Properties

Label 338.10.a.d
Level $338$
Weight $10$
Character orbit 338.a
Self dual yes
Analytic conductor $174.082$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,10,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

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: yes
Analytic conductor: \(174.082112623\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 192 q^{3} + 256 q^{4} + 1310 q^{5} + 3072 q^{6} + 5810 q^{7} + 4096 q^{8} + 17181 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 192 q^{3} + 256 q^{4} + 1310 q^{5} + 3072 q^{6} + 5810 q^{7} + 4096 q^{8} + 17181 q^{9} + 20960 q^{10} + 4498 q^{11} + 49152 q^{12} + 92960 q^{14} + 251520 q^{15} + 65536 q^{16} - 237498 q^{17} + 274896 q^{18} + 913014 q^{19} + 335360 q^{20} + 1115520 q^{21} + 71968 q^{22} + 201544 q^{23} + 786432 q^{24} - 237025 q^{25} - 480384 q^{27} + 1487360 q^{28} + 1276834 q^{29} + 4024320 q^{30} - 4163770 q^{31} + 1048576 q^{32} + 863616 q^{33} - 3799968 q^{34} + 7611100 q^{35} + 4398336 q^{36} + 18442662 q^{37} + 14608224 q^{38} + 5365760 q^{40} + 22601670 q^{41} + 17848320 q^{42} + 11726308 q^{43} + 1151488 q^{44} + 22507110 q^{45} + 3224704 q^{46} - 59291534 q^{47} + 12582912 q^{48} - 6597507 q^{49} - 3792400 q^{50} - 45599616 q^{51} + 108158694 q^{53} - 7686144 q^{54} + 5892380 q^{55} + 23797760 q^{56} + 175298688 q^{57} + 20429344 q^{58} + 14920154 q^{59} + 64389120 q^{60} - 57003746 q^{61} - 66620320 q^{62} + 99821610 q^{63} + 16777216 q^{64} + 13817856 q^{66} - 22074010 q^{67} - 60799488 q^{68} + 38696448 q^{69} + 121777600 q^{70} - 44416250 q^{71} + 70373376 q^{72} - 265794626 q^{73} + 295082592 q^{74} - 45508800 q^{75} + 233731584 q^{76} + 26133380 q^{77} + 476755484 q^{79} + 85852160 q^{80} - 430407351 q^{81} + 361626720 q^{82} + 505315830 q^{83} + 285573120 q^{84} - 311122380 q^{85} + 187620928 q^{86} + 245152128 q^{87} + 18423808 q^{88} - 890840634 q^{89} + 360113760 q^{90} + 51595264 q^{92} - 799443840 q^{93} - 948664544 q^{94} + 1196048340 q^{95} + 201326592 q^{96} + 802776958 q^{97} - 105560112 q^{98} + 77280138 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
16.0000 192.000 256.000 1310.00 3072.00 5810.00 4096.00 17181.0 20960.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.10.a.d 1
13.b even 2 1 26.10.a.b 1
39.d odd 2 1 234.10.a.c 1
52.b odd 2 1 208.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.10.a.b 1 13.b even 2 1
208.10.a.a 1 52.b odd 2 1
234.10.a.c 1 39.d odd 2 1
338.10.a.d 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} - 192 \) Copy content Toggle raw display
\( T_{5} - 1310 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T - 192 \) Copy content Toggle raw display
$5$ \( T - 1310 \) Copy content Toggle raw display
$7$ \( T - 5810 \) Copy content Toggle raw display
$11$ \( T - 4498 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 237498 \) Copy content Toggle raw display
$19$ \( T - 913014 \) Copy content Toggle raw display
$23$ \( T - 201544 \) Copy content Toggle raw display
$29$ \( T - 1276834 \) Copy content Toggle raw display
$31$ \( T + 4163770 \) Copy content Toggle raw display
$37$ \( T - 18442662 \) Copy content Toggle raw display
$41$ \( T - 22601670 \) Copy content Toggle raw display
$43$ \( T - 11726308 \) Copy content Toggle raw display
$47$ \( T + 59291534 \) Copy content Toggle raw display
$53$ \( T - 108158694 \) Copy content Toggle raw display
$59$ \( T - 14920154 \) Copy content Toggle raw display
$61$ \( T + 57003746 \) Copy content Toggle raw display
$67$ \( T + 22074010 \) Copy content Toggle raw display
$71$ \( T + 44416250 \) Copy content Toggle raw display
$73$ \( T + 265794626 \) Copy content Toggle raw display
$79$ \( T - 476755484 \) Copy content Toggle raw display
$83$ \( T - 505315830 \) Copy content Toggle raw display
$89$ \( T + 890840634 \) Copy content Toggle raw display
$97$ \( T - 802776958 \) Copy content Toggle raw display
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