Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,26,Mod(9,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.9");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.b (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: | \(39.5996779952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 1406300109694 x^{10} + \cdots + 56\!\cdots\!01 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{90}\cdot 3^{8}\cdot 5^{29} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 1406300109694 x^{10} + \cdots + 56\!\cdots\!01 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 72\!\cdots\!37 \nu^{10} + \cdots - 21\!\cdots\!77 ) / 13\!\cdots\!90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 34\!\cdots\!27 \nu^{11} + \cdots + 41\!\cdots\!47 \nu ) / 58\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 38\!\cdots\!97 \nu^{11} + \cdots - 72\!\cdots\!89 \nu ) / 19\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 22\!\cdots\!71 \nu^{11} + \cdots - 16\!\cdots\!04 ) / 34\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22\!\cdots\!71 \nu^{11} + \cdots + 26\!\cdots\!54 ) / 34\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 42\!\cdots\!49 \nu^{11} + \cdots - 51\!\cdots\!26 ) / 31\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 32\!\cdots\!79 \nu^{11} + \cdots + 17\!\cdots\!96 ) / 10\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!13 \nu^{11} + \cdots + 49\!\cdots\!88 ) / 31\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 39\!\cdots\!92 \nu^{11} + \cdots + 36\!\cdots\!33 ) / 73\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 18\!\cdots\!69 \nu^{11} + \cdots + 17\!\cdots\!44 ) / 34\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 11\!\cdots\!27 \nu^{11} + \cdots + 27\!\cdots\!77 ) / 15\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 33\beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - \beta_{9} + 3 \beta_{8} - \beta_{7} + 69 \beta_{5} - 266 \beta_{4} + 135 \beta_{3} + \cdots - 937533406440 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 202358 \beta_{11} - 1537113 \beta_{10} + 412437 \beta_{9} + 338499 \beta_{8} - 1163373 \beta_{7} + \cdots - 2104932 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 67813081218 \beta_{11} + 33906540609 \beta_{10} + 1527164057033 \beta_{9} - 2875918650702 \beta_{8} + \cdots + 71\!\cdots\!43 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64\!\cdots\!20 \beta_{11} + \cdots + 63\!\cdots\!40 ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 48\!\cdots\!10 \beta_{11} + \cdots - 30\!\cdots\!33 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 40\!\cdots\!04 \beta_{11} + \cdots - 39\!\cdots\!68 ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 33\!\cdots\!32 \beta_{11} + \cdots + 14\!\cdots\!18 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 59\!\cdots\!57 \beta_{11} + \cdots + 56\!\cdots\!22 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 21\!\cdots\!60 \beta_{11} + \cdots - 70\!\cdots\!42 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 33\!\cdots\!15 \beta_{11} + \cdots - 31\!\cdots\!10 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 4096.00i | − | 1.47469e6i | −1.67772e7 | −2.33061e8 | − | 4.93666e8i | −6.04034e9 | − | 3.57876e9i | 6.87195e10i | −1.32743e12 | −2.02205e12 | + | 9.54619e11i | |||||||||||||||||||||||||||||||||||||||||||||||
| 9.2 | − | 4096.00i | − | 1.08018e6i | −1.67772e7 | 3.02592e8 | + | 4.54380e8i | −4.42443e9 | 5.43159e10i | 6.87195e10i | −3.19509e11 | 1.86114e12 | − | 1.23942e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 9.3 | − | 4096.00i | − | 233693.i | −1.67772e7 | −3.44896e8 | + | 4.23166e8i | −9.57206e8 | − | 6.39090e10i | 6.87195e10i | 7.92676e11 | 1.73329e12 | + | 1.41270e12i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 9.4 | − | 4096.00i | − | 116655.i | −1.67772e7 | 5.03737e8 | − | 2.10410e8i | −4.77821e8 | − | 1.34370e10i | 6.87195e10i | 8.33680e11 | −8.61839e11 | − | 2.06331e12i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 9.5 | − | 4096.00i | 1.01684e6i | −1.67772e7 | −2.64029e8 | − | 4.77820e8i | 4.16497e9 | 1.16326e10i | 6.87195e10i | −1.86674e11 | −1.95715e12 | + | 1.08146e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 9.6 | − | 4096.00i | 1.08697e6i | −1.67772e7 | −2.09490e8 | + | 5.04120e8i | 4.45223e9 | 4.89841e10i | 6.87195e10i | −3.34213e11 | 2.06488e12 | + | 8.58070e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 9.7 | 4096.00i | − | 1.08697e6i | −1.67772e7 | −2.09490e8 | − | 5.04120e8i | 4.45223e9 | − | 4.89841e10i | − | 6.87195e10i | −3.34213e11 | 2.06488e12 | − | 8.58070e11i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 9.8 | 4096.00i | − | 1.01684e6i | −1.67772e7 | −2.64029e8 | + | 4.77820e8i | 4.16497e9 | − | 1.16326e10i | − | 6.87195e10i | −1.86674e11 | −1.95715e12 | − | 1.08146e12i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 9.9 | 4096.00i | 116655.i | −1.67772e7 | 5.03737e8 | + | 2.10410e8i | −4.77821e8 | 1.34370e10i | − | 6.87195e10i | 8.33680e11 | −8.61839e11 | + | 2.06331e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 9.10 | 4096.00i | 233693.i | −1.67772e7 | −3.44896e8 | − | 4.23166e8i | −9.57206e8 | 6.39090e10i | − | 6.87195e10i | 7.92676e11 | 1.73329e12 | − | 1.41270e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 9.11 | 4096.00i | 1.08018e6i | −1.67772e7 | 3.02592e8 | − | 4.54380e8i | −4.42443e9 | − | 5.43159e10i | − | 6.87195e10i | −3.19509e11 | 1.86114e12 | + | 1.23942e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 9.12 | 4096.00i | 1.47469e6i | −1.67772e7 | −2.33061e8 | + | 4.93666e8i | −6.04034e9 | 3.57876e9i | − | 6.87195e10i | −1.32743e12 | −2.02205e12 | − | 9.54619e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.26.b.a | ✓ | 12 |
| 5.b | even | 2 | 1 | inner | 10.26.b.a | ✓ | 12 |
| 5.c | odd | 4 | 1 | 50.26.a.k | 6 | ||
| 5.c | odd | 4 | 1 | 50.26.a.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.26.b.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 10.26.b.a | ✓ | 12 | 5.b | even | 2 | 1 | inner |
| 50.26.a.k | 6 | 5.c | odd | 4 | 1 | ||
| 50.26.a.l | 6 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16777216)^{6} \)
|
| $3$ |
\( T^{12} + \cdots + 23\!\cdots\!96 \)
|
| $5$ |
\( T^{12} + \cdots + 70\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots + 90\!\cdots\!36 \)
|
| $11$ |
\( (T^{6} + \cdots + 67\!\cdots\!64)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 35\!\cdots\!56 \)
|
| $17$ |
\( T^{12} + \cdots + 25\!\cdots\!96 \)
|
| $19$ |
\( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 22\!\cdots\!16 \)
|
| $29$ |
\( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots - 75\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 93\!\cdots\!16 \)
|
| $41$ |
\( (T^{6} + \cdots - 47\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 30\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots + 35\!\cdots\!76 \)
|
| $53$ |
\( T^{12} + \cdots + 47\!\cdots\!96 \)
|
| $59$ |
\( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots - 45\!\cdots\!36)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 64\!\cdots\!96 \)
|
| $71$ |
\( (T^{6} + \cdots - 14\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 87\!\cdots\!16 \)
|
| $79$ |
\( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 93\!\cdots\!76 \)
|
| $89$ |
\( (T^{6} + \cdots + 23\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 16\!\cdots\!76 \)
|
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