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Properties

Label 336.4.q.d
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,4,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (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: \(19.8246417619\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + 15 \zeta_{6} q^{5} + (7 \zeta_{6} - 21) q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + 15 \zeta_{6} q^{5} + (7 \zeta_{6} - 21) q^{7} - 9 \zeta_{6} q^{9} + (9 \zeta_{6} - 9) q^{11} - 88 q^{13} - 45 q^{15} + ( - 84 \zeta_{6} + 84) q^{17} + 104 \zeta_{6} q^{19} + ( - 63 \zeta_{6} + 42) q^{21} - 84 \zeta_{6} q^{23} + (100 \zeta_{6} - 100) q^{25} + 27 q^{27} + 51 q^{29} + ( - 185 \zeta_{6} + 185) q^{31} - 27 \zeta_{6} q^{33} + ( - 210 \zeta_{6} - 105) q^{35} - 44 \zeta_{6} q^{37} + ( - 264 \zeta_{6} + 264) q^{39} - 168 q^{41} - 326 q^{43} + ( - 135 \zeta_{6} + 135) q^{45} - 138 \zeta_{6} q^{47} + ( - 245 \zeta_{6} + 392) q^{49} + 252 \zeta_{6} q^{51} + (639 \zeta_{6} - 639) q^{53} - 135 q^{55} - 312 q^{57} + ( - 159 \zeta_{6} + 159) q^{59} - 722 \zeta_{6} q^{61} + (126 \zeta_{6} + 63) q^{63} - 1320 \zeta_{6} q^{65} + (166 \zeta_{6} - 166) q^{67} + 252 q^{69} - 1086 q^{71} + (218 \zeta_{6} - 218) q^{73} - 300 \zeta_{6} q^{75} + ( - 189 \zeta_{6} + 126) q^{77} - 583 \zeta_{6} q^{79} + (81 \zeta_{6} - 81) q^{81} + 597 q^{83} + 1260 q^{85} + (153 \zeta_{6} - 153) q^{87} + 1038 \zeta_{6} q^{89} + ( - 616 \zeta_{6} + 1848) q^{91} + 555 \zeta_{6} q^{93} + (1560 \zeta_{6} - 1560) q^{95} - 169 q^{97} + 81 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 15 q^{5} - 35 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 15 q^{5} - 35 q^{7} - 9 q^{9} - 9 q^{11} - 176 q^{13} - 90 q^{15} + 84 q^{17} + 104 q^{19} + 21 q^{21} - 84 q^{23} - 100 q^{25} + 54 q^{27} + 102 q^{29} + 185 q^{31} - 27 q^{33} - 420 q^{35} - 44 q^{37} + 264 q^{39} - 336 q^{41} - 652 q^{43} + 135 q^{45} - 138 q^{47} + 539 q^{49} + 252 q^{51} - 639 q^{53} - 270 q^{55} - 624 q^{57} + 159 q^{59} - 722 q^{61} + 252 q^{63} - 1320 q^{65} - 166 q^{67} + 504 q^{69} - 2172 q^{71} - 218 q^{73} - 300 q^{75} + 63 q^{77} - 583 q^{79} - 81 q^{81} + 1194 q^{83} + 2520 q^{85} - 153 q^{87} + 1038 q^{89} + 3080 q^{91} + 555 q^{93} - 1560 q^{95} - 338 q^{97} + 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −1.50000 + 2.59808i 0 7.50000 + 12.9904i 0 −17.5000 + 6.06218i 0 −4.50000 − 7.79423i 0
289.1 0 −1.50000 − 2.59808i 0 7.50000 − 12.9904i 0 −17.5000 − 6.06218i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.d 2
4.b odd 2 1 42.4.e.b ✓ 2
7.c even 3 1 inner 336.4.q.d 2
7.c even 3 1 2352.4.a.u 1
7.d odd 6 1 2352.4.a.q 1
12.b even 2 1 126.4.g.a 2
28.d even 2 1 294.4.e.e 2
28.f even 6 1 294.4.a.g 1
28.f even 6 1 294.4.e.e 2
28.g odd 6 1 42.4.e.b ✓ 2
28.g odd 6 1 294.4.a.a 1
84.h odd 2 1 882.4.g.l 2
84.j odd 6 1 882.4.a.h 1
84.j odd 6 1 882.4.g.l 2
84.n even 6 1 126.4.g.a 2
84.n even 6 1 882.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b ✓ 2 4.b odd 2 1
42.4.e.b ✓ 2 28.g odd 6 1
126.4.g.a 2 12.b even 2 1
126.4.g.a 2 84.n even 6 1
294.4.a.a 1 28.g odd 6 1
294.4.a.g 1 28.f even 6 1
294.4.e.e 2 28.d even 2 1
294.4.e.e 2 28.f even 6 1
336.4.q.d 2 1.a even 1 1 trivial
336.4.q.d 2 7.c even 3 1 inner
882.4.a.h 1 84.j odd 6 1
882.4.a.r 1 84.n even 6 1
882.4.g.l 2 84.h odd 2 1
882.4.g.l 2 84.j odd 6 1
2352.4.a.q 1 7.d odd 6 1
2352.4.a.u 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15T_{5} + 225 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$7$ \( T^{2} + 35T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$13$ \( (T + 88)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} - 104T + 10816 \) Copy content Toggle raw display
$23$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T - 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 185T + 34225 \) Copy content Toggle raw display
$37$ \( T^{2} + 44T + 1936 \) Copy content Toggle raw display
$41$ \( (T + 168)^{2} \) Copy content Toggle raw display
$43$ \( (T + 326)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 138T + 19044 \) Copy content Toggle raw display
$53$ \( T^{2} + 639T + 408321 \) Copy content Toggle raw display
$59$ \( T^{2} - 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} + 722T + 521284 \) Copy content Toggle raw display
$67$ \( T^{2} + 166T + 27556 \) Copy content Toggle raw display
$71$ \( (T + 1086)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 218T + 47524 \) Copy content Toggle raw display
$79$ \( T^{2} + 583T + 339889 \) Copy content Toggle raw display
$83$ \( (T - 597)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1038 T + 1077444 \) Copy content Toggle raw display
$97$ \( (T + 169)^{2} \) Copy content Toggle raw display
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