Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,2,Mod(7,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.x (of order \(12\), 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: | \(2.59513806569\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 10x^{14} + 71x^{12} - 250x^{10} + 640x^{8} - 560x^{6} + 371x^{4} - 20x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{16} - 10x^{14} + 71x^{12} - 250x^{10} + 640x^{8} - 560x^{6} + 371x^{4} - 20x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1798 \nu^{14} - 16617 \nu^{12} + 115320 \nu^{10} - 362080 \nu^{8} + 876240 \nu^{6} - 228780 \nu^{4} + \cdots + 935973 ) / 481290 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12199 \nu^{14} - 124904 \nu^{12} + 893060 \nu^{10} - 3242180 \nu^{8} + 8449500 \nu^{6} + \cdots - 579304 ) / 481290 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12199 \nu^{15} - 124904 \nu^{13} + 893060 \nu^{11} - 3242180 \nu^{9} + 8449500 \nu^{7} + \cdots - 579304 \nu ) / 481290 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19826 \nu^{14} - 195651 \nu^{12} + 1382240 \nu^{10} - 4778100 \nu^{8} + 12052600 \nu^{6} + \cdots + 252039 ) / 481290 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19826 \nu^{15} + 195651 \nu^{13} - 1382240 \nu^{11} + 4778100 \nu^{9} - 12052600 \nu^{7} + \cdots - 252039 \nu ) / 481290 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24703 \nu^{14} - 251333 \nu^{12} + 1800150 \nu^{10} - 6490460 \nu^{8} + 16903270 \nu^{6} + \cdots - 1126583 ) / 481290 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 26607 \nu^{15} - 264272 \nu^{13} + 1872480 \nu^{11} - 6536430 \nu^{9} + 16666400 \nu^{7} + \cdots - 519802 \nu ) / 481290 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26607 \nu^{14} - 264272 \nu^{12} + 1872480 \nu^{10} - 6536430 \nu^{8} + 16666400 \nu^{6} + \cdots - 38512 ) / 481290 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 51416 \nu^{14} + 511927 \nu^{12} - 3629640 \nu^{10} + 12710780 \nu^{8} - 32456560 \nu^{6} + \cdots + 1012997 ) / 481290 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 29751 \nu^{15} - 304714 \nu^{13} + 2182005 \nu^{11} - 7927460 \nu^{9} + 20687765 \nu^{7} + \cdots - 2622169 \nu ) / 240645 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 61441 \nu^{14} - 615704 \nu^{12} + 4372200 \nu^{10} - 15404520 \nu^{8} + 39356010 \nu^{6} + \cdots - 634704 ) / 481290 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 98014 \nu^{15} - 967941 \nu^{13} + 6834090 \nu^{11} - 23610440 \nu^{9} + 59486780 \nu^{7} + \cdots + 3627839 \nu ) / 481290 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 192537 \nu^{15} + 1930788 \nu^{13} - 13726410 \nu^{11} + 48537070 \nu^{9} - 124707530 \nu^{7} + \cdots + 5436898 \nu ) / 481290 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 288753 \nu^{15} - 2882112 \nu^{13} + 20445180 \nu^{11} - 71785430 \nu^{9} + 183318070 \nu^{7} + \cdots - 4188902 \nu ) / 481290 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + 2\beta_{9} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} + \beta_{11} + 5\beta_{8} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{10} + 8\beta_{9} + \beta_{5} + 2\beta_{3} - 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{15} + 5\beta_{13} + 5\beta_{11} + 23\beta_{8} - \beta_{6} + 2\beta_{4} \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{12} - 2\beta_{7} - 7\beta_{5} + 7\beta_{3} + 23\beta_{2} - 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( -21\beta_{15} - 21\beta_{14} + 22\beta_{13} - \beta_{11} + 10\beta_{6} + 10\beta_{4} - 105\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 20\beta_{12} - 105\beta_{10} - 167\beta_{9} - 10\beta_{7} - 82\beta_{5} - 41\beta_{3} + 105\beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( 10\beta_{15} - 95\beta_{14} + 10\beta_{13} - 95\beta_{11} - 482\beta_{8} + 142\beta_{6} - 71\beta_{4} - 482\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 71\beta_{12} - 482\beta_{10} - 784\beta_{9} + 71\beta_{7} - 227\beta_{5} - 454\beta_{3} + 784 \)
|
| \(\nu^{11}\) | \(=\) |
\( 411\beta_{15} - 71\beta_{14} - 340\beta_{13} - 340\beta_{11} - 2230\beta_{8} + 440\beta_{6} - 880\beta_{4} \)
|
| \(\nu^{12}\) | \(=\) |
\( -440\beta_{12} + 880\beta_{7} + 1220\beta_{5} - 1220\beta_{3} - 2230\beta_{2} + 3709 \)
|
| \(\nu^{13}\) | \(=\) |
\( 1350\beta_{15} + 1350\beta_{14} - 1790\beta_{13} + 440\beta_{11} - 2540\beta_{6} - 2540\beta_{4} + 10399\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 5080 \beta_{12} + 10399 \beta_{10} + 17658 \beta_{9} + 2540 \beta_{7} + 12860 \beta_{5} + \cdots - 10399 \beta_{2} \)
|
| \(\nu^{15}\) | \(=\) |
\( - 2540 \beta_{15} + 7859 \beta_{14} - 2540 \beta_{13} + 7859 \beta_{11} + 48855 \beta_{8} + \cdots + 48855 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(\beta_{3} + \beta_{5}\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.75074 | − | 1.01079i | 0.763462 | + | 2.84928i | 1.04340 | + | 1.80723i | 0 | 1.54340 | − | 5.76006i | 2.17356 | + | 3.76471i | − | 0.175483i | −4.93743 | + | 2.85063i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −0.827585 | − | 0.477806i | −0.0454184 | − | 0.169504i | −0.543402 | − | 0.941200i | 0 | −0.0434024 | + | 0.161980i | −1.59898 | − | 2.76951i | 2.94979i | 2.57141 | − | 1.48460i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 0.827585 | + | 0.477806i | 0.0454184 | + | 0.169504i | −0.543402 | − | 0.941200i | 0 | −0.0434024 | + | 0.161980i | 1.59898 | + | 2.76951i | − | 2.94979i | 2.57141 | − | 1.48460i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 1.75074 | + | 1.01079i | −0.763462 | − | 2.84928i | 1.04340 | + | 1.80723i | 0 | 1.54340 | − | 5.76006i | −2.17356 | − | 3.76471i | 0.175483i | −4.93743 | + | 2.85063i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.1 | −1.75074 | + | 1.01079i | 0.763462 | − | 2.84928i | 1.04340 | − | 1.80723i | 0 | 1.54340 | + | 5.76006i | 2.17356 | − | 3.76471i | 0.175483i | −4.93743 | − | 2.85063i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.2 | −0.827585 | + | 0.477806i | −0.0454184 | + | 0.169504i | −0.543402 | + | 0.941200i | 0 | −0.0434024 | − | 0.161980i | −1.59898 | + | 2.76951i | − | 2.94979i | 2.57141 | + | 1.48460i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.3 | 0.827585 | − | 0.477806i | 0.0454184 | − | 0.169504i | −0.543402 | + | 0.941200i | 0 | −0.0434024 | − | 0.161980i | 1.59898 | − | 2.76951i | 2.94979i | 2.57141 | + | 1.48460i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.4 | 1.75074 | − | 1.01079i | −0.763462 | + | 2.84928i | 1.04340 | − | 1.80723i | 0 | 1.54340 | + | 5.76006i | −2.17356 | + | 3.76471i | − | 0.175483i | −4.93743 | − | 2.85063i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.1 | −1.92597 | + | 1.11196i | −0.887132 | − | 0.237706i | 1.47291 | − | 2.55116i | 0 | 1.97291 | − | 0.528640i | −0.402292 | + | 0.696790i | 2.10344i | −1.86758 | − | 1.07825i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.2 | −0.201575 | + | 0.116380i | 2.03176 | + | 0.544409i | −0.972912 | + | 1.68513i | 0 | −0.472912 | + | 0.126716i | −1.02814 | + | 1.78079i | − | 0.918427i | 1.23360 | + | 0.712221i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.3 | 0.201575 | − | 0.116380i | −2.03176 | − | 0.544409i | −0.972912 | + | 1.68513i | 0 | −0.472912 | + | 0.126716i | 1.02814 | − | 1.78079i | 0.918427i | 1.23360 | + | 0.712221i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.4 | 1.92597 | − | 1.11196i | 0.887132 | + | 0.237706i | 1.47291 | − | 2.55116i | 0 | 1.97291 | − | 0.528640i | 0.402292 | − | 0.696790i | − | 2.10344i | −1.86758 | − | 1.07825i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 318.1 | −1.92597 | − | 1.11196i | −0.887132 | + | 0.237706i | 1.47291 | + | 2.55116i | 0 | 1.97291 | + | 0.528640i | −0.402292 | − | 0.696790i | − | 2.10344i | −1.86758 | + | 1.07825i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 318.2 | −0.201575 | − | 0.116380i | 2.03176 | − | 0.544409i | −0.972912 | − | 1.68513i | 0 | −0.472912 | − | 0.126716i | −1.02814 | − | 1.78079i | 0.918427i | 1.23360 | − | 0.712221i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 318.3 | 0.201575 | + | 0.116380i | −2.03176 | + | 0.544409i | −0.972912 | − | 1.68513i | 0 | −0.472912 | − | 0.126716i | 1.02814 | + | 1.78079i | − | 0.918427i | 1.23360 | − | 0.712221i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 318.4 | 1.92597 | + | 1.11196i | 0.887132 | − | 0.237706i | 1.47291 | + | 2.55116i | 0 | 1.97291 | + | 0.528640i | 0.402292 | + | 0.696790i | 2.10344i | −1.86758 | + | 1.07825i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 65.o | even | 12 | 1 | inner |
| 65.t | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.2.x.a | yes | 16 |
| 5.b | even | 2 | 1 | inner | 325.2.x.a | yes | 16 |
| 5.c | odd | 4 | 2 | 325.2.s.a | ✓ | 16 | |
| 13.f | odd | 12 | 1 | 325.2.s.a | ✓ | 16 | |
| 65.o | even | 12 | 1 | inner | 325.2.x.a | yes | 16 |
| 65.s | odd | 12 | 1 | 325.2.s.a | ✓ | 16 | |
| 65.t | even | 12 | 1 | inner | 325.2.x.a | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.2.s.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 325.2.s.a | ✓ | 16 | 13.f | odd | 12 | 1 | |
| 325.2.s.a | ✓ | 16 | 65.s | odd | 12 | 1 | |
| 325.2.x.a | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 325.2.x.a | yes | 16 | 5.b | even | 2 | 1 | inner |
| 325.2.x.a | yes | 16 | 65.o | even | 12 | 1 | inner |
| 325.2.x.a | yes | 16 | 65.t | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 10T_{2}^{14} + 71T_{2}^{12} - 250T_{2}^{10} + 640T_{2}^{8} - 560T_{2}^{6} + 371T_{2}^{4} - 20T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 10 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} + 6 T^{14} + \cdots + 1 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + 34 T^{14} + \cdots + 279841 \)
|
| $11$ |
\( (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{4} \)
|
| $13$ |
\( T^{16} + \cdots + 815730721 \)
|
| $17$ |
\( T^{16} + \cdots + 2655237841 \)
|
| $19$ |
\( (T^{8} + 2 T^{7} + \cdots + 267289)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 131079601 \)
|
| $29$ |
\( (T^{8} + 18 T^{7} + \cdots + 11449)^{2} \)
|
| $31$ |
\( (T^{8} - 4 T^{7} + \cdots + 43264)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 2743558264161 \)
|
| $41$ |
\( (T^{8} + 16 T^{7} + \cdots + 11449)^{2} \)
|
| $43$ |
\( T^{16} + 30 T^{14} + \cdots + 390625 \)
|
| $47$ |
\( (T^{8} - 196 T^{6} + \cdots + 456976)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( (T^{8} + 12 T^{7} + \cdots + 502681)^{2} \)
|
| $61$ |
\( (T^{8} - 18 T^{7} + \cdots + 9801)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 16881653995441 \)
|
| $71$ |
\( (T^{8} - 38 T^{7} + \cdots + 13845841)^{2} \)
|
| $73$ |
\( (T^{8} + 280 T^{6} + \cdots + 327184)^{2} \)
|
| $79$ |
\( (T^{8} + 48 T^{6} + \cdots + 2704)^{2} \)
|
| $83$ |
\( (T^{8} - 400 T^{6} + \cdots + 83759104)^{2} \)
|
| $89$ |
\( (T^{8} - 24 T^{7} + \cdots + 12780625)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 21\!\cdots\!81 \)
|
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