Properties

Label 825.4.a.b
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 3 q^{3} + 8 q^{4} - 12 q^{6} + 21 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 3 q^{3} + 8 q^{4} - 12 q^{6} + 21 q^{7} + 9 q^{9} + 11 q^{11} + 24 q^{12} - 68 q^{13} - 84 q^{14} - 64 q^{16} + 21 q^{17} - 36 q^{18} + 125 q^{19} + 63 q^{21} - 44 q^{22} + 137 q^{23} + 272 q^{26} + 27 q^{27} + 168 q^{28} - 150 q^{29} + 292 q^{31} + 256 q^{32} + 33 q^{33} - 84 q^{34} + 72 q^{36} - 349 q^{37} - 500 q^{38} - 204 q^{39} + 497 q^{41} - 252 q^{42} - 208 q^{43} + 88 q^{44} - 548 q^{46} - 369 q^{47} - 192 q^{48} + 98 q^{49} + 63 q^{51} - 544 q^{52} + 542 q^{53} - 108 q^{54} + 375 q^{57} + 600 q^{58} + 235 q^{59} + 482 q^{61} - 1168 q^{62} + 189 q^{63} - 512 q^{64} - 132 q^{66} - 734 q^{67} + 168 q^{68} + 411 q^{69} + 587 q^{71} - 518 q^{73} + 1396 q^{74} + 1000 q^{76} + 231 q^{77} + 816 q^{78} - 1045 q^{79} + 81 q^{81} - 1988 q^{82} - 608 q^{83} + 504 q^{84} + 832 q^{86} - 450 q^{87} - 770 q^{89} - 1428 q^{91} + 1096 q^{92} + 876 q^{93} + 1476 q^{94} + 768 q^{96} + 1541 q^{97} - 392 q^{98} + 99 q^{99}+O(q^{100}) \) 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
−4.00000 3.00000 8.00000 0 −12.0000 21.0000 0 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.b 1
3.b odd 2 1 2475.4.a.j 1
5.b even 2 1 825.4.a.h yes 1
5.c odd 4 2 825.4.c.c 2
15.d odd 2 1 2475.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.b 1 1.a even 1 1 trivial
825.4.a.h yes 1 5.b even 2 1
825.4.c.c 2 5.c odd 4 2
2475.4.a.c 1 15.d odd 2 1
2475.4.a.j 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2} + 4 \) Copy content Toggle raw display
\( T_{7} - 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 21 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 68 \) Copy content Toggle raw display
$17$ \( T - 21 \) Copy content Toggle raw display
$19$ \( T - 125 \) Copy content Toggle raw display
$23$ \( T - 137 \) Copy content Toggle raw display
$29$ \( T + 150 \) Copy content Toggle raw display
$31$ \( T - 292 \) Copy content Toggle raw display
$37$ \( T + 349 \) Copy content Toggle raw display
$41$ \( T - 497 \) Copy content Toggle raw display
$43$ \( T + 208 \) Copy content Toggle raw display
$47$ \( T + 369 \) Copy content Toggle raw display
$53$ \( T - 542 \) Copy content Toggle raw display
$59$ \( T - 235 \) Copy content Toggle raw display
$61$ \( T - 482 \) Copy content Toggle raw display
$67$ \( T + 734 \) Copy content Toggle raw display
$71$ \( T - 587 \) Copy content Toggle raw display
$73$ \( T + 518 \) Copy content Toggle raw display
$79$ \( T + 1045 \) Copy content Toggle raw display
$83$ \( T + 608 \) Copy content Toggle raw display
$89$ \( T + 770 \) Copy content Toggle raw display
$97$ \( T - 1541 \) Copy content Toggle raw display
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