Properties

Label 338.10.a.a
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 2)
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)
 
\(f(q)\) \(=\) \( q - 16 q^{2} - 156 q^{3} + 256 q^{4} - 870 q^{5} + 2496 q^{6} + 952 q^{7} - 4096 q^{8} + 4653 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} - 156 q^{3} + 256 q^{4} - 870 q^{5} + 2496 q^{6} + 952 q^{7} - 4096 q^{8} + 4653 q^{9} + 13920 q^{10} + 56148 q^{11} - 39936 q^{12} - 15232 q^{14} + 135720 q^{15} + 65536 q^{16} - 247662 q^{17} - 74448 q^{18} - 315380 q^{19} - 222720 q^{20} - 148512 q^{21} - 898368 q^{22} + 204504 q^{23} + 638976 q^{24} - 1196225 q^{25} + 2344680 q^{27} + 243712 q^{28} - 3840450 q^{29} - 2171520 q^{30} + 1309408 q^{31} - 1048576 q^{32} - 8759088 q^{33} + 3962592 q^{34} - 828240 q^{35} + 1191168 q^{36} - 4307078 q^{37} + 5046080 q^{38} + 3563520 q^{40} - 1512042 q^{41} + 2376192 q^{42} + 33670604 q^{43} + 14373888 q^{44} - 4048110 q^{45} - 3272064 q^{46} + 10581072 q^{47} - 10223616 q^{48} - 39447303 q^{49} + 19139600 q^{50} + 38635272 q^{51} + 16616214 q^{53} - 37514880 q^{54} - 48848760 q^{55} - 3899392 q^{56} + 49199280 q^{57} + 61447200 q^{58} - 112235100 q^{59} + 34744320 q^{60} - 33197218 q^{61} - 20950528 q^{62} + 4429656 q^{63} + 16777216 q^{64} + 140145408 q^{66} + 121372252 q^{67} - 63401472 q^{68} - 31902624 q^{69} + 13251840 q^{70} + 387172728 q^{71} - 19058688 q^{72} - 255240074 q^{73} + 68913248 q^{74} + 186611100 q^{75} - 80737280 q^{76} + 53452896 q^{77} + 492101840 q^{79} - 57016320 q^{80} - 457355079 q^{81} + 24192672 q^{82} + 457420236 q^{83} - 38019072 q^{84} + 215465940 q^{85} - 538729664 q^{86} + 599110200 q^{87} - 229982208 q^{88} + 31809510 q^{89} + 64769760 q^{90} + 52353024 q^{92} - 204267648 q^{93} - 169297152 q^{94} + 274380600 q^{95} + 163577856 q^{96} + 673532062 q^{97} + 631156848 q^{98} + 261256644 q^{99}+O(q^{100}) \) 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 −156.000 256.000 −870.000 2496.00 952.000 −4096.00 4653.00 13920.0
\(n\): 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.a 1
13.b even 2 1 2.10.a.a 1
39.d odd 2 1 18.10.a.a 1
52.b odd 2 1 16.10.a.d 1
65.d even 2 1 50.10.a.c 1
65.h odd 4 2 50.10.b.a 2
91.b odd 2 1 98.10.a.c 1
91.r even 6 2 98.10.c.c 2
91.s odd 6 2 98.10.c.b 2
104.e even 2 1 64.10.a.h 1
104.h odd 2 1 64.10.a.b 1
117.n odd 6 2 162.10.c.i 2
117.t even 6 2 162.10.c.b 2
143.d odd 2 1 242.10.a.a 1
156.h even 2 1 144.10.a.d 1
208.o odd 4 2 256.10.b.e 2
208.p even 4 2 256.10.b.g 2
260.g odd 2 1 400.10.a.b 1
260.p even 4 2 400.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.10.a.a 1 13.b even 2 1
16.10.a.d 1 52.b odd 2 1
18.10.a.a 1 39.d odd 2 1
50.10.a.c 1 65.d even 2 1
50.10.b.a 2 65.h odd 4 2
64.10.a.b 1 104.h odd 2 1
64.10.a.h 1 104.e even 2 1
98.10.a.c 1 91.b odd 2 1
98.10.c.b 2 91.s odd 6 2
98.10.c.c 2 91.r even 6 2
144.10.a.d 1 156.h even 2 1
162.10.c.b 2 117.t even 6 2
162.10.c.i 2 117.n odd 6 2
242.10.a.a 1 143.d odd 2 1
256.10.b.e 2 208.o odd 4 2
256.10.b.g 2 208.p even 4 2
338.10.a.a 1 1.a even 1 1 trivial
400.10.a.b 1 260.g odd 2 1
400.10.c.d 2 260.p even 4 2

Hecke kernels

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

\( T_{3} + 156 \) Copy content Toggle raw display
\( T_{5} + 870 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 16 \) Copy content Toggle raw display
$3$ \( T + 156 \) Copy content Toggle raw display
$5$ \( T + 870 \) Copy content Toggle raw display
$7$ \( T - 952 \) Copy content Toggle raw display
$11$ \( T - 56148 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 247662 \) Copy content Toggle raw display
$19$ \( T + 315380 \) Copy content Toggle raw display
$23$ \( T - 204504 \) Copy content Toggle raw display
$29$ \( T + 3840450 \) Copy content Toggle raw display
$31$ \( T - 1309408 \) Copy content Toggle raw display
$37$ \( T + 4307078 \) Copy content Toggle raw display
$41$ \( T + 1512042 \) Copy content Toggle raw display
$43$ \( T - 33670604 \) Copy content Toggle raw display
$47$ \( T - 10581072 \) Copy content Toggle raw display
$53$ \( T - 16616214 \) Copy content Toggle raw display
$59$ \( T + 112235100 \) Copy content Toggle raw display
$61$ \( T + 33197218 \) Copy content Toggle raw display
$67$ \( T - 121372252 \) Copy content Toggle raw display
$71$ \( T - 387172728 \) Copy content Toggle raw display
$73$ \( T + 255240074 \) Copy content Toggle raw display
$79$ \( T - 492101840 \) Copy content Toggle raw display
$83$ \( T - 457420236 \) Copy content Toggle raw display
$89$ \( T - 31809510 \) Copy content Toggle raw display
$97$ \( T - 673532062 \) Copy content Toggle raw display
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