LMFDB - Embedded newform 585.6.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,6,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("585.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	585.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:	$93.8245345906$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.34530$ of defining polynomial
Character	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.34530 q^{2} -30.1902 q^{4} +25.0000 q^{5} +48.6530 q^{7} +83.6645 q^{8} -33.6325 q^{10} -283.440 q^{11} +169.000 q^{13} -65.4530 q^{14} +853.531 q^{16} -789.522 q^{17} +83.3795 q^{19} -754.754 q^{20} +381.312 q^{22} +1935.13 q^{23} +625.000 q^{25} -227.356 q^{26} -1468.84 q^{28} -222.752 q^{29} -2789.48 q^{31} -3825.52 q^{32} +1062.15 q^{34} +1216.33 q^{35} +8369.56 q^{37} -112.170 q^{38} +2091.61 q^{40} -1218.51 q^{41} +5638.89 q^{43} +8557.10 q^{44} -2603.34 q^{46} -17776.2 q^{47} -14439.9 q^{49} -840.813 q^{50} -5102.14 q^{52} -10834.4 q^{53} -7086.00 q^{55} +4070.53 q^{56} +299.668 q^{58} +5363.36 q^{59} -18670.7 q^{61} +3752.69 q^{62} -22166.5 q^{64} +4225.00 q^{65} +13985.1 q^{67} +23835.8 q^{68} -1636.32 q^{70} -50969.5 q^{71} +42394.9 q^{73} -11259.6 q^{74} -2517.24 q^{76} -13790.2 q^{77} +106279. q^{79} +21338.3 q^{80} +1639.26 q^{82} -75513.6 q^{83} -19738.1 q^{85} -7586.01 q^{86} -23713.8 q^{88} -77017.8 q^{89} +8222.36 q^{91} -58422.0 q^{92} +23914.4 q^{94} +2084.49 q^{95} +126702. q^{97} +19426.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) $ 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

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 $

$2$	−1.34530	−0.237818	−0.118909	−	0.992905i	$-0.537940\pi$
$2$	−1.34530	−0.237818	−0.118909	+	0.992905i	$0.537940\pi$
$3$	0	0
$3$	0	0
$4$	−30.1902	−0.943443
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	48.6530	0.375288	0.187644	−	0.982237i	$-0.439915\pi$
$7$	48.6530	0.375288	0.187644	+	0.982237i	$0.439915\pi$
$8$	83.6645	0.462185
$9$	0	0
$10$	−33.6325	−0.106355
$11$	−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
$14$	−65.4530	−0.0892502
$15$	0	0
$16$	853.531	0.833527
$17$	−789.522	−0.662586	−0.331293	−	0.943528i	$-0.607485\pi$
$17$	−789.522	−0.662586	−0.331293	+	0.943528i	$0.607485\pi$
$18$	0	0
$19$	83.3795	0.0529877	0.0264939	−	0.999649i	$-0.491566\pi$
$19$	83.3795	0.0529877	0.0264939	+	0.999649i	$0.491566\pi$
$20$	−754.754	−0.421920
$21$	0	0
$22$	381.312	0.167967
$23$	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
$26$	−227.356	−0.0659588
$27$	0	0
$28$	−1468.84	−0.354063
$29$	−222.752	−0.0491843	−0.0245922	−	0.999698i	$-0.507829\pi$
$29$	−222.752	−0.0491843	−0.0245922	+	0.999698i	$0.507829\pi$
$30$	0	0
$31$	−2789.48	−0.521338	−0.260669	−	0.965428i	$-0.583943\pi$
$31$	−2789.48	−0.521338	−0.260669	+	0.965428i	$0.583943\pi$
$32$	−3825.52	−0.660413
$33$	0	0
$34$	1062.15	0.157575
$35$	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	$0.332400\pi$
$38$	−112.170	−0.0126014
$39$	0	0
$40$	2091.61	0.206696
$41$	−1218.51	−0.113206	−0.0566029	−	0.998397i	$-0.518027\pi$
$41$	−1218.51	−0.113206	−0.0566029	+	0.998397i	$0.518027\pi$
$42$	0	0
$43$	5638.89	0.465075	0.232537	−	0.972587i	$-0.425297\pi$
$43$	5638.89	0.465075	0.232537	+	0.972587i	$0.425297\pi$
$44$	8557.10	0.666338
$45$	0	0
$46$	−2603.34	−0.181399
$47$	−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
$50$	−840.813	−0.0475636
$51$	0	0
$52$	−5102.14	−0.261664
$53$	−10834.4	−0.529803	−0.264902	−	0.964275i	$-0.585339\pi$
$53$	−10834.4	−0.529803	−0.264902	+	0.964275i	$0.585339\pi$
$54$	0	0
$55$	−7086.00	−0.315860
$56$	4070.53	0.173453
$57$	0	0
$58$	299.668	0.0116969
$59$	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	$0.604094\pi$
$62$	3752.69	0.123983
$63$	0	0
$64$	−22166.5	−0.676469
$65$	4225.00	0.124035
$66$	0	0
$67$	13985.1	0.380607	0.190304	−	0.981725i	$-0.439053\pi$
$67$	13985.1	0.380607	0.190304	+	0.981725i	$0.439053\pi$
$68$	23835.8	0.625112
$69$	0	0
$70$	−1636.32	−0.0399139
$71$	−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	$0.345853\pi$
$74$	−11259.6	−0.239025
$75$	0	0
$76$	−2517.24	−0.0499909
$77$	−13790.2	−0.265060
$78$	0	0
$79$	106279.	1.91593	0.957964	−	0.286888i	$-0.0926208\pi$
$79$	106279.	1.91593	0.957964	+	0.286888i	$0.0926208\pi$
$80$	21338.3	0.372765
$81$	0	0
$82$	1639.26	0.0269224
$83$	−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
$86$	−7586.01	−0.110603
$87$	0	0
$88$	−23713.8	−0.326434
$89$	−77017.8	−1.03066	−0.515331	−	0.856991i	$-0.672331\pi$
$89$	−77017.8	−1.03066	−0.515331	+	0.856991i	$0.672331\pi$
$90$	0	0
$91$	8222.36	0.104086
$92$	−58422.0	−0.719627
$93$	0	0
$94$	23914.4	0.279151
$95$	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	$0.260397\pi$
$98$	19426.0	0.204323
$99$	0	0
$100$	−18868.9	−0.188689
$101$	177513.	1.73151	0.865757	−	0.500464i	$-0.166837\pi$
$101$	177513.	1.73151	0.865757	+	0.500464i	$0.166837\pi$
$102$	0	0
$103$	149599.	1.38943	0.694714	−	0.719286i	$-0.255531\pi$
$103$	149599.	1.38943	0.694714	+	0.719286i	$0.255531\pi$
$104$	14139.3	0.128187
$105$	0	0
$106$	14575.5	0.125997
$107$	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	$0.611591\pi$
$110$	9532.80	0.0751171
$111$	0	0
$112$	41526.9	0.312813
$113$	204137.	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$113$	204137.	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$114$	0	0
$115$	48378.4	0.341120
$116$	6724.92	0.0464026
$117$	0	0
$118$	−7215.34	−0.0477037
$119$	−38412.7	−0.248661
$120$	0	0
$121$	−80712.8	−0.501163
$122$	25117.7	0.152785
$123$	0	0
$124$	84214.9	0.491852
$125$	15625.0	0.0894427
$126$	0	0
$127$	272404.	1.49866	0.749331	−	0.662196i	$-0.230375\pi$
$127$	272404.	1.49866	0.749331	+	0.662196i	$0.230375\pi$
$128$	152237.	0.821289
$129$	0	0
$130$	−5683.90	−0.0294977
$131$	−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
$134$	−18814.1	−0.0905152
$135$	0	0
$136$	−66055.0	−0.306237
$137$	56817.3	0.258630	0.129315	−	0.991604i	$-0.458722\pi$
$137$	56817.3	0.258630	0.129315	+	0.991604i	$0.458722\pi$
$138$	0	0
$139$	−136297.	−0.598342	−0.299171	−	0.954200i	$-0.596710\pi$
$139$	−136297.	−0.598342	−0.299171	+	0.954200i	$0.596710\pi$
$140$	−36721.1	−0.158342
$141$	0	0
$142$	68569.3	0.285370
$143$	−47901.3	−0.195888
$144$	0	0
$145$	−5568.80	−0.0219959
$146$	−57033.9	−0.221437
$147$	0	0
$148$	−252678.	−0.948230
$149$	398843.	1.47176	0.735879	−	0.677113i	$-0.236769\pi$
$149$	398843.	1.47176	0.735879	+	0.677113i	$0.236769\pi$
$150$	0	0
$151$	−16131.0	−0.0575730	−0.0287865	−	0.999586i	$-0.509164\pi$
$151$	−16131.0	−0.0575730	−0.0287865	+	0.999586i	$0.509164\pi$
$152$	6975.90	0.0244901
$153$	0	0
$154$	18552.0	0.0630360
$155$	−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	$0.495107\pi$
$158$	−142977.	−0.455642
$159$	0	0
$160$	−95638.0	−0.295346
$161$	94150.2	0.286257
$162$	0	0
$163$	439257.	1.29494	0.647469	−	0.762091i	$-0.275827\pi$
$163$	439257.	1.29494	0.647469	+	0.762091i	$0.275827\pi$
$164$	36787.0	0.106803
$165$	0	0
$166$	101588.	0.286137
$167$	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
$170$	26553.6	0.0704696
$171$	0	0
$172$	−170239.	−0.438771
$173$	−121696.	−0.309144	−0.154572	−	0.987982i	$-0.549400\pi$
$173$	−121696.	−0.309144	−0.154572	+	0.987982i	$0.549400\pi$
$174$	0	0
$175$	30408.1	0.0750576
$176$	−241925.	−0.588706
$177$	0	0
$178$	103612.	0.245110
$179$	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	$0.380018\pi$
$182$	−11061.6	−0.0247536
$183$	0	0
$184$	161902.	0.352540
$185$	209239.	0.449483
$186$	0	0
$187$	223782.	0.467974
$188$	536668.	1.10742
$189$	0	0
$190$	−2804.26	−0.00563553
$191$	−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	$0.519298\pi$
$194$	−170453.	−0.325162
$195$	0	0
$196$	435942.	0.810567
$197$	367605.	0.674864	0.337432	−	0.941350i	$-0.390442\pi$
$197$	367605.	0.674864	0.337432	+	0.941350i	$0.390442\pi$
$198$	0	0
$199$	860523.	1.54039	0.770193	−	0.637810i	$-0.220160\pi$
$199$	860523.	1.54039	0.770193	+	0.637810i	$0.220160\pi$
$200$	52290.3	0.0924371
$201$	0	0
$202$	−238808.	−0.411785
$203$	−10837.6	−0.0184583
$204$	0	0
$205$	−30462.7	−0.0506272
$206$	−201256.	−0.330431
$207$	0	0
$208$	144247.	0.231179
$209$	−23633.1	−0.0374244
$210$	0	0
$211$	−457688.	−0.707723	−0.353861	−	0.935298i	$-0.615132\pi$
$211$	−457688.	−0.707723	−0.353861	+	0.935298i	$0.615132\pi$
$212$	327092.	0.499839
$213$	0	0
$214$	−149955.	−0.223834
$215$	140972.	0.207988
$216$	0	0
$217$	−135717.	−0.195652
$218$	114620.	0.163350
$219$	0	0
$220$	213927.	0.297996
$221$	−133429.	−0.183768
$222$	0	0
$223$	−925567.	−1.24637	−0.623183	−	0.782076i	$-0.714161\pi$
$223$	−925567.	−1.24637	−0.623183	+	0.782076i	$0.714161\pi$
$224$	−186123.	−0.247845
$225$	0	0
$226$	−274625.	−0.357659
$227$	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	$0.258523\pi$
$230$	−65083.5	−0.0811243
$231$	0	0
$232$	−18636.4	−0.0227323
$233$	−141596.	−0.170868	−0.0854341	−	0.996344i	$-0.527228\pi$
$233$	−141596.	−0.170868	−0.0854341	+	0.996344i	$0.527228\pi$
$234$	0	0
$235$	−444406.	−0.524941
$236$	−161921.	−0.189244
$237$	0	0
$238$	51676.6	0.0591359
$239$	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	$0.218041\pi$
$242$	108583.	0.119186
$243$	0	0
$244$	563671.	0.606110
$245$	−360997.	−0.384228
$246$	0	0
$247$	14091.1	0.0146961
$248$	−233380.	−0.240955
$249$	0	0
$250$	−21020.3	−0.0212711
$251$	−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
$254$	−366465.	−0.356408
$255$	0	0
$256$	504524.	0.481152
$257$	−1.21697e6	−1.14934	−0.574670	−	0.818385i	$-0.694870\pi$
$257$	−1.21697e6	−1.14934	−0.574670	+	0.818385i	$0.694870\pi$
$258$	0	0
$259$	407205.	0.377193
$260$	−127553.	−0.117020
$261$	0	0
$262$	56542.0	0.0508883
$263$	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
$266$	−5457.43	−0.00472916
$267$	0	0
$268$	−422211.	−0.359081
$269$	1.68629e6	1.42086	0.710432	−	0.703766i	$-0.248500\pi$
$269$	1.68629e6	1.42086	0.710432	+	0.703766i	$0.248500\pi$
$270$	0	0
$271$	1.71354e6	1.41733	0.708664	−	0.705546i	$-0.249298\pi$
$271$	1.71354e6	1.41733	0.708664	+	0.705546i	$0.249298\pi$
$272$	−673882.	−0.552283
$273$	0	0
$274$	−76436.4	−0.0615069
$275$	−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	$0.297664\pi$
$278$	183360.	0.142296
$279$	0	0
$280$	101763.	0.0775704
$281$	−2.01362e6	−1.52129	−0.760644	−	0.649169i	$-0.775117\pi$
$281$	−2.01362e6	−1.52129	−0.760644	+	0.649169i	$0.775117\pi$
$282$	0	0
$283$	451062.	0.334788	0.167394	−	0.985890i	$-0.446465\pi$
$283$	451062.	0.334788	0.167394	+	0.985890i	$0.446465\pi$
$284$	1.53878e6	1.13209
$285$	0	0
$286$	64441.7	0.0465856
$287$	−59284.1	−0.0424848
$288$	0	0
$289$	−796511.	−0.560980
$290$	7491.71	0.00523102
$291$	0	0
$292$	−1.27991e6	−0.878460
$293$	1.60140e6	1.08976	0.544879	−	0.838514i	$-0.316575\pi$
$293$	1.60140e6	1.08976	0.544879	+	0.838514i	$0.316575\pi$
$294$	0	0
$295$	134084.	0.0897062
$296$	700235.	0.464531
$297$	0	0
$298$	−536564.	−0.350010
$299$	327038.	0.211553
$300$	0	0
$301$	274349.	0.174537
$302$	21701.0	0.0136919
$303$	0	0
$304$	71167.0	0.0441667
$305$	−466767.	−0.287310
$306$	0	0
$307$	3.03553e6	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$307$	3.03553e6	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$308$	416329.	0.250069
$309$	0	0
$310$	93817.2	0.0554470
$311$	−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	$0.349906\pi$
$314$	−12774.7	−0.00731184
$315$	0	0
$316$	−3.20858e6	−1.80757
$317$	−2.88960e6	−1.61507	−0.807533	−	0.589822i	$-0.799198\pi$
$317$	−2.88960e6	−1.61507	−0.807533	+	0.589822i	$0.799198\pi$
$318$	0	0
$319$	63136.8	0.0347381
$320$	−554163.	−0.302526
$321$	0	0
$322$	−126660.	−0.0680771
$323$	−65830.0	−0.0351089
$324$	0	0
$325$	105625.	0.0554700
$326$	−590932.	−0.307960
$327$	0	0
$328$	−101946.	−0.0523221
$329$	−864868.	−0.440514
$330$	0	0
$331$	2.93760e6	1.47374	0.736872	−	0.676032i	$-0.236302\pi$
$331$	2.93760e6	1.47374	0.736872	+	0.676032i	$0.236302\pi$
$332$	2.27977e6	1.13513
$333$	0	0
$334$	−314016.	−0.154023
$335$	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	$0.564770\pi$
$338$	−38423.1	−0.0182937
$339$	0	0
$340$	595895.	0.279558
$341$	790650.	0.368212
$342$	0	0
$343$	−1.52026e6	−0.697720
$344$	471775.	0.214951
$345$	0	0
$346$	163718.	0.0735200
$347$	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	$0.667148\pi$
$350$	−40908.1	−0.0178500
$351$	0	0
$352$	1.08430e6	0.466439
$353$	−7769.01	−0.00331840	−0.00165920	−	0.999999i	$-0.500528\pi$
$353$	−7769.01	−0.00331840	−0.00165920	+	0.999999i	$0.500528\pi$
$354$	0	0
$355$	−1.27424e6	−0.536635
$356$	2.32518e6	0.972371
$357$	0	0
$358$	−377897.	−0.155835
$359$	−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
$362$	−436493.	−0.175068
$363$	0	0
$364$	−248234.	−0.0981994
$365$	1.05987e6	0.416410
$366$	0	0
$367$	−2.32781e6	−0.902156	−0.451078	−	0.892484i	$-0.648960\pi$
$367$	−2.32781e6	−0.902156	−0.451078	+	0.892484i	$0.648960\pi$
$368$	1.65170e6	0.635786
$369$	0	0
$370$	−281489.	−0.106895
$371$	−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	$0.596750\pi$
$374$	−301054.	−0.111292
$375$	0	0
$376$	−1.48724e6	−0.542515
$377$	−37645.1	−0.0136413
$378$	0	0
$379$	−473484.	−0.169320	−0.0846599	−	0.996410i	$-0.526980\pi$
$379$	−473484.	−0.169320	−0.0846599	+	0.996410i	$0.526980\pi$
$380$	−62931.0	−0.0223566
$381$	0	0
$382$	1.08088e6	0.378982
$383$	−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
$386$	84359.4	0.0288181
$387$	0	0
$388$	−3.82517e6	−1.28994
$389$	901091.	0.301922	0.150961	−	0.988540i	$-0.451763\pi$
$389$	901091.	0.301922	0.150961	+	0.988540i	$0.451763\pi$
$390$	0	0
$391$	−1.52783e6	−0.505398
$392$	−1.20811e6	−0.397091
$393$	0	0
$394$	−494539.	−0.160495
$395$	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	$0.680090\pi$
$398$	−1.15766e6	−0.366331
$399$	0	0
$400$	533457.	0.166705
$401$	−1.49612e6	−0.464629	−0.232315	−	0.972641i	$-0.574630\pi$
$401$	−1.49612e6	−0.464629	−0.232315	+	0.972641i	$0.574630\pi$
$402$	0	0
$403$	−471422.	−0.144593
$404$	−5.35914e6	−1.63358
$405$	0	0
$406$	14579.8	0.00438971
$407$	−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	$0.414832\pi$
$410$	40981.5	0.0120401
$411$	0	0
$412$	−4.51642e6	−1.31085
$413$	260944.	0.0752787
$414$	0	0
$415$	−1.88784e6	−0.538077
$416$	−646513.	−0.183166
$417$	0	0
$418$	31793.6	0.00890018
$419$	−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	$0.835817\pi$
$422$	615728.	0.168309
$423$	0	0
$424$	−906453.	−0.244867
$425$	−493452.	−0.132517
$426$	0	0
$427$	−908386.	−0.241102
$428$	−3.36517e6	−0.887967
$429$	0	0
$430$	−189650.	−0.0494632
$431$	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	$0.433329\pi$
$434$	182580.	0.0465295
$435$	0	0
$436$	2.57222e6	0.648024
$437$	161351.	0.0404172
$438$	0	0
$439$	6.21478e6	1.53909	0.769546	−	0.638592i	$-0.220483\pi$
$439$	6.21478e6	1.53909	0.769546	+	0.638592i	$0.220483\pi$
$440$	−592846.	−0.145986
$441$	0	0
$442$	179503.	0.0437034
$443$	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
$446$	1.24517e6	0.296408
$447$	0	0
$448$	−1.07847e6	−0.253871
$449$	7.08559e6	1.65867	0.829336	−	0.558750i	$-0.188719\pi$
$449$	7.08559e6	1.65867	0.829336	+	0.558750i	$0.188719\pi$
$450$	0	0
$451$	345374.	0.0799555
$452$	−6.16292e6	−1.41886
$453$	0	0
$454$	−2.07510e6	−0.472498
$455$	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	$0.533189\pi$
$458$	−1.46885e6	−0.327200
$459$	0	0
$460$	−1.46055e6	−0.321827
$461$	−7.91310e6	−1.73418	−0.867091	−	0.498150i	$-0.834013\pi$
$461$	−7.91310e6	−1.73418	−0.867091	+	0.498150i	$0.834013\pi$
$462$	0	0
$463$	−1.60654e6	−0.348289	−0.174144	−	0.984720i	$-0.555716\pi$
$463$	−1.60654e6	−0.348289	−0.174144	+	0.984720i	$0.555716\pi$
$464$	−190126.	−0.0409964
$465$	0	0
$466$	190489.	0.0406355
$467$	−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
$470$	597860.	0.124840
$471$	0	0
$472$	448723.	0.0927093
$473$	−1.59829e6	−0.328475
$474$	0	0
$475$	52112.2	0.0105975
$476$	1.15968e6	0.234597
$477$	0	0
$478$	−1.06504e6	−0.213205
$479$	−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
$482$	−1.87875e6	−0.368342
$483$	0	0
$484$	2.43673e6	0.472819
$485$	3.16756e6	0.611464
$486$	0	0
$487$	−3.52518e6	−0.673534	−0.336767	−	0.941588i	$-0.609333\pi$
$487$	−3.52518e6	−0.673534	−0.336767	+	0.941588i	$0.609333\pi$
$488$	−1.56207e6	−0.296929
$489$	0	0
$490$	485650.	0.0913761
$491$	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
$494$	−18956.8	−0.00349501
$495$	0	0
$496$	−2.38091e6	−0.434549
$497$	−2.47982e6	−0.450328
$498$	0	0
$499$	311968.	0.0560866	0.0280433	−	0.999607i	$-0.491072\pi$
$499$	311968.	0.0560866	0.0280433	+	0.999607i	$0.491072\pi$
$500$	−471721.	−0.0843841
$501$	0	0
$502$	1.24785e6	0.221006
$503$	−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
$506$	737890.	0.128120
$507$	0	0
$508$	−8.22391e6	−1.41390
$509$	6.28149e6	1.07465	0.537327	−	0.843374i	$-0.319434\pi$
$509$	6.28149e6	1.07465	0.537327	+	0.843374i	$0.319434\pi$
$510$	0	0
$511$	2.06264e6	0.349439
$512$	−5.55033e6	−0.935716
$513$	0	0
$514$	1.63720e6	0.273334
$515$	3.73998e6	0.621371
$516$	0	0
$517$	5.03850e6	0.829038
$518$	−547813.	−0.0897031
$519$	0	0
$520$	353482.	0.0573270
$521$	7.39635e6	1.19378	0.596888	−	0.802324i	$-0.296404\pi$
$521$	7.39635e6	1.19378	0.596888	+	0.802324i	$0.296404\pi$
$522$	0	0
$523$	6.26869e6	1.00213	0.501063	−	0.865411i	$-0.332943\pi$
$523$	6.26869e6	1.00213	0.501063	+	0.865411i	$0.332943\pi$
$524$	1.26887e6	0.201878
$525$	0	0
$526$	−254224.	−0.0400638
$527$	2.20236e6	0.345431
$528$	0	0
$529$	−2.69160e6	−0.418187
$530$	364388.	0.0563474
$531$	0	0
$532$	−122471.	−0.0187610
$533$	−205928.	−0.0313977
$534$	0	0
$535$	2.78664e6	0.420917
$536$	1.17005e6	0.175911
$537$	0	0
$538$	−2.26857e6	−0.337907
$539$	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	$0.566401\pi$
$542$	−2.30522e6	−0.337066
$543$	0	0
$544$	3.02033e6	0.437580
$545$	−2.13001e6	−0.307178
$546$	0	0
$547$	−9.17262e6	−1.31077	−0.655383	−	0.755297i	$-0.727493\pi$
$547$	−9.17262e6	−1.31077	−0.655383	+	0.755297i	$0.727493\pi$
$548$	−1.71532e6	−0.244003
$549$	0	0
$550$	238320.	0.0335934
$551$	−18572.9	−0.00260616
$552$	0	0
$553$	5.17079e6	0.719025
$554$	−2.03995e6	−0.282388
$555$	0	0
$556$	4.11483e6	0.564501
$557$	4.27403e6	0.583713	0.291856	−	0.956462i	$-0.405727\pi$
$557$	4.27403e6	0.583713	0.291856	+	0.956462i	$0.405727\pi$
$558$	0	0
$559$	952973.	0.128989
$560$	1.03817e6	0.139894
$561$	0	0
$562$	2.70892e6	0.361789
$563$	−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
$566$	−606813.	−0.0796185
$567$	0	0
$568$	−4.26433e6	−0.554601
$569$	4.10069e6	0.530977	0.265489	−	0.964114i	$-0.414467\pi$
$569$	4.10069e6	0.530977	0.265489	+	0.964114i	$0.414467\pi$
$570$	0	0
$571$	−1.03048e7	−1.32267	−0.661334	−	0.750092i	$-0.730009\pi$
$571$	−1.03048e7	−1.32267	−0.661334	+	0.750092i	$0.730009\pi$
$572$	1.44615e6	0.184809
$573$	0	0
$574$	79755.0	0.0101036
$575$	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	$0.456602\pi$
$578$	1.07155e6	0.133411
$579$	0	0
$580$	168123.	0.0207519
$581$	−3.67396e6	−0.451538
$582$	0	0
$583$	3.07090e6	0.374191
$584$	3.54695e6	0.430351
$585$	0	0
$586$	−2.15436e6	−0.259164
$587$	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
$590$	−180383.	−0.0213337
$591$	0	0
$592$	7.14368e6	0.837757
$593$	5.51737e6	0.644310	0.322155	−	0.946687i	$-0.395593\pi$
$593$	5.51737e6	0.644310	0.322155	+	0.946687i	$0.395593\pi$
$594$	0	0
$595$	−960317.	−0.111204
$596$	−1.20411e7	−1.38852
$597$	0	0
$598$	−439964.	−0.0503112
$599$	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	$0.260540\pi$
$602$	−369082.	−0.0415080
$603$	0	0
$604$	486997.	0.0543168
$605$	−2.01782e6	−0.224127
$606$	0	0
$607$	−1.29407e7	−1.42556	−0.712779	−	0.701388i	$-0.752564\pi$
$607$	−1.29407e7	−1.42556	−0.712779	+	0.701388i	$0.752564\pi$
$608$	−318970.	−0.0349938
$609$	0	0
$610$	627942.	0.0683274
$611$	−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	$0.460321\pi$
$614$	−4.08370e6	−0.437152
$615$	0	0
$616$	−1.15375e6	−0.122507
$617$	−8.53024e6	−0.902086	−0.451043	−	0.892502i	$-0.648948\pi$
$617$	−8.53024e6	−0.902086	−0.451043	+	0.892502i	$0.648948\pi$
$618$	0	0
$619$	8.89321e6	0.932893	0.466447	−	0.884549i	$-0.345534\pi$
$619$	8.89321e6	0.932893	0.466447	+	0.884549i	$0.345534\pi$
$620$	2.10537e6	0.219963
$621$	0	0
$622$	1.99679e6	0.206945
$623$	−3.74715e6	−0.386795
$624$	0	0
$625$	390625.	0.0400000
$626$	−2.11840e6	−0.216059
$627$	0	0
$628$	−286680.	−0.0290067
$629$	−6.60796e6	−0.665948
$630$	0	0
$631$	1.01842e7	1.01824	0.509122	−	0.860695i	$-0.329970\pi$
$631$	1.01842e7	1.01824	0.509122	+	0.860695i	$0.329970\pi$
$632$	8.89177e6	0.885514
$633$	0	0
$634$	3.88739e6	0.384091
$635$	6.81009e6	0.670222
$636$	0	0
$637$	−2.44034e6	−0.238288
$638$	−84938.0	−0.00826133
$639$	0	0
$640$	3.80593e6	0.367292
$641$	1.30240e7	1.25198	0.625991	−	0.779830i	$-0.284694\pi$
$641$	1.30240e7	1.25198	0.625991	+	0.779830i	$0.284694\pi$
$642$	0	0
$643$	−4.52103e6	−0.431231	−0.215616	−	0.976478i	$-0.569176\pi$
$643$	−4.52103e6	−0.431231	−0.215616	+	0.976478i	$0.569176\pi$
$644$	−2.84241e6	−0.270067
$645$	0	0
$646$	88561.1	0.00834952
$647$	−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
$650$	−142097.	−0.0131918
$651$	0	0
$652$	−1.32612e7	−1.22170
$653$	−1.83786e7	−1.68667	−0.843334	−	0.537389i	$-0.819411\pi$
$653$	−1.83786e7	−1.68667	−0.843334	+	0.537389i	$0.819411\pi$
$654$	0	0
$655$	−1.05073e6	−0.0956949
$656$	−1.04004e6	−0.0943601
$657$	0	0
$658$	1.16351e6	0.104762
$659$	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	$0.367907\pi$
$662$	−3.95195e6	−0.350483
$663$	0	0
$664$	−6.31780e6	−0.556091
$665$	101417.	0.00889314
$666$	0	0
$667$	−431055.	−0.0375161
$668$	−7.04689e6	−0.611021
$669$	0	0
$670$	−470353.	−0.0404796
$671$	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	$0.245003\pi$
$674$	1.13356e6	0.0961162
$675$	0	0
$676$	−862261.	−0.0725725
$677$	−7.96769e6	−0.668129	−0.334065	−	0.942550i	$-0.608420\pi$
$677$	−7.96769e6	−0.668129	−0.334065	+	0.942550i	$0.608420\pi$
$678$	0	0
$679$	6.16446e6	0.513122
$680$	−1.65137e6	−0.136954
$681$	0	0
$682$	−1.06366e6	−0.0875674
$683$	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
$686$	2.04520e6	0.165930
$687$	0	0
$688$	4.81297e6	0.387652
$689$	−1.83101e6	−0.146941
$690$	0	0
$691$	−1.37987e7	−1.09936	−0.549682	−	0.835374i	$-0.685251\pi$
$691$	−1.37987e7	−1.09936	−0.549682	+	0.835374i	$0.685251\pi$
$692$	3.67402e6	0.291660
$693$	0	0
$694$	−968344.	−0.0763187
$695$	−3.40743e6	−0.267587
$696$	0	0
$697$	962040.	0.0750086
$698$	3.06915e6	0.238440
$699$	0	0
$700$	−918027.	−0.0708126
$701$	1.54955e6	0.119099	0.0595497	−	0.998225i	$-0.481034\pi$
$701$	1.54955e6	0.119099	0.0595497	+	0.998225i	$0.481034\pi$
$702$	0	0
$703$	697849.	0.0532566
$704$	6.28288e6	0.477779
$705$	0	0
$706$	10451.7	0.000789175 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	$0.679920\pi$
$710$	1.71423e6	0.127621
$711$	0	0
$712$	−6.44366e6	−0.476357
$713$	−5.39802e6	−0.397659
$714$	0	0
$715$	−1.19753e6	−0.0876037
$716$	−8.48046e6	−0.618211
$717$	0	0
$718$	415893.	0.0301072
$719$	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
$722$	3.32175e6	0.237150
$723$	0	0
$724$	−9.79543e6	−0.694508
$725$	−139220.	−0.00983686
$726$	0	0
$727$	−1.61824e6	−0.113555	−0.0567776	−	0.998387i	$-0.518083\pi$
$727$	−1.61824e6	−0.113555	−0.0567776	+	0.998387i	$0.518083\pi$
$728$	687920.	0.0481071
$729$	0	0
$730$	−1.42585e6	−0.0990298
$731$	−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	$0.681870\pi$
$734$	3.13160e6	0.214549
$735$	0	0
$736$	−7.40290e6	−0.503741
$737$	−3.96392e6	−0.268817
$738$	0	0
$739$	−1.62586e6	−0.109515	−0.0547574	−	0.998500i	$-0.517439\pi$
$739$	−1.62586e6	−0.109515	−0.0547574	+	0.998500i	$0.517439\pi$
$740$	−6.31696e6	−0.424062
$741$	0	0
$742$	709142.	0.0472850
$743$	−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
$746$	2.16378e6	0.142353
$747$	0	0
$748$	−6.75602e6	−0.441506
$749$	5.42314e6	0.353221
$750$	0	0
$751$	7.16867e6	0.463809	0.231904	−	0.972739i	$-0.425504\pi$
$751$	7.16867e6	0.463809	0.231904	+	0.972739i	$0.425504\pi$
$752$	−1.51726e7	−0.978396
$753$	0	0
$754$	50644.0	0.00324414
$755$	−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	$0.177216\pi$
$758$	636979.	0.0402673
$759$	0	0
$760$	174397.	0.0109523
$761$	708017.	0.0443182	0.0221591	−	0.999754i	$-0.492946\pi$
$761$	708017.	0.0443182	0.0221591	+	0.999754i	$0.492946\pi$
$762$	0	0
$763$	−4.14526e6	−0.257775
$764$	2.42562e7	1.50345
$765$	0	0
$766$	488832.	0.0301014
$767$	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	$0.324269\pi$
$770$	463799.	0.0281905
$771$	0	0
$772$	1.89313e6	0.114324
$773$	−7.85325e6	−0.472716	−0.236358	−	0.971666i	$-0.575954\pi$
$773$	−7.85325e6	−0.472716	−0.236358	+	0.971666i	$0.575954\pi$
$774$	0	0
$775$	−1.74343e6	−0.104268
$776$	1.06005e7	0.631934
$777$	0	0
$778$	−1.21224e6	−0.0718024
$779$	−101599.	−0.00599852
$780$	0	0
$781$	1.44468e7	0.847507
$782$	2.05539e6	0.120193
$783$	0	0
$784$	−1.23249e7	−0.716132
$785$	237395.	0.0137498
$786$	0	0
$787$	2.05200e7	1.18097	0.590487	−	0.807047i	$-0.298936\pi$
$787$	2.05200e7	1.18097	0.590487	+	0.807047i	$0.298936\pi$
$788$	−1.10981e7	−0.636695
$789$	0	0
$790$	−3.57443e6	−0.203769
$791$	9.93187e6	0.564404
$792$	0	0
$793$	−3.15535e6	−0.178182
$794$	4.52942e6	0.254971
$795$	0	0
$796$	−2.59793e7	−1.45327
$797$	2.57349e7	1.43508	0.717540	−	0.696517i	$-0.245268\pi$
$797$	2.57349e7	1.43508	0.717540	+	0.696517i	$0.245268\pi$
$798$	0	0
$799$	1.40347e7	0.777745
$800$	−2.39095e6	−0.132083
$801$	0	0
$802$	2.01274e6	0.110497
$803$	−1.20164e7	−0.657636
$804$	0	0
$805$	2.35375e6	0.128018
$806$	634205.	0.0343868
$807$	0	0
$808$	1.48515e7	0.800280
$809$	1.45330e7	0.780699	0.390350	−	0.920667i	$-0.372354\pi$
$809$	1.45330e7	0.780699	0.390350	+	0.920667i	$0.372354\pi$
$810$	0	0
$811$	1.60281e6	0.0855717	0.0427858	−	0.999084i	$-0.486377\pi$
$811$	1.60281e6	0.0855717	0.0427858	+	0.999084i	$0.486377\pi$
$812$	327188.	0.0174143
$813$	0	0
$814$	3.19141e6	0.168819
$815$	1.09814e7	0.579114
$816$	0	0
$817$	470168.	0.0246432
$818$	−2.40652e6	−0.125750
$819$	0	0
$820$	919674.	0.0477639
$821$	−1.22945e6	−0.0636579	−0.0318289	−	0.999493i	$-0.510133\pi$
$821$	−1.22945e6	−0.0636579	−0.0318289	+	0.999493i	$0.510133\pi$
$822$	0	0
$823$	−260323.	−0.0133972	−0.00669859	−	0.999978i	$-0.502132\pi$
$823$	−260323.	−0.0133972	−0.00669859	+	0.999978i	$0.502132\pi$
$824$	1.25161e7	0.642173
$825$	0	0
$826$	−351048.	−0.0179026
$827$	−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	$0.151035\pi$
$830$	2.53971e6	0.127964
$831$	0	0
$832$	−3.74614e6	−0.187619
$833$	1.14006e7	0.569267
$834$	0	0
$835$	5.83542e6	0.289638
$836$	713486.	0.0353077
$837$	0	0
$838$	4.11881e6	0.202610
$839$	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
$842$	8.51186e6	0.413756
$843$	0	0
$844$	1.38177e7	0.667696
$845$	714025.	0.0344010
$846$	0	0
$847$	−3.92692e6	−0.188081
$848$	−9.24749e6	−0.441605
$849$	0	0
$850$	663841.	0.0315149
$851$	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	$0.628951\pi$
$854$	1.22205e6	0.0573383
$855$	0	0
$856$	9.32571e6	0.435008
$857$	2.15120e6	0.100053	0.0500264	−	0.998748i	$-0.484069\pi$
$857$	2.15120e6	0.100053	0.0500264	+	0.998748i	$0.484069\pi$
$858$	0	0
$859$	1.66088e7	0.767991	0.383996	−	0.923335i	$-0.374548\pi$
$859$	1.66088e7	0.767991	0.383996	+	0.923335i	$0.374548\pi$
$860$	−4.25598e6	−0.196225
$861$	0	0
$862$	−3.61612e6	−0.165758
$863$	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
$866$	−2.18261e6	−0.0988965
$867$	0	0
$868$	4.09731e6	0.184586
$869$	−3.01237e7	−1.35319
$870$	0	0
$871$	2.36347e6	0.105562
$872$	−7.12825e6	−0.317462
$873$	0	0
$874$	−217065.	−0.00961194
$875$	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	$0.489345\pi$
$878$	−8.36075e6	−0.366023
$879$	0	0
$880$	−6.04812e6	−0.263278
$881$	−2.77728e7	−1.20554	−0.602769	−	0.797916i	$-0.705936\pi$
$881$	−2.77728e7	−1.20554	−0.602769	+	0.797916i	$0.705936\pi$
$882$	0	0
$883$	3.15964e7	1.36376	0.681878	−	0.731466i	$-0.261164\pi$
$883$	3.15964e7	1.36376	0.681878	+	0.731466i	$0.261164\pi$
$884$	4.02825e6	0.173375
$885$	0	0
$886$	−695665.	−0.0297725
$887$	−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
$890$	2.59030e6	0.109616
$891$	0	0
$892$	2.79430e7	1.17588
$893$	−1.48217e6	−0.0621971
$894$	0	0
$895$	7.02253e6	0.293046
$896$	7.40681e6	0.308220
$897$	0	0
$898$	−9.53226e6	−0.394462
$899$	621362.	0.0256416
$900$	0	0
$901$	8.55399e6	0.351040
$902$	−464632.	−0.0190148
$903$	0	0
$904$	1.70790e7	0.695090
$905$	8.11144e6	0.329213
$906$	0	0
$907$	−1.42674e7	−0.575874	−0.287937	−	0.957649i	$-0.592969\pi$
$907$	−1.42674e7	−0.575874	−0.287937	+	0.957649i	$0.592969\pi$
$908$	−4.65678e7	−1.87444
$909$	0	0
$910$	−276539.	−0.0110701
$911$	−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
$914$	1.25024e6	0.0495024
$915$	0	0
$916$	−3.29628e7	−1.29803
$917$	−2.04485e6	−0.0803042
$918$	0	0
$919$	4.52117e7	1.76588	0.882942	−	0.469482i	$-0.155559\pi$
$919$	4.52117e7	1.76588	0.882942	+	0.469482i	$0.155559\pi$
$920$	4.04755e6	0.157660
$921$	0	0
$922$	1.06455e7	0.412419
$923$	−8.61384e6	−0.332807
$924$	0	0
$925$	5.23098e6	0.201015
$926$	2.16128e6	0.0828292
$927$	0	0
$928$	852142.	0.0324819
$929$	517735.	0.0196820	0.00984098	−	0.999952i	$-0.496867\pi$
$929$	517735.	0.0196820	0.00984098	+	0.999952i	$0.496867\pi$
$930$	0	0
$931$	−1.20399e6	−0.0455249
$932$	4.27481e6	0.161204
$933$	0	0
$934$	3.46965e6	0.130142
$935$	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	$0.659637\pi$
$938$	−915363.	−0.0339693
$939$	0	0
$940$	1.34167e7	0.495251
$941$	−4.58537e7	−1.68811	−0.844053	−	0.536259i	$-0.819837\pi$
$941$	−4.58537e7	−1.68811	−0.844053	+	0.536259i	$0.819837\pi$
$942$	0	0
$943$	−2.35798e6	−0.0863496
$944$	4.57780e6	0.167196
$945$	0	0
$946$	2.15018e6	0.0781172
$947$	−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
$950$	−70106.5	−0.00252028
$951$	0	0
$952$	−3.21378e6	−0.114927
$953$	−3.20607e7	−1.14351	−0.571755	−	0.820424i	$-0.693737\pi$
$953$	−3.20607e7	−1.14351	−0.571755	+	0.820424i	$0.693737\pi$
$954$	0	0
$955$	−2.00862e7	−0.712671
$956$	−2.39009e7	−0.845803
$957$	0	0
$958$	2.11947e6	0.0746127
$959$	2.76434e6	0.0970609
$960$	0	0
$961$	−2.08480e7	−0.728207
$962$	−1.90287e6	−0.0662935
$963$	0	0
$964$	−4.21614e7	−1.46124
$965$	−1.56767e6	−0.0541920
$966$	0	0
$967$	−3.19660e7	−1.09931	−0.549657	−	0.835391i	$-0.685242\pi$
$967$	−3.19660e7	−1.09931	−0.549657	+	0.835391i	$0.685242\pi$
$968$	−6.75280e6	−0.231630
$969$	0	0
$970$	−4.26132e6	−0.145417
$971$	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
$974$	4.74243e6	0.160178
$975$	0	0
$976$	−1.59360e7	−0.535495
$977$	3.12938e7	1.04887	0.524436	−	0.851450i	$-0.324276\pi$
$977$	3.12938e7	1.04887	0.524436	+	0.851450i	$0.324276\pi$
$978$	0	0
$979$	2.18299e7	0.727940
$980$	1.08986e7	0.362497
$981$	0	0
$982$	−4.75577e6	−0.157377
$983$	−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
$986$	−236595.	−0.00775020
$987$	0	0
$988$	−425414.	−0.0138650
$989$	1.09120e7	0.354744
$990$	0	0
$991$	−2.58041e7	−0.834650	−0.417325	−	0.908757i	$-0.637032\pi$
$991$	−2.58041e7	−0.834650	−0.417325	+	0.908757i	$0.637032\pi$
$992$	1.06712e7	0.344298
$993$	0	0
$994$	3.33610e6	0.107096
$995$	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	$0.265442\pi$
$998$	−419691.	−0.0133384
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.6.a.m.1.3		6
3.2	odd	2		65.6.a.d.1.4	✓	6
12.11	even	2		1040.6.a.q.1.6		6
15.2	even	4		325.6.b.g.274.7		12
15.8	even	4		325.6.b.g.274.6		12
15.14	odd	2		325.6.a.g.1.3		6
39.38	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	3.2	odd	2
325.6.a.g.1.3		6	15.14	odd	2
325.6.b.g.274.6		12	15.8	even	4
325.6.b.g.274.7		12	15.2	even	4
585.6.a.m.1.3		6	1.1	even	1	trivial
845.6.a.h.1.3		6	39.38	odd	2
1040.6.a.q.1.6		6	12.11	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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34530 q^{2} -30.1902 q^{4} +25.0000 q^{5} +48.6530 q^{7} +83.6645 q^{8} -33.6325 q^{10} -283.440 q^{11} +169.000 q^{13} -65.4530 q^{14} +853.531 q^{16} -789.522 q^{17} +83.3795 q^{19} -754.754 q^{20} +381.312 q^{22} +1935.13 q^{23} +625.000 q^{25} -227.356 q^{26} -1468.84 q^{28} -222.752 q^{29} -2789.48 q^{31} -3825.52 q^{32} +1062.15 q^{34} +1216.33 q^{35} +8369.56 q^{37} -112.170 q^{38} +2091.61 q^{40} -1218.51 q^{41} +5638.89 q^{43} +8557.10 q^{44} -2603.34 q^{46} -17776.2 q^{47} -14439.9 q^{49} -840.813 q^{50} -5102.14 q^{52} -10834.4 q^{53} -7086.00 q^{55} +4070.53 q^{56} +299.668 q^{58} +5363.36 q^{59} -18670.7 q^{61} +3752.69 q^{62} -22166.5 q^{64} +4225.00 q^{65} +13985.1 q^{67} +23835.8 q^{68} -1636.32 q^{70} -50969.5 q^{71} +42394.9 q^{73} -11259.6 q^{74} -2517.24 q^{76} -13790.2 q^{77} +106279. q^{79} +21338.3 q^{80} +1639.26 q^{82} -75513.6 q^{83} -19738.1 q^{85} -7586.01 q^{86} -23713.8 q^{88} -77017.8 q^{89} +8222.36 q^{91} -58422.0 q^{92} +23914.4 q^{94} +2084.49 q^{95} +126702. q^{97} +19426.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) \) 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

Introduction

L-functions

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