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Properties

Label 430.2.k.d
Level $430$
Weight $2$
Character orbit 430.k
Analytic conductor $3.434$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,2,Mod(11,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.k (of order \(7\), degree \(6\), 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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} + 12 q^{6} - 2 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} + 12 q^{6} - 2 q^{7} + 4 q^{8} - 4 q^{9} - 4 q^{10} - 9 q^{11} + 2 q^{12} - 2 q^{13} - 5 q^{14} - 2 q^{15} - 4 q^{16} + 13 q^{17} - 10 q^{18} - 3 q^{19} + 4 q^{20} - 14 q^{21} - 5 q^{22} + 11 q^{23} - 2 q^{24} - 4 q^{25} + 2 q^{26} - 13 q^{27} + 5 q^{28} + 20 q^{29} + 2 q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + q^{34} + 9 q^{35} + 24 q^{36} - 38 q^{37} - 18 q^{38} + 5 q^{39} - 4 q^{40} - 2 q^{41} - 14 q^{42} - 2 q^{43} - 2 q^{44} - 10 q^{45} - 4 q^{46} + 10 q^{47} + 2 q^{48} + 50 q^{49} - 24 q^{50} - 42 q^{51} - 2 q^{52} + 22 q^{53} - 29 q^{54} - 5 q^{55} + 9 q^{56} - 67 q^{57} + 22 q^{58} - 44 q^{59} - 2 q^{60} - 26 q^{61} - 2 q^{62} + 37 q^{63} - 4 q^{64} + 2 q^{65} - 8 q^{66} + 37 q^{67} - 15 q^{68} + 88 q^{69} - 9 q^{70} + 19 q^{71} - 10 q^{72} + 22 q^{73} - 4 q^{74} + 2 q^{75} + 18 q^{76} + 28 q^{77} + 23 q^{78} + 30 q^{79} - 24 q^{80} - 26 q^{81} + 37 q^{82} - 11 q^{83} - 14 q^{84} - 6 q^{85} + 16 q^{86} - 26 q^{87} - 5 q^{88} + 30 q^{89} - 4 q^{90} + 36 q^{91} - 10 q^{92} - 98 q^{93} + 4 q^{94} + 3 q^{95} - 2 q^{96} - 39 q^{97} + 41 q^{98} - 31 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.623490 − 0.781831i −2.05881 + 2.58167i −0.222521 + 0.974928i 0.900969 − 0.433884i 3.30207 3.39078 0.900969 − 0.433884i −1.75874 − 7.70553i −0.900969 − 0.433884i
11.2 −0.623490 − 0.781831i −1.03097 + 1.29280i −0.222521 + 0.974928i 0.900969 − 0.433884i 1.65355 −3.40711 0.900969 − 0.433884i 0.0591384 + 0.259102i −0.900969 − 0.433884i
11.3 −0.623490 − 0.781831i 0.0295669 − 0.0370758i −0.222521 + 0.974928i 0.900969 − 0.433884i −0.0474217 1.52167 0.900969 − 0.433884i 0.667062 + 2.92259i −0.900969 − 0.433884i
11.4 −0.623490 − 0.781831i 1.81324 − 2.27373i −0.222521 + 0.974928i 0.900969 − 0.433884i −2.90820 1.09852 0.900969 − 0.433884i −1.21444 − 5.32082i −0.900969 − 0.433884i
21.1 0.900969 + 0.433884i −1.69175 + 0.814703i 0.623490 + 0.781831i 0.222521 + 0.974928i −1.87770 4.12012 0.222521 + 0.974928i 0.327799 − 0.411047i −0.222521 + 0.974928i
21.2 0.900969 + 0.433884i −0.400237 + 0.192744i 0.623490 + 0.781831i 0.222521 + 0.974928i −0.444230 −4.95965 0.222521 + 0.974928i −1.74743 + 2.19121i −0.222521 + 0.974928i
21.3 0.900969 + 0.433884i 1.64572 − 0.792535i 0.623490 + 0.781831i 0.222521 + 0.974928i 1.82661 2.16285 0.222521 + 0.974928i 0.209800 − 0.263080i −0.222521 + 0.974928i
21.4 0.900969 + 0.433884i 2.24821 − 1.08268i 0.623490 + 0.781831i 0.222521 + 0.974928i 2.49532 −1.43324 0.222521 + 0.974928i 2.01177 − 2.52268i −0.222521 + 0.974928i
41.1 0.900969 − 0.433884i −1.69175 − 0.814703i 0.623490 − 0.781831i 0.222521 − 0.974928i −1.87770 4.12012 0.222521 − 0.974928i 0.327799 + 0.411047i −0.222521 − 0.974928i
41.2 0.900969 − 0.433884i −0.400237 − 0.192744i 0.623490 − 0.781831i 0.222521 − 0.974928i −0.444230 −4.95965 0.222521 − 0.974928i −1.74743 − 2.19121i −0.222521 − 0.974928i
41.3 0.900969 − 0.433884i 1.64572 + 0.792535i 0.623490 − 0.781831i 0.222521 − 0.974928i 1.82661 2.16285 0.222521 − 0.974928i 0.209800 + 0.263080i −0.222521 − 0.974928i
41.4 0.900969 − 0.433884i 2.24821 + 1.08268i 0.623490 − 0.781831i 0.222521 − 0.974928i 2.49532 −1.43324 0.222521 − 0.974928i 2.01177 + 2.52268i −0.222521 − 0.974928i
121.1 0.222521 − 0.974928i −0.273153 − 1.19676i −0.900969 − 0.433884i −0.623490 + 0.781831i −1.22754 −4.39678 −0.623490 + 0.781831i 1.34528 − 0.647852i 0.623490 + 0.781831i
121.2 0.222521 − 0.974928i −0.167393 − 0.733397i −0.900969 − 0.433884i −0.623490 + 0.781831i −0.752258 2.76599 −0.623490 + 0.781831i 2.19306 − 1.05612i 0.623490 + 0.781831i
121.3 0.222521 − 0.974928i 0.187922 + 0.823340i −0.900969 − 0.433884i −0.623490 + 0.781831i 0.844514 0.703699 −0.623490 + 0.781831i 2.06033 − 0.992204i 0.623490 + 0.781831i
121.4 0.222521 − 0.974928i 0.697666 + 3.05668i −0.900969 − 0.433884i −0.623490 + 0.781831i 3.13528 −2.56686 −0.623490 + 0.781831i −6.15362 + 2.96343i 0.623490 + 0.781831i
231.1 0.222521 + 0.974928i −0.273153 + 1.19676i −0.900969 + 0.433884i −0.623490 − 0.781831i −1.22754 −4.39678 −0.623490 − 0.781831i 1.34528 + 0.647852i 0.623490 − 0.781831i
231.2 0.222521 + 0.974928i −0.167393 + 0.733397i −0.900969 + 0.433884i −0.623490 − 0.781831i −0.752258 2.76599 −0.623490 − 0.781831i 2.19306 + 1.05612i 0.623490 − 0.781831i
231.3 0.222521 + 0.974928i 0.187922 − 0.823340i −0.900969 + 0.433884i −0.623490 − 0.781831i 0.844514 0.703699 −0.623490 − 0.781831i 2.06033 + 0.992204i 0.623490 − 0.781831i
231.4 0.222521 + 0.974928i 0.697666 − 3.05668i −0.900969 + 0.433884i −0.623490 − 0.781831i 3.13528 −2.56686 −0.623490 − 0.781831i −6.15362 − 2.96343i 0.623490 − 0.781831i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.d ✓ 24
43.e even 7 1 inner 430.2.k.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.d ✓ 24 1.a even 1 1 trivial
430.2.k.d ✓ 24 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 2 T_{3}^{23} + 10 T_{3}^{22} - 17 T_{3}^{21} + 113 T_{3}^{20} - 304 T_{3}^{19} + 772 T_{3}^{18} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display