Newspace parameters
| Level: | \( N \) | \(=\) | \( 755 = 5 \cdot 151 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 755.h (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.02870535261\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{16} - 5 x^{15} + 14 x^{14} - 22 x^{13} + 54 x^{12} - 181 x^{11} + 697 x^{10} - 1743 x^{9} + 3507 x^{8} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{16} - 5 x^{15} + 14 x^{14} - 22 x^{13} + 54 x^{12} - 181 x^{11} + 697 x^{10} - 1743 x^{9} + 3507 x^{8} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( - 14\!\cdots\!05 \nu^{15} + \cdots - 43\!\cdots\!92 ) / 12\!\cdots\!56 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 13\!\cdots\!16 \nu^{15} + \cdots - 18\!\cdots\!60 ) / 60\!\cdots\!78 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 27\!\cdots\!37 \nu^{15} + \cdots - 10\!\cdots\!76 ) / 12\!\cdots\!56 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 53\!\cdots\!33 \nu^{15} + \cdots - 61\!\cdots\!88 ) / 12\!\cdots\!56 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 89\!\cdots\!52 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 12\!\cdots\!56 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 20\!\cdots\!11 \nu^{15} + \cdots + 66\!\cdots\!88 ) / 24\!\cdots\!12 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 23\!\cdots\!65 \nu^{15} + \cdots + 23\!\cdots\!16 ) / 24\!\cdots\!12 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 14\!\cdots\!35 \nu^{15} + \cdots + 11\!\cdots\!64 ) / 12\!\cdots\!56 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 20\!\cdots\!49 \nu^{15} + \cdots + 39\!\cdots\!60 ) / 12\!\cdots\!56 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 45\!\cdots\!27 \nu^{15} + \cdots + 89\!\cdots\!20 ) / 24\!\cdots\!12 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 11\!\cdots\!10 \nu^{15} + \cdots + 98\!\cdots\!84 ) / 60\!\cdots\!78 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 23\!\cdots\!45 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 60\!\cdots\!78 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 96\!\cdots\!01 \nu^{15} + \cdots + 10\!\cdots\!36 ) / 24\!\cdots\!12 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 34\!\cdots\!05 \nu^{15} + \cdots - 37\!\cdots\!38 ) / 60\!\cdots\!78 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{15} + \beta_{14} - \beta_{10} + \beta_{8} + 3\beta_{4} + \beta_{3} + 1 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{9} + \beta_{7} - \beta_{6} - 5\beta_{2} \)
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| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{15} + \beta_{14} - \beta_{13} - 8 \beta_{11} - 7 \beta_{9} + 16 \beta_{8} + 3 \beta_{7} + \cdots - 8 \)
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| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{15} - 10 \beta_{13} - 20 \beta_{12} + 29 \beta_{11} + \beta_{10} - \beta_{9} + 21 \beta_{8} + \cdots + 38 \)
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| \(\nu^{6}\) | \(=\) |
\( 60 \beta_{15} - 13 \beta_{14} - 17 \beta_{13} - 28 \beta_{12} - 22 \beta_{11} + 60 \beta_{10} + \cdots - 80 \)
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| \(\nu^{7}\) | \(=\) |
\( 15 \beta_{15} - 15 \beta_{14} + 57 \beta_{10} + 72 \beta_{8} - 77 \beta_{7} + 26 \beta_{6} - 77 \beta_{5} + \cdots + 72 \)
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| \(\nu^{8}\) | \(=\) |
\( - 317 \beta_{14} + 119 \beta_{13} + 123 \beta_{12} + 187 \beta_{11} + 317 \beta_{10} + 121 \beta_{9} + \cdots - 123 \)
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| \(\nu^{9}\) | \(=\) |
\( - 149 \beta_{15} - 249 \beta_{14} + 554 \beta_{13} + 667 \beta_{12} - 1208 \beta_{11} + 149 \beta_{9} + \cdots - 1208 \)
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| \(\nu^{10}\) | \(=\) |
\( - 3149 \beta_{15} + 942 \beta_{13} + 73 \beta_{12} + 2825 \beta_{11} - 2160 \beta_{10} + 2160 \beta_{9} + \cdots + 4855 \)
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| \(\nu^{11}\) | \(=\) |
\( - 2782 \beta_{15} + 1516 \beta_{14} + 1931 \beta_{13} + 1499 \beta_{12} - 4392 \beta_{11} - 2782 \beta_{10} + \cdots - 8319 \)
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| \(\nu^{12}\) | \(=\) |
\( - 14855 \beta_{15} + 14855 \beta_{14} - 22444 \beta_{10} + 20399 \beta_{8} + 7102 \beta_{7} + \cdots + 20399 \)
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| \(\nu^{13}\) | \(=\) |
\( 9988 \beta_{14} - 14691 \beta_{13} - 12133 \beta_{12} + 31594 \beta_{11} - 9988 \beta_{10} + \cdots + 12133 \)
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| \(\nu^{14}\) | \(=\) |
\( 102891 \beta_{15} + 56273 \beta_{14} - 52057 \beta_{13} - 7941 \beta_{12} - 135463 \beta_{11} + \cdots - 135463 \)
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| \(\nu^{15}\) | \(=\) |
\( 138227 \beta_{15} - 191894 \beta_{13} - 189202 \beta_{12} + 386182 \beta_{11} + 75758 \beta_{10} + \cdots + 518011 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/755\mathbb{Z}\right)^\times\).
| \(n\) | \(6\) | \(152\) |
| \(\chi(n)\) | \(-1 - \beta_{4} - \beta_{8} + \beta_{12}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 321.1 |
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−1.95522 | −0.0899620 | + | 0.276875i | 1.82287 | 0.809017 | + | 0.587785i | 0.175895 | − | 0.541349i | 0.856698 | + | 0.622428i | 0.346329 | 2.35848 | + | 1.71354i | −1.58180 | − | 1.14925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 321.2 | −1.01579 | 0.962511 | − | 2.96230i | −0.968171 | 0.809017 | + | 0.587785i | −0.977709 | + | 3.00908i | −1.40130 | − | 1.01810i | 3.01504 | −5.42177 | − | 3.93914i | −0.821791 | − | 0.597066i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 321.3 | 0.488177 | −0.778897 | + | 2.39720i | −1.76168 | 0.809017 | + | 0.587785i | −0.380239 | + | 1.17026i | −2.04327 | − | 1.48452i | −1.83637 | −2.71283 | − | 1.97099i | 0.394943 | + | 0.286943i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 321.4 | 1.48283 | 0.0243824 | − | 0.0750413i | 0.198782 | 0.809017 | + | 0.587785i | 0.0361549 | − | 0.111273i | −0.457216 | − | 0.332187i | −2.67090 | 2.42201 | + | 1.75970i | 1.19963 | + | 0.871585i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.1 | −2.69752 | −1.70549 | − | 1.23911i | 5.27660 | −0.309017 | − | 0.951057i | 4.60059 | + | 3.34252i | −0.0125248 | − | 0.0385473i | −8.83868 | 0.446250 | + | 1.37342i | 0.833579 | + | 2.56549i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.2 | −2.24302 | 2.63865 | + | 1.91709i | 3.03113 | −0.309017 | − | 0.951057i | −5.91855 | − | 4.30007i | 0.681363 | + | 2.09702i | −2.31285 | 2.36019 | + | 7.26391i | 0.693131 | + | 2.13324i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.3 | 1.40610 | −1.06077 | − | 0.770696i | −0.0228706 | −0.309017 | − | 0.951057i | −1.49156 | − | 1.08368i | 1.62510 | + | 5.00155i | −2.84437 | −0.395787 | − | 1.21811i | −0.434510 | − | 1.33728i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.4 | 2.53443 | −1.99042 | − | 1.44613i | 4.42334 | −0.309017 | − | 0.951057i | −5.04459 | − | 3.66511i | 0.251146 | + | 0.772947i | 6.14180 | 0.943451 | + | 2.90364i | −0.783182 | − | 2.41039i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 366.1 | −2.69752 | −1.70549 | + | 1.23911i | 5.27660 | −0.309017 | + | 0.951057i | 4.60059 | − | 3.34252i | −0.0125248 | + | 0.0385473i | −8.83868 | 0.446250 | − | 1.37342i | 0.833579 | − | 2.56549i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 366.2 | −2.24302 | 2.63865 | − | 1.91709i | 3.03113 | −0.309017 | + | 0.951057i | −5.91855 | + | 4.30007i | 0.681363 | − | 2.09702i | −2.31285 | 2.36019 | − | 7.26391i | 0.693131 | − | 2.13324i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 366.3 | 1.40610 | −1.06077 | + | 0.770696i | −0.0228706 | −0.309017 | + | 0.951057i | −1.49156 | + | 1.08368i | 1.62510 | − | 5.00155i | −2.84437 | −0.395787 | + | 1.21811i | −0.434510 | + | 1.33728i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 366.4 | 2.53443 | −1.99042 | + | 1.44613i | 4.42334 | −0.309017 | + | 0.951057i | −5.04459 | + | 3.66511i | 0.251146 | − | 0.772947i | 6.14180 | 0.943451 | − | 2.90364i | −0.783182 | + | 2.41039i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 461.1 | −1.95522 | −0.0899620 | − | 0.276875i | 1.82287 | 0.809017 | − | 0.587785i | 0.175895 | + | 0.541349i | 0.856698 | − | 0.622428i | 0.346329 | 2.35848 | − | 1.71354i | −1.58180 | + | 1.14925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 461.2 | −1.01579 | 0.962511 | + | 2.96230i | −0.968171 | 0.809017 | − | 0.587785i | −0.977709 | − | 3.00908i | −1.40130 | + | 1.01810i | 3.01504 | −5.42177 | + | 3.93914i | −0.821791 | + | 0.597066i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 461.3 | 0.488177 | −0.778897 | − | 2.39720i | −1.76168 | 0.809017 | − | 0.587785i | −0.380239 | − | 1.17026i | −2.04327 | + | 1.48452i | −1.83637 | −2.71283 | + | 1.97099i | 0.394943 | − | 0.286943i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 461.4 | 1.48283 | 0.0243824 | + | 0.0750413i | 0.198782 | 0.809017 | − | 0.587785i | 0.0361549 | + | 0.111273i | −0.457216 | + | 0.332187i | −2.67090 | 2.42201 | − | 1.75970i | 1.19963 | − | 0.871585i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 151.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 755.2.h.d | ✓ | 16 |
| 151.d | even | 5 | 1 | inner | 755.2.h.d | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 755.2.h.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 755.2.h.d | ✓ | 16 | 151.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\):
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\( T_{2}^{8} + 2T_{2}^{7} - 12T_{2}^{6} - 21T_{2}^{5} + 46T_{2}^{4} + 60T_{2}^{3} - 71T_{2}^{2} - 47T_{2} + 31 \)
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\( T_{3}^{16} + 4 T_{3}^{15} + 14 T_{3}^{14} + 44 T_{3}^{13} + 163 T_{3}^{12} + 488 T_{3}^{11} + 1659 T_{3}^{10} + \cdots + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 2 T^{7} - 12 T^{6} + \cdots + 31)^{2} \)
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| $3$ |
\( T^{16} + 4 T^{15} + \cdots + 16 \)
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| $5$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
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| $7$ |
\( T^{16} + T^{15} + \cdots + 1 \)
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| $11$ |
\( T^{16} - 13 T^{15} + \cdots + 17547721 \)
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| $13$ |
\( T^{16} - 7 T^{15} + \cdots + 1 \)
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| $17$ |
\( T^{16} - 9 T^{15} + \cdots + 4096 \)
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| $19$ |
\( (T^{8} + 17 T^{7} + \cdots + 8629)^{2} \)
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| $23$ |
\( (T^{8} - 29 T^{7} + \cdots + 9076)^{2} \)
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| $29$ |
\( T^{16} + 16 T^{15} + \cdots + 81072016 \)
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| $31$ |
\( T^{16} + 3 T^{15} + \cdots + 95961616 \)
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| $37$ |
\( T^{16} + \cdots + 4701707761 \)
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| $41$ |
\( T^{16} + \cdots + 1968785641 \)
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| $43$ |
\( T^{16} + \cdots + 114698046241 \)
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| $47$ |
\( T^{16} + \cdots + 14011220161 \)
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| $53$ |
\( T^{16} + \cdots + 813479528761 \)
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| $59$ |
\( (T^{8} + 22 T^{7} + \cdots + 842029)^{2} \)
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| $61$ |
\( T^{16} + \cdots + 846453360841 \)
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| $67$ |
\( T^{16} + \cdots + 36893189776 \)
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| $71$ |
\( T^{16} + \cdots + 12076187956561 \)
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| $73$ |
\( T^{16} + \cdots + 164575451041 \)
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| $79$ |
\( T^{16} + \cdots + 430467843702961 \)
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| $83$ |
\( T^{16} + \cdots + 11488242503761 \)
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| $89$ |
\( T^{16} + \cdots + 5951968201 \)
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| $97$ |
\( T^{16} + \cdots + 12\!\cdots\!61 \)
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