Properties

Label 825.4.a.a
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $1$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} - 3 q^{3} + 17 q^{4} + 15 q^{6} - 3 q^{7} - 45 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - 3 q^{3} + 17 q^{4} + 15 q^{6} - 3 q^{7} - 45 q^{8} + 9 q^{9} - 11 q^{11} - 51 q^{12} - 32 q^{13} + 15 q^{14} + 89 q^{16} - 33 q^{17} - 45 q^{18} + 47 q^{19} + 9 q^{21} + 55 q^{22} - 113 q^{23} + 135 q^{24} + 160 q^{26} - 27 q^{27} - 51 q^{28} - 54 q^{29} + 178 q^{31} - 85 q^{32} + 33 q^{33} + 165 q^{34} + 153 q^{36} - 19 q^{37} - 235 q^{38} + 96 q^{39} + 139 q^{41} - 45 q^{42} + 308 q^{43} - 187 q^{44} + 565 q^{46} - 195 q^{47} - 267 q^{48} - 334 q^{49} + 99 q^{51} - 544 q^{52} - 152 q^{53} + 135 q^{54} + 135 q^{56} - 141 q^{57} + 270 q^{58} - 625 q^{59} + 320 q^{61} - 890 q^{62} - 27 q^{63} - 287 q^{64} - 165 q^{66} - 200 q^{67} - 561 q^{68} + 339 q^{69} - 947 q^{71} - 405 q^{72} + 448 q^{73} + 95 q^{74} + 799 q^{76} + 33 q^{77} - 480 q^{78} - 721 q^{79} + 81 q^{81} - 695 q^{82} - 142 q^{83} + 153 q^{84} - 1540 q^{86} + 162 q^{87} + 495 q^{88} + 404 q^{89} + 96 q^{91} - 1921 q^{92} - 534 q^{93} + 975 q^{94} + 255 q^{96} - 79 q^{97} + 1670 q^{98} - 99 q^{99}+O(q^{100}) \) 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
−5.00000 −3.00000 17.0000 0 15.0000 −3.00000 −45.0000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 825.4.a.a 1
3.b odd 2 1 2475.4.a.k 1
5.b even 2 1 825.4.a.j yes 1
5.c odd 4 2 825.4.c.b 2
15.d odd 2 1 2475.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.a 1 1.a even 1 1 trivial
825.4.a.j yes 1 5.b even 2 1
825.4.c.b 2 5.c odd 4 2
2475.4.a.a 1 15.d odd 2 1
2475.4.a.k 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 32 \) Copy content Toggle raw display
$17$ \( T + 33 \) Copy content Toggle raw display
$19$ \( T - 47 \) Copy content Toggle raw display
$23$ \( T + 113 \) Copy content Toggle raw display
$29$ \( T + 54 \) Copy content Toggle raw display
$31$ \( T - 178 \) Copy content Toggle raw display
$37$ \( T + 19 \) Copy content Toggle raw display
$41$ \( T - 139 \) Copy content Toggle raw display
$43$ \( T - 308 \) Copy content Toggle raw display
$47$ \( T + 195 \) Copy content Toggle raw display
$53$ \( T + 152 \) Copy content Toggle raw display
$59$ \( T + 625 \) Copy content Toggle raw display
$61$ \( T - 320 \) Copy content Toggle raw display
$67$ \( T + 200 \) Copy content Toggle raw display
$71$ \( T + 947 \) Copy content Toggle raw display
$73$ \( T - 448 \) Copy content Toggle raw display
$79$ \( T + 721 \) Copy content Toggle raw display
$83$ \( T + 142 \) Copy content Toggle raw display
$89$ \( T - 404 \) Copy content Toggle raw display
$97$ \( T + 79 \) Copy content Toggle raw display
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