Newspace parameters
| Level: | \( N \) | \(=\) | \( 1445 = 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1445.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(85.2577599583\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 36x^{5} + 80x^{4} + 401x^{3} - 499x^{2} - 1182x + 150 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 36x^{5} + 80x^{4} + 401x^{3} - 499x^{2} - 1182x + 150 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 26\nu^{4} - 66\nu^{3} + 119\nu^{2} + 352\nu + 34 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 26\nu^{4} - 34\nu^{3} - 167\nu^{2} + 112\nu + 182 ) / 8 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{5} + 30\nu^{4} + 58\nu^{3} - 187\nu^{2} - 272\nu + 46 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} + 6\nu^{5} - 86\nu^{4} - 166\nu^{3} + 477\nu^{2} + 656\nu - 10 ) / 16 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 16\beta _1 + 8 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 4\beta_{5} + 2\beta_{3} + 19\beta_{2} + 29\beta _1 + 183 \)
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| \(\nu^{5}\) | \(=\) |
\( 25\beta_{6} + 25\beta_{5} + 2\beta_{4} - 21\beta_{3} + 37\beta_{2} + 296\beta _1 + 278 \)
|
| \(\nu^{6}\) | \(=\) |
\( 68\beta_{6} + 120\beta_{5} - 4\beta_{4} + 44\beta_{3} + 367\beta_{2} + 747\beta _1 + 3387 \)
|
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 |
|
−4.68722 | −5.08995 | 13.9700 | 5.00000 | 23.8577 | −32.8530 | −27.9828 | −1.09243 | −23.4361 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.65226 | 5.03746 | 13.6435 | 5.00000 | −23.4356 | 6.97704 | −26.2549 | −1.62396 | −23.2613 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.59635 | 2.76477 | −1.25894 | 5.00000 | −7.17831 | 25.9819 | 24.0395 | −19.3561 | −12.9818 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.121310 | −4.89557 | −7.98528 | 5.00000 | 0.593883 | 0.443250 | 1.93918 | −3.03342 | −0.606552 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.55834 | 8.32348 | −5.57157 | 5.00000 | 12.9708 | −0.347320 | −21.1491 | 42.2804 | 7.79171 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.54165 | 1.22958 | 4.54330 | 5.00000 | 4.35475 | −9.26752 | −12.2424 | −25.4881 | 17.7083 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.95715 | −7.36978 | 7.65900 | 5.00000 | −29.1633 | −1.93436 | −1.34938 | 27.3136 | 19.7857 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 1445.4.a.n | yes | 7 |
| 17.b | even | 2 | 1 | 1445.4.a.m | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1445.4.a.m | ✓ | 7 | 17.b | even | 2 | 1 | |
| 1445.4.a.n | yes | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):
|
\( T_{2}^{7} + 3T_{2}^{6} - 36T_{2}^{5} - 80T_{2}^{4} + 401T_{2}^{3} + 499T_{2}^{2} - 1182T_{2} - 150 \)
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\( T_{3}^{7} - 104T_{3}^{5} - 26T_{3}^{4} + 2983T_{3}^{3} - 374T_{3}^{2} - 25056T_{3} + 26176 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 150 \)
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| $3$ |
\( T^{7} - 104 T^{5} + \cdots + 26176 \)
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| $5$ |
\( (T - 5)^{7} \)
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| $7$ |
\( T^{7} + 11 T^{6} + \cdots - 16436 \)
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| $11$ |
\( T^{7} - 27 T^{6} + \cdots + 38141616 \)
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| $13$ |
\( T^{7} + \cdots - 316225947136 \)
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| $17$ |
\( T^{7} \)
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| $19$ |
\( T^{7} + \cdots + 558491279296 \)
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| $23$ |
\( T^{7} + \cdots - 2578599563328 \)
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| $29$ |
\( T^{7} + \cdots + 51896708244192 \)
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| $31$ |
\( T^{7} + \cdots + 215782325088256 \)
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| $37$ |
\( T^{7} + \cdots - 12510553066624 \)
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| $41$ |
\( T^{7} + \cdots - 817092199969326 \)
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| $43$ |
\( T^{7} + \cdots - 10\!\cdots\!48 \)
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| $47$ |
\( T^{7} + \cdots + 51\!\cdots\!44 \)
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| $53$ |
\( T^{7} + \cdots - 65\!\cdots\!20 \)
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| $59$ |
\( T^{7} + \cdots - 30\!\cdots\!52 \)
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| $61$ |
\( T^{7} + \cdots + 11\!\cdots\!08 \)
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| $67$ |
\( T^{7} + \cdots - 52\!\cdots\!64 \)
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| $71$ |
\( T^{7} + \cdots - 40\!\cdots\!08 \)
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| $73$ |
\( T^{7} + \cdots + 71\!\cdots\!28 \)
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| $79$ |
\( T^{7} + \cdots + 21\!\cdots\!72 \)
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| $83$ |
\( T^{7} + \cdots - 73\!\cdots\!16 \)
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| $89$ |
\( T^{7} + \cdots - 22\!\cdots\!21 \)
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| $97$ |
\( T^{7} + \cdots - 16\!\cdots\!56 \)
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