Newspace parameters
Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1344.p (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(79.2985670477\) |
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: | \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{17}\cdot 3^{4} \) |
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 \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) :
\(\beta_{1}\) | \(=\) | \( ( - 20\!\cdots\!32 \nu^{15} + \cdots - 27\!\cdots\!76 ) / 41\!\cdots\!92 \) |
\(\beta_{2}\) | \(=\) | \( ( 45\!\cdots\!01 \nu^{15} + \cdots - 21\!\cdots\!55 ) / 10\!\cdots\!14 \) |
\(\beta_{3}\) | \(=\) | \( ( 91\!\cdots\!68 \nu^{15} + \cdots - 90\!\cdots\!27 ) / 48\!\cdots\!34 \) |
\(\beta_{4}\) | \(=\) | \( ( 22\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!12 ) / 10\!\cdots\!14 \) |
\(\beta_{5}\) | \(=\) | \( ( - 22\!\cdots\!51 \nu^{15} + \cdots + 20\!\cdots\!12 ) / 10\!\cdots\!14 \) |
\(\beta_{6}\) | \(=\) | \( ( - 63\!\cdots\!46 \nu^{15} + \cdots - 77\!\cdots\!93 ) / 27\!\cdots\!38 \) |
\(\beta_{7}\) | \(=\) | \( ( - 30\!\cdots\!71 \nu^{15} + \cdots + 20\!\cdots\!98 ) / 11\!\cdots\!06 \) |
\(\beta_{8}\) | \(=\) | \( ( 11\!\cdots\!76 \nu^{15} + \cdots + 59\!\cdots\!77 ) / 19\!\cdots\!66 \) |
\(\beta_{9}\) | \(=\) | \( ( 53\!\cdots\!25 \nu^{15} + \cdots + 25\!\cdots\!41 ) / 92\!\cdots\!46 \) |
\(\beta_{10}\) | \(=\) | \( ( - 10\!\cdots\!09 \nu^{15} + \cdots - 25\!\cdots\!35 ) / 12\!\cdots\!02 \) |
\(\beta_{11}\) | \(=\) | \( ( 39\!\cdots\!68 \nu^{15} + \cdots + 42\!\cdots\!29 ) / 45\!\cdots\!62 \) |
\(\beta_{12}\) | \(=\) | \( ( - 17\!\cdots\!50 \nu^{15} + \cdots - 17\!\cdots\!40 ) / 16\!\cdots\!28 \) |
\(\beta_{13}\) | \(=\) | \( ( - 47\!\cdots\!55 \nu^{15} + \cdots + 54\!\cdots\!94 ) / 39\!\cdots\!02 \) |
\(\beta_{14}\) | \(=\) | \( ( - 32\!\cdots\!76 \nu^{15} + \cdots - 25\!\cdots\!08 ) / 19\!\cdots\!66 \) |
\(\beta_{15}\) | \(=\) | \( ( - 10\!\cdots\!01 \nu^{15} + \cdots + 59\!\cdots\!81 ) / 42\!\cdots\!22 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} + \beta_{4} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{14} + 5\beta_{11} - \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 342\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( - 7 \beta_{15} - 7 \beta_{14} - 13 \beta_{13} - 13 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} - 28 \beta_{9} - 28 \beta_{8} + 157 \beta_{7} + 157 \beta_{6} + 41 \beta_{5} - 41 \beta_{4} + 553 \beta_{3} - 553 \beta_{2} - 97 \beta _1 + 97 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( 163 \beta_{15} + 79 \beta_{13} - 843 \beta_{10} - 198 \beta_{9} - 64 \beta_{7} + 269 \beta_{5} - 71 \beta_{4} - 71 \beta_{3} - 71 \beta_{2} - 44560 \) |
\(\nu^{5}\) | \(=\) | \( ( - 1496 \beta_{15} + 1496 \beta_{14} - 2047 \beta_{13} + 2047 \beta_{12} + 417 \beta_{11} + 417 \beta_{10} - 5238 \beta_{9} + 5238 \beta_{8} + 24568 \beta_{7} - 24568 \beta_{6} - 75344 \beta_{5} - 75344 \beta_{4} + \cdots + 5299 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( - 92451 \beta_{14} - 81136 \beta_{12} - 523953 \beta_{11} + 129075 \beta_{8} + 78256 \beta_{6} - 36503 \beta_{5} + 151104 \beta_{4} - 36503 \beta_{3} - 22029 \beta_{2} - 25601422 \beta_1 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 1044781 \beta_{15} + 1044781 \beta_{14} + 845557 \beta_{13} + 845557 \beta_{12} - 80373 \beta_{11} + 80373 \beta_{10} + 3483834 \beta_{9} + 3483834 \beta_{8} - 13191643 \beta_{7} + \cdots + 5582015 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( - 12704042 \beta_{15} - 16333579 \beta_{13} + 80518170 \beta_{10} + 20422200 \beta_{9} + 16195387 \beta_{7} - 22134178 \beta_{5} + 7004246 \beta_{4} + 1711978 \beta_{3} + \cdots + 3764173504 \) |
\(\nu^{9}\) | \(=\) | \( ( 172651005 \beta_{15} - 172651005 \beta_{14} + 52776727 \beta_{13} - 52776727 \beta_{12} + 61321509 \beta_{11} + 61321509 \beta_{10} + 567056502 \beta_{9} - 567056502 \beta_{8} + \cdots + 2274028613 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( 6902289991 \beta_{14} + 12082786564 \beta_{12} + 49433985051 \beta_{11} - 12879763521 \beta_{8} - 11882130856 \beta_{6} + 5669850665 \beta_{5} + \cdots + 2231044630090 \beta_1 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 111743533385 \beta_{15} - 111743533385 \beta_{14} + 14565283037 \beta_{13} + 14565283037 \beta_{12} + 68277298119 \beta_{11} - 68277298119 \beta_{10} + \cdots - 2315920372097 ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( 930489180369 \beta_{15} + 2139104560648 \beta_{13} - 7592699813517 \beta_{10} - 2032338076482 \beta_{9} - 2069856552361 \beta_{7} + \cdots - 332000524438189 \) |
\(\nu^{13}\) | \(=\) | \( ( - 17880852446077 \beta_{15} + 17880852446077 \beta_{14} + 9101406164066 \beta_{13} - 9101406164066 \beta_{12} - 15330070159428 \beta_{11} + \cdots - 506243338698578 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( - 497993622759701 \beta_{14} + \cdots - 19\!\cdots\!10 \beta_1 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( 11\!\cdots\!03 \beta_{15} + \cdots + 40\!\cdots\!09 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(449\) | \(577\) | \(1093\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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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223.1 |
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0 | − | 3.00000i | 0 | −10.6345 | 0 | −18.0158 | − | 4.29303i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.2 | 0 | − | 3.00000i | 0 | −10.6345 | 0 | 18.0158 | − | 4.29303i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.3 | 0 | − | 3.00000i | 0 | −10.4276 | 0 | −9.40604 | + | 15.9539i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.4 | 0 | − | 3.00000i | 0 | −10.4276 | 0 | 9.40604 | + | 15.9539i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.5 | 0 | − | 3.00000i | 0 | 4.35176 | 0 | −7.09314 | − | 17.1081i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.6 | 0 | − | 3.00000i | 0 | 4.35176 | 0 | 7.09314 | − | 17.1081i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.7 | 0 | − | 3.00000i | 0 | 16.7103 | 0 | −16.4816 | + | 8.44725i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.8 | 0 | − | 3.00000i | 0 | 16.7103 | 0 | 16.4816 | + | 8.44725i | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.9 | 0 | 3.00000i | 0 | −10.6345 | 0 | −18.0158 | + | 4.29303i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.10 | 0 | 3.00000i | 0 | −10.6345 | 0 | 18.0158 | + | 4.29303i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.11 | 0 | 3.00000i | 0 | −10.4276 | 0 | −9.40604 | − | 15.9539i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.12 | 0 | 3.00000i | 0 | −10.4276 | 0 | 9.40604 | − | 15.9539i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.13 | 0 | 3.00000i | 0 | 4.35176 | 0 | −7.09314 | + | 17.1081i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.14 | 0 | 3.00000i | 0 | 4.35176 | 0 | 7.09314 | + | 17.1081i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.15 | 0 | 3.00000i | 0 | 16.7103 | 0 | −16.4816 | − | 8.44725i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
223.16 | 0 | 3.00000i | 0 | 16.7103 | 0 | 16.4816 | − | 8.44725i | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
56.e | even | 2 | 1 | inner |
56.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1344.4.p.b | yes | 16 |
4.b | odd | 2 | 1 | inner | 1344.4.p.b | yes | 16 |
7.b | odd | 2 | 1 | 1344.4.p.a | ✓ | 16 | |
8.b | even | 2 | 1 | 1344.4.p.a | ✓ | 16 | |
8.d | odd | 2 | 1 | 1344.4.p.a | ✓ | 16 | |
28.d | even | 2 | 1 | 1344.4.p.a | ✓ | 16 | |
56.e | even | 2 | 1 | inner | 1344.4.p.b | yes | 16 |
56.h | odd | 2 | 1 | inner | 1344.4.p.b | yes | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1344.4.p.a | ✓ | 16 | 7.b | odd | 2 | 1 | |
1344.4.p.a | ✓ | 16 | 8.b | even | 2 | 1 | |
1344.4.p.a | ✓ | 16 | 8.d | odd | 2 | 1 | |
1344.4.p.a | ✓ | 16 | 28.d | even | 2 | 1 | |
1344.4.p.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
1344.4.p.b | yes | 16 | 4.b | odd | 2 | 1 | inner |
1344.4.p.b | yes | 16 | 56.e | even | 2 | 1 | inner |
1344.4.p.b | yes | 16 | 56.h | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 260T_{5}^{2} - 804T_{5} + 8064 \)
acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{2} + 9)^{8} \)
$5$
\( (T^{4} - 260 T^{2} - 804 T + 8064)^{4} \)
$7$
\( T^{16} - 196 T^{14} + \cdots + 19\!\cdots\!01 \)
$11$
\( (T^{8} - 4620 T^{6} + \cdots + 10053320832)^{2} \)
$13$
\( (T^{4} - 14 T^{3} - 4604 T^{2} + \cdots + 624672)^{4} \)
$17$
\( (T^{8} + 28080 T^{6} + \cdots + 645802409513088)^{2} \)
$19$
\( (T^{8} + 9728 T^{6} + \cdots + 1880891559936)^{2} \)
$23$
\( (T^{8} + 18576 T^{6} + \cdots + 4988879424)^{2} \)
$29$
\( (T^{8} + 123804 T^{6} + \cdots + 18\!\cdots\!28)^{2} \)
$31$
\( (T^{8} - 142620 T^{6} + \cdots + 63\!\cdots\!92)^{2} \)
$37$
\( (T^{8} + 208512 T^{6} + \cdots + 51\!\cdots\!48)^{2} \)
$41$
\( (T^{8} + 427104 T^{6} + \cdots + 10\!\cdots\!48)^{2} \)
$43$
\( (T^{8} - 168372 T^{6} + \cdots + 16\!\cdots\!92)^{2} \)
$47$
\( (T^{8} - 634464 T^{6} + \cdots + 46\!\cdots\!72)^{2} \)
$53$
\( (T^{8} + 793308 T^{6} + \cdots + 93\!\cdots\!88)^{2} \)
$59$
\( (T^{8} + 683056 T^{6} + \cdots + 28\!\cdots\!76)^{2} \)
$61$
\( (T^{4} - 46 T^{3} - 394604 T^{2} + \cdots - 1820784672)^{4} \)
$67$
\( (T^{8} - 1819044 T^{6} + \cdots + 42\!\cdots\!52)^{2} \)
$71$
\( (T^{8} + 1963872 T^{6} + \cdots + 19\!\cdots\!76)^{2} \)
$73$
\( (T^{8} + 1728096 T^{6} + \cdots + 77\!\cdots\!88)^{2} \)
$79$
\( (T^{8} + 1665844 T^{6} + \cdots + 43\!\cdots\!84)^{2} \)
$83$
\( (T^{8} + 1273408 T^{6} + \cdots + 41\!\cdots\!64)^{2} \)
$89$
\( (T^{8} + 3975840 T^{6} + \cdots + 29\!\cdots\!12)^{2} \)
$97$
\( (T^{8} + 4368480 T^{6} + \cdots + 55\!\cdots\!28)^{2} \)
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