Newspace parameters
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.36154644760\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} + 9276x^{4} + 17959899x^{2} + 616730624 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4}\cdot 5^{5} \) |
| 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
\( x^{6} + 9276x^{4} + 17959899x^{2} + 616730624 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 34300\nu^{3} + 148003099\nu ) / 21420000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 34300\nu^{3} + 469303099\nu ) / 10710000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} - 32200\nu^{2} - 32729896 ) / 2625 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -189\nu^{5} + 1088\nu^{4} - 1722700\nu^{3} + 3998400\nu^{2} - 3187265711\nu - 5815388288 ) / 476000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 42529 \nu^{5} - 277440 \nu^{4} - 387744700 \nu^{3} - 1162392000 \nu^{2} - 719011997371 \nu + 1041386413440 ) / 21420000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 30 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 5\beta_{4} - 16\beta_{3} + 2\beta_{2} - 309200 ) / 100 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} + 15\beta_{4} - 3\beta_{3} - 52064\beta_{2} + 359290\beta_1 ) / 300 \)
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| \(\nu^{4}\) | \(=\) |
\( ( -322\beta_{5} + 1610\beta_{4} + 2527\beta_{3} - 644\beta_{2} + 66832504 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -10290\beta_{5} - 51450\beta_{4} + 10290\beta_{3} + 30576421\beta_{2} - 293758502\beta_1 ) / 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
− | 152.321i | 2014.00i | −15009.6 | −14163.9 | + | 31938.8i | 306775. | 69973.1i | 1.03846e6i | −2.46189e6 | 4.86494e6 | + | 2.15745e6i | |||||||||||||||||||||||||||||||
| 4.2 | − | 127.310i | − | 2045.53i | −8015.83 | 34862.0 | + | 2311.33i | −260416. | 258383.i | − | 22427.7i | −2.58986e6 | 294255. | − | 4.43828e6i | ||||||||||||||||||||||||||||||
| 4.3 | − | 40.5284i | 298.988i | 6549.45 | −8413.14 | − | 33910.5i | 12117.5 | − | 377278.i | − | 597447.i | 1.50493e6 | −1.37434e6 | + | 340971.i | ||||||||||||||||||||||||||||||
| 4.4 | 40.5284i | − | 298.988i | 6549.45 | −8413.14 | + | 33910.5i | 12117.5 | 377278.i | 597447.i | 1.50493e6 | −1.37434e6 | − | 340971.i | ||||||||||||||||||||||||||||||||
| 4.5 | 127.310i | 2045.53i | −8015.83 | 34862.0 | − | 2311.33i | −260416. | − | 258383.i | 22427.7i | −2.58986e6 | 294255. | + | 4.43828e6i | ||||||||||||||||||||||||||||||||
| 4.6 | 152.321i | − | 2014.00i | −15009.6 | −14163.9 | − | 31938.8i | 306775. | − | 69973.1i | − | 1.03846e6i | −2.46189e6 | 4.86494e6 | − | 2.15745e6i | ||||||||||||||||||||||||||||||
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 | 5.14.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 45.14.b.b | 6 | ||
| 4.b | odd | 2 | 1 | 80.14.c.c | 6 | ||
| 5.b | even | 2 | 1 | inner | 5.14.b.a | ✓ | 6 |
| 5.c | odd | 4 | 2 | 25.14.a.e | 6 | ||
| 15.d | odd | 2 | 1 | 45.14.b.b | 6 | ||
| 20.d | odd | 2 | 1 | 80.14.c.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.14.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5.14.b.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 25.14.a.e | 6 | 5.c | odd | 4 | 2 | ||
| 45.14.b.b | 6 | 3.b | odd | 2 | 1 | ||
| 45.14.b.b | 6 | 15.d | odd | 2 | 1 | ||
| 80.14.c.c | 6 | 4.b | odd | 2 | 1 | ||
| 80.14.c.c | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(5, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 617678127104 \)
|
| $3$ |
\( T^{6} + \cdots + 15\!\cdots\!36 \)
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| $5$ |
\( T^{6} + \cdots + 18\!\cdots\!25 \)
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| $7$ |
\( T^{6} + \cdots + 46\!\cdots\!44 \)
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| $11$ |
\( (T^{3} + \cdots - 39\!\cdots\!68)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 52\!\cdots\!16 \)
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| $17$ |
\( T^{6} + \cdots + 19\!\cdots\!24 \)
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| $19$ |
\( (T^{3} + \cdots - 45\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 38\!\cdots\!96 \)
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| $29$ |
\( (T^{3} + \cdots - 48\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{3} + \cdots + 18\!\cdots\!12)^{2} \)
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| $37$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
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| $41$ |
\( (T^{3} + \cdots - 19\!\cdots\!48)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 85\!\cdots\!56 \)
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| $47$ |
\( T^{6} + \cdots + 55\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots + 41\!\cdots\!36 \)
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| $59$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots - 18\!\cdots\!68)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 19\!\cdots\!24 \)
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| $71$ |
\( (T^{3} + \cdots + 50\!\cdots\!72)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 80\!\cdots\!96 \)
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| $79$ |
\( (T^{3} + \cdots + 11\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 63\!\cdots\!76 \)
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| $89$ |
\( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 15\!\cdots\!64 \)
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