LMFDB - Embedded newform 1028.6.a.b.1.18

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			1.18
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)

$f(q)$	$=$	$q-13.0981 q^{3} +23.8997 q^{5} +161.931 q^{7} -71.4392 q^{9} +O(q^{10})$	$q-13.0981 q^{3} +23.8997 q^{5} +161.931 q^{7} -71.4392 q^{9} -675.538 q^{11} -631.834 q^{13} -313.041 q^{15} -1094.09 q^{17} -2771.65 q^{19} -2120.99 q^{21} -1124.65 q^{23} -2553.80 q^{25} +4118.56 q^{27} -613.895 q^{29} -6880.50 q^{31} +8848.27 q^{33} +3870.10 q^{35} +7549.66 q^{37} +8275.84 q^{39} +9830.37 q^{41} +5259.81 q^{43} -1707.37 q^{45} -14056.6 q^{47} +9414.70 q^{49} +14330.5 q^{51} +8795.99 q^{53} -16145.1 q^{55} +36303.4 q^{57} -4488.88 q^{59} -44278.6 q^{61} -11568.2 q^{63} -15100.6 q^{65} +54095.1 q^{67} +14730.8 q^{69} +49976.2 q^{71} -1932.23 q^{73} +33450.0 q^{75} -109391. q^{77} +40170.3 q^{79} -36585.7 q^{81} +12119.1 q^{83} -26148.3 q^{85} +8040.87 q^{87} +23452.0 q^{89} -102314. q^{91} +90121.6 q^{93} -66241.6 q^{95} -44568.0 q^{97} +48259.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−13.0981	−0.840245	−0.420122	−	0.907467i	$-0.638013\pi$
$3$	−13.0981	−0.840245	−0.420122	+	0.907467i	$0.638013\pi$
$4$	0	0
$5$	23.8997	0.427531	0.213765	−	0.976885i	$-0.431427\pi$
$5$	23.8997	0.427531	0.213765	+	0.976885i	$0.431427\pi$
$6$	0	0
$7$	161.931	1.24907	0.624533	−	0.780999i	$-0.285289\pi$
$7$	161.931	1.24907	0.624533	+	0.780999i	$0.285289\pi$
$8$	0	0
$9$	−71.4392	−0.293988
$10$	0	0
$11$	−675.538	−1.68332	−0.841662	−	0.540004i	$-0.818423\pi$
$11$	−675.538	−1.68332	−0.841662	+	0.540004i	$0.818423\pi$
$12$	0	0
$13$	−631.834	−1.03692	−0.518459	−	0.855102i	$-0.673494\pi$
$13$	−631.834	−1.03692	−0.518459	+	0.855102i	$0.673494\pi$
$14$	0	0
$15$	−313.041	−0.359231
$16$	0	0
$17$	−1094.09	−0.918184	−0.459092	−	0.888389i	$-0.651825\pi$
$17$	−1094.09	−0.918184	−0.459092	+	0.888389i	$0.651825\pi$
$18$	0	0
$19$	−2771.65	−1.76138	−0.880692	−	0.473689i	$-0.842922\pi$
$19$	−2771.65	−1.76138	−0.880692	+	0.473689i	$0.842922\pi$
$20$	0	0
$21$	−2120.99	−1.04952
$22$	0	0
$23$	−1124.65	−0.443301	−0.221651	−	0.975126i	$-0.571144\pi$
$23$	−1124.65	−0.443301	−0.221651	+	0.975126i	$0.571144\pi$
$24$	0	0
$25$	−2553.80	−0.817217
$26$	0	0
$27$	4118.56	1.08727
$28$	0	0
$29$	−613.895	−0.135550	−0.0677749	−	0.997701i	$-0.521590\pi$
$29$	−613.895	−0.135550	−0.0677749	+	0.997701i	$0.521590\pi$
$30$	0	0
$31$	−6880.50	−1.28592	−0.642962	−	0.765898i	$-0.722295\pi$
$31$	−6880.50	−1.28592	−0.642962	+	0.765898i	$0.722295\pi$
$32$	0	0
$33$	8848.27	1.41440
$34$	0	0
$35$	3870.10	0.534014
$36$	0	0
$37$	7549.66	0.906615	0.453308	−	0.891354i	$-0.350244\pi$
$37$	7549.66	0.906615	0.453308	+	0.891354i	$0.350244\pi$
$38$	0	0
$39$	8275.84	0.871266
$40$	0	0
$41$	9830.37	0.913293	0.456646	−	0.889648i	$-0.349050\pi$
$41$	9830.37	0.913293	0.456646	+	0.889648i	$0.349050\pi$
$42$	0	0
$43$	5259.81	0.433809	0.216904	−	0.976193i	$-0.430404\pi$
$43$	5259.81	0.433809	0.216904	+	0.976193i	$0.430404\pi$
$44$	0	0
$45$	−1707.37	−0.125689
$46$	0	0
$47$	−14056.6	−0.928185	−0.464093	−	0.885787i	$-0.653620\pi$
$47$	−14056.6	−0.928185	−0.464093	+	0.885787i	$0.653620\pi$
$48$	0	0
$49$	9414.70	0.560165
$50$	0	0
$51$	14330.5	0.771499
$52$	0	0
$53$	8795.99	0.430125	0.215063	−	0.976600i	$-0.431004\pi$
$53$	8795.99	0.430125	0.215063	+	0.976600i	$0.431004\pi$
$54$	0	0
$55$	−16145.1	−0.719673
$56$	0	0
$57$	36303.4	1.47999
$58$	0	0
$59$	−4488.88	−0.167883	−0.0839417	−	0.996471i	$-0.526751\pi$
$59$	−4488.88	−0.167883	−0.0839417	+	0.996471i	$0.526751\pi$
$60$	0	0
$61$	−44278.6	−1.52360	−0.761798	−	0.647815i	$-0.775683\pi$
$61$	−44278.6	−1.52360	−0.761798	+	0.647815i	$0.775683\pi$
$62$	0	0
$63$	−11568.2	−0.367211
$64$	0	0
$65$	−15100.6	−0.443315
$66$	0	0
$67$	54095.1	1.47222	0.736108	−	0.676865i	$-0.236662\pi$
$67$	54095.1	1.47222	0.736108	+	0.676865i	$0.236662\pi$
$68$	0	0
$69$	14730.8	0.372481
$70$	0	0
$71$	49976.2	1.17657	0.588284	−	0.808654i	$-0.299804\pi$
$71$	49976.2	1.17657	0.588284	+	0.808654i	$0.299804\pi$
$72$	0	0
$73$	−1932.23	−0.0424376	−0.0212188	−	0.999775i	$-0.506755\pi$
$73$	−1932.23	−0.0424376	−0.0212188	+	0.999775i	$0.506755\pi$
$74$	0	0
$75$	33450.0	0.686663
$76$	0	0
$77$	−109391.	−2.10258
$78$	0	0
$79$	40170.3	0.724165	0.362083	−	0.932146i	$-0.382066\pi$
$79$	40170.3	0.724165	0.362083	+	0.932146i	$0.382066\pi$
$80$	0	0
$81$	−36585.7	−0.619582
$82$	0	0
$83$	12119.1	0.193097	0.0965484	−	0.995328i	$-0.469220\pi$
$83$	12119.1	0.193097	0.0965484	+	0.995328i	$0.469220\pi$
$84$	0	0
$85$	−26148.3	−0.392552
$86$	0	0
$87$	8040.87	0.113895
$88$	0	0
$89$	23452.0	0.313838	0.156919	−	0.987611i	$-0.449844\pi$
$89$	23452.0	0.313838	0.156919	+	0.987611i	$0.449844\pi$
$90$	0	0
$91$	−102314.	−1.29518
$92$	0	0
$93$	90121.6	1.08049
$94$	0	0
$95$	−66241.6	−0.753046
$96$	0	0
$97$	−44568.0	−0.480943	−0.240472	−	0.970656i	$-0.577302\pi$
$97$	−44568.0	−0.480943	−0.240472	+	0.970656i	$0.577302\pi$
$98$	0	0
$99$	48259.9	0.494878
$100$	0	0
$101$	100405.	0.979385	0.489693	−	0.871895i	$-0.337109\pi$
$101$	100405.	0.979385	0.489693	+	0.871895i	$0.337109\pi$
$102$	0	0
$103$	−194484.	−1.80631	−0.903154	−	0.429317i	$-0.858754\pi$
$103$	−194484.	−1.80631	−0.903154	+	0.429317i	$0.858754\pi$
$104$	0	0
$105$	−50691.1	−0.448703
$106$	0	0
$107$	−11822.1	−0.0998241	−0.0499120	−	0.998754i	$-0.515894\pi$
$107$	−11822.1	−0.0998241	−0.0499120	+	0.998754i	$0.515894\pi$
$108$	0	0
$109$	−138556.	−1.11702	−0.558509	−	0.829498i	$-0.688627\pi$
$109$	−138556.	−1.11702	−0.558509	+	0.829498i	$0.688627\pi$
$110$	0	0
$111$	−98886.3	−0.761779
$112$	0	0
$113$	121160.	0.892611	0.446305	−	0.894881i	$-0.352740\pi$
$113$	121160.	0.892611	0.446305	+	0.894881i	$0.352740\pi$
$114$	0	0
$115$	−26878.8	−0.189525
$116$	0	0
$117$	45137.7	0.304842
$118$	0	0
$119$	−177167.	−1.14687
$120$	0	0
$121$	295300.	1.83358
$122$	0	0
$123$	−128759.	−0.767390
$124$	0	0
$125$	−135722.	−0.776916
$126$	0	0
$127$	−171250.	−0.942152	−0.471076	−	0.882093i	$-0.656134\pi$
$127$	−171250.	−0.942152	−0.471076	+	0.882093i	$0.656134\pi$
$128$	0	0
$129$	−68893.6	−0.364506
$130$	0	0
$131$	173241.	0.882010	0.441005	−	0.897505i	$-0.354622\pi$
$131$	173241.	0.882010	0.441005	+	0.897505i	$0.354622\pi$
$132$	0	0
$133$	−448816.	−2.20009
$134$	0	0
$135$	98432.4	0.464840
$136$	0	0
$137$	−192894.	−0.878044	−0.439022	−	0.898476i	$-0.644675\pi$
$137$	−192894.	−0.878044	−0.439022	+	0.898476i	$0.644675\pi$
$138$	0	0
$139$	−68111.0	−0.299006	−0.149503	−	0.988761i	$-0.547767\pi$
$139$	−68111.0	−0.299006	−0.149503	+	0.988761i	$0.547767\pi$
$140$	0	0
$141$	184115.	0.779903
$142$	0	0
$143$	426828.	1.74547
$144$	0	0
$145$	−14671.9	−0.0579517
$146$	0	0
$147$	−123315.	−0.470676
$148$	0	0
$149$	−259468.	−0.957455	−0.478728	−	0.877964i	$-0.658902\pi$
$149$	−259468.	−0.957455	−0.478728	+	0.877964i	$0.658902\pi$
$150$	0	0
$151$	238616.	0.851642	0.425821	−	0.904807i	$-0.359985\pi$
$151$	238616.	0.851642	0.425821	+	0.904807i	$0.359985\pi$
$152$	0	0
$153$	78160.7	0.269935
$154$	0	0
$155$	−164442.	−0.549772
$156$	0	0
$157$	253221.	0.819882	0.409941	−	0.912112i	$-0.365549\pi$
$157$	253221.	0.819882	0.409941	+	0.912112i	$0.365549\pi$
$158$	0	0
$159$	−115211.	−0.361411
$160$	0	0
$161$	−182116.	−0.553712
$162$	0	0
$163$	291633.	0.859741	0.429870	−	0.902891i	$-0.358559\pi$
$163$	291633.	0.859741	0.429870	+	0.902891i	$0.358559\pi$
$164$	0	0
$165$	211471.	0.604701
$166$	0	0
$167$	−304635.	−0.845257	−0.422629	−	0.906303i	$-0.638893\pi$
$167$	−304635.	−0.845257	−0.422629	+	0.906303i	$0.638893\pi$
$168$	0	0
$169$	27921.5	0.0752007
$170$	0	0
$171$	198004.	0.517827
$172$	0	0
$173$	−391694.	−0.995019	−0.497510	−	0.867458i	$-0.665752\pi$
$173$	−391694.	−0.995019	−0.497510	+	0.867458i	$0.665752\pi$
$174$	0	0
$175$	−413541.	−1.02076
$176$	0	0
$177$	58795.8	0.141063
$178$	0	0
$179$	226540.	0.528460	0.264230	−	0.964460i	$-0.414882\pi$
$179$	226540.	0.528460	0.264230	+	0.964460i	$0.414882\pi$
$180$	0	0
$181$	482685.	1.09513	0.547567	−	0.836762i	$-0.315554\pi$
$181$	482685.	1.09513	0.547567	+	0.836762i	$0.315554\pi$
$182$	0	0
$183$	579967.	1.28019
$184$	0	0
$185$	180435.	0.387606
$186$	0	0
$187$	739097.	1.54560
$188$	0	0
$189$	666924.	1.35807
$190$	0	0
$191$	129278.	0.256413	0.128206	−	0.991747i	$-0.459078\pi$
$191$	129278.	0.256413	0.128206	+	0.991747i	$0.459078\pi$
$192$	0	0
$193$	775183.	1.49800	0.748999	−	0.662571i	$-0.230535\pi$
$193$	775183.	1.49800	0.748999	+	0.662571i	$0.230535\pi$
$194$	0	0
$195$	197790.	0.372493
$196$	0	0
$197$	988936.	1.81553	0.907763	−	0.419484i	$-0.137789\pi$
$197$	988936.	1.81553	0.907763	+	0.419484i	$0.137789\pi$
$198$	0	0
$199$	230790.	0.413127	0.206563	−	0.978433i	$-0.433772\pi$
$199$	230790.	0.413127	0.206563	+	0.978433i	$0.433772\pi$
$200$	0	0
$201$	−708545.	−1.23702
$202$	0	0
$203$	−99408.7	−0.169311
$204$	0	0
$205$	234943.	0.390461
$206$	0	0
$207$	80344.3	0.130325
$208$	0	0
$209$	1.87235e6	2.96498
$210$	0	0
$211$	902851.	1.39608	0.698039	−	0.716059i	$-0.254056\pi$
$211$	902851.	1.39608	0.698039	+	0.716059i	$0.254056\pi$
$212$	0	0
$213$	−654594.	−0.988605
$214$	0	0
$215$	125708.	0.185467
$216$	0	0
$217$	−1.11417e6	−1.60620
$218$	0	0
$219$	25308.5	0.0356580
$220$	0	0
$221$	691282.	0.952082
$222$	0	0
$223$	−447138.	−0.602115	−0.301057	−	0.953606i	$-0.597340\pi$
$223$	−447138.	−0.602115	−0.301057	+	0.953606i	$0.597340\pi$
$224$	0	0
$225$	182442.	0.240253
$226$	0	0
$227$	−1.37889e6	−1.77609	−0.888043	−	0.459760i	$-0.847935\pi$
$227$	−1.37889e6	−1.77609	−0.888043	+	0.459760i	$0.847935\pi$
$228$	0	0
$229$	−451263.	−0.568645	−0.284323	−	0.958729i	$-0.591769\pi$
$229$	−451263.	−0.568645	−0.284323	+	0.958729i	$0.591769\pi$
$230$	0	0
$231$	1.43281e6	1.76668
$232$	0	0
$233$	−1.08605e6	−1.31057	−0.655283	−	0.755384i	$-0.727450\pi$
$233$	−1.08605e6	−1.31057	−0.655283	+	0.755384i	$0.727450\pi$
$234$	0	0
$235$	−335948.	−0.396828
$236$	0	0
$237$	−526156.	−0.608476
$238$	0	0
$239$	678712.	0.768583	0.384291	−	0.923212i	$-0.374446\pi$
$239$	678712.	0.768583	0.384291	+	0.923212i	$0.374446\pi$
$240$	0	0
$241$	556598.	0.617304	0.308652	−	0.951175i	$-0.400122\pi$
$241$	556598.	0.617304	0.308652	+	0.951175i	$0.400122\pi$
$242$	0	0
$243$	−521607.	−0.566666
$244$	0	0
$245$	225008.	0.239488
$246$	0	0
$247$	1.75122e6	1.82641
$248$	0	0
$249$	−158737.	−0.162249
$250$	0	0
$251$	−868413.	−0.870046	−0.435023	−	0.900419i	$-0.643260\pi$
$251$	−868413.	−0.870046	−0.435023	+	0.900419i	$0.643260\pi$
$252$	0	0
$253$	759745.	0.746219
$254$	0	0
$255$	342494.	0.329840
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	1.22252e6	1.13242
$260$	0	0
$261$	43856.2	0.0398501
$262$	0	0
$263$	−1.50675e6	−1.34324	−0.671619	−	0.740897i	$-0.734401\pi$
$263$	−1.50675e6	−1.34324	−0.671619	+	0.740897i	$0.734401\pi$
$264$	0	0
$265$	210222.	0.183892
$266$	0	0
$267$	−307178.	−0.263701
$268$	0	0
$269$	−369937.	−0.311708	−0.155854	−	0.987780i	$-0.549813\pi$
$269$	−369937.	−0.311708	−0.155854	+	0.987780i	$0.549813\pi$
$270$	0	0
$271$	316404.	0.261709	0.130854	−	0.991402i	$-0.458228\pi$
$271$	316404.	0.261709	0.130854	+	0.991402i	$0.458228\pi$
$272$	0	0
$273$	1.34012e6	1.08827
$274$	0	0
$275$	1.72519e6	1.37564
$276$	0	0
$277$	1.64650e6	1.28932	0.644662	−	0.764468i	$-0.276998\pi$
$277$	1.64650e6	1.28932	0.644662	+	0.764468i	$0.276998\pi$
$278$	0	0
$279$	491537.	0.378047
$280$	0	0
$281$	−1.62237e6	−1.22570	−0.612851	−	0.790198i	$-0.709977\pi$
$281$	−1.62237e6	−1.22570	−0.612851	+	0.790198i	$0.709977\pi$
$282$	0	0
$283$	−14423.9	−0.0107058	−0.00535288	−	0.999986i	$-0.501704\pi$
$283$	−14423.9	−0.0107058	−0.00535288	+	0.999986i	$0.501704\pi$
$284$	0	0
$285$	867640.	0.632743
$286$	0	0
$287$	1.59184e6	1.14076
$288$	0	0
$289$	−222831.	−0.156939
$290$	0	0
$291$	583757.	0.404110
$292$	0	0
$293$	689911.	0.469488	0.234744	−	0.972057i	$-0.424575\pi$
$293$	689911.	0.469488	0.234744	+	0.972057i	$0.424575\pi$
$294$	0	0
$295$	−107283.	−0.0717753
$296$	0	0
$297$	−2.78224e6	−1.83022
$298$	0	0
$299$	710594.	0.459667
$300$	0	0
$301$	851726.	0.541856
$302$	0	0
$303$	−1.31512e6	−0.822924
$304$	0	0
$305$	−1.05825e6	−0.651384
$306$	0	0
$307$	−1.99810e6	−1.20996	−0.604980	−	0.796241i	$-0.706819\pi$
$307$	−1.99810e6	−1.20996	−0.604980	+	0.796241i	$0.706819\pi$
$308$	0	0
$309$	2.54738e6	1.51774
$310$	0	0
$311$	−2.74514e6	−1.60940	−0.804698	−	0.593684i	$-0.797673\pi$
$311$	−2.74514e6	−1.60940	−0.804698	+	0.593684i	$0.797673\pi$
$312$	0	0
$313$	2.78023e6	1.60406	0.802029	−	0.597286i	$-0.203754\pi$
$313$	2.78023e6	1.60406	0.802029	+	0.597286i	$0.203754\pi$
$314$	0	0
$315$	−276477.	−0.156994
$316$	0	0
$317$	2.49769e6	1.39602	0.698009	−	0.716089i	$-0.254070\pi$
$317$	2.49769e6	1.39602	0.698009	+	0.716089i	$0.254070\pi$
$318$	0	0
$319$	414709.	0.228174
$320$	0	0
$321$	154847.	0.0838767
$322$	0	0
$323$	3.03242e6	1.61727
$324$	0	0
$325$	1.61358e6	0.847388
$326$	0	0
$327$	1.81483e6	0.938569
$328$	0	0
$329$	−2.27620e6	−1.15936
$330$	0	0
$331$	1.25129e6	0.627754	0.313877	−	0.949464i	$-0.398372\pi$
$331$	1.25129e6	0.627754	0.313877	+	0.949464i	$0.398372\pi$
$332$	0	0
$333$	−539342.	−0.266534
$334$	0	0
$335$	1.29286e6	0.629417
$336$	0	0
$337$	2.82482e6	1.35493	0.677463	−	0.735556i	$-0.263079\pi$
$337$	2.82482e6	1.35493	0.677463	+	0.735556i	$0.263079\pi$
$338$	0	0
$339$	−1.58696e6	−0.750012
$340$	0	0
$341$	4.64804e6	2.16463
$342$	0	0
$343$	−1.19704e6	−0.549383
$344$	0	0
$345$	352062.	0.159247
$346$	0	0
$347$	−1.25736e6	−0.560576	−0.280288	−	0.959916i	$-0.590430\pi$
$347$	−1.25736e6	−0.560576	−0.280288	+	0.959916i	$0.590430\pi$
$348$	0	0
$349$	1.97449e6	0.867744	0.433872	−	0.900974i	$-0.357147\pi$
$349$	1.97449e6	0.867744	0.433872	+	0.900974i	$0.357147\pi$
$350$	0	0
$351$	−2.60225e6	−1.12741
$352$	0	0
$353$	−4.27864e6	−1.82755	−0.913775	−	0.406222i	$-0.866846\pi$
$353$	−4.27864e6	−1.82755	−0.913775	+	0.406222i	$0.866846\pi$
$354$	0	0
$355$	1.19442e6	0.503019
$356$	0	0
$357$	2.32055e6	0.963653
$358$	0	0
$359$	−3.29925e6	−1.35107	−0.675537	−	0.737326i	$-0.736088\pi$
$359$	−3.29925e6	−1.35107	−0.675537	+	0.737326i	$0.736088\pi$
$360$	0	0
$361$	5.20594e6	2.10248
$362$	0	0
$363$	−3.86788e6	−1.54066
$364$	0	0
$365$	−46179.6	−0.0181434
$366$	0	0
$367$	−3.43130e6	−1.32982	−0.664910	−	0.746923i	$-0.731530\pi$
$367$	−3.43130e6	−1.32982	−0.664910	+	0.746923i	$0.731530\pi$
$368$	0	0
$369$	−702273.	−0.268498
$370$	0	0
$371$	1.42435e6	0.537255
$372$	0	0
$373$	3.50901e6	1.30591	0.652954	−	0.757397i	$-0.273529\pi$
$373$	3.50901e6	1.30591	0.652954	+	0.757397i	$0.273529\pi$
$374$	0	0
$375$	1.77770e6	0.652800
$376$	0	0
$377$	387880.	0.140554
$378$	0	0
$379$	2.68299e6	0.959447	0.479723	−	0.877420i	$-0.340737\pi$
$379$	2.68299e6	0.959447	0.479723	+	0.877420i	$0.340737\pi$
$380$	0	0
$381$	2.24305e6	0.791639
$382$	0	0
$383$	259173.	0.0902804	0.0451402	−	0.998981i	$-0.485627\pi$
$383$	259173.	0.0902804	0.0451402	+	0.998981i	$0.485627\pi$
$384$	0	0
$385$	−2.61440e6	−0.898919
$386$	0	0
$387$	−375756.	−0.127535
$388$	0	0
$389$	−2.09885e6	−0.703246	−0.351623	−	0.936142i	$-0.614370\pi$
$389$	−2.09885e6	−0.703246	−0.351623	+	0.936142i	$0.614370\pi$
$390$	0	0
$391$	1.23047e6	0.407032
$392$	0	0
$393$	−2.26914e6	−0.741104
$394$	0	0
$395$	960059.	0.309603
$396$	0	0
$397$	−5.51564e6	−1.75638	−0.878192	−	0.478309i	$-0.841250\pi$
$397$	−5.51564e6	−1.75638	−0.878192	+	0.478309i	$0.841250\pi$
$398$	0	0
$399$	5.87865e6	1.84861
$400$	0	0
$401$	4.04826e6	1.25721	0.628604	−	0.777726i	$-0.283627\pi$
$401$	4.04826e6	1.25721	0.628604	+	0.777726i	$0.283627\pi$
$402$	0	0
$403$	4.34733e6	1.33340
$404$	0	0
$405$	−874387.	−0.264890
$406$	0	0
$407$	−5.10008e6	−1.52613
$408$	0	0
$409$	1.74175e6	0.514847	0.257424	−	0.966299i	$-0.417126\pi$
$409$	1.74175e6	0.514847	0.257424	+	0.966299i	$0.417126\pi$
$410$	0	0
$411$	2.52654e6	0.737772
$412$	0	0
$413$	−726889.	−0.209697
$414$	0	0
$415$	289643.	0.0825548
$416$	0	0
$417$	892126.	0.251238
$418$	0	0
$419$	−952945.	−0.265175	−0.132587	−	0.991171i	$-0.542329\pi$
$419$	−952945.	−0.265175	−0.132587	+	0.991171i	$0.542329\pi$
$420$	0	0
$421$	3.18757e6	0.876506	0.438253	−	0.898852i	$-0.355597\pi$
$421$	3.18757e6	0.876506	0.438253	+	0.898852i	$0.355597\pi$
$422$	0	0
$423$	1.00419e6	0.272876
$424$	0	0
$425$	2.79408e6	0.750356
$426$	0	0
$427$	−7.17009e6	−1.90307
$428$	0	0
$429$	−5.59064e6	−1.46662
$430$	0	0
$431$	1.10474e6	0.286463	0.143231	−	0.989689i	$-0.454251\pi$
$431$	1.10474e6	0.286463	0.143231	+	0.989689i	$0.454251\pi$
$432$	0	0
$433$	−2.03487e6	−0.521574	−0.260787	−	0.965396i	$-0.583982\pi$
$433$	−2.03487e6	−0.521574	−0.260787	+	0.965396i	$0.583982\pi$
$434$	0	0
$435$	192174.	0.0486936
$436$	0	0
$437$	3.11714e6	0.780824
$438$	0	0
$439$	−4.65529e6	−1.15288	−0.576442	−	0.817138i	$-0.695559\pi$
$439$	−4.65529e6	−1.15288	−0.576442	+	0.817138i	$0.695559\pi$
$440$	0	0
$441$	−672578.	−0.164682
$442$	0	0
$443$	−143394.	−0.0347155	−0.0173577	−	0.999849i	$-0.505525\pi$
$443$	−143394.	−0.0347155	−0.0173577	+	0.999849i	$0.505525\pi$
$444$	0	0
$445$	560497.	0.134175
$446$	0	0
$447$	3.39855e6	0.804497
$448$	0	0
$449$	3.97353e6	0.930166	0.465083	−	0.885267i	$-0.346025\pi$
$449$	3.97353e6	0.930166	0.465083	+	0.885267i	$0.346025\pi$
$450$	0	0
$451$	−6.64078e6	−1.53737
$452$	0	0
$453$	−3.12542e6	−0.715588
$454$	0	0
$455$	−2.44526e6	−0.553729
$456$	0	0
$457$	−6.64548e6	−1.48846	−0.744228	−	0.667926i	$-0.767182\pi$
$457$	−6.64548e6	−1.48846	−0.744228	+	0.667926i	$0.767182\pi$
$458$	0	0
$459$	−4.50607e6	−0.998311
$460$	0	0
$461$	7.54179e6	1.65281	0.826403	−	0.563079i	$-0.190383\pi$
$461$	7.54179e6	1.65281	0.826403	+	0.563079i	$0.190383\pi$
$462$	0	0
$463$	1.09650e6	0.237715	0.118858	−	0.992911i	$-0.462077\pi$
$463$	1.09650e6	0.237715	0.118858	+	0.992911i	$0.462077\pi$
$464$	0	0
$465$	2.15388e6	0.461943
$466$	0	0
$467$	−68247.7	−0.0144809	−0.00724045	−	0.999974i	$-0.502305\pi$
$467$	−68247.7	−0.0144809	−0.00724045	+	0.999974i	$0.502305\pi$
$468$	0	0
$469$	8.75969e6	1.83889
$470$	0	0
$471$	−3.31673e6	−0.688902
$472$	0	0
$473$	−3.55320e6	−0.730241
$474$	0	0
$475$	7.07825e6	1.43943
$476$	0	0
$477$	−628379.	−0.126452
$478$	0	0
$479$	3.91307e6	0.779253	0.389627	−	0.920973i	$-0.372604\pi$
$479$	3.91307e6	0.779253	0.389627	+	0.920973i	$0.372604\pi$
$480$	0	0
$481$	−4.77013e6	−0.940086
$482$	0	0
$483$	2.38538e6	0.465254
$484$	0	0
$485$	−1.06516e6	−0.205618
$486$	0	0
$487$	−336411.	−0.0642758	−0.0321379	−	0.999483i	$-0.510232\pi$
$487$	−336411.	−0.0642758	−0.0321379	+	0.999483i	$0.510232\pi$
$488$	0	0
$489$	−3.81984e6	−0.722393
$490$	0	0
$491$	−6.65765e6	−1.24628	−0.623142	−	0.782108i	$-0.714144\pi$
$491$	−6.65765e6	−1.24628	−0.623142	+	0.782108i	$0.714144\pi$
$492$	0	0
$493$	671654.	0.124460
$494$	0	0
$495$	1.15340e6	0.211576
$496$	0	0
$497$	8.09270e6	1.46961
$498$	0	0
$499$	−6.50832e6	−1.17009	−0.585043	−	0.811002i	$-0.698922\pi$
$499$	−6.50832e6	−1.17009	−0.585043	+	0.811002i	$0.698922\pi$
$500$	0	0
$501$	3.99015e6	0.710223
$502$	0	0
$503$	−7.19564e6	−1.26809	−0.634044	−	0.773297i	$-0.718606\pi$
$503$	−7.19564e6	−1.26809	−0.634044	+	0.773297i	$0.718606\pi$
$504$	0	0
$505$	2.39966e6	0.418717
$506$	0	0
$507$	−365719.	−0.0631870
$508$	0	0
$509$	5.41268e6	0.926014	0.463007	−	0.886355i	$-0.346770\pi$
$509$	5.41268e6	0.926014	0.463007	+	0.886355i	$0.346770\pi$
$510$	0	0
$511$	−312888.	−0.0530073
$512$	0	0
$513$	−1.14152e7	−1.91510
$514$	0	0
$515$	−4.64812e6	−0.772252
$516$	0	0
$517$	9.49574e6	1.56244
$518$	0	0
$519$	5.13045e6	0.836060
$520$	0	0
$521$	−3.69631e6	−0.596587	−0.298294	−	0.954474i	$-0.596417\pi$
$521$	−3.69631e6	−0.596587	−0.298294	+	0.954474i	$0.596417\pi$
$522$	0	0
$523$	943960.	0.150904	0.0754518	−	0.997149i	$-0.475960\pi$
$523$	943960.	0.150904	0.0754518	+	0.997149i	$0.475960\pi$
$524$	0	0
$525$	5.41660e6	0.857687
$526$	0	0
$527$	7.52786e6	1.18072
$528$	0	0
$529$	−5.17150e6	−0.803484
$530$	0	0
$531$	320682.	0.0493558
$532$	0	0
$533$	−6.21116e6	−0.947011
$534$	0	0
$535$	−282545.	−0.0426779
$536$	0	0
$537$	−2.96725e6	−0.444036
$538$	0	0
$539$	−6.35998e6	−0.942940
$540$	0	0
$541$	4.03718e6	0.593041	0.296521	−	0.955026i	$-0.404174\pi$
$541$	4.03718e6	0.593041	0.296521	+	0.955026i	$0.404174\pi$
$542$	0	0
$543$	−6.32226e6	−0.920180
$544$	0	0
$545$	−3.31146e6	−0.477560
$546$	0	0
$547$	8.15050e6	1.16471	0.582353	−	0.812936i	$-0.302132\pi$
$547$	8.15050e6	1.16471	0.582353	+	0.812936i	$0.302132\pi$
$548$	0	0
$549$	3.16323e6	0.447919
$550$	0	0
$551$	1.70150e6	0.238755
$552$	0	0
$553$	6.50483e6	0.904530
$554$	0	0
$555$	−2.36335e6	−0.325684
$556$	0	0
$557$	9.24078e6	1.26203	0.631016	−	0.775770i	$-0.282638\pi$
$557$	9.24078e6	1.26203	0.631016	+	0.775770i	$0.282638\pi$
$558$	0	0
$559$	−3.32332e6	−0.449825
$560$	0	0
$561$	−9.68078e6	−1.29868
$562$	0	0
$563$	6.43843e6	0.856069	0.428034	−	0.903762i	$-0.359206\pi$
$563$	6.43843e6	0.856069	0.428034	+	0.903762i	$0.359206\pi$
$564$	0	0
$565$	2.89568e6	0.381618
$566$	0	0
$567$	−5.92437e6	−0.773899
$568$	0	0
$569$	−8.18164e6	−1.05940	−0.529700	−	0.848185i	$-0.677695\pi$
$569$	−8.18164e6	−1.05940	−0.529700	+	0.848185i	$0.677695\pi$
$570$	0	0
$571$	−1.00234e6	−0.128655	−0.0643273	−	0.997929i	$-0.520490\pi$
$571$	−1.00234e6	−0.128655	−0.0643273	+	0.997929i	$0.520490\pi$
$572$	0	0
$573$	−1.69329e6	−0.215450
$574$	0	0
$575$	2.87214e6	0.362273
$576$	0	0
$577$	8.36805e6	1.04637	0.523184	−	0.852220i	$-0.324744\pi$
$577$	8.36805e6	1.04637	0.523184	+	0.852220i	$0.324744\pi$
$578$	0	0
$579$	−1.01534e7	−1.25869
$580$	0	0
$581$	1.96246e6	0.241191
$582$	0	0
$583$	−5.94202e6	−0.724041
$584$	0	0
$585$	1.07878e6	0.130329
$586$	0	0
$587$	−1.22215e7	−1.46396	−0.731982	−	0.681324i	$-0.761404\pi$
$587$	−1.22215e7	−1.46396	−0.731982	+	0.681324i	$0.761404\pi$
$588$	0	0
$589$	1.90703e7	2.26501
$590$	0	0
$591$	−1.29532e7	−1.52549
$592$	0	0
$593$	−76211.4	−0.00889986	−0.00444993	−	0.999990i	$-0.501416\pi$
$593$	−76211.4	−0.00889986	−0.00444993	+	0.999990i	$0.501416\pi$
$594$	0	0
$595$	−4.23423e6	−0.490323
$596$	0	0
$597$	−3.02291e6	−0.347128
$598$	0	0
$599$	−9.75826e6	−1.11123	−0.555617	−	0.831439i	$-0.687518\pi$
$599$	−9.75826e6	−1.11123	−0.555617	+	0.831439i	$0.687518\pi$
$600$	0	0
$601$	−327914.	−0.0370317	−0.0185158	−	0.999829i	$-0.505894\pi$
$601$	−327914.	−0.0370317	−0.0185158	+	0.999829i	$0.505894\pi$
$602$	0	0
$603$	−3.86451e6	−0.432814
$604$	0	0
$605$	7.05758e6	0.783912
$606$	0	0
$607$	−6.47761e6	−0.713580	−0.356790	−	0.934185i	$-0.616129\pi$
$607$	−6.47761e6	−0.713580	−0.356790	+	0.934185i	$0.616129\pi$
$608$	0	0
$609$	1.30207e6	0.142262
$610$	0	0
$611$	8.88142e6	0.962453
$612$	0	0
$613$	−5.33193e6	−0.573104	−0.286552	−	0.958065i	$-0.592509\pi$
$613$	−5.33193e6	−0.573104	−0.286552	+	0.958065i	$0.592509\pi$
$614$	0	0
$615$	−3.07731e6	−0.328083
$616$	0	0
$617$	1.88754e7	1.99611	0.998054	−	0.0623569i	$-0.0198617\pi$
$617$	1.88754e7	1.99611	0.998054	+	0.0623569i	$0.0198617\pi$
$618$	0	0
$619$	1.06057e7	1.11253	0.556266	−	0.831004i	$-0.312234\pi$
$619$	1.06057e7	1.11253	0.556266	+	0.831004i	$0.312234\pi$
$620$	0	0
$621$	−4.63195e6	−0.481987
$622$	0	0
$623$	3.79762e6	0.392004
$624$	0	0
$625$	4.73693e6	0.485062
$626$	0	0
$627$	−2.45243e7	−2.49131
$628$	0	0
$629$	−8.25998e6	−0.832439
$630$	0	0
$631$	779899.	0.0779767	0.0389883	−	0.999240i	$-0.487586\pi$
$631$	779899.	0.0779767	0.0389883	+	0.999240i	$0.487586\pi$
$632$	0	0
$633$	−1.18257e7	−1.17305
$634$	0	0
$635$	−4.09282e6	−0.402799
$636$	0	0
$637$	−5.94853e6	−0.580846
$638$	0	0
$639$	−3.57026e6	−0.345897
$640$	0	0
$641$	6.89355e6	0.662671	0.331335	−	0.943513i	$-0.392501\pi$
$641$	6.89355e6	0.662671	0.331335	+	0.943513i	$0.392501\pi$
$642$	0	0
$643$	1.84995e7	1.76454	0.882270	−	0.470743i	$-0.156014\pi$
$643$	1.84995e7	1.76454	0.882270	+	0.470743i	$0.156014\pi$
$644$	0	0
$645$	−1.64654e6	−0.155837
$646$	0	0
$647$	1.28596e7	1.20772	0.603860	−	0.797090i	$-0.293629\pi$
$647$	1.28596e7	1.20772	0.603860	+	0.797090i	$0.293629\pi$
$648$	0	0
$649$	3.03240e6	0.282602
$650$	0	0
$651$	1.45935e7	1.34961
$652$	0	0
$653$	−9.13741e6	−0.838572	−0.419286	−	0.907854i	$-0.637720\pi$
$653$	−9.13741e6	−0.838572	−0.419286	+	0.907854i	$0.637720\pi$
$654$	0	0
$655$	4.14042e6	0.377086
$656$	0	0
$657$	138037.	0.0124762
$658$	0	0
$659$	2.71309e6	0.243361	0.121680	−	0.992569i	$-0.461172\pi$
$659$	2.71309e6	0.243361	0.121680	+	0.992569i	$0.461172\pi$
$660$	0	0
$661$	6.09418e6	0.542515	0.271257	−	0.962507i	$-0.412561\pi$
$661$	6.09418e6	0.542515	0.271257	+	0.962507i	$0.412561\pi$
$662$	0	0
$663$	−9.05449e6	−0.799982
$664$	0	0
$665$	−1.07266e7	−0.940604
$666$	0	0
$667$	690418.	0.0600894
$668$	0	0
$669$	5.85666e6	0.505924
$670$	0	0
$671$	2.99119e7	2.56470
$672$	0	0
$673$	−7.17408e6	−0.610561	−0.305280	−	0.952263i	$-0.598750\pi$
$673$	−7.17408e6	−0.610561	−0.305280	+	0.952263i	$0.598750\pi$
$674$	0	0
$675$	−1.05180e7	−0.888534
$676$	0	0
$677$	1.87917e7	1.57577	0.787886	−	0.615821i	$-0.211176\pi$
$677$	1.87917e7	1.57577	0.787886	+	0.615821i	$0.211176\pi$
$678$	0	0
$679$	−7.21695e6	−0.600730
$680$	0	0
$681$	1.80608e7	1.49235
$682$	0	0
$683$	1.69068e7	1.38678	0.693392	−	0.720561i	$-0.256116\pi$
$683$	1.69068e7	1.38678	0.693392	+	0.720561i	$0.256116\pi$
$684$	0	0
$685$	−4.61010e6	−0.375391
$686$	0	0
$687$	5.91070e6	0.477801
$688$	0	0
$689$	−5.55761e6	−0.446005
$690$	0	0
$691$	−1.05329e7	−0.839174	−0.419587	−	0.907715i	$-0.637825\pi$
$691$	−1.05329e7	−0.839174	−0.419587	+	0.907715i	$0.637825\pi$
$692$	0	0
$693$	7.81477e6	0.618135
$694$	0	0
$695$	−1.62783e6	−0.127834
$696$	0	0
$697$	−1.07553e7	−0.838570
$698$	0	0
$699$	1.42252e7	1.10120
$700$	0	0
$701$	−8.98191e6	−0.690357	−0.345178	−	0.938537i	$-0.612182\pi$
$701$	−8.98191e6	−0.690357	−0.345178	+	0.938537i	$0.612182\pi$
$702$	0	0
$703$	−2.09250e7	−1.59690
$704$	0	0
$705$	4.40028e6	0.333432
$706$	0	0
$707$	1.62588e7	1.22332
$708$	0	0
$709$	−2.10499e7	−1.57266	−0.786328	−	0.617809i	$-0.788020\pi$
$709$	−2.10499e7	−1.57266	−0.786328	+	0.617809i	$0.788020\pi$
$710$	0	0
$711$	−2.86974e6	−0.212896
$712$	0	0
$713$	7.73817e6	0.570052
$714$	0	0
$715$	1.02011e7	0.746242
$716$	0	0
$717$	−8.88985e6	−0.645798
$718$	0	0
$719$	−9.28381e6	−0.669737	−0.334868	−	0.942265i	$-0.608692\pi$
$719$	−9.28381e6	−0.669737	−0.334868	+	0.942265i	$0.608692\pi$
$720$	0	0
$721$	−3.14931e7	−2.25620
$722$	0	0
$723$	−7.29039e6	−0.518686
$724$	0	0
$725$	1.56777e6	0.110774
$726$	0	0
$727$	2.28603e7	1.60416	0.802078	−	0.597219i	$-0.203728\pi$
$727$	2.28603e7	1.60416	0.802078	+	0.597219i	$0.203728\pi$
$728$	0	0
$729$	1.57224e7	1.09572
$730$	0	0
$731$	−5.75468e6	−0.398316
$732$	0	0
$733$	−6.63736e6	−0.456284	−0.228142	−	0.973628i	$-0.573265\pi$
$733$	−6.63736e6	−0.456284	−0.228142	+	0.973628i	$0.573265\pi$
$734$	0	0
$735$	−2.94719e6	−0.201228
$736$	0	0
$737$	−3.65433e7	−2.47822
$738$	0	0
$739$	−9.98414e6	−0.672511	−0.336256	−	0.941771i	$-0.609161\pi$
$739$	−9.98414e6	−0.672511	−0.336256	+	0.941771i	$0.609161\pi$
$740$	0	0
$741$	−2.29377e7	−1.53463
$742$	0	0
$743$	−1.08666e7	−0.722143	−0.361072	−	0.932538i	$-0.617589\pi$
$743$	−1.08666e7	−0.722143	−0.361072	+	0.932538i	$0.617589\pi$
$744$	0	0
$745$	−6.20121e6	−0.409341
$746$	0	0
$747$	−865779.	−0.0567683
$748$	0	0
$749$	−1.91437e6	−0.124687
$750$	0	0
$751$	2.03923e7	1.31937	0.659685	−	0.751542i	$-0.270690\pi$
$751$	2.03923e7	1.31937	0.659685	+	0.751542i	$0.270690\pi$
$752$	0	0
$753$	1.13746e7	0.731052
$754$	0	0
$755$	5.70285e6	0.364103
$756$	0	0
$757$	−1.27905e7	−0.811235	−0.405617	−	0.914043i	$-0.632943\pi$
$757$	−1.27905e7	−0.811235	−0.405617	+	0.914043i	$0.632943\pi$
$758$	0	0
$759$	−9.95123e6	−0.627007
$760$	0	0
$761$	2.68836e7	1.68278	0.841388	−	0.540432i	$-0.181739\pi$
$761$	2.68836e7	1.68278	0.841388	+	0.540432i	$0.181739\pi$
$762$	0	0
$763$	−2.24366e7	−1.39523
$764$	0	0
$765$	1.86802e6	0.115406
$766$	0	0
$767$	2.83623e6	0.174081
$768$	0	0
$769$	−1.35295e6	−0.0825024	−0.0412512	−	0.999149i	$-0.513134\pi$
$769$	−1.35295e6	−0.0825024	−0.0412512	+	0.999149i	$0.513134\pi$
$770$	0	0
$771$	865118.	0.0524130
$772$	0	0
$773$	4.99417e6	0.300618	0.150309	−	0.988639i	$-0.451973\pi$
$773$	4.99417e6	0.300618	0.150309	+	0.988639i	$0.451973\pi$
$774$	0	0
$775$	1.75714e7	1.05088
$776$	0	0
$777$	−1.60128e7	−0.951512
$778$	0	0
$779$	−2.72463e7	−1.60866
$780$	0	0
$781$	−3.37608e7	−1.98055
$782$	0	0
$783$	−2.52836e6	−0.147379
$784$	0	0
$785$	6.05192e6	0.350525
$786$	0	0
$787$	−1.39039e7	−0.800201	−0.400101	−	0.916471i	$-0.631025\pi$
$787$	−1.39039e7	−0.800201	−0.400101	+	0.916471i	$0.631025\pi$
$788$	0	0
$789$	1.97357e7	1.12865
$790$	0	0
$791$	1.96195e7	1.11493
$792$	0	0
$793$	2.79768e7	1.57984
$794$	0	0
$795$	−2.75351e6	−0.154514
$796$	0	0
$797$	5.28883e6	0.294927	0.147463	−	0.989068i	$-0.452889\pi$
$797$	5.28883e6	0.294927	0.147463	+	0.989068i	$0.452889\pi$
$798$	0	0
$799$	1.53791e7	0.852244
$800$	0	0
$801$	−1.67540e6	−0.0922648
$802$	0	0
$803$	1.30529e6	0.0714362
$804$	0	0
$805$	−4.35252e6	−0.236729
$806$	0	0
$807$	4.84549e6	0.261911
$808$	0	0
$809$	−2.51376e7	−1.35037	−0.675185	−	0.737648i	$-0.735936\pi$
$809$	−2.51376e7	−1.35037	−0.675185	+	0.737648i	$0.735936\pi$
$810$	0	0
$811$	8.08220e6	0.431497	0.215748	−	0.976449i	$-0.430781\pi$
$811$	8.08220e6	0.431497	0.215748	+	0.976449i	$0.430781\pi$
$812$	0	0
$813$	−4.14430e6	−0.219900
$814$	0	0
$815$	6.96994e6	0.367566
$816$	0	0
$817$	−1.45783e7	−0.764104
$818$	0	0
$819$	7.30920e6	0.380768
$820$	0	0
$821$	1.25459e7	0.649598	0.324799	−	0.945783i	$-0.394703\pi$
$821$	1.25459e7	0.649598	0.324799	+	0.945783i	$0.394703\pi$
$822$	0	0
$823$	8.85247e6	0.455580	0.227790	−	0.973710i	$-0.426850\pi$
$823$	8.85247e6	0.455580	0.227790	+	0.973710i	$0.426850\pi$
$824$	0	0
$825$	−2.25968e7	−1.15588
$826$	0	0
$827$	−5.31327e6	−0.270146	−0.135073	−	0.990836i	$-0.543127\pi$
$827$	−5.31327e6	−0.270146	−0.135073	+	0.990836i	$0.543127\pi$
$828$	0	0
$829$	2.86725e7	1.44904	0.724518	−	0.689256i	$-0.242062\pi$
$829$	2.86725e7	1.44904	0.724518	+	0.689256i	$0.242062\pi$
$830$	0	0
$831$	−2.15661e7	−1.08335
$832$	0	0
$833$	−1.03005e7	−0.514334
$834$	0	0
$835$	−7.28069e6	−0.361373
$836$	0	0
$837$	−2.83378e7	−1.39814
$838$	0	0
$839$	−3.22396e7	−1.58119	−0.790597	−	0.612337i	$-0.790230\pi$
$839$	−3.22396e7	−1.58119	−0.790597	+	0.612337i	$0.790230\pi$
$840$	0	0
$841$	−2.01343e7	−0.981626
$842$	0	0
$843$	2.12500e7	1.02989
$844$	0	0
$845$	667315.	0.0321506
$846$	0	0
$847$	4.78183e7	2.29026
$848$	0	0
$849$	188926.	0.00899546
$850$	0	0
$851$	−8.49074e6	−0.401903
$852$	0	0
$853$	1.05317e7	0.495594	0.247797	−	0.968812i	$-0.420293\pi$
$853$	1.05317e7	0.495594	0.247797	+	0.968812i	$0.420293\pi$
$854$	0	0
$855$	4.73224e6	0.221387
$856$	0	0
$857$	3.08469e7	1.43469	0.717347	−	0.696716i	$-0.245356\pi$
$857$	3.08469e7	1.43469	0.717347	+	0.696716i	$0.245356\pi$
$858$	0	0
$859$	−4.38147e6	−0.202599	−0.101299	−	0.994856i	$-0.532300\pi$
$859$	−4.38147e6	−0.202599	−0.101299	+	0.994856i	$0.532300\pi$
$860$	0	0
$861$	−2.08501e7	−0.958520
$862$	0	0
$863$	−963500.	−0.0440377	−0.0220188	−	0.999758i	$-0.507009\pi$
$863$	−963500.	−0.0440377	−0.0220188	+	0.999758i	$0.507009\pi$
$864$	0	0
$865$	−9.36136e6	−0.425401
$866$	0	0
$867$	2.91867e6	0.131867
$868$	0	0
$869$	−2.71366e7	−1.21900
$870$	0	0
$871$	−3.41792e7	−1.52657
$872$	0	0
$873$	3.18390e6	0.141392
$874$	0	0
$875$	−2.19776e7	−0.970420
$876$	0	0
$877$	−3.37602e7	−1.48220	−0.741098	−	0.671396i	$-0.765695\pi$
$877$	−3.37602e7	−1.48220	−0.741098	+	0.671396i	$0.765695\pi$
$878$	0	0
$879$	−9.03654e6	−0.394485
$880$	0	0
$881$	−1.41583e7	−0.614572	−0.307286	−	0.951617i	$-0.599421\pi$
$881$	−1.41583e7	−0.614572	−0.307286	+	0.951617i	$0.599421\pi$
$882$	0	0
$883$	−4.34003e6	−0.187323	−0.0936615	−	0.995604i	$-0.529857\pi$
$883$	−4.34003e6	−0.187323	−0.0936615	+	0.995604i	$0.529857\pi$
$884$	0	0
$885$	1.40520e6	0.0603088
$886$	0	0
$887$	−4.56731e7	−1.94918	−0.974589	−	0.223999i	$-0.928089\pi$
$887$	−4.56731e7	−1.94918	−0.974589	+	0.223999i	$0.928089\pi$
$888$	0	0
$889$	−2.77307e7	−1.17681
$890$	0	0
$891$	2.47150e7	1.04296
$892$	0	0
$893$	3.89599e7	1.63489
$894$	0	0
$895$	5.41424e6	0.225933
$896$	0	0
$897$	−9.30745e6	−0.386233
$898$	0	0
$899$	4.22390e6	0.174307
$900$	0	0
$901$	−9.62358e6	−0.394934
$902$	0	0
$903$	−1.11560e7	−0.455292
$904$	0	0
$905$	1.15360e7	0.468203
$906$	0	0
$907$	−4.00976e7	−1.61845	−0.809226	−	0.587497i	$-0.800113\pi$
$907$	−4.00976e7	−1.61845	−0.809226	+	0.587497i	$0.800113\pi$
$908$	0	0
$909$	−7.17288e6	−0.287928
$910$	0	0
$911$	3.12748e6	0.124853	0.0624264	−	0.998050i	$-0.480116\pi$
$911$	3.12748e6	0.124853	0.0624264	+	0.998050i	$0.480116\pi$
$912$	0	0
$913$	−8.18691e6	−0.325045
$914$	0	0
$915$	1.38610e7	0.547322
$916$	0	0
$917$	2.80532e7	1.10169
$918$	0	0
$919$	−7.12645e6	−0.278345	−0.139173	−	0.990268i	$-0.544444\pi$
$919$	−7.12645e6	−0.278345	−0.139173	+	0.990268i	$0.544444\pi$
$920$	0	0
$921$	2.61713e7	1.01666
$922$	0	0
$923$	−3.15767e7	−1.22001
$924$	0	0
$925$	−1.92804e7	−0.740902
$926$	0	0
$927$	1.38938e7	0.531034
$928$	0	0
$929$	−1.79294e7	−0.681595	−0.340797	−	0.940137i	$-0.610697\pi$
$929$	−1.79294e7	−0.681595	−0.340797	+	0.940137i	$0.610697\pi$
$930$	0	0
$931$	−2.60942e7	−0.986666
$932$	0	0
$933$	3.59561e7	1.35229
$934$	0	0
$935$	1.76642e7	0.660792
$936$	0	0
$937$	−9.20290e6	−0.342433	−0.171217	−	0.985233i	$-0.554770\pi$
$937$	−9.20290e6	−0.342433	−0.171217	+	0.985233i	$0.554770\pi$
$938$	0	0
$939$	−3.64158e7	−1.34780
$940$	0	0
$941$	3.33675e7	1.22843	0.614213	−	0.789140i	$-0.289473\pi$
$941$	3.33675e7	1.22843	0.614213	+	0.789140i	$0.289473\pi$
$942$	0	0
$943$	−1.10557e7	−0.404864
$944$	0	0
$945$	1.59393e7	0.580616
$946$	0	0
$947$	−4.39806e7	−1.59362	−0.796812	−	0.604227i	$-0.793482\pi$
$947$	−4.39806e7	−1.59362	−0.796812	+	0.604227i	$0.793482\pi$
$948$	0	0
$949$	1.22085e6	0.0440043
$950$	0	0
$951$	−3.27151e7	−1.17300
$952$	0	0
$953$	−3.13649e7	−1.11869	−0.559347	−	0.828933i	$-0.688948\pi$
$953$	−3.13649e7	−1.11869	−0.559347	+	0.828933i	$0.688948\pi$
$954$	0	0
$955$	3.08970e6	0.109624
$956$	0	0
$957$	−5.43191e6	−0.191722
$958$	0	0
$959$	−3.12355e7	−1.09673
$960$	0	0
$961$	1.87121e7	0.653603
$962$	0	0
$963$	844561.	0.0293471
$964$	0	0
$965$	1.85266e7	0.640440
$966$	0	0
$967$	1.21293e7	0.417127	0.208564	−	0.978009i	$-0.433121\pi$
$967$	1.21293e7	0.417127	0.208564	+	0.978009i	$0.433121\pi$
$968$	0	0
$969$	−3.97191e7	−1.35891
$970$	0	0
$971$	3.72170e7	1.26676	0.633379	−	0.773842i	$-0.281668\pi$
$971$	3.72170e7	1.26676	0.633379	+	0.773842i	$0.281668\pi$
$972$	0	0
$973$	−1.10293e7	−0.373478
$974$	0	0
$975$	−2.11349e7	−0.712014
$976$	0	0
$977$	1.68541e7	0.564897	0.282448	−	0.959283i	$-0.408853\pi$
$977$	1.68541e7	0.564897	0.282448	+	0.959283i	$0.408853\pi$
$978$	0	0
$979$	−1.58427e7	−0.528291
$980$	0	0
$981$	9.89836e6	0.328391
$982$	0	0
$983$	−4.46694e7	−1.47444	−0.737219	−	0.675654i	$-0.763861\pi$
$983$	−4.46694e7	−1.47444	−0.737219	+	0.675654i	$0.763861\pi$
$984$	0	0
$985$	2.36353e7	0.776193
$986$	0	0
$987$	2.98139e7	0.974150
$988$	0	0
$989$	−5.91545e6	−0.192308
$990$	0	0
$991$	4.08174e7	1.32026	0.660132	−	0.751149i	$-0.270500\pi$
$991$	4.08174e7	1.32026	0.660132	+	0.751149i	$0.270500\pi$
$992$	0	0
$993$	−1.63896e7	−0.527467
$994$	0	0
$995$	5.51580e6	0.176624
$996$	0	0
$997$	3.61428e7	1.15155	0.575776	−	0.817607i	$-0.304700\pi$
$997$	3.61428e7	1.15155	0.575776	+	0.817607i	$0.304700\pi$
$998$	0	0
$999$	3.10937e7	0.985733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.18	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-13.0981 q^{3} +23.8997 q^{5} +161.931 q^{7} -71.4392 q^{9} +O(q^{10})\)	\(q-13.0981 q^{3} +23.8997 q^{5} +161.931 q^{7} -71.4392 q^{9} -675.538 q^{11} -631.834 q^{13} -313.041 q^{15} -1094.09 q^{17} -2771.65 q^{19} -2120.99 q^{21} -1124.65 q^{23} -2553.80 q^{25} +4118.56 q^{27} -613.895 q^{29} -6880.50 q^{31} +8848.27 q^{33} +3870.10 q^{35} +7549.66 q^{37} +8275.84 q^{39} +9830.37 q^{41} +5259.81 q^{43} -1707.37 q^{45} -14056.6 q^{47} +9414.70 q^{49} +14330.5 q^{51} +8795.99 q^{53} -16145.1 q^{55} +36303.4 q^{57} -4488.88 q^{59} -44278.6 q^{61} -11568.2 q^{63} -15100.6 q^{65} +54095.1 q^{67} +14730.8 q^{69} +49976.2 q^{71} -1932.23 q^{73} +33450.0 q^{75} -109391. q^{77} +40170.3 q^{79} -36585.7 q^{81} +12119.1 q^{83} -26148.3 q^{85} +8040.87 q^{87} +23452.0 q^{89} -102314. q^{91} +90121.6 q^{93} -66241.6 q^{95} -44568.0 q^{97} +48259.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 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

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Database

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