Properties

Label 338.3.f.c
Level $338$
Weight $3$
Character orbit 338.f
Analytic conductor $9.210$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,3,Mod(19,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.f (of order \(12\), degree \(4\), 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: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + ( - 2 \zeta_{12}^{2} + 4) q^{10} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} + 1) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} - 1) q^{14} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( - 13 \zeta_{12}^{2} - 6 \zeta_{12} - 13) q^{17} + (6 \zeta_{12}^{3} - 6) q^{18} + ( - 17 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{19} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{20} + ( - 11 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 11) q^{21} + (2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - \zeta_{12} + 9) q^{22} + ( - 21 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 21 \zeta_{12} - 4) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 19 \zeta_{12}^{3} q^{25} + ( - 15 \zeta_{12}^{3} + 30 \zeta_{12}) q^{27} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 6 \zeta_{12} - 14) q^{28} + (16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} + 16 \zeta_{12}) q^{29} + 6 \zeta_{12} q^{30} + ( - 27 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} + 27) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + (9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{33} + ( - 7 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 7) q^{34} + (2 \zeta_{12}^{3} - 21 \zeta_{12}^{2} - \zeta_{12} + 21) q^{35} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{36} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} + 6 \zeta_{12} + 7) q^{37} + (7 \zeta_{12}^{3} + 34 \zeta_{12}^{2} - 17) q^{38} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{40} + (13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 26 \zeta_{12} - 13) q^{41} + ( - \zeta_{12}^{3} + 21 \zeta_{12}^{2} - \zeta_{12}) q^{42} + ( - 20 \zeta_{12}^{2} - 39 \zeta_{12} - 20) q^{43} + (8 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 8) q^{44} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 4 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 19 \zeta_{12} - 19) q^{46} + ( - 33 \zeta_{12}^{3} - 33) q^{47} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - 25 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 25 \zeta_{12} - 14) q^{49} + ( - 19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 19 \zeta_{12}) q^{50} + ( - 39 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 6) q^{51} + (22 \zeta_{12}^{3} - 44 \zeta_{12} - 42) q^{53} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 15 \zeta_{12} - 30) q^{54} + ( - 9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 9 \zeta_{12}) q^{55} + (14 \zeta_{12}^{2} - 2 \zeta_{12} + 14) q^{56} + ( - 29 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} + 29) q^{57} + ( - 32 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 43 \zeta_{12} - 43) q^{58} + ( - 13 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{59} + ( - 6 \zeta_{12}^{3} - 6) q^{60} + (12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} - 6 \zeta_{12} + 39) q^{61} + (64 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 64 \zeta_{12} + 20) q^{62} + (18 \zeta_{12}^{3} + 18 \zeta_{12}^{2} + 24 \zeta_{12} - 42) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 9 \zeta_{12}^{3} + 18 \zeta_{12} - 3) q^{66} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 23 \zeta_{12} + 33) q^{67} + ( - 26 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 26 \zeta_{12}) q^{68} + (21 \zeta_{12}^{2} - 6 \zeta_{12} + 21) q^{69} + (20 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 20) q^{70} + (25 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 29 \zeta_{12} + 29) q^{71} + (12 \zeta_{12}^{2} - 12 \zeta_{12} - 12) q^{72} + (61 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} + 61) q^{73} + (14 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 7 \zeta_{12} - 19) q^{74} + (19 \zeta_{12}^{2} - 38) q^{75} + ( - 24 \zeta_{12}^{3} - 24 \zeta_{12}^{2} - 10 \zeta_{12} + 34) q^{76} + (15 \zeta_{12}^{3} - 64 \zeta_{12}^{2} + 32) q^{77} + (36 \zeta_{12}^{3} - 72 \zeta_{12} + 48) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + (13 \zeta_{12}^{2} + 39 \zeta_{12} + 13) q^{82} + (71 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 46 \zeta_{12} - 71) q^{83} + (2 \zeta_{12}^{3} - 20 \zeta_{12}^{2} - 22 \zeta_{12} + 22) q^{84} + ( - 12 \zeta_{12}^{3} - 33 \zeta_{12}^{2} + 45 \zeta_{12} + 45) q^{85} + (19 \zeta_{12}^{3} + 40 \zeta_{12}^{2} + 40 \zeta_{12} + 19) q^{86} + ( - 54 \zeta_{12}^{3} + 48 \zeta_{12}^{2} + 27 \zeta_{12} - 48) q^{87} + ( - 18 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 18 \zeta_{12} - 4) q^{88} + (43 \zeta_{12}^{3} + 43 \zeta_{12}^{2} - 56 \zeta_{12} + 13) q^{89} + ( - 24 \zeta_{12}^{2} + 12) q^{90} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 42) q^{92} + (47 \zeta_{12}^{3} - 47 \zeta_{12}^{2} + 17 \zeta_{12} + 64) q^{93} + 66 \zeta_{12}^{2} q^{94} + (7 \zeta_{12}^{2} + 51 \zeta_{12} + 7) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (69 \zeta_{12}^{3} + 71 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{97} + ( - 14 \zeta_{12}^{3} + 32 \zeta_{12}^{2} - 18 \zeta_{12} - 18) q^{98} + (30 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 30) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{6} - 20 q^{7} - 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{6} - 20 q^{7} - 8 q^{8} + 12 q^{9} + 12 q^{10} + 12 q^{11} - 4 q^{14} - 6 q^{15} + 8 q^{16} - 78 q^{17} - 24 q^{18} - 44 q^{19} - 12 q^{20} - 42 q^{21} + 18 q^{22} - 12 q^{23} + 12 q^{24} - 40 q^{28} - 54 q^{29} + 128 q^{31} + 8 q^{32} - 24 q^{33} + 24 q^{34} + 42 q^{35} + 2 q^{37} - 78 q^{41} + 42 q^{42} - 120 q^{43} - 36 q^{44} + 36 q^{45} - 30 q^{46} - 132 q^{47} - 42 q^{49} - 38 q^{50} - 168 q^{53} - 90 q^{54} + 6 q^{55} + 84 q^{56} + 102 q^{57} - 150 q^{58} + 72 q^{59} - 24 q^{60} + 78 q^{61} + 60 q^{62} - 132 q^{63} - 12 q^{66} + 112 q^{67} - 24 q^{68} + 126 q^{69} - 84 q^{70} + 108 q^{71} - 24 q^{72} + 136 q^{73} - 38 q^{74} - 114 q^{75} + 88 q^{76} + 192 q^{79} - 24 q^{80} - 18 q^{81} + 78 q^{82} - 192 q^{83} + 48 q^{84} + 114 q^{85} + 156 q^{86} - 96 q^{87} - 12 q^{88} + 138 q^{89} + 168 q^{92} + 162 q^{93} + 132 q^{94} + 42 q^{95} + 134 q^{97} - 8 q^{98} + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0.366025 + 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i 1.73205 1.73205i −2.36603 0.633975i −2.40192 + 8.96410i −2.00000 2.00000i 3.00000 + 5.19615i 3.00000 + 1.73205i
89.1 0.366025 1.36603i −0.866025 1.50000i −1.73205 1.00000i 1.73205 + 1.73205i −2.36603 + 0.633975i −2.40192 8.96410i −2.00000 + 2.00000i 3.00000 5.19615i 3.00000 1.73205i
249.1 −1.36603 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −1.73205 + 1.73205i −0.633975 2.36603i −7.59808 + 2.03590i −2.00000 2.00000i 3.00000 5.19615i 3.00000 1.73205i
319.1 −1.36603 + 0.366025i 0.866025 1.50000i 1.73205 1.00000i −1.73205 1.73205i −0.633975 + 2.36603i −7.59808 2.03590i −2.00000 + 2.00000i 3.00000 + 5.19615i 3.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.f.c 4
13.b even 2 1 338.3.f.f 4
13.c even 3 1 338.3.d.e 4
13.c even 3 1 338.3.f.d 4
13.d odd 4 1 26.3.f.a 4
13.d odd 4 1 338.3.f.d 4
13.e even 6 1 26.3.f.a 4
13.e even 6 1 338.3.d.d 4
13.f odd 12 1 338.3.d.d 4
13.f odd 12 1 338.3.d.e 4
13.f odd 12 1 inner 338.3.f.c 4
13.f odd 12 1 338.3.f.f 4
39.f even 4 1 234.3.bb.b 4
39.h odd 6 1 234.3.bb.b 4
52.f even 4 1 208.3.bd.c 4
52.i odd 6 1 208.3.bd.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 13.d odd 4 1
26.3.f.a 4 13.e even 6 1
208.3.bd.c 4 52.f even 4 1
208.3.bd.c 4 52.i odd 6 1
234.3.bb.b 4 39.f even 4 1
234.3.bb.b 4 39.h odd 6 1
338.3.d.d 4 13.e even 6 1
338.3.d.d 4 13.f odd 12 1
338.3.d.e 4 13.c even 3 1
338.3.d.e 4 13.f odd 12 1
338.3.f.c 4 1.a even 1 1 trivial
338.3.f.c 4 13.f odd 12 1 inner
338.3.f.d 4 13.c even 3 1
338.3.f.d 4 13.d odd 4 1
338.3.f.f 4 13.b even 2 1
338.3.f.f 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} + 20T_{7}^{3} + 221T_{7}^{2} + 1606T_{7} + 5329 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + 221 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 78 T^{3} + 2499 T^{2} + \cdots + 221841 \) Copy content Toggle raw display
$19$ \( T^{4} + 44 T^{3} + 1325 T^{2} + \cdots + 167281 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} - 381 T^{2} + \cdots + 184041 \) Copy content Toggle raw display
$29$ \( T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( T^{4} - 128 T^{3} + 8192 T^{2} + \cdots + 3602404 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + 401 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{4} + 78 T^{3} + 1521 T^{2} + \cdots + 257049 \) Copy content Toggle raw display
$43$ \( T^{4} + 120 T^{3} + 4479 T^{2} + \cdots + 103041 \) Copy content Toggle raw display
$47$ \( (T^{2} + 66 T + 2178)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 84 T + 312)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 72 T^{3} + 1305 T^{2} + \cdots + 84681 \) Copy content Toggle raw display
$61$ \( T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 \) Copy content Toggle raw display
$67$ \( T^{4} - 112 T^{3} + 4985 T^{2} + \cdots + 2399401 \) Copy content Toggle raw display
$71$ \( T^{4} - 108 T^{3} + 3357 T^{2} + \cdots + 154449 \) Copy content Toggle raw display
$73$ \( T^{4} - 136 T^{3} + 9248 T^{2} + \cdots + 4251844 \) Copy content Toggle raw display
$79$ \( (T^{2} - 96 T - 1584)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192 T^{3} + 18432 T^{2} + \cdots + 2056356 \) Copy content Toggle raw display
$89$ \( T^{4} - 138 T^{3} + \cdots + 21594609 \) Copy content Toggle raw display
$97$ \( T^{4} - 134 T^{3} + \cdots + 20043529 \) Copy content Toggle raw display
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