LMFDB - Embedded newform 1028.6.a.a.1.13

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			1.13
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)

$f(q)$	$=$	$q-15.1206 q^{3} +38.6175 q^{5} -225.318 q^{7} -14.3669 q^{9} +O(q^{10})$	$q-15.1206 q^{3} +38.6175 q^{5} -225.318 q^{7} -14.3669 q^{9} -598.145 q^{11} -286.879 q^{13} -583.920 q^{15} +236.833 q^{17} +1048.95 q^{19} +3406.95 q^{21} +4658.76 q^{23} -1633.69 q^{25} +3891.55 q^{27} -632.209 q^{29} +7782.79 q^{31} +9044.32 q^{33} -8701.23 q^{35} -7488.86 q^{37} +4337.79 q^{39} +18479.0 q^{41} +7303.40 q^{43} -554.816 q^{45} +22368.9 q^{47} +33961.4 q^{49} -3581.06 q^{51} -26777.8 q^{53} -23098.9 q^{55} -15860.7 q^{57} -50284.0 q^{59} -3094.01 q^{61} +3237.14 q^{63} -11078.6 q^{65} -63203.7 q^{67} -70443.4 q^{69} +2982.93 q^{71} +22951.8 q^{73} +24702.4 q^{75} +134773. q^{77} +63785.0 q^{79} -55351.4 q^{81} +117118. q^{83} +9145.90 q^{85} +9559.39 q^{87} -95876.4 q^{89} +64639.2 q^{91} -117681. q^{93} +40507.7 q^{95} -103344. q^{97} +8593.52 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−15.1206	−0.969988	−0.484994	−	0.874517i	$-0.661178\pi$
$3$	−15.1206	−0.969988	−0.484994	+	0.874517i	$0.661178\pi$
$4$	0	0
$5$	38.6175	0.690811	0.345405	−	0.938454i	$-0.387741\pi$
$5$	38.6175	0.690811	0.345405	+	0.938454i	$0.387741\pi$
$6$	0	0
$7$	−225.318	−1.73801	−0.869003	−	0.494806i	$-0.835239\pi$
$7$	−225.318	−1.73801	−0.869003	+	0.494806i	$0.835239\pi$
$8$	0	0
$9$	−14.3669	−0.0591232
$10$	0	0
$11$	−598.145	−1.49047	−0.745237	−	0.666799i	$-0.767664\pi$
$11$	−598.145	−1.49047	−0.745237	+	0.666799i	$0.767664\pi$
$12$	0	0
$13$	−286.879	−0.470805	−0.235402	−	0.971898i	$-0.575641\pi$
$13$	−286.879	−0.470805	−0.235402	+	0.971898i	$0.575641\pi$
$14$	0	0
$15$	−583.920	−0.670078
$16$	0	0
$17$	236.833	0.198756	0.0993780	−	0.995050i	$-0.468315\pi$
$17$	236.833	0.198756	0.0993780	+	0.995050i	$0.468315\pi$
$18$	0	0
$19$	1048.95	0.666606	0.333303	−	0.942820i	$-0.391837\pi$
$19$	1048.95	0.666606	0.333303	+	0.942820i	$0.391837\pi$
$20$	0	0
$21$	3406.95	1.68585
$22$	0	0
$23$	4658.76	1.83633	0.918166	−	0.396196i	$-0.129670\pi$
$23$	4658.76	1.83633	0.918166	+	0.396196i	$0.129670\pi$
$24$	0	0
$25$	−1633.69	−0.522781
$26$	0	0
$27$	3891.55	1.02734
$28$	0	0
$29$	−632.209	−0.139594	−0.0697968	−	0.997561i	$-0.522235\pi$
$29$	−632.209	−0.139594	−0.0697968	+	0.997561i	$0.522235\pi$
$30$	0	0
$31$	7782.79	1.45456	0.727279	−	0.686342i	$-0.240784\pi$
$31$	7782.79	1.45456	0.727279	+	0.686342i	$0.240784\pi$
$32$	0	0
$33$	9044.32	1.44574
$34$	0	0
$35$	−8701.23	−1.20063
$36$	0	0
$37$	−7488.86	−0.899314	−0.449657	−	0.893201i	$-0.648454\pi$
$37$	−7488.86	−0.899314	−0.449657	+	0.893201i	$0.648454\pi$
$38$	0	0
$39$	4337.79	0.456675
$40$	0	0
$41$	18479.0	1.71680	0.858400	−	0.512980i	$-0.171459\pi$
$41$	18479.0	1.71680	0.858400	+	0.512980i	$0.171459\pi$
$42$	0	0
$43$	7303.40	0.602357	0.301178	−	0.953568i	$-0.402620\pi$
$43$	7303.40	0.602357	0.301178	+	0.953568i	$0.402620\pi$
$44$	0	0
$45$	−554.816	−0.0408430
$46$	0	0
$47$	22368.9	1.47707	0.738533	−	0.674217i	$-0.235519\pi$
$47$	22368.9	1.47707	0.738533	+	0.674217i	$0.235519\pi$
$48$	0	0
$49$	33961.4	2.02067
$50$	0	0
$51$	−3581.06	−0.192791
$52$	0	0
$53$	−26777.8	−1.30944	−0.654720	−	0.755871i	$-0.727214\pi$
$53$	−26777.8	−1.30944	−0.654720	+	0.755871i	$0.727214\pi$
$54$	0	0
$55$	−23098.9	−1.02964
$56$	0	0
$57$	−15860.7	−0.646600
$58$	0	0
$59$	−50284.0	−1.88061	−0.940307	−	0.340328i	$-0.889462\pi$
$59$	−50284.0	−1.88061	−0.940307	+	0.340328i	$0.889462\pi$
$60$	0	0
$61$	−3094.01	−0.106463	−0.0532313	−	0.998582i	$-0.516952\pi$
$61$	−3094.01	−0.106463	−0.0532313	+	0.998582i	$0.516952\pi$
$62$	0	0
$63$	3237.14	0.102757
$64$	0	0
$65$	−11078.6	−0.325237
$66$	0	0
$67$	−63203.7	−1.72011	−0.860053	−	0.510204i	$-0.829570\pi$
$67$	−63203.7	−1.72011	−0.860053	+	0.510204i	$0.829570\pi$
$68$	0	0
$69$	−70443.4	−1.78122
$70$	0	0
$71$	2982.93	0.0702260	0.0351130	−	0.999383i	$-0.488821\pi$
$71$	2982.93	0.0702260	0.0351130	+	0.999383i	$0.488821\pi$
$72$	0	0
$73$	22951.8	0.504092	0.252046	−	0.967715i	$-0.418897\pi$
$73$	22951.8	0.504092	0.252046	+	0.967715i	$0.418897\pi$
$74$	0	0
$75$	24702.4	0.507091
$76$	0	0
$77$	134773.	2.59045
$78$	0	0
$79$	63785.0	1.14988	0.574938	−	0.818197i	$-0.305026\pi$
$79$	63785.0	1.14988	0.574938	+	0.818197i	$0.305026\pi$
$80$	0	0
$81$	−55351.4	−0.937381
$82$	0	0
$83$	117118.	1.86608	0.933039	−	0.359776i	$-0.117147\pi$
$83$	117118.	1.86608	0.933039	+	0.359776i	$0.117147\pi$
$84$	0	0
$85$	9145.90	0.137303
$86$	0	0
$87$	9559.39	0.135404
$88$	0	0
$89$	−95876.4	−1.28303	−0.641515	−	0.767110i	$-0.721694\pi$
$89$	−95876.4	−1.28303	−0.641515	+	0.767110i	$0.721694\pi$
$90$	0	0
$91$	64639.2	0.818262
$92$	0	0
$93$	−117681.	−1.41090
$94$	0	0
$95$	40507.7	0.460498
$96$	0	0
$97$	−103344.	−1.11520	−0.557602	−	0.830109i	$-0.688278\pi$
$97$	−103344.	−1.11520	−0.557602	+	0.830109i	$0.688278\pi$
$98$	0	0
$99$	8593.52	0.0881217
$100$	0	0
$101$	−113963.	−1.11163	−0.555816	−	0.831306i	$-0.687594\pi$
$101$	−113963.	−1.11163	−0.555816	+	0.831306i	$0.687594\pi$
$102$	0	0
$103$	93964.4	0.872711	0.436355	−	0.899774i	$-0.356269\pi$
$103$	93964.4	0.872711	0.436355	+	0.899774i	$0.356269\pi$
$104$	0	0
$105$	131568.	1.16460
$106$	0	0
$107$	−66213.3	−0.559095	−0.279547	−	0.960132i	$-0.590184\pi$
$107$	−66213.3	−0.559095	−0.279547	+	0.960132i	$0.590184\pi$
$108$	0	0
$109$	−132776.	−1.07042	−0.535209	−	0.844720i	$-0.679767\pi$
$109$	−132776.	−1.07042	−0.535209	+	0.844720i	$0.679767\pi$
$110$	0	0
$111$	113236.	0.872324
$112$	0	0
$113$	104397.	0.769113	0.384556	−	0.923102i	$-0.374354\pi$
$113$	104397.	0.769113	0.384556	+	0.923102i	$0.374354\pi$
$114$	0	0
$115$	179910.	1.26856
$116$	0	0
$117$	4121.58	0.0278355
$118$	0	0
$119$	−53362.8	−0.345439
$120$	0	0
$121$	196726.	1.22151
$122$	0	0
$123$	−279415.	−1.66528
$124$	0	0
$125$	−183769.	−1.05195
$126$	0	0
$127$	111812.	0.615146	0.307573	−	0.951524i	$-0.400483\pi$
$127$	111812.	0.615146	0.307573	+	0.951524i	$0.400483\pi$
$128$	0	0
$129$	−110432.	−0.584279
$130$	0	0
$131$	218978.	1.11486	0.557432	−	0.830223i	$-0.311787\pi$
$131$	218978.	1.11486	0.557432	+	0.830223i	$0.311787\pi$
$132$	0	0
$133$	−236347.	−1.15857
$134$	0	0
$135$	150282.	0.709695
$136$	0	0
$137$	319184.	1.45292	0.726458	−	0.687211i	$-0.241165\pi$
$137$	319184.	1.45292	0.726458	+	0.687211i	$0.241165\pi$
$138$	0	0
$139$	−35264.8	−0.154812	−0.0774059	−	0.997000i	$-0.524664\pi$
$139$	−35264.8	−0.154812	−0.0774059	+	0.997000i	$0.524664\pi$
$140$	0	0
$141$	−338232.	−1.43274
$142$	0	0
$143$	171595.	0.701723
$144$	0	0
$145$	−24414.3	−0.0964328
$146$	0	0
$147$	−513517.	−1.96002
$148$	0	0
$149$	−123209.	−0.454649	−0.227325	−	0.973819i	$-0.572998\pi$
$149$	−123209.	−0.454649	−0.227325	+	0.973819i	$0.572998\pi$
$150$	0	0
$151$	549101.	1.95979	0.979896	−	0.199507i	$-0.0639339\pi$
$151$	549101.	1.95979	0.979896	+	0.199507i	$0.0639339\pi$
$152$	0	0
$153$	−3402.57	−0.0117511
$154$	0	0
$155$	300552.	1.00482
$156$	0	0
$157$	85394.8	0.276492	0.138246	−	0.990398i	$-0.455854\pi$
$157$	85394.8	0.276492	0.138246	+	0.990398i	$0.455854\pi$
$158$	0	0
$159$	404897.	1.27014
$160$	0	0
$161$	−1.04970e6	−3.19156
$162$	0	0
$163$	−352680.	−1.03971	−0.519855	−	0.854255i	$-0.674014\pi$
$163$	−352680.	−1.03971	−0.519855	+	0.854255i	$0.674014\pi$
$164$	0	0
$165$	349269.	0.998734
$166$	0	0
$167$	−453826.	−1.25921	−0.629605	−	0.776915i	$-0.716783\pi$
$167$	−453826.	−1.25921	−0.629605	+	0.776915i	$0.716783\pi$
$168$	0	0
$169$	−288993.	−0.778343
$170$	0	0
$171$	−15070.1	−0.0394119
$172$	0	0
$173$	−498568.	−1.26651	−0.633256	−	0.773942i	$-0.718282\pi$
$173$	−498568.	−1.26651	−0.633256	+	0.773942i	$0.718282\pi$
$174$	0	0
$175$	368100.	0.908596
$176$	0	0
$177$	760325.	1.82417
$178$	0	0
$179$	723607.	1.68799	0.843995	−	0.536350i	$-0.180197\pi$
$179$	723607.	1.68799	0.843995	+	0.536350i	$0.180197\pi$
$180$	0	0
$181$	29421.9	0.0667536	0.0333768	−	0.999443i	$-0.489374\pi$
$181$	29421.9	0.0667536	0.0333768	+	0.999443i	$0.489374\pi$
$182$	0	0
$183$	46783.3	0.103267
$184$	0	0
$185$	−289201.	−0.621256
$186$	0	0
$187$	−141660.	−0.296241
$188$	0	0
$189$	−876837.	−1.78552
$190$	0	0
$191$	416199.	0.825500	0.412750	−	0.910844i	$-0.364568\pi$
$191$	416199.	0.825500	0.412750	+	0.910844i	$0.364568\pi$
$192$	0	0
$193$	703176.	1.35885	0.679424	−	0.733746i	$-0.262230\pi$
$193$	703176.	1.35885	0.679424	+	0.733746i	$0.262230\pi$
$194$	0	0
$195$	167515.	0.315476
$196$	0	0
$197$	329832.	0.605518	0.302759	−	0.953067i	$-0.402092\pi$
$197$	329832.	0.605518	0.302759	+	0.953067i	$0.402092\pi$
$198$	0	0
$199$	−394227.	−0.705689	−0.352845	−	0.935682i	$-0.614786\pi$
$199$	−394227.	−0.705689	−0.352845	+	0.935682i	$0.614786\pi$
$200$	0	0
$201$	955679.	1.66848
$202$	0	0
$203$	142448.	0.242615
$204$	0	0
$205$	713614.	1.18598
$206$	0	0
$207$	−66932.2	−0.108570
$208$	0	0
$209$	−627421.	−0.993559
$210$	0	0
$211$	−94376.1	−0.145934	−0.0729669	−	0.997334i	$-0.523247\pi$
$211$	−94376.1	−0.145934	−0.0729669	+	0.997334i	$0.523247\pi$
$212$	0	0
$213$	−45103.8	−0.0681184
$214$	0	0
$215$	282039.	0.416115
$216$	0	0
$217$	−1.75361e6	−2.52803
$218$	0	0
$219$	−347045.	−0.488963
$220$	0	0
$221$	−67942.5	−0.0935753
$222$	0	0
$223$	−1.04895e6	−1.41252	−0.706260	−	0.707953i	$-0.749619\pi$
$223$	−1.04895e6	−1.41252	−0.706260	+	0.707953i	$0.749619\pi$
$224$	0	0
$225$	23471.1	0.0309085
$226$	0	0
$227$	−955401.	−1.23061	−0.615306	−	0.788289i	$-0.710967\pi$
$227$	−955401.	−1.23061	−0.615306	+	0.788289i	$0.710967\pi$
$228$	0	0
$229$	62976.7	0.0793581	0.0396790	−	0.999212i	$-0.487366\pi$
$229$	62976.7	0.0793581	0.0396790	+	0.999212i	$0.487366\pi$
$230$	0	0
$231$	−2.03785e6	−2.51271
$232$	0	0
$233$	−732397.	−0.883806	−0.441903	−	0.897063i	$-0.645696\pi$
$233$	−732397.	−0.883806	−0.441903	+	0.897063i	$0.645696\pi$
$234$	0	0
$235$	863831.	1.02037
$236$	0	0
$237$	−964469.	−1.11537
$238$	0	0
$239$	−203418.	−0.230353	−0.115177	−	0.993345i	$-0.536743\pi$
$239$	−203418.	−0.230353	−0.115177	+	0.993345i	$0.536743\pi$
$240$	0	0
$241$	112789.	0.125090	0.0625449	−	0.998042i	$-0.480078\pi$
$241$	112789.	0.125090	0.0625449	+	0.998042i	$0.480078\pi$
$242$	0	0
$243$	−108698.	−0.118088
$244$	0	0
$245$	1.31150e6	1.39590
$246$	0	0
$247$	−300921.	−0.313841
$248$	0	0
$249$	−1.77090e6	−1.81007
$250$	0	0
$251$	−244119.	−0.244578	−0.122289	−	0.992495i	$-0.539024\pi$
$251$	−244119.	−0.244578	−0.122289	+	0.992495i	$0.539024\pi$
$252$	0	0
$253$	−2.78662e6	−2.73701
$254$	0	0
$255$	−138292.	−0.133182
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	1.68738e6	1.56301
$260$	0	0
$261$	9082.92	0.00825323
$262$	0	0
$263$	−1.12023e6	−0.998658	−0.499329	−	0.866413i	$-0.666420\pi$
$263$	−1.12023e6	−0.998658	−0.499329	+	0.866413i	$0.666420\pi$
$264$	0	0
$265$	−1.03409e6	−0.904575
$266$	0	0
$267$	1.44971e6	1.24452
$268$	0	0
$269$	577450.	0.486557	0.243279	−	0.969956i	$-0.421777\pi$
$269$	577450.	0.486557	0.243279	+	0.969956i	$0.421777\pi$
$270$	0	0
$271$	−181664.	−0.150261	−0.0751306	−	0.997174i	$-0.523937\pi$
$271$	−181664.	−0.150261	−0.0751306	+	0.997174i	$0.523937\pi$
$272$	0	0
$273$	−977384.	−0.793704
$274$	0	0
$275$	977183.	0.779191
$276$	0	0
$277$	23691.7	0.0185523	0.00927613	−	0.999957i	$-0.497047\pi$
$277$	23691.7	0.0185523	0.00927613	+	0.999957i	$0.497047\pi$
$278$	0	0
$279$	−111815.	−0.0859982
$280$	0	0
$281$	1.47955e6	1.11780	0.558899	−	0.829236i	$-0.311224\pi$
$281$	1.47955e6	1.11780	0.558899	+	0.829236i	$0.311224\pi$
$282$	0	0
$283$	602876.	0.447468	0.223734	−	0.974650i	$-0.428175\pi$
$283$	602876.	0.447468	0.223734	+	0.974650i	$0.428175\pi$
$284$	0	0
$285$	−612501.	−0.446678
$286$	0	0
$287$	−4.16367e6	−2.98381
$288$	0	0
$289$	−1.36377e6	−0.960496
$290$	0	0
$291$	1.56262e6	1.08173
$292$	0	0
$293$	−1.59252e6	−1.08372	−0.541858	−	0.840470i	$-0.682279\pi$
$293$	−1.59252e6	−1.08372	−0.541858	+	0.840470i	$0.682279\pi$
$294$	0	0
$295$	−1.94184e6	−1.29915
$296$	0	0
$297$	−2.32771e6	−1.53122
$298$	0	0
$299$	−1.33650e6	−0.864554
$300$	0	0
$301$	−1.64559e6	−1.04690
$302$	0	0
$303$	1.72319e6	1.07827
$304$	0	0
$305$	−119483.	−0.0735455
$306$	0	0
$307$	−322970.	−0.195576	−0.0977881	−	0.995207i	$-0.531177\pi$
$307$	−322970.	−0.195576	−0.0977881	+	0.995207i	$0.531177\pi$
$308$	0	0
$309$	−1.42080e6	−0.846519
$310$	0	0
$311$	−717073.	−0.420400	−0.210200	−	0.977658i	$-0.567412\pi$
$311$	−717073.	−0.420400	−0.210200	+	0.977658i	$0.567412\pi$
$312$	0	0
$313$	539620.	0.311335	0.155667	−	0.987810i	$-0.450247\pi$
$313$	539620.	0.311335	0.155667	+	0.987810i	$0.450247\pi$
$314$	0	0
$315$	125010.	0.0709854
$316$	0	0
$317$	85851.0	0.0479841	0.0239920	−	0.999712i	$-0.492362\pi$
$317$	85851.0	0.0479841	0.0239920	+	0.999712i	$0.492362\pi$
$318$	0	0
$319$	378153.	0.208061
$320$	0	0
$321$	1.00119e6	0.542315
$322$	0	0
$323$	248425.	0.132492
$324$	0	0
$325$	468672.	0.246128
$326$	0	0
$327$	2.00766e6	1.03829
$328$	0	0
$329$	−5.04012e6	−2.56715
$330$	0	0
$331$	1.86800e6	0.937143	0.468572	−	0.883425i	$-0.344769\pi$
$331$	1.86800e6	0.937143	0.468572	+	0.883425i	$0.344769\pi$
$332$	0	0
$333$	107592.	0.0531704
$334$	0	0
$335$	−2.44077e6	−1.18827
$336$	0	0
$337$	3.95422e6	1.89665	0.948323	−	0.317307i	$-0.102779\pi$
$337$	3.95422e6	1.89665	0.948323	+	0.317307i	$0.102779\pi$
$338$	0	0
$339$	−1.57854e6	−0.746030
$340$	0	0
$341$	−4.65524e6	−2.16798
$342$	0	0
$343$	−3.86519e6	−1.77393
$344$	0	0
$345$	−2.72035e6	−1.23049
$346$	0	0
$347$	−991145.	−0.441889	−0.220945	−	0.975286i	$-0.570914\pi$
$347$	−991145.	−0.441889	−0.220945	+	0.975286i	$0.570914\pi$
$348$	0	0
$349$	−394944.	−0.173569	−0.0867844	−	0.996227i	$-0.527659\pi$
$349$	−394944.	−0.173569	−0.0867844	+	0.996227i	$0.527659\pi$
$350$	0	0
$351$	−1.11640e6	−0.483675
$352$	0	0
$353$	−2.02668e6	−0.865664	−0.432832	−	0.901475i	$-0.642486\pi$
$353$	−2.02668e6	−0.865664	−0.432832	+	0.901475i	$0.642486\pi$
$354$	0	0
$355$	115193.	0.0485129
$356$	0	0
$357$	806879.	0.335072
$358$	0	0
$359$	2.09645e6	0.858518	0.429259	−	0.903181i	$-0.358775\pi$
$359$	2.09645e6	0.858518	0.429259	+	0.903181i	$0.358775\pi$
$360$	0	0
$361$	−1.37581e6	−0.555637
$362$	0	0
$363$	−2.97462e6	−1.18485
$364$	0	0
$365$	886341.	0.348232
$366$	0	0
$367$	−867558.	−0.336228	−0.168114	−	0.985768i	$-0.553768\pi$
$367$	−867558.	−0.336228	−0.168114	+	0.985768i	$0.553768\pi$
$368$	0	0
$369$	−265488.	−0.101503
$370$	0	0
$371$	6.03354e6	2.27582
$372$	0	0
$373$	2.76428e6	1.02875	0.514374	−	0.857566i	$-0.328024\pi$
$373$	2.76428e6	1.02875	0.514374	+	0.857566i	$0.328024\pi$
$374$	0	0
$375$	2.77870e6	1.02038
$376$	0	0
$377$	181368.	0.0657214
$378$	0	0
$379$	−5.05687e6	−1.80836	−0.904178	−	0.427156i	$-0.859516\pi$
$379$	−5.05687e6	−1.80836	−0.904178	+	0.427156i	$0.859516\pi$
$380$	0	0
$381$	−1.69066e6	−0.596685
$382$	0	0
$383$	4.52244e6	1.57535	0.787673	−	0.616093i	$-0.211285\pi$
$383$	4.52244e6	1.57535	0.787673	+	0.616093i	$0.211285\pi$
$384$	0	0
$385$	5.20459e6	1.78951
$386$	0	0
$387$	−104928.	−0.0356133
$388$	0	0
$389$	−5.32412e6	−1.78391	−0.891956	−	0.452122i	$-0.850667\pi$
$389$	−5.32412e6	−1.78391	−0.891956	+	0.452122i	$0.850667\pi$
$390$	0	0
$391$	1.10335e6	0.364982
$392$	0	0
$393$	−3.31108e6	−1.08140
$394$	0	0
$395$	2.46322e6	0.794347
$396$	0	0
$397$	3.57157e6	1.13732	0.568660	−	0.822572i	$-0.307462\pi$
$397$	3.57157e6	1.13732	0.568660	+	0.822572i	$0.307462\pi$
$398$	0	0
$399$	3.57371e6	1.12379
$400$	0	0
$401$	2.09174e6	0.649600	0.324800	−	0.945783i	$-0.394703\pi$
$401$	2.09174e6	0.649600	0.324800	+	0.945783i	$0.394703\pi$
$402$	0	0
$403$	−2.23272e6	−0.684813
$404$	0	0
$405$	−2.13753e6	−0.647553
$406$	0	0
$407$	4.47942e6	1.34041
$408$	0	0
$409$	6.41694e6	1.89679	0.948397	−	0.317086i	$-0.102705\pi$
$409$	6.41694e6	1.89679	0.948397	+	0.317086i	$0.102705\pi$
$410$	0	0
$411$	−4.82627e6	−1.40931
$412$	0	0
$413$	1.13299e7	3.26852
$414$	0	0
$415$	4.52282e6	1.28911
$416$	0	0
$417$	533225.	0.150166
$418$	0	0
$419$	2.83491e6	0.788869	0.394435	−	0.918924i	$-0.370940\pi$
$419$	2.83491e6	0.788869	0.394435	+	0.918924i	$0.370940\pi$
$420$	0	0
$421$	5.46605e6	1.50303	0.751517	−	0.659714i	$-0.229323\pi$
$421$	5.46605e6	1.50303	0.751517	+	0.659714i	$0.229323\pi$
$422$	0	0
$423$	−321373.	−0.0873290
$424$	0	0
$425$	−386912.	−0.103906
$426$	0	0
$427$	697137.	0.185033
$428$	0	0
$429$	−2.59463e6	−0.680663
$430$	0	0
$431$	−5.25969e6	−1.36385	−0.681926	−	0.731421i	$-0.738857\pi$
$431$	−5.25969e6	−1.36385	−0.681926	+	0.731421i	$0.738857\pi$
$432$	0	0
$433$	−768783.	−0.197053	−0.0985267	−	0.995134i	$-0.531413\pi$
$433$	−768783.	−0.197053	−0.0985267	+	0.995134i	$0.531413\pi$
$434$	0	0
$435$	369160.	0.0935387
$436$	0	0
$437$	4.88679e6	1.22411
$438$	0	0
$439$	−805863.	−0.199572	−0.0997860	−	0.995009i	$-0.531816\pi$
$439$	−805863.	−0.199572	−0.0997860	+	0.995009i	$0.531816\pi$
$440$	0	0
$441$	−487921.	−0.119468
$442$	0	0
$443$	−4.73042e6	−1.14522	−0.572612	−	0.819827i	$-0.694070\pi$
$443$	−4.73042e6	−1.14522	−0.572612	+	0.819827i	$0.694070\pi$
$444$	0	0
$445$	−3.70251e6	−0.886331
$446$	0	0
$447$	1.86299e6	0.441004
$448$	0	0
$449$	−4.77446e6	−1.11766	−0.558828	−	0.829284i	$-0.688749\pi$
$449$	−4.77446e6	−1.11766	−0.558828	+	0.829284i	$0.688749\pi$
$450$	0	0
$451$	−1.10531e7	−2.55885
$452$	0	0
$453$	−8.30275e6	−1.90098
$454$	0	0
$455$	2.49620e6	0.565264
$456$	0	0
$457$	264265.	0.0591902	0.0295951	−	0.999562i	$-0.490578\pi$
$457$	264265.	0.0591902	0.0295951	+	0.999562i	$0.490578\pi$
$458$	0	0
$459$	921647.	0.204189
$460$	0	0
$461$	4.25483e6	0.932459	0.466230	−	0.884664i	$-0.345612\pi$
$461$	4.25483e6	0.932459	0.466230	+	0.884664i	$0.345612\pi$
$462$	0	0
$463$	6.68172e6	1.44856	0.724279	−	0.689507i	$-0.242173\pi$
$463$	6.68172e6	1.44856	0.724279	+	0.689507i	$0.242173\pi$
$464$	0	0
$465$	−4.54453e6	−0.974668
$466$	0	0
$467$	−309263.	−0.0656199	−0.0328100	−	0.999462i	$-0.510446\pi$
$467$	−309263.	−0.0656199	−0.0328100	+	0.999462i	$0.510446\pi$
$468$	0	0
$469$	1.42410e7	2.98956
$470$	0	0
$471$	−1.29122e6	−0.268194
$472$	0	0
$473$	−4.36849e6	−0.897798
$474$	0	0
$475$	−1.71365e6	−0.348489
$476$	0	0
$477$	384716.	0.0774184
$478$	0	0
$479$	−6.10931e6	−1.21662	−0.608308	−	0.793701i	$-0.708151\pi$
$479$	−6.10931e6	−1.21662	−0.608308	+	0.793701i	$0.708151\pi$
$480$	0	0
$481$	2.14840e6	0.423402
$482$	0	0
$483$	1.58722e7	3.09577
$484$	0	0
$485$	−3.99087e6	−0.770395
$486$	0	0
$487$	−327541.	−0.0625812	−0.0312906	−	0.999510i	$-0.509962\pi$
$487$	−327541.	−0.0625812	−0.0312906	+	0.999510i	$0.509962\pi$
$488$	0	0
$489$	5.33274e6	1.00851
$490$	0	0
$491$	−3.11539e6	−0.583187	−0.291594	−	0.956542i	$-0.594186\pi$
$491$	−3.11539e6	−0.583187	−0.291594	+	0.956542i	$0.594186\pi$
$492$	0	0
$493$	−149728.	−0.0277451
$494$	0	0
$495$	331860.	0.0608754
$496$	0	0
$497$	−672110.	−0.122053
$498$	0	0
$499$	4.06007e6	0.729931	0.364966	−	0.931021i	$-0.381081\pi$
$499$	4.06007e6	0.729931	0.364966	+	0.931021i	$0.381081\pi$
$500$	0	0
$501$	6.86213e6	1.22142
$502$	0	0
$503$	−5.63782e6	−0.993554	−0.496777	−	0.867878i	$-0.665483\pi$
$503$	−5.63782e6	−0.993554	−0.496777	+	0.867878i	$0.665483\pi$
$504$	0	0
$505$	−4.40097e6	−0.767927
$506$	0	0
$507$	4.36976e6	0.754983
$508$	0	0
$509$	−2.97323e6	−0.508668	−0.254334	−	0.967116i	$-0.581856\pi$
$509$	−2.97323e6	−0.508668	−0.254334	+	0.967116i	$0.581856\pi$
$510$	0	0
$511$	−5.17146e6	−0.876114
$512$	0	0
$513$	4.08202e6	0.684829
$514$	0	0
$515$	3.62867e6	0.602878
$516$	0	0
$517$	−1.33798e7	−2.20153
$518$	0	0
$519$	7.53866e6	1.22850
$520$	0	0
$521$	−2.51643e6	−0.406153	−0.203077	−	0.979163i	$-0.565094\pi$
$521$	−2.51643e6	−0.406153	−0.203077	+	0.979163i	$0.565094\pi$
$522$	0	0
$523$	3.11984e6	0.498744	0.249372	−	0.968408i	$-0.419776\pi$
$523$	3.11984e6	0.498744	0.249372	+	0.968408i	$0.419776\pi$
$524$	0	0
$525$	−5.56590e6	−0.881327
$526$	0	0
$527$	1.84322e6	0.289102
$528$	0	0
$529$	1.52677e7	2.37211
$530$	0	0
$531$	722427.	0.111188
$532$	0	0
$533$	−5.30126e6	−0.808278
$534$	0	0
$535$	−2.55699e6	−0.386229
$536$	0	0
$537$	−1.09414e7	−1.63733
$538$	0	0
$539$	−2.03138e7	−3.01175
$540$	0	0
$541$	828768.	0.121742	0.0608709	−	0.998146i	$-0.480612\pi$
$541$	828768.	0.121742	0.0608709	+	0.998146i	$0.480612\pi$
$542$	0	0
$543$	−444878.	−0.0647502
$544$	0	0
$545$	−5.12748e6	−0.739456
$546$	0	0
$547$	−502955.	−0.0718722	−0.0359361	−	0.999354i	$-0.511441\pi$
$547$	−502955.	−0.0718722	−0.0359361	+	0.999354i	$0.511441\pi$
$548$	0	0
$549$	44451.5	0.00629441
$550$	0	0
$551$	−663153.	−0.0930540
$552$	0	0
$553$	−1.43719e7	−1.99849
$554$	0	0
$555$	4.37290e6	0.602611
$556$	0	0
$557$	2.21153e6	0.302033	0.151017	−	0.988531i	$-0.451745\pi$
$557$	2.21153e6	0.302033	0.151017	+	0.988531i	$0.451745\pi$
$558$	0	0
$559$	−2.09519e6	−0.283593
$560$	0	0
$561$	2.14199e6	0.287350
$562$	0	0
$563$	−3.78637e6	−0.503445	−0.251723	−	0.967799i	$-0.580997\pi$
$563$	−3.78637e6	−0.503445	−0.251723	+	0.967799i	$0.580997\pi$
$564$	0	0
$565$	4.03153e6	0.531311
$566$	0	0
$567$	1.24717e7	1.62917
$568$	0	0
$569$	−2.74111e6	−0.354933	−0.177466	−	0.984127i	$-0.556790\pi$
$569$	−2.74111e6	−0.354933	−0.177466	+	0.984127i	$0.556790\pi$
$570$	0	0
$571$	1.16497e7	1.49528	0.747640	−	0.664104i	$-0.231187\pi$
$571$	1.16497e7	1.49528	0.747640	+	0.664104i	$0.231187\pi$
$572$	0	0
$573$	−6.29318e6	−0.800725
$574$	0	0
$575$	−7.61097e6	−0.959999
$576$	0	0
$577$	−1.25137e7	−1.56475	−0.782376	−	0.622807i	$-0.785992\pi$
$577$	−1.25137e7	−1.56475	−0.782376	+	0.622807i	$0.785992\pi$
$578$	0	0
$579$	−1.06325e7	−1.31807
$580$	0	0
$581$	−2.63889e7	−3.24326
$582$	0	0
$583$	1.60170e7	1.95169
$584$	0	0
$585$	159165.	0.0192291
$586$	0	0
$587$	−257495.	−0.0308442	−0.0154221	−	0.999881i	$-0.504909\pi$
$587$	−257495.	−0.0308442	−0.0154221	+	0.999881i	$0.504909\pi$
$588$	0	0
$589$	8.16373e6	0.969617
$590$	0	0
$591$	−4.98726e6	−0.587345
$592$	0	0
$593$	−595042.	−0.0694882	−0.0347441	−	0.999396i	$-0.511062\pi$
$593$	−595042.	−0.0694882	−0.0347441	+	0.999396i	$0.511062\pi$
$594$	0	0
$595$	−2.06074e6	−0.238633
$596$	0	0
$597$	5.96095e6	0.684510
$598$	0	0
$599$	−1.04339e6	−0.118817	−0.0594085	−	0.998234i	$-0.518921\pi$
$599$	−1.04339e6	−0.118817	−0.0594085	+	0.998234i	$0.518921\pi$
$600$	0	0
$601$	4.07250e6	0.459912	0.229956	−	0.973201i	$-0.426142\pi$
$601$	4.07250e6	0.459912	0.229956	+	0.973201i	$0.426142\pi$
$602$	0	0
$603$	908044.	0.101698
$604$	0	0
$605$	7.59707e6	0.843835
$606$	0	0
$607$	−1.13093e6	−0.124585	−0.0622923	−	0.998058i	$-0.519841\pi$
$607$	−1.13093e6	−0.124585	−0.0622923	+	0.998058i	$0.519841\pi$
$608$	0	0
$609$	−2.15391e6	−0.235333
$610$	0	0
$611$	−6.41718e6	−0.695410
$612$	0	0
$613$	−314599.	−0.0338148	−0.0169074	−	0.999857i	$-0.505382\pi$
$613$	−314599.	−0.0338148	−0.0169074	+	0.999857i	$0.505382\pi$
$614$	0	0
$615$	−1.07903e7	−1.15039
$616$	0	0
$617$	6.83364e6	0.722668	0.361334	−	0.932436i	$-0.382321\pi$
$617$	6.83364e6	0.722668	0.361334	+	0.932436i	$0.382321\pi$
$618$	0	0
$619$	−1.46458e7	−1.53633	−0.768167	−	0.640249i	$-0.778831\pi$
$619$	−1.46458e7	−1.53633	−0.768167	+	0.640249i	$0.778831\pi$
$620$	0	0
$621$	1.81298e7	1.88653
$622$	0	0
$623$	2.16027e7	2.22991
$624$	0	0
$625$	−1.99141e6	−0.203920
$626$	0	0
$627$	9.48700e6	0.963740
$628$	0	0
$629$	−1.77361e6	−0.178744
$630$	0	0
$631$	1.01029e7	1.01012	0.505060	−	0.863084i	$-0.331470\pi$
$631$	1.01029e7	1.01012	0.505060	+	0.863084i	$0.331470\pi$
$632$	0	0
$633$	1.42702e6	0.141554
$634$	0	0
$635$	4.31789e6	0.424950
$636$	0	0
$637$	−9.74281e6	−0.951340
$638$	0	0
$639$	−42855.7	−0.00415199
$640$	0	0
$641$	−1.44147e7	−1.38568	−0.692839	−	0.721093i	$-0.743640\pi$
$641$	−1.44147e7	−1.38568	−0.692839	+	0.721093i	$0.743640\pi$
$642$	0	0
$643$	−645694.	−0.0615885	−0.0307942	−	0.999526i	$-0.509804\pi$
$643$	−645694.	−0.0615885	−0.0307942	+	0.999526i	$0.509804\pi$
$644$	0	0
$645$	−4.26460e6	−0.403626
$646$	0	0
$647$	−1.06178e7	−0.997177	−0.498589	−	0.866839i	$-0.666148\pi$
$647$	−1.06178e7	−0.997177	−0.498589	+	0.866839i	$0.666148\pi$
$648$	0	0
$649$	3.00771e7	2.80301
$650$	0	0
$651$	2.65156e7	2.45216
$652$	0	0
$653$	−1.26967e7	−1.16522	−0.582611	−	0.812751i	$-0.697969\pi$
$653$	−1.26967e7	−1.16522	−0.582611	+	0.812751i	$0.697969\pi$
$654$	0	0
$655$	8.45637e6	0.770160
$656$	0	0
$657$	−329747.	−0.0298035
$658$	0	0
$659$	1.63266e6	0.146448	0.0732239	−	0.997316i	$-0.476671\pi$
$659$	1.63266e6	0.146448	0.0732239	+	0.997316i	$0.476671\pi$
$660$	0	0
$661$	1.49309e7	1.32917	0.664587	−	0.747211i	$-0.268607\pi$
$661$	1.49309e7	1.32917	0.664587	+	0.747211i	$0.268607\pi$
$662$	0	0
$663$	1.02733e6	0.0907669
$664$	0	0
$665$	−9.12712e6	−0.800349
$666$	0	0
$667$	−2.94531e6	−0.256340
$668$	0	0
$669$	1.58608e7	1.37013
$670$	0	0
$671$	1.85067e6	0.158680
$672$	0	0
$673$	−1.87547e7	−1.59614	−0.798071	−	0.602563i	$-0.794146\pi$
$673$	−1.87547e7	−1.59614	−0.798071	+	0.602563i	$0.794146\pi$
$674$	0	0
$675$	−6.35758e6	−0.537072
$676$	0	0
$677$	−1.38417e7	−1.16069	−0.580346	−	0.814370i	$-0.697083\pi$
$677$	−1.38417e7	−1.16069	−0.580346	+	0.814370i	$0.697083\pi$
$678$	0	0
$679$	2.32852e7	1.93823
$680$	0	0
$681$	1.44462e7	1.19368
$682$	0	0
$683$	−2.18555e7	−1.79270	−0.896351	−	0.443345i	$-0.853792\pi$
$683$	−2.18555e7	−1.79270	−0.896351	+	0.443345i	$0.853792\pi$
$684$	0	0
$685$	1.23261e7	1.00369
$686$	0	0
$687$	−952246.	−0.0769764
$688$	0	0
$689$	7.68201e6	0.616491
$690$	0	0
$691$	−1.07995e7	−0.860419	−0.430209	−	0.902729i	$-0.641560\pi$
$691$	−1.07995e7	−0.860419	−0.430209	+	0.902729i	$0.641560\pi$
$692$	0	0
$693$	−1.93628e6	−0.153156
$694$	0	0
$695$	−1.36184e6	−0.106946
$696$	0	0
$697$	4.37645e6	0.341224
$698$	0	0
$699$	1.10743e7	0.857281
$700$	0	0
$701$	−2.32988e7	−1.79076	−0.895381	−	0.445301i	$-0.853097\pi$
$701$	−2.32988e7	−1.79076	−0.895381	+	0.445301i	$0.853097\pi$
$702$	0	0
$703$	−7.85541e6	−0.599488
$704$	0	0
$705$	−1.30617e7	−0.989750
$706$	0	0
$707$	2.56780e7	1.93202
$708$	0	0
$709$	8.77390e6	0.655507	0.327753	−	0.944763i	$-0.393709\pi$
$709$	8.77390e6	0.655507	0.327753	+	0.944763i	$0.393709\pi$
$710$	0	0
$711$	−916396.	−0.0679844
$712$	0	0
$713$	3.62582e7	2.67105
$714$	0	0
$715$	6.62659e6	0.484758
$716$	0	0
$717$	3.07581e6	0.223440
$718$	0	0
$719$	1.02217e7	0.737395	0.368698	−	0.929549i	$-0.379804\pi$
$719$	1.02217e7	0.737395	0.368698	+	0.929549i	$0.379804\pi$
$720$	0	0
$721$	−2.11719e7	−1.51678
$722$	0	0
$723$	−1.70543e6	−0.121336
$724$	0	0
$725$	1.03283e6	0.0729769
$726$	0	0
$727$	−4.56701e6	−0.320476	−0.160238	−	0.987078i	$-0.551226\pi$
$727$	−4.56701e6	−0.320476	−0.160238	+	0.987078i	$0.551226\pi$
$728$	0	0
$729$	1.50940e7	1.05193
$730$	0	0
$731$	1.72969e6	0.119722
$732$	0	0
$733$	378720.	0.0260351	0.0130175	−	0.999915i	$-0.495856\pi$
$733$	378720.	0.0260351	0.0130175	+	0.999915i	$0.495856\pi$
$734$	0	0
$735$	−1.98307e7	−1.35400
$736$	0	0
$737$	3.78050e7	2.56378
$738$	0	0
$739$	−1.04725e6	−0.0705408	−0.0352704	−	0.999378i	$-0.511229\pi$
$739$	−1.04725e6	−0.0705408	−0.0352704	+	0.999378i	$0.511229\pi$
$740$	0	0
$741$	4.55011e6	0.304422
$742$	0	0
$743$	−1.08180e7	−0.718908	−0.359454	−	0.933163i	$-0.617037\pi$
$743$	−1.08180e7	−0.718908	−0.359454	+	0.933163i	$0.617037\pi$
$744$	0	0
$745$	−4.75802e6	−0.314077
$746$	0	0
$747$	−1.68263e6	−0.110329
$748$	0	0
$749$	1.49191e7	0.971711
$750$	0	0
$751$	2.95876e6	0.191430	0.0957151	−	0.995409i	$-0.469486\pi$
$751$	2.95876e6	0.191430	0.0957151	+	0.995409i	$0.469486\pi$
$752$	0	0
$753$	3.69123e6	0.237238
$754$	0	0
$755$	2.12049e7	1.35385
$756$	0	0
$757$	−1.61122e7	−1.02191	−0.510957	−	0.859606i	$-0.670709\pi$
$757$	−1.61122e7	−1.02191	−0.510957	+	0.859606i	$0.670709\pi$
$758$	0	0
$759$	4.21353e7	2.65486
$760$	0	0
$761$	1.62877e7	1.01952	0.509761	−	0.860316i	$-0.329734\pi$
$761$	1.62877e7	1.01952	0.509761	+	0.860316i	$0.329734\pi$
$762$	0	0
$763$	2.99169e7	1.86039
$764$	0	0
$765$	−131399.	−0.00811778
$766$	0	0
$767$	1.44254e7	0.885402
$768$	0	0
$769$	−1.66456e7	−1.01504	−0.507521	−	0.861639i	$-0.669438\pi$
$769$	−1.66456e7	−1.01504	−0.507521	+	0.861639i	$0.669438\pi$
$770$	0	0
$771$	−998702.	−0.0605062
$772$	0	0
$773$	−1.34114e6	−0.0807281	−0.0403641	−	0.999185i	$-0.512852\pi$
$773$	−1.34114e6	−0.0807281	−0.0403641	+	0.999185i	$0.512852\pi$
$774$	0	0
$775$	−1.27147e7	−0.760415
$776$	0	0
$777$	−2.55142e7	−1.51610
$778$	0	0
$779$	1.93835e7	1.14443
$780$	0	0
$781$	−1.78423e6	−0.104670
$782$	0	0
$783$	−2.46027e6	−0.143410
$784$	0	0
$785$	3.29773e6	0.191003
$786$	0	0
$787$	1.89616e7	1.09129	0.545643	−	0.838018i	$-0.316286\pi$
$787$	1.89616e7	1.09129	0.545643	+	0.838018i	$0.316286\pi$
$788$	0	0
$789$	1.69385e7	0.968686
$790$	0	0
$791$	−2.35225e7	−1.33672
$792$	0	0
$793$	887608.	0.0501231
$794$	0	0
$795$	1.56361e7	0.877427
$796$	0	0
$797$	1.10956e7	0.618733	0.309366	−	0.950943i	$-0.399883\pi$
$797$	1.10956e7	0.618733	0.309366	+	0.950943i	$0.399883\pi$
$798$	0	0
$799$	5.29770e6	0.293576
$800$	0	0
$801$	1.37745e6	0.0758569
$802$	0	0
$803$	−1.37285e7	−0.751336
$804$	0	0
$805$	−4.05370e7	−2.20476
$806$	0	0
$807$	−8.73140e6	−0.471954
$808$	0	0
$809$	2.26659e7	1.21759	0.608797	−	0.793326i	$-0.291652\pi$
$809$	2.26659e7	1.21759	0.608797	+	0.793326i	$0.291652\pi$
$810$	0	0
$811$	−2.14208e7	−1.14363	−0.571813	−	0.820384i	$-0.693760\pi$
$811$	−2.14208e7	−1.14363	−0.571813	+	0.820384i	$0.693760\pi$
$812$	0	0
$813$	2.74688e6	0.145751
$814$	0	0
$815$	−1.36196e7	−0.718242
$816$	0	0
$817$	7.66087e6	0.401534
$818$	0	0
$819$	−928668.	−0.0483783
$820$	0	0
$821$	−2.96509e7	−1.53525	−0.767626	−	0.640898i	$-0.778562\pi$
$821$	−2.96509e7	−1.53525	−0.767626	+	0.640898i	$0.778562\pi$
$822$	0	0
$823$	−2.21189e7	−1.13832	−0.569160	−	0.822227i	$-0.692732\pi$
$823$	−2.21189e7	−1.13832	−0.569160	+	0.822227i	$0.692732\pi$
$824$	0	0
$825$	−1.47756e7	−0.755806
$826$	0	0
$827$	3.30004e7	1.67786	0.838929	−	0.544241i	$-0.183182\pi$
$827$	3.30004e7	1.67786	0.838929	+	0.544241i	$0.183182\pi$
$828$	0	0
$829$	−2.15697e7	−1.09008	−0.545039	−	0.838411i	$-0.683485\pi$
$829$	−2.15697e7	−1.09008	−0.545039	+	0.838411i	$0.683485\pi$
$830$	0	0
$831$	−358233.	−0.0179955
$832$	0	0
$833$	8.04317e6	0.401620
$834$	0	0
$835$	−1.75256e7	−0.869876
$836$	0	0
$837$	3.02871e7	1.49432
$838$	0	0
$839$	2.53249e7	1.24206	0.621031	−	0.783786i	$-0.286714\pi$
$839$	2.53249e7	1.24206	0.621031	+	0.783786i	$0.286714\pi$
$840$	0	0
$841$	−2.01115e7	−0.980514
$842$	0	0
$843$	−2.23717e7	−1.08425
$844$	0	0
$845$	−1.11602e7	−0.537687
$846$	0	0
$847$	−4.43260e7	−2.12300
$848$	0	0
$849$	−9.11586e6	−0.434039
$850$	0	0
$851$	−3.48888e7	−1.65144
$852$	0	0
$853$	−5.36742e6	−0.252576	−0.126288	−	0.991994i	$-0.540306\pi$
$853$	−5.36742e6	−0.252576	−0.126288	+	0.991994i	$0.540306\pi$
$854$	0	0
$855$	−581971.	−0.0272262
$856$	0	0
$857$	−2.95533e7	−1.37453	−0.687266	−	0.726406i	$-0.741189\pi$
$857$	−2.95533e7	−1.37453	−0.687266	+	0.726406i	$0.741189\pi$
$858$	0	0
$859$	6.60577e6	0.305450	0.152725	−	0.988269i	$-0.451195\pi$
$859$	6.60577e6	0.305450	0.152725	+	0.988269i	$0.451195\pi$
$860$	0	0
$861$	6.29572e7	2.89426
$862$	0	0
$863$	−3.74899e7	−1.71351	−0.856755	−	0.515723i	$-0.827523\pi$
$863$	−3.74899e7	−1.71351	−0.856755	+	0.515723i	$0.827523\pi$
$864$	0	0
$865$	−1.92535e7	−0.874920
$866$	0	0
$867$	2.06210e7	0.931670
$868$	0	0
$869$	−3.81527e7	−1.71386
$870$	0	0
$871$	1.81318e7	0.809835
$872$	0	0
$873$	1.48473e6	0.0659345
$874$	0	0
$875$	4.14064e7	1.82830
$876$	0	0
$877$	4.38409e7	1.92478	0.962389	−	0.271674i	$-0.0875774\pi$
$877$	4.38409e7	1.92478	0.962389	+	0.271674i	$0.0875774\pi$
$878$	0	0
$879$	2.40799e7	1.05119
$880$	0	0
$881$	3.38298e7	1.46845	0.734225	−	0.678906i	$-0.237546\pi$
$881$	3.38298e7	1.46845	0.734225	+	0.678906i	$0.237546\pi$
$882$	0	0
$883$	−1.84568e7	−0.796625	−0.398312	−	0.917250i	$-0.630404\pi$
$883$	−1.84568e7	−0.796625	−0.398312	+	0.917250i	$0.630404\pi$
$884$	0	0
$885$	2.93618e7	1.26016
$886$	0	0
$887$	2.45993e7	1.04982	0.524908	−	0.851159i	$-0.324100\pi$
$887$	2.45993e7	1.04982	0.524908	+	0.851159i	$0.324100\pi$
$888$	0	0
$889$	−2.51933e7	−1.06913
$890$	0	0
$891$	3.31082e7	1.39714
$892$	0	0
$893$	2.34638e7	0.984621
$894$	0	0
$895$	2.79439e7	1.16608
$896$	0	0
$897$	2.02088e7	0.838607
$898$	0	0
$899$	−4.92035e6	−0.203047
$900$	0	0
$901$	−6.34188e6	−0.260259
$902$	0	0
$903$	2.48823e7	1.01548
$904$	0	0
$905$	1.13620e6	0.0461141
$906$	0	0
$907$	1.60013e6	0.0645857	0.0322929	−	0.999478i	$-0.489719\pi$
$907$	1.60013e6	0.0645857	0.0322929	+	0.999478i	$0.489719\pi$
$908$	0	0
$909$	1.63730e6	0.0657233
$910$	0	0
$911$	3.75442e7	1.49881	0.749406	−	0.662111i	$-0.230339\pi$
$911$	3.75442e7	1.49881	0.749406	+	0.662111i	$0.230339\pi$
$912$	0	0
$913$	−7.00537e7	−2.78134
$914$	0	0
$915$	1.80665e6	0.0713382
$916$	0	0
$917$	−4.93397e7	−1.93764
$918$	0	0
$919$	−1.80711e7	−0.705823	−0.352911	−	0.935657i	$-0.614808\pi$
$919$	−1.80711e7	−0.705823	−0.352911	+	0.935657i	$0.614808\pi$
$920$	0	0
$921$	4.88350e6	0.189707
$922$	0	0
$923$	−855743.	−0.0330627
$924$	0	0
$925$	1.22345e7	0.470144
$926$	0	0
$927$	−1.34998e6	−0.0515975
$928$	0	0
$929$	−5.16431e6	−0.196324	−0.0981619	−	0.995170i	$-0.531296\pi$
$929$	−5.16431e6	−0.196324	−0.0981619	+	0.995170i	$0.531296\pi$
$930$	0	0
$931$	3.56236e7	1.34699
$932$	0	0
$933$	1.08426e7	0.407783
$934$	0	0
$935$	−5.47057e6	−0.204646
$936$	0	0
$937$	3.54676e6	0.131972	0.0659862	−	0.997821i	$-0.478981\pi$
$937$	3.54676e6	0.131972	0.0659862	+	0.997821i	$0.478981\pi$
$938$	0	0
$939$	−8.15939e6	−0.301991
$940$	0	0
$941$	−1.84604e7	−0.679621	−0.339810	−	0.940494i	$-0.610363\pi$
$941$	−1.84604e7	−0.679621	−0.339810	+	0.940494i	$0.610363\pi$
$942$	0	0
$943$	8.60895e7	3.15262
$944$	0	0
$945$	−3.38612e7	−1.23346
$946$	0	0
$947$	4.87096e7	1.76498	0.882489	−	0.470333i	$-0.155866\pi$
$947$	4.87096e7	1.76498	0.882489	+	0.470333i	$0.155866\pi$
$948$	0	0
$949$	−6.58440e6	−0.237329
$950$	0	0
$951$	−1.29812e6	−0.0465440
$952$	0	0
$953$	−2.49459e6	−0.0889748	−0.0444874	−	0.999010i	$-0.514165\pi$
$953$	−2.49459e6	−0.0889748	−0.0444874	+	0.999010i	$0.514165\pi$
$954$	0	0
$955$	1.60725e7	0.570264
$956$	0	0
$957$	−5.71790e6	−0.201817
$958$	0	0
$959$	−7.19181e7	−2.52518
$960$	0	0
$961$	3.19427e7	1.11574
$962$	0	0
$963$	951282.	0.0330555
$964$	0	0
$965$	2.71549e7	0.938706
$966$	0	0
$967$	−4.61026e7	−1.58548	−0.792738	−	0.609563i	$-0.791345\pi$
$967$	−4.61026e7	−1.58548	−0.792738	+	0.609563i	$0.791345\pi$
$968$	0	0
$969$	−3.75634e6	−0.128515
$970$	0	0
$971$	3.08646e7	1.05054	0.525270	−	0.850936i	$-0.323964\pi$
$971$	3.08646e7	1.05054	0.525270	+	0.850936i	$0.323964\pi$
$972$	0	0
$973$	7.94580e6	0.269064
$974$	0	0
$975$	−7.08661e6	−0.238741
$976$	0	0
$977$	1.60781e7	0.538889	0.269445	−	0.963016i	$-0.413160\pi$
$977$	1.60781e7	0.538889	0.269445	+	0.963016i	$0.413160\pi$
$978$	0	0
$979$	5.73480e7	1.91232
$980$	0	0
$981$	1.90759e6	0.0632866
$982$	0	0
$983$	−8.65901e6	−0.285815	−0.142907	−	0.989736i	$-0.545645\pi$
$983$	−8.65901e6	−0.285815	−0.142907	+	0.989736i	$0.545645\pi$
$984$	0	0
$985$	1.27373e7	0.418298
$986$	0	0
$987$	7.62098e7	2.49011
$988$	0	0
$989$	3.40248e7	1.10613
$990$	0	0
$991$	−4.92817e7	−1.59405	−0.797024	−	0.603947i	$-0.793594\pi$
$991$	−4.92817e7	−1.59405	−0.797024	+	0.603947i	$0.793594\pi$
$992$	0	0
$993$	−2.82453e7	−0.909018
$994$	0	0
$995$	−1.52241e7	−0.487498
$996$	0	0
$997$	−3.50865e7	−1.11790	−0.558948	−	0.829203i	$-0.688795\pi$
$997$	−3.50865e7	−1.11790	−0.558948	+	0.829203i	$0.688795\pi$
$998$	0	0
$999$	−2.91433e7	−0.923899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.13	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-15.1206 q^{3} +38.6175 q^{5} -225.318 q^{7} -14.3669 q^{9} +O(q^{10})\)	\(q-15.1206 q^{3} +38.6175 q^{5} -225.318 q^{7} -14.3669 q^{9} -598.145 q^{11} -286.879 q^{13} -583.920 q^{15} +236.833 q^{17} +1048.95 q^{19} +3406.95 q^{21} +4658.76 q^{23} -1633.69 q^{25} +3891.55 q^{27} -632.209 q^{29} +7782.79 q^{31} +9044.32 q^{33} -8701.23 q^{35} -7488.86 q^{37} +4337.79 q^{39} +18479.0 q^{41} +7303.40 q^{43} -554.816 q^{45} +22368.9 q^{47} +33961.4 q^{49} -3581.06 q^{51} -26777.8 q^{53} -23098.9 q^{55} -15860.7 q^{57} -50284.0 q^{59} -3094.01 q^{61} +3237.14 q^{63} -11078.6 q^{65} -63203.7 q^{67} -70443.4 q^{69} +2982.93 q^{71} +22951.8 q^{73} +24702.4 q^{75} +134773. q^{77} +63785.0 q^{79} -55351.4 q^{81} +117118. q^{83} +9145.90 q^{85} +9559.39 q^{87} -95876.4 q^{89} +64639.2 q^{91} -117681. q^{93} +40507.7 q^{95} -103344. q^{97} +8593.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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