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Properties

Label 342.2.v.b
Level $342$
Weight $2$
Character orbit 342.v
Analytic conductor $2.731$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(25,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([12, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.v (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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{3} + 6 q^{6} - 12 q^{7} - 33 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{3} + 6 q^{6} - 12 q^{7} - 33 q^{8} + 6 q^{9} - 3 q^{13} - 3 q^{14} - 3 q^{15} + 9 q^{17} + 6 q^{18} - 21 q^{19} + 6 q^{22} + 6 q^{23} - 3 q^{24} - 12 q^{25} + 3 q^{27} - 3 q^{28} - 18 q^{29} + 45 q^{33} + 9 q^{34} - 18 q^{35} + 6 q^{36} + 6 q^{37} - 12 q^{38} + 12 q^{39} - 12 q^{41} - 3 q^{43} - 12 q^{44} + 9 q^{45} - 18 q^{47} - 3 q^{48} - 57 q^{49} + 126 q^{50} - 36 q^{51} + 6 q^{52} - 24 q^{53} - 9 q^{54} + 36 q^{55} - 12 q^{56} + 48 q^{57} - 18 q^{58} - 39 q^{59} - 30 q^{60} - 54 q^{61} - 21 q^{62} - 54 q^{63} - 33 q^{64} + 24 q^{65} - 27 q^{66} + 6 q^{67} - 6 q^{68} - 15 q^{69} + 36 q^{70} - 24 q^{71} - 3 q^{72} - 15 q^{73} + 6 q^{74} + 15 q^{78} + 21 q^{79} + 138 q^{81} + 6 q^{82} + 12 q^{83} - 24 q^{84} - 36 q^{85} - 3 q^{86} + 30 q^{87} + 9 q^{89} - 42 q^{90} + 27 q^{91} - 12 q^{92} - 9 q^{93} - 6 q^{94} - 3 q^{95} - 30 q^{97} + 24 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0.173648 + 0.984808i −1.72605 + 0.144004i −0.939693 + 0.342020i −2.96434 + 2.48737i −0.441543 − 1.67483i −0.227964 − 0.394845i −0.500000 − 0.866025i 2.95853 − 0.497118i −2.96434 − 2.48737i
25.2 0.173648 + 0.984808i −1.59006 − 0.686802i −0.939693 + 0.342020i 0.283661 − 0.238020i 0.400256 − 1.68517i −0.412636 − 0.714706i −0.500000 − 0.866025i 2.05661 + 2.18412i 0.283661 + 0.238020i
25.3 0.173648 + 0.984808i −1.29297 + 1.15249i −0.939693 + 0.342020i 1.18951 − 0.998114i −1.35950 − 1.07320i −2.27893 − 3.94722i −0.500000 − 0.866025i 0.343533 − 2.98027i 1.18951 + 0.998114i
25.4 0.173648 + 0.984808i −0.387995 − 1.68803i −0.939693 + 0.342020i 2.67271 − 2.24267i 1.59501 − 0.675224i −0.502728 − 0.870751i −0.500000 − 0.866025i −2.69892 + 1.30990i 2.67271 + 2.24267i
25.5 0.173648 + 0.984808i 0.134037 + 1.72686i −0.939693 + 0.342020i −0.830023 + 0.696472i −1.67735 + 0.431866i 1.00899 + 1.74762i −0.500000 − 0.866025i −2.96407 + 0.462925i −0.830023 − 0.696472i
25.6 0.173648 + 0.984808i 0.291574 + 1.70733i −0.939693 + 0.342020i 3.36002 − 2.81939i −1.63076 + 0.583620i 0.393649 + 0.681819i −0.500000 − 0.866025i −2.82997 + 0.995628i 3.36002 + 2.81939i
25.7 0.173648 + 0.984808i 0.382500 − 1.68929i −0.939693 + 0.342020i −0.697629 + 0.585380i 1.73004 + 0.0833469i 2.20765 + 3.82376i −0.500000 − 0.866025i −2.70739 − 1.29230i −0.697629 − 0.585380i
25.8 0.173648 + 0.984808i 0.484504 − 1.66291i −0.939693 + 0.342020i −2.67378 + 2.24356i 1.72178 + 0.188382i −1.47707 − 2.55835i −0.500000 − 0.866025i −2.53051 − 1.61137i −2.67378 − 2.24356i
25.9 0.173648 + 0.984808i 1.52271 + 0.825445i −0.939693 + 0.342020i −2.26946 + 1.90431i −0.548489 + 1.64291i 0.134441 + 0.232858i −0.500000 − 0.866025i 1.63728 + 2.51382i −2.26946 − 1.90431i
25.10 0.173648 + 0.984808i 1.64462 − 0.543360i −0.939693 + 0.342020i 0.701134 − 0.588321i 0.820690 + 1.52528i −2.58015 − 4.46895i −0.500000 − 0.866025i 2.40952 − 1.78724i 0.701134 + 0.588321i
25.11 0.173648 + 0.984808i 1.71079 − 0.270545i −0.939693 + 0.342020i 1.22819 − 1.03058i 0.563511 + 1.63782i 1.56110 + 2.70390i −0.500000 − 0.866025i 2.85361 − 0.925693i 1.22819 + 1.03058i
61.1 0.766044 − 0.642788i −1.73123 − 0.0533094i 0.173648 − 0.984808i −1.61593 − 0.588149i −1.36047 + 1.07198i −0.482502 − 0.835719i −0.500000 − 0.866025i 2.99432 + 0.184582i −1.61593 + 0.588149i
61.2 0.766044 − 0.642788i −1.61841 − 0.617060i 0.173648 − 0.984808i 3.13299 + 1.14031i −1.63641 + 0.567596i 1.45844 + 2.52610i −0.500000 − 0.866025i 2.23847 + 1.99731i 3.13299 − 1.14031i
61.3 0.766044 − 0.642788i −1.43350 + 0.972152i 0.173648 − 0.984808i 1.62144 + 0.590157i −0.473237 + 1.66615i −2.04096 − 3.53505i −0.500000 − 0.866025i 1.10984 − 2.78716i 1.62144 − 0.590157i
61.4 0.766044 − 0.642788i −0.472925 + 1.66624i 0.173648 − 0.984808i −3.63267 − 1.32219i 0.708755 + 1.58040i −0.204835 − 0.354784i −0.500000 − 0.866025i −2.55268 − 1.57601i −3.63267 + 1.32219i
61.5 0.766044 − 0.642788i −0.0767973 − 1.73035i 0.173648 − 0.984808i −2.00790 − 0.730817i −1.17108 − 1.27616i −0.694843 − 1.20350i −0.500000 − 0.866025i −2.98820 + 0.265772i −2.00790 + 0.730817i
61.6 0.766044 − 0.642788i 0.349550 − 1.69641i 0.173648 − 0.984808i 1.92444 + 0.700437i −0.822662 − 1.52421i 0.703408 + 1.21834i −0.500000 − 0.866025i −2.75563 − 1.18596i 1.92444 − 0.700437i
61.7 0.766044 − 0.642788i 0.559405 + 1.63923i 0.173648 − 0.984808i −0.0414633 − 0.0150914i 1.48220 + 0.896143i 0.869695 + 1.50636i −0.500000 − 0.866025i −2.37413 + 1.83398i −0.0414633 + 0.0150914i
61.8 0.766044 − 0.642788i 1.37860 + 1.04856i 0.173648 − 0.984808i 3.86138 + 1.40543i 1.73007 − 0.0829034i −2.11856 − 3.66945i −0.500000 − 0.866025i 0.801056 + 2.89107i 3.86138 − 1.40543i
61.9 0.766044 − 0.642788i 1.38269 − 1.04315i 0.173648 − 0.984808i 0.474466 + 0.172691i 0.388676 − 1.68788i −0.414583 − 0.718079i −0.500000 − 0.866025i 0.823662 − 2.88472i 0.474466 − 0.172691i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.v even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.v.b ✓ 66
9.c even 3 1 342.2.w.b yes 66
19.e even 9 1 342.2.w.b yes 66
171.v even 9 1 inner 342.2.v.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.v.b ✓ 66 1.a even 1 1 trivial
342.2.v.b ✓ 66 171.v even 9 1 inner
342.2.w.b yes 66 9.c even 3 1
342.2.w.b yes 66 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{66} + 6 T_{5}^{64} + 6 T_{5}^{63} + 15 T_{5}^{62} + 111 T_{5}^{61} + 7113 T_{5}^{60} + \cdots + 63651292286976 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display