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Properties

Label 504.2.bf.b
Level $504$
Weight $2$
Character orbit 504.bf
Analytic conductor $4.024$
Analytic rank $0$
Dimension $180$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(115,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.115");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bf (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(90\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 180 q - 6 q^{2} - 6 q^{3} - 2 q^{4} - 6 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 180 q - 6 q^{2} - 6 q^{3} - 2 q^{4} - 6 q^{6} + 10 q^{9} - 8 q^{11} + 12 q^{12} + 10 q^{14} + 14 q^{16} - 18 q^{17} + 8 q^{18} - 6 q^{19} + 36 q^{20} - 16 q^{22} - 12 q^{24} - 78 q^{25} - 6 q^{26} + 16 q^{28} + q^{30} - 26 q^{32} - 36 q^{33} - 12 q^{34} - 12 q^{35} + 2 q^{36} - 27 q^{38} + 24 q^{40} - 42 q^{41} + 22 q^{42} + 14 q^{43} - 21 q^{44} - 12 q^{46} - 9 q^{48} + 2 q^{49} + 15 q^{50} + 9 q^{52} - 51 q^{54} + 14 q^{56} - 26 q^{57} + 19 q^{58} + 37 q^{60} - 8 q^{64} + 24 q^{65} + 6 q^{66} + 28 q^{67} + 12 q^{68} + 27 q^{70} - 28 q^{72} + 18 q^{73} + 49 q^{74} - 18 q^{75} - 12 q^{76} - 33 q^{78} - 63 q^{80} - 22 q^{81} - 54 q^{82} + 6 q^{83} + 31 q^{84} - 13 q^{86} + 29 q^{88} - 66 q^{89} - 51 q^{90} + 2 q^{91} - 60 q^{92} - 45 q^{96} - 6 q^{97} + 31 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −1.41416 − 0.0128144i 1.65702 + 0.504266i 1.99967 + 0.0362430i 2.12331 − 3.67768i −2.33682 − 0.734345i 2.60163 + 0.481180i −2.82738 − 0.0768778i 2.49143 + 1.67116i −3.04982 + 5.17360i
115.2 −1.41416 + 0.0128144i 1.65702 + 0.504266i 1.99967 − 0.0362430i −2.12331 + 3.67768i −2.34975 − 0.691877i −2.60163 − 0.481180i −2.82738 + 0.0768778i 2.49143 + 1.67116i 2.95556 − 5.22802i
115.3 −1.41229 − 0.0736946i −1.11864 − 1.32236i 1.98914 + 0.208157i 1.36131 − 2.35786i 1.48240 + 1.95000i −1.25248 + 2.33051i −2.79390 − 0.440567i −0.497287 + 2.95850i −2.09633 + 3.22967i
115.4 −1.41229 + 0.0736946i −1.11864 − 1.32236i 1.98914 − 0.208157i −1.36131 + 2.35786i 1.67730 + 1.78512i 1.25248 − 2.33051i −2.79390 + 0.440567i −0.497287 + 2.95850i 1.74881 − 3.43031i
115.5 −1.40172 − 0.187532i 0.894629 − 1.48312i 1.92966 + 0.525737i −0.356494 + 0.617465i −1.53216 + 1.91115i −0.577640 + 2.58192i −2.60626 − 1.09881i −1.39928 − 2.65368i 0.615501 − 0.798662i
115.6 −1.40172 + 0.187532i 0.894629 − 1.48312i 1.92966 − 0.525737i 0.356494 − 0.617465i −0.975890 + 2.24669i 0.577640 − 2.58192i −2.60626 + 1.09881i −1.39928 − 2.65368i −0.383911 + 0.932370i
115.7 −1.37865 − 0.315175i −0.265334 + 1.71161i 1.80133 + 0.869029i −1.58477 + 2.74491i 0.905257 − 2.27607i 2.63076 + 0.281244i −2.20950 − 1.76582i −2.85920 − 0.908295i 3.04997 − 3.28478i
115.8 −1.37865 + 0.315175i −0.265334 + 1.71161i 1.80133 − 0.869029i 1.58477 − 2.74491i −0.173654 − 2.44333i −2.63076 − 0.281244i −2.20950 + 1.76582i −2.85920 − 0.908295i −1.31972 + 4.28374i
115.9 −1.37396 − 0.335011i −1.69738 + 0.344799i 1.77554 + 0.920584i 0.237272 − 0.410968i 2.44765 + 0.0949016i −2.23531 − 1.41542i −2.13111 − 1.85967i 2.76223 − 1.17051i −0.463682 + 0.485165i
115.10 −1.37396 + 0.335011i −1.69738 + 0.344799i 1.77554 − 0.920584i −0.237272 + 0.410968i 2.21663 − 1.04238i 2.23531 + 1.41542i −2.13111 + 1.85967i 2.76223 − 1.17051i 0.188324 − 0.644143i
115.11 −1.32350 − 0.498350i 0.336489 + 1.69905i 1.50330 + 1.31913i 0.402355 − 0.696900i 0.401379 − 2.41638i 0.321531 − 2.62614i −1.33222 − 2.49503i −2.77355 + 1.14342i −0.879816 + 0.721832i
115.12 −1.32350 + 0.498350i 0.336489 + 1.69905i 1.50330 − 1.31913i −0.402355 + 0.696900i −1.29206 − 2.08100i −0.321531 + 2.62614i −1.33222 + 2.49503i −2.77355 + 1.14342i 0.185217 − 1.12286i
115.13 −1.29841 − 0.560472i 1.69247 − 0.368153i 1.37174 + 1.45545i −0.863157 + 1.49503i −2.40386 − 0.470569i 2.59622 + 0.509573i −0.965350 − 2.65859i 2.72893 − 1.24618i 1.95866 − 1.45739i
115.14 −1.29841 + 0.560472i 1.69247 − 0.368153i 1.37174 − 1.45545i 0.863157 − 1.49503i −1.99119 + 1.42660i −2.59622 − 0.509573i −0.965350 + 2.65859i 2.72893 − 1.24618i −0.282810 + 2.42494i
115.15 −1.16893 − 0.795991i −0.801348 − 1.53553i 0.732796 + 1.86092i −1.06898 + 1.85154i −0.285545 + 2.43279i −2.52157 − 0.801064i 0.624685 − 2.75858i −1.71568 + 2.46098i 2.72337 − 1.31341i
115.16 −1.16893 + 0.795991i −0.801348 − 1.53553i 0.732796 − 1.86092i 1.06898 − 1.85154i 2.15899 + 1.15706i 2.52157 + 0.801064i 0.624685 + 2.75858i −1.71568 + 2.46098i 0.224237 + 3.01522i
115.17 −1.15708 − 0.813118i 1.26585 + 1.18221i 0.677678 + 1.88169i 0.495626 − 0.858449i −0.503410 − 2.39720i −1.90467 + 1.83636i 0.745907 − 2.72830i 0.204739 + 2.99301i −1.27150 + 0.590294i
115.18 −1.15708 + 0.813118i 1.26585 + 1.18221i 0.677678 − 1.88169i −0.495626 + 0.858449i −2.42597 − 0.338635i 1.90467 − 1.83636i 0.745907 + 2.72830i 0.204739 + 2.99301i −0.124541 − 1.39630i
115.19 −1.14308 − 0.832691i −1.17641 + 1.27125i 0.613252 + 1.90366i 1.57644 − 2.73047i 2.40328 − 0.473549i 1.25267 + 2.33041i 0.884165 − 2.68668i −0.232133 − 2.99101i −4.07562 + 1.80845i
115.20 −1.14308 + 0.832691i −1.17641 + 1.27125i 0.613252 − 1.90366i −1.57644 + 2.73047i 0.286170 − 2.43272i −1.25267 − 2.33041i 0.884165 + 2.68668i −0.232133 − 2.99101i −0.471646 − 4.43382i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
63.t odd 6 1 inner
504.bf even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bf.b ✓ 180
7.d odd 6 1 504.2.cz.b yes 180
8.d odd 2 1 inner 504.2.bf.b ✓ 180
9.c even 3 1 504.2.cz.b yes 180
56.m even 6 1 504.2.cz.b yes 180
63.t odd 6 1 inner 504.2.bf.b ✓ 180
72.p odd 6 1 504.2.cz.b yes 180
504.bf even 6 1 inner 504.2.bf.b ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bf.b ✓ 180 1.a even 1 1 trivial
504.2.bf.b ✓ 180 8.d odd 2 1 inner
504.2.bf.b ✓ 180 63.t odd 6 1 inner
504.2.bf.b ✓ 180 504.bf even 6 1 inner
504.2.cz.b yes 180 7.d odd 6 1
504.2.cz.b yes 180 9.c even 3 1
504.2.cz.b yes 180 56.m even 6 1
504.2.cz.b yes 180 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{180} + 264 T_{5}^{178} + 36267 T_{5}^{176} + 3416808 T_{5}^{174} + 246532878 T_{5}^{172} + \cdots + 86\!\cdots\!04 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display