Newspace parameters
Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 245.l (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.95633484952\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(197\) |
\(\chi(n)\) | \(\zeta_{12}^{2}\) | \(\zeta_{12}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
68.1 |
|
−0.133975 | − | 0.500000i | 0.500000 | + | 0.133975i | 1.50000 | − | 0.866025i | 1.86603 | − | 1.23205i | − | 0.267949i | 0 | −1.36603 | − | 1.36603i | −2.36603 | − | 1.36603i | −0.866025 | − | 0.767949i | |||||||||||||||
117.1 | −1.86603 | + | 0.500000i | 0.500000 | − | 1.86603i | 1.50000 | − | 0.866025i | 0.133975 | − | 2.23205i | 3.73205i | 0 | 0.366025 | − | 0.366025i | −0.633975 | − | 0.366025i | 0.866025 | + | 4.23205i | |||||||||||||||||
178.1 | −1.86603 | − | 0.500000i | 0.500000 | + | 1.86603i | 1.50000 | + | 0.866025i | 0.133975 | + | 2.23205i | − | 3.73205i | 0 | 0.366025 | + | 0.366025i | −0.633975 | + | 0.366025i | 0.866025 | − | 4.23205i | ||||||||||||||||
227.1 | −0.133975 | + | 0.500000i | 0.500000 | − | 0.133975i | 1.50000 | + | 0.866025i | 1.86603 | + | 1.23205i | 0.267949i | 0 | −1.36603 | + | 1.36603i | −2.36603 | + | 1.36603i | −0.866025 | + | 0.767949i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
35.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 245.2.l.a | 4 | |
5.c | odd | 4 | 1 | 245.2.l.b | 4 | ||
7.b | odd | 2 | 1 | 35.2.k.a | ✓ | 4 | |
7.c | even | 3 | 1 | 35.2.k.b | yes | 4 | |
7.c | even | 3 | 1 | 245.2.f.b | 4 | ||
7.d | odd | 6 | 1 | 245.2.f.a | 4 | ||
7.d | odd | 6 | 1 | 245.2.l.b | 4 | ||
21.c | even | 2 | 1 | 315.2.bz.b | 4 | ||
21.h | odd | 6 | 1 | 315.2.bz.a | 4 | ||
28.d | even | 2 | 1 | 560.2.ci.a | 4 | ||
28.g | odd | 6 | 1 | 560.2.ci.b | 4 | ||
35.c | odd | 2 | 1 | 175.2.o.b | 4 | ||
35.f | even | 4 | 1 | 35.2.k.b | yes | 4 | |
35.f | even | 4 | 1 | 175.2.o.a | 4 | ||
35.j | even | 6 | 1 | 175.2.o.a | 4 | ||
35.k | even | 12 | 1 | 245.2.f.b | 4 | ||
35.k | even | 12 | 1 | inner | 245.2.l.a | 4 | |
35.l | odd | 12 | 1 | 35.2.k.a | ✓ | 4 | |
35.l | odd | 12 | 1 | 175.2.o.b | 4 | ||
35.l | odd | 12 | 1 | 245.2.f.a | 4 | ||
105.k | odd | 4 | 1 | 315.2.bz.a | 4 | ||
105.x | even | 12 | 1 | 315.2.bz.b | 4 | ||
140.j | odd | 4 | 1 | 560.2.ci.b | 4 | ||
140.w | even | 12 | 1 | 560.2.ci.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.k.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
35.2.k.a | ✓ | 4 | 35.l | odd | 12 | 1 | |
35.2.k.b | yes | 4 | 7.c | even | 3 | 1 | |
35.2.k.b | yes | 4 | 35.f | even | 4 | 1 | |
175.2.o.a | 4 | 35.f | even | 4 | 1 | ||
175.2.o.a | 4 | 35.j | even | 6 | 1 | ||
175.2.o.b | 4 | 35.c | odd | 2 | 1 | ||
175.2.o.b | 4 | 35.l | odd | 12 | 1 | ||
245.2.f.a | 4 | 7.d | odd | 6 | 1 | ||
245.2.f.a | 4 | 35.l | odd | 12 | 1 | ||
245.2.f.b | 4 | 7.c | even | 3 | 1 | ||
245.2.f.b | 4 | 35.k | even | 12 | 1 | ||
245.2.l.a | 4 | 1.a | even | 1 | 1 | trivial | |
245.2.l.a | 4 | 35.k | even | 12 | 1 | inner | |
245.2.l.b | 4 | 5.c | odd | 4 | 1 | ||
245.2.l.b | 4 | 7.d | odd | 6 | 1 | ||
315.2.bz.a | 4 | 21.h | odd | 6 | 1 | ||
315.2.bz.a | 4 | 105.k | odd | 4 | 1 | ||
315.2.bz.b | 4 | 21.c | even | 2 | 1 | ||
315.2.bz.b | 4 | 105.x | even | 12 | 1 | ||
560.2.ci.a | 4 | 28.d | even | 2 | 1 | ||
560.2.ci.a | 4 | 140.w | even | 12 | 1 | ||
560.2.ci.b | 4 | 28.g | odd | 6 | 1 | ||
560.2.ci.b | 4 | 140.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \)
$3$
\( T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1 \)
$5$
\( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \)
$7$
\( T^{4} \)
$11$
\( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \)
$13$
\( (T^{2} + 4 T + 8)^{2} \)
$17$
\( T^{4} + 4 T^{3} + 20 T^{2} + 32 T + 16 \)
$19$
\( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \)
$23$
\( T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1 \)
$29$
\( (T^{2} + 9)^{2} \)
$31$
\( T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16 \)
$37$
\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 576 \)
$41$
\( T^{4} + 42T^{2} + 9 \)
$43$
\( T^{4} + 6 T^{3} + 18 T^{2} + \cdots + 1089 \)
$47$
\( T^{4} + 18 T^{3} + 90 T^{2} + 36 T + 36 \)
$53$
\( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \)
$59$
\( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \)
$61$
\( T^{4} - 12 T^{3} + 35 T^{2} + \cdots + 169 \)
$67$
\( T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169 \)
$71$
\( (T^{2} - 6 T + 6)^{2} \)
$73$
\( T^{4} - 24 T^{3} + 144 T^{2} + \cdots + 2304 \)
$79$
\( T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484 \)
$83$
\( T^{4} - 2 T^{3} + 2 T^{2} + 26 T + 169 \)
$89$
\( T^{4} - 16 T^{3} + 267 T^{2} + \cdots + 121 \)
$97$
\( T^{4} - 4 T^{3} + 8 T^{2} + 376 T + 8836 \)
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