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Properties

Label 510.2.p.c
Level $510$
Weight $2$
Character orbit 510.p
Analytic conductor $4.072$
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] = [510,2,Mod(361,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("510.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.p (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: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{2} q^{5} - \beta_{3} q^{6} + ( - \beta_{5} + \beta_{3}) q^{7} + \beta_{4} q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{2} q^{5} - \beta_{3} q^{6} + ( - \beta_{5} + \beta_{3}) q^{7} + \beta_{4} q^{8} + \beta_{4} q^{9} + \beta_{3} q^{10} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{11} - 8 q^{13} + 8 q^{16} + 8 q^{18} - 8 q^{21} + 8 q^{22} + 8 q^{23} + 8 q^{29} - 8 q^{30} - 16 q^{31} + 8 q^{35} + 16 q^{37} - 16 q^{38} + 16 q^{41} + 8 q^{44} - 8 q^{46} - 32 q^{47} + 8 q^{50} + 8 q^{52} + 16 q^{57} + 8 q^{58} - 24 q^{61} - 16 q^{62} - 8 q^{64} - 8 q^{67} + 16 q^{69} + 32 q^{71} - 8 q^{72} + 16 q^{74} - 8 q^{81} - 16 q^{82} + 8 q^{84} + 8 q^{86} - 8 q^{88} - 64 q^{89} + 24 q^{91} - 8 q^{92} - 16 q^{95} + 8 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{7} - 18\nu^{6} + 63\nu^{5} - 115\nu^{4} + 170\nu^{3} - 152\nu^{2} + 89\nu - 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9\nu^{7} - 31\nu^{6} + 108\nu^{5} - 190\nu^{4} + 275\nu^{3} - 236\nu^{2} + 131\nu - 33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 9\nu^{7} - 32\nu^{6} + 111\nu^{5} - 200\nu^{4} + 290\nu^{3} - 253\nu^{2} + 141\nu - 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + 5\beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{7} - \beta_{6} + 3\beta_{5} - 6\beta_{4} + 12\beta_{3} + 4\beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{7} - 10\beta_{6} + 5\beta_{5} + 10\beta_{4} + 6\beta_{3} - 19\beta_{2} - 5\beta _1 + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{7} - 9\beta_{6} - 11\beta_{5} + 45\beta_{4} - 48\beta_{3} - 32\beta_{2} + 5\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 33\beta_{6} - 30\beta_{5} - 83\beta_{3} + 64\beta_{2} + 35\beta _1 - 118 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(307\) \(341\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 − 2.10607i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 − 0.0297061i
0.500000 + 2.10607i
0.500000 − 0.691860i
0.500000 − 1.44392i
0.500000 + 0.0297061i
1.00000i −0.707107 + 0.707107i −1.00000 0.707107 − 0.707107i −0.707107 − 0.707107i −1.27133 − 1.27133i − 1.00000i − 1.00000i 0.707107 + 0.707107i
361.2 1.00000i −0.707107 + 0.707107i −1.00000 0.707107 − 0.707107i −0.707107 − 0.707107i 2.68554 + 2.68554i − 1.00000i − 1.00000i 0.707107 + 0.707107i
361.3 1.00000i 0.707107 − 0.707107i −1.00000 −0.707107 + 0.707107i 0.707107 + 0.707107i −1.74912 − 1.74912i − 1.00000i − 1.00000i −0.707107 − 0.707107i
361.4 1.00000i 0.707107 − 0.707107i −1.00000 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.334904 + 0.334904i − 1.00000i − 1.00000i −0.707107 − 0.707107i
421.1 − 1.00000i −0.707107 − 0.707107i −1.00000 0.707107 + 0.707107i −0.707107 + 0.707107i −1.27133 + 1.27133i 1.00000i 1.00000i 0.707107 − 0.707107i
421.2 − 1.00000i −0.707107 − 0.707107i −1.00000 0.707107 + 0.707107i −0.707107 + 0.707107i 2.68554 − 2.68554i 1.00000i 1.00000i 0.707107 − 0.707107i
421.3 − 1.00000i 0.707107 + 0.707107i −1.00000 −0.707107 − 0.707107i 0.707107 − 0.707107i −1.74912 + 1.74912i 1.00000i 1.00000i −0.707107 + 0.707107i
421.4 − 1.00000i 0.707107 + 0.707107i −1.00000 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.334904 − 0.334904i 1.00000i 1.00000i −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
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 510.2.p.c ✓ 8
3.b odd 2 1 1530.2.q.j 8
17.c even 4 1 inner 510.2.p.c ✓ 8
17.d even 8 1 8670.2.a.bx 4
17.d even 8 1 8670.2.a.ca 4
51.f odd 4 1 1530.2.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.p.c ✓ 8 1.a even 1 1 trivial
510.2.p.c ✓ 8 17.c even 4 1 inner
1530.2.q.j 8 3.b odd 2 1
1530.2.q.j 8 51.f odd 4 1
8670.2.a.bx 4 17.d even 8 1
8670.2.a.ca 4 17.d even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 16T_{7}^{5} + 128T_{7}^{4} + 192T_{7}^{3} + 128T_{7}^{2} - 128T_{7} + 64 \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} + 128 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( T^{8} + 16 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 61504 \) Copy content Toggle raw display
$43$ \( T^{8} + 32 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$47$ \( (T^{4} + 16 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 336 T^{6} + \cdots + 13897984 \) Copy content Toggle raw display
$59$ \( T^{8} + 192 T^{6} + \cdots + 364816 \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + \cdots + 12544 \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} - 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 32 T^{7} + \cdots + 33856 \) Copy content Toggle raw display
$73$ \( T^{8} + 48 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1293984784 \) Copy content Toggle raw display
$83$ \( T^{8} + 448 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{4} + 32 T^{3} + \cdots - 496)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
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