LMFDB - Embedded newform 882.6.a.ba.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,-70,0,0,-128,0,280,-62] 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{79}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.88819$ 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)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +36.1056 q^{5} -64.0000 q^{8} -144.422 q^{10} -155.435 q^{11} +1158.87 q^{13} +256.000 q^{16} -1238.08 q^{17} -280.279 q^{19} +577.689 q^{20} +621.739 q^{22} +3482.39 q^{23} -1821.39 q^{25} -4635.48 q^{26} +5656.78 q^{29} -2314.70 q^{31} -1024.00 q^{32} +4952.32 q^{34} -2333.18 q^{37} +1121.12 q^{38} -2310.76 q^{40} -3812.61 q^{41} +3925.73 q^{43} -2486.96 q^{44} -13929.6 q^{46} +11116.5 q^{47} +7285.56 q^{50} +18541.9 q^{52} +11186.2 q^{53} -5612.06 q^{55} -22627.1 q^{58} +6010.35 q^{59} -14838.7 q^{61} +9258.81 q^{62} +4096.00 q^{64} +41841.6 q^{65} -42983.9 q^{67} -19809.3 q^{68} +19962.4 q^{71} +45550.6 q^{73} +9332.70 q^{74} -4484.47 q^{76} +108992. q^{79} +9243.02 q^{80} +15250.4 q^{82} -55829.0 q^{83} -44701.6 q^{85} -15702.9 q^{86} +9947.82 q^{88} +95545.8 q^{89} +55718.2 q^{92} -44465.8 q^{94} -10119.6 q^{95} +15004.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 70 q^{5} - 128 q^{8} + 280 q^{10} - 62 q^{11} + 1820 q^{13} + 512 q^{16} - 1694 q^{17} + 826 q^{19} - 1120 q^{20} + 248 q^{22} + 2734 q^{23} + 6312 q^{25} - 7280 q^{26} + 2852 q^{29}+ \cdots - 8316 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 70 * q^5 - 128 * q^8 + 280 * q^10 - 62 * q^11 + 1820 * q^13 + 512 * q^16 - 1694 * q^17 + 826 * q^19 - 1120 * q^20 + 248 * q^22 + 2734 * q^23 + 6312 * q^25 - 7280 * q^26 + 2852 * q^29 - 2674 * q^31 - 2048 * q^32 + 6776 * q^34 - 9146 * q^37 - 3304 * q^38 + 4480 * q^40 - 6132 * q^41 - 16040 * q^43 - 992 * q^44 - 10936 * q^46 + 25326 * q^47 - 25248 * q^50 + 29120 * q^52 - 14958 * q^53 - 15526 * q^55 - 11408 * q^58 + 1106 * q^59 - 28042 * q^61 + 10696 * q^62 + 8192 * q^64 - 28308 * q^65 - 102642 * q^67 - 27104 * q^68 + 11056 * q^71 + 35070 * q^73 + 36584 * q^74 + 13216 * q^76 + 101762 * q^79 - 17920 * q^80 + 24528 * q^82 + 44632 * q^83 + 3674 * q^85 + 64160 * q^86 + 3968 * q^88 + 75474 * q^89 + 43744 * q^92 - 101304 * q^94 - 127502 * q^95 - 8316 * q^97

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
$5$	36.1056	0.645876	0.322938	−	0.946420i	$-0.395329\pi$
$5$	36.1056	0.645876	0.322938	+	0.946420i	$0.395329\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−144.422	−0.456703
$11$	−155.435	−0.387317	−0.193658	−	0.981069i	$-0.562035\pi$
$11$	−155.435	−0.387317	−0.193658	+	0.981069i	$0.562035\pi$
$12$	0	0
$13$	1158.87	1.90185	0.950925	−	0.309422i	$-0.100136\pi$
$13$	1158.87	1.90185	0.950925	+	0.309422i	$0.100136\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−1238.08	−1.03903	−0.519513	−	0.854462i	$-0.673887\pi$
$17$	−1238.08	−1.03903	−0.519513	+	0.854462i	$0.673887\pi$
$18$	0	0
$19$	−280.279	−0.178118	−0.0890588	−	0.996026i	$-0.528386\pi$
$19$	−280.279	−0.178118	−0.0890588	+	0.996026i	$0.528386\pi$
$20$	577.689	0.322938
$21$	0	0
$22$	621.739	0.273874
$23$	3482.39	1.37264	0.686322	−	0.727298i	$-0.259224\pi$
$23$	3482.39	1.37264	0.686322	+	0.727298i	$0.259224\pi$
$24$	0	0
$25$	−1821.39	−0.582844
$26$	−4635.48	−1.34481
$27$	0	0
$28$	0	0
$29$	5656.78	1.24903	0.624517	−	0.781011i	$-0.285296\pi$
$29$	5656.78	1.24903	0.624517	+	0.781011i	$0.285296\pi$
$30$	0	0
$31$	−2314.70	−0.432604	−0.216302	−	0.976326i	$-0.569400\pi$
$31$	−2314.70	−0.432604	−0.216302	+	0.976326i	$0.569400\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	4952.32	0.734703
$35$	0	0
$36$	0	0
$37$	−2333.18	−0.280184	−0.140092	−	0.990139i	$-0.544740\pi$
$37$	−2333.18	−0.280184	−0.140092	+	0.990139i	$0.544740\pi$
$38$	1121.12	0.125948
$39$	0	0
$40$	−2310.76	−0.228352
$41$	−3812.61	−0.354211	−0.177106	−	0.984192i	$-0.556673\pi$
$41$	−3812.61	−0.354211	−0.177106	+	0.984192i	$0.556673\pi$
$42$	0	0
$43$	3925.73	0.323780	0.161890	−	0.986809i	$-0.448241\pi$
$43$	3925.73	0.323780	0.161890	+	0.986809i	$0.448241\pi$
$44$	−2486.96	−0.193658
$45$	0	0
$46$	−13929.6	−0.970606
$47$	11116.5	0.734043	0.367022	−	0.930212i	$-0.380378\pi$
$47$	11116.5	0.734043	0.367022	+	0.930212i	$0.380378\pi$
$48$	0	0
$49$	0	0
$50$	7285.56	0.412133
$51$	0	0
$52$	18541.9	0.950925
$53$	11186.2	0.547007	0.273504	−	0.961871i	$-0.411817\pi$
$53$	11186.2	0.547007	0.273504	+	0.961871i	$0.411817\pi$
$54$	0	0
$55$	−5612.06	−0.250159
$56$	0	0
$57$	0	0
$58$	−22627.1	−0.883201
$59$	6010.35	0.224786	0.112393	−	0.993664i	$-0.464148\pi$
$59$	6010.35	0.224786	0.112393	+	0.993664i	$0.464148\pi$
$60$	0	0
$61$	−14838.7	−0.510589	−0.255295	−	0.966863i	$-0.582172\pi$
$61$	−14838.7	−0.510589	−0.255295	+	0.966863i	$0.582172\pi$
$62$	9258.81	0.305897
$63$	0	0
$64$	4096.00	0.125000
$65$	41841.6	1.22836
$66$	0	0
$67$	−42983.9	−1.16982	−0.584909	−	0.811099i	$-0.698870\pi$
$67$	−42983.9	−1.16982	−0.584909	+	0.811099i	$0.698870\pi$
$68$	−19809.3	−0.519513
$69$	0	0
$70$	0	0
$71$	19962.4	0.469967	0.234984	−	0.971999i	$-0.424496\pi$
$71$	19962.4	0.469967	0.234984	+	0.971999i	$0.424496\pi$
$72$	0	0
$73$	45550.6	1.00043	0.500215	−	0.865901i	$-0.333254\pi$
$73$	45550.6	1.00043	0.500215	+	0.865901i	$0.333254\pi$
$74$	9332.70	0.198120
$75$	0	0
$76$	−4484.47	−0.0890588
$77$	0	0
$78$	0	0
$79$	108992.	1.96484	0.982419	−	0.186687i	$-0.0597750\pi$
$79$	108992.	1.96484	0.982419	+	0.186687i	$0.0597750\pi$
$80$	9243.02	0.161469
$81$	0	0
$82$	15250.4	0.250465
$83$	−55829.0	−0.889538	−0.444769	−	0.895645i	$-0.646714\pi$
$83$	−55829.0	−0.889538	−0.444769	+	0.895645i	$0.646714\pi$
$84$	0	0
$85$	−44701.6	−0.671082
$86$	−15702.9	−0.228947
$87$	0	0
$88$	9947.82	0.136937
$89$	95545.8	1.27861	0.639303	−	0.768955i	$-0.279223\pi$
$89$	95545.8	1.27861	0.639303	+	0.768955i	$0.279223\pi$
$90$	0	0
$91$	0	0
$92$	55718.2	0.686322
$93$	0	0
$94$	−44465.8	−0.519047
$95$	−10119.6	−0.115042
$96$	0	0
$97$	15004.9	0.161922	0.0809609	−	0.996717i	$-0.474201\pi$
$97$	15004.9	0.161922	0.0809609	+	0.996717i	$0.474201\pi$
$98$	0	0
$99$	0	0
$100$	−29142.2	−0.291422
$101$	−26876.1	−0.262158	−0.131079	−	0.991372i	$-0.541844\pi$
$101$	−26876.1	−0.262158	−0.131079	+	0.991372i	$0.541844\pi$
$102$	0	0
$103$	−51915.2	−0.482171	−0.241086	−	0.970504i	$-0.577503\pi$
$103$	−51915.2	−0.482171	−0.241086	+	0.970504i	$0.577503\pi$
$104$	−74167.6	−0.672405
$105$	0	0
$106$	−44744.8	−0.386793
$107$	24974.8	0.210883	0.105442	−	0.994425i	$-0.466374\pi$
$107$	24974.8	0.210883	0.105442	+	0.994425i	$0.466374\pi$
$108$	0	0
$109$	−3636.14	−0.0293140	−0.0146570	−	0.999893i	$-0.504666\pi$
$109$	−3636.14	−0.0293140	−0.0146570	+	0.999893i	$0.504666\pi$
$110$	22448.2	0.176889
$111$	0	0
$112$	0	0
$113$	−62175.0	−0.458057	−0.229028	−	0.973420i	$-0.573555\pi$
$113$	−62175.0	−0.458057	−0.229028	+	0.973420i	$0.573555\pi$
$114$	0	0
$115$	125734.	0.886557
$116$	90508.5	0.624517
$117$	0	0
$118$	−24041.4	−0.158948
$119$	0	0
$120$	0	0
$121$	−136891.	−0.849986
$122$	59354.9	0.361041
$123$	0	0
$124$	−37035.2	−0.216302
$125$	−178592.	−1.02232
$126$	0	0
$127$	63550.3	0.349630	0.174815	−	0.984601i	$-0.444067\pi$
$127$	63550.3	0.349630	0.174815	+	0.984601i	$0.444067\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−167366.	−0.868581
$131$	−136297.	−0.693918	−0.346959	−	0.937880i	$-0.612786\pi$
$131$	−136297.	−0.693918	−0.346959	+	0.937880i	$0.612786\pi$
$132$	0	0
$133$	0	0
$134$	171935.	0.827187
$135$	0	0
$136$	79237.2	0.367351
$137$	−335303.	−1.52629	−0.763144	−	0.646228i	$-0.776345\pi$
$137$	−335303.	−1.52629	−0.763144	+	0.646228i	$0.776345\pi$
$138$	0	0
$139$	58195.2	0.255476	0.127738	−	0.991808i	$-0.459228\pi$
$139$	58195.2	0.255476	0.127738	+	0.991808i	$0.459228\pi$
$140$	0	0
$141$	0	0
$142$	−79849.7	−0.332317
$143$	−180129.	−0.736618
$144$	0	0
$145$	204241.	0.806721
$146$	−182202.	−0.707411
$147$	0	0
$148$	−37330.8	−0.140092
$149$	−283058.	−1.04450	−0.522251	−	0.852792i	$-0.674908\pi$
$149$	−283058.	−1.04450	−0.522251	+	0.852792i	$0.674908\pi$
$150$	0	0
$151$	−241558.	−0.862142	−0.431071	−	0.902318i	$-0.641864\pi$
$151$	−241558.	−0.862142	−0.431071	+	0.902318i	$0.641864\pi$
$152$	17937.9	0.0629741
$153$	0	0
$154$	0	0
$155$	−83573.6	−0.279409
$156$	0	0
$157$	−214029.	−0.692984	−0.346492	−	0.938053i	$-0.612627\pi$
$157$	−214029.	−0.692984	−0.346492	+	0.938053i	$0.612627\pi$
$158$	−435968.	−1.38935
$159$	0	0
$160$	−36972.1	−0.114176
$161$	0	0
$162$	0	0
$163$	64466.0	0.190047	0.0950237	−	0.995475i	$-0.469707\pi$
$163$	64466.0	0.190047	0.0950237	+	0.995475i	$0.469707\pi$
$164$	−61001.7	−0.177106
$165$	0	0
$166$	223316.	0.628999
$167$	442694.	1.22832	0.614162	−	0.789180i	$-0.289494\pi$
$167$	442694.	1.22832	0.614162	+	0.789180i	$0.289494\pi$
$168$	0	0
$169$	971685.	2.61703
$170$	178806.	0.474527
$171$	0	0
$172$	62811.7	0.161890
$173$	78599.0	0.199665	0.0998325	−	0.995004i	$-0.468169\pi$
$173$	78599.0	0.199665	0.0998325	+	0.995004i	$0.468169\pi$
$174$	0	0
$175$	0	0
$176$	−39791.3	−0.0968292
$177$	0	0
$178$	−382183.	−0.904111
$179$	510427.	1.19070	0.595348	−	0.803468i	$-0.297014\pi$
$179$	510427.	1.19070	0.595348	+	0.803468i	$0.297014\pi$
$180$	0	0
$181$	−22051.8	−0.0500319	−0.0250160	−	0.999687i	$-0.507964\pi$
$181$	−22051.8	−0.0500319	−0.0250160	+	0.999687i	$0.507964\pi$
$182$	0	0
$183$	0	0
$184$	−222873.	−0.485303
$185$	−84240.6	−0.180964
$186$	0	0
$187$	192441.	0.402432
$188$	177863.	0.367022
$189$	0	0
$190$	40478.5	0.0813469
$191$	558152.	1.10706	0.553528	−	0.832831i	$-0.313281\pi$
$191$	558152.	1.10706	0.553528	+	0.832831i	$0.313281\pi$
$192$	0	0
$193$	−49316.3	−0.0953009	−0.0476504	−	0.998864i	$-0.515173\pi$
$193$	−49316.3	−0.0953009	−0.0476504	+	0.998864i	$0.515173\pi$
$194$	−60019.8	−0.114496
$195$	0	0
$196$	0	0
$197$	941509.	1.72846	0.864229	−	0.503099i	$-0.167807\pi$
$197$	941509.	1.72846	0.864229	+	0.503099i	$0.167807\pi$
$198$	0	0
$199$	640012.	1.14566	0.572830	−	0.819675i	$-0.305846\pi$
$199$	640012.	1.14566	0.572830	+	0.819675i	$0.305846\pi$
$200$	116569.	0.206067
$201$	0	0
$202$	107505.	0.185374
$203$	0	0
$204$	0	0
$205$	−137656.	−0.228777
$206$	207661.	0.340947
$207$	0	0
$208$	296671.	0.475462
$209$	43565.1	0.0689879
$210$	0	0
$211$	921869.	1.42549	0.712743	−	0.701425i	$-0.247452\pi$
$211$	921869.	1.42549	0.712743	+	0.701425i	$0.247452\pi$
$212$	178979.	0.273504
$213$	0	0
$214$	−99899.1	−0.149117
$215$	141741.	0.209121
$216$	0	0
$217$	0	0
$218$	14544.6	0.0207281
$219$	0	0
$220$	−89792.9	−0.125079
$221$	−1.43477e6	−1.97607
$222$	0	0
$223$	−837700.	−1.12805	−0.564023	−	0.825759i	$-0.690747\pi$
$223$	−837700.	−1.12805	−0.564023	+	0.825759i	$0.690747\pi$
$224$	0	0
$225$	0	0
$226$	248700.	0.323895
$227$	1.33069e6	1.71400	0.857002	−	0.515313i	$-0.172324\pi$
$227$	1.33069e6	1.71400	0.857002	+	0.515313i	$0.172324\pi$
$228$	0	0
$229$	1.01063e6	1.27352	0.636759	−	0.771063i	$-0.280275\pi$
$229$	1.01063e6	1.27352	0.636759	+	0.771063i	$0.280275\pi$
$230$	−502935.	−0.626891
$231$	0	0
$232$	−362034.	−0.441600
$233$	−1.48610e6	−1.79332	−0.896661	−	0.442718i	$-0.854014\pi$
$233$	−1.48610e6	−1.79332	−0.896661	+	0.442718i	$0.854014\pi$
$234$	0	0
$235$	401366.	0.474101
$236$	96165.6	0.112393
$237$	0	0
$238$	0	0
$239$	−875637.	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$239$	−875637.	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$240$	0	0
$241$	1.45294e6	1.61141	0.805706	−	0.592316i	$-0.201786\pi$
$241$	1.45294e6	1.61141	0.805706	+	0.592316i	$0.201786\pi$
$242$	547564.	0.601031
$243$	0	0
$244$	−237419.	−0.255295
$245$	0	0
$246$	0	0
$247$	−324807.	−0.338753
$248$	148141.	0.152949
$249$	0	0
$250$	714368.	0.722890
$251$	198384.	0.198757	0.0993786	−	0.995050i	$-0.468315\pi$
$251$	198384.	0.198757	0.0993786	+	0.995050i	$0.468315\pi$
$252$	0	0
$253$	−541284.	−0.531648
$254$	−254201.	−0.247226
$255$	0	0
$256$	65536.0	0.0625000
$257$	−479572.	−0.452919	−0.226460	−	0.974021i	$-0.572715\pi$
$257$	−479572.	−0.452919	−0.226460	+	0.974021i	$0.572715\pi$
$258$	0	0
$259$	0	0
$260$	669466.	0.614179
$261$	0	0
$262$	545188.	0.490674
$263$	−454943.	−0.405572	−0.202786	−	0.979223i	$-0.565000\pi$
$263$	−454943.	−0.405572	−0.202786	+	0.979223i	$0.565000\pi$
$264$	0	0
$265$	403884.	0.353299
$266$	0	0
$267$	0	0
$268$	−687742.	−0.584909
$269$	860136.	0.724747	0.362374	−	0.932033i	$-0.381966\pi$
$269$	860136.	0.724747	0.362374	+	0.932033i	$0.381966\pi$
$270$	0	0
$271$	1.30558e6	1.07989	0.539946	−	0.841700i	$-0.318445\pi$
$271$	1.30558e6	1.07989	0.539946	+	0.841700i	$0.318445\pi$
$272$	−316949.	−0.259757
$273$	0	0
$274$	1.34121e6	1.07925
$275$	283107.	0.225745
$276$	0	0
$277$	−543109.	−0.425292	−0.212646	−	0.977129i	$-0.568208\pi$
$277$	−543109.	−0.425292	−0.212646	+	0.977129i	$0.568208\pi$
$278$	−232781.	−0.180649
$279$	0	0
$280$	0	0
$281$	−998089.	−0.754055	−0.377028	−	0.926202i	$-0.623054\pi$
$281$	−998089.	−0.754055	−0.377028	+	0.926202i	$0.623054\pi$
$282$	0	0
$283$	1.77732e6	1.31917	0.659583	−	0.751632i	$-0.270733\pi$
$283$	1.77732e6	1.31917	0.659583	+	0.751632i	$0.270733\pi$
$284$	319399.	0.234984
$285$	0	0
$286$	720514.	0.520868
$287$	0	0
$288$	0	0
$289$	112986.	0.0795759
$290$	−816965.	−0.570438
$291$	0	0
$292$	728809.	0.500215
$293$	1.63987e6	1.11594	0.557969	−	0.829862i	$-0.311581\pi$
$293$	1.63987e6	1.11594	0.557969	+	0.829862i	$0.311581\pi$
$294$	0	0
$295$	217007.	0.145184
$296$	149323.	0.0990599
$297$	0	0
$298$	1.13223e6	0.738575
$299$	4.03564e6	2.61056
$300$	0	0
$301$	0	0
$302$	966231.	0.609626
$303$	0	0
$304$	−71751.5	−0.0445294
$305$	−535760.	−0.329777
$306$	0	0
$307$	2.38130e6	1.44201	0.721006	−	0.692928i	$-0.243680\pi$
$307$	2.38130e6	1.44201	0.721006	+	0.692928i	$0.243680\pi$
$308$	0	0
$309$	0	0
$310$	334294.	0.197572
$311$	1.24513e6	0.729983	0.364992	−	0.931011i	$-0.381072\pi$
$311$	1.24513e6	0.729983	0.364992	+	0.931011i	$0.381072\pi$
$312$	0	0
$313$	1.66299e6	0.959465	0.479732	−	0.877415i	$-0.340734\pi$
$313$	1.66299e6	0.959465	0.479732	+	0.877415i	$0.340734\pi$
$314$	856115.	0.490014
$315$	0	0
$316$	1.74387e6	0.982419
$317$	2.62486e6	1.46709	0.733547	−	0.679639i	$-0.237864\pi$
$317$	2.62486e6	1.46709	0.733547	+	0.679639i	$0.237864\pi$
$318$	0	0
$319$	−879260.	−0.483772
$320$	147888.	0.0807345
$321$	0	0
$322$	0	0
$323$	347008.	0.185069
$324$	0	0
$325$	−2.11075e6	−1.10848
$326$	−257864.	−0.134384
$327$	0	0
$328$	244007.	0.125233
$329$	0	0
$330$	0	0
$331$	1.73886e6	0.872355	0.436178	−	0.899861i	$-0.356332\pi$
$331$	1.73886e6	0.872355	0.436178	+	0.899861i	$0.356332\pi$
$332$	−893264.	−0.444769
$333$	0	0
$334$	−1.77078e6	−0.868556
$335$	−1.55196e6	−0.755558
$336$	0	0
$337$	853564.	0.409413	0.204706	−	0.978823i	$-0.434376\pi$
$337$	853564.	0.409413	0.204706	+	0.978823i	$0.434376\pi$
$338$	−3.88674e6	−1.85052
$339$	0	0
$340$	−715225.	−0.335541
$341$	359785.	0.167555
$342$	0	0
$343$	0	0
$344$	−251247.	−0.114473
$345$	0	0
$346$	−314396.	−0.141184
$347$	2.98500e6	1.33082	0.665411	−	0.746477i	$-0.268256\pi$
$347$	2.98500e6	1.33082	0.665411	+	0.746477i	$0.268256\pi$
$348$	0	0
$349$	2.87467e6	1.26335	0.631676	−	0.775233i	$-0.282367\pi$
$349$	2.87467e6	1.26335	0.631676	+	0.775233i	$0.282367\pi$
$350$	0	0
$351$	0	0
$352$	159165.	0.0684686
$353$	−1.47109e6	−0.628352	−0.314176	−	0.949365i	$-0.601728\pi$
$353$	−1.47109e6	−0.628352	−0.314176	+	0.949365i	$0.601728\pi$
$354$	0	0
$355$	720755.	0.303540
$356$	1.52873e6	0.639303
$357$	0	0
$358$	−2.04171e6	−0.841950
$359$	−3.07799e6	−1.26047	−0.630233	−	0.776406i	$-0.717041\pi$
$359$	−3.07799e6	−1.26047	−0.630233	+	0.776406i	$0.717041\pi$
$360$	0	0
$361$	−2.39754e6	−0.968274
$362$	88207.2	0.0353779
$363$	0	0
$364$	0	0
$365$	1.64463e6	0.646154
$366$	0	0
$367$	1.61670e6	0.626561	0.313281	−	0.949661i	$-0.398572\pi$
$367$	1.61670e6	0.626561	0.313281	+	0.949661i	$0.398572\pi$
$368$	891492.	0.343161
$369$	0	0
$370$	336962.	0.127961
$371$	0	0
$372$	0	0
$373$	−4.76119e6	−1.77192	−0.885958	−	0.463766i	$-0.846498\pi$
$373$	−4.76119e6	−1.77192	−0.885958	+	0.463766i	$0.846498\pi$
$374$	−769763.	−0.284563
$375$	0	0
$376$	−711453.	−0.259523
$377$	6.55547e6	2.37548
$378$	0	0
$379$	1.00193e6	0.358294	0.179147	−	0.983822i	$-0.442666\pi$
$379$	1.00193e6	0.358294	0.179147	+	0.983822i	$0.442666\pi$
$380$	−161914.	−0.0575209
$381$	0	0
$382$	−2.23261e6	−0.782806
$383$	2.82152e6	0.982849	0.491424	−	0.870920i	$-0.336476\pi$
$383$	2.82152e6	0.982849	0.491424	+	0.870920i	$0.336476\pi$
$384$	0	0
$385$	0	0
$386$	197265.	0.0673879
$387$	0	0
$388$	240079.	0.0809609
$389$	1.24240e6	0.416281	0.208140	−	0.978099i	$-0.433259\pi$
$389$	1.24240e6	0.416281	0.208140	+	0.978099i	$0.433259\pi$
$390$	0	0
$391$	−4.31148e6	−1.42621
$392$	0	0
$393$	0	0
$394$	−3.76603e6	−1.22220
$395$	3.93522e6	1.26904
$396$	0	0
$397$	2.74562e6	0.874308	0.437154	−	0.899387i	$-0.355986\pi$
$397$	2.74562e6	0.874308	0.437154	+	0.899387i	$0.355986\pi$
$398$	−2.56005e6	−0.810103
$399$	0	0
$400$	−466276.	−0.145711
$401$	3.71335e6	1.15320	0.576601	−	0.817026i	$-0.304379\pi$
$401$	3.71335e6	1.15320	0.576601	+	0.817026i	$0.304379\pi$
$402$	0	0
$403$	−2.68244e6	−0.822748
$404$	−430018.	−0.131079
$405$	0	0
$406$	0	0
$407$	362656.	0.108520
$408$	0	0
$409$	3.75940e6	1.11125	0.555623	−	0.831434i	$-0.312480\pi$
$409$	3.75940e6	1.11125	0.555623	+	0.831434i	$0.312480\pi$
$410$	550625.	0.161769
$411$	0	0
$412$	−830643.	−0.241086
$413$	0	0
$414$	0	0
$415$	−2.01574e6	−0.574531
$416$	−1.18668e6	−0.336203
$417$	0	0
$418$	−174260.	−0.0487818
$419$	−4.60027e6	−1.28011	−0.640056	−	0.768328i	$-0.721089\pi$
$419$	−4.60027e6	−1.28011	−0.640056	+	0.768328i	$0.721089\pi$
$420$	0	0
$421$	−404864.	−0.111328	−0.0556639	−	0.998450i	$-0.517728\pi$
$421$	−404864.	−0.111328	−0.0556639	+	0.998450i	$0.517728\pi$
$422$	−3.68748e6	−1.00797
$423$	0	0
$424$	−715917.	−0.193396
$425$	2.25503e6	0.605591
$426$	0	0
$427$	0	0
$428$	399596.	0.105442
$429$	0	0
$430$	−566963.	−0.147871
$431$	−94758.5	−0.0245711	−0.0122856	−	0.999925i	$-0.503911\pi$
$431$	−94758.5	−0.0245711	−0.0122856	+	0.999925i	$0.503911\pi$
$432$	0	0
$433$	4.31727e6	1.10660	0.553298	−	0.832983i	$-0.313369\pi$
$433$	4.31727e6	1.10660	0.553298	+	0.832983i	$0.313369\pi$
$434$	0	0
$435$	0	0
$436$	−58178.3	−0.0146570
$437$	−976041.	−0.244492
$438$	0	0
$439$	−2.22367e6	−0.550693	−0.275346	−	0.961345i	$-0.588793\pi$
$439$	−2.22367e6	−0.550693	−0.275346	+	0.961345i	$0.588793\pi$
$440$	359172.	0.0884444
$441$	0	0
$442$	5.73909e6	1.39729
$443$	4.68425e6	1.13405	0.567024	−	0.823701i	$-0.308095\pi$
$443$	4.68425e6	1.13405	0.567024	+	0.823701i	$0.308095\pi$
$444$	0	0
$445$	3.44973e6	0.825821
$446$	3.35080e6	0.797648
$447$	0	0
$448$	0	0
$449$	−897932.	−0.210198	−0.105099	−	0.994462i	$-0.533516\pi$
$449$	−897932.	−0.210198	−0.105099	+	0.994462i	$0.533516\pi$
$450$	0	0
$451$	592612.	0.137192
$452$	−994800.	−0.229028
$453$	0	0
$454$	−5.32276e6	−1.21198
$455$	0	0
$456$	0	0
$457$	5.98181e6	1.33981	0.669903	−	0.742448i	$-0.266336\pi$
$457$	5.98181e6	1.33981	0.669903	+	0.742448i	$0.266336\pi$
$458$	−4.04253e6	−0.900513
$459$	0	0
$460$	2.01174e6	0.443279
$461$	−1.62507e6	−0.356138	−0.178069	−	0.984018i	$-0.556985\pi$
$461$	−1.62507e6	−0.356138	−0.178069	+	0.984018i	$0.556985\pi$
$462$	0	0
$463$	−8.46295e6	−1.83472	−0.917359	−	0.398060i	$-0.869683\pi$
$463$	−8.46295e6	−1.83472	−0.917359	+	0.398060i	$0.869683\pi$
$464$	1.44814e6	0.312259
$465$	0	0
$466$	5.94440e6	1.26807
$467$	−7.66594e6	−1.62657	−0.813286	−	0.581864i	$-0.802324\pi$
$467$	−7.66594e6	−1.62657	−0.813286	+	0.581864i	$0.802324\pi$
$468$	0	0
$469$	0	0
$470$	−1.60546e6	−0.335240
$471$	0	0
$472$	−384662.	−0.0794740
$473$	−610195.	−0.125405
$474$	0	0
$475$	510497.	0.103815
$476$	0	0
$477$	0	0
$478$	3.50255e6	0.701156
$479$	2.40292e6	0.478521	0.239261	−	0.970955i	$-0.423095\pi$
$479$	2.40292e6	0.478521	0.239261	+	0.970955i	$0.423095\pi$
$480$	0	0
$481$	−2.70385e6	−0.532867
$482$	−5.81178e6	−1.13944
$483$	0	0
$484$	−2.19026e6	−0.424993
$485$	541762.	0.104581
$486$	0	0
$487$	−170302.	−0.0325384	−0.0162692	−	0.999868i	$-0.505179\pi$
$487$	−170302.	−0.0325384	−0.0162692	+	0.999868i	$0.505179\pi$
$488$	949678.	0.180521
$489$	0	0
$490$	0	0
$491$	5.77628e6	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$491$	5.77628e6	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$492$	0	0
$493$	−7.00355e6	−1.29778
$494$	1.29923e6	0.239534
$495$	0	0
$496$	−592564.	−0.108151
$497$	0	0
$498$	0	0
$499$	543602.	0.0977303	0.0488652	−	0.998805i	$-0.484440\pi$
$499$	543602.	0.0977303	0.0488652	+	0.998805i	$0.484440\pi$
$500$	−2.85747e6	−0.511160
$501$	0	0
$502$	−793537.	−0.140543
$503$	5.40086e6	0.951794	0.475897	−	0.879501i	$-0.342124\pi$
$503$	5.40086e6	0.951794	0.475897	+	0.879501i	$0.342124\pi$
$504$	0	0
$505$	−970378.	−0.169322
$506$	2.16514e6	0.375932
$507$	0	0
$508$	1.01680e6	0.174815
$509$	−8.34418e6	−1.42754	−0.713772	−	0.700378i	$-0.753015\pi$
$509$	−8.34418e6	−1.42754	−0.713772	+	0.700378i	$0.753015\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.91829e6	0.320262
$515$	−1.87443e6	−0.311423
$516$	0	0
$517$	−1.72788e6	−0.284307
$518$	0	0
$519$	0	0
$520$	−2.67786e6	−0.434290
$521$	1.14784e7	1.85262	0.926310	−	0.376762i	$-0.122963\pi$
$521$	1.14784e7	1.85262	0.926310	+	0.376762i	$0.122963\pi$
$522$	0	0
$523$	−4.38734e6	−0.701370	−0.350685	−	0.936494i	$-0.614051\pi$
$523$	−4.38734e6	−0.701370	−0.350685	+	0.936494i	$0.614051\pi$
$524$	−2.18075e6	−0.346959
$525$	0	0
$526$	1.81977e6	0.286782
$527$	2.86579e6	0.449487
$528$	0	0
$529$	5.69070e6	0.884151
$530$	−1.61554e6	−0.249820
$531$	0	0
$532$	0	0
$533$	−4.41832e6	−0.673657
$534$	0	0
$535$	901728.	0.136204
$536$	2.75097e6	0.413593
$537$	0	0
$538$	−3.44055e6	−0.512474
$539$	0	0
$540$	0	0
$541$	7.22306e6	1.06103	0.530515	−	0.847675i	$-0.321998\pi$
$541$	7.22306e6	1.06103	0.530515	+	0.847675i	$0.321998\pi$
$542$	−5.22232e6	−0.763599
$543$	0	0
$544$	1.26779e6	0.183676
$545$	−131285.	−0.0189332
$546$	0	0
$547$	3.54529e6	0.506622	0.253311	−	0.967385i	$-0.418480\pi$
$547$	3.54529e6	0.506622	0.253311	+	0.967385i	$0.418480\pi$
$548$	−5.36486e6	−0.763144
$549$	0	0
$550$	−1.13243e6	−0.159626
$551$	−1.58548e6	−0.222475
$552$	0	0
$553$	0	0
$554$	2.17243e6	0.300727
$555$	0	0
$556$	931124.	0.127738
$557$	−1.03663e7	−1.41575	−0.707873	−	0.706340i	$-0.750345\pi$
$557$	−1.03663e7	−1.41575	−0.707873	+	0.706340i	$0.750345\pi$
$558$	0	0
$559$	4.54941e6	0.615780
$560$	0	0
$561$	0	0
$562$	3.99235e6	0.533198
$563$	4.48282e6	0.596047	0.298023	−	0.954559i	$-0.403673\pi$
$563$	4.48282e6	0.596047	0.298023	+	0.954559i	$0.403673\pi$
$564$	0	0
$565$	−2.24486e6	−0.295848
$566$	−7.10928e6	−0.932792
$567$	0	0
$568$	−1.27760e6	−0.166158
$569$	−6.49335e6	−0.840792	−0.420396	−	0.907341i	$-0.638109\pi$
$569$	−6.49335e6	−0.840792	−0.420396	+	0.907341i	$0.638109\pi$
$570$	0	0
$571$	−497508.	−0.0638571	−0.0319286	−	0.999490i	$-0.510165\pi$
$571$	−497508.	−0.0638571	−0.0319286	+	0.999490i	$0.510165\pi$
$572$	−2.88206e6	−0.368309
$573$	0	0
$574$	0	0
$575$	−6.34279e6	−0.800038
$576$	0	0
$577$	−6.55123e6	−0.819187	−0.409594	−	0.912268i	$-0.634330\pi$
$577$	−6.55123e6	−0.819187	−0.409594	+	0.912268i	$0.634330\pi$
$578$	−451946.	−0.0562687
$579$	0	0
$580$	3.26786e6	0.403360
$581$	0	0
$582$	0	0
$583$	−1.73873e6	−0.211865
$584$	−2.91524e6	−0.353706
$585$	0	0
$586$	−6.55947e6	−0.789087
$587$	30793.5	0.00368862	0.00184431	−	0.999998i	$-0.499413\pi$
$587$	30793.5	0.00368862	0.00184431	+	0.999998i	$0.499413\pi$
$588$	0	0
$589$	648763.	0.0770544
$590$	−868028.	−0.102661
$591$	0	0
$592$	−597293.	−0.0700459
$593$	−570937.	−0.0666732	−0.0333366	−	0.999444i	$-0.510613\pi$
$593$	−570937.	−0.0666732	−0.0333366	+	0.999444i	$0.510613\pi$
$594$	0	0
$595$	0	0
$596$	−4.52893e6	−0.522251
$597$	0	0
$598$	−1.61425e7	−1.84595
$599$	−9.07111e6	−1.03298	−0.516492	−	0.856292i	$-0.672762\pi$
$599$	−9.07111e6	−1.03298	−0.516492	+	0.856292i	$0.672762\pi$
$600$	0	0
$601$	−1.36309e7	−1.53936	−0.769679	−	0.638431i	$-0.779584\pi$
$601$	−1.36309e7	−1.53936	−0.769679	+	0.638431i	$0.779584\pi$
$602$	0	0
$603$	0	0
$604$	−3.86492e6	−0.431071
$605$	−4.94253e6	−0.548985
$606$	0	0
$607$	−3.95651e6	−0.435853	−0.217927	−	0.975965i	$-0.569929\pi$
$607$	−3.95651e6	−0.435853	−0.217927	+	0.975965i	$0.569929\pi$
$608$	287006.	0.0314870
$609$	0	0
$610$	2.14304e6	0.233188
$611$	1.28825e7	1.39604
$612$	0	0
$613$	1.32310e6	0.142214	0.0711070	−	0.997469i	$-0.477347\pi$
$613$	1.32310e6	0.142214	0.0711070	+	0.997469i	$0.477347\pi$
$614$	−9.52522e6	−1.01966
$615$	0	0
$616$	0	0
$617$	−9.26370e6	−0.979651	−0.489826	−	0.871820i	$-0.662940\pi$
$617$	−9.26370e6	−0.979651	−0.489826	+	0.871820i	$0.662940\pi$
$618$	0	0
$619$	669385.	0.0702182	0.0351091	−	0.999383i	$-0.488822\pi$
$619$	669385.	0.0702182	0.0351091	+	0.999383i	$0.488822\pi$
$620$	−1.33718e6	−0.139704
$621$	0	0
$622$	−4.98051e6	−0.516176
$623$	0	0
$624$	0	0
$625$	−756327.	−0.0774479
$626$	−6.65196e6	−0.678444
$627$	0	0
$628$	−3.42446e6	−0.346492
$629$	2.88866e6	0.291118
$630$	0	0
$631$	−666246.	−0.0666133	−0.0333067	−	0.999445i	$-0.510604\pi$
$631$	−666246.	−0.0666133	−0.0333067	+	0.999445i	$0.510604\pi$
$632$	−6.97549e6	−0.694675
$633$	0	0
$634$	−1.04994e7	−1.03739
$635$	2.29452e6	0.225817
$636$	0	0
$637$	0	0
$638$	3.51704e6	0.342078
$639$	0	0
$640$	−591553.	−0.0570879
$641$	2.10877e6	0.202714	0.101357	−	0.994850i	$-0.467682\pi$
$641$	2.10877e6	0.202714	0.101357	+	0.994850i	$0.467682\pi$
$642$	0	0
$643$	1.25977e7	1.20161	0.600806	−	0.799395i	$-0.294846\pi$
$643$	1.25977e7	1.20161	0.600806	+	0.799395i	$0.294846\pi$
$644$	0	0
$645$	0	0
$646$	−1.38803e6	−0.130863
$647$	−1.71299e7	−1.60877	−0.804386	−	0.594107i	$-0.797506\pi$
$647$	−1.71299e7	−1.60877	−0.804386	+	0.594107i	$0.797506\pi$
$648$	0	0
$649$	−934217.	−0.0870635
$650$	8.44301e6	0.783815
$651$	0	0
$652$	1.03146e6	0.0950237
$653$	8.25525e6	0.757613	0.378806	−	0.925476i	$-0.376335\pi$
$653$	8.25525e6	0.757613	0.378806	+	0.925476i	$0.376335\pi$
$654$	0	0
$655$	−4.92108e6	−0.448185
$656$	−976028.	−0.0885529
$657$	0	0
$658$	0	0
$659$	1.05417e7	0.945581	0.472790	−	0.881175i	$-0.343247\pi$
$659$	1.05417e7	0.945581	0.472790	+	0.881175i	$0.343247\pi$
$660$	0	0
$661$	4.47640e6	0.398498	0.199249	−	0.979949i	$-0.436150\pi$
$661$	4.47640e6	0.398498	0.199249	+	0.979949i	$0.436150\pi$
$662$	−6.95542e6	−0.616848
$663$	0	0
$664$	3.57306e6	0.314499
$665$	0	0
$666$	0	0
$667$	1.96991e7	1.71448
$668$	7.08311e6	0.614162
$669$	0	0
$670$	6.20783e6	0.534260
$671$	2.30645e6	0.197760
$672$	0	0
$673$	−1.87165e7	−1.59289	−0.796447	−	0.604708i	$-0.793290\pi$
$673$	−1.87165e7	−1.59289	−0.796447	+	0.604708i	$0.793290\pi$
$674$	−3.41426e6	−0.289498
$675$	0	0
$676$	1.55470e7	1.30852
$677$	−4.16232e6	−0.349031	−0.174515	−	0.984654i	$-0.555836\pi$
$677$	−4.16232e6	−0.349031	−0.174515	+	0.984654i	$0.555836\pi$
$678$	0	0
$679$	0	0
$680$	2.86090e6	0.237263
$681$	0	0
$682$	−1.43914e6	−0.118479
$683$	1.28691e7	1.05559	0.527796	−	0.849371i	$-0.323019\pi$
$683$	1.28691e7	1.05559	0.527796	+	0.849371i	$0.323019\pi$
$684$	0	0
$685$	−1.21063e7	−0.985793
$686$	0	0
$687$	0	0
$688$	1.00499e6	0.0809449
$689$	1.29634e7	1.04033
$690$	0	0
$691$	−9.29037e6	−0.740181	−0.370090	−	0.928996i	$-0.620673\pi$
$691$	−9.29037e6	−0.740181	−0.370090	+	0.928996i	$0.620673\pi$
$692$	1.25758e6	0.0998325
$693$	0	0
$694$	−1.19400e7	−0.941033
$695$	2.10117e6	0.165006
$696$	0	0
$697$	4.72032e6	0.368035
$698$	−1.14987e7	−0.893325
$699$	0	0
$700$	0	0
$701$	−2.50809e7	−1.92774	−0.963870	−	0.266375i	$-0.914174\pi$
$701$	−2.50809e7	−1.92774	−0.963870	+	0.266375i	$0.914174\pi$
$702$	0	0
$703$	653940.	0.0499057
$704$	−636661.	−0.0484146
$705$	0	0
$706$	5.88437e6	0.444312
$707$	0	0
$708$	0	0
$709$	9.93780e6	0.742463	0.371231	−	0.928540i	$-0.378936\pi$
$709$	9.93780e6	0.742463	0.371231	+	0.928540i	$0.378936\pi$
$710$	−2.88302e6	−0.214635
$711$	0	0
$712$	−6.11493e6	−0.452055
$713$	−8.06069e6	−0.593811
$714$	0	0
$715$	−6.50364e6	−0.475764
$716$	8.16683e6	0.595348
$717$	0	0
$718$	1.23120e7	0.891285
$719$	1.40009e7	1.01003	0.505016	−	0.863110i	$-0.331487\pi$
$719$	1.40009e7	1.01003	0.505016	+	0.863110i	$0.331487\pi$
$720$	0	0
$721$	0	0
$722$	9.59017e6	0.684673
$723$	0	0
$724$	−352829.	−0.0250160
$725$	−1.03032e7	−0.727993
$726$	0	0
$727$	8.27315e6	0.580544	0.290272	−	0.956944i	$-0.406254\pi$
$727$	8.27315e6	0.580544	0.290272	+	0.956944i	$0.406254\pi$
$728$	0	0
$729$	0	0
$730$	−6.57852e6	−0.456900
$731$	−4.86037e6	−0.336416
$732$	0	0
$733$	−3.28591e6	−0.225889	−0.112945	−	0.993601i	$-0.536028\pi$
$733$	−3.28591e6	−0.225889	−0.112945	+	0.993601i	$0.536028\pi$
$734$	−6.46679e6	−0.443046
$735$	0	0
$736$	−3.56597e6	−0.242651
$737$	6.68119e6	0.453090
$738$	0	0
$739$	1.00913e7	0.679728	0.339864	−	0.940475i	$-0.389619\pi$
$739$	1.00913e7	0.679728	0.339864	+	0.940475i	$0.389619\pi$
$740$	−1.34785e6	−0.0904820
$741$	0	0
$742$	0	0
$743$	−1.73443e7	−1.15261	−0.576307	−	0.817233i	$-0.695507\pi$
$743$	−1.73443e7	−1.15261	−0.576307	+	0.817233i	$0.695507\pi$
$744$	0	0
$745$	−1.02200e7	−0.674619
$746$	1.90447e7	1.25293
$747$	0	0
$748$	3.07905e6	0.201216
$749$	0	0
$750$	0	0
$751$	−1.75971e7	−1.13852	−0.569262	−	0.822156i	$-0.692771\pi$
$751$	−1.75971e7	−1.13852	−0.569262	+	0.822156i	$0.692771\pi$
$752$	2.84581e6	0.183511
$753$	0	0
$754$	−2.62219e7	−1.67971
$755$	−8.72158e6	−0.556836
$756$	0	0
$757$	−4.66480e6	−0.295865	−0.147932	−	0.988997i	$-0.547262\pi$
$757$	−4.66480e6	−0.295865	−0.147932	+	0.988997i	$0.547262\pi$
$758$	−4.00772e6	−0.253352
$759$	0	0
$760$	647657.	0.0406734
$761$	−1.82717e7	−1.14371	−0.571857	−	0.820353i	$-0.693777\pi$
$761$	−1.82717e7	−1.14371	−0.571857	+	0.820353i	$0.693777\pi$
$762$	0	0
$763$	0	0
$764$	8.93044e6	0.553528
$765$	0	0
$766$	−1.12861e7	−0.694979
$767$	6.96521e6	0.427510
$768$	0	0
$769$	−2.31895e7	−1.41409	−0.707043	−	0.707171i	$-0.749971\pi$
$769$	−2.31895e7	−1.41409	−0.707043	+	0.707171i	$0.749971\pi$
$770$	0	0
$771$	0	0
$772$	−789060.	−0.0476504
$773$	−2.02149e6	−0.121681	−0.0608405	−	0.998148i	$-0.519378\pi$
$773$	−2.02149e6	−0.121681	−0.0608405	+	0.998148i	$0.519378\pi$
$774$	0	0
$775$	4.21597e6	0.252141
$776$	−960317.	−0.0572480
$777$	0	0
$778$	−4.96959e6	−0.294355
$779$	1.06859e6	0.0630913
$780$	0	0
$781$	−3.10285e6	−0.182026
$782$	1.72459e7	1.00849
$783$	0	0
$784$	0	0
$785$	−7.72763e6	−0.447582
$786$	0	0
$787$	1.57744e7	0.907856	0.453928	−	0.891038i	$-0.350022\pi$
$787$	1.57744e7	0.907856	0.453928	+	0.891038i	$0.350022\pi$
$788$	1.50641e7	0.864229
$789$	0	0
$790$	−1.57409e7	−0.897348
$791$	0	0
$792$	0	0
$793$	−1.71961e7	−0.971064
$794$	−1.09825e7	−0.618229
$795$	0	0
$796$	1.02402e7	0.572830
$797$	−1.22436e6	−0.0682750	−0.0341375	−	0.999417i	$-0.510868\pi$
$797$	−1.22436e6	−0.0682750	−0.0341375	+	0.999417i	$0.510868\pi$
$798$	0	0
$799$	−1.37631e7	−0.762690
$800$	1.86510e6	0.103033
$801$	0	0
$802$	−1.48534e7	−0.815437
$803$	−7.08014e6	−0.387483
$804$	0	0
$805$	0	0
$806$	1.07297e7	0.581771
$807$	0	0
$808$	1.72007e6	0.0926869
$809$	2.77325e7	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$809$	2.77325e7	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$810$	0	0
$811$	−1.19246e7	−0.636636	−0.318318	−	0.947984i	$-0.603118\pi$
$811$	−1.19246e7	−0.636636	−0.318318	+	0.947984i	$0.603118\pi$
$812$	0	0
$813$	0	0
$814$	−1.45063e6	−0.0767351
$815$	2.32758e6	0.122747
$816$	0	0
$817$	−1.10030e6	−0.0576709
$818$	−1.50376e7	−0.785770
$819$	0	0
$820$	−2.20250e6	−0.114388
$821$	−1.60462e7	−0.830836	−0.415418	−	0.909631i	$-0.636365\pi$
$821$	−1.60462e7	−0.830836	−0.415418	+	0.909631i	$0.636365\pi$
$822$	0	0
$823$	−1.12307e7	−0.577970	−0.288985	−	0.957334i	$-0.593318\pi$
$823$	−1.12307e7	−0.577970	−0.288985	+	0.957334i	$0.593318\pi$
$824$	3.32257e6	0.170473
$825$	0	0
$826$	0	0
$827$	2.68427e7	1.36478	0.682389	−	0.730989i	$-0.260941\pi$
$827$	2.68427e7	1.36478	0.682389	+	0.730989i	$0.260941\pi$
$828$	0	0
$829$	2.80419e7	1.41717	0.708584	−	0.705626i	$-0.249334\pi$
$829$	2.80419e7	1.41717	0.708584	+	0.705626i	$0.249334\pi$
$830$	8.06295e6	0.406255
$831$	0	0
$832$	4.74673e6	0.237731
$833$	0	0
$834$	0	0
$835$	1.59837e7	0.793344
$836$	697042.	0.0344940
$837$	0	0
$838$	1.84011e7	0.905176
$839$	3.30603e7	1.62145	0.810723	−	0.585430i	$-0.199074\pi$
$839$	3.30603e7	1.62145	0.810723	+	0.585430i	$0.199074\pi$
$840$	0	0
$841$	1.14880e7	0.560086
$842$	1.61946e6	0.0787207
$843$	0	0
$844$	1.47499e7	0.712743
$845$	3.50832e7	1.69028
$846$	0	0
$847$	0	0
$848$	2.86367e6	0.136752
$849$	0	0
$850$	−9.02010e6	−0.428217
$851$	−8.12503e6	−0.384593
$852$	0	0
$853$	2.73897e7	1.28889	0.644444	−	0.764652i	$-0.277089\pi$
$853$	2.73897e7	1.28889	0.644444	+	0.764652i	$0.277089\pi$
$854$	0	0
$855$	0	0
$856$	−1.59838e6	−0.0745585
$857$	2.96721e6	0.138005	0.0690027	−	0.997616i	$-0.478018\pi$
$857$	2.96721e6	0.138005	0.0690027	+	0.997616i	$0.478018\pi$
$858$	0	0
$859$	−3.97960e7	−1.84016	−0.920081	−	0.391729i	$-0.871877\pi$
$859$	−3.97960e7	−1.84016	−0.920081	+	0.391729i	$0.871877\pi$
$860$	2.26785e6	0.104561
$861$	0	0
$862$	379034.	0.0173744
$863$	−7.19597e6	−0.328899	−0.164450	−	0.986385i	$-0.552585\pi$
$863$	−7.19597e6	−0.328899	−0.164450	+	0.986385i	$0.552585\pi$
$864$	0	0
$865$	2.83786e6	0.128959
$866$	−1.72691e7	−0.782482
$867$	0	0
$868$	0	0
$869$	−1.69411e7	−0.761015
$870$	0	0
$871$	−4.98127e7	−2.22482
$872$	232713.	0.0103641
$873$	0	0
$874$	3.90417e6	0.172882
$875$	0	0
$876$	0	0
$877$	2.51242e7	1.10305	0.551523	−	0.834160i	$-0.314047\pi$
$877$	2.51242e7	1.10305	0.551523	+	0.834160i	$0.314047\pi$
$878$	8.89469e6	0.389399
$879$	0	0
$880$	−1.43669e6	−0.0625396
$881$	−120262.	−0.00522022	−0.00261011	−	0.999997i	$-0.500831\pi$
$881$	−120262.	−0.00522022	−0.00261011	+	0.999997i	$0.500831\pi$
$882$	0	0
$883$	2.43497e7	1.05097	0.525487	−	0.850802i	$-0.323883\pi$
$883$	2.43497e7	1.05097	0.525487	+	0.850802i	$0.323883\pi$
$884$	−2.29564e7	−0.988036
$885$	0	0
$886$	−1.87370e7	−0.801893
$887$	2.82766e7	1.20675	0.603377	−	0.797456i	$-0.293821\pi$
$887$	2.82766e7	1.20675	0.603377	+	0.797456i	$0.293821\pi$
$888$	0	0
$889$	0	0
$890$	−1.37989e7	−0.583943
$891$	0	0
$892$	−1.34032e7	−0.564023
$893$	−3.11571e6	−0.130746
$894$	0	0
$895$	1.84293e7	0.769042
$896$	0	0
$897$	0	0
$898$	3.59173e6	0.148632
$899$	−1.30938e7	−0.540337
$900$	0	0
$901$	−1.38494e7	−0.568355
$902$	−2.37045e6	−0.0970094
$903$	0	0
$904$	3.97920e6	0.161948
$905$	−796192.	−0.0323144
$906$	0	0
$907$	−2.43168e7	−0.981496	−0.490748	−	0.871302i	$-0.663276\pi$
$907$	−2.43168e7	−0.981496	−0.490748	+	0.871302i	$0.663276\pi$
$908$	2.12910e7	0.857002
$909$	0	0
$910$	0	0
$911$	3.36224e7	1.34225	0.671123	−	0.741346i	$-0.265812\pi$
$911$	3.36224e7	1.34225	0.671123	+	0.741346i	$0.265812\pi$
$912$	0	0
$913$	8.67777e6	0.344533
$914$	−2.39272e7	−0.947386
$915$	0	0
$916$	1.61701e7	0.636759
$917$	0	0
$918$	0	0
$919$	−8.13749e6	−0.317835	−0.158917	−	0.987292i	$-0.550800\pi$
$919$	−8.13749e6	−0.317835	−0.158917	+	0.987292i	$0.550800\pi$
$920$	−8.04695e6	−0.313445
$921$	0	0
$922$	6.50026e6	0.251828
$923$	2.31338e7	0.893807
$924$	0	0
$925$	4.24962e6	0.163304
$926$	3.38518e7	1.29734
$927$	0	0
$928$	−5.79254e6	−0.220800
$929$	−3.58054e7	−1.36116	−0.680580	−	0.732674i	$-0.738272\pi$
$929$	−3.58054e7	−1.36116	−0.680580	+	0.732674i	$0.738272\pi$
$930$	0	0
$931$	0	0
$932$	−2.37776e7	−0.896661
$933$	0	0
$934$	3.06638e7	1.15016
$935$	6.94818e6	0.259921
$936$	0	0
$937$	4.65849e7	1.73339	0.866694	−	0.498840i	$-0.166241\pi$
$937$	4.65849e7	1.73339	0.866694	+	0.498840i	$0.166241\pi$
$938$	0	0
$939$	0	0
$940$	6.42185e6	0.237050
$941$	2.06725e6	0.0761059	0.0380529	−	0.999276i	$-0.487884\pi$
$941$	2.06725e6	0.0761059	0.0380529	+	0.999276i	$0.487884\pi$
$942$	0	0
$943$	−1.32770e7	−0.486206
$944$	1.53865e6	0.0561966
$945$	0	0
$946$	2.44078e6	0.0886750
$947$	−2.36773e7	−0.857940	−0.428970	−	0.903319i	$-0.641123\pi$
$947$	−2.36773e7	−0.857940	−0.428970	+	0.903319i	$0.641123\pi$
$948$	0	0
$949$	5.27872e7	1.90267
$950$	−2.04199e6	−0.0734082
$951$	0	0
$952$	0	0
$953$	3.40521e6	0.121454	0.0607270	−	0.998154i	$-0.480658\pi$
$953$	3.40521e6	0.121454	0.0607270	+	0.998154i	$0.480658\pi$
$954$	0	0
$955$	2.01524e7	0.715020
$956$	−1.40102e7	−0.495792
$957$	0	0
$958$	−9.61170e6	−0.338366
$959$	0	0
$960$	0	0
$961$	−2.32713e7	−0.812854
$962$	1.08154e7	0.376794
$963$	0	0
$964$	2.32471e7	0.805706
$965$	−1.78059e6	−0.0615525
$966$	0	0
$967$	1.44238e6	0.0496036	0.0248018	−	0.999692i	$-0.492105\pi$
$967$	1.44238e6	0.0496036	0.0248018	+	0.999692i	$0.492105\pi$
$968$	8.76103e6	0.300515
$969$	0	0
$970$	−2.16705e6	−0.0739502
$971$	5.15954e6	0.175616	0.0878078	−	0.996137i	$-0.472014\pi$
$971$	5.15954e6	0.175616	0.0878078	+	0.996137i	$0.472014\pi$
$972$	0	0
$973$	0	0
$974$	681206.	0.0230081
$975$	0	0
$976$	−3.79871e6	−0.127647
$977$	6.97650e6	0.233831	0.116915	−	0.993142i	$-0.462699\pi$
$977$	6.97650e6	0.233831	0.116915	+	0.993142i	$0.462699\pi$
$978$	0	0
$979$	−1.48511e7	−0.495225
$980$	0	0
$981$	0	0
$982$	−2.31051e7	−0.764592
$983$	2.81568e7	0.929393	0.464696	−	0.885470i	$-0.346163\pi$
$983$	2.81568e7	0.929393	0.464696	+	0.885470i	$0.346163\pi$
$984$	0	0
$985$	3.39937e7	1.11637
$986$	2.80142e7	0.917669
$987$	0	0
$988$	−5.19691e6	−0.169376
$989$	1.36709e7	0.444434
$990$	0	0
$991$	−2.34043e7	−0.757028	−0.378514	−	0.925596i	$-0.623565\pi$
$991$	−2.34043e7	−0.757028	−0.378514	+	0.925596i	$0.623565\pi$
$992$	2.37025e6	0.0764743
$993$	0	0
$994$	0	0
$995$	2.31080e7	0.739954
$996$	0	0
$997$	−4.30284e7	−1.37094	−0.685468	−	0.728103i	$-0.740402\pi$
$997$	−4.30284e7	−1.37094	−0.685468	+	0.728103i	$0.740402\pi$
$998$	−2.17441e6	−0.0691058
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.ba.1.2		2
3.2	odd	2		98.6.a.h.1.2		2
7.2	even	3		126.6.g.j.109.1		4
7.4	even	3		126.6.g.j.37.1		4
7.6	odd	2		882.6.a.bi.1.1		2
12.11	even	2		784.6.a.s.1.1		2
21.2	odd	6		14.6.c.a.11.1	yes	4
21.5	even	6		98.6.c.e.67.2		4
21.11	odd	6		14.6.c.a.9.1	✓	4
21.17	even	6		98.6.c.e.79.2		4
21.20	even	2		98.6.a.g.1.1		2
84.11	even	6		112.6.i.d.65.2		4
84.23	even	6		112.6.i.d.81.2		4
84.83	odd	2		784.6.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.a.9.1	✓	4	21.11	odd	6
14.6.c.a.11.1	yes	4	21.2	odd	6
98.6.a.g.1.1		2	21.20	even	2
98.6.a.h.1.2		2	3.2	odd	2
98.6.c.e.67.2		4	21.5	even	6
98.6.c.e.79.2		4	21.17	even	6
112.6.i.d.65.2		4	84.11	even	6
112.6.i.d.81.2		4	84.23	even	6
126.6.g.j.37.1		4	7.4	even	3
126.6.g.j.109.1		4	7.2	even	3
784.6.a.s.1.1		2	12.11	even	2
784.6.a.bb.1.2		2	84.83	odd	2
882.6.a.ba.1.2		2	1.1	even	1	trivial
882.6.a.bi.1.1		2	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +36.1056 q^{5} -64.0000 q^{8} -144.422 q^{10} -155.435 q^{11} +1158.87 q^{13} +256.000 q^{16} -1238.08 q^{17} -280.279 q^{19} +577.689 q^{20} +621.739 q^{22} +3482.39 q^{23} -1821.39 q^{25} -4635.48 q^{26} +5656.78 q^{29} -2314.70 q^{31} -1024.00 q^{32} +4952.32 q^{34} -2333.18 q^{37} +1121.12 q^{38} -2310.76 q^{40} -3812.61 q^{41} +3925.73 q^{43} -2486.96 q^{44} -13929.6 q^{46} +11116.5 q^{47} +7285.56 q^{50} +18541.9 q^{52} +11186.2 q^{53} -5612.06 q^{55} -22627.1 q^{58} +6010.35 q^{59} -14838.7 q^{61} +9258.81 q^{62} +4096.00 q^{64} +41841.6 q^{65} -42983.9 q^{67} -19809.3 q^{68} +19962.4 q^{71} +45550.6 q^{73} +9332.70 q^{74} -4484.47 q^{76} +108992. q^{79} +9243.02 q^{80} +15250.4 q^{82} -55829.0 q^{83} -44701.6 q^{85} -15702.9 q^{86} +9947.82 q^{88} +95545.8 q^{89} +55718.2 q^{92} -44465.8 q^{94} -10119.6 q^{95} +15004.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 70 q^{5} - 128 q^{8} + 280 q^{10} - 62 q^{11} + 1820 q^{13} + 512 q^{16} - 1694 q^{17} + 826 q^{19} - 1120 q^{20} + 248 q^{22} + 2734 q^{23} + 6312 q^{25} - 7280 q^{26} + 2852 q^{29}+ \cdots - 8316 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 70 * q^5 - 128 * q^8 + 280 * q^10 - 62 * q^11 + 1820 * q^13 + 512 * q^16 - 1694 * q^17 + 826 * q^19 - 1120 * q^20 + 248 * q^22 + 2734 * q^23 + 6312 * q^25 - 7280 * q^26 + 2852 * q^29 - 2674 * q^31 - 2048 * q^32 + 6776 * q^34 - 9146 * q^37 - 3304 * q^38 + 4480 * q^40 - 6132 * q^41 - 16040 * q^43 - 992 * q^44 - 10936 * q^46 + 25326 * q^47 - 25248 * q^50 + 29120 * q^52 - 14958 * q^53 - 15526 * q^55 - 11408 * q^58 + 1106 * q^59 - 28042 * q^61 + 10696 * q^62 + 8192 * q^64 - 28308 * q^65 - 102642 * q^67 - 27104 * q^68 + 11056 * q^71 + 35070 * q^73 + 36584 * q^74 + 13216 * q^76 + 101762 * q^79 - 17920 * q^80 + 24528 * q^82 + 44632 * q^83 + 3674 * q^85 + 64160 * q^86 + 3968 * q^88 + 75474 * q^89 + 43744 * q^92 - 101304 * q^94 - 127502 * q^95 - 8316 * q^97

Introduction

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