Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,19,Mod(7,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.7");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.d (of order \(2\), degree \(1\), 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: | \(24.6463365252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 5 x^{17} - 332091 x^{16} - 8722796 x^{15} + 44065046710 x^{14} + 2562387593326 x^{13} + \cdots + 51\!\cdots\!09 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{139}\cdot 3^{67}\cdot 13^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 5 x^{17} - 332091 x^{16} - 8722796 x^{15} + 44065046710 x^{14} + 2562387593326 x^{13} + \cdots + 51\!\cdots\!09 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 12\!\cdots\!05 \nu^{17} + \cdots + 13\!\cdots\!31 ) / 12\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 11\!\cdots\!41 \nu^{17} + \cdots - 94\!\cdots\!19 ) / 86\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 22\!\cdots\!15 \nu^{17} + \cdots + 18\!\cdots\!81 ) / 12\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 32\!\cdots\!24 \nu^{17} + \cdots + 29\!\cdots\!35 ) / 65\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\!\cdots\!46 \nu^{17} + \cdots + 55\!\cdots\!65 ) / 19\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 67\!\cdots\!85 \nu^{17} + \cdots - 51\!\cdots\!81 ) / 28\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13\!\cdots\!37 \nu^{17} + \cdots + 40\!\cdots\!38 ) / 97\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13\!\cdots\!09 \nu^{17} + \cdots + 49\!\cdots\!00 ) / 97\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 72\!\cdots\!90 \nu^{17} + \cdots + 80\!\cdots\!03 ) / 19\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 37\!\cdots\!09 \nu^{17} + \cdots + 58\!\cdots\!37 ) / 48\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 24\!\cdots\!87 \nu^{17} + \cdots - 21\!\cdots\!13 ) / 32\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10\!\cdots\!22 \nu^{17} + \cdots - 84\!\cdots\!59 ) / 92\!\cdots\!20 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 32\!\cdots\!16 \nu^{17} + \cdots - 24\!\cdots\!67 ) / 19\!\cdots\!20 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 24\!\cdots\!27 \nu^{17} + \cdots + 16\!\cdots\!77 ) / 97\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 45\!\cdots\!85 \nu^{17} + \cdots + 31\!\cdots\!66 ) / 97\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 24\!\cdots\!81 \nu^{17} + \cdots - 21\!\cdots\!78 ) / 48\!\cdots\!80 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 12\!\cdots\!09 \nu^{17} + \cdots + 10\!\cdots\!18 ) / 13\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( - 32 \beta_{17} + 17 \beta_{16} - 49 \beta_{15} + 64 \beta_{14} - 71 \beta_{13} + \cdots + 478565313 ) / 1289945088 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 29600 \beta_{17} - 41137 \beta_{16} + 18449 \beta_{15} + 12544 \beta_{14} + \cdots + 47605348665993 ) / 1289945088 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3975712 \beta_{17} - 1037534 \beta_{16} - 1597250 \beta_{15} + 6289088 \beta_{14} + \cdots + 22\!\cdots\!61 ) / 1289945088 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 769389408 \beta_{17} - 1381848273 \beta_{16} + 537387633 \beta_{15} + 295402752 \beta_{14} + \cdots + 10\!\cdots\!19 ) / 429981696 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 437523926656 \beta_{17} - 316178763047 \beta_{16} - 51322553177 \beta_{15} + \cdots + 23\!\cdots\!34 ) / 1289945088 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 48342125681296 \beta_{17} - 96510437626505 \beta_{16} + 32657532452089 \beta_{15} + \cdots + 61\!\cdots\!61 ) / 322486272 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 45\!\cdots\!12 \beta_{17} + \cdots + 24\!\cdots\!95 ) / 1289945088 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 56\!\cdots\!56 \beta_{17} + \cdots + 67\!\cdots\!25 ) / 429981696 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 44\!\cdots\!40 \beta_{17} + \cdots + 25\!\cdots\!41 ) / 1289945088 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 15\!\cdots\!84 \beta_{17} + \cdots + 17\!\cdots\!15 ) / 1289945088 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 43\!\cdots\!12 \beta_{17} + \cdots + 25\!\cdots\!08 ) / 1289945088 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 67\!\cdots\!04 \beta_{17} + \cdots + 70\!\cdots\!07 ) / 5971968 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 41\!\cdots\!16 \beta_{17} + \cdots + 25\!\cdots\!29 ) / 1289945088 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 13\!\cdots\!84 \beta_{17} + \cdots + 13\!\cdots\!85 ) / 1289945088 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 39\!\cdots\!96 \beta_{17} + \cdots + 25\!\cdots\!13 ) / 1289945088 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 42\!\cdots\!48 \beta_{17} + \cdots + 41\!\cdots\!27 ) / 429981696 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 38\!\cdots\!92 \beta_{17} + \cdots + 25\!\cdots\!90 ) / 1289945088 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−450.888 | − | 242.578i | − | 11364.0i | 144456. | + | 218751.i | −780888. | −2.75665e6 | + | 5.12388e6i | 2.45477e7i | −1.20691e7 | − | 1.33674e8i | −1.29140e8 | 3.52093e8 | + | 1.89426e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −450.888 | + | 242.578i | 11364.0i | 144456. | − | 218751.i | −780888. | −2.75665e6 | − | 5.12388e6i | − | 2.45477e7i | −1.20691e7 | + | 1.33674e8i | −1.29140e8 | 3.52093e8 | − | 1.89426e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −429.627 | − | 278.504i | 11364.0i | 107015. | + | 239306.i | −3.52810e6 | 3.16492e6 | − | 4.88228e6i | − | 3.84156e7i | 2.06713e7 | − | 1.32616e8i | −1.29140e8 | 1.51577e9 | + | 9.82591e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −429.627 | + | 278.504i | − | 11364.0i | 107015. | − | 239306.i | −3.52810e6 | 3.16492e6 | + | 4.88228e6i | 3.84156e7i | 2.06713e7 | + | 1.32616e8i | −1.29140e8 | 1.51577e9 | − | 9.82591e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −366.100 | − | 357.931i | 11364.0i | 5914.09 | + | 262077.i | 3.16588e6 | 4.06753e6 | − | 4.16035e6i | − | 2.73339e7i | 9.16406e7 | − | 9.80633e7i | −1.29140e8 | −1.15903e9 | − | 1.13317e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | −366.100 | + | 357.931i | − | 11364.0i | 5914.09 | − | 262077.i | 3.16588e6 | 4.06753e6 | + | 4.16035e6i | 2.73339e7i | 9.16406e7 | + | 9.80633e7i | −1.29140e8 | −1.15903e9 | + | 1.13317e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | −179.406 | − | 479.539i | − | 11364.0i | −197771. | + | 172064.i | 1.99524e6 | −5.44947e6 | + | 2.03877e6i | − | 1.75473e7i | 1.17993e8 | + | 6.39695e7i | −1.29140e8 | −3.57959e8 | − | 9.56797e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.8 | −179.406 | + | 479.539i | 11364.0i | −197771. | − | 172064.i | 1.99524e6 | −5.44947e6 | − | 2.03877e6i | 1.75473e7i | 1.17993e8 | − | 6.39695e7i | −1.29140e8 | −3.57959e8 | + | 9.56797e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.9 | 20.2086 | − | 511.601i | 11364.0i | −261327. | − | 20677.5i | −429789. | 5.81383e6 | + | 229650.i | 2.05285e7i | −1.58597e7 | + | 1.33277e8i | −1.29140e8 | −8.68544e6 | + | 2.19881e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.10 | 20.2086 | + | 511.601i | − | 11364.0i | −261327. | + | 20677.5i | −429789. | 5.81383e6 | − | 229650.i | − | 2.05285e7i | −1.58597e7 | − | 1.33277e8i | −1.29140e8 | −8.68544e6 | − | 2.19881e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.11 | 148.471 | − | 490.000i | − | 11364.0i | −218057. | − | 145502.i | −3.24964e6 | −5.56836e6 | − | 1.68722e6i | − | 7.85072e7i | −1.03671e8 | + | 8.52451e7i | −1.29140e8 | −4.82477e8 | + | 1.59232e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.12 | 148.471 | + | 490.000i | 11364.0i | −218057. | + | 145502.i | −3.24964e6 | −5.56836e6 | + | 1.68722e6i | 7.85072e7i | −1.03671e8 | − | 8.52451e7i | −1.29140e8 | −4.82477e8 | − | 1.59232e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.13 | 244.545 | − | 449.824i | − | 11364.0i | −142539. | − | 220005.i | 964196. | −5.11179e6 | − | 2.77901e6i | 7.06704e7i | −1.33821e8 | + | 1.03163e7i | −1.29140e8 | 2.35790e8 | − | 4.33718e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.14 | 244.545 | + | 449.824i | 11364.0i | −142539. | + | 220005.i | 964196. | −5.11179e6 | + | 2.77901e6i | − | 7.06704e7i | −1.33821e8 | − | 1.03163e7i | −1.29140e8 | 2.35790e8 | + | 4.33718e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.15 | 421.101 | − | 291.236i | 11364.0i | 92507.4 | − | 245279.i | −1.11240e6 | 3.30960e6 | + | 4.78538e6i | − | 6.89054e6i | −3.24791e7 | − | 1.30229e8i | −1.29140e8 | −4.68431e8 | + | 3.23970e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.16 | 421.101 | + | 291.236i | − | 11364.0i | 92507.4 | + | 245279.i | −1.11240e6 | 3.30960e6 | − | 4.78538e6i | 6.89054e6i | −3.24791e7 | + | 1.30229e8i | −1.29140e8 | −4.68431e8 | − | 3.23970e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.17 | 506.695 | − | 73.5121i | − | 11364.0i | 251336. | − | 74496.4i | 2.11462e6 | −835390. | − | 5.75808e6i | − | 4.00973e7i | 1.21874e8 | − | 5.62232e7i | −1.29140e8 | 1.07147e9 | − | 1.55450e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.18 | 506.695 | + | 73.5121i | 11364.0i | 251336. | + | 74496.4i | 2.11462e6 | −835390. | + | 5.75808e6i | 4.00973e7i | 1.21874e8 | + | 5.62232e7i | −1.29140e8 | 1.07147e9 | + | 1.55450e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.19.d.a | ✓ | 18 |
| 3.b | odd | 2 | 1 | 36.19.d.e | 18 | ||
| 4.b | odd | 2 | 1 | inner | 12.19.d.a | ✓ | 18 |
| 12.b | even | 2 | 1 | 36.19.d.e | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.19.d.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 12.19.d.a | ✓ | 18 | 4.b | odd | 2 | 1 | inner |
| 36.19.d.e | 18 | 3.b | odd | 2 | 1 | ||
| 36.19.d.e | 18 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots + 58\!\cdots\!04 \)
|
| $3$ |
\( (T^{2} + 129140163)^{9} \)
|
| $5$ |
\( (T^{9} + \cdots + 55\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{18} + \cdots + 20\!\cdots\!28 \)
|
| $11$ |
\( T^{18} + \cdots + 23\!\cdots\!72 \)
|
| $13$ |
\( (T^{9} + \cdots - 50\!\cdots\!92)^{2} \)
|
| $17$ |
\( (T^{9} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{18} + \cdots + 36\!\cdots\!12 \)
|
| $23$ |
\( T^{18} + \cdots + 16\!\cdots\!28 \)
|
| $29$ |
\( (T^{9} + \cdots + 12\!\cdots\!76)^{2} \)
|
| $31$ |
\( T^{18} + \cdots + 56\!\cdots\!52 \)
|
| $37$ |
\( (T^{9} + \cdots - 80\!\cdots\!96)^{2} \)
|
| $41$ |
\( (T^{9} + \cdots - 31\!\cdots\!96)^{2} \)
|
| $43$ |
\( T^{18} + \cdots + 74\!\cdots\!32 \)
|
| $47$ |
\( T^{18} + \cdots + 20\!\cdots\!00 \)
|
| $53$ |
\( (T^{9} + \cdots + 39\!\cdots\!24)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 11\!\cdots\!32 \)
|
| $61$ |
\( (T^{9} + \cdots + 20\!\cdots\!76)^{2} \)
|
| $67$ |
\( T^{18} + \cdots + 14\!\cdots\!08 \)
|
| $71$ |
\( T^{18} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( (T^{9} + \cdots + 29\!\cdots\!72)^{2} \)
|
| $79$ |
\( T^{18} + \cdots + 67\!\cdots\!08 \)
|
| $83$ |
\( T^{18} + \cdots + 26\!\cdots\!08 \)
|
| $89$ |
\( (T^{9} + \cdots - 31\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{9} + \cdots - 47\!\cdots\!84)^{2} \)
|
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