Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 580318x^{4} + 45393344x^{3} + 72695152416x^{2} - 6623241804288x - 149217035286528 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{5}\cdot 5^{7} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 580318x^{4} + 45393344x^{3} + 72695152416x^{2} - 6623241804288x - 149217035286528 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 115\nu - 193478 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 822\nu^{4} + 227886\nu^{3} + 241652944\nu^{2} + 8908591328\nu - 5311831526144 ) / 549824 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} + 185\nu^{2} - 308302\nu - 13567280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{5} - 1478\nu^{4} + 9447278\nu^{3} - 55442192\nu^{2} - 1127935764192\nu + 42361145436544 ) / 549824 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 115\beta _1 + 193478 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 185\beta_{2} + 329577\beta _1 - 22226150 \)
|
| \(\nu^{4}\) | \(=\) |
\( 44\beta_{5} - 446\beta_{4} - 748\beta_{3} + 415700\beta_{2} - 82924884\beta _1 + 63761417968 \)
|
| \(\nu^{5}\) | \(=\) |
\( -36168\beta_{5} + 594498\beta_{4} + 65032\beta_{3} - 142211366\beta_{2} + 124388741638\beta _1 - 16034217215508 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−597.028 | 6561.00 | 225370. | 0 | −3.91710e6 | −1.98246e7 | −5.62985e7 | 4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −520.046 | 6561.00 | 139375. | 0 | −3.41202e6 | 2.58253e7 | −4.31818e6 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −168.575 | 6561.00 | −102655. | 0 | −1.10602e6 | 2.46799e6 | 3.94004e7 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −37.2667 | 6561.00 | −129683. | 0 | −244507. | −4.45476e6 | 9.71748e6 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 354.696 | 6561.00 | −5262.54 | 0 | 2.32716e6 | 1.41469e7 | −4.83574e7 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 630.218 | 6561.00 | 266103. | 0 | 4.13486e6 | 2.83896e6 | 8.50989e7 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.18.a.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 75.18.a.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 75.18.b.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.18.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 75.18.a.j | yes | 6 | 5.b | even | 2 | 1 | |
| 75.18.b.h | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 338 T_{2}^{5} - 532718 T_{2}^{4} - 171809536 T_{2}^{3} + 54300836896 T_{2}^{2} + \cdots + 436009513623552 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 436009513623552 \)
|
| $3$ |
\( (T - 6561)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 22\!\cdots\!25 \)
|
| $11$ |
\( T^{6} + \cdots - 24\!\cdots\!52 \)
|
| $13$ |
\( T^{6} + \cdots - 46\!\cdots\!63 \)
|
| $17$ |
\( T^{6} + \cdots - 15\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots - 76\!\cdots\!75 \)
|
| $23$ |
\( T^{6} + \cdots + 91\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 46\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 33\!\cdots\!91 \)
|
| $47$ |
\( T^{6} + \cdots + 33\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 29\!\cdots\!21 \)
|
| $67$ |
\( T^{6} + \cdots - 29\!\cdots\!91 \)
|
| $71$ |
\( T^{6} + \cdots + 92\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots - 33\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 10\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 66\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!69 \)
|
| show more | |
| show less |