Properties

Label 740.2.n.a
Level $740$
Weight $2$
Character orbit 740.n
Analytic conductor $5.909$
Analytic rank $0$
Dimension $216$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(223,740)]
 
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, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.n (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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{6} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{6} - 12 q^{8} - 16 q^{10} - 16 q^{12} + 8 q^{16} + 20 q^{20} - 16 q^{21} + 4 q^{22} + 16 q^{26} - 4 q^{28} - 24 q^{30} - 20 q^{32} + 16 q^{33} + 32 q^{36} - 12 q^{38} - 36 q^{40} - 40 q^{42} + 40 q^{46} - 36 q^{48} + 24 q^{50} - 20 q^{52} - 16 q^{56} - 32 q^{57} - 16 q^{58} + 20 q^{60} - 64 q^{61} + 4 q^{62} - 32 q^{65} + 52 q^{68} - 24 q^{70} + 40 q^{72} - 16 q^{73} - 24 q^{76} + 48 q^{77} + 24 q^{78} - 56 q^{80} - 136 q^{81} - 4 q^{82} + 48 q^{85} - 88 q^{86} - 64 q^{88} + 120 q^{90} + 56 q^{92} + 16 q^{96} - 80 q^{97} - 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 −1.41316 + 0.0545886i −0.951440 + 0.951440i 1.99404 0.154285i −0.914889 + 2.04034i 1.29260 1.39647i −1.13976 1.13976i −2.80947 + 0.326881i 1.18953i 1.18151 2.93327i
223.2 −1.41301 0.0583967i 0.962036 0.962036i 1.99318 + 0.165030i 2.04543 0.903457i −1.41554 + 1.30318i −1.74495 1.74495i −2.80674 0.349584i 1.14897i −2.94296 + 1.15715i
223.3 −1.41119 + 0.0924716i 0.888719 0.888719i 1.98290 0.260990i 1.22558 + 1.87028i −1.17197 + 1.33633i 0.0171664 + 0.0171664i −2.77411 + 0.551667i 1.42036i −1.90247 2.52599i
223.4 −1.40602 0.152053i −0.119582 + 0.119582i 1.95376 + 0.427577i −1.64048 + 1.51948i 0.186316 0.149951i 3.40042 + 3.40042i −2.68200 0.898254i 2.97140i 2.53758 1.88697i
223.5 −1.40438 0.166475i 2.15691 2.15691i 1.94457 + 0.467589i −1.12739 1.93106i −3.38820 + 2.67006i 0.558475 + 0.558475i −2.65308 0.980396i 6.30456i 1.26181 + 2.89963i
223.6 −1.40119 0.191494i −1.30753 + 1.30753i 1.92666 + 0.536639i 2.18104 0.493016i 2.08247 1.58171i 2.86607 + 2.86607i −2.59685 1.12088i 0.419248i −3.15046 + 0.273152i
223.7 −1.39465 + 0.234390i −0.532261 + 0.532261i 1.89012 0.653787i 0.164080 2.23004i 0.617563 0.867076i −0.702605 0.702605i −2.48283 + 1.35483i 2.43340i 0.293865 + 3.14859i
223.8 −1.39119 + 0.254135i −1.00358 + 1.00358i 1.87083 0.707102i −2.22383 + 0.233612i 1.14113 1.65122i −3.06916 3.06916i −2.42298 + 1.45916i 0.985641i 3.03441 0.890154i
223.9 −1.38440 0.288867i −2.33229 + 2.33229i 1.83311 + 0.799813i 1.44400 1.70729i 3.90254 2.55510i −2.92128 2.92128i −2.30672 1.63678i 7.87919i −2.49226 + 1.94645i
223.10 −1.35300 0.411587i 1.00904 1.00904i 1.66119 + 1.11375i −2.23607 0.000144217i −1.78053 + 0.949915i −0.740784 0.740784i −1.78918 2.19062i 0.963694i 3.02533 + 0.920531i
223.11 −1.34504 + 0.436894i −2.03010 + 2.03010i 1.61825 1.17528i −2.00579 0.988345i 1.84362 3.61750i 2.09201 + 2.09201i −1.66313 + 2.28779i 5.24262i 3.12966 + 0.453044i
223.12 −1.33397 + 0.469611i 1.90038 1.90038i 1.55893 1.25289i 2.19518 + 0.425649i −1.64260 + 3.42748i 2.19059 + 2.19059i −1.49119 + 2.40341i 4.22286i −3.12819 + 0.463080i
223.13 −1.32143 0.503803i −1.04444 + 1.04444i 1.49237 + 1.33148i −1.72951 1.41732i 1.90634 0.853962i 0.225601 + 0.225601i −1.30126 2.51132i 0.818304i 1.57137 + 2.74423i
223.14 −1.31922 + 0.509560i 0.532040 0.532040i 1.48070 1.34445i −0.341068 2.20990i −0.430773 + 0.972985i 2.43768 + 2.43768i −1.26829 + 2.52813i 2.43387i 1.57602 + 2.74156i
223.15 −1.31184 + 0.528270i −2.09162 + 2.09162i 1.44186 1.38601i 0.921582 + 2.03732i 1.63893 3.84881i 0.220841 + 0.220841i −1.15931 + 2.57992i 5.74972i −2.28523 2.18580i
223.16 −1.26502 0.632247i 0.507201 0.507201i 1.20053 + 1.59960i 1.27860 1.83444i −0.962293 + 0.320940i 2.44835 + 2.44835i −0.507338 2.78255i 2.48549i −2.77727 + 1.51221i
223.17 −1.25796 + 0.646169i 1.96497 1.96497i 1.16493 1.62571i −1.98542 + 1.02864i −1.20215 + 3.74157i 0.969540 + 0.969540i −0.414954 + 2.79782i 4.72224i 1.83291 2.57690i
223.18 −1.25640 0.649201i −0.333967 + 0.333967i 1.15708 + 1.63131i 1.91259 + 1.15845i 0.636408 0.202784i −2.67537 2.67537i −0.394699 2.80075i 2.77693i −1.65091 2.69713i
223.19 −1.20660 + 0.737648i 1.78917 1.78917i 0.911751 1.78009i 1.06004 1.96883i −0.839029 + 3.47858i −2.99225 2.99225i 0.212963 + 2.82040i 3.40226i 0.173260 + 3.15753i
223.20 −1.20163 + 0.745715i −1.00581 + 1.00581i 0.887819 1.79214i 2.14883 0.618474i 0.458562 1.95866i −1.33145 1.33145i 0.269600 + 2.81555i 0.976691i −2.12089 + 2.34559i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.108
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
20.e 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.n.a 216
4.b odd 2 1 inner 740.2.n.a 216
5.c odd 4 1 inner 740.2.n.a 216
20.e even 4 1 inner 740.2.n.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.n.a 216 1.a even 1 1 trivial
740.2.n.a 216 4.b odd 2 1 inner
740.2.n.a 216 5.c odd 4 1 inner
740.2.n.a 216 20.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).