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Properties

Label 608.2.b.c
Level $608$
Weight $2$
Character orbit 608.b
Analytic conductor $4.855$
Analytic rank $0$
Dimension $12$
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(303,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.303");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.b (of order \(2\), degree \(1\), not 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: \(12\)
Coefficient field: 12.0.319794774016000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} - \beta_{7} q^{7} + ( - \beta_{9} - \beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} - \beta_{7} q^{7} + ( - \beta_{9} - \beta_{8} - 1) q^{9} + \beta_{9} q^{11} + \beta_{11} q^{13} + (\beta_{6} + 2 \beta_{4}) q^{15} + (2 \beta_{9} + 1) q^{17} + (\beta_{10} + \beta_{9} + 2) q^{19} + (\beta_{6} + \beta_{4}) q^{21} + (\beta_{7} - \beta_{3} - \beta_{2}) q^{23} + (\beta_{9} - \beta_{8} - 4) q^{25} + ( - \beta_{10} + 2 \beta_{5} + 2 \beta_1) q^{27} + ( - \beta_{6} + \beta_{4}) q^{29} - \beta_{6} q^{31} + (\beta_{10} - \beta_{5} - 2 \beta_1) q^{33} + (\beta_{9} - 2 \beta_{8} - 4) q^{35} + ( - \beta_{11} + \beta_{6} - \beta_{4}) q^{37} + ( - \beta_{7} - \beta_{3} - 3 \beta_{2}) q^{39} + (\beta_{10} + \beta_{5} + 2 \beta_1) q^{41} + ( - 3 \beta_{9} + 2) q^{43} + (\beta_{7} - \beta_{3} - 4 \beta_{2}) q^{45} + ( - 2 \beta_{7} - \beta_{3} + \beta_{2}) q^{47} + 2 q^{49} + (2 \beta_{10} - 2 \beta_{5} - 5 \beta_1) q^{51} + (\beta_{6} + 3 \beta_{4}) q^{53} + (2 \beta_{7} + \beta_{3} + 3 \beta_{2}) q^{55} + (\beta_{10} - \beta_{9} + \beta_{8} + \cdots + 2) q^{57}+ \cdots + ( - \beta_{9} - 2 \beta_{8} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} + 12 q^{17} + 24 q^{19} - 44 q^{25} - 40 q^{35} + 24 q^{43} + 24 q^{49} + 20 q^{57} + 4 q^{73} + 36 q^{81} - 24 q^{83} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 2\nu^{7} + 4\nu^{5} + 16\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 2\nu^{6} + 4\nu^{4} + 16\nu^{2} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 2\nu^{7} + 12\nu^{5} - 16\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 2\nu^{7} + 4\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 2\nu^{7} - 4\nu^{5} + 8\nu^{3} + 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} - 2\nu^{4} + 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} - 2\nu^{6} + 8\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} - 6\nu^{7} + 4\nu^{5} + 32\nu^{3} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 2\nu^{7} - 12\nu^{5} + 96\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{5} + \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} + \beta_{6} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} + \beta_{5} + 3\beta_{4} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} - 3\beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - 6\beta_{7} - 2\beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{11} - 2\beta_{10} - 2\beta_{6} + 7\beta_{5} - 3\beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -8\beta_{9} - 6\beta_{8} - 6\beta_{7} + 2\beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4\beta_{11} - 2\beta_{10} - 10\beta_{6} - 4\beta_{5} - 4\beta_{4} + 10\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
303.1
−1.37364 + 0.336338i
1.37364 + 0.336338i
1.17117 + 0.792696i
−1.17117 + 0.792696i
−0.491416 + 1.32609i
0.491416 + 1.32609i
0.491416 − 1.32609i
−0.491416 − 1.32609i
−1.17117 − 0.792696i
1.17117 − 0.792696i
1.37364 − 0.336338i
−1.37364 − 0.336338i
0 − 2.97320i 0 − 3.04222i 0 − 2.23607i 0 −5.83991 0
303.2 0 − 2.97320i 0 3.04222i 0 2.23607i 0 −5.83991 0
303.3 0 − 1.26152i 0 − 3.51876i 0 − 2.23607i 0 1.40857 0
303.4 0 − 1.26152i 0 3.51876i 0 2.23607i 0 1.40857 0
303.5 0 − 0.754098i 0 − 2.08884i 0 2.23607i 0 2.43134 0
303.6 0 − 0.754098i 0 2.08884i 0 − 2.23607i 0 2.43134 0
303.7 0 0.754098i 0 − 2.08884i 0 2.23607i 0 2.43134 0
303.8 0 0.754098i 0 2.08884i 0 − 2.23607i 0 2.43134 0
303.9 0 1.26152i 0 − 3.51876i 0 − 2.23607i 0 1.40857 0
303.10 0 1.26152i 0 3.51876i 0 2.23607i 0 1.40857 0
303.11 0 2.97320i 0 − 3.04222i 0 − 2.23607i 0 −5.83991 0
303.12 0 2.97320i 0 3.04222i 0 2.23607i 0 −5.83991 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 303.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.b.c 12
3.b odd 2 1 5472.2.e.e 12
4.b odd 2 1 152.2.b.c ✓ 12
8.b even 2 1 152.2.b.c ✓ 12
8.d odd 2 1 inner 608.2.b.c 12
12.b even 2 1 1368.2.e.e 12
19.b odd 2 1 inner 608.2.b.c 12
24.f even 2 1 5472.2.e.e 12
24.h odd 2 1 1368.2.e.e 12
57.d even 2 1 5472.2.e.e 12
76.d even 2 1 152.2.b.c ✓ 12
152.b even 2 1 inner 608.2.b.c 12
152.g odd 2 1 152.2.b.c ✓ 12
228.b odd 2 1 1368.2.e.e 12
456.l odd 2 1 5472.2.e.e 12
456.p even 2 1 1368.2.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.b.c ✓ 12 4.b odd 2 1
152.2.b.c ✓ 12 8.b even 2 1
152.2.b.c ✓ 12 76.d even 2 1
152.2.b.c ✓ 12 152.g odd 2 1
608.2.b.c 12 1.a even 1 1 trivial
608.2.b.c 12 8.d odd 2 1 inner
608.2.b.c 12 19.b odd 2 1 inner
608.2.b.c 12 152.b even 2 1 inner
1368.2.e.e 12 12.b even 2 1
1368.2.e.e 12 24.h odd 2 1
1368.2.e.e 12 228.b odd 2 1
1368.2.e.e 12 456.p even 2 1
5472.2.e.e 12 3.b odd 2 1
5472.2.e.e 12 24.f even 2 1
5472.2.e.e 12 57.d even 2 1
5472.2.e.e 12 456.l odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 11T_{3}^{4} + 20T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 11 T^{4} + 20 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 26 T^{4} + \cdots + 500)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5)^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 7 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 51 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T^{2} - 25 T + 59)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} - 12 T^{5} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 69 T^{4} + \cdots + 320)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 71 T^{4} + \cdots - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 44 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 84 T^{4} + \cdots - 2560)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 136 T^{4} + \cdots + 70688)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} - 51 T + 10)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 106 T^{4} + \cdots + 80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 191 T^{4} + \cdots - 112360)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 103 T^{4} + \cdots + 12800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 214 T^{4} + \cdots + 50000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 155 T^{4} + \cdots + 75272)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 476 T^{4} + \cdots - 3504640)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - T^{2} - 65 T + 97)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 424 T^{4} + \cdots - 1697440)^{2} \) Copy content Toggle raw display
$83$ \( (T + 2)^{12} \) Copy content Toggle raw display
$89$ \( (T^{6} + 232 T^{4} + \cdots + 231200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 316 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
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