Newspace parameters
Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 95.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.758578819202\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} + 35x^{8} + 223x^{4} + 289 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{12} + 35x^{8} + 223x^{4} + 289 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{8} + 58\nu^{4} + 417 ) / 190 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{9} + 58\nu^{5} + 607\nu ) / 190 \) |
\(\beta_{4}\) | \(=\) | \( ( -7\nu^{8} - 216\nu^{4} - 639 ) / 190 \) |
\(\beta_{5}\) | \(=\) | \( ( -7\nu^{9} - 216\nu^{5} - 829\nu ) / 190 \) |
\(\beta_{6}\) | \(=\) | \( ( 9\nu^{10} + 332\nu^{6} + 2993\nu^{2} ) / 3230 \) |
\(\beta_{7}\) | \(=\) | \( ( 9\nu^{11} + 332\nu^{7} + 2993\nu^{3} ) / 3230 \) |
\(\beta_{8}\) | \(=\) | \( ( -27\nu^{10} - 996\nu^{6} - 5749\nu^{2} ) / 3230 \) |
\(\beta_{9}\) | \(=\) | \( ( -18\nu^{11} - 664\nu^{7} - 4371\nu^{3} ) / 1615 \) |
\(\beta_{10}\) | \(=\) | \( ( -32\nu^{10} - 1001\nu^{6} - 3464\nu^{2} ) / 1615 \) |
\(\beta_{11}\) | \(=\) | \( ( -73\nu^{11} - 2334\nu^{7} - 9921\nu^{3} ) / 3230 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{8} + 3\beta_{6} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{9} + 4\beta_{7} \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} + 7\beta_{2} - 12 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{5} + 7\beta_{3} - 18\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 9\beta_{10} - 40\beta_{8} - 56\beta_{6} \) |
\(\nu^{7}\) | \(=\) | \( 9\beta_{11} - 40\beta_{9} - 87\beta_{7} \) |
\(\nu^{8}\) | \(=\) | \( -58\beta_{4} - 216\beta_{2} + 279 \) |
\(\nu^{9}\) | \(=\) | \( -58\beta_{5} - 216\beta_{3} + 437\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( -332\beta_{10} + 1143\beta_{8} + 1427\beta_{6} \) |
\(\nu^{11}\) | \(=\) | \( -332\beta_{11} + 1143\beta_{9} + 2238\beta_{7} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(77\) |
\(\chi(n)\) | \(-1\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
18.1 |
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−1.61467 | − | 1.61467i | 1.11233 | − | 1.11233i | 3.21432i | −0.311108 | − | 2.21432i | −3.59210 | 1.21432 | − | 1.21432i | 1.96073 | − | 1.96073i | 0.525428i | −3.07306 | + | 4.07773i | ||||||||||||||||||||||||||||||||||||||||||
18.2 | −1.10924 | − | 1.10924i | −1.29790 | + | 1.29790i | 0.460811i | −2.17009 | + | 0.539189i | 2.87936 | −1.53919 | + | 1.53919i | −1.70732 | + | 1.70732i | − | 0.369102i | 3.00523 | + | 1.80905i | ||||||||||||||||||||||||||||||||||||||||||
18.3 | −0.813901 | − | 0.813901i | 2.01945 | − | 2.01945i | − | 0.675131i | 1.48119 | + | 1.67513i | −3.28726 | −2.67513 | + | 2.67513i | −2.17729 | + | 2.17729i | − | 5.15633i | 0.157845 | − | 2.56894i | |||||||||||||||||||||||||||||||||||||||||
18.4 | 0.813901 | + | 0.813901i | −2.01945 | + | 2.01945i | − | 0.675131i | 1.48119 | + | 1.67513i | −3.28726 | −2.67513 | + | 2.67513i | 2.17729 | − | 2.17729i | − | 5.15633i | −0.157845 | + | 2.56894i | |||||||||||||||||||||||||||||||||||||||||
18.5 | 1.10924 | + | 1.10924i | 1.29790 | − | 1.29790i | 0.460811i | −2.17009 | + | 0.539189i | 2.87936 | −1.53919 | + | 1.53919i | 1.70732 | − | 1.70732i | − | 0.369102i | −3.00523 | − | 1.80905i | ||||||||||||||||||||||||||||||||||||||||||
18.6 | 1.61467 | + | 1.61467i | −1.11233 | + | 1.11233i | 3.21432i | −0.311108 | − | 2.21432i | −3.59210 | 1.21432 | − | 1.21432i | −1.96073 | + | 1.96073i | 0.525428i | 3.07306 | − | 4.07773i | |||||||||||||||||||||||||||||||||||||||||||
37.1 | −1.61467 | + | 1.61467i | 1.11233 | + | 1.11233i | − | 3.21432i | −0.311108 | + | 2.21432i | −3.59210 | 1.21432 | + | 1.21432i | 1.96073 | + | 1.96073i | − | 0.525428i | −3.07306 | − | 4.07773i | |||||||||||||||||||||||||||||||||||||||||
37.2 | −1.10924 | + | 1.10924i | −1.29790 | − | 1.29790i | − | 0.460811i | −2.17009 | − | 0.539189i | 2.87936 | −1.53919 | − | 1.53919i | −1.70732 | − | 1.70732i | 0.369102i | 3.00523 | − | 1.80905i | ||||||||||||||||||||||||||||||||||||||||||
37.3 | −0.813901 | + | 0.813901i | 2.01945 | + | 2.01945i | 0.675131i | 1.48119 | − | 1.67513i | −3.28726 | −2.67513 | − | 2.67513i | −2.17729 | − | 2.17729i | 5.15633i | 0.157845 | + | 2.56894i | |||||||||||||||||||||||||||||||||||||||||||
37.4 | 0.813901 | − | 0.813901i | −2.01945 | − | 2.01945i | 0.675131i | 1.48119 | − | 1.67513i | −3.28726 | −2.67513 | − | 2.67513i | 2.17729 | + | 2.17729i | 5.15633i | −0.157845 | − | 2.56894i | |||||||||||||||||||||||||||||||||||||||||||
37.5 | 1.10924 | − | 1.10924i | 1.29790 | + | 1.29790i | − | 0.460811i | −2.17009 | − | 0.539189i | 2.87936 | −1.53919 | − | 1.53919i | 1.70732 | + | 1.70732i | 0.369102i | −3.00523 | + | 1.80905i | ||||||||||||||||||||||||||||||||||||||||||
37.6 | 1.61467 | − | 1.61467i | −1.11233 | − | 1.11233i | − | 3.21432i | −0.311108 | + | 2.21432i | −3.59210 | 1.21432 | + | 1.21432i | −1.96073 | − | 1.96073i | − | 0.525428i | 3.07306 | + | 4.07773i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.b | odd | 2 | 1 | inner |
95.g | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 95.2.g.b | ✓ | 12 |
3.b | odd | 2 | 1 | 855.2.p.f | 12 | ||
5.b | even | 2 | 1 | 475.2.g.b | 12 | ||
5.c | odd | 4 | 1 | inner | 95.2.g.b | ✓ | 12 |
5.c | odd | 4 | 1 | 475.2.g.b | 12 | ||
15.e | even | 4 | 1 | 855.2.p.f | 12 | ||
19.b | odd | 2 | 1 | inner | 95.2.g.b | ✓ | 12 |
57.d | even | 2 | 1 | 855.2.p.f | 12 | ||
95.d | odd | 2 | 1 | 475.2.g.b | 12 | ||
95.g | even | 4 | 1 | inner | 95.2.g.b | ✓ | 12 |
95.g | even | 4 | 1 | 475.2.g.b | 12 | ||
285.j | odd | 4 | 1 | 855.2.p.f | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
95.2.g.b | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
95.2.g.b | ✓ | 12 | 19.b | odd | 2 | 1 | inner |
95.2.g.b | ✓ | 12 | 95.g | even | 4 | 1 | inner |
475.2.g.b | 12 | 5.b | even | 2 | 1 | ||
475.2.g.b | 12 | 5.c | odd | 4 | 1 | ||
475.2.g.b | 12 | 95.d | odd | 2 | 1 | ||
475.2.g.b | 12 | 95.g | even | 4 | 1 | ||
855.2.p.f | 12 | 3.b | odd | 2 | 1 | ||
855.2.p.f | 12 | 15.e | even | 4 | 1 | ||
855.2.p.f | 12 | 57.d | even | 2 | 1 | ||
855.2.p.f | 12 | 285.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 35T_{2}^{8} + 223T_{2}^{4} + 289 \)
acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 35 T^{8} + 223 T^{4} + \cdots + 289 \)
$3$
\( T^{12} + 84 T^{8} + 1232 T^{4} + \cdots + 4624 \)
$5$
\( (T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125)^{2} \)
$7$
\( (T^{6} + 6 T^{5} + 18 T^{4} + 8 T^{3} + \cdots + 200)^{2} \)
$11$
\( (T^{3} + 4 T^{2} - 4 T - 20)^{4} \)
$13$
\( T^{12} + 84 T^{8} + 1232 T^{4} + \cdots + 4624 \)
$17$
\( (T^{6} - 10 T^{5} + 50 T^{4} - 80 T^{3} + \cdots + 200)^{2} \)
$19$
\( T^{12} - 70 T^{10} + 2503 T^{8} + \cdots + 47045881 \)
$23$
\( (T^{6} + 6 T^{5} + 18 T^{4} + 8 T^{3} + \cdots + 200)^{2} \)
$29$
\( (T^{6} - 132 T^{4} + 4448 T^{2} + \cdots - 13600)^{2} \)
$31$
\( (T^{6} + 148 T^{4} + 3728 T^{2} + \cdots + 13600)^{2} \)
$37$
\( T^{12} + 692 T^{8} + 49200 T^{4} + \cdots + 4624 \)
$41$
\( (T^{6} + 176 T^{4} + 6432 T^{2} + \cdots + 54400)^{2} \)
$43$
\( (T^{6} - 2 T^{5} + 2 T^{4} + 536 T^{3} + \cdots + 57800)^{2} \)
$47$
\( (T^{6} + 22 T^{5} + 242 T^{4} + \cdots + 33800)^{2} \)
$53$
\( T^{12} + 7124 T^{8} + 8604592 T^{4} + \cdots + 4624 \)
$59$
\( (T^{6} - 176 T^{4} + 6432 T^{2} + \cdots - 54400)^{2} \)
$61$
\( (T^{3} + 6 T^{2} - 88 T - 460)^{4} \)
$67$
\( T^{12} + 21428 T^{8} + \cdots + 3270467344 \)
$71$
\( (T^{6} + 140 T^{4} + 2832 T^{2} + \cdots + 13600)^{2} \)
$73$
\( (T^{6} - 14 T^{5} + 98 T^{4} + \cdots + 897800)^{2} \)
$79$
\( (T^{6} - 200 T^{4} + 6688 T^{2} + \cdots - 54400)^{2} \)
$83$
\( (T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 33800)^{2} \)
$89$
\( (T^{6} - 348 T^{4} + 20768 T^{2} + \cdots - 340000)^{2} \)
$97$
\( T^{12} + 31956 T^{8} + \cdots + 1806250000 \)
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