Properties

Label 1925.2.b.d
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + i q^{2} -2 i q^{3} + q^{4} + 2 q^{6} -i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -2 i q^{3} + q^{4} + 2 q^{6} -i q^{7} + 3 i q^{8} - q^{9} + q^{11} -2 i q^{12} -4 i q^{13} + q^{14} - q^{16} + 4 i q^{17} -i q^{18} -2 q^{21} + i q^{22} + 4 i q^{23} + 6 q^{24} + 4 q^{26} -4 i q^{27} -i q^{28} + 6 q^{29} + 10 q^{31} + 5 i q^{32} -2 i q^{33} -4 q^{34} - q^{36} -6 i q^{37} -8 q^{39} + 4 q^{41} -2 i q^{42} -12 i q^{43} + q^{44} -4 q^{46} -10 i q^{47} + 2 i q^{48} - q^{49} + 8 q^{51} -4 i q^{52} + 6 i q^{53} + 4 q^{54} + 3 q^{56} + 6 i q^{58} -2 q^{59} + 10 i q^{62} + i q^{63} -7 q^{64} + 2 q^{66} + 8 i q^{67} + 4 i q^{68} + 8 q^{69} -12 q^{71} -3 i q^{72} + 8 i q^{73} + 6 q^{74} -i q^{77} -8 i q^{78} -8 q^{79} -11 q^{81} + 4 i q^{82} -2 q^{84} + 12 q^{86} -12 i q^{87} + 3 i q^{88} + 6 q^{89} -4 q^{91} + 4 i q^{92} -20 i q^{93} + 10 q^{94} + 10 q^{96} -10 i q^{97} -i q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 4q^{6} - 2q^{9} + 2q^{11} + 2q^{14} - 2q^{16} - 4q^{21} + 12q^{24} + 8q^{26} + 12q^{29} + 20q^{31} - 8q^{34} - 2q^{36} - 16q^{39} + 8q^{41} + 2q^{44} - 8q^{46} - 2q^{49} + 16q^{51} + 8q^{54} + 6q^{56} - 4q^{59} - 14q^{64} + 4q^{66} + 16q^{69} - 24q^{71} + 12q^{74} - 16q^{79} - 22q^{81} - 4q^{84} + 24q^{86} + 12q^{89} - 8q^{91} + 20q^{94} + 20q^{96} - 2q^{99} + O(q^{100}) \)

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
1.00000i 2.00000i 1.00000 0 2.00000 1.00000i 3.00000i −1.00000 0
1849.2 1.00000i 2.00000i 1.00000 0 2.00000 1.00000i 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 1925.2.b.d 2
5.b even 2 1 inner 1925.2.b.d 2
5.c odd 4 1 77.2.a.c 1
5.c odd 4 1 1925.2.a.c 1
15.e even 4 1 693.2.a.a 1
20.e even 4 1 1232.2.a.a 1
35.f even 4 1 539.2.a.d 1
35.k even 12 2 539.2.e.b 2
35.l odd 12 2 539.2.e.a 2
40.i odd 4 1 4928.2.a.g 1
40.k even 4 1 4928.2.a.bi 1
55.e even 4 1 847.2.a.a 1
55.k odd 20 4 847.2.f.e 4
55.l even 20 4 847.2.f.k 4
105.k odd 4 1 4851.2.a.a 1
140.j odd 4 1 8624.2.a.bc 1
165.l odd 4 1 7623.2.a.n 1
385.l odd 4 1 5929.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 5.c odd 4 1
539.2.a.d 1 35.f even 4 1
539.2.e.a 2 35.l odd 12 2
539.2.e.b 2 35.k even 12 2
693.2.a.a 1 15.e even 4 1
847.2.a.a 1 55.e even 4 1
847.2.f.e 4 55.k odd 20 4
847.2.f.k 4 55.l even 20 4
1232.2.a.a 1 20.e even 4 1
1925.2.a.c 1 5.c odd 4 1
1925.2.b.d 2 1.a even 1 1 trivial
1925.2.b.d 2 5.b even 2 1 inner
4851.2.a.a 1 105.k odd 4 1
4928.2.a.g 1 40.i odd 4 1
4928.2.a.bi 1 40.k even 4 1
5929.2.a.b 1 385.l odd 4 1
7623.2.a.n 1 165.l odd 4 1
8624.2.a.bc 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 \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{3}^{2} + 4 \)
\( T_{19} \)