Properties

Label 42.6.e.b
Level $42$
Weight $6$
Character orbit 42.e
Analytic conductor $6.736$
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] = [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.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{2} + 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + (86 \zeta_{6} - 86) q^{5} + 36 q^{6} + (147 \zeta_{6} - 49) q^{7} - 64 q^{8} + (81 \zeta_{6} - 81) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{2} + 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + (86 \zeta_{6} - 86) q^{5} + 36 q^{6} + (147 \zeta_{6} - 49) q^{7} - 64 q^{8} + (81 \zeta_{6} - 81) q^{9} + 344 \zeta_{6} q^{10} - 34 \zeta_{6} q^{11} + ( - 144 \zeta_{6} + 144) q^{12} - 3 q^{13} + (196 \zeta_{6} + 392) q^{14} - 774 q^{15} + (256 \zeta_{6} - 256) q^{16} + 1904 \zeta_{6} q^{17} + 324 \zeta_{6} q^{18} + ( - 1489 \zeta_{6} + 1489) q^{19} + 1376 q^{20} + (882 \zeta_{6} - 1323) q^{21} - 136 q^{22} + ( - 224 \zeta_{6} + 224) q^{23} - 576 \zeta_{6} q^{24} - 4271 \zeta_{6} q^{25} + (12 \zeta_{6} - 12) q^{26} - 729 q^{27} + ( - 1568 \zeta_{6} + 2352) q^{28} - 6508 q^{29} + (3096 \zeta_{6} - 3096) q^{30} - 1731 \zeta_{6} q^{31} + 1024 \zeta_{6} q^{32} + ( - 306 \zeta_{6} + 306) q^{33} + 7616 q^{34} + ( - 4214 \zeta_{6} - 8428) q^{35} + 1296 q^{36} + ( - 7633 \zeta_{6} + 7633) q^{37} - 5956 \zeta_{6} q^{38} - 27 \zeta_{6} q^{39} + ( - 5504 \zeta_{6} + 5504) q^{40} + 15414 q^{41} + (5292 \zeta_{6} - 1764) q^{42} + 18491 q^{43} + (544 \zeta_{6} - 544) q^{44} - 6966 \zeta_{6} q^{45} - 896 \zeta_{6} q^{46} + (18462 \zeta_{6} - 18462) q^{47} - 2304 q^{48} + (7203 \zeta_{6} - 19208) q^{49} - 17084 q^{50} + (17136 \zeta_{6} - 17136) q^{51} + 48 \zeta_{6} q^{52} + 19956 \zeta_{6} q^{53} + (2916 \zeta_{6} - 2916) q^{54} + 2924 q^{55} + ( - 9408 \zeta_{6} + 3136) q^{56} + 13401 q^{57} + (26032 \zeta_{6} - 26032) q^{58} + 31828 \zeta_{6} q^{59} + 12384 \zeta_{6} q^{60} + ( - 57654 \zeta_{6} + 57654) q^{61} - 6924 q^{62} + ( - 3969 \zeta_{6} - 7938) q^{63} + 4096 q^{64} + ( - 258 \zeta_{6} + 258) q^{65} - 1224 \zeta_{6} q^{66} + 60563 \zeta_{6} q^{67} + ( - 30464 \zeta_{6} + 30464) q^{68} + 2016 q^{69} + (33712 \zeta_{6} - 50568) q^{70} - 44834 q^{71} + ( - 5184 \zeta_{6} + 5184) q^{72} - 20821 \zeta_{6} q^{73} - 30532 \zeta_{6} q^{74} + ( - 38439 \zeta_{6} + 38439) q^{75} - 23824 q^{76} + ( - 3332 \zeta_{6} + 4998) q^{77} - 108 q^{78} + ( - 30531 \zeta_{6} + 30531) q^{79} - 22016 \zeta_{6} q^{80} - 6561 \zeta_{6} q^{81} + ( - 61656 \zeta_{6} + 61656) q^{82} + 110602 q^{83} + (7056 \zeta_{6} + 14112) q^{84} - 163744 q^{85} + ( - 73964 \zeta_{6} + 73964) q^{86} - 58572 \zeta_{6} q^{87} + 2176 \zeta_{6} q^{88} + ( - 58992 \zeta_{6} + 58992) q^{89} - 27864 q^{90} + ( - 441 \zeta_{6} + 147) q^{91} - 3584 q^{92} + ( - 15579 \zeta_{6} + 15579) q^{93} + 73848 \zeta_{6} q^{94} + 128054 \zeta_{6} q^{95} + (9216 \zeta_{6} - 9216) q^{96} - 119846 q^{97} + (76832 \zeta_{6} - 48020) q^{98} + 2754 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9} + 344 q^{10} - 34 q^{11} + 144 q^{12} - 6 q^{13} + 980 q^{14} - 1548 q^{15} - 256 q^{16} + 1904 q^{17} + 324 q^{18} + 1489 q^{19} + 2752 q^{20} - 1764 q^{21} - 272 q^{22} + 224 q^{23} - 576 q^{24} - 4271 q^{25} - 12 q^{26} - 1458 q^{27} + 3136 q^{28} - 13016 q^{29} - 3096 q^{30} - 1731 q^{31} + 1024 q^{32} + 306 q^{33} + 15232 q^{34} - 21070 q^{35} + 2592 q^{36} + 7633 q^{37} - 5956 q^{38} - 27 q^{39} + 5504 q^{40} + 30828 q^{41} + 1764 q^{42} + 36982 q^{43} - 544 q^{44} - 6966 q^{45} - 896 q^{46} - 18462 q^{47} - 4608 q^{48} - 31213 q^{49} - 34168 q^{50} - 17136 q^{51} + 48 q^{52} + 19956 q^{53} - 2916 q^{54} + 5848 q^{55} - 3136 q^{56} + 26802 q^{57} - 26032 q^{58} + 31828 q^{59} + 12384 q^{60} + 57654 q^{61} - 13848 q^{62} - 19845 q^{63} + 8192 q^{64} + 258 q^{65} - 1224 q^{66} + 60563 q^{67} + 30464 q^{68} + 4032 q^{69} - 67424 q^{70} - 89668 q^{71} + 5184 q^{72} - 20821 q^{73} - 30532 q^{74} + 38439 q^{75} - 47648 q^{76} + 6664 q^{77} - 216 q^{78} + 30531 q^{79} - 22016 q^{80} - 6561 q^{81} + 61656 q^{82} + 221204 q^{83} + 35280 q^{84} - 327488 q^{85} + 73964 q^{86} - 58572 q^{87} + 2176 q^{88} + 58992 q^{89} - 55728 q^{90} - 147 q^{91} - 7168 q^{92} + 15579 q^{93} + 73848 q^{94} + 128054 q^{95} - 9216 q^{96} - 239692 q^{97} - 19208 q^{98} + 5508 q^{99}+O(q^{100}) \) 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 −43.0000 74.4782i 36.0000 24.5000 127.306i −64.0000 −40.5000 70.1481i 172.000 297.913i
37.1 2.00000 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i −43.0000 + 74.4782i 36.0000 24.5000 + 127.306i −64.0000 −40.5000 + 70.1481i 172.000 + 297.913i
\(n\): 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.b 2
3.b odd 2 1 126.6.g.b 2
4.b odd 2 1 336.6.q.a 2
7.b odd 2 1 294.6.e.m 2
7.c even 3 1 inner 42.6.e.b 2
7.c even 3 1 294.6.a.d 1
7.d odd 6 1 294.6.a.e 1
7.d odd 6 1 294.6.e.m 2
21.g even 6 1 882.6.a.y 1
21.h odd 6 1 126.6.g.b 2
21.h odd 6 1 882.6.a.m 1
28.g odd 6 1 336.6.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.b 2 1.a even 1 1 trivial
42.6.e.b 2 7.c even 3 1 inner
126.6.g.b 2 3.b odd 2 1
126.6.g.b 2 21.h odd 6 1
294.6.a.d 1 7.c even 3 1
294.6.a.e 1 7.d odd 6 1
294.6.e.m 2 7.b odd 2 1
294.6.e.m 2 7.d odd 6 1
336.6.q.a 2 4.b odd 2 1
336.6.q.a 2 28.g odd 6 1
882.6.a.m 1 21.h odd 6 1
882.6.a.y 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 86T_{5} + 7396 \) 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
$5$ \( T^{2} + 86T + 7396 \) Copy content Toggle raw display
$7$ \( T^{2} - 49T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} + 34T + 1156 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1904 T + 3625216 \) Copy content Toggle raw display
$19$ \( T^{2} - 1489 T + 2217121 \) Copy content Toggle raw display
$23$ \( T^{2} - 224T + 50176 \) Copy content Toggle raw display
$29$ \( (T + 6508)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1731 T + 2996361 \) Copy content Toggle raw display
$37$ \( T^{2} - 7633 T + 58262689 \) Copy content Toggle raw display
$41$ \( (T - 15414)^{2} \) Copy content Toggle raw display
$43$ \( (T - 18491)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 18462 T + 340845444 \) Copy content Toggle raw display
$53$ \( T^{2} - 19956 T + 398241936 \) Copy content Toggle raw display
$59$ \( T^{2} - 31828 T + 1013021584 \) Copy content Toggle raw display
$61$ \( T^{2} - 57654 T + 3323983716 \) Copy content Toggle raw display
$67$ \( T^{2} - 60563 T + 3667876969 \) Copy content Toggle raw display
$71$ \( (T + 44834)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 20821 T + 433514041 \) Copy content Toggle raw display
$79$ \( T^{2} - 30531 T + 932141961 \) Copy content Toggle raw display
$83$ \( (T - 110602)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 58992 T + 3480056064 \) Copy content Toggle raw display
$97$ \( (T + 119846)^{2} \) Copy content Toggle raw display
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