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Properties

Label 1020.2.bd.a
Level $1020$
Weight $2$
Character orbit 1020.bd
Analytic conductor $8.145$
Analytic rank $0$
Dimension $12$
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] = [1020,2,Mod(361,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1020.bd (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: \(8.14474100617\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 46x^{10} + 787x^{8} + 6174x^{6} + 22417x^{4} + 35600x^{2} + 18496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{2} q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{2} q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9} + (\beta_{10} + \beta_{7} + 2 \beta_1) q^{11} + ( - \beta_{8} - \beta_{7} - 2) q^{13} + \beta_{3} q^{15} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{17}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{7} + 4 q^{11} - 24 q^{13} + 4 q^{17} - 4 q^{23} - 20 q^{29} + 8 q^{31} + 16 q^{33} + 4 q^{37} + 4 q^{39} - 28 q^{41} - 8 q^{47} - 16 q^{55} + 24 q^{61} - 4 q^{63} - 4 q^{65} + 32 q^{67} + 8 q^{69} - 16 q^{71} + 28 q^{73} + 36 q^{79} - 12 q^{81} + 16 q^{89} + 12 q^{91} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 46x^{10} + 787x^{8} + 6174x^{6} + 22417x^{4} + 35600x^{2} + 18496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} - 833 \nu^{10} + 10 \nu^{9} - 36414 \nu^{8} - 2023 \nu^{7} - 565131 \nu^{6} + \cdots - 4949448 ) / 403648 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{11} - 833 \nu^{10} - 10 \nu^{9} - 36414 \nu^{8} + 2023 \nu^{7} - 565131 \nu^{6} + \cdots - 4949448 ) / 403648 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{11} + 1154\nu^{9} + 17521\nu^{7} + 110306\nu^{5} + 248315\nu^{3} + 137624\nu ) / 15232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 327 \nu^{11} + 644 \nu^{10} - 14446 \nu^{9} + 28364 \nu^{8} - 230629 \nu^{7} + 444752 \nu^{6} + \cdots + 5715808 ) / 201824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 327 \nu^{11} + 644 \nu^{10} + 14446 \nu^{9} + 28364 \nu^{8} + 230629 \nu^{7} + 444752 \nu^{6} + \cdots + 5715808 ) / 201824 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{11} + 306\nu^{9} + 4749\nu^{7} + 30546\nu^{5} + 70503\nu^{3} + 41592\nu ) / 1696 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 327 \nu^{11} + 1150 \nu^{10} - 14446 \nu^{9} + 48848 \nu^{8} - 230629 \nu^{7} + 734734 \nu^{6} + \cdots + 5853168 ) / 201824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 327 \nu^{11} - 1278 \nu^{10} + 14446 \nu^{9} - 53232 \nu^{8} + 230629 \nu^{7} - 783958 \nu^{6} + \cdots - 5908656 ) / 201824 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 327 \nu^{11} + 1810 \nu^{10} - 14446 \nu^{9} + 77760 \nu^{8} - 230629 \nu^{7} + 1191946 \nu^{6} + \cdots + 12332752 ) / 201824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1511\nu^{11} + 65386\nu^{9} + 1010205\nu^{7} + 6561954\nu^{5} + 15996495\nu^{3} + 11160440\nu ) / 201824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 327 \nu^{11} - 1810 \nu^{10} - 14446 \nu^{9} - 77760 \nu^{8} - 230629 \nu^{7} - 1191946 \nu^{6} + \cdots - 12332752 ) / 201824 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{9} + 2\beta_{8} + 4\beta_{7} + \beta_{5} + \beta_{4} + 4\beta_{2} + 4\beta _1 - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{11} + \beta_{10} - 5\beta_{9} - \beta_{6} - 6\beta_{5} + 6\beta_{4} + 8\beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17 \beta_{11} + 15 \beta_{9} - 36 \beta_{8} - 68 \beta_{7} - 11 \beta_{5} - 11 \beta_{4} - 60 \beta_{2} + \cdots + 192 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 141 \beta_{11} - 6 \beta_{10} + 141 \beta_{9} + 30 \beta_{6} + 159 \beta_{5} - 159 \beta_{4} + \cdots + 308 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 131 \beta_{11} - 76 \beta_{9} + 289 \beta_{8} + 496 \beta_{7} + 46 \beta_{5} + 46 \beta_{4} + \cdots - 1296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2153 \beta_{11} - 264 \beta_{10} - 2153 \beta_{9} - 456 \beta_{6} - 2173 \beta_{5} + \cdots - 5084 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4081 \beta_{11} + 1217 \beta_{9} - 8954 \beta_{8} - 14252 \beta_{7} - 457 \beta_{5} - 457 \beta_{4} + \cdots + 36512 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16529 \beta_{11} + 3959 \beta_{10} + 16529 \beta_{9} + 3769 \beta_{6} + 15158 \beta_{5} + \cdots + 40200 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 64761 \beta_{11} - 5463 \beta_{9} + 136236 \beta_{8} + 206460 \beta_{7} - 4509 \beta_{5} + \cdots - 526176 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 504957 \beta_{11} - 160986 \beta_{10} - 504957 \beta_{9} - 130438 \beta_{6} - 429927 \beta_{5} + \cdots - 1248372 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.77986i
3.15865i
− 3.93851i
− 0.990824i
3.67680i
− 1.68598i
− 1.77986i
− 3.15865i
3.93851i
0.990824i
− 3.67680i
1.68598i
0 −0.707107 + 0.707107i 0 0.707107 − 0.707107i 0 −2.02778 − 2.02778i 0 − 1.00000i 0
361.2 0 −0.707107 + 0.707107i 0 0.707107 − 0.707107i 0 0.133857 + 0.133857i 0 − 1.00000i 0
361.3 0 −0.707107 + 0.707107i 0 0.707107 − 0.707107i 0 0.893928 + 0.893928i 0 − 1.00000i 0
361.4 0 0.707107 − 0.707107i 0 −0.707107 + 0.707107i 0 −3.06900 − 3.06900i 0 − 1.00000i 0
361.5 0 0.707107 − 0.707107i 0 −0.707107 + 0.707107i 0 −0.903531 − 0.903531i 0 − 1.00000i 0
361.6 0 0.707107 − 0.707107i 0 −0.707107 + 0.707107i 0 2.97253 + 2.97253i 0 − 1.00000i 0
421.1 0 −0.707107 − 0.707107i 0 0.707107 + 0.707107i 0 −2.02778 + 2.02778i 0 1.00000i 0
421.2 0 −0.707107 − 0.707107i 0 0.707107 + 0.707107i 0 0.133857 − 0.133857i 0 1.00000i 0
421.3 0 −0.707107 − 0.707107i 0 0.707107 + 0.707107i 0 0.893928 − 0.893928i 0 1.00000i 0
421.4 0 0.707107 + 0.707107i 0 −0.707107 − 0.707107i 0 −3.06900 + 3.06900i 0 1.00000i 0
421.5 0 0.707107 + 0.707107i 0 −0.707107 − 0.707107i 0 −0.903531 + 0.903531i 0 1.00000i 0
421.6 0 0.707107 + 0.707107i 0 −0.707107 − 0.707107i 0 2.97253 − 2.97253i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.2.bd.a ✓ 12
3.b odd 2 1 3060.2.be.a 12
17.c even 4 1 inner 1020.2.bd.a ✓ 12
51.f odd 4 1 3060.2.be.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.2.bd.a ✓ 12 1.a even 1 1 trivial
1020.2.bd.a ✓ 12 17.c even 4 1 inner
3060.2.be.a 12 3.b odd 2 1
3060.2.be.a 12 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 4 T_{7}^{11} + 8 T_{7}^{10} - 4 T_{7}^{9} + 322 T_{7}^{8} + 1252 T_{7}^{7} + 2440 T_{7}^{6} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(1020, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 4624 \) Copy content Toggle raw display
$13$ \( (T^{6} + 12 T^{5} + \cdots + 4976)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 108 T^{10} + \cdots + 99856 \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{12} + 20 T^{11} + \cdots + 73068304 \) Copy content Toggle raw display
$31$ \( T^{12} - 8 T^{11} + \cdots + 23116864 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 317837584 \) Copy content Toggle raw display
$41$ \( T^{12} + 28 T^{11} + \cdots + 300304 \) Copy content Toggle raw display
$43$ \( T^{12} + 232 T^{10} + \cdots + 18939904 \) Copy content Toggle raw display
$47$ \( (T^{6} + 4 T^{5} + \cdots - 3554)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 432 T^{10} + \cdots + 84015556 \) Copy content Toggle raw display
$59$ \( T^{12} + 308 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$61$ \( T^{12} - 24 T^{11} + \cdots + 1498176 \) Copy content Toggle raw display
$67$ \( (T^{6} - 16 T^{5} + \cdots + 16496)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 16 T^{11} + \cdots + 18939904 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 44441699344 \) Copy content Toggle raw display
$79$ \( T^{12} - 36 T^{11} + \cdots + 4064256 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 497468416 \) Copy content Toggle raw display
$89$ \( (T^{6} - 8 T^{5} + \cdots - 784)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 157351936 \) Copy content Toggle raw display
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