Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,6,Mod(49,144)] 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("144.49"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.0952700531\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.2
Root \(0.500000 - 8.40123i\) of defining polynomial
Character \(\chi\) \(=\) 144.49
Dual form 144.6.i.b.97.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.43381 - 15.3973i) q^{3} +(-20.8014 + 36.0292i) q^{5} +(-101.661 - 176.082i) q^{7} +(-231.153 - 74.9483i) q^{9} +(235.168 + 407.323i) q^{11} +(-241.506 + 418.300i) q^{13} +(504.125 + 407.974i) q^{15} +1259.86 q^{17} -1978.94 q^{19} +(-2958.60 + 1136.75i) q^{21} +(239.119 - 414.166i) q^{23} +(697.100 + 1207.41i) q^{25} +(-1716.58 + 3376.72i) q^{27} +(580.249 + 1005.02i) q^{29} +(-1186.50 + 2055.07i) q^{31} +(6844.02 - 2629.60i) q^{33} +8458.76 q^{35} +8185.10 q^{37} +(5852.91 + 4736.60i) q^{39} +(-8758.87 + 15170.8i) q^{41} +(11435.0 + 19806.1i) q^{43} +(7508.64 - 6769.22i) q^{45} +(-8685.43 - 15043.6i) q^{47} +(-12266.3 + 21245.9i) q^{49} +(3066.26 - 19398.4i) q^{51} -5390.58 q^{53} -19567.3 q^{55} +(-4816.38 + 30470.4i) q^{57} +(-22281.8 + 38593.2i) q^{59} +(-2084.17 - 3609.88i) q^{61} +(10302.2 + 48321.1i) q^{63} +(-10047.3 - 17402.5i) q^{65} +(1228.46 - 2127.76i) q^{67} +(-5795.07 - 4689.79i) q^{69} -2184.37 q^{71} +3037.34 q^{73} +(20287.5 - 7794.83i) q^{75} +(47814.7 - 82817.5i) q^{77} +(-25266.2 - 43762.3i) q^{79} +(47814.5 + 34649.1i) q^{81} +(-25913.9 - 44884.2i) q^{83} +(-26206.9 + 45391.7i) q^{85} +(16886.8 - 6488.23i) q^{87} -20154.7 q^{89} +98206.5 q^{91} +(28754.8 + 23270.5i) q^{93} +(41164.9 - 71299.6i) q^{95} +(-40214.4 - 69653.3i) q^{97} +(-23831.7 - 111779. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9} + 315 q^{11} - 744 q^{13} - 2286 q^{15} + 2898 q^{17} - 2262 q^{19} - 11076 q^{21} + 3168 q^{23} - 2883 q^{25} - 18144 q^{27} - 5148 q^{29} + 8610 q^{31}+ \cdots - 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.43381 − 15.3973i 0.156129 − 0.987737i
\(4\) 0 0
\(5\) −20.8014 + 36.0292i −0.372108 + 0.644509i −0.989890 − 0.141840i \(-0.954698\pi\)
0.617782 + 0.786349i \(0.288031\pi\)
\(6\) 0 0
\(7\) −101.661 − 176.082i −0.784166 − 1.35822i −0.929496 − 0.368832i \(-0.879758\pi\)
0.145330 − 0.989383i \(-0.453576\pi\)
\(8\) 0 0
\(9\) −231.153 − 74.9483i −0.951247 − 0.308429i
\(10\) 0 0
\(11\) 235.168 + 407.323i 0.585998 + 1.01498i 0.994750 + 0.102333i \(0.0326308\pi\)
−0.408752 + 0.912646i \(0.634036\pi\)
\(12\) 0 0
\(13\) −241.506 + 418.300i −0.396341 + 0.686482i −0.993271 − 0.115811i \(-0.963053\pi\)
0.596931 + 0.802293i \(0.296387\pi\)
\(14\) 0 0
\(15\) 504.125 + 407.974i 0.578508 + 0.468171i
\(16\) 0 0
\(17\) 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 \(-0.716458\pi\)
−0.628810 + 0.777559i \(0.716458\pi\)
\(20\) 0 0
\(21\) −2958.60 + 1136.75i −1.46399 + 0.562492i
\(22\) 0 0
\(23\) 239.119 − 414.166i 0.0942529 − 0.163251i −0.815044 − 0.579400i \(-0.803287\pi\)
0.909297 + 0.416149i \(0.136620\pi\)
\(24\) 0 0
\(25\) 697.100 + 1207.41i 0.223072 + 0.386372i
\(26\) 0 0
\(27\) −1716.58 + 3376.72i −0.453164 + 0.891427i
\(28\) 0 0
\(29\) 580.249 + 1005.02i 0.128121 + 0.221912i 0.922949 − 0.384923i \(-0.125772\pi\)
−0.794828 + 0.606835i \(0.792439\pi\)
\(30\) 0 0
\(31\) −1186.50 + 2055.07i −0.221749 + 0.384081i −0.955339 − 0.295511i \(-0.904510\pi\)
0.733590 + 0.679592i \(0.237843\pi\)
\(32\) 0 0
\(33\) 6844.02 − 2629.60i 1.09402 − 0.420344i
\(34\) 0 0
\(35\) 8458.76 1.16718
\(36\) 0 0
\(37\) 8185.10 0.982923 0.491462 − 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(38\) 0 0
\(39\) 5852.91 + 4736.60i 0.616183 + 0.498660i
\(40\) 0 0
\(41\) −8758.87 + 15170.8i −0.813745 + 1.40945i 0.0964809 + 0.995335i \(0.469241\pi\)
−0.910226 + 0.414113i \(0.864092\pi\)
\(42\) 0 0
\(43\) 11435.0 + 19806.1i 0.943119 + 1.63353i 0.759475 + 0.650537i \(0.225456\pi\)
0.183644 + 0.982993i \(0.441211\pi\)
\(44\) 0 0
\(45\) 7508.64 − 6769.22i 0.552752 − 0.498319i
\(46\) 0 0
\(47\) −8685.43 − 15043.6i −0.573518 − 0.993362i −0.996201 − 0.0870843i \(-0.972245\pi\)
0.422683 − 0.906277i \(-0.361088\pi\)
\(48\) 0 0
\(49\) −12266.3 + 21245.9i −0.729833 + 1.26411i
\(50\) 0 0
\(51\) 3066.26 − 19398.4i 0.165076 − 1.04434i
\(52\) 0 0
\(53\) −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
\(56\) 0 0
\(57\) −4816.38 + 30470.4i −0.196351 + 1.24220i
\(58\) 0 0
\(59\) −22281.8 + 38593.2i −0.833335 + 1.44338i 0.0620431 + 0.998073i \(0.480238\pi\)
−0.895379 + 0.445306i \(0.853095\pi\)
\(60\) 0 0
\(61\) −2084.17 − 3609.88i −0.0717147 − 0.124213i 0.827938 − 0.560819i \(-0.189514\pi\)
−0.899653 + 0.436606i \(0.856180\pi\)
\(62\) 0 0
\(63\) 10302.2 + 48321.1i 0.327023 + 1.53386i
\(64\) 0 0
\(65\) −10047.3 − 17402.5i −0.294963 − 0.510891i
\(66\) 0 0
\(67\) 1228.46 − 2127.76i 0.0334330 − 0.0579076i −0.848825 − 0.528674i \(-0.822689\pi\)
0.882258 + 0.470767i \(0.156023\pi\)
\(68\) 0 0
\(69\) −5795.07 − 4689.79i −0.146533 − 0.118585i
\(70\) 0 0
\(71\) −2184.37 −0.0514256 −0.0257128 − 0.999669i \(-0.508186\pi\)
−0.0257128 + 0.999669i \(0.508186\pi\)
\(72\) 0 0
\(73\) 3037.34 0.0667093 0.0333546 − 0.999444i \(-0.489381\pi\)
0.0333546 + 0.999444i \(0.489381\pi\)
\(74\) 0 0
\(75\) 20287.5 − 7794.83i 0.416462 − 0.160012i
\(76\) 0 0
\(77\) 47814.7 − 82817.5i 0.919040 − 1.59182i
\(78\) 0 0
\(79\) −25266.2 − 43762.3i −0.455482 − 0.788918i 0.543234 − 0.839582i \(-0.317200\pi\)
−0.998716 + 0.0506633i \(0.983866\pi\)
\(80\) 0 0
\(81\) 47814.5 + 34649.1i 0.809743 + 0.586785i
\(82\) 0 0
\(83\) −25913.9 − 44884.2i −0.412893 − 0.715152i 0.582311 − 0.812966i \(-0.302148\pi\)
−0.995205 + 0.0978135i \(0.968815\pi\)
\(84\) 0 0
\(85\) −26206.9 + 45391.7i −0.393431 + 0.681442i
\(86\) 0 0
\(87\) 16886.8 − 6488.23i 0.239194 − 0.0919027i
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) 28754.8 + 23270.5i 0.344749 + 0.278996i
\(94\) 0 0
\(95\) 41164.9 − 71299.6i 0.467970 − 0.810547i
\(96\) 0 0
\(97\) −40214.4 − 69653.3i −0.433962 − 0.751644i 0.563248 − 0.826288i \(-0.309552\pi\)
−0.997210 + 0.0746433i \(0.976218\pi\)
\(98\) 0 0
\(99\) −23831.7 − 111779.i −0.244380 − 1.14623i
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 144.6.i.b.49.2 6
3.2 odd 2 432.6.i.b.145.2 6
4.3 odd 2 18.6.c.b.13.2 yes 6
9.2 odd 6 432.6.i.b.289.2 6
9.7 even 3 inner 144.6.i.b.97.2 6
12.11 even 2 54.6.c.b.37.2 6
36.7 odd 6 18.6.c.b.7.2 ✓ 6
36.11 even 6 54.6.c.b.19.2 6
36.23 even 6 162.6.a.i.1.2 3
36.31 odd 6 162.6.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.2 ✓ 6 36.7 odd 6
18.6.c.b.13.2 yes 6 4.3 odd 2
54.6.c.b.19.2 6 36.11 even 6
54.6.c.b.37.2 6 12.11 even 2
144.6.i.b.49.2 6 1.1 even 1 trivial
144.6.i.b.97.2 6 9.7 even 3 inner
162.6.a.i.1.2 3 36.23 even 6
162.6.a.j.1.2 3 36.31 odd 6
432.6.i.b.145.2 6 3.2 odd 2
432.6.i.b.289.2 6 9.2 odd 6