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Properties

Label 338.4.e.c
Level $338$
Weight $4$
Character orbit 338.e
Analytic conductor $19.943$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(23,338)]
 
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([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + 4 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{6} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 26 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + 4 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{6} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 26 \zeta_{12}^{2} q^{9} + ( - 34 \zeta_{12}^{2} + 34) q^{10} - 2 \zeta_{12} q^{11} + 4 q^{12} - 70 q^{14} - 17 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} - 19 \zeta_{12}^{2} q^{17} + 52 \zeta_{12}^{3} q^{18} + (94 \zeta_{12}^{3} - 94 \zeta_{12}) q^{19} + ( - 68 \zeta_{12}^{3} + 68 \zeta_{12}) q^{20} + 35 \zeta_{12}^{3} q^{21} - 4 \zeta_{12}^{2} q^{22} + (72 \zeta_{12}^{2} - 72) q^{23} + 8 \zeta_{12} q^{24} - 164 q^{25} + 53 q^{27} - 140 \zeta_{12} q^{28} + (246 \zeta_{12}^{2} - 246) q^{29} - 34 \zeta_{12}^{2} q^{30} + 100 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} - 38 \zeta_{12}^{3} q^{34} + 595 \zeta_{12}^{2} q^{35} + (104 \zeta_{12}^{2} - 104) q^{36} + 11 \zeta_{12} q^{37} - 188 q^{38} + 136 q^{40} - 280 \zeta_{12} q^{41} + (70 \zeta_{12}^{2} - 70) q^{42} + 241 \zeta_{12}^{2} q^{43} - 8 \zeta_{12}^{3} q^{44} + ( - 442 \zeta_{12}^{3} + 442 \zeta_{12}) q^{45} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{46} + 137 \zeta_{12}^{3} q^{47} + 16 \zeta_{12}^{2} q^{48} + ( - 882 \zeta_{12}^{2} + 882) q^{49} - 328 \zeta_{12} q^{50} - 19 q^{51} - 232 q^{53} + 106 \zeta_{12} q^{54} + (34 \zeta_{12}^{2} - 34) q^{55} - 280 \zeta_{12}^{2} q^{56} + 94 \zeta_{12}^{3} q^{57} + (492 \zeta_{12}^{3} - 492 \zeta_{12}) q^{58} + (386 \zeta_{12}^{3} - 386 \zeta_{12}) q^{59} - 68 \zeta_{12}^{3} q^{60} - 64 \zeta_{12}^{2} q^{61} + (200 \zeta_{12}^{2} - 200) q^{62} - 910 \zeta_{12} q^{63} - 64 q^{64} - 4 q^{66} - 670 \zeta_{12} q^{67} + ( - 76 \zeta_{12}^{2} + 76) q^{68} + 72 \zeta_{12}^{2} q^{69} + 1190 \zeta_{12}^{3} q^{70} + (55 \zeta_{12}^{3} - 55 \zeta_{12}) q^{71} + (208 \zeta_{12}^{3} - 208 \zeta_{12}) q^{72} - 838 \zeta_{12}^{3} q^{73} + 22 \zeta_{12}^{2} q^{74} + (164 \zeta_{12}^{2} - 164) q^{75} - 376 \zeta_{12} q^{76} + 70 q^{77} + 1016 q^{79} + 272 \zeta_{12} q^{80} + (649 \zeta_{12}^{2} - 649) q^{81} - 560 \zeta_{12}^{2} q^{82} - 420 \zeta_{12}^{3} q^{83} + (140 \zeta_{12}^{3} - 140 \zeta_{12}) q^{84} + (323 \zeta_{12}^{3} - 323 \zeta_{12}) q^{85} + 482 \zeta_{12}^{3} q^{86} + 246 \zeta_{12}^{2} q^{87} + ( - 16 \zeta_{12}^{2} + 16) q^{88} + 934 \zeta_{12} q^{89} + 884 q^{90} - 288 q^{92} + 100 \zeta_{12} q^{93} + (274 \zeta_{12}^{2} - 274) q^{94} + 1598 \zeta_{12}^{2} q^{95} + 32 \zeta_{12}^{3} q^{96} + ( - 1154 \zeta_{12}^{3} + 1154 \zeta_{12}) q^{97} + ( - 1764 \zeta_{12}^{3} + 1764 \zeta_{12}) q^{98} - 52 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 8 q^{4} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 8 q^{4} + 52 q^{9} + 68 q^{10} + 16 q^{12} - 280 q^{14} - 32 q^{16} - 38 q^{17} - 8 q^{22} - 144 q^{23} - 656 q^{25} + 212 q^{27} - 492 q^{29} - 68 q^{30} + 1190 q^{35} - 208 q^{36} - 752 q^{38} + 544 q^{40} - 140 q^{42} + 482 q^{43} + 32 q^{48} + 1764 q^{49} - 76 q^{51} - 928 q^{53} - 68 q^{55} - 560 q^{56} - 128 q^{61} - 400 q^{62} - 256 q^{64} - 16 q^{66} + 152 q^{68} + 144 q^{69} + 44 q^{74} - 328 q^{75} + 280 q^{77} + 4064 q^{79} - 1298 q^{81} - 1120 q^{82} + 492 q^{87} + 32 q^{88} + 3536 q^{90} - 1152 q^{92} - 548 q^{94} + 3196 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−1.73205 + 1.00000i 0.500000 + 0.866025i 2.00000 − 3.46410i − 17.0000i −1.73205 − 1.00000i 30.3109 + 17.5000i 8.00000i 13.0000 − 22.5167i 17.0000 + 29.4449i
23.2 1.73205 − 1.00000i 0.500000 + 0.866025i 2.00000 − 3.46410i 17.0000i 1.73205 + 1.00000i −30.3109 − 17.5000i − 8.00000i 13.0000 − 22.5167i 17.0000 + 29.4449i
147.1 −1.73205 − 1.00000i 0.500000 − 0.866025i 2.00000 + 3.46410i 17.0000i −1.73205 + 1.00000i 30.3109 − 17.5000i − 8.00000i 13.0000 + 22.5167i 17.0000 − 29.4449i
147.2 1.73205 + 1.00000i 0.500000 − 0.866025i 2.00000 + 3.46410i − 17.0000i 1.73205 − 1.00000i −30.3109 + 17.5000i 8.00000i 13.0000 + 22.5167i 17.0000 − 29.4449i
\(n\): 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
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.c 4
13.b even 2 1 inner 338.4.e.c 4
13.c even 3 1 338.4.b.b 2
13.c even 3 1 inner 338.4.e.c 4
13.d odd 4 1 338.4.c.c 2
13.d odd 4 1 338.4.c.g 2
13.e even 6 1 338.4.b.b 2
13.e even 6 1 inner 338.4.e.c 4
13.f odd 12 1 26.4.a.b ✓ 1
13.f odd 12 1 338.4.a.b 1
13.f odd 12 1 338.4.c.c 2
13.f odd 12 1 338.4.c.g 2
39.k even 12 1 234.4.a.a 1
52.l even 12 1 208.4.a.e 1
65.o even 12 1 650.4.b.d 2
65.s odd 12 1 650.4.a.c 1
65.t even 12 1 650.4.b.d 2
91.bc even 12 1 1274.4.a.f 1
104.u even 12 1 832.4.a.g 1
104.x odd 12 1 832.4.a.j 1
156.v odd 12 1 1872.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b ✓ 1 13.f odd 12 1
208.4.a.e 1 52.l even 12 1
234.4.a.a 1 39.k even 12 1
338.4.a.b 1 13.f odd 12 1
338.4.b.b 2 13.c even 3 1
338.4.b.b 2 13.e even 6 1
338.4.c.c 2 13.d odd 4 1
338.4.c.c 2 13.f odd 12 1
338.4.c.g 2 13.d odd 4 1
338.4.c.g 2 13.f odd 12 1
338.4.e.c 4 1.a even 1 1 trivial
338.4.e.c 4 13.b even 2 1 inner
338.4.e.c 4 13.c even 3 1 inner
338.4.e.c 4 13.e even 6 1 inner
650.4.a.c 1 65.s odd 12 1
650.4.b.d 2 65.o even 12 1
650.4.b.d 2 65.t even 12 1
832.4.a.g 1 104.u even 12 1
832.4.a.j 1 104.x odd 12 1
1274.4.a.f 1 91.bc even 12 1
1872.4.a.b 1 156.v odd 12 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 1225 T^{2} + 1500625 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 19 T + 361)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 8836 T^{2} + 78074896 \) Copy content Toggle raw display
$23$ \( (T^{2} + 72 T + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 246 T + 60516)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10000)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 6146560000 \) Copy content Toggle raw display
$43$ \( (T^{2} - 241 T + 58081)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 18769)^{2} \) Copy content Toggle raw display
$53$ \( (T + 232)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 22199808016 \) Copy content Toggle raw display
$61$ \( (T^{2} + 64 T + 4096)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 201511210000 \) Copy content Toggle raw display
$71$ \( T^{4} - 3025 T^{2} + 9150625 \) Copy content Toggle raw display
$73$ \( (T^{2} + 702244)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1016)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 176400)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 761004990736 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1773467504656 \) Copy content Toggle raw display
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