Newspace parameters
Level: | \( N \) | \(=\) | \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1900.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(15.1715763840\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 380) |
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 \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) | \(951\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
201.1 |
|
0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | 4.00000 | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||
501.1 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | 4.00000 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1900.2.i.b | 2 | |
5.b | even | 2 | 1 | 380.2.i.a | ✓ | 2 | |
5.c | odd | 4 | 2 | 1900.2.s.b | 4 | ||
15.d | odd | 2 | 1 | 3420.2.t.b | 2 | ||
19.c | even | 3 | 1 | inner | 1900.2.i.b | 2 | |
20.d | odd | 2 | 1 | 1520.2.q.g | 2 | ||
95.h | odd | 6 | 1 | 7220.2.a.a | 1 | ||
95.i | even | 6 | 1 | 380.2.i.a | ✓ | 2 | |
95.i | even | 6 | 1 | 7220.2.a.e | 1 | ||
95.m | odd | 12 | 2 | 1900.2.s.b | 4 | ||
285.n | odd | 6 | 1 | 3420.2.t.b | 2 | ||
380.p | odd | 6 | 1 | 1520.2.q.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.i.a | ✓ | 2 | 5.b | even | 2 | 1 | |
380.2.i.a | ✓ | 2 | 95.i | even | 6 | 1 | |
1520.2.q.g | 2 | 20.d | odd | 2 | 1 | ||
1520.2.q.g | 2 | 380.p | odd | 6 | 1 | ||
1900.2.i.b | 2 | 1.a | even | 1 | 1 | trivial | |
1900.2.i.b | 2 | 19.c | even | 3 | 1 | inner | |
1900.2.s.b | 4 | 5.c | odd | 4 | 2 | ||
1900.2.s.b | 4 | 95.m | odd | 12 | 2 | ||
3420.2.t.b | 2 | 15.d | odd | 2 | 1 | ||
3420.2.t.b | 2 | 285.n | odd | 6 | 1 | ||
7220.2.a.a | 1 | 95.h | odd | 6 | 1 | ||
7220.2.a.e | 1 | 95.i | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 2T + 4 \)
$5$
\( T^{2} \)
$7$
\( (T - 4)^{2} \)
$11$
\( (T + 3)^{2} \)
$13$
\( T^{2} - 6T + 36 \)
$17$
\( T^{2} - 2T + 4 \)
$19$
\( T^{2} - 7T + 19 \)
$23$
\( T^{2} - 4T + 16 \)
$29$
\( T^{2} + T + 1 \)
$31$
\( (T + 5)^{2} \)
$37$
\( (T - 4)^{2} \)
$41$
\( T^{2} + 2T + 4 \)
$43$
\( T^{2} \)
$47$
\( T^{2} - 6T + 36 \)
$53$
\( T^{2} + 6T + 36 \)
$59$
\( T^{2} - T + 1 \)
$61$
\( T^{2} - 7T + 49 \)
$67$
\( T^{2} + 14T + 196 \)
$71$
\( T^{2} + 15T + 225 \)
$73$
\( T^{2} - 12T + 144 \)
$79$
\( T^{2} - T + 1 \)
$83$
\( (T + 16)^{2} \)
$89$
\( T^{2} + 17T + 289 \)
$97$
\( T^{2} - 12T + 144 \)
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