LMFDB - Embedded newform 350.6.a.v.1.3

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			1.3
Root			$-19.0051$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +19.0051 q^{3} +16.0000 q^{4} -76.0204 q^{6} +49.0000 q^{7} -64.0000 q^{8} +118.194 q^{9} +539.301 q^{11} +304.081 q^{12} +554.892 q^{13} -196.000 q^{14} +256.000 q^{16} -934.878 q^{17} -472.774 q^{18} -600.732 q^{19} +931.250 q^{21} -2157.20 q^{22} +1769.65 q^{23} -1216.33 q^{24} -2219.57 q^{26} -2371.96 q^{27} +784.000 q^{28} +5705.16 q^{29} +1997.99 q^{31} -1024.00 q^{32} +10249.5 q^{33} +3739.51 q^{34} +1891.10 q^{36} +550.553 q^{37} +2402.93 q^{38} +10545.8 q^{39} -11851.5 q^{41} -3725.00 q^{42} +12570.6 q^{43} +8628.81 q^{44} -7078.61 q^{46} +15897.8 q^{47} +4865.30 q^{48} +2401.00 q^{49} -17767.4 q^{51} +8878.26 q^{52} +20424.3 q^{53} +9487.83 q^{54} -3136.00 q^{56} -11417.0 q^{57} -22820.7 q^{58} -50777.5 q^{59} -12326.3 q^{61} -7991.94 q^{62} +5791.49 q^{63} +4096.00 q^{64} -40997.8 q^{66} +1381.31 q^{67} -14958.0 q^{68} +33632.4 q^{69} +35058.3 q^{71} -7564.39 q^{72} -4412.62 q^{73} -2202.21 q^{74} -9611.71 q^{76} +26425.7 q^{77} -42183.1 q^{78} +54020.7 q^{79} -73800.3 q^{81} +47406.1 q^{82} +61812.0 q^{83} +14900.0 q^{84} -50282.5 q^{86} +108427. q^{87} -34515.2 q^{88} -95834.4 q^{89} +27189.7 q^{91} +28314.4 q^{92} +37971.9 q^{93} -63591.2 q^{94} -19461.2 q^{96} +97755.0 q^{97} -9604.00 q^{98} +63741.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	19.0051	1.21918	0.609589	−	0.792718i	$-0.291335\pi$
$3$	19.0051	1.21918	0.609589	+	0.792718i	$0.291335\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−76.0204	−0.862089
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	118.194	0.486393
$10$	0	0
$11$	539.301	1.34384	0.671922	−	0.740622i	$-0.265469\pi$
$11$	539.301	1.34384	0.671922	+	0.740622i	$0.265469\pi$
$12$	304.081	0.609589
$13$	554.892	0.910646	0.455323	−	0.890326i	$-0.349524\pi$
$13$	554.892	0.910646	0.455323	+	0.890326i	$0.349524\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−934.878	−0.784571	−0.392286	−	0.919843i	$-0.628316\pi$
$17$	−934.878	−0.784571	−0.392286	+	0.919843i	$0.628316\pi$
$18$	−472.774	−0.343932
$19$	−600.732	−0.381766	−0.190883	−	0.981613i	$-0.561135\pi$
$19$	−600.732	−0.381766	−0.190883	+	0.981613i	$0.561135\pi$
$20$	0	0
$21$	931.250	0.460806
$22$	−2157.20	−0.950242
$23$	1769.65	0.697538	0.348769	−	0.937209i	$-0.386600\pi$
$23$	1769.65	0.697538	0.348769	+	0.937209i	$0.386600\pi$
$24$	−1216.33	−0.431044
$25$	0	0
$26$	−2219.57	−0.643924
$27$	−2371.96	−0.626178
$28$	784.000	0.188982
$29$	5705.16	1.25972	0.629859	−	0.776710i	$-0.283113\pi$
$29$	5705.16	1.25972	0.629859	+	0.776710i	$0.283113\pi$
$30$	0	0
$31$	1997.99	0.373412	0.186706	−	0.982416i	$-0.440219\pi$
$31$	1997.99	0.373412	0.186706	+	0.982416i	$0.440219\pi$
$32$	−1024.00	−0.176777
$33$	10249.5	1.63839
$34$	3739.51	0.554776
$35$	0	0
$36$	1891.10	0.243197
$37$	550.553	0.0661142	0.0330571	−	0.999453i	$-0.489476\pi$
$37$	550.553	0.0661142	0.0330571	+	0.999453i	$0.489476\pi$
$38$	2402.93	0.269949
$39$	10545.8	1.11024
$40$	0	0
$41$	−11851.5	−1.10107	−0.550534	−	0.834813i	$-0.685576\pi$
$41$	−11851.5	−1.10107	−0.550534	+	0.834813i	$0.685576\pi$
$42$	−3725.00	−0.325839
$43$	12570.6	1.03678	0.518389	−	0.855145i	$-0.326532\pi$
$43$	12570.6	1.03678	0.518389	+	0.855145i	$0.326532\pi$
$44$	8628.81	0.671922
$45$	0	0
$46$	−7078.61	−0.493234
$47$	15897.8	1.04977	0.524883	−	0.851174i	$-0.324109\pi$
$47$	15897.8	1.04977	0.524883	+	0.851174i	$0.324109\pi$
$48$	4865.30	0.304794
$49$	2401.00	0.142857
$50$	0	0
$51$	−17767.4	−0.956532
$52$	8878.26	0.455323
$53$	20424.3	0.998749	0.499375	−	0.866386i	$-0.333563\pi$
$53$	20424.3	0.998749	0.499375	+	0.866386i	$0.333563\pi$
$54$	9487.83	0.442774
$55$	0	0
$56$	−3136.00	−0.133631
$57$	−11417.0	−0.465440
$58$	−22820.7	−0.890755
$59$	−50777.5	−1.89907	−0.949536	−	0.313657i	$-0.898446\pi$
$59$	−50777.5	−1.89907	−0.949536	+	0.313657i	$0.898446\pi$
$60$	0	0
$61$	−12326.3	−0.424140	−0.212070	−	0.977254i	$-0.568021\pi$
$61$	−12326.3	−0.424140	−0.212070	+	0.977254i	$0.568021\pi$
$62$	−7991.94	−0.264042
$63$	5791.49	0.183839
$64$	4096.00	0.125000
$65$	0	0
$66$	−40997.8	−1.15851
$67$	1381.31	0.0375927	0.0187964	−	0.999823i	$-0.494017\pi$
$67$	1381.31	0.0375927	0.0187964	+	0.999823i	$0.494017\pi$
$68$	−14958.0	−0.392286
$69$	33632.4	0.850423
$70$	0	0
$71$	35058.3	0.825363	0.412682	−	0.910875i	$-0.364592\pi$
$71$	35058.3	0.825363	0.412682	+	0.910875i	$0.364592\pi$
$72$	−7564.39	−0.171966
$73$	−4412.62	−0.0969145	−0.0484573	−	0.998825i	$-0.515430\pi$
$73$	−4412.62	−0.0969145	−0.0484573	+	0.998825i	$0.515430\pi$
$74$	−2202.21	−0.0467498
$75$	0	0
$76$	−9611.71	−0.190883
$77$	26425.7	0.507926
$78$	−42183.1	−0.785058
$79$	54020.7	0.973851	0.486925	−	0.873444i	$-0.338118\pi$
$79$	54020.7	0.973851	0.486925	+	0.873444i	$0.338118\pi$
$80$	0	0
$81$	−73800.3	−1.24981
$82$	47406.1	0.778573
$83$	61812.0	0.984868	0.492434	−	0.870350i	$-0.336107\pi$
$83$	61812.0	0.984868	0.492434	+	0.870350i	$0.336107\pi$
$84$	14900.0	0.230403
$85$	0	0
$86$	−50282.5	−0.733112
$87$	108427.	1.53582
$88$	−34515.2	−0.475121
$89$	−95834.4	−1.28247	−0.641234	−	0.767345i	$-0.721577\pi$
$89$	−95834.4	−1.28247	−0.641234	+	0.767345i	$0.721577\pi$
$90$	0	0
$91$	27189.7	0.344192
$92$	28314.4	0.348769
$93$	37971.9	0.455255
$94$	−63591.2	−0.742297
$95$	0	0
$96$	−19461.2	−0.215522
$97$	97755.0	1.05490	0.527448	−	0.849587i	$-0.323149\pi$
$97$	97755.0	1.05490	0.527448	+	0.849587i	$0.323149\pi$
$98$	−9604.00	−0.101015
$99$	63741.9	0.653637
$100$	0	0
$101$	−115748.	−1.12904	−0.564521	−	0.825419i	$-0.690939\pi$
$101$	−115748.	−1.12904	−0.564521	+	0.825419i	$0.690939\pi$
$102$	71069.8	0.676370
$103$	−2459.11	−0.0228394	−0.0114197	−	0.999935i	$-0.503635\pi$
$103$	−2459.11	−0.0228394	−0.0114197	+	0.999935i	$0.503635\pi$
$104$	−35513.1	−0.321962
$105$	0	0
$106$	−81697.0	−0.706222
$107$	97878.1	0.826468	0.413234	−	0.910625i	$-0.364399\pi$
$107$	97878.1	0.826468	0.413234	+	0.910625i	$0.364399\pi$
$108$	−37951.3	−0.313089
$109$	118879.	0.958380	0.479190	−	0.877711i	$-0.340931\pi$
$109$	118879.	0.958380	0.479190	+	0.877711i	$0.340931\pi$
$110$	0	0
$111$	10463.3	0.0806050
$112$	12544.0	0.0944911
$113$	202420.	1.49128	0.745639	−	0.666351i	$-0.232145\pi$
$113$	202420.	1.49128	0.745639	+	0.666351i	$0.232145\pi$
$114$	45667.9	0.329116
$115$	0	0
$116$	91282.6	0.629859
$117$	65584.6	0.442932
$118$	203110.	1.34285
$119$	−45809.0	−0.296540
$120$	0	0
$121$	129794.	0.805919
$122$	49305.4	0.299912
$123$	−225239.	−1.34240
$124$	31967.8	0.186706
$125$	0	0
$126$	−23165.9	−0.129994
$127$	218363.	1.20135	0.600675	−	0.799494i	$-0.294899\pi$
$127$	218363.	1.20135	0.600675	+	0.799494i	$0.294899\pi$
$128$	−16384.0	−0.0883883
$129$	238906.	1.26402
$130$	0	0
$131$	−60683.7	−0.308954	−0.154477	−	0.987996i	$-0.549369\pi$
$131$	−60683.7	−0.308954	−0.154477	+	0.987996i	$0.549369\pi$
$132$	163991.	0.819193
$133$	−29435.9	−0.144294
$134$	−5525.24	−0.0265821
$135$	0	0
$136$	59832.2	0.277388
$137$	−23864.9	−0.108632	−0.0543160	−	0.998524i	$-0.517298\pi$
$137$	−23864.9	−0.108632	−0.0543160	+	0.998524i	$0.517298\pi$
$138$	−134530.	−0.601340
$139$	346057.	1.51918	0.759592	−	0.650400i	$-0.225399\pi$
$139$	346057.	1.51918	0.759592	+	0.650400i	$0.225399\pi$
$140$	0	0
$141$	302139.	1.27985
$142$	−140233.	−0.583620
$143$	299253.	1.22377
$144$	30257.6	0.121598
$145$	0	0
$146$	17650.5	0.0685289
$147$	45631.2	0.174168
$148$	8808.85	0.0330571
$149$	383065.	1.41353	0.706767	−	0.707446i	$-0.250153\pi$
$149$	383065.	1.41353	0.706767	+	0.707446i	$0.250153\pi$
$150$	0	0
$151$	−365737.	−1.30535	−0.652674	−	0.757639i	$-0.726353\pi$
$151$	−365737.	−1.30535	−0.652674	+	0.757639i	$0.726353\pi$
$152$	38446.9	0.134975
$153$	−110497.	−0.381610
$154$	−105703.	−0.359158
$155$	0	0
$156$	168732.	0.555120
$157$	−186725.	−0.604579	−0.302290	−	0.953216i	$-0.597751\pi$
$157$	−186725.	−0.604579	−0.302290	+	0.953216i	$0.597751\pi$
$158$	−216083.	−0.688616
$159$	388165.	1.21765
$160$	0	0
$161$	86712.9	0.263645
$162$	295201.	0.883753
$163$	643067.	1.89578	0.947888	−	0.318603i	$-0.103214\pi$
$163$	643067.	1.89578	0.947888	+	0.318603i	$0.103214\pi$
$164$	−189624.	−0.550534
$165$	0	0
$166$	−247248.	−0.696407
$167$	−4671.53	−0.0129619	−0.00648094	−	0.999979i	$-0.502063\pi$
$167$	−4671.53	−0.0129619	−0.00648094	+	0.999979i	$0.502063\pi$
$168$	−59600.0	−0.162919
$169$	−63388.4	−0.170723
$170$	0	0
$171$	−71002.7	−0.185688
$172$	201130.	0.518389
$173$	−369779.	−0.939348	−0.469674	−	0.882840i	$-0.655629\pi$
$173$	−369779.	−0.939348	−0.469674	+	0.882840i	$0.655629\pi$
$174$	−433709.	−1.08599
$175$	0	0
$176$	138061.	0.335961
$177$	−965032.	−2.31531
$178$	383338.	0.906842
$179$	693419.	1.61757	0.808785	−	0.588105i	$-0.200126\pi$
$179$	693419.	1.61757	0.808785	+	0.588105i	$0.200126\pi$
$180$	0	0
$181$	504391.	1.14438	0.572191	−	0.820120i	$-0.306094\pi$
$181$	504391.	1.14438	0.572191	+	0.820120i	$0.306094\pi$
$182$	−108759.	−0.243380
$183$	−234263.	−0.517102
$184$	−113258.	−0.246617
$185$	0	0
$186$	−151888.	−0.321914
$187$	−504180.	−1.05434
$188$	254365.	0.524883
$189$	−116226.	−0.236673
$190$	0	0
$191$	−274513.	−0.544477	−0.272238	−	0.962230i	$-0.587764\pi$
$191$	−274513.	−0.544477	−0.272238	+	0.962230i	$0.587764\pi$
$192$	77844.9	0.152397
$193$	−796458.	−1.53911	−0.769555	−	0.638581i	$-0.779522\pi$
$193$	−796458.	−1.53911	−0.769555	+	0.638581i	$0.779522\pi$
$194$	−391020.	−0.745924
$195$	0	0
$196$	38416.0	0.0714286
$197$	−409199.	−0.751223	−0.375611	−	0.926777i	$-0.622567\pi$
$197$	−409199.	−0.751223	−0.375611	+	0.926777i	$0.622567\pi$
$198$	−254967.	−0.462191
$199$	−59417.9	−0.106362	−0.0531808	−	0.998585i	$-0.516936\pi$
$199$	−59417.9	−0.106362	−0.0531808	+	0.998585i	$0.516936\pi$
$200$	0	0
$201$	26251.9	0.0458322
$202$	462992.	0.798354
$203$	279553.	0.476128
$204$	−284279.	−0.478266
$205$	0	0
$206$	9836.43	0.0161499
$207$	209161.	0.339278
$208$	142052.	0.227662
$209$	−323975.	−0.513034
$210$	0	0
$211$	−485855.	−0.751277	−0.375639	−	0.926766i	$-0.622577\pi$
$211$	−485855.	−0.751277	−0.375639	+	0.926766i	$0.622577\pi$
$212$	326788.	0.499375
$213$	666286.	1.00626
$214$	−391513.	−0.584401
$215$	0	0
$216$	151805.	0.221387
$217$	97901.3	0.141136
$218$	−475515.	−0.677677
$219$	−83862.2	−0.118156
$220$	0	0
$221$	−518756.	−0.714467
$222$	−41853.3	−0.0569963
$223$	786879.	1.05961	0.529805	−	0.848119i	$-0.322265\pi$
$223$	786879.	1.05961	0.529805	+	0.848119i	$0.322265\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−809682.	−1.05449
$227$	−179410.	−0.231090	−0.115545	−	0.993302i	$-0.536862\pi$
$227$	−179410.	−0.231090	−0.115545	+	0.993302i	$0.536862\pi$
$228$	−182672.	−0.232720
$229$	−546161.	−0.688227	−0.344114	−	0.938928i	$-0.611821\pi$
$229$	−546161.	−0.688227	−0.344114	+	0.938928i	$0.611821\pi$
$230$	0	0
$231$	502223.	0.619251
$232$	−365130.	−0.445377
$233$	−1.25069e6	−1.50925	−0.754624	−	0.656158i	$-0.772181\pi$
$233$	−1.25069e6	−1.50925	−0.754624	+	0.656158i	$0.772181\pi$
$234$	−262338.	−0.313200
$235$	0	0
$236$	−812441.	−0.949536
$237$	1.02667e6	1.18730
$238$	183236.	0.209686
$239$	−1.56639e6	−1.77380	−0.886902	−	0.461957i	$-0.847147\pi$
$239$	−1.56639e6	−1.77380	−0.886902	+	0.461957i	$0.847147\pi$
$240$	0	0
$241$	631036.	0.699860	0.349930	−	0.936776i	$-0.386205\pi$
$241$	631036.	0.699860	0.349930	+	0.936776i	$0.386205\pi$
$242$	−519176.	−0.569871
$243$	−826196.	−0.897568
$244$	−197221.	−0.212070
$245$	0	0
$246$	900957.	0.949219
$247$	−333341.	−0.347653
$248$	−127871.	−0.132021
$249$	1.17474e6	1.20073
$250$	0	0
$251$	−207796.	−0.208186	−0.104093	−	0.994568i	$-0.533194\pi$
$251$	−207796.	−0.208186	−0.104093	+	0.994568i	$0.533194\pi$
$252$	92663.8	0.0919197
$253$	954374.	0.937383
$254$	−873451.	−0.849482
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.46104e6	−1.37984	−0.689922	−	0.723883i	$-0.742355\pi$
$257$	−1.46104e6	−1.37984	−0.689922	+	0.723883i	$0.742355\pi$
$258$	−955623.	−0.893794
$259$	26977.1	0.0249888
$260$	0	0
$261$	674314.	0.612718
$262$	242735.	0.218463
$263$	−122741.	−0.109421	−0.0547103	−	0.998502i	$-0.517424\pi$
$263$	−122741.	−0.109421	−0.0547103	+	0.998502i	$0.517424\pi$
$264$	−655965.	−0.579257
$265$	0	0
$266$	117743.	0.102031
$267$	−1.82134e6	−1.56356
$268$	22100.9	0.0187964
$269$	−2.26741e6	−1.91051	−0.955255	−	0.295784i	$-0.904419\pi$
$269$	−2.26741e6	−1.91051	−0.955255	+	0.295784i	$0.904419\pi$
$270$	0	0
$271$	146171.	0.120904	0.0604518	−	0.998171i	$-0.480746\pi$
$271$	146171.	0.120904	0.0604518	+	0.998171i	$0.480746\pi$
$272$	−239329.	−0.196143
$273$	516742.	0.419631
$274$	95459.4	0.0768144
$275$	0	0
$276$	538118.	0.425212
$277$	−2.28731e6	−1.79112	−0.895562	−	0.444937i	$-0.853226\pi$
$277$	−2.28731e6	−1.79112	−0.895562	+	0.444937i	$0.853226\pi$
$278$	−1.38423e6	−1.07423
$279$	236149.	0.181625
$280$	0	0
$281$	1.65859e6	1.25306	0.626530	−	0.779397i	$-0.284475\pi$
$281$	1.65859e6	1.25306	0.626530	+	0.779397i	$0.284475\pi$
$282$	−1.20856e6	−0.904992
$283$	2.56037e6	1.90036	0.950180	−	0.311701i	$-0.100899\pi$
$283$	2.56037e6	1.90036	0.950180	+	0.311701i	$0.100899\pi$
$284$	560933.	0.412682
$285$	0	0
$286$	−1.19701e6	−0.865334
$287$	−580724.	−0.416165
$288$	−121030.	−0.0859830
$289$	−545861.	−0.384448
$290$	0	0
$291$	1.85784e6	1.28611
$292$	−70601.8	−0.0484573
$293$	1.87766e6	1.27775	0.638877	−	0.769309i	$-0.279399\pi$
$293$	1.87766e6	1.27775	0.638877	+	0.769309i	$0.279399\pi$
$294$	−182525.	−0.123156
$295$	0	0
$296$	−35235.4	−0.0233749
$297$	−1.27920e6	−0.841486
$298$	−1.53226e6	−0.999520
$299$	981965.	0.635211
$300$	0	0
$301$	615960.	0.391865
$302$	1.46295e6	0.923021
$303$	−2.19980e6	−1.37650
$304$	−153787.	−0.0954414
$305$	0	0
$306$	441986.	0.269839
$307$	1.88424e6	1.14101	0.570505	−	0.821294i	$-0.306748\pi$
$307$	1.88424e6	1.14101	0.570505	+	0.821294i	$0.306748\pi$
$308$	422812.	0.253963
$309$	−46735.6	−0.0278453
$310$	0	0
$311$	−1.55025e6	−0.908870	−0.454435	−	0.890780i	$-0.650159\pi$
$311$	−1.55025e6	−0.908870	−0.454435	+	0.890780i	$0.650159\pi$
$312$	−674929.	−0.392529
$313$	−1.58967e6	−0.917162	−0.458581	−	0.888653i	$-0.651642\pi$
$313$	−1.58967e6	−0.917162	−0.458581	+	0.888653i	$0.651642\pi$
$314$	746900.	0.427502
$315$	0	0
$316$	864331.	0.486925
$317$	−887333.	−0.495951	−0.247975	−	0.968766i	$-0.579765\pi$
$317$	−887333.	−0.495951	−0.247975	+	0.968766i	$0.579765\pi$
$318$	−1.55266e6	−0.861010
$319$	3.07680e6	1.69286
$320$	0	0
$321$	1.86018e6	1.00761
$322$	−346852.	−0.186425
$323$	561611.	0.299522
$324$	−1.18081e6	−0.624907
$325$	0	0
$326$	−2.57227e6	−1.34052
$327$	2.25930e6	1.16843
$328$	758497.	0.389287
$329$	778993.	0.396774
$330$	0	0
$331$	1.09762e6	0.550660	0.275330	−	0.961350i	$-0.411213\pi$
$331$	1.09762e6	0.550660	0.275330	+	0.961350i	$0.411213\pi$
$332$	988993.	0.492434
$333$	65071.8	0.0321575
$334$	18686.1	0.00916544
$335$	0	0
$336$	238400.	0.115201
$337$	−473182.	−0.226962	−0.113481	−	0.993540i	$-0.536200\pi$
$337$	−473182.	−0.226962	−0.113481	+	0.993540i	$0.536200\pi$
$338$	253554.	0.120720
$339$	3.84702e6	1.81813
$340$	0	0
$341$	1.07751e6	0.501808
$342$	284011.	0.131301
$343$	117649.	0.0539949
$344$	−804520.	−0.366556
$345$	0	0
$346$	1.47912e6	0.664220
$347$	−1.18876e6	−0.529996	−0.264998	−	0.964249i	$-0.585371\pi$
$347$	−1.18876e6	−0.529996	−0.264998	+	0.964249i	$0.585371\pi$
$348$	1.73483e6	0.767909
$349$	−383069.	−0.168350	−0.0841750	−	0.996451i	$-0.526825\pi$
$349$	−383069.	−0.168350	−0.0841750	+	0.996451i	$0.526825\pi$
$350$	0	0
$351$	−1.31618e6	−0.570226
$352$	−552244.	−0.237560
$353$	−390836.	−0.166939	−0.0834696	−	0.996510i	$-0.526600\pi$
$353$	−390836.	−0.166939	−0.0834696	+	0.996510i	$0.526600\pi$
$354$	3.86013e6	1.63717
$355$	0	0
$356$	−1.53335e6	−0.641234
$357$	−870604.	−0.361535
$358$	−2.77367e6	−1.14379
$359$	−653168.	−0.267479	−0.133739	−	0.991017i	$-0.542698\pi$
$359$	−653168.	−0.267479	−0.133739	+	0.991017i	$0.542698\pi$
$360$	0	0
$361$	−2.11522e6	−0.854255
$362$	−2.01756e6	−0.809200
$363$	2.46675e6	0.982558
$364$	435035.	0.172096
$365$	0	0
$366$	937053.	0.365646
$367$	1.92975e6	0.747886	0.373943	−	0.927452i	$-0.378006\pi$
$367$	1.92975e6	0.747886	0.373943	+	0.927452i	$0.378006\pi$
$368$	453031.	0.174385
$369$	−1.40077e6	−0.535552
$370$	0	0
$371$	1.00079e6	0.377492
$372$	607551.	0.227628
$373$	732408.	0.272572	0.136286	−	0.990670i	$-0.456483\pi$
$373$	732408.	0.272572	0.136286	+	0.990670i	$0.456483\pi$
$374$	2.01672e6	0.745533
$375$	0	0
$376$	−1.01746e6	−0.371148
$377$	3.16575e6	1.14716
$378$	464904.	0.167353
$379$	−2.28899e6	−0.818552	−0.409276	−	0.912411i	$-0.634219\pi$
$379$	−2.28899e6	−0.818552	−0.409276	+	0.912411i	$0.634219\pi$
$380$	0	0
$381$	4.15000e6	1.46466
$382$	1.09805e6	0.385003
$383$	−3.46152e6	−1.20578	−0.602892	−	0.797823i	$-0.705985\pi$
$383$	−3.46152e6	−1.20578	−0.602892	+	0.797823i	$0.705985\pi$
$384$	−311379.	−0.107761
$385$	0	0
$386$	3.18583e6	1.08831
$387$	1.48577e6	0.504282
$388$	1.56408e6	0.527448
$389$	−564683.	−0.189204	−0.0946022	−	0.995515i	$-0.530158\pi$
$389$	−564683.	−0.189204	−0.0946022	+	0.995515i	$0.530158\pi$
$390$	0	0
$391$	−1.65441e6	−0.547269
$392$	−153664.	−0.0505076
$393$	−1.15330e6	−0.376670
$394$	1.63679e6	0.531195
$395$	0	0
$396$	1.01987e6	0.326819
$397$	848395.	0.270161	0.135080	−	0.990835i	$-0.456871\pi$
$397$	848395.	0.270161	0.135080	+	0.990835i	$0.456871\pi$
$398$	237672.	0.0752090
$399$	−559432.	−0.175920
$400$	0	0
$401$	3.18365e6	0.988700	0.494350	−	0.869263i	$-0.335406\pi$
$401$	3.18365e6	0.988700	0.494350	+	0.869263i	$0.335406\pi$
$402$	−105008.	−0.0324083
$403$	1.10867e6	0.340046
$404$	−1.85197e6	−0.564521
$405$	0	0
$406$	−1.11821e6	−0.336674
$407$	296914.	0.0888473
$408$	1.13712e6	0.338185
$409$	976579.	0.288668	0.144334	−	0.989529i	$-0.453896\pi$
$409$	976579.	0.288668	0.144334	+	0.989529i	$0.453896\pi$
$410$	0	0
$411$	−453554.	−0.132442
$412$	−39345.7	−0.0114197
$413$	−2.48810e6	−0.717782
$414$	−836646.	−0.239906
$415$	0	0
$416$	−568209.	−0.160981
$417$	6.57684e6	1.85215
$418$	1.29590e6	0.362770
$419$	−5.08846e6	−1.41596	−0.707980	−	0.706233i	$-0.750393\pi$
$419$	−5.08846e6	−1.41596	−0.707980	+	0.706233i	$0.750393\pi$
$420$	0	0
$421$	−6.03533e6	−1.65957	−0.829786	−	0.558082i	$-0.811537\pi$
$421$	−6.03533e6	−1.65957	−0.829786	+	0.558082i	$0.811537\pi$
$422$	1.94342e6	0.531233
$423$	1.87902e6	0.510599
$424$	−1.30715e6	−0.353111
$425$	0	0
$426$	−2.66515e6	−0.711536
$427$	−603991.	−0.160310
$428$	1.56605e6	0.413234
$429$	5.68734e6	1.49199
$430$	0	0
$431$	−7.39852e6	−1.91846	−0.959228	−	0.282634i	$-0.908792\pi$
$431$	−7.39852e6	−1.91846	−0.959228	+	0.282634i	$0.908792\pi$
$432$	−607221.	−0.156544
$433$	−1.38205e6	−0.354245	−0.177123	−	0.984189i	$-0.556679\pi$
$433$	−1.38205e6	−0.354245	−0.177123	+	0.984189i	$0.556679\pi$
$434$	−391605.	−0.0997985
$435$	0	0
$436$	1.90206e6	0.479190
$437$	−1.06309e6	−0.266296
$438$	335449.	0.0835489
$439$	−6.14636e6	−1.52215	−0.761074	−	0.648665i	$-0.775328\pi$
$439$	−6.14636e6	−1.52215	−0.761074	+	0.648665i	$0.775328\pi$
$440$	0	0
$441$	283783.	0.0694848
$442$	2.07502e6	0.505204
$443$	−1.20430e6	−0.291557	−0.145779	−	0.989317i	$-0.546569\pi$
$443$	−1.20430e6	−0.291557	−0.145779	+	0.989317i	$0.546569\pi$
$444$	167413.	0.0403025
$445$	0	0
$446$	−3.14752e6	−0.749257
$447$	7.28018e6	1.72335
$448$	200704.	0.0472456
$449$	−5.29158e6	−1.23871	−0.619355	−	0.785111i	$-0.712606\pi$
$449$	−5.29158e6	−1.23871	−0.619355	+	0.785111i	$0.712606\pi$
$450$	0	0
$451$	−6.39153e6	−1.47967
$452$	3.23873e6	0.745639
$453$	−6.95086e6	−1.59145
$454$	717640.	0.163406
$455$	0	0
$456$	730686.	0.164558
$457$	−7.03776e6	−1.57632	−0.788159	−	0.615471i	$-0.788966\pi$
$457$	−7.03776e6	−1.57632	−0.788159	+	0.615471i	$0.788966\pi$
$458$	2.18464e6	0.486650
$459$	2.21749e6	0.491281
$460$	0	0
$461$	−5.11722e6	−1.12145	−0.560727	−	0.828001i	$-0.689478\pi$
$461$	−5.11722e6	−1.12145	−0.560727	+	0.828001i	$0.689478\pi$
$462$	−2.00889e6	−0.437877
$463$	4.34892e6	0.942820	0.471410	−	0.881914i	$-0.343745\pi$
$463$	4.34892e6	0.942820	0.471410	+	0.881914i	$0.343745\pi$
$464$	1.46052e6	0.314929
$465$	0	0
$466$	5.00277e6	1.06720
$467$	8.09508e6	1.71763	0.858814	−	0.512287i	$-0.171202\pi$
$467$	8.09508e6	1.71763	0.858814	+	0.512287i	$0.171202\pi$
$468$	1.04935e6	0.221466
$469$	67684.2	0.0142087
$470$	0	0
$471$	−3.54873e6	−0.737089
$472$	3.24976e6	0.671424
$473$	6.77934e6	1.39327
$474$	−4.10667e6	−0.839545
$475$	0	0
$476$	−732944.	−0.148270
$477$	2.41402e6	0.485785
$478$	6.26557e6	1.25427
$479$	6.79155e6	1.35248	0.676239	−	0.736683i	$-0.263609\pi$
$479$	6.79155e6	1.35248	0.676239	+	0.736683i	$0.263609\pi$
$480$	0	0
$481$	305497.	0.0602067
$482$	−2.52414e6	−0.494876
$483$	1.64799e6	0.321430
$484$	2.07671e6	0.402960
$485$	0	0
$486$	3.30478e6	0.634677
$487$	−6.11477e6	−1.16831	−0.584154	−	0.811643i	$-0.698574\pi$
$487$	−6.11477e6	−1.16831	−0.584154	+	0.811643i	$0.698574\pi$
$488$	788886.	0.149956
$489$	1.22215e7	2.31129
$490$	0	0
$491$	−6.62064e6	−1.23936	−0.619679	−	0.784856i	$-0.712737\pi$
$491$	−6.62064e6	−1.23936	−0.619679	+	0.784856i	$0.712737\pi$
$492$	−3.60383e6	−0.671199
$493$	−5.33363e6	−0.988338
$494$	1.33336e6	0.245828
$495$	0	0
$496$	511484.	0.0933530
$497$	1.71786e6	0.311958
$498$	−4.69898e6	−0.849043
$499$	1.54480e6	0.277728	0.138864	−	0.990311i	$-0.455655\pi$
$499$	1.54480e6	0.277728	0.138864	+	0.990311i	$0.455655\pi$
$500$	0	0
$501$	−88782.9	−0.0158028
$502$	831183.	0.147210
$503$	−9.74305e6	−1.71702	−0.858510	−	0.512798i	$-0.828609\pi$
$503$	−9.74305e6	−1.71702	−0.858510	+	0.512798i	$0.828609\pi$
$504$	−370655.	−0.0649970
$505$	0	0
$506$	−3.81750e6	−0.662830
$507$	−1.20470e6	−0.208142
$508$	3.49380e6	0.600675
$509$	−4.73837e6	−0.810652	−0.405326	−	0.914172i	$-0.632842\pi$
$509$	−4.73837e6	−0.810652	−0.405326	+	0.914172i	$0.632842\pi$
$510$	0	0
$511$	−216218.	−0.0366303
$512$	−262144.	−0.0441942
$513$	1.42491e6	0.239053
$514$	5.84417e6	0.975698
$515$	0	0
$516$	3.82249e6	0.632008
$517$	8.57370e6	1.41072
$518$	−107908.	−0.0176698
$519$	−7.02768e6	−1.14523
$520$	0	0
$521$	−1.99960e6	−0.322737	−0.161368	−	0.986894i	$-0.551591\pi$
$521$	−1.99960e6	−0.322737	−0.161368	+	0.986894i	$0.551591\pi$
$522$	−2.69725e6	−0.433257
$523$	7.79537e6	1.24618	0.623092	−	0.782148i	$-0.285876\pi$
$523$	7.79537e6	1.24618	0.623092	+	0.782148i	$0.285876\pi$
$524$	−970939.	−0.154477
$525$	0	0
$526$	490962.	0.0773720
$527$	−1.86787e6	−0.292968
$528$	2.62386e6	0.409596
$529$	−3.30468e6	−0.513440
$530$	0	0
$531$	−6.00158e6	−0.923696
$532$	−470974.	−0.0721469
$533$	−6.57631e6	−1.00268
$534$	7.28537e6	1.10560
$535$	0	0
$536$	−88403.8	−0.0132910
$537$	1.31785e7	1.97210
$538$	9.06964e6	1.35093
$539$	1.29486e6	0.191978
$540$	0	0
$541$	1.22085e7	1.79336	0.896681	−	0.442677i	$-0.145971\pi$
$541$	1.22085e7	1.79336	0.896681	+	0.442677i	$0.145971\pi$
$542$	−584686.	−0.0854917
$543$	9.58600e6	1.39520
$544$	957315.	0.138694
$545$	0	0
$546$	−2.06697e6	−0.296724
$547$	−8.62816e6	−1.23296	−0.616481	−	0.787370i	$-0.711442\pi$
$547$	−8.62816e6	−1.23296	−0.616481	+	0.787370i	$0.711442\pi$
$548$	−381838.	−0.0543160
$549$	−1.45689e6	−0.206299
$550$	0	0
$551$	−3.42727e6	−0.480917
$552$	−2.15247e6	−0.300670
$553$	2.64701e6	0.368081
$554$	9.14924e6	1.26652
$555$	0	0
$556$	5.53691e6	0.759592
$557$	3.10873e6	0.424565	0.212283	−	0.977208i	$-0.431910\pi$
$557$	3.10873e6	0.424565	0.212283	+	0.977208i	$0.431910\pi$
$558$	−944596.	−0.128428
$559$	6.97533e6	0.944138
$560$	0	0
$561$	−9.58199e6	−1.28543
$562$	−6.63434e6	−0.886048
$563$	−4.99568e6	−0.664237	−0.332119	−	0.943238i	$-0.607763\pi$
$563$	−4.99568e6	−0.664237	−0.332119	+	0.943238i	$0.607763\pi$
$564$	4.83423e6	0.639926
$565$	0	0
$566$	−1.02415e7	−1.34376
$567$	−3.61622e6	−0.472386
$568$	−2.24373e6	−0.291810
$569$	7.11690e6	0.921531	0.460765	−	0.887522i	$-0.347575\pi$
$569$	7.11690e6	0.921531	0.460765	+	0.887522i	$0.347575\pi$
$570$	0	0
$571$	1.40799e7	1.80721	0.903606	−	0.428365i	$-0.140910\pi$
$571$	1.40799e7	1.80721	0.903606	+	0.428365i	$0.140910\pi$
$572$	4.78805e6	0.611884
$573$	−5.21714e6	−0.663814
$574$	2.32290e6	0.294273
$575$	0	0
$576$	484121.	0.0607992
$577$	2.63468e6	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$577$	2.63468e6	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$578$	2.18344e6	0.271846
$579$	−1.51368e7	−1.87645
$580$	0	0
$581$	3.02879e6	0.372245
$582$	−7.43137e6	−0.909414
$583$	1.10148e7	1.34216
$584$	282407.	0.0342645
$585$	0	0
$586$	−7.51063e6	−0.903509
$587$	−1.35001e7	−1.61711	−0.808556	−	0.588419i	$-0.799751\pi$
$587$	−1.35001e7	−1.61711	−0.808556	+	0.588419i	$0.799751\pi$
$588$	730100.	0.0870841
$589$	−1.20025e6	−0.142556
$590$	0	0
$591$	−7.77686e6	−0.915874
$592$	140942.	0.0165286
$593$	−1.38643e7	−1.61906	−0.809529	−	0.587079i	$-0.800278\pi$
$593$	−1.38643e7	−1.61906	−0.809529	+	0.587079i	$0.800278\pi$
$594$	5.11679e6	0.595020
$595$	0	0
$596$	6.12904e6	0.706767
$597$	−1.12924e6	−0.129674
$598$	−3.92786e6	−0.449162
$599$	−7.75652e6	−0.883283	−0.441642	−	0.897191i	$-0.645604\pi$
$599$	−7.75652e6	−0.883283	−0.441642	+	0.897191i	$0.645604\pi$
$600$	0	0
$601$	1.44747e7	1.63465	0.817323	−	0.576179i	$-0.195457\pi$
$601$	1.44747e7	1.63465	0.817323	+	0.576179i	$0.195457\pi$
$602$	−2.46384e6	−0.277090
$603$	163262.	0.0182849
$604$	−5.85179e6	−0.652674
$605$	0	0
$606$	8.79921e6	0.973335
$607$	−6.96646e6	−0.767433	−0.383717	−	0.923451i	$-0.625356\pi$
$607$	−6.96646e6	−0.767433	−0.383717	+	0.923451i	$0.625356\pi$
$608$	615150.	0.0674873
$609$	5.31293e6	0.580485
$610$	0	0
$611$	8.82156e6	0.955966
$612$	−1.76794e6	−0.190805
$613$	1.74134e7	1.87168	0.935841	−	0.352423i	$-0.114642\pi$
$613$	1.74134e7	1.87168	0.935841	+	0.352423i	$0.114642\pi$
$614$	−7.53694e6	−0.806816
$615$	0	0
$616$	−1.69125e6	−0.179579
$617$	4.70352e6	0.497405	0.248702	−	0.968580i	$-0.419996\pi$
$617$	4.70352e6	0.497405	0.248702	+	0.968580i	$0.419996\pi$
$618$	186942.	0.0196896
$619$	−6.54432e6	−0.686496	−0.343248	−	0.939245i	$-0.611527\pi$
$619$	−6.54432e6	−0.686496	−0.343248	+	0.939245i	$0.611527\pi$
$620$	0	0
$621$	−4.19754e6	−0.436783
$622$	6.20102e6	0.642668
$623$	−4.69589e6	−0.484727
$624$	2.69972e6	0.277560
$625$	0	0
$626$	6.35868e6	0.648531
$627$	−6.15718e6	−0.625479
$628$	−2.98760e6	−0.302290
$629$	−514700.	−0.0518713
$630$	0	0
$631$	9.47436e6	0.947276	0.473638	−	0.880720i	$-0.342940\pi$
$631$	9.47436e6	0.947276	0.473638	+	0.880720i	$0.342940\pi$
$632$	−3.45732e6	−0.344308
$633$	−9.23372e6	−0.915940
$634$	3.54933e6	0.350690
$635$	0	0
$636$	6.21064e6	0.608826
$637$	1.33229e6	0.130092
$638$	−1.23072e7	−1.19704
$639$	4.14367e6	0.401451
$640$	0	0
$641$	1.21995e7	1.17273	0.586364	−	0.810048i	$-0.300559\pi$
$641$	1.21995e7	1.17273	0.586364	+	0.810048i	$0.300559\pi$
$642$	−7.44073e6	−0.712489
$643$	−1.38685e7	−1.32282	−0.661410	−	0.750024i	$-0.730042\pi$
$643$	−1.38685e7	−1.32282	−0.661410	+	0.750024i	$0.730042\pi$
$644$	1.38741e6	0.131822
$645$	0	0
$646$	−2.24644e6	−0.211794
$647$	−938008.	−0.0880939	−0.0440469	−	0.999029i	$-0.514025\pi$
$647$	−938008.	−0.0880939	−0.0440469	+	0.999029i	$0.514025\pi$
$648$	4.72322e6	0.441876
$649$	−2.73844e7	−2.55206
$650$	0	0
$651$	1.86062e6	0.172070
$652$	1.02891e7	0.947888
$653$	−175619.	−0.0161172	−0.00805858	−	0.999968i	$-0.502565\pi$
$653$	−175619.	−0.0161172	−0.00805858	+	0.999968i	$0.502565\pi$
$654$	−9.03720e6	−0.826208
$655$	0	0
$656$	−3.03399e6	−0.275267
$657$	−521543.	−0.0471386
$658$	−3.11597e6	−0.280562
$659$	−2.00907e7	−1.80211	−0.901054	−	0.433706i	$-0.857206\pi$
$659$	−2.00907e7	−1.80211	−0.901054	+	0.433706i	$0.857206\pi$
$660$	0	0
$661$	4.14983e6	0.369425	0.184712	−	0.982793i	$-0.440865\pi$
$661$	4.14983e6	0.369425	0.184712	+	0.982793i	$0.440865\pi$
$662$	−4.39049e6	−0.389375
$663$	−9.85900e6	−0.871062
$664$	−3.95597e6	−0.348203
$665$	0	0
$666$	−260287.	−0.0227388
$667$	1.00961e7	0.878701
$668$	−74744.5	−0.00648094
$669$	1.49547e7	1.29185
$670$	0	0
$671$	−6.64760e6	−0.569979
$672$	−953600.	−0.0814597
$673$	8.38505e6	0.713621	0.356811	−	0.934177i	$-0.383864\pi$
$673$	8.38505e6	0.713621	0.356811	+	0.934177i	$0.383864\pi$
$674$	1.89273e6	0.160486
$675$	0	0
$676$	−1.01421e6	−0.0853617
$677$	2.28720e7	1.91793	0.958966	−	0.283520i	$-0.0915023\pi$
$677$	2.28720e7	1.91793	0.958966	+	0.283520i	$0.0915023\pi$
$678$	−1.53881e7	−1.28561
$679$	4.79000e6	0.398713
$680$	0	0
$681$	−3.40970e6	−0.281740
$682$	−4.31006e6	−0.354832
$683$	8.06253e6	0.661332	0.330666	−	0.943748i	$-0.392727\pi$
$683$	8.06253e6	0.661332	0.330666	+	0.943748i	$0.392727\pi$
$684$	−1.13604e6	−0.0928441
$685$	0	0
$686$	−470596.	−0.0381802
$687$	−1.03798e7	−0.839071
$688$	3.21808e6	0.259194
$689$	1.13332e7	0.909507
$690$	0	0
$691$	1.46262e7	1.16530	0.582648	−	0.812724i	$-0.302016\pi$
$691$	1.46262e7	1.16530	0.582648	+	0.812724i	$0.302016\pi$
$692$	−5.91646e6	−0.469674
$693$	3.12335e6	0.247052
$694$	4.75506e6	0.374764
$695$	0	0
$696$	−6.93934e6	−0.542994
$697$	1.10797e7	0.863867
$698$	1.53228e6	0.119041
$699$	−2.37695e7	−1.84004
$700$	0	0
$701$	1.65516e6	0.127217	0.0636086	−	0.997975i	$-0.479739\pi$
$701$	1.65516e6	0.127217	0.0636086	+	0.997975i	$0.479739\pi$
$702$	5.26472e6	0.403211
$703$	−330735.	−0.0252401
$704$	2.20898e6	0.167981
$705$	0	0
$706$	1.56335e6	0.118044
$707$	−5.67165e6	−0.426738
$708$	−1.54405e7	−1.15765
$709$	−1.05669e7	−0.789466	−0.394733	−	0.918796i	$-0.629163\pi$
$709$	−1.05669e7	−0.789466	−0.394733	+	0.918796i	$0.629163\pi$
$710$	0	0
$711$	6.38490e6	0.473674
$712$	6.13340e6	0.453421
$713$	3.53574e6	0.260469
$714$	3.48242e6	0.255644
$715$	0	0
$716$	1.10947e7	0.808785
$717$	−2.97694e7	−2.16258
$718$	2.61267e6	0.189136
$719$	−5.87623e6	−0.423913	−0.211957	−	0.977279i	$-0.567984\pi$
$719$	−5.87623e6	−0.423913	−0.211957	+	0.977279i	$0.567984\pi$
$720$	0	0
$721$	−120496.	−0.00863248
$722$	8.46088e6	0.604049
$723$	1.19929e7	0.853254
$724$	8.07026e6	0.572191
$725$	0	0
$726$	−9.86699e6	−0.694774
$727$	−1.45007e6	−0.101754	−0.0508772	−	0.998705i	$-0.516202\pi$
$727$	−1.45007e6	−0.101754	−0.0508772	+	0.998705i	$0.516202\pi$
$728$	−1.74014e6	−0.121690
$729$	2.23154e6	0.155520
$730$	0	0
$731$	−1.17520e7	−0.813426
$732$	−3.74821e6	−0.258551
$733$	−1.89152e7	−1.30032	−0.650161	−	0.759797i	$-0.725299\pi$
$733$	−1.89152e7	−1.30032	−0.650161	+	0.759797i	$0.725299\pi$
$734$	−7.71899e6	−0.528835
$735$	0	0
$736$	−1.81212e6	−0.123309
$737$	744941.	0.0505188
$738$	5.60309e6	0.378693
$739$	−3.35437e6	−0.225943	−0.112972	−	0.993598i	$-0.536037\pi$
$739$	−3.35437e6	−0.225943	−0.112972	+	0.993598i	$0.536037\pi$
$740$	0	0
$741$	−6.33518e6	−0.423851
$742$	−4.00315e6	−0.266927
$743$	−2.14209e7	−1.42353	−0.711765	−	0.702418i	$-0.752104\pi$
$743$	−2.14209e7	−1.42353	−0.711765	+	0.702418i	$0.752104\pi$
$744$	−2.43020e6	−0.160957
$745$	0	0
$746$	−2.92963e6	−0.192738
$747$	7.30579e6	0.479033
$748$	−8.06688e6	−0.527171
$749$	4.79603e6	0.312376
$750$	0	0
$751$	1.24120e6	0.0803046	0.0401523	−	0.999194i	$-0.487216\pi$
$751$	1.24120e6	0.0803046	0.0401523	+	0.999194i	$0.487216\pi$
$752$	4.06984e6	0.262442
$753$	−3.94918e6	−0.253816
$754$	−1.26630e7	−0.811162
$755$	0	0
$756$	−1.85961e6	−0.118336
$757$	1.58083e7	1.00264	0.501321	−	0.865261i	$-0.332847\pi$
$757$	1.58083e7	1.00264	0.501321	+	0.865261i	$0.332847\pi$
$758$	9.15597e6	0.578804
$759$	1.81380e7	1.14284
$760$	0	0
$761$	1.47012e7	0.920220	0.460110	−	0.887862i	$-0.347810\pi$
$761$	1.47012e7	0.920220	0.460110	+	0.887862i	$0.347810\pi$
$762$	−1.66000e7	−1.03567
$763$	5.82505e6	0.362233
$764$	−4.39221e6	−0.272238
$765$	0	0
$766$	1.38461e7	0.852618
$767$	−2.81760e7	−1.72938
$768$	1.24552e6	0.0761986
$769$	−2.04958e7	−1.24983	−0.624913	−	0.780694i	$-0.714866\pi$
$769$	−2.04958e7	−1.24983	−0.624913	+	0.780694i	$0.714866\pi$
$770$	0	0
$771$	−2.77673e7	−1.68228
$772$	−1.27433e7	−0.769555
$773$	−336883.	−0.0202783	−0.0101391	−	0.999949i	$-0.503227\pi$
$773$	−336883.	−0.0202783	−0.0101391	+	0.999949i	$0.503227\pi$
$774$	−5.94307e6	−0.356581
$775$	0	0
$776$	−6.25632e6	−0.372962
$777$	512702.	0.0304658
$778$	2.25873e6	0.133788
$779$	7.11959e6	0.420350
$780$	0	0
$781$	1.89070e7	1.10916
$782$	6.61763e6	0.386977
$783$	−1.35324e7	−0.788807
$784$	614656.	0.0357143
$785$	0	0
$786$	4.61320e6	0.266346
$787$	−3.16902e7	−1.82385	−0.911923	−	0.410361i	$-0.865403\pi$
$787$	−3.16902e7	−1.82385	−0.911923	+	0.410361i	$0.865403\pi$
$788$	−6.54718e6	−0.375611
$789$	−2.33270e6	−0.133403
$790$	0	0
$791$	9.91860e6	0.563650
$792$	−4.07948e6	−0.231096
$793$	−6.83978e6	−0.386242
$794$	−3.39358e6	−0.191032
$795$	0	0
$796$	−950687.	−0.0531808
$797$	−6.16871e6	−0.343992	−0.171996	−	0.985098i	$-0.555022\pi$
$797$	−6.16871e6	−0.343992	−0.171996	+	0.985098i	$0.555022\pi$
$798$	2.23773e6	0.124394
$799$	−1.48625e7	−0.823617
$800$	0	0
$801$	−1.13270e7	−0.623784
$802$	−1.27346e7	−0.699117
$803$	−2.37973e6	−0.130238
$804$	420031.	0.0229161
$805$	0	0
$806$	−4.43466e6	−0.240449
$807$	−4.30923e7	−2.32925
$808$	7.40788e6	0.399177
$809$	9.94168e6	0.534058	0.267029	−	0.963688i	$-0.413958\pi$
$809$	9.94168e6	0.534058	0.267029	+	0.963688i	$0.413958\pi$
$810$	0	0
$811$	1.67551e7	0.894528	0.447264	−	0.894402i	$-0.352398\pi$
$811$	1.67551e7	0.894528	0.447264	+	0.894402i	$0.352398\pi$
$812$	4.47285e6	0.238064
$813$	2.77800e6	0.147403
$814$	−1.18765e6	−0.0628245
$815$	0	0
$816$	−4.54846e6	−0.239133
$817$	−7.55158e6	−0.395806
$818$	−3.90632e6	−0.204119
$819$	3.21365e6	0.167413
$820$	0	0
$821$	2.78211e6	0.144051	0.0720256	−	0.997403i	$-0.477054\pi$
$821$	2.78211e6	0.144051	0.0720256	+	0.997403i	$0.477054\pi$
$822$	1.81422e6	0.0936503
$823$	3.40828e7	1.75402	0.877012	−	0.480468i	$-0.159533\pi$
$823$	3.40828e7	1.75402	0.877012	+	0.480468i	$0.159533\pi$
$824$	157383.	0.00807494
$825$	0	0
$826$	9.95240e6	0.507548
$827$	2.96679e7	1.50842	0.754210	−	0.656633i	$-0.228020\pi$
$827$	2.96679e7	1.50842	0.754210	+	0.656633i	$0.228020\pi$
$828$	3.34658e6	0.169639
$829$	3.66027e7	1.84981	0.924905	−	0.380198i	$-0.124144\pi$
$829$	3.66027e7	1.84981	0.924905	+	0.380198i	$0.124144\pi$
$830$	0	0
$831$	−4.34705e7	−2.18370
$832$	2.27284e6	0.113831
$833$	−2.24464e6	−0.112082
$834$	−2.63074e7	−1.30967
$835$	0	0
$836$	−5.18360e6	−0.256517
$837$	−4.73914e6	−0.233822
$838$	2.03538e7	1.00123
$839$	1.00599e6	0.0493389	0.0246695	−	0.999696i	$-0.492147\pi$
$839$	1.00599e6	0.0493389	0.0246695	+	0.999696i	$0.492147\pi$
$840$	0	0
$841$	1.20377e7	0.586887
$842$	2.41413e7	1.17349
$843$	3.15216e7	1.52770
$844$	−7.77368e6	−0.375639
$845$	0	0
$846$	−7.51608e6	−0.361048
$847$	6.35991e6	0.304609
$848$	5.22861e6	0.249687
$849$	4.86600e7	2.31688
$850$	0	0
$851$	974287.	0.0461172
$852$	1.06606e7	0.503132
$853$	7.59798e6	0.357541	0.178770	−	0.983891i	$-0.442788\pi$
$853$	7.59798e6	0.357541	0.178770	+	0.983891i	$0.442788\pi$
$854$	2.41596e6	0.113356
$855$	0	0
$856$	−6.26420e6	−0.292201
$857$	−2.32371e7	−1.08076	−0.540380	−	0.841421i	$-0.681720\pi$
$857$	−2.32371e7	−1.08076	−0.540380	+	0.841421i	$0.681720\pi$
$858$	−2.27493e7	−1.05500
$859$	−7.66200e6	−0.354290	−0.177145	−	0.984185i	$-0.556686\pi$
$859$	−7.66200e6	−0.354290	−0.177145	+	0.984185i	$0.556686\pi$
$860$	0	0
$861$	−1.10367e7	−0.507379
$862$	2.95941e7	1.35655
$863$	−5.87476e6	−0.268511	−0.134256	−	0.990947i	$-0.542864\pi$
$863$	−5.87476e6	−0.268511	−0.134256	+	0.990947i	$0.542864\pi$
$864$	2.42888e6	0.110694
$865$	0	0
$866$	5.52820e6	0.250489
$867$	−1.03741e7	−0.468710
$868$	1.56642e6	0.0705682
$869$	2.91334e7	1.30870
$870$	0	0
$871$	766477.	0.0342337
$872$	−7.60823e6	−0.338838
$873$	1.15540e7	0.513094
$874$	4.25235e6	0.188300
$875$	0	0
$876$	−1.34179e6	−0.0590780
$877$	2.00898e7	0.882018	0.441009	−	0.897503i	$-0.354621\pi$
$877$	2.00898e7	0.882018	0.441009	+	0.897503i	$0.354621\pi$
$878$	2.45854e7	1.07632
$879$	3.56851e7	1.55781
$880$	0	0
$881$	2.03816e7	0.884704	0.442352	−	0.896842i	$-0.354144\pi$
$881$	2.03816e7	0.884704	0.442352	+	0.896842i	$0.354144\pi$
$882$	−1.13513e6	−0.0491331
$883$	2.43114e7	1.04932	0.524660	−	0.851312i	$-0.324193\pi$
$883$	2.43114e7	1.04932	0.524660	+	0.851312i	$0.324193\pi$
$884$	−8.30009e6	−0.357234
$885$	0	0
$886$	4.81718e6	0.206162
$887$	−7.87521e6	−0.336088	−0.168044	−	0.985779i	$-0.553745\pi$
$887$	−7.87521e6	−0.336088	−0.168044	+	0.985779i	$0.553745\pi$
$888$	−669652.	−0.0284982
$889$	1.06998e7	0.454067
$890$	0	0
$891$	−3.98006e7	−1.67956
$892$	1.25901e7	0.529805
$893$	−9.55033e6	−0.400765
$894$	−2.91207e7	−1.21859
$895$	0	0
$896$	−802816.	−0.0334077
$897$	1.86623e7	0.774435
$898$	2.11663e7	0.875901
$899$	1.13988e7	0.470393
$900$	0	0
$901$	−1.90942e7	−0.783590
$902$	2.55661e7	1.04628
$903$	1.17064e7	0.477753
$904$	−1.29549e7	−0.527246
$905$	0	0
$906$	2.78035e7	1.12533
$907$	−1.73586e7	−0.700643	−0.350321	−	0.936630i	$-0.613928\pi$
$907$	−1.73586e7	−0.700643	−0.350321	+	0.936630i	$0.613928\pi$
$908$	−2.87056e6	−0.115545
$909$	−1.36807e7	−0.549159
$910$	0	0
$911$	−114966.	−0.00458960	−0.00229480	−	0.999997i	$-0.500730\pi$
$911$	−114966.	−0.00458960	−0.00229480	+	0.999997i	$0.500730\pi$
$912$	−2.92274e6	−0.116360
$913$	3.33353e7	1.32351
$914$	2.81510e7	1.11463
$915$	0	0
$916$	−8.73857e6	−0.344114
$917$	−2.97350e6	−0.116774
$918$	−8.86996e6	−0.347388
$919$	4.55188e7	1.77788	0.888940	−	0.458024i	$-0.151443\pi$
$919$	4.55188e7	1.77788	0.888940	+	0.458024i	$0.151443\pi$
$920$	0	0
$921$	3.58101e7	1.39109
$922$	2.04689e7	0.792988
$923$	1.94536e7	0.751614
$924$	8.03557e6	0.309626
$925$	0	0
$926$	−1.73957e7	−0.666675
$927$	−290651.	−0.0111089
$928$	−5.84209e6	−0.222689
$929$	−3.83110e6	−0.145641	−0.0728207	−	0.997345i	$-0.523200\pi$
$929$	−3.83110e6	−0.145641	−0.0728207	+	0.997345i	$0.523200\pi$
$930$	0	0
$931$	−1.44236e6	−0.0545379
$932$	−2.00111e7	−0.754624
$933$	−2.94627e7	−1.10807
$934$	−3.23803e7	−1.21455
$935$	0	0
$936$	−4.19742e6	−0.156600
$937$	2.80647e7	1.04427	0.522134	−	0.852863i	$-0.325136\pi$
$937$	2.80647e7	1.04427	0.522134	+	0.852863i	$0.325136\pi$
$938$	−270737.	−0.0100471
$939$	−3.02118e7	−1.11818
$940$	0	0
$941$	2.88859e7	1.06344	0.531718	−	0.846921i	$-0.321547\pi$
$941$	2.88859e7	1.06344	0.531718	+	0.846921i	$0.321547\pi$
$942$	1.41949e7	0.521201
$943$	−2.09731e7	−0.768038
$944$	−1.29990e7	−0.474768
$945$	0	0
$946$	−2.71174e7	−0.985189
$947$	1.60037e6	0.0579889	0.0289945	−	0.999580i	$-0.490769\pi$
$947$	1.60037e6	0.0579889	0.0289945	+	0.999580i	$0.490769\pi$
$948$	1.64267e7	0.593648
$949$	−2.44852e6	−0.0882549
$950$	0	0
$951$	−1.68638e7	−0.604652
$952$	2.93178e6	0.104843
$953$	−3.00786e7	−1.07282	−0.536409	−	0.843958i	$-0.680220\pi$
$953$	−3.00786e7	−1.07282	−0.536409	+	0.843958i	$0.680220\pi$
$954$	−9.65606e6	−0.343502
$955$	0	0
$956$	−2.50623e7	−0.886902
$957$	5.84748e7	2.06390
$958$	−2.71662e7	−0.956346
$959$	−1.16938e6	−0.0410590
$960$	0	0
$961$	−2.46372e7	−0.860564
$962$	−1.22199e6	−0.0425725
$963$	1.15686e7	0.401989
$964$	1.00966e7	0.349930
$965$	0	0
$966$	−6.59195e6	−0.227285
$967$	1.44210e7	0.495940	0.247970	−	0.968768i	$-0.420237\pi$
$967$	1.44210e7	0.495940	0.247970	+	0.968768i	$0.420237\pi$
$968$	−8.30682e6	−0.284935
$969$	1.06735e7	0.365171
$970$	0	0
$971$	−3.83778e7	−1.30627	−0.653133	−	0.757243i	$-0.726546\pi$
$971$	−3.83778e7	−1.30627	−0.653133	+	0.757243i	$0.726546\pi$
$972$	−1.32191e7	−0.448784
$973$	1.69568e7	0.574198
$974$	2.44591e7	0.826119
$975$	0	0
$976$	−3.15554e6	−0.106035
$977$	−2.03028e7	−0.680485	−0.340242	−	0.940338i	$-0.610509\pi$
$977$	−2.03028e7	−0.680485	−0.340242	+	0.940338i	$0.610509\pi$
$978$	−4.88862e7	−1.63433
$979$	−5.16836e7	−1.72344
$980$	0	0
$981$	1.40507e7	0.466149
$982$	2.64826e7	0.876358
$983$	−5.66641e7	−1.87035	−0.935177	−	0.354180i	$-0.884760\pi$
$983$	−5.66641e7	−1.87035	−0.935177	+	0.354180i	$0.884760\pi$
$984$	1.44153e7	0.474609
$985$	0	0
$986$	2.13345e7	0.698861
$987$	1.48048e7	0.483738
$988$	−5.33346e6	−0.173827
$989$	2.22456e7	0.723192
$990$	0	0
$991$	−4.65268e7	−1.50494	−0.752470	−	0.658627i	$-0.771138\pi$
$991$	−4.65268e7	−1.50494	−0.752470	+	0.658627i	$0.771138\pi$
$992$	−2.04594e6	−0.0660105
$993$	2.08604e7	0.671352
$994$	−6.87143e6	−0.220588
$995$	0	0
$996$	1.87959e7	0.600364
$997$	5.00877e7	1.59585	0.797926	−	0.602755i	$-0.205930\pi$
$997$	5.00877e7	1.59585	0.797926	+	0.602755i	$0.205930\pi$
$998$	−6.17918e6	−0.196383
$999$	−1.30589e6	−0.0413992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.v.1.3	✓	3
5.2	odd	4		350.6.c.m.99.1		6
5.3	odd	4		350.6.c.m.99.6		6
5.4	even	2		350.6.a.w.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.v.1.3	✓	3	1.1	even	1	trivial
350.6.a.w.1.1	yes	3	5.4	even	2
350.6.c.m.99.1		6	5.2	odd	4
350.6.c.m.99.6		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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +19.0051 q^{3} +16.0000 q^{4} -76.0204 q^{6} +49.0000 q^{7} -64.0000 q^{8} +118.194 q^{9} +539.301 q^{11} +304.081 q^{12} +554.892 q^{13} -196.000 q^{14} +256.000 q^{16} -934.878 q^{17} -472.774 q^{18} -600.732 q^{19} +931.250 q^{21} -2157.20 q^{22} +1769.65 q^{23} -1216.33 q^{24} -2219.57 q^{26} -2371.96 q^{27} +784.000 q^{28} +5705.16 q^{29} +1997.99 q^{31} -1024.00 q^{32} +10249.5 q^{33} +3739.51 q^{34} +1891.10 q^{36} +550.553 q^{37} +2402.93 q^{38} +10545.8 q^{39} -11851.5 q^{41} -3725.00 q^{42} +12570.6 q^{43} +8628.81 q^{44} -7078.61 q^{46} +15897.8 q^{47} +4865.30 q^{48} +2401.00 q^{49} -17767.4 q^{51} +8878.26 q^{52} +20424.3 q^{53} +9487.83 q^{54} -3136.00 q^{56} -11417.0 q^{57} -22820.7 q^{58} -50777.5 q^{59} -12326.3 q^{61} -7991.94 q^{62} +5791.49 q^{63} +4096.00 q^{64} -40997.8 q^{66} +1381.31 q^{67} -14958.0 q^{68} +33632.4 q^{69} +35058.3 q^{71} -7564.39 q^{72} -4412.62 q^{73} -2202.21 q^{74} -9611.71 q^{76} +26425.7 q^{77} -42183.1 q^{78} +54020.7 q^{79} -73800.3 q^{81} +47406.1 q^{82} +61812.0 q^{83} +14900.0 q^{84} -50282.5 q^{86} +108427. q^{87} -34515.2 q^{88} -95834.4 q^{89} +27189.7 q^{91} +28314.4 q^{92} +37971.9 q^{93} -63591.2 q^{94} -19461.2 q^{96} +97755.0 q^{97} -9604.00 q^{98} +63741.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 12 q^{2} - q^{3} + 48 q^{4} + 4 q^{6} + 147 q^{7} - 192 q^{8} - 56 q^{9} - 11 q^{11} - 16 q^{12} + 482 q^{13} - 588 q^{14} + 768 q^{16} + 1185 q^{17} + 224 q^{18} - 1811 q^{19} - 49 q^{21} + 44 q^{22}+ \cdots + 109448 q^{99}+O(q^{100}) \) 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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