LMFDB - Embedded newform 882.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +6.00000 q^{5} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} +6.00000 q^{5} -64.0000 q^{8} -24.0000 q^{10} +666.000 q^{11} -559.000 q^{13} +256.000 q^{16} +1740.00 q^{17} +1157.00 q^{19} +96.0000 q^{20} -2664.00 q^{22} +3468.00 q^{23} -3089.00 q^{25} +2236.00 q^{26} -3372.00 q^{29} +6293.00 q^{31} -1024.00 q^{32} -6960.00 q^{34} +3131.00 q^{37} -4628.00 q^{38} -384.000 q^{40} +4866.00 q^{41} -11407.0 q^{43} +10656.0 q^{44} -13872.0 q^{46} -2310.00 q^{47} +12356.0 q^{50} -8944.00 q^{52} +28296.0 q^{53} +3996.00 q^{55} +13488.0 q^{58} -20544.0 q^{59} -4630.00 q^{61} -25172.0 q^{62} +4096.00 q^{64} -3354.00 q^{65} -18745.0 q^{67} +27840.0 q^{68} +38226.0 q^{71} +70589.0 q^{73} -12524.0 q^{74} +18512.0 q^{76} -62293.0 q^{79} +1536.00 q^{80} -19464.0 q^{82} -79818.0 q^{83} +10440.0 q^{85} +45628.0 q^{86} -42624.0 q^{88} +18120.0 q^{89} +55488.0 q^{92} +9240.00 q^{94} +6942.00 q^{95} +124754. q^{97} +O(q^{100})$

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	6.00000	0.107331	0.0536656	−	0.998559i	$-0.482909\pi$
$5$	6.00000	0.107331	0.0536656	+	0.998559i	$0.482909\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−24.0000	−0.0758947
$11$	666.000	1.65956	0.829779	−	0.558092i	$-0.188466\pi$
$11$	666.000	1.65956	0.829779	+	0.558092i	$0.188466\pi$
$12$	0	0
$13$	−559.000	−0.917389	−0.458694	−	0.888594i	$-0.651683\pi$
$13$	−559.000	−0.917389	−0.458694	+	0.888594i	$0.651683\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	1740.00	1.46025	0.730125	−	0.683314i	$-0.239462\pi$
$17$	1740.00	1.46025	0.730125	+	0.683314i	$0.239462\pi$
$18$	0	0
$19$	1157.00	0.735274	0.367637	−	0.929969i	$-0.380167\pi$
$19$	1157.00	0.735274	0.367637	+	0.929969i	$0.380167\pi$
$20$	96.0000	0.0536656
$21$	0	0
$22$	−2664.00	−1.17348
$23$	3468.00	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$23$	3468.00	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$24$	0	0
$25$	−3089.00	−0.988480
$26$	2236.00	0.648692
$27$	0	0
$28$	0	0
$29$	−3372.00	−0.744548	−0.372274	−	0.928123i	$-0.621422\pi$
$29$	−3372.00	−0.744548	−0.372274	+	0.928123i	$0.621422\pi$
$30$	0	0
$31$	6293.00	1.17613	0.588063	−	0.808815i	$-0.299891\pi$
$31$	6293.00	1.17613	0.588063	+	0.808815i	$0.299891\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−6960.00	−1.03255
$35$	0	0
$36$	0	0
$37$	3131.00	0.375992	0.187996	−	0.982170i	$-0.439801\pi$
$37$	3131.00	0.375992	0.187996	+	0.982170i	$0.439801\pi$
$38$	−4628.00	−0.519917
$39$	0	0
$40$	−384.000	−0.0379473
$41$	4866.00	0.452077	0.226039	−	0.974118i	$-0.427422\pi$
$41$	4866.00	0.452077	0.226039	+	0.974118i	$0.427422\pi$
$42$	0	0
$43$	−11407.0	−0.940806	−0.470403	−	0.882452i	$-0.655892\pi$
$43$	−11407.0	−0.940806	−0.470403	+	0.882452i	$0.655892\pi$
$44$	10656.0	0.829779
$45$	0	0
$46$	−13872.0	−0.966595
$47$	−2310.00	−0.152534	−0.0762671	−	0.997087i	$-0.524300\pi$
$47$	−2310.00	−0.152534	−0.0762671	+	0.997087i	$0.524300\pi$
$48$	0	0
$49$	0	0
$50$	12356.0	0.698961
$51$	0	0
$52$	−8944.00	−0.458694
$53$	28296.0	1.38368	0.691840	−	0.722051i	$-0.256801\pi$
$53$	28296.0	1.38368	0.691840	+	0.722051i	$0.256801\pi$
$54$	0	0
$55$	3996.00	0.178122
$56$	0	0
$57$	0	0
$58$	13488.0	0.526475
$59$	−20544.0	−0.768343	−0.384171	−	0.923262i	$-0.625513\pi$
$59$	−20544.0	−0.768343	−0.384171	+	0.923262i	$0.625513\pi$
$60$	0	0
$61$	−4630.00	−0.159315	−0.0796575	−	0.996822i	$-0.525383\pi$
$61$	−4630.00	−0.159315	−0.0796575	+	0.996822i	$0.525383\pi$
$62$	−25172.0	−0.831646
$63$	0	0
$64$	4096.00	0.125000
$65$	−3354.00	−0.0984645
$66$	0	0
$67$	−18745.0	−0.510151	−0.255075	−	0.966921i	$-0.582100\pi$
$67$	−18745.0	−0.510151	−0.255075	+	0.966921i	$0.582100\pi$
$68$	27840.0	0.730125
$69$	0	0
$70$	0	0
$71$	38226.0	0.899939	0.449969	−	0.893044i	$-0.351435\pi$
$71$	38226.0	0.899939	0.449969	+	0.893044i	$0.351435\pi$
$72$	0	0
$73$	70589.0	1.55035	0.775175	−	0.631746i	$-0.217662\pi$
$73$	70589.0	1.55035	0.775175	+	0.631746i	$0.217662\pi$
$74$	−12524.0	−0.265867
$75$	0	0
$76$	18512.0	0.367637
$77$	0	0
$78$	0	0
$79$	−62293.0	−1.12298	−0.561489	−	0.827484i	$-0.689771\pi$
$79$	−62293.0	−1.12298	−0.561489	+	0.827484i	$0.689771\pi$
$80$	1536.00	0.0268328
$81$	0	0
$82$	−19464.0	−0.319667
$83$	−79818.0	−1.27176	−0.635881	−	0.771787i	$-0.719363\pi$
$83$	−79818.0	−1.27176	−0.635881	+	0.771787i	$0.719363\pi$
$84$	0	0
$85$	10440.0	0.156730
$86$	45628.0	0.665251
$87$	0	0
$88$	−42624.0	−0.586742
$89$	18120.0	0.242484	0.121242	−	0.992623i	$-0.461312\pi$
$89$	18120.0	0.242484	0.121242	+	0.992623i	$0.461312\pi$
$90$	0	0
$91$	0	0
$92$	55488.0	0.683486
$93$	0	0
$94$	9240.00	0.107858
$95$	6942.00	0.0789179
$96$	0	0
$97$	124754.	1.34625	0.673124	−	0.739530i	$-0.264952\pi$
$97$	124754.	1.34625	0.673124	+	0.739530i	$0.264952\pi$
$98$	0	0
$99$	0	0
$100$	−49424.0	−0.494240
$101$	93390.0	0.910955	0.455478	−	0.890247i	$-0.349469\pi$
$101$	93390.0	0.910955	0.455478	+	0.890247i	$0.349469\pi$
$102$	0	0
$103$	−167731.	−1.55783	−0.778915	−	0.627129i	$-0.784230\pi$
$103$	−167731.	−1.55783	−0.778915	+	0.627129i	$0.784230\pi$
$104$	35776.0	0.324346
$105$	0	0
$106$	−113184.	−0.978409
$107$	−69180.0	−0.584146	−0.292073	−	0.956396i	$-0.594345\pi$
$107$	−69180.0	−0.584146	−0.292073	+	0.956396i	$0.594345\pi$
$108$	0	0
$109$	−219559.	−1.77005	−0.885024	−	0.465546i	$-0.845858\pi$
$109$	−219559.	−1.77005	−0.885024	+	0.465546i	$0.845858\pi$
$110$	−15984.0	−0.125952
$111$	0	0
$112$	0	0
$113$	39354.0	0.289930	0.144965	−	0.989437i	$-0.453693\pi$
$113$	39354.0	0.289930	0.144965	+	0.989437i	$0.453693\pi$
$114$	0	0
$115$	20808.0	0.146719
$116$	−53952.0	−0.372274
$117$	0	0
$118$	82176.0	0.543300
$119$	0	0
$120$	0	0
$121$	282505.	1.75413
$122$	18520.0	0.112653
$123$	0	0
$124$	100688.	0.588063
$125$	−37284.0	−0.213426
$126$	0	0
$127$	317093.	1.74453	0.872263	−	0.489037i	$-0.162652\pi$
$127$	317093.	1.74453	0.872263	+	0.489037i	$0.162652\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	13416.0	0.0696249
$131$	154830.	0.788273	0.394137	−	0.919052i	$-0.371044\pi$
$131$	154830.	0.788273	0.394137	+	0.919052i	$0.371044\pi$
$132$	0	0
$133$	0	0
$134$	74980.0	0.360731
$135$	0	0
$136$	−111360.	−0.516276
$137$	−67332.0	−0.306493	−0.153246	−	0.988188i	$-0.548973\pi$
$137$	−67332.0	−0.306493	−0.153246	+	0.988188i	$0.548973\pi$
$138$	0	0
$139$	−365215.	−1.60329	−0.801644	−	0.597802i	$-0.796041\pi$
$139$	−365215.	−1.60329	−0.801644	+	0.597802i	$0.796041\pi$
$140$	0	0
$141$	0	0
$142$	−152904.	−0.636353
$143$	−372294.	−1.52246
$144$	0	0
$145$	−20232.0	−0.0799133
$146$	−282356.	−1.09626
$147$	0	0
$148$	50096.0	0.187996
$149$	168060.	0.620153	0.310076	−	0.950712i	$-0.399645\pi$
$149$	168060.	0.620153	0.310076	+	0.950712i	$0.399645\pi$
$150$	0	0
$151$	153536.	0.547984	0.273992	−	0.961732i	$-0.411656\pi$
$151$	153536.	0.547984	0.273992	+	0.961732i	$0.411656\pi$
$152$	−74048.0	−0.259959
$153$	0	0
$154$	0	0
$155$	37758.0	0.126235
$156$	0	0
$157$	202418.	0.655390	0.327695	−	0.944784i	$-0.393728\pi$
$157$	202418.	0.655390	0.327695	+	0.944784i	$0.393728\pi$
$158$	249172.	0.794066
$159$	0	0
$160$	−6144.00	−0.0189737
$161$	0	0
$162$	0	0
$163$	−179764.	−0.529949	−0.264974	−	0.964255i	$-0.585363\pi$
$163$	−179764.	−0.529949	−0.264974	+	0.964255i	$0.585363\pi$
$164$	77856.0	0.226039
$165$	0	0
$166$	319272.	0.899271
$167$	−217302.	−0.602938	−0.301469	−	0.953476i	$-0.597477\pi$
$167$	−217302.	−0.602938	−0.301469	+	0.953476i	$0.597477\pi$
$168$	0	0
$169$	−58812.0	−0.158398
$170$	−41760.0	−0.110825
$171$	0	0
$172$	−182512.	−0.470403
$173$	73980.0	0.187931	0.0939656	−	0.995575i	$-0.470046\pi$
$173$	73980.0	0.187931	0.0939656	+	0.995575i	$0.470046\pi$
$174$	0	0
$175$	0	0
$176$	170496.	0.414890
$177$	0	0
$178$	−72480.0	−0.171462
$179$	−789366.	−1.84139	−0.920695	−	0.390283i	$-0.872377\pi$
$179$	−789366.	−1.84139	−0.920695	+	0.390283i	$0.872377\pi$
$180$	0	0
$181$	−477739.	−1.08391	−0.541956	−	0.840407i	$-0.682316\pi$
$181$	−477739.	−1.08391	−0.541956	+	0.840407i	$0.682316\pi$
$182$	0	0
$183$	0	0
$184$	−221952.	−0.483297
$185$	18786.0	0.0403557
$186$	0	0
$187$	1.15884e6	2.42337
$188$	−36960.0	−0.0762671
$189$	0	0
$190$	−27768.0	−0.0558034
$191$	−358974.	−0.711999	−0.356000	−	0.934486i	$-0.615860\pi$
$191$	−358974.	−0.711999	−0.356000	+	0.934486i	$0.615860\pi$
$192$	0	0
$193$	−181933.	−0.351575	−0.175788	−	0.984428i	$-0.556247\pi$
$193$	−181933.	−0.351575	−0.175788	+	0.984428i	$0.556247\pi$
$194$	−499016.	−0.951941
$195$	0	0
$196$	0	0
$197$	−717924.	−1.31799	−0.658996	−	0.752146i	$-0.729019\pi$
$197$	−717924.	−1.31799	−0.658996	+	0.752146i	$0.729019\pi$
$198$	0	0
$199$	203096.	0.363554	0.181777	−	0.983340i	$-0.441815\pi$
$199$	203096.	0.363554	0.181777	+	0.983340i	$0.441815\pi$
$200$	197696.	0.349480
$201$	0	0
$202$	−373560.	−0.644142
$203$	0	0
$204$	0	0
$205$	29196.0	0.0485220
$206$	670924.	1.10155
$207$	0	0
$208$	−143104.	−0.229347
$209$	770562.	1.22023
$210$	0	0
$211$	1.17098e6	1.81069	0.905343	−	0.424680i	$-0.139613\pi$
$211$	1.17098e6	1.81069	0.905343	+	0.424680i	$0.139613\pi$
$212$	452736.	0.691840
$213$	0	0
$214$	276720.	0.413053
$215$	−68442.0	−0.100978
$216$	0	0
$217$	0	0
$218$	878236.	1.25161
$219$	0	0
$220$	63936.0	0.0890612
$221$	−972660.	−1.33962
$222$	0	0
$223$	1.24635e6	1.67833	0.839167	−	0.543873i	$-0.183043\pi$
$223$	1.24635e6	1.67833	0.839167	+	0.543873i	$0.183043\pi$
$224$	0	0
$225$	0	0
$226$	−157416.	−0.205011
$227$	918942.	1.18365	0.591825	−	0.806066i	$-0.298408\pi$
$227$	918942.	1.18365	0.591825	+	0.806066i	$0.298408\pi$
$228$	0	0
$229$	1.20375e6	1.51687	0.758433	−	0.651751i	$-0.225965\pi$
$229$	1.20375e6	1.51687	0.758433	+	0.651751i	$0.225965\pi$
$230$	−83232.0	−0.103746
$231$	0	0
$232$	215808.	0.263237
$233$	919062.	1.10906	0.554530	−	0.832164i	$-0.312898\pi$
$233$	919062.	1.10906	0.554530	+	0.832164i	$0.312898\pi$
$234$	0	0
$235$	−13860.0	−0.0163717
$236$	−328704.	−0.384171
$237$	0	0
$238$	0	0
$239$	625338.	0.708142	0.354071	−	0.935219i	$-0.384797\pi$
$239$	625338.	0.708142	0.354071	+	0.935219i	$0.384797\pi$
$240$	0	0
$241$	1.25382e6	1.39057	0.695286	−	0.718733i	$-0.255278\pi$
$241$	1.25382e6	1.39057	0.695286	+	0.718733i	$0.255278\pi$
$242$	−1.13002e6	−1.24036
$243$	0	0
$244$	−74080.0	−0.0796575
$245$	0	0
$246$	0	0
$247$	−646763.	−0.674532
$248$	−402752.	−0.415823
$249$	0	0
$250$	149136.	0.150915
$251$	1.51333e6	1.51618	0.758089	−	0.652152i	$-0.226133\pi$
$251$	1.51333e6	1.51618	0.758089	+	0.652152i	$0.226133\pi$
$252$	0	0
$253$	2.30969e6	2.26857
$254$	−1.26837e6	−1.23357
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.55493e6	1.46851	0.734257	−	0.678872i	$-0.237531\pi$
$257$	1.55493e6	1.46851	0.734257	+	0.678872i	$0.237531\pi$
$258$	0	0
$259$	0	0
$260$	−53664.0	−0.0492322
$261$	0	0
$262$	−619320.	−0.557393
$263$	1.11532e6	0.994280	0.497140	−	0.867670i	$-0.334384\pi$
$263$	1.11532e6	0.994280	0.497140	+	0.867670i	$0.334384\pi$
$264$	0	0
$265$	169776.	0.148512
$266$	0	0
$267$	0	0
$268$	−299920.	−0.255075
$269$	−35670.0	−0.0300554	−0.0150277	−	0.999887i	$-0.504784\pi$
$269$	−35670.0	−0.0300554	−0.0150277	+	0.999887i	$0.504784\pi$
$270$	0	0
$271$	−292768.	−0.242159	−0.121079	−	0.992643i	$-0.538636\pi$
$271$	−292768.	−0.242159	−0.121079	+	0.992643i	$0.538636\pi$
$272$	445440.	0.365062
$273$	0	0
$274$	269328.	0.216723
$275$	−2.05727e6	−1.64044
$276$	0	0
$277$	863213.	0.675956	0.337978	−	0.941154i	$-0.390257\pi$
$277$	863213.	0.675956	0.337978	+	0.941154i	$0.390257\pi$
$278$	1.46086e6	1.13370
$279$	0	0
$280$	0	0
$281$	−1.47110e6	−1.11142	−0.555709	−	0.831377i	$-0.687553\pi$
$281$	−1.47110e6	−1.11142	−0.555709	+	0.831377i	$0.687553\pi$
$282$	0	0
$283$	688841.	0.511273	0.255637	−	0.966773i	$-0.417715\pi$
$283$	688841.	0.511273	0.255637	+	0.966773i	$0.417715\pi$
$284$	611616.	0.449969
$285$	0	0
$286$	1.48918e6	1.07654
$287$	0	0
$288$	0	0
$289$	1.60774e6	1.13233
$290$	80928.0	0.0565072
$291$	0	0
$292$	1.12942e6	0.775175
$293$	−722832.	−0.491890	−0.245945	−	0.969284i	$-0.579098\pi$
$293$	−722832.	−0.491890	−0.245945	+	0.969284i	$0.579098\pi$
$294$	0	0
$295$	−123264.	−0.0824672
$296$	−200384.	−0.132933
$297$	0	0
$298$	−672240.	−0.438514
$299$	−1.93861e6	−1.25404
$300$	0	0
$301$	0	0
$302$	−614144.	−0.387483
$303$	0	0
$304$	296192.	0.183819
$305$	−27780.0	−0.0170995
$306$	0	0
$307$	−20125.0	−0.0121868	−0.00609340	−	0.999981i	$-0.501940\pi$
$307$	−20125.0	−0.0121868	−0.00609340	+	0.999981i	$0.501940\pi$
$308$	0	0
$309$	0	0
$310$	−151032.	−0.0892616
$311$	1.74356e6	1.02220	0.511099	−	0.859522i	$-0.329238\pi$
$311$	1.74356e6	1.02220	0.511099	+	0.859522i	$0.329238\pi$
$312$	0	0
$313$	1.80854e6	1.04344	0.521721	−	0.853116i	$-0.325290\pi$
$313$	1.80854e6	1.04344	0.521721	+	0.853116i	$0.325290\pi$
$314$	−809672.	−0.463431
$315$	0	0
$316$	−996688.	−0.561489
$317$	−1.02355e6	−0.572087	−0.286043	−	0.958217i	$-0.592340\pi$
$317$	−1.02355e6	−0.572087	−0.286043	+	0.958217i	$0.592340\pi$
$318$	0	0
$319$	−2.24575e6	−1.23562
$320$	24576.0	0.0134164
$321$	0	0
$322$	0	0
$323$	2.01318e6	1.07368
$324$	0	0
$325$	1.72675e6	0.906820
$326$	719056.	0.374730
$327$	0	0
$328$	−311424.	−0.159833
$329$	0	0
$330$	0	0
$331$	−1.00753e6	−0.505463	−0.252731	−	0.967536i	$-0.581329\pi$
$331$	−1.00753e6	−0.505463	−0.252731	+	0.967536i	$0.581329\pi$
$332$	−1.27709e6	−0.635881
$333$	0	0
$334$	869208.	0.426341
$335$	−112470.	−0.0547551
$336$	0	0
$337$	−1.56571e6	−0.750993	−0.375496	−	0.926824i	$-0.622528\pi$
$337$	−1.56571e6	−0.750993	−0.375496	+	0.926824i	$0.622528\pi$
$338$	235248.	0.112004
$339$	0	0
$340$	167040.	0.0783652
$341$	4.19114e6	1.95185
$342$	0	0
$343$	0	0
$344$	730048.	0.332625
$345$	0	0
$346$	−295920.	−0.132887
$347$	−757284.	−0.337625	−0.168813	−	0.985648i	$-0.553993\pi$
$347$	−757284.	−0.337625	−0.168813	+	0.985648i	$0.553993\pi$
$348$	0	0
$349$	−455638.	−0.200243	−0.100121	−	0.994975i	$-0.531923\pi$
$349$	−455638.	−0.200243	−0.100121	+	0.994975i	$0.531923\pi$
$350$	0	0
$351$	0	0
$352$	−681984.	−0.293371
$353$	3.63139e6	1.55109	0.775543	−	0.631295i	$-0.217476\pi$
$353$	3.63139e6	1.55109	0.775543	+	0.631295i	$0.217476\pi$
$354$	0	0
$355$	229356.	0.0965916
$356$	289920.	0.121242
$357$	0	0
$358$	3.15746e6	1.30206
$359$	4.02484e6	1.64821	0.824104	−	0.566438i	$-0.191679\pi$
$359$	4.02484e6	1.64821	0.824104	+	0.566438i	$0.191679\pi$
$360$	0	0
$361$	−1.13745e6	−0.459372
$362$	1.91096e6	0.766442
$363$	0	0
$364$	0	0
$365$	423534.	0.166401
$366$	0	0
$367$	2.57787e6	0.999072	0.499536	−	0.866293i	$-0.333504\pi$
$367$	2.57787e6	0.999072	0.499536	+	0.866293i	$0.333504\pi$
$368$	887808.	0.341743
$369$	0	0
$370$	−75144.0	−0.0285358
$371$	0	0
$372$	0	0
$373$	−2.53133e6	−0.942056	−0.471028	−	0.882118i	$-0.656117\pi$
$373$	−2.53133e6	−0.942056	−0.471028	+	0.882118i	$0.656117\pi$
$374$	−4.63536e6	−1.71358
$375$	0	0
$376$	147840.	0.0539290
$377$	1.88495e6	0.683040
$378$	0	0
$379$	−3.06677e6	−1.09669	−0.548344	−	0.836253i	$-0.684742\pi$
$379$	−3.06677e6	−1.09669	−0.548344	+	0.836253i	$0.684742\pi$
$380$	111072.	0.0394590
$381$	0	0
$382$	1.43590e6	0.503460
$383$	3.92520e6	1.36730	0.683652	−	0.729808i	$-0.260391\pi$
$383$	3.92520e6	1.36730	0.683652	+	0.729808i	$0.260391\pi$
$384$	0	0
$385$	0	0
$386$	727732.	0.248601
$387$	0	0
$388$	1.99606e6	0.673124
$389$	4.02669e6	1.34919	0.674597	−	0.738187i	$-0.264318\pi$
$389$	4.02669e6	1.34919	0.674597	+	0.738187i	$0.264318\pi$
$390$	0	0
$391$	6.03432e6	1.99612
$392$	0	0
$393$	0	0
$394$	2.87170e6	0.931961
$395$	−373758.	−0.120531
$396$	0	0
$397$	4.57440e6	1.45666	0.728329	−	0.685227i	$-0.240297\pi$
$397$	4.57440e6	1.45666	0.728329	+	0.685227i	$0.240297\pi$
$398$	−812384.	−0.257071
$399$	0	0
$400$	−790784.	−0.247120
$401$	2.26944e6	0.704787	0.352393	−	0.935852i	$-0.385368\pi$
$401$	2.26944e6	0.704787	0.352393	+	0.935852i	$0.385368\pi$
$402$	0	0
$403$	−3.51779e6	−1.07896
$404$	1.49424e6	0.455478
$405$	0	0
$406$	0	0
$407$	2.08525e6	0.623981
$408$	0	0
$409$	−4.04596e6	−1.19595	−0.597976	−	0.801514i	$-0.704028\pi$
$409$	−4.04596e6	−1.19595	−0.597976	+	0.801514i	$0.704028\pi$
$410$	−116784.	−0.0343102
$411$	0	0
$412$	−2.68370e6	−0.778915
$413$	0	0
$414$	0	0
$415$	−478908.	−0.136500
$416$	572416.	0.162173
$417$	0	0
$418$	−3.08225e6	−0.862833
$419$	3.91281e6	1.08881	0.544407	−	0.838821i	$-0.316755\pi$
$419$	3.91281e6	1.08881	0.544407	+	0.838821i	$0.316755\pi$
$420$	0	0
$421$	−2.78086e6	−0.764671	−0.382335	−	0.924024i	$-0.624880\pi$
$421$	−2.78086e6	−0.764671	−0.382335	+	0.924024i	$0.624880\pi$
$422$	−4.68392e6	−1.28035
$423$	0	0
$424$	−1.81094e6	−0.489204
$425$	−5.37486e6	−1.44343
$426$	0	0
$427$	0	0
$428$	−1.10688e6	−0.292073
$429$	0	0
$430$	273768.	0.0714022
$431$	4.38207e6	1.13628	0.568141	−	0.822931i	$-0.307663\pi$
$431$	4.38207e6	1.13628	0.568141	+	0.822931i	$0.307663\pi$
$432$	0	0
$433$	1.24946e6	0.320261	0.160130	−	0.987096i	$-0.448809\pi$
$433$	1.24946e6	0.320261	0.160130	+	0.987096i	$0.448809\pi$
$434$	0	0
$435$	0	0
$436$	−3.51294e6	−0.885024
$437$	4.01248e6	1.00510
$438$	0	0
$439$	−6.74421e6	−1.67020	−0.835102	−	0.550095i	$-0.814592\pi$
$439$	−6.74421e6	−1.67020	−0.835102	+	0.550095i	$0.814592\pi$
$440$	−255744.	−0.0629758
$441$	0	0
$442$	3.89064e6	0.947252
$443$	−478896.	−0.115940	−0.0579698	−	0.998318i	$-0.518463\pi$
$443$	−478896.	−0.115940	−0.0579698	+	0.998318i	$0.518463\pi$
$444$	0	0
$445$	108720.	0.0260261
$446$	−4.98541e6	−1.18676
$447$	0	0
$448$	0	0
$449$	−724506.	−0.169600	−0.0848001	−	0.996398i	$-0.527025\pi$
$449$	−724506.	−0.169600	−0.0848001	+	0.996398i	$0.527025\pi$
$450$	0	0
$451$	3.24076e6	0.750248
$452$	629664.	0.144965
$453$	0	0
$454$	−3.67577e6	−0.836967
$455$	0	0
$456$	0	0
$457$	−2.33956e6	−0.524016	−0.262008	−	0.965066i	$-0.584385\pi$
$457$	−2.33956e6	−0.524016	−0.262008	+	0.965066i	$0.584385\pi$
$458$	−4.81500e6	−1.07259
$459$	0	0
$460$	332928.	0.0733594
$461$	−2.98247e6	−0.653617	−0.326809	−	0.945091i	$-0.605973\pi$
$461$	−2.98247e6	−0.653617	−0.326809	+	0.945091i	$0.605973\pi$
$462$	0	0
$463$	4.28423e6	0.928795	0.464398	−	0.885627i	$-0.346271\pi$
$463$	4.28423e6	0.928795	0.464398	+	0.885627i	$0.346271\pi$
$464$	−863232.	−0.186137
$465$	0	0
$466$	−3.67625e6	−0.784224
$467$	5.74035e6	1.21800	0.608998	−	0.793171i	$-0.291572\pi$
$467$	5.74035e6	1.21800	0.608998	+	0.793171i	$0.291572\pi$
$468$	0	0
$469$	0	0
$470$	55440.0	0.0115765
$471$	0	0
$472$	1.31482e6	0.271650
$473$	−7.59706e6	−1.56132
$474$	0	0
$475$	−3.57397e6	−0.726804
$476$	0	0
$477$	0	0
$478$	−2.50135e6	−0.500732
$479$	−2.65051e6	−0.527826	−0.263913	−	0.964546i	$-0.585013\pi$
$479$	−2.65051e6	−0.527826	−0.263913	+	0.964546i	$0.585013\pi$
$480$	0	0
$481$	−1.75023e6	−0.344931
$482$	−5.01529e6	−0.983282
$483$	0	0
$484$	4.52008e6	0.877067
$485$	748524.	0.144495
$486$	0	0
$487$	2.80554e6	0.536036	0.268018	−	0.963414i	$-0.413631\pi$
$487$	2.80554e6	0.536036	0.268018	+	0.963414i	$0.413631\pi$
$488$	296320.	0.0563263
$489$	0	0
$490$	0	0
$491$	4.68450e6	0.876919	0.438460	−	0.898751i	$-0.355524\pi$
$491$	4.68450e6	0.876919	0.438460	+	0.898751i	$0.355524\pi$
$492$	0	0
$493$	−5.86728e6	−1.08723
$494$	2.58705e6	0.476966
$495$	0	0
$496$	1.61101e6	0.294031
$497$	0	0
$498$	0	0
$499$	1.47575e6	0.265315	0.132658	−	0.991162i	$-0.457649\pi$
$499$	1.47575e6	0.265315	0.132658	+	0.991162i	$0.457649\pi$
$500$	−596544.	−0.106713
$501$	0	0
$502$	−6.05333e6	−1.07210
$503$	−63606.0	−0.0112093	−0.00560465	−	0.999984i	$-0.501784\pi$
$503$	−63606.0	−0.0112093	−0.00560465	+	0.999984i	$0.501784\pi$
$504$	0	0
$505$	560340.	0.0977740
$506$	−9.23875e6	−1.60412
$507$	0	0
$508$	5.07349e6	0.872263
$509$	−6.21157e6	−1.06269	−0.531345	−	0.847155i	$-0.678313\pi$
$509$	−6.21157e6	−1.06269	−0.531345	+	0.847155i	$0.678313\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−6.21972e6	−1.03840
$515$	−1.00639e6	−0.167204
$516$	0	0
$517$	−1.53846e6	−0.253139
$518$	0	0
$519$	0	0
$520$	214656.	0.0348125
$521$	−1.41205e6	−0.227906	−0.113953	−	0.993486i	$-0.536351\pi$
$521$	−1.41205e6	−0.227906	−0.113953	+	0.993486i	$0.536351\pi$
$522$	0	0
$523$	5.22935e6	0.835975	0.417987	−	0.908453i	$-0.362736\pi$
$523$	5.22935e6	0.835975	0.417987	+	0.908453i	$0.362736\pi$
$524$	2.47728e6	0.394137
$525$	0	0
$526$	−4.46126e6	−0.703062
$527$	1.09498e7	1.71744
$528$	0	0
$529$	5.59068e6	0.868611
$530$	−679104.	−0.105014
$531$	0	0
$532$	0	0
$533$	−2.72009e6	−0.414730
$534$	0	0
$535$	−415080.	−0.0626971
$536$	1.19968e6	0.180365
$537$	0	0
$538$	142680.	0.0212524
$539$	0	0
$540$	0	0
$541$	4.41372e6	0.648354	0.324177	−	0.945996i	$-0.394913\pi$
$541$	4.41372e6	0.648354	0.324177	+	0.945996i	$0.394913\pi$
$542$	1.17107e6	0.171232
$543$	0	0
$544$	−1.78176e6	−0.258138
$545$	−1.31735e6	−0.189981
$546$	0	0
$547$	−1.19038e7	−1.70105	−0.850523	−	0.525938i	$-0.823714\pi$
$547$	−1.19038e7	−1.70105	−0.850523	+	0.525938i	$0.823714\pi$
$548$	−1.07731e6	−0.153246
$549$	0	0
$550$	8.22910e6	1.15997
$551$	−3.90140e6	−0.547447
$552$	0	0
$553$	0	0
$554$	−3.45285e6	−0.477973
$555$	0	0
$556$	−5.84344e6	−0.801644
$557$	−1.29133e7	−1.76360	−0.881798	−	0.471626i	$-0.843667\pi$
$557$	−1.29133e7	−1.76360	−0.881798	+	0.471626i	$0.843667\pi$
$558$	0	0
$559$	6.37651e6	0.863085
$560$	0	0
$561$	0	0
$562$	5.88442e6	0.785891
$563$	1.13698e7	1.51176	0.755881	−	0.654709i	$-0.227209\pi$
$563$	1.13698e7	1.51176	0.755881	+	0.654709i	$0.227209\pi$
$564$	0	0
$565$	236124.	0.0311185
$566$	−2.75536e6	−0.361525
$567$	0	0
$568$	−2.44646e6	−0.318176
$569$	5.69795e6	0.737799	0.368900	−	0.929469i	$-0.379735\pi$
$569$	5.69795e6	0.737799	0.368900	+	0.929469i	$0.379735\pi$
$570$	0	0
$571$	−7.04221e6	−0.903896	−0.451948	−	0.892044i	$-0.649271\pi$
$571$	−7.04221e6	−0.903896	−0.451948	+	0.892044i	$0.649271\pi$
$572$	−5.95670e6	−0.761230
$573$	0	0
$574$	0	0
$575$	−1.07127e7	−1.35122
$576$	0	0
$577$	−2.58197e6	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$577$	−2.58197e6	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$578$	−6.43097e6	−0.800676
$579$	0	0
$580$	−323712.	−0.0399566
$581$	0	0
$582$	0	0
$583$	1.88451e7	2.29630
$584$	−4.51770e6	−0.548132
$585$	0	0
$586$	2.89133e6	0.347819
$587$	−4.69459e6	−0.562345	−0.281172	−	0.959657i	$-0.590723\pi$
$587$	−4.69459e6	−0.562345	−0.281172	+	0.959657i	$0.590723\pi$
$588$	0	0
$589$	7.28100e6	0.864774
$590$	493056.	0.0583131
$591$	0	0
$592$	801536.	0.0939980
$593$	−1.34235e7	−1.56758	−0.783789	−	0.621027i	$-0.786716\pi$
$593$	−1.34235e7	−1.56758	−0.783789	+	0.621027i	$0.786716\pi$
$594$	0	0
$595$	0	0
$596$	2.68896e6	0.310076
$597$	0	0
$598$	7.75445e6	0.886743
$599$	−5.04601e6	−0.574621	−0.287310	−	0.957838i	$-0.592761\pi$
$599$	−5.04601e6	−0.574621	−0.287310	+	0.957838i	$0.592761\pi$
$600$	0	0
$601$	−1.06391e7	−1.20148	−0.600742	−	0.799443i	$-0.705128\pi$
$601$	−1.06391e7	−1.20148	−0.600742	+	0.799443i	$0.705128\pi$
$602$	0	0
$603$	0	0
$604$	2.45658e6	0.273992
$605$	1.69503e6	0.188273
$606$	0	0
$607$	1.41608e6	0.155997	0.0779986	−	0.996953i	$-0.475147\pi$
$607$	1.41608e6	0.155997	0.0779986	+	0.996953i	$0.475147\pi$
$608$	−1.18477e6	−0.129979
$609$	0	0
$610$	111120.	0.0120912
$611$	1.29129e6	0.139933
$612$	0	0
$613$	9.46303e6	1.01714	0.508568	−	0.861022i	$-0.330175\pi$
$613$	9.46303e6	1.01714	0.508568	+	0.861022i	$0.330175\pi$
$614$	80500.0	0.00861737
$615$	0	0
$616$	0	0
$617$	−1.29388e7	−1.36830	−0.684148	−	0.729343i	$-0.739826\pi$
$617$	−1.29388e7	−1.36830	−0.684148	+	0.729343i	$0.739826\pi$
$618$	0	0
$619$	−3.80376e6	−0.399013	−0.199506	−	0.979897i	$-0.563934\pi$
$619$	−3.80376e6	−0.399013	−0.199506	+	0.979897i	$0.563934\pi$
$620$	604128.	0.0631175
$621$	0	0
$622$	−6.97423e6	−0.722804
$623$	0	0
$624$	0	0
$625$	9.42942e6	0.965573
$626$	−7.23417e6	−0.737824
$627$	0	0
$628$	3.23869e6	0.327695
$629$	5.44794e6	0.549042
$630$	0	0
$631$	−9.17498e6	−0.917343	−0.458671	−	0.888606i	$-0.651674\pi$
$631$	−9.17498e6	−0.917343	−0.458671	+	0.888606i	$0.651674\pi$
$632$	3.98675e6	0.397033
$633$	0	0
$634$	4.09421e6	0.404526
$635$	1.90256e6	0.187242
$636$	0	0
$637$	0	0
$638$	8.98301e6	0.873716
$639$	0	0
$640$	−98304.0	−0.00948683
$641$	−1.02454e7	−0.984879	−0.492439	−	0.870347i	$-0.663895\pi$
$641$	−1.02454e7	−0.984879	−0.492439	+	0.870347i	$0.663895\pi$
$642$	0	0
$643$	−5.72346e6	−0.545922	−0.272961	−	0.962025i	$-0.588003\pi$
$643$	−5.72346e6	−0.545922	−0.272961	+	0.962025i	$0.588003\pi$
$644$	0	0
$645$	0	0
$646$	−8.05272e6	−0.759209
$647$	9.98794e6	0.938027	0.469013	−	0.883191i	$-0.344610\pi$
$647$	9.98794e6	0.938027	0.469013	+	0.883191i	$0.344610\pi$
$648$	0	0
$649$	−1.36823e7	−1.27511
$650$	−6.90700e6	−0.641219
$651$	0	0
$652$	−2.87622e6	−0.264974
$653$	1.19888e6	0.110025	0.0550126	−	0.998486i	$-0.482480\pi$
$653$	1.19888e6	0.110025	0.0550126	+	0.998486i	$0.482480\pi$
$654$	0	0
$655$	928980.	0.0846064
$656$	1.24570e6	0.113019
$657$	0	0
$658$	0	0
$659$	−1.18065e7	−1.05903	−0.529516	−	0.848300i	$-0.677626\pi$
$659$	−1.18065e7	−1.05903	−0.529516	+	0.848300i	$0.677626\pi$
$660$	0	0
$661$	4.72039e6	0.420218	0.210109	−	0.977678i	$-0.432618\pi$
$661$	4.72039e6	0.420218	0.210109	+	0.977678i	$0.432618\pi$
$662$	4.03013e6	0.357416
$663$	0	0
$664$	5.10835e6	0.449636
$665$	0	0
$666$	0	0
$667$	−1.16941e7	−1.01778
$668$	−3.47683e6	−0.301469
$669$	0	0
$670$	449880.	0.0387177
$671$	−3.08358e6	−0.264392
$672$	0	0
$673$	−8.70826e6	−0.741129	−0.370564	−	0.928807i	$-0.620836\pi$
$673$	−8.70826e6	−0.741129	−0.370564	+	0.928807i	$0.620836\pi$
$674$	6.26283e6	0.531032
$675$	0	0
$676$	−940992.	−0.0791989
$677$	−5.11105e6	−0.428587	−0.214293	−	0.976769i	$-0.568745\pi$
$677$	−5.11105e6	−0.428587	−0.214293	+	0.976769i	$0.568745\pi$
$678$	0	0
$679$	0	0
$680$	−668160.	−0.0554126
$681$	0	0
$682$	−1.67646e7	−1.38017
$683$	1.77198e7	1.45347	0.726736	−	0.686917i	$-0.241037\pi$
$683$	1.77198e7	1.45347	0.726736	+	0.686917i	$0.241037\pi$
$684$	0	0
$685$	−403992.	−0.0328962
$686$	0	0
$687$	0	0
$688$	−2.92019e6	−0.235202
$689$	−1.58175e7	−1.26937
$690$	0	0
$691$	−2.25993e7	−1.80053	−0.900265	−	0.435341i	$-0.856628\pi$
$691$	−2.25993e7	−1.80053	−0.900265	+	0.435341i	$0.856628\pi$
$692$	1.18368e6	0.0939656
$693$	0	0
$694$	3.02914e6	0.238737
$695$	−2.19129e6	−0.172083
$696$	0	0
$697$	8.46684e6	0.660145
$698$	1.82255e6	0.141593
$699$	0	0
$700$	0	0
$701$	818148.	0.0628835	0.0314418	−	0.999506i	$-0.489990\pi$
$701$	818148.	0.0628835	0.0314418	+	0.999506i	$0.489990\pi$
$702$	0	0
$703$	3.62257e6	0.276457
$704$	2.72794e6	0.207445
$705$	0	0
$706$	−1.45255e7	−1.09678
$707$	0	0
$708$	0	0
$709$	−5.09183e6	−0.380415	−0.190208	−	0.981744i	$-0.560916\pi$
$709$	−5.09183e6	−0.380415	−0.190208	+	0.981744i	$0.560916\pi$
$710$	−917424.	−0.0683006
$711$	0	0
$712$	−1.15968e6	−0.0857311
$713$	2.18241e7	1.60773
$714$	0	0
$715$	−2.23376e6	−0.163408
$716$	−1.26299e7	−0.920695
$717$	0	0
$718$	−1.60993e7	−1.16546
$719$	480858.	0.0346892	0.0173446	−	0.999850i	$-0.494479\pi$
$719$	480858.	0.0346892	0.0173446	+	0.999850i	$0.494479\pi$
$720$	0	0
$721$	0	0
$722$	4.54980e6	0.324825
$723$	0	0
$724$	−7.64382e6	−0.541956
$725$	1.04161e7	0.735971
$726$	0	0
$727$	−1.40783e7	−0.987905	−0.493952	−	0.869489i	$-0.664448\pi$
$727$	−1.40783e7	−0.987905	−0.493952	+	0.869489i	$0.664448\pi$
$728$	0	0
$729$	0	0
$730$	−1.69414e6	−0.117663
$731$	−1.98482e7	−1.37381
$732$	0	0
$733$	2.03932e6	0.140193	0.0700964	−	0.997540i	$-0.477669\pi$
$733$	2.03932e6	0.140193	0.0700964	+	0.997540i	$0.477669\pi$
$734$	−1.03115e7	−0.706450
$735$	0	0
$736$	−3.55123e6	−0.241649
$737$	−1.24842e7	−0.846625
$738$	0	0
$739$	−1.64957e7	−1.11112	−0.555558	−	0.831478i	$-0.687495\pi$
$739$	−1.64957e7	−1.11112	−0.555558	+	0.831478i	$0.687495\pi$
$740$	300576.	0.0201779
$741$	0	0
$742$	0	0
$743$	2.38121e7	1.58243	0.791217	−	0.611536i	$-0.209448\pi$
$743$	2.38121e7	1.58243	0.791217	+	0.611536i	$0.209448\pi$
$744$	0	0
$745$	1.00836e6	0.0665618
$746$	1.01253e7	0.666134
$747$	0	0
$748$	1.85414e7	1.21168
$749$	0	0
$750$	0	0
$751$	1.92496e6	0.124544	0.0622719	−	0.998059i	$-0.480165\pi$
$751$	1.92496e6	0.124544	0.0622719	+	0.998059i	$0.480165\pi$
$752$	−591360.	−0.0381336
$753$	0	0
$754$	−7.53979e6	−0.482982
$755$	921216.	0.0588158
$756$	0	0
$757$	8.98092e6	0.569615	0.284807	−	0.958585i	$-0.408070\pi$
$757$	8.98092e6	0.569615	0.284807	+	0.958585i	$0.408070\pi$
$758$	1.22671e7	0.775475
$759$	0	0
$760$	−444288.	−0.0279017
$761$	−1.45991e7	−0.913827	−0.456914	−	0.889511i	$-0.651045\pi$
$761$	−1.45991e7	−0.913827	−0.456914	+	0.889511i	$0.651045\pi$
$762$	0	0
$763$	0	0
$764$	−5.74358e6	−0.356000
$765$	0	0
$766$	−1.57008e7	−0.966829
$767$	1.14841e7	0.704869
$768$	0	0
$769$	2.78381e7	1.69755	0.848776	−	0.528753i	$-0.177340\pi$
$769$	2.78381e7	1.69755	0.848776	+	0.528753i	$0.177340\pi$
$770$	0	0
$771$	0	0
$772$	−2.91093e6	−0.175788
$773$	2.82857e7	1.70262	0.851312	−	0.524660i	$-0.175808\pi$
$773$	2.82857e7	1.70262	0.851312	+	0.524660i	$0.175808\pi$
$774$	0	0
$775$	−1.94391e7	−1.16258
$776$	−7.98426e6	−0.475971
$777$	0	0
$778$	−1.61068e7	−0.954024
$779$	5.62996e6	0.332401
$780$	0	0
$781$	2.54585e7	1.49350
$782$	−2.41373e7	−1.41147
$783$	0	0
$784$	0	0
$785$	1.21451e6	0.0703439
$786$	0	0
$787$	9872.00	0.000568157 0	0.000284078	−	1.00000i	$-0.499910\pi$
$787$	9872.00	0.000568157 0	0.000284078		1.00000i	$0.499910\pi$
$788$	−1.14868e7	−0.658996
$789$	0	0
$790$	1.49503e6	0.0852281
$791$	0	0
$792$	0	0
$793$	2.58817e6	0.146154
$794$	−1.82976e7	−1.03001
$795$	0	0
$796$	3.24954e6	0.181777
$797$	1.08919e7	0.607377	0.303688	−	0.952771i	$-0.401782\pi$
$797$	1.08919e7	0.607377	0.303688	+	0.952771i	$0.401782\pi$
$798$	0	0
$799$	−4.01940e6	−0.222738
$800$	3.16314e6	0.174740
$801$	0	0
$802$	−9.07776e6	−0.498360
$803$	4.70123e7	2.57290
$804$	0	0
$805$	0	0
$806$	1.40711e7	0.762943
$807$	0	0
$808$	−5.97696e6	−0.322071
$809$	5.23529e6	0.281235	0.140618	−	0.990064i	$-0.455091\pi$
$809$	5.23529e6	0.281235	0.140618	+	0.990064i	$0.455091\pi$
$810$	0	0
$811$	1.26147e7	0.673482	0.336741	−	0.941597i	$-0.390675\pi$
$811$	1.26147e7	0.673482	0.336741	+	0.941597i	$0.390675\pi$
$812$	0	0
$813$	0	0
$814$	−8.34098e6	−0.441221
$815$	−1.07858e6	−0.0568800
$816$	0	0
$817$	−1.31979e7	−0.691751
$818$	1.61839e7	0.845666
$819$	0	0
$820$	467136.	0.0242610
$821$	1.53424e7	0.794392	0.397196	−	0.917734i	$-0.369983\pi$
$821$	1.53424e7	0.794392	0.397196	+	0.917734i	$0.369983\pi$
$822$	0	0
$823$	−1.49595e7	−0.769870	−0.384935	−	0.922944i	$-0.625776\pi$
$823$	−1.49595e7	−0.769870	−0.384935	+	0.922944i	$0.625776\pi$
$824$	1.07348e7	0.550776
$825$	0	0
$826$	0	0
$827$	−2.80498e7	−1.42615	−0.713076	−	0.701087i	$-0.752699\pi$
$827$	−2.80498e7	−1.42615	−0.713076	+	0.701087i	$0.752699\pi$
$828$	0	0
$829$	1.47311e7	0.744473	0.372236	−	0.928138i	$-0.378591\pi$
$829$	1.47311e7	0.744473	0.372236	+	0.928138i	$0.378591\pi$
$830$	1.91563e6	0.0965199
$831$	0	0
$832$	−2.28966e6	−0.114674
$833$	0	0
$834$	0	0
$835$	−1.30381e6	−0.0647141
$836$	1.23290e7	0.610115
$837$	0	0
$838$	−1.56512e7	−0.769908
$839$	−2.36347e7	−1.15916	−0.579581	−	0.814914i	$-0.696784\pi$
$839$	−2.36347e7	−1.15916	−0.579581	+	0.814914i	$0.696784\pi$
$840$	0	0
$841$	−9.14076e6	−0.445649
$842$	1.11235e7	0.540704
$843$	0	0
$844$	1.87357e7	0.905343
$845$	−352872.	−0.0170010
$846$	0	0
$847$	0	0
$848$	7.24378e6	0.345920
$849$	0	0
$850$	2.14994e7	1.02066
$851$	1.08583e7	0.513971
$852$	0	0
$853$	1.43965e7	0.677459	0.338730	−	0.940884i	$-0.390003\pi$
$853$	1.43965e7	0.677459	0.338730	+	0.940884i	$0.390003\pi$
$854$	0	0
$855$	0	0
$856$	4.42752e6	0.206527
$857$	−1.24710e7	−0.580027	−0.290014	−	0.957023i	$-0.593660\pi$
$857$	−1.24710e7	−0.580027	−0.290014	+	0.957023i	$0.593660\pi$
$858$	0	0
$859$	−1.25059e7	−0.578271	−0.289136	−	0.957288i	$-0.593368\pi$
$859$	−1.25059e7	−0.578271	−0.289136	+	0.957288i	$0.593368\pi$
$860$	−1.09507e6	−0.0504890
$861$	0	0
$862$	−1.75283e7	−0.803473
$863$	−2.80289e7	−1.28109	−0.640545	−	0.767921i	$-0.721291\pi$
$863$	−2.80289e7	−1.28109	−0.640545	+	0.767921i	$0.721291\pi$
$864$	0	0
$865$	443880.	0.0201709
$866$	−4.99785e6	−0.226459
$867$	0	0
$868$	0	0
$869$	−4.14871e7	−1.86365
$870$	0	0
$871$	1.04785e7	0.468006
$872$	1.40518e7	0.625806
$873$	0	0
$874$	−1.60499e7	−0.710712
$875$	0	0
$876$	0	0
$877$	2.31173e7	1.01493	0.507467	−	0.861671i	$-0.330582\pi$
$877$	2.31173e7	1.01493	0.507467	+	0.861671i	$0.330582\pi$
$878$	2.69768e7	1.18101
$879$	0	0
$880$	1.02298e6	0.0445306
$881$	9.59891e6	0.416660	0.208330	−	0.978059i	$-0.433197\pi$
$881$	9.59891e6	0.416660	0.208330	+	0.978059i	$0.433197\pi$
$882$	0	0
$883$	443183.	0.0191285	0.00956426	−	0.999954i	$-0.496956\pi$
$883$	443183.	0.0191285	0.00956426	+	0.999954i	$0.496956\pi$
$884$	−1.55626e7	−0.669808
$885$	0	0
$886$	1.91558e6	0.0819817
$887$	2.40097e7	1.02465	0.512327	−	0.858791i	$-0.328784\pi$
$887$	2.40097e7	1.02465	0.512327	+	0.858791i	$0.328784\pi$
$888$	0	0
$889$	0	0
$890$	−434880.	−0.0184032
$891$	0	0
$892$	1.99416e7	0.839167
$893$	−2.67267e6	−0.112154
$894$	0	0
$895$	−4.73620e6	−0.197639
$896$	0	0
$897$	0	0
$898$	2.89802e6	0.119925
$899$	−2.12200e7	−0.875681
$900$	0	0
$901$	4.92350e7	2.02052
$902$	−1.29630e7	−0.530506
$903$	0	0
$904$	−2.51866e6	−0.102506
$905$	−2.86643e6	−0.116338
$906$	0	0
$907$	1.75854e7	0.709796	0.354898	−	0.934905i	$-0.384516\pi$
$907$	1.75854e7	0.709796	0.354898	+	0.934905i	$0.384516\pi$
$908$	1.47031e7	0.591825
$909$	0	0
$910$	0	0
$911$	2.95599e7	1.18007	0.590033	−	0.807379i	$-0.299115\pi$
$911$	2.95599e7	1.18007	0.590033	+	0.807379i	$0.299115\pi$
$912$	0	0
$913$	−5.31588e7	−2.11056
$914$	9.35825e6	0.370535
$915$	0	0
$916$	1.92600e7	0.758433
$917$	0	0
$918$	0	0
$919$	−3.91482e6	−0.152906	−0.0764528	−	0.997073i	$-0.524359\pi$
$919$	−3.91482e6	−0.152906	−0.0764528	+	0.997073i	$0.524359\pi$
$920$	−1.33171e6	−0.0518729
$921$	0	0
$922$	1.19299e7	0.462177
$923$	−2.13683e7	−0.825594
$924$	0	0
$925$	−9.67166e6	−0.371661
$926$	−1.71369e7	−0.656757
$927$	0	0
$928$	3.45293e6	0.131619
$929$	3.22009e6	0.122413	0.0612066	−	0.998125i	$-0.480505\pi$
$929$	3.22009e6	0.122413	0.0612066	+	0.998125i	$0.480505\pi$
$930$	0	0
$931$	0	0
$932$	1.47050e7	0.554530
$933$	0	0
$934$	−2.29614e7	−0.861254
$935$	6.95304e6	0.260103
$936$	0	0
$937$	−5.16504e7	−1.92187	−0.960936	−	0.276772i	$-0.910735\pi$
$937$	−5.16504e7	−1.92187	−0.960936	+	0.276772i	$0.910735\pi$
$938$	0	0
$939$	0	0
$940$	−221760.	−0.00818585
$941$	7.28698e6	0.268271	0.134135	−	0.990963i	$-0.457174\pi$
$941$	7.28698e6	0.268271	0.134135	+	0.990963i	$0.457174\pi$
$942$	0	0
$943$	1.68753e7	0.617977
$944$	−5.25926e6	−0.192086
$945$	0	0
$946$	3.03882e7	1.10402
$947$	3.12448e7	1.13215	0.566073	−	0.824355i	$-0.308462\pi$
$947$	3.12448e7	1.13215	0.566073	+	0.824355i	$0.308462\pi$
$948$	0	0
$949$	−3.94593e7	−1.42227
$950$	1.42959e7	0.513928
$951$	0	0
$952$	0	0
$953$	2.68273e7	0.956853	0.478426	−	0.878128i	$-0.341207\pi$
$953$	2.68273e7	0.956853	0.478426	+	0.878128i	$0.341207\pi$
$954$	0	0
$955$	−2.15384e6	−0.0764198
$956$	1.00054e7	0.354071
$957$	0	0
$958$	1.06020e7	0.373230
$959$	0	0
$960$	0	0
$961$	1.09727e7	0.383270
$962$	7.00092e6	0.243903
$963$	0	0
$964$	2.00612e7	0.695286
$965$	−1.09160e6	−0.0377350
$966$	0	0
$967$	−1.00257e7	−0.344784	−0.172392	−	0.985028i	$-0.555150\pi$
$967$	−1.00257e7	−0.344784	−0.172392	+	0.985028i	$0.555150\pi$
$968$	−1.80803e7	−0.620180
$969$	0	0
$970$	−2.99410e6	−0.102173
$971$	−115800.	−0.00394149	−0.00197075	−	0.999998i	$-0.500627\pi$
$971$	−115800.	−0.00394149	−0.00197075	+	0.999998i	$0.500627\pi$
$972$	0	0
$973$	0	0
$974$	−1.12222e7	−0.379035
$975$	0	0
$976$	−1.18528e6	−0.0398287
$977$	−8.73809e6	−0.292874	−0.146437	−	0.989220i	$-0.546781\pi$
$977$	−8.73809e6	−0.292874	−0.146437	+	0.989220i	$0.546781\pi$
$978$	0	0
$979$	1.20679e7	0.402416
$980$	0	0
$981$	0	0
$982$	−1.87380e7	−0.620075
$983$	−2.38413e7	−0.786947	−0.393473	−	0.919336i	$-0.628727\pi$
$983$	−2.38413e7	−0.786947	−0.393473	+	0.919336i	$0.628727\pi$
$984$	0	0
$985$	−4.30754e6	−0.141462
$986$	2.34691e7	0.768784
$987$	0	0
$988$	−1.03482e7	−0.337266
$989$	−3.95595e7	−1.28606
$990$	0	0
$991$	−3.33719e7	−1.07944	−0.539718	−	0.841846i	$-0.681469\pi$
$991$	−3.33719e7	−1.07944	−0.539718	+	0.841846i	$0.681469\pi$
$992$	−6.44403e6	−0.207911
$993$	0	0
$994$	0	0
$995$	1.21858e6	0.0390207
$996$	0	0
$997$	5.44687e7	1.73544	0.867718	−	0.497056i	$-0.165586\pi$
$997$	5.44687e7	1.73544	0.867718	+	0.497056i	$0.165586\pi$
$998$	−5.90301e6	−0.187606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.f.1.1		1
3.2	odd	2		294.6.a.l.1.1		1
7.2	even	3		126.6.g.c.109.1		2
7.4	even	3		126.6.g.c.37.1		2
7.6	odd	2		882.6.a.e.1.1		1
21.2	odd	6		42.6.e.a.25.1	✓	2
21.5	even	6		294.6.e.e.67.1		2
21.11	odd	6		42.6.e.a.37.1	yes	2
21.17	even	6		294.6.e.e.79.1		2
21.20	even	2		294.6.a.j.1.1		1
84.11	even	6		336.6.q.c.289.1		2
84.23	even	6		336.6.q.c.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.a.25.1	✓	2	21.2	odd	6
42.6.e.a.37.1	yes	2	21.11	odd	6
126.6.g.c.37.1		2	7.4	even	3
126.6.g.c.109.1		2	7.2	even	3
294.6.a.j.1.1		1	21.20	even	2
294.6.a.l.1.1		1	3.2	odd	2
294.6.e.e.67.1		2	21.5	even	6
294.6.e.e.79.1		2	21.17	even	6
336.6.q.c.193.1		2	84.23	even	6
336.6.q.c.289.1		2	84.11	even	6
882.6.a.e.1.1		1	7.6	odd	2
882.6.a.f.1.1		1	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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