Newspace parameters
Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 161.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.49930751092\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 16x^{6} - 2516x^{5} + 43236x^{4} + 20128x^{3} - 55968x^{2} - 6463604x + 23372755 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 2^{4}\cdot 7 \) |
Twist minimal: | yes |
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 \( x^{8} - 16x^{6} - 2516x^{5} + 43236x^{4} + 20128x^{3} - 55968x^{2} - 6463604x + 23372755 \) :
\(\beta_{1}\) | \(=\) | \( ( - 312796779 \nu^{7} - 4501364056 \nu^{6} - 8102975775 \nu^{5} + 829696059414 \nu^{4} - 2127136655513 \nu^{3} + \cdots + 12\!\cdots\!90 ) / 201720841645470 \) |
\(\beta_{2}\) | \(=\) | \( ( 5190913 \nu^{7} + 15009457 \nu^{6} + 11579205 \nu^{5} - 13251233983 \nu^{4} + 182223722761 \nu^{3} + 424559789633 \nu^{2} + \cdots - 24605864501855 ) / 3240495448120 \) |
\(\beta_{3}\) | \(=\) | \( ( 1901486953 \nu^{7} + 17906311497 \nu^{6} + 30216937765 \nu^{5} - 4804602683303 \nu^{4} + 40078148590801 \nu^{3} + \cdots - 86\!\cdots\!95 ) / 806883366581880 \) |
\(\beta_{4}\) | \(=\) | \( ( 389913038 \nu^{7} + 990048397 \nu^{6} - 9303870925 \nu^{5} - 964117175938 \nu^{4} + 14898947751346 \nu^{3} + \cdots - 20\!\cdots\!30 ) / 100860420822735 \) |
\(\beta_{5}\) | \(=\) | \( ( 5054316871 \nu^{7} + 5368063059 \nu^{6} - 302517171285 \nu^{5} - 16872010275461 \nu^{4} + 193879539447807 \nu^{3} + \cdots - 10\!\cdots\!05 ) / 806883366581880 \) |
\(\beta_{6}\) | \(=\) | \( ( 88808803 \nu^{7} + 205555327 \nu^{6} + 422480975 \nu^{5} - 222814087953 \nu^{4} + 3324397362451 \nu^{3} + \cdots - 438841756672305 ) / 9721486344360 \) |
\(\beta_{7}\) | \(=\) | \( ( - 7210546463 \nu^{7} - 20364939287 \nu^{6} + 145099265185 \nu^{5} + 18628750106953 \nu^{4} - 250873656237351 \nu^{3} + \cdots + 35\!\cdots\!25 ) / 403441683290940 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 4\beta_{4} - 9\beta_{3} + 7\beta_{2} - 8\beta_1 ) / 14 \) |
\(\nu^{2}\) | \(=\) | \( ( -4\beta_{7} + 21\beta_{6} - 9\beta_{4} - 13\beta_{3} - 140\beta_{2} - 17\beta _1 + 28 ) / 7 \) |
\(\nu^{3}\) | \(=\) | \( ( 309\beta_{7} + 126\beta_{5} + 788\beta_{4} + 1153\beta_{3} + 203\beta_{2} + 874\beta _1 + 13146 ) / 14 \) |
\(\nu^{4}\) | \(=\) | \( ( - 39 \beta_{7} - 546 \beta_{6} - 1554 \beta_{5} + 5381 \beta_{4} - 11234 \beta_{3} + 3640 \beta_{2} - 8942 \beta _1 - 150101 ) / 7 \) |
\(\nu^{5}\) | \(=\) | \( ( - 41607 \beta_{7} + 132090 \beta_{6} + 1680 \beta_{5} - 269958 \beta_{4} + 335403 \beta_{3} - 807583 \beta_{2} + 266958 \beta _1 + 175280 ) / 14 \) |
\(\nu^{6}\) | \(=\) | \( ( 530785 \beta_{7} - 967113 \beta_{6} + 139860 \beta_{5} + 1450916 \beta_{4} + 1932491 \beta_{3} + 5896268 \beta_{2} + 1394901 \beta _1 + 13216168 ) / 7 \) |
\(\nu^{7}\) | \(=\) | \( ( - 14120259 \beta_{7} - 924630 \beta_{6} - 13169730 \beta_{5} - 9511498 \beta_{4} - 100864787 \beta_{3} + 5541613 \beta_{2} - 76204388 \beta _1 - 1242867654 ) / 14 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).
\(n\) | \(24\) | \(120\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
160.1 |
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−1.00000 | − | 5.83095i | −7.00000 | −20.8933 | 5.83095i | −5.36058 | − | 17.7275i | 15.0000 | −7.00000 | 20.8933 | |||||||||||||||||||||||||||||||||||||||
160.2 | −1.00000 | − | 5.83095i | −7.00000 | −7.58101 | 5.83095i | −14.7738 | + | 11.1685i | 15.0000 | −7.00000 | 7.58101 | ||||||||||||||||||||||||||||||||||||||||
160.3 | −1.00000 | − | 5.83095i | −7.00000 | 7.58101 | 5.83095i | 14.7738 | − | 11.1685i | 15.0000 | −7.00000 | −7.58101 | ||||||||||||||||||||||||||||||||||||||||
160.4 | −1.00000 | − | 5.83095i | −7.00000 | 20.8933 | 5.83095i | 5.36058 | + | 17.7275i | 15.0000 | −7.00000 | −20.8933 | ||||||||||||||||||||||||||||||||||||||||
160.5 | −1.00000 | 5.83095i | −7.00000 | −20.8933 | − | 5.83095i | −5.36058 | + | 17.7275i | 15.0000 | −7.00000 | 20.8933 | ||||||||||||||||||||||||||||||||||||||||
160.6 | −1.00000 | 5.83095i | −7.00000 | −7.58101 | − | 5.83095i | −14.7738 | − | 11.1685i | 15.0000 | −7.00000 | 7.58101 | ||||||||||||||||||||||||||||||||||||||||
160.7 | −1.00000 | 5.83095i | −7.00000 | 7.58101 | − | 5.83095i | 14.7738 | + | 11.1685i | 15.0000 | −7.00000 | −7.58101 | ||||||||||||||||||||||||||||||||||||||||
160.8 | −1.00000 | 5.83095i | −7.00000 | 20.8933 | − | 5.83095i | 5.36058 | − | 17.7275i | 15.0000 | −7.00000 | −20.8933 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
161.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 161.4.c.b | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 161.4.c.b | ✓ | 8 |
23.b | odd | 2 | 1 | inner | 161.4.c.b | ✓ | 8 |
161.c | even | 2 | 1 | inner | 161.4.c.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
161.4.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
161.4.c.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
161.4.c.b | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
161.4.c.b | ✓ | 8 | 161.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 1)^{8} \)
$3$
\( (T^{2} + 34)^{4} \)
$5$
\( (T^{4} - 494 T^{2} + 25088)^{2} \)
$7$
\( T^{8} + 384 T^{6} + \cdots + 13841287201 \)
$11$
\( (T^{4} + 878 T^{2} + 156800)^{2} \)
$13$
\( (T^{4} + 2130 T^{2} + 1098304)^{2} \)
$17$
\( (T^{4} - 494 T^{2} + 25088)^{2} \)
$19$
\( (T^{4} - 29852 T^{2} + \cdots + 181260800)^{2} \)
$23$
\( (T^{4} + 62 T^{3} - 10626 T^{2} + \cdots + 148035889)^{2} \)
$29$
\( (T^{2} - 78 T - 34400)^{4} \)
$31$
\( (T^{4} + 109674 T^{2} + \cdots + 2371690000)^{2} \)
$37$
\( (T^{4} + 137518 T^{2} + \cdots + 648288032)^{2} \)
$41$
\( (T^{4} + 204864 T^{2} + \cdots + 1075840000)^{2} \)
$43$
\( (T^{4} + 143962 T^{2} + \cdots + 955194632)^{2} \)
$47$
\( (T^{4} + 199162 T^{2} + \cdots + 15649936)^{2} \)
$53$
\( (T^{4} + 587646 T^{2} + \cdots + 83535828768)^{2} \)
$59$
\( (T^{4} + 272932 T^{2} + 1522756)^{2} \)
$61$
\( (T^{4} - 206686 T^{2} + \cdots + 5188137248)^{2} \)
$67$
\( (T^{4} + 1031614 T^{2} + \cdots + 184044871808)^{2} \)
$71$
\( (T^{2} - 862 T + 149840)^{4} \)
$73$
\( (T^{4} + 619848 T^{2} + \cdots + 10862642176)^{2} \)
$79$
\( (T^{4} + 1933424 T^{2} + \cdots + 441709764608)^{2} \)
$83$
\( (T^{4} - 719516 T^{2} + \cdots + 29725071488)^{2} \)
$89$
\( (T^{4} - 1489338 T^{2} + \cdots + 14041528200)^{2} \)
$97$
\( (T^{4} - 2089066 T^{2} + \cdots + 279557553800)^{2} \)
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