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Properties

Label 342.4.g.j
Level $342$
Weight $4$
Character orbit 342.g
Analytic conductor $20.179$
Analytic rank $0$
Dimension $10$
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] = [342,4,Mod(163,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("342.163");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\), degree \(2\), 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: \(20.1786532220\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 145 x^{8} - 272 x^{7} + 20252 x^{6} - 29594 x^{5} + 141886 x^{4} - 151552 x^{3} + \cdots + 1052676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_1 + 2) q^{2} - 4 \beta_1 q^{4} + (\beta_{8} - \beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{2} + 2) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_1 + 2) q^{2} - 4 \beta_1 q^{4} + (\beta_{8} - \beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{2} + 2) q^{7} - 8 q^{8} + (2 \beta_{8} + 4 \beta_1) q^{10} + ( - \beta_{7} - \beta_{5} + 11) q^{11} + (\beta_{9} - \beta_{8} + \cdots - 7 \beta_1) q^{13}+ \cdots + (6 \beta_{9} - 2 \beta_{8} + \cdots + 366) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 20 q^{4} - 10 q^{5} + 22 q^{7} - 80 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 20 q^{4} - 10 q^{5} + 22 q^{7} - 80 q^{8} + 20 q^{10} + 108 q^{11} - 35 q^{13} + 22 q^{14} - 80 q^{16} - 98 q^{17} + 32 q^{19} + 80 q^{20} + 108 q^{22} - 288 q^{23} - 331 q^{25} - 140 q^{26} - 44 q^{28} + 10 q^{29} + 174 q^{31} + 160 q^{32} + 196 q^{34} - 294 q^{35} - 118 q^{37} + 230 q^{38} + 80 q^{40} - 436 q^{41} - 579 q^{43} - 216 q^{44} - 1152 q^{46} - 468 q^{47} + 1844 q^{49} - 1324 q^{50} - 140 q^{52} - 44 q^{53} + 436 q^{55} - 176 q^{56} + 40 q^{58} + 628 q^{59} - 601 q^{61} + 174 q^{62} + 640 q^{64} + 724 q^{65} - 643 q^{67} + 784 q^{68} + 588 q^{70} - 1442 q^{71} - 869 q^{73} - 118 q^{74} + 332 q^{76} - 1604 q^{77} + 1269 q^{79} - 160 q^{80} + 872 q^{82} + 1388 q^{83} + 1524 q^{85} + 1158 q^{86} - 864 q^{88} - 1760 q^{89} + 295 q^{91} - 1152 q^{92} - 1872 q^{94} + 394 q^{95} - 1346 q^{97} + 1844 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 145 x^{8} - 272 x^{7} + 20252 x^{6} - 29594 x^{5} + 141886 x^{4} - 151552 x^{3} + \cdots + 1052676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47473989997 \nu^{9} - 28313106262 \nu^{8} + 6790329845815 \nu^{7} - 9817692236909 \nu^{6} + \cdots + 14\!\cdots\!28 ) / 37\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3163356876717 \nu^{9} - 14498775870200 \nu^{8} + 480489432338428 \nu^{7} + \cdots - 82\!\cdots\!02 ) / 27\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3875626014909 \nu^{9} + 15525915037900 \nu^{8} - 514528824356006 \nu^{7} + \cdots + 81\!\cdots\!34 ) / 27\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79065806370919 \nu^{9} - 67648178374015 \nu^{8} + \cdots - 25\!\cdots\!56 ) / 51\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4704586353339 \nu^{9} + 19566637905400 \nu^{8} - 648438380184956 \nu^{7} + \cdots + 10\!\cdots\!24 ) / 27\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 158096372084533 \nu^{9} - 568024508973546 \nu^{8} + \cdots - 54\!\cdots\!92 ) / 51\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4974757530819 \nu^{9} + 21499622934200 \nu^{8} - 712497504039388 \nu^{7} + \cdots + 68\!\cdots\!75 ) / 13\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 221409360122137 \nu^{9} + 268559941764642 \nu^{8} + \cdots + 71\!\cdots\!88 ) / 51\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 227913296751726 \nu^{9} - 264681046206748 \nu^{8} + \cdots - 73\!\cdots\!24 ) / 51\!\cdots\!74 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 9 \beta_{6} + 10 \beta_{5} + 12 \beta_{4} + 2 \beta_{3} + \cdots - 347 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{7} - 172\beta_{5} + 136\beta_{3} - 42\beta_{2} + 245 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -218\beta_{9} + 1468\beta_{8} + 1311\beta_{6} - 1680\beta_{4} - 46169\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18694 \beta_{9} - 24688 \beta_{8} - 2721 \beta_{7} - 2721 \beta_{6} + 24688 \beta_{5} - 5232 \beta_{4} + \cdots - 55823 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 182955\beta_{7} - 214534\beta_{5} - 21440\beta_{3} + 230112\beta_{2} + 6404375 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2575972\beta_{9} + 3516598\beta_{8} + 461943\beta_{6} + 617328\beta_{4} - 10711727\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1763660 \beta_{9} - 31319074 \beta_{8} - 25526943 \beta_{7} - 25526943 \beta_{6} + 31319074 \beta_{5} + \cdots - 890929139 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 75819471\beta_{7} - 501103846\beta_{5} + 355531528\beta_{3} - 70680804\beta_{2} + 1897670639 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.754932 + 1.30758i
−1.11900 − 1.93816i
5.76533 + 9.98585i
1.09744 + 1.90081i
−5.99871 − 10.3901i
0.754932 − 1.30758i
−1.11900 + 1.93816i
5.76533 − 9.98585i
1.09744 − 1.90081i
−5.99871 + 10.3901i
1.00000 − 1.73205i 0 −2.00000 − 3.46410i −10.1614 + 17.6001i 0 29.5155 −8.00000 0 20.3229 + 35.2002i
163.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −6.73672 + 11.6683i 0 −21.8100 −8.00000 0 13.4734 + 23.3367i
163.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −0.997274 + 1.72733i 0 −9.20105 −8.00000 0 1.99455 + 3.45466i
163.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 4.68400 − 8.11292i 0 29.9904 −8.00000 0 −9.36800 − 16.2258i
163.5 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 8.21143 − 14.2226i 0 −17.4950 −8.00000 0 −16.4229 − 28.4452i
235.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.1614 − 17.6001i 0 29.5155 −8.00000 0 20.3229 − 35.2002i
235.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −6.73672 − 11.6683i 0 −21.8100 −8.00000 0 13.4734 − 23.3367i
235.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −0.997274 − 1.72733i 0 −9.20105 −8.00000 0 1.99455 − 3.45466i
235.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.68400 + 8.11292i 0 29.9904 −8.00000 0 −9.36800 + 16.2258i
235.5 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 8.21143 + 14.2226i 0 −17.4950 −8.00000 0 −16.4229 + 28.4452i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.4.g.j yes 10
3.b odd 2 1 342.4.g.i ✓ 10
19.c even 3 1 inner 342.4.g.j yes 10
57.h odd 6 1 342.4.g.i ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.4.g.i ✓ 10 3.b odd 2 1
342.4.g.i ✓ 10 57.h odd 6 1
342.4.g.j yes 10 1.a even 1 1 trivial
342.4.g.j yes 10 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 10 T_{5}^{9} + 528 T_{5}^{8} + 1216 T_{5}^{7} + 172252 T_{5}^{6} + 491928 T_{5}^{5} + \cdots + 7060032576 \) acting on \(S_{4}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 7060032576 \) Copy content Toggle raw display
$7$ \( (T^{5} - 11 T^{4} + \cdots + 3107697)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 54 T^{4} + \cdots - 3104568)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 797400965647521 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{5} - 87 T^{4} + \cdots - 136553099835)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 59 T^{4} + \cdots + 15338791007)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 52\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 4074122945664)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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