0.00640346
$664$	6.78544e6	0.597252
$665$	0	0
$666$	−1.78132e6	−0.155617
$667$	2.51193e6	0.218621
$668$	5.88880e6	0.510605
$669$	2.82538e6	0.244069
$670$	0	0
$671$	−9.91187e6	−0.849864
$672$	390188.	0.0333312
$673$	7.93418e6	0.675250	0.337625	−	0.941281i	$-0.390376\pi$
$673$	7.93418e6	0.675250	0.337625	+	0.941281i	$0.390376\pi$
$674$	−7.38103e6	−0.625846
$675$	0	0
$676$	−5.59252e6	−0.470696
$677$	1.73979e7	1.45890	0.729451	−	0.684033i	$-0.239776\pi$
$677$	1.73979e7	1.45890	0.729451	+	0.684033i	$0.239776\pi$
$678$	−3.57681e6	−0.298828
$679$	−4.44704e6	−0.370166
$680$	0	0
$681$	4.56571e6	0.377260
$682$	1.75240e7	1.44269
$683$	1.48842e7	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$683$	1.48842e7	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$684$	4.70924e6	0.384867
$685$	0	0
$686$	470596.	0.0381802
$687$	9.41584e6	0.761145
$688$	3.05497e6	0.246057
$689$	3.69041e6	0.296160
$690$	0	0
$691$	−1.14830e7	−0.914869	−0.457435	−	0.889243i	$-0.651232\pi$
$691$	−1.14830e7	−0.914869	−0.457435	+	0.889243i	$0.651232\pi$
$692$	801267.	0.0636080
$693$	5.26193e6	0.416210
$694$	1.07232e7	0.845138
$695$	0	0
$696$	−842012.	−0.0658863
$697$	21128.4	0.00164734
$698$	1.38609e7	1.07684
$699$	1.26807e6	0.0981638
$700$	0	0
$701$	2.06197e7	1.58485	0.792423	−	0.609972i	$-0.208819\pi$
$701$	2.06197e7	1.58485	0.792423	+	0.609972i	$0.208819\pi$
$702$	−1.95255e6	−0.149541
$703$	−3.93418e6	−0.300239
$704$	−2.40980e6	−0.183252
$705$	0	0
$706$	−948585.	−0.0716250
$707$	5.64679e6	0.424867
$708$	−6.51325e6	−0.488332
$709$	7.75134e6	0.579110	0.289555	−	0.957161i	$-0.406493\pi$
$709$	7.75134e6	0.579110	0.289555	+	0.957161i	$0.406493\pi$
$710$	0	0
$711$	341145.	0.0253084
$712$	−2.45061e6	−0.181165
$713$	1.10560e7	0.814471
$714$	96298.1	0.00706923
$715$	0	0
$716$	−5.86483e6	−0.427536
$717$	9.92719e6	0.721155
$718$	−5.33433e6	−0.386161
$719$	2.39851e7	1.73029	0.865146	−	0.501521i	$-0.167226\pi$
$719$	2.39851e7	1.73029	0.865146	+	0.501521i	$0.167226\pi$
$720$	0	0
$721$	433291.	0.0310414
$722$	496336.	0.0354351
$723$	−3.94221e6	−0.280475
$724$	3.80740e6	0.269949
$725$	0	0
$726$	5.75702e6	0.405374
$727$	−1.05562e7	−0.740750	−0.370375	−	0.928882i	$-0.620771\pi$
$727$	−1.05562e7	−0.740750	−0.370375	+	0.928882i	$0.620771\pi$
$728$	462608.	0.0323508
$729$	2.85406e6	0.198904
$730$	0	0
$731$	753965.	0.0521864
$732$	2.09620e6	0.144596
$733$	1.99297e7	1.37007	0.685033	−	0.728512i	$-0.259788\pi$
$733$	1.99297e7	1.37007	0.685033	+	0.728512i	$0.259788\pi$
$734$	−1.80585e7	−1.23721
$735$	0	0
$736$	−1.52036e6	−0.103455
$737$	4.06422e7	2.75618
$738$	−244158.	−0.0165018
$739$	−1.38830e7	−0.935133	−0.467567	−	0.883958i	$-0.654869\pi$
$739$	−1.38830e7	−0.935133	−0.467567	+	0.883958i	$0.654869\pi$
$740$	0	0
$741$	−1.84977e6	−0.123757
$742$	4.90336e6	0.326952
$743$	1.74009e7	1.15638	0.578190	−	0.815902i	$-0.303759\pi$
$743$	1.74009e7	1.15638	0.578190	+	0.815902i	$0.303759\pi$
$744$	−3.70605e6	−0.245459
$745$	0	0
$746$	−1.52445e7	−1.00292
$747$	−1.93520e7	−1.26889
$748$	−594735.	−0.0388660
$749$	−6.43877e6	−0.419371
$750$	0	0
$751$	−2.22259e7	−1.43800	−0.719001	−	0.695009i	$-0.755400\pi$
$751$	−2.22259e7	−1.43800	−0.719001	+	0.695009i	$0.755400\pi$
$752$	−1.50173e6	−0.0968386
$753$	−1.34374e7	−0.863630
$754$	−998292.	−0.0639483
$755$	0	0
$756$	−2.59431e6	−0.165089
$757$	1.78715e6	0.113350	0.0566750	−	0.998393i	$-0.481950\pi$
$757$	1.78715e6	0.113350	0.0566750	+	0.998393i	$0.481950\pi$
$758$	−3.58459e6	−0.226604
$759$	6.79274e6	0.427997
$760$	0	0
$761$	2.01942e7	1.26406	0.632028	−	0.774946i	$-0.282223\pi$
$761$	2.01942e7	1.26406	0.632028	+	0.774946i	$0.282223\pi$
$762$	−6.39335e6	−0.398879
$763$	−5.65392e6	−0.351591
$764$	2.88341e6	0.178720
$765$	0	0
$766$	6.05215e6	0.372682
$767$	−7.72213e6	−0.473967
$768$	509633.	0.0311785
$769$	1.76651e7	1.07721	0.538604	−	0.842559i	$-0.318952\pi$
$769$	1.76651e7	1.07721	0.538604	+	0.842559i	$0.318952\pi$
$770$	0	0
$771$	−814378.	−0.0493390
$772$	1.07370e7	0.648397
$773$	−8.11314e6	−0.488360	−0.244180	−	0.969730i	$-0.578519\pi$
$773$	−8.11314e6	−0.488360	−0.244180	+	0.969730i	$0.578519\pi$
$774$	−8.71277e6	−0.522762
$775$	0	0
$776$	−5.80838e6	−0.346258
$777$	929666.	0.0552426
$778$	−2.14578e7	−1.27097
$779$	−539243.	−0.0318376
$780$	0	0
$781$	−1.79605e7	−1.05364
$782$	−375223.	−0.0219418
$783$	5.59843e6	0.326334
$784$	614656.	0.0357143
$785$	0	0
$786$	5.86320e6	0.338515
$787$	−2.26545e7	−1.30382	−0.651909	−	0.758297i	$-0.726031\pi$
$787$	−2.26545e7	−1.30382	−0.651909	+	0.758297i	$0.726031\pi$
$788$	−6.91588e6	−0.396763
$789$	1.05855e7	0.605366
$790$	0	0
$791$	−5.63448e6	−0.320193
$792$	6.87273e6	0.389329
$793$	2.48526e6	0.140343
$794$	−1.13980e7	−0.641618
$795$	0	0
$796$	1.46022e7	0.816839
$797$	−2.92055e7	−1.62862	−0.814309	−	0.580431i	$-0.802884\pi$
$797$	−2.92055e7	−1.62862	−0.814309	+	0.580431i	$0.802884\pi$
$798$	−2.45774e6	−0.136625
$799$	−370627.	−0.0205385
$800$	0	0
$801$	6.98912e6	0.384894
$802$	1.12476e7	0.617481
$803$	2.71606e7	1.48645
$804$	−8.59517e6	−0.468937
$805$	0	0
$806$	−4.39390e6	−0.238239
$807$	−2.70513e6	−0.146219
$808$	7.37539e6	0.397426
$809$	−3.04067e7	−1.63342	−0.816709	−	0.577050i	$-0.804204\pi$
$809$	−3.04067e7	−1.63342	−0.816709	+	0.577050i	$0.804204\pi$
$810$	0	0
$811$	−1.67648e7	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$811$	−1.67648e7	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$812$	−1.32641e6	−0.0705970
$813$	9.19672e6	0.487985
$814$	−5.74160e6	−0.303719
$815$	0	0
$816$	125777.	0.00661266
$817$	−1.92429e7	−1.00859
$818$	1.50609e7	0.786985
$819$	−1.31936e6	−0.0687309
$820$	0	0
$821$	4.39417e6	0.227520	0.113760	−	0.993508i	$-0.463711\pi$
$821$	4.39417e6	0.227520	0.113760	+	0.993508i	$0.463711\pi$
$822$	1.27048e6	0.0655823
$823$	−2.59030e7	−1.33306	−0.666531	−	0.745477i	$-0.732222\pi$
$823$	−2.59030e7	−1.33306	−0.666531	+	0.745477i	$0.732222\pi$
$824$	565931.	0.0290366
$825$	0	0
$826$	−1.02602e7	−0.523246
$827$	2.02511e6	0.102964	0.0514820	−	0.998674i	$-0.483606\pi$
$827$	2.02511e6	0.102964	0.0514820	+	0.998674i	$0.483606\pi$
$828$	4.33606e6	0.219796
$829$	1.62217e7	0.819802	0.409901	−	0.912130i	$-0.365563\pi$
$829$	1.62217e7	0.819802	0.409901	+	0.912130i	$0.365563\pi$
$830$	0	0
$831$	4.88934e6	0.245611
$832$	604223.	0.0302614
$833$	151697.	0.00757466
$834$	−228604.	−0.0113807
$835$	0	0
$836$	1.51790e7	0.751150
$837$	2.46410e7	1.21575
$838$	−4.89026e6	−0.240559
$839$	−2.68249e7	−1.31563	−0.657815	−	0.753179i	$-0.728519\pi$
$839$	−2.68249e7	−1.31563	−0.657815	+	0.753179i	$0.728519\pi$
$840$	0	0
$841$	−1.76488e7	−0.860450
$842$	2.00725e7	0.975712
$843$	1.17607e7	0.569985
$844$	−1.57473e7	−0.760941
$845$	0	0
$846$	4.28294e6	0.205739
$847$	9.06893e6	0.434358
$848$	6.40438e6	0.305835
$849$	−9.16390e6	−0.436326
$850$	0	0
$851$	−3.62242e6	−0.171465
$852$	3.79836e6	0.179266
$853$	−3.43612e7	−1.61695	−0.808474	−	0.588532i	$-0.799706\pi$
$853$	−3.43612e7	−1.61695	−0.808474	+	0.588532i	$0.799706\pi$
$854$	3.30211e6	0.154934
$855$	0	0
$856$	−8.40982e6	−0.392285
$857$	−1.12064e6	−0.0521212	−0.0260606	−	0.999660i	$-0.508296\pi$
$857$	−1.12064e6	−0.0521212	−0.0260606	+	0.999660i	$0.508296\pi$
$858$	−2.69957e6	−0.125192
$859$	3.04895e7	1.40983	0.704916	−	0.709291i	$-0.250985\pi$
$859$	3.04895e7	1.40983	0.704916	+	0.709291i	$0.250985\pi$
$860$	0	0
$861$	127426.	0.00585799
$862$	−1.82056e7	−0.834520
$863$	−2.62444e6	−0.119953	−0.0599763	−	0.998200i	$-0.519103\pi$
$863$	−2.62444e6	−0.119953	−0.0599763	+	0.998200i	$0.519103\pi$
$864$	−3.38849e6	−0.154426
$865$	0	0
$866$	−1.50173e7	−0.680453
$867$	−1.10103e7	−0.497453
$868$	−5.83807e6	−0.263009
$869$	1.09959e6	0.0493948
$870$	0	0
$871$	−1.01905e7	−0.455143
$872$	−7.38471e6	−0.328884
$873$	1.65655e7	0.735645
$874$	9.57653e6	0.424062
$875$	0	0
$876$	−5.74404e6	−0.252905
$877$	−5.66401e6	−0.248671	−0.124335	−	0.992240i	$-0.539680\pi$
$877$	−5.66401e6	−0.248671	−0.124335	+	0.992240i	$0.539680\pi$
$878$	1.83125e7	0.801700
$879$	−3.81708e6	−0.166632
$880$	0	0
$881$	2.21548e7	0.961676	0.480838	−	0.876810i	$-0.340333\pi$
$881$	2.21548e7	0.961676	0.480838	+	0.876810i	$0.340333\pi$
$882$	−1.75300e6	−0.0758769
$883$	2.14786e7	0.927050	0.463525	−	0.886084i	$-0.346584\pi$
$883$	2.14786e7	0.927050	0.463525	+	0.886084i	$0.346584\pi$
$884$	149122.	0.00641815
$885$	0	0
$886$	−1.63811e7	−0.701067
$887$	−2.26847e7	−0.968106	−0.484053	−	0.875039i	$-0.660836\pi$
$887$	−2.26847e7	−0.968106	−0.484053	+	0.875039i	$0.660836\pi$
$888$	1.21426e6	0.0516747
$889$	−1.00713e7	−0.427397
$890$	0	0
$891$	−1.09557e7	−0.462321
$892$	5.81326e6	0.244629
$893$	9.45922e6	0.396941
$894$	8.64672e6	0.361832
$895$	0	0
$896$	802816.	0.0334077
$897$	−1.70318e6	−0.0706774
$898$	−2.34938e7	−0.972217
$899$	1.25984e7	0.519894
$900$	0	0
$901$	1.58060e6	0.0648648
$902$	−786979.	−0.0322067
$903$	4.54717e6	0.185576
$904$	−7.35931e6	−0.299513
$905$	0	0
$906$	3.13921e6	0.127058
$907$	−3.98778e7	−1.60958	−0.804790	−	0.593560i	$-0.797722\pi$
$907$	−3.98778e7	−1.60958	−0.804790	+	0.593560i	$0.797722\pi$
$908$	9.39399e6	0.378125
$909$	−2.10346e7	−0.844354
$910$	0	0
$911$	1.25677e7	0.501717	0.250858	−	0.968024i	$-0.419287\pi$
$911$	1.25677e7	0.501717	0.250858	+	0.968024i	$0.419287\pi$
$912$	−3.21011e6	−0.127801
$913$	−6.23761e7	−2.47652
$914$	−2.16220e7	−0.856113
$915$	0	0
$916$	1.93732e7	0.762891
$917$	9.23619e6	0.362718
$918$	−836275.	−0.0327524
$919$	−4.18208e6	−0.163344	−0.0816720	−	0.996659i	$-0.526026\pi$
$919$	−4.18208e6	−0.163344	−0.0816720	+	0.996659i	$0.526026\pi$
$920$	0	0
$921$	1.78753e7	0.694393
$922$	−8.11997e6	−0.314577
$923$	4.50335e6	0.173993
$924$	−3.58686e6	−0.138208
$925$	0	0
$926$	−8.82534e6	−0.338224
$927$	−1.61403e6	−0.0616898
$928$	−1.73245e6	−0.0660375
$929$	4.65328e7	1.76897	0.884484	−	0.466571i	$-0.154511\pi$
$929$	4.65328e7	1.76897	0.884484	+	0.466571i	$0.154511\pi$
$930$	0	0
$931$	−3.87163e6	−0.146393
$932$	2.60907e6	0.0983890
$933$	−1.25805e7	−0.473144
$934$	−3.69995e7	−1.38781
$935$	0	0
$936$	−1.72324e6	−0.0642919
$937$	−2.69548e7	−1.00297	−0.501484	−	0.865167i	$-0.667213\pi$
$937$	−2.69548e7	−1.00297	−0.501484	+	0.865167i	$0.667213\pi$
$938$	−1.35398e7	−0.502465
$939$	1.35620e6	0.0501947
$940$	0	0
$941$	−6.83412e6	−0.251599	−0.125799	−	0.992056i	$-0.540150\pi$
$941$	−6.83412e6	−0.251599	−0.125799	+	0.992056i	$0.540150\pi$
$942$	−6.28788e6	−0.230875
$943$	−496511.	−0.0181823
$944$	−1.34011e7	−0.489452
$945$	0	0
$946$	−2.80833e7	−1.02028
$947$	−1.70413e7	−0.617487	−0.308743	−	0.951145i	$-0.599908\pi$
$947$	−1.70413e7	−0.617487	−0.308743	+	0.951145i	$0.599908\pi$
$948$	−232546.	−0.00840403
$949$	−6.81014e6	−0.245466
$950$	0	0
$951$	2.55065e7	0.914535
$952$	198134.	0.00708545
$953$	5.08910e6	0.181513	0.0907567	−	0.995873i	$-0.471071\pi$
$953$	5.08910e6	0.181513	0.0907567	+	0.995873i	$0.471071\pi$
$954$	−1.82653e7	−0.649764
$955$	0	0
$956$	2.04253e7	0.722809
$957$	7.74032e6	0.273199
$958$	−3.49484e7	−1.23031
$959$	2.00136e6	0.0702713
$960$	0	0
$961$	2.68215e7	0.936859
$962$	1.43963e6	0.0501547
$963$	2.39848e7	0.833431
$964$	−8.11114e6	−0.281118
$965$	0	0
$966$	−2.26298e6	−0.0780257
$967$	−3.03945e7	−1.04527	−0.522635	−	0.852556i	$-0.675051\pi$
$967$	−3.03945e7	−1.04527	−0.522635	+	0.852556i	$0.675051\pi$
$968$	1.18451e7	0.406304
$969$	−792252.	−0.0271053
$970$	0	0
$971$	3.40735e7	1.15976	0.579881	−	0.814701i	$-0.303099\pi$
$971$	3.40735e7	1.15976	0.579881	+	0.814701i	$0.303099\pi$
$972$	1.51826e7	0.515443
$973$	−360116.	−0.0121944
$974$	2.30319e7	0.777915
$975$	0	0
$976$	4.31296e6	0.144928
$977$	1.28684e6	0.0431307	0.0215654	−	0.999767i	$-0.493135\pi$
$977$	1.28684e6	0.0431307	0.0215654	+	0.999767i	$0.493135\pi$
$978$	−3.73989e6	−0.125029
$979$	2.25276e7	0.751203
$980$	0	0
$981$	2.10612e7	0.698731
$982$	85111.7	0.00281650
$983$	−4.75992e7	−1.57114	−0.785571	−	0.618771i	$-0.787631\pi$
$983$	−4.75992e7	−1.57114	−0.785571	+	0.618771i	$0.787631\pi$
$984$	166433.	0.00547965
$985$	0	0
$986$	−427567.	−0.0140059
$987$	−2.23526e6	−0.0730356
$988$	−3.80591e6	−0.124041
$989$	−1.77180e7	−0.576001
$990$	0	0
$991$	2.23288e7	0.722239	0.361120	−	0.932520i	$-0.382395\pi$
$991$	2.23288e7	0.722239	0.361120	+	0.932520i	$0.382395\pi$
$992$	−7.62523e6	−0.246022
$993$	−638632.	−0.0205531
$994$	5.98349e6	0.192083
$995$	0	0
$996$	1.31915e7	0.421354
$997$	−4.81000e7	−1.53252	−0.766262	−	0.642528i	$-0.777886\pi$
$997$	−4.81000e7	−1.53252	−0.766262	+	0.642528i	$0.777886\pi$
$998$	−3.21678e7	−1.02234
$999$	−8.07344e6	−0.255944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.s.1.2	yes	2
5.2	odd	4		350.6.c.i.99.3		4
5.3	odd	4		350.6.c.i.99.2		4
5.4	even	2		350.6.a.r.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.r.1.1	✓	2	5.4	even	2
350.6.a.s.1.2	yes	2	1.1	even	1	trivial
350.6.c.i.99.2		4	5.3	odd	4
350.6.c.i.99.3		4	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +7.77639 q^{3} +16.0000 q^{4} +31.1056 q^{6} +49.0000 q^{7} +64.0000 q^{8} -182.528 q^{9} -588.329 q^{11} +124.422 q^{12} +147.515 q^{13} +196.000 q^{14} +256.000 q^{16} +63.1806 q^{17} -730.111 q^{18} -1612.51 q^{19} +381.043 q^{21} -2353.32 q^{22} -1484.73 q^{23} +497.689 q^{24} +590.061 q^{26} -3309.07 q^{27} +784.000 q^{28} -1691.84 q^{29} -7446.52 q^{31} +1024.00 q^{32} -4575.08 q^{33} +252.722 q^{34} -2920.44 q^{36} +2439.79 q^{37} -6450.03 q^{38} +1147.14 q^{39} +334.413 q^{41} +1524.17 q^{42} +11933.5 q^{43} -9413.27 q^{44} -5938.91 q^{46} -5866.15 q^{47} +1990.76 q^{48} +2401.00 q^{49} +491.317 q^{51} +2360.24 q^{52} +25017.1 q^{53} -13236.3 q^{54} +3136.00 q^{56} -12539.5 q^{57} -6767.38 q^{58} -52348.0 q^{59} +16847.5 q^{61} -29786.1 q^{62} -8943.86 q^{63} +4096.00 q^{64} -18300.3 q^{66} -69080.7 q^{67} +1010.89 q^{68} -11545.8 q^{69} +30528.0 q^{71} -11681.8 q^{72} -46165.7 q^{73} +9759.17 q^{74} -25800.1 q^{76} -28828.1 q^{77} +4588.54 q^{78} -1869.00 q^{79} +18621.6 q^{81} +1337.65 q^{82} +106022. q^{83} +6096.69 q^{84} +47734.0 q^{86} -13156.4 q^{87} -37653.1 q^{88} -38290.7 q^{89} +7228.25 q^{91} -23755.6 q^{92} -57907.0 q^{93} -23464.6 q^{94} +7963.02 q^{96} -90755.9 q^{97} +9604.00 q^{98} +107386. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 20 q^{3} + 32 q^{4} - 80 q^{6} + 98 q^{7} + 128 q^{8} + 346 q^{9} - 1070 q^{11} - 320 q^{12} - 736 q^{13} + 392 q^{14} + 512 q^{16} + 1904 q^{17} + 1384 q^{18} + 828 q^{19} - 980 q^{21}+ \cdots - 147190 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 20 * q^3 + 32 * q^4 - 80 * q^6 + 98 * q^7 + 128 * q^8 + 346 * q^9 - 1070 * q^11 - 320 * q^12 - 736 * q^13 + 392 * q^14 + 512 * q^16 + 1904 * q^17 + 1384 * q^18 + 828 * q^19 - 980 * q^21 - 4280 * q^22 - 1654 * q^23 - 1280 * q^24 - 2944 * q^26 - 11240 * q^27 + 1568 * q^28 - 2246 * q^29 + 288 * q^31 + 2048 * q^32 + 8804 * q^33 + 7616 * q^34 + 5536 * q^36 - 8986 * q^37 + 3312 * q^38 + 25688 * q^39 + 11228 * q^41 - 3920 * q^42 - 3402 * q^43 - 17120 * q^44 - 6616 * q^46 - 15572 * q^47 - 5120 * q^48 + 4802 * q^49 - 50640 * q^51 - 11776 * q^52 + 16188 * q^53 - 44960 * q^54 + 6272 * q^56 - 80328 * q^57 - 8984 * q^58 - 92928 * q^59 + 15172 * q^61 + 1152 * q^62 + 16954 * q^63 + 8192 * q^64 + 35216 * q^66 - 78966 * q^67 + 30464 * q^68 - 6844 * q^69 - 20822 * q^71 + 22144 * q^72 - 72884 * q^73 - 35944 * q^74 + 13248 * q^76 - 52430 * q^77 + 102752 * q^78 - 72106 * q^79 + 110482 * q^81 + 44912 * q^82 + 47400 * q^83 - 15680 * q^84 - 13608 * q^86 + 2236 * q^87 - 68480 * q^88 - 9920 * q^89 - 36064 * q^91 - 26464 * q^92 - 272744 * q^93 - 62288 * q^94 - 20480 * q^96 + 26792 * q^97 + 19208 * q^98 - 147190 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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