Properties

Label 338.4.c.d
Level $338$
Weight $4$
Character orbit 338.c
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(191,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.c (of order \(3\), degree \(2\), 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: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + ( - 3 \zeta_{6} + 3) q^{3} - 4 \zeta_{6} q^{4} - 2 q^{5} + 6 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + ( - 3 \zeta_{6} + 3) q^{3} - 4 \zeta_{6} q^{4} - 2 q^{5} + 6 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{10} + ( - 13 \zeta_{6} + 13) q^{11} - 12 q^{12} + 10 q^{14} + (6 \zeta_{6} - 6) q^{15} + (16 \zeta_{6} - 16) q^{16} - 27 \zeta_{6} q^{17} - 36 q^{18} + 75 \zeta_{6} q^{19} + 8 \zeta_{6} q^{20} - 15 q^{21} + 26 \zeta_{6} q^{22} + ( - 187 \zeta_{6} + 187) q^{23} + ( - 24 \zeta_{6} + 24) q^{24} - 121 q^{25} + 135 q^{27} + (20 \zeta_{6} - 20) q^{28} + ( - 13 \zeta_{6} + 13) q^{29} - 12 \zeta_{6} q^{30} + 104 q^{31} - 32 \zeta_{6} q^{32} - 39 \zeta_{6} q^{33} + 54 q^{34} + 10 \zeta_{6} q^{35} + ( - 72 \zeta_{6} + 72) q^{36} + ( - 423 \zeta_{6} + 423) q^{37} - 150 q^{38} - 16 q^{40} + ( - 195 \zeta_{6} + 195) q^{41} + ( - 30 \zeta_{6} + 30) q^{42} - 199 \zeta_{6} q^{43} - 52 q^{44} - 36 \zeta_{6} q^{45} + 374 \zeta_{6} q^{46} - 388 q^{47} + 48 \zeta_{6} q^{48} + ( - 318 \zeta_{6} + 318) q^{49} + ( - 242 \zeta_{6} + 242) q^{50} - 81 q^{51} + 618 q^{53} + (270 \zeta_{6} - 270) q^{54} + (26 \zeta_{6} - 26) q^{55} - 40 \zeta_{6} q^{56} + 225 q^{57} + 26 \zeta_{6} q^{58} + 491 \zeta_{6} q^{59} + 24 q^{60} - 175 \zeta_{6} q^{61} + (208 \zeta_{6} - 208) q^{62} + ( - 90 \zeta_{6} + 90) q^{63} + 64 q^{64} + 78 q^{66} + ( - 817 \zeta_{6} + 817) q^{67} + (108 \zeta_{6} - 108) q^{68} - 561 \zeta_{6} q^{69} - 20 q^{70} + 79 \zeta_{6} q^{71} + 144 \zeta_{6} q^{72} - 230 q^{73} + 846 \zeta_{6} q^{74} + (363 \zeta_{6} - 363) q^{75} + ( - 300 \zeta_{6} + 300) q^{76} - 65 q^{77} + 764 q^{79} + ( - 32 \zeta_{6} + 32) q^{80} + (81 \zeta_{6} - 81) q^{81} + 390 \zeta_{6} q^{82} + 732 q^{83} + 60 \zeta_{6} q^{84} + 54 \zeta_{6} q^{85} + 398 q^{86} - 39 \zeta_{6} q^{87} + ( - 104 \zeta_{6} + 104) q^{88} + (1041 \zeta_{6} - 1041) q^{89} + 72 q^{90} - 748 q^{92} + ( - 312 \zeta_{6} + 312) q^{93} + ( - 776 \zeta_{6} + 776) q^{94} - 150 \zeta_{6} q^{95} - 96 q^{96} - 97 \zeta_{6} q^{97} + 636 \zeta_{6} q^{98} + 234 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} - 5 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} - 5 q^{7} + 16 q^{8} + 18 q^{9} + 4 q^{10} + 13 q^{11} - 24 q^{12} + 20 q^{14} - 6 q^{15} - 16 q^{16} - 27 q^{17} - 72 q^{18} + 75 q^{19} + 8 q^{20} - 30 q^{21} + 26 q^{22} + 187 q^{23} + 24 q^{24} - 242 q^{25} + 270 q^{27} - 20 q^{28} + 13 q^{29} - 12 q^{30} + 208 q^{31} - 32 q^{32} - 39 q^{33} + 108 q^{34} + 10 q^{35} + 72 q^{36} + 423 q^{37} - 300 q^{38} - 32 q^{40} + 195 q^{41} + 30 q^{42} - 199 q^{43} - 104 q^{44} - 36 q^{45} + 374 q^{46} - 776 q^{47} + 48 q^{48} + 318 q^{49} + 242 q^{50} - 162 q^{51} + 1236 q^{53} - 270 q^{54} - 26 q^{55} - 40 q^{56} + 450 q^{57} + 26 q^{58} + 491 q^{59} + 48 q^{60} - 175 q^{61} - 208 q^{62} + 90 q^{63} + 128 q^{64} + 156 q^{66} + 817 q^{67} - 108 q^{68} - 561 q^{69} - 40 q^{70} + 79 q^{71} + 144 q^{72} - 460 q^{73} + 846 q^{74} - 363 q^{75} + 300 q^{76} - 130 q^{77} + 1528 q^{79} + 32 q^{80} - 81 q^{81} + 390 q^{82} + 1464 q^{83} + 60 q^{84} + 54 q^{85} + 796 q^{86} - 39 q^{87} + 104 q^{88} - 1041 q^{89} + 144 q^{90} - 1496 q^{92} + 312 q^{93} + 776 q^{94} - 150 q^{95} - 192 q^{96} - 97 q^{97} + 636 q^{98} + 468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i −2.00000 3.00000 + 5.19615i −2.50000 4.33013i 8.00000 9.00000 + 15.5885i 2.00000 3.46410i
315.1 −1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −2.00000 3.00000 5.19615i −2.50000 + 4.33013i 8.00000 9.00000 15.5885i 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.c.d 2
13.b even 2 1 26.4.c.a 2
13.c even 3 1 338.4.a.d 1
13.c even 3 1 inner 338.4.c.d 2
13.d odd 4 2 338.4.e.d 4
13.e even 6 1 26.4.c.a 2
13.e even 6 1 338.4.a.a 1
13.f odd 12 2 338.4.b.a 2
13.f odd 12 2 338.4.e.d 4
39.d odd 2 1 234.4.h.b 2
39.h odd 6 1 234.4.h.b 2
52.b odd 2 1 208.4.i.a 2
52.i odd 6 1 208.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.b even 2 1
26.4.c.a 2 13.e even 6 1
208.4.i.a 2 52.b odd 2 1
208.4.i.a 2 52.i odd 6 1
234.4.h.b 2 39.d odd 2 1
234.4.h.b 2 39.h odd 6 1
338.4.a.a 1 13.e even 6 1
338.4.a.d 1 13.c even 3 1
338.4.b.a 2 13.f odd 12 2
338.4.c.d 2 1.a even 1 1 trivial
338.4.c.d 2 13.c even 3 1 inner
338.4.e.d 4 13.d odd 4 2
338.4.e.d 4 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$11$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$23$ \( T^{2} - 187T + 34969 \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$31$ \( (T - 104)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 423T + 178929 \) Copy content Toggle raw display
$41$ \( T^{2} - 195T + 38025 \) Copy content Toggle raw display
$43$ \( T^{2} + 199T + 39601 \) Copy content Toggle raw display
$47$ \( (T + 388)^{2} \) Copy content Toggle raw display
$53$ \( (T - 618)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 491T + 241081 \) Copy content Toggle raw display
$61$ \( T^{2} + 175T + 30625 \) Copy content Toggle raw display
$67$ \( T^{2} - 817T + 667489 \) Copy content Toggle raw display
$71$ \( T^{2} - 79T + 6241 \) Copy content Toggle raw display
$73$ \( (T + 230)^{2} \) Copy content Toggle raw display
$79$ \( (T - 764)^{2} \) Copy content Toggle raw display
$83$ \( (T - 732)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1041 T + 1083681 \) Copy content Toggle raw display
$97$ \( T^{2} + 97T + 9409 \) Copy content Toggle raw display
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