Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.319401608085\)
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: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} - q^{9} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{10} + 2 \beta_{3} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{3} + 3) q^{14} + (\beta_{2} + 2 \beta_1 - 2) q^{15} + ( - 2 \beta_{3} - 2) q^{16} + (2 \beta_{2} + 4 \beta_1) q^{17} - \beta_1 q^{18} - 2 \beta_{3} q^{19} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{20} + 2 \beta_{3} q^{21} + (4 \beta_{2} + 2 \beta_1) q^{22} + (\beta_{2} + 2 \beta_1) q^{23} + 4 q^{24} + ( - 2 \beta_{2} - 4 \beta_1 - 1) q^{25} - 4 \beta_{2} q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{28} + (\beta_{3} - 2 \beta_1 - 3) q^{30} + 4 q^{31} + ( - 4 \beta_{2} - 4 \beta_1) q^{32} + ( - 2 \beta_{2} - 4 \beta_1) q^{33} + (2 \beta_{3} - 6) q^{34} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{35} + ( - \beta_{3} + 1) q^{36} + 6 \beta_{2} q^{37} + ( - 4 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{2} + 4 \beta_1 - 4) q^{40} + (4 \beta_{2} + 2 \beta_1) q^{42} - 3 \beta_{2} q^{43} + ( - 2 \beta_{3} - 6) q^{44} + (\beta_{3} + \beta_{2}) q^{45} + (\beta_{3} - 3) q^{46} + ( - 3 \beta_{2} - 6 \beta_1) q^{47} + 4 \beta_1 q^{48} + q^{49} + ( - 2 \beta_{3} - \beta_1 + 6) q^{50} - 4 \beta_{3} q^{51} - 4 \beta_{2} q^{53} + (4 \beta_{3} + 4) q^{54} + (2 \beta_{2} + 4 \beta_1 + 6) q^{55} + 4 \beta_{3} q^{56} + (2 \beta_{2} + 4 \beta_1) q^{57} + 6 \beta_{3} q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{60} - 2 \beta_{3} q^{61} + 4 \beta_1 q^{62} + (\beta_{2} + 2 \beta_1) q^{63} + 8 q^{64} + ( - 2 \beta_{3} + 6) q^{66} + 3 \beta_{2} q^{67} + (4 \beta_{2} - 4 \beta_1) q^{68} - 2 \beta_{3} q^{69} + ( - 3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 3) q^{70} - 12 q^{71} - 2 \beta_{2} q^{72} + (2 \beta_{2} + 4 \beta_1) q^{73} + ( - 6 \beta_{3} - 6) q^{74} + (4 \beta_{3} - \beta_{2}) q^{75} + (2 \beta_{3} + 6) q^{76} - 6 \beta_{2} q^{77} - 4 q^{79} + (2 \beta_{3} - 4 \beta_1 - 6) q^{80} - 5 q^{81} + 7 \beta_{2} q^{83} + ( - 2 \beta_{3} - 6) q^{84} + (4 \beta_{3} - 6 \beta_{2}) q^{85} + (3 \beta_{3} + 3) q^{86} + ( - 4 \beta_{2} - 8 \beta_1) q^{88} + 6 q^{89} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{90} + (2 \beta_{2} - 2 \beta_1) q^{92} + 4 \beta_{2} q^{93} + ( - 3 \beta_{3} + 9) q^{94} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{95} + (4 \beta_{3} - 4) q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} + \beta_1 q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{10} + 12 q^{14} - 8 q^{15} - 8 q^{16} + 12 q^{20} + 16 q^{24} - 4 q^{25} - 12 q^{30} + 16 q^{31} - 24 q^{34} + 4 q^{36} - 16 q^{40} - 24 q^{44} - 12 q^{46} + 4 q^{49} + 24 q^{50} + 16 q^{54} + 24 q^{55} + 8 q^{60} + 32 q^{64} + 24 q^{66} - 12 q^{70} - 48 q^{71} - 24 q^{74} + 24 q^{76} - 16 q^{79} - 24 q^{80} - 20 q^{81} - 24 q^{84} + 12 q^{86} + 24 q^{89} - 4 q^{90} + 36 q^{94} - 24 q^{95} - 16 q^{96}+O(q^{100}) \) 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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i 1.41421 −1.00000 + 1.73205i −1.41421 1.73205i −1.00000 1.73205i 2.44949i 2.82843 −1.00000 −1.12132 + 2.95680i
29.2 −0.707107 + 1.22474i 1.41421 −1.00000 1.73205i −1.41421 + 1.73205i −1.00000 + 1.73205i 2.44949i 2.82843 −1.00000 −1.12132 2.95680i
29.3 0.707107 1.22474i −1.41421 −1.00000 1.73205i 1.41421 + 1.73205i −1.00000 + 1.73205i 2.44949i −2.82843 −1.00000 3.12132 0.507306i
29.4 0.707107 + 1.22474i −1.41421 −1.00000 + 1.73205i 1.41421 1.73205i −1.00000 1.73205i 2.44949i −2.82843 −1.00000 3.12132 + 0.507306i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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