Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.h (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.79663404548\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 25) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
0.500000 | − | 1.53884i | 0 | −0.500000 | − | 0.363271i | 0.690983 | + | 2.12663i | 0 | 0.618034 | 1.80902 | − | 1.31433i | 0 | 3.61803 | ||||||||||||||||||||||
| 91.1 | 0.500000 | + | 0.363271i | 0 | −0.500000 | − | 1.53884i | 1.80902 | − | 1.31433i | 0 | −1.61803 | 0.690983 | − | 2.12663i | 0 | 1.38197 | |||||||||||||||||||||||
| 136.1 | 0.500000 | − | 0.363271i | 0 | −0.500000 | + | 1.53884i | 1.80902 | + | 1.31433i | 0 | −1.61803 | 0.690983 | + | 2.12663i | 0 | 1.38197 | |||||||||||||||||||||||
| 181.1 | 0.500000 | + | 1.53884i | 0 | −0.500000 | + | 0.363271i | 0.690983 | − | 2.12663i | 0 | 0.618034 | 1.80902 | + | 1.31433i | 0 | 3.61803 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.2.h.b | 4 | |
| 3.b | odd | 2 | 1 | 25.2.d.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 400.2.u.b | 4 | ||
| 15.d | odd | 2 | 1 | 125.2.d.a | 4 | ||
| 15.e | even | 4 | 2 | 125.2.e.a | 8 | ||
| 25.d | even | 5 | 1 | inner | 225.2.h.b | 4 | |
| 25.d | even | 5 | 1 | 5625.2.a.f | 2 | ||
| 25.e | even | 10 | 1 | 5625.2.a.d | 2 | ||
| 75.h | odd | 10 | 1 | 125.2.d.a | 4 | ||
| 75.h | odd | 10 | 1 | 625.2.a.c | 2 | ||
| 75.h | odd | 10 | 2 | 625.2.d.b | 4 | ||
| 75.j | odd | 10 | 1 | 25.2.d.a | ✓ | 4 | |
| 75.j | odd | 10 | 1 | 625.2.a.b | 2 | ||
| 75.j | odd | 10 | 2 | 625.2.d.h | 4 | ||
| 75.l | even | 20 | 2 | 125.2.e.a | 8 | ||
| 75.l | even | 20 | 2 | 625.2.b.a | 4 | ||
| 75.l | even | 20 | 4 | 625.2.e.c | 8 | ||
| 300.n | even | 10 | 1 | 400.2.u.b | 4 | ||
| 300.n | even | 10 | 1 | 10000.2.a.c | 2 | ||
| 300.r | even | 10 | 1 | 10000.2.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.2.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 25.2.d.a | ✓ | 4 | 75.j | odd | 10 | 1 | |
| 125.2.d.a | 4 | 15.d | odd | 2 | 1 | ||
| 125.2.d.a | 4 | 75.h | odd | 10 | 1 | ||
| 125.2.e.a | 8 | 15.e | even | 4 | 2 | ||
| 125.2.e.a | 8 | 75.l | even | 20 | 2 | ||
| 225.2.h.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.2.h.b | 4 | 25.d | even | 5 | 1 | inner | |
| 400.2.u.b | 4 | 12.b | even | 2 | 1 | ||
| 400.2.u.b | 4 | 300.n | even | 10 | 1 | ||
| 625.2.a.b | 2 | 75.j | odd | 10 | 1 | ||
| 625.2.a.c | 2 | 75.h | odd | 10 | 1 | ||
| 625.2.b.a | 4 | 75.l | even | 20 | 2 | ||
| 625.2.d.b | 4 | 75.h | odd | 10 | 2 | ||
| 625.2.d.h | 4 | 75.j | odd | 10 | 2 | ||
| 625.2.e.c | 8 | 75.l | even | 20 | 4 | ||
| 5625.2.a.d | 2 | 25.e | even | 10 | 1 | ||
| 5625.2.a.f | 2 | 25.d | even | 5 | 1 | ||
| 10000.2.a.c | 2 | 300.n | even | 10 | 1 | ||
| 10000.2.a.l | 2 | 300.r | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \)
$7$
\( (T^{2} + T - 1)^{2} \)
$11$
\( T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 \)
$13$
\( T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81 \)
$17$
\( T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16 \)
$19$
\( T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25 \)
$23$
\( T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 961 \)
$29$
\( T^{4} + 5 T^{3} + 10 T^{2} + 25 \)
$31$
\( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \)
$37$
\( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \)
$41$
\( T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16 \)
$43$
\( (T^{2} + 3 T - 9)^{2} \)
$47$
\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \)
$53$
\( T^{4} + 9 T^{3} + 61 T^{2} + 209 T + 361 \)
$59$
\( T^{4} + 90 T^{2} + 675 T + 2025 \)
$61$
\( T^{4} - 13 T^{3} + 139 T^{2} + \cdots + 1681 \)
$67$
\( T^{4} + 2 T^{3} + 64 T^{2} + \cdots + 1936 \)
$71$
\( T^{4} + 8 T^{3} + 34 T^{2} + 87 T + 841 \)
$73$
\( T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561 \)
$79$
\( T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625 \)
$83$
\( T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121 \)
$89$
\( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \)
$97$
\( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \)
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