LMFDB - Embedded newform 1028.6.a.a.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:	$1$
Dimension:	$49$
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)

$f(q)$	$=$	$q-9.49742 q^{3} -80.8595 q^{5} -210.734 q^{7} -152.799 q^{9} +O(q^{10})$	$q-9.49742 q^{3} -80.8595 q^{5} -210.734 q^{7} -152.799 q^{9} -153.192 q^{11} +433.473 q^{13} +767.956 q^{15} -2.10569 q^{17} -1849.85 q^{19} +2001.43 q^{21} -1372.25 q^{23} +3413.25 q^{25} +3759.07 q^{27} +3825.33 q^{29} +4648.33 q^{31} +1454.93 q^{33} +17039.8 q^{35} -11078.2 q^{37} -4116.88 q^{39} +3077.35 q^{41} +2699.18 q^{43} +12355.2 q^{45} +5102.76 q^{47} +27601.8 q^{49} +19.9986 q^{51} -9184.53 q^{53} +12387.0 q^{55} +17568.9 q^{57} +31854.7 q^{59} -10306.7 q^{61} +32199.9 q^{63} -35050.4 q^{65} +61186.7 q^{67} +13032.9 q^{69} +31298.7 q^{71} +32312.6 q^{73} -32417.1 q^{75} +32282.8 q^{77} +67446.7 q^{79} +1428.67 q^{81} -57426.2 q^{83} +170.265 q^{85} -36330.8 q^{87} -63929.8 q^{89} -91347.5 q^{91} -44147.1 q^{93} +149578. q^{95} -36381.2 q^{97} +23407.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−9.49742	−0.609260	−0.304630	−	0.952471i	$-0.598533\pi$
$3$	−9.49742	−0.609260	−0.304630	+	0.952471i	$0.598533\pi$
$4$	0	0
$5$	−80.8595	−1.44646	−0.723229	−	0.690608i	$-0.757343\pi$
$5$	−80.8595	−1.44646	−0.723229	+	0.690608i	$0.757343\pi$
$6$	0	0
$7$	−210.734	−1.62551	−0.812755	−	0.582606i	$-0.802033\pi$
$7$	−210.734	−1.62551	−0.812755	+	0.582606i	$0.802033\pi$
$8$	0	0
$9$	−152.799	−0.628802
$10$	0	0
$11$	−153.192	−0.381728	−0.190864	−	0.981616i	$-0.561129\pi$
$11$	−153.192	−0.381728	−0.190864	+	0.981616i	$0.561129\pi$
$12$	0	0
$13$	433.473	0.711384	0.355692	−	0.934603i	$-0.384245\pi$
$13$	433.473	0.711384	0.355692	+	0.934603i	$0.384245\pi$
$14$	0	0
$15$	767.956	0.881269
$16$	0	0
$17$	−2.10569	−0.00176714	−0.000883571	−	1.00000i	$-0.500281\pi$
$17$	−2.10569	−0.00176714	−0.000883571		1.00000i	$0.500281\pi$
$18$	0	0
$19$	−1849.85	−1.17558	−0.587792	−	0.809012i	$-0.700003\pi$
$19$	−1849.85	−1.17558	−0.587792	+	0.809012i	$0.700003\pi$
$20$	0	0
$21$	2001.43	0.990358
$22$	0	0
$23$	−1372.25	−0.540896	−0.270448	−	0.962735i	$-0.587172\pi$
$23$	−1372.25	−0.540896	−0.270448	+	0.962735i	$0.587172\pi$
$24$	0	0
$25$	3413.25	1.09224
$26$	0	0
$27$	3759.07	0.992364
$28$	0	0
$29$	3825.33	0.844645	0.422322	−	0.906446i	$-0.361215\pi$
$29$	3825.33	0.844645	0.422322	+	0.906446i	$0.361215\pi$
$30$	0	0
$31$	4648.33	0.868746	0.434373	−	0.900733i	$-0.356970\pi$
$31$	4648.33	0.868746	0.434373	+	0.900733i	$0.356970\pi$
$32$	0	0
$33$	1454.93	0.232572
$34$	0	0
$35$	17039.8	2.35123
$36$	0	0
$37$	−11078.2	−1.33034	−0.665172	−	0.746691i	$-0.731642\pi$
$37$	−11078.2	−1.33034	−0.665172	+	0.746691i	$0.731642\pi$
$38$	0	0
$39$	−4116.88	−0.433418
$40$	0	0
$41$	3077.35	0.285902	0.142951	−	0.989730i	$-0.454341\pi$
$41$	3077.35	0.285902	0.142951	+	0.989730i	$0.454341\pi$
$42$	0	0
$43$	2699.18	0.222618	0.111309	−	0.993786i	$-0.464496\pi$
$43$	2699.18	0.222618	0.111309	+	0.993786i	$0.464496\pi$
$44$	0	0
$45$	12355.2	0.909536
$46$	0	0
$47$	5102.76	0.336946	0.168473	−	0.985706i	$-0.446116\pi$
$47$	5102.76	0.336946	0.168473	+	0.985706i	$0.446116\pi$
$48$	0	0
$49$	27601.8	1.64228
$50$	0	0
$51$	19.9986	0.00107665
$52$	0	0
$53$	−9184.53	−0.449125	−0.224562	−	0.974460i	$-0.572095\pi$
$53$	−9184.53	−0.449125	−0.224562	+	0.974460i	$0.572095\pi$
$54$	0	0
$55$	12387.0	0.552154
$56$	0	0
$57$	17568.9	0.716236
$58$	0	0
$59$	31854.7	1.19136	0.595681	−	0.803221i	$-0.296882\pi$
$59$	31854.7	1.19136	0.595681	+	0.803221i	$0.296882\pi$
$60$	0	0
$61$	−10306.7	−0.354647	−0.177324	−	0.984153i	$-0.556744\pi$
$61$	−10306.7	−0.354647	−0.177324	+	0.984153i	$0.556744\pi$
$62$	0	0
$63$	32199.9	1.02212
$64$	0	0
$65$	−35050.4	−1.02899
$66$	0	0
$67$	61186.7	1.66521	0.832606	−	0.553865i	$-0.186848\pi$
$67$	61186.7	1.66521	0.832606	+	0.553865i	$0.186848\pi$
$68$	0	0
$69$	13032.9	0.329547
$70$	0	0
$71$	31298.7	0.736852	0.368426	−	0.929657i	$-0.379897\pi$
$71$	31298.7	0.736852	0.368426	+	0.929657i	$0.379897\pi$
$72$	0	0
$73$	32312.6	0.709684	0.354842	−	0.934926i	$-0.384535\pi$
$73$	32312.6	0.709684	0.354842	+	0.934926i	$0.384535\pi$
$74$	0	0
$75$	−32417.1	−0.665458
$76$	0	0
$77$	32282.8	0.620503
$78$	0	0
$79$	67446.7	1.21589	0.607943	−	0.793980i	$-0.291995\pi$
$79$	67446.7	1.21589	0.607943	+	0.793980i	$0.291995\pi$
$80$	0	0
$81$	1428.67	0.0241946
$82$	0	0
$83$	−57426.2	−0.914986	−0.457493	−	0.889213i	$-0.651253\pi$
$83$	−57426.2	−0.914986	−0.457493	+	0.889213i	$0.651253\pi$
$84$	0	0
$85$	170.265	0.00255610
$86$	0	0
$87$	−36330.8	−0.514608
$88$	0	0
$89$	−63929.8	−0.855517	−0.427758	−	0.903893i	$-0.640697\pi$
$89$	−63929.8	−0.855517	−0.427758	+	0.903893i	$0.640697\pi$
$90$	0	0
$91$	−91347.5	−1.15636
$92$	0	0
$93$	−44147.1	−0.529292
$94$	0	0
$95$	149578.	1.70043
$96$	0	0
$97$	−36381.2	−0.392598	−0.196299	−	0.980544i	$-0.562892\pi$
$97$	−36381.2	−0.392598	−0.196299	+	0.980544i	$0.562892\pi$
$98$	0	0
$99$	23407.6	0.240032
$100$	0	0
$101$	−82984.7	−0.809458	−0.404729	−	0.914437i	$-0.632634\pi$
$101$	−82984.7	−0.809458	−0.404729	+	0.914437i	$0.632634\pi$
$102$	0	0
$103$	−192637.	−1.78915	−0.894577	−	0.446915i	$-0.852523\pi$
$103$	−192637.	−1.78915	−0.894577	+	0.446915i	$0.852523\pi$
$104$	0	0
$105$	−161834.	−1.43251
$106$	0	0
$107$	−204169.	−1.72397	−0.861985	−	0.506934i	$-0.830779\pi$
$107$	−204169.	−1.72397	−0.861985	+	0.506934i	$0.830779\pi$
$108$	0	0
$109$	57361.6	0.462440	0.231220	−	0.972901i	$-0.425728\pi$
$109$	57361.6	0.462440	0.231220	+	0.972901i	$0.425728\pi$
$110$	0	0
$111$	105214.	0.810525
$112$	0	0
$113$	54884.5	0.404347	0.202173	−	0.979350i	$-0.435200\pi$
$113$	54884.5	0.404347	0.202173	+	0.979350i	$0.435200\pi$
$114$	0	0
$115$	110960.	0.782384
$116$	0	0
$117$	−66234.2	−0.447320
$118$	0	0
$119$	443.740	0.00287250
$120$	0	0
$121$	−137583.	−0.854284
$122$	0	0
$123$	−29226.9	−0.174189
$124$	0	0
$125$	−23307.8	−0.133422
$126$	0	0
$127$	193744.	1.06590	0.532952	−	0.846146i	$-0.321083\pi$
$127$	193744.	1.06590	0.532952	+	0.846146i	$0.321083\pi$
$128$	0	0
$129$	−25635.2	−0.135632
$130$	0	0
$131$	−834.375	−0.00424798	−0.00212399	−	0.999998i	$-0.500676\pi$
$131$	−834.375	−0.00424798	−0.00212399	+	0.999998i	$0.500676\pi$
$132$	0	0
$133$	389827.	1.91092
$134$	0	0
$135$	−303956.	−1.43541
$136$	0	0
$137$	−15474.4	−0.0704387	−0.0352193	−	0.999380i	$-0.511213\pi$
$137$	−15474.4	−0.0704387	−0.0352193	+	0.999380i	$0.511213\pi$
$138$	0	0
$139$	298656.	1.31110	0.655548	−	0.755153i	$-0.272438\pi$
$139$	298656.	1.31110	0.655548	+	0.755153i	$0.272438\pi$
$140$	0	0
$141$	−48463.0	−0.205288
$142$	0	0
$143$	−66404.6	−0.271555
$144$	0	0
$145$	−309314.	−1.22174
$146$	0	0
$147$	−262146.	−1.00058
$148$	0	0
$149$	243963.	0.900239	0.450119	−	0.892968i	$-0.351381\pi$
$149$	243963.	0.900239	0.450119	+	0.892968i	$0.351381\pi$
$150$	0	0
$151$	−43300.4	−0.154543	−0.0772715	−	0.997010i	$-0.524621\pi$
$151$	−43300.4	−0.154543	−0.0772715	+	0.997010i	$0.524621\pi$
$152$	0	0
$153$	321.747	0.00111118
$154$	0	0
$155$	−375861.	−1.25660
$156$	0	0
$157$	137177.	0.444152	0.222076	−	0.975029i	$-0.428717\pi$
$157$	137177.	0.444152	0.222076	+	0.975029i	$0.428717\pi$
$158$	0	0
$159$	87229.3	0.273634
$160$	0	0
$161$	289180.	0.879232
$162$	0	0
$163$	−237650.	−0.700599	−0.350299	−	0.936638i	$-0.613920\pi$
$163$	−237650.	−0.700599	−0.350299	+	0.936638i	$0.613920\pi$
$164$	0	0
$165$	−117645.	−0.336405
$166$	0	0
$167$	−85130.6	−0.236208	−0.118104	−	0.993001i	$-0.537682\pi$
$167$	−85130.6	−0.236208	−0.118104	+	0.993001i	$0.537682\pi$
$168$	0	0
$169$	−183394.	−0.493933
$170$	0	0
$171$	282656.	0.739210
$172$	0	0
$173$	607765.	1.54390	0.771952	−	0.635680i	$-0.219280\pi$
$173$	607765.	1.54390	0.771952	+	0.635680i	$0.219280\pi$
$174$	0	0
$175$	−719288.	−1.77545
$176$	0	0
$177$	−302538.	−0.725849
$178$	0	0
$179$	−343921.	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$179$	−343921.	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$180$	0	0
$181$	−229078.	−0.519742	−0.259871	−	0.965643i	$-0.583680\pi$
$181$	−229078.	−0.519742	−0.259871	+	0.965643i	$0.583680\pi$
$182$	0	0
$183$	97887.5	0.216072
$184$	0	0
$185$	895775.	1.92429
$186$	0	0
$187$	322.574	0.000674568 0
$188$	0	0
$189$	−792164.	−1.61310
$190$	0	0
$191$	53804.8	0.106718	0.0533590	−	0.998575i	$-0.483007\pi$
$191$	53804.8	0.106718	0.0533590	+	0.998575i	$0.483007\pi$
$192$	0	0
$193$	−73580.8	−0.142191	−0.0710953	−	0.997470i	$-0.522649\pi$
$193$	−73580.8	−0.142191	−0.0710953	+	0.997470i	$0.522649\pi$
$194$	0	0
$195$	332888.	0.626920
$196$	0	0
$197$	373759.	0.686160	0.343080	−	0.939306i	$-0.388530\pi$
$197$	373759.	0.686160	0.343080	+	0.939306i	$0.388530\pi$
$198$	0	0
$199$	−654786.	−1.17211	−0.586053	−	0.810273i	$-0.699319\pi$
$199$	−654786.	−1.17211	−0.586053	+	0.810273i	$0.699319\pi$
$200$	0	0
$201$	−581116.	−1.01455
$202$	0	0
$203$	−806127.	−1.37298
$204$	0	0
$205$	−248833.	−0.413545
$206$	0	0
$207$	209679.	0.340117
$208$	0	0
$209$	283383.	0.448754
$210$	0	0
$211$	725800.	1.12231	0.561153	−	0.827712i	$-0.310358\pi$
$211$	725800.	1.12231	0.561153	+	0.827712i	$0.310358\pi$
$212$	0	0
$213$	−297257.	−0.448934
$214$	0	0
$215$	−218254.	−0.322008
$216$	0	0
$217$	−979561.	−1.41215
$218$	0	0
$219$	−306887.	−0.432382
$220$	0	0
$221$	−912.758	−0.00125712
$222$	0	0
$223$	754693.	1.01627	0.508134	−	0.861278i	$-0.330336\pi$
$223$	754693.	1.01627	0.508134	+	0.861278i	$0.330336\pi$
$224$	0	0
$225$	−521541.	−0.686803
$226$	0	0
$227$	1.25000e6	1.61008	0.805038	−	0.593223i	$-0.202145\pi$
$227$	1.25000e6	1.61008	0.805038	+	0.593223i	$0.202145\pi$
$228$	0	0
$229$	−602190.	−0.758831	−0.379415	−	0.925226i	$-0.623875\pi$
$229$	−602190.	−0.758831	−0.379415	+	0.925226i	$0.623875\pi$
$230$	0	0
$231$	−306603.	−0.378048
$232$	0	0
$233$	−1.00219e6	−1.20938	−0.604688	−	0.796463i	$-0.706702\pi$
$233$	−1.00219e6	−1.20938	−0.604688	+	0.796463i	$0.706702\pi$
$234$	0	0
$235$	−412606.	−0.487378
$236$	0	0
$237$	−640570.	−0.740791
$238$	0	0
$239$	470800.	0.533141	0.266570	−	0.963815i	$-0.414110\pi$
$239$	470800.	0.533141	0.266570	+	0.963815i	$0.414110\pi$
$240$	0	0
$241$	515541.	0.571769	0.285884	−	0.958264i	$-0.407713\pi$
$241$	515541.	0.571769	0.285884	+	0.958264i	$0.407713\pi$
$242$	0	0
$243$	−927023.	−1.00710
$244$	0	0
$245$	−2.23187e6	−2.37549
$246$	0	0
$247$	−801862.	−0.836291
$248$	0	0
$249$	545400.	0.557464
$250$	0	0
$251$	1.31177e6	1.31424	0.657120	−	0.753786i	$-0.271775\pi$
$251$	1.31177e6	1.31424	0.657120	+	0.753786i	$0.271775\pi$
$252$	0	0
$253$	210218.	0.206475
$254$	0	0
$255$	−1617.08	−0.00155733
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	2.33455e6	2.16248
$260$	0	0
$261$	−584507.	−0.531114
$262$	0	0
$263$	−272930.	−0.243311	−0.121656	−	0.992572i	$-0.538820\pi$
$263$	−272930.	−0.243311	−0.121656	+	0.992572i	$0.538820\pi$
$264$	0	0
$265$	742656.	0.649640
$266$	0	0
$267$	607168.	0.521232
$268$	0	0
$269$	−1.00798e6	−0.849321	−0.424660	−	0.905353i	$-0.639606\pi$
$269$	−1.00798e6	−0.849321	−0.424660	+	0.905353i	$0.639606\pi$
$270$	0	0
$271$	1.01760e6	0.841691	0.420846	−	0.907132i	$-0.361733\pi$
$271$	1.01760e6	0.841691	0.420846	+	0.907132i	$0.361733\pi$
$272$	0	0
$273$	867566.	0.704524
$274$	0	0
$275$	−522883.	−0.416939
$276$	0	0
$277$	1.33142e6	1.04259	0.521297	−	0.853375i	$-0.325448\pi$
$277$	1.33142e6	1.04259	0.521297	+	0.853375i	$0.325448\pi$
$278$	0	0
$279$	−710260.	−0.546269
$280$	0	0
$281$	549587.	0.415213	0.207606	−	0.978212i	$-0.433433\pi$
$281$	549587.	0.415213	0.207606	+	0.978212i	$0.433433\pi$
$282$	0	0
$283$	−1.17488e6	−0.872022	−0.436011	−	0.899941i	$-0.643609\pi$
$283$	−1.17488e6	−0.872022	−0.436011	+	0.899941i	$0.643609\pi$
$284$	0	0
$285$	−1.42061e6	−1.03601
$286$	0	0
$287$	−648502.	−0.464736
$288$	0	0
$289$	−1.41985e6	−0.999997
$290$	0	0
$291$	345528.	0.239194
$292$	0	0
$293$	−1.30314e6	−0.886791	−0.443395	−	0.896326i	$-0.646226\pi$
$293$	−1.30314e6	−0.886791	−0.443395	+	0.896326i	$0.646226\pi$
$294$	0	0
$295$	−2.57575e6	−1.72325
$296$	0	0
$297$	−575859.	−0.378813
$298$	0	0
$299$	−594834.	−0.384785
$300$	0	0
$301$	−568809.	−0.361868
$302$	0	0
$303$	788141.	0.493171
$304$	0	0
$305$	833398.	0.512983
$306$	0	0
$307$	−2.68202e6	−1.62411	−0.812057	−	0.583579i	$-0.801652\pi$
$307$	−2.68202e6	−1.62411	−0.812057	+	0.583579i	$0.801652\pi$
$308$	0	0
$309$	1.82956e6	1.09006
$310$	0	0
$311$	−1.78036e6	−1.04377	−0.521887	−	0.853014i	$-0.674772\pi$
$311$	−1.78036e6	−1.04377	−0.521887	+	0.853014i	$0.674772\pi$
$312$	0	0
$313$	759034.	0.437926	0.218963	−	0.975733i	$-0.429733\pi$
$313$	759034.	0.437926	0.218963	+	0.975733i	$0.429733\pi$
$314$	0	0
$315$	−2.60367e6	−1.47846
$316$	0	0
$317$	−532871.	−0.297834	−0.148917	−	0.988850i	$-0.547579\pi$
$317$	−532871.	−0.297834	−0.148917	+	0.988850i	$0.547579\pi$
$318$	0	0
$319$	−586010.	−0.322425
$320$	0	0
$321$	1.93908e6	1.05035
$322$	0	0
$323$	3895.21	0.00207742
$324$	0	0
$325$	1.47955e6	0.777002
$326$	0	0
$327$	−544788.	−0.281746
$328$	0	0
$329$	−1.07532e6	−0.547709
$330$	0	0
$331$	1.85092e6	0.928575	0.464288	−	0.885684i	$-0.346310\pi$
$331$	1.85092e6	0.928575	0.464288	+	0.885684i	$0.346310\pi$
$332$	0	0
$333$	1.69273e6	0.836523
$334$	0	0
$335$	−4.94752e6	−2.40866
$336$	0	0
$337$	−1.39192e6	−0.667637	−0.333819	−	0.942637i	$-0.608337\pi$
$337$	−1.39192e6	−0.667637	−0.333819	+	0.942637i	$0.608337\pi$
$338$	0	0
$339$	−521261.	−0.246352
$340$	0	0
$341$	−712087.	−0.331625
$342$	0	0
$343$	−2.27483e6	−1.04403
$344$	0	0
$345$	−1.05383e6	−0.476675
$346$	0	0
$347$	999815.	0.445755	0.222877	−	0.974846i	$-0.428455\pi$
$347$	999815.	0.445755	0.222877	+	0.974846i	$0.428455\pi$
$348$	0	0
$349$	−1.76280e6	−0.774713	−0.387356	−	0.921930i	$-0.626612\pi$
$349$	−1.76280e6	−0.774713	−0.387356	+	0.921930i	$0.626612\pi$
$350$	0	0
$351$	1.62946e6	0.705951
$352$	0	0
$353$	1.98109e6	0.846188	0.423094	−	0.906086i	$-0.360944\pi$
$353$	1.98109e6	0.846188	0.423094	+	0.906086i	$0.360944\pi$
$354$	0	0
$355$	−2.53080e6	−1.06583
$356$	0	0
$357$	−4214.38	−0.00175010
$358$	0	0
$359$	−2.96190e6	−1.21293	−0.606463	−	0.795112i	$-0.707412\pi$
$359$	−2.96190e6	−1.21293	−0.606463	+	0.795112i	$0.707412\pi$
$360$	0	0
$361$	945864.	0.381997
$362$	0	0
$363$	1.30669e6	0.520481
$364$	0	0
$365$	−2.61278e6	−1.02653
$366$	0	0
$367$	754668.	0.292476	0.146238	−	0.989249i	$-0.453283\pi$
$367$	754668.	0.292476	0.146238	+	0.989249i	$0.453283\pi$
$368$	0	0
$369$	−470216.	−0.179776
$370$	0	0
$371$	1.93549e6	0.730057
$372$	0	0
$373$	446924.	0.166327	0.0831633	−	0.996536i	$-0.473498\pi$
$373$	446924.	0.166327	0.0831633	+	0.996536i	$0.473498\pi$
$374$	0	0
$375$	221364.	0.0812885
$376$	0	0
$377$	1.65818e6	0.600866
$378$	0	0
$379$	1.74706e6	0.624755	0.312378	−	0.949958i	$-0.398875\pi$
$379$	1.74706e6	0.624755	0.312378	+	0.949958i	$0.398875\pi$
$380$	0	0
$381$	−1.84006e6	−0.649413
$382$	0	0
$383$	2.79266e6	0.972796	0.486398	−	0.873737i	$-0.338311\pi$
$383$	2.79266e6	0.972796	0.486398	+	0.873737i	$0.338311\pi$
$384$	0	0
$385$	−2.61037e6	−0.897531
$386$	0	0
$387$	−412432.	−0.139983
$388$	0	0
$389$	2.22999e6	0.747186	0.373593	−	0.927593i	$-0.378126\pi$
$389$	2.22999e6	0.747186	0.373593	+	0.927593i	$0.378126\pi$
$390$	0	0
$391$	2889.53	0.000955841 0
$392$	0	0
$393$	7924.41	0.00258813
$394$	0	0
$395$	−5.45370e6	−1.75873
$396$	0	0
$397$	−2.06703e6	−0.658221	−0.329110	−	0.944291i	$-0.606749\pi$
$397$	−2.06703e6	−0.658221	−0.329110	+	0.944291i	$0.606749\pi$
$398$	0	0
$399$	−3.70235e6	−1.16425
$400$	0	0
$401$	2.42937e6	0.754453	0.377227	−	0.926121i	$-0.376878\pi$
$401$	2.42937e6	0.754453	0.377227	+	0.926121i	$0.376878\pi$
$402$	0	0
$403$	2.01493e6	0.618011
$404$	0	0
$405$	−115521.	−0.0349965
$406$	0	0
$407$	1.69709e6	0.507830
$408$	0	0
$409$	−6.46598e6	−1.91129	−0.955644	−	0.294524i	$-0.904839\pi$
$409$	−6.46598e6	−1.91129	−0.955644	+	0.294524i	$0.904839\pi$
$410$	0	0
$411$	146966.	0.0429155
$412$	0	0
$413$	−6.71287e6	−1.93657
$414$	0	0
$415$	4.64345e6	1.32349
$416$	0	0
$417$	−2.83647e6	−0.798799
$418$	0	0
$419$	−4.54032e6	−1.26343	−0.631716	−	0.775200i	$-0.717649\pi$
$419$	−4.54032e6	−1.26343	−0.631716	+	0.775200i	$0.717649\pi$
$420$	0	0
$421$	2.10412e6	0.578583	0.289291	−	0.957241i	$-0.406580\pi$
$421$	2.10412e6	0.578583	0.289291	+	0.957241i	$0.406580\pi$
$422$	0	0
$423$	−779696.	−0.211872
$424$	0	0
$425$	−7187.24	−0.00193014
$426$	0	0
$427$	2.17198e6	0.576483
$428$	0	0
$429$	630673.	0.165448
$430$	0	0
$431$	−1.05838e6	−0.274442	−0.137221	−	0.990540i	$-0.543817\pi$
$431$	−1.05838e6	−0.274442	−0.137221	+	0.990540i	$0.543817\pi$
$432$	0	0
$433$	6.62227e6	1.69741	0.848706	−	0.528865i	$-0.177382\pi$
$433$	6.62227e6	1.69741	0.848706	+	0.528865i	$0.177382\pi$
$434$	0	0
$435$	2.93769e6	0.744359
$436$	0	0
$437$	2.53847e6	0.635869
$438$	0	0
$439$	4.72637e6	1.17049	0.585243	−	0.810858i	$-0.300999\pi$
$439$	4.72637e6	1.17049	0.585243	+	0.810858i	$0.300999\pi$
$440$	0	0
$441$	−4.21753e6	−1.03267
$442$	0	0
$443$	2.59135e6	0.627361	0.313680	−	0.949529i	$-0.398438\pi$
$443$	2.59135e6	0.627361	0.313680	+	0.949529i	$0.398438\pi$
$444$	0	0
$445$	5.16933e6	1.23747
$446$	0	0
$447$	−2.31702e6	−0.548480
$448$	0	0
$449$	5.00626e6	1.17192	0.585959	−	0.810340i	$-0.300718\pi$
$449$	5.00626e6	1.17192	0.585959	+	0.810340i	$0.300718\pi$
$450$	0	0
$451$	−471425.	−0.109137
$452$	0	0
$453$	411242.	0.0941569
$454$	0	0
$455$	7.38631e6	1.67263
$456$	0	0
$457$	−2.10805e6	−0.472162	−0.236081	−	0.971733i	$-0.575863\pi$
$457$	−2.10805e6	−0.472162	−0.236081	+	0.971733i	$0.575863\pi$
$458$	0	0
$459$	−7915.42	−0.00175365
$460$	0	0
$461$	−575086.	−0.126032	−0.0630160	−	0.998013i	$-0.520072\pi$
$461$	−575086.	−0.126032	−0.0630160	+	0.998013i	$0.520072\pi$
$462$	0	0
$463$	−3.61080e6	−0.782800	−0.391400	−	0.920221i	$-0.628009\pi$
$463$	−3.61080e6	−0.782800	−0.391400	+	0.920221i	$0.628009\pi$
$464$	0	0
$465$	3.56971e6	0.765598
$466$	0	0
$467$	−4.76109e6	−1.01022	−0.505108	−	0.863056i	$-0.668547\pi$
$467$	−4.76109e6	−1.01022	−0.505108	+	0.863056i	$0.668547\pi$
$468$	0	0
$469$	−1.28941e7	−2.70682
$470$	0	0
$471$	−1.30283e6	−0.270604
$472$	0	0
$473$	−413493.	−0.0849796
$474$	0	0
$475$	−6.31402e6	−1.28402
$476$	0	0
$477$	1.40339e6	0.282411
$478$	0	0
$479$	6.32953e6	1.26047	0.630235	−	0.776404i	$-0.282958\pi$
$479$	6.32953e6	1.26047	0.630235	+	0.776404i	$0.282958\pi$
$480$	0	0
$481$	−4.80209e6	−0.946384
$482$	0	0
$483$	−2.74646e6	−0.535681
$484$	0	0
$485$	2.94176e6	0.567876
$486$	0	0
$487$	2.35889e6	0.450698	0.225349	−	0.974278i	$-0.427648\pi$
$487$	2.35889e6	0.450698	0.225349	+	0.974278i	$0.427648\pi$
$488$	0	0
$489$	2.25707e6	0.426847
$490$	0	0
$491$	2.80535e6	0.525150	0.262575	−	0.964912i	$-0.415428\pi$
$491$	2.80535e6	0.525150	0.262575	+	0.964912i	$0.415428\pi$
$492$	0	0
$493$	−8054.95	−0.00149261
$494$	0	0
$495$	−1.89272e6	−0.347196
$496$	0	0
$497$	−6.59570e6	−1.19776
$498$	0	0
$499$	−4.14772e6	−0.745690	−0.372845	−	0.927894i	$-0.621618\pi$
$499$	−4.14772e6	−0.745690	−0.372845	+	0.927894i	$0.621618\pi$
$500$	0	0
$501$	808521.	0.143912
$502$	0	0
$503$	1.16305e6	0.204964	0.102482	−	0.994735i	$-0.467322\pi$
$503$	1.16305e6	0.204964	0.102482	+	0.994735i	$0.467322\pi$
$504$	0	0
$505$	6.71010e6	1.17085
$506$	0	0
$507$	1.74177e6	0.300934
$508$	0	0
$509$	−5.53354e6	−0.946692	−0.473346	−	0.880877i	$-0.656954\pi$
$509$	−5.53354e6	−0.946692	−0.473346	+	0.880877i	$0.656954\pi$
$510$	0	0
$511$	−6.80937e6	−1.15360
$512$	0	0
$513$	−6.95373e6	−1.16661
$514$	0	0
$515$	1.55766e7	2.58793
$516$	0	0
$517$	−781701.	−0.128622
$518$	0	0
$519$	−5.77220e6	−0.940639
$520$	0	0
$521$	239348.	0.0386310	0.0193155	−	0.999813i	$-0.493851\pi$
$521$	239348.	0.0386310	0.0193155	+	0.999813i	$0.493851\pi$
$522$	0	0
$523$	4.26659e6	0.682066	0.341033	−	0.940051i	$-0.389223\pi$
$523$	4.26659e6	0.682066	0.341033	+	0.940051i	$0.389223\pi$
$524$	0	0
$525$	6.83138e6	1.08171
$526$	0	0
$527$	−9787.92	−0.00153520
$528$	0	0
$529$	−4.55327e6	−0.707431
$530$	0	0
$531$	−4.86737e6	−0.749131
$532$	0	0
$533$	1.33395e6	0.203386
$534$	0	0
$535$	1.65090e7	2.49365
$536$	0	0
$537$	3.26636e6	0.488797
$538$	0	0
$539$	−4.22837e6	−0.626905
$540$	0	0
$541$	−5.23668e6	−0.769242	−0.384621	−	0.923075i	$-0.625668\pi$
$541$	−5.23668e6	−0.769242	−0.384621	+	0.923075i	$0.625668\pi$
$542$	0	0
$543$	2.17566e6	0.316658
$544$	0	0
$545$	−4.63823e6	−0.668900
$546$	0	0
$547$	8.22393e6	1.17520	0.587599	−	0.809152i	$-0.300073\pi$
$547$	8.22393e6	1.17520	0.587599	+	0.809152i	$0.300073\pi$
$548$	0	0
$549$	1.57486e6	0.223003
$550$	0	0
$551$	−7.07631e6	−0.992951
$552$	0	0
$553$	−1.42133e7	−1.97643
$554$	0	0
$555$	−8.50755e6	−1.17239
$556$	0	0
$557$	−7.76931e6	−1.06107	−0.530536	−	0.847663i	$-0.678009\pi$
$557$	−7.76931e6	−1.06107	−0.530536	+	0.847663i	$0.678009\pi$
$558$	0	0
$559$	1.17002e6	0.158367
$560$	0	0
$561$	−3063.62	−0.000410987 0
$562$	0	0
$563$	1.12047e7	1.48981	0.744904	−	0.667172i	$-0.232495\pi$
$563$	1.12047e7	1.48981	0.744904	+	0.667172i	$0.232495\pi$
$564$	0	0
$565$	−4.43793e6	−0.584870
$566$	0	0
$567$	−301069.	−0.0393286
$568$	0	0
$569$	1.40764e7	1.82268	0.911338	−	0.411659i	$-0.135051\pi$
$569$	1.40764e7	1.82268	0.911338	+	0.411659i	$0.135051\pi$
$570$	0	0
$571$	8.33167e6	1.06940	0.534702	−	0.845041i	$-0.320424\pi$
$571$	8.33167e6	1.06940	0.534702	+	0.845041i	$0.320424\pi$
$572$	0	0
$573$	−511007.	−0.0650190
$574$	0	0
$575$	−4.68384e6	−0.590789
$576$	0	0
$577$	2.28676e6	0.285943	0.142972	−	0.989727i	$-0.454334\pi$
$577$	2.28676e6	0.285943	0.142972	+	0.989727i	$0.454334\pi$
$578$	0	0
$579$	698828.	0.0866311
$580$	0	0
$581$	1.21016e7	1.48732
$582$	0	0
$583$	1.40700e6	0.171444
$584$	0	0
$585$	5.35566e6	0.647029
$586$	0	0
$587$	1.54700e7	1.85308	0.926541	−	0.376194i	$-0.122767\pi$
$587$	1.54700e7	1.85308	0.926541	+	0.376194i	$0.122767\pi$
$588$	0	0
$589$	−8.59873e6	−1.02128
$590$	0	0
$591$	−3.54974e6	−0.418050
$592$	0	0
$593$	−4.48299e6	−0.523518	−0.261759	−	0.965133i	$-0.584302\pi$
$593$	−4.48299e6	−0.523518	−0.261759	+	0.965133i	$0.584302\pi$
$594$	0	0
$595$	−35880.5	−0.00415496
$596$	0	0
$597$	6.21878e6	0.714117
$598$	0	0
$599$	5.92510e6	0.674728	0.337364	−	0.941374i	$-0.390465\pi$
$599$	5.92510e6	0.674728	0.337364	+	0.941374i	$0.390465\pi$
$600$	0	0
$601$	−1.69335e6	−0.191232	−0.0956159	−	0.995418i	$-0.530482\pi$
$601$	−1.69335e6	−0.191232	−0.0956159	+	0.995418i	$0.530482\pi$
$602$	0	0
$603$	−9.34926e6	−1.04709
$604$	0	0
$605$	1.11249e7	1.23569
$606$	0	0
$607$	1.03964e7	1.14528	0.572639	−	0.819807i	$-0.305919\pi$
$607$	1.03964e7	1.14528	0.572639	+	0.819807i	$0.305919\pi$
$608$	0	0
$609$	7.65613e6	0.836500
$610$	0	0
$611$	2.21191e6	0.239698
$612$	0	0
$613$	1.33077e7	1.43038	0.715192	−	0.698928i	$-0.246339\pi$
$613$	1.33077e7	1.43038	0.715192	+	0.698928i	$0.246339\pi$
$614$	0	0
$615$	2.36327e6	0.251957
$616$	0	0
$617$	−1.00294e7	−1.06062	−0.530312	−	0.847803i	$-0.677925\pi$
$617$	−1.00294e7	−1.06062	−0.530312	+	0.847803i	$0.677925\pi$
$618$	0	0
$619$	−7.40894e6	−0.777194	−0.388597	−	0.921408i	$-0.627040\pi$
$619$	−7.40894e6	−0.777194	−0.388597	+	0.921408i	$0.627040\pi$
$620$	0	0
$621$	−5.15839e6	−0.536766
$622$	0	0
$623$	1.34722e7	1.39065
$624$	0	0
$625$	−8.78175e6	−0.899251
$626$	0	0
$627$	−2.69141e6	−0.273408
$628$	0	0
$629$	23327.1	0.00235090
$630$	0	0
$631$	8.36273e6	0.836131	0.418066	−	0.908417i	$-0.362708\pi$
$631$	8.36273e6	0.836131	0.418066	+	0.908417i	$0.362708\pi$
$632$	0	0
$633$	−6.89323e6	−0.683776
$634$	0	0
$635$	−1.56660e7	−1.54179
$636$	0	0
$637$	1.19646e7	1.16829
$638$	0	0
$639$	−4.78241e6	−0.463334
$640$	0	0
$641$	−6.48837e6	−0.623722	−0.311861	−	0.950128i	$-0.600952\pi$
$641$	−6.48837e6	−0.623722	−0.311861	+	0.950128i	$0.600952\pi$
$642$	0	0
$643$	3.01543e6	0.287622	0.143811	−	0.989605i	$-0.454064\pi$
$643$	3.01543e6	0.287622	0.143811	+	0.989605i	$0.454064\pi$
$644$	0	0
$645$	2.07285e6	0.196186
$646$	0	0
$647$	1.56993e7	1.47442	0.737208	−	0.675666i	$-0.236144\pi$
$647$	1.56993e7	1.47442	0.737208	+	0.675666i	$0.236144\pi$
$648$	0	0
$649$	−4.87989e6	−0.454776
$650$	0	0
$651$	9.30330e6	0.860369
$652$	0	0
$653$	9.48149e6	0.870149	0.435075	−	0.900394i	$-0.356722\pi$
$653$	9.48149e6	0.870149	0.435075	+	0.900394i	$0.356722\pi$
$654$	0	0
$655$	67467.1	0.00614453
$656$	0	0
$657$	−4.93734e6	−0.446251
$658$	0	0
$659$	4.35197e6	0.390367	0.195183	−	0.980767i	$-0.437470\pi$
$659$	4.35197e6	0.390367	0.195183	+	0.980767i	$0.437470\pi$
$660$	0	0
$661$	1.45491e7	1.29519	0.647594	−	0.761985i	$-0.275775\pi$
$661$	1.45491e7	1.29519	0.647594	+	0.761985i	$0.275775\pi$
$662$	0	0
$663$	8668.85	0.000765910 0
$664$	0	0
$665$	−3.15212e7	−2.76407
$666$	0	0
$667$	−5.24932e6	−0.456865
$668$	0	0
$669$	−7.16763e6	−0.619171
$670$	0	0
$671$	1.57891e6	0.135379
$672$	0	0
$673$	−3.23969e6	−0.275718	−0.137859	−	0.990452i	$-0.544022\pi$
$673$	−3.23969e6	−0.275718	−0.137859	+	0.990452i	$0.544022\pi$
$674$	0	0
$675$	1.28306e7	1.08390
$676$	0	0
$677$	−4.27595e6	−0.358559	−0.179279	−	0.983798i	$-0.557377\pi$
$677$	−4.27595e6	−0.358559	−0.179279	+	0.983798i	$0.557377\pi$
$678$	0	0
$679$	7.66675e6	0.638171
$680$	0	0
$681$	−1.18718e7	−0.980955
$682$	0	0
$683$	−1.17045e7	−0.960063	−0.480032	−	0.877251i	$-0.659375\pi$
$683$	−1.17045e7	−0.960063	−0.480032	+	0.877251i	$0.659375\pi$
$684$	0	0
$685$	1.25125e6	0.101887
$686$	0	0
$687$	5.71925e6	0.462325
$688$	0	0
$689$	−3.98125e6	−0.319500
$690$	0	0
$691$	−3.84903e6	−0.306659	−0.153330	−	0.988175i	$-0.549000\pi$
$691$	−3.84903e6	−0.306659	−0.153330	+	0.988175i	$0.549000\pi$
$692$	0	0
$693$	−4.93277e6	−0.390174
$694$	0	0
$695$	−2.41492e7	−1.89645
$696$	0	0
$697$	−6479.93	−0.000505229 0
$698$	0	0
$699$	9.51824e6	0.736824
$700$	0	0
$701$	4.56434e6	0.350819	0.175409	−	0.984496i	$-0.443875\pi$
$701$	4.56434e6	0.350819	0.175409	+	0.984496i	$0.443875\pi$
$702$	0	0
$703$	2.04930e7	1.56393
$704$	0	0
$705$	3.91869e6	0.296940
$706$	0	0
$707$	1.74877e7	1.31578
$708$	0	0
$709$	−1.97876e7	−1.47835	−0.739175	−	0.673514i	$-0.764784\pi$
$709$	−1.97876e7	−1.47835	−0.739175	+	0.673514i	$0.764784\pi$
$710$	0	0
$711$	−1.03058e7	−0.764552
$712$	0	0
$713$	−6.37868e6	−0.469901
$714$	0	0
$715$	5.36944e6	0.392793
$716$	0	0
$717$	−4.47139e6	−0.324821
$718$	0	0
$719$	−3.95422e6	−0.285258	−0.142629	−	0.989776i	$-0.545556\pi$
$719$	−3.95422e6	−0.285258	−0.142629	+	0.989776i	$0.545556\pi$
$720$	0	0
$721$	4.05952e7	2.90828
$722$	0	0
$723$	−4.89631e6	−0.348356
$724$	0	0
$725$	1.30568e7	0.922555
$726$	0	0
$727$	1.78132e7	1.24999	0.624995	−	0.780628i	$-0.285101\pi$
$727$	1.78132e7	1.24999	0.624995	+	0.780628i	$0.285101\pi$
$728$	0	0
$729$	8.45716e6	0.589394
$730$	0	0
$731$	−5683.62	−0.000393398 0
$732$	0	0
$733$	−3.16735e6	−0.217739	−0.108870	−	0.994056i	$-0.534723\pi$
$733$	−3.16735e6	−0.217739	−0.108870	+	0.994056i	$0.534723\pi$
$734$	0	0
$735$	2.11970e7	1.44729
$736$	0	0
$737$	−9.37331e6	−0.635659
$738$	0	0
$739$	6.55873e6	0.441783	0.220891	−	0.975298i	$-0.429103\pi$
$739$	6.55873e6	0.441783	0.220891	+	0.975298i	$0.429103\pi$
$740$	0	0
$741$	7.61563e6	0.509519
$742$	0	0
$743$	−2.07653e7	−1.37996	−0.689981	−	0.723828i	$-0.742381\pi$
$743$	−2.07653e7	−1.37996	−0.689981	+	0.723828i	$0.742381\pi$
$744$	0	0
$745$	−1.97267e7	−1.30216
$746$	0	0
$747$	8.77466e6	0.575345
$748$	0	0
$749$	4.30253e7	2.80233
$750$	0	0
$751$	1.05102e7	0.680005	0.340002	−	0.940425i	$-0.389572\pi$
$751$	1.05102e7	0.680005	0.340002	+	0.940425i	$0.389572\pi$
$752$	0	0
$753$	−1.24585e7	−0.800714
$754$	0	0
$755$	3.50125e6	0.223540
$756$	0	0
$757$	3.94148e6	0.249988	0.124994	−	0.992157i	$-0.460109\pi$
$757$	3.94148e6	0.249988	0.124994	+	0.992157i	$0.460109\pi$
$758$	0	0
$759$	−1.99653e6	−0.125797
$760$	0	0
$761$	1.37470e7	0.860493	0.430247	−	0.902711i	$-0.358427\pi$
$761$	1.37470e7	0.860493	0.430247	+	0.902711i	$0.358427\pi$
$762$	0	0
$763$	−1.20880e7	−0.751700
$764$	0	0
$765$	−26016.3	−0.00160728
$766$	0	0
$767$	1.38082e7	0.847515
$768$	0	0
$769$	−167028.	−0.0101853	−0.00509266	−	0.999987i	$-0.501621\pi$
$769$	−167028.	−0.0101853	−0.00509266	+	0.999987i	$0.501621\pi$
$770$	0	0
$771$	−627295.	−0.0380046
$772$	0	0
$773$	−1.34638e7	−0.810434	−0.405217	−	0.914221i	$-0.632804\pi$
$773$	−1.34638e7	−0.810434	−0.405217	+	0.914221i	$0.632804\pi$
$774$	0	0
$775$	1.58659e7	0.948879
$776$	0	0
$777$	−2.21722e7	−1.31752
$778$	0	0
$779$	−5.69265e6	−0.336102
$780$	0	0
$781$	−4.79471e6	−0.281277
$782$	0	0
$783$	1.43797e7	0.838195
$784$	0	0
$785$	−1.10921e7	−0.642448
$786$	0	0
$787$	−1.94128e7	−1.11725	−0.558626	−	0.829420i	$-0.688671\pi$
$787$	−1.94128e7	−1.11725	−0.558626	+	0.829420i	$0.688671\pi$
$788$	0	0
$789$	2.59213e6	0.148240
$790$	0	0
$791$	−1.15660e7	−0.657269
$792$	0	0
$793$	−4.46770e6	−0.252290
$794$	0	0
$795$	−7.05332e6	−0.395800
$796$	0	0
$797$	5.64632e6	0.314862	0.157431	−	0.987530i	$-0.449679\pi$
$797$	5.64632e6	0.314862	0.157431	+	0.987530i	$0.449679\pi$
$798$	0	0
$799$	−10744.8	−0.000595431 0
$800$	0	0
$801$	9.76841e6	0.537951
$802$	0	0
$803$	−4.95004e6	−0.270907
$804$	0	0
$805$	−2.33829e7	−1.27177
$806$	0	0
$807$	9.57322e6	0.517457
$808$	0	0
$809$	2.38883e6	0.128326	0.0641629	−	0.997939i	$-0.479562\pi$
$809$	2.38883e6	0.128326	0.0641629	+	0.997939i	$0.479562\pi$
$810$	0	0
$811$	−7.24201e6	−0.386640	−0.193320	−	0.981136i	$-0.561926\pi$
$811$	−7.24201e6	−0.386640	−0.193320	+	0.981136i	$0.561926\pi$
$812$	0	0
$813$	−9.66455e6	−0.512809
$814$	0	0
$815$	1.92163e7	1.01339
$816$	0	0
$817$	−4.99309e6	−0.261706
$818$	0	0
$819$	1.39578e7	0.727122
$820$	0	0
$821$	1.13935e7	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$821$	1.13935e7	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$822$	0	0
$823$	−3.66558e7	−1.88644	−0.943220	−	0.332169i	$-0.892220\pi$
$823$	−3.66558e7	−1.88644	−0.943220	+	0.332169i	$0.892220\pi$
$824$	0	0
$825$	4.96604e6	0.254024
$826$	0	0
$827$	5.11450e6	0.260039	0.130020	−	0.991511i	$-0.458496\pi$
$827$	5.11450e6	0.260039	0.130020	+	0.991511i	$0.458496\pi$
$828$	0	0
$829$	−6.28048e6	−0.317400	−0.158700	−	0.987327i	$-0.550730\pi$
$829$	−6.28048e6	−0.317400	−0.158700	+	0.987327i	$0.550730\pi$
$830$	0	0
$831$	−1.26451e7	−0.635211
$832$	0	0
$833$	−58120.7	−0.00290214
$834$	0	0
$835$	6.88361e6	0.341665
$836$	0	0
$837$	1.74734e7	0.862112
$838$	0	0
$839$	−1.28419e7	−0.629831	−0.314915	−	0.949120i	$-0.601976\pi$
$839$	−1.28419e7	−0.629831	−0.314915	+	0.949120i	$0.601976\pi$
$840$	0	0
$841$	−5.87799e6	−0.286575
$842$	0	0
$843$	−5.21966e6	−0.252972
$844$	0	0
$845$	1.48291e7	0.714454
$846$	0	0
$847$	2.89935e7	1.38865
$848$	0	0
$849$	1.11583e7	0.531288
$850$	0	0
$851$	1.52020e7	0.719578
$852$	0	0
$853$	−3.02638e6	−0.142413	−0.0712066	−	0.997462i	$-0.522685\pi$
$853$	−3.02638e6	−0.142413	−0.0712066	+	0.997462i	$0.522685\pi$
$854$	0	0
$855$	−2.28554e7	−1.06924
$856$	0	0
$857$	−1.05362e7	−0.490040	−0.245020	−	0.969518i	$-0.578794\pi$
$857$	−1.05362e7	−0.490040	−0.245020	+	0.969518i	$0.578794\pi$
$858$	0	0
$859$	−3.52960e6	−0.163209	−0.0816043	−	0.996665i	$-0.526004\pi$
$859$	−3.52960e6	−0.163209	−0.0816043	+	0.996665i	$0.526004\pi$
$860$	0	0
$861$	6.15910e6	0.283145
$862$	0	0
$863$	1.60888e7	0.735357	0.367678	−	0.929953i	$-0.380153\pi$
$863$	1.60888e7	0.735357	0.367678	+	0.929953i	$0.380153\pi$
$864$	0	0
$865$	−4.91436e7	−2.23319
$866$	0	0
$867$	1.34849e7	0.609258
$868$	0	0
$869$	−1.03323e7	−0.464138
$870$	0	0
$871$	2.65228e7	1.18461
$872$	0	0
$873$	5.55901e6	0.246866
$874$	0	0
$875$	4.91175e6	0.216878
$876$	0	0
$877$	−447752.	−0.0196579	−0.00982897	−	0.999952i	$-0.503129\pi$
$877$	−447752.	−0.0196579	−0.00982897	+	0.999952i	$0.503129\pi$
$878$	0	0
$879$	1.23764e7	0.540286
$880$	0	0
$881$	−1.19894e7	−0.520424	−0.260212	−	0.965551i	$-0.583792\pi$
$881$	−1.19894e7	−0.520424	−0.260212	+	0.965551i	$0.583792\pi$
$882$	0	0
$883$	−1.79015e7	−0.772657	−0.386328	−	0.922361i	$-0.626257\pi$
$883$	−1.79015e7	−0.772657	−0.386328	+	0.922361i	$0.626257\pi$
$884$	0	0
$885$	2.44630e7	1.04991
$886$	0	0
$887$	3.69634e7	1.57748	0.788738	−	0.614729i	$-0.210735\pi$
$887$	3.69634e7	1.57748	0.788738	+	0.614729i	$0.210735\pi$
$888$	0	0
$889$	−4.08284e7	−1.73264
$890$	0	0
$891$	−218861.	−0.00923578
$892$	0	0
$893$	−9.43936e6	−0.396108
$894$	0	0
$895$	2.78093e7	1.16046
$896$	0	0
$897$	5.64939e6	0.234434
$898$	0	0
$899$	1.77814e7	0.733781
$900$	0	0
$901$	19339.7	0.000793667 0
$902$	0	0
$903$	5.40222e6	0.220472
$904$	0	0
$905$	1.85232e7	0.751785
$906$	0	0
$907$	−4.42911e7	−1.78771	−0.893857	−	0.448351i	$-0.852011\pi$
$907$	−4.42911e7	−1.78771	−0.893857	+	0.448351i	$0.852011\pi$
$908$	0	0
$909$	1.26800e7	0.508989
$910$	0	0
$911$	1.32566e7	0.529220	0.264610	−	0.964356i	$-0.414757\pi$
$911$	1.32566e7	0.529220	0.264610	+	0.964356i	$0.414757\pi$
$912$	0	0
$913$	8.79723e6	0.349276
$914$	0	0
$915$	−7.91513e6	−0.312540
$916$	0	0
$917$	175831.	0.00690514
$918$	0	0
$919$	−1.06167e7	−0.414667	−0.207334	−	0.978270i	$-0.566479\pi$
$919$	−1.06167e7	−0.414667	−0.207334	+	0.978270i	$0.566479\pi$
$920$	0	0
$921$	2.54723e7	0.989507
$922$	0	0
$923$	1.35671e7	0.524184
$924$	0	0
$925$	−3.78126e7	−1.45305
$926$	0	0
$927$	2.94348e7	1.12502
$928$	0	0
$929$	−1.55999e7	−0.593037	−0.296519	−	0.955027i	$-0.595826\pi$
$929$	−1.55999e7	−0.593037	−0.296519	+	0.955027i	$0.595826\pi$
$930$	0	0
$931$	−5.10593e7	−1.93064
$932$	0	0
$933$	1.69088e7	0.635930
$934$	0	0
$935$	−26083.2	−0.000975734 0
$936$	0	0
$937$	1.98372e6	0.0738126	0.0369063	−	0.999319i	$-0.488250\pi$
$937$	1.98372e6	0.0738126	0.0369063	+	0.999319i	$0.488250\pi$
$938$	0	0
$939$	−7.20887e6	−0.266811
$940$	0	0
$941$	1.55909e7	0.573982	0.286991	−	0.957933i	$-0.407345\pi$
$941$	1.55909e7	0.573982	0.286991	+	0.957933i	$0.407345\pi$
$942$	0	0
$943$	−4.22290e6	−0.154643
$944$	0	0
$945$	6.40539e7	2.33328
$946$	0	0
$947$	1.55181e7	0.562295	0.281147	−	0.959665i	$-0.409285\pi$
$947$	1.55181e7	0.562295	0.281147	+	0.959665i	$0.409285\pi$
$948$	0	0
$949$	1.40067e7	0.504858
$950$	0	0
$951$	5.06090e6	0.181458
$952$	0	0
$953$	2.93085e7	1.04535	0.522674	−	0.852533i	$-0.324935\pi$
$953$	2.93085e7	1.04535	0.522674	+	0.852533i	$0.324935\pi$
$954$	0	0
$955$	−4.35063e6	−0.154363
$956$	0	0
$957$	5.56559e6	0.196440
$958$	0	0
$959$	3.26097e6	0.114499
$960$	0	0
$961$	−7.02219e6	−0.245281
$962$	0	0
$963$	3.11968e7	1.08404
$964$	0	0
$965$	5.94970e6	0.205673
$966$	0	0
$967$	3.26898e7	1.12421	0.562104	−	0.827067i	$-0.309992\pi$
$967$	3.26898e7	1.12421	0.562104	+	0.827067i	$0.309992\pi$
$968$	0	0
$969$	−36994.5	−0.00126569
$970$	0	0
$971$	5.20476e6	0.177155	0.0885774	−	0.996069i	$-0.471768\pi$
$971$	5.20476e6	0.177155	0.0885774	+	0.996069i	$0.471768\pi$
$972$	0	0
$973$	−6.29370e7	−2.13120
$974$	0	0
$975$	−1.40519e7	−0.473396
$976$	0	0
$977$	4.09929e7	1.37395	0.686977	−	0.726679i	$-0.258937\pi$
$977$	4.09929e7	1.37395	0.686977	+	0.726679i	$0.258937\pi$
$978$	0	0
$979$	9.79354e6	0.326575
$980$	0	0
$981$	−8.76480e6	−0.290783
$982$	0	0
$983$	5.46524e6	0.180395	0.0901977	−	0.995924i	$-0.471250\pi$
$983$	5.46524e6	0.180395	0.0901977	+	0.995924i	$0.471250\pi$
$984$	0	0
$985$	−3.02219e7	−0.992502
$986$	0	0
$987$	1.02128e7	0.333697
$988$	0	0
$989$	−3.70395e6	−0.120413
$990$	0	0
$991$	5.02507e7	1.62539	0.812696	−	0.582688i	$-0.197999\pi$
$991$	5.02507e7	1.62539	0.812696	+	0.582688i	$0.197999\pi$
$992$	0	0
$993$	−1.75789e7	−0.565744
$994$	0	0
$995$	5.29456e7	1.69540
$996$	0	0
$997$	−6.14082e7	−1.95654	−0.978269	−	0.207338i	$-0.933520\pi$
$997$	−6.14082e7	−1.95654	−0.978269	+	0.207338i	$0.933520\pi$
$998$	0	0
$999$	−4.16436e7	−1.32018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.18	✓	49

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

\(f(q)\)	\(=\)	\(q-9.49742 q^{3} -80.8595 q^{5} -210.734 q^{7} -152.799 q^{9} +O(q^{10})\)	\(q-9.49742 q^{3} -80.8595 q^{5} -210.734 q^{7} -152.799 q^{9} -153.192 q^{11} +433.473 q^{13} +767.956 q^{15} -2.10569 q^{17} -1849.85 q^{19} +2001.43 q^{21} -1372.25 q^{23} +3413.25 q^{25} +3759.07 q^{27} +3825.33 q^{29} +4648.33 q^{31} +1454.93 q^{33} +17039.8 q^{35} -11078.2 q^{37} -4116.88 q^{39} +3077.35 q^{41} +2699.18 q^{43} +12355.2 q^{45} +5102.76 q^{47} +27601.8 q^{49} +19.9986 q^{51} -9184.53 q^{53} +12387.0 q^{55} +17568.9 q^{57} +31854.7 q^{59} -10306.7 q^{61} +32199.9 q^{63} -35050.4 q^{65} +61186.7 q^{67} +13032.9 q^{69} +31298.7 q^{71} +32312.6 q^{73} -32417.1 q^{75} +32282.8 q^{77} +67446.7 q^{79} +1428.67 q^{81} -57426.2 q^{83} +170.265 q^{85} -36330.8 q^{87} -63929.8 q^{89} -91347.5 q^{91} -44147.1 q^{93} +149578. q^{95} -36381.2 q^{97} +23407.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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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Groups

Database

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