Properties

Label 240.6.f.a
Level $240$
Weight $6$
Character orbit 240.f
Analytic conductor $38.492$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.4921167551\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q -9 i q^{3} + ( -55 + 10 i ) q^{5} + 4 i q^{7} -81 q^{9} +O(q^{10})\) \( q -9 i q^{3} + ( -55 + 10 i ) q^{5} + 4 i q^{7} -81 q^{9} + 500 q^{11} + 288 i q^{13} + ( 90 + 495 i ) q^{15} + 1516 i q^{17} -1344 q^{19} + 36 q^{21} -4100 i q^{23} + ( 2925 - 1100 i ) q^{25} + 729 i q^{27} + 2646 q^{29} + 5612 q^{31} -4500 i q^{33} + ( -40 - 220 i ) q^{35} -7288 i q^{37} + 2592 q^{39} -18986 q^{41} -2404 i q^{43} + ( 4455 - 810 i ) q^{45} -8900 i q^{47} + 16791 q^{49} + 13644 q^{51} -39804 i q^{53} + ( -27500 + 5000 i ) q^{55} + 12096 i q^{57} -28300 q^{59} + 18290 q^{61} -324 i q^{63} + ( -2880 - 15840 i ) q^{65} -65956 i q^{67} -36900 q^{69} + 28800 q^{71} + 30808 i q^{73} + ( -9900 - 26325 i ) q^{75} + 2000 i q^{77} + 60228 q^{79} + 6561 q^{81} -2468 i q^{83} + ( -15160 - 83380 i ) q^{85} -23814 i q^{87} -22678 q^{89} -1152 q^{91} -50508 i q^{93} + ( 73920 - 13440 i ) q^{95} -36968 i q^{97} -40500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 110q^{5} - 162q^{9} + O(q^{10}) \) \( 2q - 110q^{5} - 162q^{9} + 1000q^{11} + 180q^{15} - 2688q^{19} + 72q^{21} + 5850q^{25} + 5292q^{29} + 11224q^{31} - 80q^{35} + 5184q^{39} - 37972q^{41} + 8910q^{45} + 33582q^{49} + 27288q^{51} - 55000q^{55} - 56600q^{59} + 36580q^{61} - 5760q^{65} - 73800q^{69} + 57600q^{71} - 19800q^{75} + 120456q^{79} + 13122q^{81} - 30320q^{85} - 45356q^{89} - 2304q^{91} + 147840q^{95} - 81000q^{99} + O(q^{100}) \)

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 9.00000i 0 −55.0000 + 10.0000i 0 4.00000i 0 −81.0000 0
49.2 0 9.00000i 0 −55.0000 10.0000i 0 4.00000i 0 −81.0000 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 240.6.f.a 2
3.b odd 2 1 720.6.f.g 2
4.b odd 2 1 30.6.c.a 2
5.b even 2 1 inner 240.6.f.a 2
12.b even 2 1 90.6.c.b 2
15.d odd 2 1 720.6.f.g 2
20.d odd 2 1 30.6.c.a 2
20.e even 4 1 150.6.a.f 1
20.e even 4 1 150.6.a.j 1
60.h even 2 1 90.6.c.b 2
60.l odd 4 1 450.6.a.f 1
60.l odd 4 1 450.6.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 4.b odd 2 1
30.6.c.a 2 20.d odd 2 1
90.6.c.b 2 12.b even 2 1
90.6.c.b 2 60.h even 2 1
150.6.a.f 1 20.e even 4 1
150.6.a.j 1 20.e even 4 1
240.6.f.a 2 1.a even 1 1 trivial
240.6.f.a 2 5.b even 2 1 inner
450.6.a.f 1 60.l odd 4 1
450.6.a.s 1 60.l odd 4 1
720.6.f.g 2 3.b odd 2 1
720.6.f.g 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16 \) acting on \(S_{6}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 81 + T^{2} \)
$5$ \( 3125 + 110 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( -500 + T )^{2} \)
$13$ \( 82944 + T^{2} \)
$17$ \( 2298256 + T^{2} \)
$19$ \( ( 1344 + T )^{2} \)
$23$ \( 16810000 + T^{2} \)
$29$ \( ( -2646 + T )^{2} \)
$31$ \( ( -5612 + T )^{2} \)
$37$ \( 53114944 + T^{2} \)
$41$ \( ( 18986 + T )^{2} \)
$43$ \( 5779216 + T^{2} \)
$47$ \( 79210000 + T^{2} \)
$53$ \( 1584358416 + T^{2} \)
$59$ \( ( 28300 + T )^{2} \)
$61$ \( ( -18290 + T )^{2} \)
$67$ \( 4350193936 + T^{2} \)
$71$ \( ( -28800 + T )^{2} \)
$73$ \( 949132864 + T^{2} \)
$79$ \( ( -60228 + T )^{2} \)
$83$ \( 6091024 + T^{2} \)
$89$ \( ( 22678 + T )^{2} \)
$97$ \( 1366633024 + T^{2} \)
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