Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{12} + 1179680 x^{10} + 508533387652 x^{8} + \cdots + 19\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{10}\cdot 5^{14} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 1179680 x^{10} + 508533387652 x^{8} + \cdots + 19\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 672015275 \nu^{10} + \cdots + 29\!\cdots\!88 ) / 87\!\cdots\!84 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 672015275 \nu^{10} + \cdots + 57\!\cdots\!80 ) / 29\!\cdots\!28 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 21\!\cdots\!41 \nu^{10} + \cdots - 53\!\cdots\!24 ) / 50\!\cdots\!04 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 829691052040949 \nu^{10} + \cdots - 13\!\cdots\!20 ) / 43\!\cdots\!92 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 31\!\cdots\!19 \nu^{10} + \cdots - 18\!\cdots\!92 ) / 13\!\cdots\!08 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 46\!\cdots\!27 \nu^{11} + \cdots - 11\!\cdots\!00 \nu ) / 40\!\cdots\!92 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 32\!\cdots\!89 \nu^{11} + \cdots - 29\!\cdots\!76 \nu ) / 50\!\cdots\!24 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 47\!\cdots\!57 \nu^{11} + \cdots - 11\!\cdots\!08 \nu ) / 20\!\cdots\!96 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 57\!\cdots\!11 \nu^{11} + \cdots + 25\!\cdots\!36 \nu ) / 12\!\cdots\!56 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 58\!\cdots\!67 \nu^{11} + \cdots - 55\!\cdots\!64 \nu ) / 19\!\cdots\!72 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 53\!\cdots\!81 \nu^{11} + \cdots - 53\!\cdots\!08 \nu ) / 15\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_{7} - 56\beta_{6} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 - 196614 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 17\beta_{8} - 319665\beta_{7} + 10453770\beta_{6} \)
|
| \(\nu^{4}\) | \(=\) |
\( -44\beta_{5} - 222\beta_{4} - 748\beta_{3} - 393076\beta_{2} + 10561012\beta _1 + 62433076912 \)
|
| \(\nu^{5}\) | \(=\) |
\( 23848 \beta_{11} - 144408 \beta_{10} - 500978 \beta_{9} - 29860806 \beta_{8} + \cdots - 1462296807788 \beta_{6} \)
|
| \(\nu^{6}\) | \(=\) |
\( 15378968 \beta_{5} + 120742368 \beta_{4} + 447282968 \beta_{3} + 147925633356 \beta_{2} + \cdots - 21\!\cdots\!96 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 15065787264 \beta_{11} + 99596851200 \beta_{10} + 209909957916 \beta_{9} + \cdots - 49\!\cdots\!44 \beta_{6} \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4808493970896 \beta_{5} - 53456290429320 \beta_{4} - 206511726379728 \beta_{3} + \cdots + 77\!\cdots\!48 \)
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| \(\nu^{9}\) | \(=\) |
\( 71\!\cdots\!96 \beta_{11} + \cdots + 56\!\cdots\!08 \beta_{6} \)
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| \(\nu^{10}\) | \(=\) |
\( 15\!\cdots\!52 \beta_{5} + \cdots - 28\!\cdots\!00 \)
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| \(\nu^{11}\) | \(=\) |
\( - 30\!\cdots\!12 \beta_{11} + \cdots - 34\!\cdots\!92 \beta_{6} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 630.218i | 6561.00i | −266103. | 0 | 4.13486e6 | − | 2.83896e6i | 8.50989e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 597.028i | − | 6561.00i | −225370. | 0 | −3.91710e6 | − | 1.98246e7i | 5.62985e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 520.046i | − | 6561.00i | −139375. | 0 | −3.41202e6 | 2.58253e7i | 4.31818e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 354.696i | 6561.00i | 5262.54 | 0 | 2.32716e6 | − | 1.41469e7i | − | 4.83574e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | − | 168.575i | − | 6561.00i | 102655. | 0 | −1.10602e6 | 2.46799e6i | − | 3.94004e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | − | 37.2667i | − | 6561.00i | 129683. | 0 | −244507. | − | 4.45476e6i | − | 9.71748e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 37.2667i | 6561.00i | 129683. | 0 | −244507. | 4.45476e6i | 9.71748e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 168.575i | 6561.00i | 102655. | 0 | −1.10602e6 | − | 2.46799e6i | 3.94004e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 354.696i | − | 6561.00i | 5262.54 | 0 | 2.32716e6 | 1.41469e7i | 4.83574e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 520.046i | 6561.00i | −139375. | 0 | −3.41202e6 | − | 2.58253e7i | − | 4.31818e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.11 | 597.028i | 6561.00i | −225370. | 0 | −3.91710e6 | 1.98246e7i | − | 5.62985e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.12 | 630.218i | − | 6561.00i | −266103. | 0 | 4.13486e6 | 2.83896e6i | − | 8.50989e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.18.b.h | 12 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.h | 12 | |
| 5.c | odd | 4 | 1 | 75.18.a.i | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 75.18.a.j | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.18.a.i | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 75.18.a.j | yes | 6 | 5.c | odd | 4 | 1 | |
| 75.18.b.h | 12 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.h | 12 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 1179680 T_{2}^{10} + 508533387652 T_{2}^{8} + \cdots + 19\!\cdots\!04 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 19\!\cdots\!04 \)
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| $3$ |
\( (T^{2} + 43046721)^{6} \)
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| $5$ |
\( T^{12} \)
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| $7$ |
\( T^{12} + \cdots + 51\!\cdots\!25 \)
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| $11$ |
\( (T^{6} + \cdots - 24\!\cdots\!52)^{2} \)
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| $13$ |
\( T^{12} + \cdots + 22\!\cdots\!69 \)
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| $17$ |
\( T^{12} + \cdots + 23\!\cdots\!56 \)
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| $19$ |
\( (T^{6} + \cdots - 76\!\cdots\!75)^{2} \)
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| $23$ |
\( T^{12} + \cdots + 83\!\cdots\!00 \)
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| $29$ |
\( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{6} + \cdots + 46\!\cdots\!25)^{2} \)
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| $37$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
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| $41$ |
\( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \)
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| $43$ |
\( T^{12} + \cdots + 11\!\cdots\!81 \)
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| $47$ |
\( T^{12} + \cdots + 11\!\cdots\!36 \)
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| $53$ |
\( T^{12} + \cdots + 14\!\cdots\!00 \)
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| $59$ |
\( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{6} + \cdots + 29\!\cdots\!21)^{2} \)
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| $67$ |
\( T^{12} + \cdots + 84\!\cdots\!81 \)
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| $71$ |
\( (T^{6} + \cdots + 92\!\cdots\!84)^{2} \)
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| $73$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
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| $79$ |
\( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{12} + \cdots + 11\!\cdots\!24 \)
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| $89$ |
\( (T^{6} + \cdots + 66\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{12} + \cdots + 18\!\cdots\!61 \)
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