Properties

Label 2475.2.c.e
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{7} + 3 i q^{8} +O(q^{10})\) \( q + i q^{2} + q^{4} + 3 i q^{7} + 3 i q^{8} + q^{11} + 2 i q^{13} -3 q^{14} - q^{16} + 3 i q^{17} + q^{19} + i q^{22} -i q^{23} -2 q^{26} + 3 i q^{28} -6 q^{29} + 4 q^{31} + 5 i q^{32} -3 q^{34} -i q^{37} + i q^{38} -5 q^{41} + 4 i q^{43} + q^{44} + q^{46} + 3 i q^{47} -2 q^{49} + 2 i q^{52} -10 i q^{53} -9 q^{56} -6 i q^{58} -11 q^{59} + 14 q^{61} + 4 i q^{62} -7 q^{64} -2 i q^{67} + 3 i q^{68} -5 q^{71} + 2 i q^{73} + q^{74} + q^{76} + 3 i q^{77} -5 q^{79} -5 i q^{82} -8 i q^{83} -4 q^{86} + 3 i q^{88} + 10 q^{89} -6 q^{91} -i q^{92} -3 q^{94} + 17 i q^{97} -2 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{11} - 6q^{14} - 2q^{16} + 2q^{19} - 4q^{26} - 12q^{29} + 8q^{31} - 6q^{34} - 10q^{41} + 2q^{44} + 2q^{46} - 4q^{49} - 18q^{56} - 22q^{59} + 28q^{61} - 14q^{64} - 10q^{71} + 2q^{74} + 2q^{76} - 10q^{79} - 8q^{86} + 20q^{89} - 12q^{91} - 6q^{94} + O(q^{100}) \)

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 3.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 3.00000i 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.e 2
3.b odd 2 1 2475.2.c.c 2
5.b even 2 1 inner 2475.2.c.e 2
5.c odd 4 1 2475.2.a.b 1
5.c odd 4 1 2475.2.a.k yes 1
15.d odd 2 1 2475.2.c.c 2
15.e even 4 1 2475.2.a.d yes 1
15.e even 4 1 2475.2.a.h yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.b 1 5.c odd 4 1
2475.2.a.d yes 1 15.e even 4 1
2475.2.a.h yes 1 15.e even 4 1
2475.2.a.k yes 1 5.c odd 4 1
2475.2.c.c 2 3.b odd 2 1
2475.2.c.c 2 15.d odd 2 1
2475.2.c.e 2 1.a even 1 1 trivial
2475.2.c.e 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 \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 9 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 5 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 11 + T )^{2} \)
$61$ \( ( -14 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 5 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( 5 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 289 + T^{2} \)
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