Properties

Label 338.8.b.a
Level $338$
Weight $8$
Character orbit 338.b
Analytic conductor $105.586$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(337,338)]
 
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([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), not 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: \(105.586138614\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 i q^{2} - 87 q^{3} - 64 q^{4} - 321 i q^{5} + 696 i q^{6} - 181 i q^{7} + 512 i q^{8} + 5382 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 i q^{2} - 87 q^{3} - 64 q^{4} - 321 i q^{5} + 696 i q^{6} - 181 i q^{7} + 512 i q^{8} + 5382 q^{9} - 2568 q^{10} + 7782 i q^{11} + 5568 q^{12} - 1448 q^{14} + 27927 i q^{15} + 4096 q^{16} - 9069 q^{17} - 43056 i q^{18} + 37150 i q^{19} + 20544 i q^{20} + 15747 i q^{21} + 62256 q^{22} - 19008 q^{23} - 44544 i q^{24} - 24916 q^{25} - 277965 q^{27} + 11584 i q^{28} + 174750 q^{29} + 223416 q^{30} - 29012 i q^{31} - 32768 i q^{32} - 677034 i q^{33} + 72552 i q^{34} - 58101 q^{35} - 344448 q^{36} + 323669 i q^{37} + 297200 q^{38} + 164352 q^{40} - 795312 i q^{41} + 125976 q^{42} + 314137 q^{43} - 498048 i q^{44} - 1727622 i q^{45} + 152064 i q^{46} - 447441 i q^{47} - 356352 q^{48} + 790782 q^{49} + 199328 i q^{50} + 789003 q^{51} - 1469232 q^{53} + 2223720 i q^{54} + 2498022 q^{55} + 92672 q^{56} - 3232050 i q^{57} - 1398000 i q^{58} + 1627770 i q^{59} - 1787328 i q^{60} - 2399608 q^{61} - 232096 q^{62} - 974142 i q^{63} - 262144 q^{64} - 5416272 q^{66} + 64066 i q^{67} + 580416 q^{68} + 1653696 q^{69} + 464808 i q^{70} + 322383 i q^{71} + 2755584 i q^{72} - 4454782 i q^{73} + 2589352 q^{74} + 2167692 q^{75} - 2377600 i q^{76} + 1408542 q^{77} + 753560 q^{79} - 1314816 i q^{80} + 12412521 q^{81} - 6362496 q^{82} + 1219092 i q^{83} - 1007808 i q^{84} + 2911149 i q^{85} - 2513096 i q^{86} - 15203250 q^{87} - 3984384 q^{88} + 3390330 i q^{89} - 13820976 q^{90} + 1216512 q^{92} + 2524044 i q^{93} - 3579528 q^{94} + 11925150 q^{95} + 2850816 i q^{96} - 1628774 i q^{97} - 6326256 i q^{98} + 41882724 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9} - 5136 q^{10} + 11136 q^{12} - 2896 q^{14} + 8192 q^{16} - 18138 q^{17} + 124512 q^{22} - 38016 q^{23} - 49832 q^{25} - 555930 q^{27} + 349500 q^{29} + 446832 q^{30} - 116202 q^{35} - 688896 q^{36} + 594400 q^{38} + 328704 q^{40} + 251952 q^{42} + 628274 q^{43} - 712704 q^{48} + 1581564 q^{49} + 1578006 q^{51} - 2938464 q^{53} + 4996044 q^{55} + 185344 q^{56} - 4799216 q^{61} - 464192 q^{62} - 524288 q^{64} - 10832544 q^{66} + 1160832 q^{68} + 3307392 q^{69} + 5178704 q^{74} + 4335384 q^{75} + 2817084 q^{77} + 1507120 q^{79} + 24825042 q^{81} - 12724992 q^{82} - 30406500 q^{87} - 7968768 q^{88} - 27641952 q^{90} + 2433024 q^{92} - 7159056 q^{94} + 23850300 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
8.00000i −87.0000 −64.0000 321.000i 696.000i 181.000i 512.000i 5382.00 −2568.00
337.2 8.00000i −87.0000 −64.0000 321.000i 696.000i 181.000i 512.000i 5382.00 −2568.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.8.b.a 2
13.b even 2 1 inner 338.8.b.a 2
13.d odd 4 1 26.8.a.b 1
13.d odd 4 1 338.8.a.a 1
39.f even 4 1 234.8.a.a 1
52.f even 4 1 208.8.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.b 1 13.d odd 4 1
208.8.a.e 1 52.f even 4 1
234.8.a.a 1 39.f even 4 1
338.8.a.a 1 13.d odd 4 1
338.8.b.a 2 1.a even 1 1 trivial
338.8.b.a 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 87 \) acting on \(S_{8}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T + 87)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 103041 \) Copy content Toggle raw display
$7$ \( T^{2} + 32761 \) Copy content Toggle raw display
$11$ \( T^{2} + 60559524 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 9069)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1380122500 \) Copy content Toggle raw display
$23$ \( (T + 19008)^{2} \) Copy content Toggle raw display
$29$ \( (T - 174750)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 841696144 \) Copy content Toggle raw display
$37$ \( T^{2} + 104761621561 \) Copy content Toggle raw display
$41$ \( T^{2} + 632521177344 \) Copy content Toggle raw display
$43$ \( (T - 314137)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 200203448481 \) Copy content Toggle raw display
$53$ \( (T + 1469232)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2649635172900 \) Copy content Toggle raw display
$61$ \( (T + 2399608)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4104452356 \) Copy content Toggle raw display
$71$ \( T^{2} + 103930798689 \) Copy content Toggle raw display
$73$ \( T^{2} + 19845082667524 \) Copy content Toggle raw display
$79$ \( (T - 753560)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1486185304464 \) Copy content Toggle raw display
$89$ \( T^{2} + 11494337508900 \) Copy content Toggle raw display
$97$ \( T^{2} + 2652904743076 \) Copy content Toggle raw display
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