↑

Properties

Label 490.2.t.a
Level $490$
Weight $2$
Character orbit 490.t
Analytic conductor $3.913$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(9,490)]
 
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([21, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.t (of order \(42\), 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: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{4} - 2 q^{5} - 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{4} - 2 q^{5} - 2 q^{6} - 12 q^{9} - 2 q^{10} + 30 q^{11} + 2 q^{14} + 2 q^{15} + 28 q^{16} - 14 q^{19} + 10 q^{20} - 6 q^{21} + 6 q^{24} - 28 q^{26} + 2 q^{29} - 6 q^{30} + 16 q^{31} - 16 q^{34} - 26 q^{35} - 66 q^{36} - 124 q^{39} - 26 q^{40} + 56 q^{41} - 30 q^{44} - 150 q^{45} + 76 q^{46} - 86 q^{49} + 8 q^{50} + 116 q^{51} - 66 q^{54} - 102 q^{55} + 26 q^{56} - 38 q^{59} - 20 q^{60} + 106 q^{61} + 56 q^{64} + 2 q^{65} + 12 q^{69} - 12 q^{70} + 68 q^{71} + 16 q^{74} - 24 q^{75} + 28 q^{76} - 2 q^{80} + 38 q^{81} - 60 q^{84} + 48 q^{85} + 94 q^{86} - 130 q^{89} - 102 q^{90} - 90 q^{91} + 68 q^{94} - 116 q^{95} - 6 q^{96} + 212 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.680173 + 0.733052i −0.452801 + 3.00414i −0.0747301 − 0.997204i −0.656908 − 2.13740i −1.89421 − 2.37526i −1.36069 − 2.26904i 0.781831 + 0.623490i −5.95309 − 1.83628i 2.01363 + 0.972253i
9.2 −0.680173 + 0.733052i −0.421296 + 2.79512i −0.0747301 − 0.997204i −1.64160 + 1.51827i −1.76241 − 2.20999i 2.63387 − 0.250413i 0.781831 + 0.623490i −4.76846 − 1.47088i 0.00359951 − 2.23607i
9.3 −0.680173 + 0.733052i −0.390367 + 2.58992i −0.0747301 − 0.997204i 2.15196 − 0.607502i −1.63303 − 2.04775i −0.807491 + 2.51952i 0.781831 + 0.623490i −3.68857 − 1.13777i −1.01838 + 1.99071i
9.4 −0.680173 + 0.733052i −0.206478 + 1.36989i −0.0747301 − 0.997204i −2.08106 + 0.818032i −0.863761 − 1.08312i −2.39528 + 1.12366i 0.781831 + 0.623490i 1.03275 + 0.318561i 0.815822 − 2.08193i
9.5 −0.680173 + 0.733052i −0.199620 + 1.32439i −0.0747301 − 0.997204i −1.69293 − 1.46082i −0.835070 − 1.04714i 2.26421 + 1.36870i 0.781831 + 0.623490i 1.15256 + 0.355518i 2.22234 − 0.247400i
9.6 −0.680173 + 0.733052i −0.172749 + 1.14612i −0.0747301 − 0.997204i 0.0531064 + 2.23544i −0.722663 − 0.906191i 0.177552 − 2.63979i 0.781831 + 0.623490i 1.58298 + 0.488284i −1.67481 − 1.48155i
9.7 −0.680173 + 0.733052i −0.0904636 + 0.600187i −0.0747301 − 0.997204i 1.04970 + 1.97437i −0.378437 − 0.474545i 0.0232975 + 2.64565i 0.781831 + 0.623490i 2.51468 + 0.775675i −2.16129 − 0.573426i
9.8 −0.680173 + 0.733052i 0.0240674 − 0.159677i −0.0747301 − 0.997204i 2.07598 − 0.830856i 0.100681 + 0.126250i −2.39121 − 1.13230i 0.781831 + 0.623490i 2.84180 + 0.876580i −0.802962 + 2.08692i
9.9 −0.680173 + 0.733052i 0.171856 − 1.14019i −0.0747301 − 0.997204i 0.202168 − 2.22691i 0.718925 + 0.901504i 1.28051 + 2.31523i 0.781831 + 0.623490i 1.59622 + 0.492370i 1.49493 + 1.66288i
9.10 −0.680173 + 0.733052i 0.187204 − 1.24202i −0.0747301 − 0.997204i 2.00831 + 0.983196i 0.783131 + 0.982016i 2.59921 − 0.494061i 0.781831 + 0.623490i 1.35916 + 0.419245i −2.08673 + 0.803455i
9.11 −0.680173 + 0.733052i 0.256662 − 1.70284i −0.0747301 − 0.997204i −1.99117 − 1.01747i 1.07370 + 1.34637i 1.98745 − 1.74644i 0.781831 + 0.623490i 0.0329200 + 0.0101545i 2.10020 − 0.767579i
9.12 −0.680173 + 0.733052i 0.306897 − 2.03613i −0.0747301 − 0.997204i −0.673580 + 2.13220i 1.28384 + 1.60989i −1.67475 − 2.04822i 0.781831 + 0.623490i −1.18491 − 0.365496i −1.10486 − 1.94404i
9.13 −0.680173 + 0.733052i 0.320103 − 2.12374i −0.0747301 − 0.997204i −1.84385 + 1.26499i 1.33909 + 1.67916i −0.701229 + 2.55113i 0.781831 + 0.623490i −1.54110 − 0.475366i 0.326836 − 2.21205i
9.14 −0.680173 + 0.733052i 0.477803 − 3.17002i −0.0747301 − 0.997204i 1.31545 − 1.80820i 1.99880 + 2.50641i 0.143595 − 2.64185i 0.781831 + 0.623490i −6.95400 − 2.14502i 0.430773 + 2.19418i
9.15 0.680173 − 0.733052i −0.477803 + 3.17002i −0.0747301 − 0.997204i −2.19418 − 0.430773i 1.99880 + 2.50641i −0.143595 + 2.64185i −0.781831 − 0.623490i −6.95400 − 2.14502i −1.80820 + 1.31545i
9.16 0.680173 − 0.733052i −0.320103 + 2.12374i −0.0747301 − 0.997204i 2.21205 − 0.326836i 1.33909 + 1.67916i 0.701229 − 2.55113i −0.781831 − 0.623490i −1.54110 − 0.475366i 1.26499 − 1.84385i
9.17 0.680173 − 0.733052i −0.306897 + 2.03613i −0.0747301 − 0.997204i 1.94404 + 1.10486i 1.28384 + 1.60989i 1.67475 + 2.04822i −0.781831 − 0.623490i −1.18491 − 0.365496i 2.13220 − 0.673580i
9.18 0.680173 − 0.733052i −0.256662 + 1.70284i −0.0747301 − 0.997204i 0.767579 − 2.10020i 1.07370 + 1.34637i −1.98745 + 1.74644i −0.781831 − 0.623490i 0.0329200 + 0.0101545i −1.01747 − 1.99117i
9.19 0.680173 − 0.733052i −0.187204 + 1.24202i −0.0747301 − 0.997204i −0.803455 + 2.08673i 0.783131 + 0.982016i −2.59921 + 0.494061i −0.781831 − 0.623490i 1.35916 + 0.419245i 0.983196 + 2.00831i
9.20 0.680173 − 0.733052i −0.171856 + 1.14019i −0.0747301 − 0.997204i −1.66288 − 1.49493i 0.718925 + 0.901504i −1.28051 − 2.31523i −0.781831 − 0.623490i 1.59622 + 0.492370i −2.22691 + 0.202168i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
49.g even 21 1 inner
245.t even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.t.a ✓ 336
5.b even 2 1 inner 490.2.t.a ✓ 336
49.g even 21 1 inner 490.2.t.a ✓ 336
245.t even 42 1 inner 490.2.t.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.t.a ✓ 336 1.a even 1 1 trivial
490.2.t.a ✓ 336 5.b even 2 1 inner
490.2.t.a ✓ 336 49.g even 21 1 inner
490.2.t.a ✓ 336 245.t even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(490, [\chi])\).