Properties

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

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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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + 2 i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + 2 i q^{7} + i q^{8} + 4 i q^{13} + 2 q^{14} + q^{16} + 6 i q^{17} + 7 q^{19} + 4 q^{26} -2 i q^{28} -6 q^{29} -10 q^{31} -i q^{32} + 6 q^{34} + 2 i q^{37} -7 i q^{38} -9 q^{41} + i q^{43} -6 i q^{47} + 3 q^{49} -4 i q^{52} -12 i q^{53} -2 q^{56} + 6 i q^{58} -9 q^{59} -4 q^{61} + 10 i q^{62} - q^{64} -13 i q^{67} -6 i q^{68} -6 q^{71} + i q^{73} + 2 q^{74} -7 q^{76} -2 q^{79} + 9 i q^{82} + 9 i q^{83} + q^{86} + 15 q^{89} -8 q^{91} -6 q^{94} + 17 i q^{97} -3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{14} + 2q^{16} + 14q^{19} + 8q^{26} - 12q^{29} - 20q^{31} + 12q^{34} - 18q^{41} + 6q^{49} - 4q^{56} - 18q^{59} - 8q^{61} - 2q^{64} - 12q^{71} + 4q^{74} - 14q^{76} - 4q^{79} + 2q^{86} + 30q^{89} - 16q^{91} - 12q^{94} + O(q^{100}) \)

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 2.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 2.00000i 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.m 2
3.b odd 2 1 4050.2.c.h 2
5.b even 2 1 inner 4050.2.c.m 2
5.c odd 4 1 4050.2.a.o 1
5.c odd 4 1 4050.2.a.u 1
9.c even 3 2 1350.2.j.b 4
9.d odd 6 2 450.2.j.b 4
15.d odd 2 1 4050.2.c.h 2
15.e even 4 1 4050.2.a.d 1
15.e even 4 1 4050.2.a.bg 1
45.h odd 6 2 450.2.j.b 4
45.j even 6 2 1350.2.j.b 4
45.k odd 12 2 1350.2.e.d 2
45.k odd 12 2 1350.2.e.h 2
45.l even 12 2 450.2.e.c 2
45.l even 12 2 450.2.e.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 45.l even 12 2
450.2.e.f yes 2 45.l even 12 2
450.2.j.b 4 9.d odd 6 2
450.2.j.b 4 45.h odd 6 2
1350.2.e.d 2 45.k odd 12 2
1350.2.e.h 2 45.k odd 12 2
1350.2.j.b 4 9.c even 3 2
1350.2.j.b 4 45.j even 6 2
4050.2.a.d 1 15.e even 4 1
4050.2.a.o 1 5.c odd 4 1
4050.2.a.u 1 5.c odd 4 1
4050.2.a.bg 1 15.e even 4 1
4050.2.c.h 2 3.b odd 2 1
4050.2.c.h 2 15.d odd 2 1
4050.2.c.m 2 1.a even 1 1 trivial
4050.2.c.m 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 \(S_{2}^{\mathrm{new}}(4050, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11} \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} + 36 \)
\( T_{19} - 7 \)
\( T_{29} + 6 \)
\( T_{41} + 9 \)
\( T_{71} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 9 + T )^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( 9 + T )^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 1 + T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( -15 + T )^{2} \)
$97$ \( 289 + T^{2} \)
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