Properties

Label 2240.4.a.bz
Level $2240$
Weight $4$
Character orbit 2240.a
Self dual yes
Analytic conductor $132.164$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,4,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

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: yes
Analytic conductor: \(132.164278413\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.11045.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 31x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + 5 q^{5} + 7 q^{7} + (\beta_{2} - 7 \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + 5 q^{5} + 7 q^{7} + (\beta_{2} - 7 \beta_1 + 8) q^{9} + (3 \beta_{2} + \beta_1 + 1) q^{11} + ( - 2 \beta_{2} + 7 \beta_1 - 2) q^{13} + ( - 5 \beta_1 + 10) q^{15} + ( - 4 \beta_{2} - 5 \beta_1 - 16) q^{17} + ( - 7 \beta_{2} - 12 \beta_1 + 53) q^{19} + ( - 7 \beta_1 + 14) q^{21} + (3 \beta_{2} - 20 \beta_1 - 17) q^{23} + 25 q^{25} + (6 \beta_{2} - 23 \beta_1 + 180) q^{27} + (9 \beta_{2} - 13 \beta_1 + 153) q^{29} + (5 \beta_{2} + 10 \beta_1 - 91) q^{31} + ( - 4 \beta_{2} - 17 \beta_1 - 26) q^{33} + 35 q^{35} + ( - 13 \beta_{2} + 22 \beta_1 - 83) q^{37} + ( - 5 \beta_{2} + 51 \beta_1 - 223) q^{39} + (13 \beta_{2} + 12 \beta_1 - 29) q^{41} + (11 \beta_{2} + 24 \beta_1 + 227) q^{43} + (5 \beta_{2} - 35 \beta_1 + 40) q^{45} + ( - 6 \beta_{2} + 25 \beta_1 - 256) q^{47} + 49 q^{49} + (9 \beta_{2} + 19 \beta_1 + 119) q^{51} + ( - 14 \beta_{2} - 22 \beta_1 - 16) q^{53} + (15 \beta_{2} + 5 \beta_1 + 5) q^{55} + (19 \beta_{2} - 64 \beta_1 + 471) q^{57} + ( - 4 \beta_{2} - 40 \beta_1 + 364) q^{59} + (33 \beta_{2} + 68 \beta_1 - 69) q^{61} + (7 \beta_{2} - 49 \beta_1 + 56) q^{63} + ( - 10 \beta_{2} + 35 \beta_1 - 10) q^{65} + ( - 20 \beta_{2} - 64 \beta_1 - 64) q^{67} + (17 \beta_{2} - 104 \beta_1 + 589) q^{69} + (24 \beta_{2} - 96 \beta_1 - 360) q^{71} + ( - 14 \beta_{2} + 92 \beta_1 - 60) q^{73} + ( - 25 \beta_1 + 50) q^{75} + (21 \beta_{2} + 7 \beta_1 + 7) q^{77} + ( - 49 \beta_{2} - 67 \beta_1 + 137) q^{79} + ( - 10 \beta_{2} - 148 \beta_1 + 863) q^{81} + (22 \beta_{2} + 40 \beta_1 + 558) q^{83} + ( - 20 \beta_{2} - 25 \beta_1 - 80) q^{85} + (4 \beta_{2} - 281 \beta_1 + 718) q^{87} + (77 \beta_{2} - 12 \beta_1 + 27) q^{89} + ( - 14 \beta_{2} + 49 \beta_1 - 14) q^{91} + ( - 15 \beta_{2} + 106 \beta_1 - 487) q^{93} + ( - 35 \beta_{2} - 60 \beta_1 + 265) q^{95} + ( - 36 \beta_{2} + 207 \beta_1 + 552) q^{97} + ( - 60 \beta_{2} - 58 \beta_1 + 444) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{3} + 15 q^{5} + 21 q^{7} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{3} + 15 q^{5} + 21 q^{7} + 25 q^{9} + 6 q^{11} - 8 q^{13} + 30 q^{15} - 52 q^{17} + 152 q^{19} + 42 q^{21} - 48 q^{23} + 75 q^{25} + 546 q^{27} + 468 q^{29} - 268 q^{31} - 82 q^{33} + 105 q^{35} - 262 q^{37} - 674 q^{39} - 74 q^{41} + 692 q^{43} + 125 q^{45} - 774 q^{47} + 147 q^{49} + 366 q^{51} - 62 q^{53} + 30 q^{55} + 1432 q^{57} + 1088 q^{59} - 174 q^{61} + 175 q^{63} - 40 q^{65} - 212 q^{67} + 1784 q^{69} - 1056 q^{71} - 194 q^{73} + 150 q^{75} + 42 q^{77} + 362 q^{79} + 2579 q^{81} + 1696 q^{83} - 260 q^{85} + 2158 q^{87} + 158 q^{89} - 56 q^{91} - 1476 q^{93} + 760 q^{95} + 1620 q^{97} + 1272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 31x - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3\nu - 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 10\nu + 39 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 10\beta _1 + 83 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.76369
−3.44861
−2.31508
0 −3.45643 0 5.00000 0 7.00000 0 −15.0531 0
1.2 0 −0.238730 0 5.00000 0 7.00000 0 −26.9430 0
1.3 0 9.69516 0 5.00000 0 7.00000 0 66.9961 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.4.a.bz 3
4.b odd 2 1 2240.4.a.br 3
8.b even 2 1 280.4.a.f 3
8.d odd 2 1 560.4.a.w 3
40.f even 2 1 1400.4.a.m 3
40.i odd 4 2 1400.4.g.l 6
56.h odd 2 1 1960.4.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.4.a.f 3 8.b even 2 1
560.4.a.w 3 8.d odd 2 1
1400.4.a.m 3 40.f even 2 1
1400.4.g.l 6 40.i odd 4 2
1960.4.a.r 3 56.h odd 2 1
2240.4.a.br 3 4.b odd 2 1
2240.4.a.bz 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3}^{3} - 6T_{3}^{2} - 35T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} - 2855T_{11} + 24620 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6 T^{2} - 35 T - 8 \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} - 2855 T + 24620 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} - 3535 T + 20410 \) Copy content Toggle raw display
$17$ \( T^{3} + 52 T^{2} - 5287 T + 10134 \) Copy content Toggle raw display
$19$ \( T^{3} - 152 T^{2} - 14420 T + 2087776 \) Copy content Toggle raw display
$23$ \( T^{3} + 48 T^{2} - 20852 T - 1469824 \) Copy content Toggle raw display
$29$ \( T^{3} - 468 T^{2} + 39685 T + 3237134 \) Copy content Toggle raw display
$31$ \( T^{3} + 268 T^{2} + 11408 T - 921984 \) Copy content Toggle raw display
$37$ \( T^{3} + 262 T^{2} + \cdots - 11050616 \) Copy content Toggle raw display
$41$ \( T^{3} + 74 T^{2} - 57896 T - 3221920 \) Copy content Toggle raw display
$43$ \( T^{3} - 692 T^{2} + 94636 T - 3560240 \) Copy content Toggle raw display
$47$ \( T^{3} + 774 T^{2} + 159037 T + 8510472 \) Copy content Toggle raw display
$53$ \( T^{3} + 62 T^{2} - 82880 T + 5800160 \) Copy content Toggle raw display
$59$ \( T^{3} - 1088 T^{2} + \cdots - 19501056 \) Copy content Toggle raw display
$61$ \( T^{3} + 174 T^{2} + \cdots - 187440544 \) Copy content Toggle raw display
$67$ \( T^{3} + 212 T^{2} + \cdots + 41574080 \) Copy content Toggle raw display
$71$ \( T^{3} + 1056 T^{2} + \cdots - 270950400 \) Copy content Toggle raw display
$73$ \( T^{3} + 194 T^{2} + \cdots + 80306840 \) Copy content Toggle raw display
$79$ \( T^{3} - 362 T^{2} + \cdots + 362644136 \) Copy content Toggle raw display
$83$ \( T^{3} - 1696 T^{2} + \cdots - 90044928 \) Copy content Toggle raw display
$89$ \( T^{3} - 158 T^{2} + \cdots + 860463840 \) Copy content Toggle raw display
$97$ \( T^{3} - 1620 T^{2} + \cdots + 2380431942 \) Copy content Toggle raw display
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