Properties

Label 240.2.f.a
Level $240$
Weight $2$
Character orbit 240.f
Analytic conductor $1.916$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} - 2 q^{11} + 6 i q^{13} + ( - 2 i - 1) q^{15} - 2 i q^{17} - 2 q^{21} + 4 i q^{23} + ( - 4 i + 3) q^{25} - i q^{27} + 8 q^{31} - 2 i q^{33} + ( - 4 i - 2) q^{35} - 2 i q^{37} - 6 q^{39} + 2 q^{41} + 4 i q^{43} + ( - i + 2) q^{45} - 8 i q^{47} + 3 q^{49} + 2 q^{51} + 6 i q^{53} + ( - 2 i + 4) q^{55} + 10 q^{59} + 2 q^{61} - 2 i q^{63} + ( - 12 i - 6) q^{65} - 8 i q^{67} - 4 q^{69} - 12 q^{71} - 4 i q^{73} + (3 i + 4) q^{75} - 4 i q^{77} + q^{81} + 4 i q^{83} + (4 i + 2) q^{85} + 10 q^{89} - 12 q^{91} + 8 i q^{93} + 8 i q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 4 q^{21} + 6 q^{25} + 16 q^{31} - 4 q^{35} - 12 q^{39} + 4 q^{41} + 4 q^{45} + 6 q^{49} + 4 q^{51} + 8 q^{55} + 20 q^{59} + 4 q^{61} - 12 q^{65} - 8 q^{69} - 24 q^{71} + 8 q^{75} + 2 q^{81} + 4 q^{85} + 20 q^{89} - 24 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 −2.00000 1.00000i 0 2.00000i 0 −1.00000 0
49.2 0 1.00000i 0 −2.00000 + 1.00000i 0 2.00000i 0 −1.00000 0
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.f.a 2
3.b odd 2 1 720.2.f.f 2
4.b odd 2 1 30.2.c.a 2
5.b even 2 1 inner 240.2.f.a 2
5.c odd 4 1 1200.2.a.g 1
5.c odd 4 1 1200.2.a.m 1
8.b even 2 1 960.2.f.i 2
8.d odd 2 1 960.2.f.h 2
12.b even 2 1 90.2.c.a 2
15.d odd 2 1 720.2.f.f 2
15.e even 4 1 3600.2.a.o 1
15.e even 4 1 3600.2.a.bg 1
16.e even 4 1 3840.2.d.j 2
16.e even 4 1 3840.2.d.x 2
16.f odd 4 1 3840.2.d.g 2
16.f odd 4 1 3840.2.d.y 2
20.d odd 2 1 30.2.c.a 2
20.e even 4 1 150.2.a.a 1
20.e even 4 1 150.2.a.c 1
24.f even 2 1 2880.2.f.e 2
24.h odd 2 1 2880.2.f.c 2
28.d even 2 1 1470.2.g.g 2
28.f even 6 2 1470.2.n.a 4
28.g odd 6 2 1470.2.n.h 4
36.f odd 6 2 810.2.i.e 4
36.h even 6 2 810.2.i.b 4
40.e odd 2 1 960.2.f.h 2
40.f even 2 1 960.2.f.i 2
40.i odd 4 1 4800.2.a.m 1
40.i odd 4 1 4800.2.a.cj 1
40.k even 4 1 4800.2.a.l 1
40.k even 4 1 4800.2.a.cg 1
60.h even 2 1 90.2.c.a 2
60.l odd 4 1 450.2.a.b 1
60.l odd 4 1 450.2.a.f 1
80.k odd 4 1 3840.2.d.g 2
80.k odd 4 1 3840.2.d.y 2
80.q even 4 1 3840.2.d.j 2
80.q even 4 1 3840.2.d.x 2
120.i odd 2 1 2880.2.f.c 2
120.m even 2 1 2880.2.f.e 2
140.c even 2 1 1470.2.g.g 2
140.j odd 4 1 7350.2.a.bg 1
140.j odd 4 1 7350.2.a.cc 1
140.p odd 6 2 1470.2.n.h 4
140.s even 6 2 1470.2.n.a 4
180.n even 6 2 810.2.i.b 4
180.p odd 6 2 810.2.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 4.b odd 2 1
30.2.c.a 2 20.d odd 2 1
90.2.c.a 2 12.b even 2 1
90.2.c.a 2 60.h even 2 1
150.2.a.a 1 20.e even 4 1
150.2.a.c 1 20.e even 4 1
240.2.f.a 2 1.a even 1 1 trivial
240.2.f.a 2 5.b even 2 1 inner
450.2.a.b 1 60.l odd 4 1
450.2.a.f 1 60.l odd 4 1
720.2.f.f 2 3.b odd 2 1
720.2.f.f 2 15.d odd 2 1
810.2.i.b 4 36.h even 6 2
810.2.i.b 4 180.n even 6 2
810.2.i.e 4 36.f odd 6 2
810.2.i.e 4 180.p odd 6 2
960.2.f.h 2 8.d odd 2 1
960.2.f.h 2 40.e odd 2 1
960.2.f.i 2 8.b even 2 1
960.2.f.i 2 40.f even 2 1
1200.2.a.g 1 5.c odd 4 1
1200.2.a.m 1 5.c odd 4 1
1470.2.g.g 2 28.d even 2 1
1470.2.g.g 2 140.c even 2 1
1470.2.n.a 4 28.f even 6 2
1470.2.n.a 4 140.s even 6 2
1470.2.n.h 4 28.g odd 6 2
1470.2.n.h 4 140.p odd 6 2
2880.2.f.c 2 24.h odd 2 1
2880.2.f.c 2 120.i odd 2 1
2880.2.f.e 2 24.f even 2 1
2880.2.f.e 2 120.m even 2 1
3600.2.a.o 1 15.e even 4 1
3600.2.a.bg 1 15.e even 4 1
3840.2.d.g 2 16.f odd 4 1
3840.2.d.g 2 80.k odd 4 1
3840.2.d.j 2 16.e even 4 1
3840.2.d.j 2 80.q even 4 1
3840.2.d.x 2 16.e even 4 1
3840.2.d.x 2 80.q even 4 1
3840.2.d.y 2 16.f odd 4 1
3840.2.d.y 2 80.k odd 4 1
4800.2.a.l 1 40.k even 4 1
4800.2.a.m 1 40.i odd 4 1
4800.2.a.cg 1 40.k even 4 1
4800.2.a.cj 1 40.i odd 4 1
7350.2.a.bg 1 140.j odd 4 1
7350.2.a.cc 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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