Newspace parameters
Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 260.r (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.07611045255\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.31678304256.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 97\nu^{7} - 173\nu^{6} + 220\nu^{5} + 316\nu^{4} + 4010\nu^{3} - 1148\nu^{2} - 1300\nu - 6130 ) / 2462 \) |
\(\beta_{3}\) | \(=\) | \( ( 140\nu^{7} - 237\nu^{6} + 216\nu^{5} + 1116\nu^{4} + 5280\nu^{3} - 1530\nu^{2} + 738\nu + 696 ) / 2462 \) |
\(\beta_{4}\) | \(=\) | \( ( 174\nu^{7} - 488\nu^{6} + 585\nu^{5} + 1176\nu^{4} + 4452\nu^{3} - 8760\nu^{2} + 2922\nu + 654 ) / 2462 \) |
\(\beta_{5}\) | \(=\) | \( ( -382\nu^{7} + 1227\nu^{6} - 1539\nu^{5} - 2412\nu^{4} - 8076\nu^{3} + 22288\nu^{2} - 5566\nu - 1266 ) / 2462 \) |
\(\beta_{6}\) | \(=\) | \( ( 452\nu^{7} - 730\nu^{6} + 416\nu^{5} + 4201\nu^{4} + 15640\nu^{3} - 4588\nu^{2} - 5144\nu + 4076 ) / 2462 \) |
\(\beta_{7}\) | \(=\) | \( ( 605\nu^{7} - 1244\nu^{6} + 1461\nu^{5} + 4471\nu^{4} + 19300\nu^{3} - 11272\nu^{2} + 12070\nu + 2656 ) / 2462 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - \beta_{2} - 2 \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{6} + 12\beta_{3} - 8\beta_{2} - 12\beta _1 - 20 \) |
\(\nu^{5}\) | \(=\) | \( 8\beta_{7} - 8\beta_{6} + 12\beta_{5} + 26\beta_{4} - 12\beta_{2} - 66\beta _1 - 26 \) |
\(\nu^{6}\) | \(=\) | \( 24\beta_{7} + 66\beta_{5} + 158\beta_{4} - 120\beta_{3} - 120\beta_1 \) |
\(\nu^{7}\) | \(=\) | \( 66\beta_{7} + 66\beta_{6} + 120\beta_{5} + 270\beta_{4} - 572\beta_{3} + 120\beta_{2} + 270 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).
\(n\) | \(41\) | \(131\) | \(157\) |
\(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
177.1 |
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0 | −1.42497 | + | 1.42497i | 0 | 1.42497 | + | 1.72321i | 0 | −4.91106 | 0 | − | 1.06111i | 0 | |||||||||||||||||||||||||||||||||||||
177.2 | 0 | −0.285451 | + | 0.285451i | 0 | 0.285451 | − | 2.21777i | 0 | 1.26613 | 0 | 2.83704i | 0 | |||||||||||||||||||||||||||||||||||||||
177.3 | 0 | 0.575868 | − | 0.575868i | 0 | −0.575868 | + | 2.16064i | 0 | 2.48849 | 0 | 2.33675i | 0 | |||||||||||||||||||||||||||||||||||||||
177.4 | 0 | 2.13456 | − | 2.13456i | 0 | −2.13456 | − | 0.666078i | 0 | −2.84356 | 0 | − | 6.11268i | 0 | ||||||||||||||||||||||||||||||||||||||
213.1 | 0 | −1.42497 | − | 1.42497i | 0 | 1.42497 | − | 1.72321i | 0 | −4.91106 | 0 | 1.06111i | 0 | |||||||||||||||||||||||||||||||||||||||
213.2 | 0 | −0.285451 | − | 0.285451i | 0 | 0.285451 | + | 2.21777i | 0 | 1.26613 | 0 | − | 2.83704i | 0 | ||||||||||||||||||||||||||||||||||||||
213.3 | 0 | 0.575868 | + | 0.575868i | 0 | −0.575868 | − | 2.16064i | 0 | 2.48849 | 0 | − | 2.33675i | 0 | ||||||||||||||||||||||||||||||||||||||
213.4 | 0 | 2.13456 | + | 2.13456i | 0 | −2.13456 | + | 0.666078i | 0 | −2.84356 | 0 | 6.11268i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 260.2.r.c | yes | 8 |
3.b | odd | 2 | 1 | 2340.2.bp.g | 8 | ||
4.b | odd | 2 | 1 | 1040.2.cd.m | 8 | ||
5.b | even | 2 | 1 | 1300.2.r.c | 8 | ||
5.c | odd | 4 | 1 | 260.2.m.c | ✓ | 8 | |
5.c | odd | 4 | 1 | 1300.2.m.c | 8 | ||
13.d | odd | 4 | 1 | 260.2.m.c | ✓ | 8 | |
15.e | even | 4 | 1 | 2340.2.u.g | 8 | ||
20.e | even | 4 | 1 | 1040.2.bg.m | 8 | ||
39.f | even | 4 | 1 | 2340.2.u.g | 8 | ||
52.f | even | 4 | 1 | 1040.2.bg.m | 8 | ||
65.f | even | 4 | 1 | inner | 260.2.r.c | yes | 8 |
65.g | odd | 4 | 1 | 1300.2.m.c | 8 | ||
65.k | even | 4 | 1 | 1300.2.r.c | 8 | ||
195.u | odd | 4 | 1 | 2340.2.bp.g | 8 | ||
260.l | odd | 4 | 1 | 1040.2.cd.m | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.m.c | ✓ | 8 | 5.c | odd | 4 | 1 | |
260.2.m.c | ✓ | 8 | 13.d | odd | 4 | 1 | |
260.2.r.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
260.2.r.c | yes | 8 | 65.f | even | 4 | 1 | inner |
1040.2.bg.m | 8 | 20.e | even | 4 | 1 | ||
1040.2.bg.m | 8 | 52.f | even | 4 | 1 | ||
1040.2.cd.m | 8 | 4.b | odd | 2 | 1 | ||
1040.2.cd.m | 8 | 260.l | odd | 4 | 1 | ||
1300.2.m.c | 8 | 5.c | odd | 4 | 1 | ||
1300.2.m.c | 8 | 65.g | odd | 4 | 1 | ||
1300.2.r.c | 8 | 5.b | even | 2 | 1 | ||
1300.2.r.c | 8 | 65.k | even | 4 | 1 | ||
2340.2.u.g | 8 | 15.e | even | 4 | 1 | ||
2340.2.u.g | 8 | 39.f | even | 4 | 1 | ||
2340.2.bp.g | 8 | 3.b | odd | 2 | 1 | ||
2340.2.bp.g | 8 | 195.u | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 8T_{3}^{5} + 32T_{3}^{4} - 20T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 2 T^{7} + 2 T^{6} + 8 T^{5} + \cdots + 4 \)
$5$
\( T^{8} + 2 T^{7} + 8 T^{6} + 22 T^{5} + \cdots + 625 \)
$7$
\( (T^{4} + 4 T^{3} - 12 T^{2} - 28 T + 44)^{2} \)
$11$
\( T^{8} - 14 T^{7} + 98 T^{6} - 332 T^{5} + \cdots + 4 \)
$13$
\( T^{8} - 6 T^{7} + 4 T^{6} + \cdots + 28561 \)
$17$
\( T^{8} - 8 T^{5} + 360 T^{4} + \cdots + 11664 \)
$19$
\( T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 86436 \)
$23$
\( T^{8} - 22 T^{7} + 242 T^{6} + \cdots + 6724 \)
$29$
\( T^{8} + 156 T^{6} + 8136 T^{4} + \cdots + 810000 \)
$31$
\( T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 22500 \)
$37$
\( (T^{4} + 10 T^{3} + 12 T^{2} - 104 T - 188)^{2} \)
$41$
\( T^{8} + 8 T^{7} + 32 T^{6} - 152 T^{5} + \cdots + 784 \)
$43$
\( T^{8} - 14 T^{7} + 98 T^{6} + \cdots + 1085764 \)
$47$
\( (T^{4} - 8 T^{3} - 168 T^{2} + 884 T + 5444)^{2} \)
$53$
\( T^{8} + 8 T^{7} + 32 T^{6} - 928 T^{5} + \cdots + 16 \)
$59$
\( T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 86436 \)
$61$
\( (T^{4} + 6 T^{3} - 168 T^{2} - 1376 T - 1452)^{2} \)
$67$
\( T^{8} + 100 T^{6} + 456 T^{4} + \cdots + 144 \)
$71$
\( T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 46321636 \)
$73$
\( T^{8} + 532 T^{6} + \cdots + 174556944 \)
$79$
\( T^{8} + 340 T^{6} + 25584 T^{4} + \cdots + 440896 \)
$83$
\( (T^{4} + 4 T^{3} - 72 T^{2} + 108 T + 108)^{2} \)
$89$
\( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 22391824 \)
$97$
\( T^{8} + 468 T^{6} + \cdots + 19289664 \)
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