Properties

Label 42.3.c.a
Level $42$
Weight $3$
Character orbit 42.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14441711031\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{3} q^{6} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{3} q^{6} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{10} + ( -6 - 3 \beta_{1} ) q^{11} -2 \beta_{2} q^{12} + ( 8 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{14} + ( -6 + 3 \beta_{1} ) q^{15} + 4 q^{16} + ( -2 \beta_{2} + 11 \beta_{3} ) q^{17} -3 \beta_{1} q^{18} + ( -2 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -4 \beta_{2} - 2 \beta_{3} ) q^{20} + ( 3 + 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -6 - 6 \beta_{1} ) q^{22} + ( 6 - 9 \beta_{1} ) q^{23} + 2 \beta_{3} q^{24} + ( 7 + 12 \beta_{1} ) q^{25} + ( -4 \beta_{2} - 8 \beta_{3} ) q^{26} + 3 \beta_{2} q^{27} + ( 4 - 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{28} + 30 q^{29} + ( 6 - 6 \beta_{1} ) q^{30} + ( -12 \beta_{2} + 12 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -22 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -6 + 9 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} ) q^{35} -6 q^{36} + ( -20 + 36 \beta_{1} ) q^{37} + ( 16 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 24 - 6 \beta_{1} ) q^{39} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{40} + ( 14 \beta_{2} + 7 \beta_{3} ) q^{41} + ( 12 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{42} + ( -32 - 30 \beta_{1} ) q^{43} + ( -12 - 6 \beta_{1} ) q^{44} + ( 6 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -18 + 6 \beta_{1} ) q^{46} + ( 28 \beta_{2} - 4 \beta_{3} ) q^{47} -4 \beta_{2} q^{48} + ( -5 - 24 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 24 + 7 \beta_{1} ) q^{50} + ( -6 - 33 \beta_{1} ) q^{51} + ( 16 \beta_{2} + 4 \beta_{3} ) q^{52} + ( -54 - 12 \beta_{1} ) q^{53} -3 \beta_{3} q^{54} + 6 \beta_{2} q^{55} + ( -12 + 4 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{56} + ( -6 + 24 \beta_{1} ) q^{57} + 30 \beta_{1} q^{58} + ( -28 \beta_{2} - 20 \beta_{3} ) q^{59} + ( -12 + 6 \beta_{1} ) q^{60} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{61} + ( -24 \beta_{2} + 12 \beta_{3} ) q^{62} + ( -6 + 9 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{63} + 8 q^{64} + ( 60 - 36 \beta_{1} ) q^{65} + ( 6 \beta_{2} - 6 \beta_{3} ) q^{66} + ( 44 - 12 \beta_{1} ) q^{67} + ( -4 \beta_{2} + 22 \beta_{3} ) q^{68} + ( -6 \beta_{2} - 9 \beta_{3} ) q^{69} + ( 18 - 6 \beta_{1} + 16 \beta_{2} + 10 \beta_{3} ) q^{70} + ( -30 + 57 \beta_{1} ) q^{71} -6 \beta_{1} q^{72} + ( 4 \beta_{2} - 26 \beta_{3} ) q^{73} + ( 72 - 20 \beta_{1} ) q^{74} + ( -7 \beta_{2} + 12 \beta_{3} ) q^{75} + ( -4 \beta_{2} - 16 \beta_{3} ) q^{76} + ( 6 + 12 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{77} + ( -12 + 24 \beta_{1} ) q^{78} + ( 32 + 72 \beta_{1} ) q^{79} + ( -8 \beta_{2} - 4 \beta_{3} ) q^{80} + 9 q^{81} + ( -14 \beta_{2} - 14 \beta_{3} ) q^{82} + ( 32 \beta_{2} - 20 \beta_{3} ) q^{83} + ( 6 + 12 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{84} + ( 54 - 60 \beta_{1} ) q^{85} + ( -60 - 32 \beta_{1} ) q^{86} -30 \beta_{2} q^{87} + ( -12 - 12 \beta_{1} ) q^{88} + ( -54 \beta_{2} + 21 \beta_{3} ) q^{89} + ( -6 \beta_{2} - 6 \beta_{3} ) q^{90} + ( -42 \beta_{1} + 28 \beta_{2} + 28 \beta_{3} ) q^{91} + ( 12 - 18 \beta_{1} ) q^{92} + ( -36 - 36 \beta_{1} ) q^{93} + ( 8 \beta_{2} - 28 \beta_{3} ) q^{94} + ( -60 + 54 \beta_{1} ) q^{95} + 4 \beta_{3} q^{96} + ( 44 \beta_{2} - 10 \beta_{3} ) q^{97} + ( -48 - 5 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} ) q^{98} + ( 18 + 9 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 8q^{7} - 12q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 8q^{7} - 12q^{9} - 24q^{11} - 24q^{14} - 24q^{15} + 16q^{16} + 12q^{21} - 24q^{22} + 24q^{23} + 28q^{25} + 16q^{28} + 120q^{29} + 24q^{30} - 24q^{35} - 24q^{36} - 80q^{37} + 96q^{39} + 48q^{42} - 128q^{43} - 48q^{44} - 72q^{46} - 20q^{49} + 96q^{50} - 24q^{51} - 216q^{53} - 48q^{56} - 24q^{57} - 48q^{60} - 24q^{63} + 32q^{64} + 240q^{65} + 176q^{67} + 72q^{70} - 120q^{71} + 288q^{74} + 24q^{77} - 48q^{78} + 128q^{79} + 36q^{81} + 24q^{84} + 216q^{85} - 240q^{86} - 48q^{88} + 48q^{92} - 144q^{93} - 240q^{95} - 192q^{98} + 72q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 5.91359i 2.44949i 6.24264 3.16693i −2.82843 −3.00000 8.36308i
13.2 −1.41421 1.73205i 2.00000 5.91359i 2.44949i 6.24264 + 3.16693i −2.82843 −3.00000 8.36308i
13.3 1.41421 1.73205i 2.00000 1.01461i 2.44949i −2.24264 + 6.63103i 2.82843 −3.00000 1.43488i
13.4 1.41421 1.73205i 2.00000 1.01461i 2.44949i −2.24264 6.63103i 2.82843 −3.00000 1.43488i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.3.c.a 4
3.b odd 2 1 126.3.c.b 4
4.b odd 2 1 336.3.f.c 4
5.b even 2 1 1050.3.f.a 4
5.c odd 4 2 1050.3.h.a 8
7.b odd 2 1 inner 42.3.c.a 4
7.c even 3 1 294.3.g.b 4
7.c even 3 1 294.3.g.c 4
7.d odd 6 1 294.3.g.b 4
7.d odd 6 1 294.3.g.c 4
8.b even 2 1 1344.3.f.f 4
8.d odd 2 1 1344.3.f.e 4
12.b even 2 1 1008.3.f.g 4
21.c even 2 1 126.3.c.b 4
21.g even 6 1 882.3.n.a 4
21.g even 6 1 882.3.n.d 4
21.h odd 6 1 882.3.n.a 4
21.h odd 6 1 882.3.n.d 4
28.d even 2 1 336.3.f.c 4
35.c odd 2 1 1050.3.f.a 4
35.f even 4 2 1050.3.h.a 8
56.e even 2 1 1344.3.f.e 4
56.h odd 2 1 1344.3.f.f 4
84.h odd 2 1 1008.3.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 1.a even 1 1 trivial
42.3.c.a 4 7.b odd 2 1 inner
126.3.c.b 4 3.b odd 2 1
126.3.c.b 4 21.c even 2 1
294.3.g.b 4 7.c even 3 1
294.3.g.b 4 7.d odd 6 1
294.3.g.c 4 7.c even 3 1
294.3.g.c 4 7.d odd 6 1
336.3.f.c 4 4.b odd 2 1
336.3.f.c 4 28.d even 2 1
882.3.n.a 4 21.g even 6 1
882.3.n.a 4 21.h odd 6 1
882.3.n.d 4 21.g even 6 1
882.3.n.d 4 21.h odd 6 1
1008.3.f.g 4 12.b even 2 1
1008.3.f.g 4 84.h odd 2 1
1050.3.f.a 4 5.b even 2 1
1050.3.f.a 4 35.c odd 2 1
1050.3.h.a 8 5.c odd 4 2
1050.3.h.a 8 35.f even 4 2
1344.3.f.e 4 8.d odd 2 1
1344.3.f.e 4 56.e even 2 1
1344.3.f.f 4 8.b even 2 1
1344.3.f.f 4 56.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( ( 1 + 3 T^{2} )^{2} \)
$5$ \( 1 - 64 T^{2} + 1986 T^{4} - 40000 T^{6} + 390625 T^{8} \)
$7$ \( 1 - 8 T + 42 T^{2} - 392 T^{3} + 2401 T^{4} \)
$11$ \( ( 1 + 12 T + 260 T^{2} + 1452 T^{3} + 14641 T^{4} )^{2} \)
$13$ \( 1 - 244 T^{2} + 53574 T^{4} - 6968884 T^{6} + 815730721 T^{8} \)
$17$ \( 1 + 320 T^{2} + 157794 T^{4} + 26726720 T^{6} + 6975757441 T^{8} \)
$19$ \( 1 - 652 T^{2} + 348486 T^{4} - 84969292 T^{6} + 16983563041 T^{8} \)
$23$ \( ( 1 - 12 T + 932 T^{2} - 6348 T^{3} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 - 30 T + 841 T^{2} )^{4} \)
$31$ \( 1 - 1252 T^{2} + 745926 T^{4} - 1156248292 T^{6} + 852891037441 T^{8} \)
$37$ \( ( 1 + 40 T + 546 T^{2} + 54760 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 14015774560 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 + 64 T + 2922 T^{2} + 118336 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 - 3940 T^{2} + 12737094 T^{4} - 19225943140 T^{6} + 23811286661761 T^{8} \)
$53$ \( ( 1 + 108 T + 8246 T^{2} + 303372 T^{3} + 7890481 T^{4} )^{2} \)
$59$ \( 1 - 4420 T^{2} + 6539622 T^{4} - 53558735620 T^{6} + 146830437604321 T^{8} \)
$61$ \( 1 - 14596 T^{2} + 80934054 T^{4} - 202093895236 T^{6} + 191707312997281 T^{8} \)
$67$ \( ( 1 - 88 T + 10626 T^{2} - 395032 T^{3} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 + 60 T + 4484 T^{2} + 302460 T^{3} + 25411681 T^{4} )^{2} \)
$73$ \( 1 - 13108 T^{2} + 98972646 T^{4} - 372244143028 T^{6} + 806460091894081 T^{8} \)
$79$ \( ( 1 - 64 T + 3138 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2} \)
$83$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 788377628452 T^{6} + 2252292232139041 T^{8} \)
$89$ \( 1 - 8896 T^{2} + 52680354 T^{4} - 558154975936 T^{6} + 3936588805702081 T^{8} \)
$97$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 2197296754420 T^{6} + 7837433594376961 T^{8} \)
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