LMFDB - Embedded newform 1028.6.a.a.1.47

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			1.47
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)

$f(q)$	$=$	$q+26.8275 q^{3} -66.6449 q^{5} +24.5548 q^{7} +476.714 q^{9} +O(q^{10})$	$q+26.8275 q^{3} -66.6449 q^{5} +24.5548 q^{7} +476.714 q^{9} -60.9349 q^{11} -585.220 q^{13} -1787.91 q^{15} +895.842 q^{17} -217.962 q^{19} +658.743 q^{21} -666.232 q^{23} +1316.54 q^{25} +6269.97 q^{27} -1747.42 q^{29} +8693.16 q^{31} -1634.73 q^{33} -1636.45 q^{35} +10356.5 q^{37} -15700.0 q^{39} -6902.23 q^{41} -13667.7 q^{43} -31770.6 q^{45} -26100.7 q^{47} -16204.1 q^{49} +24033.2 q^{51} +6634.60 q^{53} +4061.00 q^{55} -5847.37 q^{57} -1226.26 q^{59} -35244.5 q^{61} +11705.6 q^{63} +39001.9 q^{65} -4870.89 q^{67} -17873.3 q^{69} +78649.5 q^{71} -32988.7 q^{73} +35319.4 q^{75} -1496.24 q^{77} +12918.1 q^{79} +52366.0 q^{81} -42959.8 q^{83} -59703.3 q^{85} -46878.8 q^{87} -76172.0 q^{89} -14369.9 q^{91} +233216. q^{93} +14526.0 q^{95} -146220. q^{97} -29048.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	26.8275	1.72098	0.860492	−	0.509464i	$-0.170156\pi$
$3$	26.8275	1.72098	0.860492	+	0.509464i	$0.170156\pi$
$4$	0	0
$5$	−66.6449	−1.19218	−0.596090	−	0.802918i	$-0.703280\pi$
$5$	−66.6449	−1.19218	−0.596090	+	0.802918i	$0.703280\pi$
$6$	0	0
$7$	24.5548	0.189405	0.0947023	−	0.995506i	$-0.469810\pi$
$7$	24.5548	0.189405	0.0947023	+	0.995506i	$0.469810\pi$
$8$	0	0
$9$	476.714	1.96179
$10$	0	0
$11$	−60.9349	−0.151839	−0.0759197	−	0.997114i	$-0.524189\pi$
$11$	−60.9349	−0.151839	−0.0759197	+	0.997114i	$0.524189\pi$
$12$	0	0
$13$	−585.220	−0.960419	−0.480210	−	0.877154i	$-0.659439\pi$
$13$	−585.220	−0.960419	−0.480210	+	0.877154i	$0.659439\pi$
$14$	0	0
$15$	−1787.91	−2.05172
$16$	0	0
$17$	895.842	0.751812	0.375906	−	0.926658i	$-0.377332\pi$
$17$	895.842	0.751812	0.375906	+	0.926658i	$0.377332\pi$
$18$	0	0
$19$	−217.962	−0.138515	−0.0692575	−	0.997599i	$-0.522063\pi$
$19$	−217.962	−0.138515	−0.0692575	+	0.997599i	$0.522063\pi$
$20$	0	0
$21$	658.743	0.325963
$22$	0	0
$23$	−666.232	−0.262607	−0.131303	−	0.991342i	$-0.541916\pi$
$23$	−666.232	−0.262607	−0.131303	+	0.991342i	$0.541916\pi$
$24$	0	0
$25$	1316.54	0.421292
$26$	0	0
$27$	6269.97	1.65522
$28$	0	0
$29$	−1747.42	−0.385835	−0.192917	−	0.981215i	$-0.561795\pi$
$29$	−1747.42	−0.385835	−0.192917	+	0.981215i	$0.561795\pi$
$30$	0	0
$31$	8693.16	1.62470	0.812350	−	0.583170i	$-0.198188\pi$
$31$	8693.16	1.62470	0.812350	+	0.583170i	$0.198188\pi$
$32$	0	0
$33$	−1634.73	−0.261313
$34$	0	0
$35$	−1636.45	−0.225804
$36$	0	0
$37$	10356.5	1.24367	0.621837	−	0.783147i	$-0.286387\pi$
$37$	10356.5	1.24367	0.621837	+	0.783147i	$0.286387\pi$
$38$	0	0
$39$	−15700.0	−1.65287
$40$	0	0
$41$	−6902.23	−0.641254	−0.320627	−	0.947206i	$-0.603894\pi$
$41$	−6902.23	−0.641254	−0.320627	+	0.947206i	$0.603894\pi$
$42$	0	0
$43$	−13667.7	−1.12726	−0.563628	−	0.826028i	$-0.690595\pi$
$43$	−13667.7	−1.12726	−0.563628	+	0.826028i	$0.690595\pi$
$44$	0	0
$45$	−31770.6	−2.33880
$46$	0	0
$47$	−26100.7	−1.72349	−0.861744	−	0.507343i	$-0.830628\pi$
$47$	−26100.7	−1.72349	−0.861744	+	0.507343i	$0.830628\pi$
$48$	0	0
$49$	−16204.1	−0.964126
$50$	0	0
$51$	24033.2	1.29386
$52$	0	0
$53$	6634.60	0.324433	0.162216	−	0.986755i	$-0.448136\pi$
$53$	6634.60	0.324433	0.162216	+	0.986755i	$0.448136\pi$
$54$	0	0
$55$	4061.00	0.181020
$56$	0	0
$57$	−5847.37	−0.238382
$58$	0	0
$59$	−1226.26	−0.0458619	−0.0229309	−	0.999737i	$-0.507300\pi$
$59$	−1226.26	−0.0458619	−0.0229309	+	0.999737i	$0.507300\pi$
$60$	0	0
$61$	−35244.5	−1.21274	−0.606369	−	0.795183i	$-0.707375\pi$
$61$	−35244.5	−1.21274	−0.606369	+	0.795183i	$0.707375\pi$
$62$	0	0
$63$	11705.6	0.371572
$64$	0	0
$65$	39001.9	1.14499
$66$	0	0
$67$	−4870.89	−0.132563	−0.0662814	−	0.997801i	$-0.521114\pi$
$67$	−4870.89	−0.132563	−0.0662814	+	0.997801i	$0.521114\pi$
$68$	0	0
$69$	−17873.3	−0.451942
$70$	0	0
$71$	78649.5	1.85161	0.925807	−	0.377997i	$-0.123387\pi$
$71$	78649.5	1.85161	0.925807	+	0.377997i	$0.123387\pi$
$72$	0	0
$73$	−32988.7	−0.724533	−0.362267	−	0.932075i	$-0.617997\pi$
$73$	−32988.7	−0.724533	−0.362267	+	0.932075i	$0.617997\pi$
$74$	0	0
$75$	35319.4	0.725036
$76$	0	0
$77$	−1496.24	−0.0287591
$78$	0	0
$79$	12918.1	0.232880	0.116440	−	0.993198i	$-0.462852\pi$
$79$	12918.1	0.232880	0.116440	+	0.993198i	$0.462852\pi$
$80$	0	0
$81$	52366.0	0.886823
$82$	0	0
$83$	−42959.8	−0.684490	−0.342245	−	0.939611i	$-0.611187\pi$
$83$	−42959.8	−0.684490	−0.342245	+	0.939611i	$0.611187\pi$
$84$	0	0
$85$	−59703.3	−0.896295
$86$	0	0
$87$	−46878.8	−0.664016
$88$	0	0
$89$	−76172.0	−1.01934	−0.509672	−	0.860369i	$-0.670233\pi$
$89$	−76172.0	−1.01934	−0.509672	+	0.860369i	$0.670233\pi$
$90$	0	0
$91$	−14369.9	−0.181908
$92$	0	0
$93$	233216.	2.79608
$94$	0	0
$95$	14526.0	0.165135
$96$	0	0
$97$	−146220.	−1.57789	−0.788946	−	0.614462i	$-0.789373\pi$
$97$	−146220.	−1.57789	−0.788946	+	0.614462i	$0.789373\pi$
$98$	0	0
$99$	−29048.6	−0.297877
$100$	0	0
$101$	58837.7	0.573921	0.286961	−	0.957942i	$-0.407355\pi$
$101$	58837.7	0.573921	0.286961	+	0.957942i	$0.407355\pi$
$102$	0	0
$103$	−124171.	−1.15326	−0.576629	−	0.817006i	$-0.695632\pi$
$103$	−124171.	−1.15326	−0.576629	+	0.817006i	$0.695632\pi$
$104$	0	0
$105$	−43901.8	−0.388606
$106$	0	0
$107$	45318.6	0.382664	0.191332	−	0.981525i	$-0.438719\pi$
$107$	45318.6	0.382664	0.191332	+	0.981525i	$0.438719\pi$
$108$	0	0
$109$	40393.6	0.325646	0.162823	−	0.986655i	$-0.447940\pi$
$109$	40393.6	0.325646	0.162823	+	0.986655i	$0.447940\pi$
$110$	0	0
$111$	277838.	2.14034
$112$	0	0
$113$	−152175.	−1.12111	−0.560555	−	0.828117i	$-0.689412\pi$
$113$	−152175.	−1.12111	−0.560555	+	0.828117i	$0.689412\pi$
$114$	0	0
$115$	44400.9	0.313074
$116$	0	0
$117$	−278983.	−1.88414
$118$	0	0
$119$	21997.2	0.142397
$120$	0	0
$121$	−157338.	−0.976945
$122$	0	0
$123$	−185170.	−1.10359
$124$	0	0
$125$	120525.	0.689924
$126$	0	0
$127$	226565.	1.24648	0.623239	−	0.782032i	$-0.285816\pi$
$127$	226565.	1.24648	0.623239	+	0.782032i	$0.285816\pi$
$128$	0	0
$129$	−366669.	−1.93999
$130$	0	0
$131$	146655.	0.746651	0.373326	−	0.927700i	$-0.378217\pi$
$131$	146655.	0.746651	0.373326	+	0.927700i	$0.378217\pi$
$132$	0	0
$133$	−5352.00	−0.0262354
$134$	0	0
$135$	−417861.	−1.97332
$136$	0	0
$137$	−331050.	−1.50693	−0.753464	−	0.657489i	$-0.771619\pi$
$137$	−331050.	−1.50693	−0.753464	+	0.657489i	$0.771619\pi$
$138$	0	0
$139$	−398122.	−1.74775	−0.873874	−	0.486152i	$-0.838400\pi$
$139$	−398122.	−1.74775	−0.873874	+	0.486152i	$0.838400\pi$
$140$	0	0
$141$	−700218.	−2.96610
$142$	0	0
$143$	35660.3	0.145829
$144$	0	0
$145$	116456.	0.459985
$146$	0	0
$147$	−434714.	−1.65925
$148$	0	0
$149$	−279311.	−1.03068	−0.515339	−	0.856987i	$-0.672334\pi$
$149$	−279311.	−1.03068	−0.515339	+	0.856987i	$0.672334\pi$
$150$	0	0
$151$	−335569.	−1.19768	−0.598839	−	0.800870i	$-0.704371\pi$
$151$	−335569.	−1.19768	−0.598839	+	0.800870i	$0.704371\pi$
$152$	0	0
$153$	427061.	1.47490
$154$	0	0
$155$	−579354.	−1.93693
$156$	0	0
$157$	189696.	0.614199	0.307099	−	0.951677i	$-0.400642\pi$
$157$	189696.	0.614199	0.307099	+	0.951677i	$0.400642\pi$
$158$	0	0
$159$	177990.	0.558344
$160$	0	0
$161$	−16359.2	−0.0497389
$162$	0	0
$163$	−254595.	−0.750552	−0.375276	−	0.926913i	$-0.622452\pi$
$163$	−254595.	−0.750552	−0.375276	+	0.926913i	$0.622452\pi$
$164$	0	0
$165$	108946.	0.311532
$166$	0	0
$167$	−296063.	−0.821473	−0.410736	−	0.911754i	$-0.634728\pi$
$167$	−296063.	−0.821473	−0.410736	+	0.911754i	$0.634728\pi$
$168$	0	0
$169$	−28810.4	−0.0775949
$170$	0	0
$171$	−103906.	−0.271737
$172$	0	0
$173$	179981.	0.457206	0.228603	−	0.973520i	$-0.426584\pi$
$173$	179981.	0.457206	0.228603	+	0.973520i	$0.426584\pi$
$174$	0	0
$175$	32327.2	0.0797946
$176$	0	0
$177$	−32897.4	−0.0789275
$178$	0	0
$179$	−11939.6	−0.0278521	−0.0139261	−	0.999903i	$-0.504433\pi$
$179$	−11939.6	−0.0278521	−0.0139261	+	0.999903i	$0.504433\pi$
$180$	0	0
$181$	843749.	1.91433	0.957165	−	0.289544i	$-0.0935035\pi$
$181$	843749.	1.91433	0.957165	+	0.289544i	$0.0935035\pi$
$182$	0	0
$183$	−945522.	−2.08710
$184$	0	0
$185$	−690204.	−1.48268
$186$	0	0
$187$	−54588.1	−0.114155
$188$	0	0
$189$	153958.	0.313507
$190$	0	0
$191$	−62863.1	−0.124684	−0.0623422	−	0.998055i	$-0.519857\pi$
$191$	−62863.1	−0.124684	−0.0623422	+	0.998055i	$0.519857\pi$
$192$	0	0
$193$	500990.	0.968134	0.484067	−	0.875031i	$-0.339159\pi$
$193$	500990.	0.968134	0.484067	+	0.875031i	$0.339159\pi$
$194$	0	0
$195$	1.04632e6	1.97051
$196$	0	0
$197$	−678234.	−1.24513	−0.622564	−	0.782569i	$-0.713909\pi$
$197$	−678234.	−1.24513	−0.622564	+	0.782569i	$0.713909\pi$
$198$	0	0
$199$	−626432.	−1.12135	−0.560675	−	0.828036i	$-0.689458\pi$
$199$	−626432.	−1.12135	−0.560675	+	0.828036i	$0.689458\pi$
$200$	0	0
$201$	−130674.	−0.228139
$202$	0	0
$203$	−42907.4	−0.0730790
$204$	0	0
$205$	459998.	0.764490
$206$	0	0
$207$	−317602.	−0.515179
$208$	0	0
$209$	13281.5	0.0210320
$210$	0	0
$211$	−162060.	−0.250594	−0.125297	−	0.992119i	$-0.539988\pi$
$211$	−162060.	−0.250594	−0.125297	+	0.992119i	$0.539988\pi$
$212$	0	0
$213$	2.10997e6	3.18660
$214$	0	0
$215$	910879.	1.34389
$216$	0	0
$217$	213458.	0.307726
$218$	0	0
$219$	−885005.	−1.24691
$220$	0	0
$221$	−524265.	−0.722055
$222$	0	0
$223$	323366.	0.435445	0.217722	−	0.976011i	$-0.430137\pi$
$223$	323366.	0.435445	0.217722	+	0.976011i	$0.430137\pi$
$224$	0	0
$225$	627612.	0.826485
$226$	0	0
$227$	69603.7	0.0896536	0.0448268	−	0.998995i	$-0.485726\pi$
$227$	69603.7	0.0896536	0.0448268	+	0.998995i	$0.485726\pi$
$228$	0	0
$229$	962934.	1.21341	0.606705	−	0.794927i	$-0.292491\pi$
$229$	962934.	1.21341	0.606705	+	0.794927i	$0.292491\pi$
$230$	0	0
$231$	−40140.4	−0.0494940
$232$	0	0
$233$	1.48908e6	1.79692	0.898459	−	0.439058i	$-0.144688\pi$
$233$	1.48908e6	1.79692	0.898459	+	0.439058i	$0.144688\pi$
$234$	0	0
$235$	1.73948e6	2.05471
$236$	0	0
$237$	346561.	0.400783
$238$	0	0
$239$	−551336.	−0.624341	−0.312170	−	0.950026i	$-0.601056\pi$
$239$	−551336.	−0.624341	−0.312170	+	0.950026i	$0.601056\pi$
$240$	0	0
$241$	385204.	0.427216	0.213608	−	0.976919i	$-0.431478\pi$
$241$	385204.	0.427216	0.213608	+	0.976919i	$0.431478\pi$
$242$	0	0
$243$	−118754.	−0.129013
$244$	0	0
$245$	1.07992e6	1.14941
$246$	0	0
$247$	127556.	0.133032
$248$	0	0
$249$	−1.15250e6	−1.17800
$250$	0	0
$251$	−645047.	−0.646260	−0.323130	−	0.946355i	$-0.604735\pi$
$251$	−645047.	−0.646260	−0.323130	+	0.946355i	$0.604735\pi$
$252$	0	0
$253$	40596.8	0.0398741
$254$	0	0
$255$	−1.60169e6	−1.54251
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	254300.	0.235558
$260$	0	0
$261$	−833019.	−0.756926
$262$	0	0
$263$	−104432.	−0.0930991	−0.0465496	−	0.998916i	$-0.514823\pi$
$263$	−104432.	−0.0930991	−0.0465496	+	0.998916i	$0.514823\pi$
$264$	0	0
$265$	−442162.	−0.386782
$266$	0	0
$267$	−2.04350e6	−1.75427
$268$	0	0
$269$	1.22408e6	1.03141	0.515704	−	0.856767i	$-0.327531\pi$
$269$	1.22408e6	1.03141	0.515704	+	0.856767i	$0.327531\pi$
$270$	0	0
$271$	564393.	0.466829	0.233415	−	0.972377i	$-0.425010\pi$
$271$	564393.	0.466829	0.233415	+	0.972377i	$0.425010\pi$
$272$	0	0
$273$	−385510.	−0.313061
$274$	0	0
$275$	−80223.0	−0.0639687
$276$	0	0
$277$	−1.15745e6	−0.906363	−0.453182	−	0.891418i	$-0.649711\pi$
$277$	−1.15745e6	−0.906363	−0.453182	+	0.891418i	$0.649711\pi$
$278$	0	0
$279$	4.14415e6	3.18732
$280$	0	0
$281$	−133882.	−0.101148	−0.0505738	−	0.998720i	$-0.516105\pi$
$281$	−133882.	−0.101148	−0.0505738	+	0.998720i	$0.516105\pi$
$282$	0	0
$283$	−1.83132e6	−1.35925	−0.679624	−	0.733560i	$-0.737857\pi$
$283$	−1.83132e6	−1.35925	−0.679624	+	0.733560i	$0.737857\pi$
$284$	0	0
$285$	389697.	0.284194
$286$	0	0
$287$	−169483.	−0.121456
$288$	0	0
$289$	−617323.	−0.434779
$290$	0	0
$291$	−3.92272e6	−2.71553
$292$	0	0
$293$	373302.	0.254033	0.127017	−	0.991901i	$-0.459460\pi$
$293$	373302.	0.254033	0.127017	+	0.991901i	$0.459460\pi$
$294$	0	0
$295$	81723.7	0.0546756
$296$	0	0
$297$	−382060.	−0.251328
$298$	0	0
$299$	389892.	0.252213
$300$	0	0
$301$	−335606.	−0.213508
$302$	0	0
$303$	1.57847e6	0.987709
$304$	0	0
$305$	2.34887e6	1.44580
$306$	0	0
$307$	−981334.	−0.594253	−0.297126	−	0.954838i	$-0.596028\pi$
$307$	−981334.	−0.594253	−0.297126	+	0.954838i	$0.596028\pi$
$308$	0	0
$309$	−3.33119e6	−1.98474
$310$	0	0
$311$	1.84121e6	1.07945	0.539726	−	0.841841i	$-0.318528\pi$
$311$	1.84121e6	1.07945	0.539726	+	0.841841i	$0.318528\pi$
$312$	0	0
$313$	1.08218e6	0.624362	0.312181	−	0.950023i	$-0.398940\pi$
$313$	1.08218e6	0.624362	0.312181	+	0.950023i	$0.398940\pi$
$314$	0	0
$315$	−780119.	−0.442980
$316$	0	0
$317$	−1.86279e6	−1.04116	−0.520579	−	0.853814i	$-0.674284\pi$
$317$	−1.86279e6	−1.04116	−0.520579	+	0.853814i	$0.674284\pi$
$318$	0	0
$319$	106479.	0.0585850
$320$	0	0
$321$	1.21578e6	0.658558
$322$	0	0
$323$	−195260.	−0.104137
$324$	0	0
$325$	−770463.	−0.404617
$326$	0	0
$327$	1.08366e6	0.560432
$328$	0	0
$329$	−640898.	−0.326437
$330$	0	0
$331$	2.82726e6	1.41839	0.709196	−	0.705011i	$-0.249058\pi$
$331$	2.82726e6	1.41839	0.709196	+	0.705011i	$0.249058\pi$
$332$	0	0
$333$	4.93707e6	2.43983
$334$	0	0
$335$	324620.	0.158039
$336$	0	0
$337$	−1.48676e6	−0.713124	−0.356562	−	0.934272i	$-0.616051\pi$
$337$	−1.48676e6	−0.713124	−0.356562	+	0.934272i	$0.616051\pi$
$338$	0	0
$339$	−4.08248e6	−1.92941
$340$	0	0
$341$	−529717.	−0.246694
$342$	0	0
$343$	−810579.	−0.372015
$344$	0	0
$345$	1.19117e6	0.538796
$346$	0	0
$347$	−1.15723e6	−0.515936	−0.257968	−	0.966153i	$-0.583053\pi$
$347$	−1.15723e6	−0.515936	−0.257968	+	0.966153i	$0.583053\pi$
$348$	0	0
$349$	−257962.	−0.113369	−0.0566843	−	0.998392i	$-0.518053\pi$
$349$	−257962.	−0.113369	−0.0566843	+	0.998392i	$0.518053\pi$
$350$	0	0
$351$	−3.66931e6	−1.58971
$352$	0	0
$353$	−1.96816e6	−0.840666	−0.420333	−	0.907370i	$-0.638087\pi$
$353$	−1.96816e6	−0.840666	−0.420333	+	0.907370i	$0.638087\pi$
$354$	0	0
$355$	−5.24159e6	−2.20746
$356$	0	0
$357$	590130.	0.245063
$358$	0	0
$359$	241186.	0.0987680	0.0493840	−	0.998780i	$-0.484274\pi$
$359$	241186.	0.0987680	0.0493840	+	0.998780i	$0.484274\pi$
$360$	0	0
$361$	−2.42859e6	−0.980814
$362$	0	0
$363$	−4.22098e6	−1.68131
$364$	0	0
$365$	2.19853e6	0.863774
$366$	0	0
$367$	−2.50767e6	−0.971863	−0.485932	−	0.873997i	$-0.661520\pi$
$367$	−2.50767e6	−0.971863	−0.485932	+	0.873997i	$0.661520\pi$
$368$	0	0
$369$	−3.29039e6	−1.25800
$370$	0	0
$371$	162911.	0.0614491
$372$	0	0
$373$	−396816.	−0.147678	−0.0738391	−	0.997270i	$-0.523525\pi$
$373$	−396816.	−0.147678	−0.0738391	+	0.997270i	$0.523525\pi$
$374$	0	0
$375$	3.23338e6	1.18735
$376$	0	0
$377$	1.02262e6	0.370563
$378$	0	0
$379$	−2.04076e6	−0.729784	−0.364892	−	0.931050i	$-0.618894\pi$
$379$	−2.04076e6	−0.729784	−0.364892	+	0.931050i	$0.618894\pi$
$380$	0	0
$381$	6.07818e6	2.14517
$382$	0	0
$383$	619305.	0.215729	0.107864	−	0.994166i	$-0.465599\pi$
$383$	619305.	0.215729	0.107864	+	0.994166i	$0.465599\pi$
$384$	0	0
$385$	99716.9	0.0342860
$386$	0	0
$387$	−6.51557e6	−2.21144
$388$	0	0
$389$	2.74281e6	0.919013	0.459507	−	0.888174i	$-0.348026\pi$
$389$	2.74281e6	0.919013	0.459507	+	0.888174i	$0.348026\pi$
$390$	0	0
$391$	−596839.	−0.197431
$392$	0	0
$393$	3.93438e6	1.28498
$394$	0	0
$395$	−860928.	−0.277635
$396$	0	0
$397$	2.12037e6	0.675205	0.337602	−	0.941289i	$-0.390384\pi$
$397$	2.12037e6	0.675205	0.337602	+	0.941289i	$0.390384\pi$
$398$	0	0
$399$	−143581.	−0.0451507
$400$	0	0
$401$	−718946.	−0.223272	−0.111636	−	0.993749i	$-0.535609\pi$
$401$	−718946.	−0.223272	−0.111636	+	0.993749i	$0.535609\pi$
$402$	0	0
$403$	−5.08741e6	−1.56039
$404$	0	0
$405$	−3.48993e6	−1.05725
$406$	0	0
$407$	−631070.	−0.188839
$408$	0	0
$409$	4.36855e6	1.29131	0.645653	−	0.763631i	$-0.276585\pi$
$409$	4.36855e6	1.29131	0.645653	+	0.763631i	$0.276585\pi$
$410$	0	0
$411$	−8.88125e6	−2.59340
$412$	0	0
$413$	−30110.5	−0.00868645
$414$	0	0
$415$	2.86305e6	0.816035
$416$	0	0
$417$	−1.06806e7	−3.00785
$418$	0	0
$419$	1.16257e6	0.323506	0.161753	−	0.986831i	$-0.448285\pi$
$419$	1.16257e6	0.323506	0.161753	+	0.986831i	$0.448285\pi$
$420$	0	0
$421$	−3.89072e6	−1.06985	−0.534927	−	0.844898i	$-0.679661\pi$
$421$	−3.89072e6	−1.06985	−0.534927	+	0.844898i	$0.679661\pi$
$422$	0	0
$423$	−1.24426e7	−3.38112
$424$	0	0
$425$	1.17941e6	0.316732
$426$	0	0
$427$	−865421.	−0.229698
$428$	0	0
$429$	956678.	0.250970
$430$	0	0
$431$	−2.36190e6	−0.612447	−0.306223	−	0.951960i	$-0.599065\pi$
$431$	−2.36190e6	−0.612447	−0.306223	+	0.951960i	$0.599065\pi$
$432$	0	0
$433$	2.85006e6	0.730523	0.365261	−	0.930905i	$-0.380980\pi$
$433$	2.85006e6	0.730523	0.365261	+	0.930905i	$0.380980\pi$
$434$	0	0
$435$	3.12423e6	0.791626
$436$	0	0
$437$	145213.	0.0363750
$438$	0	0
$439$	−3.56512e6	−0.882903	−0.441451	−	0.897285i	$-0.645536\pi$
$439$	−3.56512e6	−0.882903	−0.441451	+	0.897285i	$0.645536\pi$
$440$	0	0
$441$	−7.72471e6	−1.89141
$442$	0	0
$443$	4.08334e6	0.988567	0.494283	−	0.869301i	$-0.335431\pi$
$443$	4.08334e6	0.988567	0.494283	+	0.869301i	$0.335431\pi$
$444$	0	0
$445$	5.07647e6	1.21524
$446$	0	0
$447$	−7.49322e6	−1.77378
$448$	0	0
$449$	6.32515e6	1.48066	0.740329	−	0.672245i	$-0.234670\pi$
$449$	6.32515e6	1.48066	0.740329	+	0.672245i	$0.234670\pi$
$450$	0	0
$451$	420587.	0.0973676
$452$	0	0
$453$	−9.00248e6	−2.06118
$454$	0	0
$455$	957683.	0.216867
$456$	0	0
$457$	7.24377e6	1.62246	0.811230	−	0.584727i	$-0.198799\pi$
$457$	7.24377e6	1.62246	0.811230	+	0.584727i	$0.198799\pi$
$458$	0	0
$459$	5.61691e6	1.24442
$460$	0	0
$461$	4.10982e6	0.900680	0.450340	−	0.892857i	$-0.351303\pi$
$461$	4.10982e6	0.900680	0.450340	+	0.892857i	$0.351303\pi$
$462$	0	0
$463$	−1.25288e6	−0.271618	−0.135809	−	0.990735i	$-0.543363\pi$
$463$	−1.25288e6	−0.271618	−0.135809	+	0.990735i	$0.543363\pi$
$464$	0	0
$465$	−1.55426e7	−3.33343
$466$	0	0
$467$	−2.78067e6	−0.590006	−0.295003	−	0.955496i	$-0.595321\pi$
$467$	−2.78067e6	−0.590006	−0.295003	+	0.955496i	$0.595321\pi$
$468$	0	0
$469$	−119604.	−0.0251080
$470$	0	0
$471$	5.08907e6	1.05703
$472$	0	0
$473$	832837.	0.171162
$474$	0	0
$475$	−286955.	−0.0583552
$476$	0	0
$477$	3.16281e6	0.636469
$478$	0	0
$479$	4.50901e6	0.897929	0.448964	−	0.893550i	$-0.351793\pi$
$479$	4.50901e6	0.897929	0.448964	+	0.893550i	$0.351793\pi$
$480$	0	0
$481$	−6.06080e6	−1.19445
$482$	0	0
$483$	−438875.	−0.0855999
$484$	0	0
$485$	9.74481e6	1.88113
$486$	0	0
$487$	4.77832e6	0.912963	0.456481	−	0.889733i	$-0.349110\pi$
$487$	4.77832e6	0.912963	0.456481	+	0.889733i	$0.349110\pi$
$488$	0	0
$489$	−6.83014e6	−1.29169
$490$	0	0
$491$	−9.07169e6	−1.69818	−0.849092	−	0.528245i	$-0.822850\pi$
$491$	−9.07169e6	−1.69818	−0.849092	+	0.528245i	$0.822850\pi$
$492$	0	0
$493$	−1.56541e6	−0.290075
$494$	0	0
$495$	1.93594e6	0.355122
$496$	0	0
$497$	1.93122e6	0.350704
$498$	0	0
$499$	−4.88442e6	−0.878136	−0.439068	−	0.898454i	$-0.644691\pi$
$499$	−4.88442e6	−0.878136	−0.439068	+	0.898454i	$0.644691\pi$
$500$	0	0
$501$	−7.94263e6	−1.41374
$502$	0	0
$503$	−2.90489e6	−0.511929	−0.255965	−	0.966686i	$-0.582393\pi$
$503$	−2.90489e6	−0.511929	−0.255965	+	0.966686i	$0.582393\pi$
$504$	0	0
$505$	−3.92123e6	−0.684217
$506$	0	0
$507$	−772912.	−0.133540
$508$	0	0
$509$	1.82619e6	0.312430	0.156215	−	0.987723i	$-0.450071\pi$
$509$	1.82619e6	0.312430	0.156215	+	0.987723i	$0.450071\pi$
$510$	0	0
$511$	−810030.	−0.137230
$512$	0	0
$513$	−1.36662e6	−0.229273
$514$	0	0
$515$	8.27534e6	1.37489
$516$	0	0
$517$	1.59045e6	0.261693
$518$	0	0
$519$	4.82844e6	0.786844
$520$	0	0
$521$	5.98142e6	0.965405	0.482703	−	0.875784i	$-0.339655\pi$
$521$	5.98142e6	0.965405	0.482703	+	0.875784i	$0.339655\pi$
$522$	0	0
$523$	233014.	0.0372501	0.0186251	−	0.999827i	$-0.494071\pi$
$523$	233014.	0.0372501	0.0186251	+	0.999827i	$0.494071\pi$
$524$	0	0
$525$	867259.	0.137325
$526$	0	0
$527$	7.78770e6	1.22147
$528$	0	0
$529$	−5.99248e6	−0.931038
$530$	0	0
$531$	−584575.	−0.0899712
$532$	0	0
$533$	4.03933e6	0.615873
$534$	0	0
$535$	−3.02025e6	−0.456204
$536$	0	0
$537$	−320310.	−0.0479330
$538$	0	0
$539$	987393.	0.146392
$540$	0	0
$541$	−5.55533e6	−0.816050	−0.408025	−	0.912971i	$-0.633782\pi$
$541$	−5.55533e6	−0.816050	−0.408025	+	0.912971i	$0.633782\pi$
$542$	0	0
$543$	2.26357e7	3.29453
$544$	0	0
$545$	−2.69202e6	−0.388228
$546$	0	0
$547$	−1.09699e7	−1.56759	−0.783797	−	0.621017i	$-0.786720\pi$
$547$	−1.09699e7	−1.56759	−0.783797	+	0.621017i	$0.786720\pi$
$548$	0	0
$549$	−1.68016e7	−2.37914
$550$	0	0
$551$	380870.	0.0534439
$552$	0	0
$553$	317202.	0.0441086
$554$	0	0
$555$	−1.85164e7	−2.55167
$556$	0	0
$557$	−4.42526e6	−0.604367	−0.302183	−	0.953250i	$-0.597715\pi$
$557$	−4.42526e6	−0.604367	−0.302183	+	0.953250i	$0.597715\pi$
$558$	0	0
$559$	7.99859e6	1.08264
$560$	0	0
$561$	−1.46446e6	−0.196458
$562$	0	0
$563$	−3.97270e6	−0.528219	−0.264110	−	0.964493i	$-0.585078\pi$
$563$	−3.97270e6	−0.528219	−0.264110	+	0.964493i	$0.585078\pi$
$564$	0	0
$565$	1.01417e7	1.33656
$566$	0	0
$567$	1.28584e6	0.167968
$568$	0	0
$569$	4.72013e6	0.611186	0.305593	−	0.952162i	$-0.401145\pi$
$569$	4.72013e6	0.611186	0.305593	+	0.952162i	$0.401145\pi$
$570$	0	0
$571$	9.01496e6	1.15711	0.578554	−	0.815644i	$-0.303617\pi$
$571$	9.01496e6	1.15711	0.578554	+	0.815644i	$0.303617\pi$
$572$	0	0
$573$	−1.68646e6	−0.214580
$574$	0	0
$575$	−877118.	−0.110634
$576$	0	0
$577$	1.42949e7	1.78749	0.893743	−	0.448579i	$-0.148070\pi$
$577$	1.42949e7	1.78749	0.893743	+	0.448579i	$0.148070\pi$
$578$	0	0
$579$	1.34403e7	1.66614
$580$	0	0
$581$	−1.05487e6	−0.129646
$582$	0	0
$583$	−404279.	−0.0492617
$584$	0	0
$585$	1.85928e7	2.24623
$586$	0	0
$587$	7.93717e6	0.950759	0.475379	−	0.879781i	$-0.342311\pi$
$587$	7.93717e6	0.950759	0.475379	+	0.879781i	$0.342311\pi$
$588$	0	0
$589$	−1.89478e6	−0.225045
$590$	0	0
$591$	−1.81953e7	−2.14285
$592$	0	0
$593$	−1.26924e7	−1.48220	−0.741098	−	0.671396i	$-0.765695\pi$
$593$	−1.26924e7	−1.48220	−0.741098	+	0.671396i	$0.765695\pi$
$594$	0	0
$595$	−1.46600e6	−0.169762
$596$	0	0
$597$	−1.68056e7	−1.92983
$598$	0	0
$599$	9.51050e6	1.08302	0.541510	−	0.840695i	$-0.317853\pi$
$599$	9.51050e6	1.08302	0.541510	+	0.840695i	$0.317853\pi$
$600$	0	0
$601$	1.22760e7	1.38635	0.693173	−	0.720771i	$-0.256212\pi$
$601$	1.22760e7	1.38635	0.693173	+	0.720771i	$0.256212\pi$
$602$	0	0
$603$	−2.32203e6	−0.260060
$604$	0	0
$605$	1.04858e7	1.16469
$606$	0	0
$607$	3.98168e6	0.438626	0.219313	−	0.975655i	$-0.429618\pi$
$607$	3.98168e6	0.438626	0.219313	+	0.975655i	$0.429618\pi$
$608$	0	0
$609$	−1.15110e6	−0.125768
$610$	0	0
$611$	1.52747e7	1.65527
$612$	0	0
$613$	−1.44806e7	−1.55645	−0.778227	−	0.627983i	$-0.783881\pi$
$613$	−1.44806e7	−1.55645	−0.778227	+	0.627983i	$0.783881\pi$
$614$	0	0
$615$	1.23406e7	1.31567
$616$	0	0
$617$	1.24870e7	1.32052	0.660262	−	0.751036i	$-0.270445\pi$
$617$	1.24870e7	1.32052	0.660262	+	0.751036i	$0.270445\pi$
$618$	0	0
$619$	1.13368e6	0.118923	0.0594615	−	0.998231i	$-0.481062\pi$
$619$	1.13368e6	0.118923	0.0594615	+	0.998231i	$0.481062\pi$
$620$	0	0
$621$	−4.17726e6	−0.434672
$622$	0	0
$623$	−1.87039e6	−0.193068
$624$	0	0
$625$	−1.21465e7	−1.24380
$626$	0	0
$627$	356309.	0.0361958
$628$	0	0
$629$	9.27775e6	0.935009
$630$	0	0
$631$	8.16076e6	0.815938	0.407969	−	0.912996i	$-0.366237\pi$
$631$	8.16076e6	0.815938	0.407969	+	0.912996i	$0.366237\pi$
$632$	0	0
$633$	−4.34768e6	−0.431269
$634$	0	0
$635$	−1.50994e7	−1.48602
$636$	0	0
$637$	9.48294e6	0.925965
$638$	0	0
$639$	3.74934e7	3.63247
$640$	0	0
$641$	3.52701e6	0.339048	0.169524	−	0.985526i	$-0.445777\pi$
$641$	3.52701e6	0.339048	0.169524	+	0.985526i	$0.445777\pi$
$642$	0	0
$643$	2.98439e6	0.284661	0.142331	−	0.989819i	$-0.454540\pi$
$643$	2.98439e6	0.284661	0.142331	+	0.989819i	$0.454540\pi$
$644$	0	0
$645$	2.44366e7	2.31282
$646$	0	0
$647$	−7.85001e6	−0.737241	−0.368620	−	0.929580i	$-0.620170\pi$
$647$	−7.85001e6	−0.737241	−0.368620	+	0.929580i	$0.620170\pi$
$648$	0	0
$649$	74721.9	0.00696364
$650$	0	0
$651$	5.72655e6	0.529591
$652$	0	0
$653$	−4.07100e6	−0.373609	−0.186805	−	0.982397i	$-0.559813\pi$
$653$	−4.07100e6	−0.373609	−0.186805	+	0.982397i	$0.559813\pi$
$654$	0	0
$655$	−9.77378e6	−0.890142
$656$	0	0
$657$	−1.57262e7	−1.42138
$658$	0	0
$659$	2.71987e6	0.243969	0.121985	−	0.992532i	$-0.461074\pi$
$659$	2.71987e6	0.243969	0.121985	+	0.992532i	$0.461074\pi$
$660$	0	0
$661$	9.69355e6	0.862937	0.431468	−	0.902128i	$-0.357996\pi$
$661$	9.69355e6	0.862937	0.431468	+	0.902128i	$0.357996\pi$
$662$	0	0
$663$	−1.40647e7	−1.24265
$664$	0	0
$665$	356684.	0.0312773
$666$	0	0
$667$	1.16419e6	0.101323
$668$	0	0
$669$	8.67511e6	0.749393
$670$	0	0
$671$	2.14762e6	0.184141
$672$	0	0
$673$	1.92130e7	1.63515	0.817577	−	0.575819i	$-0.195317\pi$
$673$	1.92130e7	1.63515	0.817577	+	0.575819i	$0.195317\pi$
$674$	0	0
$675$	8.25464e6	0.697331
$676$	0	0
$677$	−1.27727e7	−1.07105	−0.535525	−	0.844519i	$-0.679886\pi$
$677$	−1.27727e7	−1.07105	−0.535525	+	0.844519i	$0.679886\pi$
$678$	0	0
$679$	−3.59040e6	−0.298860
$680$	0	0
$681$	1.86729e6	0.154292
$682$	0	0
$683$	−5.83608e6	−0.478707	−0.239353	−	0.970933i	$-0.576935\pi$
$683$	−5.83608e6	−0.478707	−0.239353	+	0.970933i	$0.576935\pi$
$684$	0	0
$685$	2.20628e7	1.79653
$686$	0	0
$687$	2.58331e7	2.08826
$688$	0	0
$689$	−3.88270e6	−0.311592
$690$	0	0
$691$	−2.08052e6	−0.165759	−0.0828794	−	0.996560i	$-0.526412\pi$
$691$	−2.08052e6	−0.165759	−0.0828794	+	0.996560i	$0.526412\pi$
$692$	0	0
$693$	−713280.	−0.0564192
$694$	0	0
$695$	2.65328e7	2.08363
$696$	0	0
$697$	−6.18331e6	−0.482102
$698$	0	0
$699$	3.99483e7	3.09247
$700$	0	0
$701$	−1.55051e7	−1.19174	−0.595868	−	0.803083i	$-0.703192\pi$
$701$	−1.55051e7	−1.19174	−0.595868	+	0.803083i	$0.703192\pi$
$702$	0	0
$703$	−2.25731e6	−0.172267
$704$	0	0
$705$	4.66659e7	3.53612
$706$	0	0
$707$	1.44475e6	0.108703
$708$	0	0
$709$	−5.69110e6	−0.425187	−0.212594	−	0.977141i	$-0.568191\pi$
$709$	−5.69110e6	−0.425187	−0.212594	+	0.977141i	$0.568191\pi$
$710$	0	0
$711$	6.15826e6	0.456861
$712$	0	0
$713$	−5.79166e6	−0.426657
$714$	0	0
$715$	−2.37658e6	−0.173855
$716$	0	0
$717$	−1.47910e7	−1.07448
$718$	0	0
$719$	2.55130e7	1.84052	0.920258	−	0.391312i	$-0.127979\pi$
$719$	2.55130e7	1.84052	0.920258	+	0.391312i	$0.127979\pi$
$720$	0	0
$721$	−3.04898e6	−0.218432
$722$	0	0
$723$	1.03341e7	0.735233
$724$	0	0
$725$	−2.30054e6	−0.162549
$726$	0	0
$727$	1.31500e7	0.922763	0.461382	−	0.887202i	$-0.347354\pi$
$727$	1.31500e7	0.922763	0.461382	+	0.887202i	$0.347354\pi$
$728$	0	0
$729$	−1.59108e7	−1.10885
$730$	0	0
$731$	−1.22441e7	−0.847485
$732$	0	0
$733$	−2.77426e7	−1.90716	−0.953581	−	0.301138i	$-0.902634\pi$
$733$	−2.77426e7	−1.90716	−0.953581	+	0.301138i	$0.902634\pi$
$734$	0	0
$735$	2.89715e7	1.97812
$736$	0	0
$737$	296808.	0.0201283
$738$	0	0
$739$	−1.29244e7	−0.870558	−0.435279	−	0.900296i	$-0.643350\pi$
$739$	−1.29244e7	−0.870558	−0.435279	+	0.900296i	$0.643350\pi$
$740$	0	0
$741$	3.42200e6	0.228947
$742$	0	0
$743$	−3.76248e6	−0.250036	−0.125018	−	0.992155i	$-0.539899\pi$
$743$	−3.76248e6	−0.250036	−0.125018	+	0.992155i	$0.539899\pi$
$744$	0	0
$745$	1.86147e7	1.22875
$746$	0	0
$747$	−2.04796e7	−1.34282
$748$	0	0
$749$	1.11279e6	0.0724783
$750$	0	0
$751$	5.17869e6	0.335058	0.167529	−	0.985867i	$-0.446421\pi$
$751$	5.17869e6	0.335058	0.167529	+	0.985867i	$0.446421\pi$
$752$	0	0
$753$	−1.73050e7	−1.11220
$754$	0	0
$755$	2.23640e7	1.42785
$756$	0	0
$757$	−2.31723e6	−0.146970	−0.0734852	−	0.997296i	$-0.523412\pi$
$757$	−2.31723e6	−0.146970	−0.0734852	+	0.997296i	$0.523412\pi$
$758$	0	0
$759$	1.08911e6	0.0686226
$760$	0	0
$761$	2.46389e7	1.54227	0.771133	−	0.636675i	$-0.219691\pi$
$761$	2.46389e7	1.54227	0.771133	+	0.636675i	$0.219691\pi$
$762$	0	0
$763$	991854.	0.0616789
$764$	0	0
$765$	−2.84614e7	−1.75834
$766$	0	0
$767$	717630.	0.0440466
$768$	0	0
$769$	3.79150e6	0.231204	0.115602	−	0.993296i	$-0.463120\pi$
$769$	3.79150e6	0.231204	0.115602	+	0.993296i	$0.463120\pi$
$770$	0	0
$771$	1.77193e6	0.107352
$772$	0	0
$773$	308276.	0.0185563	0.00927814	−	0.999957i	$-0.497047\pi$
$773$	308276.	0.0185563	0.00927814	+	0.999957i	$0.497047\pi$
$774$	0	0
$775$	1.14449e7	0.684472
$776$	0	0
$777$	6.82224e6	0.405391
$778$	0	0
$779$	1.50442e6	0.0888232
$780$	0	0
$781$	−4.79250e6	−0.281148
$782$	0	0
$783$	−1.09563e7	−0.638642
$784$	0	0
$785$	−1.26423e7	−0.732235
$786$	0	0
$787$	−5.58241e6	−0.321281	−0.160640	−	0.987013i	$-0.551356\pi$
$787$	−5.58241e6	−0.321281	−0.160640	+	0.987013i	$0.551356\pi$
$788$	0	0
$789$	−2.80166e6	−0.160222
$790$	0	0
$791$	−3.73663e6	−0.212343
$792$	0	0
$793$	2.06258e7	1.16474
$794$	0	0
$795$	−1.18621e7	−0.665646
$796$	0	0
$797$	−193127.	−0.0107696	−0.00538478	−	0.999986i	$-0.501714\pi$
$797$	−193127.	−0.0107696	−0.00538478	+	0.999986i	$0.501714\pi$
$798$	0	0
$799$	−2.33822e7	−1.29574
$800$	0	0
$801$	−3.63123e7	−1.99974
$802$	0	0
$803$	2.01017e6	0.110013
$804$	0	0
$805$	1.09025e6	0.0592977
$806$	0	0
$807$	3.28391e7	1.77504
$808$	0	0
$809$	9.30505e6	0.499859	0.249929	−	0.968264i	$-0.419593\pi$
$809$	9.30505e6	0.499859	0.249929	+	0.968264i	$0.419593\pi$
$810$	0	0
$811$	5.54852e6	0.296227	0.148114	−	0.988970i	$-0.452680\pi$
$811$	5.54852e6	0.296227	0.148114	+	0.988970i	$0.452680\pi$
$812$	0	0
$813$	1.51412e7	0.803406
$814$	0	0
$815$	1.69674e7	0.894792
$816$	0	0
$817$	2.97903e6	0.156142
$818$	0	0
$819$	−6.85036e6	−0.356865
$820$	0	0
$821$	−1.19497e7	−0.618729	−0.309364	−	0.950944i	$-0.600116\pi$
$821$	−1.19497e7	−0.618729	−0.309364	+	0.950944i	$0.600116\pi$
$822$	0	0
$823$	−2.67335e7	−1.37580	−0.687901	−	0.725805i	$-0.741468\pi$
$823$	−2.67335e7	−1.37580	−0.687901	+	0.725805i	$0.741468\pi$
$824$	0	0
$825$	−2.15218e6	−0.110089
$826$	0	0
$827$	1.66991e7	0.849043	0.424522	−	0.905418i	$-0.360442\pi$
$827$	1.66991e7	0.849043	0.424522	+	0.905418i	$0.360442\pi$
$828$	0	0
$829$	−3.69524e7	−1.86748	−0.933741	−	0.357950i	$-0.883476\pi$
$829$	−3.69524e7	−1.86748	−0.933741	+	0.357950i	$0.883476\pi$
$830$	0	0
$831$	−3.10514e7	−1.55984
$832$	0	0
$833$	−1.45163e7	−0.724841
$834$	0	0
$835$	1.97311e7	0.979343
$836$	0	0
$837$	5.45058e7	2.68924
$838$	0	0
$839$	−3.35617e6	−0.164603	−0.0823017	−	0.996607i	$-0.526227\pi$
$839$	−3.35617e6	−0.164603	−0.0823017	+	0.996607i	$0.526227\pi$
$840$	0	0
$841$	−1.74577e7	−0.851131
$842$	0	0
$843$	−3.59171e6	−0.174073
$844$	0	0
$845$	1.92007e6	0.0925070
$846$	0	0
$847$	−3.86340e6	−0.185038
$848$	0	0
$849$	−4.91298e7	−2.33925
$850$	0	0
$851$	−6.89980e6	−0.326597
$852$	0	0
$853$	−1.01039e7	−0.475464	−0.237732	−	0.971331i	$-0.576404\pi$
$853$	−1.01039e7	−0.475464	−0.237732	+	0.971331i	$0.576404\pi$
$854$	0	0
$855$	6.92477e6	0.323959
$856$	0	0
$857$	2.35268e7	1.09423	0.547117	−	0.837056i	$-0.315725\pi$
$857$	2.35268e7	1.09423	0.547117	+	0.837056i	$0.315725\pi$
$858$	0	0
$859$	−9.27442e6	−0.428848	−0.214424	−	0.976741i	$-0.568787\pi$
$859$	−9.27442e6	−0.428848	−0.214424	+	0.976741i	$0.568787\pi$
$860$	0	0
$861$	−4.54680e6	−0.209025
$862$	0	0
$863$	1.84066e7	0.841292	0.420646	−	0.907225i	$-0.361803\pi$
$863$	1.84066e7	0.841292	0.420646	+	0.907225i	$0.361803\pi$
$864$	0	0
$865$	−1.19948e7	−0.545071
$866$	0	0
$867$	−1.65612e7	−0.748247
$868$	0	0
$869$	−787166.	−0.0353604
$870$	0	0
$871$	2.85054e6	0.127316
$872$	0	0
$873$	−6.97052e7	−3.09549
$874$	0	0
$875$	2.95946e6	0.130675
$876$	0	0
$877$	1.24762e7	0.547751	0.273875	−	0.961765i	$-0.411694\pi$
$877$	1.24762e7	0.547751	0.273875	+	0.961765i	$0.411694\pi$
$878$	0	0
$879$	1.00147e7	0.437188
$880$	0	0
$881$	9.75556e6	0.423460	0.211730	−	0.977328i	$-0.432090\pi$
$881$	9.75556e6	0.423460	0.211730	+	0.977328i	$0.432090\pi$
$882$	0	0
$883$	3.32291e7	1.43422	0.717111	−	0.696959i	$-0.245464\pi$
$883$	3.32291e7	1.43422	0.717111	+	0.696959i	$0.245464\pi$
$884$	0	0
$885$	2.19244e6	0.0940958
$886$	0	0
$887$	1.70229e7	0.726480	0.363240	−	0.931696i	$-0.381671\pi$
$887$	1.70229e7	0.726480	0.363240	+	0.931696i	$0.381671\pi$
$888$	0	0
$889$	5.56326e6	0.236089
$890$	0	0
$891$	−3.19092e6	−0.134655
$892$	0	0
$893$	5.68897e6	0.238729
$894$	0	0
$895$	795714.	0.0332047
$896$	0	0
$897$	1.04598e7	0.434054
$898$	0	0
$899$	−1.51906e7	−0.626866
$900$	0	0
$901$	5.94356e6	0.243913
$902$	0	0
$903$	−9.00347e6	−0.367443
$904$	0	0
$905$	−5.62315e7	−2.28222
$906$	0	0
$907$	−4.46500e7	−1.80220	−0.901101	−	0.433609i	$-0.857240\pi$
$907$	−4.46500e7	−1.80220	−0.901101	+	0.433609i	$0.857240\pi$
$908$	0	0
$909$	2.80488e7	1.12591
$910$	0	0
$911$	1.09561e7	0.437380	0.218690	−	0.975794i	$-0.429822\pi$
$911$	1.09561e7	0.437380	0.218690	+	0.975794i	$0.429822\pi$
$912$	0	0
$913$	2.61775e6	0.103933
$914$	0	0
$915$	6.30142e7	2.48820
$916$	0	0
$917$	3.60107e6	0.141419
$918$	0	0
$919$	4.30744e7	1.68241	0.841203	−	0.540719i	$-0.181848\pi$
$919$	4.30744e7	1.68241	0.841203	+	0.540719i	$0.181848\pi$
$920$	0	0
$921$	−2.63267e7	−1.02270
$922$	0	0
$923$	−4.60273e7	−1.77832
$924$	0	0
$925$	1.36346e7	0.523950
$926$	0	0
$927$	−5.91940e7	−2.26245
$928$	0	0
$929$	2.81585e7	1.07046	0.535229	−	0.844707i	$-0.320225\pi$
$929$	2.81585e7	1.07046	0.535229	+	0.844707i	$0.320225\pi$
$930$	0	0
$931$	3.53187e6	0.133546
$932$	0	0
$933$	4.93952e7	1.85772
$934$	0	0
$935$	3.63801e6	0.136093
$936$	0	0
$937$	−2.04232e7	−0.759930	−0.379965	−	0.925001i	$-0.624064\pi$
$937$	−2.04232e7	−0.759930	−0.379965	+	0.925001i	$0.624064\pi$
$938$	0	0
$939$	2.90320e7	1.07452
$940$	0	0
$941$	4.19830e7	1.54561	0.772805	−	0.634644i	$-0.218853\pi$
$941$	4.19830e7	1.54561	0.772805	+	0.634644i	$0.218853\pi$
$942$	0	0
$943$	4.59849e6	0.168398
$944$	0	0
$945$	−1.02605e7	−0.373756
$946$	0	0
$947$	−2.59321e7	−0.939643	−0.469821	−	0.882762i	$-0.655682\pi$
$947$	−2.59321e7	−0.939643	−0.469821	+	0.882762i	$0.655682\pi$
$948$	0	0
$949$	1.93057e7	0.695856
$950$	0	0
$951$	−4.99741e7	−1.79182
$952$	0	0
$953$	3.72243e7	1.32768	0.663842	−	0.747873i	$-0.268925\pi$
$953$	3.72243e7	1.32768	0.663842	+	0.747873i	$0.268925\pi$
$954$	0	0
$955$	4.18950e6	0.148646
$956$	0	0
$957$	2.85656e6	0.100824
$958$	0	0
$959$	−8.12887e6	−0.285419
$960$	0	0
$961$	4.69418e7	1.63965
$962$	0	0
$963$	2.16040e7	0.750705
$964$	0	0
$965$	−3.33884e7	−1.15419
$966$	0	0
$967$	3.87610e7	1.33299	0.666497	−	0.745507i	$-0.267793\pi$
$967$	3.87610e7	1.33299	0.666497	+	0.745507i	$0.267793\pi$
$968$	0	0
$969$	−5.23832e6	−0.179219
$970$	0	0
$971$	−1.92819e7	−0.656300	−0.328150	−	0.944626i	$-0.606425\pi$
$971$	−1.92819e7	−0.656300	−0.328150	+	0.944626i	$0.606425\pi$
$972$	0	0
$973$	−9.77579e6	−0.331032
$974$	0	0
$975$	−2.06696e7	−0.696339
$976$	0	0
$977$	−1.31734e7	−0.441533	−0.220766	−	0.975327i	$-0.570856\pi$
$977$	−1.31734e7	−0.441533	−0.220766	+	0.975327i	$0.570856\pi$
$978$	0	0
$979$	4.64154e6	0.154777
$980$	0	0
$981$	1.92562e7	0.638848
$982$	0	0
$983$	6.11864e6	0.201963	0.100981	−	0.994888i	$-0.467802\pi$
$983$	6.11864e6	0.201963	0.100981	+	0.994888i	$0.467802\pi$
$984$	0	0
$985$	4.52008e7	1.48442
$986$	0	0
$987$	−1.71937e7	−0.561792
$988$	0	0
$989$	9.10583e6	0.296025
$990$	0	0
$991$	2.63465e6	0.0852196	0.0426098	−	0.999092i	$-0.486433\pi$
$991$	2.63465e6	0.0852196	0.0426098	+	0.999092i	$0.486433\pi$
$992$	0	0
$993$	7.58484e7	2.44103
$994$	0	0
$995$	4.17485e7	1.33685
$996$	0	0
$997$	−995988.	−0.0317334	−0.0158667	−	0.999874i	$-0.505051\pi$
$997$	−995988.	−0.0317334	−0.0158667	+	0.999874i	$0.505051\pi$
$998$	0	0
$999$	6.49347e7	2.05856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.47	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.47	✓	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)

\(f(q)\)	\(=\)	\(q+26.8275 q^{3} -66.6449 q^{5} +24.5548 q^{7} +476.714 q^{9} +O(q^{10})\)	\(q+26.8275 q^{3} -66.6449 q^{5} +24.5548 q^{7} +476.714 q^{9} -60.9349 q^{11} -585.220 q^{13} -1787.91 q^{15} +895.842 q^{17} -217.962 q^{19} +658.743 q^{21} -666.232 q^{23} +1316.54 q^{25} +6269.97 q^{27} -1747.42 q^{29} +8693.16 q^{31} -1634.73 q^{33} -1636.45 q^{35} +10356.5 q^{37} -15700.0 q^{39} -6902.23 q^{41} -13667.7 q^{43} -31770.6 q^{45} -26100.7 q^{47} -16204.1 q^{49} +24033.2 q^{51} +6634.60 q^{53} +4061.00 q^{55} -5847.37 q^{57} -1226.26 q^{59} -35244.5 q^{61} +11705.6 q^{63} +39001.9 q^{65} -4870.89 q^{67} -17873.3 q^{69} +78649.5 q^{71} -32988.7 q^{73} +35319.4 q^{75} -1496.24 q^{77} +12918.1 q^{79} +52366.0 q^{81} -42959.8 q^{83} -59703.3 q^{85} -46878.8 q^{87} -76172.0 q^{89} -14369.9 q^{91} +233216. q^{93} +14526.0 q^{95} -146220. q^{97} -29048.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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