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Properties

Label 430.2.q.d
Level $430$
Weight $2$
Character orbit 430.q
Analytic conductor $3.434$
Analytic rank $0$
Dimension $60$
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: \(60\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 10 q^{2} - q^{3} - 10 q^{4} - 5 q^{5} - 6 q^{6} + q^{7} + 10 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 10 q^{2} - q^{3} - 10 q^{4} - 5 q^{5} - 6 q^{6} + q^{7} + 10 q^{8} + 8 q^{9} + 5 q^{10} + 3 q^{11} - q^{12} + 13 q^{13} - 15 q^{14} + q^{15} - 10 q^{16} + 38 q^{17} + 6 q^{18} + 17 q^{19} - 5 q^{20} + 4 q^{21} + 11 q^{22} - 8 q^{23} + q^{24} + 5 q^{25} + 29 q^{26} - 16 q^{27} - 6 q^{28} - 41 q^{29} - q^{30} + 15 q^{31} + 10 q^{32} + 63 q^{33} - 10 q^{34} + 2 q^{35} - 48 q^{36} + 7 q^{37} + 25 q^{38} - 64 q^{39} + 5 q^{40} - 22 q^{41} - 60 q^{42} + 6 q^{43} - 4 q^{44} + 30 q^{45} - 20 q^{46} - 13 q^{47} - q^{48} - 93 q^{49} + 30 q^{50} + 18 q^{51} - 15 q^{52} + 71 q^{53} - 5 q^{54} - 16 q^{55} - q^{56} - 94 q^{57} - 22 q^{58} + 20 q^{59} + q^{60} + 91 q^{61} + 69 q^{62} - 50 q^{63} - 10 q^{64} + 5 q^{65} + 14 q^{66} - 26 q^{67} - 18 q^{68} - 94 q^{69} - 2 q^{70} + 20 q^{71} - 36 q^{72} - 56 q^{73} - 7 q^{74} + 2 q^{75} + 38 q^{76} + 41 q^{77} - 13 q^{78} - 46 q^{79} + 30 q^{80} + 39 q^{81} + 8 q^{82} - 115 q^{83} + 4 q^{84} + 6 q^{85} + 29 q^{86} + 20 q^{87} + 11 q^{88} + 30 q^{89} - 16 q^{90} - 53 q^{91} - 8 q^{92} + 16 q^{93} + 41 q^{94} - 17 q^{95} + q^{96} + 113 q^{97} + 2 q^{98} + 139 q^{99}+O(q^{100}) \) 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 −2.40821 + 1.64189i 0.623490 − 0.781831i 0.733052 + 0.680173i −1.45733 + 2.52418i 0.522354 + 0.904743i 0.222521 − 0.974928i 2.00766 − 5.11543i 0.955573 + 0.294755i
31.2 0.900969 − 0.433884i −1.77519 + 1.21030i 0.623490 − 0.781831i 0.733052 + 0.680173i −1.07426 + 1.86067i −2.45123 − 4.24566i 0.222521 − 0.974928i 0.590433 − 1.50440i 0.955573 + 0.294755i
31.3 0.900969 − 0.433884i −0.206973 + 0.141112i 0.623490 − 0.781831i 0.733052 + 0.680173i −0.125250 + 0.216939i 0.851274 + 1.47445i 0.222521 − 0.974928i −1.07310 + 2.73421i 0.955573 + 0.294755i
31.4 0.900969 − 0.433884i 1.74260 − 1.18809i 0.623490 − 0.781831i 0.733052 + 0.680173i 1.05454 − 1.82652i −2.08827 − 3.61700i 0.222521 − 0.974928i 0.529092 − 1.34811i 0.955573 + 0.294755i
31.5 0.900969 − 0.433884i 1.82153 − 1.24190i 0.623490 − 0.781831i 0.733052 + 0.680173i 1.10230 − 1.90924i 1.71615 + 2.97246i 0.222521 − 0.974928i 0.679638 − 1.73169i 0.955573 + 0.294755i
81.1 0.222521 − 0.974928i −3.07767 − 0.949334i −0.900969 − 0.433884i −0.365341 − 0.930874i −1.61038 + 2.78926i 2.52339 + 4.37064i −0.623490 + 0.781831i 6.09208 + 4.15351i −0.988831 + 0.149042i
81.2 0.222521 − 0.974928i −2.45037 − 0.755840i −0.900969 − 0.433884i −0.365341 − 0.930874i −1.28215 + 2.22075i −2.38661 − 4.13373i −0.623490 + 0.781831i 2.95432 + 2.01422i −0.988831 + 0.149042i
81.3 0.222521 − 0.974928i −0.366458 − 0.113037i −0.900969 − 0.433884i −0.365341 − 0.930874i −0.191748 + 0.332117i −0.0855632 − 0.148200i −0.623490 + 0.781831i −2.35720 − 1.60711i −0.988831 + 0.149042i
81.4 0.222521 − 0.974928i 2.08443 + 0.642961i −0.900969 − 0.433884i −0.365341 − 0.930874i 1.09067 − 1.88910i 1.63836 + 2.83771i −0.623490 + 0.781831i 1.45273 + 0.990455i −0.988831 + 0.149042i
81.5 0.222521 − 0.974928i 2.85449 + 0.880495i −0.900969 − 0.433884i −0.365341 − 0.930874i 1.49360 − 2.58700i −1.74417 − 3.02100i −0.623490 + 0.781831i 4.89415 + 3.33678i −0.988831 + 0.149042i
101.1 −0.623490 + 0.781831i −1.04669 + 2.66692i −0.222521 − 0.974928i −0.0747301 − 0.997204i −1.43248 − 2.48113i 1.27853 − 2.21448i 0.900969 + 0.433884i −3.81773 − 3.54234i 0.826239 + 0.563320i
101.2 −0.623490 + 0.781831i −0.617972 + 1.57457i −0.222521 − 0.974928i −0.0747301 − 0.997204i −0.845746 − 1.46488i 0.932562 − 1.61524i 0.900969 + 0.433884i 0.101786 + 0.0944434i 0.826239 + 0.563320i
101.3 −0.623490 + 0.781831i −0.0969271 + 0.246966i −0.222521 − 0.974928i −0.0747301 − 0.997204i −0.132653 − 0.229762i −2.60514 + 4.51223i 0.900969 + 0.433884i 2.14756 + 1.99264i 0.826239 + 0.563320i
101.4 −0.623490 + 0.781831i 0.272539 − 0.694419i −0.222521 − 0.974928i −0.0747301 − 0.997204i 0.372993 + 0.646043i 0.186100 − 0.322335i 0.900969 + 0.433884i 1.79122 + 1.66201i 0.826239 + 0.563320i
101.5 −0.623490 + 0.781831i 1.12371 − 2.86316i −0.222521 − 0.974928i −0.0747301 − 0.997204i 1.53789 + 2.66370i 0.0651243 − 0.112799i 0.900969 + 0.433884i −4.73579 − 4.39418i 0.826239 + 0.563320i
111.1 0.900969 + 0.433884i −2.40821 − 1.64189i 0.623490 + 0.781831i 0.733052 − 0.680173i −1.45733 − 2.52418i 0.522354 − 0.904743i 0.222521 + 0.974928i 2.00766 + 5.11543i 0.955573 − 0.294755i
111.2 0.900969 + 0.433884i −1.77519 − 1.21030i 0.623490 + 0.781831i 0.733052 − 0.680173i −1.07426 − 1.86067i −2.45123 + 4.24566i 0.222521 + 0.974928i 0.590433 + 1.50440i 0.955573 − 0.294755i
111.3 0.900969 + 0.433884i −0.206973 − 0.141112i 0.623490 + 0.781831i 0.733052 − 0.680173i −0.125250 − 0.216939i 0.851274 − 1.47445i 0.222521 + 0.974928i −1.07310 − 2.73421i 0.955573 − 0.294755i
111.4 0.900969 + 0.433884i 1.74260 + 1.18809i 0.623490 + 0.781831i 0.733052 − 0.680173i 1.05454 + 1.82652i −2.08827 + 3.61700i 0.222521 + 0.974928i 0.529092 + 1.34811i 0.955573 − 0.294755i
111.5 0.900969 + 0.433884i 1.82153 + 1.24190i 0.623490 + 0.781831i 0.733052 − 0.680173i 1.10230 + 1.90924i 1.71615 − 2.97246i 0.222521 + 0.974928i 0.679638 + 1.73169i 0.955573 − 0.294755i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.5
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.d ✓ 60
43.g even 21 1 inner 430.2.q.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.q.d ✓ 60 1.a even 1 1 trivial
430.2.q.d ✓ 60 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + T_{3}^{59} - 11 T_{3}^{58} - 10 T_{3}^{57} - 20 T_{3}^{56} - 122 T_{3}^{55} + \cdots + 150381169 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display