Newspace parameters
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(32.1692786856\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} + 366974x^{4} + 33667479169x^{2} + 601699608148356 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 366974x^{4} + 33667479169x^{2} + 601699608148356 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 183487\nu ) / 24529566 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 377\nu^{4} - 305807\nu^{3} - 93704165\nu^{2} - 13196483458\nu - 3000456513120 ) / 343413924 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 377\nu^{4} - 305807\nu^{3} + 93704165\nu^{2} - 13196483458\nu + 3000456513120 ) / 343413924 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 625\nu^{4} - 305821\nu^{3} - 237327205\nu^{2} - 23501469996\nu - 15002625979524 ) / 686827848 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} + 43\nu^{4} + 1529049\nu^{3} - 16639625\nu^{2} + 96892239268\nu - 3000799927044 ) / 686827848 \)
|
| \(\nu\) | \(=\) |
\( ( 6\beta_{5} - 2\beta_{4} + 7\beta_{3} + 9\beta_{2} - 4\beta _1 + 2 ) / 300 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -754\beta_{5} - 2262\beta_{4} - 1293\beta_{3} + 539\beta_{2} - 754\beta _1 - 36697508 ) / 300 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1100922\beta_{5} + 366974\beta_{4} - 1284409\beta_{3} - 1651383\beta_{2} + 7359603748\beta _1 - 366974 ) / 300 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 37481666 \beta_{5} + 112444998 \beta_{4} + 91602957 \beta_{3} - 54121291 \beta_{2} + \cdots + 1346722753732 ) / 60 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 257490753306 \beta_{5} - 85830251102 \beta_{4} + 248893790257 \beta_{3} + 334724041359 \beta_{2} + \cdots + 85830251102 ) / 300 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 64.0000i | − | 729.000i | −4096.00 | −15564.3 | + | 31280.3i | −46656.0 | 297517.i | 262144.i | −531441. | 2.00194e6 | + | 996118.i | ||||||||||||||||||||||||||||||
| 19.2 | − | 64.0000i | − | 729.000i | −4096.00 | 14344.4 | − | 31858.1i | −46656.0 | 141453.i | 262144.i | −531441. | −2.03892e6 | − | 918042.i | |||||||||||||||||||||||||||||||
| 19.3 | − | 64.0000i | − | 729.000i | −4096.00 | 29184.9 | + | 19207.9i | −46656.0 | − | 160098.i | 262144.i | −531441. | 1.22930e6 | − | 1.86784e6i | ||||||||||||||||||||||||||||||
| 19.4 | 64.0000i | 729.000i | −4096.00 | −15564.3 | − | 31280.3i | −46656.0 | − | 297517.i | − | 262144.i | −531441. | 2.00194e6 | − | 996118.i | |||||||||||||||||||||||||||||||
| 19.5 | 64.0000i | 729.000i | −4096.00 | 14344.4 | + | 31858.1i | −46656.0 | − | 141453.i | − | 262144.i | −531441. | −2.03892e6 | + | 918042.i | |||||||||||||||||||||||||||||||
| 19.6 | 64.0000i | 729.000i | −4096.00 | 29184.9 | − | 19207.9i | −46656.0 | 160098.i | − | 262144.i | −531441. | 1.22930e6 | + | 1.86784e6i | ||||||||||||||||||||||||||||||||
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 | 30.14.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.14.c.a | 6 | ||
| 5.b | even | 2 | 1 | inner | 30.14.c.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 150.14.a.r | 3 | ||
| 5.c | odd | 4 | 1 | 150.14.a.s | 3 | ||
| 15.d | odd | 2 | 1 | 90.14.c.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.14.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 30.14.c.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 90.14.c.a | 6 | 3.b | odd | 2 | 1 | ||
| 90.14.c.a | 6 | 15.d | odd | 2 | 1 | ||
| 150.14.a.r | 3 | 5.c | odd | 4 | 1 | ||
| 150.14.a.s | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 134156846328T_{7}^{4} + 4552793367422436250128T_{7}^{2} + 45396625856112668562548436818176 \)
acting on \(S_{14}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4096)^{3} \)
|
| $3$ |
\( (T^{2} + 531441)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 18\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 45\!\cdots\!76 \)
|
| $11$ |
\( (T^{3} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 13\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots + 31\!\cdots\!56 \)
|
| $19$ |
\( (T^{3} + \cdots + 61\!\cdots\!76)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots + 19\!\cdots\!84)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 44\!\cdots\!68)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 26\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots + 34\!\cdots\!96)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 16\!\cdots\!16 \)
|
| $47$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 37\!\cdots\!96 \)
|
| $59$ |
\( (T^{3} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 22\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 78\!\cdots\!16 \)
|
| $71$ |
\( (T^{3} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 26\!\cdots\!04 \)
|
| $79$ |
\( (T^{3} + \cdots - 22\!\cdots\!08)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 97\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + \cdots + 63\!\cdots\!52)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 34\!\cdots\!44 \)
|
| show more | |
| show less |