Properties

Label 42.2.d.a
Level 42
Weight 2
Character orbit 42.d
Analytic conductor 0.335
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 42.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.335371688489\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.00000i −1.22474 1.22474i −1.00000 2.44949 −1.22474 + 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
41.2 1.00000i 1.22474 + 1.22474i −1.00000 −2.44949 1.22474 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
41.3 1.00000i −1.22474 + 1.22474i −1.00000 2.44949 −1.22474 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
41.4 1.00000i 1.22474 1.22474i −1.00000 −2.44949 1.22474 + 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.2.d.a 4
3.b odd 2 1 inner 42.2.d.a 4
4.b odd 2 1 336.2.k.b 4
5.b even 2 1 1050.2.b.b 4
5.c odd 4 1 1050.2.d.b 4
5.c odd 4 1 1050.2.d.e 4
7.b odd 2 1 inner 42.2.d.a 4
7.c even 3 2 294.2.f.b 8
7.d odd 6 2 294.2.f.b 8
8.b even 2 1 1344.2.k.c 4
8.d odd 2 1 1344.2.k.d 4
9.c even 3 2 1134.2.m.g 8
9.d odd 6 2 1134.2.m.g 8
12.b even 2 1 336.2.k.b 4
15.d odd 2 1 1050.2.b.b 4
15.e even 4 1 1050.2.d.b 4
15.e even 4 1 1050.2.d.e 4
21.c even 2 1 inner 42.2.d.a 4
21.g even 6 2 294.2.f.b 8
21.h odd 6 2 294.2.f.b 8
24.f even 2 1 1344.2.k.d 4
24.h odd 2 1 1344.2.k.c 4
28.d even 2 1 336.2.k.b 4
35.c odd 2 1 1050.2.b.b 4
35.f even 4 1 1050.2.d.b 4
35.f even 4 1 1050.2.d.e 4
56.e even 2 1 1344.2.k.d 4
56.h odd 2 1 1344.2.k.c 4
63.l odd 6 2 1134.2.m.g 8
63.o even 6 2 1134.2.m.g 8
84.h odd 2 1 336.2.k.b 4
105.g even 2 1 1050.2.b.b 4
105.k odd 4 1 1050.2.d.b 4
105.k odd 4 1 1050.2.d.e 4
168.e odd 2 1 1344.2.k.d 4
168.i even 2 1 1344.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 1.a even 1 1 trivial
42.2.d.a 4 3.b odd 2 1 inner
42.2.d.a 4 7.b odd 2 1 inner
42.2.d.a 4 21.c even 2 1 inner
294.2.f.b 8 7.c even 3 2
294.2.f.b 8 7.d odd 6 2
294.2.f.b 8 21.g even 6 2
294.2.f.b 8 21.h odd 6 2
336.2.k.b 4 4.b odd 2 1
336.2.k.b 4 12.b even 2 1
336.2.k.b 4 28.d even 2 1
336.2.k.b 4 84.h odd 2 1
1050.2.b.b 4 5.b even 2 1
1050.2.b.b 4 15.d odd 2 1
1050.2.b.b 4 35.c odd 2 1
1050.2.b.b 4 105.g even 2 1
1050.2.d.b 4 5.c odd 4 1
1050.2.d.b 4 15.e even 4 1
1050.2.d.b 4 35.f even 4 1
1050.2.d.b 4 105.k odd 4 1
1050.2.d.e 4 5.c odd 4 1
1050.2.d.e 4 15.e even 4 1
1050.2.d.e 4 35.f even 4 1
1050.2.d.e 4 105.k odd 4 1
1134.2.m.g 8 9.c even 3 2
1134.2.m.g 8 9.d odd 6 2
1134.2.m.g 8 63.l odd 6 2
1134.2.m.g 8 63.o even 6 2
1344.2.k.c 4 8.b even 2 1
1344.2.k.c 4 24.h odd 2 1
1344.2.k.c 4 56.h odd 2 1
1344.2.k.c 4 168.i even 2 1
1344.2.k.d 4 8.d odd 2 1
1344.2.k.d 4 24.f even 2 1
1344.2.k.d 4 56.e even 2 1
1344.2.k.d 4 168.e odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 10 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
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