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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,6,Mod(49,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4921167551\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{6}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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