Newspace parameters
Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 936.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.47399762919\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{14} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) :
\(\beta_{1}\) | \(=\) | \( ( 310 \nu^{14} - 2816 \nu^{12} - 9222 \nu^{10} + 18376 \nu^{8} + 130292 \nu^{6} - 53906 \nu^{4} + 290050 \nu^{2} + 217000 ) / 176275 \) |
\(\beta_{2}\) | \(=\) | \( ( 11992 \nu^{14} + 9614 \nu^{12} - 190612 \nu^{10} - 554504 \nu^{8} + 1001932 \nu^{6} + 2217000 \nu^{4} + 3496000 \nu^{2} + 5574000 ) / 881375 \) |
\(\beta_{3}\) | \(=\) | \( ( 16948 \nu^{14} + 5486 \nu^{12} - 299188 \nu^{10} - 779596 \nu^{8} + 1911868 \nu^{6} + 4019270 \nu^{4} + 6074000 \nu^{2} + 6351000 ) / 881375 \) |
\(\beta_{4}\) | \(=\) | \( ( - 17344 \nu^{14} - 7298 \nu^{12} + 306684 \nu^{10} + 805078 \nu^{8} - 1876924 \nu^{6} - 3981500 \nu^{4} - 6068600 \nu^{2} - 8342875 ) / 881375 \) |
\(\beta_{5}\) | \(=\) | \( ( - 10661 \nu^{15} - 14207 \nu^{13} + 225956 \nu^{11} + 888177 \nu^{9} - 521116 \nu^{7} - 5727895 \nu^{5} - 13225300 \nu^{3} - 26244000 \nu ) / 4406875 \) |
\(\beta_{6}\) | \(=\) | \( ( - 25832 \nu^{14} - 21274 \nu^{12} + 450392 \nu^{10} + 1381764 \nu^{8} - 2270262 \nu^{6} - 7392230 \nu^{4} - 12703750 \nu^{2} - 13499250 ) / 881375 \) |
\(\beta_{7}\) | \(=\) | \( ( - 28806 \nu^{14} - 29832 \nu^{12} + 448156 \nu^{10} + 1496702 \nu^{8} - 2047066 \nu^{6} - 6175530 \nu^{4} - 12354350 \nu^{2} - 13465750 ) / 881375 \) |
\(\beta_{8}\) | \(=\) | \( ( 15392 \nu^{15} + 52754 \nu^{13} - 145257 \nu^{11} - 987344 \nu^{9} - 868348 \nu^{7} + 1700315 \nu^{5} + 6288975 \nu^{3} + 15325500 \nu ) / 4406875 \) |
\(\beta_{9}\) | \(=\) | \( ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu ) / 400625 \) |
\(\beta_{10}\) | \(=\) | \( ( - 22194 \nu^{15} - 11523 \nu^{13} + 339409 \nu^{11} + 1024903 \nu^{9} - 1669899 \nu^{7} - 3769000 \nu^{5} - 11127550 \nu^{3} - 9814875 \nu ) / 4406875 \) |
\(\beta_{11}\) | \(=\) | \( ( - 44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000 ) / 881375 \) |
\(\beta_{12}\) | \(=\) | \( ( - 46447 \nu^{15} - 41694 \nu^{13} + 802127 \nu^{11} + 2508384 \nu^{9} - 4035597 \nu^{7} - 13232745 \nu^{5} - 19604750 \nu^{3} - 21119125 \nu ) / 4406875 \) |
\(\beta_{13}\) | \(=\) | \( ( 78923 \nu^{15} + 34611 \nu^{13} - 1339113 \nu^{11} - 3553821 \nu^{9} + 8128668 \nu^{7} + 15701920 \nu^{5} + 26400825 \nu^{3} + 32586750 \nu ) / 4406875 \) |
\(\beta_{14}\) | \(=\) | \( ( - 108813 \nu^{15} - 85331 \nu^{13} + 1829673 \nu^{11} + 5646091 \nu^{9} - 9386828 \nu^{7} - 27713410 \nu^{5} - 49911275 \nu^{3} - 50144500 \nu ) / 4406875 \) |
\(\beta_{15}\) | \(=\) | \( ( - 128244 \nu^{15} - 47208 \nu^{13} + 2180589 \nu^{11} + 5658588 \nu^{9} - 13459254 \nu^{7} - 25923185 \nu^{5} - 43314975 \nu^{3} - 53041500 \nu ) / 4406875 \) |
\(\nu\) | \(=\) | \( ( -\beta_{15} - \beta_{13} + 2\beta_{12} - 2\beta_{9} - \beta_{5} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -5\beta_{14} - \beta_{13} + 5\beta_{12} + 7\beta_{10} + 5\beta_{9} + \beta_{8} + \beta_{5} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( 2\beta_{7} - 2\beta_{6} + 8\beta_{4} + 5\beta_{3} + 5\beta_{2} + 8 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -7\beta_{15} - 12\beta_{13} - 3\beta_{9} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 11\beta_{11} - 11\beta_{7} - 6\beta_{6} + 11\beta_{3} - 15\beta_{2} + 5\beta _1 + 30 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -5\beta_{15} - 55\beta_{14} - 4\beta_{13} + 127\beta_{12} + 55\beta_{10} - 55\beta_{9} - 9\beta_{8} - 4\beta_{5} ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( 32\beta_{11} - 52\beta_{6} + 53\beta_{4} + 18\beta_{3} + 53\beta_{2} + 32\beta _1 + 17 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 137 \beta_{15} - 57 \beta_{14} - 224 \beta_{13} - 57 \beta_{12} + 247 \beta_{10} + 247 \beta_{9} + 87 \beta_{8} + 50 \beta_{5} ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( 21\beta_{11} - 36\beta_{7} + 275\beta_{4} + 275\beta_{3} + 607 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 384 \beta_{15} - 341 \beta_{14} - 623 \beta_{13} + 1421 \beta_{12} - 341 \beta_{10} - 1421 \beta_{9} - 239 \beta_{8} - 145 \beta_{5} ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( ( 752\beta_{11} - 462\beta_{7} - 752\beta_{6} - 110\beta_{4} - 177\beta_{2} + 462\beta _1 + 110 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 395 \beta_{15} - 2315 \beta_{14} - 642 \beta_{13} + 2315 \beta_{12} + 5199 \beta_{10} + 2315 \beta_{9} + 1037 \beta_{8} + 642 \beta_{5} ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( ( 639\beta_{7} - 639\beta_{6} + 5431\beta_{4} + 3360\beta_{3} + 3360\beta_{2} + 400\beta _1 + 5431 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( -4399\beta_{15} - 7109\beta_{13} + 2475\beta_{12} - 2475\beta_{10} - 3996\beta_{9} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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181.1 |
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−1.34500 | − | 0.437016i | 0 | 1.61803 | + | 1.17557i | −1.66251 | 0 | − | 3.57266i | −1.66251 | − | 2.28825i | 0 | 2.23607 | + | 0.726543i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.2 | −1.34500 | − | 0.437016i | 0 | 1.61803 | + | 1.17557i | −1.66251 | 0 | 3.57266i | −1.66251 | − | 2.28825i | 0 | 2.23607 | + | 0.726543i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.3 | −1.34500 | + | 0.437016i | 0 | 1.61803 | − | 1.17557i | −1.66251 | 0 | − | 3.57266i | −1.66251 | + | 2.28825i | 0 | 2.23607 | − | 0.726543i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.4 | −1.34500 | + | 0.437016i | 0 | 1.61803 | − | 1.17557i | −1.66251 | 0 | 3.57266i | −1.66251 | + | 2.28825i | 0 | 2.23607 | − | 0.726543i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.5 | −0.831254 | − | 1.14412i | 0 | −0.618034 | + | 1.90211i | 2.68999 | 0 | − | 4.15163i | 2.68999 | − | 0.874032i | 0 | −2.23607 | − | 3.07768i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.6 | −0.831254 | − | 1.14412i | 0 | −0.618034 | + | 1.90211i | 2.68999 | 0 | 4.15163i | 2.68999 | − | 0.874032i | 0 | −2.23607 | − | 3.07768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.7 | −0.831254 | + | 1.14412i | 0 | −0.618034 | − | 1.90211i | 2.68999 | 0 | − | 4.15163i | 2.68999 | + | 0.874032i | 0 | −2.23607 | + | 3.07768i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.8 | −0.831254 | + | 1.14412i | 0 | −0.618034 | − | 1.90211i | 2.68999 | 0 | 4.15163i | 2.68999 | + | 0.874032i | 0 | −2.23607 | + | 3.07768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.9 | 0.831254 | − | 1.14412i | 0 | −0.618034 | − | 1.90211i | −2.68999 | 0 | − | 4.15163i | −2.68999 | − | 0.874032i | 0 | −2.23607 | + | 3.07768i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.10 | 0.831254 | − | 1.14412i | 0 | −0.618034 | − | 1.90211i | −2.68999 | 0 | 4.15163i | −2.68999 | − | 0.874032i | 0 | −2.23607 | + | 3.07768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.11 | 0.831254 | + | 1.14412i | 0 | −0.618034 | + | 1.90211i | −2.68999 | 0 | − | 4.15163i | −2.68999 | + | 0.874032i | 0 | −2.23607 | − | 3.07768i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.12 | 0.831254 | + | 1.14412i | 0 | −0.618034 | + | 1.90211i | −2.68999 | 0 | 4.15163i | −2.68999 | + | 0.874032i | 0 | −2.23607 | − | 3.07768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.13 | 1.34500 | − | 0.437016i | 0 | 1.61803 | − | 1.17557i | 1.66251 | 0 | − | 3.57266i | 1.66251 | − | 2.28825i | 0 | 2.23607 | − | 0.726543i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.14 | 1.34500 | − | 0.437016i | 0 | 1.61803 | − | 1.17557i | 1.66251 | 0 | 3.57266i | 1.66251 | − | 2.28825i | 0 | 2.23607 | − | 0.726543i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.15 | 1.34500 | + | 0.437016i | 0 | 1.61803 | + | 1.17557i | 1.66251 | 0 | − | 3.57266i | 1.66251 | + | 2.28825i | 0 | 2.23607 | + | 0.726543i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
181.16 | 1.34500 | + | 0.437016i | 0 | 1.61803 | + | 1.17557i | 1.66251 | 0 | 3.57266i | 1.66251 | + | 2.28825i | 0 | 2.23607 | + | 0.726543i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
104.e | even | 2 | 1 | inner |
312.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 936.2.m.h | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
4.b | odd | 2 | 1 | 3744.2.m.h | 16 | ||
8.b | even | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
8.d | odd | 2 | 1 | 3744.2.m.h | 16 | ||
12.b | even | 2 | 1 | 3744.2.m.h | 16 | ||
13.b | even | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
24.f | even | 2 | 1 | 3744.2.m.h | 16 | ||
24.h | odd | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
39.d | odd | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
52.b | odd | 2 | 1 | 3744.2.m.h | 16 | ||
104.e | even | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
104.h | odd | 2 | 1 | 3744.2.m.h | 16 | ||
156.h | even | 2 | 1 | 3744.2.m.h | 16 | ||
312.b | odd | 2 | 1 | inner | 936.2.m.h | ✓ | 16 |
312.h | even | 2 | 1 | 3744.2.m.h | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
936.2.m.h | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
936.2.m.h | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 8.b | even | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 13.b | even | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 24.h | odd | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 39.d | odd | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 104.e | even | 2 | 1 | inner |
936.2.m.h | ✓ | 16 | 312.b | odd | 2 | 1 | inner |
3744.2.m.h | 16 | 4.b | odd | 2 | 1 | ||
3744.2.m.h | 16 | 8.d | odd | 2 | 1 | ||
3744.2.m.h | 16 | 12.b | even | 2 | 1 | ||
3744.2.m.h | 16 | 24.f | even | 2 | 1 | ||
3744.2.m.h | 16 | 52.b | odd | 2 | 1 | ||
3744.2.m.h | 16 | 104.h | odd | 2 | 1 | ||
3744.2.m.h | 16 | 156.h | even | 2 | 1 | ||
3744.2.m.h | 16 | 312.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 10T_{5}^{2} + 20 \)
acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \)
$3$
\( T^{16} \)
$5$
\( (T^{4} - 10 T^{2} + 20)^{4} \)
$7$
\( (T^{4} + 30 T^{2} + 220)^{4} \)
$11$
\( (T^{4} - 20 T^{2} + 20)^{4} \)
$13$
\( (T^{8} - 12 T^{6} + 54 T^{4} - 2028 T^{2} + \cdots + 28561)^{2} \)
$17$
\( (T^{4} - 60 T^{2} + 880)^{4} \)
$19$
\( (T^{4} - 34 T^{2} + 44)^{4} \)
$23$
\( (T^{4} - 80 T^{2} + 880)^{4} \)
$29$
\( (T^{4} + 28 T^{2} + 176)^{4} \)
$31$
\( (T^{4} + 150 T^{2} + 5500)^{4} \)
$37$
\( (T^{4} - 104 T^{2} + 704)^{4} \)
$41$
\( (T^{4} + 126 T^{2} + 324)^{4} \)
$43$
\( (T^{4} + 20 T^{2} + 80)^{4} \)
$47$
\( (T^{4} + 84 T^{2} + 1444)^{4} \)
$53$
\( (T^{4} + 128 T^{2} + 176)^{4} \)
$59$
\( (T^{4} - 100 T^{2} + 500)^{4} \)
$61$
\( (T^{4} + 180 T^{2} + 6480)^{4} \)
$67$
\( (T^{4} - 154 T^{2} + 5324)^{4} \)
$71$
\( (T^{4} + 116 T^{2} + 484)^{4} \)
$73$
\( (T^{4} + 200 T^{2} + 3520)^{4} \)
$79$
\( (T^{2} - 4 T - 76)^{8} \)
$83$
\( (T^{4} - 100 T^{2} + 500)^{4} \)
$89$
\( (T^{4} + 230 T^{2} + 12100)^{4} \)
$97$
\( (T^{4} + 160 T^{2} + 3520)^{4} \)
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