LMFDB - Embedded newform 441.6.a.bb.1.1

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			1.1
Root			$5.59680$ 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)

$f(q)$	$=$	$q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} +O(q^{10})$	$q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} -704.138 q^{10} +731.396 q^{11} -899.645 q^{13} +1442.53 q^{16} -1392.39 q^{17} +190.198 q^{19} +4796.40 q^{20} -7320.46 q^{22} -42.9770 q^{23} +1824.30 q^{25} +9004.45 q^{26} +7746.59 q^{29} -1179.22 q^{31} -2850.94 q^{32} +13936.3 q^{34} +9288.68 q^{37} -1903.67 q^{38} -25474.2 q^{40} +13453.4 q^{41} +6033.68 q^{43} +49865.0 q^{44} +430.152 q^{46} -3244.94 q^{47} -18259.2 q^{50} -61335.9 q^{52} -25675.8 q^{53} +51454.6 q^{55} -77534.8 q^{58} -26450.5 q^{59} +6432.77 q^{61} +11802.7 q^{62} -17626.3 q^{64} -63291.1 q^{65} -23815.7 q^{67} -94930.2 q^{68} +44680.7 q^{71} +39601.3 q^{73} -92969.4 q^{74} +12967.3 q^{76} +23123.0 q^{79} +101484. q^{80} -134654. q^{82} -17267.2 q^{83} -97956.3 q^{85} -60390.4 q^{86} -264839. q^{88} +49929.7 q^{89} -2930.08 q^{92} +32478.3 q^{94} +13380.7 q^{95} +16865.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.0089	−1.76934	−0.884669	−	0.466219i	$-0.845616\pi$
$2$	−10.0089	−1.76934	−0.884669	+	0.466219i	$0.845616\pi$
$3$	0	0
$3$	0	0
$4$	68.1779	2.13056
$5$	70.3512	1.25848	0.629241	−	0.777211i	$-0.283366\pi$
$5$	70.3512	1.25848	0.629241	+	0.777211i	$0.283366\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−362.101	−2.00034
$9$	0	0
$10$	−704.138	−2.22668
$11$	731.396	1.82251	0.911257	−	0.411838i	$-0.135113\pi$
$11$	731.396	1.82251	0.911257	+	0.411838i	$0.135113\pi$
$12$	0	0
$13$	−899.645	−1.47643	−0.738215	−	0.674566i	$-0.764331\pi$
$13$	−899.645	−1.47643	−0.738215	+	0.674566i	$0.764331\pi$
$14$	0	0
$15$	0	0
$16$	1442.53	1.40872
$17$	−1392.39	−1.16853	−0.584263	−	0.811564i	$-0.698616\pi$
$17$	−1392.39	−1.16853	−0.584263	+	0.811564i	$0.698616\pi$
$18$	0	0
$19$	190.198	0.120871	0.0604354	−	0.998172i	$-0.480751\pi$
$19$	190.198	0.120871	0.0604354	+	0.998172i	$0.480751\pi$
$20$	4796.40	2.68127
$21$	0	0
$22$	−7320.46	−3.22464
$23$	−42.9770	−0.0169401	−0.00847006	−	0.999964i	$-0.502696\pi$
$23$	−42.9770	−0.0169401	−0.00847006	+	0.999964i	$0.502696\pi$
$24$	0	0
$25$	1824.30	0.583775
$26$	9004.45	2.61230
$27$	0	0
$28$	0	0
$29$	7746.59	1.71047	0.855236	−	0.518239i	$-0.173412\pi$
$29$	7746.59	1.71047	0.855236	+	0.518239i	$0.173412\pi$
$30$	0	0
$31$	−1179.22	−0.220390	−0.110195	−	0.993910i	$-0.535148\pi$
$31$	−1179.22	−0.220390	−0.110195	+	0.993910i	$0.535148\pi$
$32$	−2850.94	−0.492168
$33$	0	0
$34$	13936.3	2.06752
$35$	0	0
$36$	0	0
$37$	9288.68	1.11545	0.557724	−	0.830026i	$-0.311675\pi$
$37$	9288.68	1.11545	0.557724	+	0.830026i	$0.311675\pi$
$38$	−1903.67	−0.213861
$39$	0	0
$40$	−25474.2	−2.51739
$41$	13453.4	1.24989	0.624945	−	0.780668i	$-0.285121\pi$
$41$	13453.4	1.24989	0.624945	+	0.780668i	$0.285121\pi$
$42$	0	0
$43$	6033.68	0.497635	0.248818	−	0.968550i	$-0.419958\pi$
$43$	6033.68	0.497635	0.248818	+	0.968550i	$0.419958\pi$
$44$	49865.0	3.88297
$45$	0	0
$46$	430.152	0.0299728
$47$	−3244.94	−0.214270	−0.107135	−	0.994244i	$-0.534168\pi$
$47$	−3244.94	−0.214270	−0.107135	+	0.994244i	$0.534168\pi$
$48$	0	0
$49$	0	0
$50$	−18259.2	−1.03289
$51$	0	0
$52$	−61335.9	−3.14562
$53$	−25675.8	−1.25555	−0.627775	−	0.778395i	$-0.716034\pi$
$53$	−25675.8	−1.25555	−0.627775	+	0.778395i	$0.716034\pi$
$54$	0	0
$55$	51454.6	2.29360
$56$	0	0
$57$	0	0
$58$	−77534.8	−3.02640
$59$	−26450.5	−0.989244	−0.494622	−	0.869108i	$-0.664694\pi$
$59$	−26450.5	−0.989244	−0.494622	+	0.869108i	$0.664694\pi$
$60$	0	0
$61$	6432.77	0.221347	0.110673	−	0.993857i	$-0.464699\pi$
$61$	6432.77	0.221347	0.110673	+	0.993857i	$0.464699\pi$
$62$	11802.7	0.389944
$63$	0	0
$64$	−17626.3	−0.537913
$65$	−63291.1	−1.85806
$66$	0	0
$67$	−23815.7	−0.648150	−0.324075	−	0.946031i	$-0.605053\pi$
$67$	−23815.7	−0.648150	−0.324075	+	0.946031i	$0.605053\pi$
$68$	−94930.2	−2.48961
$69$	0	0
$70$	0	0
$71$	44680.7	1.05190	0.525949	−	0.850516i	$-0.323710\pi$
$71$	44680.7	1.05190	0.525949	+	0.850516i	$0.323710\pi$
$72$	0	0
$73$	39601.3	0.869765	0.434882	−	0.900487i	$-0.356790\pi$
$73$	39601.3	0.869765	0.434882	+	0.900487i	$0.356790\pi$
$74$	−92969.4	−1.97361
$75$	0	0
$76$	12967.3	0.257523
$77$	0	0
$78$	0	0
$79$	23123.0	0.416846	0.208423	−	0.978039i	$-0.433167\pi$
$79$	23123.0	0.416846	0.208423	+	0.978039i	$0.433167\pi$
$80$	101484.	1.77285
$81$	0	0
$82$	−134654.	−2.21148
$83$	−17267.2	−0.275122	−0.137561	−	0.990493i	$-0.543926\pi$
$83$	−17267.2	−0.275122	−0.137561	+	0.990493i	$0.543926\pi$
$84$	0	0
$85$	−97956.3	−1.47057
$86$	−60390.4	−0.880485
$87$	0	0
$88$	−264839.	−3.64565
$89$	49929.7	0.668166	0.334083	−	0.942544i	$-0.391573\pi$
$89$	49929.7	0.668166	0.334083	+	0.942544i	$0.391573\pi$
$90$	0	0
$91$	0	0
$92$	−2930.08	−0.0360920
$93$	0	0
$94$	32478.3	0.379117
$95$	13380.7	0.152114
$96$	0	0
$97$	16865.0	0.181994	0.0909970	−	0.995851i	$-0.470995\pi$
$97$	16865.0	0.181994	0.0909970	+	0.995851i	$0.470995\pi$
$98$	0	0
$99$	0	0
$100$	124377.	1.24377
$101$	105449.	1.02858	0.514290	−	0.857617i	$-0.328056\pi$
$101$	105449.	1.02858	0.514290	+	0.857617i	$0.328056\pi$
$102$	0	0
$103$	20355.8	0.189058	0.0945290	−	0.995522i	$-0.469865\pi$
$103$	20355.8	0.189058	0.0945290	+	0.995522i	$0.469865\pi$
$104$	325762.	2.95337
$105$	0	0
$106$	256986.	2.22149
$107$	3775.43	0.0318792	0.0159396	−	0.999873i	$-0.494926\pi$
$107$	3775.43	0.0318792	0.0159396	+	0.999873i	$0.494926\pi$
$108$	0	0
$109$	8536.98	0.0688237	0.0344118	−	0.999408i	$-0.489044\pi$
$109$	8536.98	0.0688237	0.0344118	+	0.999408i	$0.489044\pi$
$110$	−515004.	−4.05815
$111$	0	0
$112$	0	0
$113$	100869.	0.743125	0.371562	−	0.928408i	$-0.378822\pi$
$113$	100869.	0.743125	0.371562	+	0.928408i	$0.378822\pi$
$114$	0	0
$115$	−3023.49	−0.0213188
$116$	528147.	3.64426
$117$	0	0
$118$	264740.	1.75031
$119$	0	0
$120$	0	0
$121$	373889.	2.32156
$122$	−64384.9	−0.391638
$123$	0	0
$124$	−80396.9	−0.469554
$125$	−91506.2	−0.523812
$126$	0	0
$127$	−185451.	−1.02028	−0.510142	−	0.860090i	$-0.670407\pi$
$127$	−185451.	−1.02028	−0.510142	+	0.860090i	$0.670407\pi$
$128$	267650.	1.44392
$129$	0	0
$130$	633474.	3.28754
$131$	124046.	0.631545	0.315773	−	0.948835i	$-0.397736\pi$
$131$	124046.	0.631545	0.315773	+	0.948835i	$0.397736\pi$
$132$	0	0
$133$	0	0
$134$	238368.	1.14680
$135$	0	0
$136$	504185.	2.33745
$137$	−99520.5	−0.453013	−0.226507	−	0.974010i	$-0.572731\pi$
$137$	−99520.5	−0.453013	−0.226507	+	0.974010i	$0.572731\pi$
$138$	0	0
$139$	−254481.	−1.11717	−0.558583	−	0.829449i	$-0.688655\pi$
$139$	−254481.	−1.11717	−0.558583	+	0.829449i	$0.688655\pi$
$140$	0	0
$141$	0	0
$142$	−447204.	−1.86116
$143$	−657997.	−2.69081
$144$	0	0
$145$	544982.	2.15260
$146$	−396365.	−1.53891
$147$	0	0
$148$	633283.	2.37653
$149$	427260.	1.57662	0.788309	−	0.615279i	$-0.210957\pi$
$149$	427260.	1.57662	0.788309	+	0.615279i	$0.210957\pi$
$150$	0	0
$151$	−150472.	−0.537047	−0.268523	−	0.963273i	$-0.586536\pi$
$151$	−150472.	−0.537047	−0.268523	+	0.963273i	$0.586536\pi$
$152$	−68870.8	−0.241783
$153$	0	0
$154$	0	0
$155$	−82959.8	−0.277357
$156$	0	0
$157$	−165831.	−0.536930	−0.268465	−	0.963290i	$-0.586516\pi$
$157$	−165831.	−0.536930	−0.268465	+	0.963290i	$0.586516\pi$
$158$	−231435.	−0.737542
$159$	0	0
$160$	−200567.	−0.619384
$161$	0	0
$162$	0	0
$163$	95718.2	0.282180	0.141090	−	0.989997i	$-0.454939\pi$
$163$	95718.2	0.282180	0.141090	+	0.989997i	$0.454939\pi$
$164$	917224.	2.66297
$165$	0	0
$166$	172825.	0.486784
$167$	474209.	1.31577	0.657883	−	0.753120i	$-0.271452\pi$
$167$	474209.	1.31577	0.657883	+	0.753120i	$0.271452\pi$
$168$	0	0
$169$	438068.	1.17984
$170$	980434.	2.60193
$171$	0	0
$172$	411363.	1.06024
$173$	258876.	0.657624	0.328812	−	0.944395i	$-0.393352\pi$
$173$	258876.	0.657624	0.328812	+	0.944395i	$0.393352\pi$
$174$	0	0
$175$	0	0
$176$	1.05506e6	2.56742
$177$	0	0
$178$	−499741.	−1.18221
$179$	542178.	1.26476	0.632381	−	0.774657i	$-0.282078\pi$
$179$	542178.	1.26476	0.632381	+	0.774657i	$0.282078\pi$
$180$	0	0
$181$	−653930.	−1.48366	−0.741831	−	0.670587i	$-0.766042\pi$
$181$	−653930.	−1.48366	−0.741831	+	0.670587i	$0.766042\pi$
$182$	0	0
$183$	0	0
$184$	15562.0	0.0338861
$185$	653470.	1.40377
$186$	0	0
$187$	−1.01839e6	−2.12965
$188$	−221233.	−0.456516
$189$	0	0
$190$	−133925.	−0.269141
$191$	−769535.	−1.52632	−0.763159	−	0.646210i	$-0.776353\pi$
$191$	−769535.	−1.52632	−0.763159	+	0.646210i	$0.776353\pi$
$192$	0	0
$193$	814230.	1.57345	0.786727	−	0.617301i	$-0.211774\pi$
$193$	814230.	1.57345	0.786727	+	0.617301i	$0.211774\pi$
$194$	−168800.	−0.322009
$195$	0	0
$196$	0	0
$197$	443688.	0.814539	0.407269	−	0.913308i	$-0.366481\pi$
$197$	443688.	0.814539	0.407269	+	0.913308i	$0.366481\pi$
$198$	0	0
$199$	−618941.	−1.10794	−0.553970	−	0.832536i	$-0.686888\pi$
$199$	−618941.	−1.10794	−0.553970	+	0.832536i	$0.686888\pi$
$200$	−660579.	−1.16775
$201$	0	0
$202$	−1.05542e6	−1.81991
$203$	0	0
$204$	0	0
$205$	946463.	1.57296
$206$	−203739.	−0.334508
$207$	0	0
$208$	−1.29777e6	−2.07988
$209$	139110.	0.220289
$210$	0	0
$211$	769805.	1.19035	0.595175	−	0.803596i	$-0.297083\pi$
$211$	769805.	1.19035	0.595175	+	0.803596i	$0.297083\pi$
$212$	−1.75052e6	−2.67502
$213$	0	0
$214$	−37787.9	−0.0564050
$215$	424477.	0.626264
$216$	0	0
$217$	0	0
$218$	−85445.7	−0.121772
$219$	0	0
$220$	3.50807e6	4.88665
$221$	1.25266e6	1.72525
$222$	0	0
$223$	1.45664e6	1.96151	0.980755	−	0.195244i	$-0.0625498\pi$
$223$	1.45664e6	1.96151	0.980755	+	0.195244i	$0.0625498\pi$
$224$	0	0
$225$	0	0
$226$	−1.00959e6	−1.31484
$227$	949905.	1.22353	0.611767	−	0.791038i	$-0.290459\pi$
$227$	949905.	1.22353	0.611767	+	0.791038i	$0.290459\pi$
$228$	0	0
$229$	312283.	0.393513	0.196757	−	0.980452i	$-0.436959\pi$
$229$	312283.	0.393513	0.196757	+	0.980452i	$0.436959\pi$
$230$	30261.7	0.0377202
$231$	0	0
$232$	−2.80505e6	−3.42153
$233$	−396905.	−0.478957	−0.239479	−	0.970902i	$-0.576977\pi$
$233$	−396905.	−0.478957	−0.239479	+	0.970902i	$0.576977\pi$
$234$	0	0
$235$	−228286.	−0.269655
$236$	−1.80334e6	−2.10764
$237$	0	0
$238$	0	0
$239$	87033.3	0.0985578	0.0492789	−	0.998785i	$-0.484308\pi$
$239$	87033.3	0.0985578	0.0492789	+	0.998785i	$0.484308\pi$
$240$	0	0
$241$	−1.57173e6	−1.74315	−0.871576	−	0.490260i	$-0.836902\pi$
$241$	−1.57173e6	−1.74315	−0.871576	+	0.490260i	$0.836902\pi$
$242$	−3.74222e6	−4.10762
$243$	0	0
$244$	438573.	0.471593
$245$	0	0
$246$	0	0
$247$	−171111.	−0.178457
$248$	426997.	0.440855
$249$	0	0
$250$	915875.	0.926801
$251$	385591.	0.386316	0.193158	−	0.981168i	$-0.438127\pi$
$251$	385591.	0.386316	0.193158	+	0.981168i	$0.438127\pi$
$252$	0	0
$253$	−31433.2	−0.0308736
$254$	1.85616e6	1.80523
$255$	0	0
$256$	−2.11484e6	−2.01687
$257$	441809.	0.417255	0.208627	−	0.977995i	$-0.433100\pi$
$257$	441809.	0.417255	0.208627	+	0.977995i	$0.433100\pi$
$258$	0	0
$259$	0	0
$260$	−4.31506e6	−3.95870
$261$	0	0
$262$	−1.24156e6	−1.11742
$263$	600264.	0.535122	0.267561	−	0.963541i	$-0.413782\pi$
$263$	600264.	0.535122	0.267561	+	0.963541i	$0.413782\pi$
$264$	0	0
$265$	−1.80632e6	−1.58009
$266$	0	0
$267$	0	0
$268$	−1.62370e6	−1.38092
$269$	−325609.	−0.274357	−0.137178	−	0.990546i	$-0.543803\pi$
$269$	−325609.	−0.274357	−0.137178	+	0.990546i	$0.543803\pi$
$270$	0	0
$271$	1.75592e6	1.45239	0.726193	−	0.687491i	$-0.241288\pi$
$271$	1.75592e6	1.45239	0.726193	+	0.687491i	$0.241288\pi$
$272$	−2.00857e6	−1.64613
$273$	0	0
$274$	996090.	0.801534
$275$	1.33428e6	1.06394
$276$	0	0
$277$	−342363.	−0.268094	−0.134047	−	0.990975i	$-0.542797\pi$
$277$	−342363.	−0.268094	−0.134047	+	0.990975i	$0.542797\pi$
$278$	2.54707e6	1.97664
$279$	0	0
$280$	0	0
$281$	−930671.	−0.703122	−0.351561	−	0.936165i	$-0.614349\pi$
$281$	−930671.	−0.703122	−0.351561	+	0.936165i	$0.614349\pi$
$282$	0	0
$283$	−1.64760e6	−1.22288	−0.611442	−	0.791290i	$-0.709410\pi$
$283$	−1.64760e6	−1.22288	−0.611442	+	0.791290i	$0.709410\pi$
$284$	3.04623e6	2.24113
$285$	0	0
$286$	6.58582e6	4.76096
$287$	0	0
$288$	0	0
$289$	518891.	0.365453
$290$	−5.45467e6	−3.80867
$291$	0	0
$292$	2.69993e6	1.85309
$293$	1.14378e6	0.778347	0.389174	−	0.921164i	$-0.372761\pi$
$293$	1.14378e6	0.778347	0.389174	+	0.921164i	$0.372761\pi$
$294$	0	0
$295$	−1.86082e6	−1.24495
$296$	−3.36344e6	−2.23128
$297$	0	0
$298$	−4.27640e6	−2.78957
$299$	38664.1	0.0250109
$300$	0	0
$301$	0	0
$302$	1.50605e6	0.950218
$303$	0	0
$304$	274367.	0.170274
$305$	452553.	0.278561
$306$	0	0
$307$	2.98831e6	1.80959	0.904795	−	0.425848i	$-0.140024\pi$
$307$	2.98831e6	1.80959	0.904795	+	0.425848i	$0.140024\pi$
$308$	0	0
$309$	0	0
$310$	830335.	0.490738
$311$	−2.83751e6	−1.66355	−0.831775	−	0.555113i	$-0.812675\pi$
$311$	−2.83751e6	−1.66355	−0.831775	+	0.555113i	$0.812675\pi$
$312$	0	0
$313$	−2.14749e6	−1.23899	−0.619497	−	0.784999i	$-0.712664\pi$
$313$	−2.14749e6	−1.23899	−0.619497	+	0.784999i	$0.712664\pi$
$314$	1.65979e6	0.950010
$315$	0	0
$316$	1.57648e6	0.888116
$317$	2.63357e6	1.47196	0.735982	−	0.677001i	$-0.236721\pi$
$317$	2.63357e6	1.47196	0.735982	+	0.677001i	$0.236721\pi$
$318$	0	0
$319$	5.66583e6	3.11736
$320$	−1.24003e6	−0.676953
$321$	0	0
$322$	0	0
$323$	−264829.	−0.141241
$324$	0	0
$325$	−1.64122e6	−0.861902
$326$	−958033.	−0.499271
$327$	0	0
$328$	−4.87148e6	−2.50021
$329$	0	0
$330$	0	0
$331$	3.28434e6	1.64770	0.823849	−	0.566809i	$-0.191822\pi$
$331$	3.28434e6	1.64770	0.823849	+	0.566809i	$0.191822\pi$
$332$	−1.17724e6	−0.586164
$333$	0	0
$334$	−4.74631e6	−2.32804
$335$	−1.67546e6	−0.815685
$336$	0	0
$337$	−238202.	−0.114254	−0.0571270	−	0.998367i	$-0.518194\pi$
$337$	−238202.	−0.114254	−0.0571270	+	0.998367i	$0.518194\pi$
$338$	−4.38457e6	−2.08754
$339$	0	0
$340$	−6.67846e6	−3.13313
$341$	−862479.	−0.401664
$342$	0	0
$343$	0	0
$344$	−2.18480e6	−0.995441
$345$	0	0
$346$	−2.59107e6	−1.16356
$347$	−1.06436e6	−0.474532	−0.237266	−	0.971445i	$-0.576251\pi$
$347$	−1.06436e6	−0.474532	−0.237266	+	0.971445i	$0.576251\pi$
$348$	0	0
$349$	−1.85940e6	−0.817163	−0.408582	−	0.912722i	$-0.633977\pi$
$349$	−1.85940e6	−0.817163	−0.408582	+	0.912722i	$0.633977\pi$
$350$	0	0
$351$	0	0
$352$	−2.08517e6	−0.896983
$353$	−1.66487e6	−0.711123	−0.355561	−	0.934653i	$-0.615710\pi$
$353$	−1.66487e6	−0.711123	−0.355561	+	0.934653i	$0.615710\pi$
$354$	0	0
$355$	3.14334e6	1.32379
$356$	3.40411e6	1.42357
$357$	0	0
$358$	−5.42660e6	−2.23779
$359$	−2.67457e6	−1.09526	−0.547632	−	0.836720i	$-0.684470\pi$
$359$	−2.67457e6	−1.09526	−0.547632	+	0.836720i	$0.684470\pi$
$360$	0	0
$361$	−2.43992e6	−0.985390
$362$	6.54511e6	2.62510
$363$	0	0
$364$	0	0
$365$	2.78600e6	1.09458
$366$	0	0
$367$	2.14384e6	0.830860	0.415430	−	0.909625i	$-0.363631\pi$
$367$	2.14384e6	0.830860	0.415430	+	0.909625i	$0.363631\pi$
$368$	−61995.8	−0.0238640
$369$	0	0
$370$	−6.54051e6	−2.48375
$371$	0	0
$372$	0	0
$373$	−759867.	−0.282791	−0.141395	−	0.989953i	$-0.545159\pi$
$373$	−759867.	−0.282791	−0.141395	+	0.989953i	$0.545159\pi$
$374$	1.01929e7	3.76808
$375$	0	0
$376$	1.17500e6	0.428614
$377$	−6.96918e6	−2.52539
$378$	0	0
$379$	3.03679e6	1.08597	0.542984	−	0.839743i	$-0.317294\pi$
$379$	3.03679e6	1.08597	0.542984	+	0.839743i	$0.317294\pi$
$380$	912265.	0.324087
$381$	0	0
$382$	7.70220e6	2.70057
$383$	2.50929e6	0.874086	0.437043	−	0.899441i	$-0.356026\pi$
$383$	2.50929e6	0.874086	0.437043	+	0.899441i	$0.356026\pi$
$384$	0	0
$385$	0	0
$386$	−8.14954e6	−2.78397
$387$	0	0
$388$	1.14982e6	0.387749
$389$	985137.	0.330082	0.165041	−	0.986287i	$-0.447224\pi$
$389$	985137.	0.330082	0.165041	+	0.986287i	$0.447224\pi$
$390$	0	0
$391$	59840.7	0.0197950
$392$	0	0
$393$	0	0
$394$	−4.44082e6	−1.44119
$395$	1.62673e6	0.524593
$396$	0	0
$397$	2.90510e6	0.925092	0.462546	−	0.886595i	$-0.346936\pi$
$397$	2.90510e6	0.925092	0.462546	+	0.886595i	$0.346936\pi$
$398$	6.19491e6	1.96032
$399$	0	0
$400$	2.63161e6	0.822377
$401$	4.23124e6	1.31404	0.657018	−	0.753875i	$-0.271818\pi$
$401$	4.23124e6	1.31404	0.657018	+	0.753875i	$0.271818\pi$
$402$	0	0
$403$	1.06088e6	0.325390
$404$	7.18927e6	2.19145
$405$	0	0
$406$	0	0
$407$	6.79370e6	2.03292
$408$	0	0
$409$	4.13890e6	1.22342	0.611711	−	0.791081i	$-0.290482\pi$
$409$	4.13890e6	1.22342	0.611711	+	0.791081i	$0.290482\pi$
$410$	−9.47304e6	−2.78311
$411$	0	0
$412$	1.38782e6	0.402799
$413$	0	0
$414$	0	0
$415$	−1.21477e6	−0.346236
$416$	2.56483e6	0.726651
$417$	0	0
$418$	−1.39234e6	−0.389766
$419$	−5.99852e6	−1.66920	−0.834601	−	0.550855i	$-0.814302\pi$
$419$	−5.99852e6	−1.66920	−0.834601	+	0.550855i	$0.814302\pi$
$420$	0	0
$421$	−1.00759e6	−0.277062	−0.138531	−	0.990358i	$-0.544238\pi$
$421$	−1.00759e6	−0.277062	−0.138531	+	0.990358i	$0.544238\pi$
$422$	−7.70489e6	−2.10613
$423$	0	0
$424$	9.29722e6	2.51153
$425$	−2.54013e6	−0.682156
$426$	0	0
$427$	0	0
$428$	257401.	0.0679205
$429$	0	0
$430$	−4.24854e6	−1.10807
$431$	−303149.	−0.0786073	−0.0393037	−	0.999227i	$-0.512514\pi$
$431$	−303149.	−0.0786073	−0.0393037	+	0.999227i	$0.512514\pi$
$432$	0	0
$433$	3.97130e6	1.01792	0.508960	−	0.860790i	$-0.330030\pi$
$433$	3.97130e6	1.01792	0.508960	+	0.860790i	$0.330030\pi$
$434$	0	0
$435$	0	0
$436$	582033.	0.146633
$437$	−8174.14	−0.00204757
$438$	0	0
$439$	3.74799e6	0.928192	0.464096	−	0.885785i	$-0.346379\pi$
$439$	3.74799e6	0.928192	0.464096	+	0.885785i	$0.346379\pi$
$440$	−1.86318e7	−4.58798
$441$	0	0
$442$	−1.25377e7	−3.05255
$443$	1.31077e6	0.317334	0.158667	−	0.987332i	$-0.449280\pi$
$443$	1.31077e6	0.317334	0.158667	+	0.987332i	$0.449280\pi$
$444$	0	0
$445$	3.51262e6	0.840874
$446$	−1.45794e7	−3.47057
$447$	0	0
$448$	0	0
$449$	−6.31311e6	−1.47784	−0.738920	−	0.673793i	$-0.764664\pi$
$449$	−6.31311e6	−1.47784	−0.738920	+	0.673793i	$0.764664\pi$
$450$	0	0
$451$	9.83976e6	2.27794
$452$	6.87704e6	1.58327
$453$	0	0
$454$	−9.50750e6	−2.16484
$455$	0	0
$456$	0	0
$457$	−583476.	−0.130687	−0.0653435	−	0.997863i	$-0.520814\pi$
$457$	−583476.	−0.130687	−0.0653435	+	0.997863i	$0.520814\pi$
$458$	−3.12561e6	−0.696259
$459$	0	0
$460$	−206135.	−0.0454210
$461$	2.93381e6	0.642953	0.321476	−	0.946918i	$-0.395821\pi$
$461$	2.93381e6	0.642953	0.321476	+	0.946918i	$0.395821\pi$
$462$	0	0
$463$	3.76716e6	0.816699	0.408349	−	0.912826i	$-0.366105\pi$
$463$	3.76716e6	0.816699	0.408349	+	0.912826i	$0.366105\pi$
$464$	1.11747e7	2.40958
$465$	0	0
$466$	3.97258e6	0.847438
$467$	2.90596e6	0.616592	0.308296	−	0.951290i	$-0.400241\pi$
$467$	2.90596e6	0.616592	0.308296	+	0.951290i	$0.400241\pi$
$468$	0	0
$469$	0	0
$470$	2.28489e6	0.477111
$471$	0	0
$472$	9.57774e6	1.97883
$473$	4.41301e6	0.906947
$474$	0	0
$475$	346977.	0.0705613
$476$	0	0
$477$	0	0
$478$	−871107.	−0.174382
$479$	7.06622e6	1.40718	0.703588	−	0.710608i	$-0.251580\pi$
$479$	7.06622e6	1.40718	0.703588	+	0.710608i	$0.251580\pi$
$480$	0	0
$481$	−8.35651e6	−1.64688
$482$	1.57313e7	3.08423
$483$	0	0
$484$	2.54910e7	4.94622
$485$	1.18647e6	0.229036
$486$	0	0
$487$	2.72981e6	0.521566	0.260783	−	0.965397i	$-0.416019\pi$
$487$	2.72981e6	0.521566	0.260783	+	0.965397i	$0.416019\pi$
$488$	−2.32931e6	−0.442770
$489$	0	0
$490$	0	0
$491$	1.02859e7	1.92548	0.962740	−	0.270428i	$-0.0871651\pi$
$491$	1.02859e7	1.92548	0.962740	+	0.270428i	$0.0871651\pi$
$492$	0	0
$493$	−1.07863e7	−1.99873
$494$	1.71263e6	0.315751
$495$	0	0
$496$	−1.70107e6	−0.310469
$497$	0	0
$498$	0	0
$499$	−3.63591e6	−0.653674	−0.326837	−	0.945081i	$-0.605983\pi$
$499$	−3.63591e6	−0.653674	−0.326837	+	0.945081i	$0.605983\pi$
$500$	−6.23870e6	−1.11601
$501$	0	0
$502$	−3.85934e6	−0.683524
$503$	1.65499e6	0.291658	0.145829	−	0.989310i	$-0.453415\pi$
$503$	1.65499e6	0.291658	0.145829	+	0.989310i	$0.453415\pi$
$504$	0	0
$505$	7.41845e6	1.29445
$506$	314612.	0.0546259
$507$	0	0
$508$	−1.26437e7	−2.17377
$509$	−731110.	−0.125080	−0.0625401	−	0.998042i	$-0.519920\pi$
$509$	−731110.	−0.125080	−0.0625401	+	0.998042i	$0.519920\pi$
$510$	0	0
$511$	0	0
$512$	1.26024e7	2.12460
$513$	0	0
$514$	−4.42202e6	−0.738265
$515$	1.43206e6	0.237926
$516$	0	0
$517$	−2.37334e6	−0.390511
$518$	0	0
$519$	0	0
$520$	2.29178e7	3.71675
$521$	4.73095e6	0.763579	0.381790	−	0.924249i	$-0.375308\pi$
$521$	4.73095e6	0.763579	0.381790	+	0.924249i	$0.375308\pi$
$522$	0	0
$523$	9.72563e6	1.55476	0.777380	−	0.629031i	$-0.216548\pi$
$523$	9.72563e6	1.55476	0.777380	+	0.629031i	$0.216548\pi$
$524$	8.45719e6	1.34554
$525$	0	0
$526$	−6.00798e6	−0.946812
$527$	1.64194e6	0.257531
$528$	0	0
$529$	−6.43450e6	−0.999713
$530$	1.80793e7	2.79571
$531$	0	0
$532$	0	0
$533$	−1.21033e7	−1.84538
$534$	0	0
$535$	265606.	0.0401193
$536$	8.62367e6	1.29652
$537$	0	0
$538$	3.25898e6	0.485430
$539$	0	0
$540$	0	0
$541$	8.19002e6	1.20307	0.601536	−	0.798845i	$-0.294556\pi$
$541$	8.19002e6	1.20307	0.601536	+	0.798845i	$0.294556\pi$
$542$	−1.75748e7	−2.56976
$543$	0	0
$544$	3.96962e6	0.575111
$545$	600587.	0.0866133
$546$	0	0
$547$	−4.17136e6	−0.596087	−0.298043	−	0.954552i	$-0.596334\pi$
$547$	−4.17136e6	−0.596087	−0.298043	+	0.954552i	$0.596334\pi$
$548$	−6.78510e6	−0.965172
$549$	0	0
$550$	−1.33547e7	−1.88247
$551$	1.47339e6	0.206746
$552$	0	0
$553$	0	0
$554$	3.42668e6	0.474350
$555$	0	0
$556$	−1.73500e7	−2.38019
$557$	1.26658e6	0.172980	0.0864899	−	0.996253i	$-0.472435\pi$
$557$	1.26658e6	0.172980	0.0864899	+	0.996253i	$0.472435\pi$
$558$	0	0
$559$	−5.42817e6	−0.734723
$560$	0	0
$561$	0	0
$562$	9.31499e6	1.24406
$563$	−6.37431e6	−0.847544	−0.423772	−	0.905769i	$-0.639294\pi$
$563$	−6.37431e6	−0.847544	−0.423772	+	0.905769i	$0.639294\pi$
$564$	0	0
$565$	7.09626e6	0.935208
$566$	1.64906e7	2.16369
$567$	0	0
$568$	−1.61789e7	−2.10416
$569$	−7.64040e6	−0.989317	−0.494659	−	0.869087i	$-0.664707\pi$
$569$	−7.64040e6	−0.989317	−0.494659	+	0.869087i	$0.664707\pi$
$570$	0	0
$571$	790914.	0.101517	0.0507585	−	0.998711i	$-0.483836\pi$
$571$	790914.	0.101517	0.0507585	+	0.998711i	$0.483836\pi$
$572$	−4.48608e7	−5.73294
$573$	0	0
$574$	0	0
$575$	−78402.8	−0.00988922
$576$	0	0
$577$	8.55670e6	1.06996	0.534979	−	0.844865i	$-0.320320\pi$
$577$	8.55670e6	1.06996	0.534979	+	0.844865i	$0.320320\pi$
$578$	−5.19352e6	−0.646610
$579$	0	0
$580$	3.71558e7	4.58623
$581$	0	0
$582$	0	0
$583$	−1.87792e7	−2.28826
$584$	−1.43396e7	−1.73983
$585$	0	0
$586$	−1.14480e7	−1.37716
$587$	−1.17153e7	−1.40333	−0.701663	−	0.712509i	$-0.747559\pi$
$587$	−1.17153e7	−1.40333	−0.701663	+	0.712509i	$0.747559\pi$
$588$	0	0
$589$	−224286.	−0.0266387
$590$	1.86248e7	2.20273
$591$	0	0
$592$	1.33992e7	1.57136
$593$	1.55617e7	1.81728	0.908639	−	0.417583i	$-0.137123\pi$
$593$	1.55617e7	1.81728	0.908639	+	0.417583i	$0.137123\pi$
$594$	0	0
$595$	0	0
$596$	2.91297e7	3.35908
$597$	0	0
$598$	−386984.	−0.0442528
$599$	9.18930e6	1.04644	0.523221	−	0.852197i	$-0.324730\pi$
$599$	9.18930e6	1.04644	0.523221	+	0.852197i	$0.324730\pi$
$600$	0	0
$601$	−1.61226e7	−1.82075	−0.910373	−	0.413789i	$-0.864205\pi$
$601$	−1.61226e7	−1.82075	−0.910373	+	0.413789i	$0.864205\pi$
$602$	0	0
$603$	0	0
$604$	−1.02588e7	−1.14421
$605$	2.63036e7	2.92164
$606$	0	0
$607$	−1.04056e7	−1.14629	−0.573144	−	0.819455i	$-0.694276\pi$
$607$	−1.04056e7	−1.14629	−0.573144	+	0.819455i	$0.694276\pi$
$608$	−542243.	−0.0594887
$609$	0	0
$610$	−4.52956e6	−0.492868
$611$	2.91929e6	0.316355
$612$	0	0
$613$	−1.13921e7	−1.22449	−0.612243	−	0.790670i	$-0.709732\pi$
$613$	−1.13921e7	−1.22449	−0.612243	+	0.790670i	$0.709732\pi$
$614$	−2.99097e7	−3.20178
$615$	0	0
$616$	0	0
$617$	−6.20517e6	−0.656206	−0.328103	−	0.944642i	$-0.606409\pi$
$617$	−6.20517e6	−0.656206	−0.328103	+	0.944642i	$0.606409\pi$
$618$	0	0
$619$	9.85608e6	1.03390	0.516949	−	0.856016i	$-0.327068\pi$
$619$	9.85608e6	1.03390	0.516949	+	0.856016i	$0.327068\pi$
$620$	−5.65602e6	−0.590925
$621$	0	0
$622$	2.84003e7	2.94338
$623$	0	0
$624$	0	0
$625$	−1.21385e7	−1.24298
$626$	2.14940e7	2.19220
$627$	0	0
$628$	−1.13060e7	−1.14396
$629$	−1.29335e7	−1.30343
$630$	0	0
$631$	7.26873e6	0.726750	0.363375	−	0.931643i	$-0.381624\pi$
$631$	7.26873e6	0.726750	0.363375	+	0.931643i	$0.381624\pi$
$632$	−8.37284e6	−0.833835
$633$	0	0
$634$	−2.63591e7	−2.60440
$635$	−1.30467e7	−1.28401
$636$	0	0
$637$	0	0
$638$	−5.67087e7	−5.51566
$639$	0	0
$640$	1.88295e7	1.81714
$641$	−8.30119e6	−0.797986	−0.398993	−	0.916954i	$-0.630640\pi$
$641$	−8.30119e6	−0.797986	−0.398993	+	0.916954i	$0.630640\pi$
$642$	0	0
$643$	4.14322e6	0.395194	0.197597	−	0.980283i	$-0.436686\pi$
$643$	4.14322e6	0.395194	0.197597	+	0.980283i	$0.436686\pi$
$644$	0	0
$645$	0	0
$646$	2.65065e6	0.249903
$647$	−1.47906e7	−1.38907	−0.694535	−	0.719459i	$-0.744390\pi$
$647$	−1.47906e7	−1.38907	−0.694535	+	0.719459i	$0.744390\pi$
$648$	0	0
$649$	−1.93458e7	−1.80291
$650$	1.64268e7	1.52500
$651$	0	0
$652$	6.52587e6	0.601200
$653$	6.08931e6	0.558837	0.279419	−	0.960169i	$-0.409858\pi$
$653$	6.08931e6	0.558837	0.279419	+	0.960169i	$0.409858\pi$
$654$	0	0
$655$	8.72679e6	0.794788
$656$	1.94070e7	1.76075
$657$	0	0
$658$	0	0
$659$	−4.14447e6	−0.371754	−0.185877	−	0.982573i	$-0.559513\pi$
$659$	−4.14447e6	−0.371754	−0.185877	+	0.982573i	$0.559513\pi$
$660$	0	0
$661$	−1.80378e7	−1.60576	−0.802881	−	0.596140i	$-0.796700\pi$
$661$	−1.80378e7	−1.60576	−0.802881	+	0.596140i	$0.796700\pi$
$662$	−3.28726e7	−2.91534
$663$	0	0
$664$	6.25245e6	0.550339
$665$	0	0
$666$	0	0
$667$	−332926.	−0.0289756
$668$	3.23306e7	2.80332
$669$	0	0
$670$	1.67695e7	1.44322
$671$	4.70490e6	0.403408
$672$	0	0
$673$	5.62885e6	0.479051	0.239526	−	0.970890i	$-0.423008\pi$
$673$	5.62885e6	0.479051	0.239526	+	0.970890i	$0.423008\pi$
$674$	2.38414e6	0.202154
$675$	0	0
$676$	2.98666e7	2.51373
$677$	−1.50877e7	−1.26518	−0.632588	−	0.774488i	$-0.718007\pi$
$677$	−1.50877e7	−1.26518	−0.632588	+	0.774488i	$0.718007\pi$
$678$	0	0
$679$	0	0
$680$	3.54700e7	2.94164
$681$	0	0
$682$	8.63246e6	0.710679
$683$	−7.20414e6	−0.590922	−0.295461	−	0.955355i	$-0.595473\pi$
$683$	−7.20414e6	−0.590922	−0.295461	+	0.955355i	$0.595473\pi$
$684$	0	0
$685$	−7.00139e6	−0.570109
$686$	0	0
$687$	0	0
$688$	8.70378e6	0.701031
$689$	2.30991e7	1.85373
$690$	0	0
$691$	−9.08625e6	−0.723918	−0.361959	−	0.932194i	$-0.617892\pi$
$691$	−9.08625e6	−0.723918	−0.361959	+	0.932194i	$0.617892\pi$
$692$	1.76497e7	1.40111
$693$	0	0
$694$	1.06531e7	0.839607
$695$	−1.79030e7	−1.40593
$696$	0	0
$697$	−1.87324e7	−1.46053
$698$	1.86105e7	1.44584
$699$	0	0
$700$	0	0
$701$	1.41913e7	1.09075	0.545376	−	0.838192i	$-0.316387\pi$
$701$	1.41913e7	1.09075	0.545376	+	0.838192i	$0.316387\pi$
$702$	0	0
$703$	1.76669e6	0.134825
$704$	−1.28918e7	−0.980354
$705$	0	0
$706$	1.66635e7	1.25822
$707$	0	0
$708$	0	0
$709$	−2.21070e6	−0.165164	−0.0825819	−	0.996584i	$-0.526317\pi$
$709$	−2.21070e6	−0.165164	−0.0825819	+	0.996584i	$0.526317\pi$
$710$	−3.14613e7	−2.34224
$711$	0	0
$712$	−1.80796e7	−1.33656
$713$	50679.5	0.00373343
$714$	0	0
$715$	−4.62909e7	−3.38634
$716$	3.69645e7	2.69465
$717$	0	0
$718$	2.67695e7	1.93789
$719$	−2.99082e6	−0.215758	−0.107879	−	0.994164i	$-0.534406\pi$
$719$	−2.99082e6	−0.215758	−0.107879	+	0.994164i	$0.534406\pi$
$720$	0	0
$721$	0	0
$722$	2.44209e7	1.74349
$723$	0	0
$724$	−4.45836e7	−3.16103
$725$	1.41321e7	0.998530
$726$	0	0
$727$	−8.38066e6	−0.588088	−0.294044	−	0.955792i	$-0.595001\pi$
$727$	−8.38066e6	−0.588088	−0.294044	+	0.955792i	$0.595001\pi$
$728$	0	0
$729$	0	0
$730$	−2.78847e7	−1.93669
$731$	−8.40123e6	−0.581499
$732$	0	0
$733$	−7.70245e6	−0.529504	−0.264752	−	0.964317i	$-0.585290\pi$
$733$	−7.70245e6	−0.529504	−0.264752	+	0.964317i	$0.585290\pi$
$734$	−2.14575e7	−1.47007
$735$	0	0
$736$	122525.	0.00833739
$737$	−1.74187e7	−1.18126
$738$	0	0
$739$	−7.59899e6	−0.511852	−0.255926	−	0.966696i	$-0.582380\pi$
$739$	−7.59899e6	−0.511852	−0.255926	+	0.966696i	$0.582380\pi$
$740$	4.45522e7	2.99082
$741$	0	0
$742$	0	0
$743$	2.40124e7	1.59574	0.797871	−	0.602828i	$-0.205959\pi$
$743$	2.40124e7	1.59574	0.797871	+	0.602828i	$0.205959\pi$
$744$	0	0
$745$	3.00583e7	1.98414
$746$	7.60543e6	0.500353
$747$	0	0
$748$	−6.94316e7	−4.53736
$749$	0	0
$750$	0	0
$751$	1.69479e7	1.09652	0.548259	−	0.836308i	$-0.315291\pi$
$751$	1.69479e7	1.09652	0.548259	+	0.836308i	$0.315291\pi$
$752$	−4.68094e6	−0.301848
$753$	0	0
$754$	6.97538e7	4.46827
$755$	−1.05859e7	−0.675863
$756$	0	0
$757$	4.56402e6	0.289473	0.144736	−	0.989470i	$-0.453767\pi$
$757$	4.56402e6	0.289473	0.144736	+	0.989470i	$0.453767\pi$
$758$	−3.03949e7	−1.92145
$759$	0	0
$760$	−4.84514e6	−0.304279
$761$	4.74018e6	0.296711	0.148355	−	0.988934i	$-0.452602\pi$
$761$	4.74018e6	0.296711	0.148355	+	0.988934i	$0.452602\pi$
$762$	0	0
$763$	0	0
$764$	−5.24653e7	−3.25191
$765$	0	0
$766$	−2.51152e7	−1.54655
$767$	2.37960e7	1.46055
$768$	0	0
$769$	−1.39469e7	−0.850476	−0.425238	−	0.905082i	$-0.639810\pi$
$769$	−1.39469e7	−0.850476	−0.425238	+	0.905082i	$0.639810\pi$
$770$	0	0
$771$	0	0
$772$	5.55125e7	3.35234
$773$	−1.49561e7	−0.900264	−0.450132	−	0.892962i	$-0.648623\pi$
$773$	−1.49561e7	−0.900264	−0.450132	+	0.892962i	$0.648623\pi$
$774$	0	0
$775$	−2.15125e6	−0.128658
$776$	−6.10683e6	−0.364050
$777$	0	0
$778$	−9.86012e6	−0.584028
$779$	2.55881e6	0.151075
$780$	0	0
$781$	3.26793e7	1.91710
$782$	−598940.	−0.0350240
$783$	0	0
$784$	0	0
$785$	−1.16664e7	−0.675716
$786$	0	0
$787$	2.18273e7	1.25621	0.628105	−	0.778128i	$-0.283831\pi$
$787$	2.18273e7	1.25621	0.628105	+	0.778128i	$0.283831\pi$
$788$	3.02497e7	1.73542
$789$	0	0
$790$	−1.62818e7	−0.928183
$791$	0	0
$792$	0	0
$793$	−5.78721e6	−0.326803
$794$	−2.90768e7	−1.63680
$795$	0	0
$796$	−4.21981e7	−2.36053
$797$	8.79229e6	0.490294	0.245147	−	0.969486i	$-0.421164\pi$
$797$	8.79229e6	0.490294	0.245147	+	0.969486i	$0.421164\pi$
$798$	0	0
$799$	4.51822e6	0.250381
$800$	−5.20096e6	−0.287315
$801$	0	0
$802$	−4.23501e7	−2.32497
$803$	2.89642e7	1.58516
$804$	0	0
$805$	0	0
$806$	−1.06183e7	−0.575725
$807$	0	0
$808$	−3.81831e7	−2.05751
$809$	2.87451e6	0.154416	0.0772081	−	0.997015i	$-0.475399\pi$
$809$	2.87451e6	0.154416	0.0772081	+	0.997015i	$0.475399\pi$
$810$	0	0
$811$	−660356.	−0.0352554	−0.0176277	−	0.999845i	$-0.505611\pi$
$811$	−660356.	−0.0352554	−0.0176277	+	0.999845i	$0.505611\pi$
$812$	0	0
$813$	0	0
$814$	−6.79974e7	−3.59693
$815$	6.73389e6	0.355118
$816$	0	0
$817$	1.14759e6	0.0601496
$818$	−4.14258e7	−2.16465
$819$	0	0
$820$	6.45278e7	3.35129
$821$	3.05738e7	1.58304	0.791519	−	0.611144i	$-0.209290\pi$
$821$	3.05738e7	1.58304	0.791519	+	0.611144i	$0.209290\pi$
$822$	0	0
$823$	1.11281e7	0.572693	0.286346	−	0.958126i	$-0.407559\pi$
$823$	1.11281e7	0.572693	0.286346	+	0.958126i	$0.407559\pi$
$824$	−7.37085e6	−0.378181
$825$	0	0
$826$	0	0
$827$	2.78304e7	1.41500	0.707500	−	0.706714i	$-0.249823\pi$
$827$	2.78304e7	1.41500	0.707500	+	0.706714i	$0.249823\pi$
$828$	0	0
$829$	7.06133e6	0.356862	0.178431	−	0.983952i	$-0.442898\pi$
$829$	7.06133e6	0.356862	0.178431	+	0.983952i	$0.442898\pi$
$830$	1.21585e7	0.612609
$831$	0	0
$832$	1.58574e7	0.794190
$833$	0	0
$834$	0	0
$835$	3.33612e7	1.65587
$836$	9.48422e6	0.469338
$837$	0	0
$838$	6.00385e7	2.95338
$839$	2.27299e7	1.11479	0.557395	−	0.830247i	$-0.311801\pi$
$839$	2.27299e7	1.11479	0.557395	+	0.830247i	$0.311801\pi$
$840$	0	0
$841$	3.94986e7	1.92571
$842$	1.00848e7	0.490217
$843$	0	0
$844$	5.24837e7	2.53611
$845$	3.08186e7	1.48481
$846$	0	0
$847$	0	0
$848$	−3.70382e7	−1.76872
$849$	0	0
$850$	2.54239e7	1.20696
$851$	−399200.	−0.0188958
$852$	0	0
$853$	−7.94309e6	−0.373781	−0.186890	−	0.982381i	$-0.559841\pi$
$853$	−7.94309e6	−0.373781	−0.186890	+	0.982381i	$0.559841\pi$
$854$	0	0
$855$	0	0
$856$	−1.36709e6	−0.0637693
$857$	9.91766e6	0.461272	0.230636	−	0.973040i	$-0.425919\pi$
$857$	9.91766e6	0.461272	0.230636	+	0.973040i	$0.425919\pi$
$858$	0	0
$859$	−61754.4	−0.00285552	−0.00142776	−	0.999999i	$-0.500454\pi$
$859$	−61754.4	−0.00285552	−0.00142776	+	0.999999i	$0.500454\pi$
$860$	2.89399e7	1.33429
$861$	0	0
$862$	3.03419e6	0.139083
$863$	−3.99007e7	−1.82370	−0.911851	−	0.410522i	$-0.865347\pi$
$863$	−3.99007e7	−1.82370	−0.911851	+	0.410522i	$0.865347\pi$
$864$	0	0
$865$	1.82123e7	0.827607
$866$	−3.97484e7	−1.80104
$867$	0	0
$868$	0	0
$869$	1.69120e7	0.759708
$870$	0	0
$871$	2.14256e7	0.956949
$872$	−3.09125e6	−0.137671
$873$	0	0
$874$	81814.0	0.00362284
$875$	0	0
$876$	0	0
$877$	−1.89184e7	−0.830586	−0.415293	−	0.909688i	$-0.636321\pi$
$877$	−1.89184e7	−0.830586	−0.415293	+	0.909688i	$0.636321\pi$
$878$	−3.75133e7	−1.64229
$879$	0	0
$880$	7.42250e7	3.23105
$881$	−9.12061e6	−0.395899	−0.197949	−	0.980212i	$-0.563428\pi$
$881$	−9.12061e6	−0.395899	−0.197949	+	0.980212i	$0.563428\pi$
$882$	0	0
$883$	−4.49665e7	−1.94083	−0.970414	−	0.241449i	$-0.922377\pi$
$883$	−4.49665e7	−1.94083	−0.970414	+	0.241449i	$0.922377\pi$
$884$	8.54035e7	3.67574
$885$	0	0
$886$	−1.31193e7	−0.561472
$887$	−2.82985e7	−1.20769	−0.603845	−	0.797102i	$-0.706365\pi$
$887$	−2.82985e7	−1.20769	−0.603845	+	0.797102i	$0.706365\pi$
$888$	0	0
$889$	0	0
$890$	−3.51574e7	−1.48779
$891$	0	0
$892$	9.93107e7	4.17911
$893$	−617181.	−0.0258990
$894$	0	0
$895$	3.81429e7	1.59168
$896$	0	0
$897$	0	0
$898$	6.31873e7	2.61480
$899$	−9.13496e6	−0.376971
$900$	0	0
$901$	3.57507e7	1.46714
$902$	−9.84850e7	−4.03045
$903$	0	0
$904$	−3.65247e7	−1.48650
$905$	−4.60048e7	−1.86716
$906$	0	0
$907$	−768537.	−0.0310204	−0.0155102	−	0.999880i	$-0.504937\pi$
$907$	−768537.	−0.0310204	−0.0155102	+	0.999880i	$0.504937\pi$
$908$	6.47626e7	2.60681
$909$	0	0
$910$	0	0
$911$	−3.45594e6	−0.137965	−0.0689827	−	0.997618i	$-0.521975\pi$
$911$	−3.45594e6	−0.137965	−0.0689827	+	0.997618i	$0.521975\pi$
$912$	0	0
$913$	−1.26291e7	−0.501414
$914$	5.83994e6	0.231229
$915$	0	0
$916$	2.12908e7	0.838404
$917$	0	0
$918$	0	0
$919$	5.65327e6	0.220806	0.110403	−	0.993887i	$-0.464786\pi$
$919$	5.65327e6	0.220806	0.110403	+	0.993887i	$0.464786\pi$
$920$	1.09481e6	0.0426450
$921$	0	0
$922$	−2.93641e7	−1.13760
$923$	−4.01967e7	−1.55305
$924$	0	0
$925$	1.69453e7	0.651171
$926$	−3.77051e7	−1.44502
$927$	0	0
$928$	−2.20851e7	−0.841839
$929$	−2.42253e7	−0.920937	−0.460469	−	0.887676i	$-0.652319\pi$
$929$	−2.42253e7	−0.920937	−0.460469	+	0.887676i	$0.652319\pi$
$930$	0	0
$931$	0	0
$932$	−2.70602e7	−1.02045
$933$	0	0
$934$	−2.90855e7	−1.09096
$935$	−7.16448e7	−2.68013
$936$	0	0
$937$	−3.02477e7	−1.12549	−0.562746	−	0.826630i	$-0.690255\pi$
$937$	−3.02477e7	−1.12549	−0.562746	+	0.826630i	$0.690255\pi$
$938$	0	0
$939$	0	0
$940$	−1.55640e7	−0.574517
$941$	1.38949e7	0.511541	0.255770	−	0.966738i	$-0.417671\pi$
$941$	1.38949e7	0.511541	0.255770	+	0.966738i	$0.417671\pi$
$942$	0	0
$943$	−578187.	−0.0211733
$944$	−3.81557e7	−1.39357
$945$	0	0
$946$	−4.41693e7	−1.60470
$947$	1.78680e7	0.647441	0.323721	−	0.946153i	$-0.395066\pi$
$947$	1.78680e7	0.647441	0.323721	+	0.946153i	$0.395066\pi$
$948$	0	0
$949$	−3.56271e7	−1.28415
$950$	−3.47286e6	−0.124847
$951$	0	0
$952$	0	0
$953$	−1.37348e7	−0.489880	−0.244940	−	0.969538i	$-0.578768\pi$
$953$	−1.37348e7	−0.489880	−0.244940	+	0.969538i	$0.578768\pi$
$954$	0	0
$955$	−5.41378e7	−1.92084
$956$	5.93375e6	0.209983
$957$	0	0
$958$	−7.07250e7	−2.48977
$959$	0	0
$960$	0	0
$961$	−2.72386e7	−0.951428
$962$	8.36394e7	2.91389
$963$	0	0
$964$	−1.07157e8	−3.71389
$965$	5.72821e7	1.98016
$966$	0	0
$967$	−7.63195e6	−0.262464	−0.131232	−	0.991352i	$-0.541893\pi$
$967$	−7.63195e6	−0.262464	−0.131232	+	0.991352i	$0.541893\pi$
$968$	−1.35386e8	−4.64391
$969$	0	0
$970$	−1.18753e7	−0.405242
$971$	−2.22480e7	−0.757257	−0.378629	−	0.925549i	$-0.623604\pi$
$971$	−2.22480e7	−0.757257	−0.378629	+	0.925549i	$0.623604\pi$
$972$	0	0
$973$	0	0
$974$	−2.73223e7	−0.922827
$975$	0	0
$976$	9.27949e6	0.311817
$977$	1.82385e7	0.611297	0.305648	−	0.952144i	$-0.401127\pi$
$977$	1.82385e7	0.611297	0.305648	+	0.952144i	$0.401127\pi$
$978$	0	0
$979$	3.65184e7	1.21774
$980$	0	0
$981$	0	0
$982$	−1.02951e8	−3.40683
$983$	−3.84459e7	−1.26902	−0.634508	−	0.772917i	$-0.718797\pi$
$983$	−3.84459e7	−1.26902	−0.634508	+	0.772917i	$0.718797\pi$
$984$	0	0
$985$	3.12140e7	1.02508
$986$	1.07959e8	3.53643
$987$	0	0
$988$	−1.16660e7	−0.380214
$989$	−259309.	−0.00843000
$990$	0	0
$991$	−2.49658e7	−0.807533	−0.403767	−	0.914862i	$-0.632299\pi$
$991$	−2.49658e7	−0.807533	−0.403767	+	0.914862i	$0.632299\pi$
$992$	3.36189e6	0.108469
$993$	0	0
$994$	0	0
$995$	−4.35433e7	−1.39432
$996$	0	0
$997$	−3.22763e7	−1.02836	−0.514181	−	0.857682i	$-0.671904\pi$
$997$	−3.22763e7	−1.02836	−0.514181	+	0.857682i	$0.671904\pi$
$998$	3.63914e7	1.15657
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bb.1.1		6
3.2	odd	2		147.6.a.n.1.6	✓	6
7.6	odd	2		441.6.a.ba.1.1		6
21.2	odd	6		147.6.e.q.67.1		12
21.5	even	6		147.6.e.p.67.1		12
21.11	odd	6		147.6.e.q.79.1		12
21.17	even	6		147.6.e.p.79.1		12
21.20	even	2		147.6.a.o.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.6	✓	6	3.2	odd	2
147.6.a.o.1.6	yes	6	21.20	even	2
147.6.e.p.67.1		12	21.5	even	6
147.6.e.p.79.1		12	21.17	even	6
147.6.e.q.67.1		12	21.2	odd	6
147.6.e.q.79.1		12	21.11	odd	6
441.6.a.ba.1.1		6	7.6	odd	2
441.6.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} +O(q^{10})\)	\(q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} -704.138 q^{10} +731.396 q^{11} -899.645 q^{13} +1442.53 q^{16} -1392.39 q^{17} +190.198 q^{19} +4796.40 q^{20} -7320.46 q^{22} -42.9770 q^{23} +1824.30 q^{25} +9004.45 q^{26} +7746.59 q^{29} -1179.22 q^{31} -2850.94 q^{32} +13936.3 q^{34} +9288.68 q^{37} -1903.67 q^{38} -25474.2 q^{40} +13453.4 q^{41} +6033.68 q^{43} +49865.0 q^{44} +430.152 q^{46} -3244.94 q^{47} -18259.2 q^{50} -61335.9 q^{52} -25675.8 q^{53} +51454.6 q^{55} -77534.8 q^{58} -26450.5 q^{59} +6432.77 q^{61} +11802.7 q^{62} -17626.3 q^{64} -63291.1 q^{65} -23815.7 q^{67} -94930.2 q^{68} +44680.7 q^{71} +39601.3 q^{73} -92969.4 q^{74} +12967.3 q^{76} +23123.0 q^{79} +101484. q^{80} -134654. q^{82} -17267.2 q^{83} -97956.3 q^{85} -60390.4 q^{86} -264839. q^{88} +49929.7 q^{89} -2930.08 q^{92} +32478.3 q^{94} +13380.7 q^{95} +16865.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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