Newspace parameters
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(16.4850559938\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 55) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{10}^{3} + \zeta_{10}^{2} \)
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| \(\beta_{2}\) | \(=\) |
\( -\zeta_{10}^{3} + \zeta_{10}^{2} \)
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| \(\beta_{3}\) | \(=\) |
\( \zeta_{10}^{3} - \zeta_{10}^{2} + 2\zeta_{10} - 1 \)
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| \(\zeta_{10}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
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| \(\zeta_{10}^{2}\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
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| \(\zeta_{10}^{3}\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(486\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
− | 3.07768i | −4.00000 | −5.47214 | 2.23607 | 12.3107i | − | 8.05748i | 4.53077i | 7.00000 | − | 6.88191i | |||||||||||||||||||||||||||
| 241.2 | − | 0.726543i | −4.00000 | 3.47214 | −2.23607 | 2.90617i | − | 0.277515i | − | 5.42882i | 7.00000 | 1.62460i | ||||||||||||||||||||||||||||
| 241.3 | 0.726543i | −4.00000 | 3.47214 | −2.23607 | − | 2.90617i | 0.277515i | 5.42882i | 7.00000 | − | 1.62460i | |||||||||||||||||||||||||||||
| 241.4 | 3.07768i | −4.00000 | −5.47214 | 2.23607 | − | 12.3107i | 8.05748i | − | 4.53077i | 7.00000 | 6.88191i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 605.3.c.a | 4 | |
| 11.b | odd | 2 | 1 | inner | 605.3.c.a | 4 | |
| 11.c | even | 5 | 1 | 55.3.i.a | ✓ | 4 | |
| 11.d | odd | 10 | 1 | 55.3.i.a | ✓ | 4 | |
| 55.h | odd | 10 | 1 | 275.3.x.d | 4 | ||
| 55.j | even | 10 | 1 | 275.3.x.d | 4 | ||
| 55.k | odd | 20 | 2 | 275.3.q.c | 8 | ||
| 55.l | even | 20 | 2 | 275.3.q.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.3.i.a | ✓ | 4 | 11.c | even | 5 | 1 | |
| 55.3.i.a | ✓ | 4 | 11.d | odd | 10 | 1 | |
| 275.3.q.c | 8 | 55.k | odd | 20 | 2 | ||
| 275.3.q.c | 8 | 55.l | even | 20 | 2 | ||
| 275.3.x.d | 4 | 55.h | odd | 10 | 1 | ||
| 275.3.x.d | 4 | 55.j | even | 10 | 1 | ||
| 605.3.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 605.3.c.a | 4 | 11.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(605, [\chi])\):
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\( T_{2}^{4} + 10T_{2}^{2} + 5 \)
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\( T_{3} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 10T^{2} + 5 \)
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| $3$ |
\( (T + 4)^{4} \)
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| $5$ |
\( (T^{2} - 5)^{2} \)
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| $7$ |
\( T^{4} + 65T^{2} + 5 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} + 325 T^{2} + 25205 \)
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| $17$ |
\( T^{4} + 1000 T^{2} + 242000 \)
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| $19$ |
\( T^{4} + 905 T^{2} + 59405 \)
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| $23$ |
\( (T^{2} + 53 T + 701)^{2} \)
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| $29$ |
\( T^{4} + 1640 T^{2} + 403280 \)
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| $31$ |
\( (T^{2} - 24 T - 356)^{2} \)
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| $37$ |
\( (T^{2} + 7 T - 269)^{2} \)
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| $41$ |
\( T^{4} + 3445 T^{2} + 1017005 \)
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| $43$ |
\( T^{4} + 2600 T^{2} + 1682000 \)
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| $47$ |
\( (T^{2} - 133 T + 4061)^{2} \)
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| $53$ |
\( (T^{2} + 13 T - 2269)^{2} \)
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| $59$ |
\( (T^{2} - 161 T + 6029)^{2} \)
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| $61$ |
\( T^{4} + 3940 T^{2} + 3494480 \)
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| $67$ |
\( (T^{2} - 58 T + 716)^{2} \)
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| $71$ |
\( (T^{2} - 34 T + 164)^{2} \)
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| $73$ |
\( T^{4} + 12580 T^{2} + 26083280 \)
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| $79$ |
\( T^{4} + 18500 T^{2} + 20402000 \)
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| $83$ |
\( T^{4} + 37000 T^{2} + 334562000 \)
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| $89$ |
\( (T^{2} + 39 T - 11381)^{2} \)
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| $97$ |
\( (T^{2} + 42 T - 4364)^{2} \)
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