↑

Properties

Label 418.2.j.b
Level $418$
Weight $2$
Character orbit 418.j
Analytic conductor $3.338$
Analytic rank $0$
Dimension $24$
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] = [418,2,Mod(23,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.j (of order \(9\), degree \(6\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} - 12 q^{8} - 12 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 3 q^{15} + 6 q^{17} + 30 q^{18} + 3 q^{19} - 18 q^{20} + 3 q^{21} + 3 q^{23} - 24 q^{25} + 21 q^{27} - 3 q^{28} + 24 q^{29} + 21 q^{30} + 9 q^{31} + 15 q^{34} - 9 q^{35} - 18 q^{37} - 3 q^{38} - 12 q^{39} + 6 q^{41} - 33 q^{42} - 42 q^{43} + 18 q^{46} - 33 q^{47} - 18 q^{49} - 15 q^{50} + 30 q^{51} + 21 q^{52} - 12 q^{53} - 63 q^{54} - 24 q^{56} + 90 q^{57} + 30 q^{58} + 51 q^{59} + 3 q^{60} + 21 q^{61} - 27 q^{62} - 75 q^{63} - 12 q^{64} - 36 q^{65} + 21 q^{67} + 33 q^{68} - 24 q^{69} - 9 q^{70} + 15 q^{71} - 63 q^{73} + 9 q^{74} - 42 q^{75} + 15 q^{76} - 24 q^{77} + 54 q^{78} + 9 q^{79} - 60 q^{81} + 6 q^{82} + 66 q^{83} - 9 q^{84} + 3 q^{85} + 21 q^{86} + 51 q^{87} - 12 q^{88} - 39 q^{89} + 18 q^{90} - 27 q^{91} + 3 q^{92} + 33 q^{93} - 54 q^{94} + 60 q^{95} - 6 q^{96} - 51 q^{97} + 45 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.939693 − 0.342020i −0.535511 − 3.03704i 0.766044 + 0.642788i 1.57922 − 1.32512i −0.535511 + 3.03704i 0.687863 − 1.19141i −0.500000 − 0.866025i −6.11774 + 2.22667i −1.93720 + 0.705083i
23.2 −0.939693 − 0.342020i −0.227689 − 1.29129i 0.766044 + 0.642788i −2.49024 + 2.08956i −0.227689 + 1.29129i −0.335404 + 0.580937i −0.500000 − 0.866025i 1.20349 − 0.438036i 3.05474 − 1.11183i
23.3 −0.939693 − 0.342020i 0.284478 + 1.61335i 0.766044 + 0.642788i −1.00519 + 0.843458i 0.284478 − 1.61335i −1.14094 + 1.97617i −0.500000 − 0.866025i 0.297093 − 0.108133i 1.23305 − 0.448794i
23.4 −0.939693 − 0.342020i 0.305074 + 1.73016i 0.766044 + 0.642788i −0.381914 + 0.320464i 0.305074 − 1.73016i 2.61483 − 4.52902i −0.500000 − 0.866025i −0.0813139 + 0.0295958i 0.468487 − 0.170515i
111.1 0.766044 + 0.642788i −2.76892 + 1.00780i 0.173648 + 0.984808i −0.512823 + 2.90836i −2.76892 − 1.00780i 1.03903 + 1.79965i −0.500000 + 0.866025i 4.35310 − 3.65269i −2.26231 + 1.89830i
111.2 0.766044 + 0.642788i −0.109155 + 0.0397291i 0.173648 + 0.984808i −0.0865725 + 0.490977i −0.109155 − 0.0397291i 1.14564 + 1.98431i −0.500000 + 0.866025i −2.28780 + 1.91969i −0.381912 + 0.320462i
111.3 0.766044 + 0.642788i 1.75906 − 0.640246i 0.173648 + 0.984808i 0.663091 − 3.76058i 1.75906 + 0.640246i 1.48367 + 2.56979i −0.500000 + 0.866025i 0.386249 − 0.324101i 2.92521 − 2.45454i
111.4 0.766044 + 0.642788i 2.05870 − 0.749307i 0.173648 + 0.984808i −0.584640 + 3.31566i 2.05870 + 0.749307i −0.728641 − 1.26204i −0.500000 + 0.866025i 1.37867 − 1.15684i −2.57912 + 2.16414i
177.1 0.766044 − 0.642788i −2.76892 − 1.00780i 0.173648 − 0.984808i −0.512823 − 2.90836i −2.76892 + 1.00780i 1.03903 − 1.79965i −0.500000 − 0.866025i 4.35310 + 3.65269i −2.26231 − 1.89830i
177.2 0.766044 − 0.642788i −0.109155 − 0.0397291i 0.173648 − 0.984808i −0.0865725 − 0.490977i −0.109155 + 0.0397291i 1.14564 − 1.98431i −0.500000 − 0.866025i −2.28780 − 1.91969i −0.381912 − 0.320462i
177.3 0.766044 − 0.642788i 1.75906 + 0.640246i 0.173648 − 0.984808i 0.663091 + 3.76058i 1.75906 − 0.640246i 1.48367 − 2.56979i −0.500000 − 0.866025i 0.386249 + 0.324101i 2.92521 + 2.45454i
177.4 0.766044 − 0.642788i 2.05870 + 0.749307i 0.173648 − 0.984808i −0.584640 − 3.31566i 2.05870 − 0.749307i −0.728641 + 1.26204i −0.500000 − 0.866025i 1.37867 + 1.15684i −2.57912 − 2.16414i
199.1 0.173648 + 0.984808i −2.53226 + 2.12482i −0.939693 + 0.342020i 0.0714030 + 0.0259886i −2.53226 − 2.12482i 2.39179 − 4.14270i −0.500000 − 0.866025i 1.37655 − 7.80678i −0.0131947 + 0.0748311i
199.2 0.173648 + 0.984808i −0.584832 + 0.490732i −0.939693 + 0.342020i −2.08987 − 0.760650i −0.584832 − 0.490732i −0.0399488 + 0.0691933i −0.500000 − 0.866025i −0.419734 + 2.38043i 0.386192 − 2.19020i
199.3 0.173648 + 0.984808i 0.697030 − 0.584878i −0.939693 + 0.342020i 3.92259 + 1.42771i 0.697030 + 0.584878i −2.18743 + 3.78873i −0.500000 − 0.866025i −0.377175 + 2.13907i −0.724865 + 4.11092i
199.4 0.173648 + 0.984808i 1.65402 − 1.38789i −0.939693 + 0.342020i 0.914952 + 0.333015i 1.65402 + 1.38789i 1.06954 − 1.85250i −0.500000 − 0.866025i 0.288605 − 1.63676i −0.169076 + 0.958879i
309.1 −0.939693 + 0.342020i −0.535511 + 3.03704i 0.766044 − 0.642788i 1.57922 + 1.32512i −0.535511 − 3.03704i 0.687863 + 1.19141i −0.500000 + 0.866025i −6.11774 − 2.22667i −1.93720 − 0.705083i
309.2 −0.939693 + 0.342020i −0.227689 + 1.29129i 0.766044 − 0.642788i −2.49024 − 2.08956i −0.227689 − 1.29129i −0.335404 − 0.580937i −0.500000 + 0.866025i 1.20349 + 0.438036i 3.05474 + 1.11183i
309.3 −0.939693 + 0.342020i 0.284478 − 1.61335i 0.766044 − 0.642788i −1.00519 − 0.843458i 0.284478 + 1.61335i −1.14094 − 1.97617i −0.500000 + 0.866025i 0.297093 + 0.108133i 1.23305 + 0.448794i
309.4 −0.939693 + 0.342020i 0.305074 − 1.73016i 0.766044 − 0.642788i −0.381914 − 0.320464i 0.305074 + 1.73016i 2.61483 + 4.52902i −0.500000 + 0.866025i −0.0813139 − 0.0295958i 0.468487 + 0.170515i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.j.b ✓ 24
19.e even 9 1 inner 418.2.j.b ✓ 24
19.e even 9 1 7942.2.a.bw 12
19.f odd 18 1 7942.2.a.bs 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.b ✓ 24 1.a even 1 1 trivial
418.2.j.b ✓ 24 19.e even 9 1 inner
7942.2.a.bs 12 19.f odd 18 1
7942.2.a.bw 12 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 13 T_{3}^{21} + 15 T_{3}^{20} - 63 T_{3}^{19} + 606 T_{3}^{18} - 1035 T_{3}^{17} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display