Properties

Label 338.4.c.b
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} - 11 q^{5} - 6 \zeta_{6} q^{6} + 19 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} - 11 q^{5} - 6 \zeta_{6} q^{6} + 19 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 22 \zeta_{6} + 22) q^{10} + (38 \zeta_{6} - 38) q^{11} + 12 q^{12} - 38 q^{14} + ( - 33 \zeta_{6} + 33) q^{15} + (16 \zeta_{6} - 16) q^{16} + 51 \zeta_{6} q^{17} - 36 q^{18} + 90 \zeta_{6} q^{19} + 44 \zeta_{6} q^{20} - 57 q^{21} - 76 \zeta_{6} q^{22} + ( - 52 \zeta_{6} + 52) q^{23} + (24 \zeta_{6} - 24) q^{24} - 4 q^{25} - 135 q^{27} + ( - 76 \zeta_{6} + 76) q^{28} + ( - 190 \zeta_{6} + 190) q^{29} + 66 \zeta_{6} q^{30} - 292 q^{31} - 32 \zeta_{6} q^{32} - 114 \zeta_{6} q^{33} - 102 q^{34} - 209 \zeta_{6} q^{35} + ( - 72 \zeta_{6} + 72) q^{36} + (441 \zeta_{6} - 441) q^{37} - 180 q^{38} - 88 q^{40} + ( - 312 \zeta_{6} + 312) q^{41} + ( - 114 \zeta_{6} + 114) q^{42} - 373 \zeta_{6} q^{43} + 152 q^{44} - 198 \zeta_{6} q^{45} + 104 \zeta_{6} q^{46} + 41 q^{47} - 48 \zeta_{6} q^{48} + (18 \zeta_{6} - 18) q^{49} + ( - 8 \zeta_{6} + 8) q^{50} - 153 q^{51} + 468 q^{53} + ( - 270 \zeta_{6} + 270) q^{54} + ( - 418 \zeta_{6} + 418) q^{55} + 152 \zeta_{6} q^{56} - 270 q^{57} + 380 \zeta_{6} q^{58} + 530 \zeta_{6} q^{59} - 132 q^{60} - 592 \zeta_{6} q^{61} + ( - 584 \zeta_{6} + 584) q^{62} + (342 \zeta_{6} - 342) q^{63} + 64 q^{64} + 228 q^{66} + (206 \zeta_{6} - 206) q^{67} + ( - 204 \zeta_{6} + 204) q^{68} + 156 \zeta_{6} q^{69} + 418 q^{70} - 863 \zeta_{6} q^{71} + 144 \zeta_{6} q^{72} + 322 q^{73} - 882 \zeta_{6} q^{74} + ( - 12 \zeta_{6} + 12) q^{75} + ( - 360 \zeta_{6} + 360) q^{76} - 722 q^{77} - 460 q^{79} + ( - 176 \zeta_{6} + 176) q^{80} + (81 \zeta_{6} - 81) q^{81} + 624 \zeta_{6} q^{82} - 528 q^{83} + 228 \zeta_{6} q^{84} - 561 \zeta_{6} q^{85} + 746 q^{86} + 570 \zeta_{6} q^{87} + (304 \zeta_{6} - 304) q^{88} + ( - 870 \zeta_{6} + 870) q^{89} + 396 q^{90} - 208 q^{92} + ( - 876 \zeta_{6} + 876) q^{93} + (82 \zeta_{6} - 82) q^{94} - 990 \zeta_{6} q^{95} + 96 q^{96} - 346 \zeta_{6} q^{97} - 36 \zeta_{6} q^{98} - 684 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} - 22 q^{5} - 6 q^{6} + 19 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} - 22 q^{5} - 6 q^{6} + 19 q^{7} + 16 q^{8} + 18 q^{9} + 22 q^{10} - 38 q^{11} + 24 q^{12} - 76 q^{14} + 33 q^{15} - 16 q^{16} + 51 q^{17} - 72 q^{18} + 90 q^{19} + 44 q^{20} - 114 q^{21} - 76 q^{22} + 52 q^{23} - 24 q^{24} - 8 q^{25} - 270 q^{27} + 76 q^{28} + 190 q^{29} + 66 q^{30} - 584 q^{31} - 32 q^{32} - 114 q^{33} - 204 q^{34} - 209 q^{35} + 72 q^{36} - 441 q^{37} - 360 q^{38} - 176 q^{40} + 312 q^{41} + 114 q^{42} - 373 q^{43} + 304 q^{44} - 198 q^{45} + 104 q^{46} + 82 q^{47} - 48 q^{48} - 18 q^{49} + 8 q^{50} - 306 q^{51} + 936 q^{53} + 270 q^{54} + 418 q^{55} + 152 q^{56} - 540 q^{57} + 380 q^{58} + 530 q^{59} - 264 q^{60} - 592 q^{61} + 584 q^{62} - 342 q^{63} + 128 q^{64} + 456 q^{66} - 206 q^{67} + 204 q^{68} + 156 q^{69} + 836 q^{70} - 863 q^{71} + 144 q^{72} + 644 q^{73} - 882 q^{74} + 12 q^{75} + 360 q^{76} - 1444 q^{77} - 920 q^{79} + 176 q^{80} - 81 q^{81} + 624 q^{82} - 1056 q^{83} + 228 q^{84} - 561 q^{85} + 1492 q^{86} + 570 q^{87} - 304 q^{88} + 870 q^{89} + 792 q^{90} - 416 q^{92} + 876 q^{93} - 82 q^{94} - 990 q^{95} + 192 q^{96} - 346 q^{97} - 36 q^{98} - 1368 q^{99}+O(q^{100}) \) 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 −11.0000 −3.00000 5.19615i 9.50000 + 16.4545i 8.00000 9.00000 + 15.5885i 11.0000 19.0526i
315.1 −1.00000 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −11.0000 −3.00000 + 5.19615i 9.50000 16.4545i 8.00000 9.00000 15.5885i 11.0000 + 19.0526i
\(n\): 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.b 2
13.b even 2 1 338.4.c.f 2
13.c even 3 1 338.4.a.e 1
13.c even 3 1 inner 338.4.c.b 2
13.d odd 4 2 338.4.e.b 4
13.e even 6 1 26.4.a.a 1
13.e even 6 1 338.4.c.f 2
13.f odd 12 2 338.4.b.c 2
13.f odd 12 2 338.4.e.b 4
39.h odd 6 1 234.4.a.g 1
52.i odd 6 1 208.4.a.c 1
65.l even 6 1 650.4.a.f 1
65.r odd 12 2 650.4.b.b 2
91.t odd 6 1 1274.4.a.b 1
104.p odd 6 1 832.4.a.m 1
104.s even 6 1 832.4.a.e 1
156.r even 6 1 1872.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.a 1 13.e even 6 1
208.4.a.c 1 52.i odd 6 1
234.4.a.g 1 39.h odd 6 1
338.4.a.e 1 13.c even 3 1
338.4.b.c 2 13.f odd 12 2
338.4.c.b 2 1.a even 1 1 trivial
338.4.c.b 2 13.c even 3 1 inner
338.4.c.f 2 13.b even 2 1
338.4.c.f 2 13.e even 6 1
338.4.e.b 4 13.d odd 4 2
338.4.e.b 4 13.f odd 12 2
650.4.a.f 1 65.l even 6 1
650.4.b.b 2 65.r odd 12 2
832.4.a.e 1 104.s even 6 1
832.4.a.m 1 104.p odd 6 1
1274.4.a.b 1 91.t odd 6 1
1872.4.a.c 1 156.r even 6 1

Hecke kernels

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

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5} + 11 \) 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
$5$ \( (T + 11)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
$11$ \( T^{2} + 38T + 1444 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$23$ \( T^{2} - 52T + 2704 \) Copy content Toggle raw display
$29$ \( T^{2} - 190T + 36100 \) Copy content Toggle raw display
$31$ \( (T + 292)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 441T + 194481 \) Copy content Toggle raw display
$41$ \( T^{2} - 312T + 97344 \) Copy content Toggle raw display
$43$ \( T^{2} + 373T + 139129 \) Copy content Toggle raw display
$47$ \( (T - 41)^{2} \) Copy content Toggle raw display
$53$ \( (T - 468)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 530T + 280900 \) Copy content Toggle raw display
$61$ \( T^{2} + 592T + 350464 \) Copy content Toggle raw display
$67$ \( T^{2} + 206T + 42436 \) Copy content Toggle raw display
$71$ \( T^{2} + 863T + 744769 \) Copy content Toggle raw display
$73$ \( (T - 322)^{2} \) Copy content Toggle raw display
$79$ \( (T + 460)^{2} \) Copy content Toggle raw display
$83$ \( (T + 528)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 870T + 756900 \) Copy content Toggle raw display
$97$ \( T^{2} + 346T + 119716 \) Copy content Toggle raw display
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