Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(38.6276877123\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
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 |
|
− | 36.0000i | 81.0000i | −784.000 | 0 | 2916.00 | − | 4480.00i | 9792.00i | −6561.00 | 0 | ||||||||||||||||||||||
49.2 | 36.0000i | − | 81.0000i | −784.000 | 0 | 2916.00 | 4480.00i | − | 9792.00i | −6561.00 | 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 | 75.10.b.a | 2 | |
3.b | odd | 2 | 1 | 225.10.b.a | 2 | ||
5.b | even | 2 | 1 | inner | 75.10.b.a | 2 | |
5.c | odd | 4 | 1 | 3.10.a.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 75.10.a.d | 1 | ||
15.d | odd | 2 | 1 | 225.10.b.a | 2 | ||
15.e | even | 4 | 1 | 9.10.a.c | 1 | ||
15.e | even | 4 | 1 | 225.10.a.a | 1 | ||
20.e | even | 4 | 1 | 48.10.a.e | 1 | ||
35.f | even | 4 | 1 | 147.10.a.a | 1 | ||
40.i | odd | 4 | 1 | 192.10.a.m | 1 | ||
40.k | even | 4 | 1 | 192.10.a.f | 1 | ||
45.k | odd | 12 | 2 | 81.10.c.e | 2 | ||
45.l | even | 12 | 2 | 81.10.c.a | 2 | ||
55.e | even | 4 | 1 | 363.10.a.b | 1 | ||
60.l | odd | 4 | 1 | 144.10.a.l | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3.10.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
9.10.a.c | 1 | 15.e | even | 4 | 1 | ||
48.10.a.e | 1 | 20.e | even | 4 | 1 | ||
75.10.a.d | 1 | 5.c | odd | 4 | 1 | ||
75.10.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
75.10.b.a | 2 | 5.b | even | 2 | 1 | inner | |
81.10.c.a | 2 | 45.l | even | 12 | 2 | ||
81.10.c.e | 2 | 45.k | odd | 12 | 2 | ||
144.10.a.l | 1 | 60.l | odd | 4 | 1 | ||
147.10.a.a | 1 | 35.f | even | 4 | 1 | ||
192.10.a.f | 1 | 40.k | even | 4 | 1 | ||
192.10.a.m | 1 | 40.i | odd | 4 | 1 | ||
225.10.a.a | 1 | 15.e | even | 4 | 1 | ||
225.10.b.a | 2 | 3.b | odd | 2 | 1 | ||
225.10.b.a | 2 | 15.d | odd | 2 | 1 | ||
363.10.a.b | 1 | 55.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 1296 \)
acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 1296 \)
$3$
\( T^{2} + 6561 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 20070400 \)
$11$
\( (T - 1476)^{2} \)
$13$
\( T^{2} + 22958916484 \)
$17$
\( T^{2} + 11699018244 \)
$19$
\( (T + 593084)^{2} \)
$23$
\( T^{2} + 939891470400 \)
$29$
\( (T - 6642522)^{2} \)
$31$
\( (T - 7070600)^{2} \)
$37$
\( T^{2} + 55836911208100 \)
$41$
\( (T + 4350150)^{2} \)
$43$
\( T^{2} + 18998405168656 \)
$47$
\( T^{2} + \cdots + 801413522325504 \)
$53$
\( T^{2} + \cdots + 259587199124100 \)
$59$
\( (T - 86075964)^{2} \)
$61$
\( (T - 32213918)^{2} \)
$67$
\( T^{2} + 99\!\cdots\!04 \)
$71$
\( (T + 44170488)^{2} \)
$73$
\( T^{2} + \cdots + 555103285996900 \)
$79$
\( (T - 401754760)^{2} \)
$83$
\( T^{2} + 55\!\cdots\!64 \)
$89$
\( (T + 769871034)^{2} \)
$97$
\( T^{2} + 82\!\cdots\!24 \)
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