↑

Properties

Label 348.5.j.a
Level $348$
Weight $5$
Character orbit 348.j
Analytic conductor $35.973$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,5,Mod(133,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.133");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 348.j (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: \(35.9727471532\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 108 q^{15} - 804 q^{17} + 88 q^{19} + 2184 q^{23} - 3368 q^{25} - 1068 q^{29} - 4724 q^{31} - 2612 q^{37} + 2016 q^{39} + 3468 q^{41} + 3632 q^{43} + 2592 q^{45} + 3816 q^{47} + 11656 q^{49} - 16224 q^{53} - 4392 q^{55} - 6432 q^{59} + 14524 q^{61} + 456 q^{65} + 6192 q^{69} - 12512 q^{73} - 1584 q^{75} + 6240 q^{77} - 14708 q^{79} - 29160 q^{81} - 18720 q^{83} + 7520 q^{85} - 3276 q^{87} + 5604 q^{89} - 3624 q^{95} + 24472 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 0 −3.67423 + 3.67423i 0 − 39.9719i 0 46.0876 0 − 27.0000i 0
133.2 0 −3.67423 + 3.67423i 0 − 29.0181i 0 10.2999 0 − 27.0000i 0
133.3 0 −3.67423 + 3.67423i 0 − 25.2905i 0 −91.0418 0 − 27.0000i 0
133.4 0 −3.67423 + 3.67423i 0 10.1277i 0 5.79517 0 − 27.0000i 0
133.5 0 −3.67423 + 3.67423i 0 16.0965i 0 97.0300 0 − 27.0000i 0
133.6 0 −3.67423 + 3.67423i 0 25.4969i 0 −85.2730 0 − 27.0000i 0
133.7 0 −3.67423 + 3.67423i 0 21.2364i 0 −0.107014 0 − 27.0000i 0
133.8 0 −3.67423 + 3.67423i 0 − 23.3706i 0 4.32559 0 − 27.0000i 0
133.9 0 −3.67423 + 3.67423i 0 33.3440i 0 39.6056 0 − 27.0000i 0
133.10 0 −3.67423 + 3.67423i 0 28.0011i 0 −26.7220 0 − 27.0000i 0
133.11 0 3.67423 − 3.67423i 0 − 39.6290i 0 −54.0944 0 − 27.0000i 0
133.12 0 3.67423 − 3.67423i 0 30.4508i 0 −13.2106 0 − 27.0000i 0
133.13 0 3.67423 − 3.67423i 0 − 18.2181i 0 44.0550 0 − 27.0000i 0
133.14 0 3.67423 − 3.67423i 0 0.102546i 0 27.5105 0 − 27.0000i 0
133.15 0 3.67423 − 3.67423i 0 − 20.4149i 0 37.2055 0 − 27.0000i 0
133.16 0 3.67423 − 3.67423i 0 14.4028i 0 63.1042 0 − 27.0000i 0
133.17 0 3.67423 − 3.67423i 0 12.2036i 0 −76.8093 0 − 27.0000i 0
133.18 0 3.67423 − 3.67423i 0 − 24.7361i 0 −18.1581 0 − 27.0000i 0
133.19 0 3.67423 − 3.67423i 0 36.8265i 0 −63.4004 0 − 27.0000i 0
133.20 0 3.67423 − 3.67423i 0 40.3603i 0 53.7975 0 − 27.0000i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.5.j.a ✓ 40
3.b odd 2 1 1044.5.k.c 40
29.c odd 4 1 inner 348.5.j.a ✓ 40
87.f even 4 1 1044.5.k.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.5.j.a ✓ 40 1.a even 1 1 trivial
348.5.j.a ✓ 40 29.c odd 4 1 inner
1044.5.k.c 40 3.b odd 2 1
1044.5.k.c 40 87.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(348, [\chi])\).