Properties

Label 4050.2.c.g
Level $4050$
Weight $2$
Character orbit 4050.c
Analytic conductor $32.339$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.3394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 162)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + 4 i q^{7} - i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} + 4 i q^{7} - i q^{8} - i q^{13} - 4 q^{14} + q^{16} + 3 i q^{17} + 4 q^{19} + q^{26} - 4 i q^{28} - 9 q^{29} - 4 q^{31} + i q^{32} - 3 q^{34} + i q^{37} + 4 i q^{38} + 6 q^{41} + 8 i q^{43} + 12 i q^{47} - 9 q^{49} + i q^{52} - 6 i q^{53} + 4 q^{56} - 9 i q^{58} - q^{61} - 4 i q^{62} - q^{64} + 4 i q^{67} - 3 i q^{68} - 12 q^{71} + 11 i q^{73} - q^{74} - 4 q^{76} + 16 q^{79} + 6 i q^{82} - 12 i q^{83} - 8 q^{86} + 3 q^{89} + 4 q^{91} - 12 q^{94} - 2 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 8 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{26} - 18 q^{29} - 8 q^{31} - 6 q^{34} + 12 q^{41} - 18 q^{49} + 8 q^{56} - 2 q^{61} - 2 q^{64} - 24 q^{71} - 2 q^{74} - 8 q^{76} + 32 q^{79} - 16 q^{86} + 6 q^{89} + 8 q^{91} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times\).

\(n\) \(2351\) \(3727\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 4.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 4.00000i 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 4050.2.c.g 2
3.b odd 2 1 4050.2.c.n 2
5.b even 2 1 inner 4050.2.c.g 2
5.c odd 4 1 162.2.a.a 1
5.c odd 4 1 4050.2.a.bh 1
15.d odd 2 1 4050.2.c.n 2
15.e even 4 1 162.2.a.d yes 1
15.e even 4 1 4050.2.a.r 1
20.e even 4 1 1296.2.a.c 1
35.f even 4 1 7938.2.a.n 1
40.i odd 4 1 5184.2.a.y 1
40.k even 4 1 5184.2.a.bd 1
45.k odd 12 2 162.2.c.d 2
45.l even 12 2 162.2.c.a 2
60.l odd 4 1 1296.2.a.l 1
105.k odd 4 1 7938.2.a.s 1
120.q odd 4 1 5184.2.a.h 1
120.w even 4 1 5184.2.a.c 1
180.v odd 12 2 1296.2.i.b 2
180.x even 12 2 1296.2.i.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 5.c odd 4 1
162.2.a.d yes 1 15.e even 4 1
162.2.c.a 2 45.l even 12 2
162.2.c.d 2 45.k odd 12 2
1296.2.a.c 1 20.e even 4 1
1296.2.a.l 1 60.l odd 4 1
1296.2.i.b 2 180.v odd 12 2
1296.2.i.n 2 180.x even 12 2
4050.2.a.r 1 15.e even 4 1
4050.2.a.bh 1 5.c odd 4 1
4050.2.c.g 2 1.a even 1 1 trivial
4050.2.c.g 2 5.b even 2 1 inner
4050.2.c.n 2 3.b odd 2 1
4050.2.c.n 2 15.d odd 2 1
5184.2.a.c 1 120.w even 4 1
5184.2.a.h 1 120.q odd 4 1
5184.2.a.y 1 40.i odd 4 1
5184.2.a.bd 1 40.k even 4 1
7938.2.a.n 1 35.f even 4 1
7938.2.a.s 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4050, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 9 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{29} + 9 \) Copy content Toggle raw display
\( T_{41} - 6 \) Copy content Toggle raw display
\( T_{71} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
show more
show less