Newspace parameters
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.n (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.86104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.523596960000.16 |
| Defining polynomial: |
\( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 20\nu^{6} - 51\nu^{5} + 304\nu^{4} - 193\nu^{3} + 1752\nu^{2} - 2510\nu + 2630 ) / 630 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -87\nu^{7} + 24\nu^{6} - 841\nu^{5} - 1276\nu^{4} - 10117\nu^{3} - 4640\nu^{2} - 2900\nu - 13700 ) / 21630 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 137\nu^{7} - 361\nu^{6} + 1805\nu^{5} - 1115\nu^{4} + 11191\nu^{3} - 16967\nu^{2} + 21390\nu - 10830 ) / 21630 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 472\nu^{7} - 1249\nu^{6} + 6966\nu^{5} - 8699\nu^{4} + 48092\nu^{3} - 74565\nu^{2} + 78220\nu - 131200 ) / 64890 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 661\nu^{7} - 2047\nu^{6} + 8793\nu^{5} - 5927\nu^{4} + 44711\nu^{3} - 64485\nu^{2} + 84520\nu + 126050 ) / 64890 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -977\nu^{7} + 1985\nu^{6} - 15693\nu^{5} + 4657\nu^{4} - 114889\nu^{3} + 25521\nu^{2} - 381050\nu - 55810 ) / 64890 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{6} + \beta_{5} - 5\beta_{4} - \beta_{3} + \beta _1 - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} - 9\beta_{3} - 2\beta_{2} - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - 11\beta_{6} - 24\beta_{5} + 41\beta_{4} - 2\beta_{2} - 17\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -26\beta_{7} - 42\beta_{6} - 8\beta_{5} + 72\beta_{4} + 95\beta_{3} + 13\beta_{2} - 95\beta _1 + 85 \)
|
| \(\nu^{6}\) | \(=\) |
\( -34\beta_{7} - 121\beta_{6} + 189\beta_{5} + 34\beta_{4} + 243\beta_{3} + 68\beta_{2} + 475 \)
|
| \(\nu^{7}\) | \(=\) |
\( 155\beta_{7} + 311\beta_{6} + 777\beta_{5} - 905\beta_{4} + 155\beta_{2} + 1081\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.26021 | − | 2.18275i | −1.50000 | − | 0.866025i | −1.17628 | + | 2.03737i | −1.93649 | + | 1.11803i | 4.36551i | −6.18050 | + | 3.28656i | −4.15226 | 1.50000 | + | 2.59808i | 4.88079 | + | 2.81792i | ||||||||||||||||||||||||||||
| 31.2 | −0.336732 | − | 0.583237i | −1.50000 | − | 0.866025i | 1.77322 | − | 3.07131i | 1.93649 | − | 1.11803i | 1.16647i | −6.82455 | − | 1.55742i | −5.08226 | 1.50000 | + | 2.59808i | −1.30416 | − | 0.752955i | |||||||||||||||||||||||||||||
| 31.3 | 0.836732 | + | 1.44926i | −1.50000 | − | 0.866025i | 0.599760 | − | 1.03881i | 1.93649 | − | 1.11803i | − | 2.89852i | 4.76104 | + | 5.13152i | 8.70121 | 1.50000 | + | 2.59808i | 3.24065 | + | 1.87099i | ||||||||||||||||||||||||||||
| 31.4 | 1.76021 | + | 3.04878i | −1.50000 | − | 0.866025i | −4.19671 | + | 7.26891i | −1.93649 | + | 1.11803i | − | 6.09756i | 0.244004 | + | 6.99575i | −15.4667 | 1.50000 | + | 2.59808i | −6.81728 | − | 3.93596i | ||||||||||||||||||||||||||||
| 61.1 | −1.26021 | + | 2.18275i | −1.50000 | + | 0.866025i | −1.17628 | − | 2.03737i | −1.93649 | − | 1.11803i | − | 4.36551i | −6.18050 | − | 3.28656i | −4.15226 | 1.50000 | − | 2.59808i | 4.88079 | − | 2.81792i | ||||||||||||||||||||||||||||
| 61.2 | −0.336732 | + | 0.583237i | −1.50000 | + | 0.866025i | 1.77322 | + | 3.07131i | 1.93649 | + | 1.11803i | − | 1.16647i | −6.82455 | + | 1.55742i | −5.08226 | 1.50000 | − | 2.59808i | −1.30416 | + | 0.752955i | ||||||||||||||||||||||||||||
| 61.3 | 0.836732 | − | 1.44926i | −1.50000 | + | 0.866025i | 0.599760 | + | 1.03881i | 1.93649 | + | 1.11803i | 2.89852i | 4.76104 | − | 5.13152i | 8.70121 | 1.50000 | − | 2.59808i | 3.24065 | − | 1.87099i | |||||||||||||||||||||||||||||
| 61.4 | 1.76021 | − | 3.04878i | −1.50000 | + | 0.866025i | −4.19671 | − | 7.26891i | −1.93649 | − | 1.11803i | 6.09756i | 0.244004 | − | 6.99575i | −15.4667 | 1.50000 | − | 2.59808i | −6.81728 | + | 3.93596i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.3.n.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 315.3.w.a | 8 | ||
| 5.b | even | 2 | 1 | 525.3.o.l | 8 | ||
| 5.c | odd | 4 | 2 | 525.3.s.h | 16 | ||
| 7.c | even | 3 | 1 | 735.3.h.a | 8 | ||
| 7.d | odd | 6 | 1 | inner | 105.3.n.a | ✓ | 8 |
| 7.d | odd | 6 | 1 | 735.3.h.a | 8 | ||
| 21.g | even | 6 | 1 | 315.3.w.a | 8 | ||
| 35.i | odd | 6 | 1 | 525.3.o.l | 8 | ||
| 35.k | even | 12 | 2 | 525.3.s.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.3.n.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 105.3.n.a | ✓ | 8 | 7.d | odd | 6 | 1 | inner |
| 315.3.w.a | 8 | 3.b | odd | 2 | 1 | ||
| 315.3.w.a | 8 | 21.g | even | 6 | 1 | ||
| 525.3.o.l | 8 | 5.b | even | 2 | 1 | ||
| 525.3.o.l | 8 | 35.i | odd | 6 | 1 | ||
| 525.3.s.h | 16 | 5.c | odd | 4 | 2 | ||
| 525.3.s.h | 16 | 35.k | even | 12 | 2 | ||
| 735.3.h.a | 8 | 7.c | even | 3 | 1 | ||
| 735.3.h.a | 8 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} + 13T_{2}^{6} - 2T_{2}^{5} + 91T_{2}^{4} - 50T_{2}^{3} + 190T_{2}^{2} + 100T_{2} + 100 \)
acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 2 T^{7} + 13 T^{6} - 2 T^{5} + \cdots + 100 \)
$3$
\( (T^{2} + 3 T + 3)^{4} \)
$5$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$7$
\( T^{8} + 16 T^{7} + 109 T^{6} + \cdots + 5764801 \)
$11$
\( T^{8} - 20 T^{7} + 337 T^{6} + \cdots + 196 \)
$13$
\( T^{8} + 1164 T^{6} + \cdots + 1230957225 \)
$17$
\( T^{8} + 18 T^{7} + \cdots + 138297600 \)
$19$
\( T^{8} - 846 T^{6} + 712707 T^{4} + \cdots + 9054081 \)
$23$
\( T^{8} - 62 T^{7} + \cdots + 7138560100 \)
$29$
\( (T^{4} + 50 T^{3} - 2130 T^{2} + \cdots - 1825400)^{2} \)
$31$
\( T^{8} + 126 T^{7} + \cdots + 385089749136 \)
$37$
\( T^{8} + 80 T^{7} + \cdots + 5596891350625 \)
$41$
\( T^{8} + 3342 T^{6} + \cdots + 13887679716 \)
$43$
\( (T^{4} - 176 T^{3} + 9621 T^{2} + \cdots + 762376)^{2} \)
$47$
\( T^{8} + 72 T^{7} + 2115 T^{6} + \cdots + 26010000 \)
$53$
\( T^{8} + 76 T^{7} + \cdots + 98219560000 \)
$59$
\( T^{8} + 54 T^{7} + \cdots + 2582886122496 \)
$61$
\( T^{8} + 396 T^{7} + \cdots + 84471609600 \)
$67$
\( T^{8} - 184 T^{7} + \cdots + 273278017600 \)
$71$
\( (T^{4} - 82 T^{3} - 7998 T^{2} + \cdots + 22760224)^{2} \)
$73$
\( T^{8} - 348 T^{7} + \cdots + 16\!\cdots\!36 \)
$79$
\( T^{8} + 206 T^{7} + \cdots + 4446784387600 \)
$83$
\( T^{8} + \cdots + 111959592561216 \)
$89$
\( T^{8} + \cdots + 580473805464576 \)
$97$
\( T^{8} + 30696 T^{6} + \cdots + 2211287961600 \)
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