Properties

Label 740.2.k.c
Level $740$
Weight $2$
Character orbit 740.k
Analytic conductor $5.909$
Analytic rank $0$
Dimension $216$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(179,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.k (of order \(4\), degree \(2\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(108\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{5} + 4 q^{6} - 232 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{5} + 4 q^{6} - 232 q^{9} - 16 q^{10} - 12 q^{14} + 8 q^{16} - 4 q^{20} - 36 q^{24} - 48 q^{26} + 16 q^{29} - 16 q^{34} - 72 q^{44} + 16 q^{45} - 136 q^{46} + 184 q^{49} - 32 q^{50} - 20 q^{54} + 20 q^{56} + 36 q^{60} + 16 q^{61} + 28 q^{66} - 32 q^{69} - 60 q^{70} - 92 q^{74} - 28 q^{76} - 12 q^{80} + 152 q^{81} - 32 q^{84} + 72 q^{86} - 40 q^{89} - 16 q^{90} - 48 q^{94} + 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.41417 + 0.0112112i 1.10399i 1.99975 0.0317091i −0.0702723 2.23496i 0.0123771 + 1.56123i 0.423249 −2.82763 + 0.0672617i 1.78120 0.124434 + 3.15983i
179.2 −1.41411 + 0.0169178i 1.43960i 1.99943 0.0478472i −1.78993 + 1.34020i −0.0243548 2.03576i −0.0804759 −2.82661 + 0.101487i 0.927550 2.50849 1.92548i
179.3 −1.41220 + 0.0754091i 1.91116i 1.98863 0.212986i −2.20851 0.349987i 0.144119 + 2.69894i 3.09756 −2.79228 + 0.450740i −0.652519 3.14525 + 0.327711i
179.4 −1.39238 0.247533i 3.24286i 1.87745 + 0.689322i −0.455272 + 2.18923i −0.802717 + 4.51530i −2.08644 −2.44350 1.42453i −7.51616 1.17582 2.93555i
179.5 −1.39007 0.260196i 2.69556i 1.86460 + 0.723381i 1.08099 + 1.95741i 0.701373 3.74702i 2.34449 −2.40370 1.49071i −4.26605 −0.993338 3.00221i
179.6 −1.38535 0.284249i 3.33433i 1.83841 + 0.787570i −1.65612 1.50242i 0.947781 4.61923i −2.00442 −2.32297 1.61363i −8.11779 1.86725 + 2.55213i
179.7 −1.38410 0.290269i 0.837020i 1.83149 + 0.803524i 1.69797 1.45495i −0.242961 + 1.15852i −0.279655 −2.30173 1.64379i 2.29940 −2.77250 + 1.52094i
179.8 −1.37717 + 0.321579i 1.34163i 1.79317 0.885734i 1.95385 + 1.08741i 0.431440 + 1.84765i −1.32565 −2.18467 + 1.79645i 1.20002 −3.04047 0.869228i
179.9 −1.37515 + 0.330097i 2.62444i 1.78207 0.907866i 1.29187 1.82512i −0.866322 3.60900i 3.53783 −2.15093 + 1.83671i −3.88771 −1.17405 + 2.93626i
179.10 −1.37317 0.338251i 0.235460i 1.77117 + 0.928950i −1.34759 + 1.78438i −0.0796445 + 0.323325i −4.45596 −2.11790 1.87470i 2.94456 2.45403 1.99443i
179.11 −1.36131 0.383186i 0.145516i 1.70634 + 1.04327i 0.799638 + 2.08820i −0.0557598 + 0.198093i 3.05986 −1.92309 2.07406i 2.97883 −0.288388 3.14910i
179.12 −1.36028 + 0.386818i 1.72633i 1.70074 1.05236i 1.60779 + 1.55403i −0.667777 2.34830i −3.84183 −1.90642 + 2.08939i 0.0197698 −2.78818 1.49200i
179.13 −1.31376 + 0.523479i 3.06130i 1.45194 1.37545i 1.66991 1.48707i 1.60252 + 4.02182i 0.959145 −1.18748 + 2.56708i −6.37155 −1.41542 + 2.82783i
179.14 −1.30358 + 0.548342i 1.78936i 1.39864 1.42962i −0.621142 2.14806i −0.981180 2.33257i −0.447293 −1.03932 + 2.63055i −0.201798 1.98758 + 2.45958i
179.15 −1.28091 + 0.599393i 0.924956i 1.28146 1.53554i −2.23380 + 0.100625i −0.554412 1.18479i 0.438499 −0.721042 + 2.73498i 2.14446 2.80098 1.46782i
179.16 −1.27262 0.616804i 2.09980i 1.23910 + 1.56991i 2.13596 0.661580i 1.29516 2.67223i −2.52168 −0.608578 2.76218i −1.40914 −3.12632 0.475530i
179.17 −1.25262 + 0.656462i 0.0160284i 1.13811 1.64460i −0.177395 + 2.22902i −0.0105220 0.0200775i 3.72929 −0.346011 + 2.80718i 2.99974 −1.24106 2.90857i
179.18 −1.24820 + 0.664834i 2.39692i 1.11599 1.65969i −1.98754 1.02454i 1.59355 + 2.99183i −4.93364 −0.289561 + 2.81357i −2.74521 3.16199 0.0425512i
179.19 −1.22029 0.714774i 2.58416i 0.978197 + 1.74446i 2.17123 + 0.534580i −1.84709 + 3.15342i 4.55780 0.0532110 2.82793i −3.67790 −2.26741 2.20428i
179.20 −1.21943 0.716239i 0.846684i 0.974003 + 1.74680i −0.116235 2.23304i 0.606428 1.03247i 3.61247 0.0634021 2.82772i 2.28313 −1.45765 + 2.80629i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
37.d odd 4 1 inner
148.g even 4 1 inner
185.j odd 4 1 inner
740.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.k.c 216
4.b odd 2 1 inner 740.2.k.c 216
5.b even 2 1 inner 740.2.k.c 216
20.d odd 2 1 inner 740.2.k.c 216
37.d odd 4 1 inner 740.2.k.c 216
148.g even 4 1 inner 740.2.k.c 216
185.j odd 4 1 inner 740.2.k.c 216
740.k even 4 1 inner 740.2.k.c 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.k.c 216 1.a even 1 1 trivial
740.2.k.c 216 4.b odd 2 1 inner
740.2.k.c 216 5.b even 2 1 inner
740.2.k.c 216 20.d odd 2 1 inner
740.2.k.c 216 37.d odd 4 1 inner
740.2.k.c 216 148.g even 4 1 inner
740.2.k.c 216 185.j odd 4 1 inner
740.2.k.c 216 740.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\):

\( T_{3}^{54} + 110 T_{3}^{52} + 5667 T_{3}^{50} + 181782 T_{3}^{48} + 4071449 T_{3}^{46} + 67678388 T_{3}^{44} + \cdots + 19232 \) Copy content Toggle raw display
\( T_{13}^{108} + 9598 T_{13}^{104} + 40254107 T_{13}^{100} + 97793463422 T_{13}^{96} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display