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Properties

Label 490.2.k.f
Level $490$
Weight $2$
Character orbit 490.k
Analytic conductor $3.913$
Analytic rank $0$
Dimension $24$
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(71,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.k (of order \(7\), degree \(6\), 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: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 6 q^{3} - 4 q^{4} - 4 q^{5} - 6 q^{6} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 6 q^{3} - 4 q^{4} - 4 q^{5} - 6 q^{6} + 4 q^{8} + 2 q^{9} + 4 q^{10} - q^{11} - q^{12} + 15 q^{13} + 7 q^{14} - q^{15} - 4 q^{16} - 9 q^{17} - 16 q^{18} + 20 q^{19} - 4 q^{20} + 15 q^{22} - 19 q^{23} + q^{24} - 4 q^{25} - q^{26} - 18 q^{27} + 7 q^{28} - 27 q^{29} + 8 q^{30} + 28 q^{31} + 4 q^{32} - 21 q^{33} + 9 q^{34} + 2 q^{36} + q^{37} + 8 q^{38} + 11 q^{39} + 4 q^{40} - 23 q^{41} - 35 q^{42} - 18 q^{43} + 6 q^{44} - 5 q^{45} - 16 q^{46} - 46 q^{47} - 8 q^{48} - 14 q^{49} - 24 q^{50} + 11 q^{51} - 13 q^{52} + 22 q^{53} - 17 q^{54} + 6 q^{55} + 62 q^{57} - 22 q^{58} + 23 q^{59} - q^{60} - 20 q^{61} + 28 q^{62} - 35 q^{63} - 4 q^{64} - 13 q^{65} + 14 q^{66} + 16 q^{67} + 12 q^{68} + 2 q^{69} + 5 q^{71} + 5 q^{72} + 39 q^{73} - q^{74} - q^{75} - 15 q^{76} - 21 q^{77} - 39 q^{78} + 44 q^{79} + 24 q^{80} - q^{81} + 23 q^{82} - q^{83} - 9 q^{85} - 3 q^{86} + 67 q^{87} - 6 q^{88} - 21 q^{89} + 5 q^{90} - 14 q^{91} + 16 q^{92} + 16 q^{93} - 10 q^{94} + 13 q^{95} + q^{96} - 36 q^{97} - 14 q^{98} + 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 0.900969 − 0.433884i −0.496837 + 2.17679i 0.623490 − 0.781831i −0.222521 + 0.974928i 0.496837 + 2.17679i 2.40647 + 1.09950i 0.222521 − 0.974928i −1.78864 − 0.861363i 0.222521 + 0.974928i
71.2 0.900969 − 0.433884i 0.0456104 − 0.199832i 0.623490 − 0.781831i −0.222521 + 0.974928i −0.0456104 − 0.199832i −0.110265 − 2.64345i 0.222521 − 0.974928i 2.66505 + 1.28342i 0.222521 + 0.974928i
71.3 0.900969 − 0.433884i 0.156184 − 0.684288i 0.623490 − 0.781831i −0.222521 + 0.974928i −0.156184 − 0.684288i 0.798662 + 2.52233i 0.222521 − 0.974928i 2.25905 + 1.08790i 0.222521 + 0.974928i
71.4 0.900969 − 0.433884i 0.616594 − 2.70148i 0.623490 − 0.781831i −0.222521 + 0.974928i −0.616594 − 2.70148i 1.47851 − 2.19409i 0.222521 − 0.974928i −4.21488 − 2.02978i 0.222521 + 0.974928i
141.1 −0.623490 + 0.781831i −1.95554 − 0.941737i −0.222521 − 0.974928i −0.900969 − 0.433884i 1.95554 − 0.941737i −2.49146 − 0.890295i 0.900969 + 0.433884i 1.06678 + 1.33771i 0.900969 − 0.433884i
141.2 −0.623490 + 0.781831i 0.294799 + 0.141967i −0.222521 − 0.974928i −0.900969 − 0.433884i −0.294799 + 0.141967i −1.75221 + 1.98236i 0.900969 + 0.433884i −1.80372 − 2.26179i 0.900969 − 0.433884i
141.3 −0.623490 + 0.781831i 1.58297 + 0.762318i −0.222521 − 0.974928i −0.900969 − 0.433884i −1.58297 + 0.762318i −0.603900 − 2.57591i 0.900969 + 0.433884i 0.0541958 + 0.0679593i 0.900969 − 0.433884i
141.4 −0.623490 + 0.781831i 2.60223 + 1.25317i −0.222521 − 0.974928i −0.900969 − 0.433884i −2.60223 + 1.25317i 2.30954 + 1.29075i 0.900969 + 0.433884i 3.33069 + 4.17655i 0.900969 − 0.433884i
211.1 0.222521 − 0.974928i −1.45343 + 1.82254i −0.900969 − 0.433884i 0.623490 − 0.781831i 1.45343 + 1.82254i −1.42280 − 2.23061i −0.623490 + 0.781831i −0.541643 − 2.37309i −0.623490 − 0.781831i
211.2 0.222521 − 0.974928i −0.485203 + 0.608425i −0.900969 − 0.433884i 0.623490 − 0.781831i 0.485203 + 0.608425i −1.09473 + 2.40864i −0.623490 + 0.781831i 0.532804 + 2.33436i −0.623490 − 0.781831i
211.3 0.222521 − 0.974928i 0.297754 − 0.373371i −0.900969 − 0.433884i 0.623490 − 0.781831i −0.297754 − 0.373371i 2.64500 − 0.0629581i −0.623490 + 0.781831i 0.616814 + 2.70244i −0.623490 − 0.781831i
211.4 0.222521 − 0.974928i 1.79487 − 2.25069i −0.900969 − 0.433884i 0.623490 − 0.781831i −1.79487 − 2.25069i −2.16282 − 1.52389i −0.623490 + 0.781831i −1.17651 − 5.15461i −0.623490 − 0.781831i
281.1 0.222521 + 0.974928i −1.45343 − 1.82254i −0.900969 + 0.433884i 0.623490 + 0.781831i 1.45343 − 1.82254i −1.42280 + 2.23061i −0.623490 − 0.781831i −0.541643 + 2.37309i −0.623490 + 0.781831i
281.2 0.222521 + 0.974928i −0.485203 − 0.608425i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.485203 − 0.608425i −1.09473 − 2.40864i −0.623490 − 0.781831i 0.532804 − 2.33436i −0.623490 + 0.781831i
281.3 0.222521 + 0.974928i 0.297754 + 0.373371i −0.900969 + 0.433884i 0.623490 + 0.781831i −0.297754 + 0.373371i 2.64500 + 0.0629581i −0.623490 − 0.781831i 0.616814 − 2.70244i −0.623490 + 0.781831i
281.4 0.222521 + 0.974928i 1.79487 + 2.25069i −0.900969 + 0.433884i 0.623490 + 0.781831i −1.79487 + 2.25069i −2.16282 + 1.52389i −0.623490 − 0.781831i −1.17651 + 5.15461i −0.623490 + 0.781831i
351.1 −0.623490 − 0.781831i −1.95554 + 0.941737i −0.222521 + 0.974928i −0.900969 + 0.433884i 1.95554 + 0.941737i −2.49146 + 0.890295i 0.900969 − 0.433884i 1.06678 − 1.33771i 0.900969 + 0.433884i
351.2 −0.623490 − 0.781831i 0.294799 − 0.141967i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.294799 − 0.141967i −1.75221 − 1.98236i 0.900969 − 0.433884i −1.80372 + 2.26179i 0.900969 + 0.433884i
351.3 −0.623490 − 0.781831i 1.58297 − 0.762318i −0.222521 + 0.974928i −0.900969 + 0.433884i −1.58297 − 0.762318i −0.603900 + 2.57591i 0.900969 − 0.433884i 0.0541958 − 0.0679593i 0.900969 + 0.433884i
351.4 −0.623490 − 0.781831i 2.60223 − 1.25317i −0.222521 + 0.974928i −0.900969 + 0.433884i −2.60223 − 1.25317i 2.30954 − 1.29075i 0.900969 − 0.433884i 3.33069 − 4.17655i 0.900969 + 0.433884i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.k.f ✓ 24
49.e even 7 1 inner 490.2.k.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.k.f ✓ 24 1.a even 1 1 trivial
490.2.k.f ✓ 24 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{24} - 6 T_{3}^{23} + 23 T_{3}^{22} - 58 T_{3}^{21} + 138 T_{3}^{20} - 298 T_{3}^{19} + 780 T_{3}^{18} + \cdots + 64 \) Copy content Toggle raw display
\( T_{11}^{24} + T_{11}^{23} + 14 T_{11}^{22} - 12 T_{11}^{21} + 80 T_{11}^{20} + 1035 T_{11}^{19} + \cdots + 152448409 \) Copy content Toggle raw display