LMFDB - Embedded newform 1028.6.a.b.1.5

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:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-25.3624 q^{3} +63.2925 q^{5} -82.1666 q^{7} +400.249 q^{9} +O(q^{10})$	$q-25.3624 q^{3} +63.2925 q^{5} -82.1666 q^{7} +400.249 q^{9} -607.299 q^{11} -208.834 q^{13} -1605.25 q^{15} -298.599 q^{17} -2221.86 q^{19} +2083.94 q^{21} -3798.31 q^{23} +880.939 q^{25} -3988.22 q^{27} -4877.74 q^{29} +8125.40 q^{31} +15402.5 q^{33} -5200.53 q^{35} -3013.99 q^{37} +5296.53 q^{39} -7995.44 q^{41} +13199.3 q^{43} +25332.8 q^{45} +2267.38 q^{47} -10055.7 q^{49} +7573.16 q^{51} -14726.3 q^{53} -38437.5 q^{55} +56351.5 q^{57} -41642.5 q^{59} +46768.5 q^{61} -32887.1 q^{63} -13217.7 q^{65} +25327.5 q^{67} +96334.0 q^{69} -69245.9 q^{71} -61734.1 q^{73} -22342.7 q^{75} +49899.7 q^{77} -72310.5 q^{79} +3889.98 q^{81} -93951.4 q^{83} -18899.0 q^{85} +123711. q^{87} +101944. q^{89} +17159.2 q^{91} -206079. q^{93} -140627. q^{95} +182943. q^{97} -243071. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * 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
$3$	−25.3624	−1.62700	−0.813498	−	0.581568i	$-0.802440\pi$
$3$	−25.3624	−1.62700	−0.813498	+	0.581568i	$0.802440\pi$
$4$	0	0
$5$	63.2925	1.13221	0.566105	−	0.824333i	$-0.308450\pi$
$5$	63.2925	1.13221	0.566105	+	0.824333i	$0.308450\pi$
$6$	0	0
$7$	−82.1666	−0.633797	−0.316898	−	0.948459i	$-0.602641\pi$
$7$	−82.1666	−0.633797	−0.316898	+	0.948459i	$0.602641\pi$
$8$	0	0
$9$	400.249	1.64712
$10$	0	0
$11$	−607.299	−1.51329	−0.756643	−	0.653828i	$-0.773162\pi$
$11$	−607.299	−1.51329	−0.756643	+	0.653828i	$0.773162\pi$
$12$	0	0
$13$	−208.834	−0.342723	−0.171362	−	0.985208i	$-0.554817\pi$
$13$	−208.834	−0.342723	−0.171362	+	0.985208i	$0.554817\pi$
$14$	0	0
$15$	−1605.25	−1.84210
$16$	0	0
$17$	−298.599	−0.250591	−0.125295	−	0.992119i	$-0.539988\pi$
$17$	−298.599	−0.250591	−0.125295	+	0.992119i	$0.539988\pi$
$18$	0	0
$19$	−2221.86	−1.41199	−0.705995	−	0.708217i	$-0.749500\pi$
$19$	−2221.86	−1.41199	−0.705995	+	0.708217i	$0.749500\pi$
$20$	0	0
$21$	2083.94	1.03118
$22$	0	0
$23$	−3798.31	−1.49717	−0.748584	−	0.663040i	$-0.769266\pi$
$23$	−3798.31	−1.49717	−0.748584	+	0.663040i	$0.769266\pi$
$24$	0	0
$25$	880.939	0.281901
$26$	0	0
$27$	−3988.22	−1.05286
$28$	0	0
$29$	−4877.74	−1.07702	−0.538510	−	0.842619i	$-0.681013\pi$
$29$	−4877.74	−1.07702	−0.538510	+	0.842619i	$0.681013\pi$
$30$	0	0
$31$	8125.40	1.51859	0.759295	−	0.650746i	$-0.225544\pi$
$31$	8125.40	1.51859	0.759295	+	0.650746i	$0.225544\pi$
$32$	0	0
$33$	15402.5	2.46211
$34$	0	0
$35$	−5200.53	−0.717591
$36$	0	0
$37$	−3013.99	−0.361940	−0.180970	−	0.983489i	$-0.557924\pi$
$37$	−3013.99	−0.361940	−0.180970	+	0.983489i	$0.557924\pi$
$38$	0	0
$39$	5296.53	0.557610
$40$	0	0
$41$	−7995.44	−0.742819	−0.371409	−	0.928469i	$-0.621125\pi$
$41$	−7995.44	−0.742819	−0.371409	+	0.928469i	$0.621125\pi$
$42$	0	0
$43$	13199.3	1.08863	0.544313	−	0.838882i	$-0.316791\pi$
$43$	13199.3	1.08863	0.544313	+	0.838882i	$0.316791\pi$
$44$	0	0
$45$	25332.8	1.86488
$46$	0	0
$47$	2267.38	0.149720	0.0748600	−	0.997194i	$-0.476149\pi$
$47$	2267.38	0.149720	0.0748600	+	0.997194i	$0.476149\pi$
$48$	0	0
$49$	−10055.7	−0.598302
$50$	0	0
$51$	7573.16	0.407710
$52$	0	0
$53$	−14726.3	−0.720118	−0.360059	−	0.932929i	$-0.617243\pi$
$53$	−14726.3	−0.720118	−0.360059	+	0.932929i	$0.617243\pi$
$54$	0	0
$55$	−38437.5	−1.71336
$56$	0	0
$57$	56351.5	2.29730
$58$	0	0
$59$	−41642.5	−1.55742	−0.778711	−	0.627382i	$-0.784126\pi$
$59$	−41642.5	−1.55742	−0.778711	+	0.627382i	$0.784126\pi$
$60$	0	0
$61$	46768.5	1.60927	0.804635	−	0.593769i	$-0.202361\pi$
$61$	46768.5	1.60927	0.804635	+	0.593769i	$0.202361\pi$
$62$	0	0
$63$	−32887.1	−1.04394
$64$	0	0
$65$	−13217.7	−0.388035
$66$	0	0
$67$	25327.5	0.689297	0.344648	−	0.938732i	$-0.387998\pi$
$67$	25327.5	0.689297	0.344648	+	0.938732i	$0.387998\pi$
$68$	0	0
$69$	96334.0	2.43589
$70$	0	0
$71$	−69245.9	−1.63023	−0.815114	−	0.579301i	$-0.803326\pi$
$71$	−69245.9	−1.63023	−0.815114	+	0.579301i	$0.803326\pi$
$72$	0	0
$73$	−61734.1	−1.35587	−0.677935	−	0.735121i	$-0.737125\pi$
$73$	−61734.1	−1.35587	−0.677935	+	0.735121i	$0.737125\pi$
$74$	0	0
$75$	−22342.7	−0.458651
$76$	0	0
$77$	49899.7	0.959116
$78$	0	0
$79$	−72310.5	−1.30357	−0.651784	−	0.758405i	$-0.725979\pi$
$79$	−72310.5	−1.30357	−0.651784	+	0.758405i	$0.725979\pi$
$80$	0	0
$81$	3889.98	0.0658772
$82$	0	0
$83$	−93951.4	−1.49695	−0.748476	−	0.663162i	$-0.769214\pi$
$83$	−93951.4	−1.49695	−0.748476	+	0.663162i	$0.769214\pi$
$84$	0	0
$85$	−18899.0	−0.283722
$86$	0	0
$87$	123711.	1.75231
$88$	0	0
$89$	101944.	1.36422	0.682110	−	0.731249i	$-0.261062\pi$
$89$	101944.	1.36422	0.682110	+	0.731249i	$0.261062\pi$
$90$	0	0
$91$	17159.2	0.217217
$92$	0	0
$93$	−206079.	−2.47074
$94$	0	0
$95$	−140627.	−1.59867
$96$	0	0
$97$	182943.	1.97418	0.987089	−	0.160172i	$-0.0512048\pi$
$97$	182943.	1.97418	0.987089	+	0.160172i	$0.0512048\pi$
$98$	0	0
$99$	−243071.	−2.49256
$100$	0	0
$101$	−112186.	−1.09430	−0.547151	−	0.837034i	$-0.684288\pi$
$101$	−112186.	−1.09430	−0.547151	+	0.837034i	$0.684288\pi$
$102$	0	0
$103$	−162481.	−1.50907	−0.754536	−	0.656259i	$-0.772138\pi$
$103$	−162481.	−1.50907	−0.754536	+	0.656259i	$0.772138\pi$
$104$	0	0
$105$	131898.	1.16752
$106$	0	0
$107$	−125621.	−1.06073	−0.530364	−	0.847770i	$-0.677945\pi$
$107$	−125621.	−1.06073	−0.530364	+	0.847770i	$0.677945\pi$
$108$	0	0
$109$	167703.	1.35200	0.675998	−	0.736904i	$-0.263713\pi$
$109$	167703.	1.35200	0.675998	+	0.736904i	$0.263713\pi$
$110$	0	0
$111$	76441.8	0.588875
$112$	0	0
$113$	−36562.4	−0.269364	−0.134682	−	0.990889i	$-0.543001\pi$
$113$	−36562.4	−0.269364	−0.134682	+	0.990889i	$0.543001\pi$
$114$	0	0
$115$	−240404.	−1.69511
$116$	0	0
$117$	−83585.9	−0.564505
$118$	0	0
$119$	24534.8	0.158824
$120$	0	0
$121$	207762.	1.29004
$122$	0	0
$123$	202783.	1.20856
$124$	0	0
$125$	−142032.	−0.813040
$126$	0	0
$127$	52866.4	0.290851	0.145426	−	0.989369i	$-0.453545\pi$
$127$	52866.4	0.290851	0.145426	+	0.989369i	$0.453545\pi$
$128$	0	0
$129$	−334764.	−1.77119
$130$	0	0
$131$	30684.5	0.156222	0.0781108	−	0.996945i	$-0.475111\pi$
$131$	30684.5	0.156222	0.0781108	+	0.996945i	$0.475111\pi$
$132$	0	0
$133$	182562.	0.894915
$134$	0	0
$135$	−252424.	−1.19206
$136$	0	0
$137$	−230918.	−1.05113	−0.525565	−	0.850754i	$-0.676146\pi$
$137$	−230918.	−1.05113	−0.525565	+	0.850754i	$0.676146\pi$
$138$	0	0
$139$	−46789.9	−0.205407	−0.102703	−	0.994712i	$-0.532749\pi$
$139$	−46789.9	−0.205407	−0.102703	+	0.994712i	$0.532749\pi$
$140$	0	0
$141$	−57506.1	−0.243594
$142$	0	0
$143$	126825.	0.518639
$144$	0	0
$145$	−308724.	−1.21941
$146$	0	0
$147$	255035.	0.973435
$148$	0	0
$149$	414691.	1.53024	0.765119	−	0.643889i	$-0.222680\pi$
$149$	414691.	1.53024	0.765119	+	0.643889i	$0.222680\pi$
$150$	0	0
$151$	287009.	1.02436	0.512180	−	0.858878i	$-0.328838\pi$
$151$	287009.	1.02436	0.512180	+	0.858878i	$0.328838\pi$
$152$	0	0
$153$	−119514.	−0.412753
$154$	0	0
$155$	514277.	1.71936
$156$	0	0
$157$	−42288.0	−0.136921	−0.0684603	−	0.997654i	$-0.521809\pi$
$157$	−42288.0	−0.136921	−0.0684603	+	0.997654i	$0.521809\pi$
$158$	0	0
$159$	373494.	1.17163
$160$	0	0
$161$	312094.	0.948900
$162$	0	0
$163$	−346287.	−1.02086	−0.510431	−	0.859919i	$-0.670514\pi$
$163$	−346287.	−1.02086	−0.510431	+	0.859919i	$0.670514\pi$
$164$	0	0
$165$	974866.	2.78763
$166$	0	0
$167$	−598136.	−1.65962	−0.829810	−	0.558047i	$-0.811551\pi$
$167$	−598136.	−1.65962	−0.829810	+	0.558047i	$0.811551\pi$
$168$	0	0
$169$	−327681.	−0.882541
$170$	0	0
$171$	−889296.	−2.32571
$172$	0	0
$173$	364744.	0.926559	0.463280	−	0.886212i	$-0.346673\pi$
$173$	364744.	0.926559	0.463280	+	0.886212i	$0.346673\pi$
$174$	0	0
$175$	−72383.7	−0.178668
$176$	0	0
$177$	1.05615e6	2.53392
$178$	0	0
$179$	−212773.	−0.496346	−0.248173	−	0.968716i	$-0.579830\pi$
$179$	−212773.	−0.496346	−0.248173	+	0.968716i	$0.579830\pi$
$180$	0	0
$181$	62180.1	0.141077	0.0705383	−	0.997509i	$-0.477528\pi$
$181$	62180.1	0.141077	0.0705383	+	0.997509i	$0.477528\pi$
$182$	0	0
$183$	−1.18616e6	−2.61828
$184$	0	0
$185$	−190763.	−0.409793
$186$	0	0
$187$	181339.	0.379216
$188$	0	0
$189$	327698.	0.667297
$190$	0	0
$191$	−709486.	−1.40721	−0.703607	−	0.710589i	$-0.748428\pi$
$191$	−709486.	−1.40721	−0.703607	+	0.710589i	$0.748428\pi$
$192$	0	0
$193$	−200287.	−0.387044	−0.193522	−	0.981096i	$-0.561991\pi$
$193$	−200287.	−0.387044	−0.193522	+	0.981096i	$0.561991\pi$
$194$	0	0
$195$	335231.	0.631331
$196$	0	0
$197$	−485050.	−0.890473	−0.445237	−	0.895413i	$-0.646880\pi$
$197$	−485050.	−0.890473	−0.445237	+	0.895413i	$0.646880\pi$
$198$	0	0
$199$	252763.	0.452460	0.226230	−	0.974074i	$-0.427360\pi$
$199$	252763.	0.452460	0.226230	+	0.974074i	$0.427360\pi$
$200$	0	0
$201$	−642366.	−1.12148
$202$	0	0
$203$	400787.	0.682612
$204$	0	0
$205$	−506052.	−0.841027
$206$	0	0
$207$	−1.52027e6	−2.46601
$208$	0	0
$209$	1.34933e6	2.13675
$210$	0	0
$211$	919403.	1.42167	0.710837	−	0.703357i	$-0.248316\pi$
$211$	919403.	1.42167	0.710837	+	0.703357i	$0.248316\pi$
$212$	0	0
$213$	1.75624e6	2.65237
$214$	0	0
$215$	835414.	1.23255
$216$	0	0
$217$	−667636.	−0.962478
$218$	0	0
$219$	1.56572e6	2.20600
$220$	0	0
$221$	62357.6	0.0858834
$222$	0	0
$223$	425712.	0.573263	0.286631	−	0.958041i	$-0.407465\pi$
$223$	425712.	0.573263	0.286631	+	0.958041i	$0.407465\pi$
$224$	0	0
$225$	352595.	0.464323
$226$	0	0
$227$	435571.	0.561041	0.280520	−	0.959848i	$-0.409493\pi$
$227$	435571.	0.561041	0.280520	+	0.959848i	$0.409493\pi$
$228$	0	0
$229$	−23898.9	−0.0301154	−0.0150577	−	0.999887i	$-0.504793\pi$
$229$	−23898.9	−0.0301154	−0.0150577	+	0.999887i	$0.504793\pi$
$230$	0	0
$231$	−1.26557e6	−1.56048
$232$	0	0
$233$	615057.	0.742208	0.371104	−	0.928591i	$-0.378979\pi$
$233$	615057.	0.742208	0.371104	+	0.928591i	$0.378979\pi$
$234$	0	0
$235$	143508.	0.169514
$236$	0	0
$237$	1.83396e6	2.12090
$238$	0	0
$239$	−1.63380e6	−1.85013	−0.925067	−	0.379805i	$-0.875991\pi$
$239$	−1.63380e6	−1.85013	−0.925067	+	0.379805i	$0.875991\pi$
$240$	0	0
$241$	−21467.7	−0.0238090	−0.0119045	−	0.999929i	$-0.503789\pi$
$241$	−21467.7	−0.0238090	−0.0119045	+	0.999929i	$0.503789\pi$
$242$	0	0
$243$	870477.	0.945675
$244$	0	0
$245$	−636447.	−0.677403
$246$	0	0
$247$	464000.	0.483922
$248$	0	0
$249$	2.38283e6	2.43554
$250$	0	0
$251$	1.76133e6	1.76464	0.882322	−	0.470646i	$-0.155979\pi$
$251$	1.76133e6	1.76464	0.882322	+	0.470646i	$0.155979\pi$
$252$	0	0
$253$	2.30671e6	2.26564
$254$	0	0
$255$	479324.	0.461614
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	247649.	0.229397
$260$	0	0
$261$	−1.95231e6	−1.77398
$262$	0	0
$263$	656233.	0.585018	0.292509	−	0.956263i	$-0.405510\pi$
$263$	656233.	0.585018	0.292509	+	0.956263i	$0.405510\pi$
$264$	0	0
$265$	−932064.	−0.815325
$266$	0	0
$267$	−2.58553e6	−2.21958
$268$	0	0
$269$	2.02706e6	1.70799	0.853996	−	0.520279i	$-0.174172\pi$
$269$	2.02706e6	1.70799	0.853996	+	0.520279i	$0.174172\pi$
$270$	0	0
$271$	1.13734e6	0.940737	0.470369	−	0.882470i	$-0.344121\pi$
$271$	1.13734e6	0.940737	0.470369	+	0.882470i	$0.344121\pi$
$272$	0	0
$273$	−435198.	−0.353411
$274$	0	0
$275$	−534994.	−0.426596
$276$	0	0
$277$	−1.23270e6	−0.965294	−0.482647	−	0.875815i	$-0.660325\pi$
$277$	−1.23270e6	−0.965294	−0.482647	+	0.875815i	$0.660325\pi$
$278$	0	0
$279$	3.25219e6	2.50130
$280$	0	0
$281$	−193406.	−0.146118	−0.0730592	−	0.997328i	$-0.523276\pi$
$281$	−193406.	−0.146118	−0.0730592	+	0.997328i	$0.523276\pi$
$282$	0	0
$283$	951891.	0.706514	0.353257	−	0.935526i	$-0.385074\pi$
$283$	951891.	0.706514	0.353257	+	0.935526i	$0.385074\pi$
$284$	0	0
$285$	3.56663e6	2.60103
$286$	0	0
$287$	656958.	0.470796
$288$	0	0
$289$	−1.33070e6	−0.937204
$290$	0	0
$291$	−4.63987e6	−3.21198
$292$	0	0
$293$	−148771.	−0.101239	−0.0506197	−	0.998718i	$-0.516120\pi$
$293$	−148771.	−0.101239	−0.0506197	+	0.998718i	$0.516120\pi$
$294$	0	0
$295$	−2.63566e6	−1.76333
$296$	0	0
$297$	2.42204e6	1.59327
$298$	0	0
$299$	793217.	0.513114
$300$	0	0
$301$	−1.08454e6	−0.689967
$302$	0	0
$303$	2.84531e6	1.78042
$304$	0	0
$305$	2.96010e6	1.82203
$306$	0	0
$307$	1.36271e6	0.825196	0.412598	−	0.910913i	$-0.364621\pi$
$307$	1.36271e6	0.825196	0.412598	+	0.910913i	$0.364621\pi$
$308$	0	0
$309$	4.12091e6	2.45525
$310$	0	0
$311$	3.24022e6	1.89965	0.949825	−	0.312781i	$-0.101260\pi$
$311$	3.24022e6	1.89965	0.949825	+	0.312781i	$0.101260\pi$
$312$	0	0
$313$	2.79034e6	1.60989	0.804946	−	0.593348i	$-0.202194\pi$
$313$	2.79034e6	1.60989	0.804946	+	0.593348i	$0.202194\pi$
$314$	0	0
$315$	−2.08151e6	−1.18196
$316$	0	0
$317$	2.89174e6	1.61626	0.808129	−	0.589005i	$-0.200480\pi$
$317$	2.89174e6	1.61626	0.808129	+	0.589005i	$0.200480\pi$
$318$	0	0
$319$	2.96225e6	1.62984
$320$	0	0
$321$	3.18606e6	1.72580
$322$	0	0
$323$	663443.	0.353832
$324$	0	0
$325$	−183970.	−0.0966139
$326$	0	0
$327$	−4.25335e6	−2.19969
$328$	0	0
$329$	−186303.	−0.0948920
$330$	0	0
$331$	1.61923e6	0.812342	0.406171	−	0.913797i	$-0.366864\pi$
$331$	1.61923e6	0.812342	0.406171	+	0.913797i	$0.366864\pi$
$332$	0	0
$333$	−1.20635e6	−0.596158
$334$	0	0
$335$	1.60304e6	0.780429
$336$	0	0
$337$	87783.9	0.0421056	0.0210528	−	0.999778i	$-0.493298\pi$
$337$	87783.9	0.0421056	0.0210528	+	0.999778i	$0.493298\pi$
$338$	0	0
$339$	927309.	0.438253
$340$	0	0
$341$	−4.93455e6	−2.29806
$342$	0	0
$343$	2.20721e6	1.01300
$344$	0	0
$345$	6.09722e6	2.75794
$346$	0	0
$347$	3.32234e6	1.48123	0.740613	−	0.671932i	$-0.234535\pi$
$347$	3.32234e6	1.48123	0.740613	+	0.671932i	$0.234535\pi$
$348$	0	0
$349$	−1.75607e6	−0.771754	−0.385877	−	0.922550i	$-0.626101\pi$
$349$	−1.75607e6	−0.771754	−0.385877	+	0.922550i	$0.626101\pi$
$350$	0	0
$351$	832877.	0.360839
$352$	0	0
$353$	−1.36787e6	−0.584263	−0.292132	−	0.956378i	$-0.594365\pi$
$353$	−1.36787e6	−0.584263	−0.292132	+	0.956378i	$0.594365\pi$
$354$	0	0
$355$	−4.38275e6	−1.84576
$356$	0	0
$357$	−622261.	−0.258406
$358$	0	0
$359$	−2.66092e6	−1.08967	−0.544836	−	0.838543i	$-0.683408\pi$
$359$	−2.66092e6	−1.08967	−0.544836	+	0.838543i	$0.683408\pi$
$360$	0	0
$361$	2.46054e6	0.993717
$362$	0	0
$363$	−5.26933e6	−2.09888
$364$	0	0
$365$	−3.90731e6	−1.53513
$366$	0	0
$367$	2.90557e6	1.12607	0.563036	−	0.826432i	$-0.309633\pi$
$367$	2.90557e6	1.12607	0.563036	+	0.826432i	$0.309633\pi$
$368$	0	0
$369$	−3.20017e6	−1.22351
$370$	0	0
$371$	1.21001e6	0.456409
$372$	0	0
$373$	2.73675e6	1.01850	0.509252	−	0.860618i	$-0.329922\pi$
$373$	2.73675e6	1.01850	0.509252	+	0.860618i	$0.329922\pi$
$374$	0	0
$375$	3.60227e6	1.32281
$376$	0	0
$377$	1.01864e6	0.369120
$378$	0	0
$379$	−4.40613e6	−1.57565	−0.787825	−	0.615899i	$-0.788793\pi$
$379$	−4.40613e6	−1.57565	−0.787825	+	0.615899i	$0.788793\pi$
$380$	0	0
$381$	−1.34082e6	−0.473214
$382$	0	0
$383$	−4.35009e6	−1.51531	−0.757654	−	0.652656i	$-0.773655\pi$
$383$	−4.35009e6	−1.51531	−0.757654	+	0.652656i	$0.773655\pi$
$384$	0	0
$385$	3.15828e6	1.08592
$386$	0	0
$387$	5.28300e6	1.79309
$388$	0	0
$389$	−1.93124e6	−0.647086	−0.323543	−	0.946213i	$-0.604874\pi$
$389$	−1.93124e6	−0.647086	−0.323543	+	0.946213i	$0.604874\pi$
$390$	0	0
$391$	1.13417e6	0.375177
$392$	0	0
$393$	−778232.	−0.254172
$394$	0	0
$395$	−4.57671e6	−1.47591
$396$	0	0
$397$	−2.57031e6	−0.818481	−0.409240	−	0.912427i	$-0.634206\pi$
$397$	−2.57031e6	−0.818481	−0.409240	+	0.912427i	$0.634206\pi$
$398$	0	0
$399$	−4.63021e6	−1.45602
$400$	0	0
$401$	−5.13571e6	−1.59492	−0.797461	−	0.603370i	$-0.793824\pi$
$401$	−5.13571e6	−1.59492	−0.797461	+	0.603370i	$0.793824\pi$
$402$	0	0
$403$	−1.69686e6	−0.520456
$404$	0	0
$405$	246207.	0.0745868
$406$	0	0
$407$	1.83039e6	0.547719
$408$	0	0
$409$	−106740.	−0.0315514	−0.0157757	−	0.999876i	$-0.505022\pi$
$409$	−106740.	−0.0315514	−0.0157757	+	0.999876i	$0.505022\pi$
$410$	0	0
$411$	5.85662e6	1.71018
$412$	0	0
$413$	3.42162e6	0.987089
$414$	0	0
$415$	−5.94642e6	−1.69487
$416$	0	0
$417$	1.18670e6	0.334196
$418$	0	0
$419$	−4.05926e6	−1.12957	−0.564783	−	0.825239i	$-0.691040\pi$
$419$	−4.05926e6	−1.12957	−0.564783	+	0.825239i	$0.691040\pi$
$420$	0	0
$421$	−2.66663e6	−0.733259	−0.366630	−	0.930367i	$-0.619488\pi$
$421$	−2.66663e6	−0.733259	−0.366630	+	0.930367i	$0.619488\pi$
$422$	0	0
$423$	907518.	0.246606
$424$	0	0
$425$	−263047.	−0.0706417
$426$	0	0
$427$	−3.84281e6	−1.01995
$428$	0	0
$429$	−3.21658e6	−0.843823
$430$	0	0
$431$	−315413.	−0.0817873	−0.0408937	−	0.999164i	$-0.513020\pi$
$431$	−315413.	−0.0817873	−0.0408937	+	0.999164i	$0.513020\pi$
$432$	0	0
$433$	−775798.	−0.198852	−0.0994258	−	0.995045i	$-0.531701\pi$
$433$	−775798.	−0.198852	−0.0994258	+	0.995045i	$0.531701\pi$
$434$	0	0
$435$	7.82998e6	1.98398
$436$	0	0
$437$	8.43929e6	2.11399
$438$	0	0
$439$	−5.94359e6	−1.47193	−0.735966	−	0.677019i	$-0.763272\pi$
$439$	−5.94359e6	−1.47193	−0.735966	+	0.677019i	$0.763272\pi$
$440$	0	0
$441$	−4.02477e6	−0.985473
$442$	0	0
$443$	−4.86038e6	−1.17669	−0.588344	−	0.808611i	$-0.700220\pi$
$443$	−4.86038e6	−1.17669	−0.588344	+	0.808611i	$0.700220\pi$
$444$	0	0
$445$	6.45226e6	1.54458
$446$	0	0
$447$	−1.05175e7	−2.48969
$448$	0	0
$449$	−2.82060e6	−0.660276	−0.330138	−	0.943933i	$-0.607095\pi$
$449$	−2.82060e6	−0.660276	−0.330138	+	0.943933i	$0.607095\pi$
$450$	0	0
$451$	4.85563e6	1.12410
$452$	0	0
$453$	−7.27922e6	−1.66663
$454$	0	0
$455$	1.08605e6	0.245935
$456$	0	0
$457$	3.15540e6	0.706747	0.353374	−	0.935482i	$-0.385034\pi$
$457$	3.15540e6	0.706747	0.353374	+	0.935482i	$0.385034\pi$
$458$	0	0
$459$	1.19088e6	0.263836
$460$	0	0
$461$	−4.38267e6	−0.960476	−0.480238	−	0.877138i	$-0.659450\pi$
$461$	−4.38267e6	−0.960476	−0.480238	+	0.877138i	$0.659450\pi$
$462$	0	0
$463$	1.27328e6	0.276040	0.138020	−	0.990429i	$-0.455926\pi$
$463$	1.27328e6	0.276040	0.138020	+	0.990429i	$0.455926\pi$
$464$	0	0
$465$	−1.30433e7	−2.79740
$466$	0	0
$467$	7.74440e6	1.64322	0.821610	−	0.570050i	$-0.193076\pi$
$467$	7.74440e6	1.64322	0.821610	+	0.570050i	$0.193076\pi$
$468$	0	0
$469$	−2.08108e6	−0.436874
$470$	0	0
$471$	1.07252e6	0.222769
$472$	0	0
$473$	−8.01590e6	−1.64740
$474$	0	0
$475$	−1.95732e6	−0.398041
$476$	0	0
$477$	−5.89419e6	−1.18612
$478$	0	0
$479$	−469201.	−0.0934372	−0.0467186	−	0.998908i	$-0.514876\pi$
$479$	−469201.	−0.0934372	−0.0467186	+	0.998908i	$0.514876\pi$
$480$	0	0
$481$	629424.	0.124045
$482$	0	0
$483$	−7.91544e6	−1.54386
$484$	0	0
$485$	1.15789e7	2.23519
$486$	0	0
$487$	−3.56737e6	−0.681595	−0.340797	−	0.940137i	$-0.610697\pi$
$487$	−3.56737e6	−0.681595	−0.340797	+	0.940137i	$0.610697\pi$
$488$	0	0
$489$	8.78266e6	1.66094
$490$	0	0
$491$	−1.44437e6	−0.270381	−0.135190	−	0.990820i	$-0.543165\pi$
$491$	−1.44437e6	−0.270381	−0.135190	+	0.990820i	$0.543165\pi$
$492$	0	0
$493$	1.45649e6	0.269891
$494$	0	0
$495$	−1.53846e7	−2.82210
$496$	0	0
$497$	5.68970e6	1.03323
$498$	0	0
$499$	1.67300e6	0.300776	0.150388	−	0.988627i	$-0.451948\pi$
$499$	1.67300e6	0.300776	0.150388	+	0.988627i	$0.451948\pi$
$500$	0	0
$501$	1.51701e7	2.70019
$502$	0	0
$503$	6.59839e6	1.16283	0.581417	−	0.813606i	$-0.302498\pi$
$503$	6.59839e6	1.16283	0.581417	+	0.813606i	$0.302498\pi$
$504$	0	0
$505$	−7.10056e6	−1.23898
$506$	0	0
$507$	8.31077e6	1.43589
$508$	0	0
$509$	−6.45608e6	−1.10452	−0.552261	−	0.833671i	$-0.686235\pi$
$509$	−6.45608e6	−1.10452	−0.552261	+	0.833671i	$0.686235\pi$
$510$	0	0
$511$	5.07248e6	0.859347
$512$	0	0
$513$	8.86124e6	1.48662
$514$	0	0
$515$	−1.02838e7	−1.70859
$516$	0	0
$517$	−1.37698e6	−0.226569
$518$	0	0
$519$	−9.25077e6	−1.50751
$520$	0	0
$521$	7.93220e6	1.28026	0.640132	−	0.768265i	$-0.278880\pi$
$521$	7.93220e6	1.28026	0.640132	+	0.768265i	$0.278880\pi$
$522$	0	0
$523$	4.63673e6	0.741238	0.370619	−	0.928785i	$-0.379146\pi$
$523$	4.63673e6	0.741238	0.370619	+	0.928785i	$0.379146\pi$
$524$	0	0
$525$	1.83582e6	0.290692
$526$	0	0
$527$	−2.42623e6	−0.380545
$528$	0	0
$529$	7.99079e6	1.24151
$530$	0	0
$531$	−1.66674e7	−2.56526
$532$	0	0
$533$	1.66972e6	0.254581
$534$	0	0
$535$	−7.95089e6	−1.20097
$536$	0	0
$537$	5.39643e6	0.807553
$538$	0	0
$539$	6.10679e6	0.905402
$540$	0	0
$541$	−8.97650e6	−1.31860	−0.659301	−	0.751879i	$-0.729148\pi$
$541$	−8.97650e6	−1.31860	−0.659301	+	0.751879i	$0.729148\pi$
$542$	0	0
$543$	−1.57703e6	−0.229531
$544$	0	0
$545$	1.06144e7	1.53074
$546$	0	0
$547$	3.21337e6	0.459191	0.229595	−	0.973286i	$-0.426260\pi$
$547$	3.21337e6	0.459191	0.229595	+	0.973286i	$0.426260\pi$
$548$	0	0
$549$	1.87191e7	2.65066
$550$	0	0
$551$	1.08376e7	1.52074
$552$	0	0
$553$	5.94150e6	0.826197
$554$	0	0
$555$	4.83819e6	0.666731
$556$	0	0
$557$	1.38454e7	1.89089	0.945445	−	0.325782i	$-0.105627\pi$
$557$	1.38454e7	1.89089	0.945445	+	0.325782i	$0.105627\pi$
$558$	0	0
$559$	−2.75646e6	−0.373097
$560$	0	0
$561$	−4.59918e6	−0.616983
$562$	0	0
$563$	−4.89486e6	−0.650832	−0.325416	−	0.945571i	$-0.605504\pi$
$563$	−4.89486e6	−0.650832	−0.325416	+	0.945571i	$0.605504\pi$
$564$	0	0
$565$	−2.31413e6	−0.304976
$566$	0	0
$567$	−319626.	−0.0417527
$568$	0	0
$569$	3.22650e6	0.417783	0.208891	−	0.977939i	$-0.433014\pi$
$569$	3.22650e6	0.417783	0.208891	+	0.977939i	$0.433014\pi$
$570$	0	0
$571$	−8.95775e6	−1.14976	−0.574882	−	0.818236i	$-0.694952\pi$
$571$	−8.95775e6	−1.14976	−0.574882	+	0.818236i	$0.694952\pi$
$572$	0	0
$573$	1.79942e7	2.28953
$574$	0	0
$575$	−3.34608e6	−0.422052
$576$	0	0
$577$	−4.72902e6	−0.591333	−0.295666	−	0.955291i	$-0.595542\pi$
$577$	−4.72902e6	−0.591333	−0.295666	+	0.955291i	$0.595542\pi$
$578$	0	0
$579$	5.07976e6	0.629719
$580$	0	0
$581$	7.71966e6	0.948764
$582$	0	0
$583$	8.94327e6	1.08975
$584$	0	0
$585$	−5.29036e6	−0.639139
$586$	0	0
$587$	1.36901e7	1.63987	0.819936	−	0.572456i	$-0.194009\pi$
$587$	1.36901e7	1.63987	0.819936	+	0.572456i	$0.194009\pi$
$588$	0	0
$589$	−1.80535e7	−2.14423
$590$	0	0
$591$	1.23020e7	1.44880
$592$	0	0
$593$	1.19933e7	1.40056	0.700281	−	0.713868i	$-0.253058\pi$
$593$	1.19933e7	1.40056	0.700281	+	0.713868i	$0.253058\pi$
$594$	0	0
$595$	1.55287e6	0.179822
$596$	0	0
$597$	−6.41065e6	−0.736150
$598$	0	0
$599$	5.68612e6	0.647514	0.323757	−	0.946140i	$-0.395054\pi$
$599$	5.68612e6	0.647514	0.323757	+	0.946140i	$0.395054\pi$
$600$	0	0
$601$	−1.42690e7	−1.61142	−0.805710	−	0.592310i	$-0.798216\pi$
$601$	−1.42690e7	−1.61142	−0.805710	+	0.592310i	$0.798216\pi$
$602$	0	0
$603$	1.01373e7	1.13535
$604$	0	0
$605$	1.31498e7	1.46059
$606$	0	0
$607$	1.52702e6	0.168219	0.0841093	−	0.996457i	$-0.473196\pi$
$607$	1.52702e6	0.168219	0.0841093	+	0.996457i	$0.473196\pi$
$608$	0	0
$609$	−1.01649e7	−1.11061
$610$	0	0
$611$	−473507.	−0.0513125
$612$	0	0
$613$	−1.39418e6	−0.149854	−0.0749268	−	0.997189i	$-0.523872\pi$
$613$	−1.39418e6	−0.149854	−0.0749268	+	0.997189i	$0.523872\pi$
$614$	0	0
$615$	1.28347e7	1.36835
$616$	0	0
$617$	−501546.	−0.0530393	−0.0265196	−	0.999648i	$-0.508442\pi$
$617$	−501546.	−0.0530393	−0.0265196	+	0.999648i	$0.508442\pi$
$618$	0	0
$619$	−7.85188e6	−0.823658	−0.411829	−	0.911261i	$-0.635110\pi$
$619$	−7.85188e6	−0.823658	−0.411829	+	0.911261i	$0.635110\pi$
$620$	0	0
$621$	1.51485e7	1.57630
$622$	0	0
$623$	−8.37635e6	−0.864639
$624$	0	0
$625$	−1.17425e7	−1.20243
$626$	0	0
$627$	−3.42222e7	−3.47648
$628$	0	0
$629$	899972.	0.0906989
$630$	0	0
$631$	1.61935e7	1.61908	0.809539	−	0.587067i	$-0.199717\pi$
$631$	1.61935e7	1.61908	0.809539	+	0.587067i	$0.199717\pi$
$632$	0	0
$633$	−2.33182e7	−2.31306
$634$	0	0
$635$	3.34605e6	0.329305
$636$	0	0
$637$	2.09997e6	0.205052
$638$	0	0
$639$	−2.77156e7	−2.68518
$640$	0	0
$641$	1.33189e7	1.28034	0.640169	−	0.768234i	$-0.278864\pi$
$641$	1.33189e7	1.28034	0.640169	+	0.768234i	$0.278864\pi$
$642$	0	0
$643$	−9.01202e6	−0.859597	−0.429799	−	0.902925i	$-0.641415\pi$
$643$	−9.01202e6	−0.859597	−0.429799	+	0.902925i	$0.641415\pi$
$644$	0	0
$645$	−2.11881e7	−2.00536
$646$	0	0
$647$	−2.25046e6	−0.211355	−0.105677	−	0.994400i	$-0.533701\pi$
$647$	−2.25046e6	−0.211355	−0.105677	+	0.994400i	$0.533701\pi$
$648$	0	0
$649$	2.52895e7	2.35683
$650$	0	0
$651$	1.69328e7	1.56595
$652$	0	0
$653$	−1.14532e7	−1.05110	−0.525552	−	0.850761i	$-0.676141\pi$
$653$	−1.14532e7	−1.05110	−0.525552	+	0.850761i	$0.676141\pi$
$654$	0	0
$655$	1.94210e6	0.176876
$656$	0	0
$657$	−2.47091e7	−2.23328
$658$	0	0
$659$	−951649.	−0.0853617	−0.0426808	−	0.999089i	$-0.513590\pi$
$659$	−951649.	−0.0853617	−0.0426808	+	0.999089i	$0.513590\pi$
$660$	0	0
$661$	1.35290e7	1.20438	0.602189	−	0.798354i	$-0.294296\pi$
$661$	1.35290e7	1.20438	0.602189	+	0.798354i	$0.294296\pi$
$662$	0	0
$663$	−1.58154e6	−0.139732
$664$	0	0
$665$	1.15548e7	1.01323
$666$	0	0
$667$	1.85271e7	1.61248
$668$	0	0
$669$	−1.07971e7	−0.932696
$670$	0	0
$671$	−2.84025e7	−2.43529
$672$	0	0
$673$	8.94873e6	0.761594	0.380797	−	0.924659i	$-0.375650\pi$
$673$	8.94873e6	0.761594	0.380797	+	0.924659i	$0.375650\pi$
$674$	0	0
$675$	−3.51338e6	−0.296801
$676$	0	0
$677$	−176169.	−0.0147726	−0.00738632	−	0.999973i	$-0.502351\pi$
$677$	−176169.	−0.0147726	−0.00738632	+	0.999973i	$0.502351\pi$
$678$	0	0
$679$	−1.50318e7	−1.25123
$680$	0	0
$681$	−1.10471e7	−0.912811
$682$	0	0
$683$	5.37532e6	0.440912	0.220456	−	0.975397i	$-0.429245\pi$
$683$	5.37532e6	0.440912	0.220456	+	0.975397i	$0.429245\pi$
$684$	0	0
$685$	−1.46154e7	−1.19010
$686$	0	0
$687$	606131.	0.0489976
$688$	0	0
$689$	3.07536e6	0.246801
$690$	0	0
$691$	1.06270e7	0.846669	0.423335	−	0.905973i	$-0.360859\pi$
$691$	1.06270e7	0.846669	0.423335	+	0.905973i	$0.360859\pi$
$692$	0	0
$693$	1.99723e7	1.57978
$694$	0	0
$695$	−2.96145e6	−0.232564
$696$	0	0
$697$	2.38743e6	0.186144
$698$	0	0
$699$	−1.55993e7	−1.20757
$700$	0	0
$701$	1.35541e7	1.04178	0.520888	−	0.853625i	$-0.325601\pi$
$701$	1.35541e7	1.04178	0.520888	+	0.853625i	$0.325601\pi$
$702$	0	0
$703$	6.69664e6	0.511056
$704$	0	0
$705$	−3.63971e6	−0.275799
$706$	0	0
$707$	9.21798e6	0.693565
$708$	0	0
$709$	−2.34298e7	−1.75046	−0.875232	−	0.483704i	$-0.839291\pi$
$709$	−2.34298e7	−1.75046	−0.875232	+	0.483704i	$0.839291\pi$
$710$	0	0
$711$	−2.89422e7	−2.14713
$712$	0	0
$713$	−3.08628e7	−2.27358
$714$	0	0
$715$	8.02707e6	0.587208
$716$	0	0
$717$	4.14369e7	3.01016
$718$	0	0
$719$	2.56342e7	1.84926	0.924628	−	0.380872i	$-0.124376\pi$
$719$	2.56342e7	1.84926	0.924628	+	0.380872i	$0.124376\pi$
$720$	0	0
$721$	1.33505e7	0.956445
$722$	0	0
$723$	544470.	0.0387372
$724$	0	0
$725$	−4.29699e6	−0.303612
$726$	0	0
$727$	3.64123e6	0.255513	0.127756	−	0.991806i	$-0.459222\pi$
$727$	3.64123e6	0.255513	0.127756	+	0.991806i	$0.459222\pi$
$728$	0	0
$729$	−2.30226e7	−1.60449
$730$	0	0
$731$	−3.94128e6	−0.272800
$732$	0	0
$733$	−1.64472e7	−1.13066	−0.565331	−	0.824864i	$-0.691252\pi$
$733$	−1.64472e7	−1.13066	−0.565331	+	0.824864i	$0.691252\pi$
$734$	0	0
$735$	1.61418e7	1.10213
$736$	0	0
$737$	−1.53814e7	−1.04310
$738$	0	0
$739$	−7.00621e6	−0.471924	−0.235962	−	0.971762i	$-0.575824\pi$
$739$	−7.00621e6	−0.471924	−0.235962	+	0.971762i	$0.575824\pi$
$740$	0	0
$741$	−1.17681e7	−0.787339
$742$	0	0
$743$	−1.23231e7	−0.818930	−0.409465	−	0.912326i	$-0.634285\pi$
$743$	−1.23231e7	−0.818930	−0.409465	+	0.912326i	$0.634285\pi$
$744$	0	0
$745$	2.62468e7	1.73255
$746$	0	0
$747$	−3.76040e7	−2.46566
$748$	0	0
$749$	1.03219e7	0.672286
$750$	0	0
$751$	−9.73690e6	−0.629972	−0.314986	−	0.949096i	$-0.602000\pi$
$751$	−9.73690e6	−0.629972	−0.314986	+	0.949096i	$0.602000\pi$
$752$	0	0
$753$	−4.46716e7	−2.87107
$754$	0	0
$755$	1.81655e7	1.15979
$756$	0	0
$757$	−2.09834e7	−1.33087	−0.665437	−	0.746454i	$-0.731755\pi$
$757$	−2.09834e7	−1.33087	−0.665437	+	0.746454i	$0.731755\pi$
$758$	0	0
$759$	−5.85036e7	−3.68619
$760$	0	0
$761$	−1.02169e6	−0.0639527	−0.0319763	−	0.999489i	$-0.510180\pi$
$761$	−1.02169e6	−0.0639527	−0.0319763	+	0.999489i	$0.510180\pi$
$762$	0	0
$763$	−1.37796e7	−0.856890
$764$	0	0
$765$	−7.56433e6	−0.467323
$766$	0	0
$767$	8.69638e6	0.533765
$768$	0	0
$769$	2.44093e6	0.148847	0.0744235	−	0.997227i	$-0.476288\pi$
$769$	2.44093e6	0.148847	0.0744235	+	0.997227i	$0.476288\pi$
$770$	0	0
$771$	1.67516e6	0.101489
$772$	0	0
$773$	2.46489e7	1.48371	0.741855	−	0.670560i	$-0.233946\pi$
$773$	2.46489e7	1.48371	0.741855	+	0.670560i	$0.233946\pi$
$774$	0	0
$775$	7.15798e6	0.428091
$776$	0	0
$777$	−6.28096e6	−0.373227
$778$	0	0
$779$	1.77647e7	1.04885
$780$	0	0
$781$	4.20530e7	2.46700
$782$	0	0
$783$	1.94535e7	1.13395
$784$	0	0
$785$	−2.67652e6	−0.155023
$786$	0	0
$787$	1.02955e7	0.592531	0.296266	−	0.955106i	$-0.404259\pi$
$787$	1.02955e7	0.592531	0.296266	+	0.955106i	$0.404259\pi$
$788$	0	0
$789$	−1.66436e7	−0.951822
$790$	0	0
$791$	3.00421e6	0.170722
$792$	0	0
$793$	−9.76688e6	−0.551535
$794$	0	0
$795$	2.36393e7	1.32653
$796$	0	0
$797$	9.22117e6	0.514210	0.257105	−	0.966384i	$-0.417231\pi$
$797$	9.22117e6	0.514210	0.257105	+	0.966384i	$0.417231\pi$
$798$	0	0
$799$	−677036.	−0.0375185
$800$	0	0
$801$	4.08028e7	2.24703
$802$	0	0
$803$	3.74911e7	2.05182
$804$	0	0
$805$	1.97532e7	1.07435
$806$	0	0
$807$	−5.14110e7	−2.77890
$808$	0	0
$809$	−2.40070e7	−1.28963	−0.644816	−	0.764337i	$-0.723066\pi$
$809$	−2.40070e7	−1.28963	−0.644816	+	0.764337i	$0.723066\pi$
$810$	0	0
$811$	−1.91778e7	−1.02388	−0.511938	−	0.859022i	$-0.671072\pi$
$811$	−1.91778e7	−1.02388	−0.511938	+	0.859022i	$0.671072\pi$
$812$	0	0
$813$	−2.88457e7	−1.53058
$814$	0	0
$815$	−2.19174e7	−1.15583
$816$	0	0
$817$	−2.93268e7	−1.53713
$818$	0	0
$819$	6.86796e6	0.357782
$820$	0	0
$821$	1.75497e7	0.908680	0.454340	−	0.890828i	$-0.349875\pi$
$821$	1.75497e7	0.908680	0.454340	+	0.890828i	$0.349875\pi$
$822$	0	0
$823$	2.42818e7	1.24963	0.624814	−	0.780774i	$-0.285175\pi$
$823$	2.42818e7	1.24963	0.624814	+	0.780774i	$0.285175\pi$
$824$	0	0
$825$	1.35687e7	0.694071
$826$	0	0
$827$	9.95813e6	0.506307	0.253154	−	0.967426i	$-0.418532\pi$
$827$	9.95813e6	0.506307	0.253154	+	0.967426i	$0.418532\pi$
$828$	0	0
$829$	−3.49600e7	−1.76679	−0.883395	−	0.468629i	$-0.844748\pi$
$829$	−3.49600e7	−1.76679	−0.883395	+	0.468629i	$0.844748\pi$
$830$	0	0
$831$	3.12643e7	1.57053
$832$	0	0
$833$	3.00260e6	0.149929
$834$	0	0
$835$	−3.78575e7	−1.87904
$836$	0	0
$837$	−3.24059e7	−1.59886
$838$	0	0
$839$	2.85270e7	1.39911	0.699553	−	0.714581i	$-0.253383\pi$
$839$	2.85270e7	1.39911	0.699553	+	0.714581i	$0.253383\pi$
$840$	0	0
$841$	3.28119e6	0.159971
$842$	0	0
$843$	4.90524e6	0.237734
$844$	0	0
$845$	−2.07398e7	−0.999222
$846$	0	0
$847$	−1.70711e7	−0.817621
$848$	0	0
$849$	−2.41422e7	−1.14950
$850$	0	0
$851$	1.14480e7	0.541885
$852$	0	0
$853$	2.01259e7	0.947071	0.473535	−	0.880775i	$-0.342978\pi$
$853$	2.01259e7	0.947071	0.473535	+	0.880775i	$0.342978\pi$
$854$	0	0
$855$	−5.62858e7	−2.63320
$856$	0	0
$857$	−3.79660e7	−1.76580	−0.882902	−	0.469556i	$-0.844414\pi$
$857$	−3.79660e7	−1.76580	−0.882902	+	0.469556i	$0.844414\pi$
$858$	0	0
$859$	1.92107e7	0.888299	0.444150	−	0.895953i	$-0.353506\pi$
$859$	1.92107e7	0.888299	0.444150	+	0.895953i	$0.353506\pi$
$860$	0	0
$861$	−1.66620e7	−0.765984
$862$	0	0
$863$	6.26470e6	0.286334	0.143167	−	0.989699i	$-0.454271\pi$
$863$	6.26470e6	0.286334	0.143167	+	0.989699i	$0.454271\pi$
$864$	0	0
$865$	2.30856e7	1.04906
$866$	0	0
$867$	3.37496e7	1.52483
$868$	0	0
$869$	4.39141e7	1.97267
$870$	0	0
$871$	−5.28926e6	−0.236238
$872$	0	0
$873$	7.32228e7	3.25170
$874$	0	0
$875$	1.16703e7	0.515302
$876$	0	0
$877$	1.91154e7	0.839238	0.419619	−	0.907700i	$-0.362164\pi$
$877$	1.91154e7	0.839238	0.419619	+	0.907700i	$0.362164\pi$
$878$	0	0
$879$	3.77319e6	0.164716
$880$	0	0
$881$	3.41850e7	1.48387	0.741935	−	0.670472i	$-0.233908\pi$
$881$	3.41850e7	1.48387	0.741935	+	0.670472i	$0.233908\pi$
$882$	0	0
$883$	1.42154e7	0.613560	0.306780	−	0.951780i	$-0.400748\pi$
$883$	1.42154e7	0.613560	0.306780	+	0.951780i	$0.400748\pi$
$884$	0	0
$885$	6.68464e7	2.86893
$886$	0	0
$887$	1.80799e7	0.771590	0.385795	−	0.922584i	$-0.373927\pi$
$887$	1.80799e7	0.771590	0.385795	+	0.922584i	$0.373927\pi$
$888$	0	0
$889$	−4.34385e6	−0.184340
$890$	0	0
$891$	−2.36238e6	−0.0996910
$892$	0	0
$893$	−5.03779e6	−0.211403
$894$	0	0
$895$	−1.34669e7	−0.561968
$896$	0	0
$897$	−2.01179e7	−0.834835
$898$	0	0
$899$	−3.96336e7	−1.63555
$900$	0	0
$901$	4.39725e6	0.180455
$902$	0	0
$903$	2.75064e7	1.12257
$904$	0	0
$905$	3.93553e6	0.159728
$906$	0	0
$907$	2.38869e7	0.964142	0.482071	−	0.876132i	$-0.339885\pi$
$907$	2.38869e7	0.964142	0.482071	+	0.876132i	$0.339885\pi$
$908$	0	0
$909$	−4.49026e7	−1.80244
$910$	0	0
$911$	−2.99135e7	−1.19419	−0.597093	−	0.802172i	$-0.703678\pi$
$911$	−2.99135e7	−1.19419	−0.597093	+	0.802172i	$0.703678\pi$
$912$	0	0
$913$	5.70566e7	2.26532
$914$	0	0
$915$	−7.50750e7	−2.96444
$916$	0	0
$917$	−2.52124e6	−0.0990127
$918$	0	0
$919$	1.75882e7	0.686961	0.343481	−	0.939160i	$-0.388394\pi$
$919$	1.75882e7	0.686961	0.343481	+	0.939160i	$0.388394\pi$
$920$	0	0
$921$	−3.45615e7	−1.34259
$922$	0	0
$923$	1.44609e7	0.558717
$924$	0	0
$925$	−2.65514e6	−0.102031
$926$	0	0
$927$	−6.50330e7	−2.48562
$928$	0	0
$929$	−3.61148e7	−1.37292	−0.686461	−	0.727166i	$-0.740837\pi$
$929$	−3.61148e7	−1.37292	−0.686461	+	0.727166i	$0.740837\pi$
$930$	0	0
$931$	2.23422e7	0.844796
$932$	0	0
$933$	−8.21797e7	−3.09072
$934$	0	0
$935$	1.14774e7	0.429352
$936$	0	0
$937$	−4.82298e7	−1.79460	−0.897298	−	0.441426i	$-0.854473\pi$
$937$	−4.82298e7	−1.79460	−0.897298	+	0.441426i	$0.854473\pi$
$938$	0	0
$939$	−7.07697e7	−2.61929
$940$	0	0
$941$	−1.05808e7	−0.389533	−0.194766	−	0.980850i	$-0.562395\pi$
$941$	−1.05808e7	−0.389533	−0.194766	+	0.980850i	$0.562395\pi$
$942$	0	0
$943$	3.03691e7	1.11212
$944$	0	0
$945$	2.07408e7	0.755521
$946$	0	0
$947$	−5.00847e7	−1.81481	−0.907403	−	0.420262i	$-0.861938\pi$
$947$	−5.00847e7	−1.81481	−0.907403	+	0.420262i	$0.861938\pi$
$948$	0	0
$949$	1.28922e7	0.464689
$950$	0	0
$951$	−7.33413e7	−2.62965
$952$	0	0
$953$	4.28166e6	0.152714	0.0763571	−	0.997081i	$-0.475671\pi$
$953$	4.28166e6	0.152714	0.0763571	+	0.997081i	$0.475671\pi$
$954$	0	0
$955$	−4.49051e7	−1.59326
$956$	0	0
$957$	−7.51296e7	−2.65174
$958$	0	0
$959$	1.89737e7	0.666203
$960$	0	0
$961$	3.73930e7	1.30612
$962$	0	0
$963$	−5.02799e7	−1.74714
$964$	0	0
$965$	−1.26767e7	−0.438215
$966$	0	0
$967$	6.82231e6	0.234620	0.117310	−	0.993095i	$-0.462573\pi$
$967$	6.82231e6	0.234620	0.117310	+	0.993095i	$0.462573\pi$
$968$	0	0
$969$	−1.68265e7	−0.575683
$970$	0	0
$971$	−4.32428e7	−1.47186	−0.735929	−	0.677059i	$-0.763254\pi$
$971$	−4.32428e7	−1.47186	−0.735929	+	0.677059i	$0.763254\pi$
$972$	0	0
$973$	3.84457e6	0.130186
$974$	0	0
$975$	4.66592e6	0.157190
$976$	0	0
$977$	−3.63654e7	−1.21885	−0.609427	−	0.792842i	$-0.708600\pi$
$977$	−3.63654e7	−1.21885	−0.609427	+	0.792842i	$0.708600\pi$
$978$	0	0
$979$	−6.19103e7	−2.06446
$980$	0	0
$981$	6.71231e7	2.22689
$982$	0	0
$983$	−2.36478e7	−0.780561	−0.390280	−	0.920696i	$-0.627622\pi$
$983$	−2.36478e7	−0.780561	−0.390280	+	0.920696i	$0.627622\pi$
$984$	0	0
$985$	−3.07000e7	−1.00820
$986$	0	0
$987$	4.72508e6	0.154389
$988$	0	0
$989$	−5.01348e7	−1.62985
$990$	0	0
$991$	4.34096e7	1.40411	0.702056	−	0.712122i	$-0.252266\pi$
$991$	4.34096e7	1.40411	0.702056	+	0.712122i	$0.252266\pi$
$992$	0	0
$993$	−4.10675e7	−1.32168
$994$	0	0
$995$	1.59980e7	0.512280
$996$	0	0
$997$	2.15601e7	0.686931	0.343465	−	0.939165i	$-0.388399\pi$
$997$	2.15601e7	0.686931	0.343465	+	0.939165i	$0.388399\pi$
$998$	0	0
$999$	1.20204e7	0.381071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.5	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.5	✓	57	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)

\(f(q)\)	\(=\)	\(q-25.3624 q^{3} +63.2925 q^{5} -82.1666 q^{7} +400.249 q^{9} +O(q^{10})\)	\(q-25.3624 q^{3} +63.2925 q^{5} -82.1666 q^{7} +400.249 q^{9} -607.299 q^{11} -208.834 q^{13} -1605.25 q^{15} -298.599 q^{17} -2221.86 q^{19} +2083.94 q^{21} -3798.31 q^{23} +880.939 q^{25} -3988.22 q^{27} -4877.74 q^{29} +8125.40 q^{31} +15402.5 q^{33} -5200.53 q^{35} -3013.99 q^{37} +5296.53 q^{39} -7995.44 q^{41} +13199.3 q^{43} +25332.8 q^{45} +2267.38 q^{47} -10055.7 q^{49} +7573.16 q^{51} -14726.3 q^{53} -38437.5 q^{55} +56351.5 q^{57} -41642.5 q^{59} +46768.5 q^{61} -32887.1 q^{63} -13217.7 q^{65} +25327.5 q^{67} +96334.0 q^{69} -69245.9 q^{71} -61734.1 q^{73} -22342.7 q^{75} +49899.7 q^{77} -72310.5 q^{79} +3889.98 q^{81} -93951.4 q^{83} -18899.0 q^{85} +123711. q^{87} +101944. q^{89} +17159.2 q^{91} -206079. q^{93} -140627. q^{95} +182943. q^{97} -243071. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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