Newspace parameters
| Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 414.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(129.327400550\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
| Defining polynomial: |
\( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 138) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu^{2} - 5551\nu - 110450 ) / 684 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 68\nu^{2} + 4411\nu - 207382 ) / 76 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + 36\beta_{2} + 15\beta _1 + 16758 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -32\beta_{3} + 2448\beta_{2} + 5431\beta _1 + 318838 ) / 4 \)
|
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 |
|
−8.00000 | 0 | 64.0000 | −366.876 | 0 | −561.721 | −512.000 | 0 | 2935.00 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 0 | 64.0000 | −340.987 | 0 | 1267.12 | −512.000 | 0 | 2727.89 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 0 | 64.0000 | 10.0669 | 0 | 971.172 | −512.000 | 0 | −80.5353 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 0 | 64.0000 | 427.795 | 0 | 345.429 | −512.000 | 0 | −3422.36 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 414.8.a.i | 4 | |
| 3.b | odd | 2 | 1 | 138.8.a.h | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.8.a.h | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 414.8.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 270T_{5}^{3} - 180540T_{5}^{2} - 51728000T_{5} + 538752000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(414))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 8)^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 270 T^{3} + \cdots + 538752000 \)
$7$
\( T^{4} - 2022 T^{3} + \cdots - 238777204176 \)
$11$
\( T^{4} + 4120 T^{3} + \cdots + 25210679155200 \)
$13$
\( T^{4} - 8036 T^{3} + \cdots + 10\!\cdots\!84 \)
$17$
\( T^{4} + 37182 T^{3} + \cdots - 69\!\cdots\!08 \)
$19$
\( T^{4} - 5702 T^{3} + \cdots + 12\!\cdots\!64 \)
$23$
\( (T - 12167)^{4} \)
$29$
\( T^{4} + 217716 T^{3} + \cdots - 19\!\cdots\!68 \)
$31$
\( T^{4} - 222852 T^{3} + \cdots - 32\!\cdots\!00 \)
$37$
\( T^{4} - 486428 T^{3} + \cdots - 26\!\cdots\!28 \)
$41$
\( T^{4} + 338336 T^{3} + \cdots + 11\!\cdots\!52 \)
$43$
\( T^{4} - 730974 T^{3} + \cdots - 11\!\cdots\!00 \)
$47$
\( T^{4} + 338248 T^{3} + \cdots + 25\!\cdots\!76 \)
$53$
\( T^{4} - 375502 T^{3} + \cdots + 12\!\cdots\!20 \)
$59$
\( T^{4} + 71392 T^{3} + \cdots + 23\!\cdots\!28 \)
$61$
\( T^{4} - 2101164 T^{3} + \cdots + 18\!\cdots\!72 \)
$67$
\( T^{4} - 4337162 T^{3} + \cdots + 44\!\cdots\!92 \)
$71$
\( T^{4} + 2288016 T^{3} + \cdots + 92\!\cdots\!92 \)
$73$
\( T^{4} + 1107328 T^{3} + \cdots - 25\!\cdots\!88 \)
$79$
\( T^{4} - 60610 T^{3} + \cdots + 83\!\cdots\!48 \)
$83$
\( T^{4} + 1485464 T^{3} + \cdots + 11\!\cdots\!08 \)
$89$
\( T^{4} + 1485090 T^{3} + \cdots - 10\!\cdots\!56 \)
$97$
\( T^{4} - 1935444 T^{3} + \cdots + 94\!\cdots\!60 \)
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