Properties

Label 3150.2.g.s
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: 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{7} + i q^{8} + 4 q^{11} + 6 i q^{13} + q^{14} + q^{16} + 4 i q^{17} -6 q^{19} -4 i q^{22} + 6 q^{26} -i q^{28} -6 q^{29} -4 q^{31} -i q^{32} + 4 q^{34} -8 i q^{37} + 6 i q^{38} -10 q^{41} -2 i q^{43} -4 q^{44} + 10 i q^{47} - q^{49} -6 i q^{52} -14 i q^{53} - q^{56} + 6 i q^{58} -4 q^{59} -8 q^{61} + 4 i q^{62} - q^{64} -6 i q^{67} -4 i q^{68} + 2 q^{71} -10 i q^{73} -8 q^{74} + 6 q^{76} + 4 i q^{77} -16 q^{79} + 10 i q^{82} + 8 i q^{83} -2 q^{86} + 4 i q^{88} + 2 q^{89} -6 q^{91} + 10 q^{94} -2 i q^{97} + i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 8q^{11} + 2q^{14} + 2q^{16} - 12q^{19} + 12q^{26} - 12q^{29} - 8q^{31} + 8q^{34} - 20q^{41} - 8q^{44} - 2q^{49} - 2q^{56} - 8q^{59} - 16q^{61} - 2q^{64} + 4q^{71} - 16q^{74} + 12q^{76} - 32q^{79} - 4q^{86} + 4q^{89} - 12q^{91} + 20q^{94} + O(q^{100}) \)

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.s 2
3.b odd 2 1 3150.2.g.b 2
5.b even 2 1 inner 3150.2.g.s 2
5.c odd 4 1 630.2.a.g yes 1
5.c odd 4 1 3150.2.a.s 1
15.d odd 2 1 3150.2.g.b 2
15.e even 4 1 630.2.a.e 1
15.e even 4 1 3150.2.a.bh 1
20.e even 4 1 5040.2.a.n 1
35.f even 4 1 4410.2.a.bl 1
60.l odd 4 1 5040.2.a.bp 1
105.k odd 4 1 4410.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.a.e 1 15.e even 4 1
630.2.a.g yes 1 5.c odd 4 1
3150.2.a.s 1 5.c odd 4 1
3150.2.a.bh 1 15.e even 4 1
3150.2.g.b 2 3.b odd 2 1
3150.2.g.b 2 15.d odd 2 1
3150.2.g.s 2 1.a even 1 1 trivial
3150.2.g.s 2 5.b even 2 1 inner
4410.2.a.a 1 105.k odd 4 1
4410.2.a.bl 1 35.f even 4 1
5040.2.a.n 1 20.e even 4 1
5040.2.a.bp 1 60.l odd 4 1

Hecke kernels

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

\( T_{11} - 4 \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 16 \)
\( T_{19} + 6 \)
\( T_{29} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 82 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 6 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 98 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 16 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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