Newspace parameters
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.18840615665\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.4589249536.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | no (minimal twist has level 155) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 4\nu \) |
\(\beta_{4}\) | \(=\) | \( \nu^{4} + 6\nu^{2} + 6 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} - 11\nu^{5} - 35\nu^{3} - 29\nu ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 15\nu ) / 2 \) |
\(\beta_{7}\) | \(=\) | \( \nu^{6} + 9\nu^{4} + 22\nu^{2} + 12 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 4\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} - 6\beta_{2} + 12 \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{6} - 2\beta_{5} - 6\beta_{3} + 17\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( \beta_{7} - 9\beta_{4} + 32\beta_{2} - 54 \) |
\(\nu^{7}\) | \(=\) | \( 11\beta_{6} + 18\beta_{5} + 31\beta_{3} - 76\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).
\(n\) | \(251\) | \(652\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
249.1 |
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− | 2.50350i | − | 0.704624i | −4.26753 | 0 | −1.76403 | 4.77104i | 5.67678i | 2.50350 | 0 | ||||||||||||||||||||||||||||||||||||||||
249.2 | − | 2.19117i | − | 2.27841i | −2.80122 | 0 | −4.99239 | 1.38995i | 1.75561i | −2.19117 | 0 | |||||||||||||||||||||||||||||||||||||||||
249.3 | − | 1.26438i | 1.31743i | 0.401352 | 0 | 1.66573 | − | 1.13698i | − | 3.03621i | 1.26438 | 0 | ||||||||||||||||||||||||||||||||||||||||
249.4 | − | 0.576713i | 1.89122i | 1.66740 | 0 | 1.09069 | 4.24412i | − | 2.11504i | −0.576713 | 0 | |||||||||||||||||||||||||||||||||||||||||
249.5 | 0.576713i | − | 1.89122i | 1.66740 | 0 | 1.09069 | − | 4.24412i | 2.11504i | −0.576713 | 0 | |||||||||||||||||||||||||||||||||||||||||
249.6 | 1.26438i | − | 1.31743i | 0.401352 | 0 | 1.66573 | 1.13698i | 3.03621i | 1.26438 | 0 | ||||||||||||||||||||||||||||||||||||||||||
249.7 | 2.19117i | 2.27841i | −2.80122 | 0 | −4.99239 | − | 1.38995i | − | 1.75561i | −2.19117 | 0 | |||||||||||||||||||||||||||||||||||||||||
249.8 | 2.50350i | 0.704624i | −4.26753 | 0 | −1.76403 | − | 4.77104i | − | 5.67678i | 2.50350 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 775.2.b.f | 8 | |
5.b | even | 2 | 1 | inner | 775.2.b.f | 8 | |
5.c | odd | 4 | 1 | 155.2.a.e | ✓ | 4 | |
5.c | odd | 4 | 1 | 775.2.a.e | 4 | ||
15.e | even | 4 | 1 | 1395.2.a.l | 4 | ||
15.e | even | 4 | 1 | 6975.2.a.bn | 4 | ||
20.e | even | 4 | 1 | 2480.2.a.x | 4 | ||
35.f | even | 4 | 1 | 7595.2.a.s | 4 | ||
40.i | odd | 4 | 1 | 9920.2.a.cb | 4 | ||
40.k | even | 4 | 1 | 9920.2.a.cg | 4 | ||
155.f | even | 4 | 1 | 4805.2.a.n | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
155.2.a.e | ✓ | 4 | 5.c | odd | 4 | 1 | |
775.2.a.e | 4 | 5.c | odd | 4 | 1 | ||
775.2.b.f | 8 | 1.a | even | 1 | 1 | trivial | |
775.2.b.f | 8 | 5.b | even | 2 | 1 | inner | |
1395.2.a.l | 4 | 15.e | even | 4 | 1 | ||
2480.2.a.x | 4 | 20.e | even | 4 | 1 | ||
4805.2.a.n | 4 | 155.f | even | 4 | 1 | ||
6975.2.a.bn | 4 | 15.e | even | 4 | 1 | ||
7595.2.a.s | 4 | 35.f | even | 4 | 1 | ||
9920.2.a.cb | 4 | 40.i | odd | 4 | 1 | ||
9920.2.a.cg | 4 | 40.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 13T_{2}^{6} + 52T_{2}^{4} + 64T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 13 T^{6} + 52 T^{4} + 64 T^{2} + \cdots + 16 \)
$3$
\( T^{8} + 11 T^{6} + 39 T^{4} + 49 T^{2} + \cdots + 16 \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 44 T^{6} + 544 T^{4} + \cdots + 1024 \)
$11$
\( (T^{4} + 4 T^{3} - 8 T^{2} - 12 T + 16)^{2} \)
$13$
\( T^{8} + 60 T^{6} + 1168 T^{4} + \cdots + 18496 \)
$17$
\( T^{8} + 51 T^{6} + 823 T^{4} + \cdots + 3364 \)
$19$
\( (T^{4} - 3 T^{3} - 33 T^{2} + 107 T + 44)^{2} \)
$23$
\( T^{8} + 44 T^{6} + 544 T^{4} + \cdots + 1024 \)
$29$
\( (T^{4} - 8 T^{3} - 20 T^{2} + 292 T - 584)^{2} \)
$31$
\( (T + 1)^{8} \)
$37$
\( T^{8} + 171 T^{6} + 10495 T^{4} + \cdots + 2365444 \)
$41$
\( (T^{4} + 11 T^{3} - 31 T^{2} - 359 T + 506)^{2} \)
$43$
\( T^{8} + 143 T^{6} + 5571 T^{4} + \cdots + 55696 \)
$47$
\( T^{8} + 204 T^{6} + 13040 T^{4} + \cdots + 1982464 \)
$53$
\( T^{8} + 311 T^{6} + 28011 T^{4} + \cdots + 1705636 \)
$59$
\( (T^{4} - 3 T^{3} - 97 T^{2} - 129 T - 44)^{2} \)
$61$
\( (T^{4} - 22 T^{3} + 160 T^{2} - 432 T + 352)^{2} \)
$67$
\( T^{8} + 288 T^{6} + 15552 T^{4} + \cdots + 1679616 \)
$71$
\( (T^{4} + 21 T^{3} + 159 T^{2} + 511 T + 584)^{2} \)
$73$
\( T^{8} + 191 T^{6} + 2619 T^{4} + \cdots + 1156 \)
$79$
\( (T^{4} - 2 T^{3} - 260 T^{2} - 404 T + 6592)^{2} \)
$83$
\( T^{8} + 351 T^{6} + \cdots + 11316496 \)
$89$
\( (T^{4} - 10 T^{3} - 152 T^{2} + 420 T + 3688)^{2} \)
$97$
\( T^{8} + 512 T^{6} + 68512 T^{4} + \cdots + 215296 \)
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