Properties

Label 1425.2.c.g
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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + i q^{2} - i q^{3} + q^{4} + q^{6} + 3 i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - i q^{3} + q^{4} + q^{6} + 3 i q^{8} - q^{9} - i q^{12} - 6 i q^{13} - q^{16} - 6 i q^{17} - i q^{18} + q^{19} - 4 i q^{23} + 3 q^{24} + 6 q^{26} + i q^{27} - 2 q^{29} + 8 q^{31} + 5 i q^{32} + 6 q^{34} - q^{36} - 10 i q^{37} + i q^{38} - 6 q^{39} - 2 q^{41} + 4 i q^{43} + 4 q^{46} + 12 i q^{47} + i q^{48} + 7 q^{49} - 6 q^{51} - 6 i q^{52} + 6 i q^{53} - q^{54} - i q^{57} - 2 i q^{58} + 12 q^{59} - 2 q^{61} + 8 i q^{62} - 7 q^{64} - 4 i q^{67} - 6 i q^{68} - 4 q^{69} - 3 i q^{72} - 10 i q^{73} + 10 q^{74} + q^{76} - 6 i q^{78} + q^{81} - 2 i q^{82} - 16 i q^{83} - 4 q^{86} + 2 i q^{87} + 2 q^{89} - 4 i q^{92} - 8 i q^{93} - 12 q^{94} + 5 q^{96} + 10 i q^{97} + 7 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 6 q^{24} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 12 q^{34} - 2 q^{36} - 12 q^{39} - 4 q^{41} + 8 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{69} + 20 q^{74} + 2 q^{76} + 2 q^{81} - 8 q^{86} + 4 q^{89} - 24 q^{94} + 10 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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 0 3.00000i −1.00000 0
799.2 1.00000i 1.00000i 1.00000 0 1.00000 0 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.g 2
5.b even 2 1 inner 1425.2.c.g 2
5.c odd 4 1 57.2.a.c 1
5.c odd 4 1 1425.2.a.a 1
15.e even 4 1 171.2.a.a 1
15.e even 4 1 4275.2.a.m 1
20.e even 4 1 912.2.a.b 1
35.f even 4 1 2793.2.a.i 1
40.i odd 4 1 3648.2.a.o 1
40.k even 4 1 3648.2.a.bf 1
55.e even 4 1 6897.2.a.a 1
60.l odd 4 1 2736.2.a.s 1
65.h odd 4 1 9633.2.a.h 1
95.g even 4 1 1083.2.a.a 1
105.k odd 4 1 8379.2.a.e 1
285.j odd 4 1 3249.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.c 1 5.c odd 4 1
171.2.a.a 1 15.e even 4 1
912.2.a.b 1 20.e even 4 1
1083.2.a.a 1 95.g even 4 1
1425.2.a.a 1 5.c odd 4 1
1425.2.c.g 2 1.a even 1 1 trivial
1425.2.c.g 2 5.b even 2 1 inner
2736.2.a.s 1 60.l odd 4 1
2793.2.a.i 1 35.f even 4 1
3249.2.a.g 1 285.j odd 4 1
3648.2.a.o 1 40.i odd 4 1
3648.2.a.bf 1 40.k even 4 1
4275.2.a.m 1 15.e even 4 1
6897.2.a.a 1 55.e even 4 1
8379.2.a.e 1 105.k odd 4 1
9633.2.a.h 1 65.h odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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