Properties

Label 42.6.e.a
Level $42$
Weight $6$
Character orbit 42.e
Analytic conductor $6.736$
Analytic rank $0$
Dimension $2$
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: \(2\)
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: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{2} - 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 36 q^{6} + (133 \zeta_{6} - 7) q^{7} + 64 q^{8} + (81 \zeta_{6} - 81) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{2} - 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 36 q^{6} + (133 \zeta_{6} - 7) q^{7} + 64 q^{8} + (81 \zeta_{6} - 81) q^{9} + 24 \zeta_{6} q^{10} + 666 \zeta_{6} q^{11} + (144 \zeta_{6} - 144) q^{12} - 559 q^{13} + ( - 28 \zeta_{6} - 504) q^{14} - 54 q^{15} + (256 \zeta_{6} - 256) q^{16} + 1740 \zeta_{6} q^{17} - 324 \zeta_{6} q^{18} + (1157 \zeta_{6} - 1157) q^{19} - 96 q^{20} + ( - 1134 \zeta_{6} + 1197) q^{21} - 2664 q^{22} + ( - 3468 \zeta_{6} + 3468) q^{23} - 576 \zeta_{6} q^{24} + 3089 \zeta_{6} q^{25} + ( - 2236 \zeta_{6} + 2236) q^{26} + 729 q^{27} + ( - 2016 \zeta_{6} + 2128) q^{28} + 3372 q^{29} + ( - 216 \zeta_{6} + 216) q^{30} - 6293 \zeta_{6} q^{31} - 1024 \zeta_{6} q^{32} + ( - 5994 \zeta_{6} + 5994) q^{33} - 6960 q^{34} + (42 \zeta_{6} + 756) q^{35} + 1296 q^{36} + (3131 \zeta_{6} - 3131) q^{37} - 4628 \zeta_{6} q^{38} + 5031 \zeta_{6} q^{39} + ( - 384 \zeta_{6} + 384) q^{40} - 4866 q^{41} + (4788 \zeta_{6} - 252) q^{42} - 11407 q^{43} + ( - 10656 \zeta_{6} + 10656) q^{44} + 486 \zeta_{6} q^{45} + 13872 \zeta_{6} q^{46} + (2310 \zeta_{6} - 2310) q^{47} + 2304 q^{48} + (15827 \zeta_{6} - 17640) q^{49} - 12356 q^{50} + ( - 15660 \zeta_{6} + 15660) q^{51} + 8944 \zeta_{6} q^{52} + 28296 \zeta_{6} q^{53} + (2916 \zeta_{6} - 2916) q^{54} + 3996 q^{55} + (8512 \zeta_{6} - 448) q^{56} + 10413 q^{57} + (13488 \zeta_{6} - 13488) q^{58} - 20544 \zeta_{6} q^{59} + 864 \zeta_{6} q^{60} + ( - 4630 \zeta_{6} + 4630) q^{61} + 25172 q^{62} + ( - 567 \zeta_{6} - 10206) q^{63} + 4096 q^{64} + (3354 \zeta_{6} - 3354) q^{65} + 23976 \zeta_{6} q^{66} + 18745 \zeta_{6} q^{67} + ( - 27840 \zeta_{6} + 27840) q^{68} - 31212 q^{69} + (3024 \zeta_{6} - 3192) q^{70} - 38226 q^{71} + (5184 \zeta_{6} - 5184) q^{72} - 70589 \zeta_{6} q^{73} - 12524 \zeta_{6} q^{74} + ( - 27801 \zeta_{6} + 27801) q^{75} + 18512 q^{76} + (83916 \zeta_{6} - 88578) q^{77} - 20124 q^{78} + ( - 62293 \zeta_{6} + 62293) q^{79} + 1536 \zeta_{6} q^{80} - 6561 \zeta_{6} q^{81} + ( - 19464 \zeta_{6} + 19464) q^{82} + 79818 q^{83} + ( - 1008 \zeta_{6} - 18144) q^{84} + 10440 q^{85} + ( - 45628 \zeta_{6} + 45628) q^{86} - 30348 \zeta_{6} q^{87} + 42624 \zeta_{6} q^{88} + ( - 18120 \zeta_{6} + 18120) q^{89} - 1944 q^{90} + ( - 74347 \zeta_{6} + 3913) q^{91} - 55488 q^{92} + (56637 \zeta_{6} - 56637) q^{93} - 9240 \zeta_{6} q^{94} + 6942 \zeta_{6} q^{95} + (9216 \zeta_{6} - 9216) q^{96} + 124754 q^{97} + ( - 70560 \zeta_{6} + 7252) q^{98} - 53946 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} + 6 q^{5} + 72 q^{6} + 119 q^{7} + 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} + 6 q^{5} + 72 q^{6} + 119 q^{7} + 128 q^{8} - 81 q^{9} + 24 q^{10} + 666 q^{11} - 144 q^{12} - 1118 q^{13} - 1036 q^{14} - 108 q^{15} - 256 q^{16} + 1740 q^{17} - 324 q^{18} - 1157 q^{19} - 192 q^{20} + 1260 q^{21} - 5328 q^{22} + 3468 q^{23} - 576 q^{24} + 3089 q^{25} + 2236 q^{26} + 1458 q^{27} + 2240 q^{28} + 6744 q^{29} + 216 q^{30} - 6293 q^{31} - 1024 q^{32} + 5994 q^{33} - 13920 q^{34} + 1554 q^{35} + 2592 q^{36} - 3131 q^{37} - 4628 q^{38} + 5031 q^{39} + 384 q^{40} - 9732 q^{41} + 4284 q^{42} - 22814 q^{43} + 10656 q^{44} + 486 q^{45} + 13872 q^{46} - 2310 q^{47} + 4608 q^{48} - 19453 q^{49} - 24712 q^{50} + 15660 q^{51} + 8944 q^{52} + 28296 q^{53} - 2916 q^{54} + 7992 q^{55} + 7616 q^{56} + 20826 q^{57} - 13488 q^{58} - 20544 q^{59} + 864 q^{60} + 4630 q^{61} + 50344 q^{62} - 20979 q^{63} + 8192 q^{64} - 3354 q^{65} + 23976 q^{66} + 18745 q^{67} + 27840 q^{68} - 62424 q^{69} - 3360 q^{70} - 76452 q^{71} - 5184 q^{72} - 70589 q^{73} - 12524 q^{74} + 27801 q^{75} + 37024 q^{76} - 93240 q^{77} - 40248 q^{78} + 62293 q^{79} + 1536 q^{80} - 6561 q^{81} + 19464 q^{82} + 159636 q^{83} - 37296 q^{84} + 20880 q^{85} + 45628 q^{86} - 30348 q^{87} + 42624 q^{88} + 18120 q^{89} - 3888 q^{90} - 66521 q^{91} - 110976 q^{92} - 56637 q^{93} - 9240 q^{94} + 6942 q^{95} - 9216 q^{96} + 249508 q^{97} - 56056 q^{98} - 107892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
−2.00000 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i 3.00000 + 5.19615i 36.0000 59.5000 115.181i 64.0000 −40.5000 70.1481i 12.0000 20.7846i
37.1 −2.00000 + 3.46410i −4.50000 7.79423i −8.00000 13.8564i 3.00000 5.19615i 36.0000 59.5000 + 115.181i 64.0000 −40.5000 + 70.1481i 12.0000 + 20.7846i
\(n\): 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.a 2
3.b odd 2 1 126.6.g.c 2
4.b odd 2 1 336.6.q.c 2
7.b odd 2 1 294.6.e.e 2
7.c even 3 1 inner 42.6.e.a 2
7.c even 3 1 294.6.a.l 1
7.d odd 6 1 294.6.a.j 1
7.d odd 6 1 294.6.e.e 2
21.g even 6 1 882.6.a.e 1
21.h odd 6 1 126.6.g.c 2
21.h odd 6 1 882.6.a.f 1
28.g odd 6 1 336.6.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 1.a even 1 1 trivial
42.6.e.a 2 7.c even 3 1 inner
126.6.g.c 2 3.b odd 2 1
126.6.g.c 2 21.h odd 6 1
294.6.a.j 1 7.d odd 6 1
294.6.a.l 1 7.c even 3 1
294.6.e.e 2 7.b odd 2 1
294.6.e.e 2 7.d odd 6 1
336.6.q.c 2 4.b odd 2 1
336.6.q.c 2 28.g odd 6 1
882.6.a.e 1 21.g even 6 1
882.6.a.f 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6T_{5} + 36 \) acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} - 119T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 666T + 443556 \) Copy content Toggle raw display
$13$ \( (T + 559)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1740 T + 3027600 \) Copy content Toggle raw display
$19$ \( T^{2} + 1157 T + 1338649 \) Copy content Toggle raw display
$23$ \( T^{2} - 3468 T + 12027024 \) Copy content Toggle raw display
$29$ \( (T - 3372)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6293 T + 39601849 \) Copy content Toggle raw display
$37$ \( T^{2} + 3131 T + 9803161 \) Copy content Toggle raw display
$41$ \( (T + 4866)^{2} \) Copy content Toggle raw display
$43$ \( (T + 11407)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2310 T + 5336100 \) Copy content Toggle raw display
$53$ \( T^{2} - 28296 T + 800663616 \) Copy content Toggle raw display
$59$ \( T^{2} + 20544 T + 422055936 \) Copy content Toggle raw display
$61$ \( T^{2} - 4630 T + 21436900 \) Copy content Toggle raw display
$67$ \( T^{2} - 18745 T + 351375025 \) Copy content Toggle raw display
$71$ \( (T + 38226)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 4982806921 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 3880417849 \) Copy content Toggle raw display
$83$ \( (T - 79818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18120 T + 328334400 \) Copy content Toggle raw display
$97$ \( (T - 124754)^{2} \) Copy content Toggle raw display
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