Properties

Label 3150.2.g.v
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} -i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} -i q^{7} + i q^{8} + 5 q^{11} + 6 i q^{13} - q^{14} + q^{16} + i q^{17} + 3 q^{19} -5 i q^{22} + 6 q^{26} + i q^{28} -6 q^{29} -4 q^{31} -i q^{32} + q^{34} + 8 i q^{37} -3 i q^{38} -11 q^{41} + 8 i q^{43} -5 q^{44} -2 i q^{47} - q^{49} -6 i q^{52} + 4 i q^{53} + q^{56} + 6 i q^{58} + 4 q^{59} -2 q^{61} + 4 i q^{62} - q^{64} + 9 i q^{67} -i q^{68} + 10 q^{71} + 7 i q^{73} + 8 q^{74} -3 q^{76} -5 i q^{77} + 2 q^{79} + 11 i q^{82} + 11 i q^{83} + 8 q^{86} + 5 i q^{88} -11 q^{89} + 6 q^{91} -2 q^{94} -10 i q^{97} + i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 6 q^{19} + 12 q^{26} - 12 q^{29} - 8 q^{31} + 2 q^{34} - 22 q^{41} - 10 q^{44} - 2 q^{49} + 2 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 20 q^{71} + 16 q^{74} - 6 q^{76} + 4 q^{79} + 16 q^{86} - 22 q^{89} + 12 q^{91} - 4 q^{94} + O(q^{100}) \)

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.v 2
3.b odd 2 1 350.2.c.a 2
5.b even 2 1 inner 3150.2.g.v 2
5.c odd 4 1 3150.2.a.j 1
5.c odd 4 1 3150.2.a.bq 1
12.b even 2 1 2800.2.g.a 2
15.d odd 2 1 350.2.c.a 2
15.e even 4 1 350.2.a.c 1
15.e even 4 1 350.2.a.d yes 1
21.c even 2 1 2450.2.c.r 2
60.h even 2 1 2800.2.g.a 2
60.l odd 4 1 2800.2.a.b 1
60.l odd 4 1 2800.2.a.bg 1
105.g even 2 1 2450.2.c.r 2
105.k odd 4 1 2450.2.a.a 1
105.k odd 4 1 2450.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 15.e even 4 1
350.2.a.d yes 1 15.e even 4 1
350.2.c.a 2 3.b odd 2 1
350.2.c.a 2 15.d odd 2 1
2450.2.a.a 1 105.k odd 4 1
2450.2.a.bg 1 105.k odd 4 1
2450.2.c.r 2 21.c even 2 1
2450.2.c.r 2 105.g even 2 1
2800.2.a.b 1 60.l odd 4 1
2800.2.a.bg 1 60.l odd 4 1
2800.2.g.a 2 12.b even 2 1
2800.2.g.a 2 60.h even 2 1
3150.2.a.j 1 5.c odd 4 1
3150.2.a.bq 1 5.c odd 4 1
3150.2.g.v 2 1.a even 1 1 trivial
3150.2.g.v 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{11} - 5 \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 1 \)
\( T_{19} - 3 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( -3 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 11 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 81 + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 49 + T^{2} \)
$79$ \( ( -2 + T )^{2} \)
$83$ \( 121 + T^{2} \)
$89$ \( ( 11 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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