Newspace parameters
| Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 312.q (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(50.0397517816\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{20} - 45625 x^{18} - 258130 x^{17} + 874261585 x^{16} + 8836032648 x^{15} - 9129028858830 x^{14} + \cdots + 26\!\cdots\!43 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{44}\cdot 3\cdot 13 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 45625 x^{18} - 258130 x^{17} + 874261585 x^{16} + 8836032648 x^{15} - 9129028858830 x^{14} + \cdots + 26\!\cdots\!43 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 56\!\cdots\!40 \nu^{19} + \cdots + 11\!\cdots\!22 ) / 59\!\cdots\!92 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 56\!\cdots\!40 \nu^{19} + \cdots + 11\!\cdots\!22 ) / 59\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 18\!\cdots\!04 \nu^{19} + \cdots - 27\!\cdots\!53 ) / 63\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!13 \nu^{19} + \cdots + 18\!\cdots\!65 ) / 31\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!47 \nu^{19} + \cdots + 37\!\cdots\!75 ) / 31\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29\!\cdots\!29 \nu^{19} + \cdots + 10\!\cdots\!84 ) / 31\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 18\!\cdots\!55 \nu^{19} + \cdots + 50\!\cdots\!76 ) / 19\!\cdots\!64 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!07 \nu^{19} + \cdots + 11\!\cdots\!88 ) / 63\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!97 \nu^{19} + \cdots + 17\!\cdots\!26 ) / 63\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 49\!\cdots\!68 \nu^{19} + \cdots - 37\!\cdots\!61 ) / 21\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 10\!\cdots\!13 \nu^{19} + \cdots - 41\!\cdots\!15 ) / 21\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 30\!\cdots\!07 \nu^{19} + \cdots - 16\!\cdots\!41 ) / 31\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 31\!\cdots\!19 \nu^{19} + \cdots - 72\!\cdots\!67 ) / 31\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 35\!\cdots\!32 \nu^{19} + \cdots - 91\!\cdots\!79 ) / 31\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 22\!\cdots\!44 \nu^{19} + \cdots + 21\!\cdots\!77 ) / 15\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 48\!\cdots\!96 \nu^{19} + \cdots + 50\!\cdots\!77 ) / 31\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 14\!\cdots\!90 \nu^{19} + \cdots + 14\!\cdots\!17 ) / 63\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 39\!\cdots\!65 \nu^{19} + \cdots - 12\!\cdots\!01 ) / 15\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 23\!\cdots\!92 \nu^{19} + \cdots + 16\!\cdots\!35 ) / 15\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{16} + \beta_{15} + \beta_{13} + \beta_{12} - 2\beta_{10} - 2\beta_{7} + 4\beta_{3} + 9\beta_{2} - \beta _1 + 4557 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{19} - 18 \beta_{18} - 15 \beta_{17} - \beta_{16} + 52 \beta_{15} + 68 \beta_{14} + 32 \beta_{13} + \cdots + 41902 \)
|
| \(\nu^{4}\) | \(=\) |
\( 62 \beta_{19} - 1537 \beta_{18} + 575 \beta_{17} + 5666 \beta_{16} + 9313 \beta_{15} - 873 \beta_{14} + \cdots + 33289604 \)
|
| \(\nu^{5}\) | \(=\) |
\( 50600 \beta_{19} - 295960 \beta_{18} - 289315 \beta_{17} + 46048 \beta_{16} + 626358 \beta_{15} + \cdots + 787745018 \)
|
| \(\nu^{6}\) | \(=\) |
\( 972903 \beta_{19} - 19230088 \beta_{18} + 9267040 \beta_{17} + 40177133 \beta_{16} + 84296586 \beta_{15} + \cdots + 280676315589 \)
|
| \(\nu^{7}\) | \(=\) |
\( 662797443 \beta_{19} - 3486759673 \beta_{18} - 3083419265 \beta_{17} + 519414790 \beta_{16} + \cdots + 10235486877754 \)
|
| \(\nu^{8}\) | \(=\) |
\( 14285496436 \beta_{19} - 214507860521 \beta_{18} + 109726407175 \beta_{17} + 316879703174 \beta_{16} + \cdots + 25\!\cdots\!19 \)
|
| \(\nu^{9}\) | \(=\) |
\( 7912432213164 \beta_{19} - 37505511814134 \beta_{18} - 28500878504995 \beta_{17} + 3347382841890 \beta_{16} + \cdots + 11\!\cdots\!78 \)
|
| \(\nu^{10}\) | \(=\) |
\( 201778647786939 \beta_{19} + \cdots + 23\!\cdots\!70 \)
|
| \(\nu^{11}\) | \(=\) |
\( 89\!\cdots\!84 \beta_{19} + \cdots + 12\!\cdots\!12 \)
|
| \(\nu^{12}\) | \(=\) |
\( 26\!\cdots\!06 \beta_{19} + \cdots + 21\!\cdots\!25 \)
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| \(\nu^{13}\) | \(=\) |
\( 98\!\cdots\!22 \beta_{19} + \cdots + 13\!\cdots\!84 \)
|
| \(\nu^{14}\) | \(=\) |
\( 34\!\cdots\!80 \beta_{19} + \cdots + 20\!\cdots\!65 \)
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| \(\nu^{15}\) | \(=\) |
\( 10\!\cdots\!01 \beta_{19} + \cdots + 13\!\cdots\!78 \)
|
| \(\nu^{16}\) | \(=\) |
\( 41\!\cdots\!40 \beta_{19} + \cdots + 19\!\cdots\!96 \)
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| \(\nu^{17}\) | \(=\) |
\( 11\!\cdots\!32 \beta_{19} + \cdots + 13\!\cdots\!54 \)
|
| \(\nu^{18}\) | \(=\) |
\( 49\!\cdots\!09 \beta_{19} + \cdots + 18\!\cdots\!25 \)
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| \(\nu^{19}\) | \(=\) |
\( 11\!\cdots\!89 \beta_{19} + \cdots + 13\!\cdots\!22 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(157\) | \(209\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 217.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | −93.2733 | 0 | 77.8524 | + | 134.844i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.2 | 0 | 4.50000 | − | 7.79423i | 0 | −77.8611 | 0 | −121.385 | − | 210.245i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.3 | 0 | 4.50000 | − | 7.79423i | 0 | −69.4773 | 0 | 13.6072 | + | 23.5684i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.4 | 0 | 4.50000 | − | 7.79423i | 0 | −47.9558 | 0 | 5.03066 | + | 8.71335i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.5 | 0 | 4.50000 | − | 7.79423i | 0 | −18.9393 | 0 | 73.8949 | + | 127.990i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.6 | 0 | 4.50000 | − | 7.79423i | 0 | 20.1255 | 0 | −0.0613073 | − | 0.106187i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.7 | 0 | 4.50000 | − | 7.79423i | 0 | 37.1294 | 0 | −2.34912 | − | 4.06879i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.8 | 0 | 4.50000 | − | 7.79423i | 0 | 49.5867 | 0 | −115.331 | − | 199.759i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.9 | 0 | 4.50000 | − | 7.79423i | 0 | 96.7646 | 0 | 100.937 | + | 174.827i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.10 | 0 | 4.50000 | − | 7.79423i | 0 | 98.9007 | 0 | −36.6954 | − | 63.5582i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.1 | 0 | 4.50000 | + | 7.79423i | 0 | −93.2733 | 0 | 77.8524 | − | 134.844i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.2 | 0 | 4.50000 | + | 7.79423i | 0 | −77.8611 | 0 | −121.385 | + | 210.245i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.3 | 0 | 4.50000 | + | 7.79423i | 0 | −69.4773 | 0 | 13.6072 | − | 23.5684i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.4 | 0 | 4.50000 | + | 7.79423i | 0 | −47.9558 | 0 | 5.03066 | − | 8.71335i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.5 | 0 | 4.50000 | + | 7.79423i | 0 | −18.9393 | 0 | 73.8949 | − | 127.990i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.6 | 0 | 4.50000 | + | 7.79423i | 0 | 20.1255 | 0 | −0.0613073 | + | 0.106187i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.7 | 0 | 4.50000 | + | 7.79423i | 0 | 37.1294 | 0 | −2.34912 | + | 4.06879i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.8 | 0 | 4.50000 | + | 7.79423i | 0 | 49.5867 | 0 | −115.331 | + | 199.759i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.9 | 0 | 4.50000 | + | 7.79423i | 0 | 96.7646 | 0 | 100.937 | − | 174.827i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 289.10 | 0 | 4.50000 | + | 7.79423i | 0 | 98.9007 | 0 | −36.6954 | + | 63.5582i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 312.6.q.d | ✓ | 20 |
| 13.c | even | 3 | 1 | inner | 312.6.q.d | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.6.q.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 312.6.q.d | ✓ | 20 | 13.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 5 T_{5}^{9} - 22805 T_{5}^{8} - 220315 T_{5}^{7} + 176348505 T_{5}^{6} + \cdots - 16\!\cdots\!96 \)
acting on \(S_{6}^{\mathrm{new}}(312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{2} - 9 T + 81)^{10} \)
|
| $5$ |
\( (T^{10} + \cdots - 16\!\cdots\!96)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 90\!\cdots\!24 \)
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| $11$ |
\( T^{20} + \cdots + 72\!\cdots\!00 \)
|
| $13$ |
\( T^{20} + \cdots + 49\!\cdots\!49 \)
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| $17$ |
\( T^{20} + \cdots + 50\!\cdots\!00 \)
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| $19$ |
\( T^{20} + \cdots + 25\!\cdots\!00 \)
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| $23$ |
\( T^{20} + \cdots + 25\!\cdots\!96 \)
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| $29$ |
\( T^{20} + \cdots + 18\!\cdots\!00 \)
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| $31$ |
\( (T^{10} + \cdots - 12\!\cdots\!40)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 10\!\cdots\!84 \)
|
| $41$ |
\( T^{20} + \cdots + 23\!\cdots\!00 \)
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| $43$ |
\( T^{20} + \cdots + 38\!\cdots\!44 \)
|
| $47$ |
\( (T^{10} + \cdots - 37\!\cdots\!88)^{2} \)
|
| $53$ |
\( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 23\!\cdots\!96 \)
|
| $61$ |
\( T^{20} + \cdots + 19\!\cdots\!25 \)
|
| $67$ |
\( T^{20} + \cdots + 95\!\cdots\!16 \)
|
| $71$ |
\( T^{20} + \cdots + 33\!\cdots\!76 \)
|
| $73$ |
\( (T^{10} + \cdots + 10\!\cdots\!75)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 61\!\cdots\!96)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 15\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 84\!\cdots\!96 \)
|
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