Newspace parameters
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(186.493452228\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.98344960000.9 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 127x^{4} + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{34}\cdot 3^{12} \) |
Twist minimal: | no (minimal twist has level 6) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 127x^{4} + 6561 \) :
\(\beta_{1}\) | \(=\) | \( ( -2\nu^{6} - 416\nu^{2} ) / 1377 \) |
\(\beta_{2}\) | \(=\) | \( ( 512\nu^{7} + 5184\nu^{5} + 18368\nu^{3} + 285120\nu ) / 12393 \) |
\(\beta_{3}\) | \(=\) | \( ( -512\nu^{7} + 5184\nu^{5} - 18368\nu^{3} + 285120\nu ) / 12393 \) |
\(\beta_{4}\) | \(=\) | \( ( -7616\nu^{7} + 24624\nu^{5} + 209952\nu^{4} - 1165520\nu^{3} + 8492688\nu + 13331952 ) / 12393 \) |
\(\beta_{5}\) | \(=\) | \( ( -64\nu^{7} + 648\nu^{5} - 314928\nu^{4} - 2296\nu^{3} + 35640\nu - 19997928 ) / 4131 \) |
\(\beta_{6}\) | \(=\) | \( ( -7616\nu^{7} - 22032\nu^{6} - 24624\nu^{5} - 1165520\nu^{3} - 1013472\nu^{2} - 8492688\nu ) / 12393 \) |
\(\beta_{7}\) | \(=\) | \( ( 3904\nu^{7} - 38565\nu^{6} + 13284\nu^{5} + 586204\nu^{3} - 1775448\nu^{2} + 4299804\nu ) / 12393 \) |
\(\nu\) | \(=\) | \( ( 8\beta_{7} - 14\beta_{6} + 4\beta_{5} + 18\beta_{4} - 87\beta_{3} - 87\beta_{2} - 4\beta_1 ) / 20736 \) |
\(\nu^{2}\) | \(=\) | \( ( 16\beta_{7} + 8\beta_{6} - 3\beta_{2} - 44072\beta_1 ) / 10368 \) |
\(\nu^{3}\) | \(=\) | \( ( 32\beta_{7} - 56\beta_{6} - 16\beta_{5} - 72\beta_{4} + 1077\beta_{3} - 1077\beta_{2} - 16\beta_1 ) / 10368 \) |
\(\nu^{4}\) | \(=\) | \( ( -136\beta_{5} + 51\beta_{3} - 658368 ) / 10368 \) |
\(\nu^{5}\) | \(=\) | \( ( -440\beta_{7} + 770\beta_{6} - 220\beta_{5} - 990\beta_{4} + 29571\beta_{3} + 29571\beta_{2} + 220\beta_1 ) / 20736 \) |
\(\nu^{6}\) | \(=\) | \( ( -208\beta_{7} - 104\beta_{6} + 39\beta_{2} + 126788\beta_1 ) / 648 \) |
\(\nu^{7}\) | \(=\) | \( ( - 2296 \beta_{7} + 4018 \beta_{6} + 1148 \beta_{5} + 5166 \beta_{4} - 328233 \beta_{3} + 328233 \beta_{2} + 1148 \beta_1 ) / 20736 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 |
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−90.5097 | −2152.70 | − | 385.790i | 8192.00 | 0 | 194841. | + | 34917.8i | − | 1.21591e6i | −741455. | 4.48530e6 | + | 1.66099e6i | 0 | |||||||||||||||||||||||||||||||||||
149.2 | −90.5097 | −2152.70 | + | 385.790i | 8192.00 | 0 | 194841. | − | 34917.8i | 1.21591e6i | −741455. | 4.48530e6 | − | 1.66099e6i | 0 | |||||||||||||||||||||||||||||||||||||
149.3 | −90.5097 | 829.000 | − | 2023.79i | 8192.00 | 0 | −75032.5 | + | 183173.i | − | 388872.i | −741455. | −3.40849e6 | − | 3.35545e6i | 0 | ||||||||||||||||||||||||||||||||||||
149.4 | −90.5097 | 829.000 | + | 2023.79i | 8192.00 | 0 | −75032.5 | − | 183173.i | 388872.i | −741455. | −3.40849e6 | + | 3.35545e6i | 0 | |||||||||||||||||||||||||||||||||||||
149.5 | 90.5097 | −829.000 | − | 2023.79i | 8192.00 | 0 | −75032.5 | − | 183173.i | − | 388872.i | 741455. | −3.40849e6 | + | 3.35545e6i | 0 | ||||||||||||||||||||||||||||||||||||
149.6 | 90.5097 | −829.000 | + | 2023.79i | 8192.00 | 0 | −75032.5 | + | 183173.i | 388872.i | 741455. | −3.40849e6 | − | 3.35545e6i | 0 | |||||||||||||||||||||||||||||||||||||
149.7 | 90.5097 | 2152.70 | − | 385.790i | 8192.00 | 0 | 194841. | − | 34917.8i | − | 1.21591e6i | 741455. | 4.48530e6 | − | 1.66099e6i | 0 | ||||||||||||||||||||||||||||||||||||
149.8 | 90.5097 | 2152.70 | + | 385.790i | 8192.00 | 0 | 194841. | + | 34917.8i | 1.21591e6i | 741455. | 4.48530e6 | + | 1.66099e6i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.15.b.a | 8 | |
3.b | odd | 2 | 1 | inner | 150.15.b.a | 8 | |
5.b | even | 2 | 1 | inner | 150.15.b.a | 8 | |
5.c | odd | 4 | 1 | 6.15.b.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 150.15.d.a | 4 | ||
15.d | odd | 2 | 1 | inner | 150.15.b.a | 8 | |
15.e | even | 4 | 1 | 6.15.b.a | ✓ | 4 | |
15.e | even | 4 | 1 | 150.15.d.a | 4 | ||
20.e | even | 4 | 1 | 48.15.e.d | 4 | ||
60.l | odd | 4 | 1 | 48.15.e.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6.15.b.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
6.15.b.a | ✓ | 4 | 15.e | even | 4 | 1 | |
48.15.e.d | 4 | 20.e | even | 4 | 1 | ||
48.15.e.d | 4 | 60.l | odd | 4 | 1 | ||
150.15.b.a | 8 | 1.a | even | 1 | 1 | trivial | |
150.15.b.a | 8 | 3.b | odd | 2 | 1 | inner | |
150.15.b.a | 8 | 5.b | even | 2 | 1 | inner | |
150.15.b.a | 8 | 15.d | odd | 2 | 1 | inner | |
150.15.d.a | 4 | 5.c | odd | 4 | 1 | ||
150.15.d.a | 4 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 1629655082888T_{7}^{2} + 223571300080310387289616 \)
acting on \(S_{15}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 8192)^{4} \)
$3$
\( T^{8} - 2153628 T^{6} + \cdots + 52\!\cdots\!21 \)
$5$
\( T^{8} \)
$7$
\( (T^{4} + 1629655082888 T^{2} + \cdots + 22\!\cdots\!16)^{2} \)
$11$
\( (T^{4} + \cdots + 30\!\cdots\!44)^{2} \)
$13$
\( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \)
$17$
\( (T^{4} + \cdots + 15\!\cdots\!44)^{2} \)
$19$
\( (T^{2} - 626033804 T - 11\!\cdots\!76)^{4} \)
$23$
\( (T^{4} + \cdots + 47\!\cdots\!64)^{2} \)
$29$
\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \)
$31$
\( (T^{2} - 24272242756 T - 22\!\cdots\!16)^{4} \)
$37$
\( (T^{4} + \cdots + 14\!\cdots\!56)^{2} \)
$41$
\( (T^{4} + \cdots + 11\!\cdots\!64)^{2} \)
$43$
\( (T^{4} + \cdots + 19\!\cdots\!36)^{2} \)
$47$
\( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \)
$53$
\( (T^{4} + \cdots + 88\!\cdots\!24)^{2} \)
$59$
\( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \)
$61$
\( (T^{2} + 4004253466892 T - 99\!\cdots\!04)^{4} \)
$67$
\( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \)
$71$
\( (T^{4} + \cdots + 40\!\cdots\!00)^{2} \)
$73$
\( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \)
$79$
\( (T^{2} - 40146026361596 T + 39\!\cdots\!24)^{4} \)
$83$
\( (T^{4} + \cdots + 20\!\cdots\!04)^{2} \)
$89$
\( (T^{4} + \cdots + 19\!\cdots\!64)^{2} \)
$97$
\( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \)
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