Newspace parameters
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.726638658394\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
Defining polynomial: |
\( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + 84549 \nu^{3} + 9416 \nu^{2} + 2376 \nu + 30518 ) / 118350 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} - 39747 \nu^{3} - 392048 \nu^{2} - 98928 \nu - 22604 ) / 118350 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592 ) / 118350 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842 ) / 59175 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + 39066 \nu^{3} - 46906 \nu^{2} - 20316 \nu - 3988 ) / 78900 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692 ) / 236700 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + 981276 \nu^{3} + 255184 \nu^{2} + 1362924 \nu + 532 ) / 118350 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + 11648 \nu^{2} + 50058 \nu - 136 ) / 4734 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{9} + 3\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} - 4 \)
|
\(\nu^{4}\) | \(=\) |
\( -8\beta_{9} + 2\beta_{8} - 19\beta_{7} + 9\beta_{6} - 13\beta_1 \)
|
\(\nu^{5}\) | \(=\) |
\( - 22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} - 47 \beta_{2} - 47 \beta _1 + 45 \)
|
\(\nu^{6}\) | \(=\) |
\( -23\beta_{5} + 70\beta_{4} - 78\beta_{3} - 128\beta_{2} + 154 \)
|
\(\nu^{7}\) | \(=\) |
\( 206\beta_{9} - 78\beta_{8} + 431\beta_{7} - 221\beta_{6} + 407\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( 628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + 691 \beta_{3} + 1187 \beta_{2} + 1187 \beta _1 - 1349 \)
|
\(\nu^{9}\) | \(=\) |
\( 691\beta_{5} - 1878\beta_{4} + 2036\beta_{3} + 3634\beta_{2} - 3968 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(66\) |
\(\chi(n)\) | \(1\) | \(-\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 |
|
−1.36263 | − | 2.36014i | 0.673208 | − | 1.16603i | −2.71349 | + | 4.69991i | −1.09358 | − | 1.89414i | −3.66932 | −2.19729 | − | 1.47375i | 9.33940 | 0.593582 | + | 1.02811i | −2.98028 | + | 5.16200i | ||||||||||||||||||||||||||||||||||
53.2 | −1.10666 | − | 1.91679i | −1.23721 | + | 2.14292i | −1.44940 | + | 2.51043i | 1.06140 | + | 1.83839i | 5.47671 | 2.63169 | + | 0.272389i | 1.98932 | −1.56140 | − | 2.70442i | 2.34921 | − | 4.06896i | |||||||||||||||||||||||||||||||||||
53.3 | −0.632804 | − | 1.09605i | 1.31364 | − | 2.27529i | 0.199118 | − | 0.344882i | 1.45130 | + | 2.51373i | −3.32511 | −1.29536 | + | 2.30696i | −3.03523 | −1.95130 | − | 3.37975i | 1.83678 | − | 3.18139i | |||||||||||||||||||||||||||||||||||
53.4 | 0.0978281 | + | 0.169443i | 0.129894 | − | 0.224983i | 0.980859 | − | 1.69890i | −1.96625 | − | 3.40565i | 0.0508292 | 1.12324 | + | 2.39548i | 0.775135 | 1.46625 | + | 2.53963i | 0.384710 | − | 0.666337i | |||||||||||||||||||||||||||||||||||
53.5 | 1.00426 | + | 1.73943i | −0.879528 | + | 1.52339i | −1.01709 | + | 1.76164i | −0.452861 | − | 0.784378i | −3.53311 | 0.237709 | − | 2.63505i | −0.0686323 | −0.0471392 | − | 0.0816475i | 0.909582 | − | 1.57544i | |||||||||||||||||||||||||||||||||||
79.1 | −1.36263 | + | 2.36014i | 0.673208 | + | 1.16603i | −2.71349 | − | 4.69991i | −1.09358 | + | 1.89414i | −3.66932 | −2.19729 | + | 1.47375i | 9.33940 | 0.593582 | − | 1.02811i | −2.98028 | − | 5.16200i | |||||||||||||||||||||||||||||||||||
79.2 | −1.10666 | + | 1.91679i | −1.23721 | − | 2.14292i | −1.44940 | − | 2.51043i | 1.06140 | − | 1.83839i | 5.47671 | 2.63169 | − | 0.272389i | 1.98932 | −1.56140 | + | 2.70442i | 2.34921 | + | 4.06896i | |||||||||||||||||||||||||||||||||||
79.3 | −0.632804 | + | 1.09605i | 1.31364 | + | 2.27529i | 0.199118 | + | 0.344882i | 1.45130 | − | 2.51373i | −3.32511 | −1.29536 | − | 2.30696i | −3.03523 | −1.95130 | + | 3.37975i | 1.83678 | + | 3.18139i | |||||||||||||||||||||||||||||||||||
79.4 | 0.0978281 | − | 0.169443i | 0.129894 | + | 0.224983i | 0.980859 | + | 1.69890i | −1.96625 | + | 3.40565i | 0.0508292 | 1.12324 | − | 2.39548i | 0.775135 | 1.46625 | − | 2.53963i | 0.384710 | + | 0.666337i | |||||||||||||||||||||||||||||||||||
79.5 | 1.00426 | − | 1.73943i | −0.879528 | − | 1.52339i | −1.01709 | − | 1.76164i | −0.452861 | + | 0.784378i | −3.53311 | 0.237709 | + | 2.63505i | −0.0686323 | −0.0471392 | + | 0.0816475i | 0.909582 | + | 1.57544i | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.2.e.c | ✓ | 10 |
3.b | odd | 2 | 1 | 819.2.j.h | 10 | ||
4.b | odd | 2 | 1 | 1456.2.r.p | 10 | ||
7.b | odd | 2 | 1 | 637.2.e.m | 10 | ||
7.c | even | 3 | 1 | inner | 91.2.e.c | ✓ | 10 |
7.c | even | 3 | 1 | 637.2.a.l | 5 | ||
7.d | odd | 6 | 1 | 637.2.a.k | 5 | ||
7.d | odd | 6 | 1 | 637.2.e.m | 10 | ||
13.b | even | 2 | 1 | 1183.2.e.f | 10 | ||
21.g | even | 6 | 1 | 5733.2.a.bm | 5 | ||
21.h | odd | 6 | 1 | 819.2.j.h | 10 | ||
21.h | odd | 6 | 1 | 5733.2.a.bl | 5 | ||
28.g | odd | 6 | 1 | 1456.2.r.p | 10 | ||
91.r | even | 6 | 1 | 1183.2.e.f | 10 | ||
91.r | even | 6 | 1 | 8281.2.a.bw | 5 | ||
91.s | odd | 6 | 1 | 8281.2.a.bx | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
91.2.e.c | ✓ | 10 | 7.c | even | 3 | 1 | inner |
637.2.a.k | 5 | 7.d | odd | 6 | 1 | ||
637.2.a.l | 5 | 7.c | even | 3 | 1 | ||
637.2.e.m | 10 | 7.b | odd | 2 | 1 | ||
637.2.e.m | 10 | 7.d | odd | 6 | 1 | ||
819.2.j.h | 10 | 3.b | odd | 2 | 1 | ||
819.2.j.h | 10 | 21.h | odd | 6 | 1 | ||
1183.2.e.f | 10 | 13.b | even | 2 | 1 | ||
1183.2.e.f | 10 | 91.r | even | 6 | 1 | ||
1456.2.r.p | 10 | 4.b | odd | 2 | 1 | ||
1456.2.r.p | 10 | 28.g | odd | 6 | 1 | ||
5733.2.a.bl | 5 | 21.h | odd | 6 | 1 | ||
5733.2.a.bm | 5 | 21.g | even | 6 | 1 | ||
8281.2.a.bw | 5 | 91.r | even | 6 | 1 | ||
8281.2.a.bx | 5 | 91.s | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 4T_{2}^{9} + 17T_{2}^{8} + 30T_{2}^{7} + 81T_{2}^{6} + 116T_{2}^{5} + 265T_{2}^{4} + 210T_{2}^{3} + 195T_{2}^{2} - 36T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 4 T^{9} + 17 T^{8} + 30 T^{7} + \cdots + 9 \)
$3$
\( T^{10} + 9 T^{8} + 65 T^{6} - 4 T^{5} + \cdots + 16 \)
$5$
\( T^{10} + 2 T^{9} + 19 T^{8} + \cdots + 2304 \)
$7$
\( T^{10} - T^{9} + 6 T^{8} - 17 T^{7} + \cdots + 16807 \)
$11$
\( T^{10} + 11 T^{9} + 85 T^{8} + \cdots + 1089 \)
$13$
\( (T - 1)^{10} \)
$17$
\( T^{10} - 5 T^{9} + 47 T^{8} + \cdots + 184041 \)
$19$
\( T^{10} + 9 T^{9} + 95 T^{8} + \cdots + 49729 \)
$23$
\( T^{10} + 10 T^{9} + 69 T^{8} + \cdots + 144 \)
$29$
\( (T^{5} + 3 T^{4} - 25 T^{3} - 19 T^{2} + \cdots - 108)^{2} \)
$31$
\( T^{10} - 6 T^{9} + 97 T^{8} + \cdots + 126736 \)
$37$
\( T^{10} + 4 T^{9} + 127 T^{8} + \cdots + 49505296 \)
$41$
\( (T^{5} - 14 T^{4} - 28 T^{3} + 940 T^{2} + \cdots - 1584)^{2} \)
$43$
\( (T^{5} - 2 T^{4} - 72 T^{3} + 308 T^{2} + \cdots + 64)^{2} \)
$47$
\( T^{10} + T^{9} + 125 T^{8} + \cdots + 26718561 \)
$53$
\( T^{10} + 17 T^{9} + \cdots + 398361681 \)
$59$
\( T^{10} + 11 T^{9} + 85 T^{8} + \cdots + 1089 \)
$61$
\( T^{10} - 11 T^{9} + 243 T^{8} + \cdots + 71588521 \)
$67$
\( T^{10} + 13 T^{9} + \cdots + 515244601 \)
$71$
\( (T^{5} - 15 T^{4} - 25 T^{3} + 853 T^{2} + \cdots - 6336)^{2} \)
$73$
\( T^{10} + 75 T^{8} + 84 T^{7} + \cdots + 506944 \)
$79$
\( T^{10} + 2 T^{9} + 141 T^{8} + \cdots + 1000000 \)
$83$
\( (T^{5} - 6 T^{4} - 124 T^{3} + 308 T^{2} + \cdots - 7488)^{2} \)
$89$
\( T^{10} - 4 T^{9} + 171 T^{8} + \cdots + 59166864 \)
$97$
\( (T^{5} + 12 T^{4} - 16 T^{3} - 612 T^{2} + \cdots - 2384)^{2} \)
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