Properties

Label 616.4.a.i
Level $616$
Weight $4$
Character orbit 616.a
Self dual yes
Analytic conductor $36.345$
Analytic rank $0$
Dimension $6$
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] = [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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + 7 q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + 7 q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + 11) q^{9}+ \cdots + (11 \beta_{4} + 22 \beta_{3} + \cdots - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9} - 66 q^{11} - 6 q^{13} + 126 q^{15} - 14 q^{17} + 80 q^{19} + 254 q^{23} + 220 q^{25} + 90 q^{27} + 132 q^{29} - 52 q^{31} + 14 q^{35} - 518 q^{37} + 332 q^{39} + 486 q^{41} + 428 q^{43} - 244 q^{45} + 790 q^{47} + 294 q^{49} + 640 q^{51} - 40 q^{53} - 22 q^{55} + 2276 q^{57} + 436 q^{59} + 1034 q^{61} + 420 q^{63} + 800 q^{65} + 562 q^{67} + 530 q^{69} + 2474 q^{71} - 902 q^{73} + 1986 q^{75} - 462 q^{77} + 1636 q^{79} + 570 q^{81} + 3016 q^{83} + 236 q^{85} + 4340 q^{87} + 1750 q^{89} - 42 q^{91} + 150 q^{93} + 3628 q^{95} - 1250 q^{97} - 660 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 49\nu^{4} - 73\nu^{3} - 970\nu^{2} - 180\nu + 216 ) / 372 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 59\nu^{4} + 193\nu^{3} - 1688\nu^{2} - 1644\nu + 6288 ) / 372 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 49\nu^{4} + 73\nu^{3} + 1714\nu^{2} - 564\nu - 11748 ) / 372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} - 71\nu^{3} + 111\nu^{2} + 608\nu - 308 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} - 28\nu^{4} - 100\nu^{3} + 665\nu^{2} - 92\nu - 2630 ) / 62 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + 3\beta_{4} + 2\beta_{3} + 2\beta_{2} - 2\beta _1 + 3 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 3\beta_{4} + 12\beta_{3} + 2\beta_{2} + 8\beta _1 + 313 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{5} + 19\beta_{4} + 26\beta_{3} + 36\beta_{2} - 16\beta _1 + 189 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{5} + 153\beta_{4} + 352\beta_{3} + 202\beta_{2} + 248\beta _1 + 7003 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 69\beta_{5} + 1273\beta_{4} + 1452\beta_{3} + 2342\beta_{2} - 352\beta _1 + 15723 ) / 20 \) 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
−2.26862
−4.66461
4.57505
5.61124
−3.68968
1.43662
0 −8.70461 0 −11.2312 0 7.00000 0 48.7703 0
1.2 0 −4.94660 0 4.15482 0 7.00000 0 −2.53111 0
1.3 0 −2.96505 0 3.10102 0 7.00000 0 −18.2085 0
1.4 0 3.60307 0 −19.7567 0 7.00000 0 −14.0179 0
1.5 0 3.73933 0 19.7791 0 7.00000 0 −13.0174 0
1.6 0 9.27387 0 5.95305 0 7.00000 0 59.0046 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.i 6
4.b odd 2 1 1232.4.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.i 6 1.a even 1 1 trivial
1232.4.a.bb 6 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\):

\( T_{3}^{6} - 111T_{3}^{4} - 30T_{3}^{3} + 2616T_{3}^{2} - 24T_{3} - 15952 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 483T_{5}^{4} + 1328T_{5}^{3} + 35184T_{5}^{2} - 216128T_{5} + 336624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 111 T^{4} + \cdots - 15952 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 336624 \) Copy content Toggle raw display
$7$ \( (T - 7)^{6} \) Copy content Toggle raw display
$11$ \( (T + 11)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots - 325632 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 28913335168 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 16168602112 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 41994606848 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 1667440361024 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 54893114800 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 58488779493744 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 61200631406976 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 75562012164096 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 62336281709568 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 366629128251072 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 234132668007984 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 42\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 773893110835200 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 31\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
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