Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(105.586138614\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
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| Defining polynomial: |
\( x^{9} - 3 x^{8} - 8757 x^{7} + 24639 x^{6} + 24977559 x^{5} - 155625075 x^{4} - 26856295558 x^{3} + \cdots - 82422805113727 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7\cdot 13^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3 x^{8} - 8757 x^{7} + 24639 x^{6} + 24977559 x^{5} - 155625075 x^{4} - 26856295558 x^{3} + \cdots - 82422805113727 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 59\!\cdots\!55 \nu^{8} + \cdots + 11\!\cdots\!99 ) / 12\!\cdots\!56 \)
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| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!45 \nu^{8} + \cdots - 53\!\cdots\!08 ) / 48\!\cdots\!06 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 72\!\cdots\!85 \nu^{8} + \cdots - 40\!\cdots\!71 ) / 97\!\cdots\!12 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 18\!\cdots\!09 \nu^{8} + \cdots + 14\!\cdots\!36 ) / 31\!\cdots\!39 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 12\!\cdots\!07 \nu^{8} + \cdots - 10\!\cdots\!11 ) / 63\!\cdots\!78 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 26\!\cdots\!13 \nu^{8} + \cdots + 19\!\cdots\!39 ) / 12\!\cdots\!56 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 21\!\cdots\!29 \nu^{8} + \cdots - 12\!\cdots\!90 ) / 63\!\cdots\!78 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!64 \nu^{8} + \cdots - 31\!\cdots\!41 ) / 31\!\cdots\!39 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{3} - 6\beta_{2} - 13\beta _1 + 11 ) / 13 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 48 \beta_{8} - 10 \beta_{7} + 66 \beta_{6} + 11 \beta_{5} - 11 \beta_{4} + 195 \beta_{3} - 174 \beta_{2} + \cdots + 25286 ) / 13 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 906 \beta_{8} - 585 \beta_{7} + 57 \beta_{6} + 303 \beta_{5} - 303 \beta_{4} + 11003 \beta_{3} + \cdots + 32853 ) / 13 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 222891 \beta_{8} - 75940 \beta_{7} + 229701 \beta_{6} + 53828 \beta_{5} + 2410 \beta_{4} + \cdots + 77345062 ) / 13 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 6241920 \beta_{8} - 3646918 \beta_{7} + 1375737 \beta_{6} + 2660081 \beta_{5} + 1423999 \beta_{4} + \cdots + 785612561 ) / 13 \)
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| \(\nu^{6}\) | \(=\) |
\( ( 975800181 \beta_{8} - 403012282 \beta_{7} + 810301914 \beta_{6} + 252419507 \beta_{5} + \cdots + 280636213534 ) / 13 \)
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| \(\nu^{7}\) | \(=\) |
\( ( 35918434245 \beta_{8} - 19769272601 \beta_{7} + 12470411925 \beta_{6} + 15787971727 \beta_{5} + \cdots + 6359170605309 ) / 13 \)
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| \(\nu^{8}\) | \(=\) |
\( ( 4317121996116 \beta_{8} - 1959352402988 \beta_{7} + 3132462217257 \beta_{6} + 1193239338673 \beta_{5} + \cdots + 11\!\cdots\!55 ) / 13 \)
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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 | −81.5795 | 64.0000 | 84.9813 | −652.636 | −885.572 | 512.000 | 4468.22 | 679.850 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −66.3863 | 64.0000 | 55.4903 | −531.091 | −1457.98 | 512.000 | 2220.14 | 443.922 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −36.9676 | 64.0000 | −78.2525 | −295.741 | 1715.06 | 512.000 | −820.399 | −626.020 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | −35.9225 | 64.0000 | 125.063 | −287.380 | 522.035 | 512.000 | −896.575 | 1000.51 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | −16.4445 | 64.0000 | −412.685 | −131.556 | 1047.82 | 512.000 | −1916.58 | −3301.48 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00000 | 15.0182 | 64.0000 | 212.123 | 120.146 | −458.773 | 512.000 | −1961.45 | 1696.99 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.00000 | 44.4689 | 64.0000 | −407.661 | 355.751 | −278.209 | 512.000 | −209.520 | −3261.29 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.00000 | 51.9574 | 64.0000 | −198.361 | 415.660 | −367.949 | 512.000 | 512.575 | −1586.89 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.00000 | 56.8558 | 64.0000 | 301.302 | 454.847 | −1268.43 | 512.000 | 1045.59 | 2410.41 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 338.8.a.p | yes | 9 |
| 13.b | even | 2 | 1 | 338.8.a.o | ✓ | 9 | |
| 13.d | odd | 4 | 2 | 338.8.b.j | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.8.a.o | ✓ | 9 | 13.b | even | 2 | 1 | |
| 338.8.a.p | yes | 9 | 1.a | even | 1 | 1 | trivial |
| 338.8.b.j | 18 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\):
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\( T_{3}^{9} + 69 T_{3}^{8} - 8682 T_{3}^{7} - 534745 T_{3}^{6} + 25538085 T_{3}^{5} + \cdots + 233327013346323 \)
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\( T_{5}^{9} + 318 T_{5}^{8} - 221282 T_{5}^{7} - 45527490 T_{5}^{6} + 17743146285 T_{5}^{5} + \cdots - 98\!\cdots\!00 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{9} \)
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| $3$ |
\( T^{9} + \cdots + 233327013346323 \)
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| $5$ |
\( T^{9} + \cdots - 98\!\cdots\!00 \)
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| $7$ |
\( T^{9} + \cdots - 72\!\cdots\!92 \)
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| $11$ |
\( T^{9} + \cdots - 81\!\cdots\!77 \)
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| $13$ |
\( T^{9} \)
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| $17$ |
\( T^{9} + \cdots + 21\!\cdots\!53 \)
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| $19$ |
\( T^{9} + \cdots - 56\!\cdots\!83 \)
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| $23$ |
\( T^{9} + \cdots - 38\!\cdots\!96 \)
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| $29$ |
\( T^{9} + \cdots + 52\!\cdots\!96 \)
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| $31$ |
\( T^{9} + \cdots + 49\!\cdots\!24 \)
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| $37$ |
\( T^{9} + \cdots - 12\!\cdots\!24 \)
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| $41$ |
\( T^{9} + \cdots - 15\!\cdots\!17 \)
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| $43$ |
\( T^{9} + \cdots - 30\!\cdots\!93 \)
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| $47$ |
\( T^{9} + \cdots - 27\!\cdots\!04 \)
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| $53$ |
\( T^{9} + \cdots + 12\!\cdots\!28 \)
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| $59$ |
\( T^{9} + \cdots + 56\!\cdots\!13 \)
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| $61$ |
\( T^{9} + \cdots - 76\!\cdots\!88 \)
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| $67$ |
\( T^{9} + \cdots - 15\!\cdots\!83 \)
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| $71$ |
\( T^{9} + \cdots + 75\!\cdots\!24 \)
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| $73$ |
\( T^{9} + \cdots + 81\!\cdots\!49 \)
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| $79$ |
\( T^{9} + \cdots + 15\!\cdots\!84 \)
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| $83$ |
\( T^{9} + \cdots + 41\!\cdots\!67 \)
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| $89$ |
\( T^{9} + \cdots + 18\!\cdots\!17 \)
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| $97$ |
\( T^{9} + \cdots + 70\!\cdots\!11 \)
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