Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,7,Mod(65,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.65");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.q (of order \(6\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(33.1277880413\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} + \cdots + 166668145981081 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{15} \) |
| Twist minimal: | no (minimal twist has level 36) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} + \cdots + 166668145981081 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 701831 \nu^{11} + 10189723130 \nu^{10} + 162036709048 \nu^{9} + 1882617545610 \nu^{8} + \cdots + 56\!\cdots\!58 ) / 13\!\cdots\!84 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 173906459 \nu^{11} + 3983241235 \nu^{10} + 40768097000 \nu^{9} + 75429664038 \nu^{8} + \cdots + 54\!\cdots\!37 ) / 74\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1985294795 \nu^{11} - 270259766417 \nu^{10} - 7700364166840 \nu^{9} - 79370889773274 \nu^{8} + \cdots - 24\!\cdots\!23 ) / 74\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 692857949 \nu^{11} + 4420729114 \nu^{10} - 11671686880 \nu^{9} - 833443499286 \nu^{8} + \cdots - 49\!\cdots\!86 ) / 12\!\cdots\!56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4331054153 \nu^{11} + 30507615919 \nu^{10} - 29262024280 \nu^{9} - 4925231331678 \nu^{8} + \cdots - 28\!\cdots\!79 ) / 74\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2411248673 \nu^{11} + 76142079226 \nu^{10} + 639832643000 \nu^{9} - 1258312257990 \nu^{8} + \cdots + 45\!\cdots\!94 ) / 37\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14224700771 \nu^{11} + 420603084275 \nu^{10} + 5659948730632 \nu^{9} + \cdots + 56\!\cdots\!85 ) / 74\!\cdots\!36 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 297544043 \nu^{11} - 5016247795 \nu^{10} - 58260073112 \nu^{9} - 169580696310 \nu^{8} + \cdots + 64\!\cdots\!71 ) / 91\!\cdots\!56 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10868968051 \nu^{11} - 94631178070 \nu^{10} - 3014398889792 \nu^{9} - 53062280781354 \nu^{8} + \cdots - 20\!\cdots\!86 ) / 12\!\cdots\!56 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 67584966607 \nu^{11} + 528438325153 \nu^{10} + 16784260768472 \nu^{9} + \cdots + 13\!\cdots\!15 ) / 74\!\cdots\!36 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 112212697555 \nu^{11} + 511820641633 \nu^{10} + 18178638082952 \nu^{9} + \cdots + 18\!\cdots\!27 ) / 74\!\cdots\!36 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{10} + \beta_{9} - 2\beta_{6} - 3\beta_{5} + 2\beta_{4} + 96\beta_{2} + 20 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 7 \beta_{11} - 10 \beta_{10} - \beta_{9} + 56 \beta_{8} + 21 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots + 298 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 90 \beta_{11} + 101 \beta_{10} - 124 \beta_{9} + 135 \beta_{8} - 579 \beta_{7} + 221 \beta_{6} + \cdots + 13714 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 511 \beta_{11} - 431 \beta_{10} + 523 \beta_{9} - 1154 \beta_{8} + 8328 \beta_{7} - 4171 \beta_{6} + \cdots - 1749964 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 13463 \beta_{11} - 9987 \beta_{10} - 12309 \beta_{9} - 638 \beta_{8} - 99 \beta_{7} - 20543 \beta_{6} + \cdots + 7539211 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 913359 \beta_{11} + 123399 \beta_{10} + 131769 \beta_{9} + 1130142 \beta_{8} - 1329525 \beta_{7} + \cdots + 95410692 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 15457155 \beta_{11} + 2986714 \beta_{10} - 19420262 \beta_{9} - 30983262 \beta_{8} + \cdots - 4363597204 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 109231070 \beta_{11} - 10923445 \beta_{10} + 336342518 \beta_{9} - 165978760 \beta_{8} + \cdots + 62933633128 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 649738995 \beta_{11} - 1084758172 \beta_{10} - 2521792735 \beta_{9} + 6768876603 \beta_{8} + \cdots - 1579916286002 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 20749154878 \beta_{11} + 60103835416 \beta_{10} + 25080853114 \beta_{9} - 107573629418 \beta_{8} + \cdots + 17910413305844 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) | \(1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −26.9124 | + | 2.17372i | 0 | −186.742 | + | 107.815i | 0 | 223.147 | − | 386.502i | 0 | 719.550 | − | 117.000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −19.9686 | − | 18.1729i | 0 | 114.537 | − | 66.1282i | 0 | −65.4834 | + | 113.420i | 0 | 68.4928 | + | 725.775i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | −8.85375 | + | 25.5071i | 0 | 51.3356 | − | 29.6386i | 0 | −76.8738 | + | 133.149i | 0 | −572.222 | − | 451.667i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | 9.87085 | − | 25.1310i | 0 | −166.078 | + | 95.8851i | 0 | −315.624 | + | 546.677i | 0 | −534.133 | − | 496.128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.5 | 0 | 18.8825 | − | 19.2990i | 0 | 98.8290 | − | 57.0590i | 0 | 276.118 | − | 478.250i | 0 | −15.9009 | − | 728.827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.6 | 0 | 23.9814 | + | 12.4054i | 0 | −19.8824 | + | 11.4791i | 0 | 18.7162 | − | 32.4175i | 0 | 421.213 | + | 594.996i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −26.9124 | − | 2.17372i | 0 | −186.742 | − | 107.815i | 0 | 223.147 | + | 386.502i | 0 | 719.550 | + | 117.000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −19.9686 | + | 18.1729i | 0 | 114.537 | + | 66.1282i | 0 | −65.4834 | − | 113.420i | 0 | 68.4928 | − | 725.775i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | −8.85375 | − | 25.5071i | 0 | 51.3356 | + | 29.6386i | 0 | −76.8738 | − | 133.149i | 0 | −572.222 | + | 451.667i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | 9.87085 | + | 25.1310i | 0 | −166.078 | − | 95.8851i | 0 | −315.624 | − | 546.677i | 0 | −534.133 | + | 496.128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.5 | 0 | 18.8825 | + | 19.2990i | 0 | 98.8290 | + | 57.0590i | 0 | 276.118 | + | 478.250i | 0 | −15.9009 | + | 728.827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.6 | 0 | 23.9814 | − | 12.4054i | 0 | −19.8824 | − | 11.4791i | 0 | 18.7162 | + | 32.4175i | 0 | 421.213 | − | 594.996i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.7.q.b | 12 | |
| 3.b | odd | 2 | 1 | 432.7.q.c | 12 | ||
| 4.b | odd | 2 | 1 | 36.7.g.a | ✓ | 12 | |
| 9.c | even | 3 | 1 | 432.7.q.c | 12 | ||
| 9.d | odd | 6 | 1 | inner | 144.7.q.b | 12 | |
| 12.b | even | 2 | 1 | 108.7.g.a | 12 | ||
| 36.f | odd | 6 | 1 | 108.7.g.a | 12 | ||
| 36.f | odd | 6 | 1 | 324.7.c.b | 12 | ||
| 36.h | even | 6 | 1 | 36.7.g.a | ✓ | 12 | |
| 36.h | even | 6 | 1 | 324.7.c.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.7.g.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 36.7.g.a | ✓ | 12 | 36.h | even | 6 | 1 | |
| 108.7.g.a | 12 | 12.b | even | 2 | 1 | ||
| 108.7.g.a | 12 | 36.f | odd | 6 | 1 | ||
| 144.7.q.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 144.7.q.b | 12 | 9.d | odd | 6 | 1 | inner | |
| 324.7.c.b | 12 | 36.f | odd | 6 | 1 | ||
| 324.7.c.b | 12 | 36.h | even | 6 | 1 | ||
| 432.7.q.c | 12 | 3.b | odd | 2 | 1 | ||
| 432.7.q.c | 12 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 216 T_{5}^{11} - 35586 T_{5}^{10} - 11045808 T_{5}^{9} + 1452428739 T_{5}^{8} + \cdots + 72\!\cdots\!00 \)
acting on \(S_{7}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 15\!\cdots\!21 \)
|
| $5$ |
\( T^{12} + \cdots + 72\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 13\!\cdots\!96 \)
|
| $11$ |
\( T^{12} + \cdots + 79\!\cdots\!41 \)
|
| $13$ |
\( T^{12} + \cdots + 55\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 82\!\cdots\!56 \)
|
| $19$ |
\( (T^{6} + \cdots + 80\!\cdots\!56)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 43\!\cdots\!84 \)
|
| $29$ |
\( T^{12} + \cdots + 32\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 89\!\cdots\!36 \)
|
| $37$ |
\( (T^{6} + \cdots - 71\!\cdots\!52)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 20\!\cdots\!25 \)
|
| $43$ |
\( T^{12} + \cdots + 15\!\cdots\!01 \)
|
| $47$ |
\( T^{12} + \cdots + 12\!\cdots\!16 \)
|
| $53$ |
\( T^{12} + \cdots + 61\!\cdots\!04 \)
|
| $59$ |
\( T^{12} + \cdots + 60\!\cdots\!81 \)
|
| $61$ |
\( T^{12} + \cdots + 10\!\cdots\!84 \)
|
| $67$ |
\( T^{12} + \cdots + 13\!\cdots\!41 \)
|
| $71$ |
\( T^{12} + \cdots + 52\!\cdots\!96 \)
|
| $73$ |
\( (T^{6} + \cdots - 40\!\cdots\!16)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 20\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 25\!\cdots\!56 \)
|
| $89$ |
\( T^{12} + \cdots + 32\!\cdots\!56 \)
|
| $97$ |
\( T^{12} + \cdots + 29\!\cdots\!29 \)
|
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