LMFDB - Embedded newform 1040.6.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.34530$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.6439 q^{3} -25.0000 q^{5} -48.6530 q^{7} +142.882 q^{9} -283.440 q^{11} +169.000 q^{13} -491.097 q^{15} +789.522 q^{17} -83.3795 q^{19} -955.734 q^{21} +1935.13 q^{23} +625.000 q^{25} -1966.71 q^{27} +222.752 q^{29} +2789.48 q^{31} -5567.86 q^{33} +1216.33 q^{35} +8369.56 q^{37} +3319.81 q^{39} +1218.51 q^{41} -5638.89 q^{43} -3572.04 q^{45} -17776.2 q^{47} -14439.9 q^{49} +15509.3 q^{51} +10834.4 q^{53} +7086.00 q^{55} -1637.90 q^{57} +5363.36 q^{59} -18670.7 q^{61} -6951.63 q^{63} -4225.00 q^{65} -13985.1 q^{67} +38013.5 q^{69} -50969.5 q^{71} +42394.9 q^{73} +12277.4 q^{75} +13790.2 q^{77} -106279. q^{79} -73354.1 q^{81} -75513.6 q^{83} -19738.1 q^{85} +4375.71 q^{87} +77017.8 q^{89} -8222.36 q^{91} +54796.2 q^{93} +2084.49 q^{95} +126702. q^{97} -40498.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) $ 6 * q - 38 * q^3 - 150 * q^5 - 220 * q^7 + 518 * q^9 + 170 * q^11 + 1014 * q^13 + 950 * q^15 + 728 * q^17 - 1218 * q^19 - 396 * q^21 - 8954 * q^23 + 3750 * q^25 - 13112 * q^27 + 8364 * q^29 - 2862 * q^31 - 3772 * q^33 + 5500 * q^35 + 13840 * q^37 - 6422 * q^39 + 2248 * q^41 - 3822 * q^43 - 12950 * q^45 + 10860 * q^47 + 85902 * q^49 + 55028 * q^51 + 6584 * q^53 - 4250 * q^55 + 11692 * q^57 + 28874 * q^59 + 972 * q^61 + 125236 * q^63 - 25350 * q^65 - 60040 * q^67 + 153088 * q^69 + 74174 * q^71 + 48960 * q^73 - 23750 * q^75 + 231588 * q^77 - 214508 * q^79 + 116006 * q^81 - 123260 * q^83 - 18200 * q^85 - 228144 * q^87 + 94028 * q^89 - 37180 * q^91 + 192892 * q^93 + 30450 * q^95 + 246284 * q^97 + 32270 * 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 $

$2$	0	0
$2$	0	0
$3$	19.6439	1.26015	0.630077	−	0.776532i	$-0.283023\pi$
$3$	19.6439	1.26015	0.630077	+	0.776532i	$0.283023\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−48.6530	−0.375288	−0.187644	−	0.982237i	$-0.560085\pi$
$7$	−48.6530	−0.375288	−0.187644	+	0.982237i	$0.560085\pi$
$8$	0	0
$9$	142.882	0.587990
$10$	0	0
$11$	−283.440	−0.706284	−0.353142	−	0.935570i	$-0.614887\pi$
$11$	−283.440	−0.706284	−0.353142	+	0.935570i	$0.614887\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−491.097	−0.563558
$16$	0	0
$17$	789.522	0.662586	0.331293	−	0.943528i	$-0.392515\pi$
$17$	789.522	0.662586	0.331293	+	0.943528i	$0.392515\pi$
$18$	0	0
$19$	−83.3795	−0.0529877	−0.0264939	−	0.999649i	$-0.508434\pi$
$19$	−83.3795	−0.0529877	−0.0264939	+	0.999649i	$0.508434\pi$
$20$	0	0
$21$	−955.734	−0.472921
$22$	0	0
$23$	1935.13	0.762767	0.381383	−	0.924417i	$-0.375448\pi$
$23$	1935.13	0.762767	0.381383	+	0.924417i	$0.375448\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1966.71	−0.519196
$28$	0	0
$29$	222.752	0.0491843	0.0245922	−	0.999698i	$-0.492171\pi$
$29$	222.752	0.0491843	0.0245922	+	0.999698i	$0.492171\pi$
$30$	0	0
$31$	2789.48	0.521338	0.260669	−	0.965428i	$-0.416057\pi$
$31$	2789.48	0.521338	0.260669	+	0.965428i	$0.416057\pi$
$32$	0	0
$33$	−5567.86	−0.890027
$34$	0	0
$35$	1216.33	0.167834
$36$	0	0
$37$	8369.56	1.00507	0.502537	−	0.864555i	$-0.332400\pi$
$37$	8369.56	1.00507	0.502537	+	0.864555i	$0.332400\pi$
$38$	0	0
$39$	3319.81	0.349504
$40$	0	0
$41$	1218.51	0.113206	0.0566029	−	0.998397i	$-0.481973\pi$
$41$	1218.51	0.113206	0.0566029	+	0.998397i	$0.481973\pi$
$42$	0	0
$43$	−5638.89	−0.465075	−0.232537	−	0.972587i	$-0.574703\pi$
$43$	−5638.89	−0.465075	−0.232537	+	0.972587i	$0.574703\pi$
$44$	0	0
$45$	−3572.04	−0.262957
$46$	0	0
$47$	−17776.2	−1.17380	−0.586902	−	0.809658i	$-0.699652\pi$
$47$	−17776.2	−1.17380	−0.586902	+	0.809658i	$0.699652\pi$
$48$	0	0
$49$	−14439.9	−0.859159
$50$	0	0
$51$	15509.3	0.834961
$52$	0	0
$53$	10834.4	0.529803	0.264902	−	0.964275i	$-0.414661\pi$
$53$	10834.4	0.529803	0.264902	+	0.964275i	$0.414661\pi$
$54$	0	0
$55$	7086.00	0.315860
$56$	0	0
$57$	−1637.90	−0.0667727
$58$	0	0
$59$	5363.36	0.200589	0.100295	−	0.994958i	$-0.468021\pi$
$59$	5363.36	0.200589	0.100295	+	0.994958i	$0.468021\pi$
$60$	0	0
$61$	−18670.7	−0.642445	−0.321222	−	0.947004i	$-0.604094\pi$
$61$	−18670.7	−0.642445	−0.321222	+	0.947004i	$0.604094\pi$
$62$	0	0
$63$	−6951.63	−0.220666
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−13985.1	−0.380607	−0.190304	−	0.981725i	$-0.560947\pi$
$67$	−13985.1	−0.380607	−0.190304	+	0.981725i	$0.560947\pi$
$68$	0	0
$69$	38013.5	0.961204
$70$	0	0
$71$	−50969.5	−1.19995	−0.599976	−	0.800018i	$-0.704823\pi$
$71$	−50969.5	−1.19995	−0.599976	+	0.800018i	$0.704823\pi$
$72$	0	0
$73$	42394.9	0.931122	0.465561	−	0.885016i	$-0.345853\pi$
$73$	42394.9	0.931122	0.465561	+	0.885016i	$0.345853\pi$
$74$	0	0
$75$	12277.4	0.252031
$76$	0	0
$77$	13790.2	0.265060
$78$	0	0
$79$	−106279.	−1.91593	−0.957964	−	0.286888i	$-0.907379\pi$
$79$	−106279.	−1.91593	−0.957964	+	0.286888i	$0.907379\pi$
$80$	0	0
$81$	−73354.1	−1.24226
$82$	0	0
$83$	−75513.6	−1.20318	−0.601589	−	0.798806i	$-0.705465\pi$
$83$	−75513.6	−1.20318	−0.601589	+	0.798806i	$0.705465\pi$
$84$	0	0
$85$	−19738.1	−0.296317
$86$	0	0
$87$	4375.71	0.0619799
$88$	0	0
$89$	77017.8	1.03066	0.515331	−	0.856991i	$-0.327669\pi$
$89$	77017.8	1.03066	0.515331	+	0.856991i	$0.327669\pi$
$90$	0	0
$91$	−8222.36	−0.104086
$92$	0	0
$93$	54796.2	0.656966
$94$	0	0
$95$	2084.49	0.0236968
$96$	0	0
$97$	126702.	1.36727	0.683637	−	0.729822i	$-0.260397\pi$
$97$	126702.	1.36727	0.683637	+	0.729822i	$0.260397\pi$
$98$	0	0
$99$	−40498.4	−0.415288
$100$	0	0
$101$	−177513.	−1.73151	−0.865757	−	0.500464i	$-0.833163\pi$
$101$	−177513.	−1.73151	−0.865757	+	0.500464i	$0.833163\pi$
$102$	0	0
$103$	−149599.	−1.38943	−0.694714	−	0.719286i	$-0.744469\pi$
$103$	−149599.	−1.38943	−0.694714	+	0.719286i	$0.744469\pi$
$104$	0	0
$105$	23893.3	0.211497
$106$	0	0
$107$	111466.	0.941199	0.470600	−	0.882347i	$-0.344038\pi$
$107$	111466.	0.941199	0.470600	+	0.882347i	$0.344038\pi$
$108$	0	0
$109$	−85200.4	−0.686872	−0.343436	−	0.939176i	$-0.611591\pi$
$109$	−85200.4	−0.686872	−0.343436	+	0.939176i	$0.611591\pi$
$110$	0	0
$111$	164411.	1.26655
$112$	0	0
$113$	−204137.	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$113$	−204137.	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$114$	0	0
$115$	−48378.4	−0.341120
$116$	0	0
$117$	24147.0	0.163079
$118$	0	0
$119$	−38412.7	−0.248661
$120$	0	0
$121$	−80712.8	−0.501163
$122$	0	0
$123$	23936.2	0.142657
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−272404.	−1.49866	−0.749331	−	0.662196i	$-0.769625\pi$
$127$	−272404.	−1.49866	−0.749331	+	0.662196i	$0.769625\pi$
$128$	0	0
$129$	−110770.	−0.586066
$130$	0	0
$131$	−42029.3	−0.213980	−0.106990	−	0.994260i	$-0.534121\pi$
$131$	−42029.3	−0.213980	−0.106990	+	0.994260i	$0.534121\pi$
$132$	0	0
$133$	4056.66	0.0198857
$134$	0	0
$135$	49167.8	0.232191
$136$	0	0
$137$	−56817.3	−0.258630	−0.129315	−	0.991604i	$-0.541278\pi$
$137$	−56817.3	−0.258630	−0.129315	+	0.991604i	$0.541278\pi$
$138$	0	0
$139$	136297.	0.598342	0.299171	−	0.954200i	$-0.403290\pi$
$139$	136297.	0.598342	0.299171	+	0.954200i	$0.403290\pi$
$140$	0	0
$141$	−349194.	−1.47917
$142$	0	0
$143$	−47901.3	−0.195888
$144$	0	0
$145$	−5568.80	−0.0219959
$146$	0	0
$147$	−283655.	−1.08267
$148$	0	0
$149$	−398843.	−1.47176	−0.735879	−	0.677113i	$-0.763231\pi$
$149$	−398843.	−1.47176	−0.735879	+	0.677113i	$0.763231\pi$
$150$	0	0
$151$	16131.0	0.0575730	0.0287865	−	0.999586i	$-0.490836\pi$
$151$	16131.0	0.0575730	0.0287865	+	0.999586i	$0.490836\pi$
$152$	0	0
$153$	112808.	0.389594
$154$	0	0
$155$	−69737.0	−0.233149
$156$	0	0
$157$	9495.80	0.0307456	0.0153728	−	0.999882i	$-0.495107\pi$
$157$	9495.80	0.0307456	0.0153728	+	0.999882i	$0.495107\pi$
$158$	0	0
$159$	212829.	0.667634
$160$	0	0
$161$	−94150.2	−0.286257
$162$	0	0
$163$	−439257.	−1.29494	−0.647469	−	0.762091i	$-0.724173\pi$
$163$	−439257.	−1.29494	−0.647469	+	0.762091i	$0.724173\pi$
$164$	0	0
$165$	139196.	0.398032
$166$	0	0
$167$	233417.	0.647651	0.323825	−	0.946117i	$-0.395031\pi$
$167$	233417.	0.647651	0.323825	+	0.946117i	$0.395031\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−11913.4	−0.0311563
$172$	0	0
$173$	121696.	0.309144	0.154572	−	0.987982i	$-0.450600\pi$
$173$	121696.	0.309144	0.154572	+	0.987982i	$0.450600\pi$
$174$	0	0
$175$	−30408.1	−0.0750576
$176$	0	0
$177$	105357.	0.252773
$178$	0	0
$179$	280901.	0.655271	0.327636	−	0.944804i	$-0.393748\pi$
$179$	280901.	0.655271	0.327636	+	0.944804i	$0.393748\pi$
$180$	0	0
$181$	324458.	0.736142	0.368071	−	0.929798i	$-0.380018\pi$
$181$	324458.	0.736142	0.368071	+	0.929798i	$0.380018\pi$
$182$	0	0
$183$	−366765.	−0.809580
$184$	0	0
$185$	−209239.	−0.449483
$186$	0	0
$187$	−223782.	−0.467974
$188$	0	0
$189$	95686.5	0.194848
$190$	0	0
$191$	−803448.	−1.59358	−0.796791	−	0.604256i	$-0.793471\pi$
$191$	−803448.	−1.59358	−0.796791	+	0.604256i	$0.793471\pi$
$192$	0	0
$193$	−62706.7	−0.121177	−0.0605886	−	0.998163i	$-0.519298\pi$
$193$	−62706.7	−0.121177	−0.0605886	+	0.998163i	$0.519298\pi$
$194$	0	0
$195$	−82995.4	−0.156303
$196$	0	0
$197$	−367605.	−0.674864	−0.337432	−	0.941350i	$-0.609558\pi$
$197$	−367605.	−0.674864	−0.337432	+	0.941350i	$0.609558\pi$
$198$	0	0
$199$	−860523.	−1.54039	−0.770193	−	0.637810i	$-0.779840\pi$
$199$	−860523.	−1.54039	−0.770193	+	0.637810i	$0.779840\pi$
$200$	0	0
$201$	−274721.	−0.479624
$202$	0	0
$203$	−10837.6	−0.0184583
$204$	0	0
$205$	−30462.7	−0.0506272
$206$	0	0
$207$	276495.	0.448499
$208$	0	0
$209$	23633.1	0.0374244
$210$	0	0
$211$	457688.	0.707723	0.353861	−	0.935298i	$-0.384868\pi$
$211$	457688.	0.707723	0.353861	+	0.935298i	$0.384868\pi$
$212$	0	0
$213$	−1.00124e6	−1.51213
$214$	0	0
$215$	140972.	0.207988
$216$	0	0
$217$	−135717.	−0.195652
$218$	0	0
$219$	832800.	1.17336
$220$	0	0
$221$	133429.	0.183768
$222$	0	0
$223$	925567.	1.24637	0.623183	−	0.782076i	$-0.285839\pi$
$223$	925567.	1.24637	0.623183	+	0.782076i	$0.285839\pi$
$224$	0	0
$225$	89301.0	0.117598
$226$	0	0
$227$	1.54248e6	1.98681	0.993403	−	0.114679i	$-0.0365839\pi$
$227$	1.54248e6	1.98681	0.993403	+	0.114679i	$0.0365839\pi$
$228$	0	0
$229$	1.09184e6	1.37584	0.687922	−	0.725784i	$-0.258523\pi$
$229$	1.09184e6	1.37584	0.687922	+	0.725784i	$0.258523\pi$
$230$	0	0
$231$	270893.	0.334017
$232$	0	0
$233$	141596.	0.170868	0.0854341	−	0.996344i	$-0.472772\pi$
$233$	141596.	0.170868	0.0854341	+	0.996344i	$0.472772\pi$
$234$	0	0
$235$	444406.	0.524941
$236$	0	0
$237$	−2.08773e6	−2.41437
$238$	0	0
$239$	791678.	0.896507	0.448253	−	0.893906i	$-0.352046\pi$
$239$	791678.	0.896507	0.448253	+	0.893906i	$0.352046\pi$
$240$	0	0
$241$	1.39653e6	1.54884	0.774421	−	0.632671i	$-0.218041\pi$
$241$	1.39653e6	1.54884	0.774421	+	0.632671i	$0.218041\pi$
$242$	0	0
$243$	−963047.	−1.04624
$244$	0	0
$245$	360997.	0.384228
$246$	0	0
$247$	−14091.1	−0.0146961
$248$	0	0
$249$	−1.48338e6	−1.51619
$250$	0	0
$251$	−927565.	−0.929309	−0.464654	−	0.885492i	$-0.653821\pi$
$251$	−927565.	−0.929309	−0.464654	+	0.885492i	$0.653821\pi$
$252$	0	0
$253$	−548494.	−0.538730
$254$	0	0
$255$	−387732.	−0.373406
$256$	0	0
$257$	1.21697e6	1.14934	0.574670	−	0.818385i	$-0.305130\pi$
$257$	1.21697e6	1.14934	0.574670	+	0.818385i	$0.305130\pi$
$258$	0	0
$259$	−407205.	−0.377193
$260$	0	0
$261$	31827.2	0.0289199
$262$	0	0
$263$	188972.	0.168464	0.0842321	−	0.996446i	$-0.473156\pi$
$263$	188972.	0.168464	0.0842321	+	0.996446i	$0.473156\pi$
$264$	0	0
$265$	−270860.	−0.236935
$266$	0	0
$267$	1.51293e6	1.29879
$268$	0	0
$269$	−1.68629e6	−1.42086	−0.710432	−	0.703766i	$-0.751500\pi$
$269$	−1.68629e6	−1.42086	−0.710432	+	0.703766i	$0.751500\pi$
$270$	0	0
$271$	−1.71354e6	−1.41733	−0.708664	−	0.705546i	$-0.750702\pi$
$271$	−1.71354e6	−1.41733	−0.708664	+	0.705546i	$0.750702\pi$
$272$	0	0
$273$	−161519.	−0.131165
$274$	0	0
$275$	−177150.	−0.141257
$276$	0	0
$277$	1.51635e6	1.18741	0.593706	−	0.804682i	$-0.297664\pi$
$277$	1.51635e6	1.18741	0.593706	+	0.804682i	$0.297664\pi$
$278$	0	0
$279$	398566.	0.306542
$280$	0	0
$281$	2.01362e6	1.52129	0.760644	−	0.649169i	$-0.224883\pi$
$281$	2.01362e6	1.52129	0.760644	+	0.649169i	$0.224883\pi$
$282$	0	0
$283$	−451062.	−0.334788	−0.167394	−	0.985890i	$-0.553535\pi$
$283$	−451062.	−0.334788	−0.167394	+	0.985890i	$0.553535\pi$
$284$	0	0
$285$	40947.4	0.0298617
$286$	0	0
$287$	−59284.1	−0.0424848
$288$	0	0
$289$	−796511.	−0.560980
$290$	0	0
$291$	2.48893e6	1.72298
$292$	0	0
$293$	−1.60140e6	−1.08976	−0.544879	−	0.838514i	$-0.683425\pi$
$293$	−1.60140e6	−1.08976	−0.544879	+	0.838514i	$0.683425\pi$
$294$	0	0
$295$	−134084.	−0.0897062
$296$	0	0
$297$	557444.	0.366700
$298$	0	0
$299$	327038.	0.211553
$300$	0	0
$301$	274349.	0.174537
$302$	0	0
$303$	−3.48704e6	−2.18198
$304$	0	0
$305$	466767.	0.287310
$306$	0	0
$307$	−3.03553e6	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$307$	−3.03553e6	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$308$	0	0
$309$	−2.93871e6	−1.75089
$310$	0	0
$311$	−1.48427e6	−0.870184	−0.435092	−	0.900386i	$-0.643284\pi$
$311$	−1.48427e6	−0.870184	−0.435092	+	0.900386i	$0.643284\pi$
$312$	0	0
$313$	1.57466e6	0.908504	0.454252	−	0.890873i	$-0.349906\pi$
$313$	1.57466e6	0.908504	0.454252	+	0.890873i	$0.349906\pi$
$314$	0	0
$315$	173791.	0.0986848
$316$	0	0
$317$	2.88960e6	1.61507	0.807533	−	0.589822i	$-0.200802\pi$
$317$	2.88960e6	1.61507	0.807533	+	0.589822i	$0.200802\pi$
$318$	0	0
$319$	−63136.8	−0.0347381
$320$	0	0
$321$	2.18962e6	1.18606
$322$	0	0
$323$	−65830.0	−0.0351089
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	−1.67367e6	−0.865565
$328$	0	0
$329$	864868.	0.440514
$330$	0	0
$331$	−2.93760e6	−1.47374	−0.736872	−	0.676032i	$-0.763698\pi$
$331$	−2.93760e6	−1.47374	−0.736872	+	0.676032i	$0.763698\pi$
$332$	0	0
$333$	1.19586e6	0.590974
$334$	0	0
$335$	349626.	0.170213
$336$	0	0
$337$	−842611.	−0.404159	−0.202079	−	0.979369i	$-0.564770\pi$
$337$	−842611.	−0.404159	−0.202079	+	0.979369i	$0.564770\pi$
$338$	0	0
$339$	−4.01004e6	−1.89517
$340$	0	0
$341$	−790650.	−0.368212
$342$	0	0
$343$	1.52026e6	0.697720
$344$	0	0
$345$	−950339.	−0.429864
$346$	0	0
$347$	719797.	0.320912	0.160456	−	0.987043i	$-0.448703\pi$
$347$	719797.	0.320912	0.160456	+	0.987043i	$0.448703\pi$
$348$	0	0
$349$	−2.28139e6	−1.00262	−0.501309	−	0.865269i	$-0.667148\pi$
$349$	−2.28139e6	−1.00262	−0.501309	+	0.865269i	$0.667148\pi$
$350$	0	0
$351$	−332374.	−0.143999
$352$	0	0
$353$	7769.01	0.00331840	0.00165920	−	0.999999i	$-0.499472\pi$
$353$	7769.01	0.00331840	0.00165920	+	0.999999i	$0.499472\pi$
$354$	0	0
$355$	1.27424e6	0.536635
$356$	0	0
$357$	−754573.	−0.313351
$358$	0	0
$359$	−309145.	−0.126598	−0.0632989	−	0.997995i	$-0.520162\pi$
$359$	−309145.	−0.126598	−0.0632989	+	0.997995i	$0.520162\pi$
$360$	0	0
$361$	−2.46915e6	−0.997192
$362$	0	0
$363$	−1.58551e6	−0.631543
$364$	0	0
$365$	−1.05987e6	−0.416410
$366$	0	0
$367$	2.32781e6	0.902156	0.451078	−	0.892484i	$-0.351040\pi$
$367$	2.32781e6	0.902156	0.451078	+	0.892484i	$0.351040\pi$
$368$	0	0
$369$	174103.	0.0665640
$370$	0	0
$371$	−527126.	−0.198829
$372$	0	0
$373$	−1.60840e6	−0.598579	−0.299289	−	0.954162i	$-0.596750\pi$
$373$	−1.60840e6	−0.598579	−0.299289	+	0.954162i	$0.596750\pi$
$374$	0	0
$375$	−306935.	−0.112712
$376$	0	0
$377$	37645.1	0.0136413
$378$	0	0
$379$	473484.	0.169320	0.0846599	−	0.996410i	$-0.473020\pi$
$379$	473484.	0.169320	0.0846599	+	0.996410i	$0.473020\pi$
$380$	0	0
$381$	−5.35106e6	−1.88855
$382$	0	0
$383$	−363362.	−0.126574	−0.0632868	−	0.997995i	$-0.520158\pi$
$383$	−363362.	−0.126574	−0.0632868	+	0.997995i	$0.520158\pi$
$384$	0	0
$385$	−344755.	−0.118538
$386$	0	0
$387$	−805695.	−0.273460
$388$	0	0
$389$	−901091.	−0.301922	−0.150961	−	0.988540i	$-0.548237\pi$
$389$	−901091.	−0.301922	−0.150961	+	0.988540i	$0.548237\pi$
$390$	0	0
$391$	1.52783e6	0.505398
$392$	0	0
$393$	−825618.	−0.269648
$394$	0	0
$395$	2.65697e6	0.856829
$396$	0	0
$397$	−3.36685e6	−1.07213	−0.536064	−	0.844177i	$-0.680090\pi$
$397$	−3.36685e6	−1.07213	−0.536064	+	0.844177i	$0.680090\pi$
$398$	0	0
$399$	79688.6	0.0250590
$400$	0	0
$401$	1.49612e6	0.464629	0.232315	−	0.972641i	$-0.425370\pi$
$401$	1.49612e6	0.464629	0.232315	+	0.972641i	$0.425370\pi$
$402$	0	0
$403$	471422.	0.144593
$404$	0	0
$405$	1.83385e6	0.555555
$406$	0	0
$407$	−2.37227e6	−0.709868
$408$	0	0
$409$	1.78884e6	0.528765	0.264382	−	0.964418i	$-0.414832\pi$
$409$	1.78884e6	0.528765	0.264382	+	0.964418i	$0.414832\pi$
$410$	0	0
$411$	−1.11611e6	−0.325914
$412$	0	0
$413$	−260944.	−0.0752787
$414$	0	0
$415$	1.88784e6	0.538077
$416$	0	0
$417$	2.67740e6	0.754003
$418$	0	0
$419$	−3.06163e6	−0.851956	−0.425978	−	0.904733i	$-0.640070\pi$
$419$	−3.06163e6	−0.851956	−0.425978	+	0.904733i	$0.640070\pi$
$420$	0	0
$421$	−6.32711e6	−1.73980	−0.869901	−	0.493226i	$-0.835817\pi$
$421$	−6.32711e6	−1.73980	−0.869901	+	0.493226i	$0.835817\pi$
$422$	0	0
$423$	−2.53990e6	−0.690185
$424$	0	0
$425$	493452.	0.132517
$426$	0	0
$427$	908386.	0.241102
$428$	0	0
$429$	−940968.	−0.246849
$430$	0	0
$431$	2.68796e6	0.696995	0.348498	−	0.937310i	$-0.386692\pi$
$431$	2.68796e6	0.696995	0.348498	+	0.937310i	$0.386692\pi$
$432$	0	0
$433$	1.62239e6	0.415850	0.207925	−	0.978145i	$-0.433329\pi$
$433$	1.62239e6	0.415850	0.207925	+	0.978145i	$0.433329\pi$
$434$	0	0
$435$	−109393.	−0.0277182
$436$	0	0
$437$	−161351.	−0.0404172
$438$	0	0
$439$	−6.21478e6	−1.53909	−0.769546	−	0.638592i	$-0.779517\pi$
$439$	−6.21478e6	−1.53909	−0.769546	+	0.638592i	$0.779517\pi$
$440$	0	0
$441$	−2.06319e6	−0.505177
$442$	0	0
$443$	517107.	0.125191	0.0625953	−	0.998039i	$-0.480062\pi$
$443$	517107.	0.125191	0.0625953	+	0.998039i	$0.480062\pi$
$444$	0	0
$445$	−1.92545e6	−0.460926
$446$	0	0
$447$	−7.83482e6	−1.85464
$448$	0	0
$449$	−7.08559e6	−1.65867	−0.829336	−	0.558750i	$-0.811281\pi$
$449$	−7.08559e6	−1.65867	−0.829336	+	0.558750i	$0.811281\pi$
$450$	0	0
$451$	−345374.	−0.0799555
$452$	0	0
$453$	316875.	0.0725509
$454$	0	0
$455$	205559.	0.0465488
$456$	0	0
$457$	−929335.	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$457$	−929335.	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$458$	0	0
$459$	−1.55276e6	−0.344012
$460$	0	0
$461$	7.91310e6	1.73418	0.867091	−	0.498150i	$-0.165987\pi$
$461$	7.91310e6	1.73418	0.867091	+	0.498150i	$0.165987\pi$
$462$	0	0
$463$	1.60654e6	0.348289	0.174144	−	0.984720i	$-0.444284\pi$
$463$	1.60654e6	0.348289	0.174144	+	0.984720i	$0.444284\pi$
$464$	0	0
$465$	−1.36990e6	−0.293804
$466$	0	0
$467$	−2.57909e6	−0.547236	−0.273618	−	0.961838i	$-0.588220\pi$
$467$	−2.57909e6	−0.547236	−0.273618	+	0.961838i	$0.588220\pi$
$468$	0	0
$469$	680415.	0.142837
$470$	0	0
$471$	186534.	0.0387442
$472$	0	0
$473$	1.59829e6	0.328475
$474$	0	0
$475$	−52112.2	−0.0105975
$476$	0	0
$477$	1.54803e6	0.311519
$478$	0	0
$479$	−1.57546e6	−0.313739	−0.156869	−	0.987619i	$-0.550140\pi$
$479$	−1.57546e6	−0.313739	−0.156869	+	0.987619i	$0.550140\pi$
$480$	0	0
$481$	1.41446e6	0.278758
$482$	0	0
$483$	−1.84947e6	−0.360728
$484$	0	0
$485$	−3.16756e6	−0.611464
$486$	0	0
$487$	3.52518e6	0.673534	0.336767	−	0.941588i	$-0.390667\pi$
$487$	3.52518e6	0.673534	0.336767	+	0.941588i	$0.390667\pi$
$488$	0	0
$489$	−8.62870e6	−1.63182
$490$	0	0
$491$	3.53510e6	0.661756	0.330878	−	0.943674i	$-0.392655\pi$
$491$	3.53510e6	0.661756	0.330878	+	0.943674i	$0.392655\pi$
$492$	0	0
$493$	175868.	0.0325888
$494$	0	0
$495$	1.01246e6	0.185722
$496$	0	0
$497$	2.47982e6	0.450328
$498$	0	0
$499$	−311968.	−0.0560866	−0.0280433	−	0.999607i	$-0.508928\pi$
$499$	−311968.	−0.0560866	−0.0280433	+	0.999607i	$0.508928\pi$
$500$	0	0
$501$	4.58521e6	0.816140
$502$	0	0
$503$	−6.52751e6	−1.15034	−0.575172	−	0.818032i	$-0.695065\pi$
$503$	−6.52751e6	−1.15034	−0.575172	+	0.818032i	$0.695065\pi$
$504$	0	0
$505$	4.43782e6	0.774357
$506$	0	0
$507$	561049.	0.0969350
$508$	0	0
$509$	−6.28149e6	−1.07465	−0.537327	−	0.843374i	$-0.680566\pi$
$509$	−6.28149e6	−1.07465	−0.537327	+	0.843374i	$0.680566\pi$
$510$	0	0
$511$	−2.06264e6	−0.349439
$512$	0	0
$513$	163983.	0.0275110
$514$	0	0
$515$	3.73998e6	0.621371
$516$	0	0
$517$	5.03850e6	0.829038
$518$	0	0
$519$	2.39058e6	0.389569
$520$	0	0
$521$	−7.39635e6	−1.19378	−0.596888	−	0.802324i	$-0.703596\pi$
$521$	−7.39635e6	−1.19378	−0.596888	+	0.802324i	$0.703596\pi$
$522$	0	0
$523$	−6.26869e6	−1.00213	−0.501063	−	0.865411i	$-0.667057\pi$
$523$	−6.26869e6	−1.00213	−0.501063	+	0.865411i	$0.667057\pi$
$524$	0	0
$525$	−597334.	−0.0945842
$526$	0	0
$527$	2.20236e6	0.345431
$528$	0	0
$529$	−2.69160e6	−0.418187
$530$	0	0
$531$	766327.	0.117944
$532$	0	0
$533$	205928.	0.0313977
$534$	0	0
$535$	−2.78664e6	−0.420917
$536$	0	0
$537$	5.51799e6	0.825743
$538$	0	0
$539$	4.09284e6	0.606810
$540$	0	0
$541$	−2.81963e6	−0.414189	−0.207095	−	0.978321i	$-0.566401\pi$
$541$	−2.81963e6	−0.414189	−0.207095	+	0.978321i	$0.566401\pi$
$542$	0	0
$543$	6.37360e6	0.927653
$544$	0	0
$545$	2.13001e6	0.307178
$546$	0	0
$547$	9.17262e6	1.31077	0.655383	−	0.755297i	$-0.272507\pi$
$547$	9.17262e6	1.31077	0.655383	+	0.755297i	$0.272507\pi$
$548$	0	0
$549$	−2.66770e6	−0.377751
$550$	0	0
$551$	−18572.9	−0.00260616
$552$	0	0
$553$	5.17079e6	0.719025
$554$	0	0
$555$	−4.11026e6	−0.566418
$556$	0	0
$557$	−4.27403e6	−0.583713	−0.291856	−	0.956462i	$-0.594273\pi$
$557$	−4.27403e6	−0.583713	−0.291856	+	0.956462i	$0.594273\pi$
$558$	0	0
$559$	−952973.	−0.128989
$560$	0	0
$561$	−4.39595e6	−0.589719
$562$	0	0
$563$	−9.29935e6	−1.23646	−0.618232	−	0.785995i	$-0.712151\pi$
$563$	−9.29935e6	−1.23646	−0.618232	+	0.785995i	$0.712151\pi$
$564$	0	0
$565$	5.10342e6	0.672574
$566$	0	0
$567$	3.56890e6	0.466205
$568$	0	0
$569$	−4.10069e6	−0.530977	−0.265489	−	0.964114i	$-0.585533\pi$
$569$	−4.10069e6	−0.530977	−0.265489	+	0.964114i	$0.585533\pi$
$570$	0	0
$571$	1.03048e7	1.32267	0.661334	−	0.750092i	$-0.269991\pi$
$571$	1.03048e7	1.32267	0.661334	+	0.750092i	$0.269991\pi$
$572$	0	0
$573$	−1.57828e7	−2.00816
$574$	0	0
$575$	1.20946e6	0.152553
$576$	0	0
$577$	2.17391e6	0.271833	0.135916	−	0.990720i	$-0.456602\pi$
$577$	2.17391e6	0.271833	0.135916	+	0.990720i	$0.456602\pi$
$578$	0	0
$579$	−1.23180e6	−0.152702
$580$	0	0
$581$	3.67396e6	0.451538
$582$	0	0
$583$	−3.07090e6	−0.374191
$584$	0	0
$585$	−603675.	−0.0729312
$586$	0	0
$587$	1.56597e7	1.87581	0.937905	−	0.346892i	$-0.112763\pi$
$587$	1.56597e7	1.87581	0.937905	+	0.346892i	$0.112763\pi$
$588$	0	0
$589$	−232585.	−0.0276245
$590$	0	0
$591$	−7.22119e6	−0.850433
$592$	0	0
$593$	−5.51737e6	−0.644310	−0.322155	−	0.946687i	$-0.604407\pi$
$593$	−5.51737e6	−0.644310	−0.322155	+	0.946687i	$0.604407\pi$
$594$	0	0
$595$	960317.	0.111204
$596$	0	0
$597$	−1.69040e7	−1.94113
$598$	0	0
$599$	8.76025e6	0.997584	0.498792	−	0.866722i	$-0.333777\pi$
$599$	8.76025e6	0.997584	0.498792	+	0.866722i	$0.333777\pi$
$600$	0	0
$601$	1.21014e7	1.36662	0.683310	−	0.730128i	$-0.260540\pi$
$601$	1.21014e7	1.36662	0.683310	+	0.730128i	$0.260540\pi$
$602$	0	0
$603$	−1.99821e6	−0.223794
$604$	0	0
$605$	2.01782e6	0.224127
$606$	0	0
$607$	1.29407e7	1.42556	0.712779	−	0.701388i	$-0.247436\pi$
$607$	1.29407e7	1.42556	0.712779	+	0.701388i	$0.247436\pi$
$608$	0	0
$609$	−212892.	−0.0232603
$610$	0	0
$611$	−3.00418e6	−0.325554
$612$	0	0
$613$	2.31350e6	0.248667	0.124333	−	0.992240i	$-0.460321\pi$
$613$	2.31350e6	0.248667	0.124333	+	0.992240i	$0.460321\pi$
$614$	0	0
$615$	−598406.	−0.0637981
$616$	0	0
$617$	8.53024e6	0.902086	0.451043	−	0.892502i	$-0.351052\pi$
$617$	8.53024e6	0.902086	0.451043	+	0.892502i	$0.351052\pi$
$618$	0	0
$619$	−8.89321e6	−0.932893	−0.466447	−	0.884549i	$-0.654466\pi$
$619$	−8.89321e6	−0.932893	−0.466447	+	0.884549i	$0.654466\pi$
$620$	0	0
$621$	−3.80585e6	−0.396025
$622$	0	0
$623$	−3.74715e6	−0.386795
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	464245.	0.0471605
$628$	0	0
$629$	6.60796e6	0.665948
$630$	0	0
$631$	−1.01842e7	−1.01824	−0.509122	−	0.860695i	$-0.670030\pi$
$631$	−1.01842e7	−1.01824	−0.509122	+	0.860695i	$0.670030\pi$
$632$	0	0
$633$	8.99076e6	0.891840
$634$	0	0
$635$	6.81009e6	0.670222
$636$	0	0
$637$	−2.44034e6	−0.238288
$638$	0	0
$639$	−7.28260e6	−0.705561
$640$	0	0
$641$	−1.30240e7	−1.25198	−0.625991	−	0.779830i	$-0.715306\pi$
$641$	−1.30240e7	−1.25198	−0.625991	+	0.779830i	$0.715306\pi$
$642$	0	0
$643$	4.52103e6	0.431231	0.215616	−	0.976478i	$-0.430824\pi$
$643$	4.52103e6	0.431231	0.215616	+	0.976478i	$0.430824\pi$
$644$	0	0
$645$	2.76924e6	0.262097
$646$	0	0
$647$	−2.72009e6	−0.255459	−0.127730	−	0.991809i	$-0.540769\pi$
$647$	−2.72009e6	−0.255459	−0.127730	+	0.991809i	$0.540769\pi$
$648$	0	0
$649$	−1.52019e6	−0.141673
$650$	0	0
$651$	−2.66600e6	−0.246552
$652$	0	0
$653$	1.83786e7	1.68667	0.843334	−	0.537389i	$-0.180589\pi$
$653$	1.83786e7	1.68667	0.843334	+	0.537389i	$0.180589\pi$
$654$	0	0
$655$	1.05073e6	0.0956949
$656$	0	0
$657$	6.05745e6	0.547491
$658$	0	0
$659$	1.39753e7	1.25357	0.626786	−	0.779192i	$-0.284370\pi$
$659$	1.39753e7	1.25357	0.626786	+	0.779192i	$0.284370\pi$
$660$	0	0
$661$	9.05787e6	0.806348	0.403174	−	0.915123i	$-0.367907\pi$
$661$	9.05787e6	0.806348	0.403174	+	0.915123i	$0.367907\pi$
$662$	0	0
$663$	2.62107e6	0.231576
$664$	0	0
$665$	−101417.	−0.00889314
$666$	0	0
$667$	431055.	0.0375161
$668$	0	0
$669$	1.81817e7	1.57061
$670$	0	0
$671$	5.29202e6	0.453748
$672$	0	0
$673$	1.68758e7	1.43624	0.718121	−	0.695919i	$-0.245003\pi$
$673$	1.68758e7	1.43624	0.718121	+	0.695919i	$0.245003\pi$
$674$	0	0
$675$	−1.22919e6	−0.103839
$676$	0	0
$677$	7.96769e6	0.668129	0.334065	−	0.942550i	$-0.391580\pi$
$677$	7.96769e6	0.668129	0.334065	+	0.942550i	$0.391580\pi$
$678$	0	0
$679$	−6.16446e6	−0.513122
$680$	0	0
$681$	3.03003e7	2.50368
$682$	0	0
$683$	1.29108e7	1.05901	0.529507	−	0.848306i	$-0.322377\pi$
$683$	1.29108e7	1.05901	0.529507	+	0.848306i	$0.322377\pi$
$684$	0	0
$685$	1.42043e6	0.115663
$686$	0	0
$687$	2.14479e7	1.73378
$688$	0	0
$689$	1.83101e6	0.146941
$690$	0	0
$691$	1.37987e7	1.09936	0.549682	−	0.835374i	$-0.314749\pi$
$691$	1.37987e7	1.09936	0.549682	+	0.835374i	$0.314749\pi$
$692$	0	0
$693$	1.97037e6	0.155853
$694$	0	0
$695$	−3.40743e6	−0.267587
$696$	0	0
$697$	962040.	0.0750086
$698$	0	0
$699$	2.78149e6	0.215320
$700$	0	0
$701$	−1.54955e6	−0.119099	−0.0595497	−	0.998225i	$-0.518966\pi$
$701$	−1.54955e6	−0.119099	−0.0595497	+	0.998225i	$0.518966\pi$
$702$	0	0
$703$	−697849.	−0.0532566
$704$	0	0
$705$	8.72985e6	0.661507
$706$	0	0
$707$	8.63653e6	0.649817
$708$	0	0
$709$	−1.43383e7	−1.07123	−0.535614	−	0.844463i	$-0.679920\pi$
$709$	−1.43383e7	−1.07123	−0.535614	+	0.844463i	$0.679920\pi$
$710$	0	0
$711$	−1.51853e7	−1.12655
$712$	0	0
$713$	5.39802e6	0.397659
$714$	0	0
$715$	1.19753e6	0.0876037
$716$	0	0
$717$	1.55516e7	1.12974
$718$	0	0
$719$	1.37368e7	0.990980	0.495490	−	0.868614i	$-0.334989\pi$
$719$	1.37368e7	0.990980	0.495490	+	0.868614i	$0.334989\pi$
$720$	0	0
$721$	7.27845e6	0.521436
$722$	0	0
$723$	2.74332e7	1.95178
$724$	0	0
$725$	139220.	0.00983686
$726$	0	0
$727$	1.61824e6	0.113555	0.0567776	−	0.998387i	$-0.481917\pi$
$727$	1.61824e6	0.113555	0.0567776	+	0.998387i	$0.481917\pi$
$728$	0	0
$729$	−1.09293e6	−0.0761684
$730$	0	0
$731$	−4.45203e6	−0.308152
$732$	0	0
$733$	−1.57329e7	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$733$	−1.57329e7	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$734$	0	0
$735$	7.09138e6	0.484186
$736$	0	0
$737$	3.96392e6	0.268817
$738$	0	0
$739$	1.62586e6	0.109515	0.0547574	−	0.998500i	$-0.482561\pi$
$739$	1.62586e6	0.109515	0.0547574	+	0.998500i	$0.482561\pi$
$740$	0	0
$741$	−276804.	−0.0185194
$742$	0	0
$743$	−1.87162e7	−1.24379	−0.621894	−	0.783101i	$-0.713637\pi$
$743$	−1.87162e7	−1.24379	−0.621894	+	0.783101i	$0.713637\pi$
$744$	0	0
$745$	9.97108e6	0.658190
$746$	0	0
$747$	−1.07895e7	−0.707457
$748$	0	0
$749$	−5.42314e6	−0.353221
$750$	0	0
$751$	−7.16867e6	−0.463809	−0.231904	−	0.972739i	$-0.574496\pi$
$751$	−7.16867e6	−0.463809	−0.231904	+	0.972739i	$0.574496\pi$
$752$	0	0
$753$	−1.82210e7	−1.17107
$754$	0	0
$755$	−403275.	−0.0257474
$756$	0	0
$757$	2.67712e7	1.69796	0.848982	−	0.528422i	$-0.177216\pi$
$757$	2.67712e7	1.69796	0.848982	+	0.528422i	$0.177216\pi$
$758$	0	0
$759$	−1.07746e7	−0.678883
$760$	0	0
$761$	−708017.	−0.0443182	−0.0221591	−	0.999754i	$-0.507054\pi$
$761$	−708017.	−0.0443182	−0.0221591	+	0.999754i	$0.507054\pi$
$762$	0	0
$763$	4.14526e6	0.257775
$764$	0	0
$765$	−2.82021e6	−0.174232
$766$	0	0
$767$	906409.	0.0556334
$768$	0	0
$769$	1.72011e7	1.04891	0.524456	−	0.851437i	$-0.324269\pi$
$769$	1.72011e7	1.04891	0.524456	+	0.851437i	$0.324269\pi$
$770$	0	0
$771$	2.39061e7	1.44835
$772$	0	0
$773$	7.85325e6	0.472716	0.236358	−	0.971666i	$-0.424046\pi$
$773$	7.85325e6	0.472716	0.236358	+	0.971666i	$0.424046\pi$
$774$	0	0
$775$	1.74343e6	0.104268
$776$	0	0
$777$	−7.99907e6	−0.475321
$778$	0	0
$779$	−101599.	−0.00599852
$780$	0	0
$781$	1.44468e7	0.847507
$782$	0	0
$783$	−438089.	−0.0255363
$784$	0	0
$785$	−237395.	−0.0137498
$786$	0	0
$787$	−2.05200e7	−1.18097	−0.590487	−	0.807047i	$-0.701064\pi$
$787$	−2.05200e7	−1.18097	−0.590487	+	0.807047i	$0.701064\pi$
$788$	0	0
$789$	3.71214e6	0.212291
$790$	0	0
$791$	9.93187e6	0.564404
$792$	0	0
$793$	−3.15535e6	−0.178182
$794$	0	0
$795$	−5.32073e6	−0.298575
$796$	0	0
$797$	−2.57349e7	−1.43508	−0.717540	−	0.696517i	$-0.754732\pi$
$797$	−2.57349e7	−1.43508	−0.717540	+	0.696517i	$0.754732\pi$
$798$	0	0
$799$	−1.40347e7	−0.777745
$800$	0	0
$801$	1.10044e7	0.606019
$802$	0	0
$803$	−1.20164e7	−0.657636
$804$	0	0
$805$	2.35375e6	0.128018
$806$	0	0
$807$	−3.31253e7	−1.79051
$808$	0	0
$809$	−1.45330e7	−0.780699	−0.390350	−	0.920667i	$-0.627646\pi$
$809$	−1.45330e7	−0.780699	−0.390350	+	0.920667i	$0.627646\pi$
$810$	0	0
$811$	−1.60281e6	−0.0855717	−0.0427858	−	0.999084i	$-0.513623\pi$
$811$	−1.60281e6	−0.0855717	−0.0427858	+	0.999084i	$0.513623\pi$
$812$	0	0
$813$	−3.36605e7	−1.78605
$814$	0	0
$815$	1.09814e7	0.579114
$816$	0	0
$817$	470168.	0.0246432
$818$	0	0
$819$	−1.17482e6	−0.0612017
$820$	0	0
$821$	1.22945e6	0.0636579	0.0318289	−	0.999493i	$-0.489867\pi$
$821$	1.22945e6	0.0636579	0.0318289	+	0.999493i	$0.489867\pi$
$822$	0	0
$823$	260323.	0.0133972	0.00669859	−	0.999978i	$-0.497868\pi$
$823$	260323.	0.0133972	0.00669859	+	0.999978i	$0.497868\pi$
$824$	0	0
$825$	−3.47991e6	−0.178005
$826$	0	0
$827$	−1.33246e7	−0.677471	−0.338736	−	0.940882i	$-0.609999\pi$
$827$	−1.33246e7	−0.677471	−0.338736	+	0.940882i	$0.609999\pi$
$828$	0	0
$829$	3.52026e7	1.77905	0.889526	−	0.456885i	$-0.151035\pi$
$829$	3.52026e7	1.77905	0.889526	+	0.456885i	$0.151035\pi$
$830$	0	0
$831$	2.97871e7	1.49632
$832$	0	0
$833$	−1.14006e7	−0.569267
$834$	0	0
$835$	−5.83542e6	−0.289638
$836$	0	0
$837$	−5.48610e6	−0.270676
$838$	0	0
$839$	3.45100e7	1.69255	0.846273	−	0.532750i	$-0.178841\pi$
$839$	3.45100e7	1.69255	0.846273	+	0.532750i	$0.178841\pi$
$840$	0	0
$841$	−2.04615e7	−0.997581
$842$	0	0
$843$	3.95553e7	1.91706
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	3.92692e6	0.188081
$848$	0	0
$849$	−8.86059e6	−0.421885
$850$	0	0
$851$	1.61962e7	0.766637
$852$	0	0
$853$	−1.67507e7	−0.788245	−0.394122	−	0.919058i	$-0.628951\pi$
$853$	−1.67507e7	−0.788245	−0.394122	+	0.919058i	$0.628951\pi$
$854$	0	0
$855$	297835.	0.0139335
$856$	0	0
$857$	−2.15120e6	−0.100053	−0.0500264	−	0.998748i	$-0.515931\pi$
$857$	−2.15120e6	−0.100053	−0.0500264	+	0.998748i	$0.515931\pi$
$858$	0	0
$859$	−1.66088e7	−0.767991	−0.383996	−	0.923335i	$-0.625452\pi$
$859$	−1.66088e7	−0.767991	−0.383996	+	0.923335i	$0.625452\pi$
$860$	0	0
$861$	−1.16457e6	−0.0535374
$862$	0	0
$863$	2.71816e7	1.24236	0.621181	−	0.783667i	$-0.286653\pi$
$863$	2.71816e7	1.24236	0.621181	+	0.783667i	$0.286653\pi$
$864$	0	0
$865$	−3.04240e6	−0.138253
$866$	0	0
$867$	−1.56466e7	−0.706922
$868$	0	0
$869$	3.01237e7	1.35319
$870$	0	0
$871$	−2.36347e6	−0.105562
$872$	0	0
$873$	1.81035e7	0.803944
$874$	0	0
$875$	760204.	0.0335668
$876$	0	0
$877$	1.52455e6	0.0669333	0.0334667	−	0.999440i	$-0.489345\pi$
$877$	1.52455e6	0.0669333	0.0334667	+	0.999440i	$0.489345\pi$
$878$	0	0
$879$	−3.14577e7	−1.37326
$880$	0	0
$881$	2.77728e7	1.20554	0.602769	−	0.797916i	$-0.294064\pi$
$881$	2.77728e7	1.20554	0.602769	+	0.797916i	$0.294064\pi$
$882$	0	0
$883$	−3.15964e7	−1.36376	−0.681878	−	0.731466i	$-0.738836\pi$
$883$	−3.15964e7	−1.36376	−0.681878	+	0.731466i	$0.738836\pi$
$884$	0	0
$885$	−2.63393e6	−0.113044
$886$	0	0
$887$	−9.17500e6	−0.391559	−0.195780	−	0.980648i	$-0.562724\pi$
$887$	−9.17500e6	−0.391559	−0.195780	+	0.980648i	$0.562724\pi$
$888$	0	0
$889$	1.32533e7	0.562430
$890$	0	0
$891$	2.07915e7	0.877386
$892$	0	0
$893$	1.48217e6	0.0621971
$894$	0	0
$895$	−7.02253e6	−0.293046
$896$	0	0
$897$	6.42429e6	0.266590
$898$	0	0
$899$	621362.	0.0256416
$900$	0	0
$901$	8.55399e6	0.351040
$902$	0	0
$903$	5.38928e6	0.219944
$904$	0	0
$905$	−8.11144e6	−0.329213
$906$	0	0
$907$	1.42674e7	0.575874	0.287937	−	0.957649i	$-0.407031\pi$
$907$	1.42674e7	0.575874	0.287937	+	0.957649i	$0.407031\pi$
$908$	0	0
$909$	−2.53633e7	−1.01811
$910$	0	0
$911$	−3.34037e7	−1.33352	−0.666759	−	0.745273i	$-0.732319\pi$
$911$	−3.34037e7	−1.33352	−0.666759	+	0.745273i	$0.732319\pi$
$912$	0	0
$913$	2.14036e7	0.849785
$914$	0	0
$915$	9.16912e6	0.362055
$916$	0	0
$917$	2.04485e6	0.0803042
$918$	0	0
$919$	−4.52117e7	−1.76588	−0.882942	−	0.469482i	$-0.844441\pi$
$919$	−4.52117e7	−1.76588	−0.882942	+	0.469482i	$0.844441\pi$
$920$	0	0
$921$	−5.96295e7	−2.31639
$922$	0	0
$923$	−8.61384e6	−0.332807
$924$	0	0
$925$	5.23098e6	0.201015
$926$	0	0
$927$	−2.13750e7	−0.816970
$928$	0	0
$929$	−517735.	−0.0196820	−0.00984098	−	0.999952i	$-0.503133\pi$
$929$	−517735.	−0.0196820	−0.00984098	+	0.999952i	$0.503133\pi$
$930$	0	0
$931$	1.20399e6	0.0455249
$932$	0	0
$933$	−2.91567e7	−1.09657
$934$	0	0
$935$	5.59455e6	0.209284
$936$	0	0
$937$	−2.58406e7	−0.961508	−0.480754	−	0.876856i	$-0.659637\pi$
$937$	−2.58406e7	−0.961508	−0.480754	+	0.876856i	$0.659637\pi$
$938$	0	0
$939$	3.09325e7	1.14486
$940$	0	0
$941$	4.58537e7	1.68811	0.844053	−	0.536259i	$-0.180163\pi$
$941$	4.58537e7	1.68811	0.844053	+	0.536259i	$0.180163\pi$
$942$	0	0
$943$	2.35798e6	0.0863496
$944$	0	0
$945$	−2.39216e6	−0.0871387
$946$	0	0
$947$	−604846.	−0.0219164	−0.0109582	−	0.999940i	$-0.503488\pi$
$947$	−604846.	−0.0219164	−0.0109582	+	0.999940i	$0.503488\pi$
$948$	0	0
$949$	7.16474e6	0.258247
$950$	0	0
$951$	5.67630e7	2.03523
$952$	0	0
$953$	3.20607e7	1.14351	0.571755	−	0.820424i	$-0.306263\pi$
$953$	3.20607e7	1.14351	0.571755	+	0.820424i	$0.306263\pi$
$954$	0	0
$955$	2.00862e7	0.712671
$956$	0	0
$957$	−1.24025e6	−0.0437754
$958$	0	0
$959$	2.76434e6	0.0970609
$960$	0	0
$961$	−2.08480e7	−0.728207
$962$	0	0
$963$	1.59264e7	0.553416
$964$	0	0
$965$	1.56767e6	0.0541920
$966$	0	0
$967$	3.19660e7	1.09931	0.549657	−	0.835391i	$-0.314758\pi$
$967$	3.19660e7	1.09931	0.549657	+	0.835391i	$0.314758\pi$
$968$	0	0
$969$	−1.29316e6	−0.0442427
$970$	0	0
$971$	5.47040e7	1.86196	0.930982	−	0.365066i	$-0.118954\pi$
$971$	5.47040e7	1.86196	0.930982	+	0.365066i	$0.118954\pi$
$972$	0	0
$973$	−6.63126e6	−0.224551
$974$	0	0
$975$	2.07488e6	0.0699008
$976$	0	0
$977$	−3.12938e7	−1.04887	−0.524436	−	0.851450i	$-0.675724\pi$
$977$	−3.12938e7	−1.04887	−0.524436	+	0.851450i	$0.675724\pi$
$978$	0	0
$979$	−2.18299e7	−0.727940
$980$	0	0
$981$	−1.21736e7	−0.403874
$982$	0	0
$983$	−1.21142e7	−0.399863	−0.199932	−	0.979810i	$-0.564072\pi$
$983$	−1.21142e7	−0.399863	−0.199932	+	0.979810i	$0.564072\pi$
$984$	0	0
$985$	9.19013e6	0.301808
$986$	0	0
$987$	1.69894e7	0.555116
$988$	0	0
$989$	−1.09120e7	−0.354744
$990$	0	0
$991$	2.58041e7	0.834650	0.417325	−	0.908757i	$-0.362968\pi$
$991$	2.58041e7	0.834650	0.417325	+	0.908757i	$0.362968\pi$
$992$	0	0
$993$	−5.77058e7	−1.85715
$994$	0	0
$995$	2.15131e7	0.688882
$996$	0	0
$997$	4.21820e7	1.34397	0.671984	−	0.740566i	$-0.265442\pi$
$997$	4.21820e7	1.34397	0.671984	+	0.740566i	$0.265442\pi$
$998$	0	0
$999$	−1.64605e7	−0.521831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.q.1.6		6
4.3	odd	2		65.6.a.d.1.4	✓	6
12.11	even	2		585.6.a.m.1.3		6
20.3	even	4		325.6.b.g.274.6		12
20.7	even	4		325.6.b.g.274.7		12
20.19	odd	2		325.6.a.g.1.3		6
52.51	odd	2		845.6.a.h.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.4	✓	6	4.3	odd	2
325.6.a.g.1.3		6	20.19	odd	2
325.6.b.g.274.6		12	20.3	even	4
325.6.b.g.274.7		12	20.7	even	4
585.6.a.m.1.3		6	12.11	even	2
845.6.a.h.1.3		6	52.51	odd	2
1040.6.a.q.1.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+19.6439 q^{3} -25.0000 q^{5} -48.6530 q^{7} +142.882 q^{9} -283.440 q^{11} +169.000 q^{13} -491.097 q^{15} +789.522 q^{17} -83.3795 q^{19} -955.734 q^{21} +1935.13 q^{23} +625.000 q^{25} -1966.71 q^{27} +222.752 q^{29} +2789.48 q^{31} -5567.86 q^{33} +1216.33 q^{35} +8369.56 q^{37} +3319.81 q^{39} +1218.51 q^{41} -5638.89 q^{43} -3572.04 q^{45} -17776.2 q^{47} -14439.9 q^{49} +15509.3 q^{51} +10834.4 q^{53} +7086.00 q^{55} -1637.90 q^{57} +5363.36 q^{59} -18670.7 q^{61} -6951.63 q^{63} -4225.00 q^{65} -13985.1 q^{67} +38013.5 q^{69} -50969.5 q^{71} +42394.9 q^{73} +12277.4 q^{75} +13790.2 q^{77} -106279. q^{79} -73354.1 q^{81} -75513.6 q^{83} -19738.1 q^{85} +4375.71 q^{87} +77017.8 q^{89} -8222.36 q^{91} +54796.2 q^{93} +2084.49 q^{95} +126702. q^{97} -40498.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) \) 6 * q - 38 * q^3 - 150 * q^5 - 220 * q^7 + 518 * q^9 + 170 * q^11 + 1014 * q^13 + 950 * q^15 + 728 * q^17 - 1218 * q^19 - 396 * q^21 - 8954 * q^23 + 3750 * q^25 - 13112 * q^27 + 8364 * q^29 - 2862 * q^31 - 3772 * q^33 + 5500 * q^35 + 13840 * q^37 - 6422 * q^39 + 2248 * q^41 - 3822 * q^43 - 12950 * q^45 + 10860 * q^47 + 85902 * q^49 + 55028 * q^51 + 6584 * q^53 - 4250 * q^55 + 11692 * q^57 + 28874 * q^59 + 972 * q^61 + 125236 * q^63 - 25350 * q^65 - 60040 * q^67 + 153088 * q^69 + 74174 * q^71 + 48960 * q^73 - 23750 * q^75 + 231588 * q^77 - 214508 * q^79 + 116006 * q^81 - 123260 * q^83 - 18200 * q^85 - 228144 * q^87 + 94028 * q^89 - 37180 * q^91 + 192892 * q^93 + 30450 * q^95 + 246284 * q^97 + 32270 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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