LMFDB - Embedded newform 1028.6.a.a.1.1

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			1.1
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)

$f(q)$	$=$	$q-30.6737 q^{3} +42.6130 q^{5} +15.7852 q^{7} +697.875 q^{9} +O(q^{10})$	$q-30.6737 q^{3} +42.6130 q^{5} +15.7852 q^{7} +697.875 q^{9} +262.142 q^{11} +483.035 q^{13} -1307.10 q^{15} -1479.05 q^{17} +2274.24 q^{19} -484.190 q^{21} +259.903 q^{23} -1309.14 q^{25} -13952.7 q^{27} -6866.76 q^{29} -1610.48 q^{31} -8040.87 q^{33} +672.655 q^{35} -5142.12 q^{37} -14816.4 q^{39} +7062.51 q^{41} +21284.8 q^{43} +29738.5 q^{45} -13757.0 q^{47} -16557.8 q^{49} +45367.9 q^{51} +6095.20 q^{53} +11170.7 q^{55} -69759.3 q^{57} -24369.3 q^{59} -15099.1 q^{61} +11016.1 q^{63} +20583.5 q^{65} -39095.7 q^{67} -7972.19 q^{69} +60414.6 q^{71} +9213.62 q^{73} +40156.0 q^{75} +4137.97 q^{77} -33324.6 q^{79} +258396. q^{81} +18871.3 q^{83} -63026.7 q^{85} +210629. q^{87} +30542.4 q^{89} +7624.80 q^{91} +49399.3 q^{93} +96912.1 q^{95} -22050.1 q^{97} +182942. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−30.6737	−1.96772	−0.983859	−	0.178947i	$-0.942731\pi$
$3$	−30.6737	−1.96772	−0.983859	+	0.178947i	$0.942731\pi$
$4$	0	0
$5$	42.6130	0.762284	0.381142	−	0.924517i	$-0.375531\pi$
$5$	42.6130	0.762284	0.381142	+	0.924517i	$0.375531\pi$
$6$	0	0
$7$	15.7852	0.121760	0.0608801	−	0.998145i	$-0.480609\pi$
$7$	15.7852	0.121760	0.0608801	+	0.998145i	$0.480609\pi$
$8$	0	0
$9$	697.875	2.87191
$10$	0	0
$11$	262.142	0.653214	0.326607	−	0.945160i	$-0.394095\pi$
$11$	262.142	0.653214	0.326607	+	0.945160i	$0.394095\pi$
$12$	0	0
$13$	483.035	0.792720	0.396360	−	0.918095i	$-0.370273\pi$
$13$	483.035	0.792720	0.396360	+	0.918095i	$0.370273\pi$
$14$	0	0
$15$	−1307.10	−1.49996
$16$	0	0
$17$	−1479.05	−1.24125	−0.620627	−	0.784106i	$-0.713122\pi$
$17$	−1479.05	−1.24125	−0.620627	+	0.784106i	$0.713122\pi$
$18$	0	0
$19$	2274.24	1.44528	0.722640	−	0.691224i	$-0.242928\pi$
$19$	2274.24	1.44528	0.722640	+	0.691224i	$0.242928\pi$
$20$	0	0
$21$	−484.190	−0.239590
$22$	0	0
$23$	259.903	0.102445	0.0512226	−	0.998687i	$-0.483688\pi$
$23$	259.903	0.102445	0.0512226	+	0.998687i	$0.483688\pi$
$24$	0	0
$25$	−1309.14	−0.418923
$26$	0	0
$27$	−13952.7	−3.68339
$28$	0	0
$29$	−6866.76	−1.51620	−0.758101	−	0.652137i	$-0.773873\pi$
$29$	−6866.76	−1.51620	−0.758101	+	0.652137i	$0.773873\pi$
$30$	0	0
$31$	−1610.48	−0.300989	−0.150495	−	0.988611i	$-0.548087\pi$
$31$	−1610.48	−0.300989	−0.150495	+	0.988611i	$0.548087\pi$
$32$	0	0
$33$	−8040.87	−1.28534
$34$	0	0
$35$	672.655	0.0928158
$36$	0	0
$37$	−5142.12	−0.617502	−0.308751	−	0.951143i	$-0.599911\pi$
$37$	−5142.12	−0.617502	−0.308751	+	0.951143i	$0.599911\pi$
$38$	0	0
$39$	−14816.4	−1.55985
$40$	0	0
$41$	7062.51	0.656145	0.328072	−	0.944653i	$-0.393601\pi$
$41$	7062.51	0.656145	0.328072	+	0.944653i	$0.393601\pi$
$42$	0	0
$43$	21284.8	1.75549	0.877746	−	0.479126i	$-0.159046\pi$
$43$	21284.8	1.75549	0.877746	+	0.479126i	$0.159046\pi$
$44$	0	0
$45$	29738.5	2.18921
$46$	0	0
$47$	−13757.0	−0.908403	−0.454201	−	0.890899i	$-0.650075\pi$
$47$	−13757.0	−0.908403	−0.454201	+	0.890899i	$0.650075\pi$
$48$	0	0
$49$	−16557.8	−0.985174
$50$	0	0
$51$	45367.9	2.44244
$52$	0	0
$53$	6095.20	0.298056	0.149028	−	0.988833i	$-0.452386\pi$
$53$	6095.20	0.298056	0.149028	+	0.988833i	$0.452386\pi$
$54$	0	0
$55$	11170.7	0.497934
$56$	0	0
$57$	−69759.3	−2.84390
$58$	0	0
$59$	−24369.3	−0.911408	−0.455704	−	0.890131i	$-0.650613\pi$
$59$	−24369.3	−0.911408	−0.455704	+	0.890131i	$0.650613\pi$
$60$	0	0
$61$	−15099.1	−0.519549	−0.259774	−	0.965669i	$-0.583648\pi$
$61$	−15099.1	−0.519549	−0.259774	+	0.965669i	$0.583648\pi$
$62$	0	0
$63$	11016.1	0.349684
$64$	0	0
$65$	20583.5	0.604278
$66$	0	0
$67$	−39095.7	−1.06400	−0.532000	−	0.846744i	$-0.678559\pi$
$67$	−39095.7	−1.06400	−0.532000	+	0.846744i	$0.678559\pi$
$68$	0	0
$69$	−7972.19	−0.201583
$70$	0	0
$71$	60414.6	1.42232	0.711158	−	0.703033i	$-0.248171\pi$
$71$	60414.6	1.42232	0.711158	+	0.703033i	$0.248171\pi$
$72$	0	0
$73$	9213.62	0.202359	0.101180	−	0.994868i	$-0.467738\pi$
$73$	9213.62	0.202359	0.101180	+	0.994868i	$0.467738\pi$
$74$	0	0
$75$	40156.0	0.824323
$76$	0	0
$77$	4137.97	0.0795354
$78$	0	0
$79$	−33324.6	−0.600755	−0.300377	−	0.953820i	$-0.597113\pi$
$79$	−33324.6	−0.600755	−0.300377	+	0.953820i	$0.597113\pi$
$80$	0	0
$81$	258396.	4.37596
$82$	0	0
$83$	18871.3	0.300682	0.150341	−	0.988634i	$-0.451963\pi$
$83$	18871.3	0.300682	0.150341	+	0.988634i	$0.451963\pi$
$84$	0	0
$85$	−63026.7	−0.946187
$86$	0	0
$87$	210629.	2.98346
$88$	0	0
$89$	30542.4	0.408722	0.204361	−	0.978896i	$-0.434488\pi$
$89$	30542.4	0.408722	0.204361	+	0.978896i	$0.434488\pi$
$90$	0	0
$91$	7624.80	0.0965217
$92$	0	0
$93$	49399.3	0.592262
$94$	0	0
$95$	96912.1	1.10171
$96$	0	0
$97$	−22050.1	−0.237948	−0.118974	−	0.992897i	$-0.537961\pi$
$97$	−22050.1	−0.237948	−0.118974	+	0.992897i	$0.537961\pi$
$98$	0	0
$99$	182942.	1.87597
$100$	0	0
$101$	−3189.80	−0.0311143	−0.0155571	−	0.999879i	$-0.504952\pi$
$101$	−3189.80	−0.0311143	−0.0155571	+	0.999879i	$0.504952\pi$
$102$	0	0
$103$	−94060.7	−0.873605	−0.436803	−	0.899557i	$-0.643889\pi$
$103$	−94060.7	−0.873605	−0.436803	+	0.899557i	$0.643889\pi$
$104$	0	0
$105$	−20632.8	−0.182635
$106$	0	0
$107$	19212.8	0.162230	0.0811150	−	0.996705i	$-0.474152\pi$
$107$	19212.8	0.162230	0.0811150	+	0.996705i	$0.474152\pi$
$108$	0	0
$109$	−23974.3	−0.193277	−0.0966385	−	0.995320i	$-0.530809\pi$
$109$	−23974.3	−0.193277	−0.0966385	+	0.995320i	$0.530809\pi$
$110$	0	0
$111$	157728.	1.21507
$112$	0	0
$113$	45669.7	0.336459	0.168230	−	0.985748i	$-0.446195\pi$
$113$	45669.7	0.336459	0.168230	+	0.985748i	$0.446195\pi$
$114$	0	0
$115$	11075.2	0.0780924
$116$	0	0
$117$	337098.	2.27662
$118$	0	0
$119$	−23347.1	−0.151135
$120$	0	0
$121$	−92332.4	−0.573312
$122$	0	0
$123$	−216633.	−1.29111
$124$	0	0
$125$	−188952.	−1.08162
$126$	0	0
$127$	59510.3	0.327403	0.163701	−	0.986510i	$-0.447657\pi$
$127$	59510.3	0.327403	0.163701	+	0.986510i	$0.447657\pi$
$128$	0	0
$129$	−652884.	−3.45431
$130$	0	0
$131$	359513.	1.83036	0.915179	−	0.403047i	$-0.132049\pi$
$131$	359513.	1.83036	0.915179	+	0.403047i	$0.132049\pi$
$132$	0	0
$133$	35899.3	0.175978
$134$	0	0
$135$	−594565.	−2.80779
$136$	0	0
$137$	−219800.	−1.00052	−0.500260	−	0.865875i	$-0.666762\pi$
$137$	−219800.	−1.00052	−0.500260	+	0.865875i	$0.666762\pi$
$138$	0	0
$139$	392419.	1.72271	0.861356	−	0.508001i	$-0.169616\pi$
$139$	392419.	1.72271	0.861356	+	0.508001i	$0.169616\pi$
$140$	0	0
$141$	421977.	1.78748
$142$	0	0
$143$	126624.	0.517816
$144$	0	0
$145$	−292613.	−1.15578
$146$	0	0
$147$	507889.	1.93854
$148$	0	0
$149$	−206226.	−0.760988	−0.380494	−	0.924783i	$-0.624246\pi$
$149$	−206226.	−0.760988	−0.380494	+	0.924783i	$0.624246\pi$
$150$	0	0
$151$	−115373.	−0.411778	−0.205889	−	0.978575i	$-0.566009\pi$
$151$	−115373.	−0.411778	−0.205889	+	0.978575i	$0.566009\pi$
$152$	0	0
$153$	−1.03219e6	−3.56477
$154$	0	0
$155$	−68627.3	−0.229439
$156$	0	0
$157$	−234516.	−0.759317	−0.379659	−	0.925127i	$-0.623959\pi$
$157$	−234516.	−0.759317	−0.379659	+	0.925127i	$0.623959\pi$
$158$	0	0
$159$	−186962.	−0.586490
$160$	0	0
$161$	4102.63	0.0124738
$162$	0	0
$163$	−30498.4	−0.0899101	−0.0449551	−	0.998989i	$-0.514314\pi$
$163$	−30498.4	−0.0899101	−0.0449551	+	0.998989i	$0.514314\pi$
$164$	0	0
$165$	−342645.	−0.979794
$166$	0	0
$167$	−14304.8	−0.0396910	−0.0198455	−	0.999803i	$-0.506317\pi$
$167$	−14304.8	−0.0396910	−0.0198455	+	0.999803i	$0.506317\pi$
$168$	0	0
$169$	−137971.	−0.371595
$170$	0	0
$171$	1.58713e6	4.15072
$172$	0	0
$173$	30238.8	0.0768155	0.0384077	−	0.999262i	$-0.487771\pi$
$173$	30238.8	0.0768155	0.0384077	+	0.999262i	$0.487771\pi$
$174$	0	0
$175$	−20665.0	−0.0510082
$176$	0	0
$177$	747496.	1.79339
$178$	0	0
$179$	246828.	0.575786	0.287893	−	0.957663i	$-0.407045\pi$
$179$	246828.	0.575786	0.287893	+	0.957663i	$0.407045\pi$
$180$	0	0
$181$	582265.	1.32107	0.660533	−	0.750797i	$-0.270331\pi$
$181$	582265.	1.32107	0.660533	+	0.750797i	$0.270331\pi$
$182$	0	0
$183$	463145.	1.02233
$184$	0	0
$185$	−219121.	−0.470712
$186$	0	0
$187$	−387721.	−0.810804
$188$	0	0
$189$	−220246.	−0.448491
$190$	0	0
$191$	492714.	0.977263	0.488632	−	0.872490i	$-0.337496\pi$
$191$	492714.	0.977263	0.488632	+	0.872490i	$0.337496\pi$
$192$	0	0
$193$	−361032.	−0.697674	−0.348837	−	0.937183i	$-0.613423\pi$
$193$	−361032.	−0.697674	−0.348837	+	0.937183i	$0.613423\pi$
$194$	0	0
$195$	−631373.	−1.18905
$196$	0	0
$197$	−375677.	−0.689683	−0.344841	−	0.938661i	$-0.612067\pi$
$197$	−375677.	−0.689683	−0.344841	+	0.938661i	$0.612067\pi$
$198$	0	0
$199$	−922462.	−1.65126	−0.825631	−	0.564211i	$-0.809181\pi$
$199$	−922462.	−1.65126	−0.825631	+	0.564211i	$0.809181\pi$
$200$	0	0
$201$	1.19921e6	2.09365
$202$	0	0
$203$	−108393.	−0.184613
$204$	0	0
$205$	300955.	0.500168
$206$	0	0
$207$	181380.	0.294214
$208$	0	0
$209$	596174.	0.944077
$210$	0	0
$211$	−571761.	−0.884114	−0.442057	−	0.896987i	$-0.645751\pi$
$211$	−571761.	−0.884114	−0.442057	+	0.896987i	$0.645751\pi$
$212$	0	0
$213$	−1.85314e6	−2.79871
$214$	0	0
$215$	907009.	1.33818
$216$	0	0
$217$	−25421.8	−0.0366485
$218$	0	0
$219$	−282615.	−0.398186
$220$	0	0
$221$	−714432.	−0.983966
$222$	0	0
$223$	822314.	1.10733	0.553663	−	0.832741i	$-0.313230\pi$
$223$	822314.	1.10733	0.553663	+	0.832741i	$0.313230\pi$
$224$	0	0
$225$	−913612.	−1.20311
$226$	0	0
$227$	−1.23922e6	−1.59618	−0.798092	−	0.602535i	$-0.794157\pi$
$227$	−1.23922e6	−1.59618	−0.798092	+	0.602535i	$0.794157\pi$
$228$	0	0
$229$	−1.40911e6	−1.77565	−0.887823	−	0.460185i	$-0.847783\pi$
$229$	−1.40911e6	−1.77565	−0.887823	+	0.460185i	$0.847783\pi$
$230$	0	0
$231$	−126927.	−0.156503
$232$	0	0
$233$	−901826.	−1.08826	−0.544131	−	0.839001i	$-0.683140\pi$
$233$	−901826.	−1.08826	−0.544131	+	0.839001i	$0.683140\pi$
$234$	0	0
$235$	−586226.	−0.692461
$236$	0	0
$237$	1.02219e6	1.18212
$238$	0	0
$239$	−415575.	−0.470603	−0.235302	−	0.971922i	$-0.575608\pi$
$239$	−415575.	−0.470603	−0.235302	+	0.971922i	$0.575608\pi$
$240$	0	0
$241$	−641776.	−0.711772	−0.355886	−	0.934529i	$-0.615821\pi$
$241$	−641776.	−0.711772	−0.355886	+	0.934529i	$0.615821\pi$
$242$	0	0
$243$	−4.53547e6	−4.92727
$244$	0	0
$245$	−705578.	−0.750983
$246$	0	0
$247$	1.09854e6	1.14570
$248$	0	0
$249$	−578853.	−0.591657
$250$	0	0
$251$	1.85317e6	1.85665	0.928327	−	0.371764i	$-0.121247\pi$
$251$	1.85317e6	1.85665	0.928327	+	0.371764i	$0.121247\pi$
$252$	0	0
$253$	68131.6	0.0669187
$254$	0	0
$255$	1.93326e6	1.86183
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−81169.5	−0.0751871
$260$	0	0
$261$	−4.79214e6	−4.35440
$262$	0	0
$263$	−42055.3	−0.0374914	−0.0187457	−	0.999824i	$-0.505967\pi$
$263$	−42055.3	−0.0374914	−0.0187457	+	0.999824i	$0.505967\pi$
$264$	0	0
$265$	259734.	0.227203
$266$	0	0
$267$	−936847.	−0.804249
$268$	0	0
$269$	679225.	0.572312	0.286156	−	0.958183i	$-0.407622\pi$
$269$	679225.	0.572312	0.286156	+	0.958183i	$0.407622\pi$
$270$	0	0
$271$	−678652.	−0.561338	−0.280669	−	0.959805i	$-0.590556\pi$
$271$	−678652.	−0.561338	−0.280669	+	0.959805i	$0.590556\pi$
$272$	0	0
$273$	−233881.	−0.189928
$274$	0	0
$275$	−343180.	−0.273647
$276$	0	0
$277$	1.42189e6	1.11344	0.556719	−	0.830701i	$-0.312060\pi$
$277$	1.42189e6	1.11344	0.556719	+	0.830701i	$0.312060\pi$
$278$	0	0
$279$	−1.12391e6	−0.864415
$280$	0	0
$281$	−1.31465e6	−0.993219	−0.496609	−	0.867974i	$-0.665422\pi$
$281$	−1.31465e6	−0.993219	−0.496609	+	0.867974i	$0.665422\pi$
$282$	0	0
$283$	−544373.	−0.404046	−0.202023	−	0.979381i	$-0.564752\pi$
$283$	−544373.	−0.404046	−0.202023	+	0.979381i	$0.564752\pi$
$284$	0	0
$285$	−2.97265e6	−2.16786
$286$	0	0
$287$	111483.	0.0798923
$288$	0	0
$289$	767730.	0.540710
$290$	0	0
$291$	676359.	0.468214
$292$	0	0
$293$	2.28066e6	1.55200	0.776001	−	0.630732i	$-0.217245\pi$
$293$	2.28066e6	1.55200	0.776001	+	0.630732i	$0.217245\pi$
$294$	0	0
$295$	−1.03845e6	−0.694752
$296$	0	0
$297$	−3.65759e6	−2.40604
$298$	0	0
$299$	125542.	0.0812104
$300$	0	0
$301$	335985.	0.213749
$302$	0	0
$303$	97842.8	0.0612241
$304$	0	0
$305$	−643417.	−0.396044
$306$	0	0
$307$	−1.09630e6	−0.663873	−0.331937	−	0.943302i	$-0.607702\pi$
$307$	−1.09630e6	−0.663873	−0.331937	+	0.943302i	$0.607702\pi$
$308$	0	0
$309$	2.88519e6	1.71901
$310$	0	0
$311$	−2.15637e6	−1.26422	−0.632111	−	0.774878i	$-0.717811\pi$
$311$	−2.15637e6	−1.26422	−0.632111	+	0.774878i	$0.717811\pi$
$312$	0	0
$313$	1.08140e6	0.623913	0.311956	−	0.950096i	$-0.399016\pi$
$313$	1.08140e6	0.623913	0.311956	+	0.950096i	$0.399016\pi$
$314$	0	0
$315$	469428.	0.266559
$316$	0	0
$317$	−1.69327e6	−0.946409	−0.473205	−	0.880953i	$-0.656903\pi$
$317$	−1.69327e6	−0.946409	−0.473205	+	0.880953i	$0.656903\pi$
$318$	0	0
$319$	−1.80007e6	−0.990404
$320$	0	0
$321$	−589327.	−0.319223
$322$	0	0
$323$	−3.36371e6	−1.79396
$324$	0	0
$325$	−632358.	−0.332089
$326$	0	0
$327$	735381.	0.380315
$328$	0	0
$329$	−217157.	−0.110607
$330$	0	0
$331$	540876.	0.271349	0.135674	−	0.990753i	$-0.456680\pi$
$331$	540876.	0.271349	0.135674	+	0.990753i	$0.456680\pi$
$332$	0	0
$333$	−3.58856e6	−1.77341
$334$	0	0
$335$	−1.66598e6	−0.811070
$336$	0	0
$337$	−245388.	−0.117701	−0.0588503	−	0.998267i	$-0.518743\pi$
$337$	−245388.	−0.117701	−0.0588503	+	0.998267i	$0.518743\pi$
$338$	0	0
$339$	−1.40086e6	−0.662057
$340$	0	0
$341$	−422175.	−0.196610
$342$	0	0
$343$	−526671.	−0.241715
$344$	0	0
$345$	−339718.	−0.153664
$346$	0	0
$347$	−3.15873e6	−1.40828	−0.704140	−	0.710061i	$-0.748667\pi$
$347$	−3.15873e6	−1.40828	−0.704140	+	0.710061i	$0.748667\pi$
$348$	0	0
$349$	1.32791e6	0.583585	0.291792	−	0.956482i	$-0.405748\pi$
$349$	1.32791e6	0.583585	0.291792	+	0.956482i	$0.405748\pi$
$350$	0	0
$351$	−6.73962e6	−2.91990
$352$	0	0
$353$	−2.26058e6	−0.965567	−0.482784	−	0.875740i	$-0.660374\pi$
$353$	−2.26058e6	−0.965567	−0.482784	+	0.875740i	$0.660374\pi$
$354$	0	0
$355$	2.57444e6	1.08421
$356$	0	0
$357$	716142.	0.297391
$358$	0	0
$359$	4.70387e6	1.92628	0.963139	−	0.269004i	$-0.0866946\pi$
$359$	4.70387e6	1.92628	0.963139	+	0.269004i	$0.0866946\pi$
$360$	0	0
$361$	2.69606e6	1.08884
$362$	0	0
$363$	2.83217e6	1.12812
$364$	0	0
$365$	392619.	0.154255
$366$	0	0
$367$	−2.02004e6	−0.782878	−0.391439	−	0.920204i	$-0.628023\pi$
$367$	−2.02004e6	−0.782878	−0.391439	+	0.920204i	$0.628023\pi$
$368$	0	0
$369$	4.92875e6	1.88439
$370$	0	0
$371$	96214.0	0.0362914
$372$	0	0
$373$	−3.37180e6	−1.25484	−0.627422	−	0.778680i	$-0.715890\pi$
$373$	−3.37180e6	−1.25484	−0.627422	+	0.778680i	$0.715890\pi$
$374$	0	0
$375$	5.79584e6	2.12833
$376$	0	0
$377$	−3.31688e6	−1.20192
$378$	0	0
$379$	1.49023e6	0.532911	0.266456	−	0.963847i	$-0.414147\pi$
$379$	1.49023e6	0.532911	0.266456	+	0.963847i	$0.414147\pi$
$380$	0	0
$381$	−1.82540e6	−0.644236
$382$	0	0
$383$	776541.	0.270500	0.135250	−	0.990811i	$-0.456816\pi$
$383$	776541.	0.270500	0.135250	+	0.990811i	$0.456816\pi$
$384$	0	0
$385$	176331.	0.0606286
$386$	0	0
$387$	1.48541e7	5.04162
$388$	0	0
$389$	−1.33234e6	−0.446419	−0.223209	−	0.974771i	$-0.571653\pi$
$389$	−1.33234e6	−0.446419	−0.223209	+	0.974771i	$0.571653\pi$
$390$	0	0
$391$	−384410.	−0.127161
$392$	0	0
$393$	−1.10276e7	−3.60163
$394$	0	0
$395$	−1.42006e6	−0.457946
$396$	0	0
$397$	4.81062e6	1.53188	0.765940	−	0.642912i	$-0.222274\pi$
$397$	4.81062e6	1.53188	0.765940	+	0.642912i	$0.222274\pi$
$398$	0	0
$399$	−1.10116e6	−0.346274
$400$	0	0
$401$	2.76622e6	0.859065	0.429533	−	0.903051i	$-0.358678\pi$
$401$	2.76622e6	0.859065	0.429533	+	0.903051i	$0.358678\pi$
$402$	0	0
$403$	−777918.	−0.238600
$404$	0	0
$405$	1.10110e7	3.33573
$406$	0	0
$407$	−1.34797e6	−0.403361
$408$	0	0
$409$	−961950.	−0.284344	−0.142172	−	0.989842i	$-0.545409\pi$
$409$	−961950.	−0.284344	−0.142172	+	0.989842i	$0.545409\pi$
$410$	0	0
$411$	6.74207e6	1.96874
$412$	0	0
$413$	−384675.	−0.110973
$414$	0	0
$415$	804163.	0.229205
$416$	0	0
$417$	−1.20369e7	−3.38981
$418$	0	0
$419$	−6.39194e6	−1.77868	−0.889340	−	0.457247i	$-0.848836\pi$
$419$	−6.39194e6	−1.77868	−0.889340	+	0.457247i	$0.848836\pi$
$420$	0	0
$421$	4.94962e6	1.36103	0.680513	−	0.732736i	$-0.261757\pi$
$421$	4.94962e6	1.36103	0.680513	+	0.732736i	$0.261757\pi$
$422$	0	0
$423$	−9.60065e6	−2.60885
$424$	0	0
$425$	1.93628e6	0.519990
$426$	0	0
$427$	−238342.	−0.0632604
$428$	0	0
$429$	−3.88402e6	−1.01892
$430$	0	0
$431$	6.89399e6	1.78763	0.893815	−	0.448436i	$-0.148019\pi$
$431$	6.89399e6	1.78763	0.893815	+	0.448436i	$0.148019\pi$
$432$	0	0
$433$	768820.	0.197063	0.0985315	−	0.995134i	$-0.468585\pi$
$433$	768820.	0.197063	0.0985315	+	0.995134i	$0.468585\pi$
$434$	0	0
$435$	8.97552e6	2.27424
$436$	0	0
$437$	591082.	0.148062
$438$	0	0
$439$	6.61957e6	1.63934	0.819669	−	0.572837i	$-0.194157\pi$
$439$	6.61957e6	1.63934	0.819669	+	0.572837i	$0.194157\pi$
$440$	0	0
$441$	−1.15553e7	−2.82933
$442$	0	0
$443$	−3.31212e6	−0.801856	−0.400928	−	0.916110i	$-0.631312\pi$
$443$	−3.31212e6	−0.801856	−0.400928	+	0.916110i	$0.631312\pi$
$444$	0	0
$445$	1.30150e6	0.311562
$446$	0	0
$447$	6.32571e6	1.49741
$448$	0	0
$449$	4.55844e6	1.06709	0.533545	−	0.845772i	$-0.320860\pi$
$449$	4.55844e6	1.06709	0.533545	+	0.845772i	$0.320860\pi$
$450$	0	0
$451$	1.85138e6	0.428603
$452$	0	0
$453$	3.53892e6	0.810263
$454$	0	0
$455$	324915.	0.0735770
$456$	0	0
$457$	−7.61644e6	−1.70593	−0.852965	−	0.521968i	$-0.825198\pi$
$457$	−7.61644e6	−1.70593	−0.852965	+	0.521968i	$0.825198\pi$
$458$	0	0
$459$	2.06367e7	4.57202
$460$	0	0
$461$	−7.52566e6	−1.64927	−0.824637	−	0.565663i	$-0.808620\pi$
$461$	−7.52566e6	−1.64927	−0.824637	+	0.565663i	$0.808620\pi$
$462$	0	0
$463$	5.28610e6	1.14600	0.572998	−	0.819557i	$-0.305780\pi$
$463$	5.28610e6	1.14600	0.572998	+	0.819557i	$0.305780\pi$
$464$	0	0
$465$	2.10505e6	0.451472
$466$	0	0
$467$	−6.74152e6	−1.43043	−0.715213	−	0.698907i	$-0.753670\pi$
$467$	−6.74152e6	−1.43043	−0.715213	+	0.698907i	$0.753670\pi$
$468$	0	0
$469$	−617133.	−0.129553
$470$	0	0
$471$	7.19347e6	1.49412
$472$	0	0
$473$	5.57965e6	1.14671
$474$	0	0
$475$	−2.97729e6	−0.605462
$476$	0	0
$477$	4.25368e6	0.855991
$478$	0	0
$479$	2.98864e6	0.595161	0.297580	−	0.954697i	$-0.403820\pi$
$479$	2.98864e6	0.595161	0.297580	+	0.954697i	$0.403820\pi$
$480$	0	0
$481$	−2.48382e6	−0.489506
$482$	0	0
$483$	−125843.	−0.0245448
$484$	0	0
$485$	−939622.	−0.181384
$486$	0	0
$487$	3.54969e6	0.678217	0.339108	−	0.940747i	$-0.389875\pi$
$487$	3.54969e6	0.678217	0.339108	+	0.940747i	$0.389875\pi$
$488$	0	0
$489$	935499.	0.176918
$490$	0	0
$491$	2.45461e6	0.459492	0.229746	−	0.973251i	$-0.426210\pi$
$491$	2.45461e6	0.459492	0.229746	+	0.973251i	$0.426210\pi$
$492$	0	0
$493$	1.01563e7	1.88199
$494$	0	0
$495$	7.79572e6	1.43002
$496$	0	0
$497$	953657.	0.173181
$498$	0	0
$499$	−1.17637e6	−0.211492	−0.105746	−	0.994393i	$-0.533723\pi$
$499$	−1.17637e6	−0.211492	−0.105746	+	0.994393i	$0.533723\pi$
$500$	0	0
$501$	438782.	0.0781006
$502$	0	0
$503$	2.86060e6	0.504123	0.252062	−	0.967711i	$-0.418891\pi$
$503$	2.86060e6	0.504123	0.252062	+	0.967711i	$0.418891\pi$
$504$	0	0
$505$	−135927.	−0.0237179
$506$	0	0
$507$	4.23206e6	0.731194
$508$	0	0
$509$	4.37634e6	0.748716	0.374358	−	0.927284i	$-0.377863\pi$
$509$	4.37634e6	0.748716	0.374358	+	0.927284i	$0.377863\pi$
$510$	0	0
$511$	145439.	0.0246393
$512$	0	0
$513$	−3.17317e7	−5.32354
$514$	0	0
$515$	−4.00821e6	−0.665935
$516$	0	0
$517$	−3.60629e6	−0.593381
$518$	0	0
$519$	−927534.	−0.151151
$520$	0	0
$521$	8.47254e6	1.36747	0.683737	−	0.729728i	$-0.260354\pi$
$521$	8.47254e6	1.36747	0.683737	+	0.729728i	$0.260354\pi$
$522$	0	0
$523$	3.53316e6	0.564818	0.282409	−	0.959294i	$-0.408866\pi$
$523$	3.53316e6	0.564818	0.282409	+	0.959294i	$0.408866\pi$
$524$	0	0
$525$	633871.	0.100370
$526$	0	0
$527$	2.38198e6	0.373604
$528$	0	0
$529$	−6.36879e6	−0.989505
$530$	0	0
$531$	−1.70067e7	−2.61748
$532$	0	0
$533$	3.41144e6	0.520139
$534$	0	0
$535$	818714.	0.123665
$536$	0	0
$537$	−7.57111e6	−1.13298
$538$	0	0
$539$	−4.34051e6	−0.643530
$540$	0	0
$541$	127828.	0.0187774	0.00938868	−	0.999956i	$-0.497011\pi$
$541$	127828.	0.0187774	0.00938868	+	0.999956i	$0.497011\pi$
$542$	0	0
$543$	−1.78602e7	−2.59948
$544$	0	0
$545$	−1.02162e6	−0.147332
$546$	0	0
$547$	−1.24652e7	−1.78128	−0.890641	−	0.454708i	$-0.849744\pi$
$547$	−1.24652e7	−1.78128	−0.890641	+	0.454708i	$0.849744\pi$
$548$	0	0
$549$	−1.05373e7	−1.49210
$550$	0	0
$551$	−1.56167e7	−2.19134
$552$	0	0
$553$	−526036.	−0.0731480
$554$	0	0
$555$	6.72125e6	0.926228
$556$	0	0
$557$	6.01017e6	0.820822	0.410411	−	0.911901i	$-0.365385\pi$
$557$	6.01017e6	0.820822	0.410411	+	0.911901i	$0.365385\pi$
$558$	0	0
$559$	1.02813e7	1.39161
$560$	0	0
$561$	1.18928e7	1.59543
$562$	0	0
$563$	−4.33237e6	−0.576043	−0.288021	−	0.957624i	$-0.592997\pi$
$563$	−4.33237e6	−0.576043	−0.288021	+	0.957624i	$0.592997\pi$
$564$	0	0
$565$	1.94612e6	0.256477
$566$	0	0
$567$	4.07884e6	0.532818
$568$	0	0
$569$	2.52482e6	0.326926	0.163463	−	0.986549i	$-0.447734\pi$
$569$	2.52482e6	0.326926	0.163463	+	0.986549i	$0.447734\pi$
$570$	0	0
$571$	−8.57825e6	−1.10105	−0.550527	−	0.834818i	$-0.685573\pi$
$571$	−8.57825e6	−1.10105	−0.550527	+	0.834818i	$0.685573\pi$
$572$	0	0
$573$	−1.51134e7	−1.92298
$574$	0	0
$575$	−340248.	−0.0429167
$576$	0	0
$577$	−7.80910e6	−0.976475	−0.488238	−	0.872711i	$-0.662360\pi$
$577$	−7.80910e6	−0.976475	−0.488238	+	0.872711i	$0.662360\pi$
$578$	0	0
$579$	1.10742e7	1.37282
$580$	0	0
$581$	297888.	0.0366111
$582$	0	0
$583$	1.59781e6	0.194694
$584$	0	0
$585$	1.43647e7	1.73543
$586$	0	0
$587$	−487623.	−0.0584102	−0.0292051	−	0.999573i	$-0.509298\pi$
$587$	−487623.	−0.0584102	−0.0292051	+	0.999573i	$0.509298\pi$
$588$	0	0
$589$	−3.66262e6	−0.435014
$590$	0	0
$591$	1.15234e7	1.35710
$592$	0	0
$593$	−9.29708e6	−1.08570	−0.542850	−	0.839830i	$-0.682655\pi$
$593$	−9.29708e6	−1.08570	−0.542850	+	0.839830i	$0.682655\pi$
$594$	0	0
$595$	−994889.	−0.115208
$596$	0	0
$597$	2.82953e7	3.24922
$598$	0	0
$599$	−9.41110e6	−1.07170	−0.535850	−	0.844313i	$-0.680009\pi$
$599$	−9.41110e6	−1.07170	−0.535850	+	0.844313i	$0.680009\pi$
$600$	0	0
$601$	−623336.	−0.0703941	−0.0351970	−	0.999380i	$-0.511206\pi$
$601$	−623336.	−0.0703941	−0.0351970	+	0.999380i	$0.511206\pi$
$602$	0	0
$603$	−2.72839e7	−3.05571
$604$	0	0
$605$	−3.93456e6	−0.437026
$606$	0	0
$607$	7.01038e6	0.772272	0.386136	−	0.922442i	$-0.373810\pi$
$607$	7.01038e6	0.772272	0.386136	+	0.922442i	$0.373810\pi$
$608$	0	0
$609$	3.32482e6	0.363266
$610$	0	0
$611$	−6.64510e6	−0.720109
$612$	0	0
$613$	−6.10281e6	−0.655962	−0.327981	−	0.944684i	$-0.606368\pi$
$613$	−6.10281e6	−0.655962	−0.327981	+	0.944684i	$0.606368\pi$
$614$	0	0
$615$	−9.23138e6	−0.984190
$616$	0	0
$617$	2.69527e6	0.285029	0.142515	−	0.989793i	$-0.454481\pi$
$617$	2.69527e6	0.285029	0.142515	+	0.989793i	$0.454481\pi$
$618$	0	0
$619$	−1.67882e7	−1.76108	−0.880538	−	0.473975i	$-0.842819\pi$
$619$	−1.67882e7	−1.76108	−0.880538	+	0.473975i	$0.842819\pi$
$620$	0	0
$621$	−3.62634e6	−0.377346
$622$	0	0
$623$	482118.	0.0497660
$624$	0	0
$625$	−3.96074e6	−0.405580
$626$	0	0
$627$	−1.82869e7	−1.85768
$628$	0	0
$629$	7.60546e6	0.766476
$630$	0	0
$631$	8.12492e6	0.812355	0.406178	−	0.913794i	$-0.366861\pi$
$631$	8.12492e6	0.812355	0.406178	+	0.913794i	$0.366861\pi$
$632$	0	0
$633$	1.75380e7	1.73969
$634$	0	0
$635$	2.53591e6	0.249574
$636$	0	0
$637$	−7.99800e6	−0.780968
$638$	0	0
$639$	4.21618e7	4.08476
$640$	0	0
$641$	−1.77596e6	−0.170721	−0.0853607	−	0.996350i	$-0.527204\pi$
$641$	−1.77596e6	−0.170721	−0.0853607	+	0.996350i	$0.527204\pi$
$642$	0	0
$643$	3.55584e6	0.339168	0.169584	−	0.985516i	$-0.445758\pi$
$643$	3.55584e6	0.339168	0.169584	+	0.985516i	$0.445758\pi$
$644$	0	0
$645$	−2.78213e7	−2.63317
$646$	0	0
$647$	−4.87136e6	−0.457498	−0.228749	−	0.973485i	$-0.573464\pi$
$647$	−4.87136e6	−0.457498	−0.228749	+	0.973485i	$0.573464\pi$
$648$	0	0
$649$	−6.38822e6	−0.595345
$650$	0	0
$651$	779779.	0.0721139
$652$	0	0
$653$	−1.59768e7	−1.46625	−0.733124	−	0.680094i	$-0.761939\pi$
$653$	−1.59768e7	−1.46625	−0.733124	+	0.680094i	$0.761939\pi$
$654$	0	0
$655$	1.53199e7	1.39525
$656$	0	0
$657$	6.42995e6	0.581158
$658$	0	0
$659$	−2.78708e6	−0.249997	−0.124999	−	0.992157i	$-0.539893\pi$
$659$	−2.78708e6	−0.249997	−0.124999	+	0.992157i	$0.539893\pi$
$660$	0	0
$661$	1.99966e6	0.178013	0.0890065	−	0.996031i	$-0.471631\pi$
$661$	1.99966e6	0.178013	0.0890065	+	0.996031i	$0.471631\pi$
$662$	0	0
$663$	2.19143e7	1.93617
$664$	0	0
$665$	1.52978e6	0.134145
$666$	0	0
$667$	−1.78469e6	−0.155328
$668$	0	0
$669$	−2.52234e7	−2.17890
$670$	0	0
$671$	−3.95811e6	−0.339377
$672$	0	0
$673$	9.52984e6	0.811050	0.405525	−	0.914084i	$-0.367089\pi$
$673$	9.52984e6	0.811050	0.405525	+	0.914084i	$0.367089\pi$
$674$	0	0
$675$	1.82659e7	1.54306
$676$	0	0
$677$	4.46624e6	0.374516	0.187258	−	0.982311i	$-0.440040\pi$
$677$	4.46624e6	0.374516	0.187258	+	0.982311i	$0.440040\pi$
$678$	0	0
$679$	−348066.	−0.0289726
$680$	0	0
$681$	3.80114e7	3.14084
$682$	0	0
$683$	1.33394e7	1.09417	0.547086	−	0.837076i	$-0.315737\pi$
$683$	1.33394e7	1.09417	0.547086	+	0.837076i	$0.315737\pi$
$684$	0	0
$685$	−9.36632e6	−0.762680
$686$	0	0
$687$	4.32226e7	3.49397
$688$	0	0
$689$	2.94419e6	0.236275
$690$	0	0
$691$	1.26929e7	1.01126	0.505632	−	0.862749i	$-0.331259\pi$
$691$	1.26929e7	1.01126	0.505632	+	0.862749i	$0.331259\pi$
$692$	0	0
$693$	2.88778e6	0.228419
$694$	0	0
$695$	1.67221e7	1.31320
$696$	0	0
$697$	−1.04458e7	−0.814442
$698$	0	0
$699$	2.76623e7	2.14139
$700$	0	0
$701$	−2.01284e7	−1.54708	−0.773542	−	0.633745i	$-0.781517\pi$
$701$	−2.01284e7	−1.54708	−0.773542	+	0.633745i	$0.781517\pi$
$702$	0	0
$703$	−1.16944e7	−0.892463
$704$	0	0
$705$	1.79817e7	1.36257
$706$	0	0
$707$	−50351.6	−0.00378848
$708$	0	0
$709$	−1.23820e7	−0.925069	−0.462534	−	0.886601i	$-0.653060\pi$
$709$	−1.23820e7	−0.925069	−0.462534	+	0.886601i	$0.653060\pi$
$710$	0	0
$711$	−2.32564e7	−1.72531
$712$	0	0
$713$	−418569.	−0.0308349
$714$	0	0
$715$	5.39582e6	0.394723
$716$	0	0
$717$	1.27472e7	0.926014
$718$	0	0
$719$	−1.82048e6	−0.131330	−0.0656648	−	0.997842i	$-0.520917\pi$
$719$	−1.82048e6	−0.131330	−0.0656648	+	0.997842i	$0.520917\pi$
$720$	0	0
$721$	−1.48477e6	−0.106370
$722$	0	0
$723$	1.96856e7	1.40057
$724$	0	0
$725$	8.98952e6	0.635172
$726$	0	0
$727$	−1.74182e7	−1.22227	−0.611137	−	0.791525i	$-0.709287\pi$
$727$	−1.74182e7	−1.22227	−0.611137	+	0.791525i	$0.709287\pi$
$728$	0	0
$729$	7.63291e7	5.31951
$730$	0	0
$731$	−3.14813e7	−2.17901
$732$	0	0
$733$	−1.81309e7	−1.24641	−0.623204	−	0.782059i	$-0.714169\pi$
$733$	−1.81309e7	−1.24641	−0.623204	+	0.782059i	$0.714169\pi$
$734$	0	0
$735$	2.16427e7	1.47772
$736$	0	0
$737$	−1.02486e7	−0.695019
$738$	0	0
$739$	−2.50945e7	−1.69032	−0.845158	−	0.534517i	$-0.820494\pi$
$739$	−2.50945e7	−1.69032	−0.845158	+	0.534517i	$0.820494\pi$
$740$	0	0
$741$	−3.36961e7	−2.25442
$742$	0	0
$743$	−1.98777e7	−1.32097	−0.660486	−	0.750839i	$-0.729650\pi$
$743$	−1.98777e7	−1.32097	−0.660486	+	0.750839i	$0.729650\pi$
$744$	0	0
$745$	−8.78790e6	−0.580089
$746$	0	0
$747$	1.31698e7	0.863532
$748$	0	0
$749$	303278.	0.0197532
$750$	0	0
$751$	−2.59846e7	−1.68119	−0.840593	−	0.541668i	$-0.817793\pi$
$751$	−2.59846e7	−1.68119	−0.840593	+	0.541668i	$0.817793\pi$
$752$	0	0
$753$	−5.68435e7	−3.65337
$754$	0	0
$755$	−4.91640e6	−0.313892
$756$	0	0
$757$	−2.58790e7	−1.64138	−0.820688	−	0.571376i	$-0.806410\pi$
$757$	−2.58790e7	−1.64138	−0.820688	+	0.571376i	$0.806410\pi$
$758$	0	0
$759$	−2.08985e6	−0.131677
$760$	0	0
$761$	5.87755e6	0.367904	0.183952	−	0.982935i	$-0.441111\pi$
$761$	5.87755e6	0.367904	0.183952	+	0.982935i	$0.441111\pi$
$762$	0	0
$763$	−378440.	−0.0235335
$764$	0	0
$765$	−4.39847e7	−2.71737
$766$	0	0
$767$	−1.17712e7	−0.722492
$768$	0	0
$769$	1.25320e7	0.764197	0.382099	−	0.924122i	$-0.375201\pi$
$769$	1.25320e7	0.764197	0.382099	+	0.924122i	$0.375201\pi$
$770$	0	0
$771$	−2.02597e6	−0.122743
$772$	0	0
$773$	−5.91699e6	−0.356166	−0.178083	−	0.984016i	$-0.556989\pi$
$773$	−5.91699e6	−0.356166	−0.178083	+	0.984016i	$0.556989\pi$
$774$	0	0
$775$	2.10834e6	0.126091
$776$	0	0
$777$	2.48977e6	0.147947
$778$	0	0
$779$	1.60618e7	0.948313
$780$	0	0
$781$	1.58372e7	0.929076
$782$	0	0
$783$	9.58097e7	5.58477
$784$	0	0
$785$	−9.99342e6	−0.578815
$786$	0	0
$787$	−2.77791e7	−1.59875	−0.799375	−	0.600832i	$-0.794836\pi$
$787$	−2.77791e7	−1.59875	−0.799375	+	0.600832i	$0.794836\pi$
$788$	0	0
$789$	1.28999e6	0.0737725
$790$	0	0
$791$	720907.	0.0409673
$792$	0	0
$793$	−7.29339e6	−0.411857
$794$	0	0
$795$	−7.96701e6	−0.447072
$796$	0	0
$797$	−1.89874e7	−1.05882	−0.529408	−	0.848367i	$-0.677586\pi$
$797$	−1.89874e7	−1.05882	−0.529408	+	0.848367i	$0.677586\pi$
$798$	0	0
$799$	2.03473e7	1.12756
$800$	0	0
$801$	2.13147e7	1.17381
$802$	0	0
$803$	2.41528e6	0.132184
$804$	0	0
$805$	174825.	0.00950854
$806$	0	0
$807$	−2.08343e7	−1.12615
$808$	0	0
$809$	3.25907e6	0.175074	0.0875372	−	0.996161i	$-0.472100\pi$
$809$	3.25907e6	0.175074	0.0875372	+	0.996161i	$0.472100\pi$
$810$	0	0
$811$	1.90762e7	1.01845	0.509225	−	0.860633i	$-0.329932\pi$
$811$	1.90762e7	1.01845	0.509225	+	0.860633i	$0.329932\pi$
$812$	0	0
$813$	2.08168e7	1.10455
$814$	0	0
$815$	−1.29963e6	−0.0685370
$816$	0	0
$817$	4.84068e7	2.53718
$818$	0	0
$819$	5.32116e6	0.277202
$820$	0	0
$821$	2.74012e7	1.41877	0.709383	−	0.704823i	$-0.248974\pi$
$821$	2.74012e7	1.41877	0.709383	+	0.704823i	$0.248974\pi$
$822$	0	0
$823$	−3.03826e7	−1.56360	−0.781799	−	0.623530i	$-0.785698\pi$
$823$	−3.03826e7	−1.56360	−0.781799	+	0.623530i	$0.785698\pi$
$824$	0	0
$825$	1.05266e7	0.538459
$826$	0	0
$827$	−3.69665e7	−1.87951	−0.939754	−	0.341852i	$-0.888946\pi$
$827$	−3.69665e7	−1.87951	−0.939754	+	0.341852i	$0.888946\pi$
$828$	0	0
$829$	−3.24802e7	−1.64147	−0.820735	−	0.571309i	$-0.806436\pi$
$829$	−3.24802e7	−1.64147	−0.820735	+	0.571309i	$0.806436\pi$
$830$	0	0
$831$	−4.36146e7	−2.19093
$832$	0	0
$833$	2.44898e7	1.22285
$834$	0	0
$835$	−609571.	−0.0302558
$836$	0	0
$837$	2.24705e7	1.10866
$838$	0	0
$839$	2.38680e7	1.17061	0.585304	−	0.810814i	$-0.300975\pi$
$839$	2.38680e7	1.17061	0.585304	+	0.810814i	$0.300975\pi$
$840$	0	0
$841$	2.66413e7	1.29887
$842$	0	0
$843$	4.03252e7	1.95437
$844$	0	0
$845$	−5.87933e6	−0.283261
$846$	0	0
$847$	−1.45749e6	−0.0698065
$848$	0	0
$849$	1.66979e7	0.795048
$850$	0	0
$851$	−1.33645e6	−0.0632602
$852$	0	0
$853$	2.11690e7	0.996157	0.498079	−	0.867132i	$-0.334039\pi$
$853$	2.11690e7	0.996157	0.498079	+	0.867132i	$0.334039\pi$
$854$	0	0
$855$	6.76325e7	3.16402
$856$	0	0
$857$	3.18917e7	1.48329	0.741644	−	0.670793i	$-0.234046\pi$
$857$	3.18917e7	1.48329	0.741644	+	0.670793i	$0.234046\pi$
$858$	0	0
$859$	1.78730e7	0.826447	0.413223	−	0.910630i	$-0.364403\pi$
$859$	1.78730e7	0.826447	0.413223	+	0.910630i	$0.364403\pi$
$860$	0	0
$861$	−3.41960e6	−0.157205
$862$	0	0
$863$	1.45652e7	0.665718	0.332859	−	0.942977i	$-0.391987\pi$
$863$	1.45652e7	0.665718	0.332859	+	0.942977i	$0.391987\pi$
$864$	0	0
$865$	1.28856e6	0.0585552
$866$	0	0
$867$	−2.35491e7	−1.06396
$868$	0	0
$869$	−8.73579e6	−0.392421
$870$	0	0
$871$	−1.88846e7	−0.843454
$872$	0	0
$873$	−1.53882e7	−0.683366
$874$	0	0
$875$	−2.98264e6	−0.131699
$876$	0	0
$877$	−3.51755e7	−1.54433	−0.772167	−	0.635419i	$-0.780827\pi$
$877$	−3.51755e7	−1.54433	−0.772167	+	0.635419i	$0.780827\pi$
$878$	0	0
$879$	−6.99563e7	−3.05390
$880$	0	0
$881$	−1.23867e7	−0.537670	−0.268835	−	0.963186i	$-0.586639\pi$
$881$	−1.23867e7	−0.537670	−0.268835	+	0.963186i	$0.586639\pi$
$882$	0	0
$883$	−3.02182e7	−1.30427	−0.652134	−	0.758104i	$-0.726126\pi$
$883$	−3.02182e7	−1.30427	−0.652134	+	0.758104i	$0.726126\pi$
$884$	0	0
$885$	3.18530e7	1.36708
$886$	0	0
$887$	−5.98801e6	−0.255549	−0.127774	−	0.991803i	$-0.540783\pi$
$887$	−5.98801e6	−0.255549	−0.127774	+	0.991803i	$0.540783\pi$
$888$	0	0
$889$	939382.	0.0398646
$890$	0	0
$891$	6.77366e7	2.85844
$892$	0	0
$893$	−3.12867e7	−1.31290
$894$	0	0
$895$	1.05181e7	0.438912
$896$	0	0
$897$	−3.85084e6	−0.159799
$898$	0	0
$899$	1.10588e7	0.456361
$900$	0	0
$901$	−9.01510e6	−0.369963
$902$	0	0
$903$	−1.03059e7	−0.420598
$904$	0	0
$905$	2.48120e7	1.00703
$906$	0	0
$907$	2.80541e7	1.13234	0.566172	−	0.824287i	$-0.308424\pi$
$907$	2.80541e7	1.13234	0.566172	+	0.824287i	$0.308424\pi$
$908$	0	0
$909$	−2.22608e6	−0.0893575
$910$	0	0
$911$	3.79856e7	1.51643	0.758216	−	0.652003i	$-0.226071\pi$
$911$	3.79856e7	1.51643	0.758216	+	0.652003i	$0.226071\pi$
$912$	0	0
$913$	4.94697e6	0.196410
$914$	0	0
$915$	1.97360e7	0.779302
$916$	0	0
$917$	5.67499e6	0.222865
$918$	0	0
$919$	1.40414e7	0.548430	0.274215	−	0.961668i	$-0.411582\pi$
$919$	1.40414e7	0.548430	0.274215	+	0.961668i	$0.411582\pi$
$920$	0	0
$921$	3.36277e7	1.30632
$922$	0	0
$923$	2.91823e7	1.12750
$924$	0	0
$925$	6.73174e6	0.258686
$926$	0	0
$927$	−6.56426e7	−2.50892
$928$	0	0
$929$	−9.52540e6	−0.362113	−0.181057	−	0.983473i	$-0.557952\pi$
$929$	−9.52540e6	−0.362113	−0.181057	+	0.983473i	$0.557952\pi$
$930$	0	0
$931$	−3.76565e7	−1.42385
$932$	0	0
$933$	6.61439e7	2.48763
$934$	0	0
$935$	−1.65220e7	−0.618063
$936$	0	0
$937$	−1.16334e7	−0.432871	−0.216435	−	0.976297i	$-0.569443\pi$
$937$	−1.16334e7	−0.432871	−0.216435	+	0.976297i	$0.569443\pi$
$938$	0	0
$939$	−3.31704e7	−1.22768
$940$	0	0
$941$	−4.06317e7	−1.49586	−0.747930	−	0.663777i	$-0.768952\pi$
$941$	−4.06317e7	−1.49586	−0.747930	+	0.663777i	$0.768952\pi$
$942$	0	0
$943$	1.83557e6	0.0672189
$944$	0	0
$945$	−9.38533e6	−0.341877
$946$	0	0
$947$	−2.15696e7	−0.781569	−0.390785	−	0.920482i	$-0.627796\pi$
$947$	−2.15696e7	−0.781569	−0.390785	+	0.920482i	$0.627796\pi$
$948$	0	0
$949$	4.45050e6	0.160414
$950$	0	0
$951$	5.19389e7	1.86227
$952$	0	0
$953$	3.24618e7	1.15782	0.578909	−	0.815392i	$-0.303479\pi$
$953$	3.24618e7	1.15782	0.578909	+	0.815392i	$0.303479\pi$
$954$	0	0
$955$	2.09960e7	0.744952
$956$	0	0
$957$	5.52147e7	1.94884
$958$	0	0
$959$	−3.46959e6	−0.121823
$960$	0	0
$961$	−2.60355e7	−0.909405
$962$	0	0
$963$	1.34081e7	0.465910
$964$	0	0
$965$	−1.53846e7	−0.531825
$966$	0	0
$967$	3.23769e6	0.111345	0.0556723	−	0.998449i	$-0.482270\pi$
$967$	3.23769e6	0.111345	0.0556723	+	0.998449i	$0.482270\pi$
$968$	0	0
$969$	1.03177e8	3.53000
$970$	0	0
$971$	−3.96040e7	−1.34800	−0.674002	−	0.738729i	$-0.735426\pi$
$971$	−3.96040e7	−1.34800	−0.674002	+	0.738729i	$0.735426\pi$
$972$	0	0
$973$	6.19441e6	0.209758
$974$	0	0
$975$	1.93967e7	0.653457
$976$	0	0
$977$	4.08812e7	1.37021	0.685106	−	0.728443i	$-0.259756\pi$
$977$	4.08812e7	1.37021	0.685106	+	0.728443i	$0.259756\pi$
$978$	0	0
$979$	8.00645e6	0.266983
$980$	0	0
$981$	−1.67311e7	−0.555075
$982$	0	0
$983$	−3.98100e7	−1.31404	−0.657020	−	0.753873i	$-0.728184\pi$
$983$	−3.98100e7	−1.31404	−0.657020	+	0.753873i	$0.728184\pi$
$984$	0	0
$985$	−1.60087e7	−0.525734
$986$	0	0
$987$	6.66100e6	0.217644
$988$	0	0
$989$	5.53199e6	0.179842
$990$	0	0
$991$	−4.88139e7	−1.57892	−0.789458	−	0.613805i	$-0.789638\pi$
$991$	−4.88139e7	−1.57892	−0.789458	+	0.613805i	$0.789638\pi$
$992$	0	0
$993$	−1.65907e7	−0.533938
$994$	0	0
$995$	−3.93088e7	−1.25873
$996$	0	0
$997$	2.44813e7	0.780003	0.390002	−	0.920814i	$-0.372474\pi$
$997$	2.44813e7	0.780003	0.390002	+	0.920814i	$0.372474\pi$
$998$	0	0
$999$	7.17464e7	2.27450

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-30.6737 q^{3} +42.6130 q^{5} +15.7852 q^{7} +697.875 q^{9} +O(q^{10})\)	\(q-30.6737 q^{3} +42.6130 q^{5} +15.7852 q^{7} +697.875 q^{9} +262.142 q^{11} +483.035 q^{13} -1307.10 q^{15} -1479.05 q^{17} +2274.24 q^{19} -484.190 q^{21} +259.903 q^{23} -1309.14 q^{25} -13952.7 q^{27} -6866.76 q^{29} -1610.48 q^{31} -8040.87 q^{33} +672.655 q^{35} -5142.12 q^{37} -14816.4 q^{39} +7062.51 q^{41} +21284.8 q^{43} +29738.5 q^{45} -13757.0 q^{47} -16557.8 q^{49} +45367.9 q^{51} +6095.20 q^{53} +11170.7 q^{55} -69759.3 q^{57} -24369.3 q^{59} -15099.1 q^{61} +11016.1 q^{63} +20583.5 q^{65} -39095.7 q^{67} -7972.19 q^{69} +60414.6 q^{71} +9213.62 q^{73} +40156.0 q^{75} +4137.97 q^{77} -33324.6 q^{79} +258396. q^{81} +18871.3 q^{83} -63026.7 q^{85} +210629. q^{87} +30542.4 q^{89} +7624.80 q^{91} +49399.3 q^{93} +96912.1 q^{95} -22050.1 q^{97} +182942. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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