Properties

Label 1925.2.b.l
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1925,2,Mod(1849,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1925.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - 2 q^{9} + q^{11} + (3 \beta_{3} + \beta_1) q^{12} + ( - 3 \beta_{3} - 2 \beta_1) q^{13} + \beta_{2} q^{14} + 3 \beta_{2} q^{16} + (2 \beta_{3} + 5 \beta_1) q^{17} - 2 \beta_1 q^{18} + (3 \beta_{2} + 6) q^{19} + (2 \beta_{2} + 1) q^{21} + \beta_1 q^{22} + ( - 2 \beta_{3} - \beta_1) q^{23} - 5 q^{24} + (\beta_{2} + 2) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + ( - \beta_{3} - \beta_1) q^{28} + ( - 2 \beta_{2} + 3) q^{29} + 6 \beta_{2} q^{31} + (5 \beta_{3} + \beta_1) q^{32} + (\beta_{3} + 2 \beta_1) q^{33} + (3 \beta_{2} - 5) q^{34} + ( - 2 \beta_{2} - 2) q^{36} + (8 \beta_{3} + 2 \beta_1) q^{37} + (3 \beta_{3} + 3 \beta_1) q^{38} + (4 \beta_{2} + 7) q^{39} + ( - 3 \beta_{2} - 9) q^{41} + (2 \beta_{3} - \beta_1) q^{42} + (9 \beta_{3} + 6 \beta_1) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{2} + 1) q^{46} + ( - 8 \beta_{3} - 7 \beta_1) q^{47} + (6 \beta_{3} - 3 \beta_1) q^{48} - q^{49} + (\beta_{2} - 12) q^{51} + ( - 5 \beta_{3} - 3 \beta_1) q^{52} + ( - 5 \beta_{3} + 2 \beta_1) q^{53} + (\beta_{2} - 2) q^{54} + (2 \beta_{2} + 1) q^{56} + (12 \beta_{3} + 9 \beta_1) q^{57} + ( - 2 \beta_{3} + 5 \beta_1) q^{58} + ( - 7 \beta_{2} - 4) q^{59} + (\beta_{2} + 4) q^{61} + (6 \beta_{3} - 6 \beta_1) q^{62} + 2 \beta_{3} q^{63} + (2 \beta_{2} - 1) q^{64} + (\beta_{2} - 2) q^{66} + (4 \beta_{3} - 3 \beta_1) q^{67} + (7 \beta_{3} + 2 \beta_1) q^{68} + (3 \beta_{2} + 4) q^{69} + ( - \beta_{2} - 5) q^{71} + ( - 2 \beta_{3} - 4 \beta_1) q^{72} + (6 \beta_{3} - 2 \beta_1) q^{73} + ( - 6 \beta_{2} - 2) q^{74} + (6 \beta_{2} + 9) q^{76} - \beta_{3} q^{77} + (4 \beta_{3} + 3 \beta_1) q^{78} + (3 \beta_{2} + 14) q^{79} - 11 q^{81} + ( - 3 \beta_{3} - 6 \beta_1) q^{82} + (3 \beta_{3} - 3 \beta_1) q^{83} + (\beta_{2} + 3) q^{84} + ( - 3 \beta_{2} - 6) q^{86} + ( - \beta_{3} + 8 \beta_1) q^{87} + (\beta_{3} + 2 \beta_1) q^{88} + (2 \beta_{2} + 6) q^{89} + ( - 2 \beta_{2} - 3) q^{91} + ( - 3 \beta_{3} - 2 \beta_1) q^{92} + (12 \beta_{3} - 6 \beta_1) q^{93} + (\beta_{2} + 7) q^{94} + ( - 9 \beta_{2} - 7) q^{96} + ( - 9 \beta_{3} + 4 \beta_1) q^{97} - \beta_1 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9} + 4 q^{11} - 2 q^{14} - 6 q^{16} + 18 q^{19} - 20 q^{24} + 6 q^{26} + 16 q^{29} - 12 q^{31} - 26 q^{34} - 4 q^{36} + 20 q^{39} - 30 q^{41} + 2 q^{44} + 2 q^{46} - 4 q^{49} - 50 q^{51} - 10 q^{54} - 2 q^{59} + 14 q^{61} - 8 q^{64} - 10 q^{66} + 10 q^{69} - 18 q^{71} + 4 q^{74} + 24 q^{76} + 50 q^{79} - 44 q^{81} + 10 q^{84} - 18 q^{86} + 20 q^{89} - 8 q^{91} + 26 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 2.23607i −0.618034 0 −3.61803 1.00000i 2.23607i −2.00000 0
1849.2 0.618034i 2.23607i 1.61803 0 −1.38197 1.00000i 2.23607i −2.00000 0
1849.3 0.618034i 2.23607i 1.61803 0 −1.38197 1.00000i 2.23607i −2.00000 0
1849.4 1.61803i 2.23607i −0.618034 0 −3.61803 1.00000i 2.23607i −2.00000 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.l 4
5.b even 2 1 inner 1925.2.b.l 4
5.c odd 4 1 1925.2.a.o 2
5.c odd 4 1 1925.2.a.t yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.2.a.o 2 5.c odd 4 1
1925.2.a.t yes 2 5.c odd 4 1
1925.2.b.l 4 1.a even 1 1 trivial
1925.2.b.l 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{19}^{2} - 9T_{19} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) 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
$13$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 63T^{2} + 961 \) Copy content Toggle raw display
$19$ \( (T^{2} - 9 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 11)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} + 15 T + 45)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 162T^{2} + 81 \) Copy content Toggle raw display
$47$ \( T^{4} + 163T^{2} + 1681 \) Copy content Toggle raw display
$53$ \( T^{4} + 82T^{2} + 961 \) Copy content Toggle raw display
$59$ \( (T^{2} + T - 61)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 83T^{2} + 361 \) Copy content Toggle raw display
$71$ \( (T^{2} + 9 T + 19)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T^{2} - 25 T + 145)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 63T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 282 T^{2} + 10201 \) Copy content Toggle raw display
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