Newspace parameters
Level: | \( N \) | \(=\) | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1200.bg (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(32.6976317232\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 60) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 in terms of a root \(\nu\) of \( x^{4} + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 3 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 3\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 3\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).
\(n\) | \(401\) | \(577\) | \(751\) | \(901\) |
\(\chi(n)\) | \(1\) | \(-\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 |
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0 | −1.22474 | − | 1.22474i | 0 | 0 | 0 | 7.44949 | − | 7.44949i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||
193.2 | 0 | 1.22474 | + | 1.22474i | 0 | 0 | 0 | 2.55051 | − | 2.55051i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||
1057.1 | 0 | −1.22474 | + | 1.22474i | 0 | 0 | 0 | 7.44949 | + | 7.44949i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
1057.2 | 0 | 1.22474 | − | 1.22474i | 0 | 0 | 0 | 2.55051 | + | 2.55051i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1200.3.bg.o | 4 | |
4.b | odd | 2 | 1 | 300.3.k.a | 4 | ||
5.b | even | 2 | 1 | 240.3.bg.d | 4 | ||
5.c | odd | 4 | 1 | 240.3.bg.d | 4 | ||
5.c | odd | 4 | 1 | inner | 1200.3.bg.o | 4 | |
12.b | even | 2 | 1 | 900.3.l.b | 4 | ||
15.d | odd | 2 | 1 | 720.3.bh.f | 4 | ||
15.e | even | 4 | 1 | 720.3.bh.f | 4 | ||
20.d | odd | 2 | 1 | 60.3.k.a | ✓ | 4 | |
20.e | even | 4 | 1 | 60.3.k.a | ✓ | 4 | |
20.e | even | 4 | 1 | 300.3.k.a | 4 | ||
40.e | odd | 2 | 1 | 960.3.bg.b | 4 | ||
40.f | even | 2 | 1 | 960.3.bg.a | 4 | ||
40.i | odd | 4 | 1 | 960.3.bg.a | 4 | ||
40.k | even | 4 | 1 | 960.3.bg.b | 4 | ||
60.h | even | 2 | 1 | 180.3.l.b | 4 | ||
60.l | odd | 4 | 1 | 180.3.l.b | 4 | ||
60.l | odd | 4 | 1 | 900.3.l.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.3.k.a | ✓ | 4 | 20.d | odd | 2 | 1 | |
60.3.k.a | ✓ | 4 | 20.e | even | 4 | 1 | |
180.3.l.b | 4 | 60.h | even | 2 | 1 | ||
180.3.l.b | 4 | 60.l | odd | 4 | 1 | ||
240.3.bg.d | 4 | 5.b | even | 2 | 1 | ||
240.3.bg.d | 4 | 5.c | odd | 4 | 1 | ||
300.3.k.a | 4 | 4.b | odd | 2 | 1 | ||
300.3.k.a | 4 | 20.e | even | 4 | 1 | ||
720.3.bh.f | 4 | 15.d | odd | 2 | 1 | ||
720.3.bh.f | 4 | 15.e | even | 4 | 1 | ||
900.3.l.b | 4 | 12.b | even | 2 | 1 | ||
900.3.l.b | 4 | 60.l | odd | 4 | 1 | ||
960.3.bg.a | 4 | 40.f | even | 2 | 1 | ||
960.3.bg.a | 4 | 40.i | odd | 4 | 1 | ||
960.3.bg.b | 4 | 40.e | odd | 2 | 1 | ||
960.3.bg.b | 4 | 40.k | even | 4 | 1 | ||
1200.3.bg.o | 4 | 1.a | even | 1 | 1 | trivial | |
1200.3.bg.o | 4 | 5.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 20T_{7}^{3} + 200T_{7}^{2} - 760T_{7} + 1444 \)
acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 9 \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 1444 \)
$11$
\( (T^{2} - 8 T - 134)^{2} \)
$13$
\( T^{4} + 90000 \)
$17$
\( T^{4} - 40 T^{3} + 800 T^{2} + \cdots + 35344 \)
$19$
\( T^{4} + 1400 T^{2} + 250000 \)
$23$
\( T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 8464 \)
$29$
\( T^{4} + 1268 T^{2} + 111556 \)
$31$
\( (T^{2} + 8 T - 584)^{2} \)
$37$
\( T^{4} + 11664 \)
$41$
\( (T^{2} - 100 T + 1900)^{2} \)
$43$
\( T^{4} - 120 T^{3} + 7200 T^{2} + \cdots + 3069504 \)
$47$
\( T^{4} + 944784 \)
$53$
\( T^{4} - 200 T^{3} + \cdots + 20866624 \)
$59$
\( T^{4} + 8108 T^{2} + \cdots + 1833316 \)
$61$
\( (T^{2} + 156 T + 5484)^{2} \)
$67$
\( T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 53824 \)
$71$
\( (T^{2} - 40 T - 9200)^{2} \)
$73$
\( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 9132484 \)
$79$
\( T^{4} + 2352T^{2} + 576 \)
$83$
\( T^{4} + 240 T^{3} + \cdots + 13927824 \)
$89$
\( T^{4} + 43400 T^{2} + \cdots + 462250000 \)
$97$
\( T^{4} + 300 T^{3} + \cdots + 109872324 \)
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