Properties

Label 338.4.e.d
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( - 3 \zeta_{12}^{2} + 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 2 \zeta_{12}^{3} q^{5} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{6} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{2} + ( - 3 \zeta_{12}^{2} + 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 2 \zeta_{12}^{3} q^{5} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{6} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9} + (4 \zeta_{12}^{2} - 4) q^{10} + 13 \zeta_{12} q^{11} + 12 q^{12} + 10 q^{14} + 6 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} + 27 \zeta_{12}^{2} q^{17} + 36 \zeta_{12}^{3} q^{18} + ( - 75 \zeta_{12}^{3} + 75 \zeta_{12}) q^{19} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{20} - 15 \zeta_{12}^{3} q^{21} + 26 \zeta_{12}^{2} q^{22} + (187 \zeta_{12}^{2} - 187) q^{23} + 24 \zeta_{12} q^{24} + 121 q^{25} + 135 q^{27} + 20 \zeta_{12} q^{28} + ( - 13 \zeta_{12}^{2} + 13) q^{29} + 12 \zeta_{12}^{2} q^{30} - 104 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + ( - 39 \zeta_{12}^{3} + 39 \zeta_{12}) q^{33} + 54 \zeta_{12}^{3} q^{34} + 10 \zeta_{12}^{2} q^{35} + (72 \zeta_{12}^{2} - 72) q^{36} + 423 \zeta_{12} q^{37} + 150 q^{38} - 16 q^{40} - 195 \zeta_{12} q^{41} + ( - 30 \zeta_{12}^{2} + 30) q^{42} + 199 \zeta_{12}^{2} q^{43} + 52 \zeta_{12}^{3} q^{44} + (36 \zeta_{12}^{3} - 36 \zeta_{12}) q^{45} + (374 \zeta_{12}^{3} - 374 \zeta_{12}) q^{46} - 388 \zeta_{12}^{3} q^{47} + 48 \zeta_{12}^{2} q^{48} + (318 \zeta_{12}^{2} - 318) q^{49} + 242 \zeta_{12} q^{50} + 81 q^{51} + 618 q^{53} + 270 \zeta_{12} q^{54} + (26 \zeta_{12}^{2} - 26) q^{55} + 40 \zeta_{12}^{2} q^{56} - 225 \zeta_{12}^{3} q^{57} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}) q^{58} + (491 \zeta_{12}^{3} - 491 \zeta_{12}) q^{59} + 24 \zeta_{12}^{3} q^{60} - 175 \zeta_{12}^{2} q^{61} + ( - 208 \zeta_{12}^{2} + 208) q^{62} + 90 \zeta_{12} q^{63} - 64 q^{64} + 78 q^{66} - 817 \zeta_{12} q^{67} + (108 \zeta_{12}^{2} - 108) q^{68} + 561 \zeta_{12}^{2} q^{69} + 20 \zeta_{12}^{3} q^{70} + ( - 79 \zeta_{12}^{3} + 79 \zeta_{12}) q^{71} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{72} - 230 \zeta_{12}^{3} q^{73} + 846 \zeta_{12}^{2} q^{74} + ( - 363 \zeta_{12}^{2} + 363) q^{75} + 300 \zeta_{12} q^{76} + 65 q^{77} + 764 q^{79} - 32 \zeta_{12} q^{80} + (81 \zeta_{12}^{2} - 81) q^{81} - 390 \zeta_{12}^{2} q^{82} - 732 \zeta_{12}^{3} q^{83} + ( - 60 \zeta_{12}^{3} + 60 \zeta_{12}) q^{84} + (54 \zeta_{12}^{3} - 54 \zeta_{12}) q^{85} + 398 \zeta_{12}^{3} q^{86} - 39 \zeta_{12}^{2} q^{87} + (104 \zeta_{12}^{2} - 104) q^{88} - 1041 \zeta_{12} q^{89} - 72 q^{90} - 748 q^{92} - 312 \zeta_{12} q^{93} + ( - 776 \zeta_{12}^{2} + 776) q^{94} + 150 \zeta_{12}^{2} q^{95} + 96 \zeta_{12}^{3} q^{96} + (97 \zeta_{12}^{3} - 97 \zeta_{12}) q^{97} + (636 \zeta_{12}^{3} - 636 \zeta_{12}) q^{98} + 234 \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 \( 4 q + 6 q^{3} + 8 q^{4} + 36 q^{9} - 8 q^{10} + 48 q^{12} + 40 q^{14} - 32 q^{16} + 54 q^{17} + 52 q^{22} - 374 q^{23} + 484 q^{25} + 540 q^{27} + 26 q^{29} + 24 q^{30} + 20 q^{35} - 144 q^{36} + 600 q^{38} - 64 q^{40} + 60 q^{42} + 398 q^{43} + 96 q^{48} - 636 q^{49} + 324 q^{51} + 2472 q^{53} - 52 q^{55} + 80 q^{56} - 350 q^{61} + 416 q^{62} - 256 q^{64} + 312 q^{66} - 216 q^{68} + 1122 q^{69} + 1692 q^{74} + 726 q^{75} + 260 q^{77} + 3056 q^{79} - 162 q^{81} - 780 q^{82} - 78 q^{87} - 208 q^{88} - 288 q^{90} - 2992 q^{92} + 1552 q^{94} + 300 q^{95}+O(q^{100}) \) 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 2.00000i −5.19615 3.00000i −4.33013 2.50000i 8.00000i 9.00000 15.5885i −2.00000 3.46410i
23.2 1.73205 1.00000i 1.50000 + 2.59808i 2.00000 3.46410i 2.00000i 5.19615 + 3.00000i 4.33013 + 2.50000i 8.00000i 9.00000 15.5885i −2.00000 3.46410i
147.1 −1.73205 1.00000i 1.50000 2.59808i 2.00000 + 3.46410i 2.00000i −5.19615 + 3.00000i −4.33013 + 2.50000i 8.00000i 9.00000 + 15.5885i −2.00000 + 3.46410i
147.2 1.73205 + 1.00000i 1.50000 2.59808i 2.00000 + 3.46410i 2.00000i 5.19615 3.00000i 4.33013 2.50000i 8.00000i 9.00000 + 15.5885i −2.00000 + 3.46410i
\(n\): 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.d 4
13.b even 2 1 inner 338.4.e.d 4
13.c even 3 1 338.4.b.a 2
13.c even 3 1 inner 338.4.e.d 4
13.d odd 4 1 26.4.c.a 2
13.d odd 4 1 338.4.c.d 2
13.e even 6 1 338.4.b.a 2
13.e even 6 1 inner 338.4.e.d 4
13.f odd 12 1 26.4.c.a 2
13.f odd 12 1 338.4.a.a 1
13.f odd 12 1 338.4.a.d 1
13.f odd 12 1 338.4.c.d 2
39.f even 4 1 234.4.h.b 2
39.k even 12 1 234.4.h.b 2
52.f even 4 1 208.4.i.a 2
52.l even 12 1 208.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.d odd 4 1
26.4.c.a 2 13.f odd 12 1
208.4.i.a 2 52.f even 4 1
208.4.i.a 2 52.l even 12 1
234.4.h.b 2 39.f even 4 1
234.4.h.b 2 39.k even 12 1
338.4.a.a 1 13.f odd 12 1
338.4.a.d 1 13.f odd 12 1
338.4.b.a 2 13.c even 3 1
338.4.b.a 2 13.e even 6 1
338.4.c.d 2 13.d odd 4 1
338.4.c.d 2 13.f odd 12 1
338.4.e.d 4 1.a even 1 1 trivial
338.4.e.d 4 13.b even 2 1 inner
338.4.e.d 4 13.c even 3 1 inner
338.4.e.d 4 13.e even 6 1 inner

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
\( T_{5}^{2} + 4 \) 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
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$11$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27 T + 729)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 5625 T^{2} + 31640625 \) Copy content Toggle raw display
$23$ \( (T^{2} + 187 T + 34969)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10816)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 32015587041 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 1445900625 \) Copy content Toggle raw display
$43$ \( (T^{2} - 199 T + 39601)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 150544)^{2} \) Copy content Toggle raw display
$53$ \( (T - 618)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 58120048561 \) Copy content Toggle raw display
$61$ \( (T^{2} + 175 T + 30625)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 445541565121 \) Copy content Toggle raw display
$71$ \( T^{4} - 6241 T^{2} + 38950081 \) Copy content Toggle raw display
$73$ \( (T^{2} + 52900)^{2} \) Copy content Toggle raw display
$79$ \( (T - 764)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 535824)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 1174364509761 \) Copy content Toggle raw display
$97$ \( T^{4} - 9409 T^{2} + 88529281 \) Copy content Toggle raw display
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