LMFDB - Embedded newform 441.6.a.bb.1.3

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			1.3
Root			$-5.10089$ 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)

$f(q)$	$=$	$q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +O(q^{10})$	$q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +184.313 q^{10} -481.911 q^{11} -512.622 q^{13} +57.6111 q^{16} +590.744 q^{17} -2451.02 q^{19} +1121.77 q^{20} +1629.02 q^{22} -1774.90 q^{23} -151.993 q^{25} +1732.83 q^{26} +4246.44 q^{29} -9767.37 q^{31} -5881.64 q^{32} -1996.91 q^{34} -9969.65 q^{37} +8285.24 q^{38} -9689.98 q^{40} -3377.54 q^{41} -18223.4 q^{43} +9914.53 q^{44} +5999.76 q^{46} +1320.64 q^{47} +513.788 q^{50} +10546.4 q^{52} -34837.0 q^{53} +26276.3 q^{55} -14354.4 q^{58} +11592.5 q^{59} -31406.1 q^{61} +33016.9 q^{62} +18038.3 q^{64} +27950.9 q^{65} +27555.9 q^{67} -12153.6 q^{68} +22868.5 q^{71} -15936.4 q^{73} +33700.7 q^{74} +50425.6 q^{76} +87165.3 q^{79} -3141.26 q^{80} +11417.2 q^{82} +90307.5 q^{83} -32210.5 q^{85} +61601.2 q^{86} -85643.0 q^{88} -126570. q^{89} +36515.7 q^{92} -4464.20 q^{94} +133642. q^{95} +16573.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.38033	−0.597564	−0.298782	−	0.954321i	$-0.596580\pi$
$2$	−3.38033	−0.597564	−0.298782	+	0.954321i	$0.596580\pi$
$3$	0	0
$3$	0	0
$4$	−20.5734	−0.642918
$5$	−54.5253	−0.975378	−0.487689	−	0.873017i	$-0.662160\pi$
$5$	−54.5253	−0.975378	−0.487689	+	0.873017i	$0.662160\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	177.715	0.981748
$9$	0	0
$10$	184.313	0.582850
$11$	−481.911	−1.20084	−0.600420	−	0.799685i	$-0.705000\pi$
$11$	−481.911	−1.20084	−0.600420	+	0.799685i	$0.705000\pi$
$12$	0	0
$13$	−512.622	−0.841277	−0.420638	−	0.907228i	$-0.638194\pi$
$13$	−512.622	−0.841277	−0.420638	+	0.907228i	$0.638194\pi$
$14$	0	0
$15$	0	0
$16$	57.6111	0.0562608
$17$	590.744	0.495766	0.247883	−	0.968790i	$-0.420265\pi$
$17$	590.744	0.495766	0.247883	+	0.968790i	$0.420265\pi$
$18$	0	0
$19$	−2451.02	−1.55762	−0.778811	−	0.627259i	$-0.784177\pi$
$19$	−2451.02	−1.55762	−0.778811	+	0.627259i	$0.784177\pi$
$20$	1121.77	0.627088
$21$	0	0
$22$	1629.02	0.717579
$23$	−1774.90	−0.699609	−0.349804	−	0.936823i	$-0.613752\pi$
$23$	−1774.90	−0.699609	−0.349804	+	0.936823i	$0.613752\pi$
$24$	0	0
$25$	−151.993	−0.0486379
$26$	1732.83	0.502716
$27$	0	0
$28$	0	0
$29$	4246.44	0.937627	0.468814	−	0.883297i	$-0.344682\pi$
$29$	4246.44	0.937627	0.468814	+	0.883297i	$0.344682\pi$
$30$	0	0
$31$	−9767.37	−1.82547	−0.912733	−	0.408558i	$-0.866032\pi$
$31$	−9767.37	−1.82547	−0.912733	+	0.408558i	$0.866032\pi$
$32$	−5881.64	−1.01537
$33$	0	0
$34$	−1996.91	−0.296252
$35$	0	0
$36$	0	0
$37$	−9969.65	−1.19722	−0.598612	−	0.801039i	$-0.704281\pi$
$37$	−9969.65	−1.19722	−0.598612	+	0.801039i	$0.704281\pi$
$38$	8285.24	0.930778
$39$	0	0
$40$	−9689.98	−0.957575
$41$	−3377.54	−0.313791	−0.156896	−	0.987615i	$-0.550149\pi$
$41$	−3377.54	−0.313791	−0.156896	+	0.987615i	$0.550149\pi$
$42$	0	0
$43$	−18223.4	−1.50300	−0.751500	−	0.659733i	$-0.770669\pi$
$43$	−18223.4	−1.50300	−0.751500	+	0.659733i	$0.770669\pi$
$44$	9914.53	0.772042
$45$	0	0
$46$	5999.76	0.418061
$47$	1320.64	0.0872046	0.0436023	−	0.999049i	$-0.486117\pi$
$47$	1320.64	0.0872046	0.0436023	+	0.999049i	$0.486117\pi$
$48$	0	0
$49$	0	0
$50$	513.788	0.0290642
$51$	0	0
$52$	10546.4	0.540872
$53$	−34837.0	−1.70353	−0.851767	−	0.523921i	$-0.824469\pi$
$53$	−34837.0	−1.70353	−0.851767	+	0.523921i	$0.824469\pi$
$54$	0	0
$55$	26276.3	1.17127
$56$	0	0
$57$	0	0
$58$	−14354.4	−0.560292
$59$	11592.5	0.433558	0.216779	−	0.976221i	$-0.430445\pi$
$59$	11592.5	0.433558	0.216779	+	0.976221i	$0.430445\pi$
$60$	0	0
$61$	−31406.1	−1.08066	−0.540331	−	0.841452i	$-0.681701\pi$
$61$	−31406.1	−1.08066	−0.540331	+	0.841452i	$0.681701\pi$
$62$	33016.9	1.09083
$63$	0	0
$64$	18038.3	0.550486
$65$	27950.9	0.820563
$66$	0	0
$67$	27555.9	0.749942	0.374971	−	0.927036i	$-0.377653\pi$
$67$	27555.9	0.749942	0.374971	+	0.927036i	$0.377653\pi$
$68$	−12153.6	−0.318737
$69$	0	0
$70$	0	0
$71$	22868.5	0.538384	0.269192	−	0.963086i	$-0.413243\pi$
$71$	22868.5	0.538384	0.269192	+	0.963086i	$0.413243\pi$
$72$	0	0
$73$	−15936.4	−0.350012	−0.175006	−	0.984567i	$-0.555995\pi$
$73$	−15936.4	−0.350012	−0.175006	+	0.984567i	$0.555995\pi$
$74$	33700.7	0.715418
$75$	0	0
$76$	50425.6	1.00142
$77$	0	0
$78$	0	0
$79$	87165.3	1.57136	0.785680	−	0.618633i	$-0.212313\pi$
$79$	87165.3	1.57136	0.785680	+	0.618633i	$0.212313\pi$
$80$	−3141.26	−0.0548755
$81$	0	0
$82$	11417.2	0.187510
$83$	90307.5	1.43889	0.719447	−	0.694548i	$-0.244395\pi$
$83$	90307.5	1.43889	0.719447	+	0.694548i	$0.244395\pi$
$84$	0	0
$85$	−32210.5	−0.483559
$86$	61601.2	0.898138
$87$	0	0
$88$	−85643.0	−1.17892
$89$	−126570.	−1.69377	−0.846884	−	0.531777i	$-0.821524\pi$
$89$	−126570.	−1.69377	−0.846884	+	0.531777i	$0.821524\pi$
$90$	0	0
$91$	0	0
$92$	36515.7	0.449791
$93$	0	0
$94$	−4464.20	−0.0521103
$95$	133642.	1.51927
$96$	0	0
$97$	16573.1	0.178844	0.0894219	−	0.995994i	$-0.471498\pi$
$97$	16573.1	0.178844	0.0894219	+	0.995994i	$0.471498\pi$
$98$	0	0
$99$	0	0
$100$	3127.02	0.0312702
$101$	−76610.1	−0.747279	−0.373639	−	0.927574i	$-0.621890\pi$
$101$	−76610.1	−0.747279	−0.373639	+	0.927574i	$0.621890\pi$
$102$	0	0
$103$	−80435.8	−0.747062	−0.373531	−	0.927618i	$-0.621853\pi$
$103$	−80435.8	−0.747062	−0.373531	+	0.927618i	$0.621853\pi$
$104$	−91100.8	−0.825922
$105$	0	0
$106$	117760.	1.01797
$107$	−148042.	−1.25005	−0.625023	−	0.780606i	$-0.714910\pi$
$107$	−148042.	−1.25005	−0.625023	+	0.780606i	$0.714910\pi$
$108$	0	0
$109$	−124982.	−1.00759	−0.503793	−	0.863825i	$-0.668062\pi$
$109$	−124982.	−1.00759	−0.503793	+	0.863825i	$0.668062\pi$
$110$	−88822.7	−0.699910
$111$	0	0
$112$	0	0
$113$	194474.	1.43273	0.716366	−	0.697724i	$-0.245804\pi$
$113$	194474.	1.43273	0.716366	+	0.697724i	$0.245804\pi$
$114$	0	0
$115$	96777.1	0.682383
$116$	−87363.6	−0.602817
$117$	0	0
$118$	−39186.5	−0.259078
$119$	0	0
$120$	0	0
$121$	71187.4	0.442018
$122$	106163.	0.645765
$123$	0	0
$124$	200948.	1.17362
$125$	178679.	1.02282
$126$	0	0
$127$	239766.	1.31910	0.659551	−	0.751660i	$-0.270746\pi$
$127$	239766.	1.31910	0.659551	+	0.751660i	$0.270746\pi$
$128$	127237.	0.686417
$129$	0	0
$130$	−94483.2	−0.490339
$131$	−349576.	−1.77977	−0.889883	−	0.456188i	$-0.849214\pi$
$131$	−349576.	−1.77977	−0.889883	+	0.456188i	$0.849214\pi$
$132$	0	0
$133$	0	0
$134$	−93148.1	−0.448138
$135$	0	0
$136$	104984.	0.486717
$137$	280966.	1.27894	0.639472	−	0.768814i	$-0.279153\pi$
$137$	280966.	1.27894	0.639472	+	0.768814i	$0.279153\pi$
$138$	0	0
$139$	−50042.9	−0.219688	−0.109844	−	0.993949i	$-0.535035\pi$
$139$	−50042.9	−0.219688	−0.109844	+	0.993949i	$0.535035\pi$
$140$	0	0
$141$	0	0
$142$	−77303.2	−0.321719
$143$	247038.	1.01024
$144$	0	0
$145$	−231538.	−0.914541
$146$	53870.3	0.209155
$147$	0	0
$148$	205109.	0.769717
$149$	328813.	1.21334	0.606671	−	0.794953i	$-0.292505\pi$
$149$	328813.	1.21334	0.606671	+	0.794953i	$0.292505\pi$
$150$	0	0
$151$	511304.	1.82489	0.912445	−	0.409200i	$-0.134192\pi$
$151$	511304.	1.82489	0.912445	+	0.409200i	$0.134192\pi$
$152$	−435583.	−1.52919
$153$	0	0
$154$	0	0
$155$	532569.	1.78052
$156$	0	0
$157$	75442.1	0.244267	0.122134	−	0.992514i	$-0.461026\pi$
$157$	75442.1	0.244267	0.122134	+	0.992514i	$0.461026\pi$
$158$	−294647.	−0.938988
$159$	0	0
$160$	320698.	0.990367
$161$	0	0
$162$	0	0
$163$	340753.	1.00455	0.502274	−	0.864709i	$-0.332497\pi$
$163$	340753.	1.00455	0.502274	+	0.864709i	$0.332497\pi$
$164$	69487.4	0.201742
$165$	0	0
$166$	−305269.	−0.859831
$167$	−132392.	−0.367342	−0.183671	−	0.982988i	$-0.558798\pi$
$167$	−132392.	−0.367342	−0.183671	+	0.982988i	$0.558798\pi$
$168$	0	0
$169$	−108512.	−0.292253
$170$	108882.	0.288957
$171$	0	0
$172$	374917.	0.966306
$173$	449322.	1.14141	0.570706	−	0.821154i	$-0.306670\pi$
$173$	449322.	1.14141	0.570706	+	0.821154i	$0.306670\pi$
$174$	0	0
$175$	0	0
$176$	−27763.4	−0.0675602
$177$	0	0
$178$	427847.	1.01213
$179$	93666.8	0.218501	0.109250	−	0.994014i	$-0.465155\pi$
$179$	93666.8	0.218501	0.109250	+	0.994014i	$0.465155\pi$
$180$	0	0
$181$	−399795.	−0.907071	−0.453535	−	0.891238i	$-0.649837\pi$
$181$	−399795.	−0.907071	−0.453535	+	0.891238i	$0.649837\pi$
$182$	0	0
$183$	0	0
$184$	−315428.	−0.686839
$185$	543598.	1.16775
$186$	0	0
$187$	−284686.	−0.595336
$188$	−27170.0	−0.0560654
$189$	0	0
$190$	−451755.	−0.907861
$191$	−27822.7	−0.0551844	−0.0275922	−	0.999619i	$-0.508784\pi$
$191$	−27822.7	−0.0551844	−0.0275922	+	0.999619i	$0.508784\pi$
$192$	0	0
$193$	−66464.3	−0.128438	−0.0642192	−	0.997936i	$-0.520456\pi$
$193$	−66464.3	−0.128438	−0.0642192	+	0.997936i	$0.520456\pi$
$194$	−56022.5	−0.106871
$195$	0	0
$196$	0	0
$197$	492506.	0.904162	0.452081	−	0.891977i	$-0.350682\pi$
$197$	492506.	0.904162	0.452081	+	0.891977i	$0.350682\pi$
$198$	0	0
$199$	−349206.	−0.625099	−0.312549	−	0.949902i	$-0.601183\pi$
$199$	−349206.	−0.625099	−0.312549	+	0.949902i	$0.601183\pi$
$200$	−27011.6	−0.0477501
$201$	0	0
$202$	258967.	0.446547
$203$	0	0
$204$	0	0
$205$	184161.	0.306065
$206$	271900.	0.446417
$207$	0	0
$208$	−29532.7	−0.0473309
$209$	1.18117e6	1.87046
$210$	0	0
$211$	−218250.	−0.337480	−0.168740	−	0.985661i	$-0.553970\pi$
$211$	−218250.	−0.337480	−0.168740	+	0.985661i	$0.553970\pi$
$212$	716714.	1.09523
$213$	0	0
$214$	500431.	0.746982
$215$	993638.	1.46599
$216$	0	0
$217$	0	0
$218$	422481.	0.602096
$219$	0	0
$220$	−540593.	−0.753032
$221$	−302828.	−0.417076
$222$	0	0
$223$	−1.24807e6	−1.68065	−0.840324	−	0.542084i	$-0.817635\pi$
$223$	−1.24807e6	−1.68065	−0.840324	+	0.542084i	$0.817635\pi$
$224$	0	0
$225$	0	0
$226$	−657386.	−0.856149
$227$	−520973.	−0.671044	−0.335522	−	0.942032i	$-0.608913\pi$
$227$	−520973.	−0.671044	−0.335522	+	0.942032i	$0.608913\pi$
$228$	0	0
$229$	525651.	0.662383	0.331191	−	0.943564i	$-0.392549\pi$
$229$	525651.	0.662383	0.331191	+	0.943564i	$0.392549\pi$
$230$	−327139.	−0.407767
$231$	0	0
$232$	754658.	0.920514
$233$	−210205.	−0.253661	−0.126831	−	0.991924i	$-0.540480\pi$
$233$	−210205.	−0.253661	−0.126831	+	0.991924i	$0.540480\pi$
$234$	0	0
$235$	−72008.2	−0.0850575
$236$	−238497.	−0.278742
$237$	0	0
$238$	0	0
$239$	−805791.	−0.912489	−0.456244	−	0.889855i	$-0.650806\pi$
$239$	−805791.	−0.912489	−0.456244	+	0.889855i	$0.650806\pi$
$240$	0	0
$241$	−1.13795e6	−1.26206	−0.631030	−	0.775758i	$-0.717368\pi$
$241$	−1.13795e6	−1.26206	−0.631030	+	0.775758i	$0.717368\pi$
$242$	−240637.	−0.264134
$243$	0	0
$244$	646130.	0.694777
$245$	0	0
$246$	0	0
$247$	1.25644e6	1.31039
$248$	−1.73581e6	−1.79215
$249$	0	0
$250$	−603994.	−0.611199
$251$	−1.16566e6	−1.16785	−0.583927	−	0.811806i	$-0.698484\pi$
$251$	−1.16566e6	−1.16785	−0.583927	+	0.811806i	$0.698484\pi$
$252$	0	0
$253$	855346.	0.840118
$254$	−810489.	−0.788248
$255$	0	0
$256$	−1.00733e6	−0.960664
$257$	−1.02881e6	−0.971635	−0.485817	−	0.874060i	$-0.661478\pi$
$257$	−1.02881e6	−0.971635	−0.485817	+	0.874060i	$0.661478\pi$
$258$	0	0
$259$	0	0
$260$	−575043.	−0.527554
$261$	0	0
$262$	1.18168e6	1.06352
$263$	657345.	0.586009	0.293004	−	0.956111i	$-0.405345\pi$
$263$	657345.	0.586009	0.293004	+	0.956111i	$0.405345\pi$
$264$	0	0
$265$	1.89950e6	1.66159
$266$	0	0
$267$	0	0
$268$	−566918.	−0.482151
$269$	5475.86	0.00461393	0.00230697	−	0.999997i	$-0.499266\pi$
$269$	5475.86	0.00461393	0.00230697	+	0.999997i	$0.499266\pi$
$270$	0	0
$271$	769147.	0.636189	0.318094	−	0.948059i	$-0.396957\pi$
$271$	769147.	0.636189	0.318094	+	0.948059i	$0.396957\pi$
$272$	34033.4	0.0278922
$273$	0	0
$274$	−949757.	−0.764251
$275$	73247.3	0.0584063
$276$	0	0
$277$	−1.17384e6	−0.919196	−0.459598	−	0.888127i	$-0.652007\pi$
$277$	−1.17384e6	−0.919196	−0.459598	+	0.888127i	$0.652007\pi$
$278$	169162.	0.131277
$279$	0	0
$280$	0	0
$281$	−837649.	−0.632843	−0.316422	−	0.948619i	$-0.602481\pi$
$281$	−837649.	−0.632843	−0.316422	+	0.948619i	$0.602481\pi$
$282$	0	0
$283$	597221.	0.443271	0.221635	−	0.975130i	$-0.428861\pi$
$283$	597221.	0.443271	0.221635	+	0.975130i	$0.428861\pi$
$284$	−470483.	−0.346137
$285$	0	0
$286$	−835071.	−0.603682
$287$	0	0
$288$	0	0
$289$	−1.07088e6	−0.754216
$290$	782677.	0.546496
$291$	0	0
$292$	327866.	0.225029
$293$	−851321.	−0.579328	−0.289664	−	0.957128i	$-0.593543\pi$
$293$	−851321.	−0.579328	−0.289664	+	0.957128i	$0.593543\pi$
$294$	0	0
$295$	−632084.	−0.422883
$296$	−1.77176e6	−1.17537
$297$	0	0
$298$	−1.11150e6	−0.725049
$299$	909855.	0.588565
$300$	0	0
$301$	0	0
$302$	−1.72838e6	−1.09049
$303$	0	0
$304$	−141206.	−0.0876331
$305$	1.71243e6	1.05405
$306$	0	0
$307$	600906.	0.363882	0.181941	−	0.983309i	$-0.441762\pi$
$307$	600906.	0.363882	0.181941	+	0.983309i	$0.441762\pi$
$308$	0	0
$309$	0	0
$310$	−1.80026e6	−1.06397
$311$	−465866.	−0.273124	−0.136562	−	0.990632i	$-0.543605\pi$
$311$	−465866.	−0.273124	−0.136562	+	0.990632i	$0.543605\pi$
$312$	0	0
$313$	2.97366e6	1.71565	0.857827	−	0.513938i	$-0.171814\pi$
$313$	2.97366e6	1.71565	0.857827	+	0.513938i	$0.171814\pi$
$314$	−255019.	−0.145965
$315$	0	0
$316$	−1.79328e6	−1.01026
$317$	−2.55521e6	−1.42816	−0.714082	−	0.700062i	$-0.753156\pi$
$317$	−2.55521e6	−1.42816	−0.714082	+	0.700062i	$0.753156\pi$
$318$	0	0
$319$	−2.04641e6	−1.12594
$320$	−983544.	−0.536932
$321$	0	0
$322$	0	0
$323$	−1.44792e6	−0.772216
$324$	0	0
$325$	77915.2	0.0409179
$326$	−1.15186e6	−0.600281
$327$	0	0
$328$	−600241.	−0.308064
$329$	0	0
$330$	0	0
$331$	2.56138e6	1.28500	0.642502	−	0.766284i	$-0.277896\pi$
$331$	2.56138e6	1.28500	0.642502	+	0.766284i	$0.277896\pi$
$332$	−1.85793e6	−0.925090
$333$	0	0
$334$	447528.	0.219510
$335$	−1.50249e6	−0.731477
$336$	0	0
$337$	−1.21525e6	−0.582894	−0.291447	−	0.956587i	$-0.594137\pi$
$337$	−1.21525e6	−0.582894	−0.291447	+	0.956587i	$0.594137\pi$
$338$	366805.	0.174640
$339$	0	0
$340$	662678.	0.310889
$341$	4.70701e6	2.19209
$342$	0	0
$343$	0	0
$344$	−3.23858e6	−1.47557
$345$	0	0
$346$	−1.51886e6	−0.682066
$347$	−1.25318e6	−0.558714	−0.279357	−	0.960187i	$-0.590121\pi$
$347$	−1.25318e6	−0.558714	−0.279357	+	0.960187i	$0.590121\pi$
$348$	0	0
$349$	−450403.	−0.197942	−0.0989709	−	0.995090i	$-0.531555\pi$
$349$	−450403.	−0.197942	−0.0989709	+	0.995090i	$0.531555\pi$
$350$	0	0
$351$	0	0
$352$	2.83443e6	1.21929
$353$	545281.	0.232907	0.116454	−	0.993196i	$-0.462847\pi$
$353$	545281.	0.232907	0.116454	+	0.993196i	$0.462847\pi$
$354$	0	0
$355$	−1.24691e6	−0.525128
$356$	2.60396e6	1.08895
$357$	0	0
$358$	−316625.	−0.130568
$359$	1.01159e6	0.414254	0.207127	−	0.978314i	$-0.433589\pi$
$359$	1.01159e6	0.414254	0.207127	+	0.978314i	$0.433589\pi$
$360$	0	0
$361$	3.53138e6	1.42619
$362$	1.35144e6	0.542033
$363$	0	0
$364$	0	0
$365$	868937.	0.341394
$366$	0	0
$367$	1.09038e6	0.422584	0.211292	−	0.977423i	$-0.432233\pi$
$367$	1.09038e6	0.422584	0.211292	+	0.977423i	$0.432233\pi$
$368$	−102254.	−0.0393605
$369$	0	0
$370$	−1.83754e6	−0.697803
$371$	0	0
$372$	0	0
$373$	−3.13003e6	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$373$	−3.13003e6	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$374$	962333.	0.355751
$375$	0	0
$376$	234698.	0.0856130
$377$	−2.17682e6	−0.788804
$378$	0	0
$379$	−1.27933e6	−0.457492	−0.228746	−	0.973486i	$-0.573463\pi$
$379$	−1.27933e6	−0.457492	−0.228746	+	0.973486i	$0.573463\pi$
$380$	−2.74947e6	−0.976766
$381$	0	0
$382$	94050.0	0.0329762
$383$	1.30666e6	0.455161	0.227580	−	0.973759i	$-0.426919\pi$
$383$	1.30666e6	0.455161	0.227580	+	0.973759i	$0.426919\pi$
$384$	0	0
$385$	0	0
$386$	224671.	0.0767501
$387$	0	0
$388$	−340964.	−0.114982
$389$	36488.2	0.0122258	0.00611292	−	0.999981i	$-0.498054\pi$
$389$	36488.2	0.0122258	0.00611292	+	0.999981i	$0.498054\pi$
$390$	0	0
$391$	−1.04851e6	−0.346842
$392$	0	0
$393$	0	0
$394$	−1.66483e6	−0.540294
$395$	−4.75271e6	−1.53267
$396$	0	0
$397$	−640204.	−0.203865	−0.101932	−	0.994791i	$-0.532503\pi$
$397$	−640204.	−0.203865	−0.101932	+	0.994791i	$0.532503\pi$
$398$	1.18043e6	0.373536
$399$	0	0
$400$	−8756.50	−0.00273641
$401$	2.69149e6	0.835858	0.417929	−	0.908480i	$-0.362756\pi$
$401$	2.69149e6	0.835858	0.417929	+	0.908480i	$0.362756\pi$
$402$	0	0
$403$	5.00697e6	1.53572
$404$	1.57613e6	0.480439
$405$	0	0
$406$	0	0
$407$	4.80449e6	1.43768
$408$	0	0
$409$	−2.92572e6	−0.864817	−0.432409	−	0.901678i	$-0.642336\pi$
$409$	−2.92572e6	−0.864817	−0.432409	+	0.901678i	$0.642336\pi$
$410$	−622526.	−0.182893
$411$	0	0
$412$	1.65484e6	0.480299
$413$	0	0
$414$	0	0
$415$	−4.92404e6	−1.40347
$416$	3.01506e6	0.854205
$417$	0	0
$418$	−3.99275e6	−1.11772
$419$	−5.52810e6	−1.53830	−0.769150	−	0.639068i	$-0.779320\pi$
$419$	−5.52810e6	−1.53830	−0.769150	+	0.639068i	$0.779320\pi$
$420$	0	0
$421$	5.77477e6	1.58792	0.793961	−	0.607968i	$-0.208015\pi$
$421$	5.77477e6	1.58792	0.793961	+	0.607968i	$0.208015\pi$
$422$	737756.	0.201666
$423$	0	0
$424$	−6.19106e6	−1.67244
$425$	−89789.1	−0.0241130
$426$	0	0
$427$	0	0
$428$	3.04573e6	0.803677
$429$	0	0
$430$	−3.35882e6	−0.876024
$431$	−2.20964e6	−0.572965	−0.286483	−	0.958085i	$-0.592486\pi$
$431$	−2.20964e6	−0.572965	−0.286483	+	0.958085i	$0.592486\pi$
$432$	0	0
$433$	−4.97215e6	−1.27446	−0.637228	−	0.770676i	$-0.719919\pi$
$433$	−4.97215e6	−1.27446	−0.637228	+	0.770676i	$0.719919\pi$
$434$	0	0
$435$	0	0
$436$	2.57130e6	0.647794
$437$	4.35032e6	1.08973
$438$	0	0
$439$	2.86405e6	0.709282	0.354641	−	0.935002i	$-0.384603\pi$
$439$	2.86405e6	0.709282	0.354641	+	0.935002i	$0.384603\pi$
$440$	4.66971e6	1.14990
$441$	0	0
$442$	1.02366e6	0.249230
$443$	5.66234e6	1.37084	0.685419	−	0.728149i	$-0.259619\pi$
$443$	5.66234e6	1.37084	0.685419	+	0.728149i	$0.259619\pi$
$444$	0	0
$445$	6.90124e6	1.65206
$446$	4.21889e6	1.00429
$447$	0	0
$448$	0	0
$449$	−7.54480e6	−1.76617	−0.883084	−	0.469215i	$-0.844537\pi$
$449$	−7.54480e6	−1.76617	−0.883084	+	0.469215i	$0.844537\pi$
$450$	0	0
$451$	1.62767e6	0.376813
$452$	−4.00098e6	−0.921129
$453$	0	0
$454$	1.76106e6	0.400991
$455$	0	0
$456$	0	0
$457$	−589014.	−0.131928	−0.0659638	−	0.997822i	$-0.521012\pi$
$457$	−589014.	−0.131928	−0.0659638	+	0.997822i	$0.521012\pi$
$458$	−1.77687e6	−0.395816
$459$	0	0
$460$	−1.99103e6	−0.438716
$461$	−5.94951e6	−1.30385	−0.651927	−	0.758282i	$-0.726039\pi$
$461$	−5.94951e6	−1.30385	−0.651927	+	0.758282i	$0.726039\pi$
$462$	0	0
$463$	−2.16083e6	−0.468456	−0.234228	−	0.972182i	$-0.575256\pi$
$463$	−2.16083e6	−0.468456	−0.234228	+	0.972182i	$0.575256\pi$
$464$	244642.	0.0527517
$465$	0	0
$466$	710563.	0.151579
$467$	505766.	0.107314	0.0536571	−	0.998559i	$-0.482912\pi$
$467$	505766.	0.107314	0.0536571	+	0.998559i	$0.482912\pi$
$468$	0	0
$469$	0	0
$470$	243412.	0.0508272
$471$	0	0
$472$	2.06016e6	0.425644
$473$	8.78208e6	1.80486
$474$	0	0
$475$	372538.	0.0757594
$476$	0	0
$477$	0	0
$478$	2.72384e6	0.545270
$479$	−547067.	−0.108944	−0.0544718	−	0.998515i	$-0.517348\pi$
$479$	−547067.	−0.108944	−0.0544718	+	0.998515i	$0.517348\pi$
$480$	0	0
$481$	5.11066e6	1.00720
$482$	3.84665e6	0.754162
$483$	0	0
$484$	−1.46456e6	−0.284181
$485$	−903652.	−0.174440
$486$	0	0
$487$	2.37889e6	0.454519	0.227259	−	0.973834i	$-0.427024\pi$
$487$	2.37889e6	0.454519	0.227259	+	0.973834i	$0.427024\pi$
$488$	−5.58135e6	−1.06094
$489$	0	0
$490$	0	0
$491$	7.19719e6	1.34728	0.673642	−	0.739058i	$-0.264729\pi$
$491$	7.19719e6	1.34728	0.673642	+	0.739058i	$0.264729\pi$
$492$	0	0
$493$	2.50856e6	0.464844
$494$	−4.24720e6	−0.783042
$495$	0	0
$496$	−562709.	−0.102702
$497$	0	0
$498$	0	0
$499$	2.22236e6	0.399543	0.199771	−	0.979843i	$-0.435980\pi$
$499$	2.22236e6	0.399543	0.199771	+	0.979843i	$0.435980\pi$
$500$	−3.67603e6	−0.657588
$501$	0	0
$502$	3.94032e6	0.697867
$503$	−305857.	−0.0539013	−0.0269506	−	0.999637i	$-0.508580\pi$
$503$	−305857.	−0.0539013	−0.0269506	+	0.999637i	$0.508580\pi$
$504$	0	0
$505$	4.17719e6	0.728879
$506$	−2.89135e6	−0.502024
$507$	0	0
$508$	−4.93280e6	−0.848074
$509$	−9.06420e6	−1.55073	−0.775363	−	0.631516i	$-0.782433\pi$
$509$	−9.06420e6	−1.55073	−0.775363	+	0.631516i	$0.782433\pi$
$510$	0	0
$511$	0	0
$512$	−666475.	−0.112359
$513$	0	0
$514$	3.47772e6	0.580614
$515$	4.38579e6	0.728668
$516$	0	0
$517$	−636431.	−0.104719
$518$	0	0
$519$	0	0
$520$	4.96730e6	0.805586
$521$	8.44649e6	1.36327	0.681635	−	0.731692i	$-0.261269\pi$
$521$	8.44649e6	1.36327	0.681635	+	0.731692i	$0.261269\pi$
$522$	0	0
$523$	−3.91585e6	−0.625997	−0.312998	−	0.949754i	$-0.601333\pi$
$523$	−3.91585e6	−0.625997	−0.312998	+	0.949754i	$0.601333\pi$
$524$	7.19195e6	1.14424
$525$	0	0
$526$	−2.22204e6	−0.350178
$527$	−5.77001e6	−0.905003
$528$	0	0
$529$	−3.28606e6	−0.510548
$530$	−6.42092e6	−0.992905
$531$	0	0
$532$	0	0
$533$	1.73140e6	0.263985
$534$	0	0
$535$	8.07204e6	1.21927
$536$	4.89711e6	0.736254
$537$	0	0
$538$	−18510.2	−0.00275712
$539$	0	0
$540$	0	0
$541$	−2.94802e6	−0.433049	−0.216524	−	0.976277i	$-0.569472\pi$
$541$	−2.94802e6	−0.433049	−0.216524	+	0.976277i	$0.569472\pi$
$542$	−2.59997e6	−0.380163
$543$	0	0
$544$	−3.47454e6	−0.503385
$545$	6.81469e6	0.982776
$546$	0	0
$547$	−5.50670e6	−0.786906	−0.393453	−	0.919345i	$-0.628720\pi$
$547$	−5.50670e6	−0.786906	−0.393453	+	0.919345i	$0.628720\pi$
$548$	−5.78041e6	−0.822256
$549$	0	0
$550$	−247600.	−0.0349015
$551$	−1.04081e7	−1.46047
$552$	0	0
$553$	0	0
$554$	3.96796e6	0.549278
$555$	0	0
$556$	1.02955e6	0.141241
$557$	4.97235e6	0.679085	0.339542	−	0.940591i	$-0.389728\pi$
$557$	4.97235e6	0.679085	0.339542	+	0.940591i	$0.389728\pi$
$558$	0	0
$559$	9.34174e6	1.26444
$560$	0	0
$561$	0	0
$562$	2.83153e6	0.378164
$563$	1.31397e7	1.74709	0.873544	−	0.486745i	$-0.161816\pi$
$563$	1.31397e7	1.74709	0.873544	+	0.486745i	$0.161816\pi$
$564$	0	0
$565$	−1.06037e7	−1.39746
$566$	−2.01880e6	−0.264882
$567$	0	0
$568$	4.06409e6	0.528558
$569$	1.38589e7	1.79452	0.897261	−	0.441500i	$-0.145553\pi$
$569$	1.38589e7	1.79452	0.897261	+	0.441500i	$0.145553\pi$
$570$	0	0
$571$	−1.06977e7	−1.37309	−0.686547	−	0.727085i	$-0.740874\pi$
$571$	−1.06977e7	−1.37309	−0.686547	+	0.727085i	$0.740874\pi$
$572$	−5.08241e6	−0.649501
$573$	0	0
$574$	0	0
$575$	269774.	0.0340275
$576$	0	0
$577$	5.61898e6	0.702616	0.351308	−	0.936260i	$-0.385737\pi$
$577$	5.61898e6	0.702616	0.351308	+	0.936260i	$0.385737\pi$
$578$	3.61993e6	0.450692
$579$	0	0
$580$	4.76353e6	0.587975
$581$	0	0
$582$	0	0
$583$	1.67883e7	2.04567
$584$	−2.83214e6	−0.343624
$585$	0	0
$586$	2.87775e6	0.346185
$587$	−6.86832e6	−0.822726	−0.411363	−	0.911471i	$-0.634947\pi$
$587$	−6.86832e6	−0.822726	−0.411363	+	0.911471i	$0.634947\pi$
$588$	0	0
$589$	2.39400e7	2.84338
$590$	2.13665e6	0.252699
$591$	0	0
$592$	−574362.	−0.0673568
$593$	4.07737e6	0.476149	0.238075	−	0.971247i	$-0.423484\pi$
$593$	4.07737e6	0.476149	0.238075	+	0.971247i	$0.423484\pi$
$594$	0	0
$595$	0	0
$596$	−6.76479e6	−0.780079
$597$	0	0
$598$	−3.07561e6	−0.351705
$599$	−582993.	−0.0663890	−0.0331945	−	0.999449i	$-0.510568\pi$
$599$	−582993.	−0.0663890	−0.0331945	+	0.999449i	$0.510568\pi$
$600$	0	0
$601$	6.87250e6	0.776119	0.388060	−	0.921634i	$-0.373145\pi$
$601$	6.87250e6	0.776119	0.388060	+	0.921634i	$0.373145\pi$
$602$	0	0
$603$	0	0
$604$	−1.05192e7	−1.17325
$605$	−3.88151e6	−0.431134
$606$	0	0
$607$	2.40563e6	0.265006	0.132503	−	0.991183i	$-0.457699\pi$
$607$	2.40563e6	0.265006	0.132503	+	0.991183i	$0.457699\pi$
$608$	1.44160e7	1.58156
$609$	0	0
$610$	−5.78858e6	−0.629865
$611$	−676989.	−0.0733632
$612$	0	0
$613$	−1.16245e7	−1.24946	−0.624731	−	0.780840i	$-0.714791\pi$
$613$	−1.16245e7	−1.24946	−0.624731	+	0.780840i	$0.714791\pi$
$614$	−2.03126e6	−0.217443
$615$	0	0
$616$	0	0
$617$	1.37370e7	1.45271	0.726353	−	0.687322i	$-0.241214\pi$
$617$	1.37370e7	1.45271	0.726353	+	0.687322i	$0.241214\pi$
$618$	0	0
$619$	−1.64008e7	−1.72044	−0.860220	−	0.509923i	$-0.829674\pi$
$619$	−1.64008e7	−1.72044	−0.860220	+	0.509923i	$0.829674\pi$
$620$	−1.09567e7	−1.14473
$621$	0	0
$622$	1.57478e6	0.163209
$623$	0	0
$624$	0	0
$625$	−9.26754e6	−0.948996
$626$	−1.00519e7	−1.02521
$627$	0	0
$628$	−1.55210e6	−0.157044
$629$	−5.88951e6	−0.593543
$630$	0	0
$631$	−7.28252e6	−0.728129	−0.364065	−	0.931374i	$-0.618611\pi$
$631$	−7.28252e6	−0.728129	−0.364065	+	0.931374i	$0.618611\pi$
$632$	1.54906e7	1.54268
$633$	0	0
$634$	8.63745e6	0.853419
$635$	−1.30733e7	−1.28662
$636$	0	0
$637$	0	0
$638$	6.91754e6	0.672821
$639$	0	0
$640$	−6.93763e6	−0.669516
$641$	107514.	0.0103352	0.00516762	−	0.999987i	$-0.498355\pi$
$641$	107514.	0.0103352	0.00516762	+	0.999987i	$0.498355\pi$
$642$	0	0
$643$	−9.75913e6	−0.930858	−0.465429	−	0.885085i	$-0.654100\pi$
$643$	−9.75913e6	−0.930858	−0.465429	+	0.885085i	$0.654100\pi$
$644$	0	0
$645$	0	0
$646$	4.89445e6	0.461448
$647$	−6.75588e6	−0.634485	−0.317243	−	0.948344i	$-0.602757\pi$
$647$	−6.75588e6	−0.634485	−0.317243	+	0.948344i	$0.602757\pi$
$648$	0	0
$649$	−5.58655e6	−0.520633
$650$	−263379.	−0.0244511
$651$	0	0
$652$	−7.01043e6	−0.645841
$653$	−2.20346e6	−0.202219	−0.101110	−	0.994875i	$-0.532239\pi$
$653$	−2.20346e6	−0.202219	−0.101110	+	0.994875i	$0.532239\pi$
$654$	0	0
$655$	1.90607e7	1.73595
$656$	−194584.	−0.0176542
$657$	0	0
$658$	0	0
$659$	1.39630e7	1.25247	0.626233	−	0.779636i	$-0.284596\pi$
$659$	1.39630e7	1.25247	0.626233	+	0.779636i	$0.284596\pi$
$660$	0	0
$661$	51052.8	0.00454482	0.00227241	−	0.999997i	$-0.499277\pi$
$661$	51052.8	0.00454482	0.00227241	+	0.999997i	$0.499277\pi$
$662$	−8.65832e6	−0.767872
$663$	0	0
$664$	1.60490e7	1.41263
$665$	0	0
$666$	0	0
$667$	−7.53703e6	−0.655972
$668$	2.72375e6	0.236170
$669$	0	0
$670$	5.07893e6	0.437104
$671$	1.51350e7	1.29770
$672$	0	0
$673$	6.71688e6	0.571650	0.285825	−	0.958282i	$-0.407732\pi$
$673$	6.71688e6	0.571650	0.285825	+	0.958282i	$0.407732\pi$
$674$	4.10793e6	0.348316
$675$	0	0
$676$	2.23245e6	0.187895
$677$	1.45655e7	1.22139	0.610693	−	0.791867i	$-0.290891\pi$
$677$	1.45655e7	1.22139	0.610693	+	0.791867i	$0.290891\pi$
$678$	0	0
$679$	0	0
$680$	−5.72429e6	−0.474733
$681$	0	0
$682$	−1.59112e7	−1.30991
$683$	1.62565e6	0.133344	0.0666722	−	0.997775i	$-0.478762\pi$
$683$	1.62565e6	0.133344	0.0666722	+	0.997775i	$0.478762\pi$
$684$	0	0
$685$	−1.53197e7	−1.24745
$686$	0	0
$687$	0	0
$688$	−1.04987e6	−0.0845600
$689$	1.78582e7	1.43314
$690$	0	0
$691$	2.81556e6	0.224321	0.112160	−	0.993690i	$-0.464223\pi$
$691$	2.81556e6	0.224321	0.112160	+	0.993690i	$0.464223\pi$
$692$	−9.24407e6	−0.733834
$693$	0	0
$694$	4.23616e6	0.333867
$695$	2.72860e6	0.214278
$696$	0	0
$697$	−1.99526e6	−0.155567
$698$	1.52251e6	0.118283
$699$	0	0
$700$	0	0
$701$	−1.40996e7	−1.08371	−0.541855	−	0.840472i	$-0.682278\pi$
$701$	−1.40996e7	−1.08371	−0.541855	+	0.840472i	$0.682278\pi$
$702$	0	0
$703$	2.44358e7	1.86482
$704$	−8.69287e6	−0.661046
$705$	0	0
$706$	−1.84323e6	−0.139177
$707$	0	0
$708$	0	0
$709$	−8.44943e6	−0.631265	−0.315633	−	0.948881i	$-0.602217\pi$
$709$	−8.44943e6	−0.631265	−0.315633	+	0.948881i	$0.602217\pi$
$710$	4.21498e6	0.313797
$711$	0	0
$712$	−2.24933e7	−1.66285
$713$	1.73361e7	1.27711
$714$	0	0
$715$	−1.34698e7	−0.985365
$716$	−1.92704e6	−0.140478
$717$	0	0
$718$	−3.41950e6	−0.247543
$719$	1.82657e7	1.31769	0.658846	−	0.752278i	$-0.271045\pi$
$719$	1.82657e7	1.31769	0.658846	+	0.752278i	$0.271045\pi$
$720$	0	0
$721$	0	0
$722$	−1.19372e7	−0.852237
$723$	0	0
$724$	8.22513e6	0.583172
$725$	−645431.	−0.0456042
$726$	0	0
$727$	−1.37476e6	−0.0964700	−0.0482350	−	0.998836i	$-0.515360\pi$
$727$	−1.37476e6	−0.0964700	−0.0482350	+	0.998836i	$0.515360\pi$
$728$	0	0
$729$	0	0
$730$	−2.93729e6	−0.204005
$731$	−1.07654e7	−0.745136
$732$	0	0
$733$	−1.64625e7	−1.13171	−0.565857	−	0.824503i	$-0.691455\pi$
$733$	−1.64625e7	−1.13171	−0.565857	+	0.824503i	$0.691455\pi$
$734$	−3.68585e6	−0.252521
$735$	0	0
$736$	1.04393e7	0.710360
$737$	−1.32795e7	−0.900561
$738$	0	0
$739$	2.72997e6	0.183885	0.0919425	−	0.995764i	$-0.470692\pi$
$739$	2.72997e6	0.183885	0.0919425	+	0.995764i	$0.470692\pi$
$740$	−1.11836e7	−0.750765
$741$	0	0
$742$	0	0
$743$	2.34396e7	1.55768	0.778839	−	0.627224i	$-0.215809\pi$
$743$	2.34396e7	1.55768	0.778839	+	0.627224i	$0.215809\pi$
$744$	0	0
$745$	−1.79286e7	−1.18347
$746$	1.05805e7	0.696082
$747$	0	0
$748$	5.85695e6	0.382752
$749$	0	0
$750$	0	0
$751$	1.79407e7	1.16075	0.580376	−	0.814349i	$-0.302906\pi$
$751$	1.79407e7	1.16075	0.580376	+	0.814349i	$0.302906\pi$
$752$	76083.4	0.00490620
$753$	0	0
$754$	7.35837e6	0.471361
$755$	−2.78790e7	−1.77996
$756$	0	0
$757$	−2.30823e7	−1.46399	−0.731997	−	0.681308i	$-0.761412\pi$
$757$	−2.30823e7	−1.46399	−0.731997	+	0.681308i	$0.761412\pi$
$758$	4.32455e6	0.273381
$759$	0	0
$760$	2.37503e7	1.49154
$761$	1.91587e7	1.19924	0.599618	−	0.800286i	$-0.295319\pi$
$761$	1.91587e7	1.19924	0.599618	+	0.800286i	$0.295319\pi$
$762$	0	0
$763$	0	0
$764$	572407.	0.0354790
$765$	0	0
$766$	−4.41693e6	−0.271988
$767$	−5.94257e6	−0.364742
$768$	0	0
$769$	−1.73017e7	−1.05505	−0.527526	−	0.849539i	$-0.676880\pi$
$769$	−1.73017e7	−1.05505	−0.527526	+	0.849539i	$0.676880\pi$
$770$	0	0
$771$	0	0
$772$	1.36739e6	0.0825753
$773$	3.12287e7	1.87977	0.939885	−	0.341491i	$-0.110932\pi$
$773$	3.12287e7	1.87977	0.939885	+	0.341491i	$0.110932\pi$
$774$	0	0
$775$	1.48458e6	0.0887868
$776$	2.94529e6	0.175580
$777$	0	0
$778$	−123342.	−0.00730572
$779$	8.27840e6	0.488768
$780$	0	0
$781$	−1.10206e7	−0.646514
$782$	3.54432e6	0.207260
$783$	0	0
$784$	0	0
$785$	−4.11350e6	−0.238253
$786$	0	0
$787$	2.48691e7	1.43128	0.715638	−	0.698471i	$-0.246136\pi$
$787$	2.48691e7	1.43128	0.715638	+	0.698471i	$0.246136\pi$
$788$	−1.01325e7	−0.581302
$789$	0	0
$790$	1.60657e7	0.915868
$791$	0	0
$792$	0	0
$793$	1.60995e7	0.909136
$794$	2.16410e6	0.121822
$795$	0	0
$796$	7.18433e6	0.401887
$797$	584574.	0.0325982	0.0162991	−	0.999867i	$-0.494812\pi$
$797$	584574.	0.0325982	0.0162991	+	0.999867i	$0.494812\pi$
$798$	0	0
$799$	780159.	0.0432331
$800$	893970.	0.0493853
$801$	0	0
$802$	−9.09814e6	−0.499478
$803$	7.67993e6	0.420309
$804$	0	0
$805$	0	0
$806$	−1.69252e7	−0.917691
$807$	0	0
$808$	−1.36148e7	−0.733639
$809$	1.10552e7	0.593878	0.296939	−	0.954897i	$-0.404034\pi$
$809$	1.10552e7	0.593878	0.296939	+	0.954897i	$0.404034\pi$
$810$	0	0
$811$	−1.62780e7	−0.869060	−0.434530	−	0.900657i	$-0.643086\pi$
$811$	−1.62780e7	−0.869060	−0.434530	+	0.900657i	$0.643086\pi$
$812$	0	0
$813$	0	0
$814$	−1.62408e7	−0.859103
$815$	−1.85796e7	−0.979813
$816$	0	0
$817$	4.46659e7	2.34111
$818$	9.88989e6	0.516783
$819$	0	0
$820$	−3.78882e6	−0.196775
$821$	−3.28574e7	−1.70128	−0.850638	−	0.525752i	$-0.823784\pi$
$821$	−3.28574e7	−1.70128	−0.850638	+	0.525752i	$0.823784\pi$
$822$	0	0
$823$	1.06046e7	0.545752	0.272876	−	0.962049i	$-0.412025\pi$
$823$	1.06046e7	0.545752	0.272876	+	0.962049i	$0.412025\pi$
$824$	−1.42947e7	−0.733426
$825$	0	0
$826$	0	0
$827$	−1.57582e7	−0.801203	−0.400602	−	0.916252i	$-0.631199\pi$
$827$	−1.57582e7	−0.801203	−0.400602	+	0.916252i	$0.631199\pi$
$828$	0	0
$829$	2.70135e7	1.36520	0.682598	−	0.730794i	$-0.260850\pi$
$829$	2.70135e7	1.36520	0.682598	+	0.730794i	$0.260850\pi$
$830$	1.66449e7	0.838660
$831$	0	0
$832$	−9.24684e6	−0.463111
$833$	0	0
$834$	0	0
$835$	7.21871e6	0.358297
$836$	−2.43007e7	−1.20255
$837$	0	0
$838$	1.86868e7	0.919232
$839$	−3.50490e7	−1.71898	−0.859489	−	0.511154i	$-0.829218\pi$
$839$	−3.50490e7	−1.71898	−0.859489	+	0.511154i	$0.829218\pi$
$840$	0	0
$841$	−2.47888e6	−0.120855
$842$	−1.95206e7	−0.948885
$843$	0	0
$844$	4.49013e6	0.216972
$845$	5.91663e6	0.285057
$846$	0	0
$847$	0	0
$848$	−2.00699e6	−0.0958422
$849$	0	0
$850$	303517.	0.0144091
$851$	1.76952e7	0.837589
$852$	0	0
$853$	538313.	0.0253316	0.0126658	−	0.999920i	$-0.495968\pi$
$853$	538313.	0.0253316	0.0126658	+	0.999920i	$0.495968\pi$
$854$	0	0
$855$	0	0
$856$	−2.63094e7	−1.22723
$857$	−3.68275e6	−0.171285	−0.0856426	−	0.996326i	$-0.527294\pi$
$857$	−3.68275e6	−0.171285	−0.0856426	+	0.996326i	$0.527294\pi$
$858$	0	0
$859$	1.88883e7	0.873394	0.436697	−	0.899609i	$-0.356148\pi$
$859$	1.88883e7	0.873394	0.436697	+	0.899609i	$0.356148\pi$
$860$	−2.04425e7	−0.942513
$861$	0	0
$862$	7.46931e6	0.342383
$863$	−2.78544e7	−1.27311	−0.636557	−	0.771230i	$-0.719642\pi$
$863$	−2.78544e7	−1.27311	−0.636557	+	0.771230i	$0.719642\pi$
$864$	0	0
$865$	−2.44994e7	−1.11331
$866$	1.68075e7	0.761568
$867$	0	0
$868$	0	0
$869$	−4.20059e7	−1.88695
$870$	0	0
$871$	−1.41258e7	−0.630909
$872$	−2.22113e7	−0.989195
$873$	0	0
$874$	−1.47055e7	−0.651180
$875$	0	0
$876$	0	0
$877$	−1.75052e7	−0.768544	−0.384272	−	0.923220i	$-0.625548\pi$
$877$	−1.75052e7	−0.768544	−0.384272	+	0.923220i	$0.625548\pi$
$878$	−9.68143e6	−0.423841
$879$	0	0
$880$	1.51381e6	0.0658968
$881$	−5.66292e6	−0.245811	−0.122905	−	0.992418i	$-0.539221\pi$
$881$	−5.66292e6	−0.245811	−0.122905	+	0.992418i	$0.539221\pi$
$882$	0	0
$883$	1.43933e7	0.621239	0.310620	−	0.950534i	$-0.399463\pi$
$883$	1.43933e7	0.621239	0.310620	+	0.950534i	$0.399463\pi$
$884$	6.23020e6	0.268146
$885$	0	0
$886$	−1.91406e7	−0.819163
$887$	5.50193e6	0.234804	0.117402	−	0.993084i	$-0.462543\pi$
$887$	5.50193e6	0.234804	0.117402	+	0.993084i	$0.462543\pi$
$888$	0	0
$889$	0	0
$890$	−2.33285e7	−0.987214
$891$	0	0
$892$	2.56770e7	1.08052
$893$	−3.23691e6	−0.135832
$894$	0	0
$895$	−5.10721e6	−0.213121
$896$	0	0
$897$	0	0
$898$	2.55039e7	1.05540
$899$	−4.14766e7	−1.71161
$900$	0	0
$901$	−2.05797e7	−0.844554
$902$	−5.50208e6	−0.225170
$903$	0	0
$904$	3.45610e7	1.40658
$905$	2.17990e7	0.884737
$906$	0	0
$907$	6.01555e6	0.242805	0.121402	−	0.992603i	$-0.461261\pi$
$907$	6.01555e6	0.242805	0.121402	+	0.992603i	$0.461261\pi$
$908$	1.07182e7	0.431426
$909$	0	0
$910$	0	0
$911$	1.47513e7	0.588892	0.294446	−	0.955668i	$-0.404865\pi$
$911$	1.47513e7	0.588892	0.294446	+	0.955668i	$0.404865\pi$
$912$	0	0
$913$	−4.35202e7	−1.72788
$914$	1.99106e6	0.0788351
$915$	0	0
$916$	−1.08144e7	−0.425858
$917$	0	0
$918$	0	0
$919$	−3.54001e7	−1.38266	−0.691331	−	0.722538i	$-0.742976\pi$
$919$	−3.54001e7	−1.38266	−0.691331	+	0.722538i	$0.742976\pi$
$920$	1.71988e7	0.669928
$921$	0	0
$922$	2.01113e7	0.779135
$923$	−1.17229e7	−0.452930
$924$	0	0
$925$	1.51532e6	0.0582305
$926$	7.30434e6	0.279932
$927$	0	0
$928$	−2.49760e7	−0.952036
$929$	−3.83266e7	−1.45700	−0.728502	−	0.685044i	$-0.759783\pi$
$929$	−3.83266e7	−1.45700	−0.728502	+	0.685044i	$0.759783\pi$
$930$	0	0
$931$	0	0
$932$	4.32463e6	0.163083
$933$	0	0
$934$	−1.70966e6	−0.0641271
$935$	1.55226e7	0.580677
$936$	0	0
$937$	−1.48693e7	−0.553274	−0.276637	−	0.960974i	$-0.589220\pi$
$937$	−1.48693e7	−0.553274	−0.276637	+	0.960974i	$0.589220\pi$
$938$	0	0
$939$	0	0
$940$	1.48145e6	0.0546849
$941$	−3.50894e6	−0.129182	−0.0645909	−	0.997912i	$-0.520574\pi$
$941$	−3.50894e6	−0.129182	−0.0645909	+	0.997912i	$0.520574\pi$
$942$	0	0
$943$	5.99481e6	0.219531
$944$	667856.	0.0243923
$945$	0	0
$946$	−2.96863e7	−1.07852
$947$	−3.89774e7	−1.41234	−0.706168	−	0.708044i	$-0.749578\pi$
$947$	−3.89774e7	−1.41234	−0.706168	+	0.708044i	$0.749578\pi$
$948$	0	0
$949$	8.16935e6	0.294457
$950$	−1.25930e6	−0.0452711
$951$	0	0
$952$	0	0
$953$	−7.11438e6	−0.253749	−0.126875	−	0.991919i	$-0.540495\pi$
$953$	−7.11438e6	−0.253749	−0.126875	+	0.991919i	$0.540495\pi$
$954$	0	0
$955$	1.51704e6	0.0538256
$956$	1.65778e7	0.586655
$957$	0	0
$958$	1.84927e6	0.0651008
$959$	0	0
$960$	0	0
$961$	6.67724e7	2.33232
$962$	−1.72757e7	−0.601865
$963$	0	0
$964$	2.34114e7	0.811401
$965$	3.62398e6	0.125276
$966$	0	0
$967$	−1.99357e7	−0.685592	−0.342796	−	0.939410i	$-0.611374\pi$
$967$	−1.99357e7	−0.685592	−0.342796	+	0.939410i	$0.611374\pi$
$968$	1.26511e7	0.433950
$969$	0	0
$970$	3.05464e6	0.104239
$971$	4.95888e7	1.68786	0.843928	−	0.536456i	$-0.180237\pi$
$971$	4.95888e7	1.68786	0.843928	+	0.536456i	$0.180237\pi$
$972$	0	0
$973$	0	0
$974$	−8.04143e6	−0.271604
$975$	0	0
$976$	−1.80934e6	−0.0607989
$977$	355348.	0.0119102	0.00595508	−	0.999982i	$-0.498104\pi$
$977$	355348.	0.0119102	0.00595508	+	0.999982i	$0.498104\pi$
$978$	0	0
$979$	6.09953e7	2.03395
$980$	0	0
$981$	0	0
$982$	−2.43289e7	−0.805088
$983$	−991754.	−0.0327356	−0.0163678	−	0.999866i	$-0.505210\pi$
$983$	−991754.	−0.0327356	−0.0163678	+	0.999866i	$0.505210\pi$
$984$	0	0
$985$	−2.68541e7	−0.881900
$986$	−8.47976e6	−0.277774
$987$	0	0
$988$	−2.58493e7	−0.842474
$989$	3.23448e7	1.05151
$990$	0	0
$991$	1.06674e7	0.345045	0.172523	−	0.985006i	$-0.444808\pi$
$991$	1.06674e7	0.345045	0.172523	+	0.985006i	$0.444808\pi$
$992$	5.74481e7	1.85352
$993$	0	0
$994$	0	0
$995$	1.90405e7	0.609707
$996$	0	0
$997$	−5.58283e7	−1.77876	−0.889379	−	0.457171i	$-0.848863\pi$
$997$	−5.58283e7	−1.77876	−0.889379	+	0.457171i	$0.848863\pi$
$998$	−7.51232e6	−0.238752
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bb.1.3		6
3.2	odd	2		147.6.a.n.1.4	✓	6
7.6	odd	2		441.6.a.ba.1.3		6
21.2	odd	6		147.6.e.q.67.3		12
21.5	even	6		147.6.e.p.67.3		12
21.11	odd	6		147.6.e.q.79.3		12
21.17	even	6		147.6.e.p.79.3		12
21.20	even	2		147.6.a.o.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.4	✓	6	3.2	odd	2
147.6.a.o.1.4	yes	6	21.20	even	2
147.6.e.p.67.3		12	21.5	even	6
147.6.e.p.79.3		12	21.17	even	6
147.6.e.q.67.3		12	21.2	odd	6
147.6.e.q.79.3		12	21.11	odd	6
441.6.a.ba.1.3		6	7.6	odd	2
441.6.a.bb.1.3		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)

\(f(q)\)	\(=\)	\(q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +O(q^{10})\)	\(q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +184.313 q^{10} -481.911 q^{11} -512.622 q^{13} +57.6111 q^{16} +590.744 q^{17} -2451.02 q^{19} +1121.77 q^{20} +1629.02 q^{22} -1774.90 q^{23} -151.993 q^{25} +1732.83 q^{26} +4246.44 q^{29} -9767.37 q^{31} -5881.64 q^{32} -1996.91 q^{34} -9969.65 q^{37} +8285.24 q^{38} -9689.98 q^{40} -3377.54 q^{41} -18223.4 q^{43} +9914.53 q^{44} +5999.76 q^{46} +1320.64 q^{47} +513.788 q^{50} +10546.4 q^{52} -34837.0 q^{53} +26276.3 q^{55} -14354.4 q^{58} +11592.5 q^{59} -31406.1 q^{61} +33016.9 q^{62} +18038.3 q^{64} +27950.9 q^{65} +27555.9 q^{67} -12153.6 q^{68} +22868.5 q^{71} -15936.4 q^{73} +33700.7 q^{74} +50425.6 q^{76} +87165.3 q^{79} -3141.26 q^{80} +11417.2 q^{82} +90307.5 q^{83} -32210.5 q^{85} +61601.2 q^{86} -85643.0 q^{88} -126570. q^{89} +36515.7 q^{92} -4464.20 q^{94} +133642. q^{95} +16573.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) 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

L-functions

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