Properties

Label 6025.2.a.a
Level $6025$
Weight $2$
Character orbit 6025.a
Self dual yes
Analytic conductor $48.110$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6025,2,Mod(1,6025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)

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: yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} - q^{4} - \beta q^{6} + (2 \beta - 2) q^{7} + 3 q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta q^{3} - q^{4} - \beta q^{6} + (2 \beta - 2) q^{7} + 3 q^{8} + (\beta - 2) q^{9} + (\beta + 2) q^{11} - \beta q^{12} + ( - \beta + 2) q^{13} + ( - 2 \beta + 2) q^{14} - q^{16} + (3 \beta - 1) q^{17} + ( - \beta + 2) q^{18} + (\beta - 1) q^{19} + 2 q^{21} + ( - \beta - 2) q^{22} + ( - 4 \beta - 2) q^{23} + 3 \beta q^{24} + (\beta - 2) q^{26} + ( - 4 \beta + 1) q^{27} + ( - 2 \beta + 2) q^{28} + ( - 5 \beta + 1) q^{29} + (4 \beta - 8) q^{31} - 5 q^{32} + (3 \beta + 1) q^{33} + ( - 3 \beta + 1) q^{34} + ( - \beta + 2) q^{36} + ( - 4 \beta + 2) q^{37} + ( - \beta + 1) q^{38} + (\beta - 1) q^{39} + ( - 5 \beta + 4) q^{41} - 2 q^{42} - 4 \beta q^{43} + ( - \beta - 2) q^{44} + (4 \beta + 2) q^{46} + ( - 3 \beta - 3) q^{47} - \beta q^{48} + ( - 4 \beta + 1) q^{49} + (2 \beta + 3) q^{51} + (\beta - 2) q^{52} + 4 q^{53} + (4 \beta - 1) q^{54} + (6 \beta - 6) q^{56} + q^{57} + (5 \beta - 1) q^{58} + (2 \beta - 6) q^{59} + 3 \beta q^{61} + ( - 4 \beta + 8) q^{62} + ( - 4 \beta + 6) q^{63} + 7 q^{64} + ( - 3 \beta - 1) q^{66} + ( - 11 \beta + 5) q^{67} + ( - 3 \beta + 1) q^{68} + ( - 6 \beta - 4) q^{69} + ( - 11 \beta + 3) q^{71} + (3 \beta - 6) q^{72} + (5 \beta - 7) q^{73} + (4 \beta - 2) q^{74} + ( - \beta + 1) q^{76} + (4 \beta - 2) q^{77} + ( - \beta + 1) q^{78} + (14 \beta - 8) q^{79} + ( - 6 \beta + 2) q^{81} + (5 \beta - 4) q^{82} + ( - \beta + 15) q^{83} - 2 q^{84} + 4 \beta q^{86} + ( - 4 \beta - 5) q^{87} + (3 \beta + 6) q^{88} + (4 \beta + 10) q^{89} + (4 \beta - 6) q^{91} + (4 \beta + 2) q^{92} + ( - 4 \beta + 4) q^{93} + (3 \beta + 3) q^{94} - 5 \beta q^{96} + ( - 2 \beta + 16) q^{97} + (4 \beta - 1) q^{98} + (\beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9} + 5 q^{11} - q^{12} + 3 q^{13} + 2 q^{14} - 2 q^{16} + q^{17} + 3 q^{18} - q^{19} + 4 q^{21} - 5 q^{22} - 8 q^{23} + 3 q^{24} - 3 q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} - 12 q^{31} - 10 q^{32} + 5 q^{33} - q^{34} + 3 q^{36} + q^{38} - q^{39} + 3 q^{41} - 4 q^{42} - 4 q^{43} - 5 q^{44} + 8 q^{46} - 9 q^{47} - q^{48} - 2 q^{49} + 8 q^{51} - 3 q^{52} + 8 q^{53} + 2 q^{54} - 6 q^{56} + 2 q^{57} + 3 q^{58} - 10 q^{59} + 3 q^{61} + 12 q^{62} + 8 q^{63} + 14 q^{64} - 5 q^{66} - q^{67} - q^{68} - 14 q^{69} - 5 q^{71} - 9 q^{72} - 9 q^{73} + q^{76} + q^{78} - 2 q^{79} - 2 q^{81} - 3 q^{82} + 29 q^{83} - 4 q^{84} + 4 q^{86} - 14 q^{87} + 15 q^{88} + 24 q^{89} - 8 q^{91} + 8 q^{92} + 4 q^{93} + 9 q^{94} - 5 q^{96} + 30 q^{97} + 2 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
−1.00000 −0.618034 −1.00000 0 0.618034 −3.23607 3.00000 −2.61803 0
1.2 −1.00000 1.61803 −1.00000 0 −1.61803 1.23607 3.00000 −0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(241\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.a 2
5.b even 2 1 6025.2.a.d 2
5.c odd 4 2 1205.2.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.b 4 5.c odd 4 2
6025.2.a.a 2 1.a even 1 1 trivial
6025.2.a.d 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 151 \) Copy content Toggle raw display
$71$ \( T^{2} + 5T - 145 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 244 \) Copy content Toggle raw display
$83$ \( T^{2} - 29T + 209 \) Copy content Toggle raw display
$89$ \( T^{2} - 24T + 124 \) Copy content Toggle raw display
$97$ \( T^{2} - 30T + 220 \) Copy content Toggle raw display
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