Properties

Label 1110.2.h.d
Level $1110$
Weight $2$
Character orbit 1110.h
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(961,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.h (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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{73})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 37x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 37x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 19\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 18\beta_{2} - 19\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
4.77200i
3.77200i
3.77200i
4.77200i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.h.d 4
3.b odd 2 1 3330.2.h.j 4
37.b even 2 1 inner 1110.2.h.d 4
111.d odd 2 1 3330.2.h.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.d 4 1.a even 1 1 trivial
1110.2.h.d 4 37.b even 2 1 inner
3330.2.h.j 4 3.b odd 2 1
3330.2.h.j 4 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 37T_{13}^{2} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 37T^{2} + 324 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 61T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 61T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 61T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{4} + 61T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( (T^{2} - 7 T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 97T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 69)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 148T^{2} + 5184 \) Copy content Toggle raw display
$61$ \( T^{4} + 181T^{2} + 2916 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 72)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 9 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 164T^{2} + 4096 \) Copy content Toggle raw display
$83$ \( (T^{2} - 5 T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 369 T^{2} + 20736 \) Copy content Toggle raw display
$97$ \( T^{4} + 121T^{2} + 576 \) Copy content Toggle raw display
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