LMFDB - Embedded newform 1028.6.a.a.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.7147 q^{3} +60.4053 q^{5} +191.832 q^{7} +36.3820 q^{9} +O(q^{10})$	$q-16.7147 q^{3} +60.4053 q^{5} +191.832 q^{7} +36.3820 q^{9} -24.7765 q^{11} +577.915 q^{13} -1009.66 q^{15} +831.771 q^{17} +471.826 q^{19} -3206.41 q^{21} -2313.04 q^{23} +523.802 q^{25} +3453.56 q^{27} -6776.80 q^{29} -8117.95 q^{31} +414.132 q^{33} +11587.6 q^{35} -5173.99 q^{37} -9659.68 q^{39} -8551.40 q^{41} -8837.76 q^{43} +2197.66 q^{45} -12379.5 q^{47} +19992.3 q^{49} -13902.8 q^{51} -22852.6 q^{53} -1496.63 q^{55} -7886.44 q^{57} +8714.95 q^{59} +14281.9 q^{61} +6979.21 q^{63} +34909.1 q^{65} +10649.1 q^{67} +38661.8 q^{69} +20101.3 q^{71} -87268.7 q^{73} -8755.20 q^{75} -4752.91 q^{77} -99538.8 q^{79} -66566.2 q^{81} +38721.6 q^{83} +50243.4 q^{85} +113272. q^{87} -132561. q^{89} +110862. q^{91} +135689. q^{93} +28500.8 q^{95} -1549.74 q^{97} -901.416 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

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
$3$	−16.7147	−1.07225	−0.536125	−	0.844139i	$-0.680112\pi$
$3$	−16.7147	−1.07225	−0.536125	+	0.844139i	$0.680112\pi$
$4$	0	0
$5$	60.4053	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$5$	60.4053	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$6$	0	0
$7$	191.832	1.47970	0.739852	−	0.672770i	$-0.234895\pi$
$7$	191.832	1.47970	0.739852	+	0.672770i	$0.234895\pi$
$8$	0	0
$9$	36.3820	0.149720
$10$	0	0
$11$	−24.7765	−0.0617387	−0.0308693	−	0.999523i	$-0.509828\pi$
$11$	−24.7765	−0.0617387	−0.0308693	+	0.999523i	$0.509828\pi$
$12$	0	0
$13$	577.915	0.948430	0.474215	−	0.880409i	$-0.342732\pi$
$13$	577.915	0.948430	0.474215	+	0.880409i	$0.342732\pi$
$14$	0	0
$15$	−1009.66	−1.15863
$16$	0	0
$17$	831.771	0.698042	0.349021	−	0.937115i	$-0.386514\pi$
$17$	831.771	0.698042	0.349021	+	0.937115i	$0.386514\pi$
$18$	0	0
$19$	471.826	0.299846	0.149923	−	0.988698i	$-0.452097\pi$
$19$	471.826	0.299846	0.149923	+	0.988698i	$0.452097\pi$
$20$	0	0
$21$	−3206.41	−1.58661
$22$	0	0
$23$	−2313.04	−0.911724	−0.455862	−	0.890051i	$-0.650669\pi$
$23$	−2313.04	−0.911724	−0.455862	+	0.890051i	$0.650669\pi$
$24$	0	0
$25$	523.802	0.167617
$26$	0	0
$27$	3453.56	0.911713
$28$	0	0
$29$	−6776.80	−1.49634	−0.748169	−	0.663509i	$-0.769067\pi$
$29$	−6776.80	−1.49634	−0.748169	+	0.663509i	$0.769067\pi$
$30$	0	0
$31$	−8117.95	−1.51720	−0.758599	−	0.651558i	$-0.774116\pi$
$31$	−8117.95	−1.51720	−0.758599	+	0.651558i	$0.774116\pi$
$32$	0	0
$33$	414.132	0.0661993
$34$	0	0
$35$	11587.6	1.59891
$36$	0	0
$37$	−5173.99	−0.621329	−0.310664	−	0.950520i	$-0.600552\pi$
$37$	−5173.99	−0.621329	−0.310664	+	0.950520i	$0.600552\pi$
$38$	0	0
$39$	−9659.68	−1.01695
$40$	0	0
$41$	−8551.40	−0.794470	−0.397235	−	0.917717i	$-0.630030\pi$
$41$	−8551.40	−0.794470	−0.397235	+	0.917717i	$0.630030\pi$
$42$	0	0
$43$	−8837.76	−0.728905	−0.364453	−	0.931222i	$-0.618744\pi$
$43$	−8837.76	−0.728905	−0.364453	+	0.931222i	$0.618744\pi$
$44$	0	0
$45$	2197.66	0.161782
$46$	0	0
$47$	−12379.5	−0.817448	−0.408724	−	0.912658i	$-0.634026\pi$
$47$	−12379.5	−0.817448	−0.408724	+	0.912658i	$0.634026\pi$
$48$	0	0
$49$	19992.3	1.18952
$50$	0	0
$51$	−13902.8	−0.748476
$52$	0	0
$53$	−22852.6	−1.11750	−0.558748	−	0.829337i	$-0.688718\pi$
$53$	−22852.6	−1.11750	−0.558748	+	0.829337i	$0.688718\pi$
$54$	0	0
$55$	−1496.63	−0.0667126
$56$	0	0
$57$	−7886.44	−0.321510
$58$	0	0
$59$	8714.95	0.325938	0.162969	−	0.986631i	$-0.447893\pi$
$59$	8714.95	0.325938	0.162969	+	0.986631i	$0.447893\pi$
$60$	0	0
$61$	14281.9	0.491430	0.245715	−	0.969342i	$-0.420977\pi$
$61$	14281.9	0.491430	0.245715	+	0.969342i	$0.420977\pi$
$62$	0	0
$63$	6979.21	0.221541
$64$	0	0
$65$	34909.1	1.02484
$66$	0	0
$67$	10649.1	0.289820	0.144910	−	0.989445i	$-0.453711\pi$
$67$	10649.1	0.289820	0.144910	+	0.989445i	$0.453711\pi$
$68$	0	0
$69$	38661.8	0.977596
$70$	0	0
$71$	20101.3	0.473237	0.236619	−	0.971603i	$-0.423961\pi$
$71$	20101.3	0.473237	0.236619	+	0.971603i	$0.423961\pi$
$72$	0	0
$73$	−87268.7	−1.91669	−0.958344	−	0.285617i	$-0.907801\pi$
$73$	−87268.7	−1.91669	−0.958344	+	0.285617i	$0.907801\pi$
$74$	0	0
$75$	−8755.20	−0.179727
$76$	0	0
$77$	−4752.91	−0.0913550
$78$	0	0
$79$	−99538.8	−1.79442	−0.897211	−	0.441602i	$-0.854410\pi$
$79$	−99538.8	−1.79442	−0.897211	+	0.441602i	$0.854410\pi$
$80$	0	0
$81$	−66566.2	−1.12730
$82$	0	0
$83$	38721.6	0.616962	0.308481	−	0.951230i	$-0.400179\pi$
$83$	38721.6	0.616962	0.308481	+	0.951230i	$0.400179\pi$
$84$	0	0
$85$	50243.4	0.754278
$86$	0	0
$87$	113272.	1.60445
$88$	0	0
$89$	−132561.	−1.77395	−0.886973	−	0.461821i	$-0.847196\pi$
$89$	−132561.	−1.77395	−0.886973	+	0.461821i	$0.847196\pi$
$90$	0	0
$91$	110862.	1.40340
$92$	0	0
$93$	135689.	1.62682
$94$	0	0
$95$	28500.8	0.324002
$96$	0	0
$97$	−1549.74	−0.0167236	−0.00836182	−	0.999965i	$-0.502662\pi$
$97$	−1549.74	−0.0167236	−0.00836182	+	0.999965i	$0.502662\pi$
$98$	0	0
$99$	−901.416	−0.00924352
$100$	0	0
$101$	107029.	1.04400	0.521999	−	0.852946i	$-0.325186\pi$
$101$	107029.	1.04400	0.521999	+	0.852946i	$0.325186\pi$
$102$	0	0
$103$	165453.	1.53667	0.768336	−	0.640046i	$-0.221085\pi$
$103$	165453.	1.53667	0.768336	+	0.640046i	$0.221085\pi$
$104$	0	0
$105$	−193684.	−1.71444
$106$	0	0
$107$	−118995.	−1.00478	−0.502390	−	0.864641i	$-0.667546\pi$
$107$	−118995.	−1.00478	−0.502390	+	0.864641i	$0.667546\pi$
$108$	0	0
$109$	−42154.9	−0.339845	−0.169923	−	0.985457i	$-0.554352\pi$
$109$	−42154.9	−0.339845	−0.169923	+	0.985457i	$0.554352\pi$
$110$	0	0
$111$	86481.8	0.666220
$112$	0	0
$113$	193479.	1.42540	0.712701	−	0.701468i	$-0.247472\pi$
$113$	193479.	1.42540	0.712701	+	0.701468i	$0.247472\pi$
$114$	0	0
$115$	−139720.	−0.985175
$116$	0	0
$117$	21025.7	0.141999
$118$	0	0
$119$	159560.	1.03290
$120$	0	0
$121$	−160437.	−0.996188
$122$	0	0
$123$	142934.	0.851870
$124$	0	0
$125$	−157126.	−0.899443
$126$	0	0
$127$	233427.	1.28423	0.642113	−	0.766610i	$-0.278058\pi$
$127$	233427.	1.28423	0.642113	+	0.766610i	$0.278058\pi$
$128$	0	0
$129$	147721.	0.781569
$130$	0	0
$131$	67515.3	0.343735	0.171867	−	0.985120i	$-0.445020\pi$
$131$	67515.3	0.343735	0.171867	+	0.985120i	$0.445020\pi$
$132$	0	0
$133$	90511.1	0.443683
$134$	0	0
$135$	208614.	0.985163
$136$	0	0
$137$	198437.	0.903279	0.451639	−	0.892201i	$-0.350839\pi$
$137$	198437.	0.903279	0.451639	+	0.892201i	$0.350839\pi$
$138$	0	0
$139$	−7724.99	−0.0339126	−0.0169563	−	0.999856i	$-0.505398\pi$
$139$	−7724.99	−0.0339126	−0.0169563	+	0.999856i	$0.505398\pi$
$140$	0	0
$141$	206921.	0.876508
$142$	0	0
$143$	−14318.7	−0.0585548
$144$	0	0
$145$	−409355.	−1.61689
$146$	0	0
$147$	−334166.	−1.27547
$148$	0	0
$149$	234439.	0.865094	0.432547	−	0.901611i	$-0.357615\pi$
$149$	234439.	0.865094	0.432547	+	0.901611i	$0.357615\pi$
$150$	0	0
$151$	36792.3	0.131315	0.0656575	−	0.997842i	$-0.479086\pi$
$151$	36792.3	0.131315	0.0656575	+	0.997842i	$0.479086\pi$
$152$	0	0
$153$	30261.5	0.104511
$154$	0	0
$155$	−490367.	−1.63943
$156$	0	0
$157$	394869.	1.27851	0.639254	−	0.768995i	$-0.279243\pi$
$157$	394869.	1.27851	0.639254	+	0.768995i	$0.279243\pi$
$158$	0	0
$159$	381975.	1.19824
$160$	0	0
$161$	−443714.	−1.34908
$162$	0	0
$163$	−533583.	−1.57301	−0.786507	−	0.617581i	$-0.788113\pi$
$163$	−533583.	−1.57301	−0.786507	+	0.617581i	$0.788113\pi$
$164$	0	0
$165$	25015.7	0.0715325
$166$	0	0
$167$	−160379.	−0.444995	−0.222498	−	0.974933i	$-0.571421\pi$
$167$	−160379.	−0.444995	−0.222498	+	0.974933i	$0.571421\pi$
$168$	0	0
$169$	−37307.6	−0.100480
$170$	0	0
$171$	17166.0	0.0448929
$172$	0	0
$173$	371042.	0.942557	0.471278	−	0.881985i	$-0.343793\pi$
$173$	371042.	0.942557	0.471278	+	0.881985i	$0.343793\pi$
$174$	0	0
$175$	100482.	0.248023
$176$	0	0
$177$	−145668.	−0.349487
$178$	0	0
$179$	662689.	1.54588	0.772942	−	0.634476i	$-0.218784\pi$
$179$	662689.	1.54588	0.772942	+	0.634476i	$0.218784\pi$
$180$	0	0
$181$	−711041.	−1.61324	−0.806619	−	0.591072i	$-0.798705\pi$
$181$	−711041.	−1.61324	−0.806619	+	0.591072i	$0.798705\pi$
$182$	0	0
$183$	−238718.	−0.526935
$184$	0	0
$185$	−312537.	−0.671385
$186$	0	0
$187$	−20608.3	−0.0430962
$188$	0	0
$189$	662502.	1.34907
$190$	0	0
$191$	−271885.	−0.539264	−0.269632	−	0.962963i	$-0.586902\pi$
$191$	−271885.	−0.539264	−0.269632	+	0.962963i	$0.586902\pi$
$192$	0	0
$193$	551151.	1.06507	0.532534	−	0.846409i	$-0.321240\pi$
$193$	551151.	1.06507	0.532534	+	0.846409i	$0.321240\pi$
$194$	0	0
$195$	−583496.	−1.09888
$196$	0	0
$197$	−395592.	−0.726242	−0.363121	−	0.931742i	$-0.618289\pi$
$197$	−395592.	−0.726242	−0.363121	+	0.931742i	$0.618289\pi$
$198$	0	0
$199$	−39911.3	−0.0714436	−0.0357218	−	0.999362i	$-0.511373\pi$
$199$	−39911.3	−0.0714436	−0.0357218	+	0.999362i	$0.511373\pi$
$200$	0	0
$201$	−177997.	−0.310759
$202$	0	0
$203$	−1.30000e6	−2.21414
$204$	0	0
$205$	−516550.	−0.858475
$206$	0	0
$207$	−84152.9	−0.136503
$208$	0	0
$209$	−11690.2	−0.0185121
$210$	0	0
$211$	−1.05359e6	−1.62917	−0.814584	−	0.580045i	$-0.803035\pi$
$211$	−1.05359e6	−1.62917	−0.814584	+	0.580045i	$0.803035\pi$
$212$	0	0
$213$	−335988.	−0.507429
$214$	0	0
$215$	−533848.	−0.787628
$216$	0	0
$217$	−1.55728e6	−2.24500
$218$	0	0
$219$	1.45867e6	2.05517
$220$	0	0
$221$	480693.	0.662044
$222$	0	0
$223$	36679.4	0.0493924	0.0246962	−	0.999695i	$-0.492138\pi$
$223$	36679.4	0.0493924	0.0246962	+	0.999695i	$0.492138\pi$
$224$	0	0
$225$	19056.9	0.0250956
$226$	0	0
$227$	−91913.9	−0.118390	−0.0591952	−	0.998246i	$-0.518853\pi$
$227$	−91913.9	−0.118390	−0.0591952	+	0.998246i	$0.518853\pi$
$228$	0	0
$229$	−1.28918e6	−1.62452	−0.812262	−	0.583292i	$-0.801764\pi$
$229$	−1.28918e6	−1.62452	−0.812262	+	0.583292i	$0.801764\pi$
$230$	0	0
$231$	79443.5	0.0979554
$232$	0	0
$233$	97510.1	0.117668	0.0588342	−	0.998268i	$-0.481262\pi$
$233$	97510.1	0.117668	0.0588342	+	0.998268i	$0.481262\pi$
$234$	0	0
$235$	−747790.	−0.883304
$236$	0	0
$237$	1.66376e6	1.92407
$238$	0	0
$239$	−933731.	−1.05737	−0.528685	−	0.848818i	$-0.677315\pi$
$239$	−933731.	−1.05737	−0.528685	+	0.848818i	$0.677315\pi$
$240$	0	0
$241$	−844095.	−0.936157	−0.468079	−	0.883687i	$-0.655054\pi$
$241$	−844095.	−0.936157	−0.468079	+	0.883687i	$0.655054\pi$
$242$	0	0
$243$	273419.	0.297039
$244$	0	0
$245$	1.20764e6	1.28536
$246$	0	0
$247$	272675.	0.284383
$248$	0	0
$249$	−647221.	−0.661538
$250$	0	0
$251$	−1.27690e6	−1.27930	−0.639649	−	0.768667i	$-0.720920\pi$
$251$	−1.27690e6	−1.27930	−0.639649	+	0.768667i	$0.720920\pi$
$252$	0	0
$253$	57308.9	0.0562886
$254$	0	0
$255$	−839805.	−0.808775
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−992535.	−0.919383
$260$	0	0
$261$	−246553.	−0.224032
$262$	0	0
$263$	1.77974e6	1.58660	0.793301	−	0.608830i	$-0.208361\pi$
$263$	1.77974e6	1.58660	0.793301	+	0.608830i	$0.208361\pi$
$264$	0	0
$265$	−1.38042e6	−1.20752
$266$	0	0
$267$	2.21572e6	1.90211
$268$	0	0
$269$	−256020.	−0.215722	−0.107861	−	0.994166i	$-0.534400\pi$
$269$	−256020.	−0.215722	−0.107861	+	0.994166i	$0.534400\pi$
$270$	0	0
$271$	−2.23817e6	−1.85127	−0.925637	−	0.378412i	$-0.876470\pi$
$271$	−2.23817e6	−1.85127	−0.925637	+	0.378412i	$0.876470\pi$
$272$	0	0
$273$	−1.85303e6	−1.50479
$274$	0	0
$275$	−12977.9	−0.0103484
$276$	0	0
$277$	−1.90818e6	−1.49424	−0.747121	−	0.664688i	$-0.768564\pi$
$277$	−1.90818e6	−1.49424	−0.747121	+	0.664688i	$0.768564\pi$
$278$	0	0
$279$	−295347.	−0.227155
$280$	0	0
$281$	−2.23948e6	−1.69192	−0.845961	−	0.533244i	$-0.820973\pi$
$281$	−2.23948e6	−1.69192	−0.845961	+	0.533244i	$0.820973\pi$
$282$	0	0
$283$	2.11005e6	1.56612	0.783062	−	0.621944i	$-0.213657\pi$
$283$	2.11005e6	1.56612	0.783062	+	0.621944i	$0.213657\pi$
$284$	0	0
$285$	−476383.	−0.347411
$286$	0	0
$287$	−1.64043e6	−1.17558
$288$	0	0
$289$	−728014.	−0.512737
$290$	0	0
$291$	25903.6	0.0179319
$292$	0	0
$293$	−1.22239e6	−0.831839	−0.415919	−	0.909402i	$-0.636540\pi$
$293$	−1.22239e6	−0.831839	−0.415919	+	0.909402i	$0.636540\pi$
$294$	0	0
$295$	526429.	0.352197
$296$	0	0
$297$	−85567.1	−0.0562880
$298$	0	0
$299$	−1.33674e6	−0.864706
$300$	0	0
$301$	−1.69536e6	−1.07856
$302$	0	0
$303$	−1.78897e6	−1.11943
$304$	0	0
$305$	862702.	0.531021
$306$	0	0
$307$	431755.	0.261452	0.130726	−	0.991419i	$-0.458269\pi$
$307$	431755.	0.261452	0.130726	+	0.991419i	$0.458269\pi$
$308$	0	0
$309$	−2.76550e6	−1.64770
$310$	0	0
$311$	−362769.	−0.212681	−0.106341	−	0.994330i	$-0.533913\pi$
$311$	−362769.	−0.212681	−0.106341	+	0.994330i	$0.533913\pi$
$312$	0	0
$313$	376992.	0.217506	0.108753	−	0.994069i	$-0.465314\pi$
$313$	376992.	0.217506	0.108753	+	0.994069i	$0.465314\pi$
$314$	0	0
$315$	421581.	0.239389
$316$	0	0
$317$	3.09863e6	1.73190	0.865949	−	0.500133i	$-0.166715\pi$
$317$	3.09863e6	1.73190	0.865949	+	0.500133i	$0.166715\pi$
$318$	0	0
$319$	167905.	0.0923819
$320$	0	0
$321$	1.98898e6	1.07738
$322$	0	0
$323$	392451.	0.209305
$324$	0	0
$325$	302713.	0.158973
$326$	0	0
$327$	704607.	0.364399
$328$	0	0
$329$	−2.37479e6	−1.20958
$330$	0	0
$331$	2.12004e6	1.06359	0.531796	−	0.846873i	$-0.321517\pi$
$331$	2.12004e6	1.06359	0.531796	+	0.846873i	$0.321517\pi$
$332$	0	0
$333$	−188240.	−0.0930254
$334$	0	0
$335$	643265.	0.313168
$336$	0	0
$337$	2.21356e6	1.06174	0.530868	−	0.847455i	$-0.321866\pi$
$337$	2.21356e6	1.06174	0.530868	+	0.847455i	$0.321866\pi$
$338$	0	0
$339$	−3.23394e6	−1.52839
$340$	0	0
$341$	201134.	0.0936698
$342$	0	0
$343$	611048.	0.280440
$344$	0	0
$345$	2.33538e6	1.05635
$346$	0	0
$347$	274344.	0.122313	0.0611565	−	0.998128i	$-0.480521\pi$
$347$	274344.	0.122313	0.0611565	+	0.998128i	$0.480521\pi$
$348$	0	0
$349$	−449521.	−0.197554	−0.0987772	−	0.995110i	$-0.531493\pi$
$349$	−449521.	−0.197554	−0.0987772	+	0.995110i	$0.531493\pi$
$350$	0	0
$351$	1.99587e6	0.864696
$352$	0	0
$353$	1.87809e6	0.802196	0.401098	−	0.916035i	$-0.368629\pi$
$353$	1.87809e6	0.802196	0.401098	+	0.916035i	$0.368629\pi$
$354$	0	0
$355$	1.21423e6	0.511363
$356$	0	0
$357$	−2.66700e6	−1.10752
$358$	0	0
$359$	1.93866e6	0.793899	0.396950	−	0.917840i	$-0.370069\pi$
$359$	1.93866e6	0.793899	0.396950	+	0.917840i	$0.370069\pi$
$360$	0	0
$361$	−2.25348e6	−0.910093
$362$	0	0
$363$	2.68166e6	1.06816
$364$	0	0
$365$	−5.27149e6	−2.07110
$366$	0	0
$367$	4.37388e6	1.69512	0.847562	−	0.530697i	$-0.178070\pi$
$367$	4.37388e6	1.69512	0.847562	+	0.530697i	$0.178070\pi$
$368$	0	0
$369$	−311117.	−0.118948
$370$	0	0
$371$	−4.38385e6	−1.65356
$372$	0	0
$373$	2.35893e6	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$373$	2.35893e6	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$374$	0	0
$375$	2.62632e6	0.964428
$376$	0	0
$377$	−3.91641e6	−1.41917
$378$	0	0
$379$	4.27098e6	1.52732	0.763660	−	0.645619i	$-0.223401\pi$
$379$	4.27098e6	1.52732	0.763660	+	0.645619i	$0.223401\pi$
$380$	0	0
$381$	−3.90167e6	−1.37701
$382$	0	0
$383$	2.30397e6	0.802564	0.401282	−	0.915955i	$-0.368565\pi$
$383$	2.30397e6	0.802564	0.401282	+	0.915955i	$0.368565\pi$
$384$	0	0
$385$	−287101.	−0.0987148
$386$	0	0
$387$	−321535.	−0.109132
$388$	0	0
$389$	−5.85135e6	−1.96057	−0.980285	−	0.197591i	$-0.936688\pi$
$389$	−5.85135e6	−1.96057	−0.980285	+	0.197591i	$0.936688\pi$
$390$	0	0
$391$	−1.92392e6	−0.636421
$392$	0	0
$393$	−1.12850e6	−0.368570
$394$	0	0
$395$	−6.01267e6	−1.93899
$396$	0	0
$397$	−4.65153e6	−1.48122	−0.740610	−	0.671935i	$-0.765463\pi$
$397$	−4.65153e6	−1.48122	−0.740610	+	0.671935i	$0.765463\pi$
$398$	0	0
$399$	−1.51287e6	−0.475739
$400$	0	0
$401$	−1.17087e6	−0.363620	−0.181810	−	0.983334i	$-0.558196\pi$
$401$	−1.17087e6	−0.363620	−0.181810	+	0.983334i	$0.558196\pi$
$402$	0	0
$403$	−4.69148e6	−1.43896
$404$	0	0
$405$	−4.02095e6	−1.21812
$406$	0	0
$407$	128193.	0.0383600
$408$	0	0
$409$	−3.01250e6	−0.890469	−0.445234	−	0.895414i	$-0.646880\pi$
$409$	−3.01250e6	−0.890469	−0.445234	+	0.895414i	$0.646880\pi$
$410$	0	0
$411$	−3.31682e6	−0.968540
$412$	0	0
$413$	1.67180e6	0.482292
$414$	0	0
$415$	2.33899e6	0.666666
$416$	0	0
$417$	129121.	0.0363628
$418$	0	0
$419$	4.70782e6	1.31004	0.655020	−	0.755611i	$-0.272660\pi$
$419$	4.70782e6	1.31004	0.655020	+	0.755611i	$0.272660\pi$
$420$	0	0
$421$	1.12847e6	0.310303	0.155152	−	0.987891i	$-0.450413\pi$
$421$	1.12847e6	0.310303	0.155152	+	0.987891i	$0.450413\pi$
$422$	0	0
$423$	−450392.	−0.122388
$424$	0	0
$425$	435683.	0.117003
$426$	0	0
$427$	2.73972e6	0.727170
$428$	0	0
$429$	239333.	0.0627854
$430$	0	0
$431$	3.62247e6	0.939315	0.469658	−	0.882849i	$-0.344377\pi$
$431$	3.62247e6	0.939315	0.469658	+	0.882849i	$0.344377\pi$
$432$	0	0
$433$	1.07816e6	0.276353	0.138176	−	0.990408i	$-0.455876\pi$
$433$	1.07816e6	0.276353	0.138176	+	0.990408i	$0.455876\pi$
$434$	0	0
$435$	6.84225e6	1.73371
$436$	0	0
$437$	−1.09135e6	−0.273376
$438$	0	0
$439$	−5.86689e6	−1.45294	−0.726468	−	0.687200i	$-0.758840\pi$
$439$	−5.86689e6	−1.45294	−0.726468	+	0.687200i	$0.758840\pi$
$440$	0	0
$441$	727361.	0.178096
$442$	0	0
$443$	6.92229e6	1.67587	0.837936	−	0.545769i	$-0.183762\pi$
$443$	6.92229e6	1.67587	0.837936	+	0.545769i	$0.183762\pi$
$444$	0	0
$445$	−8.00738e6	−1.91686
$446$	0	0
$447$	−3.91858e6	−0.927597
$448$	0	0
$449$	978126.	0.228970	0.114485	−	0.993425i	$-0.463478\pi$
$449$	978126.	0.228970	0.114485	+	0.993425i	$0.463478\pi$
$450$	0	0
$451$	211873.	0.0490495
$452$	0	0
$453$	−614973.	−0.140802
$454$	0	0
$455$	6.69667e6	1.51646
$456$	0	0
$457$	−1.06922e6	−0.239485	−0.119742	−	0.992805i	$-0.538207\pi$
$457$	−1.06922e6	−0.239485	−0.119742	+	0.992805i	$0.538207\pi$
$458$	0	0
$459$	2.87257e6	0.636414
$460$	0	0
$461$	925032.	0.202724	0.101362	−	0.994850i	$-0.467680\pi$
$461$	925032.	0.202724	0.101362	+	0.994850i	$0.467680\pi$
$462$	0	0
$463$	7.20506e6	1.56202	0.781008	−	0.624521i	$-0.214706\pi$
$463$	7.20506e6	1.56202	0.781008	+	0.624521i	$0.214706\pi$
$464$	0	0
$465$	8.19636e6	1.75788
$466$	0	0
$467$	6.13247e6	1.30120	0.650599	−	0.759421i	$-0.274518\pi$
$467$	6.13247e6	1.30120	0.650599	+	0.759421i	$0.274518\pi$
$468$	0	0
$469$	2.04284e6	0.428847
$470$	0	0
$471$	−6.60012e6	−1.37088
$472$	0	0
$473$	218968.	0.0450017
$474$	0	0
$475$	247143.	0.0502591
$476$	0	0
$477$	−831423.	−0.167312
$478$	0	0
$479$	5.48402e6	1.09209	0.546047	−	0.837754i	$-0.316132\pi$
$479$	5.48402e6	1.09209	0.546047	+	0.837754i	$0.316132\pi$
$480$	0	0
$481$	−2.99013e6	−0.589287
$482$	0	0
$483$	7.41655e6	1.44655
$484$	0	0
$485$	−93612.8	−0.0180710
$486$	0	0
$487$	−836762.	−0.159875	−0.0799373	−	0.996800i	$-0.525472\pi$
$487$	−836762.	−0.159875	−0.0799373	+	0.996800i	$0.525472\pi$
$488$	0	0
$489$	8.91869e6	1.68667
$490$	0	0
$491$	−2.03193e6	−0.380370	−0.190185	−	0.981748i	$-0.560909\pi$
$491$	−2.03193e6	−0.380370	−0.190185	+	0.981748i	$0.560909\pi$
$492$	0	0
$493$	−5.63675e6	−1.04451
$494$	0	0
$495$	−54450.3	−0.00998821
$496$	0	0
$497$	3.85607e6	0.700251
$498$	0	0
$499$	1.14177e6	0.205272	0.102636	−	0.994719i	$-0.467272\pi$
$499$	1.14177e6	0.205272	0.102636	+	0.994719i	$0.467272\pi$
$500$	0	0
$501$	2.68069e6	0.477146
$502$	0	0
$503$	−3.16443e6	−0.557668	−0.278834	−	0.960339i	$-0.589948\pi$
$503$	−3.16443e6	−0.557668	−0.278834	+	0.960339i	$0.589948\pi$
$504$	0	0
$505$	6.46515e6	1.12811
$506$	0	0
$507$	623586.	0.107740
$508$	0	0
$509$	3.60476e6	0.616712	0.308356	−	0.951271i	$-0.400221\pi$
$509$	3.60476e6	0.616712	0.308356	+	0.951271i	$0.400221\pi$
$510$	0	0
$511$	−1.67409e7	−2.83613
$512$	0	0
$513$	1.62948e6	0.273373
$514$	0	0
$515$	9.99424e6	1.66047
$516$	0	0
$517$	306721.	0.0504682
$518$	0	0
$519$	−6.20186e6	−1.01066
$520$	0	0
$521$	5.46755e6	0.882466	0.441233	−	0.897392i	$-0.354541\pi$
$521$	5.46755e6	0.882466	0.441233	+	0.897392i	$0.354541\pi$
$522$	0	0
$523$	6.27023e6	1.00237	0.501186	−	0.865339i	$-0.332897\pi$
$523$	6.27023e6	1.00237	0.501186	+	0.865339i	$0.332897\pi$
$524$	0	0
$525$	−1.67952e6	−0.265943
$526$	0	0
$527$	−6.75228e6	−1.05907
$528$	0	0
$529$	−1.08620e6	−0.168760
$530$	0	0
$531$	317067.	0.0487995
$532$	0	0
$533$	−4.94198e6	−0.753499
$534$	0	0
$535$	−7.18796e6	−1.08573
$536$	0	0
$537$	−1.10767e7	−1.65757
$538$	0	0
$539$	−495339.	−0.0734397
$540$	0	0
$541$	3.78331e6	0.555749	0.277874	−	0.960617i	$-0.410370\pi$
$541$	3.78331e6	0.555749	0.277874	+	0.960617i	$0.410370\pi$
$542$	0	0
$543$	1.18849e7	1.72979
$544$	0	0
$545$	−2.54638e6	−0.367224
$546$	0	0
$547$	8.50897e6	1.21593	0.607965	−	0.793964i	$-0.291986\pi$
$547$	8.50897e6	1.21593	0.607965	+	0.793964i	$0.291986\pi$
$548$	0	0
$549$	519603.	0.0735768
$550$	0	0
$551$	−3.19747e6	−0.448670
$552$	0	0
$553$	−1.90947e7	−2.65521
$554$	0	0
$555$	5.22396e6	0.719892
$556$	0	0
$557$	−9.79807e6	−1.33814	−0.669072	−	0.743198i	$-0.733308\pi$
$557$	−9.79807e6	−1.33814	−0.669072	+	0.743198i	$0.733308\pi$
$558$	0	0
$559$	−5.10747e6	−0.691316
$560$	0	0
$561$	344463.	0.0462099
$562$	0	0
$563$	−6.03730e6	−0.802734	−0.401367	−	0.915917i	$-0.631465\pi$
$563$	−6.03730e6	−0.802734	−0.401367	+	0.915917i	$0.631465\pi$
$564$	0	0
$565$	1.16871e7	1.54024
$566$	0	0
$567$	−1.27695e7	−1.66808
$568$	0	0
$569$	1.16111e7	1.50346	0.751732	−	0.659468i	$-0.229219\pi$
$569$	1.16111e7	1.50346	0.751732	+	0.659468i	$0.229219\pi$
$570$	0	0
$571$	−5.30453e6	−0.680858	−0.340429	−	0.940270i	$-0.610572\pi$
$571$	−5.30453e6	−0.680858	−0.340429	+	0.940270i	$0.610572\pi$
$572$	0	0
$573$	4.54448e6	0.578226
$574$	0	0
$575$	−1.21157e6	−0.152820
$576$	0	0
$577$	8.76133e6	1.09555	0.547773	−	0.836627i	$-0.315476\pi$
$577$	8.76133e6	1.09555	0.547773	+	0.836627i	$0.315476\pi$
$578$	0	0
$579$	−9.21233e6	−1.14202
$580$	0	0
$581$	7.42803e6	0.912921
$582$	0	0
$583$	566206.	0.0689928
$584$	0	0
$585$	1.27006e6	0.153439
$586$	0	0
$587$	−1.24243e7	−1.48825	−0.744124	−	0.668041i	$-0.767133\pi$
$587$	−1.24243e7	−1.48825	−0.744124	+	0.668041i	$0.767133\pi$
$588$	0	0
$589$	−3.83026e6	−0.454925
$590$	0	0
$591$	6.61220e6	0.778713
$592$	0	0
$593$	−1.50979e7	−1.76312	−0.881558	−	0.472076i	$-0.843505\pi$
$593$	−1.50979e7	−1.76312	−0.881558	+	0.472076i	$0.843505\pi$
$594$	0	0
$595$	9.63827e6	1.11611
$596$	0	0
$597$	667107.	0.0766054
$598$	0	0
$599$	1.29733e7	1.47735	0.738676	−	0.674060i	$-0.235451\pi$
$599$	1.29733e7	1.47735	0.738676	+	0.674060i	$0.235451\pi$
$600$	0	0
$601$	5.13379e6	0.579765	0.289883	−	0.957062i	$-0.406384\pi$
$601$	5.13379e6	0.579765	0.289883	+	0.957062i	$0.406384\pi$
$602$	0	0
$603$	387437.	0.0433918
$604$	0	0
$605$	−9.69125e6	−1.07644
$606$	0	0
$607$	5.64889e6	0.622288	0.311144	−	0.950363i	$-0.399288\pi$
$607$	5.64889e6	0.622288	0.311144	+	0.950363i	$0.399288\pi$
$608$	0	0
$609$	2.17292e7	2.37411
$610$	0	0
$611$	−7.15432e6	−0.775292
$612$	0	0
$613$	−3.87971e6	−0.417012	−0.208506	−	0.978021i	$-0.566860\pi$
$613$	−3.87971e6	−0.417012	−0.208506	+	0.978021i	$0.566860\pi$
$614$	0	0
$615$	8.63399e6	0.920500
$616$	0	0
$617$	−1.02038e7	−1.07907	−0.539533	−	0.841964i	$-0.681399\pi$
$617$	−1.02038e7	−1.07907	−0.539533	+	0.841964i	$0.681399\pi$
$618$	0	0
$619$	−1.25799e6	−0.131963	−0.0659815	−	0.997821i	$-0.521018\pi$
$619$	−1.25799e6	−0.131963	−0.0659815	+	0.997821i	$0.521018\pi$
$620$	0	0
$621$	−7.98822e6	−0.831230
$622$	0	0
$623$	−2.54294e7	−2.62492
$624$	0	0
$625$	−1.11281e7	−1.13952
$626$	0	0
$627$	195398.	0.0198496
$628$	0	0
$629$	−4.30358e6	−0.433714
$630$	0	0
$631$	−4.15745e6	−0.415675	−0.207837	−	0.978163i	$-0.566642\pi$
$631$	−4.15745e6	−0.415675	−0.207837	+	0.978163i	$0.566642\pi$
$632$	0	0
$633$	1.76105e7	1.74688
$634$	0	0
$635$	1.41002e7	1.38769
$636$	0	0
$637$	1.15539e7	1.12818
$638$	0	0
$639$	731326.	0.0708531
$640$	0	0
$641$	−1.00234e7	−0.963541	−0.481771	−	0.876297i	$-0.660006\pi$
$641$	−1.00234e7	−0.963541	−0.481771	+	0.876297i	$0.660006\pi$
$642$	0	0
$643$	−1.40797e7	−1.34297	−0.671484	−	0.741019i	$-0.734343\pi$
$643$	−1.40797e7	−1.34297	−0.671484	+	0.741019i	$0.734343\pi$
$644$	0	0
$645$	8.92312e6	0.844534
$646$	0	0
$647$	455312.	0.0427611	0.0213805	−	0.999771i	$-0.493194\pi$
$647$	455312.	0.0427611	0.0213805	+	0.999771i	$0.493194\pi$
$648$	0	0
$649$	−215926.	−0.0201230
$650$	0	0
$651$	2.60295e7	2.40721
$652$	0	0
$653$	4.53250e6	0.415964	0.207982	−	0.978133i	$-0.433311\pi$
$653$	4.53250e6	0.415964	0.207982	+	0.978133i	$0.433311\pi$
$654$	0	0
$655$	4.07828e6	0.371427
$656$	0	0
$657$	−3.17501e6	−0.286967
$658$	0	0
$659$	−6.56278e6	−0.588673	−0.294337	−	0.955702i	$-0.595099\pi$
$659$	−6.56278e6	−0.588673	−0.294337	+	0.955702i	$0.595099\pi$
$660$	0	0
$661$	−1.42036e7	−1.26443	−0.632215	−	0.774793i	$-0.717854\pi$
$661$	−1.42036e7	−1.26443	−0.632215	+	0.774793i	$0.717854\pi$
$662$	0	0
$663$	−8.03465e6	−0.709877
$664$	0	0
$665$	5.46735e6	0.479427
$666$	0	0
$667$	1.56750e7	1.36425
$668$	0	0
$669$	−613086.	−0.0529610
$670$	0	0
$671$	−353855.	−0.0303402
$672$	0	0
$673$	−1.35703e7	−1.15492	−0.577462	−	0.816418i	$-0.695957\pi$
$673$	−1.35703e7	−1.15492	−0.577462	+	0.816418i	$0.695957\pi$
$674$	0	0
$675$	1.80898e6	0.152818
$676$	0	0
$677$	−1.24089e7	−1.04055	−0.520275	−	0.853999i	$-0.674171\pi$
$677$	−1.24089e7	−1.04055	−0.520275	+	0.853999i	$0.674171\pi$
$678$	0	0
$679$	−297290.	−0.0247460
$680$	0	0
$681$	1.53632e6	0.126944
$682$	0	0
$683$	−2.14232e7	−1.75725	−0.878624	−	0.477513i	$-0.841538\pi$
$683$	−2.14232e7	−1.75725	−0.878624	+	0.477513i	$0.841538\pi$
$684$	0	0
$685$	1.19867e7	0.976049
$686$	0	0
$687$	2.15484e7	1.74190
$688$	0	0
$689$	−1.32069e7	−1.05987
$690$	0	0
$691$	−1.08375e7	−0.863446	−0.431723	−	0.902006i	$-0.642094\pi$
$691$	−1.08375e7	−0.863446	−0.431723	+	0.902006i	$0.642094\pi$
$692$	0	0
$693$	−172920.	−0.0136777
$694$	0	0
$695$	−466631.	−0.0366447
$696$	0	0
$697$	−7.11281e6	−0.554573
$698$	0	0
$699$	−1.62985e6	−0.126170
$700$	0	0
$701$	−1.86363e7	−1.43240	−0.716202	−	0.697893i	$-0.754121\pi$
$701$	−1.86363e7	−1.43240	−0.716202	+	0.697893i	$0.754121\pi$
$702$	0	0
$703$	−2.44122e6	−0.186303
$704$	0	0
$705$	1.24991e7	0.947122
$706$	0	0
$707$	2.05316e7	1.54481
$708$	0	0
$709$	1.62073e7	1.21086	0.605431	−	0.795897i	$-0.293001\pi$
$709$	1.62073e7	1.21086	0.605431	+	0.795897i	$0.293001\pi$
$710$	0	0
$711$	−3.62142e6	−0.268661
$712$	0	0
$713$	1.87771e7	1.38327
$714$	0	0
$715$	−864924.	−0.0632722
$716$	0	0
$717$	1.56071e7	1.13377
$718$	0	0
$719$	−4.93915e6	−0.356311	−0.178156	−	0.984002i	$-0.557013\pi$
$719$	−4.93915e6	−0.356311	−0.178156	+	0.984002i	$0.557013\pi$
$720$	0	0
$721$	3.17391e7	2.27382
$722$	0	0
$723$	1.41088e7	1.00379
$724$	0	0
$725$	−3.54970e6	−0.250811
$726$	0	0
$727$	−6.02918e6	−0.423080	−0.211540	−	0.977369i	$-0.567848\pi$
$727$	−6.02918e6	−0.423080	−0.211540	+	0.977369i	$0.567848\pi$
$728$	0	0
$729$	1.16055e7	0.808804
$730$	0	0
$731$	−7.35099e6	−0.508807
$732$	0	0
$733$	−1.33239e7	−0.915952	−0.457976	−	0.888965i	$-0.651425\pi$
$733$	−1.33239e7	−0.915952	−0.457976	+	0.888965i	$0.651425\pi$
$734$	0	0
$735$	−2.01854e7	−1.37822
$736$	0	0
$737$	−263848.	−0.0178931
$738$	0	0
$739$	7.00525e6	0.471859	0.235930	−	0.971770i	$-0.424187\pi$
$739$	7.00525e6	0.471859	0.235930	+	0.971770i	$0.424187\pi$
$740$	0	0
$741$	−4.55769e6	−0.304929
$742$	0	0
$743$	1.70727e7	1.13456	0.567282	−	0.823524i	$-0.307995\pi$
$743$	1.70727e7	1.13456	0.567282	+	0.823524i	$0.307995\pi$
$744$	0	0
$745$	1.41613e7	0.934789
$746$	0	0
$747$	1.40877e6	0.0923716
$748$	0	0
$749$	−2.28271e7	−1.48678
$750$	0	0
$751$	2.34322e7	1.51605	0.758025	−	0.652225i	$-0.226164\pi$
$751$	2.34322e7	1.51605	0.758025	+	0.652225i	$0.226164\pi$
$752$	0	0
$753$	2.13430e7	1.37173
$754$	0	0
$755$	2.22245e6	0.141894
$756$	0	0
$757$	1.21852e7	0.772847	0.386423	−	0.922321i	$-0.373710\pi$
$757$	1.21852e7	0.772847	0.386423	+	0.922321i	$0.373710\pi$
$758$	0	0
$759$	−957902.	−0.0603555
$760$	0	0
$761$	−2.81174e7	−1.76000	−0.880001	−	0.474973i	$-0.842458\pi$
$761$	−2.81174e7	−1.76000	−0.880001	+	0.474973i	$0.842458\pi$
$762$	0	0
$763$	−8.08663e6	−0.502871
$764$	0	0
$765$	1.82795e6	0.112931
$766$	0	0
$767$	5.03650e6	0.309129
$768$	0	0
$769$	8.77556e6	0.535130	0.267565	−	0.963540i	$-0.413781\pi$
$769$	8.77556e6	0.535130	0.267565	+	0.963540i	$0.413781\pi$
$770$	0	0
$771$	−1.10399e6	−0.0668851
$772$	0	0
$773$	9.98374e6	0.600959	0.300479	−	0.953788i	$-0.402853\pi$
$773$	9.98374e6	0.600959	0.300479	+	0.953788i	$0.402853\pi$
$774$	0	0
$775$	−4.25220e6	−0.254307
$776$	0	0
$777$	1.65899e7	0.985808
$778$	0	0
$779$	−4.03477e6	−0.238218
$780$	0	0
$781$	−498040.	−0.0292171
$782$	0	0
$783$	−2.34041e7	−1.36423
$784$	0	0
$785$	2.38522e7	1.38151
$786$	0	0
$787$	−2.52484e7	−1.45311	−0.726553	−	0.687110i	$-0.758879\pi$
$787$	−2.52484e7	−1.45311	−0.726553	+	0.687110i	$0.758879\pi$
$788$	0	0
$789$	−2.97479e7	−1.70123
$790$	0	0
$791$	3.71153e7	2.10917
$792$	0	0
$793$	8.25372e6	0.466087
$794$	0	0
$795$	2.30733e7	1.29477
$796$	0	0
$797$	−2.17795e7	−1.21451	−0.607255	−	0.794507i	$-0.707730\pi$
$797$	−2.17795e7	−1.21451	−0.607255	+	0.794507i	$0.707730\pi$
$798$	0	0
$799$	−1.02969e7	−0.570613
$800$	0	0
$801$	−4.82283e6	−0.265595
$802$	0	0
$803$	2.16221e6	0.118334
$804$	0	0
$805$	−2.68027e7	−1.45777
$806$	0	0
$807$	4.27931e6	0.231308
$808$	0	0
$809$	−1.72504e7	−0.926676	−0.463338	−	0.886182i	$-0.653348\pi$
$809$	−1.72504e7	−0.926676	−0.463338	+	0.886182i	$0.653348\pi$
$810$	0	0
$811$	1.84146e7	0.983129	0.491564	−	0.870841i	$-0.336425\pi$
$811$	1.84146e7	0.983129	0.491564	+	0.870841i	$0.336425\pi$
$812$	0	0
$813$	3.74105e7	1.98503
$814$	0	0
$815$	−3.22312e7	−1.69974
$816$	0	0
$817$	−4.16988e6	−0.218559
$818$	0	0
$819$	4.03339e6	0.210117
$820$	0	0
$821$	−3.82112e6	−0.197848	−0.0989242	−	0.995095i	$-0.531540\pi$
$821$	−3.82112e6	−0.197848	−0.0989242	+	0.995095i	$0.531540\pi$
$822$	0	0
$823$	8.52678e6	0.438819	0.219409	−	0.975633i	$-0.429587\pi$
$823$	8.52678e6	0.438819	0.219409	+	0.975633i	$0.429587\pi$
$824$	0	0
$825$	216923.	0.0110961
$826$	0	0
$827$	−105036.	−0.00534043	−0.00267022	−	0.999996i	$-0.500850\pi$
$827$	−105036.	−0.00534043	−0.00267022	+	0.999996i	$0.500850\pi$
$828$	0	0
$829$	1.02885e7	0.519953	0.259977	−	0.965615i	$-0.416285\pi$
$829$	1.02885e7	0.519953	0.259977	+	0.965615i	$0.416285\pi$
$830$	0	0
$831$	3.18948e7	1.60220
$832$	0	0
$833$	1.66291e7	0.830338
$834$	0	0
$835$	−9.68773e6	−0.480846
$836$	0	0
$837$	−2.80359e7	−1.38325
$838$	0	0
$839$	−6.89634e6	−0.338231	−0.169116	−	0.985596i	$-0.554091\pi$
$839$	−6.89634e6	−0.338231	−0.169116	+	0.985596i	$0.554091\pi$
$840$	0	0
$841$	2.54138e7	1.23903
$842$	0	0
$843$	3.74322e7	1.81416
$844$	0	0
$845$	−2.25358e6	−0.108575
$846$	0	0
$847$	−3.07769e7	−1.47406
$848$	0	0
$849$	−3.52689e7	−1.67928
$850$	0	0
$851$	1.19676e7	0.566480
$852$	0	0
$853$	1.83296e7	0.862544	0.431272	−	0.902222i	$-0.358065\pi$
$853$	1.83296e7	0.862544	0.431272	+	0.902222i	$0.358065\pi$
$854$	0	0
$855$	1.03692e6	0.0485096
$856$	0	0
$857$	4.71571e6	0.219329	0.109664	−	0.993969i	$-0.465022\pi$
$857$	4.71571e6	0.219329	0.109664	+	0.993969i	$0.465022\pi$
$858$	0	0
$859$	−1.03087e7	−0.476673	−0.238336	−	0.971183i	$-0.576602\pi$
$859$	−1.03087e7	−0.476673	−0.238336	+	0.971183i	$0.576602\pi$
$860$	0	0
$861$	2.74193e7	1.26052
$862$	0	0
$863$	3.48832e7	1.59437	0.797186	−	0.603734i	$-0.206321\pi$
$863$	3.48832e7	1.59437	0.797186	+	0.603734i	$0.206321\pi$
$864$	0	0
$865$	2.24129e7	1.01849
$866$	0	0
$867$	1.21685e7	0.549783
$868$	0	0
$869$	2.46622e6	0.110785
$870$	0	0
$871$	6.15430e6	0.274874
$872$	0	0
$873$	−56382.8	−0.00250386
$874$	0	0
$875$	−3.01418e7	−1.33091
$876$	0	0
$877$	1.73535e7	0.761881	0.380940	−	0.924600i	$-0.375600\pi$
$877$	1.73535e7	0.761881	0.380940	+	0.924600i	$0.375600\pi$
$878$	0	0
$879$	2.04318e7	0.891939
$880$	0	0
$881$	9.23892e6	0.401034	0.200517	−	0.979690i	$-0.435738\pi$
$881$	9.23892e6	0.401034	0.200517	+	0.979690i	$0.435738\pi$
$882$	0	0
$883$	−3.03072e7	−1.30811	−0.654056	−	0.756447i	$-0.726934\pi$
$883$	−3.03072e7	−1.30811	−0.654056	+	0.756447i	$0.726934\pi$
$884$	0	0
$885$	−8.79912e6	−0.377643
$886$	0	0
$887$	2.57983e7	1.10099	0.550493	−	0.834840i	$-0.314440\pi$
$887$	2.57983e7	1.10099	0.550493	+	0.834840i	$0.314440\pi$
$888$	0	0
$889$	4.47786e7	1.90028
$890$	0	0
$891$	1.64927e6	0.0695983
$892$	0	0
$893$	−5.84099e6	−0.245108
$894$	0	0
$895$	4.00299e7	1.67043
$896$	0	0
$897$	2.23432e7	0.927181
$898$	0	0
$899$	5.50137e7	2.27024
$900$	0	0
$901$	−1.90081e7	−0.780059
$902$	0	0
$903$	2.83375e7	1.15649
$904$	0	0
$905$	−4.29507e7	−1.74321
$906$	0	0
$907$	2.70407e7	1.09144	0.545721	−	0.837967i	$-0.316256\pi$
$907$	2.70407e7	1.09144	0.545721	+	0.837967i	$0.316256\pi$
$908$	0	0
$909$	3.89394e6	0.156308
$910$	0	0
$911$	−3.82336e7	−1.52633	−0.763166	−	0.646203i	$-0.776356\pi$
$911$	−3.82336e7	−1.52633	−0.763166	+	0.646203i	$0.776356\pi$
$912$	0	0
$913$	−959385.	−0.0380904
$914$	0	0
$915$	−1.44198e7	−0.569387
$916$	0	0
$917$	1.29516e7	0.508626
$918$	0	0
$919$	3.00956e7	1.17548	0.587739	−	0.809050i	$-0.300018\pi$
$919$	3.00956e7	1.17548	0.587739	+	0.809050i	$0.300018\pi$
$920$	0	0
$921$	−7.21667e6	−0.280342
$922$	0	0
$923$	1.16169e7	0.448833
$924$	0	0
$925$	−2.71015e6	−0.104145
$926$	0	0
$927$	6.01950e6	0.230071
$928$	0	0
$929$	2.12493e7	0.807801	0.403901	−	0.914803i	$-0.367654\pi$
$929$	2.12493e7	0.807801	0.403901	+	0.914803i	$0.367654\pi$
$930$	0	0
$931$	9.43290e6	0.356674
$932$	0	0
$933$	6.06358e6	0.228047
$934$	0	0
$935$	−1.24485e6	−0.0465682
$936$	0	0
$937$	−8.27995e6	−0.308091	−0.154045	−	0.988064i	$-0.549230\pi$
$937$	−8.27995e6	−0.308091	−0.154045	+	0.988064i	$0.549230\pi$
$938$	0	0
$939$	−6.30132e6	−0.233221
$940$	0	0
$941$	−3.84844e7	−1.41681	−0.708404	−	0.705807i	$-0.750585\pi$
$941$	−3.84844e7	−1.41681	−0.708404	+	0.705807i	$0.750585\pi$
$942$	0	0
$943$	1.97797e7	0.724337
$944$	0	0
$945$	4.00187e7	1.45775
$946$	0	0
$947$	757025.	0.0274306	0.0137153	−	0.999906i	$-0.495634\pi$
$947$	757025.	0.0274306	0.0137153	+	0.999906i	$0.495634\pi$
$948$	0	0
$949$	−5.04339e7	−1.81784
$950$	0	0
$951$	−5.17928e7	−1.85703
$952$	0	0
$953$	−1.23216e7	−0.439476	−0.219738	−	0.975559i	$-0.570520\pi$
$953$	−1.23216e7	−0.439476	−0.219738	+	0.975559i	$0.570520\pi$
$954$	0	0
$955$	−1.64233e7	−0.582709
$956$	0	0
$957$	−2.80649e6	−0.0990565
$958$	0	0
$959$	3.80665e7	1.33658
$960$	0	0
$961$	3.72720e7	1.30189
$962$	0	0
$963$	−4.32929e6	−0.150436
$964$	0	0
$965$	3.32924e7	1.15087
$966$	0	0
$967$	−1.78885e7	−0.615186	−0.307593	−	0.951518i	$-0.599524\pi$
$967$	−1.78885e7	−0.615186	−0.307593	+	0.951518i	$0.599524\pi$
$968$	0	0
$969$	−6.55971e6	−0.224427
$970$	0	0
$971$	3.00603e6	0.102317	0.0511583	−	0.998691i	$-0.483709\pi$
$971$	3.00603e6	0.102317	0.0511583	+	0.998691i	$0.483709\pi$
$972$	0	0
$973$	−1.48190e6	−0.0501806
$974$	0	0
$975$	−5.05976e6	−0.170458
$976$	0	0
$977$	1.39526e7	0.467647	0.233824	−	0.972279i	$-0.424876\pi$
$977$	1.39526e7	0.467647	0.233824	+	0.972279i	$0.424876\pi$
$978$	0	0
$979$	3.28439e6	0.109521
$980$	0	0
$981$	−1.53368e6	−0.0508817
$982$	0	0
$983$	3.60837e7	1.19104	0.595522	−	0.803339i	$-0.296945\pi$
$983$	3.60837e7	1.19104	0.595522	+	0.803339i	$0.296945\pi$
$984$	0	0
$985$	−2.38958e7	−0.784750
$986$	0	0
$987$	3.96939e7	1.29697
$988$	0	0
$989$	2.04421e7	0.664560
$990$	0	0
$991$	4.54924e7	1.47148	0.735741	−	0.677263i	$-0.236834\pi$
$991$	4.54924e7	1.47148	0.735741	+	0.677263i	$0.236834\pi$
$992$	0	0
$993$	−3.54359e7	−1.14044
$994$	0	0
$995$	−2.41086e6	−0.0771994
$996$	0	0
$997$	1.62907e7	0.519043	0.259521	−	0.965737i	$-0.416435\pi$
$997$	1.62907e7	0.519043	0.259521	+	0.965737i	$0.416435\pi$
$998$	0	0
$999$	−1.78687e7	−0.566473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.12	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.12	✓	49	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-16.7147 q^{3} +60.4053 q^{5} +191.832 q^{7} +36.3820 q^{9} +O(q^{10})\)	\(q-16.7147 q^{3} +60.4053 q^{5} +191.832 q^{7} +36.3820 q^{9} -24.7765 q^{11} +577.915 q^{13} -1009.66 q^{15} +831.771 q^{17} +471.826 q^{19} -3206.41 q^{21} -2313.04 q^{23} +523.802 q^{25} +3453.56 q^{27} -6776.80 q^{29} -8117.95 q^{31} +414.132 q^{33} +11587.6 q^{35} -5173.99 q^{37} -9659.68 q^{39} -8551.40 q^{41} -8837.76 q^{43} +2197.66 q^{45} -12379.5 q^{47} +19992.3 q^{49} -13902.8 q^{51} -22852.6 q^{53} -1496.63 q^{55} -7886.44 q^{57} +8714.95 q^{59} +14281.9 q^{61} +6979.21 q^{63} +34909.1 q^{65} +10649.1 q^{67} +38661.8 q^{69} +20101.3 q^{71} -87268.7 q^{73} -8755.20 q^{75} -4752.91 q^{77} -99538.8 q^{79} -66566.2 q^{81} +38721.6 q^{83} +50243.4 q^{85} +113272. q^{87} -132561. q^{89} +110862. q^{91} +135689. q^{93} +28500.8 q^{95} -1549.74 q^{97} -901.416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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