Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-7}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 56) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
209.1 |
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0 | 0 | 0 | 0 | 0 | −7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||
209.2 | 0 | 0 | 0 | 0 | 0 | −7.00000 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
8.b | even | 2 | 1 | inner |
56.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.3.h.a | 2 | |
3.b | odd | 2 | 1 | 2016.3.l.b | 2 | ||
4.b | odd | 2 | 1 | 56.3.h.b | ✓ | 2 | |
7.b | odd | 2 | 1 | CM | 224.3.h.a | 2 | |
8.b | even | 2 | 1 | inner | 224.3.h.a | 2 | |
8.d | odd | 2 | 1 | 56.3.h.b | ✓ | 2 | |
12.b | even | 2 | 1 | 504.3.l.b | 2 | ||
21.c | even | 2 | 1 | 2016.3.l.b | 2 | ||
24.f | even | 2 | 1 | 504.3.l.b | 2 | ||
24.h | odd | 2 | 1 | 2016.3.l.b | 2 | ||
28.d | even | 2 | 1 | 56.3.h.b | ✓ | 2 | |
28.f | even | 6 | 2 | 392.3.j.b | 4 | ||
28.g | odd | 6 | 2 | 392.3.j.b | 4 | ||
56.e | even | 2 | 1 | 56.3.h.b | ✓ | 2 | |
56.h | odd | 2 | 1 | inner | 224.3.h.a | 2 | |
56.k | odd | 6 | 2 | 392.3.j.b | 4 | ||
56.m | even | 6 | 2 | 392.3.j.b | 4 | ||
84.h | odd | 2 | 1 | 504.3.l.b | 2 | ||
168.e | odd | 2 | 1 | 504.3.l.b | 2 | ||
168.i | even | 2 | 1 | 2016.3.l.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.3.h.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
56.3.h.b | ✓ | 2 | 8.d | odd | 2 | 1 | |
56.3.h.b | ✓ | 2 | 28.d | even | 2 | 1 | |
56.3.h.b | ✓ | 2 | 56.e | even | 2 | 1 | |
224.3.h.a | 2 | 1.a | even | 1 | 1 | trivial | |
224.3.h.a | 2 | 7.b | odd | 2 | 1 | CM | |
224.3.h.a | 2 | 8.b | even | 2 | 1 | inner | |
224.3.h.a | 2 | 56.h | odd | 2 | 1 | inner | |
392.3.j.b | 4 | 28.f | even | 6 | 2 | ||
392.3.j.b | 4 | 28.g | odd | 6 | 2 | ||
392.3.j.b | 4 | 56.k | odd | 6 | 2 | ||
392.3.j.b | 4 | 56.m | even | 6 | 2 | ||
504.3.l.b | 2 | 12.b | even | 2 | 1 | ||
504.3.l.b | 2 | 24.f | even | 2 | 1 | ||
504.3.l.b | 2 | 84.h | odd | 2 | 1 | ||
504.3.l.b | 2 | 168.e | odd | 2 | 1 | ||
2016.3.l.b | 2 | 3.b | odd | 2 | 1 | ||
2016.3.l.b | 2 | 21.c | even | 2 | 1 | ||
2016.3.l.b | 2 | 24.h | odd | 2 | 1 | ||
2016.3.l.b | 2 | 168.i | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( (T + 7)^{2} \)
$11$
\( T^{2} + 448 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( (T + 18)^{2} \)
$29$
\( T^{2} + 448 \)
$31$
\( T^{2} \)
$37$
\( T^{2} + 4032 \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 4032 \)
$47$
\( T^{2} \)
$53$
\( T^{2} + 11200 \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} + 4032 \)
$71$
\( (T - 114)^{2} \)
$73$
\( T^{2} \)
$79$
\( (T + 94)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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