↑

Properties

Label 70.4.g.a
Level $70$
Weight $4$
Character orbit 70.g
Analytic conductor $4.130$
Analytic rank $0$
Dimension $24$
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] = [70,4,Mod(13,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 70.g (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: \(4.13013370040\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{7} + 40 q^{11} + 504 q^{15} - 384 q^{16} - 16 q^{18} - 432 q^{21} - 192 q^{22} + 336 q^{23} + 520 q^{25} - 16 q^{28} - 384 q^{30} - 628 q^{35} - 896 q^{36} - 456 q^{37} + 1280 q^{42} - 1264 q^{43} + 1792 q^{46} - 448 q^{50} + 2184 q^{51} - 2256 q^{53} - 384 q^{56} + 3584 q^{57} + 992 q^{58} + 352 q^{60} - 60 q^{63} - 3312 q^{65} - 1088 q^{67} + 2016 q^{70} - 2688 q^{71} - 64 q^{72} + 3128 q^{77} - 80 q^{78} - 3208 q^{81} + 5072 q^{85} + 2080 q^{86} + 768 q^{88} - 5472 q^{91} + 1344 q^{92} + 7752 q^{93} - 1976 q^{95} - 4064 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41421 + 1.41421i −6.82881 + 6.82881i − 4.00000i −10.1197 − 4.75308i − 19.3148i 9.00041 + 16.1862i 5.65685 + 5.65685i − 66.2652i 21.0333 − 7.58953i
13.2 −1.41421 + 1.41421i −1.99723 + 1.99723i − 4.00000i 5.65906 − 9.64235i − 5.64901i −2.03365 − 18.4083i 5.65685 + 5.65685i 19.0222i 5.63323 + 21.6395i
13.3 −1.41421 + 1.41421i −1.78083 + 1.78083i − 4.00000i 11.1084 − 1.26609i − 5.03694i 1.76212 + 18.4362i 5.65685 + 5.65685i 20.6573i −13.9192 + 17.5002i
13.4 −1.41421 + 1.41421i 1.78083 − 1.78083i − 4.00000i −11.1084 + 1.26609i 5.03694i −18.4362 − 1.76212i 5.65685 + 5.65685i 20.6573i 13.9192 − 17.5002i
13.5 −1.41421 + 1.41421i 1.99723 − 1.99723i − 4.00000i −5.65906 + 9.64235i 5.64901i 18.4083 + 2.03365i 5.65685 + 5.65685i 19.0222i −5.63323 − 21.6395i
13.6 −1.41421 + 1.41421i 6.82881 − 6.82881i − 4.00000i 10.1197 + 4.75308i 19.3148i −16.1862 − 9.00041i 5.65685 + 5.65685i − 66.2652i −21.0333 + 7.58953i
13.7 1.41421 − 1.41421i −5.70899 + 5.70899i − 4.00000i −3.33731 − 10.6706i 16.1475i −3.44728 − 18.1966i −5.65685 − 5.65685i − 38.1851i −19.8102 − 10.3709i
13.8 1.41421 − 1.41421i −3.53853 + 3.53853i − 4.00000i −7.91406 + 7.89732i 10.0085i −11.0854 + 14.8362i −5.65685 − 5.65685i 1.95766i −0.0236651 + 22.3607i
13.9 1.41421 − 1.41421i −3.17701 + 3.17701i − 4.00000i 10.4115 + 4.07432i 8.98595i 18.3796 − 2.27797i −5.65685 − 5.65685i 6.81319i 20.4861 − 8.96216i
13.10 1.41421 − 1.41421i 3.17701 − 3.17701i − 4.00000i −10.4115 − 4.07432i − 8.98595i 2.27797 − 18.3796i −5.65685 − 5.65685i 6.81319i −20.4861 + 8.96216i
13.11 1.41421 − 1.41421i 3.53853 − 3.53853i − 4.00000i 7.91406 − 7.89732i − 10.0085i −14.8362 + 11.0854i −5.65685 − 5.65685i 1.95766i 0.0236651 − 22.3607i
13.12 1.41421 − 1.41421i 5.70899 − 5.70899i − 4.00000i 3.33731 + 10.6706i − 16.1475i 18.1966 + 3.44728i −5.65685 − 5.65685i − 38.1851i 19.8102 + 10.3709i
27.1 −1.41421 − 1.41421i −6.82881 − 6.82881i 4.00000i −10.1197 + 4.75308i 19.3148i 9.00041 − 16.1862i 5.65685 − 5.65685i 66.2652i 21.0333 + 7.58953i
27.2 −1.41421 − 1.41421i −1.99723 − 1.99723i 4.00000i 5.65906 + 9.64235i 5.64901i −2.03365 + 18.4083i 5.65685 − 5.65685i − 19.0222i 5.63323 − 21.6395i
27.3 −1.41421 − 1.41421i −1.78083 − 1.78083i 4.00000i 11.1084 + 1.26609i 5.03694i 1.76212 − 18.4362i 5.65685 − 5.65685i − 20.6573i −13.9192 − 17.5002i
27.4 −1.41421 − 1.41421i 1.78083 + 1.78083i 4.00000i −11.1084 − 1.26609i − 5.03694i −18.4362 + 1.76212i 5.65685 − 5.65685i − 20.6573i 13.9192 + 17.5002i
27.5 −1.41421 − 1.41421i 1.99723 + 1.99723i 4.00000i −5.65906 − 9.64235i − 5.64901i 18.4083 − 2.03365i 5.65685 − 5.65685i − 19.0222i −5.63323 + 21.6395i
27.6 −1.41421 − 1.41421i 6.82881 + 6.82881i 4.00000i 10.1197 − 4.75308i − 19.3148i −16.1862 + 9.00041i 5.65685 − 5.65685i 66.2652i −21.0333 − 7.58953i
27.7 1.41421 + 1.41421i −5.70899 − 5.70899i 4.00000i −3.33731 + 10.6706i − 16.1475i −3.44728 + 18.1966i −5.65685 + 5.65685i 38.1851i −19.8102 + 10.3709i
27.8 1.41421 + 1.41421i −3.53853 − 3.53853i 4.00000i −7.91406 − 7.89732i − 10.0085i −11.0854 − 14.8362i −5.65685 + 5.65685i − 1.95766i −0.0236651 − 22.3607i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.4.g.a ✓ 24
5.b even 2 1 350.4.g.b 24
5.c odd 4 1 inner 70.4.g.a ✓ 24
5.c odd 4 1 350.4.g.b 24
7.b odd 2 1 inner 70.4.g.a ✓ 24
35.c odd 2 1 350.4.g.b 24
35.f even 4 1 inner 70.4.g.a ✓ 24
35.f even 4 1 350.4.g.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.g.a ✓ 24 1.a even 1 1 trivial
70.4.g.a ✓ 24 5.c odd 4 1 inner
70.4.g.a ✓ 24 7.b odd 2 1 inner
70.4.g.a ✓ 24 35.f even 4 1 inner
350.4.g.b 24 5.b even 2 1
350.4.g.b 24 5.c odd 4 1
350.4.g.b 24 35.c odd 2 1
350.4.g.b 24 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(70, [\chi])\).