Properties

Label 1925.2.b.e
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 2 q^{4} -i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 2 q^{4} -i q^{7} -6 q^{9} - q^{11} + 6 i q^{12} + 4 i q^{13} + 4 q^{16} + 2 i q^{17} + 6 q^{19} + 3 q^{21} + 5 i q^{23} -9 i q^{27} -2 i q^{28} -10 q^{29} + q^{31} -3 i q^{33} -12 q^{36} -5 i q^{37} -12 q^{39} -2 q^{41} + 8 i q^{43} -2 q^{44} + 8 i q^{47} + 12 i q^{48} - q^{49} -6 q^{51} + 8 i q^{52} + 6 i q^{53} + 18 i q^{57} -3 q^{59} -2 q^{61} + 6 i q^{63} + 8 q^{64} -3 i q^{67} + 4 i q^{68} -15 q^{69} + q^{71} -10 i q^{73} + 12 q^{76} + i q^{77} -6 q^{79} + 9 q^{81} -12 i q^{83} + 6 q^{84} -30 i q^{87} + 15 q^{89} + 4 q^{91} + 10 i q^{92} + 3 i q^{93} -5 i q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 12 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{4} - 12 q^{9} - 2 q^{11} + 8 q^{16} + 12 q^{19} + 6 q^{21} - 20 q^{29} + 2 q^{31} - 24 q^{36} - 24 q^{39} - 4 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} - 6 q^{59} - 4 q^{61} + 16 q^{64} - 30 q^{69} + 2 q^{71} + 24 q^{76} - 12 q^{79} + 18 q^{81} + 12 q^{84} + 30 q^{89} + 8 q^{91} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 3.00000i 2.00000 0 0 1.00000i 0 −6.00000 0
1849.2 0 3.00000i 2.00000 0 0 1.00000i 0 −6.00000 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 1925.2.b.e 2
5.b even 2 1 inner 1925.2.b.e 2
5.c odd 4 1 77.2.a.a 1
5.c odd 4 1 1925.2.a.h 1
15.e even 4 1 693.2.a.c 1
20.e even 4 1 1232.2.a.l 1
35.f even 4 1 539.2.a.c 1
35.k even 12 2 539.2.e.c 2
35.l odd 12 2 539.2.e.f 2
40.i odd 4 1 4928.2.a.bj 1
40.k even 4 1 4928.2.a.a 1
55.e even 4 1 847.2.a.b 1
55.k odd 20 4 847.2.f.i 4
55.l even 20 4 847.2.f.h 4
105.k odd 4 1 4851.2.a.j 1
140.j odd 4 1 8624.2.a.a 1
165.l odd 4 1 7623.2.a.j 1
385.l odd 4 1 5929.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 5.c odd 4 1
539.2.a.c 1 35.f even 4 1
539.2.e.c 2 35.k even 12 2
539.2.e.f 2 35.l odd 12 2
693.2.a.c 1 15.e even 4 1
847.2.a.b 1 55.e even 4 1
847.2.f.h 4 55.l even 20 4
847.2.f.i 4 55.k odd 20 4
1232.2.a.l 1 20.e even 4 1
1925.2.a.h 1 5.c odd 4 1
1925.2.b.e 2 1.a even 1 1 trivial
1925.2.b.e 2 5.b even 2 1 inner
4851.2.a.j 1 105.k odd 4 1
4928.2.a.a 1 40.k even 4 1
4928.2.a.bj 1 40.i odd 4 1
5929.2.a.f 1 385.l odd 4 1
7623.2.a.j 1 165.l odd 4 1
8624.2.a.a 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2} \)
\( T_{3}^{2} + 9 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 25 + T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( 25 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 3 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( ( -1 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 6 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -15 + T )^{2} \)
$97$ \( 25 + T^{2} \)
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