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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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.q (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{2} + 5q^{3} - 6q^{4} + 3q^{5} + 5q^{6} + q^{7} - 6q^{8} + 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{2} + 5q^{3} - 6q^{4} + 3q^{5} + 5q^{6} + q^{7} - 6q^{8} + 12q^{9} + 3q^{10} + 9q^{11} - 16q^{12} + 21q^{13} - 6q^{14} + 5q^{15} - 6q^{16} - 37q^{17} + 19q^{18} + 7q^{19} + 3q^{20} - 8q^{21} + 16q^{22} + 13q^{23} + 5q^{24} + 3q^{25} - 21q^{26} - 28q^{27} - 6q^{28} + 4q^{29} - 2q^{30} + 13q^{31} - 6q^{32} - 54q^{33} - 23q^{34} - 2q^{35} - 9q^{36} + 17q^{37} + 28q^{38} + 8q^{39} + 3q^{40} + 10q^{41} - 50q^{42} + 14q^{43} - 26q^{44} + 4q^{45} - 8q^{46} + 13q^{47} - 2q^{48} + 5q^{49} - 18q^{50} + 17q^{51} - 7q^{52} - 71q^{53} + 7q^{54} - 29q^{55} + q^{56} - 70q^{57} + 25q^{58} + 8q^{59} - 2q^{60} + 25q^{61} - 15q^{62} + 88q^{63} - 6q^{64} + 7q^{65} + 44q^{66} + 26q^{67} - 9q^{68} + 78q^{69} - 2q^{70} - 60q^{71} - 23q^{72} + 7q^{73} + 17q^{74} - 10q^{75} - 35q^{76} - 5q^{77} + 57q^{78} + 52q^{79} - 18q^{80} + q^{81} - 4q^{82} - 101q^{83} - 8q^{84} - 24q^{85} + 49q^{86} - 18q^{87} + 16q^{88} + 14q^{89} - 3q^{90} + 35q^{91} + 6q^{92} + 28q^{93} - 15q^{94} + 7q^{95} - 2q^{96} + 11q^{97} + 12q^{98} + 78q^{99} + O(q^{100}) \)

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.

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 411.3
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 \(T_{3}^{36} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database