Newspace parameters
Level: | \( N \) | \(=\) | \( 1156 = 2^{2} \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1156.h (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.23070647366\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
Coefficient field: | 16.0.229607785695641627262976.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 799x^{8} + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{8} \) |
Twist minimal: | no (minimal twist has level 68) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 \( x^{16} + 799x^{8} + 6561 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{10} + 1159\nu^{2} ) / 1953 \) |
\(\beta_{2}\) | \(=\) | \( ( 2\nu^{8} + 799 ) / 217 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{10} - 601\nu^{2} ) / 279 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{12} + 880\nu^{4} ) / 2511 \) |
\(\beta_{5}\) | \(=\) | \( ( -40\nu^{14} - 32689\nu^{6} ) / 158193 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{12} - 718\nu^{4} ) / 567 \) |
\(\beta_{7}\) | \(=\) | \( ( 19\nu^{13} - 81\nu^{9} + 14209\nu^{5} - 41148\nu ) / 52731 \) |
\(\beta_{8}\) | \(=\) | \( ( 19\nu^{13} + 81\nu^{9} + 14209\nu^{5} + 41148\nu ) / 52731 \) |
\(\beta_{9}\) | \(=\) | \( ( \nu^{13} + 880\nu^{5} + 2511\nu ) / 2511 \) |
\(\beta_{10}\) | \(=\) | \( ( \nu^{13} + 880\nu^{5} - 2511\nu ) / 2511 \) |
\(\beta_{11}\) | \(=\) | \( ( 22\nu^{14} + 16849\nu^{6} ) / 22599 \) |
\(\beta_{12}\) | \(=\) | \( ( -97\nu^{15} + 324\nu^{11} - 75316\nu^{7} + 217323\nu^{3} ) / 474579 \) |
\(\beta_{13}\) | \(=\) | \( ( -97\nu^{15} - 324\nu^{11} - 75316\nu^{7} - 217323\nu^{3} ) / 474579 \) |
\(\beta_{14}\) | \(=\) | \( ( -40\nu^{15} + 81\nu^{11} - 32689\nu^{7} + 93879\nu^{3} ) / 158193 \) |
\(\beta_{15}\) | \(=\) | \( ( -40\nu^{15} - 81\nu^{11} - 32689\nu^{7} - 93879\nu^{3} ) / 158193 \) |
\(\nu\) | \(=\) | \( ( -\beta_{10} + \beta_{9} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 7\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -4\beta_{15} + 4\beta_{14} + 3\beta_{13} - 3\beta_{12} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 7\beta_{6} + 31\beta_{4} ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 19\beta_{10} + 19\beta_{9} - 21\beta_{8} - 21\beta_{7} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -20\beta_{11} - 77\beta_{5} \) |
\(\nu^{7}\) | \(=\) | \( ( -97\beta_{15} - 97\beta_{14} + 120\beta_{13} + 120\beta_{12} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 217\beta_{2} - 799 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 508\beta_{10} - 508\beta_{9} + 651\beta_{8} - 651\beta_{7} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( -1159\beta_{3} - 4207\beta_1 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 2683\beta_{15} - 2683\beta_{14} - 3477\beta_{13} + 3477\beta_{12} ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( -3080\beta_{6} - 11129\beta_{4} \) |
\(\nu^{13}\) | \(=\) | \( ( -14209\beta_{10} - 14209\beta_{9} + 18480\beta_{8} + 18480\beta_{7} ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( 32689\beta_{11} + 117943\beta_{5} ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( 75316\beta_{15} + 75316\beta_{14} - 98067\beta_{13} - 98067\beta_{12} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).
\(n\) | \(579\) | \(581\) |
\(\chi(n)\) | \(1\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
733.1 |
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0 | −3.00872 | + | 1.24625i | 0 | −0.541196 | − | 1.30656i | 0 | 1.24625 | − | 3.00872i | 0 | 5.37794 | − | 5.37794i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
733.2 | 0 | −1.70216 | + | 0.705057i | 0 | 0.541196 | + | 1.30656i | 0 | 0.705057 | − | 1.70216i | 0 | 0.278917 | − | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
733.3 | 0 | 1.70216 | − | 0.705057i | 0 | −0.541196 | − | 1.30656i | 0 | −0.705057 | + | 1.70216i | 0 | 0.278917 | − | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
733.4 | 0 | 3.00872 | − | 1.24625i | 0 | 0.541196 | + | 1.30656i | 0 | −1.24625 | + | 3.00872i | 0 | 5.37794 | − | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
757.1 | 0 | −3.00872 | − | 1.24625i | 0 | −0.541196 | + | 1.30656i | 0 | 1.24625 | + | 3.00872i | 0 | 5.37794 | + | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
757.2 | 0 | −1.70216 | − | 0.705057i | 0 | 0.541196 | − | 1.30656i | 0 | 0.705057 | + | 1.70216i | 0 | 0.278917 | + | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
757.3 | 0 | 1.70216 | + | 0.705057i | 0 | −0.541196 | + | 1.30656i | 0 | −0.705057 | − | 1.70216i | 0 | 0.278917 | + | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
757.4 | 0 | 3.00872 | + | 1.24625i | 0 | 0.541196 | − | 1.30656i | 0 | −1.24625 | − | 3.00872i | 0 | 5.37794 | + | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
977.1 | 0 | −1.24625 | + | 3.00872i | 0 | 1.30656 | + | 0.541196i | 0 | −3.00872 | + | 1.24625i | 0 | −5.37794 | − | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
977.2 | 0 | −0.705057 | + | 1.70216i | 0 | −1.30656 | − | 0.541196i | 0 | −1.70216 | + | 0.705057i | 0 | −0.278917 | − | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
977.3 | 0 | 0.705057 | − | 1.70216i | 0 | 1.30656 | + | 0.541196i | 0 | 1.70216 | − | 0.705057i | 0 | −0.278917 | − | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
977.4 | 0 | 1.24625 | − | 3.00872i | 0 | −1.30656 | − | 0.541196i | 0 | 3.00872 | − | 1.24625i | 0 | −5.37794 | − | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1001.1 | 0 | −1.24625 | − | 3.00872i | 0 | 1.30656 | − | 0.541196i | 0 | −3.00872 | − | 1.24625i | 0 | −5.37794 | + | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1001.2 | 0 | −0.705057 | − | 1.70216i | 0 | −1.30656 | + | 0.541196i | 0 | −1.70216 | − | 0.705057i | 0 | −0.278917 | + | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1001.3 | 0 | 0.705057 | + | 1.70216i | 0 | 1.30656 | − | 0.541196i | 0 | 1.70216 | + | 0.705057i | 0 | −0.278917 | + | 0.278917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1001.4 | 0 | 1.24625 | + | 3.00872i | 0 | −1.30656 | + | 0.541196i | 0 | 3.00872 | + | 1.24625i | 0 | −5.37794 | + | 5.37794i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1156.2.h.e | 16 | |
17.b | even | 2 | 1 | inner | 1156.2.h.e | 16 | |
17.c | even | 4 | 2 | inner | 1156.2.h.e | 16 | |
17.d | even | 8 | 4 | inner | 1156.2.h.e | 16 | |
17.e | odd | 16 | 2 | 68.2.e.a | ✓ | 4 | |
17.e | odd | 16 | 2 | 1156.2.a.h | 4 | ||
17.e | odd | 16 | 2 | 1156.2.b.a | 4 | ||
17.e | odd | 16 | 2 | 1156.2.e.c | 4 | ||
51.i | even | 16 | 2 | 612.2.k.e | 4 | ||
68.i | even | 16 | 2 | 272.2.o.g | 4 | ||
68.i | even | 16 | 2 | 4624.2.a.bq | 4 | ||
85.o | even | 16 | 1 | 1700.2.m.a | 4 | ||
85.o | even | 16 | 1 | 1700.2.m.b | 4 | ||
85.p | odd | 16 | 2 | 1700.2.o.c | 4 | ||
85.r | even | 16 | 1 | 1700.2.m.a | 4 | ||
85.r | even | 16 | 1 | 1700.2.m.b | 4 | ||
136.q | odd | 16 | 2 | 1088.2.o.t | 4 | ||
136.s | even | 16 | 2 | 1088.2.o.s | 4 | ||
204.t | odd | 16 | 2 | 2448.2.be.u | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
68.2.e.a | ✓ | 4 | 17.e | odd | 16 | 2 | |
272.2.o.g | 4 | 68.i | even | 16 | 2 | ||
612.2.k.e | 4 | 51.i | even | 16 | 2 | ||
1088.2.o.s | 4 | 136.s | even | 16 | 2 | ||
1088.2.o.t | 4 | 136.q | odd | 16 | 2 | ||
1156.2.a.h | 4 | 17.e | odd | 16 | 2 | ||
1156.2.b.a | 4 | 17.e | odd | 16 | 2 | ||
1156.2.e.c | 4 | 17.e | odd | 16 | 2 | ||
1156.2.h.e | 16 | 1.a | even | 1 | 1 | trivial | |
1156.2.h.e | 16 | 17.b | even | 2 | 1 | inner | |
1156.2.h.e | 16 | 17.c | even | 4 | 2 | inner | |
1156.2.h.e | 16 | 17.d | even | 8 | 4 | inner | |
1700.2.m.a | 4 | 85.o | even | 16 | 1 | ||
1700.2.m.a | 4 | 85.r | even | 16 | 1 | ||
1700.2.m.b | 4 | 85.o | even | 16 | 1 | ||
1700.2.m.b | 4 | 85.r | even | 16 | 1 | ||
1700.2.o.c | 4 | 85.p | odd | 16 | 2 | ||
2448.2.be.u | 4 | 204.t | odd | 16 | 2 | ||
4624.2.a.bq | 4 | 68.i | even | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} + 12784T_{3}^{8} + 1679616 \)
acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 12784 T^{8} + \cdots + 1679616 \)
$5$
\( (T^{8} + 16)^{2} \)
$7$
\( T^{16} + 12784 T^{8} + \cdots + 1679616 \)
$11$
\( T^{16} + 226544 T^{8} + 256 \)
$13$
\( (T^{4} + 28 T^{2} + 144)^{4} \)
$17$
\( T^{16} \)
$19$
\( (T^{8} + 1904 T^{4} + 256)^{2} \)
$23$
\( T^{16} + 1879792 T^{8} + \cdots + 1679616 \)
$29$
\( T^{16} + 15598624 T^{8} + \cdots + 11019960576 \)
$31$
\( T^{16} + 1879792 T^{8} + \cdots + 1679616 \)
$37$
\( (T^{8} + 104976)^{2} \)
$41$
\( (T^{8} + 16)^{2} \)
$43$
\( (T^{8} + 12784 T^{4} + 1679616)^{2} \)
$47$
\( (T^{2} + 16)^{8} \)
$53$
\( (T^{8} + 7936 T^{4} + 5308416)^{2} \)
$59$
\( (T^{8} + 5488 T^{4} + 20736)^{2} \)
$61$
\( T^{16} + 366060064 T^{8} + \cdots + 11\!\cdots\!56 \)
$67$
\( (T^{2} + 4 T - 48)^{8} \)
$71$
\( T^{16} + 244873712 T^{8} + \cdots + 20047612231936 \)
$73$
\( (T^{8} + 92236816)^{2} \)
$79$
\( T^{16} + 226544 T^{8} + 256 \)
$83$
\( (T^{8} + 100976 T^{4} + \cdots + 21381376)^{2} \)
$89$
\( (T^{4} + 252 T^{2} + 11664)^{4} \)
$97$
\( T^{16} + 179128864 T^{8} + \cdots + 1679616 \)
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