LMFDB - Embedded newform 1024.6.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,6,Mod(1,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, 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("1024.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1024.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:	$164.233031488$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 276 x^{16} + 31669 x^{14} - 1944052 x^{12} + 68219476 x^{10} - 1345935104 x^{8} + \cdots - 10616832 $ x^18 - 276x^16 + 31669x^14 - 1944052x^12 + 68219476x^10 - 1345935104x^8 + 13450597696x^6 - 50390014976x^4 + 13479874560x^2 - 10616832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{84} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$7.25216$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.47482 q^{3} +68.7282 q^{5} -106.338 q^{7} -201.077 q^{9} +O(q^{10})$	$q-6.47482 q^{3} +68.7282 q^{5} -106.338 q^{7} -201.077 q^{9} -261.607 q^{11} +5.15805 q^{13} -445.003 q^{15} -2064.90 q^{17} +1781.37 q^{19} +688.520 q^{21} -3861.52 q^{23} +1598.56 q^{25} +2875.32 q^{27} -2771.80 q^{29} -2760.10 q^{31} +1693.86 q^{33} -7308.42 q^{35} -9097.12 q^{37} -33.3975 q^{39} -11296.7 q^{41} +18282.7 q^{43} -13819.6 q^{45} +25827.2 q^{47} -5499.20 q^{49} +13369.9 q^{51} +30832.5 q^{53} -17979.8 q^{55} -11534.0 q^{57} +19944.4 q^{59} -8784.25 q^{61} +21382.1 q^{63} +354.503 q^{65} +40529.9 q^{67} +25002.6 q^{69} -29423.1 q^{71} -2857.88 q^{73} -10350.4 q^{75} +27818.8 q^{77} -23387.3 q^{79} +30244.5 q^{81} +34754.7 q^{83} -141917. q^{85} +17946.9 q^{87} -19340.2 q^{89} -548.497 q^{91} +17871.2 q^{93} +122430. q^{95} +44218.7 q^{97} +52603.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 196 q^{7} + 1134 q^{9}+O(q^{10}) $ 18 * q + 196 * q^7 + 1134 * q^9	$ 18 q + 196 q^{7} + 1134 q^{9} + 1796 q^{15} + 4 q^{17} + 6348 q^{23} + 6250 q^{25} + 11536 q^{31} - 4 q^{33} + 23364 q^{39} + 44176 q^{47} + 14410 q^{49} + 35900 q^{55} + 972 q^{57} + 12156 q^{63} + 27684 q^{65} + 12756 q^{71} - 10072 q^{73} - 52864 q^{79} + 13126 q^{81} + 31692 q^{87} - 70232 q^{89} + 250380 q^{95} - 4 q^{97}+O(q^{100}) $ 18 * q + 196 * q^7 + 1134 * q^9 + 1796 * q^15 + 4 * q^17 + 6348 * q^23 + 6250 * q^25 + 11536 * q^31 - 4 * q^33 + 23364 * q^39 + 44176 * q^47 + 14410 * q^49 + 35900 * q^55 + 972 * q^57 + 12156 * q^63 + 27684 * q^65 + 12756 * q^71 - 10072 * q^73 - 52864 * q^79 + 13126 * q^81 + 31692 * q^87 - 70232 * q^89 + 250380 * q^95 - 4 * q^97

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
$3$	−6.47482	−0.415360	−0.207680	−	0.978197i	$-0.566591\pi$
$3$	−6.47482	−0.415360	−0.207680	+	0.978197i	$0.566591\pi$
$4$	0	0
$5$	68.7282	1.22945	0.614723	−	0.788743i	$-0.289268\pi$
$5$	68.7282	1.22945	0.614723	+	0.788743i	$0.289268\pi$
$6$	0	0
$7$	−106.338	−0.820246	−0.410123	−	0.912030i	$-0.634514\pi$
$7$	−106.338	−0.820246	−0.410123	+	0.912030i	$0.634514\pi$
$8$	0	0
$9$	−201.077	−0.827476
$10$	0	0
$11$	−261.607	−0.651881	−0.325940	−	0.945390i	$-0.605681\pi$
$11$	−261.607	−0.651881	−0.325940	+	0.945390i	$0.605681\pi$
$12$	0	0
$13$	5.15805	0.00846500	0.00423250	−	0.999991i	$-0.498653\pi$
$13$	5.15805	0.00846500	0.00423250	+	0.999991i	$0.498653\pi$
$14$	0	0
$15$	−445.003	−0.510663
$16$	0	0
$17$	−2064.90	−1.73291	−0.866456	−	0.499253i	$-0.833608\pi$
$17$	−2064.90	−1.73291	−0.866456	+	0.499253i	$0.833608\pi$
$18$	0	0
$19$	1781.37	1.13206	0.566030	−	0.824385i	$-0.308479\pi$
$19$	1781.37	1.13206	0.566030	+	0.824385i	$0.308479\pi$
$20$	0	0
$21$	688.520	0.340697
$22$	0	0
$23$	−3861.52	−1.52208	−0.761042	−	0.648703i	$-0.775312\pi$
$23$	−3861.52	−1.52208	−0.761042	+	0.648703i	$0.775312\pi$
$24$	0	0
$25$	1598.56	0.511539
$26$	0	0
$27$	2875.32	0.759060
$28$	0	0
$29$	−2771.80	−0.612021	−0.306011	−	0.952028i	$-0.598994\pi$
$29$	−2771.80	−0.612021	−0.306011	+	0.952028i	$0.598994\pi$
$30$	0	0
$31$	−2760.10	−0.515847	−0.257924	−	0.966165i	$-0.583038\pi$
$31$	−2760.10	−0.515847	−0.257924	+	0.966165i	$0.583038\pi$
$32$	0	0
$33$	1693.86	0.270765
$34$	0	0
$35$	−7308.42	−1.00845
$36$	0	0
$37$	−9097.12	−1.09244	−0.546222	−	0.837640i	$-0.683935\pi$
$37$	−9097.12	−1.09244	−0.546222	+	0.837640i	$0.683935\pi$
$38$	0	0
$39$	−33.3975	−0.00351602
$40$	0	0
$41$	−11296.7	−1.04952	−0.524761	−	0.851250i	$-0.675845\pi$
$41$	−11296.7	−1.04952	−0.524761	+	0.851250i	$0.675845\pi$
$42$	0	0
$43$	18282.7	1.50789	0.753945	−	0.656938i	$-0.228149\pi$
$43$	18282.7	1.50789	0.753945	+	0.656938i	$0.228149\pi$
$44$	0	0
$45$	−13819.6	−1.01734
$46$	0	0
$47$	25827.2	1.70542	0.852711	−	0.522382i	$-0.174957\pi$
$47$	25827.2	1.70542	0.852711	+	0.522382i	$0.174957\pi$
$48$	0	0
$49$	−5499.20	−0.327197
$50$	0	0
$51$	13369.9	0.719783
$52$	0	0
$53$	30832.5	1.50771	0.753856	−	0.657040i	$-0.228192\pi$
$53$	30832.5	1.50771	0.753856	+	0.657040i	$0.228192\pi$
$54$	0	0
$55$	−17979.8	−0.801452
$56$	0	0
$57$	−11534.0	−0.470212
$58$	0	0
$59$	19944.4	0.745918	0.372959	−	0.927848i	$-0.378343\pi$
$59$	19944.4	0.745918	0.372959	+	0.927848i	$0.378343\pi$
$60$	0	0
$61$	−8784.25	−0.302259	−0.151130	−	0.988514i	$-0.548291\pi$
$61$	−8784.25	−0.302259	−0.151130	+	0.988514i	$0.548291\pi$
$62$	0	0
$63$	21382.1	0.678734
$64$	0	0
$65$	354.503	0.0104073
$66$	0	0
$67$	40529.9	1.10303	0.551517	−	0.834164i	$-0.314049\pi$
$67$	40529.9	1.10303	0.551517	+	0.834164i	$0.314049\pi$
$68$	0	0
$69$	25002.6	0.632213
$70$	0	0
$71$	−29423.1	−0.692696	−0.346348	−	0.938106i	$-0.612578\pi$
$71$	−29423.1	−0.692696	−0.346348	+	0.938106i	$0.612578\pi$
$72$	0	0
$73$	−2857.88	−0.0627678	−0.0313839	−	0.999507i	$-0.509991\pi$
$73$	−2857.88	−0.0627678	−0.0313839	+	0.999507i	$0.509991\pi$
$74$	0	0
$75$	−10350.4	−0.212473
$76$	0	0
$77$	27818.8	0.534702
$78$	0	0
$79$	−23387.3	−0.421612	−0.210806	−	0.977528i	$-0.567609\pi$
$79$	−23387.3	−0.421612	−0.210806	+	0.977528i	$0.567609\pi$
$80$	0	0
$81$	30244.5	0.512193
$82$	0	0
$83$	34754.7	0.553755	0.276878	−	0.960905i	$-0.410700\pi$
$83$	34754.7	0.553755	0.276878	+	0.960905i	$0.410700\pi$
$84$	0	0
$85$	−141917.	−2.13052
$86$	0	0
$87$	17946.9	0.254209
$88$	0	0
$89$	−19340.2	−0.258813	−0.129407	−	0.991592i	$-0.541307\pi$
$89$	−19340.2	−0.258813	−0.129407	+	0.991592i	$0.541307\pi$
$90$	0	0
$91$	−548.497	−0.00694338
$92$	0	0
$93$	17871.2	0.214262
$94$	0	0
$95$	122430.	1.39181
$96$	0	0
$97$	44218.7	0.477174	0.238587	−	0.971121i	$-0.423316\pi$
$97$	44218.7	0.477174	0.238587	+	0.971121i	$0.423316\pi$
$98$	0	0
$99$	52603.1	0.539416
$100$	0	0
$101$	−14310.6	−0.139590	−0.0697949	−	0.997561i	$-0.522234\pi$
$101$	−14310.6	−0.139590	−0.0697949	+	0.997561i	$0.522234\pi$
$102$	0	0
$103$	5350.67	0.0496953	0.0248476	−	0.999691i	$-0.492090\pi$
$103$	5350.67	0.0496953	0.0248476	+	0.999691i	$0.492090\pi$
$104$	0	0
$105$	47320.7	0.418869
$106$	0	0
$107$	17447.7	0.147326	0.0736628	−	0.997283i	$-0.476531\pi$
$107$	17447.7	0.147326	0.0736628	+	0.997283i	$0.476531\pi$
$108$	0	0
$109$	−36552.7	−0.294681	−0.147341	−	0.989086i	$-0.547071\pi$
$109$	−36552.7	−0.294681	−0.147341	+	0.989086i	$0.547071\pi$
$110$	0	0
$111$	58902.2	0.453758
$112$	0	0
$113$	159991.	1.17869	0.589344	−	0.807882i	$-0.299386\pi$
$113$	159991.	1.17869	0.589344	+	0.807882i	$0.299386\pi$
$114$	0	0
$115$	−265395.	−1.87132
$116$	0	0
$117$	−1037.16	−0.00700459
$118$	0	0
$119$	219578.	1.42141
$120$	0	0
$121$	−92612.6	−0.575052
$122$	0	0
$123$	73144.0	0.435929
$124$	0	0
$125$	−104909.	−0.600537
$126$	0	0
$127$	−139544.	−0.767717	−0.383858	−	0.923392i	$-0.625405\pi$
$127$	−139544.	−0.767717	−0.383858	+	0.923392i	$0.625405\pi$
$128$	0	0
$129$	−118377.	−0.626317
$130$	0	0
$131$	−204426.	−1.04078	−0.520389	−	0.853929i	$-0.674213\pi$
$131$	−204426.	−1.04078	−0.520389	+	0.853929i	$0.674213\pi$
$132$	0	0
$133$	−189427.	−0.928567
$134$	0	0
$135$	197615.	0.933224
$136$	0	0
$137$	34191.5	0.155638	0.0778192	−	0.996967i	$-0.475204\pi$
$137$	34191.5	0.155638	0.0778192	+	0.996967i	$0.475204\pi$
$138$	0	0
$139$	188562.	0.827783	0.413892	−	0.910326i	$-0.364169\pi$
$139$	188562.	0.827783	0.413892	+	0.910326i	$0.364169\pi$
$140$	0	0
$141$	−167226.	−0.708364
$142$	0	0
$143$	−1349.38	−0.00551817
$144$	0	0
$145$	−190501.	−0.752448
$146$	0	0
$147$	35606.4	0.135905
$148$	0	0
$149$	117665.	0.434192	0.217096	−	0.976150i	$-0.430342\pi$
$149$	117665.	0.434192	0.217096	+	0.976150i	$0.430342\pi$
$150$	0	0
$151$	−136797.	−0.488242	−0.244121	−	0.969745i	$-0.578499\pi$
$151$	−136797.	−0.488242	−0.244121	+	0.969745i	$0.578499\pi$
$152$	0	0
$153$	415203.	1.43394
$154$	0	0
$155$	−189697.	−0.634206
$156$	0	0
$157$	183083.	0.592789	0.296394	−	0.955066i	$-0.404216\pi$
$157$	183083.	0.592789	0.296394	+	0.955066i	$0.404216\pi$
$158$	0	0
$159$	−199635.	−0.626243
$160$	0	0
$161$	410627.	1.24848
$162$	0	0
$163$	−654846.	−1.93050	−0.965250	−	0.261327i	$-0.915840\pi$
$163$	−654846.	−1.93050	−0.965250	+	0.261327i	$0.915840\pi$
$164$	0	0
$165$	116416.	0.332891
$166$	0	0
$167$	211138.	0.585836	0.292918	−	0.956138i	$-0.405374\pi$
$167$	211138.	0.585836	0.292918	+	0.956138i	$0.405374\pi$
$168$	0	0
$169$	−371266.	−0.999928
$170$	0	0
$171$	−358191.	−0.936752
$172$	0	0
$173$	501977.	1.27517	0.637585	−	0.770380i	$-0.279933\pi$
$173$	501977.	1.27517	0.637585	+	0.770380i	$0.279933\pi$
$174$	0	0
$175$	−169988.	−0.419587
$176$	0	0
$177$	−129136.	−0.309825
$178$	0	0
$179$	−79492.6	−0.185436	−0.0927180	−	0.995692i	$-0.529555\pi$
$179$	−79492.6	−0.185436	−0.0927180	+	0.995692i	$0.529555\pi$
$180$	0	0
$181$	223217.	0.506443	0.253222	−	0.967408i	$-0.418510\pi$
$181$	223217.	0.506443	0.253222	+	0.967408i	$0.418510\pi$
$182$	0	0
$183$	56876.4	0.125547
$184$	0	0
$185$	−625228.	−1.34310
$186$	0	0
$187$	540193.	1.12965
$188$	0	0
$189$	−305756.	−0.622616
$190$	0	0
$191$	234025.	0.464171	0.232086	−	0.972695i	$-0.425445\pi$
$191$	234025.	0.464171	0.232086	+	0.972695i	$0.425445\pi$
$192$	0	0
$193$	−162458.	−0.313940	−0.156970	−	0.987603i	$-0.550173\pi$
$193$	−162458.	−0.313940	−0.156970	+	0.987603i	$0.550173\pi$
$194$	0	0
$195$	−2295.35	−0.00432276
$196$	0	0
$197$	−39944.3	−0.0733312	−0.0366656	−	0.999328i	$-0.511674\pi$
$197$	−39944.3	−0.0733312	−0.0366656	+	0.999328i	$0.511674\pi$
$198$	0	0
$199$	792440.	1.41851	0.709257	−	0.704950i	$-0.249030\pi$
$199$	792440.	1.41851	0.709257	+	0.704950i	$0.249030\pi$
$200$	0	0
$201$	−262424.	−0.458156
$202$	0	0
$203$	294748.	0.502008
$204$	0	0
$205$	−776400.	−1.29033
$206$	0	0
$207$	776461.	1.25949
$208$	0	0
$209$	−466018.	−0.737968
$210$	0	0
$211$	−374272.	−0.578738	−0.289369	−	0.957218i	$-0.593445\pi$
$211$	−374272.	−0.578738	−0.289369	+	0.957218i	$0.593445\pi$
$212$	0	0
$213$	190509.	0.287718
$214$	0	0
$215$	1.25654e6	1.85387
$216$	0	0
$217$	293504.	0.423121
$218$	0	0
$219$	18504.3	0.0260712
$220$	0	0
$221$	−10650.9	−0.0146691
$222$	0	0
$223$	1.22374e6	1.64788	0.823942	−	0.566674i	$-0.191770\pi$
$223$	1.22374e6	1.64788	0.823942	+	0.566674i	$0.191770\pi$
$224$	0	0
$225$	−321433.	−0.423286
$226$	0	0
$227$	−190381.	−0.245222	−0.122611	−	0.992455i	$-0.539127\pi$
$227$	−190381.	−0.245222	−0.122611	+	0.992455i	$0.539127\pi$
$228$	0	0
$229$	447449.	0.563839	0.281919	−	0.959438i	$-0.409029\pi$
$229$	447449.	0.563839	0.281919	+	0.959438i	$0.409029\pi$
$230$	0	0
$231$	−180122.	−0.222094
$232$	0	0
$233$	561597.	0.677697	0.338848	−	0.940841i	$-0.389963\pi$
$233$	561597.	0.677697	0.338848	+	0.940841i	$0.389963\pi$
$234$	0	0
$235$	1.77505e6	2.09673
$236$	0	0
$237$	151429.	0.175121
$238$	0	0
$239$	1.42167e6	1.60992	0.804962	−	0.593327i	$-0.202186\pi$
$239$	1.42167e6	1.60992	0.804962	+	0.593327i	$0.202186\pi$
$240$	0	0
$241$	1.07355e6	1.19063	0.595317	−	0.803491i	$-0.297027\pi$
$241$	1.07355e6	1.19063	0.595317	+	0.803491i	$0.297027\pi$
$242$	0	0
$243$	−894530.	−0.971805
$244$	0	0
$245$	−377950.	−0.402272
$246$	0	0
$247$	9188.38	0.00958289
$248$	0	0
$249$	−225030.	−0.230008
$250$	0	0
$251$	1.14905e6	1.15121	0.575604	−	0.817729i	$-0.304767\pi$
$251$	1.14905e6	1.15121	0.575604	+	0.817729i	$0.304767\pi$
$252$	0	0
$253$	1.01020e6	0.992217
$254$	0	0
$255$	918886.	0.884934
$256$	0	0
$257$	−1.97617e6	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$257$	−1.97617e6	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$258$	0	0
$259$	967370.	0.896073
$260$	0	0
$261$	557344.	0.506433
$262$	0	0
$263$	−723548.	−0.645027	−0.322514	−	0.946565i	$-0.604528\pi$
$263$	−723548.	−0.645027	−0.322514	+	0.946565i	$0.604528\pi$
$264$	0	0
$265$	2.11906e6	1.85365
$266$	0	0
$267$	125225.	0.107501
$268$	0	0
$269$	1.85963e6	1.56692	0.783459	−	0.621443i	$-0.213453\pi$
$269$	1.85963e6	1.56692	0.783459	+	0.621443i	$0.213453\pi$
$270$	0	0
$271$	−324667.	−0.268543	−0.134272	−	0.990945i	$-0.542869\pi$
$271$	−324667.	−0.268543	−0.134272	+	0.990945i	$0.542869\pi$
$272$	0	0
$273$	3551.42	0.00288400
$274$	0	0
$275$	−418195.	−0.333462
$276$	0	0
$277$	−815185.	−0.638347	−0.319173	−	0.947696i	$-0.603405\pi$
$277$	−815185.	−0.638347	−0.319173	+	0.947696i	$0.603405\pi$
$278$	0	0
$279$	554992.	0.426851
$280$	0	0
$281$	1.64048e6	1.23938	0.619690	−	0.784847i	$-0.287258\pi$
$281$	1.64048e6	1.23938	0.619690	+	0.784847i	$0.287258\pi$
$282$	0	0
$283$	1.03757e6	0.770107	0.385054	−	0.922894i	$-0.374183\pi$
$283$	1.03757e6	0.770107	0.385054	+	0.922894i	$0.374183\pi$
$284$	0	0
$285$	−792713.	−0.578101
$286$	0	0
$287$	1.20127e6	0.860866
$288$	0	0
$289$	2.84395e6	2.00299
$290$	0	0
$291$	−286308.	−0.198199
$292$	0	0
$293$	−447216.	−0.304333	−0.152166	−	0.988355i	$-0.548625\pi$
$293$	−447216.	−0.304333	−0.152166	+	0.988355i	$0.548625\pi$
$294$	0	0
$295$	1.37074e6	0.917066
$296$	0	0
$297$	−752204.	−0.494817
$298$	0	0
$299$	−19917.9	−0.0128844
$300$	0	0
$301$	−1.94415e6	−1.23684
$302$	0	0
$303$	92658.5	0.0579801
$304$	0	0
$305$	−603725.	−0.371612
$306$	0	0
$307$	135971.	0.0823380	0.0411690	−	0.999152i	$-0.486892\pi$
$307$	135971.	0.0823380	0.0411690	+	0.999152i	$0.486892\pi$
$308$	0	0
$309$	−34644.6	−0.0206414
$310$	0	0
$311$	−810338.	−0.475078	−0.237539	−	0.971378i	$-0.576341\pi$
$311$	−810338.	−0.475078	−0.237539	+	0.971378i	$0.576341\pi$
$312$	0	0
$313$	−1.82461e6	−1.05271	−0.526355	−	0.850265i	$-0.676442\pi$
$313$	−1.82461e6	−1.05271	−0.526355	+	0.850265i	$0.676442\pi$
$314$	0	0
$315$	1.46955e6	0.834467
$316$	0	0
$317$	1.02813e6	0.574644	0.287322	−	0.957834i	$-0.407235\pi$
$317$	1.02813e6	0.574644	0.287322	+	0.957834i	$0.407235\pi$
$318$	0	0
$319$	725123.	0.398965
$320$	0	0
$321$	−112971.	−0.0611932
$322$	0	0
$323$	−3.67834e6	−1.96176
$324$	0	0
$325$	8245.44	0.00433018
$326$	0	0
$327$	236672.	0.122399
$328$	0	0
$329$	−2.74641e6	−1.39887
$330$	0	0
$331$	3.57679e6	1.79442	0.897209	−	0.441607i	$-0.145591\pi$
$331$	3.57679e6	1.79442	0.897209	+	0.441607i	$0.145591\pi$
$332$	0	0
$333$	1.82922e6	0.903972
$334$	0	0
$335$	2.78555e6	1.35612
$336$	0	0
$337$	−1.70272e6	−0.816712	−0.408356	−	0.912823i	$-0.633898\pi$
$337$	−1.70272e6	−0.816712	−0.408356	+	0.912823i	$0.633898\pi$
$338$	0	0
$339$	−1.03591e6	−0.489580
$340$	0	0
$341$	722063.	0.336271
$342$	0	0
$343$	2.37200e6	1.08863
$344$	0	0
$345$	1.71839e6	0.777272
$346$	0	0
$347$	3.01239e6	1.34304	0.671518	−	0.740988i	$-0.265642\pi$
$347$	3.01239e6	1.34304	0.671518	+	0.740988i	$0.265642\pi$
$348$	0	0
$349$	−579499.	−0.254677	−0.127338	−	0.991859i	$-0.540643\pi$
$349$	−579499.	−0.254677	−0.127338	+	0.991859i	$0.540643\pi$
$350$	0	0
$351$	14831.0	0.00642545
$352$	0	0
$353$	−908761.	−0.388162	−0.194081	−	0.980986i	$-0.562172\pi$
$353$	−908761.	−0.388162	−0.194081	+	0.980986i	$0.562172\pi$
$354$	0	0
$355$	−2.02220e6	−0.851633
$356$	0	0
$357$	−1.42173e6	−0.590399
$358$	0	0
$359$	2.57742e6	1.05548	0.527740	−	0.849406i	$-0.323040\pi$
$359$	2.57742e6	1.05548	0.527740	+	0.849406i	$0.323040\pi$
$360$	0	0
$361$	697168.	0.281559
$362$	0	0
$363$	599650.	0.238853
$364$	0	0
$365$	−196417.	−0.0771696
$366$	0	0
$367$	1.72383e6	0.668080	0.334040	−	0.942559i	$-0.391588\pi$
$367$	1.72383e6	0.668080	0.334040	+	0.942559i	$0.391588\pi$
$368$	0	0
$369$	2.27150e6	0.868454
$370$	0	0
$371$	−3.27867e6	−1.23669
$372$	0	0
$373$	3.28021e6	1.22076	0.610379	−	0.792110i	$-0.291017\pi$
$373$	3.28021e6	1.22076	0.610379	+	0.792110i	$0.291017\pi$
$374$	0	0
$375$	679270.	0.249439
$376$	0	0
$377$	−14297.1	−0.00518076
$378$	0	0
$379$	−3.84121e6	−1.37363	−0.686816	−	0.726831i	$-0.740992\pi$
$379$	−3.84121e6	−1.37363	−0.686816	+	0.726831i	$0.740992\pi$
$380$	0	0
$381$	903521.	0.318879
$382$	0	0
$383$	−1.85005e6	−0.644447	−0.322224	−	0.946664i	$-0.604430\pi$
$383$	−1.85005e6	−0.644447	−0.322224	+	0.946664i	$0.604430\pi$
$384$	0	0
$385$	1.91194e6	0.657388
$386$	0	0
$387$	−3.67623e6	−1.24774
$388$	0	0
$389$	−2.08510e6	−0.698640	−0.349320	−	0.937003i	$-0.613587\pi$
$389$	−2.08510e6	−0.698640	−0.349320	+	0.937003i	$0.613587\pi$
$390$	0	0
$391$	7.97365e6	2.63764
$392$	0	0
$393$	1.32362e6	0.432298
$394$	0	0
$395$	−1.60737e6	−0.518350
$396$	0	0
$397$	−2.52241e6	−0.803230	−0.401615	−	0.915809i	$-0.631551\pi$
$397$	−2.52241e6	−0.803230	−0.401615	+	0.915809i	$0.631551\pi$
$398$	0	0
$399$	1.22651e6	0.385690
$400$	0	0
$401$	284391.	0.0883192	0.0441596	−	0.999024i	$-0.485939\pi$
$401$	284391.	0.0883192	0.0441596	+	0.999024i	$0.485939\pi$
$402$	0	0
$403$	−14236.7	−0.00436665
$404$	0	0
$405$	2.07865e6	0.629714
$406$	0	0
$407$	2.37987e6	0.712144
$408$	0	0
$409$	−2.21457e6	−0.654608	−0.327304	−	0.944919i	$-0.606140\pi$
$409$	−2.21457e6	−0.654608	−0.327304	+	0.944919i	$0.606140\pi$
$410$	0	0
$411$	−221384.	−0.0646459
$412$	0	0
$413$	−2.12085e6	−0.611836
$414$	0	0
$415$	2.38862e6	0.680812
$416$	0	0
$417$	−1.22090e6	−0.343828
$418$	0	0
$419$	1.69935e6	0.472876	0.236438	−	0.971647i	$-0.424020\pi$
$419$	1.69935e6	0.472876	0.236438	+	0.971647i	$0.424020\pi$
$420$	0	0
$421$	3.43559e6	0.944705	0.472352	−	0.881410i	$-0.343405\pi$
$421$	3.43559e6	0.944705	0.472352	+	0.881410i	$0.343405\pi$
$422$	0	0
$423$	−5.19324e6	−1.41120
$424$	0	0
$425$	−3.30086e6	−0.886452
$426$	0	0
$427$	934100.	0.247927
$428$	0	0
$429$	8737.02	0.00229203
$430$	0	0
$431$	−1.37294e6	−0.356006	−0.178003	−	0.984030i	$-0.556964\pi$
$431$	−1.37294e6	−0.356006	−0.178003	+	0.984030i	$0.556964\pi$
$432$	0	0
$433$	−3.17249e6	−0.813167	−0.406584	−	0.913614i	$-0.633280\pi$
$433$	−3.17249e6	−0.813167	−0.406584	+	0.913614i	$0.633280\pi$
$434$	0	0
$435$	1.23346e6	0.312537
$436$	0	0
$437$	−6.87878e6	−1.72309
$438$	0	0
$439$	6.08336e6	1.50655	0.753273	−	0.657708i	$-0.228474\pi$
$439$	6.08336e6	1.50655	0.753273	+	0.657708i	$0.228474\pi$
$440$	0	0
$441$	1.10576e6	0.270748
$442$	0	0
$443$	9109.31	0.00220534	0.00110267	−	0.999999i	$-0.499649\pi$
$443$	9109.31	0.00220534	0.00110267	+	0.999999i	$0.499649\pi$
$444$	0	0
$445$	−1.32922e6	−0.318197
$446$	0	0
$447$	−761860.	−0.180346
$448$	0	0
$449$	−2.31061e6	−0.540892	−0.270446	−	0.962735i	$-0.587171\pi$
$449$	−2.31061e6	−0.540892	−0.270446	+	0.962735i	$0.587171\pi$
$450$	0	0
$451$	2.95530e6	0.684163
$452$	0	0
$453$	885739.	0.202796
$454$	0	0
$455$	−37697.2	−0.00853652
$456$	0	0
$457$	−186457.	−0.0417627	−0.0208814	−	0.999782i	$-0.506647\pi$
$457$	−186457.	−0.0417627	−0.0208814	+	0.999782i	$0.506647\pi$
$458$	0	0
$459$	−5.93724e6	−1.31539
$460$	0	0
$461$	−5.04945e6	−1.10660	−0.553302	−	0.832981i	$-0.686632\pi$
$461$	−5.04945e6	−1.10660	−0.553302	+	0.832981i	$0.686632\pi$
$462$	0	0
$463$	−6.84013e6	−1.48290	−0.741450	−	0.671008i	$-0.765862\pi$
$463$	−6.84013e6	−1.48290	−0.741450	+	0.671008i	$0.765862\pi$
$464$	0	0
$465$	1.22825e6	0.263424
$466$	0	0
$467$	−8.34645e6	−1.77096	−0.885482	−	0.464674i	$-0.846171\pi$
$467$	−8.34645e6	−1.77096	−0.885482	+	0.464674i	$0.846171\pi$
$468$	0	0
$469$	−4.30988e6	−0.904759
$470$	0	0
$471$	−1.18543e6	−0.246221
$472$	0	0
$473$	−4.78289e6	−0.982964
$474$	0	0
$475$	2.84762e6	0.579092
$476$	0	0
$477$	−6.19969e6	−1.24760
$478$	0	0
$479$	6.99964e6	1.39392	0.696958	−	0.717111i	$-0.254536\pi$
$479$	6.99964e6	1.39392	0.696958	+	0.717111i	$0.254536\pi$
$480$	0	0
$481$	−46923.4	−0.00924755
$482$	0	0
$483$	−2.65873e6	−0.518570
$484$	0	0
$485$	3.03907e6	0.586660
$486$	0	0
$487$	2.66337e6	0.508873	0.254436	−	0.967090i	$-0.418110\pi$
$487$	2.66337e6	0.508873	0.254436	+	0.967090i	$0.418110\pi$
$488$	0	0
$489$	4.24001e6	0.801853
$490$	0	0
$491$	2.02363e6	0.378815	0.189407	−	0.981899i	$-0.439343\pi$
$491$	2.02363e6	0.378815	0.189407	+	0.981899i	$0.439343\pi$
$492$	0	0
$493$	5.72349e6	1.06058
$494$	0	0
$495$	3.61532e6	0.663183
$496$	0	0
$497$	3.12880e6	0.568181
$498$	0	0
$499$	3.46064e6	0.622165	0.311082	−	0.950383i	$-0.399308\pi$
$499$	3.46064e6	0.622165	0.311082	+	0.950383i	$0.399308\pi$
$500$	0	0
$501$	−1.36708e6	−0.243333
$502$	0	0
$503$	−3.84992e6	−0.678472	−0.339236	−	0.940701i	$-0.610168\pi$
$503$	−3.84992e6	−0.678472	−0.339236	+	0.940701i	$0.610168\pi$
$504$	0	0
$505$	−983540.	−0.171618
$506$	0	0
$507$	2.40388e6	0.415330
$508$	0	0
$509$	5.50234e6	0.941354	0.470677	−	0.882306i	$-0.344010\pi$
$509$	5.50234e6	0.941354	0.470677	+	0.882306i	$0.344010\pi$
$510$	0	0
$511$	303901.	0.0514850
$512$	0	0
$513$	5.12199e6	0.859302
$514$	0	0
$515$	367742.	0.0610977
$516$	0	0
$517$	−6.75657e6	−1.11173
$518$	0	0
$519$	−3.25021e6	−0.529655
$520$	0	0
$521$	−1.24942e6	−0.201658	−0.100829	−	0.994904i	$-0.532150\pi$
$521$	−1.24942e6	−0.201658	−0.100829	+	0.994904i	$0.532150\pi$
$522$	0	0
$523$	1.10927e7	1.77331	0.886655	−	0.462431i	$-0.153023\pi$
$523$	1.10927e7	1.77331	0.886655	+	0.462431i	$0.153023\pi$
$524$	0	0
$525$	1.10064e6	0.174280
$526$	0	0
$527$	5.69934e6	0.893918
$528$	0	0
$529$	8.47497e6	1.31674
$530$	0	0
$531$	−4.01036e6	−0.617229
$532$	0	0
$533$	−58268.9	−0.00888420
$534$	0	0
$535$	1.19915e6	0.181129
$536$	0	0
$537$	514700.	0.0770227
$538$	0	0
$539$	1.43863e6	0.213294
$540$	0	0
$541$	1.03144e7	1.51513	0.757565	−	0.652759i	$-0.226389\pi$
$541$	1.03144e7	1.51513	0.757565	+	0.652759i	$0.226389\pi$
$542$	0	0
$543$	−1.44529e6	−0.210356
$544$	0	0
$545$	−2.51220e6	−0.362295
$546$	0	0
$547$	4.66243e6	0.666261	0.333130	−	0.942881i	$-0.391895\pi$
$547$	4.66243e6	0.666261	0.333130	+	0.942881i	$0.391895\pi$
$548$	0	0
$549$	1.76631e6	0.250112
$550$	0	0
$551$	−4.93759e6	−0.692845
$552$	0	0
$553$	2.48697e6	0.345825
$554$	0	0
$555$	4.04824e6	0.557871
$556$	0	0
$557$	5.86078e6	0.800419	0.400210	−	0.916424i	$-0.368937\pi$
$557$	5.86078e6	0.800419	0.400210	+	0.916424i	$0.368937\pi$
$558$	0	0
$559$	94303.1	0.0127643
$560$	0	0
$561$	−3.49765e6	−0.469212
$562$	0	0
$563$	5.74194e6	0.763462	0.381731	−	0.924273i	$-0.375328\pi$
$563$	5.74194e6	0.763462	0.381731	+	0.924273i	$0.375328\pi$
$564$	0	0
$565$	1.09959e7	1.44913
$566$	0	0
$567$	−3.21614e6	−0.420124
$568$	0	0
$569$	1.21386e7	1.57177	0.785885	−	0.618373i	$-0.212208\pi$
$569$	1.21386e7	1.57177	0.785885	+	0.618373i	$0.212208\pi$
$570$	0	0
$571$	2.86336e6	0.367524	0.183762	−	0.982971i	$-0.441172\pi$
$571$	2.86336e6	0.367524	0.183762	+	0.982971i	$0.441172\pi$
$572$	0	0
$573$	−1.51527e6	−0.192798
$574$	0	0
$575$	−6.17286e6	−0.778605
$576$	0	0
$577$	731916.	0.0915212	0.0457606	−	0.998952i	$-0.485429\pi$
$577$	731916.	0.0915212	0.0457606	+	0.998952i	$0.485429\pi$
$578$	0	0
$579$	1.05188e6	0.130398
$580$	0	0
$581$	−3.69575e6	−0.454215
$582$	0	0
$583$	−8.06600e6	−0.982848
$584$	0	0
$585$	−71282.3	−0.00861177
$586$	0	0
$587$	−5.63029e6	−0.674428	−0.337214	−	0.941428i	$-0.609485\pi$
$587$	−5.63029e6	−0.674428	−0.337214	+	0.941428i	$0.609485\pi$
$588$	0	0
$589$	−4.91675e6	−0.583970
$590$	0	0
$591$	258632.	0.0304588
$592$	0	0
$593$	−3.74221e6	−0.437010	−0.218505	−	0.975836i	$-0.570118\pi$
$593$	−3.74221e6	−0.437010	−0.218505	+	0.975836i	$0.570118\pi$
$594$	0	0
$595$	1.50912e7	1.74755
$596$	0	0
$597$	−5.13091e6	−0.589194
$598$	0	0
$599$	−9.02706e6	−1.02797	−0.513984	−	0.857800i	$-0.671831\pi$
$599$	−9.02706e6	−1.02797	−0.513984	+	0.857800i	$0.671831\pi$
$600$	0	0
$601$	−3.36352e6	−0.379846	−0.189923	−	0.981799i	$-0.560824\pi$
$601$	−3.36352e6	−0.379846	−0.189923	+	0.981799i	$0.560824\pi$
$602$	0	0
$603$	−8.14963e6	−0.912734
$604$	0	0
$605$	−6.36509e6	−0.706995
$606$	0	0
$607$	−8.35767e6	−0.920690	−0.460345	−	0.887740i	$-0.652274\pi$
$607$	−8.35767e6	−0.920690	−0.460345	+	0.887740i	$0.652274\pi$
$608$	0	0
$609$	−1.90844e6	−0.208514
$610$	0	0
$611$	133218.	0.0144364
$612$	0	0
$613$	862250.	0.0926792	0.0463396	−	0.998926i	$-0.485244\pi$
$613$	862250.	0.0926792	0.0463396	+	0.998926i	$0.485244\pi$
$614$	0	0
$615$	5.02705e6	0.535952
$616$	0	0
$617$	−4.89131e6	−0.517264	−0.258632	−	0.965976i	$-0.583272\pi$
$617$	−4.89131e6	−0.517264	−0.258632	+	0.965976i	$0.583272\pi$
$618$	0	0
$619$	−1.06883e7	−1.12120	−0.560599	−	0.828088i	$-0.689429\pi$
$619$	−1.06883e7	−1.12120	−0.560599	+	0.828088i	$0.689429\pi$
$620$	0	0
$621$	−1.11031e7	−1.15535
$622$	0	0
$623$	2.05660e6	0.212291
$624$	0	0
$625$	−1.22057e7	−1.24987
$626$	0	0
$627$	3.01739e6	0.306522
$628$	0	0
$629$	1.87846e7	1.89311
$630$	0	0
$631$	−1.92664e7	−1.92631	−0.963155	−	0.268947i	$-0.913324\pi$
$631$	−1.92664e7	−1.92631	−0.963155	+	0.268947i	$0.913324\pi$
$632$	0	0
$633$	2.42335e6	0.240384
$634$	0	0
$635$	−9.59058e6	−0.943867
$636$	0	0
$637$	−28365.2	−0.00276973
$638$	0	0
$639$	5.91630e6	0.573190
$640$	0	0
$641$	1.91377e7	1.83969	0.919843	−	0.392287i	$-0.128316\pi$
$641$	1.91377e7	1.83969	0.919843	+	0.392287i	$0.128316\pi$
$642$	0	0
$643$	4.09191e6	0.390300	0.195150	−	0.980773i	$-0.437481\pi$
$643$	4.09191e6	0.390300	0.195150	+	0.980773i	$0.437481\pi$
$644$	0	0
$645$	−8.13585e6	−0.770023
$646$	0	0
$647$	−410001.	−0.0385056	−0.0192528	−	0.999815i	$-0.506129\pi$
$647$	−410001.	−0.0385056	−0.0192528	+	0.999815i	$0.506129\pi$
$648$	0	0
$649$	−5.21760e6	−0.486250
$650$	0	0
$651$	−1.90039e6	−0.175748
$652$	0	0
$653$	1.48941e7	1.36688	0.683442	−	0.730005i	$-0.260482\pi$
$653$	1.48941e7	1.36688	0.683442	+	0.730005i	$0.260482\pi$
$654$	0	0
$655$	−1.40498e7	−1.27958
$656$	0	0
$657$	574653.	0.0519388
$658$	0	0
$659$	1.77484e7	1.59201	0.796006	−	0.605289i	$-0.206942\pi$
$659$	1.77484e7	1.59201	0.796006	+	0.605289i	$0.206942\pi$
$660$	0	0
$661$	−1.42137e7	−1.26533	−0.632665	−	0.774426i	$-0.718039\pi$
$661$	−1.42137e7	−1.26533	−0.632665	+	0.774426i	$0.718039\pi$
$662$	0	0
$663$	68962.4	0.00609296
$664$	0	0
$665$	−1.30190e7	−1.14162
$666$	0	0
$667$	1.07033e7	0.931548
$668$	0	0
$669$	−7.92349e6	−0.684465
$670$	0	0
$671$	2.29802e6	0.197037
$672$	0	0
$673$	−1.57084e7	−1.33689	−0.668445	−	0.743762i	$-0.733040\pi$
$673$	−1.57084e7	−1.33689	−0.668445	+	0.743762i	$0.733040\pi$
$674$	0	0
$675$	4.59636e6	0.388289
$676$	0	0
$677$	1.09788e7	0.920628	0.460314	−	0.887756i	$-0.347737\pi$
$677$	1.09788e7	0.920628	0.460314	+	0.887756i	$0.347737\pi$
$678$	0	0
$679$	−4.70213e6	−0.391400
$680$	0	0
$681$	1.23268e6	0.101855
$682$	0	0
$683$	−1.11206e7	−0.912171	−0.456085	−	0.889936i	$-0.650749\pi$
$683$	−1.11206e7	−0.912171	−0.456085	+	0.889936i	$0.650749\pi$
$684$	0	0
$685$	2.34992e6	0.191349
$686$	0	0
$687$	−2.89715e6	−0.234196
$688$	0	0
$689$	159035.	0.0127628
$690$	0	0
$691$	−8.04418e6	−0.640895	−0.320448	−	0.947266i	$-0.603833\pi$
$691$	−8.04418e6	−0.640895	−0.320448	+	0.947266i	$0.603833\pi$
$692$	0	0
$693$	−5.59372e6	−0.442453
$694$	0	0
$695$	1.29595e7	1.01772
$696$	0	0
$697$	2.33265e7	1.81873
$698$	0	0
$699$	−3.63624e6	−0.281488
$700$	0	0
$701$	1.41874e7	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$701$	1.41874e7	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$702$	0	0
$703$	−1.62053e7	−1.23671
$704$	0	0
$705$	−1.14932e7	−0.870896
$706$	0	0
$707$	1.52176e6	0.114498
$708$	0	0
$709$	1.39399e7	1.04146	0.520732	−	0.853720i	$-0.325659\pi$
$709$	1.39399e7	1.04146	0.520732	+	0.853720i	$0.325659\pi$
$710$	0	0
$711$	4.70265e6	0.348874
$712$	0	0
$713$	1.06582e7	0.785162
$714$	0	0
$715$	−92740.6	−0.00678430
$716$	0	0
$717$	−9.20508e6	−0.668698
$718$	0	0
$719$	1.06685e6	0.0769630	0.0384815	−	0.999259i	$-0.487748\pi$
$719$	1.06685e6	0.0769630	0.0384815	+	0.999259i	$0.487748\pi$
$720$	0	0
$721$	−568980.	−0.0407623
$722$	0	0
$723$	−6.95102e6	−0.494542
$724$	0	0
$725$	−4.43088e6	−0.313073
$726$	0	0
$727$	1.74630e7	1.22541	0.612707	−	0.790310i	$-0.290081\pi$
$727$	1.74630e7	1.22541	0.612707	+	0.790310i	$0.290081\pi$
$728$	0	0
$729$	−1.55748e6	−0.108544
$730$	0	0
$731$	−3.77520e7	−2.61304
$732$	0	0
$733$	−1.27972e7	−0.879740	−0.439870	−	0.898061i	$-0.644976\pi$
$733$	−1.27972e7	−0.879740	−0.439870	+	0.898061i	$0.644976\pi$
$734$	0	0
$735$	2.44716e6	0.167088
$736$	0	0
$737$	−1.06029e7	−0.719047
$738$	0	0
$739$	8.59958e6	0.579250	0.289625	−	0.957140i	$-0.406469\pi$
$739$	8.59958e6	0.579250	0.289625	+	0.957140i	$0.406469\pi$
$740$	0	0
$741$	−59493.1	−0.00398035
$742$	0	0
$743$	−1.29381e7	−0.859804	−0.429902	−	0.902875i	$-0.641452\pi$
$743$	−1.29381e7	−0.859804	−0.429902	+	0.902875i	$0.641452\pi$
$744$	0	0
$745$	8.08690e6	0.533816
$746$	0	0
$747$	−6.98835e6	−0.458219
$748$	0	0
$749$	−1.85535e6	−0.120843
$750$	0	0
$751$	−1.81776e7	−1.17608	−0.588040	−	0.808832i	$-0.700100\pi$
$751$	−1.81776e7	−1.17608	−0.588040	+	0.808832i	$0.700100\pi$
$752$	0	0
$753$	−7.43988e6	−0.478166
$754$	0	0
$755$	−9.40183e6	−0.600268
$756$	0	0
$757$	2.16737e7	1.37465	0.687327	−	0.726348i	$-0.258784\pi$
$757$	2.16737e7	1.37465	0.687327	+	0.726348i	$0.258784\pi$
$758$	0	0
$759$	−6.54087e6	−0.412127
$760$	0	0
$761$	−3.17565e7	−1.98779	−0.993895	−	0.110327i	$-0.964810\pi$
$761$	−3.17565e7	−1.98779	−0.993895	+	0.110327i	$0.964810\pi$
$762$	0	0
$763$	3.88694e6	0.241711
$764$	0	0
$765$	2.85361e7	1.76296
$766$	0	0
$767$	102874.	0.00631420
$768$	0	0
$769$	−6.64122e6	−0.404979	−0.202489	−	0.979284i	$-0.564903\pi$
$769$	−6.64122e6	−0.404979	−0.202489	+	0.979284i	$0.564903\pi$
$770$	0	0
$771$	1.27954e7	0.775206
$772$	0	0
$773$	−1.36850e7	−0.823754	−0.411877	−	0.911239i	$-0.635127\pi$
$773$	−1.36850e7	−0.823754	−0.411877	+	0.911239i	$0.635127\pi$
$774$	0	0
$775$	−4.41219e6	−0.263876
$776$	0	0
$777$	−6.26355e6	−0.372193
$778$	0	0
$779$	−2.01235e7	−1.18812
$780$	0	0
$781$	7.69730e6	0.451555
$782$	0	0
$783$	−7.96980e6	−0.464561
$784$	0	0
$785$	1.25830e7	0.728802
$786$	0	0
$787$	−8.16437e6	−0.469879	−0.234939	−	0.972010i	$-0.575489\pi$
$787$	−8.16437e6	−0.469879	−0.234939	+	0.972010i	$0.575489\pi$
$788$	0	0
$789$	4.68484e6	0.267918
$790$	0	0
$791$	−1.70131e7	−0.966814
$792$	0	0
$793$	−45309.6	−0.00255863
$794$	0	0
$795$	−1.37205e7	−0.769933
$796$	0	0
$797$	−1.97249e7	−1.09994	−0.549969	−	0.835185i	$-0.685360\pi$
$797$	−1.97249e7	−1.09994	−0.549969	+	0.835185i	$0.685360\pi$
$798$	0	0
$799$	−5.33305e7	−2.95535
$800$	0	0
$801$	3.88887e6	0.214162
$802$	0	0
$803$	747642.	0.0409171
$804$	0	0
$805$	2.82216e7	1.53494
$806$	0	0
$807$	−1.20408e7	−0.650835
$808$	0	0
$809$	1.01170e7	0.543478	0.271739	−	0.962371i	$-0.412401\pi$
$809$	1.01170e7	0.543478	0.271739	+	0.962371i	$0.412401\pi$
$810$	0	0
$811$	−1.66699e6	−0.0889980	−0.0444990	−	0.999009i	$-0.514169\pi$
$811$	−1.66699e6	−0.0889980	−0.0444990	+	0.999009i	$0.514169\pi$
$812$	0	0
$813$	2.10216e6	0.111542
$814$	0	0
$815$	−4.50063e7	−2.37345
$816$	0	0
$817$	3.25682e7	1.70702
$818$	0	0
$819$	110290.	0.00574548
$820$	0	0
$821$	−2.23666e7	−1.15809	−0.579044	−	0.815296i	$-0.696574\pi$
$821$	−2.23666e7	−1.15809	−0.579044	+	0.815296i	$0.696574\pi$
$822$	0	0
$823$	−4.45762e6	−0.229405	−0.114703	−	0.993400i	$-0.536592\pi$
$823$	−4.45762e6	−0.229405	−0.114703	+	0.993400i	$0.536592\pi$
$824$	0	0
$825$	2.70774e6	0.138507
$826$	0	0
$827$	−3.11739e6	−0.158499	−0.0792496	−	0.996855i	$-0.525252\pi$
$827$	−3.11739e6	−0.158499	−0.0792496	+	0.996855i	$0.525252\pi$
$828$	0	0
$829$	−9.54315e6	−0.482287	−0.241143	−	0.970489i	$-0.577522\pi$
$829$	−9.54315e6	−0.482287	−0.241143	+	0.970489i	$0.577522\pi$
$830$	0	0
$831$	5.27818e6	0.265144
$832$	0	0
$833$	1.13553e7	0.567004
$834$	0	0
$835$	1.45111e7	0.720254
$836$	0	0
$837$	−7.93617e6	−0.391559
$838$	0	0
$839$	−2.39429e7	−1.17428	−0.587140	−	0.809485i	$-0.699746\pi$
$839$	−2.39429e7	−1.17428	−0.587140	+	0.809485i	$0.699746\pi$
$840$	0	0
$841$	−1.28283e7	−0.625430
$842$	0	0
$843$	−1.06218e7	−0.514789
$844$	0	0
$845$	−2.55165e7	−1.22936
$846$	0	0
$847$	9.84825e6	0.471683
$848$	0	0
$849$	−6.71808e6	−0.319872
$850$	0	0
$851$	3.51287e7	1.66279
$852$	0	0
$853$	−1.41679e7	−0.666703	−0.333351	−	0.942803i	$-0.608179\pi$
$853$	−1.41679e7	−0.666703	−0.333351	+	0.942803i	$0.608179\pi$
$854$	0	0
$855$	−2.46178e7	−1.15169
$856$	0	0
$857$	1.85006e7	0.860465	0.430232	−	0.902718i	$-0.358432\pi$
$857$	1.85006e7	0.860465	0.430232	+	0.902718i	$0.358432\pi$
$858$	0	0
$859$	7.87548e6	0.364161	0.182081	−	0.983284i	$-0.441717\pi$
$859$	7.87548e6	0.364161	0.182081	+	0.983284i	$0.441717\pi$
$860$	0	0
$861$	−7.77800e6	−0.357569
$862$	0	0
$863$	1.36785e7	0.625191	0.312596	−	0.949886i	$-0.398802\pi$
$863$	1.36785e7	0.625191	0.312596	+	0.949886i	$0.398802\pi$
$864$	0	0
$865$	3.44999e7	1.56775
$866$	0	0
$867$	−1.84141e7	−0.831961
$868$	0	0
$869$	6.11830e6	0.274841
$870$	0	0
$871$	209055.	0.00933719
$872$	0	0
$873$	−8.89135e6	−0.394850
$874$	0	0
$875$	1.11559e7	0.492588
$876$	0	0
$877$	4.48776e7	1.97029	0.985145	−	0.171723i	$-0.0549334\pi$
$877$	4.48776e7	1.97029	0.985145	+	0.171723i	$0.0549334\pi$
$878$	0	0
$879$	2.89565e6	0.126408
$880$	0	0
$881$	−1.66529e7	−0.722853	−0.361427	−	0.932401i	$-0.617710\pi$
$881$	−1.66529e7	−0.722853	−0.361427	+	0.932401i	$0.617710\pi$
$882$	0	0
$883$	2.51533e7	1.08566	0.542830	−	0.839843i	$-0.317353\pi$
$883$	2.51533e7	1.08566	0.542830	+	0.839843i	$0.317353\pi$
$884$	0	0
$885$	−8.87531e6	−0.380913
$886$	0	0
$887$	−1.26187e6	−0.0538523	−0.0269261	−	0.999637i	$-0.508572\pi$
$887$	−1.26187e6	−0.0538523	−0.0269261	+	0.999637i	$0.508572\pi$
$888$	0	0
$889$	1.48388e7	0.629716
$890$	0	0
$891$	−7.91217e6	−0.333889
$892$	0	0
$893$	4.60076e7	1.93064
$894$	0	0
$895$	−5.46338e6	−0.227984
$896$	0	0
$897$	128965.	0.00535168
$898$	0	0
$899$	7.65045e6	0.315709
$900$	0	0
$901$	−6.36659e7	−2.61273
$902$	0	0
$903$	1.25880e7	0.513734
$904$	0	0
$905$	1.53413e7	0.622645
$906$	0	0
$907$	−2.51361e7	−1.01456	−0.507282	−	0.861780i	$-0.669350\pi$
$907$	−2.51361e7	−1.01456	−0.507282	+	0.861780i	$0.669350\pi$
$908$	0	0
$909$	2.87752e6	0.115507
$910$	0	0
$911$	−1.64034e6	−0.0654843	−0.0327422	−	0.999464i	$-0.510424\pi$
$911$	−1.64034e6	−0.0654843	−0.0327422	+	0.999464i	$0.510424\pi$
$912$	0	0
$913$	−9.09207e6	−0.360982
$914$	0	0
$915$	3.90901e6	0.154353
$916$	0	0
$917$	2.17383e7	0.853694
$918$	0	0
$919$	−1.33815e7	−0.522657	−0.261328	−	0.965250i	$-0.584161\pi$
$919$	−1.33815e7	−0.522657	−0.261328	+	0.965250i	$0.584161\pi$
$920$	0	0
$921$	−880388.	−0.0341999
$922$	0	0
$923$	−151766.	−0.00586368
$924$	0	0
$925$	−1.45423e7	−0.558828
$926$	0	0
$927$	−1.07589e6	−0.0411216
$928$	0	0
$929$	1.74486e7	0.663318	0.331659	−	0.943399i	$-0.392392\pi$
$929$	1.74486e7	0.663318	0.331659	+	0.943399i	$0.392392\pi$
$930$	0	0
$931$	−9.79610e6	−0.370407
$932$	0	0
$933$	5.24679e6	0.197329
$934$	0	0
$935$	3.71265e7	1.38885
$936$	0	0
$937$	−3.22416e7	−1.19969	−0.599843	−	0.800117i	$-0.704770\pi$
$937$	−3.22416e7	−1.19969	−0.599843	+	0.800117i	$0.704770\pi$
$938$	0	0
$939$	1.18140e7	0.437253
$940$	0	0
$941$	−2.60381e7	−0.958594	−0.479297	−	0.877653i	$-0.659108\pi$
$941$	−2.60381e7	−0.958594	−0.479297	+	0.877653i	$0.659108\pi$
$942$	0	0
$943$	4.36224e7	1.59746
$944$	0	0
$945$	−2.10140e7	−0.765473
$946$	0	0
$947$	1.01512e7	0.367826	0.183913	−	0.982943i	$-0.441124\pi$
$947$	1.01512e7	0.367826	0.183913	+	0.982943i	$0.441124\pi$
$948$	0	0
$949$	−14741.1	−0.000531329 0
$950$	0	0
$951$	−6.65694e6	−0.238684
$952$	0	0
$953$	−3.08295e6	−0.109960	−0.0549799	−	0.998487i	$-0.517509\pi$
$953$	−3.08295e6	−0.109960	−0.0549799	+	0.998487i	$0.517509\pi$
$954$	0	0
$955$	1.60841e7	0.570674
$956$	0	0
$957$	−4.69504e6	−0.165714
$958$	0	0
$959$	−3.63586e6	−0.127662
$960$	0	0
$961$	−2.10110e7	−0.733902
$962$	0	0
$963$	−3.50832e6	−0.121908
$964$	0	0
$965$	−1.11654e7	−0.385973
$966$	0	0
$967$	−3.51323e7	−1.20820	−0.604102	−	0.796907i	$-0.706468\pi$
$967$	−3.51323e7	−1.20820	−0.604102	+	0.796907i	$0.706468\pi$
$968$	0	0
$969$	2.38166e7	0.814837
$970$	0	0
$971$	4.10839e7	1.39837	0.699187	−	0.714939i	$-0.253545\pi$
$971$	4.10839e7	1.39837	0.699187	+	0.714939i	$0.253545\pi$
$972$	0	0
$973$	−2.00513e7	−0.678986
$974$	0	0
$975$	−53387.8	−0.00179858
$976$	0	0
$977$	−650099.	−0.0217893	−0.0108947	−	0.999941i	$-0.503468\pi$
$977$	−650099.	−0.0217893	−0.0108947	+	0.999941i	$0.503468\pi$
$978$	0	0
$979$	5.05955e6	0.168715
$980$	0	0
$981$	7.34989e6	0.243842
$982$	0	0
$983$	−1.51329e7	−0.499502	−0.249751	−	0.968310i	$-0.580349\pi$
$983$	−1.51329e7	−0.499502	−0.249751	+	0.968310i	$0.580349\pi$
$984$	0	0
$985$	−2.74529e6	−0.0901568
$986$	0	0
$987$	1.77825e7	0.581033
$988$	0	0
$989$	−7.05990e7	−2.29513
$990$	0	0
$991$	2.88015e7	0.931605	0.465802	−	0.884889i	$-0.345766\pi$
$991$	2.88015e7	0.931605	0.465802	+	0.884889i	$0.345766\pi$
$992$	0	0
$993$	−2.31591e7	−0.745329
$994$	0	0
$995$	5.44630e7	1.74399
$996$	0	0
$997$	1.25413e7	0.399581	0.199791	−	0.979839i	$-0.435974\pi$
$997$	1.25413e7	0.399581	0.199791	+	0.979839i	$0.435974\pi$
$998$	0	0
$999$	−2.61571e7	−0.829232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.6.a.l.1.7		18
4.3	odd	2		1024.6.a.k.1.12		18
8.3	odd	2		1024.6.a.k.1.7		18
8.5	even	2	inner	1024.6.a.l.1.12		18
32.3	odd	8		128.6.e.b.33.6		18
32.5	even	8		64.6.e.a.49.6		18
32.11	odd	8		128.6.e.b.97.6		18
32.13	even	8		64.6.e.a.17.6		18
32.19	odd	8		16.6.e.a.13.2	yes	18
32.21	even	8		128.6.e.a.97.4		18
32.27	odd	8		16.6.e.a.5.2	✓	18
32.29	even	8		128.6.e.a.33.4		18
96.5	odd	8		576.6.k.a.433.2		18
96.59	even	8		144.6.k.a.37.8		18
96.77	odd	8		576.6.k.a.145.2		18
96.83	even	8		144.6.k.a.109.8		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.6.e.a.5.2	✓	18	32.27	odd	8
16.6.e.a.13.2	yes	18	32.19	odd	8
64.6.e.a.17.6		18	32.13	even	8
64.6.e.a.49.6		18	32.5	even	8
128.6.e.a.33.4		18	32.29	even	8
128.6.e.a.97.4		18	32.21	even	8
128.6.e.b.33.6		18	32.3	odd	8
128.6.e.b.97.6		18	32.11	odd	8
144.6.k.a.37.8		18	96.59	even	8
144.6.k.a.109.8		18	96.83	even	8
576.6.k.a.145.2		18	96.77	odd	8
576.6.k.a.433.2		18	96.5	odd	8
1024.6.a.k.1.7		18	8.3	odd	2
1024.6.a.k.1.12		18	4.3	odd	2
1024.6.a.l.1.7		18	1.1	even	1	trivial
1024.6.a.l.1.12		18	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-6.47482 q^{3} +68.7282 q^{5} -106.338 q^{7} -201.077 q^{9} +O(q^{10})\)	\(q-6.47482 q^{3} +68.7282 q^{5} -106.338 q^{7} -201.077 q^{9} -261.607 q^{11} +5.15805 q^{13} -445.003 q^{15} -2064.90 q^{17} +1781.37 q^{19} +688.520 q^{21} -3861.52 q^{23} +1598.56 q^{25} +2875.32 q^{27} -2771.80 q^{29} -2760.10 q^{31} +1693.86 q^{33} -7308.42 q^{35} -9097.12 q^{37} -33.3975 q^{39} -11296.7 q^{41} +18282.7 q^{43} -13819.6 q^{45} +25827.2 q^{47} -5499.20 q^{49} +13369.9 q^{51} +30832.5 q^{53} -17979.8 q^{55} -11534.0 q^{57} +19944.4 q^{59} -8784.25 q^{61} +21382.1 q^{63} +354.503 q^{65} +40529.9 q^{67} +25002.6 q^{69} -29423.1 q^{71} -2857.88 q^{73} -10350.4 q^{75} +27818.8 q^{77} -23387.3 q^{79} +30244.5 q^{81} +34754.7 q^{83} -141917. q^{85} +17946.9 q^{87} -19340.2 q^{89} -548.497 q^{91} +17871.2 q^{93} +122430. q^{95} +44218.7 q^{97} +52603.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 196 q^{7} + 1134 q^{9}+O(q^{10}) \) 18 * q + 196 * q^7 + 1134 * q^9	\( 18 q + 196 q^{7} + 1134 q^{9} + 1796 q^{15} + 4 q^{17} + 6348 q^{23} + 6250 q^{25} + 11536 q^{31} - 4 q^{33} + 23364 q^{39} + 44176 q^{47} + 14410 q^{49} + 35900 q^{55} + 972 q^{57} + 12156 q^{63} + 27684 q^{65} + 12756 q^{71} - 10072 q^{73} - 52864 q^{79} + 13126 q^{81} + 31692 q^{87} - 70232 q^{89} + 250380 q^{95} - 4 q^{97}+O(q^{100}) \) 18 * q + 196 * q^7 + 1134 * q^9 + 1796 * q^15 + 4 * q^17 + 6348 * q^23 + 6250 * q^25 + 11536 * q^31 - 4 * q^33 + 23364 * q^39 + 44176 * q^47 + 14410 * q^49 + 35900 * q^55 + 972 * q^57 + 12156 * q^63 + 27684 * q^65 + 12756 * q^71 - 10072 * q^73 - 52864 * q^79 + 13126 * q^81 + 31692 * q^87 - 70232 * q^89 + 250380 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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