Properties

Label 42.3.g.a
Level $42$
Weight $3$
Character orbit 42.g
Analytic conductor $1.144$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14441711031\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -5 - 2 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -5 - 2 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} + ( -8 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{10} + 6 \beta_{2} q^{11} + ( -2 + 2 \beta_{2} ) q^{12} + ( -1 - 16 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{13} + ( 8 - 5 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{14} + ( 6 + 6 \beta_{3} ) q^{15} + ( -4 - 4 \beta_{2} ) q^{16} + ( -16 + 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{18} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 4 - 8 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{20} + ( -5 - 10 \beta_{2} - 6 \beta_{3} ) q^{21} + 6 \beta_{3} q^{22} + ( 12 + 18 \beta_{1} + 12 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{24} + ( 24 \beta_{1} - 11 \beta_{2} + 24 \beta_{3} ) q^{25} + ( 16 - \beta_{1} - 16 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 3 + 6 \beta_{2} ) q^{27} + ( 10 + 8 \beta_{1} + 4 \beta_{3} ) q^{28} + 24 \beta_{3} q^{29} + ( -12 + 6 \beta_{1} - 12 \beta_{2} ) q^{30} + ( 34 + 6 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -6 + 6 \beta_{2} ) q^{33} + ( 4 - 16 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -20 - 2 \beta_{1} + 14 \beta_{2} - 22 \beta_{3} ) q^{35} -6 q^{36} + ( 11 + 12 \beta_{1} + 11 \beta_{2} ) q^{37} + ( 8 - 7 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} ) q^{38} + ( -24 \beta_{1} - 3 \beta_{2} - 24 \beta_{3} ) q^{39} + ( 8 + 4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{40} + ( -26 - 8 \beta_{1} - 52 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 12 - 5 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} ) q^{42} + ( 7 + 6 \beta_{3} ) q^{43} + ( -12 - 12 \beta_{2} ) q^{44} + ( 12 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{45} + ( 12 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} ) q^{46} + ( -22 + 2 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} ) q^{47} + ( -4 - 8 \beta_{2} ) q^{48} + ( -20 \beta_{1} + \beta_{2} + 20 \beta_{3} ) q^{49} + ( -48 - 11 \beta_{3} ) q^{50} + ( -24 + 6 \beta_{1} - 24 \beta_{2} ) q^{51} + ( 4 + 16 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} ) q^{52} + ( -18 \beta_{1} - 60 \beta_{2} - 18 \beta_{3} ) q^{53} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{54} + ( 12 - 24 \beta_{1} + 24 \beta_{2} - 12 \beta_{3} ) q^{55} + ( -8 + 10 \beta_{1} + 8 \beta_{2} ) q^{56} + ( -21 - 6 \beta_{3} ) q^{57} + ( -48 - 48 \beta_{2} ) q^{58} + ( -8 - 14 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} ) q^{59} + ( -12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -12 - 8 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} ) q^{61} + ( 12 + 34 \beta_{1} + 24 \beta_{2} + 17 \beta_{3} ) q^{62} + ( 6 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} ) q^{63} + 8 q^{64} + ( 90 - 42 \beta_{1} + 90 \beta_{2} ) q^{65} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{66} + ( 42 \beta_{1} - 55 \beta_{2} + 42 \beta_{3} ) q^{67} + ( 16 + 4 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{68} + ( 12 + 36 \beta_{1} + 24 \beta_{2} + 18 \beta_{3} ) q^{69} + ( 44 - 20 \beta_{1} + 40 \beta_{2} + 14 \beta_{3} ) q^{70} + ( 78 - 42 \beta_{3} ) q^{71} -6 \beta_{1} q^{72} + ( -22 + 40 \beta_{1} - 11 \beta_{2} - 40 \beta_{3} ) q^{73} + ( 11 \beta_{1} + 24 \beta_{2} + 11 \beta_{3} ) q^{74} + ( 11 + 24 \beta_{1} - 11 \beta_{2} + 48 \beta_{3} ) q^{75} + ( -14 + 8 \beta_{1} - 28 \beta_{2} + 4 \beta_{3} ) q^{76} + ( 30 + 24 \beta_{1} + 12 \beta_{3} ) q^{77} + ( 48 - 3 \beta_{3} ) q^{78} + ( -5 - 66 \beta_{1} - 5 \beta_{2} ) q^{79} + ( -16 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{80} + 9 \beta_{2} q^{81} + ( 8 - 26 \beta_{1} - 8 \beta_{2} - 52 \beta_{3} ) q^{82} + ( 10 + 76 \beta_{1} + 20 \beta_{2} + 38 \beta_{3} ) q^{83} + ( 20 + 12 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -72 - 60 \beta_{3} ) q^{85} + ( -12 + 7 \beta_{1} - 12 \beta_{2} ) q^{86} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{87} + ( -12 \beta_{1} - 12 \beta_{3} ) q^{88} + ( -12 + 12 \beta_{2} ) q^{89} + ( -12 + 12 \beta_{1} - 24 \beta_{2} + 6 \beta_{3} ) q^{90} + ( -101 + 34 \beta_{1} - 91 \beta_{2} + 80 \beta_{3} ) q^{91} + ( -24 + 36 \beta_{3} ) q^{92} + ( 51 + 18 \beta_{1} + 51 \beta_{2} ) q^{93} + ( -8 - 22 \beta_{1} - 4 \beta_{2} + 22 \beta_{3} ) q^{94} + ( -54 \beta_{1} + 66 \beta_{2} - 54 \beta_{3} ) q^{95} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{96} + ( -12 + 8 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -40 - 80 \beta_{2} + \beta_{3} ) q^{98} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 4 q^{4} + 12 q^{5} - 10 q^{7} + 6 q^{9} + O(q^{10}) \) \( 4 q + 6 q^{3} - 4 q^{4} + 12 q^{5} - 10 q^{7} + 6 q^{9} - 24 q^{10} - 12 q^{11} - 12 q^{12} + 24 q^{14} + 24 q^{15} - 8 q^{16} - 48 q^{17} - 42 q^{19} + 24 q^{23} + 22 q^{25} + 96 q^{26} + 40 q^{28} - 24 q^{30} + 102 q^{31} - 36 q^{33} - 108 q^{35} - 24 q^{36} + 22 q^{37} + 24 q^{38} + 6 q^{39} + 48 q^{40} + 24 q^{42} + 28 q^{43} - 24 q^{44} + 36 q^{45} - 72 q^{46} - 132 q^{47} - 2 q^{49} - 192 q^{50} - 48 q^{51} + 12 q^{52} + 120 q^{53} - 48 q^{56} - 84 q^{57} - 96 q^{58} - 24 q^{59} - 24 q^{60} - 72 q^{61} + 30 q^{63} + 32 q^{64} + 180 q^{65} + 110 q^{67} + 96 q^{68} + 96 q^{70} + 312 q^{71} - 66 q^{73} - 48 q^{74} + 66 q^{75} + 120 q^{77} + 192 q^{78} - 10 q^{79} - 48 q^{80} - 18 q^{81} + 48 q^{82} + 60 q^{84} - 288 q^{85} - 24 q^{86} - 72 q^{89} - 222 q^{91} - 96 q^{92} + 102 q^{93} - 24 q^{94} - 132 q^{95} - 72 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 7.24264 + 4.18154i 2.44949i −6.74264 + 1.88064i 2.82843 1.50000 2.59808i −10.2426 + 5.91359i
19.2 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i −1.24264 0.717439i 2.44949i 1.74264 + 6.77962i −2.82843 1.50000 2.59808i −1.75736 + 1.01461i
31.1 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 7.24264 4.18154i 2.44949i −6.74264 1.88064i 2.82843 1.50000 + 2.59808i −10.2426 5.91359i
31.2 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.24264 + 0.717439i 2.44949i 1.74264 6.77962i −2.82843 1.50000 + 2.59808i −1.75736 1.01461i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.3.g.a 4
3.b odd 2 1 126.3.n.a 4
4.b odd 2 1 336.3.bh.e 4
5.b even 2 1 1050.3.p.a 4
5.c odd 4 2 1050.3.q.a 8
7.b odd 2 1 294.3.g.a 4
7.c even 3 1 294.3.c.a 4
7.c even 3 1 294.3.g.a 4
7.d odd 6 1 inner 42.3.g.a 4
7.d odd 6 1 294.3.c.a 4
12.b even 2 1 1008.3.cg.h 4
21.c even 2 1 882.3.n.e 4
21.g even 6 1 126.3.n.a 4
21.g even 6 1 882.3.c.b 4
21.h odd 6 1 882.3.c.b 4
21.h odd 6 1 882.3.n.e 4
28.f even 6 1 336.3.bh.e 4
28.f even 6 1 2352.3.f.e 4
28.g odd 6 1 2352.3.f.e 4
35.i odd 6 1 1050.3.p.a 4
35.k even 12 2 1050.3.q.a 8
84.j odd 6 1 1008.3.cg.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 1.a even 1 1 trivial
42.3.g.a 4 7.d odd 6 1 inner
126.3.n.a 4 3.b odd 2 1
126.3.n.a 4 21.g even 6 1
294.3.c.a 4 7.c even 3 1
294.3.c.a 4 7.d odd 6 1
294.3.g.a 4 7.b odd 2 1
294.3.g.a 4 7.c even 3 1
336.3.bh.e 4 4.b odd 2 1
336.3.bh.e 4 28.f even 6 1
882.3.c.b 4 21.g even 6 1
882.3.c.b 4 21.h odd 6 1
882.3.n.e 4 21.c even 2 1
882.3.n.e 4 21.h odd 6 1
1008.3.cg.h 4 12.b even 2 1
1008.3.cg.h 4 84.j odd 6 1
1050.3.p.a 4 5.b even 2 1
1050.3.p.a 4 35.i odd 6 1
1050.3.q.a 8 5.c odd 4 2
1050.3.q.a 8 35.k even 12 2
2352.3.f.e 4 28.f even 6 1
2352.3.f.e 4 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T^{2} + T^{4} \)
$3$ \( ( 3 - 3 T + T^{2} )^{2} \)
$5$ \( 144 + 144 T + 36 T^{2} - 12 T^{3} + T^{4} \)
$7$ \( 2401 + 490 T + 51 T^{2} + 10 T^{3} + T^{4} \)
$11$ \( ( 36 + 6 T + T^{2} )^{2} \)
$13$ \( 145161 + 774 T^{2} + T^{4} \)
$17$ \( 28224 + 8064 T + 936 T^{2} + 48 T^{3} + T^{4} \)
$19$ \( 15129 + 5166 T + 711 T^{2} + 42 T^{3} + T^{4} \)
$23$ \( 254016 + 12096 T + 1080 T^{2} - 24 T^{3} + T^{4} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( 423801 - 66402 T + 4119 T^{2} - 102 T^{3} + T^{4} \)
$37$ \( 27889 + 3674 T + 651 T^{2} - 22 T^{3} + T^{4} \)
$41$ \( 3732624 + 4248 T^{2} + T^{4} \)
$43$ \( ( -23 - 14 T + T^{2} )^{2} \)
$47$ \( 2039184 + 188496 T + 7236 T^{2} + 132 T^{3} + T^{4} \)
$53$ \( 8714304 - 354240 T + 11448 T^{2} - 120 T^{3} + T^{4} \)
$59$ \( 1272384 - 27072 T - 936 T^{2} + 24 T^{3} + T^{4} \)
$61$ \( 2304 + 3456 T + 1776 T^{2} + 72 T^{3} + T^{4} \)
$67$ \( 253009 + 55330 T + 12603 T^{2} - 110 T^{3} + T^{4} \)
$71$ \( ( 2556 - 156 T + T^{2} )^{2} \)
$73$ \( 85322169 - 609642 T - 7785 T^{2} + 66 T^{3} + T^{4} \)
$79$ \( 75463969 - 86870 T + 8787 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 69956496 + 17928 T^{2} + T^{4} \)
$89$ \( ( 432 + 36 T + T^{2} )^{2} \)
$97$ \( 112896 + 1056 T^{2} + T^{4} \)
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