Properties

Label 3150.2.d.b
Level 3150
Weight 2
Character orbit 3150.d
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{8} + \beta_{2} q^{11} + ( \beta_{1} + \beta_{6} ) q^{13} + ( -\beta_{1} - \beta_{2} ) q^{14} + q^{16} + \beta_{3} q^{17} + ( -2 \beta_{3} - \beta_{5} ) q^{19} -\beta_{2} q^{22} - q^{23} + ( -\beta_{1} - \beta_{6} ) q^{26} + ( \beta_{1} + \beta_{2} ) q^{28} + ( 3 \beta_{2} + \beta_{4} ) q^{29} + ( \beta_{3} - 2 \beta_{5} ) q^{31} - q^{32} -\beta_{3} q^{34} -2 \beta_{2} q^{37} + ( 2 \beta_{3} + \beta_{5} ) q^{38} + ( -\beta_{1} + 2 \beta_{6} ) q^{41} + ( -\beta_{2} - \beta_{4} ) q^{43} + \beta_{2} q^{44} + q^{46} + ( -2 \beta_{3} + \beta_{5} ) q^{47} + ( 3 - 2 \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{6} ) q^{52} + ( -1 - \beta_{7} ) q^{53} + ( -\beta_{1} - \beta_{2} ) q^{56} + ( -3 \beta_{2} - \beta_{4} ) q^{58} + ( -\beta_{1} - 3 \beta_{6} ) q^{59} + ( \beta_{3} - \beta_{5} ) q^{61} + ( -\beta_{3} + 2 \beta_{5} ) q^{62} + q^{64} + \beta_{3} q^{68} + ( 4 \beta_{2} - 2 \beta_{4} ) q^{71} + ( -2 \beta_{1} + 3 \beta_{6} ) q^{73} + 2 \beta_{2} q^{74} + ( -2 \beta_{3} - \beta_{5} ) q^{76} + ( -2 - \beta_{5} ) q^{77} + ( 6 + \beta_{7} ) q^{79} + ( \beta_{1} - 2 \beta_{6} ) q^{82} + ( 5 \beta_{3} + \beta_{5} ) q^{83} + ( \beta_{2} + \beta_{4} ) q^{86} -\beta_{2} q^{88} + ( -4 \beta_{1} - 2 \beta_{6} ) q^{89} + ( 5 + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{91} - q^{92} + ( 2 \beta_{3} - \beta_{5} ) q^{94} + ( -4 \beta_{1} - 3 \beta_{6} ) q^{97} + ( -3 + 2 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + 8q^{16} - 8q^{23} - 8q^{32} + 8q^{46} + 24q^{49} - 8q^{53} + 8q^{64} - 16q^{77} + 48q^{79} + 40q^{91} - 8q^{92} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( -\nu^{6} - 6 \nu^{2} \)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 40 \nu^{2} \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 10 \nu^{7} - 5 \nu^{5} + 65 \nu^{3} - 25 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 5 \beta_{6} + 5 \beta_{5} + 5 \beta_{2}\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{4} + 5 \beta_{3}\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} + 10 \beta_{2}\)\()/10\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{7} - 25 \beta_{6} - 25 \beta_{5} - 55 \beta_{2}\)\()/20\)
\(\nu^{6}\)\(=\)\((\)\(9 \beta_{4} - 20 \beta_{3}\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{7} + 65 \beta_{6} - 65 \beta_{5} - 145 \beta_{2}\)\()/20\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3149.1
−1.14412 1.14412i
1.14412 1.14412i
−1.14412 + 1.14412i
1.14412 + 1.14412i
−0.437016 + 0.437016i
0.437016 + 0.437016i
−0.437016 0.437016i
0.437016 0.437016i
−1.00000 0 1.00000 0 0 −2.23607 1.41421i −1.00000 0 0
3149.2 −1.00000 0 1.00000 0 0 −2.23607 1.41421i −1.00000 0 0
3149.3 −1.00000 0 1.00000 0 0 −2.23607 + 1.41421i −1.00000 0 0
3149.4 −1.00000 0 1.00000 0 0 −2.23607 + 1.41421i −1.00000 0 0
3149.5 −1.00000 0 1.00000 0 0 2.23607 1.41421i −1.00000 0 0
3149.6 −1.00000 0 1.00000 0 0 2.23607 1.41421i −1.00000 0 0
3149.7 −1.00000 0 1.00000 0 0 2.23607 + 1.41421i −1.00000 0 0
3149.8 −1.00000 0 1.00000 0 0 2.23607 + 1.41421i −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.d.b 8
3.b odd 2 1 3150.2.d.e 8
5.b even 2 1 3150.2.d.e 8
5.c odd 4 1 3150.2.b.a 8
5.c odd 4 1 3150.2.b.d yes 8
7.b odd 2 1 inner 3150.2.d.b 8
15.d odd 2 1 inner 3150.2.d.b 8
15.e even 4 1 3150.2.b.a 8
15.e even 4 1 3150.2.b.d yes 8
21.c even 2 1 3150.2.d.e 8
35.c odd 2 1 3150.2.d.e 8
35.f even 4 1 3150.2.b.a 8
35.f even 4 1 3150.2.b.d yes 8
105.g even 2 1 inner 3150.2.d.b 8
105.k odd 4 1 3150.2.b.a 8
105.k odd 4 1 3150.2.b.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.b.a 8 5.c odd 4 1
3150.2.b.a 8 15.e even 4 1
3150.2.b.a 8 35.f even 4 1
3150.2.b.a 8 105.k odd 4 1
3150.2.b.d yes 8 5.c odd 4 1
3150.2.b.d yes 8 15.e even 4 1
3150.2.b.d yes 8 35.f even 4 1
3150.2.b.d yes 8 105.k odd 4 1
3150.2.d.b 8 1.a even 1 1 trivial
3150.2.d.b 8 7.b odd 2 1 inner
3150.2.d.b 8 15.d odd 2 1 inner
3150.2.d.b 8 105.g even 2 1 inner
3150.2.d.e 8 3.b odd 2 1
3150.2.d.e 8 5.b even 2 1
3150.2.d.e 8 21.c even 2 1
3150.2.d.e 8 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{2} + 2 \)
\( T_{13}^{4} - 30 T_{13}^{2} + 25 \)
\( T_{23} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 22 T^{2} + 259 T^{4} + 3718 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 29 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 16 T^{2} - 14 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + T + 23 T^{2} )^{8} \)
$29$ \( ( 1 - 30 T^{2} + 107 T^{4} - 25230 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 34 T^{2} + 1411 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 66 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 74 T^{2} + 3931 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 118 T^{2} + 6979 T^{4} - 218182 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 128 T^{2} + 7714 T^{4} - 282752 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 2 T + 57 T^{2} + 106 T^{3} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 46 T^{2} + 5691 T^{4} + 160126 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 214 T^{2} + 18691 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{8} \)
$71$ \( ( 1 - 20 T^{2} - 2618 T^{4} - 100820 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 72 T^{2} + 4754 T^{4} + 383688 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 12 T + 144 T^{2} - 948 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 62 T^{2} + 9739 T^{4} - 427118 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 116 T^{2} + 6406 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 48 T^{2} - 9406 T^{4} + 451632 T^{6} + 88529281 T^{8} )^{2} \)
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