↑

Properties

Label 260.2.bj.c
Level $260$
Weight $2$
Character orbit 260.bj
Analytic conductor $2.076$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,2,Mod(3,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.bj (of order \(12\), degree \(4\), 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: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 24 q^{5} - 4 q^{6} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 24 q^{5} - 4 q^{6} - 24 q^{8} - 16 q^{10} + 20 q^{12} - 12 q^{13} - 28 q^{16} - 4 q^{18} + 30 q^{20} - 32 q^{21} - 28 q^{22} - 24 q^{25} - 12 q^{26} + 14 q^{28} - 4 q^{30} + 4 q^{32} - 28 q^{33} + 4 q^{36} + 20 q^{40} + 24 q^{41} - 56 q^{42} - 4 q^{46} + 12 q^{48} + 20 q^{50} - 2 q^{52} + 24 q^{53} - 20 q^{56} - 24 q^{57} - 42 q^{58} + 88 q^{60} - 32 q^{61} - 128 q^{66} - 32 q^{68} + 108 q^{70} + 2 q^{72} - 8 q^{73} + 60 q^{76} - 72 q^{77} - 120 q^{78} - 64 q^{80} - 32 q^{81} - 42 q^{82} - 48 q^{85} - 24 q^{86} - 42 q^{88} - 56 q^{90} - 84 q^{92} + 8 q^{93} + 160 q^{96} + 68 q^{97} + 98 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41378 + 0.0350218i −1.66173 − 0.445260i 1.99755 − 0.0990263i 1.78102 + 1.35202i 2.36492 + 0.571303i −1.76644 + 0.473317i −2.82062 + 0.209959i −0.0349771 − 0.0201940i −2.56533 − 1.84908i
3.2 −1.40734 + 0.139220i 0.160249 + 0.0429387i 1.96124 − 0.391860i 1.36197 − 1.77343i −0.231504 − 0.0381197i 3.90057 − 1.04516i −2.70558 + 0.824524i −2.57424 − 1.48624i −1.66986 + 2.68544i
3.3 −1.38499 − 0.285996i 2.68406 + 0.719191i 1.83641 + 0.792206i −2.23362 − 0.104689i −3.51171 − 1.76370i 1.57219 − 0.421268i −2.31685 − 1.62241i 4.08885 + 2.36070i 3.06360 + 0.783800i
3.4 −1.34244 − 0.444817i −2.68406 − 0.719191i 1.60428 + 1.19428i −2.23362 − 0.104689i 3.28327 + 2.15938i −1.57219 + 0.421268i −1.62241 − 2.31685i 4.08885 + 2.36070i 2.95192 + 1.13409i
3.5 −1.34050 + 0.450613i −0.804148 − 0.215471i 1.59390 − 1.20810i −2.02083 + 0.957199i 1.17506 − 0.0735202i 1.98521 − 0.531935i −1.59224 + 2.33769i −1.99785 − 1.15346i 2.27761 − 2.19374i
3.6 −1.29282 + 0.573250i 3.02314 + 0.810048i 1.34277 − 1.48222i 2.23524 − 0.0606684i −4.37274 + 0.685770i −2.22941 + 0.597369i −0.886276 + 2.68599i 5.88512 + 3.39778i −2.85499 + 1.35979i
3.7 −1.20686 − 0.737220i 1.66173 + 0.445260i 0.913014 + 1.77944i 1.78102 + 1.35202i −1.67722 − 1.76243i 1.76644 − 0.473317i 0.209959 − 2.82062i −0.0349771 − 0.0201940i −1.15271 − 2.94470i
3.8 −1.14919 − 0.824240i −0.160249 − 0.0429387i 0.641258 + 1.89441i 1.36197 − 1.77343i 0.148765 + 0.181428i −3.90057 + 1.04516i 0.824524 − 2.70558i −2.57424 − 1.48624i −3.02688 + 0.915413i
3.9 −1.11075 + 0.875350i −2.57196 − 0.689155i 0.467524 − 1.94459i 0.222626 − 2.22496i 3.46005 − 1.48589i −1.04514 + 0.280044i 1.18289 + 2.56919i 3.54197 + 2.04496i 1.70034 + 2.66624i
3.10 −1.10569 + 0.881731i 0.840154 + 0.225119i 0.445099 − 1.94984i −0.905596 + 2.04448i −1.12744 + 0.491879i −2.80535 + 0.751691i 1.22710 + 2.54838i −1.94290 − 1.12173i −0.801374 − 3.05905i
3.11 −0.935603 − 1.06049i 0.804148 + 0.215471i −0.249293 + 1.98440i −2.02083 + 0.957199i −0.523858 − 1.05439i −1.98521 + 0.531935i 2.33769 − 1.59224i −1.99785 − 1.15346i 2.90580 + 1.24752i
3.12 −0.924149 + 1.07049i 1.38963 + 0.372351i −0.291896 − 1.97858i −0.609763 − 2.15132i −1.68282 + 1.14348i 0.0945931 − 0.0253462i 2.38781 + 1.51604i −0.805646 − 0.465140i 2.86648 + 1.33540i
3.13 −0.832990 − 1.14286i −3.02314 − 0.810048i −0.612255 + 1.90398i 2.23524 − 0.0606684i 1.59247 + 4.12979i 2.22941 − 0.597369i 2.68599 − 0.886276i 5.88512 + 3.39778i −1.93127 − 2.50403i
3.14 −0.594250 + 1.28330i 0.960123 + 0.257264i −1.29373 − 1.52521i 1.38665 + 1.75420i −0.900701 + 1.07925i 4.50882 − 1.20813i 2.72610 − 0.753900i −1.74242 − 1.00599i −3.07518 + 0.737057i
3.15 −0.524261 − 1.31345i 2.57196 + 0.689155i −1.45030 + 1.37718i 0.222626 − 2.22496i −0.443209 − 3.73944i 1.04514 − 0.280044i 2.56919 + 1.18289i 3.54197 + 2.04496i −3.03908 + 0.874051i
3.16 −0.516690 − 1.31645i −0.840154 − 0.225119i −1.46606 + 1.36039i −0.905596 + 2.04448i 0.137742 + 1.22233i 2.80535 − 0.751691i 2.54838 + 1.22710i −1.94290 − 1.12173i 3.15936 + 0.135807i
3.17 −0.358012 + 1.36815i −2.61303 − 0.700158i −1.74365 − 0.979628i −0.297936 + 2.21613i 1.89342 − 3.32434i −3.10919 + 0.833104i 1.96452 − 2.03486i 3.73961 + 2.15907i −2.92533 − 1.20102i
3.18 −0.265092 − 1.38915i −1.38963 − 0.372351i −1.85945 + 0.736503i −0.609763 − 2.15132i −0.148869 + 2.02911i −0.0945931 + 0.0253462i 1.51604 + 2.38781i −0.805646 − 0.465140i −2.82686 + 1.41735i
3.19 −0.0733540 + 1.41231i 2.99344 + 0.802089i −1.98924 − 0.207197i −1.68089 + 1.47466i −1.35238 + 4.16882i −0.986808 + 0.264414i 0.438545 − 2.79422i 5.71924 + 3.30200i −1.95938 − 2.48210i
3.20 −0.0703644 + 1.41246i −0.124346 − 0.0333185i −1.99010 − 0.198774i −1.85457 − 1.24923i 0.0558107 − 0.173290i −0.462161 + 0.123836i 0.420793 − 2.79695i −2.58372 − 1.49171i 1.89498 − 2.53161i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.c even 3 1 inner
20.e even 4 1 inner
52.j odd 6 1 inner
65.q odd 12 1 inner
260.bj even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.bj.c ✓ 144
4.b odd 2 1 inner 260.2.bj.c ✓ 144
5.c odd 4 1 inner 260.2.bj.c ✓ 144
13.c even 3 1 inner 260.2.bj.c ✓ 144
20.e even 4 1 inner 260.2.bj.c ✓ 144
52.j odd 6 1 inner 260.2.bj.c ✓ 144
65.q odd 12 1 inner 260.2.bj.c ✓ 144
260.bj even 12 1 inner 260.2.bj.c ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bj.c ✓ 144 1.a even 1 1 trivial
260.2.bj.c ✓ 144 4.b odd 2 1 inner
260.2.bj.c ✓ 144 5.c odd 4 1 inner
260.2.bj.c ✓ 144 13.c even 3 1 inner
260.2.bj.c ✓ 144 20.e even 4 1 inner
260.2.bj.c ✓ 144 52.j odd 6 1 inner
260.2.bj.c ✓ 144 65.q odd 12 1 inner
260.2.bj.c ✓ 144 260.bj even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3}^{144} - 478 T_{3}^{140} + 129567 T_{3}^{136} - 23966422 T_{3}^{132} + 3345451151 T_{3}^{128} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
\( T_{17}^{72} - 80 T_{17}^{69} - 3662 T_{17}^{68} + 472 T_{17}^{67} + 3200 T_{17}^{66} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display