Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,9,Mod(7,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.f (of order \(4\), degree \(2\), 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: | \(30.5533957546\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + \cdots + 66\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{20}\cdot 5^{8} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + \cdots + 66\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 96\!\cdots\!78 \nu^{15} + \cdots + 52\!\cdots\!00 ) / 28\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 65\!\cdots\!74 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 78\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39\!\cdots\!03 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 31\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!51 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 24\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 59\!\cdots\!66 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 52\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 23\!\cdots\!17 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 20\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 12\!\cdots\!57 \nu^{15} + \cdots + 95\!\cdots\!00 ) / 81\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 28\!\cdots\!79 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 40\!\cdots\!91 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 31\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 48\!\cdots\!57 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 31\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 10\!\cdots\!17 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 57\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 72\!\cdots\!26 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 38\!\cdots\!67 \nu^{15} + \cdots + 39\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 12\!\cdots\!02 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 61\!\cdots\!43 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} - 27\beta_{2} ) / 27 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 27\beta_{9} + 4\beta_{8} + 69\beta_{3} + 69\beta_{2} - 10934\beta_1 ) / 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 85 \beta_{6} + \cdots + 776 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 252 \beta_{14} - 54 \beta_{13} - 333 \beta_{12} + 333 \beta_{11} + 252 \beta_{10} + 4492 \beta_{7} + \cdots - 7495850 ) / 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 23787 \beta_{15} + 13815 \beta_{14} - 23787 \beta_{13} - 35649 \beta_{12} + 56979 \beta_{9} + \cdots + 25855038 ) / 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2538 \beta_{15} - 6274 \beta_{14} - 11011 \beta_{12} - 11011 \beta_{11} - 6274 \beta_{10} + \cdots + 206761164 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 18229455 \beta_{15} + 18229455 \beta_{13} - 36089793 \beta_{11} + 2015847 \beta_{10} + \cdots - 26691404094 ) / 27 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 63307764 \beta_{14} + 69910938 \beta_{13} + 205029171 \beta_{12} - 205029171 \beta_{11} + \cdots + 4287483101386 ) / 27 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 505840601 \beta_{15} + 193668291 \beta_{14} + 505840601 \beta_{13} + 1245732943 \beta_{12} + \cdots - 948332962462 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 65295387198 \beta_{15} - 20529981126 \beta_{14} + 118354382001 \beta_{12} + \cdots - 33\!\cdots\!20 \beta_1 ) / 27 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 10217224989423 \beta_{15} - 10217224989423 \beta_{13} + 30136591664217 \beta_{11} + \cdots + 23\!\cdots\!70 ) / 27 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2925541320320 \beta_{14} - 2149201323270 \beta_{13} - 1815338707165 \beta_{12} + 1815338707165 \beta_{11} + \cdots - 98\!\cdots\!54 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 76\!\cdots\!79 \beta_{15} + \cdots + 21\!\cdots\!34 ) / 27 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 49\!\cdots\!18 \beta_{15} + \cdots + 21\!\cdots\!52 \beta_1 ) / 27 \)
|
| \(\nu^{15}\) | \(=\) |
\( 21\!\cdots\!57 \beta_{15} + \cdots - 71\!\cdots\!78 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−19.4129 | − | 19.4129i | 33.0681 | − | 33.0681i | 497.724i | 0 | −1283.90 | −1858.12 | − | 1858.12i | 4692.57 | − | 4692.57i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −17.1630 | − | 17.1630i | −33.0681 | + | 33.0681i | 333.134i | 0 | 1135.09 | −1268.31 | − | 1268.31i | 1323.85 | − | 1323.85i | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −6.62030 | − | 6.62030i | −33.0681 | + | 33.0681i | − | 168.343i | 0 | 437.842 | 216.619 | + | 216.619i | −2809.28 | + | 2809.28i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −4.09140 | − | 4.09140i | 33.0681 | − | 33.0681i | − | 222.521i | 0 | −270.590 | 2635.24 | + | 2635.24i | −1957.82 | + | 1957.82i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −3.41326 | − | 3.41326i | 33.0681 | − | 33.0681i | − | 232.699i | 0 | −225.740 | −2986.31 | − | 2986.31i | −1668.06 | + | 1668.06i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | 11.1104 | + | 11.1104i | −33.0681 | + | 33.0681i | − | 9.11845i | 0 | −734.799 | 1159.26 | + | 1159.26i | 2945.57 | − | 2945.57i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | 18.3444 | + | 18.3444i | 33.0681 | − | 33.0681i | 417.032i | 0 | 1213.23 | 1693.91 | + | 1693.91i | −2954.03 | + | 2954.03i | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.8 | 21.2461 | + | 21.2461i | −33.0681 | + | 33.0681i | 646.792i | 0 | −1405.14 | −1862.28 | − | 1862.28i | −8302.80 | + | 8302.80i | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.1 | −19.4129 | + | 19.4129i | 33.0681 | + | 33.0681i | − | 497.724i | 0 | −1283.90 | −1858.12 | + | 1858.12i | 4692.57 | + | 4692.57i | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −17.1630 | + | 17.1630i | −33.0681 | − | 33.0681i | − | 333.134i | 0 | 1135.09 | −1268.31 | + | 1268.31i | 1323.85 | + | 1323.85i | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −6.62030 | + | 6.62030i | −33.0681 | − | 33.0681i | 168.343i | 0 | 437.842 | 216.619 | − | 216.619i | −2809.28 | − | 2809.28i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −4.09140 | + | 4.09140i | 33.0681 | + | 33.0681i | 222.521i | 0 | −270.590 | 2635.24 | − | 2635.24i | −1957.82 | − | 1957.82i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | −3.41326 | + | 3.41326i | 33.0681 | + | 33.0681i | 232.699i | 0 | −225.740 | −2986.31 | + | 2986.31i | −1668.06 | − | 1668.06i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 11.1104 | − | 11.1104i | −33.0681 | − | 33.0681i | 9.11845i | 0 | −734.799 | 1159.26 | − | 1159.26i | 2945.57 | + | 2945.57i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 18.3444 | − | 18.3444i | 33.0681 | + | 33.0681i | − | 417.032i | 0 | 1213.23 | 1693.91 | − | 1693.91i | −2954.03 | − | 2954.03i | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 21.2461 | − | 21.2461i | −33.0681 | − | 33.0681i | − | 646.792i | 0 | −1405.14 | −1862.28 | + | 1862.28i | −8302.80 | − | 8302.80i | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.9.f.e | 16 | |
| 5.b | even | 2 | 1 | 15.9.f.a | ✓ | 16 | |
| 5.c | odd | 4 | 1 | 15.9.f.a | ✓ | 16 | |
| 5.c | odd | 4 | 1 | inner | 75.9.f.e | 16 | |
| 15.d | odd | 2 | 1 | 45.9.g.c | 16 | ||
| 15.e | even | 4 | 1 | 45.9.g.c | 16 | ||
| 20.d | odd | 2 | 1 | 240.9.bg.d | 16 | ||
| 20.e | even | 4 | 1 | 240.9.bg.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.9.f.a | ✓ | 16 | 5.b | even | 2 | 1 | |
| 15.9.f.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
| 45.9.g.c | 16 | 15.d | odd | 2 | 1 | ||
| 45.9.g.c | 16 | 15.e | even | 4 | 1 | ||
| 75.9.f.e | 16 | 1.a | even | 1 | 1 | trivial | |
| 75.9.f.e | 16 | 5.c | odd | 4 | 1 | inner | |
| 240.9.bg.d | 16 | 20.d | odd | 2 | 1 | ||
| 240.9.bg.d | 16 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 5820 T_{2}^{13} + 1126741 T_{2}^{12} + 3704100 T_{2}^{11} + 16936200 T_{2}^{10} + \cdots + 45\!\cdots\!56 \)
acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 45\!\cdots\!56 \)
|
| $3$ |
\( (T^{4} + 4782969)^{4} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 55\!\cdots\!00 \)
|
| $11$ |
\( (T^{8} + \cdots + 79\!\cdots\!76)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 23\!\cdots\!16 \)
|
| $17$ |
\( T^{16} + \cdots + 16\!\cdots\!56 \)
|
| $19$ |
\( T^{16} + \cdots + 82\!\cdots\!00 \)
|
| $23$ |
\( T^{16} + \cdots + 43\!\cdots\!00 \)
|
| $29$ |
\( T^{16} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( (T^{8} + \cdots - 47\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 10\!\cdots\!00 \)
|
| $41$ |
\( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 68\!\cdots\!16 \)
|
| $47$ |
\( T^{16} + \cdots + 78\!\cdots\!16 \)
|
| $53$ |
\( T^{16} + \cdots + 16\!\cdots\!00 \)
|
| $59$ |
\( T^{16} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( (T^{8} + \cdots + 86\!\cdots\!36)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 54\!\cdots\!96 \)
|
| $71$ |
\( (T^{8} + \cdots + 36\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 81\!\cdots\!00 \)
|
| $79$ |
\( T^{16} + \cdots + 43\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 78\!\cdots\!56 \)
|
| $89$ |
\( T^{16} + \cdots + 97\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 60\!\cdots\!36 \)
|
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