Properties

Label 616.4.a.e
Level $616$
Weight $4$
Character orbit 616.a
Self dual yes
Analytic conductor $36.345$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,4,Mod(1,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, 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("616.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 616.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: \(36.3451765635\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 40x^{2} - 41x + 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + ( - \beta_{2} - 5) q^{5} - 7 q^{7} + (3 \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + ( - \beta_{2} - 5) q^{5} - 7 q^{7} + (3 \beta_{2} + 10) q^{9} + 11 q^{11} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 21) q^{15}+ \cdots + (33 \beta_{2} + 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} - 19 q^{5} - 28 q^{7} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} - 19 q^{5} - 28 q^{7} + 37 q^{9} + 44 q^{11} + 2 q^{13} + 71 q^{15} + 8 q^{17} - 6 q^{19} + 21 q^{21} + 159 q^{23} - 119 q^{25} - 117 q^{27} + 144 q^{29} - 183 q^{31} - 33 q^{33} + 133 q^{35} - 475 q^{37} - 64 q^{39} - 768 q^{41} - 152 q^{43} - 1048 q^{45} - 228 q^{47} + 196 q^{49} - 882 q^{51} + 396 q^{53} - 209 q^{55} - 1286 q^{57} - 733 q^{59} - 1012 q^{61} - 259 q^{63} + 420 q^{65} - 171 q^{67} - 321 q^{69} - 1019 q^{71} - 1836 q^{73} - 1016 q^{75} - 308 q^{77} + 1374 q^{79} - 956 q^{81} - 1248 q^{83} + 46 q^{85} + 2238 q^{87} - 1401 q^{89} - 14 q^{91} - 1859 q^{93} + 238 q^{95} - 2559 q^{97} + 407 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 40x^{2} - 41x + 194 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 25\nu - 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 39\nu + 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 5\nu^{2} + 23\nu - 69 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 8\beta _1 + 78 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 51\beta_{2} + 86\beta _1 + 114 ) / 4 \) 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
1.80678
−4.79958
−3.46450
6.45730
0 −8.36227 0 −15.9758 0 −7.00000 0 42.9275 0
1.2 0 −4.22993 0 1.36923 0 −7.00000 0 −9.10770 0
1.3 0 2.43230 0 5.36131 0 −7.00000 0 −21.0839 0
1.4 0 7.15990 0 −9.75471 0 −7.00000 0 24.2641 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(11\) \( -1 \)

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 616.4.a.e 4
4.b odd 2 1 1232.4.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.e 4 1.a even 1 1 trivial
1232.4.a.t 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\):

\( T_{3}^{4} + 3T_{3}^{3} - 68T_{3}^{2} - 120T_{3} + 616 \) Copy content Toggle raw display
\( T_{5}^{4} + 19T_{5}^{3} - 10T_{5}^{2} - 860T_{5} + 1144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 616 \) Copy content Toggle raw display
$5$ \( T^{4} + 19 T^{3} + \cdots + 1144 \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 63616 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots + 37838944 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 8981504 \) Copy content Toggle raw display
$23$ \( T^{4} - 159 T^{3} + \cdots - 27228096 \) Copy content Toggle raw display
$29$ \( T^{4} - 144 T^{3} + \cdots + 32288592 \) Copy content Toggle raw display
$31$ \( T^{4} + 183 T^{3} + \cdots - 967624 \) Copy content Toggle raw display
$37$ \( T^{4} + 475 T^{3} + \cdots - 672375512 \) Copy content Toggle raw display
$41$ \( T^{4} + 768 T^{3} + \cdots + 40685664 \) Copy content Toggle raw display
$43$ \( T^{4} + 152 T^{3} + \cdots - 162100224 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10623846656 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 2515760528 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 28943918424 \) Copy content Toggle raw display
$61$ \( T^{4} + 1012 T^{3} + \cdots + 772785216 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 327391058816 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 16928024256 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 54340975104 \) Copy content Toggle raw display
$79$ \( T^{4} - 1374 T^{3} + \cdots - 954747648 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 31070871552 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 94585872472 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 13654212712 \) Copy content Toggle raw display
show more
show less