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Properties

Label 608.2.i.c
Level $608$
Weight $2$
Character orbit 608.i
Analytic conductor $4.855$
Analytic rank $0$
Dimension $8$
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] = [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: 8.0.41342275584.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} + 2x^{5} + 81x^{4} - 8x^{3} + 208x^{2} + 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{7} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{7} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{9} + (\beta_{6} - \beta_{5} + 1) q^{11} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{13}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 2 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 2 q^{5} + 8 q^{7} + 4 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{17} - 2 q^{19} - 8 q^{21} - 2 q^{23} - 2 q^{25} + 8 q^{27} - 10 q^{29} - 24 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 12 q^{39} + 8 q^{41} + 18 q^{43} + 16 q^{45} + 6 q^{47} + 32 q^{49} - 18 q^{51} - 10 q^{53} - 20 q^{55} + 10 q^{57} + 8 q^{59} + 18 q^{61} - 8 q^{63} - 36 q^{65} - 4 q^{67} + 52 q^{69} + 6 q^{71} + 4 q^{75} + 16 q^{77} - 14 q^{79} + 4 q^{81} + 4 q^{83} - 22 q^{85} + 60 q^{87} - 2 q^{89} + 40 q^{91} - 16 q^{93} - 50 q^{95} - 12 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} + 2x^{5} + 81x^{4} - 8x^{3} + 208x^{2} + 128x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 212\nu^{6} + 901\nu^{5} - 3304\nu^{4} + 4081\nu^{3} - 9010\nu^{2} + 23992\nu - 15264 ) / 22304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{7} - 142\nu^{6} + 255\nu^{5} + 462\nu^{4} - 1797\nu^{3} + 1632\nu^{2} + 1536\nu + 34384 ) / 11152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -77\nu^{7} + 244\nu^{6} - 1037\nu^{5} + 696\nu^{4} - 4697\nu^{3} + 10370\nu^{2} - 10576\nu + 17568 ) / 22304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -45\nu^{7} + 18\nu^{6} - 425\nu^{5} - 770\nu^{4} - 4877\nu^{3} - 2720\nu^{2} - 2560\nu - 9856 ) / 11152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 10\nu^{6} + 85\nu^{5} + 154\nu^{4} + 473\nu^{3} + 544\nu^{2} + 512\nu + 1728 ) / 1088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 241\nu^{7} - 900\nu^{6} + 3825\nu^{5} - 5944\nu^{4} + 17325\nu^{3} - 38250\nu^{2} + 22384\nu - 64800 ) / 22304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} + 5\beta_{4} + \beta_{3} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} - 7\beta_{5} + 2\beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{7} - 41\beta_{4} + 8\beta_{2} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 52\beta_{6} + 69\beta_{5} - 96\beta_{4} - 36\beta_{3} + 52\beta_{2} - 69\beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 144\beta_{6} + 201\beta_{5} - 157\beta_{3} + 433 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 502\beta_{7} + 1306\beta_{4} - 628\beta_{2} + 791\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(1\)

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
−0.564469 + 0.977689i
1.77158 − 3.06846i
−1.10890 + 1.92067i
0.901794 − 1.56195i
−0.564469 − 0.977689i
1.77158 + 3.06846i
−1.10890 − 1.92067i
0.901794 + 1.56195i
0 −1.20711 + 2.09077i 0 −1.27158 + 2.20243i 0 5.01077 0 −1.41421 − 2.44949i 0
353.2 0 −1.20711 + 2.09077i 0 1.06447 − 1.84371i 0 −1.59656 0 −1.41421 − 2.44949i 0
353.3 0 0.207107 − 0.358719i 0 −0.401794 + 0.695928i 0 −2.55066 0 1.41421 + 2.44949i 0
353.4 0 0.207107 − 0.358719i 0 1.60890 − 2.78670i 0 3.13644 0 1.41421 + 2.44949i 0
577.1 0 −1.20711 − 2.09077i 0 −1.27158 − 2.20243i 0 5.01077 0 −1.41421 + 2.44949i 0
577.2 0 −1.20711 − 2.09077i 0 1.06447 + 1.84371i 0 −1.59656 0 −1.41421 + 2.44949i 0
577.3 0 0.207107 + 0.358719i 0 −0.401794 − 0.695928i 0 −2.55066 0 1.41421 − 2.44949i 0
577.4 0 0.207107 + 0.358719i 0 1.60890 + 2.78670i 0 3.13644 0 1.41421 − 2.44949i 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
19.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) 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
$3$ \( (T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 14 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 9 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 8464 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 33856 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + \cdots + 42436 \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + \cdots + 1119364 \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots - 392)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} - 38 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + \cdots + 25921 \) Copy content Toggle raw display
$43$ \( T^{8} - 18 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + \cdots + 31684 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 795664 \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + \cdots + 47089 \) Copy content Toggle raw display
$61$ \( T^{8} - 18 T^{7} + \cdots + 103684 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 27636049 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 11262736 \) Copy content Toggle raw display
$73$ \( T^{8} + 98 T^{6} + \cdots + 124609 \) Copy content Toggle raw display
$79$ \( T^{8} + 14 T^{7} + \cdots + 33362176 \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + \cdots + 1052)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 2 T^{7} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( T^{8} + 12 T^{7} + \cdots + 508369 \) Copy content Toggle raw display
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