↑

Properties

Label 490.2.q.a
Level $490$
Weight $2$
Character orbit 490.q
Analytic conductor $3.913$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(48\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{2} + 8 q^{3} + 4 q^{4} + 4 q^{5} + 5 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{2} + 8 q^{3} + 4 q^{4} + 4 q^{5} + 5 q^{6} - 8 q^{8} - 16 q^{9} + 4 q^{10} + q^{11} + q^{12} + 8 q^{13} + 7 q^{14} - 2 q^{15} + 4 q^{16} + 11 q^{17} - 9 q^{18} + 22 q^{19} - 8 q^{20} + 12 q^{22} + 23 q^{23} + q^{24} + 4 q^{25} + 10 q^{26} - 22 q^{27} - 14 q^{28} + 22 q^{29} + q^{30} + 32 q^{31} + 4 q^{32} - 52 q^{33} - q^{34} + 7 q^{35} + 11 q^{36} - 41 q^{37} - 6 q^{38} - 34 q^{39} + 4 q^{40} - 12 q^{41} - 7 q^{42} + 35 q^{43} - 6 q^{44} + 5 q^{45} - 26 q^{46} - 15 q^{47} - 2 q^{48} - 42 q^{49} + 48 q^{50} + 13 q^{51} + 10 q^{52} - 40 q^{53} - 38 q^{54} + 12 q^{55} - 14 q^{56} + 28 q^{57} - 25 q^{58} - 9 q^{59} + q^{60} + 49 q^{61} + 13 q^{62} - 8 q^{64} + 10 q^{65} - 24 q^{66} - q^{67} + 4 q^{68} + 92 q^{69} - 27 q^{71} + 5 q^{72} - 39 q^{73} + 8 q^{74} + q^{75} + 12 q^{76} + 21 q^{77} + 19 q^{78} + 45 q^{79} - 24 q^{80} + 31 q^{81} - 15 q^{82} + 24 q^{83} - 28 q^{84} - q^{85} + 7 q^{86} - 86 q^{87} - 6 q^{88} - 23 q^{89} - 10 q^{90} + 63 q^{91} + 10 q^{92} - 45 q^{93} + 27 q^{94} - 41 q^{95} + q^{96} - 136 q^{97} + 70 q^{98} - 34 q^{99}+O(q^{100}) \) 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 −2.29227 + 0.345504i 0.0747301 − 0.997204i 0.365341 − 0.930874i 1.44535 − 1.81241i −0.725784 + 2.54426i 0.623490 + 0.781831i 2.26839 − 0.699706i 0.365341 + 0.930874i
11.2 −0.733052 + 0.680173i −0.0534245 + 0.00805244i 0.0747301 − 0.997204i 0.365341 − 0.930874i 0.0336859 − 0.0422407i 1.67757 + 2.04592i 0.623490 + 0.781831i −2.86393 + 0.883405i 0.365341 + 0.930874i
11.3 −0.733052 + 0.680173i 0.691235 − 0.104187i 0.0747301 − 0.997204i 0.365341 − 0.930874i −0.435846 + 0.546533i −2.40567 + 1.10126i 0.623490 + 0.781831i −2.39977 + 0.740230i 0.365341 + 0.930874i
11.4 −0.733052 + 0.680173i 2.37698 − 0.358272i 0.0747301 − 0.997204i 0.365341 − 0.930874i −1.49876 + 1.87939i −0.126847 − 2.64271i 0.623490 + 0.781831i 2.65494 − 0.818941i 0.365341 + 0.930874i
51.1 0.955573 + 0.294755i −0.740318 + 1.88630i 0.826239 + 0.563320i −0.988831 + 0.149042i −1.26342 + 1.58428i −1.10690 + 2.40307i 0.623490 + 0.781831i −0.810896 − 0.752402i −0.988831 − 0.149042i
51.2 0.955573 + 0.294755i −0.0866614 + 0.220810i 0.826239 + 0.563320i −0.988831 + 0.149042i −0.147896 + 0.185456i −2.38414 + 1.14711i 0.623490 + 0.781831i 2.15791 + 2.00225i −0.988831 − 0.149042i
51.3 0.955573 + 0.294755i 0.421999 − 1.07524i 0.826239 + 0.563320i −0.988831 + 0.149042i 0.720182 − 0.903079i 2.55840 + 0.674216i 0.623490 + 0.781831i 1.22111 + 1.13302i −0.988831 − 0.149042i
51.4 0.955573 + 0.294755i 1.12750 − 2.87283i 0.826239 + 0.563320i −0.988831 + 0.149042i 1.92419 − 2.41286i −1.21399 − 2.35079i 0.623490 + 0.781831i −4.78272 − 4.43771i −0.988831 − 0.149042i
81.1 0.0747301 + 0.997204i −1.63896 − 0.505551i −0.988831 + 0.149042i −0.733052 + 0.680173i 0.381658 − 1.67215i 2.60740 − 0.448823i −0.222521 − 0.974928i −0.0481202 − 0.0328078i −0.733052 − 0.680173i
81.2 0.0747301 + 0.997204i −0.0437759 − 0.0135031i −0.988831 + 0.149042i −0.733052 + 0.680173i 0.0101939 − 0.0446626i −0.958059 − 2.46620i −0.222521 − 0.974928i −2.47698 − 1.68878i −0.733052 − 0.680173i
81.3 0.0747301 + 0.997204i 0.496222 + 0.153064i −0.988831 + 0.149042i −0.733052 + 0.680173i −0.115554 + 0.506273i −0.898856 + 2.48838i −0.222521 − 0.974928i −2.25591 − 1.53805i −0.733052 − 0.680173i
81.4 0.0747301 + 0.997204i 2.58748 + 0.798132i −0.988831 + 0.149042i −0.733052 + 0.680173i −0.602537 + 2.63989i 2.51325 − 0.826772i −0.222521 − 0.974928i 3.57932 + 2.44034i −0.733052 − 0.680173i
121.1 0.0747301 − 0.997204i −1.63896 + 0.505551i −0.988831 − 0.149042i −0.733052 − 0.680173i 0.381658 + 1.67215i 2.60740 + 0.448823i −0.222521 + 0.974928i −0.0481202 + 0.0328078i −0.733052 + 0.680173i
121.2 0.0747301 − 0.997204i −0.0437759 + 0.0135031i −0.988831 − 0.149042i −0.733052 − 0.680173i 0.0101939 + 0.0446626i −0.958059 + 2.46620i −0.222521 + 0.974928i −2.47698 + 1.68878i −0.733052 + 0.680173i
121.3 0.0747301 − 0.997204i 0.496222 − 0.153064i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.115554 − 0.506273i −0.898856 − 2.48838i −0.222521 + 0.974928i −2.25591 + 1.53805i −0.733052 + 0.680173i
121.4 0.0747301 − 0.997204i 2.58748 − 0.798132i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.602537 − 2.63989i 2.51325 + 0.826772i −0.222521 + 0.974928i 3.57932 − 2.44034i −0.733052 + 0.680173i
151.1 0.826239 + 0.563320i −1.26371 − 1.17255i 0.365341 + 0.930874i 0.955573 − 0.294755i −0.383603 − 1.68068i −2.63600 − 0.226895i −0.222521 + 0.974928i −0.00210584 − 0.0281004i 0.955573 + 0.294755i
151.2 0.826239 + 0.563320i −0.314462 − 0.291778i 0.365341 + 0.930874i 0.955573 − 0.294755i −0.0954562 − 0.418221i 2.31837 − 1.27482i −0.222521 + 0.974928i −0.210438 − 2.80811i 0.955573 + 0.294755i
151.3 0.826239 + 0.563320i 1.07663 + 0.998970i 0.365341 + 0.930874i 0.955573 − 0.294755i 0.326816 + 1.43188i −0.896339 + 2.48929i −0.222521 + 0.974928i −0.0629915 − 0.840563i 0.955573 + 0.294755i
151.4 0.826239 + 0.563320i 1.90250 + 1.76527i 0.365341 + 0.930874i 0.955573 − 0.294755i 0.577513 + 2.53025i 1.16671 − 2.37461i −0.222521 + 0.974928i 0.279168 + 3.72524i 0.955573 + 0.294755i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

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.a ✓ 48
49.g even 21 1 inner 490.2.q.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.q.a ✓ 48 1.a even 1 1 trivial
490.2.q.a ✓ 48 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 8 T_{3}^{47} + 34 T_{3}^{46} - 90 T_{3}^{45} + 109 T_{3}^{44} + 179 T_{3}^{43} - 1148 T_{3}^{42} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display