Newspace parameters
| Level: | \( N \) | \(=\) | \( 446 = 2 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 446.c (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.56132793015\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{14} + 16 x^{12} - 12 x^{11} + 175 x^{10} - 149 x^{9} + 1070 x^{8} - 1093 x^{7} + 4783 x^{6} + \cdots + 13689 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} + 16 x^{12} - 12 x^{11} + 175 x^{10} - 149 x^{9} + 1070 x^{8} - 1093 x^{7} + 4783 x^{6} + \cdots + 13689 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21698860657301 \nu^{13} - 35673861573020 \nu^{12} - 205948553802311 \nu^{11} + \cdots - 81\!\cdots\!65 ) / 78\!\cdots\!03 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43198485694955 \nu^{13} - 912255338616716 \nu^{12} + 438516109272337 \nu^{11} + \cdots - 10\!\cdots\!25 ) / 78\!\cdots\!03 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 57372722230321 \nu^{13} - 105559355141485 \nu^{12} + 728523903537619 \nu^{11} + \cdots - 33\!\cdots\!39 ) / 78\!\cdots\!03 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 69\!\cdots\!45 \nu^{13} + \cdots - 13\!\cdots\!75 ) / 91\!\cdots\!51 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 30\!\cdots\!85 \nu^{13} + \cdots - 10\!\cdots\!88 ) / 91\!\cdots\!51 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 45\!\cdots\!95 \nu^{13} + \cdots - 77\!\cdots\!42 ) / 91\!\cdots\!51 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 417026483583693 \nu^{13} - 836902699143030 \nu^{12} + \cdots - 13\!\cdots\!83 ) / 78\!\cdots\!03 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 489846925297178 \nu^{13} - 517300174563425 \nu^{12} + \cdots - 74\!\cdots\!63 ) / 78\!\cdots\!03 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 61\!\cdots\!25 \nu^{13} + \cdots - 45\!\cdots\!22 ) / 91\!\cdots\!51 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 71\!\cdots\!93 \nu^{13} + \cdots + 11\!\cdots\!81 ) / 91\!\cdots\!51 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 29\!\cdots\!20 \nu^{13} + \cdots - 18\!\cdots\!15 ) / 30\!\cdots\!17 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 10\!\cdots\!10 \nu^{13} + 671327122309198 \nu^{12} + \cdots + 16\!\cdots\!59 ) / 78\!\cdots\!03 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 5\beta_{5} - \beta_{4} - \beta_{2} - \beta _1 - 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{4} + 6\beta_{2} + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{12} - \beta_{10} - 11\beta_{6} + 32\beta_{5} + 11\beta_1 \)
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| \(\nu^{5}\) | \(=\) |
\( - 12 \beta_{12} + 11 \beta_{10} - 11 \beta_{9} + 12 \beta_{8} - \beta_{7} + 18 \beta_{6} - 39 \beta_{5} + \cdots - 39 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{13} + 17\beta_{9} - 31\beta_{8} + 103\beta_{4} + 2\beta_{3} + 106\beta_{2} + 238 \)
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| \(\nu^{7}\) | \(=\) |
\( 123\beta_{12} - 101\beta_{10} + 16\beta_{7} - 220\beta_{6} + 425\beta_{5} - 16\beta_{3} + 378\beta_1 \)
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| \(\nu^{8}\) | \(=\) |
\( 16 \beta_{13} - 353 \beta_{12} - 16 \beta_{11} + 204 \beta_{10} - 204 \beta_{9} + 353 \beta_{8} + \cdots - 1946 \)
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| \(\nu^{9}\) | \(=\) |
\( -6\beta_{13} + 905\beta_{9} - 1207\beta_{8} + 2345\beta_{4} + 187\beta_{3} + 3359\beta_{2} + 4351 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3618 \beta_{12} + 175 \beta_{11} - 2158 \beta_{10} + 489 \beta_{7} - 8669 \beta_{6} + 16884 \beta_{5} + \cdots + 9520 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 139 \beta_{13} - 11630 \beta_{12} - 139 \beta_{11} + 8180 \beta_{10} - 8180 \beta_{9} + 11630 \beta_{8} + \cdots - 43106 \)
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| \(\nu^{12}\) | \(=\) |
\( -1671\beta_{13} + 21595\beta_{9} - 35483\beta_{8} + 80330\beta_{4} + 5399\beta_{3} + 90186\beta_{2} + 151840 \)
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| \(\nu^{13}\) | \(=\) |
\( 111052 \beta_{12} + 2057 \beta_{11} - 74931 \beta_{10} + 19287 \beta_{7} - 229665 \beta_{6} + \cdots + 285216 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/446\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 39.1 |
|
1.00000 | −1.54240 | + | 2.67152i | 1.00000 | −2.02734 | − | 3.51145i | −1.54240 | + | 2.67152i | −0.565541 | 1.00000 | −3.25801 | − | 5.64305i | −2.02734 | − | 3.51145i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.2 | 1.00000 | −1.24410 | + | 2.15484i | 1.00000 | 1.45788 | + | 2.52512i | −1.24410 | + | 2.15484i | −3.18805 | 1.00000 | −1.59555 | − | 2.76357i | 1.45788 | + | 2.52512i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.3 | 1.00000 | −1.22256 | + | 2.11753i | 1.00000 | 0.904347 | + | 1.56637i | −1.22256 | + | 2.11753i | 4.85106 | 1.00000 | −1.48930 | − | 2.57954i | 0.904347 | + | 1.56637i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.4 | 1.00000 | −0.723729 | + | 1.25354i | 1.00000 | −0.806833 | − | 1.39748i | −0.723729 | + | 1.25354i | 0.193964 | 1.00000 | 0.452433 | + | 0.783637i | −0.806833 | − | 1.39748i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.5 | 1.00000 | 0.686757 | − | 1.18950i | 1.00000 | −1.36639 | − | 2.36665i | 0.686757 | − | 1.18950i | 2.10820 | 1.00000 | 0.556729 | + | 0.964283i | −1.36639 | − | 2.36665i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.6 | 1.00000 | 1.05851 | − | 1.83340i | 1.00000 | −2.03924 | − | 3.53208i | 1.05851 | − | 1.83340i | −4.20578 | 1.00000 | −0.740900 | − | 1.28328i | −2.03924 | − | 3.53208i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 39.7 | 1.00000 | 1.48752 | − | 2.57645i | 1.00000 | 0.877575 | + | 1.52000i | 1.48752 | − | 2.57645i | −2.19386 | 1.00000 | −2.92540 | − | 5.06695i | 0.877575 | + | 1.52000i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.1 | 1.00000 | −1.54240 | − | 2.67152i | 1.00000 | −2.02734 | + | 3.51145i | −1.54240 | − | 2.67152i | −0.565541 | 1.00000 | −3.25801 | + | 5.64305i | −2.02734 | + | 3.51145i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.2 | 1.00000 | −1.24410 | − | 2.15484i | 1.00000 | 1.45788 | − | 2.52512i | −1.24410 | − | 2.15484i | −3.18805 | 1.00000 | −1.59555 | + | 2.76357i | 1.45788 | − | 2.52512i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.3 | 1.00000 | −1.22256 | − | 2.11753i | 1.00000 | 0.904347 | − | 1.56637i | −1.22256 | − | 2.11753i | 4.85106 | 1.00000 | −1.48930 | + | 2.57954i | 0.904347 | − | 1.56637i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.4 | 1.00000 | −0.723729 | − | 1.25354i | 1.00000 | −0.806833 | + | 1.39748i | −0.723729 | − | 1.25354i | 0.193964 | 1.00000 | 0.452433 | − | 0.783637i | −0.806833 | + | 1.39748i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.5 | 1.00000 | 0.686757 | + | 1.18950i | 1.00000 | −1.36639 | + | 2.36665i | 0.686757 | + | 1.18950i | 2.10820 | 1.00000 | 0.556729 | − | 0.964283i | −1.36639 | + | 2.36665i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.6 | 1.00000 | 1.05851 | + | 1.83340i | 1.00000 | −2.03924 | + | 3.53208i | 1.05851 | + | 1.83340i | −4.20578 | 1.00000 | −0.740900 | + | 1.28328i | −2.03924 | + | 3.53208i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 183.7 | 1.00000 | 1.48752 | + | 2.57645i | 1.00000 | 0.877575 | − | 1.52000i | 1.48752 | + | 2.57645i | −2.19386 | 1.00000 | −2.92540 | + | 5.06695i | 0.877575 | − | 1.52000i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 223.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 446.2.c.d | ✓ | 14 |
| 223.c | even | 3 | 1 | inner | 446.2.c.d | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 446.2.c.d | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 446.2.c.d | ✓ | 14 | 223.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(446, [\chi])\):
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\( T_{3}^{14} + 3 T_{3}^{13} + 24 T_{3}^{12} + 47 T_{3}^{11} + 301 T_{3}^{10} + 517 T_{3}^{9} + \cdots + 55225 \)
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\( T_{5}^{14} + 6 T_{5}^{13} + 47 T_{5}^{12} + 128 T_{5}^{11} + 680 T_{5}^{10} + 1259 T_{5}^{9} + \cdots + 455625 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{14} \)
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| $3$ |
\( T^{14} + 3 T^{13} + \cdots + 55225 \)
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| $5$ |
\( T^{14} + 6 T^{13} + \cdots + 455625 \)
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| $7$ |
\( (T^{7} + 3 T^{6} - 26 T^{5} + \cdots - 33)^{2} \)
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| $11$ |
\( T^{14} + 5 T^{13} + \cdots + 17313921 \)
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| $13$ |
\( (T^{7} - 7 T^{6} + \cdots + 135)^{2} \)
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| $17$ |
\( (T^{7} - 36 T^{5} + \cdots + 1091)^{2} \)
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| $19$ |
\( T^{14} + 4 T^{13} + \cdots + 1010025 \)
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| $23$ |
\( T^{14} - 2 T^{13} + \cdots + 8555625 \)
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| $29$ |
\( T^{14} - 2 T^{13} + \cdots + 19881 \)
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| $31$ |
\( T^{14} + \cdots + 9255402025 \)
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| $37$ |
\( T^{14} + 5 T^{13} + \cdots + 6305121 \)
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| $41$ |
\( (T^{7} - 22 T^{6} + \cdots + 116211)^{2} \)
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| $43$ |
\( T^{14} + \cdots + 6380015625 \)
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| $47$ |
\( T^{14} + \cdots + 4565046740409 \)
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| $53$ |
\( T^{14} + \cdots + 146019515625 \)
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| $59$ |
\( (T^{7} - 10 T^{6} + \cdots - 405)^{2} \)
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| $61$ |
\( T^{14} + \cdots + 215731451961 \)
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| $67$ |
\( T^{14} + \cdots + 757625625 \)
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| $71$ |
\( T^{14} + 18 T^{13} + \cdots + 1010025 \)
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| $73$ |
\( T^{14} + \cdots + 81524525625 \)
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| $79$ |
\( T^{14} + \cdots + 21412761561 \)
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| $83$ |
\( T^{14} + 15 T^{13} + \cdots + 5517801 \)
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| $89$ |
\( T^{14} + \cdots + 572103140625 \)
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| $97$ |
\( T^{14} + \cdots + 870729730641 \)
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