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Properties

Label 608.2.i.d
Level $608$
Weight $2$
Character orbit 608.i
Analytic conductor $4.855$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,2,Mod(353,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.i (of order \(3\), degree \(2\), 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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.39075800976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 12x^{6} - 13x^{5} + 125x^{4} - 116x^{3} + 232x^{2} + 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{5} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{5} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1 - 2) q^{9} + (\beta_{6} - \beta_{2}) q^{11} + ( - \beta_{5} + 2 \beta_{3} - \beta_1 + 1) q^{13} + (\beta_{7} + \beta_{6} - 2 \beta_{4}) q^{15} + ( - 2 \beta_{3} + \beta_1 + 2) q^{17} + ( - \beta_{7} + \beta_{4}) q^{19} + (\beta_{7} + \beta_{6}) q^{23} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{25} + (2 \beta_{6} - 3 \beta_{2}) q^{27} + ( - 3 \beta_{5} - 3 \beta_1 + 3) q^{29} + 2 \beta_{2} q^{31} + (3 \beta_{3} - 5 \beta_1 - 3) q^{33} + ( - 2 \beta_{5} + 4) q^{37} + ( - \beta_{6} + 4 \beta_{2}) q^{39} + (\beta_{3} - 1) q^{41} + ( - \beta_{7} - 2 \beta_{4} + 2 \beta_{2}) q^{43} + (4 \beta_{5} - 12) q^{45} + (\beta_{7} + \beta_{6} - 2 \beta_{4}) q^{47} - 7 q^{49} + (\beta_{7} + \beta_{6} - 4 \beta_{4}) q^{51} + (\beta_{5} - 2 \beta_{3} + \beta_1 - 1) q^{53} + (4 \beta_{4} - 4 \beta_{2}) q^{55} + ( - 5 \beta_{5} - 2 \beta_{3} + \cdots + 10) q^{57}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + 10 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 12 q^{9} + 10 q^{13} + 10 q^{17} + 2 q^{25} + 6 q^{29} - 22 q^{33} + 24 q^{37} - 4 q^{41} - 80 q^{45} - 56 q^{49} - 10 q^{53} + 46 q^{57} + 18 q^{61} + 44 q^{65} + 4 q^{69} + 28 q^{73} - 32 q^{81} - 22 q^{85} + 18 q^{89} + 48 q^{93} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 12x^{6} - 13x^{5} + 125x^{4} - 116x^{3} + 232x^{2} + 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 83\nu^{7} - 325\nu^{6} + 2470\nu^{5} - 3543\nu^{4} + 26065\nu^{3} - 38870\nu^{2} + 139144\nu - 11440 ) / 47664 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1067 \nu^{7} - 1445 \nu^{6} - 10864 \nu^{5} - 873 \nu^{4} - 106543 \nu^{3} - 31816 \nu^{2} + \cdots + 330448 ) / 238320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 557\nu^{7} - 865\nu^{6} + 6574\nu^{5} - 10377\nu^{4} + 69373\nu^{3} - 103454\nu^{2} + 120040\nu + 48992 ) / 79440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1138 \nu^{7} - 3140 \nu^{6} + 12941 \nu^{5} - 35178 \nu^{4} + 140612 \nu^{3} - 305041 \nu^{2} + \cdots + 80128 ) / 119160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 154\nu^{7} + 55\nu^{6} + 1568\nu^{5} + 126\nu^{4} + 14456\nu^{3} + 4592\nu^{2} + 2240\nu + 37684 ) / 14895 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 649\nu^{7} + 19\nu^{6} + 6608\nu^{5} + 531\nu^{4} + 71561\nu^{3} + 19352\nu^{2} + 9440\nu + 229456 ) / 47664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5429 \nu^{7} - 9055 \nu^{6} + 68818 \nu^{5} - 118329 \nu^{4} + 726211 \nu^{3} - 1082978 \nu^{2} + \cdots - 318736 ) / 238320 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} - 11\beta_{3} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{6} - 7\beta_{5} - 4\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} - 37\beta_{4} + 99\beta_{3} + 37\beta_{2} - 9\beta _1 - 99 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -75\beta_{7} - 75\beta_{6} + 149\beta_{5} + 79\beta_{4} + 25\beta_{3} + 149\beta _1 - 149 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{6} - 55\beta_{5} - 200\beta_{2} + 532 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 725\beta_{7} - 849\beta_{4} + 7\beta_{3} + 849\beta_{2} - 1525\beta _1 - 7 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
−1.63248 + 2.82754i
1.51772 − 2.62877i
−0.236942 + 0.410396i
0.851703 − 1.47519i
−1.63248 − 2.82754i
1.51772 + 2.62877i
−0.236942 − 0.410396i
0.851703 + 1.47519i
0 −1.59084 + 2.75542i 0 1.28078 − 2.21837i 0 0 0 −3.56155 − 6.16879i 0
353.2 0 −0.684999 + 1.18645i 0 −0.780776 + 1.35234i 0 0 0 0.561553 + 0.972638i 0
353.3 0 0.684999 − 1.18645i 0 −0.780776 + 1.35234i 0 0 0 0.561553 + 0.972638i 0
353.4 0 1.59084 − 2.75542i 0 1.28078 − 2.21837i 0 0 0 −3.56155 − 6.16879i 0
577.1 0 −1.59084 − 2.75542i 0 1.28078 + 2.21837i 0 0 0 −3.56155 + 6.16879i 0
577.2 0 −0.684999 − 1.18645i 0 −0.780776 − 1.35234i 0 0 0 0.561553 − 0.972638i 0
577.3 0 0.684999 + 1.18645i 0 −0.780776 − 1.35234i 0 0 0 0.561553 − 0.972638i 0
577.4 0 1.59084 + 2.75542i 0 1.28078 + 2.21837i 0 0 0 −3.56155 + 6.16879i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.i.d ✓ 8
4.b odd 2 1 inner 608.2.i.d ✓ 8
8.b even 2 1 1216.2.i.p 8
8.d odd 2 1 1216.2.i.p 8
19.c even 3 1 inner 608.2.i.d ✓ 8
76.g odd 6 1 inner 608.2.i.d ✓ 8
152.k odd 6 1 1216.2.i.p 8
152.p even 6 1 1216.2.i.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.d ✓ 8 1.a even 1 1 trivial
608.2.i.d ✓ 8 4.b odd 2 1 inner
608.2.i.d ✓ 8 19.c even 3 1 inner
608.2.i.d ✓ 8 76.g odd 6 1 inner
1216.2.i.p 8 8.b even 2 1
1216.2.i.p 8 8.d odd 2 1
1216.2.i.p 8 152.k odd 6 1
1216.2.i.p 8 152.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12T_{3}^{6} + 125T_{3}^{4} + 228T_{3}^{2} + 361 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 12 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + 5 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 37 T^{2} + 304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} + 23 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T^{3} + 23 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 35 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 27 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 48 T^{2} + 304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 79 T^{6} + \cdots + 1478656 \) Copy content Toggle raw display
$47$ \( T^{8} + 71 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$53$ \( (T^{4} + 5 T^{3} + 23 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( (T^{4} - 9 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 228 T^{6} + \cdots + 47045881 \) Copy content Toggle raw display
$71$ \( T^{8} + 27 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$73$ \( (T^{4} - 14 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 71 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$83$ \( (T^{4} - 41 T^{2} + 76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} + \cdots + 324)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 6 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
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