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Properties

Label 310.2.p.a
Level $310$
Weight $2$
Character orbit 310.p
Analytic conductor $2.475$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,2,Mod(37,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.p (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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{7} + 8 q^{10} - 64 q^{16} - 4 q^{20} + 48 q^{21} + 12 q^{22} - 20 q^{25} - 4 q^{28} - 8 q^{31} - 16 q^{33} - 48 q^{35} + 32 q^{36} - 72 q^{37} - 8 q^{38} - 8 q^{41} - 48 q^{42} + 48 q^{43} + 28 q^{45} + 40 q^{47} + 8 q^{50} - 16 q^{51} + 12 q^{53} + 24 q^{57} - 20 q^{62} - 24 q^{63} - 12 q^{65} + 16 q^{66} - 4 q^{67} + 40 q^{70} - 48 q^{71} + 48 q^{73} + 12 q^{75} + 16 q^{76} - 8 q^{78} - 48 q^{81} - 8 q^{82} - 72 q^{83} + 24 q^{86} - 8 q^{87} - 12 q^{88} + 32 q^{90} - 100 q^{93} - 24 q^{95} + 88 q^{97} - 8 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.707107 − 0.707107i −0.714932 + 2.66816i 1.00000i 0.501503 − 2.17910i 2.39221 − 1.38114i 1.04061 − 3.88359i 0.707107 − 0.707107i −4.00989 − 2.31511i −1.89548 + 1.18624i
37.2 −0.707107 − 0.707107i −0.566409 + 2.11387i 1.00000i 2.00082 + 0.998364i 1.89524 − 1.09422i −0.937496 + 3.49878i 0.707107 − 0.707107i −1.54954 − 0.894625i −0.708841 − 2.12074i
37.3 −0.707107 − 0.707107i −0.554043 + 2.06772i 1.00000i −0.861736 + 2.06335i 1.85387 − 1.07033i 0.253159 − 0.944803i 0.707107 − 0.707107i −1.37042 − 0.791211i 2.06835 − 0.849669i
37.4 −0.707107 − 0.707107i 0.0166417 − 0.0621075i 1.00000i −1.71772 + 1.43159i −0.0556841 + 0.0321492i 0.739876 − 2.76125i 0.707107 − 0.707107i 2.59450 + 1.49793i 2.22690 + 0.202322i
37.5 −0.707107 − 0.707107i 0.113482 − 0.423519i 1.00000i 2.23322 − 0.112901i −0.379717 + 0.219229i 0.0237450 − 0.0886174i 0.707107 − 0.707107i 2.43159 + 1.40388i −1.65895 − 1.49929i
37.6 −0.707107 − 0.707107i 0.455793 − 1.70104i 1.00000i 0.814817 + 2.08232i −1.52511 + 0.880525i −0.503191 + 1.87794i 0.707107 − 0.707107i −0.0877240 − 0.0506475i 0.896263 − 2.04859i
37.7 −0.707107 − 0.707107i 0.584762 − 2.18236i 1.00000i −2.21667 + 0.293879i −1.95665 + 1.12967i −1.25515 + 4.68430i 0.707107 − 0.707107i −1.82267 − 1.05232i 1.77523 + 1.35962i
37.8 −0.707107 − 0.707107i 0.664706 − 2.48072i 1.00000i 0.629435 − 2.14565i −2.22415 + 1.28411i 0.272430 − 1.01672i 0.707107 − 0.707107i −3.11405 − 1.79790i −1.96228 + 1.07213i
37.9 0.707107 + 0.707107i −0.793649 + 2.96194i 1.00000i −1.44998 + 1.70222i −2.65560 + 1.53321i 0.351828 − 1.31304i −0.707107 + 0.707107i −5.54512 − 3.20148i −2.22894 + 0.178358i
37.10 0.707107 + 0.707107i −0.645923 + 2.41062i 1.00000i −0.906206 − 2.04421i −2.16130 + 1.24783i −0.914294 + 3.41219i −0.707107 + 0.707107i −2.79578 − 1.61414i 0.804690 − 2.08626i
37.11 0.707107 + 0.707107i −0.206124 + 0.769265i 1.00000i 1.13081 + 1.92906i −0.689704 + 0.398201i 0.143140 − 0.534205i −0.707107 + 0.707107i 2.04880 + 1.18287i −0.564447 + 2.16365i
37.12 0.707107 + 0.707107i −0.168634 + 0.629352i 1.00000i 1.07113 − 1.96282i −0.564262 + 0.325777i 1.04782 − 3.91050i −0.707107 + 0.707107i 2.23043 + 1.28774i 2.14533 − 0.630522i
37.13 0.707107 + 0.707107i 0.231532 − 0.864088i 1.00000i 1.91474 − 1.15489i 0.774720 − 0.447285i −1.06631 + 3.97951i −0.707107 + 0.707107i 1.90504 + 1.09987i 2.17056 + 0.537296i
37.14 0.707107 + 0.707107i 0.236699 − 0.883371i 1.00000i −1.93078 + 1.12787i 0.792009 − 0.457266i −0.713262 + 2.66193i −0.707107 + 0.707107i 1.87376 + 1.08181i −2.16279 − 0.567743i
37.15 0.707107 + 0.707107i 0.602519 − 2.24863i 1.00000i −1.67001 − 1.48696i 2.01607 − 1.16398i 0.573758 − 2.14129i −0.707107 + 0.707107i −2.09523 − 1.20968i −0.129434 − 2.23232i
37.16 0.707107 + 0.707107i 0.743581 − 2.77508i 1.00000i 2.18868 + 0.457888i 2.48807 − 1.43649i 0.211296 − 0.788567i −0.707107 + 0.707107i −4.55009 − 2.62700i 1.22386 + 1.87141i
57.1 −0.707107 − 0.707107i −2.66816 + 0.714932i 1.00000i 1.63641 + 1.52387i 2.39221 + 1.38114i −3.88359 + 1.04061i 0.707107 − 0.707107i 4.00989 − 2.31511i −0.0795793 − 2.23465i
57.2 −0.707107 − 0.707107i −2.11387 + 0.566409i 1.00000i −1.86502 + 1.23358i 1.89524 + 1.09422i 3.49878 − 0.937496i 0.707107 − 0.707107i 1.54954 − 0.894625i 2.19104 + 0.446496i
57.3 −0.707107 − 0.707107i −2.06772 + 0.554043i 1.00000i −1.35605 − 1.77796i 1.85387 + 1.07033i −0.944803 + 0.253159i 0.707107 − 0.707107i 1.37042 − 0.791211i −0.298339 + 2.21608i
57.4 −0.707107 − 0.707107i 0.0621075 − 0.0166417i 1.00000i −0.380935 − 2.20338i −0.0556841 − 0.0321492i −2.76125 + 0.739876i 0.707107 − 0.707107i −2.59450 + 1.49793i −1.28866 + 1.82739i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.e odd 6 1 inner
155.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.p.a ✓ 64
5.c odd 4 1 inner 310.2.p.a ✓ 64
31.e odd 6 1 inner 310.2.p.a ✓ 64
155.p even 12 1 inner 310.2.p.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.p.a ✓ 64 1.a even 1 1 trivial
310.2.p.a ✓ 64 5.c odd 4 1 inner
310.2.p.a ✓ 64 31.e odd 6 1 inner
310.2.p.a ✓ 64 155.p even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(310, [\chi])\).