LMFDB - Embedded newform 441.6.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 $ x^6 - 2x^5 - 59x^4 + 122x^3 + 941x^2 - 1856*x - 2338
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.27213$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +O(q^{10})$	$q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +42.6416 q^{10} +3.66659 q^{11} +780.229 q^{13} +197.776 q^{16} +50.0832 q^{17} -1063.38 q^{19} -309.532 q^{20} +11.3357 q^{22} -4102.45 q^{23} -2934.76 q^{25} +2412.18 q^{26} +1487.11 q^{29} -5519.97 q^{31} +5997.49 q^{32} +154.838 q^{34} +6143.27 q^{37} -3287.59 q^{38} -2321.49 q^{40} +10757.9 q^{41} +17696.7 q^{43} -82.2850 q^{44} -12683.3 q^{46} +29468.5 q^{47} -9073.19 q^{50} -17509.8 q^{52} +19255.9 q^{53} +50.5718 q^{55} +4597.58 q^{58} +6619.18 q^{59} -36750.3 q^{61} -17065.7 q^{62} +12213.2 q^{64} +10761.4 q^{65} +46909.2 q^{67} -1123.96 q^{68} +41693.7 q^{71} -29451.1 q^{73} +18992.7 q^{74} +23864.3 q^{76} +22124.4 q^{79} +2727.84 q^{80} +33259.5 q^{82} -3896.35 q^{83} +690.778 q^{85} +54711.6 q^{86} -617.138 q^{88} +20530.8 q^{89} +92066.6 q^{92} +91105.4 q^{94} -14666.8 q^{95} -17742.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8	$ 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) $ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 - 864 * q^10 - 604 * q^11 - 1352 * q^13 + 4578 * q^16 + 3028 * q^17 - 1728 * q^19 + 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 + 14172 * q^26 + 5320 * q^29 - 3976 * q^31 + 37326 * q^32 + 16336 * q^34 + 22680 * q^37 + 52744 * q^38 - 100600 * q^40 + 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 + 51552 * q^47 + 40622 * q^50 - 119296 * q^52 - 80884 * q^53 - 11656 * q^55 - 70464 * q^58 + 8872 * q^59 - 50896 * q^61 + 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 + 37348 * q^68 + 110852 * q^71 - 64232 * q^73 + 27464 * q^74 + 194864 * q^76 + 111696 * q^79 - 308940 * q^80 + 189640 * q^82 + 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 - 35012 * q^89 + 449260 * q^92 + 121016 * q^94 + 119080 * q^95 - 70952 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.09163	0.546527	0.273264	−	0.961939i	$-0.411897\pi$
$2$	3.09163	0.546527	0.273264	+	0.961939i	$0.411897\pi$
$3$	0	0
$3$	0	0
$4$	−22.4418	−0.701308
$5$	13.7926	0.246730	0.123365	−	0.992361i	$-0.460631\pi$
$5$	13.7926	0.246730	0.123365	+	0.992361i	$0.460631\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−168.314	−0.929811
$9$	0	0
$10$	42.6416	0.134845
$11$	3.66659	0.00913651	0.00456826	−	0.999990i	$-0.498546\pi$
$11$	3.66659	0.00913651	0.00456826	+	0.999990i	$0.498546\pi$
$12$	0	0
$13$	780.229	1.28045	0.640226	−	0.768186i	$-0.278841\pi$
$13$	780.229	1.28045	0.640226	+	0.768186i	$0.278841\pi$
$14$	0	0
$15$	0	0
$16$	197.776	0.193140
$17$	50.0832	0.0420310	0.0210155	−	0.999779i	$-0.493310\pi$
$17$	50.0832	0.0420310	0.0210155	+	0.999779i	$0.493310\pi$
$18$	0	0
$19$	−1063.38	−0.675781	−0.337891	−	0.941185i	$-0.609713\pi$
$19$	−1063.38	−0.675781	−0.337891	+	0.941185i	$0.609713\pi$
$20$	−309.532	−0.173033
$21$	0	0
$22$	11.3357	0.00499336
$23$	−4102.45	−1.61705	−0.808526	−	0.588460i	$-0.799734\pi$
$23$	−4102.45	−1.61705	−0.808526	+	0.588460i	$0.799734\pi$
$24$	0	0
$25$	−2934.76	−0.939124
$26$	2412.18	0.699803
$27$	0	0
$28$	0	0
$29$	1487.11	0.328358	0.164179	−	0.986431i	$-0.447503\pi$
$29$	1487.11	0.328358	0.164179	+	0.986431i	$0.447503\pi$
$30$	0	0
$31$	−5519.97	−1.03165	−0.515825	−	0.856694i	$-0.672515\pi$
$31$	−5519.97	−1.03165	−0.515825	+	0.856694i	$0.672515\pi$
$32$	5997.49	1.03537
$33$	0	0
$34$	154.838	0.0229711
$35$	0	0
$36$	0	0
$37$	6143.27	0.737726	0.368863	−	0.929484i	$-0.379747\pi$
$37$	6143.27	0.737726	0.368863	+	0.929484i	$0.379747\pi$
$38$	−3287.59	−0.369333
$39$	0	0
$40$	−2321.49	−0.229412
$41$	10757.9	0.999468	0.499734	−	0.866179i	$-0.333431\pi$
$41$	10757.9	0.999468	0.499734	+	0.866179i	$0.333431\pi$
$42$	0	0
$43$	17696.7	1.45956	0.729779	−	0.683683i	$-0.239623\pi$
$43$	17696.7	1.45956	0.729779	+	0.683683i	$0.239623\pi$
$44$	−82.2850	−0.00640751
$45$	0	0
$46$	−12683.3	−0.883763
$47$	29468.5	1.94587	0.972933	−	0.231089i	$-0.0742289\pi$
$47$	29468.5	1.94587	0.972933	+	0.231089i	$0.0742289\pi$
$48$	0	0
$49$	0	0
$50$	−9073.19	−0.513257
$51$	0	0
$52$	−17509.8	−0.897991
$53$	19255.9	0.941616	0.470808	−	0.882236i	$-0.343962\pi$
$53$	19255.9	0.941616	0.470808	+	0.882236i	$0.343962\pi$
$54$	0	0
$55$	50.5718	0.00225425
$56$	0	0
$57$	0	0
$58$	4597.58	0.179457
$59$	6619.18	0.247557	0.123778	−	0.992310i	$-0.460499\pi$
$59$	6619.18	0.247557	0.123778	+	0.992310i	$0.460499\pi$
$60$	0	0
$61$	−36750.3	−1.26455	−0.632275	−	0.774744i	$-0.717879\pi$
$61$	−36750.3	−1.26455	−0.632275	+	0.774744i	$0.717879\pi$
$62$	−17065.7	−0.563825
$63$	0	0
$64$	12213.2	0.372717
$65$	10761.4	0.315926
$66$	0	0
$67$	46909.2	1.27665	0.638323	−	0.769768i	$-0.279628\pi$
$67$	46909.2	1.27665	0.638323	+	0.769768i	$0.279628\pi$
$68$	−1123.96	−0.0294767
$69$	0	0
$70$	0	0
$71$	41693.7	0.981577	0.490789	−	0.871279i	$-0.336709\pi$
$71$	41693.7	0.981577	0.490789	+	0.871279i	$0.336709\pi$
$72$	0	0
$73$	−29451.1	−0.646837	−0.323418	−	0.946256i	$-0.604832\pi$
$73$	−29451.1	−0.646837	−0.323418	+	0.946256i	$0.604832\pi$
$74$	18992.7	0.403188
$75$	0	0
$76$	23864.3	0.473931
$77$	0	0
$78$	0	0
$79$	22124.4	0.398844	0.199422	−	0.979914i	$-0.436094\pi$
$79$	22124.4	0.398844	0.199422	+	0.979914i	$0.436094\pi$
$80$	2727.84	0.0476535
$81$	0	0
$82$	33259.5	0.546237
$83$	−3896.35	−0.0620816	−0.0310408	−	0.999518i	$-0.509882\pi$
$83$	−3896.35	−0.0620816	−0.0310408	+	0.999518i	$0.509882\pi$
$84$	0	0
$85$	690.778	0.0103703
$86$	54711.6	0.797689
$87$	0	0
$88$	−617.138	−0.00849523
$89$	20530.8	0.274746	0.137373	−	0.990519i	$-0.456134\pi$
$89$	20530.8	0.274746	0.137373	+	0.990519i	$0.456134\pi$
$90$	0	0
$91$	0	0
$92$	92066.6	1.13405
$93$	0	0
$94$	91105.4	1.06347
$95$	−14666.8	−0.166735
$96$	0	0
$97$	−17742.9	−0.191468	−0.0957338	−	0.995407i	$-0.530520\pi$
$97$	−17742.9	−0.191468	−0.0957338	+	0.995407i	$0.530520\pi$
$98$	0	0
$99$	0	0
$100$	65861.5	0.658615
$101$	−142085.	−1.38594	−0.692971	−	0.720965i	$-0.743699\pi$
$101$	−142085.	−1.38594	−0.692971	+	0.720965i	$0.743699\pi$
$102$	0	0
$103$	210538.	1.95540	0.977702	−	0.209996i	$-0.0673450\pi$
$103$	210538.	1.95540	0.977702	+	0.209996i	$0.0673450\pi$
$104$	−131323.	−1.19058
$105$	0	0
$106$	59532.0	0.514619
$107$	202671.	1.71132	0.855660	−	0.517538i	$-0.173151\pi$
$107$	202671.	1.71132	0.855660	+	0.517538i	$0.173151\pi$
$108$	0	0
$109$	139538.	1.12493	0.562466	−	0.826821i	$-0.309853\pi$
$109$	139538.	1.12493	0.562466	+	0.826821i	$0.309853\pi$
$110$	156.349	0.00123201
$111$	0	0
$112$	0	0
$113$	206316.	1.51998	0.759989	−	0.649936i	$-0.225204\pi$
$113$	206316.	1.51998	0.759989	+	0.649936i	$0.225204\pi$
$114$	0	0
$115$	−56583.5	−0.398975
$116$	−33373.4	−0.230280
$117$	0	0
$118$	20464.0	0.135296
$119$	0	0
$120$	0	0
$121$	−161038.	−0.999917
$122$	−113618.	−0.691111
$123$	0	0
$124$	123878.	0.723504
$125$	−83580.0	−0.478440
$126$	0	0
$127$	89874.2	0.494454	0.247227	−	0.968958i	$-0.420481\pi$
$127$	89874.2	0.494454	0.247227	+	0.968958i	$0.420481\pi$
$128$	−154161.	−0.831668
$129$	0	0
$130$	33270.2	0.172662
$131$	320115.	1.62978	0.814889	−	0.579617i	$-0.196798\pi$
$131$	320115.	1.62978	0.814889	+	0.579617i	$0.196798\pi$
$132$	0	0
$133$	0	0
$134$	145026.	0.697722
$135$	0	0
$136$	−8429.69	−0.0390809
$137$	−258971.	−1.17883	−0.589414	−	0.807831i	$-0.700641\pi$
$137$	−258971.	−1.17883	−0.589414	+	0.807831i	$0.700641\pi$
$138$	0	0
$139$	−282366.	−1.23958	−0.619791	−	0.784767i	$-0.712783\pi$
$139$	−282366.	−1.23958	−0.619791	+	0.784767i	$0.712783\pi$
$140$	0	0
$141$	0	0
$142$	128901.	0.536459
$143$	2860.78	0.0116989
$144$	0	0
$145$	20511.1	0.0810156
$146$	−91051.9	−0.353514
$147$	0	0
$148$	−137866.	−0.517373
$149$	319669.	1.17960	0.589800	−	0.807549i	$-0.299207\pi$
$149$	319669.	1.17960	0.589800	+	0.807549i	$0.299207\pi$
$150$	0	0
$151$	−84046.3	−0.299969	−0.149985	−	0.988688i	$-0.547922\pi$
$151$	−84046.3	−0.299969	−0.149985	+	0.988688i	$0.547922\pi$
$152$	178982.	0.628349
$153$	0	0
$154$	0	0
$155$	−76134.8	−0.254539
$156$	0	0
$157$	−126195.	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$157$	−126195.	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$158$	68400.2	0.217979
$159$	0	0
$160$	82721.1	0.255456
$161$	0	0
$162$	0	0
$163$	−235790.	−0.695114	−0.347557	−	0.937659i	$-0.612989\pi$
$163$	−235790.	−0.695114	−0.347557	+	0.937659i	$0.612989\pi$
$164$	−241428.	−0.700935
$165$	0	0
$166$	−12046.1	−0.0339293
$167$	149843.	0.415761	0.207881	−	0.978154i	$-0.433343\pi$
$167$	149843.	0.415761	0.207881	+	0.978154i	$0.433343\pi$
$168$	0	0
$169$	237464.	0.639559
$170$	2135.63	0.00566765
$171$	0	0
$172$	−397147.	−1.02360
$173$	704460.	1.78954	0.894769	−	0.446530i	$-0.147340\pi$
$173$	704460.	1.78954	0.894769	+	0.446530i	$0.147340\pi$
$174$	0	0
$175$	0	0
$176$	725.162	0.00176463
$177$	0	0
$178$	63473.6	0.150156
$179$	557573.	1.30067	0.650337	−	0.759645i	$-0.274627\pi$
$179$	557573.	1.30067	0.650337	+	0.759645i	$0.274627\pi$
$180$	0	0
$181$	−443092.	−1.00530	−0.502652	−	0.864489i	$-0.667642\pi$
$181$	−443092.	−1.00530	−0.502652	+	0.864489i	$0.667642\pi$
$182$	0	0
$183$	0	0
$184$	690500.	1.50355
$185$	84731.7	0.182019
$186$	0	0
$187$	183.634	0.000384017 0
$188$	−661327.	−1.36465
$189$	0	0
$190$	−45344.4	−0.0911254
$191$	−157481.	−0.312353	−0.156177	−	0.987729i	$-0.549917\pi$
$191$	−157481.	−0.312353	−0.156177	+	0.987729i	$0.549917\pi$
$192$	0	0
$193$	−778040.	−1.50352	−0.751759	−	0.659437i	$-0.770795\pi$
$193$	−778040.	−1.50352	−0.751759	+	0.659437i	$0.770795\pi$
$194$	−54854.4	−0.104642
$195$	0	0
$196$	0	0
$197$	340283.	0.624704	0.312352	−	0.949966i	$-0.398883\pi$
$197$	340283.	0.624704	0.312352	+	0.949966i	$0.398883\pi$
$198$	0	0
$199$	−296509.	−0.530768	−0.265384	−	0.964143i	$-0.585499\pi$
$199$	−296509.	−0.530768	−0.265384	+	0.964143i	$0.585499\pi$
$200$	493961.	0.873209
$201$	0	0
$202$	−439274.	−0.757456
$203$	0	0
$204$	0	0
$205$	148380.	0.246599
$206$	650904.	1.06868
$207$	0	0
$208$	154310.	0.247307
$209$	−3898.99	−0.00617429
$210$	0	0
$211$	506728.	0.783554	0.391777	−	0.920060i	$-0.371860\pi$
$211$	506728.	0.783554	0.391777	+	0.920060i	$0.371860\pi$
$212$	−432138.	−0.660363
$213$	0	0
$214$	626582.	0.935283
$215$	244084.	0.360116
$216$	0	0
$217$	0	0
$218$	431399.	0.614806
$219$	0	0
$220$	−1134.93	−0.00158092
$221$	39076.3	0.0538187
$222$	0	0
$223$	462362.	0.622615	0.311308	−	0.950309i	$-0.399233\pi$
$223$	462362.	0.622615	0.311308	+	0.950309i	$0.399233\pi$
$224$	0	0
$225$	0	0
$226$	637852.	0.830709
$227$	−98226.6	−0.126521	−0.0632607	−	0.997997i	$-0.520150\pi$
$227$	−98226.6	−0.126521	−0.0632607	+	0.997997i	$0.520150\pi$
$228$	0	0
$229$	−501403.	−0.631827	−0.315914	−	0.948788i	$-0.602311\pi$
$229$	−501403.	−0.631827	−0.315914	+	0.948788i	$0.602311\pi$
$230$	−174935.	−0.218051
$231$	0	0
$232$	−250301.	−0.305311
$233$	613916.	0.740831	0.370415	−	0.928866i	$-0.379215\pi$
$233$	613916.	0.740831	0.370415	+	0.928866i	$0.379215\pi$
$234$	0	0
$235$	406447.	0.480103
$236$	−148547.	−0.173613
$237$	0	0
$238$	0	0
$239$	−1.10020e6	−1.24588	−0.622940	−	0.782269i	$-0.714062\pi$
$239$	−1.10020e6	−1.24588	−0.622940	+	0.782269i	$0.714062\pi$
$240$	0	0
$241$	−89883.5	−0.0996868	−0.0498434	−	0.998757i	$-0.515872\pi$
$241$	−89883.5	−0.0996868	−0.0498434	+	0.998757i	$0.515872\pi$
$242$	−497868.	−0.546482
$243$	0	0
$244$	824744.	0.886839
$245$	0	0
$246$	0	0
$247$	−829683.	−0.865306
$248$	929087.	0.959240
$249$	0	0
$250$	−258398.	−0.261480
$251$	1.01050e6	1.01240	0.506202	−	0.862415i	$-0.331049\pi$
$251$	1.01050e6	1.01240	0.506202	+	0.862415i	$0.331049\pi$
$252$	0	0
$253$	−15042.0	−0.0147742
$254$	277858.	0.270233
$255$	0	0
$256$	−867430.	−0.827246
$257$	−1.21233e6	−1.14495	−0.572475	−	0.819922i	$-0.694017\pi$
$257$	−1.21233e6	−1.14495	−0.572475	+	0.819922i	$0.694017\pi$
$258$	0	0
$259$	0	0
$260$	−241505.	−0.221561
$261$	0	0
$262$	989677.	0.890718
$263$	−384901.	−0.343131	−0.171566	−	0.985173i	$-0.554883\pi$
$263$	−384901.	−0.343131	−0.171566	+	0.985173i	$0.554883\pi$
$264$	0	0
$265$	265589.	0.232325
$266$	0	0
$267$	0	0
$268$	−1.05273e6	−0.895322
$269$	−1.57455e6	−1.32671	−0.663355	−	0.748305i	$-0.730868\pi$
$269$	−1.57455e6	−1.32671	−0.663355	+	0.748305i	$0.730868\pi$
$270$	0	0
$271$	−332270.	−0.274832	−0.137416	−	0.990513i	$-0.543880\pi$
$271$	−332270.	−0.274832	−0.137416	+	0.990513i	$0.543880\pi$
$272$	9905.24	0.00811788
$273$	0	0
$274$	−800643.	−0.644262
$275$	−10760.6	−0.00858032
$276$	0	0
$277$	2.23543e6	1.75050	0.875251	−	0.483669i	$-0.160696\pi$
$277$	2.23543e6	1.75050	0.875251	+	0.483669i	$0.160696\pi$
$278$	−872970.	−0.677465
$279$	0	0
$280$	0	0
$281$	−69723.6	−0.0526761	−0.0263381	−	0.999653i	$-0.508385\pi$
$281$	−69723.6	−0.0526761	−0.0263381	+	0.999653i	$0.508385\pi$
$282$	0	0
$283$	476452.	0.353633	0.176817	−	0.984244i	$-0.443420\pi$
$283$	476452.	0.353633	0.176817	+	0.984244i	$0.443420\pi$
$284$	−935684.	−0.688388
$285$	0	0
$286$	8844.46	0.00639376
$287$	0	0
$288$	0	0
$289$	−1.41735e6	−0.998233
$290$	63412.6	0.0442773
$291$	0	0
$292$	660938.	0.453632
$293$	2.27589e6	1.54875	0.774377	−	0.632724i	$-0.218063\pi$
$293$	2.27589e6	1.54875	0.774377	+	0.632724i	$0.218063\pi$
$294$	0	0
$295$	91295.8	0.0610796
$296$	−1.03400e6	−0.685946
$297$	0	0
$298$	988297.	0.644684
$299$	−3.20085e6	−2.07056
$300$	0	0
$301$	0	0
$302$	−259840.	−0.163941
$303$	0	0
$304$	−210312.	−0.130521
$305$	−506882.	−0.312002
$306$	0	0
$307$	1.61790e6	0.979731	0.489866	−	0.871798i	$-0.337046\pi$
$307$	1.61790e6	0.979731	0.489866	+	0.871798i	$0.337046\pi$
$308$	0	0
$309$	0	0
$310$	−235380.	−0.139112
$311$	−1.02667e6	−0.601910	−0.300955	−	0.953638i	$-0.597305\pi$
$311$	−1.02667e6	−0.601910	−0.300955	+	0.953638i	$0.597305\pi$
$312$	0	0
$313$	−317214.	−0.183017	−0.0915085	−	0.995804i	$-0.529169\pi$
$313$	−317214.	−0.183017	−0.0915085	+	0.995804i	$0.529169\pi$
$314$	−390149.	−0.223309
$315$	0	0
$316$	−496512.	−0.279712
$317$	−1.57126e6	−0.878212	−0.439106	−	0.898435i	$-0.644705\pi$
$317$	−1.57126e6	−0.878212	−0.439106	+	0.898435i	$0.644705\pi$
$318$	0	0
$319$	5452.61	0.00300005
$320$	168452.	0.0919602
$321$	0	0
$322$	0	0
$323$	−53257.7	−0.0284038
$324$	0	0
$325$	−2.28979e6	−1.20250
$326$	−728974.	−0.379899
$327$	0	0
$328$	−1.81071e6	−0.929317
$329$	0	0
$330$	0	0
$331$	1.53832e6	0.771751	0.385875	−	0.922551i	$-0.373899\pi$
$331$	1.53832e6	0.771751	0.385875	+	0.922551i	$0.373899\pi$
$332$	87441.4	0.0435383
$333$	0	0
$334$	463257.	0.227225
$335$	647000.	0.314987
$336$	0	0
$337$	−2.86811e6	−1.37569	−0.687846	−	0.725857i	$-0.741444\pi$
$337$	−2.86811e6	−1.37569	−0.687846	+	0.725857i	$0.741444\pi$
$338$	734149.	0.349537
$339$	0	0
$340$	−15502.3	−0.00727277
$341$	−20239.5	−0.00942569
$342$	0	0
$343$	0	0
$344$	−2.97860e6	−1.35711
$345$	0	0
$346$	2.17793e6	0.978031
$347$	111966.	0.0499185	0.0249593	−	0.999688i	$-0.492054\pi$
$347$	111966.	0.0499185	0.0249593	+	0.999688i	$0.492054\pi$
$348$	0	0
$349$	3.75314e6	1.64942	0.824711	−	0.565555i	$-0.191338\pi$
$349$	3.75314e6	1.64942	0.824711	+	0.565555i	$0.191338\pi$
$350$	0	0
$351$	0	0
$352$	21990.3	0.00945965
$353$	3.22923e6	1.37931	0.689655	−	0.724138i	$-0.257762\pi$
$353$	3.22923e6	1.37931	0.689655	+	0.724138i	$0.257762\pi$
$354$	0	0
$355$	575065.	0.242184
$356$	−460749.	−0.192681
$357$	0	0
$358$	1.72381e6	0.710855
$359$	−1.83659e6	−0.752100	−0.376050	−	0.926599i	$-0.622718\pi$
$359$	−1.83659e6	−0.752100	−0.376050	+	0.926599i	$0.622718\pi$
$360$	0	0
$361$	−1.34531e6	−0.543319
$362$	−1.36988e6	−0.549427
$363$	0	0
$364$	0	0
$365$	−406208.	−0.159594
$366$	0	0
$367$	339025.	0.131391	0.0656957	−	0.997840i	$-0.479073\pi$
$367$	339025.	0.131391	0.0656957	+	0.997840i	$0.479073\pi$
$368$	−811366.	−0.312318
$369$	0	0
$370$	261959.	0.0994784
$371$	0	0
$372$	0	0
$373$	−706622.	−0.262975	−0.131488	−	0.991318i	$-0.541975\pi$
$373$	−706622.	−0.262975	−0.131488	+	0.991318i	$0.541975\pi$
$374$	567.729	0.000209876 0
$375$	0	0
$376$	−4.95995e6	−1.80929
$377$	1.16028e6	0.420447
$378$	0	0
$379$	648296.	0.231833	0.115917	−	0.993259i	$-0.463019\pi$
$379$	648296.	0.231833	0.115917	+	0.993259i	$0.463019\pi$
$380$	329151.	0.116933
$381$	0	0
$382$	−486873.	−0.170709
$383$	−3.07022e6	−1.06948	−0.534741	−	0.845016i	$-0.679591\pi$
$383$	−3.07022e6	−1.06948	−0.534741	+	0.845016i	$0.679591\pi$
$384$	0	0
$385$	0	0
$386$	−2.40541e6	−0.821714
$387$	0	0
$388$	398183.	0.134278
$389$	4.59998e6	1.54128	0.770641	−	0.637270i	$-0.219936\pi$
$389$	4.59998e6	1.54128	0.770641	+	0.637270i	$0.219936\pi$
$390$	0	0
$391$	−205464.	−0.0679663
$392$	0	0
$393$	0	0
$394$	1.05203e6	0.341418
$395$	305153.	0.0984066
$396$	0	0
$397$	−2.64159e6	−0.841181	−0.420590	−	0.907251i	$-0.638177\pi$
$397$	−2.64159e6	−0.841181	−0.420590	+	0.907251i	$0.638177\pi$
$398$	−916694.	−0.290079
$399$	0	0
$400$	−580425.	−0.181383
$401$	−4.89677e6	−1.52072	−0.760359	−	0.649503i	$-0.774977\pi$
$401$	−4.89677e6	−1.52072	−0.760359	+	0.649503i	$0.774977\pi$
$402$	0	0
$403$	−4.30684e6	−1.32098
$404$	3.18865e6	0.971972
$405$	0	0
$406$	0	0
$407$	22524.8	0.00674025
$408$	0	0
$409$	−96336.3	−0.0284762	−0.0142381	−	0.999899i	$-0.504532\pi$
$409$	−96336.3	−0.0284762	−0.0142381	+	0.999899i	$0.504532\pi$
$410$	458735.	0.134773
$411$	0	0
$412$	−4.72485e6	−1.37134
$413$	0	0
$414$	0	0
$415$	−53740.9	−0.0153174
$416$	4.67941e6	1.32574
$417$	0	0
$418$	−12054.2	−0.00337442
$419$	1.09657e6	0.305142	0.152571	−	0.988292i	$-0.451245\pi$
$419$	1.09657e6	0.305142	0.152571	+	0.988292i	$0.451245\pi$
$420$	0	0
$421$	−1.93660e6	−0.532517	−0.266259	−	0.963902i	$-0.585788\pi$
$421$	−1.93660e6	−0.532517	−0.266259	+	0.963902i	$0.585788\pi$
$422$	1.56661e6	0.428234
$423$	0	0
$424$	−3.24103e6	−0.875526
$425$	−146982.	−0.0394723
$426$	0	0
$427$	0	0
$428$	−4.54830e6	−1.20016
$429$	0	0
$430$	754616.	0.196813
$431$	3.07330e6	0.796914	0.398457	−	0.917187i	$-0.369546\pi$
$431$	3.07330e6	0.796914	0.398457	+	0.917187i	$0.369546\pi$
$432$	0	0
$433$	−3.80919e6	−0.976366	−0.488183	−	0.872741i	$-0.662340\pi$
$433$	−3.80919e6	−0.976366	−0.488183	+	0.872741i	$0.662340\pi$
$434$	0	0
$435$	0	0
$436$	−3.13149e6	−0.788923
$437$	4.36248e6	1.09277
$438$	0	0
$439$	−533019.	−0.132002	−0.0660011	−	0.997820i	$-0.521024\pi$
$439$	−533019.	−0.132002	−0.0660011	+	0.997820i	$0.521024\pi$
$440$	−8511.94	−0.00209603
$441$	0	0
$442$	120809.	0.0294134
$443$	4.66745e6	1.12998	0.564990	−	0.825098i	$-0.308880\pi$
$443$	4.66745e6	1.12998	0.564990	+	0.825098i	$0.308880\pi$
$444$	0	0
$445$	283173.	0.0677879
$446$	1.42945e6	0.340276
$447$	0	0
$448$	0	0
$449$	6.10134e6	1.42827	0.714133	−	0.700010i	$-0.246821\pi$
$449$	6.10134e6	1.42827	0.714133	+	0.700010i	$0.246821\pi$
$450$	0	0
$451$	39444.9	0.00913166
$452$	−4.63012e6	−1.06597
$453$	0	0
$454$	−303680.	−0.0691475
$455$	0	0
$456$	0	0
$457$	5.94498e6	1.33156	0.665779	−	0.746149i	$-0.268099\pi$
$457$	5.94498e6	1.33156	0.665779	+	0.746149i	$0.268099\pi$
$458$	−1.55015e6	−0.345311
$459$	0	0
$460$	1.26984e6	0.279804
$461$	2.09415e6	0.458940	0.229470	−	0.973316i	$-0.426301\pi$
$461$	2.09415e6	0.458940	0.229470	+	0.973316i	$0.426301\pi$
$462$	0	0
$463$	2.41123e6	0.522741	0.261371	−	0.965239i	$-0.415826\pi$
$463$	2.41123e6	0.522741	0.261371	+	0.965239i	$0.415826\pi$
$464$	294114.	0.0634191
$465$	0	0
$466$	1.89800e6	0.404884
$467$	4.51833e6	0.958706	0.479353	−	0.877622i	$-0.340871\pi$
$467$	4.51833e6	0.958706	0.479353	+	0.877622i	$0.340871\pi$
$468$	0	0
$469$	0	0
$470$	1.25658e6	0.262389
$471$	0	0
$472$	−1.11410e6	−0.230181
$473$	64886.6	0.0133353
$474$	0	0
$475$	3.12078e6	0.634643
$476$	0	0
$477$	0	0
$478$	−3.40140e6	−0.680908
$479$	−8.35928e6	−1.66468	−0.832339	−	0.554267i	$-0.812999\pi$
$479$	−8.35928e6	−1.66468	−0.832339	+	0.554267i	$0.812999\pi$
$480$	0	0
$481$	4.79315e6	0.944624
$482$	−277886.	−0.0544815
$483$	0	0
$484$	3.61398e6	0.701249
$485$	−244721.	−0.0472407
$486$	0	0
$487$	598304.	0.114314	0.0571570	−	0.998365i	$-0.481796\pi$
$487$	598304.	0.114314	0.0571570	+	0.998365i	$0.481796\pi$
$488$	6.18558e6	1.17579
$489$	0	0
$490$	0	0
$491$	−2.76760e6	−0.518084	−0.259042	−	0.965866i	$-0.583407\pi$
$491$	−2.76760e6	−0.518084	−0.259042	+	0.965866i	$0.583407\pi$
$492$	0	0
$493$	74479.1	0.0138012
$494$	−2.56507e6	−0.472914
$495$	0	0
$496$	−1.09172e6	−0.199253
$497$	0	0
$498$	0	0
$499$	3.03921e6	0.546398	0.273199	−	0.961958i	$-0.411918\pi$
$499$	3.03921e6	0.546398	0.273199	+	0.961958i	$0.411918\pi$
$500$	1.87569e6	0.335533
$501$	0	0
$502$	3.12410e6	0.553307
$503$	−9.37896e6	−1.65286	−0.826428	−	0.563043i	$-0.809631\pi$
$503$	−9.37896e6	−1.65286	−0.826428	+	0.563043i	$0.809631\pi$
$504$	0	0
$505$	−1.95973e6	−0.341953
$506$	−46504.3	−0.00807452
$507$	0	0
$508$	−2.01694e6	−0.346764
$509$	−5.90139e6	−1.00962	−0.504812	−	0.863229i	$-0.668438\pi$
$509$	−5.90139e6	−1.00962	−0.504812	+	0.863229i	$0.668438\pi$
$510$	0	0
$511$	0	0
$512$	2.25139e6	0.379555
$513$	0	0
$514$	−3.74806e6	−0.625747
$515$	2.90386e6	0.482456
$516$	0	0
$517$	108049.	0.0177784
$518$	0	0
$519$	0	0
$520$	−1.81129e6	−0.293751
$521$	−9.37426e6	−1.51301	−0.756506	−	0.653986i	$-0.773095\pi$
$521$	−9.37426e6	−1.51301	−0.756506	+	0.653986i	$0.773095\pi$
$522$	0	0
$523$	6.46830e6	1.03404	0.517018	−	0.855975i	$-0.327042\pi$
$523$	6.46830e6	1.03404	0.517018	+	0.855975i	$0.327042\pi$
$524$	−7.18398e6	−1.14298
$525$	0	0
$526$	−1.18997e6	−0.187531
$527$	−276458.	−0.0433613
$528$	0	0
$529$	1.03938e7	1.61486
$530$	821102.	0.126972
$531$	0	0
$532$	0	0
$533$	8.39364e6	1.27977
$534$	0	0
$535$	2.79536e6	0.422234
$536$	−7.89546e6	−1.18704
$537$	0	0
$538$	−4.86792e6	−0.725083
$539$	0	0
$540$	0	0
$541$	−2.46089e6	−0.361492	−0.180746	−	0.983530i	$-0.557851\pi$
$541$	−2.46089e6	−0.361492	−0.180746	+	0.983530i	$0.557851\pi$
$542$	−1.02725e6	−0.150203
$543$	0	0
$544$	300373.	0.0435175
$545$	1.92459e6	0.277554
$546$	0	0
$547$	503726.	0.0719824	0.0359912	−	0.999352i	$-0.488541\pi$
$547$	503726.	0.0719824	0.0359912	+	0.999352i	$0.488541\pi$
$548$	5.81180e6	0.826721
$549$	0	0
$550$	−33267.7	−0.00468938
$551$	−1.58137e6	−0.221898
$552$	0	0
$553$	0	0
$554$	6.91113e6	0.956697
$555$	0	0
$556$	6.33681e6	0.869328
$557$	−632648.	−0.0864021	−0.0432011	−	0.999066i	$-0.513756\pi$
$557$	−632648.	−0.0864021	−0.0432011	+	0.999066i	$0.513756\pi$
$558$	0	0
$559$	1.38075e7	1.86890
$560$	0	0
$561$	0	0
$562$	−215559.	−0.0287889
$563$	−1.00780e7	−1.33999	−0.669995	−	0.742366i	$-0.733704\pi$
$563$	−1.00780e7	−1.33999	−0.669995	+	0.742366i	$0.733704\pi$
$564$	0	0
$565$	2.84564e6	0.375024
$566$	1.47301e6	0.193270
$567$	0	0
$568$	−7.01763e6	−0.912682
$569$	−2.01685e6	−0.261151	−0.130576	−	0.991438i	$-0.541683\pi$
$569$	−2.01685e6	−0.261151	−0.130576	+	0.991438i	$0.541683\pi$
$570$	0	0
$571$	4.09727e6	0.525902	0.262951	−	0.964809i	$-0.415304\pi$
$571$	4.09727e6	0.525902	0.262951	+	0.964809i	$0.415304\pi$
$572$	−64201.1	−0.00820451
$573$	0	0
$574$	0	0
$575$	1.20397e7	1.51861
$576$	0	0
$577$	−8.14322e6	−1.01826	−0.509128	−	0.860691i	$-0.670032\pi$
$577$	−8.14322e6	−1.01826	−0.509128	+	0.860691i	$0.670032\pi$
$578$	−4.38191e6	−0.545562
$579$	0	0
$580$	−460307.	−0.0568169
$581$	0	0
$582$	0	0
$583$	70603.4	0.00860309
$584$	4.95703e6	0.601436
$585$	0	0
$586$	7.03621e6	0.846437
$587$	8.81465e6	1.05587	0.527934	−	0.849285i	$-0.322967\pi$
$587$	8.81465e6	1.05587	0.527934	+	0.849285i	$0.322967\pi$
$588$	0	0
$589$	5.86985e6	0.697170
$590$	282253.	0.0333816
$591$	0	0
$592$	1.21499e6	0.142485
$593$	1.15739e7	1.35158	0.675792	−	0.737093i	$-0.263802\pi$
$593$	1.15739e7	1.35158	0.675792	+	0.737093i	$0.263802\pi$
$594$	0	0
$595$	0	0
$596$	−7.17396e6	−0.827263
$597$	0	0
$598$	−9.89584e6	−1.13162
$599$	1.23551e7	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$599$	1.23551e7	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$600$	0	0
$601$	−8.33752e6	−0.941566	−0.470783	−	0.882249i	$-0.656029\pi$
$601$	−8.33752e6	−0.941566	−0.470783	+	0.882249i	$0.656029\pi$
$602$	0	0
$603$	0	0
$604$	1.88616e6	0.210371
$605$	−2.22113e6	−0.246709
$606$	0	0
$607$	−8.67189e6	−0.955305	−0.477653	−	0.878549i	$-0.658512\pi$
$607$	−8.67189e6	−0.955305	−0.477653	+	0.878549i	$0.658512\pi$
$608$	−6.37764e6	−0.699682
$609$	0	0
$610$	−1.56709e6	−0.170518
$611$	2.29921e7	2.49159
$612$	0	0
$613$	5.24031e6	0.563256	0.281628	−	0.959524i	$-0.409126\pi$
$613$	5.24031e6	0.563256	0.281628	+	0.959524i	$0.409126\pi$
$614$	5.00196e6	0.535450
$615$	0	0
$616$	0	0
$617$	6.95644e6	0.735655	0.367828	−	0.929894i	$-0.380102\pi$
$617$	6.95644e6	0.735655	0.367828	+	0.929894i	$0.380102\pi$
$618$	0	0
$619$	1.82763e7	1.91717	0.958586	−	0.284802i	$-0.0919278\pi$
$619$	1.82763e7	1.91717	0.958586	+	0.284802i	$0.0919278\pi$
$620$	1.70861e6	0.178510
$621$	0	0
$622$	−3.17409e6	−0.328960
$623$	0	0
$624$	0	0
$625$	8.01835e6	0.821079
$626$	−980707.	−0.100024
$627$	0	0
$628$	2.83205e6	0.286551
$629$	307674.	0.0310074
$630$	0	0
$631$	1.90708e7	1.90676	0.953379	−	0.301777i	$-0.0975798\pi$
$631$	1.90708e7	1.90676	0.953379	+	0.301777i	$0.0975798\pi$
$632$	−3.72384e6	−0.370850
$633$	0	0
$634$	−4.85774e6	−0.479967
$635$	1.23960e6	0.121997
$636$	0	0
$637$	0	0
$638$	16857.4	0.00163961
$639$	0	0
$640$	−2.12628e6	−0.205197
$641$	−3.20624e6	−0.308213	−0.154106	−	0.988054i	$-0.549250\pi$
$641$	−3.20624e6	−0.308213	−0.154106	+	0.988054i	$0.549250\pi$
$642$	0	0
$643$	4.35153e6	0.415063	0.207532	−	0.978228i	$-0.433457\pi$
$643$	4.35153e6	0.415063	0.207532	+	0.978228i	$0.433457\pi$
$644$	0	0
$645$	0	0
$646$	−164653.	−0.0155234
$647$	5.59976e6	0.525907	0.262953	−	0.964809i	$-0.415303\pi$
$647$	5.59976e6	0.525907	0.262953	+	0.964809i	$0.415303\pi$
$648$	0	0
$649$	24269.8	0.00226180
$650$	−7.07917e6	−0.657202
$651$	0	0
$652$	5.29156e6	0.487489
$653$	6.74495e6	0.619007	0.309504	−	0.950898i	$-0.399837\pi$
$653$	6.74495e6	0.619007	0.309504	+	0.950898i	$0.399837\pi$
$654$	0	0
$655$	4.41523e6	0.402115
$656$	2.12766e6	0.193038
$657$	0	0
$658$	0	0
$659$	−2.09622e6	−0.188029	−0.0940143	−	0.995571i	$-0.529970\pi$
$659$	−2.09622e6	−0.188029	−0.0940143	+	0.995571i	$0.529970\pi$
$660$	0	0
$661$	−1.94185e7	−1.72867	−0.864335	−	0.502917i	$-0.832260\pi$
$661$	−1.94185e7	−1.72867	−0.864335	+	0.502917i	$0.832260\pi$
$662$	4.75591e6	0.421783
$663$	0	0
$664$	655810.	0.0577242
$665$	0	0
$666$	0	0
$667$	−6.10079e6	−0.530972
$668$	−3.36274e6	−0.291577
$669$	0	0
$670$	2.00028e6	0.172149
$671$	−134748.	−0.0115536
$672$	0	0
$673$	1.27627e7	1.08619	0.543094	−	0.839672i	$-0.317253\pi$
$673$	1.27627e7	1.08619	0.543094	+	0.839672i	$0.317253\pi$
$674$	−8.86713e6	−0.751853
$675$	0	0
$676$	−5.32913e6	−0.448528
$677$	3.86502e6	0.324101	0.162050	−	0.986782i	$-0.448189\pi$
$677$	3.86502e6	0.324101	0.162050	+	0.986782i	$0.448189\pi$
$678$	0	0
$679$	0	0
$680$	−116267.	−0.00964241
$681$	0	0
$682$	−62572.8	−0.00515140
$683$	2.06418e7	1.69316	0.846578	−	0.532265i	$-0.178659\pi$
$683$	2.06418e7	1.69316	0.846578	+	0.532265i	$0.178659\pi$
$684$	0	0
$685$	−3.57189e6	−0.290852
$686$	0	0
$687$	0	0
$688$	3.49998e6	0.281900
$689$	1.50240e7	1.20570
$690$	0	0
$691$	−1.62329e7	−1.29330	−0.646651	−	0.762786i	$-0.723831\pi$
$691$	−1.62329e7	−1.29330	−0.646651	+	0.762786i	$0.723831\pi$
$692$	−1.58094e7	−1.25502
$693$	0	0
$694$	346156.	0.0272818
$695$	−3.89456e6	−0.305842
$696$	0	0
$697$	538791.	0.0420086
$698$	1.16033e7	0.901454
$699$	0	0
$700$	0	0
$701$	8.27590e6	0.636092	0.318046	−	0.948075i	$-0.396973\pi$
$701$	8.27590e6	0.636092	0.318046	+	0.948075i	$0.396973\pi$
$702$	0	0
$703$	−6.53266e6	−0.498542
$704$	44780.7	0.00340533
$705$	0	0
$706$	9.98356e6	0.753830
$707$	0	0
$708$	0	0
$709$	2.38973e7	1.78539	0.892696	−	0.450660i	$-0.148811\pi$
$709$	2.38973e7	1.78539	0.892696	+	0.450660i	$0.148811\pi$
$710$	1.77789e6	0.132360
$711$	0	0
$712$	−3.45562e6	−0.255462
$713$	2.26454e7	1.66823
$714$	0	0
$715$	39457.6	0.00288646
$716$	−1.25130e7	−0.912173
$717$	0	0
$718$	−5.67804e6	−0.411043
$719$	784349.	0.0565832	0.0282916	−	0.999600i	$-0.490993\pi$
$719$	784349.	0.0565832	0.0282916	+	0.999600i	$0.490993\pi$
$720$	0	0
$721$	0	0
$722$	−4.15920e6	−0.296939
$723$	0	0
$724$	9.94381e6	0.705028
$725$	−4.36431e6	−0.308369
$726$	0	0
$727$	−1.61766e7	−1.13514	−0.567571	−	0.823324i	$-0.692117\pi$
$727$	−1.61766e7	−1.13514	−0.567571	+	0.823324i	$0.692117\pi$
$728$	0	0
$729$	0	0
$730$	−1.25584e6	−0.0872224
$731$	886307.	0.0613467
$732$	0	0
$733$	−1.36978e7	−0.941654	−0.470827	−	0.882226i	$-0.656044\pi$
$733$	−1.36978e7	−0.941654	−0.470827	+	0.882226i	$0.656044\pi$
$734$	1.04814e6	0.0718090
$735$	0	0
$736$	−2.46044e7	−1.67424
$737$	171997.	0.0116641
$738$	0	0
$739$	−2.53712e7	−1.70895	−0.854477	−	0.519490i	$-0.826122\pi$
$739$	−2.53712e7	−1.70895	−0.854477	+	0.519490i	$0.826122\pi$
$740$	−1.90154e6	−0.127651
$741$	0	0
$742$	0	0
$743$	−2.45336e7	−1.63038	−0.815192	−	0.579191i	$-0.803368\pi$
$743$	−2.45336e7	−1.63038	−0.815192	+	0.579191i	$0.803368\pi$
$744$	0	0
$745$	4.40907e6	0.291042
$746$	−2.18461e6	−0.143723
$747$	0	0
$748$	−4121.10	−0.000269314 0
$749$	0	0
$750$	0	0
$751$	−1.84907e7	−1.19634	−0.598169	−	0.801370i	$-0.704105\pi$
$751$	−1.84907e7	−1.19634	−0.598169	+	0.801370i	$0.704105\pi$
$752$	5.82815e6	0.375825
$753$	0	0
$754$	3.58716e6	0.229786
$755$	−1.15922e6	−0.0740113
$756$	0	0
$757$	−2.94413e7	−1.86732	−0.933658	−	0.358166i	$-0.883402\pi$
$757$	−2.94413e7	−1.86732	−0.933658	+	0.358166i	$0.883402\pi$
$758$	2.00429e6	0.126703
$759$	0	0
$760$	2.46863e6	0.155032
$761$	−1.86236e7	−1.16574	−0.582870	−	0.812565i	$-0.698070\pi$
$761$	−1.86236e7	−1.16574	−0.582870	+	0.812565i	$0.698070\pi$
$762$	0	0
$763$	0	0
$764$	3.53417e6	0.219056
$765$	0	0
$766$	−9.49199e6	−0.584501
$767$	5.16448e6	0.316984
$768$	0	0
$769$	1.13326e7	0.691056	0.345528	−	0.938408i	$-0.387700\pi$
$769$	1.13326e7	0.691056	0.345528	+	0.938408i	$0.387700\pi$
$770$	0	0
$771$	0	0
$772$	1.74607e7	1.05443
$773$	−8.22020e6	−0.494804	−0.247402	−	0.968913i	$-0.579577\pi$
$773$	−8.22020e6	−0.494804	−0.247402	+	0.968913i	$0.579577\pi$
$774$	0	0
$775$	1.61998e7	0.968848
$776$	2.98637e6	0.178029
$777$	0	0
$778$	1.42214e7	0.842352
$779$	−1.14398e7	−0.675422
$780$	0	0
$781$	152874.	0.00896819
$782$	−635217.	−0.0371454
$783$	0	0
$784$	0	0
$785$	−1.74056e6	−0.100813
$786$	0	0
$787$	5.43081e6	0.312556	0.156278	−	0.987713i	$-0.450050\pi$
$787$	5.43081e6	0.312556	0.156278	+	0.987713i	$0.450050\pi$
$788$	−7.63657e6	−0.438110
$789$	0	0
$790$	943418.	0.0537819
$791$	0	0
$792$	0	0
$793$	−2.86736e7	−1.61920
$794$	−8.16681e6	−0.459728
$795$	0	0
$796$	6.65421e6	0.372232
$797$	1.78388e7	0.994765	0.497382	−	0.867531i	$-0.334295\pi$
$797$	1.78388e7	0.994765	0.497382	+	0.867531i	$0.334295\pi$
$798$	0	0
$799$	1.47587e6	0.0817866
$800$	−1.76012e7	−0.972339
$801$	0	0
$802$	−1.51390e7	−0.831115
$803$	−107985.	−0.00590983
$804$	0	0
$805$	0	0
$806$	−1.33151e7	−0.721951
$807$	0	0
$808$	2.39149e7	1.28867
$809$	−1.18431e7	−0.636199	−0.318099	−	0.948057i	$-0.603045\pi$
$809$	−1.18431e7	−0.636199	−0.318099	+	0.948057i	$0.603045\pi$
$810$	0	0
$811$	−618053.	−0.0329969	−0.0164985	−	0.999864i	$-0.505252\pi$
$811$	−618053.	−0.0329969	−0.0164985	+	0.999864i	$0.505252\pi$
$812$	0	0
$813$	0	0
$814$	69638.4	0.00368373
$815$	−3.25216e6	−0.171505
$816$	0	0
$817$	−1.88184e7	−0.986342
$818$	−297836.	−0.0155630
$819$	0	0
$820$	−3.32992e6	−0.172941
$821$	−4.31256e6	−0.223294	−0.111647	−	0.993748i	$-0.535613\pi$
$821$	−4.31256e6	−0.223294	−0.111647	+	0.993748i	$0.535613\pi$
$822$	0	0
$823$	2.86620e7	1.47505	0.737524	−	0.675321i	$-0.235995\pi$
$823$	2.86620e7	1.47505	0.737524	+	0.675321i	$0.235995\pi$
$824$	−3.54364e7	−1.81816
$825$	0	0
$826$	0	0
$827$	−1.86894e7	−0.950235	−0.475117	−	0.879922i	$-0.657594\pi$
$827$	−1.86894e7	−0.950235	−0.475117	+	0.879922i	$0.657594\pi$
$828$	0	0
$829$	−1.32486e7	−0.669550	−0.334775	−	0.942298i	$-0.608660\pi$
$829$	−1.32486e7	−0.669550	−0.334775	+	0.942298i	$0.608660\pi$
$830$	−166147.	−0.00837137
$831$	0	0
$832$	9.52907e6	0.477246
$833$	0	0
$834$	0	0
$835$	2.06672e6	0.102581
$836$	87500.6	0.00433007
$837$	0	0
$838$	3.39019e6	0.166769
$839$	−2.52930e7	−1.24049	−0.620247	−	0.784406i	$-0.712968\pi$
$839$	−2.52930e7	−1.24049	−0.620247	+	0.784406i	$0.712968\pi$
$840$	0	0
$841$	−1.82997e7	−0.892181
$842$	−5.98723e6	−0.291035
$843$	0	0
$844$	−1.13719e7	−0.549512
$845$	3.27525e6	0.157798
$846$	0	0
$847$	0	0
$848$	3.80835e6	0.181864
$849$	0	0
$850$	−454414.	−0.0215727
$851$	−2.52025e7	−1.19294
$852$	0	0
$853$	−4.05806e7	−1.90962	−0.954808	−	0.297223i	$-0.903939\pi$
$853$	−4.05806e7	−1.90962	−0.954808	+	0.297223i	$0.903939\pi$
$854$	0	0
$855$	0	0
$856$	−3.41123e7	−1.59120
$857$	1.94469e7	0.904480	0.452240	−	0.891896i	$-0.350625\pi$
$857$	1.94469e7	0.904480	0.452240	+	0.891896i	$0.350625\pi$
$858$	0	0
$859$	2.03242e7	0.939788	0.469894	−	0.882723i	$-0.344292\pi$
$859$	2.03242e7	0.939788	0.469894	+	0.882723i	$0.344292\pi$
$860$	−5.47769e6	−0.252552
$861$	0	0
$862$	9.50149e6	0.435536
$863$	2.61202e7	1.19385	0.596926	−	0.802297i	$-0.296389\pi$
$863$	2.61202e7	1.19385	0.596926	+	0.802297i	$0.296389\pi$
$864$	0	0
$865$	9.71634e6	0.441532
$866$	−1.17766e7	−0.533611
$867$	0	0
$868$	0	0
$869$	81120.9	0.00364404
$870$	0	0
$871$	3.65999e7	1.63469
$872$	−2.34862e7	−1.04597
$873$	0	0
$874$	1.34872e7	0.597231
$875$	0	0
$876$	0	0
$877$	−1.62956e7	−0.715436	−0.357718	−	0.933830i	$-0.616445\pi$
$877$	−1.62956e7	−0.715436	−0.357718	+	0.933830i	$0.616445\pi$
$878$	−1.64789e6	−0.0721428
$879$	0	0
$880$	10001.9	0.000435387 0
$881$	2.93722e7	1.27496	0.637479	−	0.770467i	$-0.279977\pi$
$881$	2.93722e7	1.27496	0.637479	+	0.770467i	$0.279977\pi$
$882$	0	0
$883$	1.21821e7	0.525800	0.262900	−	0.964823i	$-0.415321\pi$
$883$	1.21821e7	0.525800	0.262900	+	0.964823i	$0.415321\pi$
$884$	−876945.	−0.0377435
$885$	0	0
$886$	1.44300e7	0.617565
$887$	−2.38021e7	−1.01579	−0.507897	−	0.861418i	$-0.669577\pi$
$887$	−2.38021e7	−1.01579	−0.507897	+	0.861418i	$0.669577\pi$
$888$	0	0
$889$	0	0
$890$	875466.	0.0370480
$891$	0	0
$892$	−1.03763e7	−0.436645
$893$	−3.13363e7	−1.31498
$894$	0	0
$895$	7.69038e6	0.320915
$896$	0	0
$897$	0	0
$898$	1.88631e7	0.780587
$899$	−8.20879e6	−0.338750
$900$	0	0
$901$	964396.	0.0395771
$902$	121949.	0.00499070
$903$	0	0
$904$	−3.47259e7	−1.41329
$905$	−6.11140e6	−0.248039
$906$	0	0
$907$	2.10870e7	0.851133	0.425566	−	0.904927i	$-0.360075\pi$
$907$	2.10870e7	0.851133	0.425566	+	0.904927i	$0.360075\pi$
$908$	2.20439e6	0.0887305
$909$	0	0
$910$	0	0
$911$	−1.71528e7	−0.684761	−0.342381	−	0.939561i	$-0.611233\pi$
$911$	−1.71528e7	−0.684761	−0.342381	+	0.939561i	$0.611233\pi$
$912$	0	0
$913$	−14286.3	−0.000567210 0
$914$	1.83797e7	0.727733
$915$	0	0
$916$	1.12524e7	0.443105
$917$	0	0
$918$	0	0
$919$	7.35541e6	0.287289	0.143644	−	0.989629i	$-0.454118\pi$
$919$	7.35541e6	0.287289	0.143644	+	0.989629i	$0.454118\pi$
$920$	9.52379e6	0.370971
$921$	0	0
$922$	6.47433e6	0.250823
$923$	3.25306e7	1.25686
$924$	0	0
$925$	−1.80290e7	−0.692817
$926$	7.45463e6	0.285692
$927$	0	0
$928$	8.91891e6	0.339971
$929$	−3.34199e7	−1.27047	−0.635236	−	0.772318i	$-0.719097\pi$
$929$	−3.34199e7	−1.27047	−0.635236	+	0.772318i	$0.719097\pi$
$930$	0	0
$931$	0	0
$932$	−1.37774e7	−0.519550
$933$	0	0
$934$	1.39690e7	0.523959
$935$	2532.80	9.47483e−5 0
$936$	0	0
$937$	2.87696e7	1.07049	0.535247	−	0.844696i	$-0.320219\pi$
$937$	2.87696e7	1.07049	0.535247	+	0.844696i	$0.320219\pi$
$938$	0	0
$939$	0	0
$940$	−9.12142e6	−0.336700
$941$	2.16379e7	0.796601	0.398300	−	0.917255i	$-0.369600\pi$
$941$	2.16379e7	0.796601	0.398300	+	0.917255i	$0.369600\pi$
$942$	0	0
$943$	−4.41339e7	−1.61619
$944$	1.30911e6	0.0478132
$945$	0	0
$946$	200605.	0.00728809
$947$	6.31254e6	0.228733	0.114367	−	0.993439i	$-0.463516\pi$
$947$	6.31254e6	0.228733	0.114367	+	0.993439i	$0.463516\pi$
$948$	0	0
$949$	−2.29786e7	−0.828244
$950$	9.64829e6	0.346850
$951$	0	0
$952$	0	0
$953$	5.59599e6	0.199593	0.0997963	−	0.995008i	$-0.468181\pi$
$953$	5.59599e6	0.199593	0.0997963	+	0.995008i	$0.468181\pi$
$954$	0	0
$955$	−2.17208e6	−0.0770668
$956$	2.46905e7	0.873746
$957$	0	0
$958$	−2.58438e7	−0.909792
$959$	0	0
$960$	0	0
$961$	1.84091e6	0.0643021
$962$	1.48186e7	0.516263
$963$	0	0
$964$	2.01715e6	0.0699111
$965$	−1.07312e7	−0.370963
$966$	0	0
$967$	1.33277e7	0.458340	0.229170	−	0.973386i	$-0.426399\pi$
$967$	1.33277e7	0.458340	0.229170	+	0.973386i	$0.426399\pi$
$968$	2.71048e7	0.929734
$969$	0	0
$970$	−756585.	−0.0258183
$971$	−3.63257e7	−1.23642	−0.618210	−	0.786013i	$-0.712142\pi$
$971$	−3.63257e7	−1.23642	−0.618210	+	0.786013i	$0.712142\pi$
$972$	0	0
$973$	0	0
$974$	1.84973e6	0.0624758
$975$	0	0
$976$	−7.26831e6	−0.244236
$977$	9.03739e6	0.302905	0.151453	−	0.988465i	$-0.451605\pi$
$977$	9.03739e6	0.302905	0.151453	+	0.988465i	$0.451605\pi$
$978$	0	0
$979$	75278.0	0.00251022
$980$	0	0
$981$	0	0
$982$	−8.55639e6	−0.283147
$983$	1.98539e7	0.655332	0.327666	−	0.944794i	$-0.393738\pi$
$983$	1.98539e7	0.655332	0.327666	+	0.944794i	$0.393738\pi$
$984$	0	0
$985$	4.69339e6	0.154133
$986$	230261.	0.00754273
$987$	0	0
$988$	1.86196e7	0.606846
$989$	−7.25999e7	−2.36018
$990$	0	0
$991$	−4.55557e7	−1.47353	−0.736764	−	0.676150i	$-0.763647\pi$
$991$	−4.55557e7	−1.47353	−0.736764	+	0.676150i	$0.763647\pi$
$992$	−3.31060e7	−1.06814
$993$	0	0
$994$	0	0
$995$	−4.08963e6	−0.130956
$996$	0	0
$997$	4.51665e7	1.43906	0.719529	−	0.694462i	$-0.244358\pi$
$997$	4.51665e7	1.43906	0.719529	+	0.694462i	$0.244358\pi$
$998$	9.39610e6	0.298622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bb.1.4		6
3.2	odd	2		147.6.a.n.1.3	✓	6
7.6	odd	2		441.6.a.ba.1.4		6
21.2	odd	6		147.6.e.q.67.4		12
21.5	even	6		147.6.e.p.67.4		12
21.11	odd	6		147.6.e.q.79.4		12
21.17	even	6		147.6.e.p.79.4		12
21.20	even	2		147.6.a.o.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.3	✓	6	3.2	odd	2
147.6.a.o.1.3	yes	6	21.20	even	2
147.6.e.p.67.4		12	21.5	even	6
147.6.e.p.79.4		12	21.17	even	6
147.6.e.q.67.4		12	21.2	odd	6
147.6.e.q.79.4		12	21.11	odd	6
441.6.a.ba.1.4		6	7.6	odd	2
441.6.a.bb.1.4		6	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +O(q^{10})\)	\(q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +42.6416 q^{10} +3.66659 q^{11} +780.229 q^{13} +197.776 q^{16} +50.0832 q^{17} -1063.38 q^{19} -309.532 q^{20} +11.3357 q^{22} -4102.45 q^{23} -2934.76 q^{25} +2412.18 q^{26} +1487.11 q^{29} -5519.97 q^{31} +5997.49 q^{32} +154.838 q^{34} +6143.27 q^{37} -3287.59 q^{38} -2321.49 q^{40} +10757.9 q^{41} +17696.7 q^{43} -82.2850 q^{44} -12683.3 q^{46} +29468.5 q^{47} -9073.19 q^{50} -17509.8 q^{52} +19255.9 q^{53} +50.5718 q^{55} +4597.58 q^{58} +6619.18 q^{59} -36750.3 q^{61} -17065.7 q^{62} +12213.2 q^{64} +10761.4 q^{65} +46909.2 q^{67} -1123.96 q^{68} +41693.7 q^{71} -29451.1 q^{73} +18992.7 q^{74} +23864.3 q^{76} +22124.4 q^{79} +2727.84 q^{80} +33259.5 q^{82} -3896.35 q^{83} +690.778 q^{85} +54711.6 q^{86} -617.138 q^{88} +20530.8 q^{89} +92066.6 q^{92} +91105.4 q^{94} -14666.8 q^{95} -17742.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 - 864 * q^10 - 604 * q^11 - 1352 * q^13 + 4578 * q^16 + 3028 * q^17 - 1728 * q^19 + 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 + 14172 * q^26 + 5320 * q^29 - 3976 * q^31 + 37326 * q^32 + 16336 * q^34 + 22680 * q^37 + 52744 * q^38 - 100600 * q^40 + 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 + 51552 * q^47 + 40622 * q^50 - 119296 * q^52 - 80884 * q^53 - 11656 * q^55 - 70464 * q^58 + 8872 * q^59 - 50896 * q^61 + 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 + 37348 * q^68 + 110852 * q^71 - 64232 * q^73 + 27464 * q^74 + 194864 * q^76 + 111696 * q^79 - 308940 * q^80 + 189640 * q^82 + 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 - 35012 * q^89 + 449260 * q^92 + 121016 * q^94 + 119080 * q^95 - 70952 * q^97

Introduction

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