Properties

Label 338.10.a.b
Level $338$
Weight $10$
Character orbit 338.a
Self dual yes
Analytic conductor $174.082$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 16 q^{2} + 75 q^{3} + 256 q^{4} + 1979 q^{5} - 1200 q^{6} + 10115 q^{7} - 4096 q^{8} - 14058 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + 75 q^{3} + 256 q^{4} + 1979 q^{5} - 1200 q^{6} + 10115 q^{7} - 4096 q^{8} - 14058 q^{9} - 31664 q^{10} - 18850 q^{11} + 19200 q^{12} - 161840 q^{14} + 148425 q^{15} + 65536 q^{16} - 142403 q^{17} + 224928 q^{18} - 83302 q^{19} + 506624 q^{20} + 758625 q^{21} + 301600 q^{22} - 536544 q^{23} - 307200 q^{24} + 1963316 q^{25} - 2530575 q^{27} + 2589440 q^{28} - 2600442 q^{29} - 2374800 q^{30} + 2214004 q^{31} - 1048576 q^{32} - 1413750 q^{33} + 2278448 q^{34} + 20017585 q^{35} - 3598848 q^{36} - 18099241 q^{37} + 1332832 q^{38} - 8105984 q^{40} - 26812240 q^{41} - 12138000 q^{42} - 42253475 q^{43} - 4825600 q^{44} - 27820782 q^{45} + 8584704 q^{46} - 35914993 q^{47} + 4915200 q^{48} + 61959618 q^{49} - 31413056 q^{50} - 10680225 q^{51} - 66514064 q^{53} + 40489200 q^{54} - 37304150 q^{55} - 41431040 q^{56} - 6247650 q^{57} + 41607072 q^{58} + 108164002 q^{59} + 37996800 q^{60} - 207449912 q^{61} - 35424064 q^{62} - 142196670 q^{63} + 16777216 q^{64} + 22620000 q^{66} - 193015514 q^{67} - 36455168 q^{68} - 40240800 q^{69} - 320281360 q^{70} + 201833497 q^{71} + 57581568 q^{72} + 121628110 q^{73} + 289587856 q^{74} + 147248700 q^{75} - 21325312 q^{76} - 190667750 q^{77} + 112871912 q^{79} + 129695744 q^{80} + 86910489 q^{81} + 428995840 q^{82} - 308254212 q^{83} + 194208000 q^{84} - 281815537 q^{85} + 676055600 q^{86} - 195033150 q^{87} + 77209600 q^{88} + 6374870 q^{89} + 445132512 q^{90} - 137355264 q^{92} + 166050300 q^{93} + 574639888 q^{94} - 164854658 q^{95} - 78643200 q^{96} - 871266886 q^{97} - 991353888 q^{98} + 264993300 q^{99}+O(q^{100}) \) 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 75.0000 256.000 1979.00 −1200.00 10115.0 −4096.00 −14058.0 −31664.0
\(n\): 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.b 1
13.b even 2 1 26.10.a.c 1
39.d odd 2 1 234.10.a.a 1
52.b odd 2 1 208.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.10.a.c 1 13.b even 2 1
208.10.a.b 1 52.b odd 2 1
234.10.a.a 1 39.d odd 2 1
338.10.a.b 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} - 75 \) Copy content Toggle raw display
\( T_{5} - 1979 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 16 \) Copy content Toggle raw display
$3$ \( T - 75 \) Copy content Toggle raw display
$5$ \( T - 1979 \) Copy content Toggle raw display
$7$ \( T - 10115 \) Copy content Toggle raw display
$11$ \( T + 18850 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 142403 \) Copy content Toggle raw display
$19$ \( T + 83302 \) Copy content Toggle raw display
$23$ \( T + 536544 \) Copy content Toggle raw display
$29$ \( T + 2600442 \) Copy content Toggle raw display
$31$ \( T - 2214004 \) Copy content Toggle raw display
$37$ \( T + 18099241 \) Copy content Toggle raw display
$41$ \( T + 26812240 \) Copy content Toggle raw display
$43$ \( T + 42253475 \) Copy content Toggle raw display
$47$ \( T + 35914993 \) Copy content Toggle raw display
$53$ \( T + 66514064 \) Copy content Toggle raw display
$59$ \( T - 108164002 \) Copy content Toggle raw display
$61$ \( T + 207449912 \) Copy content Toggle raw display
$67$ \( T + 193015514 \) Copy content Toggle raw display
$71$ \( T - 201833497 \) Copy content Toggle raw display
$73$ \( T - 121628110 \) Copy content Toggle raw display
$79$ \( T - 112871912 \) Copy content Toggle raw display
$83$ \( T + 308254212 \) Copy content Toggle raw display
$89$ \( T - 6374870 \) Copy content Toggle raw display
$97$ \( T + 871266886 \) Copy content Toggle raw display
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