Properties

Label 6025.2.a.e
Level $6025$
Weight $2$
Character orbit 6025.a
Self dual yes
Analytic conductor $48.110$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.38569.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{4} - \beta_{2} - 2) q^{6} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{4} - \beta_{2} - 2) q^{6} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{9} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{11} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{12} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 1) q^{14} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{16} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{17} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{18} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{21} + (\beta_{4} - \beta_{3} - \beta_1 + 3) q^{22} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{23} + (\beta_{3} - 3 \beta_1 - 3) q^{24} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 3) q^{26} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{27} + (3 \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{28} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} + (\beta_{4} - 2 \beta_1 - 3) q^{31} + (\beta_{4} - 2 \beta_{2} + 3 \beta_1 + 3) q^{32} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{33} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{34} + (3 \beta_{4} - \beta_{2} - 3) q^{36} + (2 \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{37} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 5) q^{38} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{39} + (\beta_{2} + 2) q^{41} + ( - 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 5) q^{42} + (\beta_{4} + 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 7) q^{43} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{44} + (\beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 1) q^{46} + ( - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 1) q^{47} + ( - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{48} + ( - 2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 2) q^{49} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{51} + ( - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 2) q^{52} + ( - \beta_{4} - \beta_{2} - \beta_1 + 6) q^{53} + (2 \beta_{4} + \beta_{2} + 3 \beta_1 + 7) q^{54} + (\beta_{4} + 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 4) q^{56} + ( - 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{57} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{58} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{59} + ( - 2 \beta_{4} - \beta_{2} + 5 \beta_1 - 3) q^{61} + ( - 4 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 3) q^{62} + ( - 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 1) q^{63} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{64} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{66} + ( - 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 8 \beta_1 + 2) q^{68} + (3 \beta_{2} - 7 \beta_1 + 2) q^{69} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 1) q^{71} + (2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{72} + (2 \beta_{4} - \beta_{2} + \beta_1 + 6) q^{73} + (2 \beta_{4} + 3 \beta_{2} + 2 \beta_1 + 3) q^{74} + (\beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 2) q^{76} + ( - \beta_{4} + 4 \beta_{3} - \beta_1) q^{77} + ( - \beta_{4} - 3 \beta_{3} - 4 \beta_1) q^{78} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{79} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{81} + (3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{82} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{83} + (6 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 9 \beta_1 + 4) q^{84} + (2 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 10 \beta_1 + 2) q^{86} + ( - 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{87} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{88} + (3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 7 \beta_1 + 4) q^{89} + (3 \beta_{3} + \beta_{2} + 2) q^{91} + (3 \beta_{4} + 3 \beta_{3} + 9 \beta_1 + 5) q^{92} + (5 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + \beta_1 - 2) q^{93} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{94} + ( - 3 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{96} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 8) q^{97} + ( - 5 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 8) q^{98} + ( - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 5 q^{3} + 3 q^{4} - 8 q^{6} + 10 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} + 5 q^{3} + 3 q^{4} - 8 q^{6} + 10 q^{7} + 6 q^{9} - 3 q^{11} - 8 q^{12} + q^{13} - q^{14} + 15 q^{16} + 5 q^{17} - 4 q^{18} + 3 q^{19} + 11 q^{21} + 13 q^{22} + 8 q^{23} - 14 q^{24} + 14 q^{26} + 14 q^{27} + 17 q^{28} + 9 q^{29} - 16 q^{31} + 16 q^{32} - 23 q^{33} - 10 q^{34} - 17 q^{36} - 7 q^{37} + 22 q^{38} - 19 q^{39} + 9 q^{41} - 17 q^{42} + 32 q^{43} - 8 q^{44} + 5 q^{46} + 7 q^{47} + 6 q^{48} + 9 q^{49} - 8 q^{51} + 10 q^{52} + 32 q^{53} + 32 q^{54} - 18 q^{56} - 3 q^{57} - 11 q^{58} - 8 q^{59} - 12 q^{61} - 17 q^{62} + 11 q^{63} - 16 q^{64} + 15 q^{66} + 5 q^{67} + 2 q^{68} + 7 q^{69} - 11 q^{71} - 7 q^{72} + 29 q^{73} + 10 q^{74} - 8 q^{76} + 5 q^{77} - 2 q^{78} + 16 q^{79} - 15 q^{81} + 2 q^{82} + 10 q^{83} + 5 q^{84} + 14 q^{86} + 37 q^{87} - 10 q^{88} + 9 q^{89} + 12 q^{91} + 25 q^{92} - 15 q^{93} - 11 q^{94} - 3 q^{96} + 43 q^{97} - 30 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 5x^{3} + 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 4\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 5\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + \nu^{3} - 5\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} + 5\beta_{2} + 7 \) 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
0.275834
−1.95408
0.790734
−1.15098
2.03850
−2.34953 1.80652 3.52030 0 −4.24447 4.19975 −3.57200 0.263500 0
1.2 −0.442330 2.59927 −1.80434 0 −1.14974 −1.77251 1.68277 3.75621 0
1.3 0.526087 2.87779 −1.72323 0 1.51397 4.16547 −1.95874 5.28166 0
1.4 0.717838 −0.928169 −1.48471 0 −0.666275 1.52425 −2.50146 −2.13850 0
1.5 2.54794 −1.35541 4.49198 0 −3.45349 1.88303 6.34942 −1.16287 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.e 5
5.b even 2 1 1205.2.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.a 5 5.b even 2 1
6025.2.a.e 5 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{5} - T_{2}^{4} - 6T_{2}^{3} + 5T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{5} - 5T_{3}^{4} + 2T_{3}^{3} + 17T_{3}^{2} - 9T_{3} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} - 6 T^{3} + 5 T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{5} - 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 10 T^{4} + 28 T^{3} + 3 T^{2} + \cdots + 89 \) Copy content Toggle raw display
$11$ \( T^{5} + 3 T^{4} - 25 T^{3} - 68 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{5} - T^{4} - 17 T^{3} + 18 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{5} - 5 T^{4} - 52 T^{3} + 159 T^{2} + \cdots - 361 \) Copy content Toggle raw display
$19$ \( T^{5} - 3 T^{4} - 23 T^{3} + 58 T^{2} + \cdots - 91 \) Copy content Toggle raw display
$23$ \( T^{5} - 8 T^{4} - 10 T^{3} + 121 T^{2} + \cdots - 259 \) Copy content Toggle raw display
$29$ \( T^{5} - 9 T^{4} - 7 T^{3} + 262 T^{2} + \cdots + 479 \) Copy content Toggle raw display
$31$ \( T^{5} + 16 T^{4} + 75 T^{3} + 79 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$37$ \( T^{5} + 7 T^{4} - 49 T^{3} + \cdots + 3757 \) Copy content Toggle raw display
$41$ \( T^{5} - 9 T^{4} + 27 T^{3} - 27 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$43$ \( T^{5} - 32 T^{4} + 341 T^{3} + \cdots + 2953 \) Copy content Toggle raw display
$47$ \( T^{5} - 7 T^{4} - 25 T^{3} + 177 T^{2} + \cdots - 593 \) Copy content Toggle raw display
$53$ \( T^{5} - 32 T^{4} + 390 T^{3} + \cdots - 5687 \) Copy content Toggle raw display
$59$ \( T^{5} + 8 T^{4} - 4 T^{3} - 181 T^{2} + \cdots - 421 \) Copy content Toggle raw display
$61$ \( T^{5} + 12 T^{4} - 79 T^{3} + \cdots + 20479 \) Copy content Toggle raw display
$67$ \( T^{5} - 5 T^{4} - 183 T^{3} + \cdots - 23093 \) Copy content Toggle raw display
$71$ \( T^{5} + 11 T^{4} - 62 T^{3} + \cdots - 289 \) Copy content Toggle raw display
$73$ \( T^{5} - 29 T^{4} + 311 T^{3} + \cdots - 2251 \) Copy content Toggle raw display
$79$ \( T^{5} - 16 T^{4} + 29 T^{3} + 145 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{5} - 10 T^{4} - 117 T^{3} + \cdots - 9649 \) Copy content Toggle raw display
$89$ \( T^{5} - 9 T^{4} - 293 T^{3} + \cdots + 20647 \) Copy content Toggle raw display
$97$ \( T^{5} - 43 T^{4} + 631 T^{3} + \cdots - 2219 \) Copy content Toggle raw display
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