Newspace parameters
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
Defining polynomial: |
\( x^{12} + 6x^{10} + 29x^{8} + 40x^{6} + 43x^{4} + 7x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 95) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 6x^{10} + 29x^{8} + 40x^{6} + 43x^{4} + 7x^{2} + 1 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 36\nu^{10} + 174\nu^{8} + 841\nu^{6} + 258\nu^{4} + 42\nu^{2} - 2207 ) / 1205 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -138\nu^{10} - 667\nu^{8} - 3023\nu^{6} - 989\nu^{4} - 161\nu^{2} + 2636 ) / 1205 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -138\nu^{11} - 667\nu^{9} - 3023\nu^{7} - 989\nu^{5} - 161\nu^{3} + 2636\nu ) / 1205 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -174\nu^{11} - 841\nu^{9} - 3864\nu^{7} - 1247\nu^{5} - 203\nu^{3} + 7253\nu ) / 1205 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -203\nu^{10} - 1182\nu^{8} - 5713\nu^{6} - 7279\nu^{4} - 8471\nu^{2} - 174 ) / 1205 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -203\nu^{11} - 1182\nu^{9} - 5713\nu^{7} - 7279\nu^{5} - 8471\nu^{3} - 174\nu ) / 1205 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 74\nu^{10} + 438\nu^{8} + 2117\nu^{6} + 2860\nu^{4} + 3139\nu^{2} + 511 ) / 241 \)
|
\(\beta_{9}\) | \(=\) |
\( ( -429\nu^{10} - 2676\nu^{8} - 12934\nu^{6} - 19342\nu^{4} - 19178\nu^{2} - 3122 ) / 1205 \)
|
\(\beta_{10}\) | \(=\) |
\( ( -429\nu^{11} - 2676\nu^{9} - 12934\nu^{7} - 19342\nu^{5} - 19178\nu^{3} - 3122\nu ) / 1205 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 1002\nu^{11} + 6048\nu^{9} + 29232\nu^{7} + 40921\nu^{5} + 43344\nu^{3} + 7056\nu ) / 1205 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{8} - 2\beta_{6} - \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} - 3\beta_{7} + \beta_{5} - \beta_{4} - \beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( \beta_{9} + 5\beta_{8} + 7\beta_{6} - 7 \)
|
\(\nu^{5}\) | \(=\) |
\( 5\beta_{11} + 6\beta_{10} + 12\beta_{7} - 12\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( 6\beta_{3} + 23\beta_{2} + 29 \)
|
\(\nu^{7}\) | \(=\) |
\( -23\beta_{5} + 29\beta_{4} + 75\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( -29\beta_{9} - 104\beta_{8} - 127\beta_{6} - 29\beta_{3} - 104\beta_{2} \)
|
\(\nu^{9}\) | \(=\) |
\( -104\beta_{11} - 133\beta_{10} - 231\beta_{7} + 104\beta_{5} - 133\beta_{4} - 104\beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( 133\beta_{9} + 468\beta_{8} + 566\beta_{6} - 566 \)
|
\(\nu^{11}\) | \(=\) |
\( 468\beta_{11} + 601\beta_{10} + 1034\beta_{7} - 1034\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) |
\(\chi(n)\) | \(1\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
|
−1.22810 | − | 2.12713i | −0.780522 | − | 1.35190i | −2.01647 | + | 3.49262i | 0 | −1.91712 | + | 3.32055i | 4.50527 | 4.99330 | 0.281570 | − | 0.487693i | 0 | ||||||||||||||||||||||||||||||||||||||||||||
26.2 | −0.431391 | − | 0.747190i | −1.53957 | − | 2.66661i | 0.627804 | − | 1.08739i | 0 | −1.32831 | + | 2.30070i | 0.566520 | −2.80888 | −3.24054 | + | 5.61278i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
26.3 | −0.235942 | − | 0.408663i | 0.520111 | + | 0.900858i | 0.888663 | − | 1.53921i | 0 | 0.245432 | − | 0.425100i | −1.17540 | −1.78246 | 0.958970 | − | 1.66098i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
26.4 | 0.235942 | + | 0.408663i | −0.520111 | − | 0.900858i | 0.888663 | − | 1.53921i | 0 | 0.245432 | − | 0.425100i | 1.17540 | 1.78246 | 0.958970 | − | 1.66098i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
26.5 | 0.431391 | + | 0.747190i | 1.53957 | + | 2.66661i | 0.627804 | − | 1.08739i | 0 | −1.32831 | + | 2.30070i | −0.566520 | 2.80888 | −3.24054 | + | 5.61278i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
26.6 | 1.22810 | + | 2.12713i | 0.780522 | + | 1.35190i | −2.01647 | + | 3.49262i | 0 | −1.91712 | + | 3.32055i | −4.50527 | −4.99330 | 0.281570 | − | 0.487693i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.1 | −1.22810 | + | 2.12713i | −0.780522 | + | 1.35190i | −2.01647 | − | 3.49262i | 0 | −1.91712 | − | 3.32055i | 4.50527 | 4.99330 | 0.281570 | + | 0.487693i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.2 | −0.431391 | + | 0.747190i | −1.53957 | + | 2.66661i | 0.627804 | + | 1.08739i | 0 | −1.32831 | − | 2.30070i | 0.566520 | −2.80888 | −3.24054 | − | 5.61278i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.3 | −0.235942 | + | 0.408663i | 0.520111 | − | 0.900858i | 0.888663 | + | 1.53921i | 0 | 0.245432 | + | 0.425100i | −1.17540 | −1.78246 | 0.958970 | + | 1.66098i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.4 | 0.235942 | − | 0.408663i | −0.520111 | + | 0.900858i | 0.888663 | + | 1.53921i | 0 | 0.245432 | + | 0.425100i | 1.17540 | 1.78246 | 0.958970 | + | 1.66098i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.5 | 0.431391 | − | 0.747190i | 1.53957 | − | 2.66661i | 0.627804 | + | 1.08739i | 0 | −1.32831 | − | 2.30070i | −0.566520 | 2.80888 | −3.24054 | − | 5.61278i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
201.6 | 1.22810 | − | 2.12713i | 0.780522 | − | 1.35190i | −2.01647 | − | 3.49262i | 0 | −1.91712 | − | 3.32055i | −4.50527 | −4.99330 | 0.281570 | + | 0.487693i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.e.g | 12 | |
5.b | even | 2 | 1 | inner | 475.2.e.g | 12 | |
5.c | odd | 4 | 2 | 95.2.i.b | ✓ | 12 | |
15.e | even | 4 | 2 | 855.2.be.d | 12 | ||
19.c | even | 3 | 1 | inner | 475.2.e.g | 12 | |
19.c | even | 3 | 1 | 9025.2.a.bu | 6 | ||
19.d | odd | 6 | 1 | 9025.2.a.bt | 6 | ||
95.h | odd | 6 | 1 | 9025.2.a.bt | 6 | ||
95.i | even | 6 | 1 | inner | 475.2.e.g | 12 | |
95.i | even | 6 | 1 | 9025.2.a.bu | 6 | ||
95.l | even | 12 | 2 | 1805.2.b.g | 6 | ||
95.m | odd | 12 | 2 | 95.2.i.b | ✓ | 12 | |
95.m | odd | 12 | 2 | 1805.2.b.f | 6 | ||
285.v | even | 12 | 2 | 855.2.be.d | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.i.b | ✓ | 12 | 5.c | odd | 4 | 2 | |
95.2.i.b | ✓ | 12 | 95.m | odd | 12 | 2 | |
475.2.e.g | 12 | 1.a | even | 1 | 1 | trivial | |
475.2.e.g | 12 | 5.b | even | 2 | 1 | inner | |
475.2.e.g | 12 | 19.c | even | 3 | 1 | inner | |
475.2.e.g | 12 | 95.i | even | 6 | 1 | inner | |
855.2.be.d | 12 | 15.e | even | 4 | 2 | ||
855.2.be.d | 12 | 285.v | even | 12 | 2 | ||
1805.2.b.f | 6 | 95.m | odd | 12 | 2 | ||
1805.2.b.g | 6 | 95.l | even | 12 | 2 | ||
9025.2.a.bt | 6 | 19.d | odd | 6 | 1 | ||
9025.2.a.bt | 6 | 95.h | odd | 6 | 1 | ||
9025.2.a.bu | 6 | 19.c | even | 3 | 1 | ||
9025.2.a.bu | 6 | 95.i | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 7T_{2}^{10} + 43T_{2}^{8} + 40T_{2}^{6} + 29T_{2}^{4} + 6T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 7 T^{10} + 43 T^{8} + 40 T^{6} + \cdots + 1 \)
$3$
\( T^{12} + 13 T^{10} + 133 T^{8} + \cdots + 625 \)
$5$
\( T^{12} \)
$7$
\( (T^{6} - 22 T^{4} + 35 T^{2} - 9)^{2} \)
$11$
\( (T^{3} - T^{2} - 4 T + 3)^{4} \)
$13$
\( T^{12} + 31 T^{10} + 722 T^{8} + \cdots + 81 \)
$17$
\( T^{12} + 35 T^{10} + 1027 T^{8} + \cdots + 6561 \)
$19$
\( (T^{6} - 6 T^{5} + 30 T^{4} - 115 T^{3} + \cdots + 6859)^{2} \)
$23$
\( T^{12} + 12 T^{10} + 137 T^{8} + 82 T^{6} + \cdots + 1 \)
$29$
\( (T^{6} - 6 T^{5} + 63 T^{4} + 108 T^{3} + \cdots + 729)^{2} \)
$31$
\( (T^{3} - 15 T^{2} + 70 T - 97)^{4} \)
$37$
\( (T^{6} - 98 T^{4} + 887 T^{2} - 729)^{2} \)
$41$
\( (T^{6} + 6 T^{5} + 77 T^{4} - 240 T^{3} + \cdots + 9)^{2} \)
$43$
\( T^{12} + 127 T^{10} + 13613 T^{8} + \cdots + 194481 \)
$47$
\( T^{12} + 214 T^{10} + \cdots + 47562811921 \)
$53$
\( T^{12} + 99 T^{10} + \cdots + 131079601 \)
$59$
\( (T^{6} + 10 T^{5} + 155 T^{4} + \cdots + 84681)^{2} \)
$61$
\( (T^{6} - T^{5} + 42 T^{4} - 185 T^{3} + \cdots + 12769)^{2} \)
$67$
\( T^{12} + 76 T^{10} + \cdots + 207360000 \)
$71$
\( (T^{6} - T^{5} + 125 T^{4} + 1078 T^{3} + \cdots + 227529)^{2} \)
$73$
\( T^{12} + 236 T^{10} + \cdots + 9845600625 \)
$79$
\( (T^{6} + 12 T^{5} + 176 T^{4} + \cdots + 200704)^{2} \)
$83$
\( (T^{6} - 459 T^{4} + 56302 T^{2} + \cdots - 966289)^{2} \)
$89$
\( (T^{6} + 18 T^{5} + 296 T^{4} + 648 T^{3} + \cdots + 5184)^{2} \)
$97$
\( T^{12} + 529 T^{10} + \cdots + 1534548635361 \)
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