Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(209.608008215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 2844.00i | 59049.0i | −5.99118e6 | 0 | 1.67935e8 | 3.63304e8i | 1.10746e10i | −3.48678e9 | 0 | |||||||||||||||||||||||
| 49.2 | 2844.00i | − | 59049.0i | −5.99118e6 | 0 | 1.67935e8 | − | 3.63304e8i | − | 1.10746e10i | −3.48678e9 | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.22.b.a | 2 | |
| 5.b | even | 2 | 1 | inner | 75.22.b.a | 2 | |
| 5.c | odd | 4 | 1 | 3.22.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 75.22.a.c | 1 | ||
| 15.e | even | 4 | 1 | 9.22.a.d | 1 | ||
| 20.e | even | 4 | 1 | 48.22.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.22.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 9.22.a.d | 1 | 15.e | even | 4 | 1 | ||
| 48.22.a.e | 1 | 20.e | even | 4 | 1 | ||
| 75.22.a.c | 1 | 5.c | odd | 4 | 1 | ||
| 75.22.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.a | 2 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8088336 \)
acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 8088336 \)
$3$
\( T^{2} + 3486784401 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 13\!\cdots\!00 \)
$11$
\( (T - 14581833156)^{2} \)
$13$
\( T^{2} + 12\!\cdots\!04 \)
$17$
\( T^{2} + 73\!\cdots\!24 \)
$19$
\( (T - 29202939273796)^{2} \)
$23$
\( T^{2} + 24\!\cdots\!00 \)
$29$
\( (T + 24\!\cdots\!58)^{2} \)
$31$
\( (T - 22\!\cdots\!00)^{2} \)
$37$
\( T^{2} + 94\!\cdots\!00 \)
$41$
\( (T + 10\!\cdots\!30)^{2} \)
$43$
\( T^{2} + 27\!\cdots\!76 \)
$47$
\( T^{2} + 44\!\cdots\!64 \)
$53$
\( T^{2} + 18\!\cdots\!00 \)
$59$
\( (T + 55\!\cdots\!16)^{2} \)
$61$
\( (T + 71\!\cdots\!02)^{2} \)
$67$
\( T^{2} + 24\!\cdots\!44 \)
$71$
\( (T - 26\!\cdots\!32)^{2} \)
$73$
\( T^{2} + 18\!\cdots\!00 \)
$79$
\( (T - 16\!\cdots\!40)^{2} \)
$83$
\( T^{2} + 29\!\cdots\!84 \)
$89$
\( (T - 31\!\cdots\!86)^{2} \)
$97$
\( T^{2} + 90\!\cdots\!24 \)
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