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: \(\Q(i, \sqrt{5})\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

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:
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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 \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{2} + 5 \)
\( T_{3}^{4} + 12 T_{3}^{2} + 16 \)
\( T_{19}^{2} + 4 T_{19} - 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} ) \)
$5$ \( \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 56 T^{2} + 1342 T^{4} - 16184 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 44 T^{2} + 1222 T^{4} - 23276 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 - 76 T^{2} + 2902 T^{4} - 104044 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 10 T + 142 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 28 T^{2} + 1174 T^{4} - 125692 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 264 T^{2} + 27902 T^{4} - 1406856 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 36 T^{2} + 11222 T^{4} + 248004 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{4} \)
$97$ \( 1 + 4 T^{2} + 7302 T^{4} + 37636 T^{6} + 88529281 T^{8} \)
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