Properties

Label 2475.2.c.d
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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} -4 i q^{7} + 3 i q^{8} +O(q^{10})\) \( q + i q^{2} + q^{4} -4 i q^{7} + 3 i q^{8} - q^{11} -2 i q^{13} + 4 q^{14} - q^{16} -2 i q^{17} -i q^{22} -8 i q^{23} + 2 q^{26} -4 i q^{28} -6 q^{29} -8 q^{31} + 5 i q^{32} + 2 q^{34} -6 i q^{37} + 2 q^{41} - q^{44} + 8 q^{46} + 8 i q^{47} -9 q^{49} -2 i q^{52} -6 i q^{53} + 12 q^{56} -6 i q^{58} -4 q^{59} + 6 q^{61} -8 i q^{62} -7 q^{64} + 4 i q^{67} -2 i q^{68} -14 i q^{73} + 6 q^{74} + 4 i q^{77} + 4 q^{79} + 2 i q^{82} -12 i q^{83} -3 i q^{88} -6 q^{89} -8 q^{91} -8 i q^{92} -8 q^{94} -2 i q^{97} -9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{11} + 8q^{14} - 2q^{16} + 4q^{26} - 12q^{29} - 16q^{31} + 4q^{34} + 4q^{41} - 2q^{44} + 16q^{46} - 18q^{49} + 24q^{56} - 8q^{59} + 12q^{61} - 14q^{64} + 12q^{74} + 8q^{79} - 12q^{89} - 16q^{91} - 16q^{94} + O(q^{100}) \)

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 4.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 4.00000i 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.d 2
3.b odd 2 1 825.2.c.a 2
5.b even 2 1 inner 2475.2.c.d 2
5.c odd 4 1 99.2.a.b 1
5.c odd 4 1 2475.2.a.g 1
15.d odd 2 1 825.2.c.a 2
15.e even 4 1 33.2.a.a 1
15.e even 4 1 825.2.a.a 1
20.e even 4 1 1584.2.a.o 1
35.f even 4 1 4851.2.a.b 1
40.i odd 4 1 6336.2.a.x 1
40.k even 4 1 6336.2.a.n 1
45.k odd 12 2 891.2.e.g 2
45.l even 12 2 891.2.e.e 2
55.e even 4 1 1089.2.a.j 1
60.l odd 4 1 528.2.a.g 1
105.k odd 4 1 1617.2.a.j 1
120.q odd 4 1 2112.2.a.j 1
120.w even 4 1 2112.2.a.bb 1
165.l odd 4 1 363.2.a.b 1
165.l odd 4 1 9075.2.a.q 1
165.u odd 20 4 363.2.e.g 4
165.v even 20 4 363.2.e.e 4
195.s even 4 1 5577.2.a.a 1
255.o even 4 1 9537.2.a.m 1
660.q even 4 1 5808.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 15.e even 4 1
99.2.a.b 1 5.c odd 4 1
363.2.a.b 1 165.l odd 4 1
363.2.e.e 4 165.v even 20 4
363.2.e.g 4 165.u odd 20 4
528.2.a.g 1 60.l odd 4 1
825.2.a.a 1 15.e even 4 1
825.2.c.a 2 3.b odd 2 1
825.2.c.a 2 15.d odd 2 1
891.2.e.e 2 45.l even 12 2
891.2.e.g 2 45.k odd 12 2
1089.2.a.j 1 55.e even 4 1
1584.2.a.o 1 20.e even 4 1
1617.2.a.j 1 105.k odd 4 1
2112.2.a.j 1 120.q odd 4 1
2112.2.a.bb 1 120.w even 4 1
2475.2.a.g 1 5.c odd 4 1
2475.2.c.d 2 1.a even 1 1 trivial
2475.2.c.d 2 5.b even 2 1 inner
4851.2.a.b 1 35.f even 4 1
5577.2.a.a 1 195.s even 4 1
5808.2.a.t 1 660.q even 4 1
6336.2.a.n 1 40.k even 4 1
6336.2.a.x 1 40.i odd 4 1
9075.2.a.q 1 165.l odd 4 1
9537.2.a.m 1 255.o even 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 16 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ 1
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 43 T^{2} )^{2} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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