Newspace parameters
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(14.4554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 116.1 |
|
−0.500000 | − | 0.866025i | 4.00000 | − | 6.92820i | 3.50000 | − | 6.06218i | 2.50000 | + | 4.33013i | −8.00000 | 0 | −15.0000 | −18.5000 | − | 32.0429i | 2.50000 | − | 4.33013i | ||||||||||||
| 226.1 | −0.500000 | + | 0.866025i | 4.00000 | + | 6.92820i | 3.50000 | + | 6.06218i | 2.50000 | − | 4.33013i | −8.00000 | 0 | −15.0000 | −18.5000 | + | 32.0429i | 2.50000 | + | 4.33013i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.e.e | 2 | |
| 7.b | odd | 2 | 1 | 245.4.e.b | 2 | ||
| 7.c | even | 3 | 1 | 35.4.a.a | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 245.4.e.e | 2 | |
| 7.d | odd | 6 | 1 | 245.4.a.d | 1 | ||
| 7.d | odd | 6 | 1 | 245.4.e.b | 2 | ||
| 21.g | even | 6 | 1 | 2205.4.a.i | 1 | ||
| 21.h | odd | 6 | 1 | 315.4.a.c | 1 | ||
| 28.g | odd | 6 | 1 | 560.4.a.p | 1 | ||
| 35.i | odd | 6 | 1 | 1225.4.a.e | 1 | ||
| 35.j | even | 6 | 1 | 175.4.a.a | 1 | ||
| 35.l | odd | 12 | 2 | 175.4.b.a | 2 | ||
| 56.k | odd | 6 | 1 | 2240.4.a.b | 1 | ||
| 56.p | even | 6 | 1 | 2240.4.a.bk | 1 | ||
| 105.o | odd | 6 | 1 | 1575.4.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
| 175.4.a.a | 1 | 35.j | even | 6 | 1 | ||
| 175.4.b.a | 2 | 35.l | odd | 12 | 2 | ||
| 245.4.a.d | 1 | 7.d | odd | 6 | 1 | ||
| 245.4.e.b | 2 | 7.b | odd | 2 | 1 | ||
| 245.4.e.b | 2 | 7.d | odd | 6 | 1 | ||
| 245.4.e.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 245.4.e.e | 2 | 7.c | even | 3 | 1 | inner | |
| 315.4.a.c | 1 | 21.h | odd | 6 | 1 | ||
| 560.4.a.p | 1 | 28.g | odd | 6 | 1 | ||
| 1225.4.a.e | 1 | 35.i | odd | 6 | 1 | ||
| 1575.4.a.g | 1 | 105.o | odd | 6 | 1 | ||
| 2205.4.a.i | 1 | 21.g | even | 6 | 1 | ||
| 2240.4.a.b | 1 | 56.k | odd | 6 | 1 | ||
| 2240.4.a.bk | 1 | 56.p | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{2} + T_{2} + 1 \)
|
|
\( T_{3}^{2} - 8T_{3} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T + 1 \)
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| $3$ |
\( T^{2} - 8T + 64 \)
|
| $5$ |
\( T^{2} - 5T + 25 \)
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| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 12T + 144 \)
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| $13$ |
\( (T + 78)^{2} \)
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| $17$ |
\( T^{2} - 94T + 8836 \)
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| $19$ |
\( T^{2} + 40T + 1600 \)
|
| $23$ |
\( T^{2} + 32T + 1024 \)
|
| $29$ |
\( (T + 50)^{2} \)
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| $31$ |
\( T^{2} - 248T + 61504 \)
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| $37$ |
\( T^{2} - 434T + 188356 \)
|
| $41$ |
\( (T - 402)^{2} \)
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| $43$ |
\( (T + 68)^{2} \)
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| $47$ |
\( T^{2} + 536T + 287296 \)
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| $53$ |
\( T^{2} + 22T + 484 \)
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| $59$ |
\( T^{2} - 560T + 313600 \)
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| $61$ |
\( T^{2} - 278T + 77284 \)
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| $67$ |
\( T^{2} - 164T + 26896 \)
|
| $71$ |
\( (T - 672)^{2} \)
|
| $73$ |
\( T^{2} + 82T + 6724 \)
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| $79$ |
\( T^{2} - 1000 T + 1000000 \)
|
| $83$ |
\( (T + 448)^{2} \)
|
| $89$ |
\( T^{2} - 870T + 756900 \)
|
| $97$ |
\( (T - 1026)^{2} \)
|
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