Newspace parameters
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.36806021607\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{12} - 484x^{10} + 194194x^{8} - 39867800x^{6} + 5398720873x^{4} - 310089434788x^{2} + 9371104623076 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 484x^{10} + 194194x^{8} - 39867800x^{6} + 5398720873x^{4} - 310089434788x^{2} + 9371104623076 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 349147121 \nu^{10} + 169990205666 \nu^{8} - 61997289967221 \nu^{6} + \cdots - 14\!\cdots\!48 ) / 39\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 577504845699127 \nu^{11} + \cdots + 53\!\cdots\!40 ) / 44\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 577504845699127 \nu^{11} + \cdots + 12\!\cdots\!20 ) / 44\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 577504845699127 \nu^{11} + \cdots + 86\!\cdots\!24 ) / 44\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3044261635951 \nu^{11} + \cdots - 34\!\cdots\!42 \nu ) / 86\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 63\!\cdots\!97 \nu^{11} + \cdots + 60\!\cdots\!68 ) / 44\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 40\!\cdots\!09 \nu^{11} + \cdots + 28\!\cdots\!48 \nu ) / 22\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!65 \nu^{11} + \cdots - 32\!\cdots\!36 ) / 44\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 17\!\cdots\!93 \nu^{11} + \cdots + 10\!\cdots\!72 ) / 44\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 461224318369921 \nu^{11} + \cdots + 55\!\cdots\!92 \nu ) / 17\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 19\!\cdots\!67 \nu^{11} + \cdots - 87\!\cdots\!80 ) / 44\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( 9\beta_{7} + \beta_{6} + 54\beta_{5} - 7\beta_{4} ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{9} - 16\beta_{8} + 8\beta_{7} + 48\beta_{4} - 15\beta_{3} - 9\beta_{2} + 6\beta _1 + 1452 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 108 \beta_{11} + 108 \beta_{10} + 102 \beta_{9} + 408 \beta_{8} - 573 \beta_{7} + 55 \beta_{6} + \cdots + 54 \beta_{2} ) / 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1632 \beta_{9} - 3264 \beta_{8} + 1632 \beta_{7} - 112 \beta_{6} + 7264 \beta_{4} - 2337 \beta_{3} + \cdots - 457392 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 32796 \beta_{11} - 31860 \beta_{10} + 55794 \beta_{9} + 223176 \beta_{8} - 619233 \beta_{7} + \cdots + 16398 \beta_{2} ) / 54 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 135680 \beta_{9} + 271360 \beta_{8} - 135680 \beta_{7} - 418040 \beta_{6} - 6240880 \beta_{4} + \cdots - 145767480 ) / 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6058116 \beta_{11} - 27571428 \beta_{10} + 1628442 \beta_{9} + 6513768 \beta_{8} + \cdots + 3029058 \beta_{2} ) / 54 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 105542016 \beta_{9} + 211084032 \beta_{8} - 105542016 \beta_{7} - 126681632 \beta_{6} + \cdots + 11513807496 ) / 18 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 115352964 \beta_{11} - 3406638924 \beta_{10} - 5179046022 \beta_{9} - 20716184088 \beta_{8} + \cdots - 57676482 \beta_{2} ) / 54 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 1475567008 \beta_{9} + 2951134016 \beta_{8} - 1475567008 \beta_{7} + 10127357464 \beta_{6} + \cdots + 12452817789624 ) / 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 550266690924 \beta_{11} + 1808171237268 \beta_{10} - 1181191737750 \beta_{9} + \cdots - 275133345462 \beta_{2} ) / 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 20.1 |
|
− | 10.2458i | −4.60012 | + | 14.8943i | −72.9763 | −73.2992 | 152.604 | + | 47.1319i | 55.7250 | − | 117.054i | 419.835i | −200.678 | − | 137.031i | 751.009i | |||||||||||||||||||||||||||||||||||||||||||||
| 20.2 | − | 10.2458i | 4.60012 | − | 14.8943i | −72.9763 | 73.2992 | −152.604 | − | 47.1319i | 55.7250 | + | 117.054i | 419.835i | −200.678 | − | 137.031i | − | 751.009i | |||||||||||||||||||||||||||||||||||||||||||||
| 20.3 | − | 5.57294i | −13.5283 | − | 7.74498i | 0.942331 | −46.2136 | −43.1623 | + | 75.3925i | −120.146 | + | 48.7024i | − | 183.586i | 123.031 | + | 209.553i | 257.545i | |||||||||||||||||||||||||||||||||||||||||||||
| 20.4 | − | 5.57294i | 13.5283 | + | 7.74498i | 0.942331 | 46.2136 | 43.1623 | − | 75.3925i | −120.146 | − | 48.7024i | − | 183.586i | 123.031 | + | 209.553i | − | 257.545i | ||||||||||||||||||||||||||||||||||||||||||||
| 20.5 | − | 2.99433i | −9.94100 | + | 12.0074i | 23.0340 | 61.8023 | 35.9539 | + | 29.7666i | 92.4210 | + | 90.9140i | − | 164.790i | −45.3530 | − | 238.730i | − | 185.056i | ||||||||||||||||||||||||||||||||||||||||||||
| 20.6 | − | 2.99433i | 9.94100 | − | 12.0074i | 23.0340 | −61.8023 | −35.9539 | − | 29.7666i | 92.4210 | − | 90.9140i | − | 164.790i | −45.3530 | − | 238.730i | 185.056i | |||||||||||||||||||||||||||||||||||||||||||||
| 20.7 | 2.99433i | −9.94100 | − | 12.0074i | 23.0340 | 61.8023 | 35.9539 | − | 29.7666i | 92.4210 | − | 90.9140i | 164.790i | −45.3530 | + | 238.730i | 185.056i | |||||||||||||||||||||||||||||||||||||||||||||||
| 20.8 | 2.99433i | 9.94100 | + | 12.0074i | 23.0340 | −61.8023 | −35.9539 | + | 29.7666i | 92.4210 | + | 90.9140i | 164.790i | −45.3530 | + | 238.730i | − | 185.056i | ||||||||||||||||||||||||||||||||||||||||||||||
| 20.9 | 5.57294i | −13.5283 | + | 7.74498i | 0.942331 | −46.2136 | −43.1623 | − | 75.3925i | −120.146 | − | 48.7024i | 183.586i | 123.031 | − | 209.553i | − | 257.545i | ||||||||||||||||||||||||||||||||||||||||||||||
| 20.10 | 5.57294i | 13.5283 | − | 7.74498i | 0.942331 | 46.2136 | 43.1623 | + | 75.3925i | −120.146 | + | 48.7024i | 183.586i | 123.031 | − | 209.553i | 257.545i | |||||||||||||||||||||||||||||||||||||||||||||||
| 20.11 | 10.2458i | −4.60012 | − | 14.8943i | −72.9763 | −73.2992 | 152.604 | − | 47.1319i | 55.7250 | + | 117.054i | − | 419.835i | −200.678 | + | 137.031i | − | 751.009i | |||||||||||||||||||||||||||||||||||||||||||||
| 20.12 | 10.2458i | 4.60012 | + | 14.8943i | −72.9763 | 73.2992 | −152.604 | + | 47.1319i | 55.7250 | − | 117.054i | − | 419.835i | −200.678 | + | 137.031i | 751.009i | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.6.c.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 21.6.c.a | ✓ | 12 |
| 4.b | odd | 2 | 1 | 336.6.k.d | 12 | ||
| 7.b | odd | 2 | 1 | inner | 21.6.c.a | ✓ | 12 |
| 12.b | even | 2 | 1 | 336.6.k.d | 12 | ||
| 21.c | even | 2 | 1 | inner | 21.6.c.a | ✓ | 12 |
| 28.d | even | 2 | 1 | 336.6.k.d | 12 | ||
| 84.h | odd | 2 | 1 | 336.6.k.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.c.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 21.6.c.a | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
| 21.6.c.a | ✓ | 12 | 7.b | odd | 2 | 1 | inner |
| 21.6.c.a | ✓ | 12 | 21.c | even | 2 | 1 | inner |
| 336.6.k.d | 12 | 4.b | odd | 2 | 1 | ||
| 336.6.k.d | 12 | 12.b | even | 2 | 1 | ||
| 336.6.k.d | 12 | 28.d | even | 2 | 1 | ||
| 336.6.k.d | 12 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + 145 T^{4} + \cdots + 29232)^{2} \)
|
| $3$ |
\( T^{12} + \cdots + 205891132094649 \)
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| $5$ |
\( (T^{6} - 11328 T^{4} + \cdots - 43827584976)^{2} \)
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| $7$ |
\( (T^{6} + \cdots + 4747561509943)^{2} \)
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| $11$ |
\( (T^{6} + \cdots + 17\!\cdots\!88)^{2} \)
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| $13$ |
\( (T^{6} + \cdots + 545717977444800)^{2} \)
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| $17$ |
\( (T^{6} + \cdots - 14\!\cdots\!56)^{2} \)
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| $19$ |
\( (T^{6} + \cdots + 246289511999088)^{2} \)
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| $23$ |
\( (T^{6} + \cdots + 97\!\cdots\!52)^{2} \)
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| $29$ |
\( (T^{6} + \cdots + 23\!\cdots\!68)^{2} \)
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| $31$ |
\( (T^{6} + \cdots + 43\!\cdots\!08)^{2} \)
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| $37$ |
\( (T^{3} - 6866 T^{2} + \cdots + 403376576936)^{4} \)
|
| $41$ |
\( (T^{6} + \cdots - 19\!\cdots\!96)^{2} \)
|
| $43$ |
\( (T^{3} + 18460 T^{2} + \cdots - 624577978816)^{4} \)
|
| $47$ |
\( (T^{6} + \cdots - 52\!\cdots\!76)^{2} \)
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| $53$ |
\( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \)
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| $59$ |
\( (T^{6} + \cdots - 10\!\cdots\!44)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 35\!\cdots\!72)^{2} \)
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| $67$ |
\( (T^{3} + \cdots + 14656583988736)^{4} \)
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| $71$ |
\( (T^{6} + \cdots + 30\!\cdots\!72)^{2} \)
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| $73$ |
\( (T^{6} + \cdots + 48\!\cdots\!52)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 53317412578496)^{4} \)
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| $83$ |
\( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{6} + \cdots - 16\!\cdots\!04)^{2} \)
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| $97$ |
\( (T^{6} + \cdots + 29\!\cdots\!12)^{2} \)
|
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