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Properties

Label 1150.3.d.a
Level $1150$
Weight $3$
Character orbit 1150.d
Analytic conductor $31.335$
Analytic rank $0$
Dimension $4$
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] = [1150,3,Mod(551,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.551");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1150.d (of order \(2\), degree \(1\), 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: \(31.3352304014\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.613376.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 58x^{2} + 599 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 46)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + 2 q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{3} q^{7} - 2 \beta_{2} q^{8} + ( - 2 \beta_{2} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + 2 q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{3} q^{7} - 2 \beta_{2} q^{8} + ( - 2 \beta_{2} - 6) q^{9} - \beta_{3} q^{11} + ( - 2 \beta_{2} + 2) q^{12} + (8 \beta_{2} + 1) q^{13} + (\beta_{3} + \beta_1) q^{14} + 4 q^{16} + ( - 2 \beta_{3} + \beta_1) q^{17} + (6 \beta_{2} + 4) q^{18} + ( - 2 \beta_{3} + 2 \beta_1) q^{19} + \beta_1 q^{21} + (\beta_{3} + \beta_1) q^{22} + ( - \beta_{3} - 7 \beta_{2} + \cdots - 13) q^{23}+ \cdots + (8 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} + 8 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{4} + 8 q^{6} - 24 q^{9} + 8 q^{12} + 4 q^{13} + 16 q^{16} + 16 q^{18} - 52 q^{23} + 16 q^{24} - 64 q^{26} - 44 q^{27} + 84 q^{29} + 116 q^{31} - 48 q^{36} - 60 q^{39} + 20 q^{41} + 56 q^{46} + 28 q^{47} + 16 q^{48} - 148 q^{49} + 8 q^{52} - 104 q^{54} + 80 q^{58} - 184 q^{59} + 24 q^{62} + 32 q^{64} + 4 q^{69} - 100 q^{71} + 32 q^{72} + 148 q^{73} - 344 q^{77} - 56 q^{78} + 68 q^{81} + 48 q^{82} + 164 q^{87} - 104 q^{92} + 140 q^{93} + 136 q^{94} + 32 q^{96} + 400 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 58x^{2} + 599 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 29\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 29 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 51\nu ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11\beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -29\beta_{3} + 51\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1
3.66656i
− 3.66656i
6.67505i
− 6.67505i
−1.41421 −0.414214 2.00000 0 0.585786 − 12.5184i −2.82843 −8.82843 0
551.2 −1.41421 −0.414214 2.00000 0 0.585786 12.5184i −2.82843 −8.82843 0
551.3 1.41421 2.41421 2.00000 0 3.41421 − 3.91016i 2.82843 −3.17157 0
551.4 1.41421 2.41421 2.00000 0 3.41421 3.91016i 2.82843 −3.17157 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1150.3.d.a 4
5.b even 2 1 46.3.b.a ✓ 4
5.c odd 4 2 1150.3.c.a 8
15.d odd 2 1 414.3.b.a 4
20.d odd 2 1 368.3.f.c 4
23.b odd 2 1 inner 1150.3.d.a 4
40.e odd 2 1 1472.3.f.c 4
40.f even 2 1 1472.3.f.f 4
115.c odd 2 1 46.3.b.a ✓ 4
115.e even 4 2 1150.3.c.a 8
345.h even 2 1 414.3.b.a 4
460.g even 2 1 368.3.f.c 4
920.b even 2 1 1472.3.f.c 4
920.p odd 2 1 1472.3.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.3.b.a ✓ 4 5.b even 2 1
46.3.b.a ✓ 4 115.c odd 2 1
368.3.f.c 4 20.d odd 2 1
368.3.f.c 4 460.g even 2 1
414.3.b.a 4 15.d odd 2 1
414.3.b.a 4 345.h even 2 1
1150.3.c.a 8 5.c odd 4 2
1150.3.c.a 8 115.e even 4 2
1150.3.d.a 4 1.a even 1 1 trivial
1150.3.d.a 4 23.b odd 2 1 inner
1472.3.f.c 4 40.e odd 2 1
1472.3.f.c 4 920.b even 2 1
1472.3.f.f 4 40.f even 2 1
1472.3.f.f 4 920.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 1 \) acting on \(S_{3}^{\mathrm{new}}(1150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 172T^{2} + 2396 \) Copy content Toggle raw display
$11$ \( T^{4} + 172T^{2} + 2396 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 127)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 692 T^{2} + 117404 \) Copy content Toggle raw display
$19$ \( T^{4} + 928 T^{2} + 153344 \) Copy content Toggle raw display
$23$ \( T^{4} + 52 T^{3} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( (T^{2} - 42 T + 241)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 58 T + 823)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4148 T^{2} + 4027676 \) Copy content Toggle raw display
$41$ \( (T^{2} - 10 T - 47)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 2992 T^{2} + 1878464 \) Copy content Toggle raw display
$47$ \( (T^{2} - 14 T - 529)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 6704 T^{2} + 11079104 \) Copy content Toggle raw display
$59$ \( (T^{2} + 92 T + 1724)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 2752 T^{2} + 613376 \) Copy content Toggle raw display
$67$ \( T^{4} + 4716 T^{2} + 194076 \) Copy content Toggle raw display
$71$ \( (T^{2} + 50 T - 6817)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 74 T - 5831)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 14432 T^{2} + 7513856 \) Copy content Toggle raw display
$83$ \( T^{4} + 18700 T^{2} + 73377500 \) Copy content Toggle raw display
$89$ \( T^{4} + 17200 T^{2} + 23960000 \) Copy content Toggle raw display
$97$ \( T^{4} + 5684 T^{2} + 5752796 \) Copy content Toggle raw display
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