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 1.2
Root \(2.72791\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +41.6029 q^{5} -203.321 q^{7} +64.0000 q^{8} +166.412 q^{10} +470.336 q^{11} +483.011 q^{13} -813.286 q^{14} +256.000 q^{16} +1259.86 q^{17} +1978.94 q^{19} +665.646 q^{20} +1881.34 q^{22} +478.238 q^{23} -1394.20 q^{25} +1932.04 q^{26} -3253.14 q^{28} -1160.50 q^{29} -2372.99 q^{31} +1024.00 q^{32} +5039.44 q^{34} -8458.76 q^{35} +8185.10 q^{37} +7915.77 q^{38} +2662.58 q^{40} +17517.7 q^{41} +22870.1 q^{43} +7525.37 q^{44} +1912.95 q^{46} -17370.9 q^{47} +24532.6 q^{49} -5576.80 q^{50} +7728.18 q^{52} -5390.58 q^{53} +19567.3 q^{55} -13012.6 q^{56} -4642.00 q^{58} -44563.6 q^{59} +4168.33 q^{61} -9491.97 q^{62} +4096.00 q^{64} +20094.7 q^{65} +2456.92 q^{67} +20157.8 q^{68} -33835.0 q^{70} +2184.37 q^{71} +3037.34 q^{73} +32740.4 q^{74} +31663.1 q^{76} -95629.4 q^{77} -50532.3 q^{79} +10650.3 q^{80} +70070.9 q^{82} -51827.9 q^{83} +52413.8 q^{85} +91480.3 q^{86} +30101.5 q^{88} -20154.7 q^{89} -98206.5 q^{91} +7651.81 q^{92} -69483.5 q^{94} +82329.7 q^{95} +80428.7 q^{97} +98130.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8} + 216 q^{10} + 315 q^{11} + 744 q^{13} + 528 q^{14} + 768 q^{16} + 1449 q^{17} + 1131 q^{19} + 864 q^{20} + 1260 q^{22} + 3168 q^{23} + 2883 q^{25}+ \cdots + 211428 q^{98}+O(q^{100}) \) 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
\(5\) 41.6029 0.744215 0.372108 − 0.928190i \(-0.378635\pi\)
0.372108 + 0.928190i \(0.378635\pi\)
\(6\) 0 0
\(7\) −203.321 −1.56833 −0.784166 − 0.620551i \(-0.786909\pi\)
−0.784166 + 0.620551i \(0.786909\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 166.412 0.526239
\(11\) 470.336 1.17200 0.585998 − 0.810312i \(-0.300703\pi\)
0.585998 + 0.810312i \(0.300703\pi\)
\(12\) 0 0
\(13\) 483.011 0.792682 0.396341 − 0.918103i \(-0.370280\pi\)
0.396341 + 0.918103i \(0.370280\pi\)
\(14\) −813.286 −1.10898
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1259.86 1.05730 0.528652 − 0.848839i \(-0.322698\pi\)
0.528652 + 0.848839i \(0.322698\pi\)
\(18\) 0 0
\(19\) 1978.94 1.25762 0.628810 − 0.777559i \(-0.283542\pi\)
0.628810 + 0.777559i \(0.283542\pi\)
\(20\) 665.646 0.372108
\(21\) 0 0
\(22\) 1881.34 0.828727
\(23\) 478.238 0.188506 0.0942529 − 0.995548i \(-0.469954\pi\)
0.0942529 + 0.995548i \(0.469954\pi\)
\(24\) 0 0
\(25\) −1394.20 −0.446144
\(26\) 1932.04 0.560511
\(27\) 0 0
\(28\) −3253.14 −0.784166
\(29\) −1160.50 −0.256242 −0.128121 − 0.991759i \(-0.540895\pi\)
−0.128121 + 0.991759i \(0.540895\pi\)
\(30\) 0 0
\(31\) −2372.99 −0.443498 −0.221749 − 0.975104i \(-0.571177\pi\)
−0.221749 + 0.975104i \(0.571177\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 5039.44 0.747627
\(35\) −8458.76 −1.16718
\(36\) 0 0
\(37\) 8185.10 0.982923 0.491462 − 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(38\) 7915.77 0.889271
\(39\) 0 0
\(40\) 2662.58 0.263120
\(41\) 17517.7 1.62749 0.813745 − 0.581222i \(-0.197425\pi\)
0.813745 + 0.581222i \(0.197425\pi\)
\(42\) 0 0
\(43\) 22870.1 1.88624 0.943119 − 0.332456i \(-0.107877\pi\)
0.943119 + 0.332456i \(0.107877\pi\)
\(44\) 7525.37 0.585998
\(45\) 0 0
\(46\) 1912.95 0.133294
\(47\) −17370.9 −1.14704 −0.573518 − 0.819193i \(-0.694422\pi\)
−0.573518 + 0.819193i \(0.694422\pi\)
\(48\) 0 0
\(49\) 24532.6 1.45967
\(50\) −5576.80 −0.315471
\(51\) 0 0
\(52\) 7728.18 0.396341
\(53\) −5390.58 −0.263600 −0.131800 − 0.991276i \(-0.542076\pi\)
−0.131800 + 0.991276i \(0.542076\pi\)
\(54\) 0 0
\(55\) 19567.3 0.872218
\(56\) −13012.6 −0.554489
\(57\) 0 0
\(58\) −4642.00 −0.181190
\(59\) −44563.6 −1.66667 −0.833335 − 0.552768i \(-0.813572\pi\)
−0.833335 + 0.552768i \(0.813572\pi\)
\(60\) 0 0
\(61\) 4168.33 0.143429 0.0717147 − 0.997425i \(-0.477153\pi\)
0.0717147 + 0.997425i \(0.477153\pi\)
\(62\) −9491.97 −0.313601
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 20094.7 0.589926
\(66\) 0 0
\(67\) 2456.92 0.0668659 0.0334330 − 0.999441i \(-0.489356\pi\)
0.0334330 + 0.999441i \(0.489356\pi\)
\(68\) 20157.8 0.528652
\(69\) 0 0
\(70\) −33835.0 −0.825318
\(71\) 2184.37 0.0514256 0.0257128 − 0.999669i \(-0.491814\pi\)
0.0257128 + 0.999669i \(0.491814\pi\)
\(72\) 0 0
\(73\) 3037.34 0.0667093 0.0333546 − 0.999444i \(-0.489381\pi\)
0.0333546 + 0.999444i \(0.489381\pi\)
\(74\) 32740.4 0.695032
\(75\) 0 0
\(76\) 31663.1 0.628810
\(77\) −95629.4 −1.83808
\(78\) 0 0
\(79\) −50532.3 −0.910964 −0.455482 − 0.890245i \(-0.650533\pi\)
−0.455482 + 0.890245i \(0.650533\pi\)
\(80\) 10650.3 0.186054
\(81\) 0 0
\(82\) 70070.9 1.15081
\(83\) −51827.9 −0.825787 −0.412893 − 0.910779i \(-0.635482\pi\)
−0.412893 + 0.910779i \(0.635482\pi\)
\(84\) 0 0
\(85\) 52413.8 0.786862
\(86\) 91480.3 1.33377
\(87\) 0 0
\(88\) 30101.5 0.414363
\(89\) −20154.7 −0.269713 −0.134857 − 0.990865i \(-0.543057\pi\)
−0.134857 + 0.990865i \(0.543057\pi\)
\(90\) 0 0
\(91\) −98206.5 −1.24319
\(92\) 7651.81 0.0942529
\(93\) 0 0
\(94\) −69483.5 −0.811077
\(95\) 82329.7 0.935939
\(96\) 0 0
\(97\) 80428.7 0.867924 0.433962 − 0.900931i \(-0.357115\pi\)
0.433962 + 0.900931i \(0.357115\pi\)
\(98\) 98130.4 1.03214
\(99\) 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.j.1.2 3
3.2 odd 2 162.6.a.i.1.2 3
9.2 odd 6 54.6.c.b.37.2 6
9.4 even 3 18.6.c.b.7.2 ✓ 6
9.5 odd 6 54.6.c.b.19.2 6
9.7 even 3 18.6.c.b.13.2 yes 6
36.7 odd 6 144.6.i.b.49.2 6
36.11 even 6 432.6.i.b.145.2 6
36.23 even 6 432.6.i.b.289.2 6
36.31 odd 6 144.6.i.b.97.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.2 ✓ 6 9.4 even 3
18.6.c.b.13.2 yes 6 9.7 even 3
54.6.c.b.19.2 6 9.5 odd 6
54.6.c.b.37.2 6 9.2 odd 6
144.6.i.b.49.2 6 36.7 odd 6
144.6.i.b.97.2 6 36.31 odd 6
162.6.a.i.1.2 3 3.2 odd 2
162.6.a.j.1.2 3 1.1 even 1 trivial
432.6.i.b.145.2 6 36.11 even 6
432.6.i.b.289.2 6 36.23 even 6