LMFDB - Embedded newform 350.6.a.w.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 = [3,12,1] 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:	$3$
Coefficient field:	3.3.1378776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 336x + 840 $ x^3 - x^2 - 336*x + 840
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52911$ 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)

$f(q)$	$=$	\(q+4.00000 q^{2} +2.52911 q^{3} +16.0000 q^{4} +10.1164 q^{6} -49.0000 q^{7} +64.0000 q^{8} -236.604 q^{9} -267.715 q^{11} +40.4658 q^{12} +896.081 q^{13} -196.000 q^{14} +256.000 q^{16} +61.1910 q^{17} -946.414 q^{18} +1622.37 q^{19} -123.926 q^{21} -1070.86 q^{22} -4280.88 q^{23} +161.863 q^{24} +3584.33 q^{26} -1212.97 q^{27} -784.000 q^{28} -7422.33 q^{29} -8936.12 q^{31} +1024.00 q^{32} -677.080 q^{33} +244.764 q^{34} -3785.66 q^{36} -640.734 q^{37} +6489.48 q^{38} +2266.29 q^{39} +3879.45 q^{41} -495.705 q^{42} -19769.9 q^{43} -4283.44 q^{44} -17123.5 q^{46} +2063.78 q^{47} +647.452 q^{48} +2401.00 q^{49} +154.759 q^{51} +14337.3 q^{52} +19772.5 q^{53} -4851.88 q^{54} -3136.00 q^{56} +4103.15 q^{57} -29689.3 q^{58} +46458.9 q^{59} -53924.4 q^{61} -35744.5 q^{62} +11593.6 q^{63} +4096.00 q^{64} -2708.32 q^{66} -44624.4 q^{67} +979.055 q^{68} -10826.8 q^{69} -50547.9 q^{71} -15142.6 q^{72} -24073.0 q^{73} -2562.94 q^{74} +25957.9 q^{76} +13118.0 q^{77} +9065.15 q^{78} +1991.87 q^{79} +54426.9 q^{81} +15517.8 q^{82} -50516.4 q^{83} -1982.82 q^{84} -79079.6 q^{86} -18771.9 q^{87} -17133.8 q^{88} +60371.0 q^{89} -43908.0 q^{91} -68494.1 q^{92} -22600.4 q^{93} +8255.14 q^{94} +2589.81 q^{96} -121045. q^{97} +9604.00 q^{98} +63342.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} + q^{3} + 48 q^{4} + 4 q^{6} - 147 q^{7} + 192 q^{8} - 56 q^{9} - 11 q^{11} + 16 q^{12} - 482 q^{13} - 588 q^{14} + 768 q^{16} - 1185 q^{17} - 224 q^{18} - 1811 q^{19} - 49 q^{21} - 44 q^{22}+ \cdots + 109448 q^{99}+O(q^{100}) $ 3 * q + 12 * q^2 + q^3 + 48 * q^4 + 4 * q^6 - 147 * q^7 + 192 * q^8 - 56 * q^9 - 11 * q^11 + 16 * q^12 - 482 * q^13 - 588 * q^14 + 768 * q^16 - 1185 * q^17 - 224 * q^18 - 1811 * q^19 - 49 * q^21 - 44 * q^22 - 3350 * q^23 + 64 * q^24 - 1928 * q^26 - 1997 * q^27 - 2352 * q^28 + 1924 * q^29 - 6560 * q^31 + 3072 * q^32 - 15865 * q^33 - 4740 * q^34 - 896 * q^36 + 5194 * q^37 - 7244 * q^38 - 1574 * q^39 - 26981 * q^41 - 196 * q^42 - 18938 * q^43 - 176 * q^44 - 13400 * q^46 + 2878 * q^47 + 256 * q^48 + 7203 * q^49 - 55729 * q^51 - 7712 * q^52 - 6416 * q^53 - 7988 * q^54 - 9408 * q^56 - 33983 * q^57 + 7696 * q^58 + 17480 * q^59 - 40538 * q^61 - 26240 * q^62 + 2744 * q^63 + 12288 * q^64 - 63460 * q^66 - 43761 * q^67 - 18960 * q^68 + 70000 * q^69 - 65476 * q^71 - 3584 * q^72 - 59143 * q^73 + 20776 * q^74 - 28976 * q^76 + 539 * q^77 - 6296 * q^78 + 133128 * q^79 - 89693 * q^81 - 107924 * q^82 - 97205 * q^83 - 784 * q^84 - 75752 * q^86 - 63566 * q^87 - 704 * q^88 + 61353 * q^89 + 23618 * q^91 - 53600 * q^92 - 53964 * q^93 + 11512 * q^94 + 1024 * q^96 - 288942 * q^97 + 28812 * q^98 + 109448 * 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
$3$	2.52911	0.162242	0.0811212	−	0.996704i	$-0.474150\pi$
$3$	2.52911	0.162242	0.0811212	+	0.996704i	$0.474150\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	10.1164	0.114723
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	−236.604	−0.973677
$10$	0	0
$11$	−267.715	−0.667100	−0.333550	−	0.942732i	$-0.608247\pi$
$11$	−267.715	−0.667100	−0.333550	+	0.942732i	$0.608247\pi$
$12$	40.4658	0.0811212
$13$	896.081	1.47058	0.735291	−	0.677752i	$-0.237046\pi$
$13$	896.081	1.47058	0.735291	+	0.677752i	$0.237046\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	61.1910	0.0513529	0.0256765	−	0.999670i	$-0.491826\pi$
$17$	61.1910	0.0513529	0.0256765	+	0.999670i	$0.491826\pi$
$18$	−946.414	−0.688494
$19$	1622.37	1.03102	0.515508	−	0.856885i	$-0.327603\pi$
$19$	1622.37	1.03102	0.515508	+	0.856885i	$0.327603\pi$
$20$	0	0
$21$	−123.926	−0.0613219
$22$	−1070.86	−0.471711
$23$	−4280.88	−1.68738	−0.843691	−	0.536829i	$-0.819622\pi$
$23$	−4280.88	−1.68738	−0.843691	+	0.536829i	$0.819622\pi$
$24$	161.863	0.0573614
$25$	0	0
$26$	3584.33	1.03986
$27$	−1212.97	−0.320214
$28$	−784.000	−0.188982
$29$	−7422.33	−1.63887	−0.819437	−	0.573170i	$-0.805714\pi$
$29$	−7422.33	−1.63887	−0.819437	+	0.573170i	$0.805714\pi$
$30$	0	0
$31$	−8936.12	−1.67011	−0.835055	−	0.550167i	$-0.814564\pi$
$31$	−8936.12	−1.67011	−0.835055	+	0.550167i	$0.814564\pi$
$32$	1024.00	0.176777
$33$	−677.080	−0.108232
$34$	244.764	0.0363120
$35$	0	0
$36$	−3785.66	−0.486839
$37$	−640.734	−0.0769438	−0.0384719	−	0.999260i	$-0.512249\pi$
$37$	−640.734	−0.0769438	−0.0384719	+	0.999260i	$0.512249\pi$
$38$	6489.48	0.729039
$39$	2266.29	0.238591
$40$	0	0
$41$	3879.45	0.360422	0.180211	−	0.983628i	$-0.442322\pi$
$41$	3879.45	0.360422	0.180211	+	0.983628i	$0.442322\pi$
$42$	−495.705	−0.0433611
$43$	−19769.9	−1.63055	−0.815274	−	0.579075i	$-0.803414\pi$
$43$	−19769.9	−1.63055	−0.815274	+	0.579075i	$0.803414\pi$
$44$	−4283.44	−0.333550
$45$	0	0
$46$	−17123.5	−1.19316
$47$	2063.78	0.136276	0.0681380	−	0.997676i	$-0.478294\pi$
$47$	2063.78	0.136276	0.0681380	+	0.997676i	$0.478294\pi$
$48$	647.452	0.0405606
$49$	2401.00	0.142857
$50$	0	0
$51$	154.759	0.00833162
$52$	14337.3	0.735291
$53$	19772.5	0.966878	0.483439	−	0.875378i	$-0.339388\pi$
$53$	19772.5	0.966878	0.483439	+	0.875378i	$0.339388\pi$
$54$	−4851.88	−0.226426
$55$	0	0
$56$	−3136.00	−0.133631
$57$	4103.15	0.167275
$58$	−29689.3	−1.15886
$59$	46458.9	1.73756	0.868778	−	0.495202i	$-0.164906\pi$
$59$	46458.9	1.73756	0.868778	+	0.495202i	$0.164906\pi$
$60$	0	0
$61$	−53924.4	−1.85550	−0.927750	−	0.373203i	$-0.878259\pi$
$61$	−53924.4	−1.85550	−0.927750	+	0.373203i	$0.878259\pi$
$62$	−35744.5	−1.18095
$63$	11593.6	0.368015
$64$	4096.00	0.125000
$65$	0	0
$66$	−2708.32	−0.0765315
$67$	−44624.4	−1.21447	−0.607234	−	0.794523i	$-0.707721\pi$
$67$	−44624.4	−1.21447	−0.607234	+	0.794523i	$0.707721\pi$
$68$	979.055	0.0256765
$69$	−10826.8	−0.273765
$70$	0	0
$71$	−50547.9	−1.19003	−0.595014	−	0.803716i	$-0.702853\pi$
$71$	−50547.9	−1.19003	−0.595014	+	0.803716i	$0.702853\pi$
$72$	−15142.6	−0.344247
$73$	−24073.0	−0.528717	−0.264358	−	0.964425i	$-0.585160\pi$
$73$	−24073.0	−0.528717	−0.264358	+	0.964425i	$0.585160\pi$
$74$	−2562.94	−0.0544075
$75$	0	0
$76$	25957.9	0.515508
$77$	13118.0	0.252140
$78$	9065.15	0.168709
$79$	1991.87	0.0359082	0.0179541	−	0.999839i	$-0.494285\pi$
$79$	1991.87	0.0359082	0.0179541	+	0.999839i	$0.494285\pi$
$80$	0	0
$81$	54426.9	0.921725
$82$	15517.8	0.254857
$83$	−50516.4	−0.804891	−0.402445	−	0.915444i	$-0.631840\pi$
$83$	−50516.4	−0.804891	−0.402445	+	0.915444i	$0.631840\pi$
$84$	−1982.82	−0.0306609
$85$	0	0
$86$	−79079.6	−1.15297
$87$	−18771.9	−0.265895
$88$	−17133.8	−0.235855
$89$	60371.0	0.807892	0.403946	−	0.914783i	$-0.367638\pi$
$89$	60371.0	0.807892	0.403946	+	0.914783i	$0.367638\pi$
$90$	0	0
$91$	−43908.0	−0.555828
$92$	−68494.1	−0.843691
$93$	−22600.4	−0.270963
$94$	8255.14	0.0963617
$95$	0	0
$96$	2589.81	0.0286807
$97$	−121045.	−1.30623	−0.653113	−	0.757260i	$-0.726537\pi$
$97$	−121045.	−1.30623	−0.653113	+	0.757260i	$0.726537\pi$
$98$	9604.00	0.101015
$99$	63342.3	0.649540
$100$	0	0
$101$	−40068.2	−0.390838	−0.195419	−	0.980720i	$-0.562607\pi$
$101$	−40068.2	−0.390838	−0.195419	+	0.980720i	$0.562607\pi$
$102$	619.035	0.00589135
$103$	−141505.	−1.31426	−0.657128	−	0.753779i	$-0.728229\pi$
$103$	−141505.	−1.31426	−0.657128	+	0.753779i	$0.728229\pi$
$104$	57349.2	0.519929
$105$	0	0
$106$	79090.0	0.683686
$107$	73248.0	0.618495	0.309248	−	0.950982i	$-0.399923\pi$
$107$	73248.0	0.618495	0.309248	+	0.950982i	$0.399923\pi$
$108$	−19407.5	−0.160107
$109$	181497.	1.46320	0.731601	−	0.681733i	$-0.238774\pi$
$109$	181497.	1.46320	0.731601	+	0.681733i	$0.238774\pi$
$110$	0	0
$111$	−1620.49	−0.0124835
$112$	−12544.0	−0.0944911
$113$	61085.2	0.450028	0.225014	−	0.974356i	$-0.427757\pi$
$113$	61085.2	0.450028	0.225014	+	0.974356i	$0.427757\pi$
$114$	16412.6	0.118281
$115$	0	0
$116$	−118757.	−0.819437
$117$	−212016.	−1.43187
$118$	185836.	1.22864
$119$	−2998.36	−0.0194096
$120$	0	0
$121$	−89379.7	−0.554978
$122$	−215698.	−1.31204
$123$	9811.56	0.0584757
$124$	−142978.	−0.835055
$125$	0	0
$126$	46374.3	0.260226
$127$	−165980.	−0.913158	−0.456579	−	0.889683i	$-0.650925\pi$
$127$	−165980.	−0.913158	−0.456579	+	0.889683i	$0.650925\pi$
$128$	16384.0	0.0883883
$129$	−50000.3	−0.264544
$130$	0	0
$131$	162690.	0.828291	0.414146	−	0.910211i	$-0.364080\pi$
$131$	162690.	0.828291	0.414146	+	0.910211i	$0.364080\pi$
$132$	−10833.3	−0.0541160
$133$	−79496.1	−0.389688
$134$	−178498.	−0.858758
$135$	0	0
$136$	3916.22	0.0181560
$137$	−224926.	−1.02385	−0.511926	−	0.859029i	$-0.671068\pi$
$137$	−224926.	−1.02385	−0.511926	+	0.859029i	$0.671068\pi$
$138$	−43307.2	−0.193581
$139$	372472.	1.63515	0.817574	−	0.575824i	$-0.195319\pi$
$139$	372472.	1.63515	0.817574	+	0.575824i	$0.195319\pi$
$140$	0	0
$141$	5219.54	0.0221098
$142$	−202191.	−0.841476
$143$	−239894.	−0.981025
$144$	−60570.5	−0.243419
$145$	0	0
$146$	−96292.0	−0.373859
$147$	6072.39	0.0231775
$148$	−10251.7	−0.0384719
$149$	166217.	0.613350	0.306675	−	0.951814i	$-0.400783\pi$
$149$	166217.	0.613350	0.306675	+	0.951814i	$0.400783\pi$
$150$	0	0
$151$	289233.	1.03230	0.516149	−	0.856499i	$-0.327365\pi$
$151$	289233.	1.03230	0.516149	+	0.856499i	$0.327365\pi$
$152$	103832.	0.364519
$153$	−14478.0	−0.0500012
$154$	52472.1	0.178290
$155$	0	0
$156$	36260.6	0.119295
$157$	259507.	0.840232	0.420116	−	0.907470i	$-0.361989\pi$
$157$	259507.	0.840232	0.420116	+	0.907470i	$0.361989\pi$
$158$	7967.49	0.0253909
$159$	50006.8	0.156869
$160$	0	0
$161$	209763.	0.637770
$162$	217708.	0.651758
$163$	353857.	1.04318	0.521590	−	0.853196i	$-0.325339\pi$
$163$	353857.	1.04318	0.521590	+	0.853196i	$0.325339\pi$
$164$	62071.2	0.180211
$165$	0	0
$166$	−202065.	−0.569144
$167$	−116741.	−0.323917	−0.161959	−	0.986798i	$-0.551781\pi$
$167$	−116741.	−0.323917	−0.161959	+	0.986798i	$0.551781\pi$
$168$	−7931.29	−0.0216806
$169$	431669.	1.16261
$170$	0	0
$171$	−383858.	−1.00388
$172$	−316319.	−0.815274
$173$	206069.	0.523476	0.261738	−	0.965139i	$-0.415704\pi$
$173$	206069.	0.523476	0.261738	+	0.965139i	$0.415704\pi$
$174$	−75087.6	−0.188016
$175$	0	0
$176$	−68535.0	−0.166775
$177$	117500.	0.281905
$178$	241484.	0.571266
$179$	−35189.4	−0.0820879	−0.0410440	−	0.999157i	$-0.513068\pi$
$179$	−35189.4	−0.0820879	−0.0410440	+	0.999157i	$0.513068\pi$
$180$	0	0
$181$	−338848.	−0.768791	−0.384395	−	0.923169i	$-0.625590\pi$
$181$	−338848.	−0.768791	−0.384395	+	0.923169i	$0.625590\pi$
$182$	−175632.	−0.393029
$183$	−136381.	−0.301041
$184$	−273976.	−0.596580
$185$	0	0
$186$	−90401.7	−0.191600
$187$	−16381.7	−0.0342575
$188$	33020.5	0.0681380
$189$	59435.5	0.121030
$190$	0	0
$191$	68297.5	0.135463	0.0677316	−	0.997704i	$-0.478424\pi$
$191$	68297.5	0.135463	0.0677316	+	0.997704i	$0.478424\pi$
$192$	10359.2	0.0202803
$193$	−699065.	−1.35090	−0.675452	−	0.737404i	$-0.736051\pi$
$193$	−699065.	−1.35090	−0.675452	+	0.737404i	$0.736051\pi$
$194$	−484181.	−0.923641
$195$	0	0
$196$	38416.0	0.0714286
$197$	407750.	0.748563	0.374281	−	0.927315i	$-0.377889\pi$
$197$	407750.	0.748563	0.374281	+	0.927315i	$0.377889\pi$
$198$	253369.	0.459294
$199$	−191571.	−0.342924	−0.171462	−	0.985191i	$-0.554849\pi$
$199$	−191571.	−0.342924	−0.171462	+	0.985191i	$0.554849\pi$
$200$	0	0
$201$	−112860.	−0.197038
$202$	−160273.	−0.276364
$203$	363694.	0.619436
$204$	2476.14	0.00416581
$205$	0	0
$206$	−566022.	−0.929319
$207$	1.01287e6	1.64297
$208$	229397.	0.367645
$209$	−434332.	−0.687791
$210$	0	0
$211$	868386.	1.34279	0.671393	−	0.741101i	$-0.265696\pi$
$211$	868386.	1.34279	0.671393	+	0.741101i	$0.265696\pi$
$212$	316360.	0.483439
$213$	−127841.	−0.193073
$214$	292992.	0.437342
$215$	0	0
$216$	−77630.1	−0.113213
$217$	437870.	0.631242
$218$	725989.	1.03464
$219$	−60883.2	−0.0857803
$220$	0	0
$221$	54832.1	0.0755186
$222$	−6481.94	−0.00882720
$223$	−1.13512e6	−1.52855	−0.764275	−	0.644891i	$-0.776903\pi$
$223$	−1.13512e6	−1.52855	−0.764275	+	0.644891i	$0.776903\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	244341.	0.318218
$227$	567988.	0.731602	0.365801	−	0.930693i	$-0.380795\pi$
$227$	567988.	0.731602	0.365801	+	0.930693i	$0.380795\pi$
$228$	65650.4	0.0836373
$229$	482333.	0.607797	0.303899	−	0.952704i	$-0.401712\pi$
$229$	482333.	0.607797	0.303899	+	0.952704i	$0.401712\pi$
$230$	0	0
$231$	33176.9	0.0409078
$232$	−475029.	−0.579429
$233$	−1.37986e6	−1.66512	−0.832562	−	0.553932i	$-0.813127\pi$
$233$	−1.37986e6	−1.66512	−0.832562	+	0.553932i	$0.813127\pi$
$234$	−848064.	−1.01249
$235$	0	0
$236$	743342.	0.868778
$237$	5037.66	0.00582584
$238$	−11993.4	−0.0137246
$239$	−819638.	−0.928170	−0.464085	−	0.885791i	$-0.653617\pi$
$239$	−819638.	−0.928170	−0.464085	+	0.885791i	$0.653617\pi$
$240$	0	0
$241$	248460.	0.275558	0.137779	−	0.990463i	$-0.456004\pi$
$241$	248460.	0.275558	0.137779	+	0.990463i	$0.456004\pi$
$242$	−357519.	−0.392429
$243$	432403.	0.469757
$244$	−862791.	−0.927750
$245$	0	0
$246$	39246.2	0.0413486
$247$	1.45377e6	1.51619
$248$	−571912.	−0.590473
$249$	−127761.	−0.130587
$250$	0	0
$251$	−166104.	−0.166416	−0.0832081	−	0.996532i	$-0.526517\pi$
$251$	−166104.	−0.166416	−0.0832081	+	0.996532i	$0.526517\pi$
$252$	185497.	0.184008
$253$	1.14606e6	1.12565
$254$	−663919.	−0.645700
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.58823e6	1.49996	0.749980	−	0.661460i	$-0.230063\pi$
$257$	1.58823e6	1.49996	0.749980	+	0.661460i	$0.230063\pi$
$258$	−200001.	−0.187061
$259$	31396.0	0.0290820
$260$	0	0
$261$	1.75615e6	1.59573
$262$	650761.	0.585690
$263$	128781.	0.114806	0.0574029	−	0.998351i	$-0.481718\pi$
$263$	128781.	0.114806	0.0574029	+	0.998351i	$0.481718\pi$
$264$	−43333.1	−0.0382658
$265$	0	0
$266$	−317984.	−0.275551
$267$	152685.	0.131074
$268$	−713991.	−0.607234
$269$	1.79127e6	1.50932	0.754658	−	0.656118i	$-0.227803\pi$
$269$	1.79127e6	1.50932	0.754658	+	0.656118i	$0.227803\pi$
$270$	0	0
$271$	−1.24380e6	−1.02879	−0.514395	−	0.857553i	$-0.671983\pi$
$271$	−1.24380e6	−1.02879	−0.514395	+	0.857553i	$0.671983\pi$
$272$	15664.9	0.0128382
$273$	−111048.	−0.0901788
$274$	−899703.	−0.723973
$275$	0	0
$276$	−173229.	−0.136882
$277$	−335893.	−0.263028	−0.131514	−	0.991314i	$-0.541984\pi$
$277$	−335893.	−0.263028	−0.131514	+	0.991314i	$0.541984\pi$
$278$	1.48989e6	1.15622
$279$	2.11432e6	1.62615
$280$	0	0
$281$	689796.	0.521141	0.260570	−	0.965455i	$-0.416089\pi$
$281$	689796.	0.521141	0.260570	+	0.965455i	$0.416089\pi$
$282$	20878.1	0.0156340
$283$	292861.	0.217368	0.108684	−	0.994076i	$-0.465336\pi$
$283$	292861.	0.217368	0.108684	+	0.994076i	$0.465336\pi$
$284$	−808766.	−0.595014
$285$	0	0
$286$	−959577.	−0.693689
$287$	−190093.	−0.136227
$288$	−242282.	−0.172123
$289$	−1.41611e6	−0.997363
$290$	0	0
$291$	−306137.	−0.211925
$292$	−385168.	−0.264358
$293$	2.30017e6	1.56528	0.782639	−	0.622476i	$-0.213873\pi$
$293$	2.30017e6	1.56528	0.782639	+	0.622476i	$0.213873\pi$
$294$	24289.6	0.0163890
$295$	0	0
$296$	−41007.0	−0.0272037
$297$	324730.	0.213615
$298$	664866.	0.433704
$299$	−3.83602e6	−2.48143
$300$	0	0
$301$	968726.	0.616289
$302$	1.15693e6	0.729944
$303$	−101337.	−0.0634105
$304$	415326.	0.257754
$305$	0	0
$306$	−57912.0	−0.0353562
$307$	−481066.	−0.291312	−0.145656	−	0.989335i	$-0.546529\pi$
$307$	−481066.	−0.291312	−0.145656	+	0.989335i	$0.546529\pi$
$308$	209888.	0.126070
$309$	−357883.	−0.213228
$310$	0	0
$311$	1.06065e6	0.621827	0.310913	−	0.950438i	$-0.399365\pi$
$311$	1.06065e6	0.621827	0.310913	+	0.950438i	$0.399365\pi$
$312$	145042.	0.0843546
$313$	−1.64246e6	−0.947618	−0.473809	−	0.880628i	$-0.657121\pi$
$313$	−1.64246e6	−0.947618	−0.473809	+	0.880628i	$0.657121\pi$
$314$	1.03803e6	0.594134
$315$	0	0
$316$	31870.0	0.0179541
$317$	1.24413e6	0.695372	0.347686	−	0.937611i	$-0.386967\pi$
$317$	1.24413e6	0.695372	0.347686	+	0.937611i	$0.386967\pi$
$318$	200027.	0.110923
$319$	1.98707e6	1.09329
$320$	0	0
$321$	185252.	0.100346
$322$	839052.	0.450972
$323$	99274.3	0.0529457
$324$	870831.	0.460863
$325$	0	0
$326$	1.41543e6	0.737639
$327$	459027.	0.237393
$328$	248285.	0.127428
$329$	−101125.	−0.0515075
$330$	0	0
$331$	617637.	0.309858	0.154929	−	0.987926i	$-0.450485\pi$
$331$	617637.	0.309858	0.154929	+	0.987926i	$0.450485\pi$
$332$	−808262.	−0.402445
$333$	151600.	0.0749184
$334$	−466966.	−0.229044
$335$	0	0
$336$	−31725.1	−0.0153305
$337$	1.66929e6	0.800678	0.400339	−	0.916367i	$-0.368892\pi$
$337$	1.66929e6	0.800678	0.400339	+	0.916367i	$0.368892\pi$
$338$	1.72668e6	0.822089
$339$	154491.	0.0730137
$340$	0	0
$341$	2.39233e6	1.11413
$342$	−1.53543e6	−0.709849
$343$	−117649.	−0.0539949
$344$	−1.26527e6	−0.576486
$345$	0	0
$346$	824275.	0.370154
$347$	−1.75377e6	−0.781895	−0.390947	−	0.920413i	$-0.627853\pi$
$347$	−1.75377e6	−0.781895	−0.390947	+	0.920413i	$0.627853\pi$
$348$	−300350.	−0.132947
$349$	−313728.	−0.137876	−0.0689382	−	0.997621i	$-0.521961\pi$
$349$	−313728.	−0.137876	−0.0689382	+	0.997621i	$0.521961\pi$
$350$	0	0
$351$	−1.08692e6	−0.470901
$352$	−274140.	−0.117928
$353$	395659.	0.168999	0.0844995	−	0.996424i	$-0.473071\pi$
$353$	395659.	0.168999	0.0844995	+	0.996424i	$0.473071\pi$
$354$	469999.	0.199337
$355$	0	0
$356$	965936.	0.403946
$357$	−7583.17	−0.00314906
$358$	−140758.	−0.0580449
$359$	3.45067e6	1.41308	0.706541	−	0.707673i	$-0.250255\pi$
$359$	3.45067e6	1.41308	0.706541	+	0.707673i	$0.250255\pi$
$360$	0	0
$361$	155983.	0.0629953
$362$	−1.35539e6	−0.543617
$363$	−226051.	−0.0900410
$364$	−702528.	−0.277914
$365$	0	0
$366$	−545523.	−0.212868
$367$	−641978.	−0.248803	−0.124401	−	0.992232i	$-0.539701\pi$
$367$	−641978.	−0.248803	−0.124401	+	0.992232i	$0.539701\pi$
$368$	−1.09590e6	−0.421845
$369$	−917892.	−0.350934
$370$	0	0
$371$	−968852.	−0.365446
$372$	−361607.	−0.135481
$373$	3.04138e6	1.13188	0.565938	−	0.824448i	$-0.308514\pi$
$373$	3.04138e6	1.13188	0.565938	+	0.824448i	$0.308514\pi$
$374$	−65526.9	−0.0242237
$375$	0	0
$376$	132082.	0.0481809
$377$	−6.65101e6	−2.41010
$378$	237742.	0.0855809
$379$	−2.52318e6	−0.902297	−0.451148	−	0.892449i	$-0.648985\pi$
$379$	−2.52318e6	−0.902297	−0.451148	+	0.892449i	$0.648985\pi$
$380$	0	0
$381$	−419781.	−0.148153
$382$	273190.	0.0957869
$383$	4.97184e6	1.73189	0.865945	−	0.500140i	$-0.166718\pi$
$383$	4.97184e6	1.73189	0.865945	+	0.500140i	$0.166718\pi$
$384$	41436.9	0.0143403
$385$	0	0
$386$	−2.79626e6	−0.955234
$387$	4.67763e6	1.58763
$388$	−1.93672e6	−0.653113
$389$	2.87669e6	0.963870	0.481935	−	0.876207i	$-0.339934\pi$
$389$	2.87669e6	0.963870	0.481935	+	0.876207i	$0.339934\pi$
$390$	0	0
$391$	−261951.	−0.0866519
$392$	153664.	0.0505076
$393$	411461.	0.134384
$394$	1.63100e6	0.529314
$395$	0	0
$396$	1.01348e6	0.324770
$397$	−4.49103e6	−1.43011	−0.715055	−	0.699068i	$-0.753598\pi$
$397$	−4.49103e6	−1.43011	−0.715055	+	0.699068i	$0.753598\pi$
$398$	−766286.	−0.242484
$399$	−201054.	−0.0632239
$400$	0	0
$401$	−2.93396e6	−0.911156	−0.455578	−	0.890196i	$-0.650567\pi$
$401$	−2.93396e6	−0.911156	−0.455578	+	0.890196i	$0.650567\pi$
$402$	−451440.	−0.139327
$403$	−8.00749e6	−2.45603
$404$	−641091.	−0.195419
$405$	0	0
$406$	1.45478e6	0.438007
$407$	171534.	0.0513292
$408$	9904.55	0.00294567
$409$	3.00941e6	0.889556	0.444778	−	0.895641i	$-0.353283\pi$
$409$	3.00941e6	0.889556	0.444778	+	0.895641i	$0.353283\pi$
$410$	0	0
$411$	−568862.	−0.166112
$412$	−2.26409e6	−0.657128
$413$	−2.27649e6	−0.656735
$414$	4.05149e6	1.16175
$415$	0	0
$416$	917587.	0.259965
$417$	942023.	0.265290
$418$	−1.73733e6	−0.486342
$419$	1.41273e6	0.393120	0.196560	−	0.980492i	$-0.437023\pi$
$419$	1.41273e6	0.393120	0.196560	+	0.980492i	$0.437023\pi$
$420$	0	0
$421$	1.75531e6	0.482667	0.241334	−	0.970442i	$-0.422415\pi$
$421$	1.75531e6	0.482667	0.241334	+	0.970442i	$0.422415\pi$
$422$	3.47355e6	0.949493
$423$	−488299.	−0.132689
$424$	1.26544e6	0.341843
$425$	0	0
$426$	−511364.	−0.136523
$427$	2.64230e6	0.701313
$428$	1.17197e6	0.309248
$429$	−606719.	−0.159164
$430$	0	0
$431$	−2.08219e6	−0.539917	−0.269958	−	0.962872i	$-0.587010\pi$
$431$	−2.08219e6	−0.539917	−0.269958	+	0.962872i	$0.587010\pi$
$432$	−310520.	−0.0800536
$433$	−1.51218e6	−0.387599	−0.193800	−	0.981041i	$-0.562081\pi$
$433$	−1.51218e6	−0.387599	−0.193800	+	0.981041i	$0.562081\pi$
$434$	1.75148e6	0.446356
$435$	0	0
$436$	2.90396e6	0.731601
$437$	−6.94517e6	−1.73972
$438$	−243533.	−0.0606558
$439$	−3.94138e6	−0.976084	−0.488042	−	0.872820i	$-0.662289\pi$
$439$	−3.94138e6	−0.976084	−0.488042	+	0.872820i	$0.662289\pi$
$440$	0	0
$441$	−568085.	−0.139097
$442$	219328.	0.0533997
$443$	−2.71502e6	−0.657301	−0.328651	−	0.944452i	$-0.606594\pi$
$443$	−2.71502e6	−0.657301	−0.328651	+	0.944452i	$0.606594\pi$
$444$	−25927.8	−0.00624177
$445$	0	0
$446$	−4.54048e6	−1.08085
$447$	420380.	0.0995115
$448$	−200704.	−0.0472456
$449$	−5.32421e6	−1.24635	−0.623174	−	0.782084i	$-0.714157\pi$
$449$	−5.32421e6	−1.24635	−0.623174	+	0.782084i	$0.714157\pi$
$450$	0	0
$451$	−1.03859e6	−0.240437
$452$	977363.	0.225014
$453$	731501.	0.167482
$454$	2.27195e6	0.517321
$455$	0	0
$456$	262602.	0.0591405
$457$	−6.65930e6	−1.49155	−0.745776	−	0.666197i	$-0.767921\pi$
$457$	−6.65930e6	−1.49155	−0.745776	+	0.666197i	$0.767921\pi$
$458$	1.92933e6	0.429778
$459$	−74222.8	−0.0164439
$460$	0	0
$461$	−8.90090e6	−1.95066	−0.975330	−	0.220754i	$-0.929148\pi$
$461$	−8.90090e6	−1.95066	−0.975330	+	0.220754i	$0.929148\pi$
$462$	132708.	0.0289262
$463$	−1.42794e6	−0.309570	−0.154785	−	0.987948i	$-0.549468\pi$
$463$	−1.42794e6	−0.309570	−0.154785	+	0.987948i	$0.549468\pi$
$464$	−1.90012e6	−0.409718
$465$	0	0
$466$	−5.51946e6	−1.17742
$467$	−2.22536e6	−0.472180	−0.236090	−	0.971731i	$-0.575866\pi$
$467$	−2.22536e6	−0.472180	−0.236090	+	0.971731i	$0.575866\pi$
$468$	−3.39226e6	−0.715936
$469$	2.18660e6	0.459025
$470$	0	0
$471$	656321.	0.136321
$472$	2.97337e6	0.614319
$473$	5.29270e6	1.08774
$474$	20150.7	0.00411949
$475$	0	0
$476$	−47973.7	−0.00970479
$477$	−4.67824e6	−0.941427
$478$	−3.27855e6	−0.656315
$479$	−1.25431e6	−0.249785	−0.124893	−	0.992170i	$-0.539859\pi$
$479$	−1.25431e6	−0.249785	−0.124893	+	0.992170i	$0.539859\pi$
$480$	0	0
$481$	−574150.	−0.113152
$482$	993839.	0.194849
$483$	530514.	0.103473
$484$	−1.43008e6	−0.277489
$485$	0	0
$486$	1.72961e6	0.332168
$487$	−7.51934e6	−1.43667	−0.718336	−	0.695697i	$-0.755096\pi$
$487$	−7.51934e6	−1.43667	−0.718336	+	0.695697i	$0.755096\pi$
$488$	−3.45116e6	−0.656018
$489$	894944.	0.169248
$490$	0	0
$491$	−2.33106e6	−0.436365	−0.218183	−	0.975908i	$-0.570013\pi$
$491$	−2.33106e6	−0.436365	−0.218183	+	0.975908i	$0.570013\pi$
$492$	156985.	0.0292378
$493$	−454180.	−0.0841609
$494$	5.81510e6	1.07211
$495$	0	0
$496$	−2.28765e6	−0.417527
$497$	2.47685e6	0.449788
$498$	−511046.	−0.0923393
$499$	4.03781e6	0.725930	0.362965	−	0.931803i	$-0.381765\pi$
$499$	4.03781e6	0.725930	0.362965	+	0.931803i	$0.381765\pi$
$500$	0	0
$501$	−295252.	−0.0525531
$502$	−664416.	−0.117674
$503$	−6.49570e6	−1.14474	−0.572369	−	0.819996i	$-0.693975\pi$
$503$	−6.49570e6	−1.14474	−0.572369	+	0.819996i	$0.693975\pi$
$504$	741989.	0.130113
$505$	0	0
$506$	4.58422e6	0.795956
$507$	1.09174e6	0.188625
$508$	−2.65568e6	−0.456579
$509$	−1.16552e7	−1.99400	−0.997000	−	0.0773970i	$-0.975339\pi$
$509$	−1.16552e7	−1.99400	−0.997000	+	0.0773970i	$0.975339\pi$
$510$	0	0
$511$	1.17958e6	0.199836
$512$	262144.	0.0441942
$513$	−1.96789e6	−0.330146
$514$	6.35291e6	1.06063
$515$	0	0
$516$	−800004.	−0.132272
$517$	−552506.	−0.0909097
$518$	125584.	0.0205641
$519$	521171.	0.0849301
$520$	0	0
$521$	−4.37636e6	−0.706349	−0.353174	−	0.935558i	$-0.614898\pi$
$521$	−4.37636e6	−0.706349	−0.353174	+	0.935558i	$0.614898\pi$
$522$	7.02460e6	1.12835
$523$	9.49630e6	1.51810	0.759050	−	0.651033i	$-0.225664\pi$
$523$	9.49630e6	1.51810	0.759050	+	0.651033i	$0.225664\pi$
$524$	2.60304e6	0.414146
$525$	0	0
$526$	515126.	0.0811800
$527$	−546810.	−0.0857650
$528$	−173333.	−0.0270580
$529$	1.18896e7	1.84726
$530$	0	0
$531$	−1.09923e7	−1.69182
$532$	−1.27194e6	−0.194844
$533$	3.47631e6	0.530029
$534$	610740.	0.0926836
$535$	0	0
$536$	−2.85596e6	−0.429379
$537$	−88997.8	−0.0133181
$538$	7.16508e6	1.06725
$539$	−642783.	−0.0953000
$540$	0	0
$541$	9.30701e6	1.36715	0.683576	−	0.729879i	$-0.260424\pi$
$541$	9.30701e6	1.36715	0.683576	+	0.729879i	$0.260424\pi$
$542$	−4.97519e6	−0.727464
$543$	−856983.	−0.124730
$544$	62659.5	0.00907800
$545$	0	0
$546$	−444192.	−0.0637661
$547$	1.51435e6	0.216401	0.108200	−	0.994129i	$-0.465491\pi$
$547$	1.51435e6	0.216401	0.108200	+	0.994129i	$0.465491\pi$
$548$	−3.59881e6	−0.511926
$549$	1.27587e7	1.80666
$550$	0	0
$551$	−1.20418e7	−1.68971
$552$	−692916.	−0.0967905
$553$	−97601.8	−0.0135720
$554$	−1.34357e6	−0.185989
$555$	0	0
$556$	5.95956e6	0.817574
$557$	1.22656e7	1.67514	0.837572	−	0.546327i	$-0.183974\pi$
$557$	1.22656e7	1.67514	0.837572	+	0.546327i	$0.183974\pi$
$558$	8.45728e6	1.14986
$559$	−1.77154e7	−2.39785
$560$	0	0
$561$	−41431.2	−0.00555802
$562$	2.75918e6	0.368502
$563$	1.38325e6	0.183920	0.0919601	−	0.995763i	$-0.470687\pi$
$563$	1.38325e6	0.183920	0.0919601	+	0.995763i	$0.470687\pi$
$564$	83512.6	0.0110549
$565$	0	0
$566$	1.17144e6	0.153702
$567$	−2.66692e6	−0.348379
$568$	−3.23506e6	−0.420738
$569$	−1.90452e6	−0.246607	−0.123303	−	0.992369i	$-0.539349\pi$
$569$	−1.90452e6	−0.246607	−0.123303	+	0.992369i	$0.539349\pi$
$570$	0	0
$571$	−705632.	−0.0905708	−0.0452854	−	0.998974i	$-0.514420\pi$
$571$	−705632.	−0.0905708	−0.0452854	+	0.998974i	$0.514420\pi$
$572$	−3.83831e6	−0.490512
$573$	172732.	0.0219779
$574$	−760373.	−0.0963267
$575$	0	0
$576$	−969128.	−0.121710
$577$	9.59475e6	1.19976	0.599879	−	0.800090i	$-0.295215\pi$
$577$	9.59475e6	1.19976	0.599879	+	0.800090i	$0.295215\pi$
$578$	−5.66445e6	−0.705242
$579$	−1.76801e6	−0.219174
$580$	0	0
$581$	2.47530e6	0.304220
$582$	−1.22455e6	−0.149854
$583$	−5.29339e6	−0.645004
$584$	−1.54067e6	−0.186930
$585$	0	0
$586$	9.20069e6	1.10682
$587$	−1.21853e7	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$587$	−1.21853e7	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$588$	97158.3	0.0115887
$589$	−1.44977e7	−1.72191
$590$	0	0
$591$	1.03124e6	0.121449
$592$	−164028.	−0.0192359
$593$	1.50682e7	1.75965	0.879824	−	0.475300i	$-0.157660\pi$
$593$	1.50682e7	1.75965	0.879824	+	0.475300i	$0.157660\pi$
$594$	1.29892e6	0.151049
$595$	0	0
$596$	2.65947e6	0.306675
$597$	−484505.	−0.0556368
$598$	−1.53441e7	−1.75464
$599$	−3.77218e6	−0.429561	−0.214781	−	0.976662i	$-0.568904\pi$
$599$	−3.77218e6	−0.429561	−0.214781	+	0.976662i	$0.568904\pi$
$600$	0	0
$601$	9.60700e6	1.08493	0.542465	−	0.840078i	$-0.317491\pi$
$601$	9.60700e6	1.08493	0.542465	+	0.840078i	$0.317491\pi$
$602$	3.87490e6	0.435782
$603$	1.05583e7	1.18250
$604$	4.62772e6	0.516149
$605$	0	0
$606$	−405347.	−0.0448380
$607$	−8.41593e6	−0.927108	−0.463554	−	0.886069i	$-0.653426\pi$
$607$	−8.41593e6	−0.927108	−0.463554	+	0.886069i	$0.653426\pi$
$608$	1.66131e6	0.182260
$609$	919823.	0.100499
$610$	0	0
$611$	1.84932e6	0.200405
$612$	−231648.	−0.0250006
$613$	−1.30421e7	−1.40184	−0.700918	−	0.713242i	$-0.747226\pi$
$613$	−1.30421e7	−1.40184	−0.700918	+	0.713242i	$0.747226\pi$
$614$	−1.92427e6	−0.205989
$615$	0	0
$616$	839554.	0.0891450
$617$	−1.09403e7	−1.15695	−0.578477	−	0.815699i	$-0.696353\pi$
$617$	−1.09403e7	−1.15695	−0.578477	+	0.815699i	$0.696353\pi$
$618$	−1.43153e6	−0.150775
$619$	−1.16825e7	−1.22549	−0.612743	−	0.790282i	$-0.709934\pi$
$619$	−1.16825e7	−1.22549	−0.612743	+	0.790282i	$0.709934\pi$
$620$	0	0
$621$	5.19258e6	0.540324
$622$	4.24258e6	0.439698
$623$	−2.95818e6	−0.305355
$624$	580170.	0.0596477
$625$	0	0
$626$	−6.56983e6	−0.670067
$627$	−1.09847e6	−0.111589
$628$	4.15211e6	0.420116
$629$	−39207.1	−0.00395128
$630$	0	0
$631$	−1.05048e7	−1.05030	−0.525150	−	0.851010i	$-0.675991\pi$
$631$	−1.05048e7	−1.05030	−0.525150	+	0.851010i	$0.675991\pi$
$632$	127480.	0.0126955
$633$	2.19624e6	0.217857
$634$	4.97652e6	0.491702
$635$	0	0
$636$	800109.	0.0784343
$637$	2.15149e6	0.210083
$638$	7.94828e6	0.773074
$639$	1.19598e7	1.15870
$640$	0	0
$641$	1.89782e7	1.82435	0.912177	−	0.409796i	$-0.134400\pi$
$641$	1.89782e7	1.82435	0.912177	+	0.409796i	$0.134400\pi$
$642$	741009.	0.0709555
$643$	−1.00231e7	−0.956041	−0.478021	−	0.878349i	$-0.658646\pi$
$643$	−1.00231e7	−0.956041	−0.478021	+	0.878349i	$0.658646\pi$
$644$	3.35621e6	0.318885
$645$	0	0
$646$	397097.	0.0374383
$647$	9.92873e6	0.932466	0.466233	−	0.884662i	$-0.345611\pi$
$647$	9.92873e6	0.932466	0.466233	+	0.884662i	$0.345611\pi$
$648$	3.48332e6	0.325879
$649$	−1.24377e7	−1.15912
$650$	0	0
$651$	1.10742e6	0.102414
$652$	5.66172e6	0.521590
$653$	2.10468e7	1.93154	0.965768	−	0.259406i	$-0.0835269\pi$
$653$	2.10468e7	1.93154	0.965768	+	0.259406i	$0.0835269\pi$
$654$	1.83611e6	0.167862
$655$	0	0
$656$	993140.	0.0901054
$657$	5.69576e6	0.514800
$658$	−404502.	−0.0364213
$659$	−390852.	−0.0350589	−0.0175295	−	0.999846i	$-0.505580\pi$
$659$	−390852.	−0.0350589	−0.0175295	+	0.999846i	$0.505580\pi$
$660$	0	0
$661$	2.05713e7	1.83129	0.915647	−	0.401982i	$-0.131679\pi$
$661$	2.05713e7	1.83129	0.915647	+	0.401982i	$0.131679\pi$
$662$	2.47055e6	0.219103
$663$	138676.	0.0122523
$664$	−3.23305e6	−0.284572
$665$	0	0
$666$	606400.	0.0529753
$667$	3.17741e7	2.76541
$668$	−1.86786e6	−0.161959
$669$	−2.87084e6	−0.247996
$670$	0	0
$671$	1.44364e7	1.23780
$672$	−126901.	−0.0108403
$673$	1.65065e6	0.140481	0.0702405	−	0.997530i	$-0.477623\pi$
$673$	1.65065e6	0.140481	0.0702405	+	0.997530i	$0.477623\pi$
$674$	6.67717e6	0.566165
$675$	0	0
$676$	6.90670e6	0.581305
$677$	−1.71440e7	−1.43760	−0.718802	−	0.695214i	$-0.755309\pi$
$677$	−1.71440e7	−1.43760	−0.718802	+	0.695214i	$0.755309\pi$
$678$	617964.	0.0516285
$679$	5.93122e6	0.493707
$680$	0	0
$681$	1.43650e6	0.118697
$682$	9.56933e6	0.787809
$683$	41131.0	0.00337379	0.00168689	−	0.999999i	$-0.499463\pi$
$683$	41131.0	0.00337379	0.00168689	+	0.999999i	$0.499463\pi$
$684$	−6.14173e6	−0.501939
$685$	0	0
$686$	−470596.	−0.0381802
$687$	1.21987e6	0.0986105
$688$	−5.06110e6	−0.407637
$689$	1.77178e7	1.42187
$690$	0	0
$691$	−1.31496e7	−1.04765	−0.523825	−	0.851826i	$-0.675495\pi$
$691$	−1.31496e7	−1.04765	−0.523825	+	0.851826i	$0.675495\pi$
$692$	3.29710e6	0.261738
$693$	−3.10377e6	−0.245503
$694$	−7.01507e6	−0.552883
$695$	0	0
$696$	−1.20140e6	−0.0940080
$697$	237387.	0.0185087
$698$	−1.25491e6	−0.0974934
$699$	−3.48983e6	−0.270154
$700$	0	0
$701$	1.01963e7	0.783698	0.391849	−	0.920029i	$-0.371836\pi$
$701$	1.01963e7	0.783698	0.391849	+	0.920029i	$0.371836\pi$
$702$	−4.34768e6	−0.332977
$703$	−1.03951e6	−0.0793303
$704$	−1.09656e6	−0.0833875
$705$	0	0
$706$	1.58264e6	0.119500
$707$	1.96334e6	0.147723
$708$	1.87999e6	0.140953
$709$	1.15049e7	0.859540	0.429770	−	0.902939i	$-0.358595\pi$
$709$	1.15049e7	0.859540	0.429770	+	0.902939i	$0.358595\pi$
$710$	0	0
$711$	−471284.	−0.0349630
$712$	3.86375e6	0.285633
$713$	3.82545e7	2.81811
$714$	−30332.7	−0.00222672
$715$	0	0
$716$	−563030.	−0.0410440
$717$	−2.07295e6	−0.150589
$718$	1.38027e7	0.999199
$719$	−2.27054e7	−1.63798	−0.818988	−	0.573811i	$-0.805464\pi$
$719$	−2.27054e7	−1.63798	−0.818988	+	0.573811i	$0.805464\pi$
$720$	0	0
$721$	6.93377e6	0.496742
$722$	623930.	0.0445444
$723$	628382.	0.0447073
$724$	−5.42156e6	−0.384395
$725$	0	0
$726$	−904204.	−0.0636686
$727$	4.77668e6	0.335189	0.167595	−	0.985856i	$-0.446400\pi$
$727$	4.77668e6	0.335189	0.167595	+	0.985856i	$0.446400\pi$
$728$	−2.81011e6	−0.196515
$729$	−1.21322e7	−0.845510
$730$	0	0
$731$	−1.20974e6	−0.0837334
$732$	−2.18209e6	−0.150520
$733$	345377.	0.0237429	0.0118714	−	0.999930i	$-0.496221\pi$
$733$	345377.	0.0237429	0.0118714	+	0.999930i	$0.496221\pi$
$734$	−2.56791e6	−0.175930
$735$	0	0
$736$	−4.38362e6	−0.298290
$737$	1.19466e7	0.810171
$738$	−3.67157e6	−0.248148
$739$	1.90895e6	0.128583	0.0642916	−	0.997931i	$-0.479521\pi$
$739$	1.90895e6	0.128583	0.0642916	+	0.997931i	$0.479521\pi$
$740$	0	0
$741$	3.67676e6	0.245991
$742$	−3.87541e6	−0.258409
$743$	−4.62625e6	−0.307438	−0.153719	−	0.988115i	$-0.549125\pi$
$743$	−4.62625e6	−0.307438	−0.153719	+	0.988115i	$0.549125\pi$
$744$	−1.44643e6	−0.0957998
$745$	0	0
$746$	1.21655e7	0.800357
$747$	1.19524e7	0.783704
$748$	−262108.	−0.0171288
$749$	−3.58915e6	−0.233769
$750$	0	0
$751$	−1.30711e7	−0.845689	−0.422844	−	0.906202i	$-0.638968\pi$
$751$	−1.30711e7	−0.845689	−0.422844	+	0.906202i	$0.638968\pi$
$752$	528329.	0.0340690
$753$	−420095.	−0.0269998
$754$	−2.66041e7	−1.70420
$755$	0	0
$756$	950969.	0.0605148
$757$	−1.55732e7	−0.987732	−0.493866	−	0.869538i	$-0.664417\pi$
$757$	−1.55732e7	−0.987732	−0.493866	+	0.869538i	$0.664417\pi$
$758$	−1.00927e7	−0.638020
$759$	2.89850e6	0.182629
$760$	0	0
$761$	−2.70655e7	−1.69416	−0.847081	−	0.531463i	$-0.821642\pi$
$761$	−2.70655e7	−1.69416	−0.847081	+	0.531463i	$0.821642\pi$
$762$	−1.67912e6	−0.104760
$763$	−8.89337e6	−0.553038
$764$	1.09276e6	0.0677316
$765$	0	0
$766$	1.98874e7	1.22463
$767$	4.16310e7	2.55522
$768$	165748.	0.0101402
$769$	2.46671e6	0.150419	0.0752095	−	0.997168i	$-0.476037\pi$
$769$	2.46671e6	0.150419	0.0752095	+	0.997168i	$0.476037\pi$
$770$	0	0
$771$	4.01680e6	0.243357
$772$	−1.11850e7	−0.675452
$773$	−1.56465e6	−0.0941821	−0.0470911	−	0.998891i	$-0.514995\pi$
$773$	−1.56465e6	−0.0941821	−0.0470911	+	0.998891i	$0.514995\pi$
$774$	1.87105e7	1.12262
$775$	0	0
$776$	−7.74690e6	−0.461821
$777$	79403.8	0.00471834
$778$	1.15067e7	0.681559
$779$	6.29390e6	0.371601
$780$	0	0
$781$	1.35324e7	0.793867
$782$	−1.04780e6	−0.0612722
$783$	9.00307e6	0.524791
$784$	614656.	0.0357143
$785$	0	0
$786$	1.64584e6	0.0950238
$787$	8.04368e6	0.462933	0.231466	−	0.972843i	$-0.425648\pi$
$787$	8.04368e6	0.462933	0.231466	+	0.972843i	$0.425648\pi$
$788$	6.52400e6	0.374281
$789$	325702.	0.0186264
$790$	0	0
$791$	−2.99317e6	−0.170095
$792$	4.05391e6	0.229647
$793$	−4.83207e7	−2.72866
$794$	−1.79641e7	−1.01124
$795$	0	0
$796$	−3.06514e6	−0.171462
$797$	9.65319e6	0.538301	0.269151	−	0.963098i	$-0.413257\pi$
$797$	9.65319e6	0.538301	0.269151	+	0.963098i	$0.413257\pi$
$798$	−804217.	−0.0447060
$799$	126285.	0.00699817
$800$	0	0
$801$	−1.42840e7	−0.786627
$802$	−1.17358e7	−0.644285
$803$	6.44470e6	0.352707
$804$	−1.80576e6	−0.0985191
$805$	0	0
$806$	−3.20300e7	−1.73668
$807$	4.53032e6	0.244875
$808$	−2.56436e6	−0.138182
$809$	3.45796e7	1.85758	0.928792	−	0.370602i	$-0.120849\pi$
$809$	3.45796e7	1.85758	0.928792	+	0.370602i	$0.120849\pi$
$810$	0	0
$811$	2.20981e6	0.117978	0.0589892	−	0.998259i	$-0.481212\pi$
$811$	2.20981e6	0.117978	0.0589892	+	0.998259i	$0.481212\pi$
$812$	5.81911e6	0.309718
$813$	−3.14570e6	−0.166913
$814$	686136.	0.0362952
$815$	0	0
$816$	39618.2	0.00208291
$817$	−3.20741e7	−1.68112
$818$	1.20376e7	0.629011
$819$	1.03888e7	0.541197
$820$	0	0
$821$	−1.19800e7	−0.620297	−0.310149	−	0.950688i	$-0.600379\pi$
$821$	−1.19800e7	−0.620297	−0.310149	+	0.950688i	$0.600379\pi$
$822$	−2.27545e6	−0.117459
$823$	−2.56594e7	−1.32053	−0.660264	−	0.751034i	$-0.729555\pi$
$823$	−2.56594e7	−1.32053	−0.660264	+	0.751034i	$0.729555\pi$
$824$	−9.05635e6	−0.464660
$825$	0	0
$826$	−9.10594e6	−0.464381
$827$	−6.48745e6	−0.329845	−0.164923	−	0.986307i	$-0.552737\pi$
$827$	−6.48745e6	−0.329845	−0.164923	+	0.986307i	$0.552737\pi$
$828$	1.62059e7	0.821483
$829$	−1.60180e7	−0.809510	−0.404755	−	0.914425i	$-0.632643\pi$
$829$	−1.60180e7	−0.809510	−0.404755	+	0.914425i	$0.632643\pi$
$830$	0	0
$831$	−849511.	−0.0426743
$832$	3.67035e6	0.183823
$833$	146919.	0.00733613
$834$	3.76809e6	0.187589
$835$	0	0
$836$	−6.94932e6	−0.343896
$837$	1.08393e7	0.534793
$838$	5.65094e6	0.277978
$839$	−2.14692e7	−1.05296	−0.526479	−	0.850188i	$-0.676488\pi$
$839$	−2.14692e7	−1.05296	−0.526479	+	0.850188i	$0.676488\pi$
$840$	0	0
$841$	3.45799e7	1.68591
$842$	7.02123e6	0.341297
$843$	1.74457e6	0.0845511
$844$	1.38942e7	0.671393
$845$	0	0
$846$	−1.95320e6	−0.0938252
$847$	4.37961e6	0.209762
$848$	5.06176e6	0.241719
$849$	740677.	0.0352663
$850$	0	0
$851$	2.74290e6	0.129833
$852$	−2.04546e6	−0.0965365
$853$	−5.16759e6	−0.243173	−0.121586	−	0.992581i	$-0.538798\pi$
$853$	−5.16759e6	−0.243173	−0.121586	+	0.992581i	$0.538798\pi$
$854$	1.05692e7	0.495903
$855$	0	0
$856$	4.68787e6	0.218671
$857$	2.40841e7	1.12015	0.560077	−	0.828440i	$-0.310771\pi$
$857$	2.40841e7	1.12015	0.560077	+	0.828440i	$0.310771\pi$
$858$	−2.42688e6	−0.112546
$859$	−3.85746e7	−1.78369	−0.891844	−	0.452343i	$-0.850588\pi$
$859$	−3.85746e7	−1.78369	−0.891844	+	0.452343i	$0.850588\pi$
$860$	0	0
$861$	−480766.	−0.0221017
$862$	−8.32875e6	−0.381779
$863$	−1.26527e7	−0.578304	−0.289152	−	0.957283i	$-0.593373\pi$
$863$	−1.26527e7	−0.578304	−0.289152	+	0.957283i	$0.593373\pi$
$864$	−1.24208e6	−0.0566064
$865$	0	0
$866$	−6.04871e6	−0.274074
$867$	−3.58150e6	−0.161815
$868$	7.00592e6	0.315621
$869$	−533254.	−0.0239544
$870$	0	0
$871$	−3.99871e7	−1.78597
$872$	1.16158e7	0.517320
$873$	2.86397e7	1.27184
$874$	−2.77807e7	−1.23017
$875$	0	0
$876$	−974132.	−0.0428901
$877$	−1.61444e7	−0.708797	−0.354399	−	0.935094i	$-0.615314\pi$
$877$	−1.61444e7	−0.708797	−0.354399	+	0.935094i	$0.615314\pi$
$878$	−1.57655e7	−0.690196
$879$	5.81739e6	0.253955
$880$	0	0
$881$	3.71933e7	1.61445	0.807225	−	0.590244i	$-0.200969\pi$
$881$	3.71933e7	1.61445	0.807225	+	0.590244i	$0.200969\pi$
$882$	−2.27234e6	−0.0983563
$883$	−1.42372e7	−0.614503	−0.307251	−	0.951628i	$-0.599409\pi$
$883$	−1.42372e7	−0.614503	−0.307251	+	0.951628i	$0.599409\pi$
$884$	877313.	0.0377593
$885$	0	0
$886$	−1.08601e7	−0.464782
$887$	−3.30742e7	−1.41150	−0.705749	−	0.708462i	$-0.749389\pi$
$887$	−3.30742e7	−1.41150	−0.705749	+	0.708462i	$0.749389\pi$
$888$	−103711.	−0.00441360
$889$	8.13301e6	0.345141
$890$	0	0
$891$	−1.45709e7	−0.614883
$892$	−1.81619e7	−0.764275
$893$	3.34822e6	0.140503
$894$	1.68152e6	0.0703652
$895$	0	0
$896$	−802816.	−0.0334077
$897$	−9.70170e6	−0.402594
$898$	−2.12968e7	−0.881300
$899$	6.63269e7	2.73710
$900$	0	0
$901$	1.20990e6	0.0496520
$902$	−4.15435e6	−0.170015
$903$	2.45001e6	0.0999883
$904$	3.90945e6	0.159109
$905$	0	0
$906$	2.92600e6	0.118428
$907$	1.41066e7	0.569382	0.284691	−	0.958619i	$-0.408109\pi$
$907$	1.41066e7	0.569382	0.284691	+	0.958619i	$0.408109\pi$
$908$	9.08781e6	0.365801
$909$	9.48028e6	0.380550
$910$	0	0
$911$	−4.56582e7	−1.82273	−0.911366	−	0.411597i	$-0.864971\pi$
$911$	−4.56582e7	−1.82273	−0.911366	+	0.411597i	$0.864971\pi$
$912$	1.05041e6	0.0418187
$913$	1.35240e7	0.536943
$914$	−2.66372e7	−1.05469
$915$	0	0
$916$	7.71734e6	0.303899
$917$	−7.97182e6	−0.313065
$918$	−296891.	−0.0116276
$919$	2.92150e7	1.14108	0.570541	−	0.821269i	$-0.306734\pi$
$919$	2.92150e7	1.14108	0.570541	+	0.821269i	$0.306734\pi$
$920$	0	0
$921$	−1.21667e6	−0.0472632
$922$	−3.56036e7	−1.37932
$923$	−4.52950e7	−1.75003
$924$	530831.	0.0204539
$925$	0	0
$926$	−5.71177e6	−0.218899
$927$	3.34807e7	1.27966
$928$	−7.60047e6	−0.289715
$929$	1.54242e7	0.586357	0.293179	−	0.956058i	$-0.405287\pi$
$929$	1.54242e7	0.586357	0.293179	+	0.956058i	$0.405287\pi$
$930$	0	0
$931$	3.89531e6	0.147288
$932$	−2.20778e7	−0.832562
$933$	2.68249e6	0.100887
$934$	−8.90142e6	−0.333881
$935$	0	0
$936$	−1.35690e7	−0.506243
$937$	1.30699e6	0.0486322	0.0243161	−	0.999704i	$-0.492259\pi$
$937$	1.30699e6	0.0486322	0.0243161	+	0.999704i	$0.492259\pi$
$938$	8.74639e6	0.324580
$939$	−4.15396e6	−0.153744
$940$	0	0
$941$	1.48116e7	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$941$	1.48116e7	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$942$	2.62528e6	0.0963937
$943$	−1.66075e7	−0.608169
$944$	1.18935e7	0.434389
$945$	0	0
$946$	2.11708e7	0.769147
$947$	−2.03359e7	−0.736867	−0.368434	−	0.929654i	$-0.620106\pi$
$947$	−2.03359e7	−0.736867	−0.368434	+	0.929654i	$0.620106\pi$
$948$	80602.6	0.00291292
$949$	−2.15714e7	−0.777521
$950$	0	0
$951$	3.14654e6	0.112819
$952$	−191895.	−0.00686232
$953$	−1.41872e7	−0.506015	−0.253008	−	0.967464i	$-0.581420\pi$
$953$	−1.41872e7	−0.506015	−0.253008	+	0.967464i	$0.581420\pi$
$954$	−1.87130e7	−0.665690
$955$	0	0
$956$	−1.31142e7	−0.464085
$957$	5.02552e6	0.177378
$958$	−5.01725e6	−0.176625
$959$	1.10214e7	0.386980
$960$	0	0
$961$	5.12252e7	1.78927
$962$	−2.29660e6	−0.0800106
$963$	−1.73307e7	−0.602215
$964$	3.97536e6	0.137779
$965$	0	0
$966$	2.12205e6	0.0731668
$967$	−4.99631e7	−1.71824	−0.859118	−	0.511777i	$-0.828987\pi$
$967$	−4.99631e7	−1.71824	−0.859118	+	0.511777i	$0.828987\pi$
$968$	−5.72030e6	−0.196214
$969$	251076.	0.00859004
$970$	0	0
$971$	−1.87628e7	−0.638629	−0.319314	−	0.947649i	$-0.603453\pi$
$971$	−1.87628e7	−0.638629	−0.319314	+	0.947649i	$0.603453\pi$
$972$	6.91845e6	0.234879
$973$	−1.82511e7	−0.618028
$974$	−3.00774e7	−1.01588
$975$	0	0
$976$	−1.38046e7	−0.463875
$977$	−4.77178e7	−1.59935	−0.799676	−	0.600432i	$-0.794995\pi$
$977$	−4.77178e7	−1.59935	−0.799676	+	0.600432i	$0.794995\pi$
$978$	3.57978e6	0.119676
$979$	−1.61622e7	−0.538945
$980$	0	0
$981$	−4.29429e7	−1.42469
$982$	−9.32425e6	−0.308557
$983$	3.67695e6	0.121368	0.0606840	−	0.998157i	$-0.480672\pi$
$983$	3.67695e6	0.121368	0.0606840	+	0.998157i	$0.480672\pi$
$984$	627940.	0.0206743
$985$	0	0
$986$	−1.81672e6	−0.0595108
$987$	−255757.	−0.00835671
$988$	2.32604e7	0.758097
$989$	8.46326e7	2.75136
$990$	0	0
$991$	3.89025e7	1.25833	0.629164	−	0.777273i	$-0.283397\pi$
$991$	3.89025e7	1.25833	0.629164	+	0.777273i	$0.283397\pi$
$992$	−9.15059e6	−0.295236
$993$	1.56207e6	0.0502722
$994$	9.90738e6	0.318048
$995$	0	0
$996$	−2.04418e6	−0.0652937
$997$	−1.84228e7	−0.586974	−0.293487	−	0.955963i	$-0.594816\pi$
$997$	−1.84228e7	−0.586974	−0.293487	+	0.955963i	$0.594816\pi$
$998$	1.61512e7	0.513310
$999$	777191.	0.0246385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.w.1.2	yes	3
5.2	odd	4		350.6.c.m.99.5		6
5.3	odd	4		350.6.c.m.99.2		6
5.4	even	2		350.6.a.v.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.v.1.2	✓	3	5.4	even	2
350.6.a.w.1.2	yes	3	1.1	even	1	trivial
350.6.c.m.99.2		6	5.3	odd	4
350.6.c.m.99.5		6	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +2.52911 q^{3} +16.0000 q^{4} +10.1164 q^{6} -49.0000 q^{7} +64.0000 q^{8} -236.604 q^{9} -267.715 q^{11} +40.4658 q^{12} +896.081 q^{13} -196.000 q^{14} +256.000 q^{16} +61.1910 q^{17} -946.414 q^{18} +1622.37 q^{19} -123.926 q^{21} -1070.86 q^{22} -4280.88 q^{23} +161.863 q^{24} +3584.33 q^{26} -1212.97 q^{27} -784.000 q^{28} -7422.33 q^{29} -8936.12 q^{31} +1024.00 q^{32} -677.080 q^{33} +244.764 q^{34} -3785.66 q^{36} -640.734 q^{37} +6489.48 q^{38} +2266.29 q^{39} +3879.45 q^{41} -495.705 q^{42} -19769.9 q^{43} -4283.44 q^{44} -17123.5 q^{46} +2063.78 q^{47} +647.452 q^{48} +2401.00 q^{49} +154.759 q^{51} +14337.3 q^{52} +19772.5 q^{53} -4851.88 q^{54} -3136.00 q^{56} +4103.15 q^{57} -29689.3 q^{58} +46458.9 q^{59} -53924.4 q^{61} -35744.5 q^{62} +11593.6 q^{63} +4096.00 q^{64} -2708.32 q^{66} -44624.4 q^{67} +979.055 q^{68} -10826.8 q^{69} -50547.9 q^{71} -15142.6 q^{72} -24073.0 q^{73} -2562.94 q^{74} +25957.9 q^{76} +13118.0 q^{77} +9065.15 q^{78} +1991.87 q^{79} +54426.9 q^{81} +15517.8 q^{82} -50516.4 q^{83} -1982.82 q^{84} -79079.6 q^{86} -18771.9 q^{87} -17133.8 q^{88} +60371.0 q^{89} -43908.0 q^{91} -68494.1 q^{92} -22600.4 q^{93} +8255.14 q^{94} +2589.81 q^{96} -121045. q^{97} +9604.00 q^{98} +63342.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} + q^{3} + 48 q^{4} + 4 q^{6} - 147 q^{7} + 192 q^{8} - 56 q^{9} - 11 q^{11} + 16 q^{12} - 482 q^{13} - 588 q^{14} + 768 q^{16} - 1185 q^{17} - 224 q^{18} - 1811 q^{19} - 49 q^{21} - 44 q^{22}+ \cdots + 109448 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + q^3 + 48 * q^4 + 4 * q^6 - 147 * q^7 + 192 * q^8 - 56 * q^9 - 11 * q^11 + 16 * q^12 - 482 * q^13 - 588 * q^14 + 768 * q^16 - 1185 * q^17 - 224 * q^18 - 1811 * q^19 - 49 * q^21 - 44 * q^22 - 3350 * q^23 + 64 * q^24 - 1928 * q^26 - 1997 * q^27 - 2352 * q^28 + 1924 * q^29 - 6560 * q^31 + 3072 * q^32 - 15865 * q^33 - 4740 * q^34 - 896 * q^36 + 5194 * q^37 - 7244 * q^38 - 1574 * q^39 - 26981 * q^41 - 196 * q^42 - 18938 * q^43 - 176 * q^44 - 13400 * q^46 + 2878 * q^47 + 256 * q^48 + 7203 * q^49 - 55729 * q^51 - 7712 * q^52 - 6416 * q^53 - 7988 * q^54 - 9408 * q^56 - 33983 * q^57 + 7696 * q^58 + 17480 * q^59 - 40538 * q^61 - 26240 * q^62 + 2744 * q^63 + 12288 * q^64 - 63460 * q^66 - 43761 * q^67 - 18960 * q^68 + 70000 * q^69 - 65476 * q^71 - 3584 * q^72 - 59143 * q^73 + 20776 * q^74 - 28976 * q^76 + 539 * q^77 - 6296 * q^78 + 133128 * q^79 - 89693 * q^81 - 107924 * q^82 - 97205 * q^83 - 784 * q^84 - 75752 * q^86 - 63566 * q^87 - 704 * q^88 + 61353 * q^89 + 23618 * q^91 - 53600 * q^92 - 53964 * q^93 + 11512 * q^94 + 1024 * q^96 - 288942 * q^97 + 28812 * q^98 + 109448 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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