Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,2,Mod(301,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.301");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.n (of order \(5\), degree \(4\), 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: | \(6.58765816676\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} + 9 x^{14} - 15 x^{13} + 44 x^{12} - 61 x^{11} + 208 x^{10} - 281 x^{9} + 851 x^{8} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} + 9 x^{14} - 15 x^{13} + 44 x^{12} - 61 x^{11} + 208 x^{10} - 281 x^{9} + 851 x^{8} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 67\!\cdots\!19 \nu^{15} + \cdots + 82\!\cdots\!50 ) / 26\!\cdots\!30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39\!\cdots\!75 \nu^{15} + \cdots - 15\!\cdots\!13 ) / 52\!\cdots\!06 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36\!\cdots\!42 \nu^{15} + \cdots + 61\!\cdots\!25 ) / 26\!\cdots\!30 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!96 \nu^{15} + \cdots + 62\!\cdots\!30 ) / 52\!\cdots\!06 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 89\!\cdots\!27 \nu^{15} + \cdots - 35\!\cdots\!50 ) / 26\!\cdots\!30 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!84 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 26\!\cdots\!30 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 14\!\cdots\!26 \nu^{15} + \cdots + 77\!\cdots\!85 ) / 26\!\cdots\!30 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 22\!\cdots\!23 \nu^{15} + \cdots + 17\!\cdots\!75 ) / 26\!\cdots\!30 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 22\!\cdots\!81 \nu^{15} + \cdots - 44\!\cdots\!35 ) / 26\!\cdots\!30 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 24\!\cdots\!57 \nu^{15} + \cdots + 38\!\cdots\!80 ) / 26\!\cdots\!30 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 38\!\cdots\!79 \nu^{15} + \cdots + 19\!\cdots\!50 ) / 26\!\cdots\!30 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 42\!\cdots\!13 \nu^{15} + \cdots + 22\!\cdots\!05 ) / 26\!\cdots\!30 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 72\!\cdots\!21 \nu^{15} + \cdots - 11\!\cdots\!90 ) / 26\!\cdots\!30 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 73\!\cdots\!70 \nu^{15} + \cdots - 14\!\cdots\!90 ) / 26\!\cdots\!30 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{13} + \beta_{10} - 3\beta_{9} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - \beta_{14} - \beta_{12} + \beta_{7} - 4\beta_{6} - \beta_{5} - 4\beta_{4} + 4\beta_{2} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - 6\beta_{14} - \beta_{12} - 13\beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{11} - \beta_{8} - 6 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 25 \beta_{13} + 8 \beta_{12} - 2 \beta_{10} + 10 \beta_{9} - 52 \beta_{8} - 8 \beta_{7} + 11 \beta_{6} + \cdots - 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 11 \beta_{15} + 21 \beta_{14} + 12 \beta_{13} + 42 \beta_{12} + 15 \beta_{11} - 12 \beta_{10} + \cdots - 15 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 65 \beta_{15} + 194 \beta_{14} + 23 \beta_{13} + 65 \beta_{12} + 314 \beta_{11} - 129 \beta_{10} + \cdots - 236 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 246 \beta_{15} + 349 \beta_{14} + 88 \beta_{12} + 231 \beta_{11} - 77 \beta_{10} + 140 \beta_{9} + \cdots + 101 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 323 \beta_{15} + 512 \beta_{14} - 189 \beta_{13} + 549 \beta_{11} + 549 \beta_{8} + 114 \beta_{7} + \cdots + 1114 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 771 \beta_{13} - 626 \beta_{12} + 540 \beta_{10} - 1088 \beta_{9} + 674 \beta_{8} + 626 \beta_{7} + \cdots + 1088 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1989 \beta_{15} - 3353 \beta_{14} - 3801 \beta_{13} - 2846 \beta_{12} - 3663 \beta_{11} + 3801 \beta_{10} + \cdots + 3663 \)
|
| \(\nu^{13}\) | \(=\) |
\( 8636 \beta_{15} - 14015 \beta_{14} - 3617 \beta_{13} - 8636 \beta_{12} - 12434 \beta_{11} + 5379 \beta_{10} + \cdots + 4704 \)
|
| \(\nu^{14}\) | \(=\) |
\( 18225 \beta_{15} - 39793 \beta_{14} - 12253 \beta_{12} - 51642 \beta_{11} + 9237 \beta_{10} + \cdots - 11347 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 27462 \beta_{15} - 51062 \beta_{14} + 23600 \beta_{13} - 52266 \beta_{11} - 52266 \beta_{8} + \cdots - 31651 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(-\beta_{8}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 301.1 |
|
−1.63575 | − | 1.18844i | −0.309017 | + | 0.951057i | 0.645245 | + | 1.98586i | 0 | 1.63575 | − | 1.18844i | −0.404805 | − | 1.24586i | 0.0550187 | − | 0.169330i | −0.809017 | − | 0.587785i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 301.2 | −0.553874 | − | 0.402413i | −0.309017 | + | 0.951057i | −0.473194 | − | 1.45634i | 0 | 0.553874 | − | 0.402413i | 1.30626 | + | 4.02026i | −0.747082 | + | 2.29928i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 301.3 | 1.03207 | + | 0.749844i | −0.309017 | + | 0.951057i | −0.115127 | − | 0.354326i | 0 | −1.03207 | + | 0.749844i | −0.810118 | − | 2.49329i | 0.935302 | − | 2.87856i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 301.4 | 1.65755 | + | 1.20428i | −0.309017 | + | 0.951057i | 0.679144 | + | 2.09019i | 0 | −1.65755 | + | 1.20428i | 1.14473 | + | 3.52312i | −0.125205 | + | 0.385340i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 526.1 | −0.647198 | + | 1.99187i | 0.809017 | − | 0.587785i | −1.93065 | − | 1.40270i | 0 | 0.647198 | + | 1.99187i | −2.09875 | − | 1.52483i | 0.654736 | − | 0.475694i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 526.2 | −0.0581136 | + | 0.178855i | 0.809017 | − | 0.587785i | 1.58942 | + | 1.15478i | 0 | 0.0581136 | + | 0.178855i | −1.33395 | − | 0.969174i | −0.603192 | + | 0.438245i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 526.3 | 0.434899 | − | 1.33848i | 0.809017 | − | 0.587785i | 0.0156350 | + | 0.0113595i | 0 | −0.434899 | − | 1.33848i | 1.75053 | + | 1.27183i | 2.29917 | − | 1.67044i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 526.4 | 0.770412 | − | 2.37108i | 0.809017 | − | 0.587785i | −3.41047 | − | 2.47785i | 0 | −0.770412 | − | 2.37108i | −1.55389 | − | 1.12897i | −4.46875 | + | 3.24673i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 676.1 | −0.647198 | − | 1.99187i | 0.809017 | + | 0.587785i | −1.93065 | + | 1.40270i | 0 | 0.647198 | − | 1.99187i | −2.09875 | + | 1.52483i | 0.654736 | + | 0.475694i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 676.2 | −0.0581136 | − | 0.178855i | 0.809017 | + | 0.587785i | 1.58942 | − | 1.15478i | 0 | 0.0581136 | − | 0.178855i | −1.33395 | + | 0.969174i | −0.603192 | − | 0.438245i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 676.3 | 0.434899 | + | 1.33848i | 0.809017 | + | 0.587785i | 0.0156350 | − | 0.0113595i | 0 | −0.434899 | + | 1.33848i | 1.75053 | − | 1.27183i | 2.29917 | + | 1.67044i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 676.4 | 0.770412 | + | 2.37108i | 0.809017 | + | 0.587785i | −3.41047 | + | 2.47785i | 0 | −0.770412 | + | 2.37108i | −1.55389 | + | 1.12897i | −4.46875 | − | 3.24673i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 751.1 | −1.63575 | + | 1.18844i | −0.309017 | − | 0.951057i | 0.645245 | − | 1.98586i | 0 | 1.63575 | + | 1.18844i | −0.404805 | + | 1.24586i | 0.0550187 | + | 0.169330i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 751.2 | −0.553874 | + | 0.402413i | −0.309017 | − | 0.951057i | −0.473194 | + | 1.45634i | 0 | 0.553874 | + | 0.402413i | 1.30626 | − | 4.02026i | −0.747082 | − | 2.29928i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 751.3 | 1.03207 | − | 0.749844i | −0.309017 | − | 0.951057i | −0.115127 | + | 0.354326i | 0 | −1.03207 | − | 0.749844i | −0.810118 | + | 2.49329i | 0.935302 | + | 2.87856i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 751.4 | 1.65755 | − | 1.20428i | −0.309017 | − | 0.951057i | 0.679144 | − | 2.09019i | 0 | −1.65755 | − | 1.20428i | 1.14473 | − | 3.52312i | −0.125205 | − | 0.385340i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.2.n.n | yes | 16 |
| 5.b | even | 2 | 1 | 825.2.n.m | ✓ | 16 | |
| 5.c | odd | 4 | 2 | 825.2.bx.j | 32 | ||
| 11.c | even | 5 | 1 | inner | 825.2.n.n | yes | 16 |
| 11.c | even | 5 | 1 | 9075.2.a.dv | 8 | ||
| 11.d | odd | 10 | 1 | 9075.2.a.dt | 8 | ||
| 55.h | odd | 10 | 1 | 9075.2.a.dw | 8 | ||
| 55.j | even | 10 | 1 | 825.2.n.m | ✓ | 16 | |
| 55.j | even | 10 | 1 | 9075.2.a.du | 8 | ||
| 55.k | odd | 20 | 2 | 825.2.bx.j | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.2.n.m | ✓ | 16 | 5.b | even | 2 | 1 | |
| 825.2.n.m | ✓ | 16 | 55.j | even | 10 | 1 | |
| 825.2.n.n | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 825.2.n.n | yes | 16 | 11.c | even | 5 | 1 | inner |
| 825.2.bx.j | 32 | 5.c | odd | 4 | 2 | ||
| 825.2.bx.j | 32 | 55.k | odd | 20 | 2 | ||
| 9075.2.a.dt | 8 | 11.d | odd | 10 | 1 | ||
| 9075.2.a.du | 8 | 55.j | even | 10 | 1 | ||
| 9075.2.a.dv | 8 | 11.c | even | 5 | 1 | ||
| 9075.2.a.dw | 8 | 55.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{16} - 2 T_{2}^{15} + 9 T_{2}^{14} - 10 T_{2}^{13} + 34 T_{2}^{12} - 44 T_{2}^{11} + 143 T_{2}^{10} + \cdots + 25 \)
|
|
\( T_{13}^{16} + 11 T_{13}^{15} + 116 T_{13}^{14} + 798 T_{13}^{13} + 5523 T_{13}^{12} + 28322 T_{13}^{11} + \cdots + 3933296656 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 2 T^{15} + \cdots + 25 \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + 4 T^{15} + \cdots + 913936 \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( T^{16} + \cdots + 3933296656 \)
|
| $17$ |
\( T^{16} + \cdots + 983700496 \)
|
| $19$ |
\( T^{16} - 8 T^{15} + \cdots + 15840400 \)
|
| $23$ |
\( (T^{8} + 7 T^{7} + \cdots + 3169)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 2484922801 \)
|
| $31$ |
\( T^{16} + \cdots + 941814721 \)
|
| $37$ |
\( T^{16} + \cdots + 294263936521 \)
|
| $41$ |
\( T^{16} + \cdots + 5078356897441 \)
|
| $43$ |
\( (T^{8} - 11 T^{7} + \cdots + 5045)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 21461214390625 \)
|
| $53$ |
\( T^{16} + \cdots + 256128016 \)
|
| $59$ |
\( T^{16} + \cdots + 3563493025 \)
|
| $61$ |
\( T^{16} + \cdots + 32428806400 \)
|
| $67$ |
\( (T^{8} + 6 T^{7} + \cdots + 45905)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 15356166400 \)
|
| $73$ |
\( T^{16} + \cdots + 989565363361 \)
|
| $79$ |
\( T^{16} + \cdots + 2392184995561 \)
|
| $83$ |
\( T^{16} + \cdots + 248885245456 \)
|
| $89$ |
\( (T^{8} - 21 T^{7} + \cdots - 60824801)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 20\!\cdots\!41 \)
|
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