Properties

Label 40.2.d.a
Level $40$
Weight $2$
Character orbit 40.d
Analytic conductor $0.319$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 40.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.319401608085\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 2.73205i 1.73205 + 1.00000i 1.00000i 1.00000 3.73205i 0.732051 −2.00000 2.00000i −4.46410 −0.366025 + 1.36603i
21.2 −1.36603 + 0.366025i 2.73205i 1.73205 1.00000i 1.00000i 1.00000 + 3.73205i 0.732051 −2.00000 + 2.00000i −4.46410 −0.366025 1.36603i
21.3 0.366025 1.36603i 0.732051i −1.73205 1.00000i 1.00000i 1.00000 + 0.267949i −2.73205 −2.00000 + 2.00000i 2.46410 1.36603 + 0.366025i
21.4 0.366025 + 1.36603i 0.732051i −1.73205 + 1.00000i 1.00000i 1.00000 0.267949i −2.73205 −2.00000 2.00000i 2.46410 1.36603 0.366025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.2.d.a 4
3.b odd 2 1 360.2.k.e 4
4.b odd 2 1 160.2.d.a 4
5.b even 2 1 200.2.d.f 4
5.c odd 4 1 200.2.f.c 4
5.c odd 4 1 200.2.f.e 4
8.b even 2 1 inner 40.2.d.a 4
8.d odd 2 1 160.2.d.a 4
12.b even 2 1 1440.2.k.e 4
15.d odd 2 1 1800.2.k.j 4
15.e even 4 1 1800.2.d.l 4
15.e even 4 1 1800.2.d.p 4
16.e even 4 1 1280.2.a.a 2
16.e even 4 1 1280.2.a.o 2
16.f odd 4 1 1280.2.a.d 2
16.f odd 4 1 1280.2.a.n 2
20.d odd 2 1 800.2.d.e 4
20.e even 4 1 800.2.f.c 4
20.e even 4 1 800.2.f.e 4
24.f even 2 1 1440.2.k.e 4
24.h odd 2 1 360.2.k.e 4
40.e odd 2 1 800.2.d.e 4
40.f even 2 1 200.2.d.f 4
40.i odd 4 1 200.2.f.c 4
40.i odd 4 1 200.2.f.e 4
40.k even 4 1 800.2.f.c 4
40.k even 4 1 800.2.f.e 4
60.h even 2 1 7200.2.k.j 4
60.l odd 4 1 7200.2.d.n 4
60.l odd 4 1 7200.2.d.o 4
80.k odd 4 1 6400.2.a.be 2
80.k odd 4 1 6400.2.a.cj 2
80.q even 4 1 6400.2.a.z 2
80.q even 4 1 6400.2.a.ce 2
120.i odd 2 1 1800.2.k.j 4
120.m even 2 1 7200.2.k.j 4
120.q odd 4 1 7200.2.d.n 4
120.q odd 4 1 7200.2.d.o 4
120.w even 4 1 1800.2.d.l 4
120.w even 4 1 1800.2.d.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.d.a 4 1.a even 1 1 trivial
40.2.d.a 4 8.b even 2 1 inner
160.2.d.a 4 4.b odd 2 1
160.2.d.a 4 8.d odd 2 1
200.2.d.f 4 5.b even 2 1
200.2.d.f 4 40.f even 2 1
200.2.f.c 4 5.c odd 4 1
200.2.f.c 4 40.i odd 4 1
200.2.f.e 4 5.c odd 4 1
200.2.f.e 4 40.i odd 4 1
360.2.k.e 4 3.b odd 2 1
360.2.k.e 4 24.h odd 2 1
800.2.d.e 4 20.d odd 2 1
800.2.d.e 4 40.e odd 2 1
800.2.f.c 4 20.e even 4 1
800.2.f.c 4 40.k even 4 1
800.2.f.e 4 20.e even 4 1
800.2.f.e 4 40.k even 4 1
1280.2.a.a 2 16.e even 4 1
1280.2.a.d 2 16.f odd 4 1
1280.2.a.n 2 16.f odd 4 1
1280.2.a.o 2 16.e even 4 1
1440.2.k.e 4 12.b even 2 1
1440.2.k.e 4 24.f even 2 1
1800.2.d.l 4 15.e even 4 1
1800.2.d.l 4 120.w even 4 1
1800.2.d.p 4 15.e even 4 1
1800.2.d.p 4 120.w even 4 1
1800.2.k.j 4 15.d odd 2 1
1800.2.k.j 4 120.i odd 2 1
6400.2.a.z 2 80.q even 4 1
6400.2.a.be 2 80.k odd 4 1
6400.2.a.ce 2 80.q even 4 1
6400.2.a.cj 2 80.k odd 4 1
7200.2.d.n 4 60.l odd 4 1
7200.2.d.n 4 120.q odd 4 1
7200.2.d.o 4 60.l odd 4 1
7200.2.d.o 4 120.q odd 4 1
7200.2.k.j 4 60.h even 2 1
7200.2.k.j 4 120.m even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 4 + 8 T^{2} + T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -2 + 2 T + T^{2} )^{2} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( 16 + 56 T^{2} + T^{4} \)
$23$ \( ( -26 + 2 T + T^{2} )^{2} \)
$29$ \( ( 48 + T^{2} )^{2} \)
$31$ \( ( -8 + 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( -8 + 4 T + T^{2} )^{2} \)
$43$ \( 2116 + 104 T^{2} + T^{4} \)
$47$ \( ( 22 - 10 T + T^{2} )^{2} \)
$53$ \( 2704 + 152 T^{2} + T^{4} \)
$59$ \( 16 + 56 T^{2} + T^{4} \)
$61$ \( 1936 + 104 T^{2} + T^{4} \)
$67$ \( 6084 + 168 T^{2} + T^{4} \)
$71$ \( ( -8 - 4 T + T^{2} )^{2} \)
$73$ \( ( 4 - 8 T + T^{2} )^{2} \)
$79$ \( ( 16 + 16 T + T^{2} )^{2} \)
$83$ \( 36 + 24 T^{2} + T^{4} \)
$89$ \( ( -44 - 4 T + T^{2} )^{2} \)
$97$ \( ( -92 + 8 T + T^{2} )^{2} \)
show more
show less