Properties

Label 2475.2.c.g
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
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 99)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} -2 i q^{7} + 3 i q^{8} +O(q^{10})\) \( q + i q^{2} + q^{4} -2 i q^{7} + 3 i q^{8} + q^{11} + 2 i q^{13} + 2 q^{14} - q^{16} -2 i q^{17} + 6 q^{19} + i q^{22} + 4 i q^{23} -2 q^{26} -2 i q^{28} -6 q^{29} + 4 q^{31} + 5 i q^{32} + 2 q^{34} -6 i q^{37} + 6 i q^{38} + 10 q^{41} -6 i q^{43} + q^{44} -4 q^{46} + 8 i q^{47} + 3 q^{49} + 2 i q^{52} + 6 q^{56} -6 i q^{58} + 4 q^{59} -6 q^{61} + 4 i q^{62} -7 q^{64} + 8 i q^{67} -2 i q^{68} + 2 i q^{73} + 6 q^{74} + 6 q^{76} -2 i q^{77} + 10 q^{79} + 10 i q^{82} + 12 i q^{83} + 6 q^{86} + 3 i q^{88} + 4 q^{91} + 4 i q^{92} -8 q^{94} + 2 i q^{97} + 3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{11} + 4q^{14} - 2q^{16} + 12q^{19} - 4q^{26} - 12q^{29} + 8q^{31} + 4q^{34} + 20q^{41} + 2q^{44} - 8q^{46} + 6q^{49} + 12q^{56} + 8q^{59} - 12q^{61} - 14q^{64} + 12q^{74} + 12q^{76} + 20q^{79} + 12q^{86} + 8q^{91} - 16q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 2.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 2.00000i 3.00000i 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 2475.2.c.g 2
3.b odd 2 1 2475.2.c.b 2
5.b even 2 1 inner 2475.2.c.g 2
5.c odd 4 1 99.2.a.c yes 1
5.c odd 4 1 2475.2.a.c 1
15.d odd 2 1 2475.2.c.b 2
15.e even 4 1 99.2.a.a 1
15.e even 4 1 2475.2.a.j 1
20.e even 4 1 1584.2.a.r 1
35.f even 4 1 4851.2.a.o 1
40.i odd 4 1 6336.2.a.b 1
40.k even 4 1 6336.2.a.f 1
45.k odd 12 2 891.2.e.c 2
45.l even 12 2 891.2.e.j 2
55.e even 4 1 1089.2.a.d 1
60.l odd 4 1 1584.2.a.b 1
105.k odd 4 1 4851.2.a.g 1
120.q odd 4 1 6336.2.a.cm 1
120.w even 4 1 6336.2.a.cl 1
165.l odd 4 1 1089.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 15.e even 4 1
99.2.a.c yes 1 5.c odd 4 1
891.2.e.c 2 45.k odd 12 2
891.2.e.j 2 45.l even 12 2
1089.2.a.d 1 55.e even 4 1
1089.2.a.h 1 165.l odd 4 1
1584.2.a.b 1 60.l odd 4 1
1584.2.a.r 1 20.e even 4 1
2475.2.a.c 1 5.c odd 4 1
2475.2.a.j 1 15.e even 4 1
2475.2.c.b 2 3.b odd 2 1
2475.2.c.b 2 15.d odd 2 1
2475.2.c.g 2 1.a even 1 1 trivial
2475.2.c.g 2 5.b even 2 1 inner
4851.2.a.g 1 105.k odd 4 1
4851.2.a.o 1 35.f even 4 1
6336.2.a.b 1 40.i odd 4 1
6336.2.a.f 1 40.k even 4 1
6336.2.a.cl 1 120.w even 4 1
6336.2.a.cm 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 4 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 + T^{2} \)
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