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Properties

Label 490.2.q.c
Level $490$
Weight $2$
Character orbit 490.q
Analytic conductor $3.913$
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] = [490,2,Mod(11,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.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: \(3.91266969904\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + q^{7} + 10 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + q^{7} + 10 q^{8} + 31 q^{9} + 5 q^{10} - 5 q^{11} - 8 q^{13} + 3 q^{14} + 5 q^{16} - 9 q^{17} + 32 q^{18} - 20 q^{19} + 10 q^{20} - 5 q^{21} + 4 q^{22} + 16 q^{23} + 5 q^{25} + 10 q^{26} - 24 q^{27} + 9 q^{28} - 13 q^{29} - 20 q^{31} - 5 q^{32} + 50 q^{33} - 11 q^{34} + 3 q^{35} - 20 q^{36} + 37 q^{37} - 22 q^{38} + 62 q^{39} + 5 q^{40} - 42 q^{41} - 8 q^{42} - 29 q^{43} - 26 q^{44} - 3 q^{45} - 9 q^{46} - 81 q^{47} - q^{49} - 60 q^{50} - 11 q^{51} + 4 q^{52} - 98 q^{53} + 9 q^{54} - 10 q^{55} - 15 q^{56} + 64 q^{57} - 24 q^{58} + 31 q^{59} + 20 q^{61} + 23 q^{62} + 125 q^{63} - 10 q^{64} - 4 q^{65} + 34 q^{66} + 31 q^{67} + 26 q^{68} - 12 q^{69} - 5 q^{70} + 57 q^{71} - 3 q^{72} - 89 q^{73} + 12 q^{74} - 16 q^{76} + 51 q^{77} - 51 q^{78} - 21 q^{79} + 30 q^{80} - 47 q^{81} - 42 q^{82} + 38 q^{83} + 88 q^{84} - 11 q^{85} + 10 q^{86} + 36 q^{87} + 26 q^{88} + 6 q^{89} - 6 q^{90} - 25 q^{91} + 3 q^{92} - 127 q^{93} + 25 q^{94} + 41 q^{95} + 16 q^{97} - 31 q^{98} - 182 q^{99}+O(q^{100}) \) 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 0.733052 − 0.680173i −3.26805 + 0.492579i 0.0747301 − 0.997204i −0.365341 + 0.930874i −2.06061 + 2.58392i 2.52013 − 0.805572i −0.623490 − 0.781831i 7.57079 − 2.33528i 0.365341 + 0.930874i
11.2 0.733052 − 0.680173i −0.947935 + 0.142878i 0.0747301 − 0.997204i −0.365341 + 0.930874i −0.597704 + 0.749497i 1.45200 + 2.21172i −0.623490 − 0.781831i −1.98855 + 0.613387i 0.365341 + 0.930874i
11.3 0.733052 − 0.680173i −0.477182 + 0.0719236i 0.0747301 − 0.997204i −0.365341 + 0.930874i −0.300879 + 0.377290i −2.61010 + 0.432869i −0.623490 − 0.781831i −2.64419 + 0.815624i 0.365341 + 0.930874i
11.4 0.733052 − 0.680173i 1.85987 − 0.280331i 0.0747301 − 0.997204i −0.365341 + 0.930874i 1.17271 − 1.47053i 0.539711 − 2.59012i −0.623490 − 0.781831i 0.513823 − 0.158493i 0.365341 + 0.930874i
11.5 0.733052 − 0.680173i 2.83329 − 0.427050i 0.0747301 − 0.997204i −0.365341 + 0.930874i 1.78648 − 2.24018i −0.100161 + 2.64385i −0.623490 − 0.781831i 4.97846 − 1.53565i 0.365341 + 0.930874i
51.1 −0.955573 − 0.294755i −0.969450 + 2.47012i 0.826239 + 0.563320i 0.988831 − 0.149042i 1.65446 − 2.07463i −2.27791 − 1.34578i −0.623490 − 0.781831i −2.96250 − 2.74879i −0.988831 − 0.149042i
51.2 −0.955573 − 0.294755i −0.468104 + 1.19271i 0.826239 + 0.563320i 0.988831 − 0.149042i 0.798865 − 1.00174i 2.63253 − 0.264187i −0.623490 − 0.781831i 0.995721 + 0.923894i −0.988831 − 0.149042i
51.3 −0.955573 − 0.294755i −0.0113799 + 0.0289954i 0.826239 + 0.563320i 0.988831 − 0.149042i 0.0194208 − 0.0243530i −2.30399 + 1.30062i −0.623490 − 0.781831i 2.19844 + 2.03986i −0.988831 − 0.149042i
51.4 −0.955573 − 0.294755i 0.538297 − 1.37156i 0.826239 + 0.563320i 0.988831 − 0.149042i −0.918656 + 1.15196i 2.28605 + 1.33190i −0.623490 − 0.781831i 0.607746 + 0.563906i −0.988831 − 0.149042i
51.5 −0.955573 − 0.294755i 0.910637 − 2.32026i 0.826239 + 0.563320i 0.988831 − 0.149042i −1.55409 + 1.94877i −0.223715 − 2.63628i −0.623490 − 0.781831i −2.35521 − 2.18532i −0.988831 − 0.149042i
81.1 −0.0747301 − 0.997204i −2.33879 − 0.721421i −0.988831 + 0.149042i 0.733052 − 0.680173i −0.544626 + 2.38616i 1.46983 + 2.19991i 0.222521 + 0.974928i 2.47078 + 1.68455i −0.733052 − 0.680173i
81.2 −0.0747301 − 0.997204i −2.26122 − 0.697495i −0.988831 + 0.149042i 0.733052 − 0.680173i −0.526563 + 2.30702i 1.55041 − 2.14388i 0.222521 + 0.974928i 2.14792 + 1.46443i −0.733052 − 0.680173i
81.3 −0.0747301 − 0.997204i 0.0274658 + 0.00847207i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.00639586 − 0.0280221i −1.83528 + 1.90571i 0.222521 + 0.974928i −2.47803 − 1.68949i −0.733052 − 0.680173i
81.4 −0.0747301 − 0.997204i 1.93823 + 0.597864i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.451348 − 1.97749i 1.92374 + 1.81637i 0.222521 + 0.974928i 0.920566 + 0.627631i −0.733052 − 0.680173i
81.5 −0.0747301 − 0.997204i 2.63432 + 0.812581i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.613445 − 2.68768i −1.60402 − 2.10408i 0.222521 + 0.974928i 3.80065 + 2.59124i −0.733052 − 0.680173i
121.1 −0.0747301 + 0.997204i −2.33879 + 0.721421i −0.988831 − 0.149042i 0.733052 + 0.680173i −0.544626 − 2.38616i 1.46983 − 2.19991i 0.222521 − 0.974928i 2.47078 − 1.68455i −0.733052 + 0.680173i
121.2 −0.0747301 + 0.997204i −2.26122 + 0.697495i −0.988831 − 0.149042i 0.733052 + 0.680173i −0.526563 − 2.30702i 1.55041 + 2.14388i 0.222521 − 0.974928i 2.14792 − 1.46443i −0.733052 + 0.680173i
121.3 −0.0747301 + 0.997204i 0.0274658 − 0.00847207i −0.988831 − 0.149042i 0.733052 + 0.680173i 0.00639586 + 0.0280221i −1.83528 − 1.90571i 0.222521 − 0.974928i −2.47803 + 1.68949i −0.733052 + 0.680173i
121.4 −0.0747301 + 0.997204i 1.93823 − 0.597864i −0.988831 − 0.149042i 0.733052 + 0.680173i 0.451348 + 1.97749i 1.92374 − 1.81637i 0.222521 − 0.974928i 0.920566 − 0.627631i −0.733052 + 0.680173i
121.5 −0.0747301 + 0.997204i 2.63432 − 0.812581i −0.988831 − 0.149042i 0.733052 + 0.680173i 0.613445 + 2.68768i −1.60402 + 2.10408i 0.222521 − 0.974928i 3.80065 − 2.59124i −0.733052 + 0.680173i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.5
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.q.c ✓ 60
49.g even 21 1 inner 490.2.q.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.q.c ✓ 60 1.a even 1 1 trivial
490.2.q.c ✓ 60 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 23 T_{3}^{58} + 8 T_{3}^{57} + 247 T_{3}^{56} - 119 T_{3}^{55} - 1328 T_{3}^{54} + \cdots + 6889 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display