Properties

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

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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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} -4 q^{11} -3 i q^{13} - q^{14} + q^{16} -7 i q^{17} + 6 q^{19} -4 i q^{22} + 9 i q^{23} + 3 q^{26} -i q^{28} + 3 q^{29} -7 q^{31} + i q^{32} + 7 q^{34} + 10 i q^{37} + 6 i q^{38} + q^{41} + 13 i q^{43} + 4 q^{44} -9 q^{46} + 2 i q^{47} - q^{49} + 3 i q^{52} -i q^{53} + q^{56} + 3 i q^{58} -11 q^{59} + 13 q^{61} -7 i q^{62} - q^{64} + 7 i q^{68} -8 q^{71} + 8 i q^{73} -10 q^{74} -6 q^{76} -4 i q^{77} -4 q^{79} + i q^{82} + 7 i q^{83} -13 q^{86} + 4 i q^{88} -14 q^{89} + 3 q^{91} -9 i q^{92} -2 q^{94} -8 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 8q^{11} - 2q^{14} + 2q^{16} + 12q^{19} + 6q^{26} + 6q^{29} - 14q^{31} + 14q^{34} + 2q^{41} + 8q^{44} - 18q^{46} - 2q^{49} + 2q^{56} - 22q^{59} + 26q^{61} - 2q^{64} - 16q^{71} - 20q^{74} - 12q^{76} - 8q^{79} - 26q^{86} - 28q^{89} + 6q^{91} - 4q^{94} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
2899.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.g.d 2
3.b odd 2 1 3150.2.g.u 2
5.b even 2 1 inner 3150.2.g.d 2
5.c odd 4 1 3150.2.a.b 1
5.c odd 4 1 3150.2.a.bi yes 1
15.d odd 2 1 3150.2.g.u 2
15.e even 4 1 3150.2.a.u yes 1
15.e even 4 1 3150.2.a.bf yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.a.b 1 5.c odd 4 1
3150.2.a.u yes 1 15.e even 4 1
3150.2.a.bf yes 1 15.e even 4 1
3150.2.a.bi yes 1 5.c odd 4 1
3150.2.g.d 2 1.a even 1 1 trivial
3150.2.g.d 2 5.b even 2 1 inner
3150.2.g.u 2 3.b odd 2 1
3150.2.g.u 2 15.d odd 2 1

Hecke kernels

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

\( T_{11} + 4 \)
\( T_{13}^{2} + 9 \)
\( T_{17}^{2} + 49 \)
\( T_{19} - 6 \)
\( T_{29} - 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 17 T^{2} + 169 T^{4} \)
$17$ \( 1 + 15 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 6 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 35 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 3 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - T + 41 T^{2} )^{2} \)
$43$ \( 1 + 83 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 90 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 105 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 11 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{2} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 82 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 117 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 14 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \)
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