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Properties

Label 429.2.bb.b
Level $429$
Weight $2$
Character orbit 429.bb
Analytic conductor $3.426$
Analytic rank $0$
Dimension $56$
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] = [429,2,Mod(25,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bb (of order \(10\), degree \(4\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 14 q^{3} + 20 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 14 q^{3} + 20 q^{4} - 14 q^{9} + 60 q^{12} - 3 q^{13} - 42 q^{14} - 16 q^{16} - 6 q^{17} - 40 q^{22} + 12 q^{23} + 4 q^{25} - 24 q^{26} + 14 q^{27} - 22 q^{29} - 10 q^{30} - 10 q^{35} + 20 q^{36} - 26 q^{38} + 3 q^{39} - 18 q^{42} + 6 q^{48} - 14 q^{49} - 24 q^{51} + 39 q^{52} + 10 q^{53} - 64 q^{55} - 20 q^{56} + 2 q^{61} - 8 q^{62} + 24 q^{64} - 58 q^{65} + 30 q^{66} + 18 q^{68} - 2 q^{69} - 86 q^{74} - 14 q^{75} + 52 q^{77} + 14 q^{78} - 24 q^{79} - 14 q^{81} + 62 q^{82} - 28 q^{87} - 46 q^{88} - 10 q^{90} - 19 q^{91} + 58 q^{92} + 86 q^{94} + 156 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −1.59022 − 2.18875i −0.309017 − 0.951057i −1.64379 + 5.05905i 0.714239 − 0.983066i −1.59022 + 2.18875i 1.83379 + 0.595836i 8.54091 − 2.77511i −0.809017 + 0.587785i −3.28748
25.2 −1.28832 − 1.77322i −0.309017 − 0.951057i −0.866514 + 2.66686i −1.37901 + 1.89804i −1.28832 + 1.77322i −1.10854 − 0.360186i 1.67618 − 0.544625i −0.809017 + 0.587785i 5.14227
25.3 −1.20071 − 1.65263i −0.309017 − 0.951057i −0.671457 + 2.06653i −0.326611 + 0.449541i −1.20071 + 1.65263i −1.66332 − 0.540445i 0.335871 − 0.109131i −0.809017 + 0.587785i 1.13509
25.4 −0.812362 − 1.11812i −0.309017 − 0.951057i 0.0277724 − 0.0854748i 2.36802 − 3.25930i −0.812362 + 1.11812i −1.14719 − 0.372746i −2.74699 + 0.892552i −0.809017 + 0.587785i −5.56798
25.5 −0.774047 − 1.06538i −0.309017 − 0.951057i 0.0821387 − 0.252797i −1.70947 + 2.35289i −0.774047 + 1.06538i 4.34559 + 1.41197i −2.83777 + 0.922049i −0.809017 + 0.587785i 3.82995
25.6 −0.386931 − 0.532564i −0.309017 − 0.951057i 0.484125 − 1.48998i −0.663105 + 0.912685i −0.386931 + 0.532564i −3.82419 − 1.24256i −2.23297 + 0.725535i −0.809017 + 0.587785i 0.742639
25.7 −0.0523187 − 0.0720105i −0.309017 − 0.951057i 0.615586 − 1.89458i −1.59536 + 2.19582i −0.0523187 + 0.0720105i −0.897995 − 0.291776i −0.337943 + 0.109804i −0.809017 + 0.587785i 0.241589
25.8 0.0523187 + 0.0720105i −0.309017 − 0.951057i 0.615586 − 1.89458i 1.59536 − 2.19582i 0.0523187 − 0.0720105i 0.897995 + 0.291776i 0.337943 − 0.109804i −0.809017 + 0.587785i 0.241589
25.9 0.386931 + 0.532564i −0.309017 − 0.951057i 0.484125 − 1.48998i 0.663105 − 0.912685i 0.386931 − 0.532564i 3.82419 + 1.24256i 2.23297 − 0.725535i −0.809017 + 0.587785i 0.742639
25.10 0.774047 + 1.06538i −0.309017 − 0.951057i 0.0821387 − 0.252797i 1.70947 − 2.35289i 0.774047 − 1.06538i −4.34559 − 1.41197i 2.83777 − 0.922049i −0.809017 + 0.587785i 3.82995
25.11 0.812362 + 1.11812i −0.309017 − 0.951057i 0.0277724 − 0.0854748i −2.36802 + 3.25930i 0.812362 − 1.11812i 1.14719 + 0.372746i 2.74699 − 0.892552i −0.809017 + 0.587785i −5.56798
25.12 1.20071 + 1.65263i −0.309017 − 0.951057i −0.671457 + 2.06653i 0.326611 − 0.449541i 1.20071 − 1.65263i 1.66332 + 0.540445i −0.335871 + 0.109131i −0.809017 + 0.587785i 1.13509
25.13 1.28832 + 1.77322i −0.309017 − 0.951057i −0.866514 + 2.66686i 1.37901 − 1.89804i 1.28832 − 1.77322i 1.10854 + 0.360186i −1.67618 + 0.544625i −0.809017 + 0.587785i 5.14227
25.14 1.59022 + 2.18875i −0.309017 − 0.951057i −1.64379 + 5.05905i −0.714239 + 0.983066i 1.59022 − 2.18875i −1.83379 − 0.595836i −8.54091 + 2.77511i −0.809017 + 0.587785i −3.28748
64.1 −2.52137 − 0.819244i 0.809017 − 0.587785i 4.06813 + 2.95567i 1.72603 − 0.560820i −2.52137 + 0.819244i 2.49852 − 3.43892i −4.71928 − 6.49553i 0.309017 − 0.951057i −4.81141
64.2 −2.29028 − 0.744157i 0.809017 − 0.587785i 3.07358 + 2.23309i 0.810545 − 0.263362i −2.29028 + 0.744157i −1.77414 + 2.44189i −2.54665 − 3.50517i 0.309017 − 0.951057i −2.05236
64.3 −1.97843 − 0.642831i 0.809017 − 0.587785i 1.88292 + 1.36802i −3.50059 + 1.13741i −1.97843 + 0.642831i 2.06240 − 2.83865i −0.400343 − 0.551025i 0.309017 − 0.951057i 7.65683
64.4 −1.54582 − 0.502269i 0.809017 − 0.587785i 0.519265 + 0.377268i −1.17463 + 0.381659i −1.54582 + 0.502269i 0.262950 − 0.361920i 1.29754 + 1.78591i 0.309017 − 0.951057i 2.00746
64.5 −1.15287 − 0.374592i 0.809017 − 0.587785i −0.429233 − 0.311856i −0.0540551 + 0.0175636i −1.15287 + 0.374592i −1.35564 + 1.86588i 1.80306 + 2.48171i 0.309017 − 0.951057i 0.0688979
64.6 −0.975059 − 0.316816i 0.809017 − 0.587785i −0.767667 − 0.557743i 3.70008 − 1.20223i −0.975059 + 0.316816i 1.98998 − 2.73897i 1.77706 + 2.44591i 0.309017 − 0.951057i −3.98868
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.b even 2 1 inner
143.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bb.b ✓ 56
11.c even 5 1 inner 429.2.bb.b ✓ 56
13.b even 2 1 inner 429.2.bb.b ✓ 56
143.n even 10 1 inner 429.2.bb.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bb.b ✓ 56 1.a even 1 1 trivial
429.2.bb.b ✓ 56 11.c even 5 1 inner
429.2.bb.b ✓ 56 13.b even 2 1 inner
429.2.bb.b ✓ 56 143.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 24 T_{2}^{54} + 335 T_{2}^{52} - 3616 T_{2}^{50} + 33913 T_{2}^{48} - 267476 T_{2}^{46} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display