Properties

Label 1925.2.b.h
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 3 q^{4} + ( - \beta_{3} - 5) q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 3 q^{4} + ( - \beta_{3} - 5) q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9} - q^{11} + ( - 3 \beta_{2} - 3 \beta_1) q^{12} + ( - \beta_{2} + \beta_1) q^{13} + \beta_{3} q^{14} - q^{16} + ( - \beta_{2} + \beta_1) q^{17} + ( - 3 \beta_{2} - 10 \beta_1) q^{18} + (2 \beta_{3} - 2) q^{19} + (\beta_{3} + 1) q^{21} - \beta_{2} q^{22} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{3} + 5) q^{24} + ( - \beta_{3} + 5) q^{26} + ( - 2 \beta_{2} - 10 \beta_1) q^{27} + 3 \beta_1 q^{28} + (2 \beta_{3} - 4) q^{29} + ( - \beta_{3} - 5) q^{31} - 3 \beta_{2} q^{32} + ( - \beta_{2} - \beta_1) q^{33} + ( - \beta_{3} + 5) q^{34} + (6 \beta_{3} + 9) q^{36} + ( - 2 \beta_{2} + 4 \beta_1) q^{37} + ( - 2 \beta_{2} + 10 \beta_1) q^{38} + 4 q^{39} + (\beta_{3} - 9) q^{41} + (\beta_{2} + 5 \beta_1) q^{42} + 8 \beta_1 q^{43} + 3 q^{44} + (2 \beta_{3} + 10) q^{46} + ( - \beta_{2} - 5 \beta_1) q^{47} + ( - \beta_{2} - \beta_1) q^{48} - q^{49} + 4 q^{51} + (3 \beta_{2} - 3 \beta_1) q^{52} + (2 \beta_{2} + 4 \beta_1) q^{53} + (10 \beta_{3} + 10) q^{54} - \beta_{3} q^{56} + 8 \beta_1 q^{57} + ( - 4 \beta_{2} + 10 \beta_1) q^{58} + ( - \beta_{3} - 1) q^{59} + (\beta_{3} - 5) q^{61} + ( - 5 \beta_{2} - 5 \beta_1) q^{62} + (2 \beta_{2} + 3 \beta_1) q^{63} + 13 q^{64} + (\beta_{3} + 5) q^{66} + (2 \beta_{2} - 10 \beta_1) q^{67} + (3 \beta_{2} - 3 \beta_1) q^{68} + (4 \beta_{3} + 12) q^{69} + (2 \beta_{3} - 6) q^{71} + (3 \beta_{2} + 10 \beta_1) q^{72} + ( - \beta_{2} - 3 \beta_1) q^{73} + ( - 4 \beta_{3} + 10) q^{74} + ( - 6 \beta_{3} + 6) q^{76} + \beta_1 q^{77} + 4 \beta_{2} q^{78} - 4 \beta_{3} q^{79} + (6 \beta_{3} + 11) q^{81} + ( - 9 \beta_{2} + 5 \beta_1) q^{82} + (6 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} - 3) q^{84} - 8 \beta_{3} q^{86} + ( - 2 \beta_{2} + 6 \beta_1) q^{87} + \beta_{2} q^{88} - 2 q^{89} + ( - \beta_{3} + 1) q^{91} + (6 \beta_{2} + 6 \beta_1) q^{92} + ( - 6 \beta_{2} - 10 \beta_1) q^{93} + (5 \beta_{3} + 5) q^{94} + (3 \beta_{3} + 15) q^{96} + (6 \beta_{2} - 4 \beta_1) q^{97} - \beta_{2} q^{98} + (2 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{16} - 8 q^{19} + 4 q^{21} + 20 q^{24} + 20 q^{26} - 16 q^{29} - 20 q^{31} + 20 q^{34} + 36 q^{36} + 16 q^{39} - 36 q^{41} + 12 q^{44} + 40 q^{46} - 4 q^{49} + 16 q^{51} + 40 q^{54} - 4 q^{59} - 20 q^{61} + 52 q^{64} + 20 q^{66} + 48 q^{69} - 24 q^{71} + 40 q^{74} + 24 q^{76} + 44 q^{81} - 12 q^{84} - 8 q^{89} + 4 q^{91} + 20 q^{94} + 60 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

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
0.618034i
1.61803i
1.61803i
0.618034i
2.23607i 3.23607i −3.00000 0 −7.23607 1.00000i 2.23607i −7.47214 0
1849.2 2.23607i 1.23607i −3.00000 0 −2.76393 1.00000i 2.23607i 1.47214 0
1849.3 2.23607i 1.23607i −3.00000 0 −2.76393 1.00000i 2.23607i 1.47214 0
1849.4 2.23607i 3.23607i −3.00000 0 −7.23607 1.00000i 2.23607i −7.47214 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.h 4
5.b even 2 1 inner 1925.2.b.h 4
5.c odd 4 1 77.2.a.d 2
5.c odd 4 1 1925.2.a.r 2
15.e even 4 1 693.2.a.h 2
20.e even 4 1 1232.2.a.m 2
35.f even 4 1 539.2.a.f 2
35.k even 12 2 539.2.e.j 4
35.l odd 12 2 539.2.e.i 4
40.i odd 4 1 4928.2.a.bm 2
40.k even 4 1 4928.2.a.bv 2
55.e even 4 1 847.2.a.f 2
55.k odd 20 2 847.2.f.a 4
55.k odd 20 2 847.2.f.n 4
55.l even 20 2 847.2.f.b 4
55.l even 20 2 847.2.f.m 4
105.k odd 4 1 4851.2.a.y 2
140.j odd 4 1 8624.2.a.ce 2
165.l odd 4 1 7623.2.a.bl 2
385.l odd 4 1 5929.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 5.c odd 4 1
539.2.a.f 2 35.f even 4 1
539.2.e.i 4 35.l odd 12 2
539.2.e.j 4 35.k even 12 2
693.2.a.h 2 15.e even 4 1
847.2.a.f 2 55.e even 4 1
847.2.f.a 4 55.k odd 20 2
847.2.f.b 4 55.l even 20 2
847.2.f.m 4 55.l even 20 2
847.2.f.n 4 55.k odd 20 2
1232.2.a.m 2 20.e even 4 1
1925.2.a.r 2 5.c odd 4 1
1925.2.b.h 4 1.a even 1 1 trivial
1925.2.b.h 4 5.b even 2 1 inner
4851.2.a.y 2 105.k odd 4 1
4928.2.a.bm 2 40.i odd 4 1
4928.2.a.bv 2 40.k even 4 1
5929.2.a.m 2 385.l odd 4 1
7623.2.a.bl 2 165.l odd 4 1
8624.2.a.ce 2 140.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 \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{3}^{4} + 12T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} + 18 T + 76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 240T^{2} + 6400 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 392 T^{2} + 26896 \) Copy content Toggle raw display
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