LMFDB - Embedded newform 240.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,6,Mod(1,240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	240.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.4921167551$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} -25.0000 q^{5} +28.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} -25.0000 q^{5} +28.0000 q^{7} +81.0000 q^{9} +208.000 q^{11} -422.000 q^{13} +225.000 q^{15} -146.000 q^{17} +2012.00 q^{19} -252.000 q^{21} +1096.00 q^{23} +625.000 q^{25} -729.000 q^{27} -1462.00 q^{29} +80.0000 q^{31} -1872.00 q^{33} -700.000 q^{35} -15750.0 q^{37} +3798.00 q^{39} -2358.00 q^{41} -2812.00 q^{43} -2025.00 q^{45} +7960.00 q^{47} -16023.0 q^{49} +1314.00 q^{51} -7590.00 q^{53} -5200.00 q^{55} -18108.0 q^{57} -18064.0 q^{59} -19658.0 q^{61} +2268.00 q^{63} +10550.0 q^{65} -31868.0 q^{67} -9864.00 q^{69} -57216.0 q^{71} +9906.00 q^{73} -5625.00 q^{75} +5824.00 q^{77} -7872.00 q^{79} +6561.00 q^{81} -109996. q^{83} +3650.00 q^{85} +13158.0 q^{87} +62466.0 q^{89} -11816.0 q^{91} -720.000 q^{93} -50300.0 q^{95} -97598.0 q^{97} +16848.0 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	28.0000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	28.0000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	208.000	0.518300	0.259150	−	0.965837i	$-0.416558\pi$
$11$	208.000	0.518300	0.259150	+	0.965837i	$0.416558\pi$
$12$	0	0
$13$	−422.000	−0.692555	−0.346277	−	0.938132i	$-0.612554\pi$
$13$	−422.000	−0.692555	−0.346277	+	0.938132i	$0.612554\pi$
$14$	0	0
$15$	225.000	0.258199
$16$	0	0
$17$	−146.000	−0.122527	−0.0612633	−	0.998122i	$-0.519513\pi$
$17$	−146.000	−0.122527	−0.0612633	+	0.998122i	$0.519513\pi$
$18$	0	0
$19$	2012.00	1.27863	0.639314	−	0.768946i	$-0.279219\pi$
$19$	2012.00	1.27863	0.639314	+	0.768946i	$0.279219\pi$
$20$	0	0
$21$	−252.000	−0.124696
$22$	0	0
$23$	1096.00	0.432007	0.216004	−	0.976393i	$-0.430698\pi$
$23$	1096.00	0.432007	0.216004	+	0.976393i	$0.430698\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−1462.00	−0.322814	−0.161407	−	0.986888i	$-0.551603\pi$
$29$	−1462.00	−0.322814	−0.161407	+	0.986888i	$0.551603\pi$
$30$	0	0
$31$	80.0000	0.0149515	0.00747577	−	0.999972i	$-0.497620\pi$
$31$	80.0000	0.0149515	0.00747577	+	0.999972i	$0.497620\pi$
$32$	0	0
$33$	−1872.00	−0.299241
$34$	0	0
$35$	−700.000	−0.0965891
$36$	0	0
$37$	−15750.0	−1.89137	−0.945684	−	0.325086i	$-0.894607\pi$
$37$	−15750.0	−1.89137	−0.945684	+	0.325086i	$0.894607\pi$
$38$	0	0
$39$	3798.00	0.399847
$40$	0	0
$41$	−2358.00	−0.219071	−0.109535	−	0.993983i	$-0.534936\pi$
$41$	−2358.00	−0.219071	−0.109535	+	0.993983i	$0.534936\pi$
$42$	0	0
$43$	−2812.00	−0.231923	−0.115962	−	0.993254i	$-0.536995\pi$
$43$	−2812.00	−0.231923	−0.115962	+	0.993254i	$0.536995\pi$
$44$	0	0
$45$	−2025.00	−0.149071
$46$	0	0
$47$	7960.00	0.525616	0.262808	−	0.964848i	$-0.415351\pi$
$47$	7960.00	0.525616	0.262808	+	0.964848i	$0.415351\pi$
$48$	0	0
$49$	−16023.0	−0.953353
$50$	0	0
$51$	1314.00	0.0707408
$52$	0	0
$53$	−7590.00	−0.371152	−0.185576	−	0.982630i	$-0.559415\pi$
$53$	−7590.00	−0.371152	−0.185576	+	0.982630i	$0.559415\pi$
$54$	0	0
$55$	−5200.00	−0.231791
$56$	0	0
$57$	−18108.0	−0.738216
$58$	0	0
$59$	−18064.0	−0.675591	−0.337796	−	0.941220i	$-0.609681\pi$
$59$	−18064.0	−0.675591	−0.337796	+	0.941220i	$0.609681\pi$
$60$	0	0
$61$	−19658.0	−0.676417	−0.338209	−	0.941071i	$-0.609821\pi$
$61$	−19658.0	−0.676417	−0.338209	+	0.941071i	$0.609821\pi$
$62$	0	0
$63$	2268.00	0.0719932
$64$	0	0
$65$	10550.0	0.309720
$66$	0	0
$67$	−31868.0	−0.867297	−0.433648	−	0.901082i	$-0.642774\pi$
$67$	−31868.0	−0.867297	−0.433648	+	0.901082i	$0.642774\pi$
$68$	0	0
$69$	−9864.00	−0.249419
$70$	0	0
$71$	−57216.0	−1.34701	−0.673506	−	0.739182i	$-0.735213\pi$
$71$	−57216.0	−1.34701	−0.673506	+	0.739182i	$0.735213\pi$
$72$	0	0
$73$	9906.00	0.217566	0.108783	−	0.994066i	$-0.465305\pi$
$73$	9906.00	0.217566	0.108783	+	0.994066i	$0.465305\pi$
$74$	0	0
$75$	−5625.00	−0.115470
$76$	0	0
$77$	5824.00	0.111942
$78$	0	0
$79$	−7872.00	−0.141911	−0.0709557	−	0.997479i	$-0.522605\pi$
$79$	−7872.00	−0.141911	−0.0709557	+	0.997479i	$0.522605\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−109996.	−1.75260	−0.876298	−	0.481770i	$-0.839994\pi$
$83$	−109996.	−1.75260	−0.876298	+	0.481770i	$0.839994\pi$
$84$	0	0
$85$	3650.00	0.0547956
$86$	0	0
$87$	13158.0	0.186377
$88$	0	0
$89$	62466.0	0.835928	0.417964	−	0.908464i	$-0.362744\pi$
$89$	62466.0	0.835928	0.417964	+	0.908464i	$0.362744\pi$
$90$	0	0
$91$	−11816.0	−0.149578
$92$	0	0
$93$	−720.000	−0.00863227
$94$	0	0
$95$	−50300.0	−0.571820
$96$	0	0
$97$	−97598.0	−1.05320	−0.526601	−	0.850113i	$-0.676534\pi$
$97$	−97598.0	−1.05320	−0.526601	+	0.850113i	$0.676534\pi$
$98$	0	0
$99$	16848.0	0.172767
$100$	0	0
$101$	11490.0	0.112077	0.0560385	−	0.998429i	$-0.482153\pi$
$101$	11490.0	0.112077	0.0560385	+	0.998429i	$0.482153\pi$
$102$	0	0
$103$	−34372.0	−0.319236	−0.159618	−	0.987179i	$-0.551026\pi$
$103$	−34372.0	−0.319236	−0.159618	+	0.987179i	$0.551026\pi$
$104$	0	0
$105$	6300.00	0.0557657
$106$	0	0
$107$	−126276.	−1.06626	−0.533128	−	0.846035i	$-0.678984\pi$
$107$	−126276.	−1.06626	−0.533128	+	0.846035i	$0.678984\pi$
$108$	0	0
$109$	−78586.0	−0.633547	−0.316774	−	0.948501i	$-0.602600\pi$
$109$	−78586.0	−0.633547	−0.316774	+	0.948501i	$0.602600\pi$
$110$	0	0
$111$	141750.	1.09198
$112$	0	0
$113$	267318.	1.96939	0.984696	−	0.174282i	$-0.0557603\pi$
$113$	267318.	1.96939	0.984696	+	0.174282i	$0.0557603\pi$
$114$	0	0
$115$	−27400.0	−0.193199
$116$	0	0
$117$	−34182.0	−0.230852
$118$	0	0
$119$	−4088.00	−0.0264633
$120$	0	0
$121$	−117787.	−0.731365
$122$	0	0
$123$	21222.0	0.126480
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−38268.0	−0.210536	−0.105268	−	0.994444i	$-0.533570\pi$
$127$	−38268.0	−0.210536	−0.105268	+	0.994444i	$0.533570\pi$
$128$	0	0
$129$	25308.0	0.133901
$130$	0	0
$131$	286848.	1.46041	0.730203	−	0.683230i	$-0.239425\pi$
$131$	286848.	1.46041	0.730203	+	0.683230i	$0.239425\pi$
$132$	0	0
$133$	56336.0	0.276158
$134$	0	0
$135$	18225.0	0.0860663
$136$	0	0
$137$	−77410.0	−0.352367	−0.176184	−	0.984357i	$-0.556375\pi$
$137$	−77410.0	−0.352367	−0.176184	+	0.984357i	$0.556375\pi$
$138$	0	0
$139$	69036.0	0.303067	0.151533	−	0.988452i	$-0.451579\pi$
$139$	69036.0	0.303067	0.151533	+	0.988452i	$0.451579\pi$
$140$	0	0
$141$	−71640.0	−0.303464
$142$	0	0
$143$	−87776.0	−0.358951
$144$	0	0
$145$	36550.0	0.144367
$146$	0	0
$147$	144207.	0.550418
$148$	0	0
$149$	182698.	0.674168	0.337084	−	0.941475i	$-0.390559\pi$
$149$	182698.	0.674168	0.337084	+	0.941475i	$0.390559\pi$
$150$	0	0
$151$	415984.	1.48468	0.742342	−	0.670021i	$-0.233715\pi$
$151$	415984.	1.48468	0.742342	+	0.670021i	$0.233715\pi$
$152$	0	0
$153$	−11826.0	−0.0408422
$154$	0	0
$155$	−2000.00	−0.00668653
$156$	0	0
$157$	−372086.	−1.20474	−0.602371	−	0.798216i	$-0.705777\pi$
$157$	−372086.	−1.20474	−0.602371	+	0.798216i	$0.705777\pi$
$158$	0	0
$159$	68310.0	0.214285
$160$	0	0
$161$	30688.0	0.0933048
$162$	0	0
$163$	−391324.	−1.15363	−0.576816	−	0.816874i	$-0.695705\pi$
$163$	−391324.	−1.15363	−0.576816	+	0.816874i	$0.695705\pi$
$164$	0	0
$165$	46800.0	0.133825
$166$	0	0
$167$	7272.00	0.0201773	0.0100886	−	0.999949i	$-0.496789\pi$
$167$	7272.00	0.0201773	0.0100886	+	0.999949i	$0.496789\pi$
$168$	0	0
$169$	−193209.	−0.520368
$170$	0	0
$171$	162972.	0.426209
$172$	0	0
$173$	133858.	0.340039	0.170020	−	0.985441i	$-0.445617\pi$
$173$	133858.	0.340039	0.170020	+	0.985441i	$0.445617\pi$
$174$	0	0
$175$	17500.0	0.0431959
$176$	0	0
$177$	162576.	0.390053
$178$	0	0
$179$	462632.	1.07920	0.539601	−	0.841921i	$-0.318575\pi$
$179$	462632.	1.07920	0.539601	+	0.841921i	$0.318575\pi$
$180$	0	0
$181$	−563074.	−1.27752	−0.638762	−	0.769404i	$-0.720553\pi$
$181$	−563074.	−1.27752	−0.638762	+	0.769404i	$0.720553\pi$
$182$	0	0
$183$	176922.	0.390530
$184$	0	0
$185$	393750.	0.845846
$186$	0	0
$187$	−30368.0	−0.0635056
$188$	0	0
$189$	−20412.0	−0.0415653
$190$	0	0
$191$	791768.	1.57042	0.785208	−	0.619233i	$-0.212556\pi$
$191$	791768.	1.57042	0.785208	+	0.619233i	$0.212556\pi$
$192$	0	0
$193$	591706.	1.14344	0.571719	−	0.820449i	$-0.306277\pi$
$193$	591706.	1.14344	0.571719	+	0.820449i	$0.306277\pi$
$194$	0	0
$195$	−94950.0	−0.178817
$196$	0	0
$197$	−205414.	−0.377107	−0.188553	−	0.982063i	$-0.560380\pi$
$197$	−205414.	−0.377107	−0.188553	+	0.982063i	$0.560380\pi$
$198$	0	0
$199$	−100592.	−0.180066	−0.0900328	−	0.995939i	$-0.528697\pi$
$199$	−100592.	−0.180066	−0.0900328	+	0.995939i	$0.528697\pi$
$200$	0	0
$201$	286812.	0.500734
$202$	0	0
$203$	−40936.0	−0.0697213
$204$	0	0
$205$	58950.0	0.0979714
$206$	0	0
$207$	88776.0	0.144002
$208$	0	0
$209$	418496.	0.662713
$210$	0	0
$211$	358996.	0.555116	0.277558	−	0.960709i	$-0.410475\pi$
$211$	358996.	0.555116	0.277558	+	0.960709i	$0.410475\pi$
$212$	0	0
$213$	514944.	0.777698
$214$	0	0
$215$	70300.0	0.103719
$216$	0	0
$217$	2240.00	0.00322923
$218$	0	0
$219$	−89154.0	−0.125612
$220$	0	0
$221$	61612.0	0.0848564
$222$	0	0
$223$	−1.31668e6	−1.77304	−0.886522	−	0.462687i	$-0.846885\pi$
$223$	−1.31668e6	−1.77304	−0.886522	+	0.462687i	$0.846885\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	599436.	0.772108	0.386054	−	0.922476i	$-0.373838\pi$
$227$	599436.	0.772108	0.386054	+	0.922476i	$0.373838\pi$
$228$	0	0
$229$	−642794.	−0.809996	−0.404998	−	0.914317i	$-0.632728\pi$
$229$	−642794.	−0.809996	−0.404998	+	0.914317i	$0.632728\pi$
$230$	0	0
$231$	−52416.0	−0.0646300
$232$	0	0
$233$	909414.	1.09742	0.548709	−	0.836014i	$-0.315120\pi$
$233$	909414.	1.09742	0.548709	+	0.836014i	$0.315120\pi$
$234$	0	0
$235$	−199000.	−0.235063
$236$	0	0
$237$	70848.0	0.0819326
$238$	0	0
$239$	−876360.	−0.992402	−0.496201	−	0.868208i	$-0.665272\pi$
$239$	−876360.	−0.992402	−0.496201	+	0.868208i	$0.665272\pi$
$240$	0	0
$241$	1.57037e6	1.74164	0.870822	−	0.491598i	$-0.163587\pi$
$241$	1.57037e6	1.74164	0.870822	+	0.491598i	$0.163587\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	400575.	0.426352
$246$	0	0
$247$	−849064.	−0.885519
$248$	0	0
$249$	989964.	1.01186
$250$	0	0
$251$	925752.	0.927492	0.463746	−	0.885968i	$-0.346505\pi$
$251$	925752.	0.927492	0.463746	+	0.885968i	$0.346505\pi$
$252$	0	0
$253$	227968.	0.223910
$254$	0	0
$255$	−32850.0	−0.0316362
$256$	0	0
$257$	−537354.	−0.507490	−0.253745	−	0.967271i	$-0.581662\pi$
$257$	−537354.	−0.507490	−0.253745	+	0.967271i	$0.581662\pi$
$258$	0	0
$259$	−441000.	−0.408497
$260$	0	0
$261$	−118422.	−0.107605
$262$	0	0
$263$	581112.	0.518049	0.259024	−	0.965871i	$-0.416599\pi$
$263$	581112.	0.518049	0.259024	+	0.965871i	$0.416599\pi$
$264$	0	0
$265$	189750.	0.165984
$266$	0	0
$267$	−562194.	−0.482623
$268$	0	0
$269$	−2.23737e6	−1.88520	−0.942598	−	0.333931i	$-0.891625\pi$
$269$	−2.23737e6	−1.88520	−0.942598	+	0.333931i	$0.891625\pi$
$270$	0	0
$271$	−1.53842e6	−1.27249	−0.636243	−	0.771489i	$-0.719512\pi$
$271$	−1.53842e6	−1.27249	−0.636243	+	0.771489i	$0.719512\pi$
$272$	0	0
$273$	106344.	0.0863588
$274$	0	0
$275$	130000.	0.103660
$276$	0	0
$277$	−1.95017e6	−1.52712	−0.763558	−	0.645739i	$-0.776549\pi$
$277$	−1.95017e6	−1.52712	−0.763558	+	0.645739i	$0.776549\pi$
$278$	0	0
$279$	6480.00	0.00498384
$280$	0	0
$281$	−104638.	−0.0790540	−0.0395270	−	0.999219i	$-0.512585\pi$
$281$	−104638.	−0.0790540	−0.0395270	+	0.999219i	$0.512585\pi$
$282$	0	0
$283$	−1.52378e6	−1.13098	−0.565492	−	0.824754i	$-0.691314\pi$
$283$	−1.52378e6	−1.13098	−0.565492	+	0.824754i	$0.691314\pi$
$284$	0	0
$285$	452700.	0.330140
$286$	0	0
$287$	−66024.0	−0.0473148
$288$	0	0
$289$	−1.39854e6	−0.984987
$290$	0	0
$291$	878382.	0.608066
$292$	0	0
$293$	−2.13252e6	−1.45119	−0.725594	−	0.688123i	$-0.758435\pi$
$293$	−2.13252e6	−1.45119	−0.725594	+	0.688123i	$0.758435\pi$
$294$	0	0
$295$	451600.	0.302134
$296$	0	0
$297$	−151632.	−0.0997470
$298$	0	0
$299$	−462512.	−0.299189
$300$	0	0
$301$	−78736.0	−0.0500907
$302$	0	0
$303$	−103410.	−0.0647077
$304$	0	0
$305$	491450.	0.302503
$306$	0	0
$307$	−1.12234e6	−0.679639	−0.339820	−	0.940491i	$-0.610366\pi$
$307$	−1.12234e6	−0.679639	−0.339820	+	0.940491i	$0.610366\pi$
$308$	0	0
$309$	309348.	0.184311
$310$	0	0
$311$	1.04821e6	0.614535	0.307267	−	0.951623i	$-0.400585\pi$
$311$	1.04821e6	0.614535	0.307267	+	0.951623i	$0.400585\pi$
$312$	0	0
$313$	−1.44225e6	−0.832110	−0.416055	−	0.909339i	$-0.636588\pi$
$313$	−1.44225e6	−0.832110	−0.416055	+	0.909339i	$0.636588\pi$
$314$	0	0
$315$	−56700.0	−0.0321964
$316$	0	0
$317$	−2.44558e6	−1.36689	−0.683446	−	0.730001i	$-0.739519\pi$
$317$	−2.44558e6	−1.36689	−0.683446	+	0.730001i	$0.739519\pi$
$318$	0	0
$319$	−304096.	−0.167315
$320$	0	0
$321$	1.13648e6	0.615603
$322$	0	0
$323$	−293752.	−0.156666
$324$	0	0
$325$	−263750.	−0.138511
$326$	0	0
$327$	707274.	0.365779
$328$	0	0
$329$	222880.	0.113522
$330$	0	0
$331$	2.56688e6	1.28776	0.643882	−	0.765125i	$-0.277323\pi$
$331$	2.56688e6	1.28776	0.643882	+	0.765125i	$0.277323\pi$
$332$	0	0
$333$	−1.27575e6	−0.630456
$334$	0	0
$335$	796700.	0.387867
$336$	0	0
$337$	1.39844e6	0.670764	0.335382	−	0.942082i	$-0.391135\pi$
$337$	1.39844e6	0.670764	0.335382	+	0.942082i	$0.391135\pi$
$338$	0	0
$339$	−2.40586e6	−1.13703
$340$	0	0
$341$	16640.0	0.00774939
$342$	0	0
$343$	−919240.	−0.421885
$344$	0	0
$345$	246600.	0.111544
$346$	0	0
$347$	−1.07907e6	−0.481089	−0.240544	−	0.970638i	$-0.577326\pi$
$347$	−1.07907e6	−0.481089	−0.240544	+	0.970638i	$0.577326\pi$
$348$	0	0
$349$	−4.12708e6	−1.81376	−0.906879	−	0.421390i	$-0.861542\pi$
$349$	−4.12708e6	−1.81376	−0.906879	+	0.421390i	$0.861542\pi$
$350$	0	0
$351$	307638.	0.133282
$352$	0	0
$353$	−827906.	−0.353626	−0.176813	−	0.984244i	$-0.556579\pi$
$353$	−827906.	−0.353626	−0.176813	+	0.984244i	$0.556579\pi$
$354$	0	0
$355$	1.43040e6	0.602402
$356$	0	0
$357$	36792.0	0.0152786
$358$	0	0
$359$	−246696.	−0.101024	−0.0505122	−	0.998723i	$-0.516085\pi$
$359$	−246696.	−0.101024	−0.0505122	+	0.998723i	$0.516085\pi$
$360$	0	0
$361$	1.57204e6	0.634888
$362$	0	0
$363$	1.06008e6	0.422254
$364$	0	0
$365$	−247650.	−0.0972985
$366$	0	0
$367$	−1.91808e6	−0.743363	−0.371681	−	0.928360i	$-0.621219\pi$
$367$	−1.91808e6	−0.743363	−0.371681	+	0.928360i	$0.621219\pi$
$368$	0	0
$369$	−190998.	−0.0730235
$370$	0	0
$371$	−212520.	−0.0801614
$372$	0	0
$373$	−48542.0	−0.0180653	−0.00903266	−	0.999959i	$-0.502875\pi$
$373$	−48542.0	−0.0180653	−0.00903266	+	0.999959i	$0.502875\pi$
$374$	0	0
$375$	140625.	0.0516398
$376$	0	0
$377$	616964.	0.223566
$378$	0	0
$379$	994364.	0.355588	0.177794	−	0.984068i	$-0.443104\pi$
$379$	994364.	0.355588	0.177794	+	0.984068i	$0.443104\pi$
$380$	0	0
$381$	344412.	0.121553
$382$	0	0
$383$	4.79206e6	1.66926	0.834632	−	0.550808i	$-0.185680\pi$
$383$	4.79206e6	1.66926	0.834632	+	0.550808i	$0.185680\pi$
$384$	0	0
$385$	−145600.	−0.0500622
$386$	0	0
$387$	−227772.	−0.0773077
$388$	0	0
$389$	2.19364e6	0.735007	0.367504	−	0.930022i	$-0.380213\pi$
$389$	2.19364e6	0.735007	0.367504	+	0.930022i	$0.380213\pi$
$390$	0	0
$391$	−160016.	−0.0529324
$392$	0	0
$393$	−2.58163e6	−0.843166
$394$	0	0
$395$	196800.	0.0634647
$396$	0	0
$397$	−839006.	−0.267171	−0.133585	−	0.991037i	$-0.542649\pi$
$397$	−839006.	−0.267171	−0.133585	+	0.991037i	$0.542649\pi$
$398$	0	0
$399$	−507024.	−0.159440
$400$	0	0
$401$	3.08524e6	0.958139	0.479069	−	0.877777i	$-0.340974\pi$
$401$	3.08524e6	0.958139	0.479069	+	0.877777i	$0.340974\pi$
$402$	0	0
$403$	−33760.0	−0.0103548
$404$	0	0
$405$	−164025.	−0.0496904
$406$	0	0
$407$	−3.27600e6	−0.980297
$408$	0	0
$409$	486970.	0.143944	0.0719721	−	0.997407i	$-0.477071\pi$
$409$	486970.	0.143944	0.0719721	+	0.997407i	$0.477071\pi$
$410$	0	0
$411$	696690.	0.203439
$412$	0	0
$413$	−505792.	−0.145914
$414$	0	0
$415$	2.74990e6	0.783784
$416$	0	0
$417$	−621324.	−0.174976
$418$	0	0
$419$	−4.24474e6	−1.18118	−0.590589	−	0.806972i	$-0.701105\pi$
$419$	−4.24474e6	−1.18118	−0.590589	+	0.806972i	$0.701105\pi$
$420$	0	0
$421$	6.73109e6	1.85089	0.925445	−	0.378883i	$-0.123692\pi$
$421$	6.73109e6	1.85089	0.925445	+	0.378883i	$0.123692\pi$
$422$	0	0
$423$	644760.	0.175205
$424$	0	0
$425$	−91250.0	−0.0245053
$426$	0	0
$427$	−550424.	−0.146092
$428$	0	0
$429$	789984.	0.207241
$430$	0	0
$431$	−1.28378e6	−0.332886	−0.166443	−	0.986051i	$-0.553228\pi$
$431$	−1.28378e6	−0.332886	−0.166443	+	0.986051i	$0.553228\pi$
$432$	0	0
$433$	3.77700e6	0.968116	0.484058	−	0.875036i	$-0.339162\pi$
$433$	3.77700e6	0.968116	0.484058	+	0.875036i	$0.339162\pi$
$434$	0	0
$435$	−328950.	−0.0833502
$436$	0	0
$437$	2.20515e6	0.552376
$438$	0	0
$439$	4.73298e6	1.17212	0.586061	−	0.810267i	$-0.300678\pi$
$439$	4.73298e6	1.17212	0.586061	+	0.810267i	$0.300678\pi$
$440$	0	0
$441$	−1.29786e6	−0.317784
$442$	0	0
$443$	−18636.0	−0.00451173	−0.00225587	−	0.999997i	$-0.500718\pi$
$443$	−18636.0	−0.00451173	−0.00225587	+	0.999997i	$0.500718\pi$
$444$	0	0
$445$	−1.56165e6	−0.373838
$446$	0	0
$447$	−1.64428e6	−0.389231
$448$	0	0
$449$	5.76604e6	1.34978	0.674888	−	0.737920i	$-0.264192\pi$
$449$	5.76604e6	1.34978	0.674888	+	0.737920i	$0.264192\pi$
$450$	0	0
$451$	−490464.	−0.113544
$452$	0	0
$453$	−3.74386e6	−0.857183
$454$	0	0
$455$	295400.	0.0668932
$456$	0	0
$457$	194170.	0.0434902	0.0217451	−	0.999764i	$-0.493078\pi$
$457$	194170.	0.0434902	0.0217451	+	0.999764i	$0.493078\pi$
$458$	0	0
$459$	106434.	0.0235803
$460$	0	0
$461$	4.57442e6	1.00250	0.501249	−	0.865303i	$-0.332874\pi$
$461$	4.57442e6	1.00250	0.501249	+	0.865303i	$0.332874\pi$
$462$	0	0
$463$	1.45763e6	0.316005	0.158003	−	0.987439i	$-0.449495\pi$
$463$	1.45763e6	0.316005	0.158003	+	0.987439i	$0.449495\pi$
$464$	0	0
$465$	18000.0	0.00386047
$466$	0	0
$467$	−7.72788e6	−1.63972	−0.819858	−	0.572568i	$-0.805947\pi$
$467$	−7.72788e6	−1.63972	−0.819858	+	0.572568i	$0.805947\pi$
$468$	0	0
$469$	−892304.	−0.187319
$470$	0	0
$471$	3.34877e6	0.695558
$472$	0	0
$473$	−584896.	−0.120206
$474$	0	0
$475$	1.25750e6	0.255725
$476$	0	0
$477$	−614790.	−0.123717
$478$	0	0
$479$	−6.99255e6	−1.39251	−0.696253	−	0.717797i	$-0.745151\pi$
$479$	−6.99255e6	−1.39251	−0.696253	+	0.717797i	$0.745151\pi$
$480$	0	0
$481$	6.64650e6	1.30988
$482$	0	0
$483$	−276192.	−0.0538695
$484$	0	0
$485$	2.43995e6	0.471006
$486$	0	0
$487$	−4.65200e6	−0.888826	−0.444413	−	0.895822i	$-0.646588\pi$
$487$	−4.65200e6	−0.888826	−0.444413	+	0.895822i	$0.646588\pi$
$488$	0	0
$489$	3.52192e6	0.666050
$490$	0	0
$491$	−310536.	−0.0581311	−0.0290655	−	0.999578i	$-0.509253\pi$
$491$	−310536.	−0.0581311	−0.0290655	+	0.999578i	$0.509253\pi$
$492$	0	0
$493$	213452.	0.0395533
$494$	0	0
$495$	−421200.	−0.0772637
$496$	0	0
$497$	−1.60205e6	−0.290927
$498$	0	0
$499$	1.01933e7	1.83257	0.916287	−	0.400521i	$-0.131171\pi$
$499$	1.01933e7	1.83257	0.916287	+	0.400521i	$0.131171\pi$
$500$	0	0
$501$	−65448.0	−0.0116494
$502$	0	0
$503$	9.20746e6	1.62263	0.811315	−	0.584609i	$-0.198752\pi$
$503$	9.20746e6	1.62263	0.811315	+	0.584609i	$0.198752\pi$
$504$	0	0
$505$	−287250.	−0.0501224
$506$	0	0
$507$	1.73888e6	0.300435
$508$	0	0
$509$	−2.71596e6	−0.464653	−0.232326	−	0.972638i	$-0.574634\pi$
$509$	−2.71596e6	−0.464653	−0.232326	+	0.972638i	$0.574634\pi$
$510$	0	0
$511$	277368.	0.0469899
$512$	0	0
$513$	−1.46675e6	−0.246072
$514$	0	0
$515$	859300.	0.142767
$516$	0	0
$517$	1.65568e6	0.272427
$518$	0	0
$519$	−1.20472e6	−0.196322
$520$	0	0
$521$	2.45851e6	0.396806	0.198403	−	0.980121i	$-0.436424\pi$
$521$	2.45851e6	0.396806	0.198403	+	0.980121i	$0.436424\pi$
$522$	0	0
$523$	4.36396e6	0.697633	0.348816	−	0.937191i	$-0.386584\pi$
$523$	4.36396e6	0.697633	0.348816	+	0.937191i	$0.386584\pi$
$524$	0	0
$525$	−157500.	−0.0249392
$526$	0	0
$527$	−11680.0	−0.00183196
$528$	0	0
$529$	−5.23513e6	−0.813370
$530$	0	0
$531$	−1.46318e6	−0.225197
$532$	0	0
$533$	995076.	0.151718
$534$	0	0
$535$	3.15690e6	0.476844
$536$	0	0
$537$	−4.16369e6	−0.623078
$538$	0	0
$539$	−3.33278e6	−0.494123
$540$	0	0
$541$	9.77959e6	1.43657	0.718286	−	0.695748i	$-0.244927\pi$
$541$	9.77959e6	1.43657	0.718286	+	0.695748i	$0.244927\pi$
$542$	0	0
$543$	5.06767e6	0.737579
$544$	0	0
$545$	1.96465e6	0.283331
$546$	0	0
$547$	1.16311e7	1.66208	0.831041	−	0.556211i	$-0.187745\pi$
$547$	1.16311e7	1.66208	0.831041	+	0.556211i	$0.187745\pi$
$548$	0	0
$549$	−1.59230e6	−0.225472
$550$	0	0
$551$	−2.94154e6	−0.412759
$552$	0	0
$553$	−220416.	−0.0306500
$554$	0	0
$555$	−3.54375e6	−0.488349
$556$	0	0
$557$	−7.66144e6	−1.04634	−0.523169	−	0.852229i	$-0.675250\pi$
$557$	−7.66144e6	−1.04634	−0.523169	+	0.852229i	$0.675250\pi$
$558$	0	0
$559$	1.18666e6	0.160619
$560$	0	0
$561$	273312.	0.0366650
$562$	0	0
$563$	−801956.	−0.106630	−0.0533150	−	0.998578i	$-0.516979\pi$
$563$	−801956.	−0.106630	−0.0533150	+	0.998578i	$0.516979\pi$
$564$	0	0
$565$	−6.68295e6	−0.880739
$566$	0	0
$567$	183708.	0.0239977
$568$	0	0
$569$	−1.44510e6	−0.187119	−0.0935595	−	0.995614i	$-0.529825\pi$
$569$	−1.44510e6	−0.187119	−0.0935595	+	0.995614i	$0.529825\pi$
$570$	0	0
$571$	7.87432e6	1.01070	0.505351	−	0.862914i	$-0.331363\pi$
$571$	7.87432e6	1.01070	0.505351	+	0.862914i	$0.331363\pi$
$572$	0	0
$573$	−7.12591e6	−0.906680
$574$	0	0
$575$	685000.	0.0864014
$576$	0	0
$577$	−1.41341e6	−0.176737	−0.0883685	−	0.996088i	$-0.528165\pi$
$577$	−1.41341e6	−0.176737	−0.0883685	+	0.996088i	$0.528165\pi$
$578$	0	0
$579$	−5.32535e6	−0.660164
$580$	0	0
$581$	−3.07989e6	−0.378525
$582$	0	0
$583$	−1.57872e6	−0.192368
$584$	0	0
$585$	854550.	0.103240
$586$	0	0
$587$	1.47164e6	0.176282	0.0881409	−	0.996108i	$-0.471907\pi$
$587$	1.47164e6	0.176282	0.0881409	+	0.996108i	$0.471907\pi$
$588$	0	0
$589$	160960.	0.0191174
$590$	0	0
$591$	1.84873e6	0.217723
$592$	0	0
$593$	−1.26374e7	−1.47578	−0.737891	−	0.674920i	$-0.764178\pi$
$593$	−1.26374e7	−1.47578	−0.737891	+	0.674920i	$0.764178\pi$
$594$	0	0
$595$	102200.	0.0118347
$596$	0	0
$597$	905328.	0.103961
$598$	0	0
$599$	6.09122e6	0.693645	0.346823	−	0.937931i	$-0.387261\pi$
$599$	6.09122e6	0.693645	0.346823	+	0.937931i	$0.387261\pi$
$600$	0	0
$601$	−2.71775e6	−0.306919	−0.153459	−	0.988155i	$-0.549041\pi$
$601$	−2.71775e6	−0.306919	−0.153459	+	0.988155i	$0.549041\pi$
$602$	0	0
$603$	−2.58131e6	−0.289099
$604$	0	0
$605$	2.94468e6	0.327076
$606$	0	0
$607$	−5.27989e6	−0.581639	−0.290819	−	0.956778i	$-0.593928\pi$
$607$	−5.27989e6	−0.581639	−0.290819	+	0.956778i	$0.593928\pi$
$608$	0	0
$609$	368424.	0.0402536
$610$	0	0
$611$	−3.35912e6	−0.364018
$612$	0	0
$613$	1.42000e7	1.52629	0.763147	−	0.646225i	$-0.223653\pi$
$613$	1.42000e7	1.52629	0.763147	+	0.646225i	$0.223653\pi$
$614$	0	0
$615$	−530550.	−0.0565638
$616$	0	0
$617$	−6.66129e6	−0.704442	−0.352221	−	0.935917i	$-0.614574\pi$
$617$	−6.66129e6	−0.704442	−0.352221	+	0.935917i	$0.614574\pi$
$618$	0	0
$619$	1.60605e7	1.68474	0.842371	−	0.538898i	$-0.181159\pi$
$619$	1.60605e7	1.68474	0.842371	+	0.538898i	$0.181159\pi$
$620$	0	0
$621$	−798984.	−0.0831398
$622$	0	0
$623$	1.74905e6	0.180543
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−3.76646e6	−0.382618
$628$	0	0
$629$	2.29950e6	0.231743
$630$	0	0
$631$	−1.06716e7	−1.06698	−0.533492	−	0.845805i	$-0.679120\pi$
$631$	−1.06716e7	−1.06698	−0.533492	+	0.845805i	$0.679120\pi$
$632$	0	0
$633$	−3.23096e6	−0.320496
$634$	0	0
$635$	956700.	0.0941546
$636$	0	0
$637$	6.76171e6	0.660249
$638$	0	0
$639$	−4.63450e6	−0.449004
$640$	0	0
$641$	−1.95269e7	−1.87710	−0.938550	−	0.345143i	$-0.887830\pi$
$641$	−1.95269e7	−1.87710	−0.938550	+	0.345143i	$0.887830\pi$
$642$	0	0
$643$	−1.96547e7	−1.87473	−0.937366	−	0.348345i	$-0.886744\pi$
$643$	−1.96547e7	−1.87473	−0.937366	+	0.348345i	$0.886744\pi$
$644$	0	0
$645$	−632700.	−0.0598823
$646$	0	0
$647$	−4.53461e6	−0.425872	−0.212936	−	0.977066i	$-0.568303\pi$
$647$	−4.53461e6	−0.425872	−0.212936	+	0.977066i	$0.568303\pi$
$648$	0	0
$649$	−3.75731e6	−0.350159
$650$	0	0
$651$	−20160.0	−0.00186440
$652$	0	0
$653$	1.34804e7	1.23714	0.618569	−	0.785730i	$-0.287713\pi$
$653$	1.34804e7	1.23714	0.618569	+	0.785730i	$0.287713\pi$
$654$	0	0
$655$	−7.17120e6	−0.653113
$656$	0	0
$657$	802386.	0.0725220
$658$	0	0
$659$	−4.11792e6	−0.369372	−0.184686	−	0.982798i	$-0.559127\pi$
$659$	−4.11792e6	−0.369372	−0.184686	+	0.982798i	$0.559127\pi$
$660$	0	0
$661$	2.45230e6	0.218308	0.109154	−	0.994025i	$-0.465186\pi$
$661$	2.45230e6	0.218308	0.109154	+	0.994025i	$0.465186\pi$
$662$	0	0
$663$	−554508.	−0.0489919
$664$	0	0
$665$	−1.40840e6	−0.123501
$666$	0	0
$667$	−1.60235e6	−0.139458
$668$	0	0
$669$	1.18502e7	1.02367
$670$	0	0
$671$	−4.08886e6	−0.350587
$672$	0	0
$673$	−7.01727e6	−0.597215	−0.298607	−	0.954376i	$-0.596522\pi$
$673$	−7.01727e6	−0.597215	−0.298607	+	0.954376i	$0.596522\pi$
$674$	0	0
$675$	−455625.	−0.0384900
$676$	0	0
$677$	−2.70895e6	−0.227159	−0.113579	−	0.993529i	$-0.536232\pi$
$677$	−2.70895e6	−0.227159	−0.113579	+	0.993529i	$0.536232\pi$
$678$	0	0
$679$	−2.73274e6	−0.227470
$680$	0	0
$681$	−5.39492e6	−0.445777
$682$	0	0
$683$	7.63014e6	0.625865	0.312933	−	0.949775i	$-0.398689\pi$
$683$	7.63014e6	0.625865	0.312933	+	0.949775i	$0.398689\pi$
$684$	0	0
$685$	1.93525e6	0.157583
$686$	0	0
$687$	5.78515e6	0.467652
$688$	0	0
$689$	3.20298e6	0.257043
$690$	0	0
$691$	7.85801e6	0.626062	0.313031	−	0.949743i	$-0.398656\pi$
$691$	7.85801e6	0.626062	0.313031	+	0.949743i	$0.398656\pi$
$692$	0	0
$693$	471744.	0.0373141
$694$	0	0
$695$	−1.72590e6	−0.135536
$696$	0	0
$697$	344268.	0.0268420
$698$	0	0
$699$	−8.18473e6	−0.633594
$700$	0	0
$701$	−1.70566e7	−1.31098	−0.655492	−	0.755202i	$-0.727539\pi$
$701$	−1.70566e7	−1.31098	−0.655492	+	0.755202i	$0.727539\pi$
$702$	0	0
$703$	−3.16890e7	−2.41836
$704$	0	0
$705$	1.79100e6	0.135713
$706$	0	0
$707$	321720.	0.0242064
$708$	0	0
$709$	−1.43982e7	−1.07571	−0.537853	−	0.843039i	$-0.680764\pi$
$709$	−1.43982e7	−1.07571	−0.537853	+	0.843039i	$0.680764\pi$
$710$	0	0
$711$	−637632.	−0.0473038
$712$	0	0
$713$	87680.0	0.00645917
$714$	0	0
$715$	2.19440e6	0.160528
$716$	0	0
$717$	7.88724e6	0.572964
$718$	0	0
$719$	−5.44136e6	−0.392541	−0.196271	−	0.980550i	$-0.562883\pi$
$719$	−5.44136e6	−0.392541	−0.196271	+	0.980550i	$0.562883\pi$
$720$	0	0
$721$	−962416.	−0.0689485
$722$	0	0
$723$	−1.41333e7	−1.00554
$724$	0	0
$725$	−913750.	−0.0645628
$726$	0	0
$727$	1.10291e7	0.773931	0.386966	−	0.922094i	$-0.373523\pi$
$727$	1.10291e7	0.773931	0.386966	+	0.922094i	$0.373523\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	410552.	0.0284168
$732$	0	0
$733$	1.42718e7	0.981110	0.490555	−	0.871410i	$-0.336794\pi$
$733$	1.42718e7	0.981110	0.490555	+	0.871410i	$0.336794\pi$
$734$	0	0
$735$	−3.60518e6	−0.246155
$736$	0	0
$737$	−6.62854e6	−0.449520
$738$	0	0
$739$	−7.45338e6	−0.502044	−0.251022	−	0.967981i	$-0.580767\pi$
$739$	−7.45338e6	−0.502044	−0.251022	+	0.967981i	$0.580767\pi$
$740$	0	0
$741$	7.64158e6	0.511255
$742$	0	0
$743$	1.23197e7	0.818705	0.409352	−	0.912376i	$-0.365755\pi$
$743$	1.23197e7	0.818705	0.409352	+	0.912376i	$0.365755\pi$
$744$	0	0
$745$	−4.56745e6	−0.301497
$746$	0	0
$747$	−8.90968e6	−0.584198
$748$	0	0
$749$	−3.53573e6	−0.230290
$750$	0	0
$751$	−1.15966e7	−0.750293	−0.375147	−	0.926965i	$-0.622408\pi$
$751$	−1.15966e7	−0.750293	−0.375147	+	0.926965i	$0.622408\pi$
$752$	0	0
$753$	−8.33177e6	−0.535488
$754$	0	0
$755$	−1.03996e7	−0.663971
$756$	0	0
$757$	2.28940e7	1.45205	0.726024	−	0.687669i	$-0.241366\pi$
$757$	2.28940e7	1.45205	0.726024	+	0.687669i	$0.241366\pi$
$758$	0	0
$759$	−2.05171e6	−0.129274
$760$	0	0
$761$	−2.49753e7	−1.56333	−0.781663	−	0.623700i	$-0.785629\pi$
$761$	−2.49753e7	−1.56333	−0.781663	+	0.623700i	$0.785629\pi$
$762$	0	0
$763$	−2.20041e6	−0.136833
$764$	0	0
$765$	295650.	0.0182652
$766$	0	0
$767$	7.62301e6	0.467884
$768$	0	0
$769$	−810206.	−0.0494060	−0.0247030	−	0.999695i	$-0.507864\pi$
$769$	−810206.	−0.0494060	−0.0247030	+	0.999695i	$0.507864\pi$
$770$	0	0
$771$	4.83619e6	0.293000
$772$	0	0
$773$	2.80973e7	1.69128	0.845641	−	0.533752i	$-0.179218\pi$
$773$	2.80973e7	1.69128	0.845641	+	0.533752i	$0.179218\pi$
$774$	0	0
$775$	50000.0	0.00299031
$776$	0	0
$777$	3.96900e6	0.235846
$778$	0	0
$779$	−4.74430e6	−0.280110
$780$	0	0
$781$	−1.19009e7	−0.698157
$782$	0	0
$783$	1.06580e6	0.0621256
$784$	0	0
$785$	9.30215e6	0.538777
$786$	0	0
$787$	1.48469e6	0.0854475	0.0427238	−	0.999087i	$-0.486396\pi$
$787$	1.48469e6	0.0854475	0.0427238	+	0.999087i	$0.486396\pi$
$788$	0	0
$789$	−5.23001e6	−0.299095
$790$	0	0
$791$	7.48490e6	0.425349
$792$	0	0
$793$	8.29568e6	0.468456
$794$	0	0
$795$	−1.70775e6	−0.0958311
$796$	0	0
$797$	1.74602e7	0.973652	0.486826	−	0.873499i	$-0.338155\pi$
$797$	1.74602e7	0.973652	0.486826	+	0.873499i	$0.338155\pi$
$798$	0	0
$799$	−1.16216e6	−0.0644019
$800$	0	0
$801$	5.05975e6	0.278643
$802$	0	0
$803$	2.06045e6	0.112765
$804$	0	0
$805$	−767200.	−0.0417272
$806$	0	0
$807$	2.01363e7	1.08842
$808$	0	0
$809$	1.25217e7	0.672652	0.336326	−	0.941746i	$-0.390816\pi$
$809$	1.25217e7	0.672652	0.336326	+	0.941746i	$0.390816\pi$
$810$	0	0
$811$	1.13119e7	0.603928	0.301964	−	0.953319i	$-0.402358\pi$
$811$	1.13119e7	0.603928	0.301964	+	0.953319i	$0.402358\pi$
$812$	0	0
$813$	1.38458e7	0.734670
$814$	0	0
$815$	9.78310e6	0.515920
$816$	0	0
$817$	−5.65774e6	−0.296543
$818$	0	0
$819$	−957096.	−0.0498593
$820$	0	0
$821$	−2.02692e7	−1.04949	−0.524744	−	0.851260i	$-0.675839\pi$
$821$	−2.02692e7	−1.04949	−0.524744	+	0.851260i	$0.675839\pi$
$822$	0	0
$823$	−1.33585e7	−0.687478	−0.343739	−	0.939065i	$-0.611694\pi$
$823$	−1.33585e7	−0.687478	−0.343739	+	0.939065i	$0.611694\pi$
$824$	0	0
$825$	−1.17000e6	−0.0598482
$826$	0	0
$827$	3.61014e7	1.83552	0.917762	−	0.397132i	$-0.129994\pi$
$827$	3.61014e7	1.83552	0.917762	+	0.397132i	$0.129994\pi$
$828$	0	0
$829$	−3.55687e6	−0.179755	−0.0898777	−	0.995953i	$-0.528648\pi$
$829$	−3.55687e6	−0.179755	−0.0898777	+	0.995953i	$0.528648\pi$
$830$	0	0
$831$	1.75515e7	0.881681
$832$	0	0
$833$	2.33936e6	0.116811
$834$	0	0
$835$	−181800.	−0.00902356
$836$	0	0
$837$	−58320.0	−0.00287742
$838$	0	0
$839$	2.53678e7	1.24417	0.622083	−	0.782951i	$-0.286287\pi$
$839$	2.53678e7	1.24417	0.622083	+	0.782951i	$0.286287\pi$
$840$	0	0
$841$	−1.83737e7	−0.895791
$842$	0	0
$843$	941742.	0.0456418
$844$	0	0
$845$	4.83022e6	0.232716
$846$	0	0
$847$	−3.29804e6	−0.157960
$848$	0	0
$849$	1.37140e7	0.652973
$850$	0	0
$851$	−1.72620e7	−0.817085
$852$	0	0
$853$	3.15809e6	0.148611	0.0743057	−	0.997236i	$-0.476326\pi$
$853$	3.15809e6	0.148611	0.0743057	+	0.997236i	$0.476326\pi$
$854$	0	0
$855$	−4.07430e6	−0.190607
$856$	0	0
$857$	777654.	0.0361688	0.0180844	−	0.999836i	$-0.494243\pi$
$857$	777654.	0.0361688	0.0180844	+	0.999836i	$0.494243\pi$
$858$	0	0
$859$	2.79844e7	1.29400	0.646999	−	0.762491i	$-0.276024\pi$
$859$	2.79844e7	1.29400	0.646999	+	0.762491i	$0.276024\pi$
$860$	0	0
$861$	594216.	0.0273172
$862$	0	0
$863$	−3.06091e6	−0.139902	−0.0699510	−	0.997550i	$-0.522284\pi$
$863$	−3.06091e6	−0.139902	−0.0699510	+	0.997550i	$0.522284\pi$
$864$	0	0
$865$	−3.34645e6	−0.152070
$866$	0	0
$867$	1.25869e7	0.568683
$868$	0	0
$869$	−1.63738e6	−0.0735528
$870$	0	0
$871$	1.34483e7	0.600651
$872$	0	0
$873$	−7.90544e6	−0.351067
$874$	0	0
$875$	−437500.	−0.0193178
$876$	0	0
$877$	1.05543e6	0.0463374	0.0231687	−	0.999732i	$-0.492625\pi$
$877$	1.05543e6	0.0463374	0.0231687	+	0.999732i	$0.492625\pi$
$878$	0	0
$879$	1.91927e7	0.837844
$880$	0	0
$881$	−3.19129e7	−1.38524	−0.692622	−	0.721300i	$-0.743545\pi$
$881$	−3.19129e7	−1.38524	−0.692622	+	0.721300i	$0.743545\pi$
$882$	0	0
$883$	2.51981e6	0.108759	0.0543796	−	0.998520i	$-0.482682\pi$
$883$	2.51981e6	0.108759	0.0543796	+	0.998520i	$0.482682\pi$
$884$	0	0
$885$	−4.06440e6	−0.174437
$886$	0	0
$887$	2.96868e7	1.26693	0.633467	−	0.773769i	$-0.281631\pi$
$887$	2.96868e7	1.26693	0.633467	+	0.773769i	$0.281631\pi$
$888$	0	0
$889$	−1.07150e6	−0.0454715
$890$	0	0
$891$	1.36469e6	0.0575889
$892$	0	0
$893$	1.60155e7	0.672067
$894$	0	0
$895$	−1.15658e7	−0.482634
$896$	0	0
$897$	4.16261e6	0.172737
$898$	0	0
$899$	−116960.	−0.00482656
$900$	0	0
$901$	1.10814e6	0.0454760
$902$	0	0
$903$	708624.	0.0289199
$904$	0	0
$905$	1.40768e7	0.571326
$906$	0	0
$907$	4.15782e7	1.67821	0.839107	−	0.543966i	$-0.183078\pi$
$907$	4.15782e7	1.67821	0.839107	+	0.543966i	$0.183078\pi$
$908$	0	0
$909$	930690.	0.0373590
$910$	0	0
$911$	−1.80284e7	−0.719718	−0.359859	−	0.933007i	$-0.617175\pi$
$911$	−1.80284e7	−0.719718	−0.359859	+	0.933007i	$0.617175\pi$
$912$	0	0
$913$	−2.28792e7	−0.908371
$914$	0	0
$915$	−4.42305e6	−0.174650
$916$	0	0
$917$	8.03174e6	0.315418
$918$	0	0
$919$	−1.21974e6	−0.0476406	−0.0238203	−	0.999716i	$-0.507583\pi$
$919$	−1.21974e6	−0.0476406	−0.0238203	+	0.999716i	$0.507583\pi$
$920$	0	0
$921$	1.01011e7	0.392390
$922$	0	0
$923$	2.41452e7	0.932880
$924$	0	0
$925$	−9.84375e6	−0.378274
$926$	0	0
$927$	−2.78413e6	−0.106412
$928$	0	0
$929$	−5.53889e6	−0.210564	−0.105282	−	0.994442i	$-0.533574\pi$
$929$	−5.53889e6	−0.210564	−0.105282	+	0.994442i	$0.533574\pi$
$930$	0	0
$931$	−3.22383e7	−1.21898
$932$	0	0
$933$	−9.43387e6	−0.354802
$934$	0	0
$935$	759200.	0.0284006
$936$	0	0
$937$	1.82003e6	0.0677218	0.0338609	−	0.999427i	$-0.489220\pi$
$937$	1.82003e6	0.0677218	0.0338609	+	0.999427i	$0.489220\pi$
$938$	0	0
$939$	1.29803e7	0.480419
$940$	0	0
$941$	3.10093e7	1.14161	0.570806	−	0.821085i	$-0.306631\pi$
$941$	3.10093e7	1.14161	0.570806	+	0.821085i	$0.306631\pi$
$942$	0	0
$943$	−2.58437e6	−0.0946401
$944$	0	0
$945$	510300.	0.0185886
$946$	0	0
$947$	−3.71308e7	−1.34542	−0.672712	−	0.739904i	$-0.734871\pi$
$947$	−3.71308e7	−1.34542	−0.672712	+	0.739904i	$0.734871\pi$
$948$	0	0
$949$	−4.18033e6	−0.150676
$950$	0	0
$951$	2.20102e7	0.789175
$952$	0	0
$953$	4.26930e7	1.52273	0.761367	−	0.648321i	$-0.224529\pi$
$953$	4.26930e7	1.52273	0.761367	+	0.648321i	$0.224529\pi$
$954$	0	0
$955$	−1.97942e7	−0.702311
$956$	0	0
$957$	2.73686e6	0.0965992
$958$	0	0
$959$	−2.16748e6	−0.0761042
$960$	0	0
$961$	−2.86228e7	−0.999776
$962$	0	0
$963$	−1.02284e7	−0.355419
$964$	0	0
$965$	−1.47926e7	−0.511361
$966$	0	0
$967$	1.35482e7	0.465924	0.232962	−	0.972486i	$-0.425158\pi$
$967$	1.35482e7	0.465924	0.232962	+	0.972486i	$0.425158\pi$
$968$	0	0
$969$	2.64377e6	0.0904511
$970$	0	0
$971$	−3.78272e7	−1.28752	−0.643762	−	0.765225i	$-0.722628\pi$
$971$	−3.78272e7	−1.28752	−0.643762	+	0.765225i	$0.722628\pi$
$972$	0	0
$973$	1.93301e6	0.0654563
$974$	0	0
$975$	2.37375e6	0.0799693
$976$	0	0
$977$	3.48282e7	1.16733	0.583667	−	0.811993i	$-0.301617\pi$
$977$	3.48282e7	1.16733	0.583667	+	0.811993i	$0.301617\pi$
$978$	0	0
$979$	1.29929e7	0.433262
$980$	0	0
$981$	−6.36547e6	−0.211182
$982$	0	0
$983$	1.86899e7	0.616910	0.308455	−	0.951239i	$-0.400188\pi$
$983$	1.86899e7	0.616910	0.308455	+	0.951239i	$0.400188\pi$
$984$	0	0
$985$	5.13535e6	0.168647
$986$	0	0
$987$	−2.00592e6	−0.0655422
$988$	0	0
$989$	−3.08195e6	−0.100192
$990$	0	0
$991$	−4.39920e7	−1.42295	−0.711475	−	0.702711i	$-0.751973\pi$
$991$	−4.39920e7	−1.42295	−0.711475	+	0.702711i	$0.751973\pi$
$992$	0	0
$993$	−2.31020e7	−0.743491
$994$	0	0
$995$	2.51480e6	0.0805278
$996$	0	0
$997$	−3.25274e7	−1.03636	−0.518181	−	0.855271i	$-0.673391\pi$
$997$	−3.25274e7	−1.03636	−0.518181	+	0.855271i	$0.673391\pi$
$998$	0	0
$999$	1.14818e7	0.363994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.6.a.c.1.1		1
3.2	odd	2		720.6.a.s.1.1		1
4.3	odd	2		120.6.a.e.1.1	✓	1
8.3	odd	2		960.6.a.j.1.1		1
8.5	even	2		960.6.a.y.1.1		1
12.11	even	2		360.6.a.g.1.1		1
20.3	even	4		600.6.f.d.49.2		2
20.7	even	4		600.6.f.d.49.1		2
20.19	odd	2		600.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.e.1.1	✓	1	4.3	odd	2
240.6.a.c.1.1		1	1.1	even	1	trivial
360.6.a.g.1.1		1	12.11	even	2
600.6.a.b.1.1		1	20.19	odd	2
600.6.f.d.49.1		2	20.7	even	4
600.6.f.d.49.2		2	20.3	even	4
720.6.a.s.1.1		1	3.2	odd	2
960.6.a.j.1.1		1	8.3	odd	2
960.6.a.y.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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