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Properties

Label 430.2.q.b
Level $430$
Weight $2$
Character orbit 430.q
Analytic conductor $3.434$
Analytic rank $0$
Dimension $48$
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(31,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.q (of order \(21\), degree \(12\), 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: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{2} - q^{3} - 8 q^{4} - 4 q^{5} - 8 q^{6} - 7 q^{7} - 8 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{2} - q^{3} - 8 q^{4} - 4 q^{5} - 8 q^{6} - 7 q^{7} - 8 q^{8} - q^{9} - 4 q^{10} - 9 q^{11} - q^{12} - 11 q^{13} + 7 q^{14} - 6 q^{15} - 8 q^{16} - 11 q^{17} - 15 q^{18} + 31 q^{19} - 4 q^{20} + 4 q^{21} + 12 q^{22} + 4 q^{23} + 6 q^{24} + 4 q^{25} + 31 q^{26} + 44 q^{27} + 13 q^{29} + q^{30} - q^{31} - 8 q^{32} - 30 q^{33} - 25 q^{34} + 14 q^{35} - 36 q^{36} - 27 q^{37} - 4 q^{38} + 4 q^{39} - 4 q^{40} + 16 q^{41} + 4 q^{42} - 7 q^{43} - 30 q^{44} + 12 q^{45} - 52 q^{46} - 43 q^{47} - q^{48} - 23 q^{49} - 24 q^{50} - 7 q^{51} - 25 q^{52} + 73 q^{53} - 61 q^{54} + 27 q^{55} + 7 q^{56} + 136 q^{57} - 36 q^{58} + 58 q^{59} + q^{60} - 61 q^{61} + 27 q^{62} + 32 q^{63} - 8 q^{64} - q^{65} + 12 q^{66} + 39 q^{67} + 17 q^{68} - 34 q^{69} + 14 q^{70} + 60 q^{71} + 27 q^{72} + 3 q^{73} + 29 q^{74} - 12 q^{75} + 17 q^{76} - 27 q^{77} - 17 q^{78} - 2 q^{79} + 24 q^{80} - 165 q^{81} + 2 q^{82} + 74 q^{83} + 4 q^{84} - 8 q^{85} + 14 q^{86} + 88 q^{87} + 12 q^{88} + 23 q^{89} - 2 q^{90} - 39 q^{91} + 4 q^{92} - 46 q^{93} + 55 q^{94} - 31 q^{95} - q^{96} - 85 q^{97} + 33 q^{98} - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.900969 + 0.433884i −2.80831 + 1.91467i 0.623490 − 0.781831i 0.733052 + 0.680173i 1.69945 − 2.94354i −0.338333 − 0.586011i −0.222521 + 0.974928i 3.12461 − 7.96138i −0.955573 − 0.294755i
31.2 −0.900969 + 0.433884i −0.993471 + 0.677337i 0.623490 − 0.781831i 0.733052 + 0.680173i 0.601201 − 1.04131i −0.895490 − 1.55103i −0.222521 + 0.974928i −0.567823 + 1.44679i −0.955573 − 0.294755i
31.3 −0.900969 + 0.433884i 0.953461 − 0.650059i 0.623490 − 0.781831i 0.733052 + 0.680173i −0.576989 + 0.999374i 1.53817 + 2.66418i −0.222521 + 0.974928i −0.609511 + 1.55301i −0.955573 − 0.294755i
31.4 −0.900969 + 0.433884i 2.64426 − 1.80282i 0.623490 − 0.781831i 0.733052 + 0.680173i −1.60018 + 2.77159i −0.952135 − 1.64915i −0.222521 + 0.974928i 2.64590 − 6.74164i −0.955573 − 0.294755i
81.1 −0.222521 + 0.974928i −1.77929 − 0.548838i −0.900969 − 0.433884i −0.365341 − 0.930874i 0.931007 − 1.61255i 1.00410 + 1.73915i 0.623490 − 0.781831i 0.385931 + 0.263123i 0.988831 − 0.149042i
81.2 −0.222521 + 0.974928i 0.269177 + 0.0830302i −0.900969 − 0.433884i −0.365341 − 0.930874i −0.140846 + 0.243953i −0.703462 − 1.21843i 0.623490 − 0.781831i −2.41315 − 1.64526i 0.988831 − 0.149042i
81.3 −0.222521 + 0.974928i 1.03154 + 0.318188i −0.900969 − 0.433884i −0.365341 − 0.930874i −0.539749 + 0.934873i −1.69214 − 2.93087i 0.623490 − 0.781831i −1.51589 − 1.03351i 0.988831 − 0.149042i
81.4 −0.222521 + 0.974928i 3.15603 + 0.973506i −0.900969 − 0.433884i −0.365341 − 0.930874i −1.65138 + 2.86027i 0.781939 + 1.35436i 0.623490 − 0.781831i 6.53408 + 4.45486i 0.988831 − 0.149042i
101.1 0.623490 − 0.781831i −1.07289 + 2.73367i −0.222521 − 0.974928i −0.0747301 − 0.997204i 1.46834 + 2.54323i −1.49192 + 2.58408i −0.900969 − 0.433884i −4.12272 − 3.82532i −0.826239 − 0.563320i
101.2 0.623490 − 0.781831i −0.320555 + 0.816762i −0.222521 − 0.974928i −0.0747301 − 0.997204i 0.438707 + 0.759863i −0.337906 + 0.585270i −0.900969 − 0.433884i 1.63481 + 1.51688i −0.826239 − 0.563320i
101.3 0.623490 − 0.781831i 0.808920 − 2.06109i −0.222521 − 0.974928i −0.0747301 − 0.997204i −1.10708 − 1.91751i 1.63742 − 2.83609i −0.900969 − 0.433884i −1.39460 − 1.29400i −0.826239 − 0.563320i
101.4 0.623490 − 0.781831i 1.11246 − 2.83449i −0.222521 − 0.974928i −0.0747301 − 0.997204i −1.52249 − 2.63703i −2.19740 + 3.80600i −0.900969 − 0.433884i −4.59762 − 4.26597i −0.826239 − 0.563320i
111.1 −0.900969 − 0.433884i −2.80831 − 1.91467i 0.623490 + 0.781831i 0.733052 − 0.680173i 1.69945 + 2.94354i −0.338333 + 0.586011i −0.222521 − 0.974928i 3.12461 + 7.96138i −0.955573 + 0.294755i
111.2 −0.900969 − 0.433884i −0.993471 − 0.677337i 0.623490 + 0.781831i 0.733052 − 0.680173i 0.601201 + 1.04131i −0.895490 + 1.55103i −0.222521 − 0.974928i −0.567823 − 1.44679i −0.955573 + 0.294755i
111.3 −0.900969 − 0.433884i 0.953461 + 0.650059i 0.623490 + 0.781831i 0.733052 − 0.680173i −0.576989 − 0.999374i 1.53817 − 2.66418i −0.222521 − 0.974928i −0.609511 − 1.55301i −0.955573 + 0.294755i
111.4 −0.900969 − 0.433884i 2.64426 + 1.80282i 0.623490 + 0.781831i 0.733052 − 0.680173i −1.60018 − 2.77159i −0.952135 + 1.64915i −0.222521 − 0.974928i 2.64590 + 6.74164i −0.955573 + 0.294755i
181.1 0.623490 + 0.781831i −2.67211 − 0.402755i −0.222521 + 0.974928i −0.826239 − 0.563320i −1.35114 − 2.34025i 0.0929644 − 0.161019i −0.900969 + 0.433884i 4.11122 + 1.26814i −0.0747301 − 0.997204i
181.2 0.623490 + 0.781831i −0.933730 − 0.140737i −0.222521 + 0.974928i −0.826239 − 0.563320i −0.472138 − 0.817768i −0.0334660 + 0.0579648i −0.900969 + 0.433884i −2.01467 − 0.621444i −0.0747301 − 0.997204i
181.3 0.623490 + 0.781831i −0.399412 − 0.0602017i −0.222521 + 0.974928i −0.826239 − 0.563320i −0.201962 − 0.349808i −2.60651 + 4.51462i −0.900969 + 0.433884i −2.71081 − 0.836175i −0.0747301 − 0.997204i
181.4 0.623490 + 0.781831i 2.57635 + 0.388322i −0.222521 + 0.974928i −0.826239 − 0.563320i 1.30272 + 2.25638i 1.51139 − 2.61780i −0.900969 + 0.433884i 3.62005 + 1.11664i −0.0747301 − 0.997204i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.q.b ✓ 48
43.g even 21 1 inner 430.2.q.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.q.b ✓ 48 1.a even 1 1 trivial
430.2.q.b ✓ 48 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + T_{3}^{47} - 5 T_{3}^{46} - 24 T_{3}^{45} + 35 T_{3}^{44} - 63 T_{3}^{43} + \cdots + 2832474841 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display