LMFDB - Embedded newform 350.6.a.v.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:	$0$
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	$-0.525850\pi$
$3$	−2.52911	−0.162242	−0.0811212	+	0.996704i	$0.525850\pi$
$4$	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	$-0.762954\pi$
$13$	−896.081	−1.47058	−0.735291	+	0.677752i	$0.762954\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−61.1910	−0.0513529	−0.0256765	−	0.999670i	$-0.508174\pi$
$17$	−61.1910	−0.0513529	−0.0256765	+	0.999670i	$0.508174\pi$
$18$	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	$-0.180378\pi$
$23$	4280.88	1.68738	0.843691	+	0.536829i	$0.180378\pi$
$24$	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	$-0.487751\pi$
$37$	640.734	0.0769438	0.0384719	+	0.999260i	$0.487751\pi$
$38$	−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	$-0.196586\pi$
$43$	19769.9	1.63055	0.815274	+	0.579075i	$0.196586\pi$
$44$	−4283.44	−0.333550
$45$	0	0
$46$	−17123.5	−1.19316
$47$	−2063.78	−0.136276	−0.0681380	−	0.997676i	$-0.521706\pi$
$47$	−2063.78	−0.136276	−0.0681380	+	0.997676i	$0.521706\pi$
$48$	−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	$-0.660612\pi$
$53$	−19772.5	−0.966878	−0.483439	+	0.875378i	$0.660612\pi$
$54$	−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	$-0.292279\pi$
$67$	44624.4	1.21447	0.607234	+	0.794523i	$0.292279\pi$
$68$	−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	$-0.414840\pi$
$73$	24073.0	0.528717	0.264358	+	0.964425i	$0.414840\pi$
$74$	−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	$-0.368160\pi$
$83$	50516.4	0.804891	0.402445	+	0.915444i	$0.368160\pi$
$84$	−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	$-0.273463\pi$
$97$	121045.	1.30623	0.653113	+	0.757260i	$0.273463\pi$
$98$	−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	$-0.271771\pi$
$103$	141505.	1.31426	0.657128	+	0.753779i	$0.271771\pi$
$104$	57349.2	0.519929
$105$	0	0
$106$	79090.0	0.683686
$107$	−73248.0	−0.618495	−0.309248	−	0.950982i	$-0.600077\pi$
$107$	−73248.0	−0.618495	−0.309248	+	0.950982i	$0.600077\pi$
$108$	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	$-0.572243\pi$
$113$	−61085.2	−0.450028	−0.225014	+	0.974356i	$0.572243\pi$
$114$	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	$-0.349075\pi$
$127$	165980.	0.913158	0.456579	+	0.889683i	$0.349075\pi$
$128$	−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	$-0.328932\pi$
$137$	224926.	1.02385	0.511926	+	0.859029i	$0.328932\pi$
$138$	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	$-0.638011\pi$
$157$	−259507.	−0.840232	−0.420116	+	0.907470i	$0.638011\pi$
$158$	−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	$-0.674661\pi$
$163$	−353857.	−1.04318	−0.521590	+	0.853196i	$0.674661\pi$
$164$	62071.2	0.180211
$165$	0	0
$166$	−202065.	−0.569144
$167$	116741.	0.323917	0.161959	−	0.986798i	$-0.448219\pi$
$167$	116741.	0.323917	0.161959	+	0.986798i	$0.448219\pi$
$168$	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	$-0.584296\pi$
$173$	−206069.	−0.523476	−0.261738	+	0.965139i	$0.584296\pi$
$174$	−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	$-0.263949\pi$
$193$	699065.	1.35090	0.675452	+	0.737404i	$0.263949\pi$
$194$	−484181.	−0.923641
$195$	0	0
$196$	38416.0	0.0714286
$197$	−407750.	−0.748563	−0.374281	−	0.927315i	$-0.622111\pi$
$197$	−407750.	−0.748563	−0.374281	+	0.927315i	$0.622111\pi$
$198$	−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	$-0.223097\pi$
$223$	1.13512e6	1.52855	0.764275	+	0.644891i	$0.223097\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	244341.	0.318218
$227$	−567988.	−0.731602	−0.365801	−	0.930693i	$-0.619205\pi$
$227$	−567988.	−0.731602	−0.365801	+	0.930693i	$0.619205\pi$
$228$	−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	$-0.186873\pi$
$233$	1.37986e6	1.66512	0.832562	+	0.553932i	$0.186873\pi$
$234$	−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	$-0.769937\pi$
$257$	−1.58823e6	−1.49996	−0.749980	+	0.661460i	$0.769937\pi$
$258$	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	$-0.518282\pi$
$263$	−128781.	−0.114806	−0.0574029	+	0.998351i	$0.518282\pi$
$264$	−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	$-0.458016\pi$
$277$	335893.	0.263028	0.131514	+	0.991314i	$0.458016\pi$
$278$	−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	$-0.534664\pi$
$283$	−292861.	−0.217368	−0.108684	+	0.994076i	$0.534664\pi$
$284$	−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	$-0.786127\pi$
$293$	−2.30017e6	−1.56528	−0.782639	+	0.622476i	$0.786127\pi$
$294$	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	$-0.453471\pi$
$307$	481066.	0.291312	0.145656	+	0.989335i	$0.453471\pi$
$308$	−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	$-0.342879\pi$
$313$	1.64246e6	0.947618	0.473809	+	0.880628i	$0.342879\pi$
$314$	1.03803e6	0.594134
$315$	0	0
$316$	31870.0	0.0179541
$317$	−1.24413e6	−0.695372	−0.347686	−	0.937611i	$-0.613033\pi$
$317$	−1.24413e6	−0.695372	−0.347686	+	0.937611i	$0.613033\pi$
$318$	−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	$-0.631108\pi$
$337$	−1.66929e6	−0.800678	−0.400339	+	0.916367i	$0.631108\pi$
$338$	−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	$-0.372147\pi$
$347$	1.75377e6	0.781895	0.390947	+	0.920413i	$0.372147\pi$
$348$	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	$-0.526929\pi$
$353$	−395659.	−0.168999	−0.0844995	+	0.996424i	$0.526929\pi$
$354$	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	$-0.460299\pi$
$367$	641978.	0.248803	0.124401	+	0.992232i	$0.460299\pi$
$368$	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	$-0.691486\pi$
$373$	−3.04138e6	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$374$	−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	$-0.833282\pi$
$383$	−4.97184e6	−1.73189	−0.865945	+	0.500140i	$0.833282\pi$
$384$	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	$-0.246402\pi$
$397$	4.49103e6	1.43011	0.715055	+	0.699068i	$0.246402\pi$
$398$	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	$-0.437919\pi$
$433$	1.51218e6	0.387599	0.193800	+	0.981041i	$0.437919\pi$
$434$	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	$-0.393406\pi$
$443$	2.71502e6	0.657301	0.328651	+	0.944452i	$0.393406\pi$
$444$	−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	$-0.232079\pi$
$457$	6.65930e6	1.49155	0.745776	+	0.666197i	$0.232079\pi$
$458$	−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	$-0.450532\pi$
$463$	1.42794e6	0.309570	0.154785	+	0.987948i	$0.450532\pi$
$464$	−1.90012e6	−0.409718
$465$	0	0
$466$	−5.51946e6	−1.17742
$467$	2.22536e6	0.472180	0.236090	−	0.971731i	$-0.424134\pi$
$467$	2.22536e6	0.472180	0.236090	+	0.971731i	$0.424134\pi$
$468$	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	$-0.244904\pi$
$487$	7.51934e6	1.43667	0.718336	+	0.695697i	$0.244904\pi$
$488$	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	$-0.306025\pi$
$503$	6.49570e6	1.14474	0.572369	+	0.819996i	$0.306025\pi$
$504$	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	$-0.774336\pi$
$523$	−9.49630e6	−1.51810	−0.759050	+	0.651033i	$0.774336\pi$
$524$	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	$-0.534509\pi$
$547$	−1.51435e6	−0.216401	−0.108200	+	0.994129i	$0.534509\pi$
$548$	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	$-0.816026\pi$
$557$	−1.22656e7	−1.67514	−0.837572	+	0.546327i	$0.816026\pi$
$558$	−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	$-0.529313\pi$
$563$	−1.38325e6	−0.183920	−0.0919601	+	0.995763i	$0.529313\pi$
$564$	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	$-0.704785\pi$
$577$	−9.59475e6	−1.19976	−0.599879	+	0.800090i	$0.704785\pi$
$578$	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	$-0.239608\pi$
$587$	1.21853e7	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$588$	−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	$-0.842340\pi$
$593$	−1.50682e7	−1.75965	−0.879824	+	0.475300i	$0.842340\pi$
$594$	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	$-0.346574\pi$
$607$	8.41593e6	0.927108	0.463554	+	0.886069i	$0.346574\pi$
$608$	−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	$-0.252774\pi$
$613$	1.30421e7	1.40184	0.700918	+	0.713242i	$0.252774\pi$
$614$	−1.92427e6	−0.205989
$615$	0	0
$616$	839554.	0.0891450
$617$	1.09403e7	1.15695	0.578477	−	0.815699i	$-0.303647\pi$
$617$	1.09403e7	1.15695	0.578477	+	0.815699i	$0.303647\pi$
$618$	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	$-0.341354\pi$
$643$	1.00231e7	0.956041	0.478021	+	0.878349i	$0.341354\pi$
$644$	3.35621e6	0.318885
$645$	0	0
$646$	397097.	0.0374383
$647$	−9.92873e6	−0.932466	−0.466233	−	0.884662i	$-0.654389\pi$
$647$	−9.92873e6	−0.932466	−0.466233	+	0.884662i	$0.654389\pi$
$648$	−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	$-0.916473\pi$
$653$	−2.10468e7	−1.93154	−0.965768	+	0.259406i	$0.916473\pi$
$654$	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	$-0.522377\pi$
$673$	−1.65065e6	−0.140481	−0.0702405	+	0.997530i	$0.522377\pi$
$674$	6.67717e6	0.566165
$675$	0	0
$676$	6.90670e6	0.581305
$677$	1.71440e7	1.43760	0.718802	−	0.695214i	$-0.244691\pi$
$677$	1.71440e7	1.43760	0.718802	+	0.695214i	$0.244691\pi$
$678$	−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	$-0.500537\pi$
$683$	−41131.0	−0.00337379	−0.00168689	+	0.999999i	$0.500537\pi$
$684$	−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	$-0.553600\pi$
$727$	−4.77668e6	−0.335189	−0.167595	+	0.985856i	$0.553600\pi$
$728$	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	$-0.503779\pi$
$733$	−345377.	−0.0237429	−0.0118714	+	0.999930i	$0.503779\pi$
$734$	−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	$-0.450875\pi$
$743$	4.62625e6	0.307438	0.153719	+	0.988115i	$0.450875\pi$
$744$	−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	$-0.335583\pi$
$757$	1.55732e7	0.987732	0.493866	+	0.869538i	$0.335583\pi$
$758$	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	$-0.485005\pi$
$773$	1.56465e6	0.0941821	0.0470911	+	0.998891i	$0.485005\pi$
$774$	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	$-0.574352\pi$
$787$	−8.04368e6	−0.462933	−0.231466	+	0.972843i	$0.574352\pi$
$788$	−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	$-0.586743\pi$
$797$	−9.65319e6	−0.538301	−0.269151	+	0.963098i	$0.586743\pi$
$798$	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	$-0.270445\pi$
$823$	2.56594e7	1.32053	0.660264	+	0.751034i	$0.270445\pi$
$824$	−9.05635e6	−0.464660
$825$	0	0
$826$	−9.10594e6	−0.464381
$827$	6.48745e6	0.329845	0.164923	−	0.986307i	$-0.447263\pi$
$827$	6.48745e6	0.329845	0.164923	+	0.986307i	$0.447263\pi$
$828$	−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	$-0.461202\pi$
$853$	5.16759e6	0.243173	0.121586	+	0.992581i	$0.461202\pi$
$854$	1.05692e7	0.495903
$855$	0	0
$856$	4.68787e6	0.218671
$857$	−2.40841e7	−1.12015	−0.560077	−	0.828440i	$-0.689229\pi$
$857$	−2.40841e7	−1.12015	−0.560077	+	0.828440i	$0.689229\pi$
$858$	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	$-0.406627\pi$
$863$	1.26527e7	0.578304	0.289152	+	0.957283i	$0.406627\pi$
$864$	−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	$-0.384686\pi$
$877$	1.61444e7	0.708797	0.354399	+	0.935094i	$0.384686\pi$
$878$	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	$-0.400591\pi$
$883$	1.42372e7	0.614503	0.307251	+	0.951628i	$0.400591\pi$
$884$	877313.	0.0377593
$885$	0	0
$886$	−1.08601e7	−0.464782
$887$	3.30742e7	1.41150	0.705749	−	0.708462i	$-0.250611\pi$
$887$	3.30742e7	1.41150	0.705749	+	0.708462i	$0.250611\pi$
$888$	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	$-0.591891\pi$
$907$	−1.41066e7	−0.569382	−0.284691	+	0.958619i	$0.591891\pi$
$908$	−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	$-0.507741\pi$
$937$	−1.30699e6	−0.0486322	−0.0243161	+	0.999704i	$0.507741\pi$
$938$	−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	$-0.379894\pi$
$947$	2.03359e7	0.736867	0.368434	+	0.929654i	$0.379894\pi$
$948$	−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	$-0.418580\pi$
$953$	1.41872e7	0.506015	0.253008	+	0.967464i	$0.418580\pi$
$954$	−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	$-0.171013\pi$
$967$	4.99631e7	1.71824	0.859118	+	0.511777i	$0.171013\pi$
$968$	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	$-0.205005\pi$
$977$	4.77178e7	1.59935	0.799676	+	0.600432i	$0.205005\pi$
$978$	−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	$-0.519328\pi$
$983$	−3.67695e6	−0.121368	−0.0606840	+	0.998157i	$0.519328\pi$
$984$	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	$-0.405184\pi$
$997$	1.84228e7	0.586974	0.293487	+	0.955963i	$0.405184\pi$
$998$	−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.v.1.2	✓	3
5.2	odd	4		350.6.c.m.99.2		6
5.3	odd	4		350.6.c.m.99.5		6
5.4	even	2		350.6.a.w.1.2	yes	3

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