Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,6,Mod(1,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.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: | \(52.1247414392\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5^{2} \) |
| 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
\( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8351315 \nu^{10} - 62345059 \nu^{9} + 2150889153 \nu^{8} + 15801380623 \nu^{7} + \cdots + 69560356278400 ) / 6153930604800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 14779564 \nu^{10} - 63760673 \nu^{9} + 3564258525 \nu^{8} + 19680597269 \nu^{7} + \cdots - 266861708622400 ) / 3076965302400 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1457477 \nu^{10} + 4958145 \nu^{9} - 365671467 \nu^{8} - 1356282429 \nu^{7} + \cdots + 2986895958400 ) / 93241372800 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5033511 \nu^{10} - 5003705 \nu^{9} + 1243018371 \nu^{8} + 1816203137 \nu^{7} + \cdots + 16275665106400 ) / 256413775200 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 136014289 \nu^{10} - 83517389 \nu^{9} + 35623272087 \nu^{8} + 35249071601 \nu^{7} + \cdots + 39\!\cdots\!00 ) / 6153930604800 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 9140405 \nu^{10} - 17080841 \nu^{9} - 2343707873 \nu^{8} + 3438961437 \nu^{7} + \cdots - 64143233480960 ) / 410262040320 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 85970113 \nu^{10} - 45374565 \nu^{9} - 21662686333 \nu^{8} + 2665182089 \nu^{7} + \cdots - 541179137126400 ) / 2051310201600 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 308037113 \nu^{10} + 24595469 \nu^{9} + 76517962005 \nu^{8} + 25107723103 \nu^{7} + \cdots + 14\!\cdots\!00 ) / 6153930604800 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{2} + 85\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5 \beta_{10} + 6 \beta_{8} + 3 \beta_{7} - 8 \beta_{6} + \beta_{5} - \beta_{4} - 4 \beta_{3} + \cdots + 4181 \)
|
| \(\nu^{5}\) | \(=\) |
\( 20 \beta_{10} - 133 \beta_{9} + 157 \beta_{8} + 12 \beta_{7} - 179 \beta_{6} - 24 \beta_{5} + \cdots + 2303 \)
|
| \(\nu^{6}\) | \(=\) |
\( 885 \beta_{10} - 146 \beta_{9} + 1110 \beta_{8} + 461 \beta_{7} - 1724 \beta_{6} + 37 \beta_{5} + \cdots + 415083 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4704 \beta_{10} - 15449 \beta_{9} + 20835 \beta_{8} + 2822 \beta_{7} - 26521 \beta_{6} - 4944 \beta_{5} + \cdots + 539521 \)
|
| \(\nu^{8}\) | \(=\) |
\( 124967 \beta_{10} - 35680 \beta_{9} + 162426 \beta_{8} + 61441 \beta_{7} - 279588 \beta_{6} + \cdots + 43863969 \)
|
| \(\nu^{9}\) | \(=\) |
\( 802652 \beta_{10} - 1767337 \beta_{9} + 2651997 \beta_{8} + 474136 \beta_{7} - 3670911 \beta_{6} + \cdots + 97501055 \)
|
| \(\nu^{10}\) | \(=\) |
\( 16504085 \beta_{10} - 6232822 \beta_{9} + 22193634 \beta_{8} + 7926229 \beta_{7} - 40588944 \beta_{6} + \cdots + 4800223251 \)
|
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 |
|
−11.2252 | −19.1897 | 94.0044 | 0 | 215.408 | −70.9909 | −696.009 | 125.246 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.27946 | 5.85765 | 54.1083 | 0 | −54.3558 | 90.4001 | −205.153 | −208.688 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.07859 | −25.6203 | 4.94922 | 0 | 155.735 | −137.301 | 164.431 | 413.398 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.83767 | 12.7661 | 2.07840 | 0 | −74.5243 | −231.930 | 174.672 | −80.0268 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.89241 | 3.45730 | −23.6340 | 0 | −9.99993 | 148.288 | 160.916 | −231.047 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.979695 | −20.7822 | −31.0402 | 0 | 20.3602 | 233.896 | 61.7601 | 188.898 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.96479 | 29.7675 | −28.1396 | 0 | 58.4868 | −176.972 | −118.162 | 643.103 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.78432 | −8.20788 | −17.6789 | 0 | −31.0612 | −88.9969 | −188.001 | −175.631 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.84001 | 12.3562 | 2.10573 | 0 | 72.1603 | 135.041 | −174.583 | −90.3246 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 9.60154 | −25.9222 | 60.1895 | 0 | −248.893 | −181.554 | 270.662 | 428.963 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 10.1023 | 24.5176 | 70.0572 | 0 | 247.685 | 72.1186 | 384.466 | 358.110 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 325.6.a.j | ✓ | 11 |
| 5.b | even | 2 | 1 | 325.6.a.k | yes | 11 | |
| 5.c | odd | 4 | 2 | 325.6.b.i | 22 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.6.a.j | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 325.6.a.k | yes | 11 | 5.b | even | 2 | 1 | |
| 325.6.b.i | 22 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 5 T_{2}^{10} - 257 T_{2}^{9} - 1165 T_{2}^{8} + 22234 T_{2}^{7} + 90282 T_{2}^{6} + \cdots - 44115200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 5 T^{10} + \cdots - 44115200 \)
|
| $3$ |
\( T^{11} + \cdots + 5068375276032 \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} + \cdots - 19\!\cdots\!28 \)
|
| $11$ |
\( T^{11} + \cdots + 58\!\cdots\!12 \)
|
| $13$ |
\( (T + 169)^{11} \)
|
| $17$ |
\( T^{11} + \cdots - 14\!\cdots\!08 \)
|
| $19$ |
\( T^{11} + \cdots - 54\!\cdots\!60 \)
|
| $23$ |
\( T^{11} + \cdots - 22\!\cdots\!24 \)
|
| $29$ |
\( T^{11} + \cdots + 33\!\cdots\!60 \)
|
| $31$ |
\( T^{11} + \cdots - 29\!\cdots\!80 \)
|
| $37$ |
\( T^{11} + \cdots + 17\!\cdots\!44 \)
|
| $41$ |
\( T^{11} + \cdots + 22\!\cdots\!72 \)
|
| $43$ |
\( T^{11} + \cdots + 86\!\cdots\!40 \)
|
| $47$ |
\( T^{11} + \cdots + 16\!\cdots\!80 \)
|
| $53$ |
\( T^{11} + \cdots - 13\!\cdots\!36 \)
|
| $59$ |
\( T^{11} + \cdots + 27\!\cdots\!40 \)
|
| $61$ |
\( T^{11} + \cdots - 99\!\cdots\!24 \)
|
| $67$ |
\( T^{11} + \cdots + 27\!\cdots\!64 \)
|
| $71$ |
\( T^{11} + \cdots + 64\!\cdots\!72 \)
|
| $73$ |
\( T^{11} + \cdots - 11\!\cdots\!92 \)
|
| $79$ |
\( T^{11} + \cdots - 18\!\cdots\!20 \)
|
| $83$ |
\( T^{11} + \cdots - 18\!\cdots\!88 \)
|
| $89$ |
\( T^{11} + \cdots - 24\!\cdots\!00 \)
|
| $97$ |
\( T^{11} + \cdots + 74\!\cdots\!40 \)
|
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