LMFDB - Embedded newform 1028.6.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.0123 q^{3} -46.6500 q^{5} +25.6441 q^{7} +157.493 q^{9} +O(q^{10})$	$q-20.0123 q^{3} -46.6500 q^{5} +25.6441 q^{7} +157.493 q^{9} +355.126 q^{11} -865.327 q^{13} +933.575 q^{15} -1622.19 q^{17} -1546.22 q^{19} -513.198 q^{21} +4747.10 q^{23} -948.780 q^{25} +1711.19 q^{27} +2021.99 q^{29} -907.323 q^{31} -7106.89 q^{33} -1196.30 q^{35} -12016.0 q^{37} +17317.2 q^{39} -5584.98 q^{41} -3324.49 q^{43} -7347.06 q^{45} +10791.9 q^{47} -16149.4 q^{49} +32463.8 q^{51} +17915.4 q^{53} -16566.6 q^{55} +30943.5 q^{57} -6407.30 q^{59} -53869.2 q^{61} +4038.77 q^{63} +40367.5 q^{65} -20558.1 q^{67} -95000.5 q^{69} -32671.6 q^{71} -87763.2 q^{73} +18987.3 q^{75} +9106.87 q^{77} -46594.4 q^{79} -72515.7 q^{81} -41672.5 q^{83} +75675.0 q^{85} -40464.8 q^{87} -95377.3 q^{89} -22190.5 q^{91} +18157.6 q^{93} +72131.2 q^{95} -122369. q^{97} +55929.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−20.0123	−1.28379	−0.641896	−	0.766792i	$-0.721852\pi$
$3$	−20.0123	−1.28379	−0.641896	+	0.766792i	$0.721852\pi$
$4$	0	0
$5$	−46.6500	−0.834500	−0.417250	−	0.908792i	$-0.637006\pi$
$5$	−46.6500	−0.834500	−0.417250	+	0.908792i	$0.637006\pi$
$6$	0	0
$7$	25.6441	0.197807	0.0989036	−	0.995097i	$-0.468466\pi$
$7$	25.6441	0.197807	0.0989036	+	0.995097i	$0.468466\pi$
$8$	0	0
$9$	157.493	0.648120
$10$	0	0
$11$	355.126	0.884912	0.442456	−	0.896790i	$-0.354107\pi$
$11$	355.126	0.884912	0.442456	+	0.896790i	$0.354107\pi$
$12$	0	0
$13$	−865.327	−1.42011	−0.710055	−	0.704147i	$-0.751330\pi$
$13$	−865.327	−1.42011	−0.710055	+	0.704147i	$0.751330\pi$
$14$	0	0
$15$	933.575	1.07132
$16$	0	0
$17$	−1622.19	−1.36138	−0.680689	−	0.732572i	$-0.738320\pi$
$17$	−1622.19	−1.36138	−0.680689	+	0.732572i	$0.738320\pi$
$18$	0	0
$19$	−1546.22	−0.982625	−0.491313	−	0.870983i	$-0.663483\pi$
$19$	−1546.22	−0.982625	−0.491313	+	0.870983i	$0.663483\pi$
$20$	0	0
$21$	−513.198	−0.253943
$22$	0	0
$23$	4747.10	1.87115	0.935575	−	0.353128i	$-0.114882\pi$
$23$	4747.10	1.87115	0.935575	+	0.353128i	$0.114882\pi$
$24$	0	0
$25$	−948.780	−0.303610
$26$	0	0
$27$	1711.19	0.451740
$28$	0	0
$29$	2021.99	0.446463	0.223231	−	0.974766i	$-0.428339\pi$
$29$	2021.99	0.446463	0.223231	+	0.974766i	$0.428339\pi$
$30$	0	0
$31$	−907.323	−0.169573	−0.0847866	−	0.996399i	$-0.527021\pi$
$31$	−907.323	−0.169573	−0.0847866	+	0.996399i	$0.527021\pi$
$32$	0	0
$33$	−7106.89	−1.13604
$34$	0	0
$35$	−1196.30	−0.165070
$36$	0	0
$37$	−12016.0	−1.44297	−0.721483	−	0.692432i	$-0.756539\pi$
$37$	−12016.0	−1.44297	−0.721483	+	0.692432i	$0.756539\pi$
$38$	0	0
$39$	17317.2	1.82312
$40$	0	0
$41$	−5584.98	−0.518874	−0.259437	−	0.965760i	$-0.583537\pi$
$41$	−5584.98	−0.518874	−0.259437	+	0.965760i	$0.583537\pi$
$42$	0	0
$43$	−3324.49	−0.274192	−0.137096	−	0.990558i	$-0.543777\pi$
$43$	−3324.49	−0.274192	−0.137096	+	0.990558i	$0.543777\pi$
$44$	0	0
$45$	−7347.06	−0.540857
$46$	0	0
$47$	10791.9	0.712614	0.356307	−	0.934369i	$-0.384036\pi$
$47$	10791.9	0.712614	0.356307	+	0.934369i	$0.384036\pi$
$48$	0	0
$49$	−16149.4	−0.960872
$50$	0	0
$51$	32463.8	1.74773
$52$	0	0
$53$	17915.4	0.876068	0.438034	−	0.898958i	$-0.355675\pi$
$53$	17915.4	0.876068	0.438034	+	0.898958i	$0.355675\pi$
$54$	0	0
$55$	−16566.6	−0.738459
$56$	0	0
$57$	30943.5	1.26149
$58$	0	0
$59$	−6407.30	−0.239632	−0.119816	−	0.992796i	$-0.538230\pi$
$59$	−6407.30	−0.239632	−0.119816	+	0.992796i	$0.538230\pi$
$60$	0	0
$61$	−53869.2	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$61$	−53869.2	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$62$	0	0
$63$	4038.77	0.128203
$64$	0	0
$65$	40367.5	1.18508
$66$	0	0
$67$	−20558.1	−0.559495	−0.279748	−	0.960074i	$-0.590251\pi$
$67$	−20558.1	−0.559495	−0.279748	+	0.960074i	$0.590251\pi$
$68$	0	0
$69$	−95000.5	−2.40217
$70$	0	0
$71$	−32671.6	−0.769174	−0.384587	−	0.923089i	$-0.625656\pi$
$71$	−32671.6	−0.769174	−0.384587	+	0.923089i	$0.625656\pi$
$72$	0	0
$73$	−87763.2	−1.92755	−0.963774	−	0.266719i	$-0.914060\pi$
$73$	−87763.2	−1.92755	−0.963774	+	0.266719i	$0.914060\pi$
$74$	0	0
$75$	18987.3	0.389772
$76$	0	0
$77$	9106.87	0.175042
$78$	0	0
$79$	−46594.4	−0.839974	−0.419987	−	0.907530i	$-0.637965\pi$
$79$	−46594.4	−0.839974	−0.419987	+	0.907530i	$0.637965\pi$
$80$	0	0
$81$	−72515.7	−1.22806
$82$	0	0
$83$	−41672.5	−0.663979	−0.331990	−	0.943283i	$-0.607720\pi$
$83$	−41672.5	−0.663979	−0.331990	+	0.943283i	$0.607720\pi$
$84$	0	0
$85$	75675.0	1.13607
$86$	0	0
$87$	−40464.8	−0.573165
$88$	0	0
$89$	−95377.3	−1.27635	−0.638175	−	0.769891i	$-0.720310\pi$
$89$	−95377.3	−1.27635	−0.638175	+	0.769891i	$0.720310\pi$
$90$	0	0
$91$	−22190.5	−0.280908
$92$	0	0
$93$	18157.6	0.217697
$94$	0	0
$95$	72131.2	0.820001
$96$	0	0
$97$	−122369.	−1.32051	−0.660257	−	0.751039i	$-0.729553\pi$
$97$	−122369.	−1.32051	−0.660257	+	0.751039i	$0.729553\pi$
$98$	0	0
$99$	55929.9	0.573530
$100$	0	0
$101$	−100938.	−0.984582	−0.492291	−	0.870431i	$-0.663840\pi$
$101$	−100938.	−0.984582	−0.492291	+	0.870431i	$0.663840\pi$
$102$	0	0
$103$	−98634.8	−0.916087	−0.458044	−	0.888930i	$-0.651450\pi$
$103$	−98634.8	−0.916087	−0.458044	+	0.888930i	$0.651450\pi$
$104$	0	0
$105$	23940.7	0.211916
$106$	0	0
$107$	−297.441	−0.00251155	−0.00125577	−	0.999999i	$-0.500400\pi$
$107$	−297.441	−0.00251155	−0.00125577	+	0.999999i	$0.500400\pi$
$108$	0	0
$109$	221000.	1.78166	0.890831	−	0.454335i	$-0.150123\pi$
$109$	221000.	1.78166	0.890831	+	0.454335i	$0.150123\pi$
$110$	0	0
$111$	240468.	1.85247
$112$	0	0
$113$	−15622.7	−0.115096	−0.0575481	−	0.998343i	$-0.518328\pi$
$113$	−15622.7	−0.115096	−0.0575481	+	0.998343i	$0.518328\pi$
$114$	0	0
$115$	−221452.	−1.56147
$116$	0	0
$117$	−136283.	−0.920402
$118$	0	0
$119$	−41599.5	−0.269291
$120$	0	0
$121$	−34936.9	−0.216931
$122$	0	0
$123$	111769.	0.666126
$124$	0	0
$125$	190042.	1.08786
$126$	0	0
$127$	81637.7	0.449140	0.224570	−	0.974458i	$-0.427902\pi$
$127$	81637.7	0.449140	0.224570	+	0.974458i	$0.427902\pi$
$128$	0	0
$129$	66530.9	0.352005
$130$	0	0
$131$	386594.	1.96824	0.984118	−	0.177515i	$-0.0568058\pi$
$131$	386594.	1.96824	0.984118	+	0.177515i	$0.0568058\pi$
$132$	0	0
$133$	−39651.5	−0.194370
$134$	0	0
$135$	−79826.9	−0.376977
$136$	0	0
$137$	−137262.	−0.624809	−0.312405	−	0.949949i	$-0.601134\pi$
$137$	−137262.	−0.624809	−0.312405	+	0.949949i	$0.601134\pi$
$138$	0	0
$139$	201603.	0.885032	0.442516	−	0.896761i	$-0.354086\pi$
$139$	201603.	0.885032	0.442516	+	0.896761i	$0.354086\pi$
$140$	0	0
$141$	−215972.	−0.914848
$142$	0	0
$143$	−307300.	−1.25667
$144$	0	0
$145$	−94326.0	−0.372573
$146$	0	0
$147$	323187.	1.23356
$148$	0	0
$149$	119896.	0.442423	0.221212	−	0.975226i	$-0.428999\pi$
$149$	119896.	0.442423	0.221212	+	0.975226i	$0.428999\pi$
$150$	0	0
$151$	116373.	0.415348	0.207674	−	0.978198i	$-0.433411\pi$
$151$	116373.	0.415348	0.207674	+	0.978198i	$0.433411\pi$
$152$	0	0
$153$	−255484.	−0.882338
$154$	0	0
$155$	42326.6	0.141509
$156$	0	0
$157$	−79780.7	−0.258314	−0.129157	−	0.991624i	$-0.541227\pi$
$157$	−79780.7	−0.258314	−0.129157	+	0.991624i	$0.541227\pi$
$158$	0	0
$159$	−358530.	−1.12469
$160$	0	0
$161$	121735.	0.370127
$162$	0	0
$163$	−35668.8	−0.105152	−0.0525762	−	0.998617i	$-0.516743\pi$
$163$	−35668.8	−0.105152	−0.0525762	+	0.998617i	$0.516743\pi$
$164$	0	0
$165$	331536.	0.948028
$166$	0	0
$167$	−48100.2	−0.133461	−0.0667307	−	0.997771i	$-0.521257\pi$
$167$	−48100.2	−0.133461	−0.0667307	+	0.997771i	$0.521257\pi$
$168$	0	0
$169$	377497.	1.01671
$170$	0	0
$171$	−243520.	−0.636860
$172$	0	0
$173$	383116.	0.973229	0.486614	−	0.873617i	$-0.338232\pi$
$173$	383116.	0.973229	0.486614	+	0.873617i	$0.338232\pi$
$174$	0	0
$175$	−24330.6	−0.0600562
$176$	0	0
$177$	128225.	0.307638
$178$	0	0
$179$	−268170.	−0.625573	−0.312787	−	0.949823i	$-0.601262\pi$
$179$	−268170.	−0.625573	−0.312787	+	0.949823i	$0.601262\pi$
$180$	0	0
$181$	186813.	0.423849	0.211924	−	0.977286i	$-0.432027\pi$
$181$	186813.	0.423849	0.211924	+	0.977286i	$0.432027\pi$
$182$	0	0
$183$	1.07805e6	2.37964
$184$	0	0
$185$	560547.	1.20416
$186$	0	0
$187$	−576080.	−1.20470
$188$	0	0
$189$	43881.9	0.0893574
$190$	0	0
$191$	447992.	0.888559	0.444280	−	0.895888i	$-0.353460\pi$
$191$	447992.	0.888559	0.444280	+	0.895888i	$0.353460\pi$
$192$	0	0
$193$	−448421.	−0.866548	−0.433274	−	0.901262i	$-0.642642\pi$
$193$	−448421.	−0.866548	−0.433274	+	0.901262i	$0.642642\pi$
$194$	0	0
$195$	−807847.	−1.52140
$196$	0	0
$197$	865678.	1.58924	0.794622	−	0.607104i	$-0.207669\pi$
$197$	865678.	1.58924	0.794622	+	0.607104i	$0.207669\pi$
$198$	0	0
$199$	465338.	0.832983	0.416492	−	0.909140i	$-0.363260\pi$
$199$	465338.	0.832983	0.416492	+	0.909140i	$0.363260\pi$
$200$	0	0
$201$	411416.	0.718275
$202$	0	0
$203$	51852.2	0.0883135
$204$	0	0
$205$	260539.	0.433001
$206$	0	0
$207$	747636.	1.21273
$208$	0	0
$209$	−549103.	−0.869537
$210$	0	0
$211$	−753065.	−1.16446	−0.582232	−	0.813023i	$-0.697821\pi$
$211$	−753065.	−1.16446	−0.582232	+	0.813023i	$0.697821\pi$
$212$	0	0
$213$	653835.	0.987459
$214$	0	0
$215$	155088.	0.228813
$216$	0	0
$217$	−23267.5	−0.0335428
$218$	0	0
$219$	1.75635e6	2.47457
$220$	0	0
$221$	1.40372e6	1.93331
$222$	0	0
$223$	−381228.	−0.513360	−0.256680	−	0.966496i	$-0.582629\pi$
$223$	−381228.	−0.513360	−0.256680	+	0.966496i	$0.582629\pi$
$224$	0	0
$225$	−149427.	−0.196776
$226$	0	0
$227$	−1.44018e6	−1.85504	−0.927518	−	0.373780i	$-0.878062\pi$
$227$	−1.44018e6	−1.85504	−0.927518	+	0.373780i	$0.878062\pi$
$228$	0	0
$229$	−409587.	−0.516128	−0.258064	−	0.966128i	$-0.583085\pi$
$229$	−409587.	−0.516128	−0.258064	+	0.966128i	$0.583085\pi$
$230$	0	0
$231$	−182250.	−0.224717
$232$	0	0
$233$	−407009.	−0.491150	−0.245575	−	0.969378i	$-0.578977\pi$
$233$	−407009.	−0.491150	−0.245575	+	0.969378i	$0.578977\pi$
$234$	0	0
$235$	−503443.	−0.594676
$236$	0	0
$237$	932462.	1.07835
$238$	0	0
$239$	−1.29321e6	−1.46444	−0.732222	−	0.681066i	$-0.761517\pi$
$239$	−1.29321e6	−1.46444	−0.732222	+	0.681066i	$0.761517\pi$
$240$	0	0
$241$	−1.32370e6	−1.46807	−0.734037	−	0.679110i	$-0.762366\pi$
$241$	−1.32370e6	−1.46807	−0.734037	+	0.679110i	$0.762366\pi$
$242$	0	0
$243$	1.03539e6	1.12483
$244$	0	0
$245$	753368.	0.801848
$246$	0	0
$247$	1.33799e6	1.39544
$248$	0	0
$249$	833964.	0.852411
$250$	0	0
$251$	−405086.	−0.405848	−0.202924	−	0.979195i	$-0.565044\pi$
$251$	−405086.	−0.405848	−0.202924	+	0.979195i	$0.565044\pi$
$252$	0	0
$253$	1.68582e6	1.65580
$254$	0	0
$255$	−1.51443e6	−1.45848
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−308140.	−0.285429
$260$	0	0
$261$	318451.	0.289362
$262$	0	0
$263$	2.13453e6	1.90288	0.951442	−	0.307827i	$-0.0996017\pi$
$263$	2.13453e6	1.90288	0.951442	+	0.307827i	$0.0996017\pi$
$264$	0	0
$265$	−835755.	−0.731079
$266$	0	0
$267$	1.90872e6	1.63857
$268$	0	0
$269$	254436.	0.214386	0.107193	−	0.994238i	$-0.465814\pi$
$269$	254436.	0.214386	0.107193	+	0.994238i	$0.465814\pi$
$270$	0	0
$271$	−465611.	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$271$	−465611.	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$272$	0	0
$273$	444084.	0.360627
$274$	0	0
$275$	−336936.	−0.268668
$276$	0	0
$277$	236139.	0.184914	0.0924569	−	0.995717i	$-0.470528\pi$
$277$	236139.	0.184914	0.0924569	+	0.995717i	$0.470528\pi$
$278$	0	0
$279$	−142897.	−0.109904
$280$	0	0
$281$	−235735.	−0.178098	−0.0890488	−	0.996027i	$-0.528383\pi$
$281$	−235735.	−0.178098	−0.0890488	+	0.996027i	$0.528383\pi$
$282$	0	0
$283$	2.55826e6	1.89880	0.949398	−	0.314076i	$-0.101695\pi$
$283$	2.55826e6	1.89880	0.949398	+	0.314076i	$0.101695\pi$
$284$	0	0
$285$	−1.44351e6	−1.05271
$286$	0	0
$287$	−143222.	−0.102637
$288$	0	0
$289$	1.21164e6	0.853352
$290$	0	0
$291$	2.44890e6	1.69527
$292$	0	0
$293$	−1.61990e6	−1.10235	−0.551174	−	0.834390i	$-0.685820\pi$
$293$	−1.61990e6	−1.10235	−0.551174	+	0.834390i	$0.685820\pi$
$294$	0	0
$295$	298900.	0.199973
$296$	0	0
$297$	607687.	0.399750
$298$	0	0
$299$	−4.10779e6	−2.65724
$300$	0	0
$301$	−85253.6	−0.0542371
$302$	0	0
$303$	2.02001e6	1.26400
$304$	0	0
$305$	2.51300e6	1.54683
$306$	0	0
$307$	1.41117e6	0.854544	0.427272	−	0.904123i	$-0.359475\pi$
$307$	1.41117e6	0.854544	0.427272	+	0.904123i	$0.359475\pi$
$308$	0	0
$309$	1.97391e6	1.17607
$310$	0	0
$311$	1.61309e6	0.945710	0.472855	−	0.881140i	$-0.343223\pi$
$311$	1.61309e6	0.945710	0.472855	+	0.881140i	$0.343223\pi$
$312$	0	0
$313$	1.73334e6	1.00005	0.500025	−	0.866011i	$-0.333324\pi$
$313$	1.73334e6	1.00005	0.500025	+	0.866011i	$0.333324\pi$
$314$	0	0
$315$	−188409.	−0.106985
$316$	0	0
$317$	−2.03600e6	−1.13797	−0.568985	−	0.822348i	$-0.692664\pi$
$317$	−2.03600e6	−1.13797	−0.568985	+	0.822348i	$0.692664\pi$
$318$	0	0
$319$	718062.	0.395080
$320$	0	0
$321$	5952.49	0.00322430
$322$	0	0
$323$	2.50826e6	1.33773
$324$	0	0
$325$	821005.	0.431159
$326$	0	0
$327$	−4.42272e6	−2.28728
$328$	0	0
$329$	276749.	0.140960
$330$	0	0
$331$	−659611.	−0.330916	−0.165458	−	0.986217i	$-0.552910\pi$
$331$	−659611.	−0.330916	−0.165458	+	0.986217i	$0.552910\pi$
$332$	0	0
$333$	−1.89244e6	−0.935216
$334$	0	0
$335$	959036.	0.466899
$336$	0	0
$337$	1.25312e6	0.601058	0.300529	−	0.953773i	$-0.402837\pi$
$337$	1.25312e6	0.601058	0.300529	+	0.953773i	$0.402837\pi$
$338$	0	0
$339$	312647.	0.147759
$340$	0	0
$341$	−322213.	−0.150057
$342$	0	0
$343$	−845136.	−0.387875
$344$	0	0
$345$	4.43177e6	2.00461
$346$	0	0
$347$	3.65445e6	1.62929	0.814646	−	0.579959i	$-0.196931\pi$
$347$	3.65445e6	1.62929	0.814646	+	0.579959i	$0.196931\pi$
$348$	0	0
$349$	2.65576e6	1.16715	0.583573	−	0.812061i	$-0.301654\pi$
$349$	2.65576e6	1.16715	0.583573	+	0.812061i	$0.301654\pi$
$350$	0	0
$351$	−1.48074e6	−0.641520
$352$	0	0
$353$	3.82572e6	1.63409	0.817045	−	0.576574i	$-0.195611\pi$
$353$	3.82572e6	1.63409	0.817045	+	0.576574i	$0.195611\pi$
$354$	0	0
$355$	1.52413e6	0.641876
$356$	0	0
$357$	832504.	0.345713
$358$	0	0
$359$	−1.56498e6	−0.640876	−0.320438	−	0.947270i	$-0.603830\pi$
$359$	−1.56498e6	−0.640876	−0.320438	+	0.947270i	$0.603830\pi$
$360$	0	0
$361$	−85295.1	−0.0344474
$362$	0	0
$363$	699168.	0.278494
$364$	0	0
$365$	4.09415e6	1.60854
$366$	0	0
$367$	892996.	0.346086	0.173043	−	0.984914i	$-0.444640\pi$
$367$	892996.	0.346086	0.173043	+	0.984914i	$0.444640\pi$
$368$	0	0
$369$	−879597.	−0.336293
$370$	0	0
$371$	459425.	0.173293
$372$	0	0
$373$	700441.	0.260675	0.130338	−	0.991470i	$-0.458394\pi$
$373$	700441.	0.260675	0.130338	+	0.991470i	$0.458394\pi$
$374$	0	0
$375$	−3.80318e6	−1.39659
$376$	0	0
$377$	−1.74969e6	−0.634026
$378$	0	0
$379$	−1.00838e6	−0.360601	−0.180301	−	0.983612i	$-0.557707\pi$
$379$	−1.00838e6	−0.360601	−0.180301	+	0.983612i	$0.557707\pi$
$380$	0	0
$381$	−1.63376e6	−0.576601
$382$	0	0
$383$	−166319.	−0.0579355	−0.0289678	−	0.999580i	$-0.509222\pi$
$383$	−166319.	−0.0579355	−0.0289678	+	0.999580i	$0.509222\pi$
$384$	0	0
$385$	−424835.	−0.146073
$386$	0	0
$387$	−523585.	−0.177709
$388$	0	0
$389$	−4.48343e6	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$389$	−4.48343e6	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$390$	0	0
$391$	−7.70069e6	−2.54734
$392$	0	0
$393$	−7.73665e6	−2.52680
$394$	0	0
$395$	2.17363e6	0.700958
$396$	0	0
$397$	2.42594e6	0.772508	0.386254	−	0.922392i	$-0.373769\pi$
$397$	2.42594e6	0.772508	0.386254	+	0.922392i	$0.373769\pi$
$398$	0	0
$399$	793518.	0.249531
$400$	0	0
$401$	−2.57056e6	−0.798303	−0.399151	−	0.916885i	$-0.630695\pi$
$401$	−2.57056e6	−0.798303	−0.399151	+	0.916885i	$0.630695\pi$
$402$	0	0
$403$	785130.	0.240813
$404$	0	0
$405$	3.38286e6	1.02482
$406$	0	0
$407$	−4.26719e6	−1.27690
$408$	0	0
$409$	−3.29484e6	−0.973927	−0.486964	−	0.873422i	$-0.661896\pi$
$409$	−3.29484e6	−0.973927	−0.486964	+	0.873422i	$0.661896\pi$
$410$	0	0
$411$	2.74692e6	0.802125
$412$	0	0
$413$	−164309.	−0.0474010
$414$	0	0
$415$	1.94402e6	0.554091
$416$	0	0
$417$	−4.03454e6	−1.13620
$418$	0	0
$419$	1.90557e6	0.530262	0.265131	−	0.964212i	$-0.414585\pi$
$419$	1.90557e6	0.530262	0.265131	+	0.964212i	$0.414585\pi$
$420$	0	0
$421$	3.12775e6	0.860057	0.430029	−	0.902815i	$-0.358503\pi$
$421$	3.12775e6	0.860057	0.430029	+	0.902815i	$0.358503\pi$
$422$	0	0
$423$	1.69966e6	0.461860
$424$	0	0
$425$	1.53910e6	0.413328
$426$	0	0
$427$	−1.38143e6	−0.366655
$428$	0	0
$429$	6.14978e6	1.61330
$430$	0	0
$431$	2.46542e6	0.639290	0.319645	−	0.947537i	$-0.396436\pi$
$431$	2.46542e6	0.639290	0.319645	+	0.947537i	$0.396436\pi$
$432$	0	0
$433$	405224.	0.103866	0.0519332	−	0.998651i	$-0.483462\pi$
$433$	405224.	0.103866	0.0519332	+	0.998651i	$0.483462\pi$
$434$	0	0
$435$	1.88768e6	0.478306
$436$	0	0
$437$	−7.34007e6	−1.83864
$438$	0	0
$439$	3.68273e6	0.912028	0.456014	−	0.889973i	$-0.349277\pi$
$439$	3.68273e6	0.912028	0.456014	+	0.889973i	$0.349277\pi$
$440$	0	0
$441$	−2.54342e6	−0.622761
$442$	0	0
$443$	−753529.	−0.182428	−0.0912139	−	0.995831i	$-0.529075\pi$
$443$	−753529.	−0.182428	−0.0912139	+	0.995831i	$0.529075\pi$
$444$	0	0
$445$	4.44935e6	1.06511
$446$	0	0
$447$	−2.39939e6	−0.567979
$448$	0	0
$449$	7.73018e6	1.80956	0.904782	−	0.425876i	$-0.140034\pi$
$449$	7.73018e6	1.80956	0.904782	+	0.425876i	$0.140034\pi$
$450$	0	0
$451$	−1.98337e6	−0.459158
$452$	0	0
$453$	−2.32890e6	−0.533220
$454$	0	0
$455$	1.03519e6	0.234418
$456$	0	0
$457$	3.81523e6	0.854535	0.427267	−	0.904125i	$-0.359476\pi$
$457$	3.81523e6	0.854535	0.427267	+	0.904125i	$0.359476\pi$
$458$	0	0
$459$	−2.77587e6	−0.614989
$460$	0	0
$461$	−3.99711e6	−0.875979	−0.437989	−	0.898980i	$-0.644309\pi$
$461$	−3.99711e6	−0.875979	−0.437989	+	0.898980i	$0.644309\pi$
$462$	0	0
$463$	−4.96198e6	−1.07573	−0.537864	−	0.843031i	$-0.680769\pi$
$463$	−4.96198e6	−1.07573	−0.537864	+	0.843031i	$0.680769\pi$
$464$	0	0
$465$	−847053.	−0.181668
$466$	0	0
$467$	5.49707e6	1.16638	0.583189	−	0.812336i	$-0.301805\pi$
$467$	5.49707e6	1.16638	0.583189	+	0.812336i	$0.301805\pi$
$468$	0	0
$469$	−527194.	−0.110672
$470$	0	0
$471$	1.59660e6	0.331622
$472$	0	0
$473$	−1.18061e6	−0.242636
$474$	0	0
$475$	1.46703e6	0.298335
$476$	0	0
$477$	2.82156e6	0.567798
$478$	0	0
$479$	5.18291e6	1.03213	0.516066	−	0.856549i	$-0.327396\pi$
$479$	5.18291e6	1.03213	0.516066	+	0.856549i	$0.327396\pi$
$480$	0	0
$481$	1.03978e7	2.04917
$482$	0	0
$483$	−2.43620e6	−0.475166
$484$	0	0
$485$	5.70853e6	1.10197
$486$	0	0
$487$	4.12597e6	0.788323	0.394161	−	0.919041i	$-0.371035\pi$
$487$	4.12597e6	0.788323	0.394161	+	0.919041i	$0.371035\pi$
$488$	0	0
$489$	713815.	0.134994
$490$	0	0
$491$	4.19231e6	0.784783	0.392392	−	0.919798i	$-0.371648\pi$
$491$	4.19231e6	0.784783	0.392392	+	0.919798i	$0.371648\pi$
$492$	0	0
$493$	−3.28006e6	−0.607805
$494$	0	0
$495$	−2.60913e6	−0.478611
$496$	0	0
$497$	−837834.	−0.152148
$498$	0	0
$499$	−4.82211e6	−0.866933	−0.433466	−	0.901170i	$-0.642710\pi$
$499$	−4.82211e6	−0.866933	−0.433466	+	0.901170i	$0.642710\pi$
$500$	0	0
$501$	962597.	0.171337
$502$	0	0
$503$	−7.17990e6	−1.26532	−0.632658	−	0.774432i	$-0.718036\pi$
$503$	−7.17990e6	−1.26532	−0.632658	+	0.774432i	$0.718036\pi$
$504$	0	0
$505$	4.70876e6	0.821634
$506$	0	0
$507$	−7.55460e6	−1.30524
$508$	0	0
$509$	−1.83118e6	−0.313284	−0.156642	−	0.987655i	$-0.550067\pi$
$509$	−1.83118e6	−0.313284	−0.156642	+	0.987655i	$0.550067\pi$
$510$	0	0
$511$	−2.25061e6	−0.381283
$512$	0	0
$513$	−2.64588e6	−0.443891
$514$	0	0
$515$	4.60131e6	0.764475
$516$	0	0
$517$	3.83249e6	0.630601
$518$	0	0
$519$	−7.66704e6	−1.24942
$520$	0	0
$521$	−7.11618e6	−1.14856	−0.574279	−	0.818660i	$-0.694717\pi$
$521$	−7.11618e6	−1.14856	−0.574279	+	0.818660i	$0.694717\pi$
$522$	0	0
$523$	−6.46434e6	−1.03340	−0.516701	−	0.856166i	$-0.672840\pi$
$523$	−6.46434e6	−1.03340	−0.516701	+	0.856166i	$0.672840\pi$
$524$	0	0
$525$	486912.	0.0770996
$526$	0	0
$527$	1.47185e6	0.230854
$528$	0	0
$529$	1.60986e7	2.50120
$530$	0	0
$531$	−1.00911e6	−0.155310
$532$	0	0
$533$	4.83283e6	0.736858
$534$	0	0
$535$	13875.6	0.00209589
$536$	0	0
$537$	5.36671e6	0.803105
$538$	0	0
$539$	−5.73506e6	−0.850288
$540$	0	0
$541$	−6.80476e6	−0.999585	−0.499792	−	0.866145i	$-0.666590\pi$
$541$	−6.80476e6	−0.999585	−0.499792	+	0.866145i	$0.666590\pi$
$542$	0	0
$543$	−3.73857e6	−0.544133
$544$	0	0
$545$	−1.03096e7	−1.48680
$546$	0	0
$547$	−2.49977e6	−0.357217	−0.178609	−	0.983920i	$-0.557160\pi$
$547$	−2.49977e6	−0.357217	−0.178609	+	0.983920i	$0.557160\pi$
$548$	0	0
$549$	−8.48404e6	−1.20136
$550$	0	0
$551$	−3.12645e6	−0.438705
$552$	0	0
$553$	−1.19487e6	−0.166153
$554$	0	0
$555$	−1.12178e7	−1.54588
$556$	0	0
$557$	−3.87724e6	−0.529523	−0.264762	−	0.964314i	$-0.585293\pi$
$557$	−3.87724e6	−0.529523	−0.264762	+	0.964314i	$0.585293\pi$
$558$	0	0
$559$	2.87677e6	0.389382
$560$	0	0
$561$	1.15287e7	1.54658
$562$	0	0
$563$	−4.04614e6	−0.537985	−0.268992	−	0.963142i	$-0.586691\pi$
$563$	−4.04614e6	−0.537985	−0.268992	+	0.963142i	$0.586691\pi$
$564$	0	0
$565$	728800.	0.0960478
$566$	0	0
$567$	−1.85960e6	−0.242919
$568$	0	0
$569$	−8.71308e6	−1.12821	−0.564106	−	0.825702i	$-0.690779\pi$
$569$	−8.71308e6	−1.12821	−0.564106	+	0.825702i	$0.690779\pi$
$570$	0	0
$571$	−1.49475e7	−1.91858	−0.959288	−	0.282431i	$-0.908859\pi$
$571$	−1.49475e7	−1.91858	−0.959288	+	0.282431i	$0.908859\pi$
$572$	0	0
$573$	−8.96535e6	−1.14072
$574$	0	0
$575$	−4.50395e6	−0.568099
$576$	0	0
$577$	4.35431e6	0.544477	0.272239	−	0.962230i	$-0.412236\pi$
$577$	4.35431e6	0.544477	0.272239	+	0.962230i	$0.412236\pi$
$578$	0	0
$579$	8.97395e6	1.11247
$580$	0	0
$581$	−1.06865e6	−0.131340
$582$	0	0
$583$	6.36223e6	0.775243
$584$	0	0
$585$	6.35760e6	0.768075
$586$	0	0
$587$	1.33215e7	1.59573	0.797863	−	0.602839i	$-0.205964\pi$
$587$	1.33215e7	1.59573	0.797863	+	0.602839i	$0.205964\pi$
$588$	0	0
$589$	1.40292e6	0.166627
$590$	0	0
$591$	−1.73242e7	−2.04026
$592$	0	0
$593$	5.02792e6	0.587153	0.293577	−	0.955936i	$-0.405154\pi$
$593$	5.02792e6	0.587153	0.293577	+	0.955936i	$0.405154\pi$
$594$	0	0
$595$	1.94062e6	0.224723
$596$	0	0
$597$	−9.31251e6	−1.06938
$598$	0	0
$599$	−9.30412e6	−1.05952	−0.529759	−	0.848148i	$-0.677718\pi$
$599$	−9.30412e6	−1.05952	−0.529759	+	0.848148i	$0.677718\pi$
$600$	0	0
$601$	−4.08369e6	−0.461175	−0.230588	−	0.973052i	$-0.574065\pi$
$601$	−4.08369e6	−0.461175	−0.230588	+	0.973052i	$0.574065\pi$
$602$	0	0
$603$	−3.23777e6	−0.362620
$604$	0	0
$605$	1.62980e6	0.181029
$606$	0	0
$607$	7.22425e6	0.795831	0.397915	−	0.917422i	$-0.369734\pi$
$607$	7.22425e6	0.795831	0.397915	+	0.917422i	$0.369734\pi$
$608$	0	0
$609$	−1.03768e6	−0.113376
$610$	0	0
$611$	−9.33854e6	−1.01199
$612$	0	0
$613$	1.70032e7	1.82759	0.913796	−	0.406173i	$-0.133137\pi$
$613$	1.70032e7	1.82759	0.913796	+	0.406173i	$0.133137\pi$
$614$	0	0
$615$	−5.21400e6	−0.555883
$616$	0	0
$617$	−8.71076e6	−0.921178	−0.460589	−	0.887614i	$-0.652362\pi$
$617$	−8.71076e6	−0.921178	−0.460589	+	0.887614i	$0.652362\pi$
$618$	0	0
$619$	−4.36437e6	−0.457820	−0.228910	−	0.973448i	$-0.573516\pi$
$619$	−4.36437e6	−0.457820	−0.228910	+	0.973448i	$0.573516\pi$
$620$	0	0
$621$	8.12318e6	0.845273
$622$	0	0
$623$	−2.44586e6	−0.252471
$624$	0	0
$625$	−5.90050e6	−0.604211
$626$	0	0
$627$	1.09888e7	1.11630
$628$	0	0
$629$	1.94922e7	1.96442
$630$	0	0
$631$	−4.32576e6	−0.432503	−0.216252	−	0.976338i	$-0.569383\pi$
$631$	−4.32576e6	−0.432503	−0.216252	+	0.976338i	$0.569383\pi$
$632$	0	0
$633$	1.50706e7	1.49493
$634$	0	0
$635$	−3.80839e6	−0.374807
$636$	0	0
$637$	1.39745e7	1.36454
$638$	0	0
$639$	−5.14556e6	−0.498518
$640$	0	0
$641$	8.14209e6	0.782692	0.391346	−	0.920244i	$-0.372010\pi$
$641$	8.14209e6	0.782692	0.391346	+	0.920244i	$0.372010\pi$
$642$	0	0
$643$	1.23162e7	1.17476	0.587382	−	0.809310i	$-0.300159\pi$
$643$	1.23162e7	1.17476	0.587382	+	0.809310i	$0.300159\pi$
$644$	0	0
$645$	−3.10366e6	−0.293748
$646$	0	0
$647$	2.65117e6	0.248987	0.124494	−	0.992220i	$-0.460269\pi$
$647$	2.65117e6	0.248987	0.124494	+	0.992220i	$0.460269\pi$
$648$	0	0
$649$	−2.27540e6	−0.212053
$650$	0	0
$651$	465636.	0.0430620
$652$	0	0
$653$	2.43665e6	0.223620	0.111810	−	0.993730i	$-0.464335\pi$
$653$	2.43665e6	0.223620	0.111810	+	0.993730i	$0.464335\pi$
$654$	0	0
$655$	−1.80346e7	−1.64249
$656$	0	0
$657$	−1.38221e7	−1.24928
$658$	0	0
$659$	8.12429e6	0.728739	0.364369	−	0.931255i	$-0.381285\pi$
$659$	8.12429e6	0.728739	0.364369	+	0.931255i	$0.381285\pi$
$660$	0	0
$661$	−1.31213e7	−1.16808	−0.584042	−	0.811723i	$-0.698530\pi$
$661$	−1.31213e7	−1.16808	−0.584042	+	0.811723i	$0.698530\pi$
$662$	0	0
$663$	−2.80918e7	−2.48196
$664$	0	0
$665$	1.84974e6	0.162202
$666$	0	0
$667$	9.59861e6	0.835399
$668$	0	0
$669$	7.62925e6	0.659048
$670$	0	0
$671$	−1.91303e7	−1.64027
$672$	0	0
$673$	1.97323e7	1.67935	0.839674	−	0.543091i	$-0.182746\pi$
$673$	1.97323e7	1.67935	0.839674	+	0.543091i	$0.182746\pi$
$674$	0	0
$675$	−1.62354e6	−0.137153
$676$	0	0
$677$	−2.72216e6	−0.228267	−0.114133	−	0.993465i	$-0.536409\pi$
$677$	−2.72216e6	−0.228267	−0.114133	+	0.993465i	$0.536409\pi$
$678$	0	0
$679$	−3.13805e6	−0.261207
$680$	0	0
$681$	2.88214e7	2.38148
$682$	0	0
$683$	−2.14126e7	−1.75637	−0.878186	−	0.478318i	$-0.841246\pi$
$683$	−2.14126e7	−1.75637	−0.878186	+	0.478318i	$0.841246\pi$
$684$	0	0
$685$	6.40325e6	0.521403
$686$	0	0
$687$	8.19679e6	0.662601
$688$	0	0
$689$	−1.55027e7	−1.24411
$690$	0	0
$691$	−857776.	−0.0683406	−0.0341703	−	0.999416i	$-0.510879\pi$
$691$	−857776.	−0.0683406	−0.0341703	+	0.999416i	$0.510879\pi$
$692$	0	0
$693$	1.43427e6	0.113448
$694$	0	0
$695$	−9.40475e6	−0.738559
$696$	0	0
$697$	9.05989e6	0.706385
$698$	0	0
$699$	8.14519e6	0.630534
$700$	0	0
$701$	2.01148e7	1.54604	0.773020	−	0.634381i	$-0.218745\pi$
$701$	2.01148e7	1.54604	0.773020	+	0.634381i	$0.218745\pi$
$702$	0	0
$703$	1.85794e7	1.41790
$704$	0	0
$705$	1.00751e7	0.763440
$706$	0	0
$707$	−2.58847e6	−0.194758
$708$	0	0
$709$	1.49168e7	1.11445	0.557224	−	0.830362i	$-0.311866\pi$
$709$	1.49168e7	1.11445	0.557224	+	0.830362i	$0.311866\pi$
$710$	0	0
$711$	−7.33830e6	−0.544404
$712$	0	0
$713$	−4.30715e6	−0.317297
$714$	0	0
$715$	1.43355e7	1.04869
$716$	0	0
$717$	2.58801e7	1.88004
$718$	0	0
$719$	2.22057e7	1.60192	0.800962	−	0.598716i	$-0.204322\pi$
$719$	2.22057e7	1.60192	0.800962	+	0.598716i	$0.204322\pi$
$720$	0	0
$721$	−2.52940e6	−0.181209
$722$	0	0
$723$	2.64904e7	1.88470
$724$	0	0
$725$	−1.91843e6	−0.135550
$726$	0	0
$727$	−7.66282e6	−0.537715	−0.268858	−	0.963180i	$-0.586646\pi$
$727$	−7.66282e6	−0.537715	−0.268858	+	0.963180i	$0.586646\pi$
$728$	0	0
$729$	−3.09924e6	−0.215991
$730$	0	0
$731$	5.39296e6	0.373279
$732$	0	0
$733$	−2.59191e7	−1.78180	−0.890901	−	0.454198i	$-0.849926\pi$
$733$	−2.59191e7	−1.78180	−0.890901	+	0.454198i	$0.849926\pi$
$734$	0	0
$735$	−1.50767e7	−1.02941
$736$	0	0
$737$	−7.30072e6	−0.495104
$738$	0	0
$739$	−1.34375e7	−0.905122	−0.452561	−	0.891733i	$-0.649490\pi$
$739$	−1.34375e7	−0.905122	−0.452561	+	0.891733i	$0.649490\pi$
$740$	0	0
$741$	−2.67762e7	−1.79145
$742$	0	0
$743$	−5.51804e6	−0.366701	−0.183351	−	0.983048i	$-0.558694\pi$
$743$	−5.51804e6	−0.366701	−0.183351	+	0.983048i	$0.558694\pi$
$744$	0	0
$745$	−5.59313e6	−0.369202
$746$	0	0
$747$	−6.56314e6	−0.430338
$748$	0	0
$749$	−7627.60	−0.000496802 0
$750$	0	0
$751$	2.09277e7	1.35401	0.677005	−	0.735979i	$-0.263278\pi$
$751$	2.09277e7	1.35401	0.677005	+	0.735979i	$0.263278\pi$
$752$	0	0
$753$	8.10671e6	0.521024
$754$	0	0
$755$	−5.42882e6	−0.346608
$756$	0	0
$757$	−9.68422e6	−0.614221	−0.307111	−	0.951674i	$-0.599362\pi$
$757$	−9.68422e6	−0.614221	−0.307111	+	0.951674i	$0.599362\pi$
$758$	0	0
$759$	−3.37371e7	−2.12571
$760$	0	0
$761$	−8.11833e6	−0.508165	−0.254083	−	0.967183i	$-0.581774\pi$
$761$	−8.11833e6	−0.508165	−0.254083	+	0.967183i	$0.581774\pi$
$762$	0	0
$763$	5.66733e6	0.352425
$764$	0	0
$765$	1.19183e7	0.736311
$766$	0	0
$767$	5.54441e6	0.340304
$768$	0	0
$769$	−2.20461e7	−1.34436	−0.672180	−	0.740388i	$-0.734642\pi$
$769$	−2.20461e7	−1.34436	−0.672180	+	0.740388i	$0.734642\pi$
$770$	0	0
$771$	1.32179e6	0.0800807
$772$	0	0
$773$	−2.34174e7	−1.40958	−0.704791	−	0.709415i	$-0.748959\pi$
$773$	−2.34174e7	−1.40958	−0.704791	+	0.709415i	$0.748959\pi$
$774$	0	0
$775$	860850.	0.0514841
$776$	0	0
$777$	6.16659e6	0.366431
$778$	0	0
$779$	8.63563e6	0.509859
$780$	0	0
$781$	−1.16025e7	−0.680652
$782$	0	0
$783$	3.46001e6	0.201685
$784$	0	0
$785$	3.72177e6	0.215563
$786$	0	0
$787$	2.19277e7	1.26199	0.630996	−	0.775786i	$-0.282647\pi$
$787$	2.19277e7	1.26199	0.630996	+	0.775786i	$0.282647\pi$
$788$	0	0
$789$	−4.27169e7	−2.44291
$790$	0	0
$791$	−400631.	−0.0227669
$792$	0	0
$793$	4.66145e7	2.63231
$794$	0	0
$795$	1.67254e7	0.938553
$796$	0	0
$797$	9.21108e6	0.513647	0.256823	−	0.966458i	$-0.417324\pi$
$797$	9.21108e6	0.513647	0.256823	+	0.966458i	$0.417324\pi$
$798$	0	0
$799$	−1.75065e7	−0.970138
$800$	0	0
$801$	−1.50213e7	−0.827229
$802$	0	0
$803$	−3.11670e7	−1.70571
$804$	0	0
$805$	−5.67893e6	−0.308871
$806$	0	0
$807$	−5.09185e6	−0.275228
$808$	0	0
$809$	−2.80028e7	−1.50429	−0.752143	−	0.659000i	$-0.770980\pi$
$809$	−2.80028e7	−1.50429	−0.752143	+	0.659000i	$0.770980\pi$
$810$	0	0
$811$	2.51635e7	1.34344	0.671721	−	0.740804i	$-0.265555\pi$
$811$	2.51635e7	1.34344	0.671721	+	0.740804i	$0.265555\pi$
$812$	0	0
$813$	9.31795e6	0.494418
$814$	0	0
$815$	1.66395e6	0.0877497
$816$	0	0
$817$	5.14041e6	0.269428
$818$	0	0
$819$	−3.49486e6	−0.182062
$820$	0	0
$821$	−3.98384e6	−0.206274	−0.103137	−	0.994667i	$-0.532888\pi$
$821$	−3.98384e6	−0.206274	−0.103137	+	0.994667i	$0.532888\pi$
$822$	0	0
$823$	−5.72933e6	−0.294852	−0.147426	−	0.989073i	$-0.547099\pi$
$823$	−5.72933e6	−0.294852	−0.147426	+	0.989073i	$0.547099\pi$
$824$	0	0
$825$	6.74288e6	0.344914
$826$	0	0
$827$	3.03962e7	1.54545	0.772725	−	0.634741i	$-0.218893\pi$
$827$	3.03962e7	1.54545	0.772725	+	0.634741i	$0.218893\pi$
$828$	0	0
$829$	−5.65058e6	−0.285566	−0.142783	−	0.989754i	$-0.545605\pi$
$829$	−5.65058e6	−0.285566	−0.142783	+	0.989754i	$0.545605\pi$
$830$	0	0
$831$	−4.72570e6	−0.237391
$832$	0	0
$833$	2.61973e7	1.30811
$834$	0	0
$835$	2.24387e6	0.111374
$836$	0	0
$837$	−1.55260e6	−0.0766030
$838$	0	0
$839$	−7.10714e6	−0.348570	−0.174285	−	0.984695i	$-0.555761\pi$
$839$	−7.10714e6	−0.348570	−0.174285	+	0.984695i	$0.555761\pi$
$840$	0	0
$841$	−1.64227e7	−0.800671
$842$	0	0
$843$	4.71760e6	0.228640
$844$	0	0
$845$	−1.76102e7	−0.848445
$846$	0	0
$847$	−895924.	−0.0429104
$848$	0	0
$849$	−5.11967e7	−2.43766
$850$	0	0
$851$	−5.70412e7	−2.70001
$852$	0	0
$853$	−2.34607e7	−1.10400	−0.552000	−	0.833844i	$-0.686135\pi$
$853$	−2.34607e7	−1.10400	−0.552000	+	0.833844i	$0.686135\pi$
$854$	0	0
$855$	1.13602e7	0.531459
$856$	0	0
$857$	−1.50817e7	−0.701452	−0.350726	−	0.936478i	$-0.614065\pi$
$857$	−1.50817e7	−0.701452	−0.350726	+	0.936478i	$0.614065\pi$
$858$	0	0
$859$	−1.85730e7	−0.858813	−0.429406	−	0.903111i	$-0.641277\pi$
$859$	−1.85730e7	−0.858813	−0.429406	+	0.903111i	$0.641277\pi$
$860$	0	0
$861$	2.86620e6	0.131765
$862$	0	0
$863$	1.71547e7	0.784071	0.392036	−	0.919950i	$-0.371771\pi$
$863$	1.71547e7	0.784071	0.392036	+	0.919950i	$0.371771\pi$
$864$	0	0
$865$	−1.78723e7	−0.812159
$866$	0	0
$867$	−2.42477e7	−1.09553
$868$	0	0
$869$	−1.65469e7	−0.743303
$870$	0	0
$871$	1.77895e7	0.794545
$872$	0	0
$873$	−1.92723e7	−0.855853
$874$	0	0
$875$	4.87345e6	0.215187
$876$	0	0
$877$	2.68148e7	1.17727	0.588635	−	0.808399i	$-0.299665\pi$
$877$	2.68148e7	1.17727	0.588635	+	0.808399i	$0.299665\pi$
$878$	0	0
$879$	3.24180e7	1.41519
$880$	0	0
$881$	−2.78346e7	−1.20822	−0.604109	−	0.796902i	$-0.706471\pi$
$881$	−2.78346e7	−1.20822	−0.604109	+	0.796902i	$0.706471\pi$
$882$	0	0
$883$	−2.56085e7	−1.10530	−0.552652	−	0.833412i	$-0.686384\pi$
$883$	−2.56085e7	−1.10530	−0.552652	+	0.833412i	$0.686384\pi$
$884$	0	0
$885$	−5.98169e6	−0.256724
$886$	0	0
$887$	5.64183e6	0.240775	0.120387	−	0.992727i	$-0.461586\pi$
$887$	5.64183e6	0.240775	0.120387	+	0.992727i	$0.461586\pi$
$888$	0	0
$889$	2.09352e6	0.0888430
$890$	0	0
$891$	−2.57522e7	−1.08673
$892$	0	0
$893$	−1.66867e7	−0.700233
$894$	0	0
$895$	1.25101e7	0.522041
$896$	0	0
$897$	8.22065e7	3.41134
$898$	0	0
$899$	−1.83460e6	−0.0757081
$900$	0	0
$901$	−2.90622e7	−1.19266
$902$	0	0
$903$	1.70612e6	0.0696291
$904$	0	0
$905$	−8.71483e6	−0.353702
$906$	0	0
$907$	2.66110e7	1.07410	0.537049	−	0.843551i	$-0.319539\pi$
$907$	2.66110e7	1.07410	0.537049	+	0.843551i	$0.319539\pi$
$908$	0	0
$909$	−1.58971e7	−0.638128
$910$	0	0
$911$	2.01224e7	0.803311	0.401655	−	0.915791i	$-0.368435\pi$
$911$	2.01224e7	0.803311	0.401655	+	0.915791i	$0.368435\pi$
$912$	0	0
$913$	−1.47990e7	−0.587563
$914$	0	0
$915$	−5.02909e7	−1.98581
$916$	0	0
$917$	9.91386e6	0.389331
$918$	0	0
$919$	−4.42635e7	−1.72885	−0.864425	−	0.502762i	$-0.832317\pi$
$919$	−4.42635e7	−1.72885	−0.864425	+	0.502762i	$0.832317\pi$
$920$	0	0
$921$	−2.82409e7	−1.09706
$922$	0	0
$923$	2.82716e7	1.09231
$924$	0	0
$925$	1.14006e7	0.438099
$926$	0	0
$927$	−1.55343e7	−0.593735
$928$	0	0
$929$	−2.21889e6	−0.0843522	−0.0421761	−	0.999110i	$-0.513429\pi$
$929$	−2.21889e6	−0.0843522	−0.0421761	+	0.999110i	$0.513429\pi$
$930$	0	0
$931$	2.49705e7	0.944178
$932$	0	0
$933$	−3.22817e7	−1.21409
$934$	0	0
$935$	2.68741e7	1.00532
$936$	0	0
$937$	1.81865e7	0.676707	0.338353	−	0.941019i	$-0.390130\pi$
$937$	1.81865e7	0.676707	0.338353	+	0.941019i	$0.390130\pi$
$938$	0	0
$939$	−3.46881e7	−1.28386
$940$	0	0
$941$	4.97838e7	1.83279	0.916397	−	0.400271i	$-0.131084\pi$
$941$	4.97838e7	1.83279	0.916397	+	0.400271i	$0.131084\pi$
$942$	0	0
$943$	−2.65125e7	−0.970892
$944$	0	0
$945$	−2.04709e6	−0.0745688
$946$	0	0
$947$	2.56304e7	0.928711	0.464355	−	0.885649i	$-0.346286\pi$
$947$	2.56304e7	0.928711	0.464355	+	0.885649i	$0.346286\pi$
$948$	0	0
$949$	7.59438e7	2.73733
$950$	0	0
$951$	4.07452e7	1.46091
$952$	0	0
$953$	−1.69089e7	−0.603093	−0.301546	−	0.953451i	$-0.597503\pi$
$953$	−1.69089e7	−0.603093	−0.301546	+	0.953451i	$0.597503\pi$
$954$	0	0
$955$	−2.08988e7	−0.741503
$956$	0	0
$957$	−1.43701e7	−0.507200
$958$	0	0
$959$	−3.51995e6	−0.123592
$960$	0	0
$961$	−2.78059e7	−0.971245
$962$	0	0
$963$	−46845.0	−0.00162779
$964$	0	0
$965$	2.09188e7	0.723134
$966$	0	0
$967$	−5.55454e7	−1.91021	−0.955107	−	0.296260i	$-0.904261\pi$
$967$	−5.55454e7	−1.91021	−0.955107	+	0.296260i	$0.904261\pi$
$968$	0	0
$969$	−5.01962e7	−1.71736
$970$	0	0
$971$	2.51896e7	0.857380	0.428690	−	0.903452i	$-0.358975\pi$
$971$	2.51896e7	0.857380	0.428690	+	0.903452i	$0.358975\pi$
$972$	0	0
$973$	5.16991e6	0.175066
$974$	0	0
$975$	−1.64302e7	−0.553518
$976$	0	0
$977$	1.55162e7	0.520054	0.260027	−	0.965601i	$-0.416269\pi$
$977$	1.55162e7	0.520054	0.260027	+	0.965601i	$0.416269\pi$
$978$	0	0
$979$	−3.38709e7	−1.12946
$980$	0	0
$981$	3.48059e7	1.15473
$982$	0	0
$983$	1.83149e7	0.604533	0.302266	−	0.953224i	$-0.402257\pi$
$983$	1.83149e7	0.604533	0.302266	+	0.953224i	$0.402257\pi$
$984$	0	0
$985$	−4.03838e7	−1.32622
$986$	0	0
$987$	−5.53839e6	−0.180963
$988$	0	0
$989$	−1.57817e7	−0.513054
$990$	0	0
$991$	5.01962e7	1.62363	0.811814	−	0.583917i	$-0.198480\pi$
$991$	5.01962e7	1.62363	0.811814	+	0.583917i	$0.198480\pi$
$992$	0	0
$993$	1.32003e7	0.424827
$994$	0	0
$995$	−2.17080e7	−0.695125
$996$	0	0
$997$	−5.08428e6	−0.161991	−0.0809956	−	0.996714i	$-0.525810\pi$
$997$	−5.08428e6	−0.161991	−0.0809956	+	0.996714i	$0.525810\pi$
$998$	0	0
$999$	−2.05617e7	−0.651845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.12	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-20.0123 q^{3} -46.6500 q^{5} +25.6441 q^{7} +157.493 q^{9} +O(q^{10})\)	\(q-20.0123 q^{3} -46.6500 q^{5} +25.6441 q^{7} +157.493 q^{9} +355.126 q^{11} -865.327 q^{13} +933.575 q^{15} -1622.19 q^{17} -1546.22 q^{19} -513.198 q^{21} +4747.10 q^{23} -948.780 q^{25} +1711.19 q^{27} +2021.99 q^{29} -907.323 q^{31} -7106.89 q^{33} -1196.30 q^{35} -12016.0 q^{37} +17317.2 q^{39} -5584.98 q^{41} -3324.49 q^{43} -7347.06 q^{45} +10791.9 q^{47} -16149.4 q^{49} +32463.8 q^{51} +17915.4 q^{53} -16566.6 q^{55} +30943.5 q^{57} -6407.30 q^{59} -53869.2 q^{61} +4038.77 q^{63} +40367.5 q^{65} -20558.1 q^{67} -95000.5 q^{69} -32671.6 q^{71} -87763.2 q^{73} +18987.3 q^{75} +9106.87 q^{77} -46594.4 q^{79} -72515.7 q^{81} -41672.5 q^{83} +75675.0 q^{85} -40464.8 q^{87} -95377.3 q^{89} -22190.5 q^{91} +18157.6 q^{93} +72131.2 q^{95} -122369. q^{97} +55929.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 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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Groups

Database

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