Newspace parameters
Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 363.e (of order \(5\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.89856959337\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{10})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 33) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
\(n\) | \(122\) | \(244\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
124.1 |
|
0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | −0.309017 | + | 0.951057i | 1.61803 | + | 1.17557i | −0.809017 | − | 0.587785i | −1.23607 | + | 3.80423i | 0.927051 | + | 2.85317i | −0.809017 | + | 0.587785i | 2.00000 | ||||||||||||||
130.1 | −0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | 0.809017 | + | 0.587785i | −0.618034 | − | 1.90211i | 0.309017 | + | 0.951057i | 3.23607 | + | 2.35114i | −2.42705 | + | 1.76336i | 0.309017 | − | 0.951057i | 2.00000 | |||||||||||||||
148.1 | −0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | 0.809017 | − | 0.587785i | −0.618034 | + | 1.90211i | 0.309017 | − | 0.951057i | 3.23607 | − | 2.35114i | −2.42705 | − | 1.76336i | 0.309017 | + | 0.951057i | 2.00000 | |||||||||||||||
202.1 | 0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | −0.309017 | − | 0.951057i | 1.61803 | − | 1.17557i | −0.809017 | + | 0.587785i | −1.23607 | − | 3.80423i | 0.927051 | − | 2.85317i | −0.809017 | − | 0.587785i | 2.00000 | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 363.2.e.g | 4 | |
11.b | odd | 2 | 1 | 363.2.e.e | 4 | ||
11.c | even | 5 | 1 | 363.2.a.b | 1 | ||
11.c | even | 5 | 3 | inner | 363.2.e.g | 4 | |
11.d | odd | 10 | 1 | 33.2.a.a | ✓ | 1 | |
11.d | odd | 10 | 3 | 363.2.e.e | 4 | ||
33.f | even | 10 | 1 | 99.2.a.b | 1 | ||
33.h | odd | 10 | 1 | 1089.2.a.j | 1 | ||
44.g | even | 10 | 1 | 528.2.a.g | 1 | ||
44.h | odd | 10 | 1 | 5808.2.a.t | 1 | ||
55.h | odd | 10 | 1 | 825.2.a.a | 1 | ||
55.j | even | 10 | 1 | 9075.2.a.q | 1 | ||
55.l | even | 20 | 2 | 825.2.c.a | 2 | ||
77.l | even | 10 | 1 | 1617.2.a.j | 1 | ||
88.k | even | 10 | 1 | 2112.2.a.j | 1 | ||
88.p | odd | 10 | 1 | 2112.2.a.bb | 1 | ||
99.o | odd | 30 | 2 | 891.2.e.e | 2 | ||
99.p | even | 30 | 2 | 891.2.e.g | 2 | ||
132.n | odd | 10 | 1 | 1584.2.a.o | 1 | ||
143.l | odd | 10 | 1 | 5577.2.a.a | 1 | ||
165.r | even | 10 | 1 | 2475.2.a.g | 1 | ||
165.u | odd | 20 | 2 | 2475.2.c.d | 2 | ||
187.l | odd | 10 | 1 | 9537.2.a.m | 1 | ||
231.r | odd | 10 | 1 | 4851.2.a.b | 1 | ||
264.r | odd | 10 | 1 | 6336.2.a.n | 1 | ||
264.u | even | 10 | 1 | 6336.2.a.x | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.2.a.a | ✓ | 1 | 11.d | odd | 10 | 1 | |
99.2.a.b | 1 | 33.f | even | 10 | 1 | ||
363.2.a.b | 1 | 11.c | even | 5 | 1 | ||
363.2.e.e | 4 | 11.b | odd | 2 | 1 | ||
363.2.e.e | 4 | 11.d | odd | 10 | 3 | ||
363.2.e.g | 4 | 1.a | even | 1 | 1 | trivial | |
363.2.e.g | 4 | 11.c | even | 5 | 3 | inner | |
528.2.a.g | 1 | 44.g | even | 10 | 1 | ||
825.2.a.a | 1 | 55.h | odd | 10 | 1 | ||
825.2.c.a | 2 | 55.l | even | 20 | 2 | ||
891.2.e.e | 2 | 99.o | odd | 30 | 2 | ||
891.2.e.g | 2 | 99.p | even | 30 | 2 | ||
1089.2.a.j | 1 | 33.h | odd | 10 | 1 | ||
1584.2.a.o | 1 | 132.n | odd | 10 | 1 | ||
1617.2.a.j | 1 | 77.l | even | 10 | 1 | ||
2112.2.a.j | 1 | 88.k | even | 10 | 1 | ||
2112.2.a.bb | 1 | 88.p | odd | 10 | 1 | ||
2475.2.a.g | 1 | 165.r | even | 10 | 1 | ||
2475.2.c.d | 2 | 165.u | odd | 20 | 2 | ||
4851.2.a.b | 1 | 231.r | odd | 10 | 1 | ||
5577.2.a.a | 1 | 143.l | odd | 10 | 1 | ||
5808.2.a.t | 1 | 44.h | odd | 10 | 1 | ||
6336.2.a.n | 1 | 264.r | odd | 10 | 1 | ||
6336.2.a.x | 1 | 264.u | even | 10 | 1 | ||
9075.2.a.q | 1 | 55.j | even | 10 | 1 | ||
9537.2.a.m | 1 | 187.l | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$3$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$5$
\( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \)
$7$
\( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$17$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$19$
\( T^{4} \)
$23$
\( (T - 8)^{4} \)
$29$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$31$
\( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \)
$37$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$41$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$43$
\( T^{4} \)
$47$
\( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \)
$53$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$59$
\( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \)
$61$
\( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$67$
\( (T + 4)^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416 \)
$79$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$83$
\( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \)
$89$
\( (T + 6)^{4} \)
$97$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
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