Properties

Label 1375.2.a.b
Level $1375$
Weight $2$
Character orbit 1375.a
Self dual yes
Analytic conductor $10.979$
Analytic rank $1$
Dimension $2$
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] = [1375,2,Mod(1,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, 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("1375.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1375.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: \(10.9794302779\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + \beta q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + \beta q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9} + q^{11} + ( - \beta + 1) q^{12} - q^{13} + (\beta + 1) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 5) q^{17} - 2 \beta q^{18} + ( - \beta - 2) q^{19} - \beta q^{21} + \beta q^{22} + ( - 4 \beta + 1) q^{23} + (2 \beta - 1) q^{24} - \beta q^{26} + 5 q^{27} + q^{28} + (\beta - 3) q^{29} - 3 q^{31} + (\beta - 5) q^{32} - q^{33} + (\beta - 4) q^{34} + ( - 2 \beta + 2) q^{36} + (3 \beta - 6) q^{37} + ( - 3 \beta - 1) q^{38} + q^{39} - 3 q^{41} + ( - \beta - 1) q^{42} - q^{43} + (\beta - 1) q^{44} + ( - 3 \beta - 4) q^{46} + 3 q^{47} + 3 \beta q^{48} + (\beta - 6) q^{49} + (4 \beta - 5) q^{51} + ( - \beta + 1) q^{52} + ( - 2 \beta + 10) q^{53} + 5 \beta q^{54} + ( - \beta - 2) q^{56} + (\beta + 2) q^{57} + ( - 2 \beta + 1) q^{58} + ( - \beta - 7) q^{59} + ( - 5 \beta + 2) q^{61} - 3 \beta q^{62} - 2 \beta q^{63} + (2 \beta + 1) q^{64} - \beta q^{66} + ( - 6 \beta + 1) q^{67} + (5 \beta - 9) q^{68} + (4 \beta - 1) q^{69} + ( - 5 \beta - 8) q^{71} + (4 \beta - 2) q^{72} + (3 \beta - 5) q^{73} + ( - 3 \beta + 3) q^{74} + ( - 2 \beta + 1) q^{76} + \beta q^{77} + \beta q^{78} + (7 \beta - 6) q^{79} + q^{81} - 3 \beta q^{82} + (2 \beta - 12) q^{83} - q^{84} - \beta q^{86} + ( - \beta + 3) q^{87} + ( - 2 \beta + 1) q^{88} + ( - \beta + 3) q^{89} - \beta q^{91} + (\beta - 5) q^{92} + 3 q^{93} + 3 \beta q^{94} + ( - \beta + 5) q^{96} + (4 \beta - 4) q^{97} + ( - 5 \beta + 1) q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} + 2 q^{11} + q^{12} - 2 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} + q^{22} - 2 q^{23} - q^{26} + 10 q^{27} + 2 q^{28} - 5 q^{29} - 6 q^{31} - 9 q^{32} - 2 q^{33} - 7 q^{34} + 2 q^{36} - 9 q^{37} - 5 q^{38} + 2 q^{39} - 6 q^{41} - 3 q^{42} - 2 q^{43} - q^{44} - 11 q^{46} + 6 q^{47} + 3 q^{48} - 11 q^{49} - 6 q^{51} + q^{52} + 18 q^{53} + 5 q^{54} - 5 q^{56} + 5 q^{57} - 15 q^{59} - q^{61} - 3 q^{62} - 2 q^{63} + 4 q^{64} - q^{66} - 4 q^{67} - 13 q^{68} + 2 q^{69} - 21 q^{71} - 7 q^{73} + 3 q^{74} + q^{77} + q^{78} - 5 q^{79} + 2 q^{81} - 3 q^{82} - 22 q^{83} - 2 q^{84} - q^{86} + 5 q^{87} + 5 q^{89} - q^{91} - 9 q^{92} + 6 q^{93} + 3 q^{94} + 9 q^{96} - 4 q^{97} - 3 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 −1.00000 −1.61803 0 0.618034 −0.618034 2.23607 −2.00000 0
1.2 1.61803 −1.00000 0.618034 0 −1.61803 1.61803 −2.23607 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)

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 1375.2.a.b yes 2
5.b even 2 1 1375.2.a.a 2
5.c odd 4 2 1375.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.2.a.a 2 5.b even 2 1
1375.2.a.b yes 2 1.a even 1 1 trivial
1375.2.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1375))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T - 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} + 21T + 79 \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T - 55 \) Copy content Toggle raw display
$83$ \( T^{2} + 22T + 116 \) Copy content Toggle raw display
$89$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
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