Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4225,2,Mod(1,4225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4225 = 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4225.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: | \(33.7367948540\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} + 2 x^{8} - 171 x^{7} + 6 x^{6} + 260 x^{5} + 27 x^{4} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} + 2 x^{8} - 171 x^{7} + 6 x^{6} + 260 x^{5} + 27 x^{4} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13 \nu^{11} - 69 \nu^{10} - 40 \nu^{9} + 611 \nu^{8} - 134 \nu^{7} - 2059 \nu^{6} + 480 \nu^{5} + \cdots + 44 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12 \nu^{11} + 61 \nu^{10} + 59 \nu^{9} - 606 \nu^{8} + 44 \nu^{7} + 2186 \nu^{6} - 442 \nu^{5} + \cdots - 60 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27 \nu^{11} - 139 \nu^{10} - 117 \nu^{9} + 1332 \nu^{8} - 169 \nu^{7} - 4677 \nu^{6} + 1054 \nu^{5} + \cdots + 121 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 33 \nu^{11} + 173 \nu^{10} + 122 \nu^{9} - 1607 \nu^{8} + 317 \nu^{7} + 5539 \nu^{6} - 1492 \nu^{5} + \cdots - 151 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 37 \nu^{11} + 191 \nu^{10} + 158 \nu^{9} - 1830 \nu^{8} + 257 \nu^{7} + 6431 \nu^{6} - 1553 \nu^{5} + \cdots - 164 ) / 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 39 \nu^{11} + 207 \nu^{10} + 127 \nu^{9} - 1882 \nu^{8} + 472 \nu^{7} + 6366 \nu^{6} - 1923 \nu^{5} + \cdots - 160 ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 41 \nu^{11} + 216 \nu^{10} + 145 \nu^{9} - 1990 \nu^{8} + 421 \nu^{7} + 6833 \nu^{6} - 1873 \nu^{5} + \cdots - 177 ) / 7 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 68 \nu^{11} - 355 \nu^{10} - 262 \nu^{9} + 3322 \nu^{8} - 590 \nu^{7} - 11510 \nu^{6} + 2927 \nu^{5} + \cdots + 319 ) / 7 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 160 \nu^{11} - 839 \nu^{10} - 593 \nu^{9} + 7807 \nu^{8} - 1562 \nu^{7} - 26888 \nu^{6} + 7326 \nu^{5} + \cdots + 702 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{5} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{10} + 3\beta_{9} + \beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 8\beta_{2} + 12\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{11} + 10 \beta_{10} + 12 \beta_{9} - 2 \beta_{8} + \beta_{7} - 5 \beta_{6} - 10 \beta_{5} + \cdots + 28 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 3 \beta_{11} + 25 \beta_{10} + 37 \beta_{9} - 7 \beta_{8} + 7 \beta_{7} - 24 \beta_{6} - 26 \beta_{5} + \cdots + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 14 \beta_{11} + 86 \beta_{10} + 121 \beta_{9} - 32 \beta_{8} + 11 \beta_{7} - 71 \beta_{6} + \cdots + 230 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 43 \beta_{11} + 234 \beta_{10} + 371 \beta_{9} - 106 \beta_{8} + 43 \beta_{7} - 252 \beta_{6} + \cdots + 697 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 149 \beta_{11} + 714 \beta_{10} + 1151 \beta_{9} - 370 \beta_{8} + 85 \beta_{7} - 779 \beta_{6} + \cdots + 1851 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 455 \beta_{11} + 2000 \beta_{10} + 3499 \beta_{9} - 1176 \beta_{8} + 259 \beta_{7} - 2533 \beta_{6} + \cdots + 5382 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1435 \beta_{11} + 5834 \beta_{10} + 10648 \beta_{9} - 3763 \beta_{8} + 565 \beta_{7} - 7875 \beta_{6} + \cdots + 14928 \)
|
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 |
|
−2.73335 | −1.07706 | 5.47121 | 0 | 2.94398 | −4.31040 | −9.48805 | −1.83995 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.72192 | 2.59984 | 5.40885 | 0 | −7.07655 | 1.46276 | −9.27861 | 3.75915 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.20799 | −3.30963 | 2.87523 | 0 | 7.30764 | −3.12554 | −1.93251 | 7.95366 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.95335 | −2.06718 | 1.81556 | 0 | 4.03792 | −1.53866 | 0.360280 | 1.27323 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.23178 | 2.02961 | −0.482720 | 0 | −2.50004 | 2.20188 | 3.05816 | 1.11933 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.22263 | 0.0895352 | −0.505174 | 0 | −0.109469 | 3.47868 | 3.06290 | −2.99198 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.837905 | 0.531386 | −1.29791 | 0 | −0.445251 | −1.15977 | 2.76334 | −2.71763 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.0552442 | 2.11312 | −1.99695 | 0 | 0.116737 | −4.11530 | −0.220808 | 1.46526 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.595328 | −1.48507 | −1.64558 | 0 | −0.884102 | −0.0123504 | −2.17032 | −0.794576 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.45570 | −1.57064 | 0.119054 | 0 | −2.28637 | 0.947023 | −2.73809 | −0.533102 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.83678 | 2.76961 | 1.37375 | 0 | 5.08716 | −4.65087 | −1.15029 | 4.67076 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.96588 | −1.62353 | 1.86469 | 0 | −3.19166 | −1.17744 | −0.266009 | −0.364158 | 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 | 4225.2.a.bw | ✓ | 12 |
| 5.b | even | 2 | 1 | 4225.2.a.bz | yes | 12 | |
| 13.b | even | 2 | 1 | 4225.2.a.by | yes | 12 | |
| 65.d | even | 2 | 1 | 4225.2.a.bx | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4225.2.a.bw | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 4225.2.a.bx | yes | 12 | 65.d | even | 2 | 1 | |
| 4225.2.a.by | yes | 12 | 13.b | even | 2 | 1 | |
| 4225.2.a.bz | yes | 12 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):
|
\( T_{2}^{12} + 7 T_{2}^{11} + 6 T_{2}^{10} - 57 T_{2}^{9} - 121 T_{2}^{8} + 115 T_{2}^{7} + 461 T_{2}^{6} + \cdots - 7 \)
|
|
\( T_{3}^{12} + T_{3}^{11} - 23 T_{3}^{10} - 23 T_{3}^{9} + 190 T_{3}^{8} + 211 T_{3}^{7} - 686 T_{3}^{6} + \cdots + 41 \)
|
|
\( T_{7}^{12} + 12 T_{7}^{11} + 26 T_{7}^{10} - 187 T_{7}^{9} - 811 T_{7}^{8} + 301 T_{7}^{7} + 4992 T_{7}^{6} + \cdots + 71 \)
|
|
\( T_{11}^{12} + 3 T_{11}^{11} - 70 T_{11}^{10} - 181 T_{11}^{9} + 1676 T_{11}^{8} + 3619 T_{11}^{7} + \cdots + 13637 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 7 T^{11} + \cdots - 7 \)
|
| $3$ |
\( T^{12} + T^{11} + \cdots + 41 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + 12 T^{11} + \cdots + 71 \)
|
| $11$ |
\( T^{12} + 3 T^{11} + \cdots + 13637 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} - 2 T^{11} + \cdots - 857493 \)
|
| $19$ |
\( T^{12} - 139 T^{10} + \cdots + 13629931 \)
|
| $23$ |
\( T^{12} + 3 T^{11} + \cdots - 23233 \)
|
| $29$ |
\( T^{12} - 9 T^{11} + \cdots + 5429717 \)
|
| $31$ |
\( T^{12} - 4 T^{11} + \cdots - 181117 \)
|
| $37$ |
\( T^{12} + 17 T^{11} + \cdots - 54866624 \)
|
| $41$ |
\( T^{12} + 9 T^{11} + \cdots - 8114317 \)
|
| $43$ |
\( T^{12} + \cdots + 136367497 \)
|
| $47$ |
\( T^{12} + \cdots - 13460383391 \)
|
| $53$ |
\( T^{12} - 23 T^{11} + \cdots - 7877521 \)
|
| $59$ |
\( T^{12} + \cdots - 204358217 \)
|
| $61$ |
\( T^{12} + \cdots - 99827166223 \)
|
| $67$ |
\( T^{12} + 43 T^{11} + \cdots - 21853503 \)
|
| $71$ |
\( T^{12} + \cdots - 539717669 \)
|
| $73$ |
\( T^{12} + 21 T^{11} + \cdots - 66187687 \)
|
| $79$ |
\( T^{12} + \cdots + 164671459 \)
|
| $83$ |
\( T^{12} + \cdots - 1676222759 \)
|
| $89$ |
\( T^{12} + \cdots + 1280077267 \)
|
| $97$ |
\( T^{12} + \cdots - 3175912193 \)
|
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