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Properties

Label 430.2.q.a
Level $430$
Weight $2$
Character orbit 430.q
Analytic conductor $3.434$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{2} + 5 q^{3} - 6 q^{4} + 3 q^{5} + 5 q^{6} + q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{2} + 5 q^{3} - 6 q^{4} + 3 q^{5} + 5 q^{6} + q^{7} - 6 q^{8} + 12 q^{9} + 3 q^{10} + 9 q^{11} - 16 q^{12} + 21 q^{13} - 6 q^{14} + 5 q^{15} - 6 q^{16} - 37 q^{17} + 19 q^{18} + 7 q^{19} + 3 q^{20} - 8 q^{21} + 16 q^{22} + 13 q^{23} + 5 q^{24} + 3 q^{25} - 21 q^{26} - 28 q^{27} - 6 q^{28} + 4 q^{29} - 2 q^{30} + 13 q^{31} - 6 q^{32} - 54 q^{33} - 23 q^{34} - 2 q^{35} - 9 q^{36} + 17 q^{37} + 28 q^{38} + 8 q^{39} + 3 q^{40} + 10 q^{41} - 50 q^{42} + 14 q^{43} - 26 q^{44} + 4 q^{45} - 8 q^{46} + 13 q^{47} - 2 q^{48} + 5 q^{49} - 18 q^{50} + 17 q^{51} - 7 q^{52} - 71 q^{53} + 7 q^{54} - 29 q^{55} + q^{56} - 70 q^{57} + 25 q^{58} + 8 q^{59} - 2 q^{60} + 25 q^{61} - 15 q^{62} + 88 q^{63} - 6 q^{64} + 7 q^{65} + 44 q^{66} + 26 q^{67} - 9 q^{68} + 78 q^{69} - 2 q^{70} - 60 q^{71} - 23 q^{72} + 7 q^{73} + 17 q^{74} - 10 q^{75} - 35 q^{76} - 5 q^{77} + 57 q^{78} + 52 q^{79} - 18 q^{80} + q^{81} - 4 q^{82} - 101 q^{83} - 8 q^{84} - 24 q^{85} + 49 q^{86} - 18 q^{87} + 16 q^{88} + 14 q^{89} - 3 q^{90} + 35 q^{91} + 6 q^{92} + 28 q^{93} - 15 q^{94} + 7 q^{95} - 2 q^{96} + 11 q^{97} + 12 q^{98} + 78 q^{99}+O(q^{100}) \) 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 −1.94282 + 1.32459i 0.623490 − 0.781831i −0.733052 − 0.680173i 1.17570 − 2.03638i 1.33082 + 2.30504i −0.222521 + 0.974928i 0.923984 − 2.35427i 0.955573 + 0.294755i
31.2 −0.900969 + 0.433884i −0.529721 + 0.361157i 0.623490 − 0.781831i −0.733052 − 0.680173i 0.320562 − 0.555229i −1.04610 − 1.81189i −0.222521 + 0.974928i −0.945854 + 2.41000i 0.955573 + 0.294755i
31.3 −0.900969 + 0.433884i 0.492510 − 0.335788i 0.623490 − 0.781831i −0.733052 − 0.680173i −0.298043 + 0.516226i −0.982941 − 1.70250i −0.222521 + 0.974928i −0.966210 + 2.46186i 0.955573 + 0.294755i
81.1 −0.222521 + 0.974928i −1.36754 − 0.421830i −0.900969 − 0.433884i 0.365341 + 0.930874i 0.715559 − 1.23938i −0.610035 − 1.05661i 0.623490 − 0.781831i −0.786497 − 0.536224i −0.988831 + 0.149042i
81.2 −0.222521 + 0.974928i 1.14570 + 0.353401i −0.900969 − 0.433884i 0.365341 + 0.930874i −0.599482 + 1.03833i 1.95703 + 3.38968i 0.623490 − 0.781831i −1.29098 − 0.880178i −0.988831 + 0.149042i
81.3 −0.222521 + 0.974928i 2.38912 + 0.736945i −0.900969 − 0.433884i 0.365341 + 0.930874i −1.25010 + 2.16523i 0.287026 + 0.497143i 0.623490 − 0.781831i 2.68608 + 1.83134i −0.988831 + 0.149042i
101.1 0.623490 − 0.781831i −0.305653 + 0.778792i −0.222521 − 0.974928i 0.0747301 + 0.997204i 0.418312 + 0.724538i −1.41697 + 2.45427i −0.900969 − 0.433884i 1.68606 + 1.56444i 0.826239 + 0.563320i
101.2 0.623490 − 0.781831i −0.0251063 + 0.0639697i −0.222521 − 0.974928i 0.0747301 + 0.997204i 0.0343600 + 0.0595134i 1.80548 − 3.12719i −0.900969 − 0.433884i 2.19569 + 2.03731i 0.826239 + 0.563320i
101.3 0.623490 − 0.781831i 0.850532 − 2.16712i −0.222521 − 0.974928i 0.0747301 + 0.997204i −1.16402 − 2.01615i 0.822841 − 1.42520i −0.900969 − 0.433884i −1.77385 − 1.64589i 0.826239 + 0.563320i
111.1 −0.900969 − 0.433884i −1.94282 − 1.32459i 0.623490 + 0.781831i −0.733052 + 0.680173i 1.17570 + 2.03638i 1.33082 − 2.30504i −0.222521 − 0.974928i 0.923984 + 2.35427i 0.955573 − 0.294755i
111.2 −0.900969 − 0.433884i −0.529721 − 0.361157i 0.623490 + 0.781831i −0.733052 + 0.680173i 0.320562 + 0.555229i −1.04610 + 1.81189i −0.222521 − 0.974928i −0.945854 − 2.41000i 0.955573 − 0.294755i
111.3 −0.900969 − 0.433884i 0.492510 + 0.335788i 0.623490 + 0.781831i −0.733052 + 0.680173i −0.298043 − 0.516226i −0.982941 + 1.70250i −0.222521 − 0.974928i −0.966210 − 2.46186i 0.955573 − 0.294755i
181.1 0.623490 + 0.781831i −2.81926 − 0.424935i −0.222521 + 0.974928i 0.826239 + 0.563320i −1.42555 − 2.46913i 1.01287 − 1.75433i −0.900969 + 0.433884i 4.90094 + 1.51174i 0.0747301 + 0.997204i
181.2 0.623490 + 0.781831i 1.26688 + 0.190952i −0.222521 + 0.974928i 0.826239 + 0.563320i 0.640596 + 1.10954i −1.20270 + 2.08314i −0.900969 + 0.433884i −1.29819 − 0.400439i 0.0747301 + 0.997204i
181.3 0.623490 + 0.781831i 2.82366 + 0.425598i −0.222521 + 0.974928i 0.826239 + 0.563320i 1.42778 + 2.47298i 0.0470137 − 0.0814302i −0.900969 + 0.433884i 4.92521 + 1.51922i 0.0747301 + 0.997204i
271.1 −0.222521 + 0.974928i −0.861302 + 0.799171i −0.900969 − 0.433884i −0.988831 − 0.149042i −0.587477 − 1.01754i −1.53300 + 2.65524i 0.623490 − 0.781831i −0.121024 + 1.61496i 0.365341 − 0.930874i
271.2 −0.222521 + 0.974928i −0.387472 + 0.359522i −0.900969 − 0.433884i −0.988831 − 0.149042i −0.264287 − 0.457759i 1.56569 − 2.71185i 0.623490 − 0.781831i −0.203311 + 2.71300i 0.365341 − 0.930874i
271.3 −0.222521 + 0.974928i 2.06188 − 1.91315i −0.900969 − 0.433884i −0.988831 − 0.149042i 1.40637 + 2.43590i −0.0872858 + 0.151183i 0.623490 − 0.781831i 0.367035 − 4.89775i 0.365341 − 0.930874i
281.1 0.623490 + 0.781831i −0.305653 − 0.778792i −0.222521 + 0.974928i 0.0747301 − 0.997204i 0.418312 − 0.724538i −1.41697 − 2.45427i −0.900969 + 0.433884i 1.68606 − 1.56444i 0.826239 − 0.563320i
281.2 0.623490 + 0.781831i −0.0251063 − 0.0639697i −0.222521 + 0.974928i 0.0747301 − 0.997204i 0.0343600 − 0.0595134i 1.80548 + 3.12719i −0.900969 + 0.433884i 2.19569 − 2.03731i 0.826239 − 0.563320i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

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.a ✓ 36
43.g even 21 1 inner 430.2.q.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.q.a ✓ 36 1.a even 1 1 trivial
430.2.q.a ✓ 36 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 5 T_{3}^{35} + 2 T_{3}^{34} + 33 T_{3}^{33} - 129 T_{3}^{32} + 326 T_{3}^{31} + 60 T_{3}^{30} + \cdots + 1849 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display