Properties

Label 1005.2.a.c
Level $1005$
Weight $2$
Character orbit 1005.a
Self dual yes
Analytic conductor $8.025$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1005,2,Mod(1,1005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1005.a (trivial)

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: yes
Analytic conductor: \(8.02496540314\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} - \beta_1 - 2) q^{7} + ( - 3 \beta_{3} - 2 \beta_{2} - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} - \beta_1 - 2) q^{7} + ( - 3 \beta_{3} - 2 \beta_{2} - 2) q^{8} + q^{9} + (\beta_1 - 1) q^{10} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{13} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{14} + q^{15} + (4 \beta_{3} - 3 \beta_1 + 2) q^{16} + (2 \beta_{2} + \beta_1 - 3) q^{17} + (\beta_1 - 1) q^{18} + (\beta_{3} - 2 \beta_1 - 4) q^{19} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{20} + ( - \beta_{3} - \beta_1 - 2) q^{21} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{22} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{23} + ( - 3 \beta_{3} - 2 \beta_{2} - 2) q^{24} + q^{25} + ( - \beta_{3} + \beta_1 + 4) q^{26} + q^{27} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{28} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 - 5) q^{29} + (\beta_1 - 1) q^{30} + ( - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{31} + ( - 5 \beta_{3} + \beta_{2} + 6 \beta_1 - 4) q^{32} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{33} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{34} + ( - \beta_{3} - \beta_1 - 2) q^{35} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{36} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{37} + ( - 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{38} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 3 \beta_{3} - 2 \beta_{2} - 2) q^{40} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 + 1) q^{41} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{42} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{43} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 2) q^{44} + q^{45} + (2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 9) q^{46} + (4 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{47} + (4 \beta_{3} - 3 \beta_1 + 2) q^{48} + (3 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 2) q^{49} + (\beta_1 - 1) q^{50} + (2 \beta_{2} + \beta_1 - 3) q^{51} + ( - \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{52} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{53} + (\beta_1 - 1) q^{54} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{55} + (7 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 9) q^{56} + (\beta_{3} - 2 \beta_1 - 4) q^{57} + (\beta_{3} + 3 \beta_{2} - 6 \beta_1 + 7) q^{58} + (5 \beta_{2} + 5 \beta_1 - 3) q^{59} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{60} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{61} + (8 \beta_{3} - 4 \beta_1 - 3) q^{62} + ( - \beta_{3} - \beta_1 - 2) q^{63} + (8 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 11) q^{64} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{65} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{66} - q^{67} + ( - 5 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{68} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{69} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{70} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{71} + ( - 3 \beta_{3} - 2 \beta_{2} - 2) q^{72} + ( - 4 \beta_{2} + 5 \beta_1 + 3) q^{73} + ( - 6 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 7) q^{74} + q^{75} + (3 \beta_{3} - \beta_{2} + 4) q^{76} + (5 \beta_{3} + 8 \beta_{2} + 10 \beta_1 + 2) q^{77} + ( - \beta_{3} + \beta_1 + 4) q^{78} + ( - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{79} + (4 \beta_{3} - 3 \beta_1 + 2) q^{80} + q^{81} + (2 \beta_{3} + 2 \beta_1 + 3) q^{82} + (5 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{83} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{84} + (2 \beta_{2} + \beta_1 - 3) q^{85} + (3 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 1) q^{86} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 - 5) q^{87} + (3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 3) q^{88} + ( - \beta_{3} + 7 \beta_{2} + \beta_1 + 6) q^{89} + (\beta_1 - 1) q^{90} + ( - 4 \beta_{3} - 11 \beta_{2} - 11 \beta_1 - 4) q^{91} + ( - 7 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 13) q^{92} + ( - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{93} + ( - 9 \beta_{3} + \beta_1 + 2) q^{94} + (\beta_{3} - 2 \beta_1 - 4) q^{95} + ( - 5 \beta_{3} + \beta_{2} + 6 \beta_1 - 4) q^{96} + ( - 2 \beta_{3} + \beta_{2} + 6 \beta_1 - 1) q^{97} + (4 \beta_{2} + 5 \beta_1 + 8) q^{98} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 9 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 9 q^{7} - 9 q^{8} + 4 q^{9} - 4 q^{10} - 5 q^{11} + 4 q^{12} - 5 q^{13} + 2 q^{14} + 4 q^{15} + 12 q^{16} - 14 q^{17} - 4 q^{18} - 15 q^{19} + 4 q^{20} - 9 q^{21} + q^{22} - 10 q^{23} - 9 q^{24} + 4 q^{25} + 15 q^{26} + 4 q^{27} - 5 q^{28} - 17 q^{29} - 4 q^{30} + q^{31} - 22 q^{32} - 5 q^{33} + 14 q^{34} - 9 q^{35} + 4 q^{36} + 4 q^{37} - 2 q^{38} - 5 q^{39} - 9 q^{40} + q^{41} + 2 q^{42} - 19 q^{43} - 9 q^{44} + 4 q^{45} + 34 q^{46} - 7 q^{47} + 12 q^{48} + 9 q^{49} - 4 q^{50} - 14 q^{51} + 4 q^{52} - 11 q^{53} - 4 q^{54} - 5 q^{55} + 36 q^{56} - 15 q^{57} + 26 q^{58} - 17 q^{59} + 4 q^{60} - 11 q^{61} - 4 q^{62} - 9 q^{63} + 47 q^{64} - 5 q^{65} + q^{66} - 4 q^{67} - 7 q^{68} - 10 q^{69} + 2 q^{70} + 22 q^{71} - 9 q^{72} + 16 q^{73} - 30 q^{74} + 4 q^{75} + 20 q^{76} + 5 q^{77} + 15 q^{78} + 5 q^{79} + 12 q^{80} + 4 q^{81} + 14 q^{82} - 12 q^{83} - 5 q^{84} - 14 q^{85} + 9 q^{86} - 17 q^{87} + 17 q^{88} + 16 q^{89} - 4 q^{90} - 9 q^{91} - 56 q^{92} + q^{93} - q^{94} - 15 q^{95} - 22 q^{96} - 7 q^{97} + 28 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 3\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.76401
−0.693822
0.396339
2.06150
−2.76401 1.00000 5.63978 1.00000 −2.76401 −2.54486 −10.0604 1.00000 −2.76401
1.2 −1.69382 1.00000 0.869035 1.00000 −1.69382 0.959901 1.91565 1.00000 −1.69382
1.3 −0.603661 1.00000 −1.63559 1.00000 −0.603661 −2.68018 2.19467 1.00000 −0.603661
1.4 1.06150 1.00000 −0.873220 1.00000 1.06150 −4.73486 −3.04992 1.00000 1.06150
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1005.2.a.c 4
3.b odd 2 1 3015.2.a.i 4
5.b even 2 1 5025.2.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1005.2.a.c 4 1.a even 1 1 trivial
3015.2.a.i 4 3.b odd 2 1
5025.2.a.v 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 2 T^{2} - 5 T - 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 T^{3} + 22 T^{2} + 2 T - 31 \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} - 22 T^{2} - 114 T - 19 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} - 29 T^{2} - 113 T + 3 \) Copy content Toggle raw display
$17$ \( T^{4} + 14 T^{3} + 60 T^{2} + 93 T + 43 \) Copy content Toggle raw display
$19$ \( T^{4} + 15 T^{3} + 61 T^{2} + \cdots - 301 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} - 12 T^{2} + \cdots + 331 \) Copy content Toggle raw display
$29$ \( T^{4} + 17 T^{3} + 57 T^{2} + \cdots - 1559 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} - 114 T^{2} - 92 T + 1869 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} - 58 T^{2} - 76 T + 89 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} - 60 T^{2} + 16 T + 147 \) Copy content Toggle raw display
$43$ \( T^{4} + 19 T^{3} + 98 T^{2} + \cdots - 991 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} - 87 T^{2} - 157 T + 47 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + 17 T^{2} + \cdots - 267 \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} - T^{2} - 607 T + 441 \) Copy content Toggle raw display
$61$ \( T^{4} + 11 T^{3} - 75 T^{2} + \cdots + 2437 \) Copy content Toggle raw display
$67$ \( (T + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 22 T^{3} + 74 T^{2} + \cdots - 4599 \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} - 154 T^{2} + \cdots + 11589 \) Copy content Toggle raw display
$79$ \( T^{4} - 5 T^{3} - 112 T^{2} + \cdots - 457 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} - 66 T^{2} + \cdots + 411 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} - 120 T^{2} + \cdots - 4801 \) Copy content Toggle raw display
$97$ \( T^{4} + 7 T^{3} - 145 T^{2} + \cdots + 207 \) Copy content Toggle raw display
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