Properties

Label 338.4.b.a
Level $338$
Weight $4$
Character orbit 338.b
Analytic conductor $19.943$
Analytic rank $0$
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] = [338,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(19.9426455819\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 3 q^{3} - 4 q^{4} - 2 i q^{5} - 6 i q^{6} - 5 i q^{7} - 8 i q^{8} - 18 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 3 q^{3} - 4 q^{4} - 2 i q^{5} - 6 i q^{6} - 5 i q^{7} - 8 i q^{8} - 18 q^{9} + 4 q^{10} + 13 i q^{11} + 12 q^{12} + 10 q^{14} + 6 i q^{15} + 16 q^{16} - 27 q^{17} - 36 i q^{18} - 75 i q^{19} + 8 i q^{20} + 15 i q^{21} - 26 q^{22} + 187 q^{23} + 24 i q^{24} + 121 q^{25} + 135 q^{27} + 20 i q^{28} - 13 q^{29} - 12 q^{30} + 104 i q^{31} + 32 i q^{32} - 39 i q^{33} - 54 i q^{34} - 10 q^{35} + 72 q^{36} + 423 i q^{37} + 150 q^{38} - 16 q^{40} - 195 i q^{41} - 30 q^{42} - 199 q^{43} - 52 i q^{44} + 36 i q^{45} + 374 i q^{46} + 388 i q^{47} - 48 q^{48} + 318 q^{49} + 242 i q^{50} + 81 q^{51} + 618 q^{53} + 270 i q^{54} + 26 q^{55} - 40 q^{56} + 225 i q^{57} - 26 i q^{58} + 491 i q^{59} - 24 i q^{60} + 175 q^{61} - 208 q^{62} + 90 i q^{63} - 64 q^{64} + 78 q^{66} - 817 i q^{67} + 108 q^{68} - 561 q^{69} - 20 i q^{70} - 79 i q^{71} + 144 i q^{72} + 230 i q^{73} - 846 q^{74} - 363 q^{75} + 300 i q^{76} + 65 q^{77} + 764 q^{79} - 32 i q^{80} + 81 q^{81} + 390 q^{82} + 732 i q^{83} - 60 i q^{84} + 54 i q^{85} - 398 i q^{86} + 39 q^{87} + 104 q^{88} - 1041 i q^{89} - 72 q^{90} - 748 q^{92} - 312 i q^{93} - 776 q^{94} - 150 q^{95} - 96 i q^{96} + 97 i q^{97} + 636 i q^{98} - 234 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{4} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{4} - 36 q^{9} + 8 q^{10} + 24 q^{12} + 20 q^{14} + 32 q^{16} - 54 q^{17} - 52 q^{22} + 374 q^{23} + 242 q^{25} + 270 q^{27} - 26 q^{29} - 24 q^{30} - 20 q^{35} + 144 q^{36} + 300 q^{38} - 32 q^{40} - 60 q^{42} - 398 q^{43} - 96 q^{48} + 636 q^{49} + 162 q^{51} + 1236 q^{53} + 52 q^{55} - 80 q^{56} + 350 q^{61} - 416 q^{62} - 128 q^{64} + 156 q^{66} + 216 q^{68} - 1122 q^{69} - 1692 q^{74} - 726 q^{75} + 130 q^{77} + 1528 q^{79} + 162 q^{81} + 780 q^{82} + 78 q^{87} + 208 q^{88} - 144 q^{90} - 1496 q^{92} - 1552 q^{94} - 300 q^{95}+O(q^{100}) \) 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
2.00000i −3.00000 −4.00000 2.00000i 6.00000i 5.00000i 8.00000i −18.0000 4.00000
337.2 2.00000i −3.00000 −4.00000 2.00000i 6.00000i 5.00000i 8.00000i −18.0000 4.00000
\(n\): 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 338.4.b.a 2
13.b even 2 1 inner 338.4.b.a 2
13.c even 3 2 338.4.e.d 4
13.d odd 4 1 338.4.a.a 1
13.d odd 4 1 338.4.a.d 1
13.e even 6 2 338.4.e.d 4
13.f odd 12 2 26.4.c.a 2
13.f odd 12 2 338.4.c.d 2
39.k even 12 2 234.4.h.b 2
52.l even 12 2 208.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.f odd 12 2
208.4.i.a 2 52.l even 12 2
234.4.h.b 2 39.k even 12 2
338.4.a.a 1 13.d odd 4 1
338.4.a.d 1 13.d odd 4 1
338.4.b.a 2 1.a even 1 1 trivial
338.4.b.a 2 13.b even 2 1 inner
338.4.c.d 2 13.f odd 12 2
338.4.e.d 4 13.c even 3 2
338.4.e.d 4 13.e even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3 \) acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 25 \) Copy content Toggle raw display
$11$ \( T^{2} + 169 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 5625 \) Copy content Toggle raw display
$23$ \( (T - 187)^{2} \) Copy content Toggle raw display
$29$ \( (T + 13)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 10816 \) Copy content Toggle raw display
$37$ \( T^{2} + 178929 \) Copy content Toggle raw display
$41$ \( T^{2} + 38025 \) Copy content Toggle raw display
$43$ \( (T + 199)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 150544 \) Copy content Toggle raw display
$53$ \( (T - 618)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 241081 \) Copy content Toggle raw display
$61$ \( (T - 175)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 667489 \) Copy content Toggle raw display
$71$ \( T^{2} + 6241 \) Copy content Toggle raw display
$73$ \( T^{2} + 52900 \) Copy content Toggle raw display
$79$ \( (T - 764)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 535824 \) Copy content Toggle raw display
$89$ \( T^{2} + 1083681 \) Copy content Toggle raw display
$97$ \( T^{2} + 9409 \) Copy content Toggle raw display
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