Properties

Label 338.4.c.a
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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + (4 \zeta_{6} - 4) q^{3} - 4 \zeta_{6} q^{4} - 18 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 8 q^{8} + 11 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + (4 \zeta_{6} - 4) q^{3} - 4 \zeta_{6} q^{4} - 18 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 8 q^{8} + 11 \zeta_{6} q^{9} + ( - 36 \zeta_{6} + 36) q^{10} + ( - 48 \zeta_{6} + 48) q^{11} + 16 q^{12} + 40 q^{14} + ( - 72 \zeta_{6} + 72) q^{15} + (16 \zeta_{6} - 16) q^{16} - 66 \zeta_{6} q^{17} - 22 q^{18} + 16 \zeta_{6} q^{19} + 72 \zeta_{6} q^{20} + 80 q^{21} + 96 \zeta_{6} q^{22} + (168 \zeta_{6} - 168) q^{23} + (32 \zeta_{6} - 32) q^{24} + 199 q^{25} - 152 q^{27} + (80 \zeta_{6} - 80) q^{28} + (6 \zeta_{6} - 6) q^{29} + 144 \zeta_{6} q^{30} + 20 q^{31} - 32 \zeta_{6} q^{32} + 192 \zeta_{6} q^{33} + 132 q^{34} + 360 \zeta_{6} q^{35} + ( - 44 \zeta_{6} + 44) q^{36} + (254 \zeta_{6} - 254) q^{37} - 32 q^{38} - 144 q^{40} + ( - 390 \zeta_{6} + 390) q^{41} + (160 \zeta_{6} - 160) q^{42} + 124 \zeta_{6} q^{43} - 192 q^{44} - 198 \zeta_{6} q^{45} - 336 \zeta_{6} q^{46} - 468 q^{47} - 64 \zeta_{6} q^{48} + (57 \zeta_{6} - 57) q^{49} + (398 \zeta_{6} - 398) q^{50} + 264 q^{51} + 558 q^{53} + ( - 304 \zeta_{6} + 304) q^{54} + (864 \zeta_{6} - 864) q^{55} - 160 \zeta_{6} q^{56} - 64 q^{57} - 12 \zeta_{6} q^{58} + 96 \zeta_{6} q^{59} - 288 q^{60} + 826 \zeta_{6} q^{61} + (40 \zeta_{6} - 40) q^{62} + ( - 220 \zeta_{6} + 220) q^{63} + 64 q^{64} - 384 q^{66} + ( - 160 \zeta_{6} + 160) q^{67} + (264 \zeta_{6} - 264) q^{68} - 672 \zeta_{6} q^{69} - 720 q^{70} + 420 \zeta_{6} q^{71} + 88 \zeta_{6} q^{72} + 362 q^{73} - 508 \zeta_{6} q^{74} + (796 \zeta_{6} - 796) q^{75} + ( - 64 \zeta_{6} + 64) q^{76} - 960 q^{77} + 776 q^{79} + ( - 288 \zeta_{6} + 288) q^{80} + ( - 311 \zeta_{6} + 311) q^{81} + 780 \zeta_{6} q^{82} - 320 \zeta_{6} q^{84} + 1188 \zeta_{6} q^{85} - 248 q^{86} - 24 \zeta_{6} q^{87} + ( - 384 \zeta_{6} + 384) q^{88} + (1626 \zeta_{6} - 1626) q^{89} + 396 q^{90} + 672 q^{92} + (80 \zeta_{6} - 80) q^{93} + ( - 936 \zeta_{6} + 936) q^{94} - 288 \zeta_{6} q^{95} + 128 q^{96} + 1294 \zeta_{6} q^{97} - 114 \zeta_{6} q^{98} + 528 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 36 q^{5} - 8 q^{6} - 20 q^{7} + 16 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 36 q^{5} - 8 q^{6} - 20 q^{7} + 16 q^{8} + 11 q^{9} + 36 q^{10} + 48 q^{11} + 32 q^{12} + 80 q^{14} + 72 q^{15} - 16 q^{16} - 66 q^{17} - 44 q^{18} + 16 q^{19} + 72 q^{20} + 160 q^{21} + 96 q^{22} - 168 q^{23} - 32 q^{24} + 398 q^{25} - 304 q^{27} - 80 q^{28} - 6 q^{29} + 144 q^{30} + 40 q^{31} - 32 q^{32} + 192 q^{33} + 264 q^{34} + 360 q^{35} + 44 q^{36} - 254 q^{37} - 64 q^{38} - 288 q^{40} + 390 q^{41} - 160 q^{42} + 124 q^{43} - 384 q^{44} - 198 q^{45} - 336 q^{46} - 936 q^{47} - 64 q^{48} - 57 q^{49} - 398 q^{50} + 528 q^{51} + 1116 q^{53} + 304 q^{54} - 864 q^{55} - 160 q^{56} - 128 q^{57} - 12 q^{58} + 96 q^{59} - 576 q^{60} + 826 q^{61} - 40 q^{62} + 220 q^{63} + 128 q^{64} - 768 q^{66} + 160 q^{67} - 264 q^{68} - 672 q^{69} - 1440 q^{70} + 420 q^{71} + 88 q^{72} + 724 q^{73} - 508 q^{74} - 796 q^{75} + 64 q^{76} - 1920 q^{77} + 1552 q^{79} + 288 q^{80} + 311 q^{81} + 780 q^{82} - 320 q^{84} + 1188 q^{85} - 496 q^{86} - 24 q^{87} + 384 q^{88} - 1626 q^{89} + 792 q^{90} + 1344 q^{92} - 80 q^{93} + 936 q^{94} - 288 q^{95} + 256 q^{96} + 1294 q^{97} - 114 q^{98} + 1056 q^{99}+O(q^{100}) \) 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 −2.00000 + 3.46410i −2.00000 3.46410i −18.0000 −4.00000 6.92820i −10.0000 17.3205i 8.00000 5.50000 + 9.52628i 18.0000 31.1769i
315.1 −1.00000 1.73205i −2.00000 3.46410i −2.00000 + 3.46410i −18.0000 −4.00000 + 6.92820i −10.0000 + 17.3205i 8.00000 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.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.a 2
13.b even 2 1 338.4.c.e 2
13.c even 3 1 26.4.a.c 1
13.c even 3 1 inner 338.4.c.a 2
13.d odd 4 2 338.4.e.a 4
13.e even 6 1 338.4.a.c 1
13.e even 6 1 338.4.c.e 2
13.f odd 12 2 338.4.b.d 2
13.f odd 12 2 338.4.e.a 4
39.i odd 6 1 234.4.a.e 1
52.j odd 6 1 208.4.a.b 1
65.n even 6 1 650.4.a.b 1
65.q odd 12 2 650.4.b.f 2
91.n odd 6 1 1274.4.a.d 1
104.n odd 6 1 832.4.a.o 1
104.r even 6 1 832.4.a.d 1
156.p even 6 1 1872.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 13.c even 3 1
208.4.a.b 1 52.j odd 6 1
234.4.a.e 1 39.i odd 6 1
338.4.a.c 1 13.e even 6 1
338.4.b.d 2 13.f odd 12 2
338.4.c.a 2 1.a even 1 1 trivial
338.4.c.a 2 13.c even 3 1 inner
338.4.c.e 2 13.b even 2 1
338.4.c.e 2 13.e even 6 1
338.4.e.a 4 13.d odd 4 2
338.4.e.a 4 13.f odd 12 2
650.4.a.b 1 65.n even 6 1
650.4.b.f 2 65.q odd 12 2
832.4.a.d 1 104.r even 6 1
832.4.a.o 1 104.n odd 6 1
1274.4.a.d 1 91.n odd 6 1
1872.4.a.q 1 156.p even 6 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
\( T_{5} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$5$ \( (T + 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$11$ \( T^{2} - 48T + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 66T + 4356 \) Copy content Toggle raw display
$19$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$23$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 254T + 64516 \) Copy content Toggle raw display
$41$ \( T^{2} - 390T + 152100 \) Copy content Toggle raw display
$43$ \( T^{2} - 124T + 15376 \) Copy content Toggle raw display
$47$ \( (T + 468)^{2} \) Copy content Toggle raw display
$53$ \( (T - 558)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$61$ \( T^{2} - 826T + 682276 \) Copy content Toggle raw display
$67$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$71$ \( T^{2} - 420T + 176400 \) Copy content Toggle raw display
$73$ \( (T - 362)^{2} \) Copy content Toggle raw display
$79$ \( (T - 776)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1626 T + 2643876 \) Copy content Toggle raw display
$97$ \( T^{2} - 1294 T + 1674436 \) Copy content Toggle raw display
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