Properties

Label 338.4.e.b
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + (3 \zeta_{12}^{2} - 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 11 \zeta_{12}^{3} q^{5} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{6} + (19 \zeta_{12}^{3} - 19 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{2} + (3 \zeta_{12}^{2} - 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 11 \zeta_{12}^{3} q^{5} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{6} + (19 \zeta_{12}^{3} - 19 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9} + (22 \zeta_{12}^{2} - 22) q^{10} - 38 \zeta_{12} q^{11} - 12 q^{12} - 38 q^{14} - 33 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} - 51 \zeta_{12}^{2} q^{17} + 36 \zeta_{12}^{3} q^{18} + ( - 90 \zeta_{12}^{3} + 90 \zeta_{12}) q^{19} + (44 \zeta_{12}^{3} - 44 \zeta_{12}) q^{20} - 57 \zeta_{12}^{3} q^{21} - 76 \zeta_{12}^{2} q^{22} + (52 \zeta_{12}^{2} - 52) q^{23} - 24 \zeta_{12} q^{24} + 4 q^{25} - 135 q^{27} - 76 \zeta_{12} q^{28} + ( - 190 \zeta_{12}^{2} + 190) q^{29} - 66 \zeta_{12}^{2} q^{30} + 292 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + ( - 114 \zeta_{12}^{3} + 114 \zeta_{12}) q^{33} - 102 \zeta_{12}^{3} q^{34} - 209 \zeta_{12}^{2} q^{35} + (72 \zeta_{12}^{2} - 72) q^{36} - 441 \zeta_{12} q^{37} + 180 q^{38} - 88 q^{40} - 312 \zeta_{12} q^{41} + ( - 114 \zeta_{12}^{2} + 114) q^{42} + 373 \zeta_{12}^{2} q^{43} - 152 \zeta_{12}^{3} q^{44} + (198 \zeta_{12}^{3} - 198 \zeta_{12}) q^{45} + (104 \zeta_{12}^{3} - 104 \zeta_{12}) q^{46} + 41 \zeta_{12}^{3} q^{47} - 48 \zeta_{12}^{2} q^{48} + ( - 18 \zeta_{12}^{2} + 18) q^{49} + 8 \zeta_{12} q^{50} + 153 q^{51} + 468 q^{53} - 270 \zeta_{12} q^{54} + ( - 418 \zeta_{12}^{2} + 418) q^{55} - 152 \zeta_{12}^{2} q^{56} + 270 \zeta_{12}^{3} q^{57} + ( - 380 \zeta_{12}^{3} + 380 \zeta_{12}) q^{58} + (530 \zeta_{12}^{3} - 530 \zeta_{12}) q^{59} - 132 \zeta_{12}^{3} q^{60} - 592 \zeta_{12}^{2} q^{61} + (584 \zeta_{12}^{2} - 584) q^{62} - 342 \zeta_{12} q^{63} - 64 q^{64} + 228 q^{66} + 206 \zeta_{12} q^{67} + ( - 204 \zeta_{12}^{2} + 204) q^{68} - 156 \zeta_{12}^{2} q^{69} - 418 \zeta_{12}^{3} q^{70} + (863 \zeta_{12}^{3} - 863 \zeta_{12}) q^{71} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{72} + 322 \zeta_{12}^{3} q^{73} - 882 \zeta_{12}^{2} q^{74} + (12 \zeta_{12}^{2} - 12) q^{75} + 360 \zeta_{12} q^{76} + 722 q^{77} - 460 q^{79} - 176 \zeta_{12} q^{80} + (81 \zeta_{12}^{2} - 81) q^{81} - 624 \zeta_{12}^{2} q^{82} + 528 \zeta_{12}^{3} q^{83} + ( - 228 \zeta_{12}^{3} + 228 \zeta_{12}) q^{84} + ( - 561 \zeta_{12}^{3} + 561 \zeta_{12}) q^{85} + 746 \zeta_{12}^{3} q^{86} + 570 \zeta_{12}^{2} q^{87} + ( - 304 \zeta_{12}^{2} + 304) q^{88} + 870 \zeta_{12} q^{89} - 396 q^{90} - 208 q^{92} - 876 \zeta_{12} q^{93} + (82 \zeta_{12}^{2} - 82) q^{94} + 990 \zeta_{12}^{2} q^{95} - 96 \zeta_{12}^{3} q^{96} + (346 \zeta_{12}^{3} - 346 \zeta_{12}) q^{97} + ( - 36 \zeta_{12}^{3} + 36 \zeta_{12}) q^{98} - 684 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 8 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 8 q^{4} + 36 q^{9} - 44 q^{10} - 48 q^{12} - 152 q^{14} - 32 q^{16} - 102 q^{17} - 152 q^{22} - 104 q^{23} + 16 q^{25} - 540 q^{27} + 380 q^{29} - 132 q^{30} - 418 q^{35} - 144 q^{36} + 720 q^{38} - 352 q^{40} + 228 q^{42} + 746 q^{43} - 96 q^{48} + 36 q^{49} + 612 q^{51} + 1872 q^{53} + 836 q^{55} - 304 q^{56} - 1184 q^{61} - 1168 q^{62} - 256 q^{64} + 912 q^{66} + 408 q^{68} - 312 q^{69} - 1764 q^{74} - 24 q^{75} + 2888 q^{77} - 1840 q^{79} - 162 q^{81} - 1248 q^{82} + 1140 q^{87} + 608 q^{88} - 1584 q^{90} - 832 q^{92} - 164 q^{94} + 1980 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 −1.50000 2.59808i 2.00000 3.46410i 11.0000i 5.19615 + 3.00000i 16.4545 + 9.50000i 8.00000i 9.00000 15.5885i −11.0000 19.0526i
23.2 1.73205 1.00000i −1.50000 2.59808i 2.00000 3.46410i 11.0000i −5.19615 3.00000i −16.4545 9.50000i 8.00000i 9.00000 15.5885i −11.0000 19.0526i
147.1 −1.73205 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 11.0000i 5.19615 3.00000i 16.4545 9.50000i 8.00000i 9.00000 + 15.5885i −11.0000 + 19.0526i
147.2 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 11.0000i −5.19615 + 3.00000i −16.4545 + 9.50000i 8.00000i 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.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.b 4
13.b even 2 1 inner 338.4.e.b 4
13.c even 3 1 338.4.b.c 2
13.c even 3 1 inner 338.4.e.b 4
13.d odd 4 1 338.4.c.b 2
13.d odd 4 1 338.4.c.f 2
13.e even 6 1 338.4.b.c 2
13.e even 6 1 inner 338.4.e.b 4
13.f odd 12 1 26.4.a.a 1
13.f odd 12 1 338.4.a.e 1
13.f odd 12 1 338.4.c.b 2
13.f odd 12 1 338.4.c.f 2
39.k even 12 1 234.4.a.g 1
52.l even 12 1 208.4.a.c 1
65.o even 12 1 650.4.b.b 2
65.s odd 12 1 650.4.a.f 1
65.t even 12 1 650.4.b.b 2
91.bc even 12 1 1274.4.a.b 1
104.u even 12 1 832.4.a.m 1
104.x odd 12 1 832.4.a.e 1
156.v odd 12 1 1872.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.a 1 13.f odd 12 1
208.4.a.c 1 52.l even 12 1
234.4.a.g 1 39.k even 12 1
338.4.a.e 1 13.f odd 12 1
338.4.b.c 2 13.c even 3 1
338.4.b.c 2 13.e even 6 1
338.4.c.b 2 13.d odd 4 1
338.4.c.b 2 13.f odd 12 1
338.4.c.f 2 13.d odd 4 1
338.4.c.f 2 13.f odd 12 1
338.4.e.b 4 1.a even 1 1 trivial
338.4.e.b 4 13.b even 2 1 inner
338.4.e.b 4 13.c even 3 1 inner
338.4.e.b 4 13.e even 6 1 inner
650.4.a.f 1 65.s odd 12 1
650.4.b.b 2 65.o even 12 1
650.4.b.b 2 65.t even 12 1
832.4.a.e 1 104.x odd 12 1
832.4.a.m 1 104.u even 12 1
1274.4.a.b 1 91.bc even 12 1
1872.4.a.c 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 \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 361 T^{2} + 130321 \) Copy content Toggle raw display
$11$ \( T^{4} - 1444 T^{2} + 2085136 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 51 T + 2601)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 8100 T^{2} + 65610000 \) Copy content Toggle raw display
$23$ \( (T^{2} + 52 T + 2704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 190 T + 36100)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 85264)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 37822859361 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 9475854336 \) Copy content Toggle raw display
$43$ \( (T^{2} - 373 T + 139129)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
$53$ \( (T - 468)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 78904810000 \) Copy content Toggle raw display
$61$ \( (T^{2} + 592 T + 350464)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 1800814096 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 554680863361 \) Copy content Toggle raw display
$73$ \( (T^{2} + 103684)^{2} \) Copy content Toggle raw display
$79$ \( (T + 460)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 278784)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 572897610000 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 14331920656 \) Copy content Toggle raw display
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