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Properties

Label 338.2.h.a
Level $338$
Weight $2$
Character orbit 338.h
Analytic conductor $2.699$
Analytic rank $0$
Dimension $192$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,2,Mod(25,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.h (of order \(26\), degree \(12\), 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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{3} + 16 q^{4} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{3} + 16 q^{4} - 10 q^{9} - 6 q^{10} - 2 q^{12} - 56 q^{13} + 8 q^{14} - 52 q^{15} - 16 q^{16} - 8 q^{17} - 24 q^{22} + 8 q^{23} + 18 q^{25} + 6 q^{26} - 4 q^{27} + 6 q^{29} - 26 q^{30} - 52 q^{31} - 8 q^{35} + 10 q^{36} - 14 q^{38} + 56 q^{39} - 20 q^{40} - 2 q^{43} + 130 q^{45} + 52 q^{47} + 2 q^{48} - 18 q^{49} - 138 q^{51} + 4 q^{52} - 34 q^{53} - 78 q^{54} + 8 q^{55} - 8 q^{56} - 78 q^{57} - 52 q^{58} + 52 q^{60} + 26 q^{61} - 8 q^{62} + 130 q^{63} + 16 q^{64} - 18 q^{65} + 8 q^{66} + 52 q^{67} + 8 q^{68} - 28 q^{69} - 78 q^{71} + 40 q^{74} - 82 q^{75} - 26 q^{76} - 20 q^{77} - 6 q^{78} - 4 q^{79} + 8 q^{81} - 4 q^{82} - 130 q^{83} + 26 q^{85} + 78 q^{86} - 32 q^{87} - 2 q^{88} - 2 q^{90} - 34 q^{91} - 8 q^{92} - 52 q^{93} - 40 q^{94} - 92 q^{95} + 78 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.935016 − 0.354605i −1.80064 − 2.60868i 0.748511 + 0.663123i −0.607764 − 0.0737959i 0.758579 + 3.07768i 1.71128 + 3.26057i −0.464723 − 0.885456i −2.49909 + 6.58955i 0.542101 + 0.284517i
25.2 −0.935016 − 0.354605i −1.40238 − 2.03170i 0.748511 + 0.663123i 2.83628 + 0.344387i 0.590799 + 2.39697i −1.00069 − 1.90665i −0.464723 − 0.885456i −1.09732 + 2.89341i −2.52985 − 1.32777i
25.3 −0.935016 − 0.354605i −0.748725 − 1.08472i 0.748511 + 0.663123i −0.417775 − 0.0507270i 0.315425 + 1.27973i −0.0119482 − 0.0227653i −0.464723 − 0.885456i 0.447796 − 1.18074i 0.372638 + 0.195576i
25.4 −0.935016 − 0.354605i −0.172883 − 0.250464i 0.748511 + 0.663123i −3.81662 − 0.463422i 0.0728325 + 0.295493i 1.70788 + 3.25409i −0.464723 − 0.885456i 1.03097 − 2.71845i 3.40427 + 1.78670i
25.5 −0.935016 − 0.354605i 0.291244 + 0.421940i 0.748511 + 0.663123i 4.11894 + 0.500130i −0.122696 − 0.497797i 2.09974 + 4.00073i −0.464723 − 0.885456i 0.970605 − 2.55927i −3.67393 − 1.92823i
25.6 −0.935016 − 0.354605i 0.518075 + 0.750561i 0.748511 + 0.663123i 0.927916 + 0.112669i −0.218256 − 0.885499i −0.232423 − 0.442846i −0.464723 − 0.885456i 0.768874 − 2.02735i −0.827664 − 0.434391i
25.7 −0.935016 − 0.354605i 1.14493 + 1.65871i 0.748511 + 0.663123i −3.25668 − 0.395433i −0.482338 − 1.95692i −1.58550 − 3.02092i −0.464723 − 0.885456i −0.376658 + 0.993165i 2.90483 + 1.52457i
25.8 −0.935016 − 0.354605i 1.60232 + 2.32137i 0.748511 + 0.663123i −0.0729627 − 0.00885928i −0.675030 − 2.73871i 0.593925 + 1.13163i −0.464723 − 0.885456i −1.75748 + 4.63411i 0.0650798 + 0.0341565i
25.9 0.935016 + 0.354605i −1.58787 − 2.30043i 0.748511 + 0.663123i 2.72199 + 0.330510i −0.668942 − 2.71400i 1.80830 + 3.44542i 0.464723 + 0.885456i −1.70682 + 4.50051i 2.42791 + 1.27426i
25.10 0.935016 + 0.354605i −1.43793 − 2.08320i 0.748511 + 0.663123i −3.04641 − 0.369901i −0.605774 − 2.45772i −0.188470 − 0.359099i 0.464723 + 0.885456i −1.20826 + 3.18593i −2.71727 − 1.42613i
25.11 0.935016 + 0.354605i −0.619250 − 0.897138i 0.748511 + 0.663123i −0.866056 − 0.105158i −0.260879 − 1.05843i −1.97803 − 3.76883i 0.464723 + 0.885456i 0.642428 − 1.69394i −0.772487 − 0.405432i
25.12 0.935016 + 0.354605i −0.480787 − 0.696540i 0.748511 + 0.663123i 3.26929 + 0.396964i −0.202547 − 0.821765i −0.884971 − 1.68617i 0.464723 + 0.885456i 0.809803 − 2.13527i 2.91608 + 1.53048i
25.13 0.935016 + 0.354605i −0.348712 − 0.505197i 0.748511 + 0.663123i −1.59753 − 0.193975i −0.146906 − 0.596022i 1.88394 + 3.58956i 0.464723 + 0.885456i 0.930191 − 2.45271i −1.42493 − 0.747862i
25.14 0.935016 + 0.354605i 0.918643 + 1.33088i 0.748511 + 0.663123i −0.410173 − 0.0498040i 0.387008 + 1.57015i 1.39218 + 2.65259i 0.464723 + 0.885456i 0.136467 − 0.359833i −0.365857 − 0.192017i
25.15 0.935016 + 0.354605i 1.09194 + 1.58194i 0.748511 + 0.663123i 2.12391 + 0.257889i 0.460014 + 1.86635i −1.04108 − 1.98361i 0.464723 + 0.885456i −0.246404 + 0.649714i 1.89444 + 0.994279i
25.16 0.935016 + 0.354605i 1.89590 + 2.74669i 0.748511 + 0.663123i −1.90636 − 0.231474i 0.798710 + 3.24049i −1.21447 − 2.31398i 0.464723 + 0.885456i −2.88604 + 7.60986i −1.70039 − 0.892435i
51.1 −0.663123 + 0.748511i −1.02010 + 2.68977i −0.120537 − 0.992709i −0.603235 + 2.44742i −1.33687 − 2.54720i −3.79332 − 2.61834i 0.822984 + 0.568065i −3.94874 − 3.49828i −1.43190 − 2.07447i
51.2 −0.663123 + 0.748511i −0.865169 + 2.28126i −0.120537 − 0.992709i 0.807218 − 3.27501i −1.13384 − 2.16035i 1.31824 + 0.909915i 0.822984 + 0.568065i −2.21011 − 1.95799i 1.91610 + 2.77595i
51.3 −0.663123 + 0.748511i −0.562514 + 1.48323i −0.120537 − 0.992709i −0.0720759 + 0.292423i −0.737195 − 1.40461i 2.40171 + 1.65778i 0.822984 + 0.568065i 0.361993 + 0.320698i −0.171087 − 0.247862i
51.4 −0.663123 + 0.748511i −0.0970675 + 0.255946i −0.120537 − 0.992709i 0.397512 − 1.61277i −0.127211 − 0.242380i −1.39835 − 0.965212i 0.822984 + 0.568065i 2.18945 + 1.93968i 0.943576 + 1.36701i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.h.a ✓ 192
169.h even 26 1 inner 338.2.h.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.h.a ✓ 192 1.a even 1 1 trivial
338.2.h.a ✓ 192 169.h even 26 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(338, [\chi])\).