Newspace parameters
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.g (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.19775603032\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
Coefficient field: | 8.0.1064390625.3 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 27571 \nu^{7} + 156 \nu^{6} - 50800 \nu^{5} + 197116 \nu^{4} - 261151 \nu^{3} + 1281772 \nu^{2} - 1105295 \nu + 4276691 ) / 4238558 \) |
\(\beta_{3}\) | \(=\) | \( ( - 45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + 1275844 \nu^{2} - 687257 \nu + 5880747 ) / 4238558 \) |
\(\beta_{4}\) | \(=\) | \( ( 2894\nu^{7} + 7027\nu^{6} - 3119\nu^{5} - 38495\nu^{4} - 16673\nu^{3} + 24798\nu^{2} + 23050\nu - 523849 ) / 223082 \) |
\(\beta_{5}\) | \(=\) | \( ( 60697 \nu^{7} + 29865 \nu^{6} - 107003 \nu^{5} - 736723 \nu^{4} + 989612 \nu^{3} + 38090 \nu^{2} + 939005 \nu - 13281190 ) / 4238558 \) |
\(\beta_{6}\) | \(=\) | \( ( - 72436 \nu^{7} - 26109 \nu^{6} + 188067 \nu^{5} + 265983 \nu^{4} - 1082509 \nu^{3} + 501044 \nu^{2} + 3422970 \nu + 7026371 ) / 4238558 \) |
\(\beta_{7}\) | \(=\) | \( ( 5808 \nu^{7} + 2617 \nu^{6} - 8495 \nu^{5} - 68083 \nu^{4} + 17029 \nu^{3} - 19146 \nu^{2} + 101760 \nu - 868243 ) / 223082 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} - \beta_{6} + 4\beta_{3} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( -3\beta_{7} + 5\beta_{5} - \beta_{2} + 5 \) |
\(\nu^{4}\) | \(=\) | \( -8\beta_{6} - 7\beta_{5} + \beta_{4} + 7\beta_{3} - 4\beta_{2} + 8\beta _1 - 12 \) |
\(\nu^{5}\) | \(=\) | \( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 4\beta_{4} + 39\beta_{3} - 39\beta_{2} - 4\beta _1 + 12 \) |
\(\nu^{6}\) | \(=\) | \( -39\beta_{7} + 16\beta_{5} + 39\beta_{4} - 26\beta_{3} + 6\beta _1 + 26 \) |
\(\nu^{7}\) | \(=\) | \( 71\beta_{7} - 100\beta_{6} - 101\beta_{5} + 164\beta_{3} - 101\beta_{2} + 71\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(1\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
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−0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | −2.12459 | − | 0.697217i | 0.809017 | + | 0.587785i | −4.25729 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | −0.00655751 | + | 2.23606i | ||||||||||||||||||||||||||
31.2 | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | 0.00655751 | + | 2.23606i | 0.809017 | + | 0.587785i | 2.63925 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 2.12459 | − | 0.697217i | |||||||||||||||||||||||||||
61.1 | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | −2.05984 | − | 0.870094i | −0.309017 | − | 0.951057i | 2.92807 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −2.17787 | + | 0.506822i | |||||||||||||||||||||||||||
61.2 | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | 2.17787 | + | 0.506822i | −0.309017 | − | 0.951057i | −2.31003 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 2.05984 | − | 0.870094i | |||||||||||||||||||||||||||
91.1 | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | −2.05984 | + | 0.870094i | −0.309017 | + | 0.951057i | 2.92807 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −2.17787 | − | 0.506822i | |||||||||||||||||||||||||||
91.2 | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | 2.17787 | − | 0.506822i | −0.309017 | + | 0.951057i | −2.31003 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 2.05984 | + | 0.870094i | |||||||||||||||||||||||||||
121.1 | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | −2.12459 | + | 0.697217i | 0.809017 | − | 0.587785i | −4.25729 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | −0.00655751 | − | 2.23606i | |||||||||||||||||||||||||||
121.2 | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | 0.00655751 | − | 2.23606i | 0.809017 | − | 0.587785i | 2.63925 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 2.12459 | + | 0.697217i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.2.g.c | ✓ | 8 |
3.b | odd | 2 | 1 | 450.2.h.d | 8 | ||
5.b | even | 2 | 1 | 750.2.g.d | 8 | ||
5.c | odd | 4 | 2 | 750.2.h.e | 16 | ||
25.d | even | 5 | 1 | inner | 150.2.g.c | ✓ | 8 |
25.d | even | 5 | 1 | 3750.2.a.l | 4 | ||
25.e | even | 10 | 1 | 750.2.g.d | 8 | ||
25.e | even | 10 | 1 | 3750.2.a.q | 4 | ||
25.f | odd | 20 | 2 | 750.2.h.e | 16 | ||
25.f | odd | 20 | 2 | 3750.2.c.h | 8 | ||
75.j | odd | 10 | 1 | 450.2.h.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.2.g.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
150.2.g.c | ✓ | 8 | 25.d | even | 5 | 1 | inner |
450.2.h.d | 8 | 3.b | odd | 2 | 1 | ||
450.2.h.d | 8 | 75.j | odd | 10 | 1 | ||
750.2.g.d | 8 | 5.b | even | 2 | 1 | ||
750.2.g.d | 8 | 25.e | even | 10 | 1 | ||
750.2.h.e | 16 | 5.c | odd | 4 | 2 | ||
750.2.h.e | 16 | 25.f | odd | 20 | 2 | ||
3750.2.a.l | 4 | 25.d | even | 5 | 1 | ||
3750.2.a.q | 4 | 25.e | even | 10 | 1 | ||
3750.2.c.h | 8 | 25.f | odd | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + T_{7}^{3} - 19T_{7}^{2} - 4T_{7} + 76 \)
acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$3$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$5$
\( T^{8} + 4 T^{7} + T^{6} - 16 T^{5} + \cdots + 625 \)
$7$
\( (T^{4} + T^{3} - 19 T^{2} - 4 T + 76)^{2} \)
$11$
\( T^{8} + 5 T^{7} + 60 T^{6} + 320 T^{5} + \cdots + 400 \)
$13$
\( T^{8} - 6 T^{7} + 32 T^{6} - 143 T^{5} + \cdots + 256 \)
$17$
\( T^{8} + 2 T^{7} + 78 T^{6} + \cdots + 5776 \)
$19$
\( (T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16)^{2} \)
$23$
\( (T^{4} + 10 T^{3} + 60 T^{2} + 200 T + 400)^{2} \)
$29$
\( T^{8} + 18 T^{7} + 218 T^{6} + \cdots + 1860496 \)
$31$
\( T^{8} - 9 T^{7} + 42 T^{6} + \cdots + 1296 \)
$37$
\( T^{8} - 21 T^{7} + 332 T^{6} + \cdots + 11182336 \)
$41$
\( T^{8} - 2 T^{7} + 118 T^{6} + \cdots + 1936 \)
$43$
\( (T^{4} + 16 T^{3} - 4 T^{2} - 784 T - 1424)^{2} \)
$47$
\( T^{8} + 10 T^{7} + 160 T^{6} + \cdots + 774400 \)
$53$
\( T^{8} - 7 T^{7} + 168 T^{6} + \cdots + 85359121 \)
$59$
\( T^{8} + 25 T^{7} + 330 T^{6} + \cdots + 2310400 \)
$61$
\( T^{8} - 10 T^{7} + 60 T^{6} + \cdots + 400 \)
$67$
\( T^{8} + 2 T^{7} + 168 T^{6} + \cdots + 6885376 \)
$71$
\( T^{8} - 40 T^{6} - 40 T^{5} + \cdots + 102400 \)
$73$
\( T^{8} + 24 T^{7} + 372 T^{6} + \cdots + 1296 \)
$79$
\( T^{8} + 6 T^{7} + 192 T^{6} + \cdots + 331776 \)
$83$
\( T^{8} - 11 T^{7} + 162 T^{6} + \cdots + 1008016 \)
$89$
\( T^{8} - 9 T^{7} + 12 T^{6} + \cdots + 1296 \)
$97$
\( T^{8} - T^{7} + 222 T^{6} + \cdots + 130321 \)
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