LMFDB - Embedded newform 882.6.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,6,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,8,0,32,0,0,0,128,0,0,-952] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.458529075$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} +94.9526 q^{5} +64.0000 q^{8} +379.810 q^{10} -476.000 q^{11} +963.091 q^{13} +256.000 q^{16} +895.268 q^{17} +637.539 q^{19} +1519.24 q^{20} -1904.00 q^{22} -3696.00 q^{23} +5891.00 q^{25} +3852.36 q^{26} -1394.00 q^{29} +1926.18 q^{31} +1024.00 q^{32} +3581.07 q^{34} +12090.0 q^{37} +2550.16 q^{38} +6076.97 q^{40} -15219.5 q^{41} +9724.00 q^{43} -7616.00 q^{44} -14784.0 q^{46} +29272.5 q^{47} +23564.0 q^{50} +15409.5 q^{52} -4310.00 q^{53} -45197.4 q^{55} -5576.00 q^{58} +20848.9 q^{59} +9291.79 q^{61} +7704.73 q^{62} +4096.00 q^{64} +91448.0 q^{65} +20236.0 q^{67} +14324.3 q^{68} -29792.0 q^{71} +11285.8 q^{73} +48360.0 q^{74} +10200.6 q^{76} -33176.0 q^{79} +24307.9 q^{80} -60878.2 q^{82} +3540.38 q^{83} +85008.0 q^{85} +38896.0 q^{86} -30464.0 q^{88} +70753.3 q^{89} -59136.0 q^{92} +117090. q^{94} +60536.0 q^{95} -17769.7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8} - 952 q^{11} + 512 q^{16} - 3808 q^{22} - 7392 q^{23} + 11782 q^{25} - 2788 q^{29} + 2048 q^{32} + 24180 q^{37} + 19448 q^{43} - 15232 q^{44} - 29568 q^{46} + 47128 q^{50}+ \cdots + 121072 q^{95}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8 - 952 * q^11 + 512 * q^16 - 3808 * q^22 - 7392 * q^23 + 11782 * q^25 - 2788 * q^29 + 2048 * q^32 + 24180 * q^37 + 19448 * q^43 - 15232 * q^44 - 29568 * q^46 + 47128 * q^50 - 8620 * q^53 - 11152 * q^58 + 8192 * q^64 + 182896 * q^65 + 40472 * q^67 - 59584 * q^71 + 96720 * q^74 - 66352 * q^79 + 170016 * q^85 + 77792 * q^86 - 60928 * q^88 - 118272 * q^92 + 121072 * q^95

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
$5$	94.9526	1.69856	0.849282	−	0.527939i	$-0.177035\pi$
$5$	94.9526	1.69856	0.849282	+	0.527939i	$0.177035\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	379.810	1.20107
$11$	−476.000	−1.18611	−0.593055	−	0.805162i	$-0.702078\pi$
$11$	−476.000	−1.18611	−0.593055	+	0.805162i	$0.702078\pi$
$12$	0	0
$13$	963.091	1.58055	0.790276	−	0.612751i	$-0.209937\pi$
$13$	963.091	1.58055	0.790276	+	0.612751i	$0.209937\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	895.268	0.751330	0.375665	−	0.926756i	$-0.377414\pi$
$17$	895.268	0.751330	0.375665	+	0.926756i	$0.377414\pi$
$18$	0	0
$19$	637.539	0.405156	0.202578	−	0.979266i	$-0.435068\pi$
$19$	637.539	0.405156	0.202578	+	0.979266i	$0.435068\pi$
$20$	1519.24	0.849282
$21$	0	0
$22$	−1904.00	−0.838707
$23$	−3696.00	−1.45684	−0.728421	−	0.685130i	$-0.759745\pi$
$23$	−3696.00	−1.45684	−0.728421	+	0.685130i	$0.759745\pi$
$24$	0	0
$25$	5891.00	1.88512
$26$	3852.36	1.11762
$27$	0	0
$28$	0	0
$29$	−1394.00	−0.307799	−0.153900	−	0.988086i	$-0.549183\pi$
$29$	−1394.00	−0.307799	−0.153900	+	0.988086i	$0.549183\pi$
$30$	0	0
$31$	1926.18	0.359992	0.179996	−	0.983667i	$-0.442391\pi$
$31$	1926.18	0.359992	0.179996	+	0.983667i	$0.442391\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	3581.07	0.531270
$35$	0	0
$36$	0	0
$37$	12090.0	1.45185	0.725925	−	0.687773i	$-0.241412\pi$
$37$	12090.0	1.45185	0.725925	+	0.687773i	$0.241412\pi$
$38$	2550.16	0.286489
$39$	0	0
$40$	6076.97	0.600533
$41$	−15219.5	−1.41398	−0.706988	−	0.707225i	$-0.749947\pi$
$41$	−15219.5	−1.41398	−0.706988	+	0.707225i	$0.749947\pi$
$42$	0	0
$43$	9724.00	0.801999	0.400999	−	0.916078i	$-0.368663\pi$
$43$	9724.00	0.801999	0.400999	+	0.916078i	$0.368663\pi$
$44$	−7616.00	−0.593055
$45$	0	0
$46$	−14784.0	−1.03014
$47$	29272.5	1.93293	0.966464	−	0.256802i	$-0.0826688\pi$
$47$	29272.5	1.93293	0.966464	+	0.256802i	$0.0826688\pi$
$48$	0	0
$49$	0	0
$50$	23564.0	1.33298
$51$	0	0
$52$	15409.5	0.790276
$53$	−4310.00	−0.210760	−0.105380	−	0.994432i	$-0.533606\pi$
$53$	−4310.00	−0.210760	−0.105380	+	0.994432i	$0.533606\pi$
$54$	0	0
$55$	−45197.4	−2.01469
$56$	0	0
$57$	0	0
$58$	−5576.00	−0.217647
$59$	20848.9	0.779745	0.389873	−	0.920869i	$-0.372519\pi$
$59$	20848.9	0.779745	0.389873	+	0.920869i	$0.372519\pi$
$60$	0	0
$61$	9291.79	0.319724	0.159862	−	0.987139i	$-0.448895\pi$
$61$	9291.79	0.319724	0.159862	+	0.987139i	$0.448895\pi$
$62$	7704.73	0.254553
$63$	0	0
$64$	4096.00	0.125000
$65$	91448.0	2.68467
$66$	0	0
$67$	20236.0	0.550729	0.275364	−	0.961340i	$-0.411202\pi$
$67$	20236.0	0.550729	0.275364	+	0.961340i	$0.411202\pi$
$68$	14324.3	0.375665
$69$	0	0
$70$	0	0
$71$	−29792.0	−0.701381	−0.350690	−	0.936491i	$-0.614053\pi$
$71$	−29792.0	−0.701381	−0.350690	+	0.936491i	$0.614053\pi$
$72$	0	0
$73$	11285.8	0.247871	0.123935	−	0.992290i	$-0.460448\pi$
$73$	11285.8	0.247871	0.123935	+	0.992290i	$0.460448\pi$
$74$	48360.0	1.02661
$75$	0	0
$76$	10200.6	0.202578
$77$	0	0
$78$	0	0
$79$	−33176.0	−0.598076	−0.299038	−	0.954241i	$-0.596666\pi$
$79$	−33176.0	−0.598076	−0.299038	+	0.954241i	$0.596666\pi$
$80$	24307.9	0.424641
$81$	0	0
$82$	−60878.2	−0.999832
$83$	3540.38	0.0564098	0.0282049	−	0.999602i	$-0.491021\pi$
$83$	3540.38	0.0564098	0.0282049	+	0.999602i	$0.491021\pi$
$84$	0	0
$85$	85008.0	1.27618
$86$	38896.0	0.567099
$87$	0	0
$88$	−30464.0	−0.419353
$89$	70753.3	0.946829	0.473414	−	0.880840i	$-0.343021\pi$
$89$	70753.3	0.946829	0.473414	+	0.880840i	$0.343021\pi$
$90$	0	0
$91$	0	0
$92$	−59136.0	−0.728421
$93$	0	0
$94$	117090.	1.36679
$95$	60536.0	0.688184
$96$	0	0
$97$	−17769.7	−0.191757	−0.0958784	−	0.995393i	$-0.530566\pi$
$97$	−17769.7	−0.191757	−0.0958784	+	0.995393i	$0.530566\pi$
$98$	0	0
$99$	0	0
$100$	94256.0	0.942560
$101$	−89920.1	−0.877109	−0.438554	−	0.898705i	$-0.644509\pi$
$101$	−89920.1	−0.877109	−0.438554	+	0.898705i	$0.644509\pi$
$102$	0	0
$103$	−8111.67	−0.0753385	−0.0376693	−	0.999290i	$-0.511993\pi$
$103$	−8111.67	−0.0753385	−0.0376693	+	0.999290i	$0.511993\pi$
$104$	61637.8	0.558810
$105$	0	0
$106$	−17240.0	−0.149030
$107$	−125908.	−1.06315	−0.531574	−	0.847012i	$-0.678399\pi$
$107$	−125908.	−1.06315	−0.531574	+	0.847012i	$0.678399\pi$
$108$	0	0
$109$	89170.0	0.718874	0.359437	−	0.933169i	$-0.382969\pi$
$109$	89170.0	0.718874	0.359437	+	0.933169i	$0.382969\pi$
$110$	−180790.	−1.42460
$111$	0	0
$112$	0	0
$113$	29702.0	0.218821	0.109411	−	0.993997i	$-0.465104\pi$
$113$	29702.0	0.218821	0.109411	+	0.993997i	$0.465104\pi$
$114$	0	0
$115$	−350945.	−2.47454
$116$	−22304.0	−0.153900
$117$	0	0
$118$	83395.5	0.551363
$119$	0	0
$120$	0	0
$121$	65525.0	0.406859
$122$	37167.2	0.226079
$123$	0	0
$124$	30818.9	0.179996
$125$	262639.	1.50343
$126$	0	0
$127$	−243112.	−1.33751	−0.668755	−	0.743483i	$-0.733173\pi$
$127$	−243112.	−1.33751	−0.668755	+	0.743483i	$0.733173\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	365792.	1.89835
$131$	−107147.	−0.545510	−0.272755	−	0.962084i	$-0.587935\pi$
$131$	−107147.	−0.545510	−0.272755	+	0.962084i	$0.587935\pi$
$132$	0	0
$133$	0	0
$134$	80944.0	0.389424
$135$	0	0
$136$	57297.1	0.265635
$137$	332842.	1.51508	0.757542	−	0.652786i	$-0.226400\pi$
$137$	332842.	1.51508	0.757542	+	0.652786i	$0.226400\pi$
$138$	0	0
$139$	−186582.	−0.819092	−0.409546	−	0.912290i	$-0.634313\pi$
$139$	−186582.	−0.819092	−0.409546	+	0.912290i	$0.634313\pi$
$140$	0	0
$141$	0	0
$142$	−119168.	−0.495951
$143$	−458431.	−1.87471
$144$	0	0
$145$	−132364.	−0.522817
$146$	45143.2	0.175271
$147$	0	0
$148$	193440.	0.725925
$149$	202330.	0.746611	0.373306	−	0.927708i	$-0.378224\pi$
$149$	202330.	0.746611	0.373306	+	0.927708i	$0.378224\pi$
$150$	0	0
$151$	345760.	1.23405	0.617024	−	0.786944i	$-0.288338\pi$
$151$	345760.	1.23405	0.617024	+	0.786944i	$0.288338\pi$
$152$	40802.5	0.143244
$153$	0	0
$154$	0	0
$155$	182896.	0.611470
$156$	0	0
$157$	−82866.5	−0.268306	−0.134153	−	0.990961i	$-0.542831\pi$
$157$	−82866.5	−0.268306	−0.134153	+	0.990961i	$0.542831\pi$
$158$	−132704.	−0.422904
$159$	0	0
$160$	97231.5	0.300267
$161$	0	0
$162$	0	0
$163$	127476.	0.375802	0.187901	−	0.982188i	$-0.439832\pi$
$163$	127476.	0.375802	0.187901	+	0.982188i	$0.439832\pi$
$164$	−243513.	−0.706988
$165$	0	0
$166$	14161.5	0.0398877
$167$	−671261.	−1.86252	−0.931258	−	0.364360i	$-0.881288\pi$
$167$	−671261.	−1.86252	−0.931258	+	0.364360i	$0.881288\pi$
$168$	0	0
$169$	556251.	1.49815
$170$	340032.	0.902397
$171$	0	0
$172$	155584.	0.400999
$173$	−347784.	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$173$	−347784.	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$174$	0	0
$175$	0	0
$176$	−121856.	−0.296528
$177$	0	0
$178$	283013.	0.669509
$179$	625236.	1.45852	0.729258	−	0.684238i	$-0.239865\pi$
$179$	625236.	1.45852	0.729258	+	0.684238i	$0.239865\pi$
$180$	0	0
$181$	241492.	0.547906	0.273953	−	0.961743i	$-0.411669\pi$
$181$	241492.	0.547906	0.273953	+	0.961743i	$0.411669\pi$
$182$	0	0
$183$	0	0
$184$	−236544.	−0.515071
$185$	1.14798e6	2.46606
$186$	0	0
$187$	−426147.	−0.891160
$188$	468361.	0.966464
$189$	0	0
$190$	242144.	0.486620
$191$	−212952.	−0.422375	−0.211188	−	0.977446i	$-0.567733\pi$
$191$	−212952.	−0.422375	−0.211188	+	0.977446i	$0.567733\pi$
$192$	0	0
$193$	135002.	0.260884	0.130442	−	0.991456i	$-0.458360\pi$
$193$	135002.	0.260884	0.130442	+	0.991456i	$0.458360\pi$
$194$	−71078.8	−0.135593
$195$	0	0
$196$	0	0
$197$	−548838.	−1.00758	−0.503789	−	0.863827i	$-0.668061\pi$
$197$	−548838.	−1.00758	−0.503789	+	0.863827i	$0.668061\pi$
$198$	0	0
$199$	631869.	1.13108	0.565541	−	0.824720i	$-0.308667\pi$
$199$	631869.	1.13108	0.565541	+	0.824720i	$0.308667\pi$
$200$	377024.	0.666491
$201$	0	0
$202$	−359681.	−0.620210
$203$	0	0
$204$	0	0
$205$	−1.44514e6	−2.40173
$206$	−32446.7	−0.0532724
$207$	0	0
$208$	246551.	0.395138
$209$	−303469.	−0.480560
$210$	0	0
$211$	−159940.	−0.247315	−0.123658	−	0.992325i	$-0.539462\pi$
$211$	−159940.	−0.247315	−0.123658	+	0.992325i	$0.539462\pi$
$212$	−68960.0	−0.105380
$213$	0	0
$214$	−503632.	−0.751759
$215$	923319.	1.36225
$216$	0	0
$217$	0	0
$218$	356680.	0.508320
$219$	0	0
$220$	−723159.	−1.00734
$221$	862224.	1.18752
$222$	0	0
$223$	876765.	1.18065	0.590325	−	0.807166i	$-0.299000\pi$
$223$	876765.	1.18065	0.590325	+	0.807166i	$0.299000\pi$
$224$	0	0
$225$	0	0
$226$	118808.	0.154730
$227$	218513.	0.281458	0.140729	−	0.990048i	$-0.455055\pi$
$227$	218513.	0.281458	0.140729	+	0.990048i	$0.455055\pi$
$228$	0	0
$229$	1.42786e6	1.79927	0.899634	−	0.436644i	$-0.143833\pi$
$229$	1.42786e6	1.79927	0.899634	+	0.436644i	$0.143833\pi$
$230$	−1.40378e6	−1.74976
$231$	0	0
$232$	−89216.0	−0.108824
$233$	429418.	0.518192	0.259096	−	0.965852i	$-0.416575\pi$
$233$	429418.	0.518192	0.259096	+	0.965852i	$0.416575\pi$
$234$	0	0
$235$	2.77950e6	3.28320
$236$	333582.	0.389873
$237$	0	0
$238$	0	0
$239$	338328.	0.383127	0.191564	−	0.981480i	$-0.438644\pi$
$239$	338328.	0.383127	0.191564	+	0.981480i	$0.438644\pi$
$240$	0	0
$241$	536645.	0.595175	0.297587	−	0.954695i	$-0.403818\pi$
$241$	536645.	0.595175	0.297587	+	0.954695i	$0.403818\pi$
$242$	262100.	0.287693
$243$	0	0
$244$	148669.	0.159862
$245$	0	0
$246$	0	0
$247$	614008.	0.640371
$248$	123276.	0.127276
$249$	0	0
$250$	1.05056e6	1.06309
$251$	548813.	0.549844	0.274922	−	0.961466i	$-0.411348\pi$
$251$	548813.	0.549844	0.274922	+	0.961466i	$0.411348\pi$
$252$	0	0
$253$	1.75930e6	1.72798
$254$	−972448.	−0.945763
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.69591e6	1.60166	0.800828	−	0.598894i	$-0.204393\pi$
$257$	1.69591e6	1.60166	0.800828	+	0.598894i	$0.204393\pi$
$258$	0	0
$259$	0	0
$260$	1.46317e6	1.34233
$261$	0	0
$262$	−428589.	−0.385734
$263$	−1.96616e6	−1.75279	−0.876394	−	0.481594i	$-0.840058\pi$
$263$	−1.96616e6	−1.75279	−0.876394	+	0.481594i	$0.840058\pi$
$264$	0	0
$265$	−409246.	−0.357989
$266$	0	0
$267$	0	0
$268$	323776.	0.275364
$269$	431858.	0.363882	0.181941	−	0.983309i	$-0.441762\pi$
$269$	431858.	0.363882	0.181941	+	0.983309i	$0.441762\pi$
$270$	0	0
$271$	−590768.	−0.488645	−0.244323	−	0.969694i	$-0.578566\pi$
$271$	−590768.	−0.488645	−0.244323	+	0.969694i	$0.578566\pi$
$272$	229188.	0.187832
$273$	0	0
$274$	1.33137e6	1.07133
$275$	−2.80412e6	−2.23596
$276$	0	0
$277$	828262.	0.648587	0.324294	−	0.945956i	$-0.394873\pi$
$277$	828262.	0.648587	0.324294	+	0.945956i	$0.394873\pi$
$278$	−746328.	−0.579185
$279$	0	0
$280$	0	0
$281$	719130.	0.543302	0.271651	−	0.962396i	$-0.412430\pi$
$281$	719130.	0.543302	0.271651	+	0.962396i	$0.412430\pi$
$282$	0	0
$283$	−2.12413e6	−1.57658	−0.788288	−	0.615306i	$-0.789033\pi$
$283$	−2.12413e6	−1.57658	−0.788288	+	0.615306i	$0.789033\pi$
$284$	−476672.	−0.350690
$285$	0	0
$286$	−1.83372e6	−1.32562
$287$	0	0
$288$	0	0
$289$	−618353.	−0.435504
$290$	−529456.	−0.369687
$291$	0	0
$292$	180573.	0.123935
$293$	313710.	0.213481	0.106740	−	0.994287i	$-0.465959\pi$
$293$	313710.	0.213481	0.106740	+	0.994287i	$0.465959\pi$
$294$	0	0
$295$	1.97966e6	1.32445
$296$	773760.	0.513307
$297$	0	0
$298$	809320.	0.527934
$299$	−3.55958e6	−2.30261
$300$	0	0
$301$	0	0
$302$	1.38304e6	0.872604
$303$	0	0
$304$	163210.	0.101289
$305$	882280.	0.543071
$306$	0	0
$307$	2.82279e6	1.70936	0.854679	−	0.519157i	$-0.173754\pi$
$307$	2.82279e6	1.70936	0.854679	+	0.519157i	$0.173754\pi$
$308$	0	0
$309$	0	0
$310$	731584.	0.432374
$311$	441557.	0.258872	0.129436	−	0.991588i	$-0.458683\pi$
$311$	441557.	0.258872	0.129436	+	0.991588i	$0.458683\pi$
$312$	0	0
$313$	−1.42339e6	−0.821229	−0.410615	−	0.911809i	$-0.634686\pi$
$313$	−1.42339e6	−0.821229	−0.410615	+	0.911809i	$0.634686\pi$
$314$	−331466.	−0.189721
$315$	0	0
$316$	−530816.	−0.299038
$317$	−1.25360e6	−0.700665	−0.350332	−	0.936625i	$-0.613931\pi$
$317$	−1.25360e6	−0.700665	−0.350332	+	0.936625i	$0.613931\pi$
$318$	0	0
$319$	663544.	0.365084
$320$	388926.	0.212321
$321$	0	0
$322$	0	0
$323$	570768.	0.304406
$324$	0	0
$325$	5.67357e6	2.97953
$326$	509904.	0.265732
$327$	0	0
$328$	−974051.	−0.499916
$329$	0	0
$330$	0	0
$331$	−3.38223e6	−1.69681	−0.848404	−	0.529349i	$-0.822436\pi$
$331$	−3.38223e6	−1.69681	−0.848404	+	0.529349i	$0.822436\pi$
$332$	56646.0	0.0282049
$333$	0	0
$334$	−2.68504e6	−1.31700
$335$	1.92146e6	0.935448
$336$	0	0
$337$	1.94529e6	0.933058	0.466529	−	0.884506i	$-0.345504\pi$
$337$	1.94529e6	0.933058	0.466529	+	0.884506i	$0.345504\pi$
$338$	2.22500e6	1.05935
$339$	0	0
$340$	1.36013e6	0.638091
$341$	−916862.	−0.426991
$342$	0	0
$343$	0	0
$344$	622336.	0.283549
$345$	0	0
$346$	−1.39114e6	−0.624712
$347$	2.08232e6	0.928378	0.464189	−	0.885736i	$-0.346346\pi$
$347$	2.08232e6	0.928378	0.464189	+	0.885736i	$0.346346\pi$
$348$	0	0
$349$	−3.15765e6	−1.38772	−0.693858	−	0.720112i	$-0.744090\pi$
$349$	−3.15765e6	−1.38772	−0.693858	+	0.720112i	$0.744090\pi$
$350$	0	0
$351$	0	0
$352$	−487424.	−0.209677
$353$	−2.55021e6	−1.08928	−0.544640	−	0.838670i	$-0.683334\pi$
$353$	−2.55021e6	−1.08928	−0.544640	+	0.838670i	$0.683334\pi$
$354$	0	0
$355$	−2.82883e6	−1.19134
$356$	1.13205e6	0.473414
$357$	0	0
$358$	2.50094e6	1.03133
$359$	−2.43467e6	−0.997021	−0.498511	−	0.866884i	$-0.666119\pi$
$359$	−2.43467e6	−0.997021	−0.498511	+	0.866884i	$0.666119\pi$
$360$	0	0
$361$	−2.06964e6	−0.835848
$362$	965967.	0.387428
$363$	0	0
$364$	0	0
$365$	1.07162e6	0.421024
$366$	0	0
$367$	−4.33141e6	−1.67867	−0.839333	−	0.543617i	$-0.817054\pi$
$367$	−4.33141e6	−1.67867	−0.839333	+	0.543617i	$0.817054\pi$
$368$	−946176.	−0.364210
$369$	0	0
$370$	4.59191e6	1.74377
$371$	0	0
$372$	0	0
$373$	−3.66435e6	−1.36372	−0.681859	−	0.731484i	$-0.738828\pi$
$373$	−3.66435e6	−1.36372	−0.681859	+	0.731484i	$0.738828\pi$
$374$	−1.70459e6	−0.630145
$375$	0	0
$376$	1.87344e6	0.683393
$377$	−1.34255e6	−0.486493
$378$	0	0
$379$	−1.24352e6	−0.444689	−0.222344	−	0.974968i	$-0.571371\pi$
$379$	−1.24352e6	−0.444689	−0.222344	+	0.974968i	$0.571371\pi$
$380$	968576.	0.344092
$381$	0	0
$382$	−851808.	−0.298664
$383$	−1.50527e6	−0.524346	−0.262173	−	0.965021i	$-0.584439\pi$
$383$	−1.50527e6	−0.524346	−0.262173	+	0.965021i	$0.584439\pi$
$384$	0	0
$385$	0	0
$386$	540008.	0.184473
$387$	0	0
$388$	−284315.	−0.0958784
$389$	3.85930e6	1.29311	0.646554	−	0.762868i	$-0.276210\pi$
$389$	3.85930e6	1.29311	0.646554	+	0.762868i	$0.276210\pi$
$390$	0	0
$391$	−3.30891e6	−1.09457
$392$	0	0
$393$	0	0
$394$	−2.19535e6	−0.712465
$395$	−3.15015e6	−1.01587
$396$	0	0
$397$	3.32013e6	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$397$	3.32013e6	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$398$	2.52748e6	0.799796
$399$	0	0
$400$	1.50810e6	0.471280
$401$	−3.69337e6	−1.14699	−0.573497	−	0.819207i	$-0.694414\pi$
$401$	−3.69337e6	−1.14699	−0.573497	+	0.819207i	$0.694414\pi$
$402$	0	0
$403$	1.85509e6	0.568986
$404$	−1.43872e6	−0.438554
$405$	0	0
$406$	0	0
$407$	−5.75484e6	−1.72206
$408$	0	0
$409$	−4.04213e6	−1.19482	−0.597410	−	0.801936i	$-0.703803\pi$
$409$	−4.04213e6	−1.19482	−0.597410	+	0.801936i	$0.703803\pi$
$410$	−5.78054e6	−1.69828
$411$	0	0
$412$	−129787.	−0.0376693
$413$	0	0
$414$	0	0
$415$	336168.	0.0958156
$416$	986205.	0.279405
$417$	0	0
$418$	−1.21387e6	−0.339808
$419$	2.42383e6	0.674476	0.337238	−	0.941419i	$-0.390507\pi$
$419$	2.42383e6	0.674476	0.337238	+	0.941419i	$0.390507\pi$
$420$	0	0
$421$	−6.65639e6	−1.83035	−0.915174	−	0.403058i	$-0.867947\pi$
$421$	−6.65639e6	−1.83035	−0.915174	+	0.403058i	$0.867947\pi$
$422$	−639760.	−0.174878
$423$	0	0
$424$	−275840.	−0.0745148
$425$	5.27402e6	1.41635
$426$	0	0
$427$	0	0
$428$	−2.01453e6	−0.531574
$429$	0	0
$430$	3.69328e6	0.963254
$431$	610520.	0.158309	0.0791547	−	0.996862i	$-0.474778\pi$
$431$	610520.	0.158309	0.0791547	+	0.996862i	$0.474778\pi$
$432$	0	0
$433$	29977.9	0.00768390	0.00384195	−	0.999993i	$-0.498777\pi$
$433$	29977.9	0.00768390	0.00384195	+	0.999993i	$0.498777\pi$
$434$	0	0
$435$	0	0
$436$	1.42672e6	0.359437
$437$	−2.35634e6	−0.590249
$438$	0	0
$439$	369827.	0.0915877	0.0457939	−	0.998951i	$-0.485418\pi$
$439$	369827.	0.0915877	0.0457939	+	0.998951i	$0.485418\pi$
$440$	−2.89264e6	−0.712299
$441$	0	0
$442$	3.44890e6	0.839701
$443$	639036.	0.154709	0.0773546	−	0.997004i	$-0.475353\pi$
$443$	639036.	0.154709	0.0773546	+	0.997004i	$0.475353\pi$
$444$	0	0
$445$	6.71821e6	1.60825
$446$	3.50706e6	0.834846
$447$	0	0
$448$	0	0
$449$	−1.90682e6	−0.446368	−0.223184	−	0.974776i	$-0.571645\pi$
$449$	−1.90682e6	−0.446368	−0.223184	+	0.974776i	$0.571645\pi$
$450$	0	0
$451$	7.24451e6	1.67713
$452$	475232.	0.109411
$453$	0	0
$454$	874052.	0.199021
$455$	0	0
$456$	0	0
$457$	6.60039e6	1.47836	0.739178	−	0.673510i	$-0.235214\pi$
$457$	6.60039e6	1.47836	0.739178	+	0.673510i	$0.235214\pi$
$458$	5.71143e6	1.27227
$459$	0	0
$460$	−5.61512e6	−1.23727
$461$	−4.93085e6	−1.08061	−0.540305	−	0.841469i	$-0.681691\pi$
$461$	−4.93085e6	−1.08061	−0.540305	+	0.841469i	$0.681691\pi$
$462$	0	0
$463$	257576.	0.0558410	0.0279205	−	0.999610i	$-0.491111\pi$
$463$	257576.	0.0558410	0.0279205	+	0.999610i	$0.491111\pi$
$464$	−356864.	−0.0769499
$465$	0	0
$466$	1.71767e6	0.366417
$467$	−541976.	−0.114997	−0.0574987	−	0.998346i	$-0.518312\pi$
$467$	−541976.	−0.114997	−0.0574987	+	0.998346i	$0.518312\pi$
$468$	0	0
$469$	0	0
$470$	1.11180e7	2.32157
$471$	0	0
$472$	1.33433e6	0.275682
$473$	−4.62862e6	−0.951260
$474$	0	0
$475$	3.75574e6	0.763769
$476$	0	0
$477$	0	0
$478$	1.35331e6	0.270912
$479$	6.28828e6	1.25226	0.626128	−	0.779720i	$-0.284639\pi$
$479$	6.28828e6	1.25226	0.626128	+	0.779720i	$0.284639\pi$
$480$	0	0
$481$	1.16438e7	2.29473
$482$	2.14658e6	0.420852
$483$	0	0
$484$	1.04840e6	0.203429
$485$	−1.68728e6	−0.325711
$486$	0	0
$487$	−6.18478e6	−1.18169	−0.590843	−	0.806787i	$-0.701205\pi$
$487$	−6.18478e6	−1.18169	−0.590843	+	0.806787i	$0.701205\pi$
$488$	594675.	0.113039
$489$	0	0
$490$	0	0
$491$	4.93753e6	0.924286	0.462143	−	0.886806i	$-0.347081\pi$
$491$	4.93753e6	0.924286	0.462143	+	0.886806i	$0.347081\pi$
$492$	0	0
$493$	−1.24800e6	−0.231259
$494$	2.45603e6	0.452811
$495$	0	0
$496$	493103.	0.0899980
$497$	0	0
$498$	0	0
$499$	−742212.	−0.133437	−0.0667186	−	0.997772i	$-0.521253\pi$
$499$	−742212.	−0.133437	−0.0667186	+	0.997772i	$0.521253\pi$
$500$	4.20222e6	0.751717
$501$	0	0
$502$	2.19525e6	0.388799
$503$	−2.92872e6	−0.516128	−0.258064	−	0.966128i	$-0.583085\pi$
$503$	−2.92872e6	−0.516128	−0.258064	+	0.966128i	$0.583085\pi$
$504$	0	0
$505$	−8.53815e6	−1.48983
$506$	7.03718e6	1.22186
$507$	0	0
$508$	−3.88979e6	−0.668755
$509$	6.05761e6	1.03635	0.518176	−	0.855274i	$-0.326611\pi$
$509$	6.05761e6	1.03635	0.518176	+	0.855274i	$0.326611\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	6.78363e6	1.13254
$515$	−770224.	−0.127967
$516$	0	0
$517$	−1.39337e7	−2.29267
$518$	0	0
$519$	0	0
$520$	5.85267e6	0.949174
$521$	4.99920e6	0.806875	0.403438	−	0.915007i	$-0.367815\pi$
$521$	4.99920e6	0.806875	0.403438	+	0.915007i	$0.367815\pi$
$522$	0	0
$523$	−9.19002e6	−1.46914	−0.734568	−	0.678535i	$-0.762615\pi$
$523$	−9.19002e6	−1.46914	−0.734568	+	0.678535i	$0.762615\pi$
$524$	−1.71436e6	−0.272755
$525$	0	0
$526$	−7.86464e6	−1.23941
$527$	1.72445e6	0.270473
$528$	0	0
$529$	7.22407e6	1.12239
$530$	−1.63698e6	−0.253136
$531$	0	0
$532$	0	0
$533$	−1.46578e7	−2.23486
$534$	0	0
$535$	−1.19553e7	−1.80583
$536$	1.29510e6	0.194712
$537$	0	0
$538$	1.72743e6	0.257303
$539$	0	0
$540$	0	0
$541$	−7.53883e6	−1.10742	−0.553708	−	0.832711i	$-0.686788\pi$
$541$	−7.53883e6	−1.10742	−0.553708	+	0.832711i	$0.686788\pi$
$542$	−2.36307e6	−0.345524
$543$	0	0
$544$	916754.	0.132818
$545$	8.46693e6	1.22105
$546$	0	0
$547$	3.73311e6	0.533460	0.266730	−	0.963771i	$-0.414057\pi$
$547$	3.73311e6	0.533460	0.266730	+	0.963771i	$0.414057\pi$
$548$	5.32547e6	0.757542
$549$	0	0
$550$	−1.12165e7	−1.58106
$551$	−888729.	−0.124707
$552$	0	0
$553$	0	0
$554$	3.31305e6	0.458620
$555$	0	0
$556$	−2.98531e6	−0.409546
$557$	7.95391e6	1.08628	0.543141	−	0.839642i	$-0.317235\pi$
$557$	7.95391e6	1.08628	0.543141	+	0.839642i	$0.317235\pi$
$558$	0	0
$559$	9.36510e6	1.26760
$560$	0	0
$561$	0	0
$562$	2.87652e6	0.384173
$563$	−8.85242e6	−1.17704	−0.588520	−	0.808483i	$-0.700289\pi$
$563$	−8.85242e6	−1.17704	−0.588520	+	0.808483i	$0.700289\pi$
$564$	0	0
$565$	2.82028e6	0.371682
$566$	−8.49652e6	−1.11481
$567$	0	0
$568$	−1.90669e6	−0.247976
$569$	−6.43774e6	−0.833591	−0.416795	−	0.909000i	$-0.636847\pi$
$569$	−6.43774e6	−0.833591	−0.416795	+	0.909000i	$0.636847\pi$
$570$	0	0
$571$	−1.06947e7	−1.37271	−0.686353	−	0.727269i	$-0.740789\pi$
$571$	−1.06947e7	−1.37271	−0.686353	+	0.727269i	$0.740789\pi$
$572$	−7.33490e6	−0.937355
$573$	0	0
$574$	0	0
$575$	−2.17731e7	−2.74632
$576$	0	0
$577$	1.30088e7	1.62666	0.813330	−	0.581802i	$-0.197652\pi$
$577$	1.30088e7	1.62666	0.813330	+	0.581802i	$0.197652\pi$
$578$	−2.47341e6	−0.307948
$579$	0	0
$580$	−2.11782e6	−0.261409
$581$	0	0
$582$	0	0
$583$	2.05156e6	0.249984
$584$	722291.	0.0876355
$585$	0	0
$586$	1.25484e6	0.150954
$587$	−1.27410e7	−1.52619	−0.763096	−	0.646285i	$-0.776322\pi$
$587$	−1.27410e7	−1.52619	−0.763096	+	0.646285i	$0.776322\pi$
$588$	0	0
$589$	1.22802e6	0.145853
$590$	7.91862e6	0.936526
$591$	0	0
$592$	3.09504e6	0.362963
$593$	−2.71592e6	−0.317161	−0.158580	−	0.987346i	$-0.550692\pi$
$593$	−2.71592e6	−0.317161	−0.158580	+	0.987346i	$0.550692\pi$
$594$	0	0
$595$	0	0
$596$	3.23728e6	0.373306
$597$	0	0
$598$	−1.42383e7	−1.62819
$599$	1.04748e7	1.19284	0.596418	−	0.802674i	$-0.296590\pi$
$599$	1.04748e7	1.19284	0.596418	+	0.802674i	$0.296590\pi$
$600$	0	0
$601$	754738.	0.0852334	0.0426167	−	0.999091i	$-0.486431\pi$
$601$	754738.	0.0852334	0.0426167	+	0.999091i	$0.486431\pi$
$602$	0	0
$603$	0	0
$604$	5.53216e6	0.617024
$605$	6.22177e6	0.691076
$606$	0	0
$607$	9.17503e6	1.01073	0.505366	−	0.862905i	$-0.331358\pi$
$607$	9.17503e6	1.01073	0.505366	+	0.862905i	$0.331358\pi$
$608$	652840.	0.0716222
$609$	0	0
$610$	3.52912e6	0.384009
$611$	2.81921e7	3.05509
$612$	0	0
$613$	−8.85554e6	−0.951840	−0.475920	−	0.879489i	$-0.657885\pi$
$613$	−8.85554e6	−0.951840	−0.475920	+	0.879489i	$0.657885\pi$
$614$	1.12912e7	1.20870
$615$	0	0
$616$	0	0
$617$	−7.50753e6	−0.793934	−0.396967	−	0.917833i	$-0.629937\pi$
$617$	−7.50753e6	−0.793934	−0.396967	+	0.917833i	$0.629937\pi$
$618$	0	0
$619$	−1.01449e7	−1.06420	−0.532098	−	0.846683i	$-0.678596\pi$
$619$	−1.01449e7	−1.06420	−0.532098	+	0.846683i	$0.678596\pi$
$620$	2.92634e6	0.305735
$621$	0	0
$622$	1.76623e6	0.183050
$623$	0	0
$624$	0	0
$625$	6.52888e6	0.668557
$626$	−5.69358e6	−0.580697
$627$	0	0
$628$	−1.32586e6	−0.134153
$629$	1.08238e7	1.09082
$630$	0	0
$631$	−9.28258e6	−0.928101	−0.464050	−	0.885809i	$-0.653604\pi$
$631$	−9.28258e6	−0.928101	−0.464050	+	0.885809i	$0.653604\pi$
$632$	−2.12326e6	−0.211452
$633$	0	0
$634$	−5.01439e6	−0.495445
$635$	−2.30841e7	−2.27185
$636$	0	0
$637$	0	0
$638$	2.65418e6	0.258154
$639$	0	0
$640$	1.55570e6	0.150133
$641$	−1.70740e7	−1.64130	−0.820652	−	0.571428i	$-0.806390\pi$
$641$	−1.70740e7	−1.64130	−0.820652	+	0.571428i	$0.806390\pi$
$642$	0	0
$643$	2.73150e6	0.260540	0.130270	−	0.991479i	$-0.458416\pi$
$643$	2.73150e6	0.260540	0.130270	+	0.991479i	$0.458416\pi$
$644$	0	0
$645$	0	0
$646$	2.28307e6	0.215248
$647$	−5.66533e6	−0.532065	−0.266033	−	0.963964i	$-0.585713\pi$
$647$	−5.66533e6	−0.532065	−0.266033	+	0.963964i	$0.585713\pi$
$648$	0	0
$649$	−9.92407e6	−0.924864
$650$	2.26943e7	2.10685
$651$	0	0
$652$	2.03962e6	0.187901
$653$	−8.13187e6	−0.746290	−0.373145	−	0.927773i	$-0.621721\pi$
$653$	−8.13187e6	−0.746290	−0.373145	+	0.927773i	$0.621721\pi$
$654$	0	0
$655$	−1.01739e7	−0.926584
$656$	−3.89620e6	−0.353494
$657$	0	0
$658$	0	0
$659$	1.99012e6	0.178512	0.0892558	−	0.996009i	$-0.471551\pi$
$659$	1.99012e6	0.178512	0.0892558	+	0.996009i	$0.471551\pi$
$660$	0	0
$661$	3.18755e6	0.283761	0.141881	−	0.989884i	$-0.454685\pi$
$661$	3.18755e6	0.283761	0.141881	+	0.989884i	$0.454685\pi$
$662$	−1.35289e7	−1.19982
$663$	0	0
$664$	226584.	0.0199439
$665$	0	0
$666$	0	0
$667$	5.15222e6	0.448415
$668$	−1.07402e7	−0.931258
$669$	0	0
$670$	7.68584e6	0.661462
$671$	−4.42289e6	−0.379228
$672$	0	0
$673$	−1.72276e7	−1.46618	−0.733090	−	0.680131i	$-0.761923\pi$
$673$	−1.72276e7	−1.46618	−0.733090	+	0.680131i	$0.761923\pi$
$674$	7.78114e6	0.659772
$675$	0	0
$676$	8.90002e6	0.749073
$677$	−1.17235e7	−0.983072	−0.491536	−	0.870857i	$-0.663564\pi$
$677$	−1.17235e7	−0.983072	−0.491536	+	0.870857i	$0.663564\pi$
$678$	0	0
$679$	0	0
$680$	5.44051e6	0.451198
$681$	0	0
$682$	−3.66745e6	−0.301928
$683$	−6.40383e6	−0.525276	−0.262638	−	0.964894i	$-0.584593\pi$
$683$	−6.40383e6	−0.525276	−0.262638	+	0.964894i	$0.584593\pi$
$684$	0	0
$685$	3.16042e7	2.57347
$686$	0	0
$687$	0	0
$688$	2.48934e6	0.200500
$689$	−4.15092e6	−0.333117
$690$	0	0
$691$	−1.14960e7	−0.915904	−0.457952	−	0.888977i	$-0.651417\pi$
$691$	−1.14960e7	−0.915904	−0.457952	+	0.888977i	$0.651417\pi$
$692$	−5.56455e6	−0.441738
$693$	0	0
$694$	8.32930e6	0.656462
$695$	−1.77164e7	−1.39128
$696$	0	0
$697$	−1.36256e7	−1.06236
$698$	−1.26306e7	−0.981263
$699$	0	0
$700$	0	0
$701$	1.71167e7	1.31560	0.657802	−	0.753191i	$-0.271486\pi$
$701$	1.71167e7	1.31560	0.657802	+	0.753191i	$0.271486\pi$
$702$	0	0
$703$	7.70785e6	0.588227
$704$	−1.94970e6	−0.148264
$705$	0	0
$706$	−1.02008e7	−0.770237
$707$	0	0
$708$	0	0
$709$	1.69689e7	1.26776	0.633882	−	0.773429i	$-0.281460\pi$
$709$	1.69689e7	1.26776	0.633882	+	0.773429i	$0.281460\pi$
$710$	−1.13153e7	−0.842405
$711$	0	0
$712$	4.52821e6	0.334755
$713$	−7.11917e6	−0.524452
$714$	0	0
$715$	−4.35292e7	−3.18432
$716$	1.00038e7	0.729258
$717$	0	0
$718$	−9.73869e6	−0.705000
$719$	2.40621e7	1.73585	0.867923	−	0.496698i	$-0.165454\pi$
$719$	2.40621e7	1.73585	0.867923	+	0.496698i	$0.165454\pi$
$720$	0	0
$721$	0	0
$722$	−8.27857e6	−0.591034
$723$	0	0
$724$	3.86387e6	0.273953
$725$	−8.21205e6	−0.580239
$726$	0	0
$727$	−3.24345e6	−0.227599	−0.113800	−	0.993504i	$-0.536302\pi$
$727$	−3.24345e6	−0.227599	−0.113800	+	0.993504i	$0.536302\pi$
$728$	0	0
$729$	0	0
$730$	4.28646e6	0.297709
$731$	8.70558e6	0.602566
$732$	0	0
$733$	−2.59315e7	−1.78266	−0.891328	−	0.453359i	$-0.850226\pi$
$733$	−2.59315e7	−1.78266	−0.891328	+	0.453359i	$0.850226\pi$
$734$	−1.73257e7	−1.18700
$735$	0	0
$736$	−3.78470e6	−0.257536
$737$	−9.63234e6	−0.653225
$738$	0	0
$739$	−5.53387e6	−0.372750	−0.186375	−	0.982479i	$-0.559674\pi$
$739$	−5.53387e6	−0.372750	−0.186375	+	0.982479i	$0.559674\pi$
$740$	1.83676e7	1.23303
$741$	0	0
$742$	0	0
$743$	−1.09491e7	−0.727622	−0.363811	−	0.931473i	$-0.618525\pi$
$743$	−1.09491e7	−0.727622	−0.363811	+	0.931473i	$0.618525\pi$
$744$	0	0
$745$	1.92118e7	1.26817
$746$	−1.46574e7	−0.964294
$747$	0	0
$748$	−6.81836e6	−0.445580
$749$	0	0
$750$	0	0
$751$	4.93260e6	0.319136	0.159568	−	0.987187i	$-0.448990\pi$
$751$	4.93260e6	0.319136	0.159568	+	0.987187i	$0.448990\pi$
$752$	7.49377e6	0.483232
$753$	0	0
$754$	−5.37019e6	−0.344003
$755$	3.28308e7	2.09611
$756$	0	0
$757$	7.69782e6	0.488234	0.244117	−	0.969746i	$-0.421502\pi$
$757$	7.69782e6	0.488234	0.244117	+	0.969746i	$0.421502\pi$
$758$	−4.97410e6	−0.314442
$759$	0	0
$760$	3.87430e6	0.243310
$761$	7.27427e6	0.455331	0.227666	−	0.973739i	$-0.426891\pi$
$761$	7.27427e6	0.455331	0.227666	+	0.973739i	$0.426891\pi$
$762$	0	0
$763$	0	0
$764$	−3.40723e6	−0.211188
$765$	0	0
$766$	−6.02108e6	−0.370768
$767$	2.00794e7	1.23243
$768$	0	0
$769$	−1.72394e7	−1.05125	−0.525626	−	0.850716i	$-0.676169\pi$
$769$	−1.72394e7	−1.05125	−0.525626	+	0.850716i	$0.676169\pi$
$770$	0	0
$771$	0	0
$772$	2.16003e6	0.130442
$773$	3.96168e6	0.238468	0.119234	−	0.992866i	$-0.461956\pi$
$773$	3.96168e6	0.238468	0.119234	+	0.992866i	$0.461956\pi$
$774$	0	0
$775$	1.13471e7	0.678628
$776$	−1.13726e6	−0.0677963
$777$	0	0
$778$	1.54372e7	0.914365
$779$	−9.70306e6	−0.572882
$780$	0	0
$781$	1.41810e7	0.831915
$782$	−1.32356e7	−0.773977
$783$	0	0
$784$	0	0
$785$	−7.86839e6	−0.455734
$786$	0	0
$787$	−3.41283e6	−0.196416	−0.0982082	−	0.995166i	$-0.531311\pi$
$787$	−3.41283e6	−0.196416	−0.0982082	+	0.995166i	$0.531311\pi$
$788$	−8.78141e6	−0.503789
$789$	0	0
$790$	−1.26006e7	−0.718329
$791$	0	0
$792$	0	0
$793$	8.94884e6	0.505340
$794$	1.32805e7	0.747590
$795$	0	0
$796$	1.01099e7	0.565541
$797$	−5.38044e6	−0.300035	−0.150017	−	0.988683i	$-0.547933\pi$
$797$	−5.38044e6	−0.300035	−0.150017	+	0.988683i	$0.547933\pi$
$798$	0	0
$799$	2.62068e7	1.45227
$800$	6.03238e6	0.333245
$801$	0	0
$802$	−1.47735e7	−0.811048
$803$	−5.37204e6	−0.294002
$804$	0	0
$805$	0	0
$806$	7.42035e6	0.402334
$807$	0	0
$808$	−5.75489e6	−0.310105
$809$	1.99148e7	1.06980	0.534901	−	0.844915i	$-0.320349\pi$
$809$	1.99148e7	1.06980	0.534901	+	0.844915i	$0.320349\pi$
$810$	0	0
$811$	−2.24825e7	−1.20031	−0.600153	−	0.799886i	$-0.704893\pi$
$811$	−2.24825e7	−1.20031	−0.600153	+	0.799886i	$0.704893\pi$
$812$	0	0
$813$	0	0
$814$	−2.30194e7	−1.21768
$815$	1.21042e7	0.638324
$816$	0	0
$817$	6.19943e6	0.324935
$818$	−1.61685e7	−0.844865
$819$	0	0
$820$	−2.31222e7	−1.20086
$821$	1.88281e7	0.974872	0.487436	−	0.873159i	$-0.337932\pi$
$821$	1.88281e7	0.974872	0.487436	+	0.873159i	$0.337932\pi$
$822$	0	0
$823$	−9.23000e6	−0.475009	−0.237505	−	0.971386i	$-0.576329\pi$
$823$	−9.23000e6	−0.475009	−0.237505	+	0.971386i	$0.576329\pi$
$824$	−519147.	−0.0266362
$825$	0	0
$826$	0	0
$827$	−3.44089e7	−1.74947	−0.874736	−	0.484600i	$-0.838965\pi$
$827$	−3.44089e7	−1.74947	−0.874736	+	0.484600i	$0.838965\pi$
$828$	0	0
$829$	−2.51500e7	−1.27102	−0.635509	−	0.772093i	$-0.719210\pi$
$829$	−2.51500e7	−1.27102	−0.635509	+	0.772093i	$0.719210\pi$
$830$	1.34467e6	0.0677518
$831$	0	0
$832$	3.94482e6	0.197569
$833$	0	0
$834$	0	0
$835$	−6.37380e7	−3.16360
$836$	−4.85550e6	−0.240280
$837$	0	0
$838$	9.69531e6	0.476927
$839$	−3.27543e7	−1.60644	−0.803219	−	0.595684i	$-0.796881\pi$
$839$	−3.27543e7	−1.60644	−0.803219	+	0.595684i	$0.796881\pi$
$840$	0	0
$841$	−1.85679e7	−0.905260
$842$	−2.66256e7	−1.29425
$843$	0	0
$844$	−2.55904e6	−0.123658
$845$	5.28175e7	2.54470
$846$	0	0
$847$	0	0
$848$	−1.10336e6	−0.0526899
$849$	0	0
$850$	2.10961e7	1.00151
$851$	−4.46846e7	−2.11512
$852$	0	0
$853$	−1.70918e7	−0.804295	−0.402148	−	0.915575i	$-0.631736\pi$
$853$	−1.70918e7	−0.804295	−0.402148	+	0.915575i	$0.631736\pi$
$854$	0	0
$855$	0	0
$856$	−8.05811e6	−0.375880
$857$	7.19282e6	0.334539	0.167270	−	0.985911i	$-0.446505\pi$
$857$	7.19282e6	0.334539	0.167270	+	0.985911i	$0.446505\pi$
$858$	0	0
$859$	1.49120e7	0.689531	0.344765	−	0.938689i	$-0.387958\pi$
$859$	1.49120e7	0.689531	0.344765	+	0.938689i	$0.387958\pi$
$860$	1.47731e7	0.681123
$861$	0	0
$862$	2.44208e6	0.111942
$863$	−3.24943e7	−1.48518	−0.742592	−	0.669744i	$-0.766404\pi$
$863$	−3.24943e7	−1.48518	−0.742592	+	0.669744i	$0.766404\pi$
$864$	0	0
$865$	−3.30230e7	−1.50064
$866$	119912.	0.00543333
$867$	0	0
$868$	0	0
$869$	1.57918e7	0.709384
$870$	0	0
$871$	1.94891e7	0.870455
$872$	5.70688e6	0.254160
$873$	0	0
$874$	−9.42538e6	−0.417369
$875$	0	0
$876$	0	0
$877$	−1.72512e7	−0.757391	−0.378695	−	0.925521i	$-0.623627\pi$
$877$	−1.72512e7	−0.757391	−0.378695	+	0.925521i	$0.623627\pi$
$878$	1.47931e6	0.0647623
$879$	0	0
$880$	−1.15705e7	−0.503671
$881$	3.45237e7	1.49857	0.749286	−	0.662247i	$-0.230397\pi$
$881$	3.45237e7	1.49857	0.749286	+	0.662247i	$0.230397\pi$
$882$	0	0
$883$	−3.99893e7	−1.72600	−0.863002	−	0.505200i	$-0.831419\pi$
$883$	−3.99893e7	−1.72600	−0.863002	+	0.505200i	$0.831419\pi$
$884$	1.37956e7	0.593758
$885$	0	0
$886$	2.55614e6	0.109396
$887$	3.71547e7	1.58564	0.792819	−	0.609457i	$-0.208612\pi$
$887$	3.71547e7	1.58564	0.792819	+	0.609457i	$0.208612\pi$
$888$	0	0
$889$	0	0
$890$	2.68728e7	1.13720
$891$	0	0
$892$	1.40282e7	0.590325
$893$	1.86624e7	0.783138
$894$	0	0
$895$	5.93678e7	2.47738
$896$	0	0
$897$	0	0
$898$	−7.62727e6	−0.315630
$899$	−2.68510e6	−0.110805
$900$	0	0
$901$	−3.85860e6	−0.158350
$902$	2.89780e7	1.18591
$903$	0	0
$904$	1.90093e6	0.0773650
$905$	2.29303e7	0.930653
$906$	0	0
$907$	1.78366e7	0.719934	0.359967	−	0.932965i	$-0.382788\pi$
$907$	1.78366e7	0.719934	0.359967	+	0.932965i	$0.382788\pi$
$908$	3.49621e6	0.140729
$909$	0	0
$910$	0	0
$911$	600440.	0.0239703	0.0119852	−	0.999928i	$-0.496185\pi$
$911$	600440.	0.0239703	0.0119852	+	0.999928i	$0.496185\pi$
$912$	0	0
$913$	−1.68522e6	−0.0669082
$914$	2.64015e7	1.04536
$915$	0	0
$916$	2.28457e7	0.899634
$917$	0	0
$918$	0	0
$919$	4.14543e7	1.61913	0.809564	−	0.587032i	$-0.199704\pi$
$919$	4.14543e7	1.61913	0.809564	+	0.587032i	$0.199704\pi$
$920$	−2.24605e7	−0.874882
$921$	0	0
$922$	−1.97234e7	−0.764107
$923$	−2.86924e7	−1.10857
$924$	0	0
$925$	7.12222e7	2.73691
$926$	1.03030e6	0.0394855
$927$	0	0
$928$	−1.42746e6	−0.0544118
$929$	−4.87049e7	−1.85154	−0.925771	−	0.378085i	$-0.876583\pi$
$929$	−4.87049e7	−1.85154	−0.925771	+	0.378085i	$0.876583\pi$
$930$	0	0
$931$	0	0
$932$	6.87069e6	0.259096
$933$	0	0
$934$	−2.16790e6	−0.0813154
$935$	−4.04638e7	−1.51369
$936$	0	0
$937$	−2.45567e7	−0.913737	−0.456868	−	0.889534i	$-0.651029\pi$
$937$	−2.45567e7	−0.913737	−0.456868	+	0.889534i	$0.651029\pi$
$938$	0	0
$939$	0	0
$940$	4.44721e7	1.64160
$941$	8.38235e6	0.308597	0.154299	−	0.988024i	$-0.450688\pi$
$941$	8.38235e6	0.308597	0.154299	+	0.988024i	$0.450688\pi$
$942$	0	0
$943$	5.62515e7	2.05994
$944$	5.33731e6	0.194936
$945$	0	0
$946$	−1.85145e7	−0.672642
$947$	−1.11502e7	−0.404026	−0.202013	−	0.979383i	$-0.564748\pi$
$947$	−1.11502e7	−0.404026	−0.202013	+	0.979383i	$0.564748\pi$
$948$	0	0
$949$	1.08692e7	0.391773
$950$	1.50230e7	0.540066
$951$	0	0
$952$	0	0
$953$	1.57312e7	0.561088	0.280544	−	0.959841i	$-0.409485\pi$
$953$	1.57312e7	0.561088	0.280544	+	0.959841i	$0.409485\pi$
$954$	0	0
$955$	−2.02204e7	−0.717431
$956$	5.41325e6	0.191564
$957$	0	0
$958$	2.51531e7	0.885478
$959$	0	0
$960$	0	0
$961$	−2.49190e7	−0.870406
$962$	4.65751e7	1.62262
$963$	0	0
$964$	8.58632e6	0.297587
$965$	1.28188e7	0.443128
$966$	0	0
$967$	5.38684e7	1.85254	0.926271	−	0.376857i	$-0.122995\pi$
$967$	5.38684e7	1.85254	0.926271	+	0.376857i	$0.122995\pi$
$968$	4.19360e6	0.143846
$969$	0	0
$970$	−6.74912e6	−0.230313
$971$	−3.06393e7	−1.04287	−0.521436	−	0.853291i	$-0.674603\pi$
$971$	−3.06393e7	−1.04287	−0.521436	+	0.853291i	$0.674603\pi$
$972$	0	0
$973$	0	0
$974$	−2.47391e7	−0.835578
$975$	0	0
$976$	2.37870e6	0.0799309
$977$	−1.15906e7	−0.388480	−0.194240	−	0.980954i	$-0.562224\pi$
$977$	−1.15906e7	−0.388480	−0.194240	+	0.980954i	$0.562224\pi$
$978$	0	0
$979$	−3.36786e7	−1.12304
$980$	0	0
$981$	0	0
$982$	1.97501e7	0.653569
$983$	−609460.	−0.0201169	−0.0100585	−	0.999949i	$-0.503202\pi$
$983$	−609460.	−0.0201169	−0.0100585	+	0.999949i	$0.503202\pi$
$984$	0	0
$985$	−5.21136e7	−1.71144
$986$	−4.99201e6	−0.163525
$987$	0	0
$988$	9.82413e6	0.320185
$989$	−3.59399e7	−1.16839
$990$	0	0
$991$	2.18372e7	0.706339	0.353170	−	0.935559i	$-0.385104\pi$
$991$	2.18372e7	0.706339	0.353170	+	0.935559i	$0.385104\pi$
$992$	1.97241e6	0.0636382
$993$	0	0
$994$	0	0
$995$	5.99976e7	1.92122
$996$	0	0
$997$	−3.46465e7	−1.10388	−0.551939	−	0.833885i	$-0.686112\pi$
$997$	−3.46465e7	−1.10388	−0.551939	+	0.833885i	$0.686112\pi$
$998$	−2.96885e6	−0.0943543
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bo.1.2		2
3.2	odd	2		98.6.a.e.1.2	yes	2
7.6	odd	2	inner	882.6.a.bo.1.1		2
12.11	even	2		784.6.a.y.1.1		2
21.2	odd	6		98.6.c.g.67.1		4
21.5	even	6		98.6.c.g.67.2		4
21.11	odd	6		98.6.c.g.79.1		4
21.17	even	6		98.6.c.g.79.2		4
21.20	even	2		98.6.a.e.1.1	✓	2
84.83	odd	2		784.6.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.6.a.e.1.1	✓	2	21.20	even	2
98.6.a.e.1.2	yes	2	3.2	odd	2
98.6.c.g.67.1		4	21.2	odd	6
98.6.c.g.67.2		4	21.5	even	6
98.6.c.g.79.1		4	21.11	odd	6
98.6.c.g.79.2		4	21.17	even	6
784.6.a.y.1.1		2	12.11	even	2
784.6.a.y.1.2		2	84.83	odd	2
882.6.a.bo.1.1		2	7.6	odd	2	inner
882.6.a.bo.1.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} +94.9526 q^{5} +64.0000 q^{8} +379.810 q^{10} -476.000 q^{11} +963.091 q^{13} +256.000 q^{16} +895.268 q^{17} +637.539 q^{19} +1519.24 q^{20} -1904.00 q^{22} -3696.00 q^{23} +5891.00 q^{25} +3852.36 q^{26} -1394.00 q^{29} +1926.18 q^{31} +1024.00 q^{32} +3581.07 q^{34} +12090.0 q^{37} +2550.16 q^{38} +6076.97 q^{40} -15219.5 q^{41} +9724.00 q^{43} -7616.00 q^{44} -14784.0 q^{46} +29272.5 q^{47} +23564.0 q^{50} +15409.5 q^{52} -4310.00 q^{53} -45197.4 q^{55} -5576.00 q^{58} +20848.9 q^{59} +9291.79 q^{61} +7704.73 q^{62} +4096.00 q^{64} +91448.0 q^{65} +20236.0 q^{67} +14324.3 q^{68} -29792.0 q^{71} +11285.8 q^{73} +48360.0 q^{74} +10200.6 q^{76} -33176.0 q^{79} +24307.9 q^{80} -60878.2 q^{82} +3540.38 q^{83} +85008.0 q^{85} +38896.0 q^{86} -30464.0 q^{88} +70753.3 q^{89} -59136.0 q^{92} +117090. q^{94} +60536.0 q^{95} -17769.7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 128 q^{8} - 952 q^{11} + 512 q^{16} - 3808 q^{22} - 7392 q^{23} + 11782 q^{25} - 2788 q^{29} + 2048 q^{32} + 24180 q^{37} + 19448 q^{43} - 15232 q^{44} - 29568 q^{46} + 47128 q^{50}+ \cdots + 121072 q^{95}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 128 * q^8 - 952 * q^11 + 512 * q^16 - 3808 * q^22 - 7392 * q^23 + 11782 * q^25 - 2788 * q^29 + 2048 * q^32 + 24180 * q^37 + 19448 * q^43 - 15232 * q^44 - 29568 * q^46 + 47128 * q^50 - 8620 * q^53 - 11152 * q^58 + 8192 * q^64 + 182896 * q^65 + 40472 * q^67 - 59584 * q^71 + 96720 * q^74 - 66352 * q^79 + 170016 * q^85 + 77792 * q^86 - 60928 * q^88 - 118272 * q^92 + 121072 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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