Properties

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

Related objects

Downloads

Learn more

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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} - 273 q^{3} + 256 q^{4} - 1015 q^{5} - 4368 q^{6} - 3955 q^{7} + 4096 q^{8} + 54846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} - 273 q^{3} + 256 q^{4} - 1015 q^{5} - 4368 q^{6} - 3955 q^{7} + 4096 q^{8} + 54846 q^{9} - 16240 q^{10} + 50998 q^{11} - 69888 q^{12} - 63280 q^{14} + 277095 q^{15} + 65536 q^{16} + 509757 q^{17} + 877536 q^{18} + 626574 q^{19} - 259840 q^{20} + 1079715 q^{21} + 815968 q^{22} + 653524 q^{23} - 1118208 q^{24} - 922900 q^{25} - 9599499 q^{27} - 1012480 q^{28} - 4943006 q^{29} + 4433520 q^{30} - 4071700 q^{31} + 1048576 q^{32} - 13922454 q^{33} + 8156112 q^{34} + 4014325 q^{35} + 14040576 q^{36} - 2348883 q^{37} + 10025184 q^{38} - 4157440 q^{40} + 13350960 q^{41} + 17275440 q^{42} - 7834847 q^{43} + 13055488 q^{44} - 55668690 q^{45} + 10456384 q^{46} + 39637681 q^{47} - 17891328 q^{48} - 24711582 q^{49} - 14766400 q^{50} - 139163661 q^{51} + 73200924 q^{53} - 153591984 q^{54} - 51762970 q^{55} - 16199680 q^{56} - 171054702 q^{57} - 79088096 q^{58} + 141141614 q^{59} + 70936320 q^{60} - 132061256 q^{61} - 65147200 q^{62} - 216915930 q^{63} + 16777216 q^{64} - 222759264 q^{66} + 185673110 q^{67} + 130497792 q^{68} - 178412052 q^{69} + 64229200 q^{70} - 224452625 q^{71} + 224649216 q^{72} + 172523674 q^{73} - 37582128 q^{74} + 251951700 q^{75} + 160402944 q^{76} - 201697090 q^{77} - 643288156 q^{79} - 66519040 q^{80} + 1541129409 q^{81} + 213615360 q^{82} - 720077280 q^{83} + 276407040 q^{84} - 517403355 q^{85} - 125357552 q^{86} + 1349440638 q^{87} + 208887808 q^{88} + 73028106 q^{89} - 890699040 q^{90} + 167302144 q^{92} + 1111574100 q^{93} + 634202896 q^{94} - 635972610 q^{95} - 286261248 q^{96} + 15879778 q^{97} - 395385312 q^{98} + 2797036308 q^{99}+O(q^{100}) \) 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 −273.000 256.000 −1015.00 −4368.00 −3955.00 4096.00 54846.0 −16240.0
\(n\): 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.c 1
13.b even 2 1 26.10.a.a 1
39.d odd 2 1 234.10.a.b 1
52.b odd 2 1 208.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.10.a.a 1 13.b even 2 1
208.10.a.c 1 52.b odd 2 1
234.10.a.b 1 39.d odd 2 1
338.10.a.c 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} + 273 \) Copy content Toggle raw display
\( T_{5} + 1015 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T + 273 \) Copy content Toggle raw display
$5$ \( T + 1015 \) Copy content Toggle raw display
$7$ \( T + 3955 \) Copy content Toggle raw display
$11$ \( T - 50998 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 509757 \) Copy content Toggle raw display
$19$ \( T - 626574 \) Copy content Toggle raw display
$23$ \( T - 653524 \) Copy content Toggle raw display
$29$ \( T + 4943006 \) Copy content Toggle raw display
$31$ \( T + 4071700 \) Copy content Toggle raw display
$37$ \( T + 2348883 \) Copy content Toggle raw display
$41$ \( T - 13350960 \) Copy content Toggle raw display
$43$ \( T + 7834847 \) Copy content Toggle raw display
$47$ \( T - 39637681 \) Copy content Toggle raw display
$53$ \( T - 73200924 \) Copy content Toggle raw display
$59$ \( T - 141141614 \) Copy content Toggle raw display
$61$ \( T + 132061256 \) Copy content Toggle raw display
$67$ \( T - 185673110 \) Copy content Toggle raw display
$71$ \( T + 224452625 \) Copy content Toggle raw display
$73$ \( T - 172523674 \) Copy content Toggle raw display
$79$ \( T + 643288156 \) Copy content Toggle raw display
$83$ \( T + 720077280 \) Copy content Toggle raw display
$89$ \( T - 73028106 \) Copy content Toggle raw display
$97$ \( T - 15879778 \) Copy content Toggle raw display
show more
show less