Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,6,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-12,0,48,-54,0,132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.9821788097\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.125628.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 63x + 159 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.81933\) of defining polynomial
Character \(\chi\) \(=\) 162.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +66.0868 q^{5} +114.190 q^{7} -64.0000 q^{8} -264.347 q^{10} +385.625 q^{11} +1032.57 q^{13} -456.762 q^{14} +256.000 q^{16} -959.020 q^{17} -464.576 q^{19} +1057.39 q^{20} -1542.50 q^{22} -2303.41 q^{23} +1242.47 q^{25} -4130.30 q^{26} +1827.05 q^{28} -7098.27 q^{29} +7763.06 q^{31} -1024.00 q^{32} +3836.08 q^{34} +7546.48 q^{35} +9317.57 q^{37} +1858.31 q^{38} -4229.56 q^{40} +13323.3 q^{41} -2112.58 q^{43} +6169.99 q^{44} +9213.62 q^{46} -2494.80 q^{47} -3767.56 q^{49} -4969.88 q^{50} +16521.2 q^{52} +10044.2 q^{53} +25484.7 q^{55} -7308.18 q^{56} +28393.1 q^{58} -5440.66 q^{59} +34188.8 q^{61} -31052.2 q^{62} +4096.00 q^{64} +68239.6 q^{65} +53585.2 q^{67} -15344.3 q^{68} -30185.9 q^{70} +970.010 q^{71} -72400.3 q^{73} -37270.3 q^{74} -7433.22 q^{76} +44034.6 q^{77} +32197.7 q^{79} +16918.2 q^{80} -53293.1 q^{82} +36093.2 q^{83} -63378.6 q^{85} +8450.34 q^{86} -24680.0 q^{88} -42622.2 q^{89} +117910. q^{91} -36854.5 q^{92} +9979.21 q^{94} -30702.4 q^{95} -43853.9 q^{97} +15070.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} + 48 q^{4} - 54 q^{5} + 132 q^{7} - 192 q^{8} + 216 q^{10} - 315 q^{11} + 744 q^{13} - 528 q^{14} + 768 q^{16} - 1449 q^{17} + 1131 q^{19} - 864 q^{20} + 1260 q^{22} - 3168 q^{23} + 2883 q^{25}+ \cdots - 211428 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 66.0868 1.18220 0.591099 − 0.806599i \(-0.298694\pi\)
0.591099 + 0.806599i \(0.298694\pi\)
\(6\) 0 0
\(7\) 114.190 0.880814 0.440407 − 0.897798i \(-0.354834\pi\)
0.440407 + 0.897798i \(0.354834\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) −264.347 −0.835940
\(11\) 385.625 0.960911 0.480455 − 0.877019i \(-0.340471\pi\)
0.480455 + 0.877019i \(0.340471\pi\)
\(12\) 0 0
\(13\) 1032.57 1.69458 0.847292 − 0.531128i \(-0.178232\pi\)
0.847292 + 0.531128i \(0.178232\pi\)
\(14\) −456.762 −0.622830
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −959.020 −0.804832 −0.402416 − 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(18\) 0 0
\(19\) −464.576 −0.295239 −0.147619 − 0.989044i \(-0.547161\pi\)
−0.147619 + 0.989044i \(0.547161\pi\)
\(20\) 1057.39 0.591099
\(21\) 0 0
\(22\) −1542.50 −0.679466
\(23\) −2303.41 −0.907927 −0.453963 − 0.891020i \(-0.649990\pi\)
−0.453963 + 0.891020i \(0.649990\pi\)
\(24\) 0 0
\(25\) 1242.47 0.397590
\(26\) −4130.30 −1.19825
\(27\) 0 0
\(28\) 1827.05 0.440407
\(29\) −7098.27 −1.56732 −0.783659 − 0.621191i \(-0.786649\pi\)
−0.783659 + 0.621191i \(0.786649\pi\)
\(30\) 0 0
\(31\) 7763.06 1.45087 0.725435 − 0.688291i \(-0.241639\pi\)
0.725435 + 0.688291i \(0.241639\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) 3836.08 0.569102
\(35\) 7546.48 1.04130
\(36\) 0 0
\(37\) 9317.57 1.11892 0.559459 − 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(38\) 1858.31 0.208765
\(39\) 0 0
\(40\) −4229.56 −0.417970
\(41\) 13323.3 1.23780 0.618901 − 0.785469i \(-0.287578\pi\)
0.618901 + 0.785469i \(0.287578\pi\)
\(42\) 0 0
\(43\) −2112.58 −0.174238 −0.0871190 − 0.996198i \(-0.527766\pi\)
−0.0871190 + 0.996198i \(0.527766\pi\)
\(44\) 6169.99 0.480455
\(45\) 0 0
\(46\) 9213.62 0.642001
\(47\) −2494.80 −0.164737 −0.0823686 − 0.996602i \(-0.526248\pi\)
−0.0823686 + 0.996602i \(0.526248\pi\)
\(48\) 0 0
\(49\) −3767.56 −0.224166
\(50\) −4969.88 −0.281139
\(51\) 0 0
\(52\) 16521.2 0.847292
\(53\) 10044.2 0.491163 0.245582 − 0.969376i \(-0.421021\pi\)
0.245582 + 0.969376i \(0.421021\pi\)
\(54\) 0 0
\(55\) 25484.7 1.13599
\(56\) −7308.18 −0.311415
\(57\) 0 0
\(58\) 28393.1 1.10826
\(59\) −5440.66 −0.203480 −0.101740 − 0.994811i \(-0.532441\pi\)
−0.101740 + 0.994811i \(0.532441\pi\)
\(60\) 0 0
\(61\) 34188.8 1.17641 0.588206 − 0.808711i \(-0.299834\pi\)
0.588206 + 0.808711i \(0.299834\pi\)
\(62\) −31052.2 −1.02592
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 68239.6 2.00333
\(66\) 0 0
\(67\) 53585.2 1.45834 0.729169 − 0.684334i \(-0.239907\pi\)
0.729169 + 0.684334i \(0.239907\pi\)
\(68\) −15344.3 −0.402416
\(69\) 0 0
\(70\) −30185.9 −0.736308
\(71\) 970.010 0.0228365 0.0114183 − 0.999935i \(-0.496365\pi\)
0.0114183 + 0.999935i \(0.496365\pi\)
\(72\) 0 0
\(73\) −72400.3 −1.59013 −0.795066 − 0.606523i \(-0.792564\pi\)
−0.795066 + 0.606523i \(0.792564\pi\)
\(74\) −37270.3 −0.791195
\(75\) 0 0
\(76\) −7433.22 −0.147619
\(77\) 44034.6 0.846384
\(78\) 0 0
\(79\) 32197.7 0.580440 0.290220 − 0.956960i \(-0.406271\pi\)
0.290220 + 0.956960i \(0.406271\pi\)
\(80\) 16918.2 0.295549
\(81\) 0 0
\(82\) −53293.1 −0.875258
\(83\) 36093.2 0.575082 0.287541 − 0.957768i \(-0.407162\pi\)
0.287541 + 0.957768i \(0.407162\pi\)
\(84\) 0 0
\(85\) −63378.6 −0.951470
\(86\) 8450.34 0.123205
\(87\) 0 0
\(88\) −24680.0 −0.339733
\(89\) −42622.2 −0.570375 −0.285188 − 0.958472i \(-0.592056\pi\)
−0.285188 + 0.958472i \(0.592056\pi\)
\(90\) 0 0
\(91\) 117910. 1.49261
\(92\) −36854.5 −0.453963
\(93\) 0 0
\(94\) 9979.21 0.116487
\(95\) −30702.4 −0.349030
\(96\) 0 0
\(97\) −43853.9 −0.473237 −0.236618 − 0.971603i \(-0.576039\pi\)
−0.236618 + 0.971603i \(0.576039\pi\)
\(98\) 15070.2 0.158509
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 162.6.a.i.1.3 3
3.2 odd 2 162.6.a.j.1.1 3
9.2 odd 6 18.6.c.b.13.3 yes 6
9.4 even 3 54.6.c.b.19.1 6
9.5 odd 6 18.6.c.b.7.3 ✓ 6
9.7 even 3 54.6.c.b.37.1 6
36.7 odd 6 432.6.i.b.145.1 6
36.11 even 6 144.6.i.b.49.1 6
36.23 even 6 144.6.i.b.97.1 6
36.31 odd 6 432.6.i.b.289.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.3 ✓ 6 9.5 odd 6
18.6.c.b.13.3 yes 6 9.2 odd 6
54.6.c.b.19.1 6 9.4 even 3
54.6.c.b.37.1 6 9.7 even 3
144.6.i.b.49.1 6 36.11 even 6
144.6.i.b.97.1 6 36.23 even 6
162.6.a.i.1.3 3 1.1 even 1 trivial
162.6.a.j.1.1 3 3.2 odd 2
432.6.i.b.145.1 6 36.7 odd 6
432.6.i.b.289.1 6 36.31 odd 6