Properties

Label 531.4.a.b
Level $531$
Weight $4$
Character orbit 531.a
Self dual yes
Analytic conductor $31.330$
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] = [531,4,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(31.3300142130\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 59)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{2} + (4 \beta + 1) q^{4} + ( - 2 \beta - 3) q^{5} + (2 \beta - 11) q^{7} + (\beta + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2) q^{2} + (4 \beta + 1) q^{4} + ( - 2 \beta - 3) q^{5} + (2 \beta - 11) q^{7} + (\beta + 6) q^{8} + ( - 7 \beta - 16) q^{10} + (2 \beta + 24) q^{11} + ( - 22 \beta - 16) q^{13} + ( - 7 \beta - 12) q^{14} + ( - 24 \beta + 9) q^{16} + ( - 24 \beta + 42) q^{17} + ( - 20 \beta - 1) q^{19} + ( - 14 \beta - 43) q^{20} + (28 \beta + 58) q^{22} + (2 \beta - 82) q^{23} + (12 \beta - 96) q^{25} + ( - 60 \beta - 142) q^{26} + ( - 42 \beta + 29) q^{28} + (74 \beta - 57) q^{29} + (12 \beta - 62) q^{31} + ( - 47 \beta - 150) q^{32} + ( - 6 \beta - 36) q^{34} + (16 \beta + 13) q^{35} + (6 \beta - 146) q^{37} + ( - 41 \beta - 102) q^{38} + ( - 15 \beta - 28) q^{40} + ( - 40 \beta + 209) q^{41} + (72 \beta - 146) q^{43} + (98 \beta + 64) q^{44} + ( - 78 \beta - 154) q^{46} + ( - 102 \beta - 24) q^{47} + ( - 44 \beta - 202) q^{49} + ( - 72 \beta - 132) q^{50} + ( - 86 \beta - 456) q^{52} + ( - 34 \beta - 163) q^{53} + ( - 54 \beta - 92) q^{55} + (\beta - 56) q^{56} + (91 \beta + 256) q^{58} - 59 q^{59} + (270 \beta - 134) q^{61} + ( - 38 \beta - 64) q^{62} + ( - 52 \beta - 607) q^{64} + (98 \beta + 268) q^{65} + (102 \beta + 460) q^{67} + (144 \beta - 438) q^{68} + (45 \beta + 106) q^{70} + ( - 236 \beta - 328) q^{71} + (252 \beta + 116) q^{73} + ( - 134 \beta - 262) q^{74} + ( - 24 \beta - 401) q^{76} + (26 \beta - 244) q^{77} + ( - 486 \beta - 57) q^{79} + (54 \beta + 213) q^{80} + (129 \beta + 218) q^{82} + (566 \beta - 4) q^{83} + ( - 12 \beta + 114) q^{85} + ( - 2 \beta + 68) q^{86} + (36 \beta + 154) q^{88} + (406 \beta + 426) q^{89} + (210 \beta - 44) q^{91} + ( - 326 \beta - 42) q^{92} + ( - 228 \beta - 558) q^{94} + (62 \beta + 203) q^{95} + ( - 300 \beta - 240) q^{97} + ( - 290 \beta - 624) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{4} - 6 q^{5} - 22 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 2 q^{4} - 6 q^{5} - 22 q^{7} + 12 q^{8} - 32 q^{10} + 48 q^{11} - 32 q^{13} - 24 q^{14} + 18 q^{16} + 84 q^{17} - 2 q^{19} - 86 q^{20} + 116 q^{22} - 164 q^{23} - 192 q^{25} - 284 q^{26} + 58 q^{28} - 114 q^{29} - 124 q^{31} - 300 q^{32} - 72 q^{34} + 26 q^{35} - 292 q^{37} - 204 q^{38} - 56 q^{40} + 418 q^{41} - 292 q^{43} + 128 q^{44} - 308 q^{46} - 48 q^{47} - 404 q^{49} - 264 q^{50} - 912 q^{52} - 326 q^{53} - 184 q^{55} - 112 q^{56} + 512 q^{58} - 118 q^{59} - 268 q^{61} - 128 q^{62} - 1214 q^{64} + 536 q^{65} + 920 q^{67} - 876 q^{68} + 212 q^{70} - 656 q^{71} + 232 q^{73} - 524 q^{74} - 802 q^{76} - 488 q^{77} - 114 q^{79} + 426 q^{80} + 436 q^{82} - 8 q^{83} + 228 q^{85} + 136 q^{86} + 308 q^{88} + 852 q^{89} - 88 q^{91} - 84 q^{92} - 1116 q^{94} + 406 q^{95} - 480 q^{97} - 1248 q^{98}+O(q^{100}) \) 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
−0.236068 0 −7.94427 1.47214 0 −15.4721 3.76393 0 −0.347524
1.2 4.23607 0 9.94427 −7.47214 0 −6.52786 8.23607 0 −31.6525
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

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 531.4.a.b 2
3.b odd 2 1 59.4.a.a 2
12.b even 2 1 944.4.a.f 2
15.d odd 2 1 1475.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.4.a.a 2 3.b odd 2 1
531.4.a.b 2 1.a even 1 1 trivial
944.4.a.f 2 12.b even 2 1
1475.4.a.b 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 4T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} + 22T + 101 \) Copy content Toggle raw display
$11$ \( T^{2} - 48T + 556 \) Copy content Toggle raw display
$13$ \( T^{2} + 32T - 2164 \) Copy content Toggle raw display
$17$ \( T^{2} - 84T - 1116 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 1999 \) Copy content Toggle raw display
$23$ \( T^{2} + 164T + 6704 \) Copy content Toggle raw display
$29$ \( T^{2} + 114T - 24131 \) Copy content Toggle raw display
$31$ \( T^{2} + 124T + 3124 \) Copy content Toggle raw display
$37$ \( T^{2} + 292T + 21136 \) Copy content Toggle raw display
$41$ \( T^{2} - 418T + 35681 \) Copy content Toggle raw display
$43$ \( T^{2} + 292T - 4604 \) Copy content Toggle raw display
$47$ \( T^{2} + 48T - 51444 \) Copy content Toggle raw display
$53$ \( T^{2} + 326T + 20789 \) Copy content Toggle raw display
$59$ \( (T + 59)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 268T - 346544 \) Copy content Toggle raw display
$67$ \( T^{2} - 920T + 159580 \) Copy content Toggle raw display
$71$ \( T^{2} + 656T - 170896 \) Copy content Toggle raw display
$73$ \( T^{2} - 232T - 304064 \) Copy content Toggle raw display
$79$ \( T^{2} + 114 T - 1177731 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 1601764 \) Copy content Toggle raw display
$89$ \( T^{2} - 852T - 642704 \) Copy content Toggle raw display
$97$ \( T^{2} + 480T - 392400 \) Copy content Toggle raw display
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