Properties

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

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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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 630)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 8 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 8 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\( \nu^{6} + 6 \nu^{2} \)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + 3 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{3} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-8 \beta_{7} - 5 \beta_{5} + 8 \beta_{4} - 5 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{6} - 9 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(8 \beta_{7} + 13 \beta_{5} + 8 \beta_{4} - 13 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.14412 + 1.14412i
0.437016 0.437016i
−0.437016 + 0.437016i
1.14412 1.14412i
0.437016 + 0.437016i
−1.14412 1.14412i
1.14412 + 1.14412i
−0.437016 0.437016i
1.00000i 0 −1.00000 0 0 −2.12132 1.58114i 1.00000i 0 0
251.2 1.00000i 0 −1.00000 0 0 −2.12132 + 1.58114i 1.00000i 0 0
251.3 1.00000i 0 −1.00000 0 0 2.12132 1.58114i 1.00000i 0 0
251.4 1.00000i 0 −1.00000 0 0 2.12132 + 1.58114i 1.00000i 0 0
251.5 1.00000i 0 −1.00000 0 0 −2.12132 1.58114i 1.00000i 0 0
251.6 1.00000i 0 −1.00000 0 0 −2.12132 + 1.58114i 1.00000i 0 0
251.7 1.00000i 0 −1.00000 0 0 2.12132 1.58114i 1.00000i 0 0
251.8 1.00000i 0 −1.00000 0 0 2.12132 + 1.58114i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

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
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c 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.b.c 8
3.b odd 2 1 inner 3150.2.b.c 8
5.b even 2 1 inner 3150.2.b.c 8
5.c odd 4 1 630.2.d.a 4
5.c odd 4 1 630.2.d.d yes 4
7.b odd 2 1 inner 3150.2.b.c 8
15.d odd 2 1 inner 3150.2.b.c 8
15.e even 4 1 630.2.d.a 4
15.e even 4 1 630.2.d.d yes 4
20.e even 4 1 5040.2.k.a 4
20.e even 4 1 5040.2.k.d 4
21.c even 2 1 inner 3150.2.b.c 8
35.c odd 2 1 inner 3150.2.b.c 8
35.f even 4 1 630.2.d.a 4
35.f even 4 1 630.2.d.d yes 4
60.l odd 4 1 5040.2.k.a 4
60.l odd 4 1 5040.2.k.d 4
105.g even 2 1 inner 3150.2.b.c 8
105.k odd 4 1 630.2.d.a 4
105.k odd 4 1 630.2.d.d yes 4
140.j odd 4 1 5040.2.k.a 4
140.j odd 4 1 5040.2.k.d 4
420.w even 4 1 5040.2.k.a 4
420.w even 4 1 5040.2.k.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.a 4 5.c odd 4 1
630.2.d.a 4 15.e even 4 1
630.2.d.a 4 35.f even 4 1
630.2.d.a 4 105.k odd 4 1
630.2.d.d yes 4 5.c odd 4 1
630.2.d.d yes 4 15.e even 4 1
630.2.d.d yes 4 35.f even 4 1
630.2.d.d yes 4 105.k odd 4 1
3150.2.b.c 8 1.a even 1 1 trivial
3150.2.b.c 8 3.b odd 2 1 inner
3150.2.b.c 8 5.b even 2 1 inner
3150.2.b.c 8 7.b odd 2 1 inner
3150.2.b.c 8 15.d odd 2 1 inner
3150.2.b.c 8 21.c even 2 1 inner
3150.2.b.c 8 35.c odd 2 1 inner
3150.2.b.c 8 105.g even 2 1 inner
5040.2.k.a 4 20.e even 4 1
5040.2.k.a 4 60.l odd 4 1
5040.2.k.a 4 140.j odd 4 1
5040.2.k.a 4 420.w even 4 1
5040.2.k.d 4 20.e even 4 1
5040.2.k.d 4 60.l odd 4 1
5040.2.k.d 4 140.j odd 4 1
5040.2.k.d 4 420.w even 4 1

Hecke kernels

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

\( T_{11}^{2} + 2 \)
\( T_{13}^{2} + 10 \)
\( T_{17}^{2} - 20 \)
\( T_{43}^{2} - 72 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 - 4 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 16 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 14 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{8} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 50 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 + 56 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 8 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 74 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 28 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{4}( 1 + 8 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 106 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{8} \)
$83$ \( ( 1 + 86 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 88 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 34 T^{2} + 9409 T^{4} )^{4} \)
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