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Properties

Label 490.2.q.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} + 5 q^{6} + 4 q^{7} + 10 q^{8} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} + 5 q^{6} + 4 q^{7} + 10 q^{8} + 41 q^{9} - 5 q^{10} + 4 q^{11} - q^{12} - 10 q^{13} - 2 q^{14} + 2 q^{15} + 5 q^{16} + 3 q^{17} + 43 q^{18} - 21 q^{19} - 10 q^{20} - 2 q^{21} - 6 q^{22} + 28 q^{23} + q^{24} + 5 q^{25} + 2 q^{26} + 82 q^{27} + 15 q^{28} + 18 q^{29} + q^{30} - 20 q^{31} - 5 q^{32} - 90 q^{33} - 15 q^{34} - 5 q^{35} - 19 q^{36} + 6 q^{37} - 21 q^{38} + 68 q^{39} - 5 q^{40} - 10 q^{41} + q^{42} + 23 q^{43} - 3 q^{44} + 6 q^{45} - 35 q^{46} + 76 q^{47} + 2 q^{48} + 12 q^{49} - 60 q^{50} - 41 q^{51} + 5 q^{52} + 8 q^{53} - 50 q^{54} - 15 q^{55} - 4 q^{56} - 48 q^{57} - 5 q^{58} - 2 q^{59} - q^{60} + 21 q^{61} - 33 q^{62} - 103 q^{63} - 10 q^{64} + 5 q^{65} + 34 q^{66} - 29 q^{67} + 24 q^{68} + 40 q^{69} - q^{70} - 53 q^{71} - 6 q^{72} + 83 q^{73} + 8 q^{74} - q^{75} + 14 q^{76} + 24 q^{77} + 31 q^{78} + 9 q^{79} - 30 q^{80} - 52 q^{81} + 16 q^{82} - 2 q^{83} - 74 q^{84} + 15 q^{85} + q^{86} - 16 q^{87} + 3 q^{88} + 27 q^{89} + 12 q^{90} - 32 q^{91} - 28 q^{92} + 115 q^{93} + q^{94} + 28 q^{95} + q^{96} - 136 q^{97} + q^{98} + 88 q^{99}+O(q^{100}) \) 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 −2.88536 + 0.434898i 0.0747301 − 0.997204i 0.365341 − 0.930874i −1.81931 + 2.28135i −1.91617 + 1.82436i −0.623490 − 0.781831i 5.26944 − 1.62541i −0.365341 − 0.930874i
11.2 0.733052 − 0.680173i −1.77355 + 0.267320i 0.0747301 − 0.997204i 0.365341 − 0.930874i −1.11828 + 1.40228i 2.43836 + 1.02684i −0.623490 − 0.781831i 0.207310 − 0.0639468i −0.365341 − 0.930874i
11.3 0.733052 − 0.680173i −1.15786 + 0.174520i 0.0747301 − 0.997204i 0.365341 − 0.930874i −0.730070 + 0.915478i −1.07869 − 2.41587i −0.623490 − 0.781831i −1.55653 + 0.480126i −0.365341 − 0.930874i
11.4 0.733052 − 0.680173i 1.87814 − 0.283084i 0.0747301 − 0.997204i 0.365341 − 0.930874i 1.18423 − 1.48498i 2.56182 + 0.661108i −0.623490 − 0.781831i 0.580563 − 0.179080i −0.365341 − 0.930874i
11.5 0.733052 − 0.680173i 3.21611 − 0.484751i 0.0747301 − 0.997204i 0.365341 − 0.930874i 2.02786 − 2.54286i −2.51418 − 0.823954i −0.623490 − 0.781831i 7.24167 − 2.23376i −0.365341 − 0.930874i
51.1 −0.955573 − 0.294755i −1.18540 + 3.02036i 0.826239 + 0.563320i −0.988831 + 0.149042i 2.02300 − 2.53677i 2.22116 − 1.43751i −0.623490 − 0.781831i −5.51822 − 5.12016i 0.988831 + 0.149042i
51.2 −0.955573 − 0.294755i −0.618640 + 1.57627i 0.826239 + 0.563320i −0.988831 + 0.149042i 1.05577 − 1.32389i 1.56034 + 2.13666i −0.623490 − 0.781831i 0.0972466 + 0.0902317i 0.988831 + 0.149042i
51.3 −0.955573 − 0.294755i 0.169726 − 0.432456i 0.826239 + 0.563320i −0.988831 + 0.149042i −0.289655 + 0.363215i 1.60699 − 2.10181i −0.623490 − 0.781831i 2.04094 + 1.89372i 0.988831 + 0.149042i
51.4 −0.955573 − 0.294755i 0.286220 − 0.729276i 0.826239 + 0.563320i −0.988831 + 0.149042i −0.488462 + 0.612512i −2.44102 + 1.02050i −0.623490 − 0.781831i 1.74923 + 1.62305i 0.988831 + 0.149042i
51.5 −0.955573 − 0.294755i 0.625575 − 1.59394i 0.826239 + 0.563320i −0.988831 + 0.149042i −1.06760 + 1.33873i 0.486819 + 2.60058i −0.623490 − 0.781831i 0.0498569 + 0.0462604i 0.988831 + 0.149042i
81.1 −0.0747301 − 0.997204i −3.04070 − 0.937933i −0.988831 + 0.149042i −0.733052 + 0.680173i −0.708078 + 3.10229i −2.20280 + 1.46550i 0.222521 + 0.974928i 5.88744 + 4.01399i 0.733052 + 0.680173i
81.2 −0.0747301 − 0.997204i −1.94576 − 0.600187i −0.988831 + 0.149042i −0.733052 + 0.680173i −0.453102 + 1.98517i 0.0985503 − 2.64392i 0.222521 + 0.974928i 0.947038 + 0.645680i 0.733052 + 0.680173i
81.3 −0.0747301 − 0.997204i −0.211630 − 0.0652792i −0.988831 + 0.149042i −0.733052 + 0.680173i −0.0492815 + 0.215917i 2.54929 − 0.707898i 0.222521 + 0.974928i −2.43819 − 1.66233i 0.733052 + 0.680173i
81.4 −0.0747301 − 0.997204i 0.740061 + 0.228279i −0.988831 + 0.149042i −0.733052 + 0.680173i 0.172335 − 0.755051i −2.57052 − 0.626432i 0.222521 + 0.974928i −1.98314 − 1.35208i 0.733052 + 0.680173i
81.5 −0.0747301 − 0.997204i 3.05706 + 0.942979i −0.988831 + 0.149042i −0.733052 + 0.680173i 0.711888 − 3.11898i −0.766285 + 2.53235i 0.222521 + 0.974928i 5.97771 + 4.07553i 0.733052 + 0.680173i
121.1 −0.0747301 + 0.997204i −3.04070 + 0.937933i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.708078 − 3.10229i −2.20280 − 1.46550i 0.222521 − 0.974928i 5.88744 − 4.01399i 0.733052 − 0.680173i
121.2 −0.0747301 + 0.997204i −1.94576 + 0.600187i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.453102 − 1.98517i 0.0985503 + 2.64392i 0.222521 − 0.974928i 0.947038 − 0.645680i 0.733052 − 0.680173i
121.3 −0.0747301 + 0.997204i −0.211630 + 0.0652792i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.0492815 − 0.215917i 2.54929 + 0.707898i 0.222521 − 0.974928i −2.43819 + 1.66233i 0.733052 − 0.680173i
121.4 −0.0747301 + 0.997204i 0.740061 − 0.228279i −0.988831 − 0.149042i −0.733052 − 0.680173i 0.172335 + 0.755051i −2.57052 + 0.626432i 0.222521 − 0.974928i −1.98314 + 1.35208i 0.733052 − 0.680173i
121.5 −0.0747301 + 0.997204i 3.05706 − 0.942979i −0.988831 − 0.149042i −0.733052 − 0.680173i 0.711888 + 3.11898i −0.766285 − 2.53235i 0.222521 − 0.974928i 5.97771 − 4.07553i 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.b ✓ 60
49.g even 21 1 inner 490.2.q.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.q.b ✓ 60 1.a even 1 1 trivial
490.2.q.b ✓ 60 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 8 T_{3}^{59} + 4 T_{3}^{58} - 170 T_{3}^{57} - 607 T_{3}^{56} + 639 T_{3}^{55} + \cdots + 194389282816 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display