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Properties

Label 330.2.l.c
Level $330$
Weight $2$
Character orbit 330.l
Analytic conductor $2.635$
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] = [330,2,Mod(43,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.l (of order \(4\), 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: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.822083584.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 8x^{4} + 28x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{2} q^{3} + \beta_{4} q^{4} + (\beta_{7} + \beta_{6} - 1) q^{5} - \beta_{4} q^{6} + (\beta_{7} + \beta_{5} - \beta_{4} - 1) q^{7} + \beta_{6} q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{2} q^{3} + \beta_{4} q^{4} + (\beta_{7} + \beta_{6} - 1) q^{5} - \beta_{4} q^{6} + (\beta_{7} + \beta_{5} - \beta_{4} - 1) q^{7} + \beta_{6} q^{8} + \beta_{4} q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{10} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{11}+ \cdots + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{7} - 8 q^{10} - 8 q^{13} + 8 q^{15} - 8 q^{16} - 8 q^{17} - 16 q^{19} - 8 q^{22} + 8 q^{23} + 8 q^{24} + 16 q^{25} + 8 q^{28} + 16 q^{29} + 8 q^{33} + 16 q^{35} - 8 q^{36} + 16 q^{37} - 16 q^{38} - 8 q^{42} - 16 q^{43} + 8 q^{44} - 8 q^{47} - 8 q^{50} - 8 q^{52} - 24 q^{53} + 8 q^{54} + 16 q^{55} + 16 q^{57} + 8 q^{63} - 16 q^{65} - 8 q^{66} - 24 q^{67} + 8 q^{68} - 8 q^{70} + 32 q^{71} - 32 q^{73} + 16 q^{74} + 8 q^{75} + 24 q^{77} + 8 q^{78} + 32 q^{79} + 8 q^{80} - 8 q^{81} + 8 q^{82} + 16 q^{83} + 24 q^{85} + 48 q^{86} - 8 q^{88} - 32 q^{91} - 8 q^{92} - 16 q^{94} + 32 q^{95} - 8 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 8x^{4} + 28x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 15\nu^{4} - 107\nu^{2} + 49 ) / 231 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 15\nu^{5} - 107\nu^{3} + 49\nu ) / 231 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 15\nu^{4} - 30\nu^{2} - 28 ) / 77 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 15\nu^{5} - 30\nu^{3} - 28\nu ) / 77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} + 29\nu^{4} - 58\nu^{2} - 203 ) / 231 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{7} + 29\nu^{5} - 58\nu^{3} - 203\nu ) / 231 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 3\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + 11\beta_{4} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{7} + 11\beta_{5} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -45\beta_{6} + 29\beta_{4} - 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -45\beta_{7} + 29\beta_{5} - 29\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(-\beta_{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.94107 − 0.804019i
1.94107 + 0.804019i
−0.481906 + 1.16342i
0.481906 − 1.16342i
−1.94107 + 0.804019i
1.94107 − 0.804019i
−0.481906 − 1.16342i
0.481906 + 1.16342i
−0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −2.23397 + 0.0969123i − 1.00000i −2.13705 − 2.13705i 0.707107 − 0.707107i 1.00000i 1.64818 + 1.51113i
43.2 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.64818 − 1.51113i − 1.00000i 0.137055 + 0.137055i 0.707107 − 0.707107i 1.00000i −2.23397 − 0.0969123i
43.3 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −2.18901 − 0.456316i − 1.00000i −2.64533 − 2.64533i −0.707107 + 0.707107i 1.00000i −1.22520 − 1.87053i
43.4 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.22520 + 1.87053i − 1.00000i 0.645329 + 0.645329i −0.707107 + 0.707107i 1.00000i −2.18901 + 0.456316i
307.1 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −2.23397 − 0.0969123i 1.00000i −2.13705 + 2.13705i 0.707107 + 0.707107i − 1.00000i 1.64818 − 1.51113i
307.2 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 1.64818 + 1.51113i 1.00000i 0.137055 − 0.137055i 0.707107 + 0.707107i − 1.00000i −2.23397 + 0.0969123i
307.3 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −2.18901 + 0.456316i 1.00000i −2.64533 + 2.64533i −0.707107 − 0.707107i − 1.00000i −1.22520 + 1.87053i
307.4 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −1.22520 − 1.87053i 1.00000i 0.645329 − 0.645329i −0.707107 − 0.707107i − 1.00000i −2.18901 − 0.456316i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.l.c ✓ 8
3.b odd 2 1 990.2.m.f 8
5.b even 2 1 1650.2.l.f 8
5.c odd 4 1 330.2.l.d yes 8
5.c odd 4 1 1650.2.l.c 8
11.b odd 2 1 330.2.l.d yes 8
15.e even 4 1 990.2.m.g 8
33.d even 2 1 990.2.m.g 8
55.d odd 2 1 1650.2.l.c 8
55.e even 4 1 inner 330.2.l.c ✓ 8
55.e even 4 1 1650.2.l.f 8
165.l odd 4 1 990.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.l.c ✓ 8 1.a even 1 1 trivial
330.2.l.c ✓ 8 55.e even 4 1 inner
330.2.l.d yes 8 5.c odd 4 1
330.2.l.d yes 8 11.b odd 2 1
990.2.m.f 8 3.b odd 2 1
990.2.m.f 8 165.l odd 4 1
990.2.m.g 8 15.e even 4 1
990.2.m.g 8 33.d even 2 1
1650.2.l.c 8 5.c odd 4 1
1650.2.l.c 8 55.d odd 2 1
1650.2.l.f 8 5.b even 2 1
1650.2.l.f 8 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 8T_{7}^{7} + 32T_{7}^{6} + 48T_{7}^{5} + 12T_{7}^{4} - 80T_{7}^{3} + 128T_{7}^{2} - 32T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 2116 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 26896 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 56644 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 40 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 32 T^{2} + 224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 136 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$43$ \( T^{8} + 16 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} + \cdots + 315844 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots + 122500 \) Copy content Toggle raw display
$59$ \( T^{8} + 256 T^{6} + \cdots + 2458624 \) Copy content Toggle raw display
$61$ \( T^{8} + 392 T^{6} + \cdots + 19909444 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + \cdots + 8464 \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + \cdots + 4850)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots + 50176 \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} + \cdots - 158)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots + 4129024 \) Copy content Toggle raw display
$89$ \( T^{8} + 272 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$97$ \( T^{8} + 512 T^{5} + \cdots + 3211264 \) Copy content Toggle raw display
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