Properties

Label 42.6.e.c
Level $42$
Weight $6$
Character orbit 42.e
Analytic conductor $6.736$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [42,6,Mod(25,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.e (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: \(6.73612043215\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{9601})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + ( - 9 \beta_{2} + 9) q^{3} + (16 \beta_{2} - 16) q^{4} + (27 \beta_{2} - \beta_1) q^{5} - 36 q^{6} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} + 64 q^{8} - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{2} q^{2} + ( - 9 \beta_{2} + 9) q^{3} + (16 \beta_{2} - 16) q^{4} + (27 \beta_{2} - \beta_1) q^{5} - 36 q^{6} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} + 64 q^{8} - 81 \beta_{2} q^{9} + (4 \beta_{3} - 108 \beta_{2} + \cdots + 104) q^{10}+ \cdots + ( - 405 \beta_{3} + 7938) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 18 q^{3} - 32 q^{4} + 53 q^{5} - 144 q^{6} + 6 q^{7} + 256 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} + 18 q^{3} - 32 q^{4} + 53 q^{5} - 144 q^{6} + 6 q^{7} + 256 q^{8} - 162 q^{9} + 212 q^{10} - 191 q^{11} + 288 q^{12} - 758 q^{13} - 36 q^{14} + 954 q^{15} - 512 q^{16} + 340 q^{17} - 648 q^{18} + 1769 q^{19} - 1696 q^{20} - 27 q^{21} + 1528 q^{22} - 3236 q^{23} + 1152 q^{24} + 45 q^{25} + 1516 q^{26} - 2916 q^{27} + 48 q^{28} + 8918 q^{29} - 1908 q^{30} - 1994 q^{31} - 2048 q^{32} + 1719 q^{33} - 2720 q^{34} - 4562 q^{35} + 5184 q^{36} - 20587 q^{37} + 7076 q^{38} - 3411 q^{39} + 3392 q^{40} + 17628 q^{41} - 216 q^{42} + 31706 q^{43} - 3056 q^{44} + 4293 q^{45} - 12944 q^{46} - 33912 q^{47} - 9216 q^{48} + 9598 q^{49} - 360 q^{50} - 3060 q^{51} + 6064 q^{52} - 49239 q^{53} + 5832 q^{54} + 37882 q^{55} + 384 q^{56} + 31842 q^{57} - 17836 q^{58} + 56735 q^{59} - 7632 q^{60} - 67508 q^{61} + 15952 q^{62} - 729 q^{63} + 16384 q^{64} + 42762 q^{65} + 6876 q^{66} - 75723 q^{67} + 5440 q^{68} - 58248 q^{69} + 95692 q^{70} - 17984 q^{71} - 10368 q^{72} + 3201 q^{73} - 82348 q^{74} - 405 q^{75} - 56608 q^{76} + 120299 q^{77} + 27288 q^{78} - 26612 q^{79} + 13568 q^{80} - 13122 q^{81} - 35256 q^{82} - 1898 q^{83} + 1296 q^{84} + 210040 q^{85} - 63412 q^{86} + 40131 q^{87} - 12224 q^{88} - 176562 q^{89} - 34344 q^{90} + 210085 q^{91} + 103552 q^{92} + 17946 q^{93} - 135648 q^{94} - 234098 q^{95} + 18432 q^{96} - 258846 q^{97} + 211156 q^{98} + 30942 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2401\nu^{2} - 2401\nu + 5760000 ) / 5762400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4801 ) / 2401 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2400\beta_{2} + \beta _1 - 2401 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2401\beta_{3} - 4801 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
24.7462 + 42.8616i
−24.2462 41.9956i
24.7462 42.8616i
−24.2462 + 41.9956i
−2.00000 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i −11.2462 19.4789i −36.0000 −96.4847 86.5893i 64.0000 −40.5000 70.1481i −44.9847 + 77.9158i
25.2 −2.00000 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i 37.7462 + 65.3783i −36.0000 99.4847 + 83.1252i 64.0000 −40.5000 70.1481i 150.985 261.513i
37.1 −2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i −11.2462 + 19.4789i −36.0000 −96.4847 + 86.5893i 64.0000 −40.5000 + 70.1481i −44.9847 77.9158i
37.2 −2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 37.7462 65.3783i −36.0000 99.4847 83.1252i 64.0000 −40.5000 + 70.1481i 150.985 + 261.513i
\(n\): 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 42.6.e.c 4
3.b odd 2 1 126.6.g.h 4
4.b odd 2 1 336.6.q.f 4
7.b odd 2 1 294.6.e.s 4
7.c even 3 1 inner 42.6.e.c 4
7.c even 3 1 294.6.a.r 2
7.d odd 6 1 294.6.a.w 2
7.d odd 6 1 294.6.e.s 4
21.g even 6 1 882.6.a.bb 2
21.h odd 6 1 126.6.g.h 4
21.h odd 6 1 882.6.a.bh 2
28.g odd 6 1 336.6.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.c 4 1.a even 1 1 trivial
42.6.e.c 4 7.c even 3 1 inner
126.6.g.h 4 3.b odd 2 1
126.6.g.h 4 21.h odd 6 1
294.6.a.r 2 7.c even 3 1
294.6.a.w 2 7.d odd 6 1
294.6.e.s 4 7.b odd 2 1
294.6.e.s 4 7.d odd 6 1
336.6.q.f 4 4.b odd 2 1
336.6.q.f 4 28.g odd 6 1
882.6.a.bb 2 21.g even 6 1
882.6.a.bh 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 53T_{5}^{3} + 4507T_{5}^{2} + 89994T_{5} + 2883204 \) acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 53 T^{3} + \cdots + 2883204 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 2589384996 \) Copy content Toggle raw display
$13$ \( (T^{2} + 379 T - 254520)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 867133440000 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 8227948033600 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 6653923430400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4459 T - 3960660)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8814 T + 8966160)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 15853 T + 44661910)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8992 T - 4289674884)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 46\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} + 949 T - 7511023590)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{2} + 129423 T - 5231869258)^{2} \) Copy content Toggle raw display
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