Properties

Label 475.4.a.a
Level $475$
Weight $4$
Character orbit 475.a
Self dual yes
Analytic conductor $28.026$
Analytic rank $1$
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] = [475,4,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(28.0259072527\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} - 4 q^{3} + 17 q^{4} + 20 q^{6} + 32 q^{7} - 45 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - 4 q^{3} + 17 q^{4} + 20 q^{6} + 32 q^{7} - 45 q^{8} - 11 q^{9} - 12 q^{11} - 68 q^{12} + 42 q^{13} - 160 q^{14} + 89 q^{16} - 114 q^{17} + 55 q^{18} + 19 q^{19} - 128 q^{21} + 60 q^{22} - 160 q^{23} + 180 q^{24} - 210 q^{26} + 152 q^{27} + 544 q^{28} + 214 q^{29} - 144 q^{31} - 85 q^{32} + 48 q^{33} + 570 q^{34} - 187 q^{36} - 94 q^{37} - 95 q^{38} - 168 q^{39} - 6 q^{41} + 640 q^{42} + 308 q^{43} - 204 q^{44} + 800 q^{46} - 184 q^{47} - 356 q^{48} + 681 q^{49} + 456 q^{51} + 714 q^{52} + 274 q^{53} - 760 q^{54} - 1440 q^{56} - 76 q^{57} - 1070 q^{58} + 276 q^{59} - 826 q^{61} + 720 q^{62} - 352 q^{63} - 287 q^{64} - 240 q^{66} - 52 q^{67} - 1938 q^{68} + 640 q^{69} - 344 q^{71} + 495 q^{72} + 166 q^{73} + 470 q^{74} + 323 q^{76} - 384 q^{77} + 840 q^{78} - 688 q^{79} - 311 q^{81} + 30 q^{82} - 996 q^{83} - 2176 q^{84} - 1540 q^{86} - 856 q^{87} + 540 q^{88} + 1578 q^{89} + 1344 q^{91} - 2720 q^{92} + 576 q^{93} + 920 q^{94} + 340 q^{96} - 786 q^{97} - 3405 q^{98} + 132 q^{99}+O(q^{100}) \) 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 −4.00000 17.0000 0 20.0000 32.0000 −45.0000 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 475.4.a.a 1
5.b even 2 1 95.4.a.d 1
5.c odd 4 2 475.4.b.a 2
15.d odd 2 1 855.4.a.a 1
20.d odd 2 1 1520.4.a.c 1
95.d odd 2 1 1805.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.a.d 1 5.b even 2 1
475.4.a.a 1 1.a even 1 1 trivial
475.4.b.a 2 5.c odd 4 2
855.4.a.a 1 15.d odd 2 1
1520.4.a.c 1 20.d odd 2 1
1805.4.a.a 1 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(475))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 32 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T - 42 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T + 160 \) Copy content Toggle raw display
$29$ \( T - 214 \) Copy content Toggle raw display
$31$ \( T + 144 \) Copy content Toggle raw display
$37$ \( T + 94 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 308 \) Copy content Toggle raw display
$47$ \( T + 184 \) Copy content Toggle raw display
$53$ \( T - 274 \) Copy content Toggle raw display
$59$ \( T - 276 \) Copy content Toggle raw display
$61$ \( T + 826 \) Copy content Toggle raw display
$67$ \( T + 52 \) Copy content Toggle raw display
$71$ \( T + 344 \) Copy content Toggle raw display
$73$ \( T - 166 \) Copy content Toggle raw display
$79$ \( T + 688 \) Copy content Toggle raw display
$83$ \( T + 996 \) Copy content Toggle raw display
$89$ \( T - 1578 \) Copy content Toggle raw display
$97$ \( T + 786 \) Copy content Toggle raw display
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