Properties

Label 1075.2.a.j
Level $1075$
Weight $2$
Character orbit 1075.a
Self dual yes
Analytic conductor $8.584$
Analytic rank $1$
Dimension $3$
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] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 215)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
2.65109
−0.273891
−1.37720
−2.65109 1.37720 5.02830 0 −3.65109 −2.65109 −8.02830 −1.10331 0
1.2 0.273891 −2.65109 −1.92498 0 −0.726109 0.273891 −1.07502 4.02830 0
1.3 1.37720 0.273891 −0.103312 0 0.377203 1.37720 −2.89669 −2.92498 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(43\) \(-1\)

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 1075.2.a.j 3
3.b odd 2 1 9675.2.a.bt 3
5.b even 2 1 1075.2.a.k 3
5.c odd 4 2 215.2.b.a 6
15.d odd 2 1 9675.2.a.br 3
15.e even 4 2 1935.2.b.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.b.a 6 5.c odd 4 2
1075.2.a.j 3 1.a even 1 1 trivial
1075.2.a.k 3 5.b even 2 1
1935.2.b.c 6 15.e even 4 2
9675.2.a.br 3 15.d odd 2 1
9675.2.a.bt 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1075))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{3} + T_{3}^{2} - 4T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 4T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 4T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 4T + 1 \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$17$ \( (T + 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 40 T + 200 \) Copy content Toggle raw display
$23$ \( T^{3} - 52T + 104 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 40 T - 200 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 88 T + 25 \) Copy content Toggle raw display
$37$ \( T^{3} + 9 T^{2} + 14 T - 25 \) Copy content Toggle raw display
$41$ \( T^{3} - 11 T^{2} - 42 T + 515 \) Copy content Toggle raw display
$43$ \( (T - 1)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} + 26 T^{2} + 208 T + 520 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} + 4 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 21 T^{2} + 108 T + 5 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} + 16 T - 40 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} - 4 T - 248 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 16 T - 8 \) Copy content Toggle raw display
$73$ \( T^{3} + T^{2} - 186 T - 961 \) Copy content Toggle raw display
$79$ \( T^{3} + 3 T^{2} - 88 T - 25 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} - 220 T - 248 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 116 T + 40 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 40 T - 8 \) Copy content Toggle raw display
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