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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + \beta_{3} q^{6} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + \beta_{3} q^{6} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} - 3 q^{9} + (2 \beta_{3} + 2 \beta_{2}) q^{10} + ( - 3 \beta_1 - 6) q^{11} - 2 \beta_{2} q^{12} + (2 \beta_{3} + 8 \beta_{2}) q^{13} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{14} + (3 \beta_1 - 6) q^{15} + 4 q^{16} + (11 \beta_{3} - 2 \beta_{2}) q^{17} - 3 \beta_1 q^{18} + ( - 8 \beta_{3} - 2 \beta_{2}) q^{19} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{20} + ( - 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 3) q^{21} + ( - 6 \beta_1 - 6) q^{22} + ( - 9 \beta_1 + 6) q^{23} + 2 \beta_{3} q^{24} + (12 \beta_1 + 7) q^{25} + ( - 8 \beta_{3} - 4 \beta_{2}) q^{26} + 3 \beta_{2} q^{27} + ( - 4 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 4) q^{28} + 30 q^{29} + ( - 6 \beta_1 + 6) q^{30} + (12 \beta_{3} - 12 \beta_{2}) q^{31} + 4 \beta_1 q^{32} + ( - 3 \beta_{3} + 6 \beta_{2}) q^{33} + (2 \beta_{3} - 22 \beta_{2}) q^{34} + ( - 8 \beta_{3} - 10 \beta_{2} + 9 \beta_1 - 6) q^{35} - 6 q^{36} + (36 \beta_1 - 20) q^{37} + (2 \beta_{3} + 16 \beta_{2}) q^{38} + ( - 6 \beta_1 + 24) q^{39} + (4 \beta_{3} + 4 \beta_{2}) q^{40} + (7 \beta_{3} + 14 \beta_{2}) q^{41} + (2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 12) q^{42} + ( - 30 \beta_1 - 32) q^{43} + ( - 6 \beta_1 - 12) q^{44} + (3 \beta_{3} + 6 \beta_{2}) q^{45} + (6 \beta_1 - 18) q^{46} + ( - 4 \beta_{3} + 28 \beta_{2}) q^{47} - 4 \beta_{2} q^{48} + ( - 2 \beta_{3} - 20 \beta_{2} - 24 \beta_1 - 5) q^{49} + (7 \beta_1 + 24) q^{50} + ( - 33 \beta_1 - 6) q^{51} + (4 \beta_{3} + 16 \beta_{2}) q^{52} + ( - 12 \beta_1 - 54) q^{53} - 3 \beta_{3} q^{54} + 6 \beta_{2} q^{55} + ( - 2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 - 12) q^{56} + (24 \beta_1 - 6) q^{57} + 30 \beta_1 q^{58} + ( - 20 \beta_{3} - 28 \beta_{2}) q^{59} + (6 \beta_1 - 12) q^{60} + (4 \beta_{3} + 4 \beta_{2}) q^{61} + (12 \beta_{3} - 24 \beta_{2}) q^{62} + (6 \beta_{3} - 3 \beta_{2} + 9 \beta_1 - 6) q^{63} + 8 q^{64} + ( - 36 \beta_1 + 60) q^{65} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{66} + ( - 12 \beta_1 + 44) q^{67} + (22 \beta_{3} - 4 \beta_{2}) q^{68} + ( - 9 \beta_{3} - 6 \beta_{2}) q^{69} + (10 \beta_{3} + 16 \beta_{2} - 6 \beta_1 + 18) q^{70} + (57 \beta_1 - 30) q^{71} - 6 \beta_1 q^{72} + ( - 26 \beta_{3} + 4 \beta_{2}) q^{73} + ( - 20 \beta_1 + 72) q^{74} + (12 \beta_{3} - 7 \beta_{2}) q^{75} + ( - 16 \beta_{3} - 4 \beta_{2}) q^{76} + (15 \beta_{3} - 18 \beta_{2} + 12 \beta_1 + 6) q^{77} + (24 \beta_1 - 12) q^{78} + (72 \beta_1 + 32) q^{79} + ( - 4 \beta_{3} - 8 \beta_{2}) q^{80} + 9 q^{81} + ( - 14 \beta_{3} - 14 \beta_{2}) q^{82} + ( - 20 \beta_{3} + 32 \beta_{2}) q^{83} + ( - 6 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 6) q^{84} + ( - 60 \beta_1 + 54) q^{85} + ( - 32 \beta_1 - 60) q^{86} - 30 \beta_{2} q^{87} + ( - 12 \beta_1 - 12) q^{88} + (21 \beta_{3} - 54 \beta_{2}) q^{89} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{90} + (28 \beta_{3} + 28 \beta_{2} - 42 \beta_1) q^{91} + ( - 18 \beta_1 + 12) q^{92} + ( - 36 \beta_1 - 36) q^{93} + ( - 28 \beta_{3} + 8 \beta_{2}) q^{94} + (54 \beta_1 - 60) q^{95} + 4 \beta_{3} q^{96} + ( - 10 \beta_{3} + 44 \beta_{2}) q^{97} + (20 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 48) q^{98} + (9 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 8 q^{7} - 12 q^{9} - 24 q^{11} - 24 q^{14} - 24 q^{15} + 16 q^{16} + 12 q^{21} - 24 q^{22} + 24 q^{23} + 28 q^{25} + 16 q^{28} + 120 q^{29} + 24 q^{30} - 24 q^{35} - 24 q^{36} - 80 q^{37} + 96 q^{39} + 48 q^{42} - 128 q^{43} - 48 q^{44} - 72 q^{46} - 20 q^{49} + 96 q^{50} - 24 q^{51} - 216 q^{53} - 48 q^{56} - 24 q^{57} - 48 q^{60} - 24 q^{63} + 32 q^{64} + 240 q^{65} + 176 q^{67} + 72 q^{70} - 120 q^{71} + 288 q^{74} + 24 q^{77} - 48 q^{78} + 128 q^{79} + 36 q^{81} + 24 q^{84} + 216 q^{85} - 240 q^{86} - 48 q^{88} + 48 q^{92} - 144 q^{93} - 240 q^{95} - 192 q^{98} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

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

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

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

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 36T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$17$ \( T^{4} + 1476 T^{2} + 509796 \) Copy content Toggle raw display
$19$ \( T^{4} + 792 T^{2} + 138384 \) Copy content Toggle raw display
$23$ \( (T^{2} - 12 T - 126)^{2} \) Copy content Toggle raw display
$29$ \( (T - 30)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2592 T^{2} + 186624 \) Copy content Toggle raw display
$37$ \( (T^{2} + 40 T - 2192)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 86436 \) Copy content Toggle raw display
$43$ \( (T^{2} + 64 T - 776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4896 T^{2} + \cdots + 5089536 \) Copy content Toggle raw display
$53$ \( (T^{2} + 108 T + 2628)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 9504 T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 288T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} - 88 T + 1648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60 T - 5598)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8208 T^{2} + \cdots + 16064064 \) Copy content Toggle raw display
$79$ \( (T^{2} - 64 T - 9344)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + \cdots + 451584 \) Copy content Toggle raw display
$89$ \( T^{4} + 22788 T^{2} + \cdots + 37234404 \) Copy content Toggle raw display
$97$ \( T^{4} + 12816 T^{2} + \cdots + 27123264 \) Copy content Toggle raw display
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