Newspace parameters
Level: | \( N \) | \(=\) | \( 869 = 11 \cdot 79 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 869.l (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.93899993565\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | 8.0.484000000.9 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \) |
\(\beta_{3}\) | \(=\) | \( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \) |
\(\beta_{4}\) | \(=\) | \( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \) |
\(\beta_{5}\) | \(=\) | \( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \) |
\(\beta_{6}\) | \(=\) | \( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \) |
\(\beta_{7}\) | \(=\) | \( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 4\beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( 7\beta_{5} + 10\beta_{3} - 7 \) |
\(\nu^{5}\) | \(=\) | \( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \) |
\(\nu^{7}\) | \(=\) | \( -38\beta_{7} - 75\beta_{6} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/869\mathbb{Z}\right)^\times\).
\(n\) | \(475\) | \(793\) |
\(\chi(n)\) | \(1 + \beta_{2} + \beta_{3} - \beta_{5}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
315.1 |
|
0.690983 | − | 0.951057i | −2.79197 | − | 0.907165i | 0.190983 | + | 0.587785i | 2.30902 | − | 1.67760i | −2.79197 | + | 2.02848i | 0.771681 | + | 2.37499i | 2.92705 | + | 0.951057i | 4.54508 | + | 3.30220i | − | 3.35520i | |||||||||||||||||||||||||
315.2 | 0.690983 | − | 0.951057i | 2.79197 | + | 0.907165i | 0.190983 | + | 0.587785i | 2.30902 | − | 1.67760i | 2.79197 | − | 2.02848i | −0.771681 | − | 2.37499i | 2.92705 | + | 0.951057i | 4.54508 | + | 3.30220i | − | 3.35520i | ||||||||||||||||||||||||||
552.1 | 1.80902 | + | 0.587785i | −1.48490 | − | 2.04378i | 1.30902 | + | 0.951057i | 1.19098 | + | 3.66547i | −1.48490 | − | 4.57004i | 1.07448 | + | 0.780656i | −0.427051 | − | 0.587785i | −1.04508 | + | 3.21644i | 7.33094i | |||||||||||||||||||||||||||
552.2 | 1.80902 | + | 0.587785i | 1.48490 | + | 2.04378i | 1.30902 | + | 0.951057i | 1.19098 | + | 3.66547i | 1.48490 | + | 4.57004i | −1.07448 | − | 0.780656i | −0.427051 | − | 0.587785i | −1.04508 | + | 3.21644i | 7.33094i | |||||||||||||||||||||||||||
710.1 | 1.80902 | − | 0.587785i | −1.48490 | + | 2.04378i | 1.30902 | − | 0.951057i | 1.19098 | − | 3.66547i | −1.48490 | + | 4.57004i | 1.07448 | − | 0.780656i | −0.427051 | + | 0.587785i | −1.04508 | − | 3.21644i | − | 7.33094i | ||||||||||||||||||||||||||
710.2 | 1.80902 | − | 0.587785i | 1.48490 | − | 2.04378i | 1.30902 | − | 0.951057i | 1.19098 | − | 3.66547i | 1.48490 | − | 4.57004i | −1.07448 | + | 0.780656i | −0.427051 | + | 0.587785i | −1.04508 | − | 3.21644i | − | 7.33094i | ||||||||||||||||||||||||||
789.1 | 0.690983 | + | 0.951057i | −2.79197 | + | 0.907165i | 0.190983 | − | 0.587785i | 2.30902 | + | 1.67760i | −2.79197 | − | 2.02848i | 0.771681 | − | 2.37499i | 2.92705 | − | 0.951057i | 4.54508 | − | 3.30220i | 3.35520i | |||||||||||||||||||||||||||
789.2 | 0.690983 | + | 0.951057i | 2.79197 | − | 0.907165i | 0.190983 | − | 0.587785i | 2.30902 | + | 1.67760i | 2.79197 | + | 2.02848i | −0.771681 | + | 2.37499i | 2.92705 | − | 0.951057i | 4.54508 | − | 3.30220i | 3.35520i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
79.b | odd | 2 | 1 | inner |
869.l | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 869.2.l.a | ✓ | 8 |
11.d | odd | 10 | 1 | inner | 869.2.l.a | ✓ | 8 |
79.b | odd | 2 | 1 | inner | 869.2.l.a | ✓ | 8 |
869.l | even | 10 | 1 | inner | 869.2.l.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
869.2.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
869.2.l.a | ✓ | 8 | 11.d | odd | 10 | 1 | inner |
869.2.l.a | ✓ | 8 | 79.b | odd | 2 | 1 | inner |
869.2.l.a | ✓ | 8 | 869.l | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} + 10T_{2}^{2} - 10T_{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 5 T^{3} + 10 T^{2} - 10 T + 5)^{2} \)
$3$
\( T^{8} - 10 T^{6} + 60 T^{4} + \cdots + 3025 \)
$5$
\( (T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121)^{2} \)
$7$
\( T^{8} + 9 T^{6} + 31 T^{4} - 11 T^{2} + \cdots + 121 \)
$11$
\( (T^{4} + T^{3} - 9 T^{2} + 11 T + 121)^{2} \)
$13$
\( (T^{4} + 10 T^{3} + 60 T^{2} + 120 T + 80)^{2} \)
$17$
\( T^{8} + 54 T^{6} + 1296 T^{4} + \cdots + 793881 \)
$19$
\( (T^{4} + 320 T + 1280)^{2} \)
$23$
\( (T^{2} + 3 T - 29)^{4} \)
$29$
\( T^{8} + 5 T^{6} + 400 T^{4} + \cdots + 75625 \)
$31$
\( (T^{4} + 10 T^{2} + 25 T + 25)^{2} \)
$37$
\( T^{8} + 25 T^{6} + 5875 T^{4} + \cdots + 1890625 \)
$41$
\( T^{8} + 69 T^{6} + \cdots + 111746041 \)
$43$
\( (T^{4} - 272 T^{2} + 18491)^{2} \)
$47$
\( T^{8} - 175 T^{6} + 11500 T^{4} + \cdots + 1890625 \)
$53$
\( T^{8} \)
$59$
\( T^{8} + 5 T^{6} + 235 T^{4} + \cdots + 3025 \)
$61$
\( T^{8} + 134 T^{6} + 6856 T^{4} + \cdots + 121 \)
$67$
\( (T^{2} + 6 T - 71)^{4} \)
$71$
\( T^{8} - 100 T^{6} + \cdots + 484000000 \)
$73$
\( (T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5)^{2} \)
$79$
\( T^{8} - 40 T^{7} + 799 T^{6} + \cdots + 38950081 \)
$83$
\( (T^{4} + 5 T^{3} - 15 T^{2} + 435 T + 4205)^{2} \)
$89$
\( (T^{2} - 5)^{4} \)
$97$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \)
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