Newspace parameters
Level: | \( N \) | \(=\) | \( 1156 = 2^{2} \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1156.g (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.576919154604\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
Defining polynomial: |
\( x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 68) |
Projective image: | \(D_{2}\) |
Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Artin image: | $\OD_{32}$ |
Artin field: | Galois closure of 16.8.732780301186512843008.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).
\(n\) | \(579\) | \(581\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
155.1 |
|
0.707107 | + | 0.707107i | 0 | 1.00000i | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 0 | ||||||||||||||||||||||||
179.1 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 0 | ||||||||||||||||||||||||
399.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 0 | ||||||||||||||||||||||||
423.1 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 0 | 0 | 0 | 0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
17.b | even | 2 | 1 | RM by \(\Q(\sqrt{17}) \) |
68.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-17}) \) |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
68.f | odd | 4 | 2 | inner |
68.g | odd | 8 | 4 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1156.1.g.a | 4 | |
4.b | odd | 2 | 1 | CM | 1156.1.g.a | 4 | |
17.b | even | 2 | 1 | RM | 1156.1.g.a | 4 | |
17.c | even | 4 | 2 | inner | 1156.1.g.a | 4 | |
17.d | even | 8 | 4 | inner | 1156.1.g.a | 4 | |
17.e | odd | 16 | 2 | 68.1.d.a | ✓ | 1 | |
17.e | odd | 16 | 2 | 1156.1.c.a | 1 | ||
17.e | odd | 16 | 4 | 1156.1.f.a | 2 | ||
51.i | even | 16 | 2 | 612.1.e.a | 1 | ||
68.d | odd | 2 | 1 | CM | 1156.1.g.a | 4 | |
68.f | odd | 4 | 2 | inner | 1156.1.g.a | 4 | |
68.g | odd | 8 | 4 | inner | 1156.1.g.a | 4 | |
68.i | even | 16 | 2 | 68.1.d.a | ✓ | 1 | |
68.i | even | 16 | 2 | 1156.1.c.a | 1 | ||
68.i | even | 16 | 4 | 1156.1.f.a | 2 | ||
85.o | even | 16 | 2 | 1700.1.d.b | 2 | ||
85.p | odd | 16 | 2 | 1700.1.h.d | 1 | ||
85.r | even | 16 | 2 | 1700.1.d.b | 2 | ||
119.p | even | 16 | 2 | 3332.1.g.a | 1 | ||
119.s | even | 48 | 4 | 3332.1.o.d | 2 | ||
119.t | odd | 48 | 4 | 3332.1.o.c | 2 | ||
136.q | odd | 16 | 2 | 1088.1.g.a | 1 | ||
136.s | even | 16 | 2 | 1088.1.g.a | 1 | ||
204.t | odd | 16 | 2 | 612.1.e.a | 1 | ||
340.bc | odd | 16 | 2 | 1700.1.d.b | 2 | ||
340.bg | even | 16 | 2 | 1700.1.h.d | 1 | ||
340.bj | odd | 16 | 2 | 1700.1.d.b | 2 | ||
476.bf | odd | 16 | 2 | 3332.1.g.a | 1 | ||
476.bk | odd | 48 | 4 | 3332.1.o.d | 2 | ||
476.bm | even | 48 | 4 | 3332.1.o.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
68.1.d.a | ✓ | 1 | 17.e | odd | 16 | 2 | |
68.1.d.a | ✓ | 1 | 68.i | even | 16 | 2 | |
612.1.e.a | 1 | 51.i | even | 16 | 2 | ||
612.1.e.a | 1 | 204.t | odd | 16 | 2 | ||
1088.1.g.a | 1 | 136.q | odd | 16 | 2 | ||
1088.1.g.a | 1 | 136.s | even | 16 | 2 | ||
1156.1.c.a | 1 | 17.e | odd | 16 | 2 | ||
1156.1.c.a | 1 | 68.i | even | 16 | 2 | ||
1156.1.f.a | 2 | 17.e | odd | 16 | 4 | ||
1156.1.f.a | 2 | 68.i | even | 16 | 4 | ||
1156.1.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
1156.1.g.a | 4 | 4.b | odd | 2 | 1 | CM | |
1156.1.g.a | 4 | 17.b | even | 2 | 1 | RM | |
1156.1.g.a | 4 | 17.c | even | 4 | 2 | inner | |
1156.1.g.a | 4 | 17.d | even | 8 | 4 | inner | |
1156.1.g.a | 4 | 68.d | odd | 2 | 1 | CM | |
1156.1.g.a | 4 | 68.f | odd | 4 | 2 | inner | |
1156.1.g.a | 4 | 68.g | odd | 8 | 4 | inner | |
1700.1.d.b | 2 | 85.o | even | 16 | 2 | ||
1700.1.d.b | 2 | 85.r | even | 16 | 2 | ||
1700.1.d.b | 2 | 340.bc | odd | 16 | 2 | ||
1700.1.d.b | 2 | 340.bj | odd | 16 | 2 | ||
1700.1.h.d | 1 | 85.p | odd | 16 | 2 | ||
1700.1.h.d | 1 | 340.bg | even | 16 | 2 | ||
3332.1.g.a | 1 | 119.p | even | 16 | 2 | ||
3332.1.g.a | 1 | 476.bf | odd | 16 | 2 | ||
3332.1.o.c | 2 | 119.t | odd | 48 | 4 | ||
3332.1.o.c | 2 | 476.bm | even | 48 | 4 | ||
3332.1.o.d | 2 | 119.s | even | 48 | 4 | ||
3332.1.o.d | 2 | 476.bk | odd | 48 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 4)^{2} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} + 16 \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( (T^{2} + 4)^{2} \)
$97$
\( T^{4} \)
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