Properties

Label 1425.2.c.d
Level $1425$
Weight $2$
Character orbit 1425.c
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3786822880\)
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 285)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} -2 i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} -2 i q^{7} + 3 i q^{8} - q^{9} -2 q^{11} + i q^{12} + 4 i q^{13} + 2 q^{14} - q^{16} + 2 i q^{17} -i q^{18} + q^{19} + 2 q^{21} -2 i q^{22} + 4 i q^{23} -3 q^{24} -4 q^{26} -i q^{27} -2 i q^{28} -4 q^{29} + 5 i q^{32} -2 i q^{33} -2 q^{34} - q^{36} + i q^{38} -4 q^{39} + 2 i q^{42} + 10 i q^{43} -2 q^{44} -4 q^{46} + 12 i q^{47} -i q^{48} + 3 q^{49} -2 q^{51} + 4 i q^{52} + 2 i q^{53} + q^{54} + 6 q^{56} + i q^{57} -4 i q^{58} -4 q^{59} + 2 q^{61} + 2 i q^{63} -7 q^{64} + 2 q^{66} -16 i q^{67} + 2 i q^{68} -4 q^{69} -3 i q^{72} + 2 i q^{73} + q^{76} + 4 i q^{77} -4 i q^{78} + 8 q^{79} + q^{81} + 12 i q^{83} + 2 q^{84} -10 q^{86} -4 i q^{87} -6 i q^{88} + 8 q^{91} + 4 i q^{92} -12 q^{94} -5 q^{96} -16 i q^{97} + 3 i q^{98} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 2.00000i 3.00000i −1.00000 0
799.2 1.00000i 1.00000i 1.00000 0 −1.00000 2.00000i 3.00000i −1.00000 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 1425.2.c.d 2
5.b even 2 1 inner 1425.2.c.d 2
5.c odd 4 1 285.2.a.b 1
5.c odd 4 1 1425.2.a.d 1
15.e even 4 1 855.2.a.b 1
15.e even 4 1 4275.2.a.o 1
20.e even 4 1 4560.2.a.v 1
95.g even 4 1 5415.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.b 1 5.c odd 4 1
855.2.a.b 1 15.e even 4 1
1425.2.a.d 1 5.c odd 4 1
1425.2.c.d 2 1.a even 1 1 trivial
1425.2.c.d 2 5.b even 2 1 inner
4275.2.a.o 1 15.e even 4 1
4560.2.a.v 1 20.e even 4 1
5415.2.a.c 1 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1425, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 4 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 256 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 256 + T^{2} \)
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