Properties

Label 338.4.e.a
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (4 \beta_{2} - 4) q^{3} + 4 \beta_{2} q^{4} + 9 \beta_{3} q^{5} + (4 \beta_{3} - 4 \beta_1) q^{6} + ( - 10 \beta_{3} + 10 \beta_1) q^{7} + 4 \beta_{3} q^{8} + 11 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (4 \beta_{2} - 4) q^{3} + 4 \beta_{2} q^{4} + 9 \beta_{3} q^{5} + (4 \beta_{3} - 4 \beta_1) q^{6} + ( - 10 \beta_{3} + 10 \beta_1) q^{7} + 4 \beta_{3} q^{8} + 11 \beta_{2} q^{9} + (36 \beta_{2} - 36) q^{10} + 24 \beta_1 q^{11} - 16 q^{12} + 40 q^{14} - 36 \beta_1 q^{15} + (16 \beta_{2} - 16) q^{16} + 66 \beta_{2} q^{17} + 11 \beta_{3} q^{18} + ( - 8 \beta_{3} + 8 \beta_1) q^{19} + (36 \beta_{3} - 36 \beta_1) q^{20} + 40 \beta_{3} q^{21} + 96 \beta_{2} q^{22} + ( - 168 \beta_{2} + 168) q^{23} - 16 \beta_1 q^{24} - 199 q^{25} - 152 q^{27} + 40 \beta_1 q^{28} + (6 \beta_{2} - 6) q^{29} - 144 \beta_{2} q^{30} - 10 \beta_{3} q^{31} + (16 \beta_{3} - 16 \beta_1) q^{32} + (96 \beta_{3} - 96 \beta_1) q^{33} + 66 \beta_{3} q^{34} + 360 \beta_{2} q^{35} + (44 \beta_{2} - 44) q^{36} - 127 \beta_1 q^{37} + 32 q^{38} - 144 q^{40} - 195 \beta_1 q^{41} + (160 \beta_{2} - 160) q^{42} - 124 \beta_{2} q^{43} + 96 \beta_{3} q^{44} + (99 \beta_{3} - 99 \beta_1) q^{45} + ( - 168 \beta_{3} + 168 \beta_1) q^{46} - 234 \beta_{3} q^{47} - 64 \beta_{2} q^{48} + ( - 57 \beta_{2} + 57) q^{49} - 199 \beta_1 q^{50} - 264 q^{51} + 558 q^{53} - 152 \beta_1 q^{54} + (864 \beta_{2} - 864) q^{55} + 160 \beta_{2} q^{56} + 32 \beta_{3} q^{57} + (6 \beta_{3} - 6 \beta_1) q^{58} + (48 \beta_{3} - 48 \beta_1) q^{59} - 144 \beta_{3} q^{60} + 826 \beta_{2} q^{61} + ( - 40 \beta_{2} + 40) q^{62} + 110 \beta_1 q^{63} - 64 q^{64} - 384 q^{66} - 80 \beta_1 q^{67} + (264 \beta_{2} - 264) q^{68} + 672 \beta_{2} q^{69} + 360 \beta_{3} q^{70} + ( - 210 \beta_{3} + 210 \beta_1) q^{71} + (44 \beta_{3} - 44 \beta_1) q^{72} + 181 \beta_{3} q^{73} - 508 \beta_{2} q^{74} + ( - 796 \beta_{2} + 796) q^{75} + 32 \beta_1 q^{76} + 960 q^{77} + 776 q^{79} - 144 \beta_1 q^{80} + ( - 311 \beta_{2} + 311) q^{81} - 780 \beta_{2} q^{82} + (160 \beta_{3} - 160 \beta_1) q^{84} + (594 \beta_{3} - 594 \beta_1) q^{85} - 124 \beta_{3} q^{86} - 24 \beta_{2} q^{87} + (384 \beta_{2} - 384) q^{88} - 813 \beta_1 q^{89} - 396 q^{90} + 672 q^{92} + 40 \beta_1 q^{93} + ( - 936 \beta_{2} + 936) q^{94} + 288 \beta_{2} q^{95} - 64 \beta_{3} q^{96} + ( - 647 \beta_{3} + 647 \beta_1) q^{97} + ( - 57 \beta_{3} + 57 \beta_1) q^{98} + 264 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} + 8 q^{4} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} + 8 q^{4} + 22 q^{9} - 72 q^{10} - 64 q^{12} + 160 q^{14} - 32 q^{16} + 132 q^{17} + 192 q^{22} + 336 q^{23} - 796 q^{25} - 608 q^{27} - 12 q^{29} - 288 q^{30} + 720 q^{35} - 88 q^{36} + 128 q^{38} - 576 q^{40} - 320 q^{42} - 248 q^{43} - 128 q^{48} + 114 q^{49} - 1056 q^{51} + 2232 q^{53} - 1728 q^{55} + 320 q^{56} + 1652 q^{61} + 80 q^{62} - 256 q^{64} - 1536 q^{66} - 528 q^{68} + 1344 q^{69} - 1016 q^{74} + 1592 q^{75} + 3840 q^{77} + 3104 q^{79} + 622 q^{81} - 1560 q^{82} - 48 q^{87} - 768 q^{88} - 1584 q^{90} + 2688 q^{92} + 1872 q^{94} + 576 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\beta_{2}\)

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 −2.00000 3.46410i 2.00000 3.46410i 18.0000i 6.92820 + 4.00000i −17.3205 10.0000i 8.00000i 5.50000 9.52628i −18.0000 31.1769i
23.2 1.73205 1.00000i −2.00000 3.46410i 2.00000 3.46410i 18.0000i −6.92820 4.00000i 17.3205 + 10.0000i 8.00000i 5.50000 9.52628i −18.0000 31.1769i
147.1 −1.73205 1.00000i −2.00000 + 3.46410i 2.00000 + 3.46410i 18.0000i 6.92820 4.00000i −17.3205 + 10.0000i 8.00000i 5.50000 + 9.52628i −18.0000 + 31.1769i
147.2 1.73205 + 1.00000i −2.00000 + 3.46410i 2.00000 + 3.46410i 18.0000i −6.92820 + 4.00000i 17.3205 10.0000i 8.00000i 5.50000 + 9.52628i −18.0000 + 31.1769i
\(n\): 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.a 4
13.b even 2 1 inner 338.4.e.a 4
13.c even 3 1 338.4.b.d 2
13.c even 3 1 inner 338.4.e.a 4
13.d odd 4 1 338.4.c.a 2
13.d odd 4 1 338.4.c.e 2
13.e even 6 1 338.4.b.d 2
13.e even 6 1 inner 338.4.e.a 4
13.f odd 12 1 26.4.a.c 1
13.f odd 12 1 338.4.a.c 1
13.f odd 12 1 338.4.c.a 2
13.f odd 12 1 338.4.c.e 2
39.k even 12 1 234.4.a.e 1
52.l even 12 1 208.4.a.b 1
65.o even 12 1 650.4.b.f 2
65.s odd 12 1 650.4.a.b 1
65.t even 12 1 650.4.b.f 2
91.bc even 12 1 1274.4.a.d 1
104.u even 12 1 832.4.a.o 1
104.x odd 12 1 832.4.a.d 1
156.v odd 12 1 1872.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 13.f odd 12 1
208.4.a.b 1 52.l even 12 1
234.4.a.e 1 39.k even 12 1
338.4.a.c 1 13.f odd 12 1
338.4.b.d 2 13.c even 3 1
338.4.b.d 2 13.e even 6 1
338.4.c.a 2 13.d odd 4 1
338.4.c.a 2 13.f odd 12 1
338.4.c.e 2 13.d odd 4 1
338.4.c.e 2 13.f odd 12 1
338.4.e.a 4 1.a even 1 1 trivial
338.4.e.a 4 13.b even 2 1 inner
338.4.e.a 4 13.c even 3 1 inner
338.4.e.a 4 13.e even 6 1 inner
650.4.a.b 1 65.s odd 12 1
650.4.b.f 2 65.o even 12 1
650.4.b.f 2 65.t even 12 1
832.4.a.d 1 104.x odd 12 1
832.4.a.o 1 104.u even 12 1
1274.4.a.d 1 91.bc even 12 1
1872.4.a.q 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 \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{2} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 400 T^{2} + 160000 \) Copy content Toggle raw display
$11$ \( T^{4} - 2304 T^{2} + 5308416 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 66 T + 4356)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$23$ \( (T^{2} - 168 T + 28224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 4162314256 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 23134410000 \) Copy content Toggle raw display
$43$ \( (T^{2} + 124 T + 15376)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 219024)^{2} \) Copy content Toggle raw display
$53$ \( (T - 558)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 9216 T^{2} + 84934656 \) Copy content Toggle raw display
$61$ \( (T^{2} - 826 T + 682276)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25600 T^{2} + 655360000 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 31116960000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 131044)^{2} \) Copy content Toggle raw display
$79$ \( (T - 776)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 6990080303376 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 2803735918096 \) Copy content Toggle raw display
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