Newspace parameters
Level: | \( N \) | \(=\) | \( 85 = 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 85.j (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.678728417181\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).
\(n\) | \(52\) | \(71\) |
\(\chi(n)\) | \(-1\) | \(i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 |
|
1.00000 | 1.00000 | + | 1.00000i | −1.00000 | −1.00000 | + | 2.00000i | 1.00000 | + | 1.00000i | 3.00000 | − | 3.00000i | −3.00000 | − | 1.00000i | −1.00000 | + | 2.00000i | |||||||||||||
64.1 | 1.00000 | 1.00000 | − | 1.00000i | −1.00000 | −1.00000 | − | 2.00000i | 1.00000 | − | 1.00000i | 3.00000 | + | 3.00000i | −3.00000 | 1.00000i | −1.00000 | − | 2.00000i | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.j | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 85.2.j.b | yes | 2 |
3.b | odd | 2 | 1 | 765.2.t.a | 2 | ||
5.b | even | 2 | 1 | 85.2.j.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 425.2.e.a | 2 | ||
5.c | odd | 4 | 1 | 425.2.e.b | 2 | ||
15.d | odd | 2 | 1 | 765.2.t.b | 2 | ||
17.c | even | 4 | 1 | 85.2.j.a | ✓ | 2 | |
17.d | even | 8 | 2 | 1445.2.b.a | 4 | ||
51.f | odd | 4 | 1 | 765.2.t.b | 2 | ||
85.f | odd | 4 | 1 | 425.2.e.a | 2 | ||
85.i | odd | 4 | 1 | 425.2.e.b | 2 | ||
85.j | even | 4 | 1 | inner | 85.2.j.b | yes | 2 |
85.k | odd | 8 | 2 | 7225.2.a.i | 2 | ||
85.m | even | 8 | 2 | 1445.2.b.a | 4 | ||
85.n | odd | 8 | 2 | 7225.2.a.p | 2 | ||
255.i | odd | 4 | 1 | 765.2.t.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
85.2.j.a | ✓ | 2 | 5.b | even | 2 | 1 | |
85.2.j.a | ✓ | 2 | 17.c | even | 4 | 1 | |
85.2.j.b | yes | 2 | 1.a | even | 1 | 1 | trivial |
85.2.j.b | yes | 2 | 85.j | even | 4 | 1 | inner |
425.2.e.a | 2 | 5.c | odd | 4 | 1 | ||
425.2.e.a | 2 | 85.f | odd | 4 | 1 | ||
425.2.e.b | 2 | 5.c | odd | 4 | 1 | ||
425.2.e.b | 2 | 85.i | odd | 4 | 1 | ||
765.2.t.a | 2 | 3.b | odd | 2 | 1 | ||
765.2.t.a | 2 | 255.i | odd | 4 | 1 | ||
765.2.t.b | 2 | 15.d | odd | 2 | 1 | ||
765.2.t.b | 2 | 51.f | odd | 4 | 1 | ||
1445.2.b.a | 4 | 17.d | even | 8 | 2 | ||
1445.2.b.a | 4 | 85.m | even | 8 | 2 | ||
7225.2.a.i | 2 | 85.k | odd | 8 | 2 | ||
7225.2.a.p | 2 | 85.n | odd | 8 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(85, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} - 2T + 2 \)
$5$
\( T^{2} + 2T + 5 \)
$7$
\( T^{2} - 6T + 18 \)
$11$
\( T^{2} + 6T + 18 \)
$13$
\( T^{2} \)
$17$
\( T^{2} - 2T + 17 \)
$19$
\( T^{2} + 36 \)
$23$
\( T^{2} + 2T + 2 \)
$29$
\( T^{2} + 6T + 18 \)
$31$
\( T^{2} + 2T + 2 \)
$37$
\( T^{2} + 6T + 18 \)
$41$
\( T^{2} + 6T + 18 \)
$43$
\( (T - 12)^{2} \)
$47$
\( T^{2} + 4 \)
$53$
\( (T + 2)^{2} \)
$59$
\( T^{2} + 36 \)
$61$
\( T^{2} - 2T + 2 \)
$67$
\( T^{2} + 36 \)
$71$
\( T^{2} - 6T + 18 \)
$73$
\( T^{2} + 6T + 18 \)
$79$
\( T^{2} + 14T + 98 \)
$83$
\( (T - 4)^{2} \)
$89$
\( (T - 6)^{2} \)
$97$
\( T^{2} + 6T + 18 \)
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