Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,34,Mod(1,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
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: | yes |
| Analytic conductor: | \(172.457072203\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + \cdots + 13\!\cdots\!48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{47}\cdot 3^{12}\cdot 5^{30}\cdot 7^{2}\cdot 11^{3} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + \cdots + 13\!\cdots\!48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 29\!\cdots\!81 \nu^{10} + \cdots - 15\!\cdots\!48 ) / 30\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 29\!\cdots\!81 \nu^{10} + \cdots - 51\!\cdots\!08 ) / 38\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!87 \nu^{10} + \cdots - 27\!\cdots\!28 ) / 30\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 65\!\cdots\!73 \nu^{10} + \cdots - 33\!\cdots\!80 ) / 10\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10\!\cdots\!23 \nu^{10} + \cdots + 13\!\cdots\!16 ) / 10\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 83\!\cdots\!47 \nu^{10} + \cdots - 61\!\cdots\!84 ) / 15\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 17\!\cdots\!37 \nu^{10} + \cdots - 60\!\cdots\!04 ) / 15\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26\!\cdots\!99 \nu^{10} + \cdots - 61\!\cdots\!84 ) / 61\!\cdots\!48 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 52\!\cdots\!87 \nu^{10} + \cdots + 14\!\cdots\!56 ) / 30\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 8\beta_{2} + 8185\beta _1 + 13074496202 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} + \beta_{5} - 21\beta_{4} + 18185\beta_{3} - 1311661\beta_{2} + 22152168592\beta _1 + 107025408036041 \)
|
| \(\nu^{4}\) | \(=\) |
\( 26 \beta_{10} + 57 \beta_{9} - 259 \beta_{8} - 1245 \beta_{7} + 75517 \beta_{6} - 7678 \beta_{5} + \cdots + 28\!\cdots\!68 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 522454 \beta_{10} + 820053 \beta_{9} - 12720239 \beta_{8} + 720283927 \beta_{7} + \cdots + 59\!\cdots\!62 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1021661423510 \beta_{10} + 2937492549339 \beta_{9} - 12067437892993 \beta_{8} - 56702122654215 \beta_{7} + \cdots + 74\!\cdots\!70 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 18\!\cdots\!78 \beta_{10} + \cdots + 23\!\cdots\!66 \)
|
| \(\nu^{8}\) | \(=\) |
\( 32\!\cdots\!82 \beta_{10} + \cdots + 20\!\cdots\!22 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 50\!\cdots\!06 \beta_{10} + \cdots + 80\!\cdots\!46 \)
|
| \(\nu^{10}\) | \(=\) |
\( 97\!\cdots\!18 \beta_{10} + \cdots + 58\!\cdots\!54 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−171393. | −3.66048e7 | 2.07856e10 | 0 | 6.27381e12 | −4.77732e13 | −2.09024e15 | −4.21915e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −140551. | 1.29136e8 | 1.11647e10 | 0 | −1.81502e13 | 4.00085e13 | −3.61884e14 | 1.11170e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −104621. | 2.08087e7 | 2.35554e9 | 0 | −2.17702e12 | 5.34780e13 | 6.52247e14 | −5.12606e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −100295. | −1.18201e8 | 1.46921e9 | 0 | 1.18550e13 | −1.07417e13 | 7.14175e14 | 8.41244e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −34748.6 | 4.98225e7 | −7.38247e9 | 0 | −1.73126e12 | −1.47813e14 | 5.55018e14 | −3.07678e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −642.905 | −5.85306e7 | −8.58952e9 | 0 | 3.76296e10 | 1.03914e14 | 1.10448e13 | −2.13323e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 41215.3 | 1.10688e8 | −6.89123e9 | 0 | 4.56203e12 | 7.02715e13 | −6.38061e14 | 6.69270e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 61242.8 | −1.16397e8 | −4.83925e9 | 0 | −7.12847e12 | −1.52684e14 | −8.22441e14 | 7.98915e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 109239. | −1.96178e7 | 3.34318e9 | 0 | −2.14302e12 | 7.24227e12 | −5.73150e14 | −5.17420e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 153994. | 7.94763e7 | 1.51243e10 | 0 | 1.22389e13 | −4.85062e13 | 1.00626e15 | 7.57418e14 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 177167. | −1.20911e8 | 2.27982e10 | 0 | −2.14214e13 | 1.11259e14 | 2.51723e15 | 9.06036e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.34.a.d | ✓ | 11 |
| 5.b | even | 2 | 1 | 25.34.a.e | yes | 11 | |
| 5.c | odd | 4 | 2 | 25.34.b.d | 22 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.34.a.d | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 25.34.a.e | yes | 11 | 5.b | even | 2 | 1 | |
| 25.34.b.d | 22 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 9393 T_{2}^{10} - 71869629370 T_{2}^{9} - 944957445671160 T_{2}^{8} + \cdots - 42\!\cdots\!32 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots - 42\!\cdots\!32 \)
|
| $3$ |
\( T^{11} + \cdots - 82\!\cdots\!49 \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} + \cdots + 70\!\cdots\!08 \)
|
| $11$ |
\( T^{11} + \cdots + 13\!\cdots\!57 \)
|
| $13$ |
\( T^{11} + \cdots - 12\!\cdots\!04 \)
|
| $17$ |
\( T^{11} + \cdots + 39\!\cdots\!13 \)
|
| $19$ |
\( T^{11} + \cdots - 26\!\cdots\!75 \)
|
| $23$ |
\( T^{11} + \cdots - 46\!\cdots\!84 \)
|
| $29$ |
\( T^{11} + \cdots - 16\!\cdots\!00 \)
|
| $31$ |
\( T^{11} + \cdots - 24\!\cdots\!08 \)
|
| $37$ |
\( T^{11} + \cdots + 23\!\cdots\!48 \)
|
| $41$ |
\( T^{11} + \cdots - 20\!\cdots\!53 \)
|
| $43$ |
\( T^{11} + \cdots + 18\!\cdots\!56 \)
|
| $47$ |
\( T^{11} + \cdots - 24\!\cdots\!72 \)
|
| $53$ |
\( T^{11} + \cdots - 26\!\cdots\!24 \)
|
| $59$ |
\( T^{11} + \cdots + 23\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots + 26\!\cdots\!32 \)
|
| $67$ |
\( T^{11} + \cdots + 46\!\cdots\!13 \)
|
| $71$ |
\( T^{11} + \cdots - 61\!\cdots\!88 \)
|
| $73$ |
\( T^{11} + \cdots - 80\!\cdots\!09 \)
|
| $79$ |
\( T^{11} + \cdots - 92\!\cdots\!00 \)
|
| $83$ |
\( T^{11} + \cdots + 17\!\cdots\!61 \)
|
| $89$ |
\( T^{11} + \cdots - 12\!\cdots\!75 \)
|
| $97$ |
\( T^{11} + \cdots - 85\!\cdots\!72 \)
|
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