Properties

Label 1000.2.k.b
Level $1000$
Weight $2$
Character orbit 1000.k
Analytic conductor $7.985$
Analytic rank $0$
Dimension $8$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(307,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.k (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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1024000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + 2 \beta_1 q^{4} - \beta_{3} q^{7} + (2 \beta_1 - 2) q^{8} - 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + 2 \beta_1 q^{4} - \beta_{3} q^{7} + (2 \beta_1 - 2) q^{8} - 3 \beta_1 q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 1) q^{11} + (\beta_{4} - \beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{14} - 4 q^{16} + ( - 3 \beta_1 + 3) q^{18} + (\beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{22} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_1 - 2) q^{23} + ( - \beta_{7} + \beta_{4} + 2) q^{26} + (2 \beta_{2} - 2 \beta_1) q^{28} + ( - 4 \beta_1 - 4) q^{32} + 6 q^{36} + (\beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_1 + 2) q^{37} + ( - \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1 - 2) q^{38} + ( - \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{44} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} - 4) q^{46} + (3 \beta_{7} - \beta_{6} - \beta_{5} - \beta_1) q^{47} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{4} + 9 \beta_1) q^{49} + ( - 2 \beta_{7} + 2 \beta_1 + 2) q^{52} + (\beta_{6} - \beta_{5} - 3 \beta_{2} + 2) q^{53} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{56} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_1) q^{59} + ( - 3 \beta_{2} + 3 \beta_1) q^{63} - 8 \beta_1 q^{64} + (6 \beta_1 + 6) q^{72} + (2 \beta_{5} - \beta_{3} + \beta_{2} + 5 \beta_1) q^{74} + ( - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{76} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{3} - 2 \beta_1 + 1) q^{77} - 9 q^{81} + ( - \beta_{6} - \beta_{5} + 4 \beta_{3} - 3 \beta_1 - 2) q^{82} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 2) q^{88} + ( - 2 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{89} + ( - \beta_{7} + \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 7) q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_1 - 4) q^{92} + (3 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{94} + ( - \beta_{6} + \beta_{5} - 4 \beta_{4} + 9 \beta_1 - 8) q^{98} + (3 \beta_{7} - 3 \beta_{5} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8} - 4 q^{11} + 8 q^{13} - 32 q^{16} + 24 q^{18} - 4 q^{22} - 12 q^{23} + 16 q^{26} + 8 q^{28} - 32 q^{32} + 48 q^{36} + 16 q^{37} - 12 q^{38} - 4 q^{41} - 24 q^{46} - 4 q^{47} + 16 q^{52} + 8 q^{53} + 16 q^{56} - 12 q^{63} + 48 q^{72} - 24 q^{76} - 8 q^{77} - 72 q^{81} - 4 q^{82} + 8 q^{88} + 32 q^{91} - 24 q^{92} - 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 15x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20\nu^{2} ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} + 3\nu^{6} - 5\nu^{4} + 75\nu^{3} + 35\nu^{2} - 25 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 5\nu^{5} - 5\nu^{4} - 15\nu^{2} - 100\nu - 25 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 2\nu^{4} - 10\nu^{3} - 10\nu^{2} + 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{6} - 135\nu^{2} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{4} - 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} - 2\nu^{5} - 2\nu^{4} - 10\nu^{2} - 15\nu - 15 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{3} + \beta _1 + 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 13\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - \beta_{5} + 5\beta_{4} + 5\beta_{2} - 3\beta _1 - 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{3} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{5} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{6} + 4\beta_{5} - 15\beta_{4} - 10\beta_{2} + 7\beta _1 + 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.34500 + 1.34500i
−0.831254 + 0.831254i
1.34500 1.34500i
0.831254 0.831254i
−1.34500 1.34500i
−0.831254 0.831254i
1.34500 + 1.34500i
0.831254 + 0.831254i
1.00000 1.00000i 0 2.00000i 0 0 −3.47677 + 3.47677i −2.00000 2.00000i 3.00000i 0
307.2 1.00000 1.00000i 0 2.00000i 0 0 −2.38947 + 2.38947i −2.00000 2.00000i 3.00000i 0
307.3 1.00000 1.00000i 0 2.00000i 0 0 0.240706 0.240706i −2.00000 2.00000i 3.00000i 0
307.4 1.00000 1.00000i 0 2.00000i 0 0 3.62554 3.62554i −2.00000 2.00000i 3.00000i 0
443.1 1.00000 + 1.00000i 0 2.00000i 0 0 −3.47677 3.47677i −2.00000 + 2.00000i 3.00000i 0
443.2 1.00000 + 1.00000i 0 2.00000i 0 0 −2.38947 2.38947i −2.00000 + 2.00000i 3.00000i 0
443.3 1.00000 + 1.00000i 0 2.00000i 0 0 0.240706 + 0.240706i −2.00000 + 2.00000i 3.00000i 0
443.4 1.00000 + 1.00000i 0 2.00000i 0 0 3.62554 + 3.62554i −2.00000 + 2.00000i 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.c odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.k.b yes 8
5.b even 2 1 1000.2.k.a 8
5.c odd 4 1 1000.2.k.a 8
5.c odd 4 1 inner 1000.2.k.b yes 8
8.d odd 2 1 1000.2.k.a 8
40.e odd 2 1 CM 1000.2.k.b yes 8
40.k even 4 1 1000.2.k.a 8
40.k even 4 1 inner 1000.2.k.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.k.a 8 5.b even 2 1
1000.2.k.a 8 5.c odd 4 1
1000.2.k.a 8 8.d odd 2 1
1000.2.k.a 8 40.k even 4 1
1000.2.k.b yes 8 1.a even 1 1 trivial
1000.2.k.b yes 8 5.c odd 4 1 inner
1000.2.k.b yes 8 40.e odd 2 1 CM
1000.2.k.b yes 8 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 671T_{7}^{4} + 2800T_{7}^{3} + 5832T_{7}^{2} - 3132T_{7} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + 8 T^{6} + 671 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 51 T^{2} - 102 T + 401)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 44521 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 154 T^{6} + 7091 T^{4} + \cdots + 101761 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 203401 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 203401 \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} - 201 T^{2} - 402 T + 7601)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 85174441 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 43414921 \) Copy content Toggle raw display
$59$ \( T^{8} + 394 T^{6} + 44611 T^{4} + \cdots + 3996001 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 694 T^{6} + \cdots + 84621601 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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