Properties

Label 1089.3.c.g
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
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] = [1089,3,Mod(604,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (3 \beta_{2} + 2) q^{5} + (7 \beta_{3} - 9 \beta_1) q^{7} + (\beta_{3} + 4 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (3 \beta_{2} + 2) q^{5} + (7 \beta_{3} - 9 \beta_1) q^{7} + (\beta_{3} + 4 \beta_1) q^{8} + (3 \beta_{3} - 4 \beta_1) q^{10} + ( - 11 \beta_{3} + 10 \beta_1) q^{13} + ( - 9 \beta_{2} + 11) q^{14} + (8 \beta_{2} - 1) q^{16} + (18 \beta_{3} - 15 \beta_1) q^{17} + ( - 8 \beta_{3} + 4 \beta_1) q^{19} + (8 \beta_{2} + 13) q^{20} + (11 \beta_{2} + 9) q^{23} + (12 \beta_{2} + 6) q^{25} + (10 \beta_{2} - 9) q^{26} + (19 \beta_{3} - 7 \beta_1) q^{28} + ( - 5 \beta_{3} + 26 \beta_1) q^{29} + ( - 10 \beta_{2} - 20) q^{31} + (12 \beta_{3} - \beta_1) q^{32} + ( - 15 \beta_{2} + 12) q^{34} + (29 \beta_{3} + 15 \beta_1) q^{35} + (7 \beta_{2} + 14) q^{37} + 4 \beta_{2} q^{38} + (20 \beta_{3} - 19 \beta_1) q^{40} + (29 \beta_{3} - 40 \beta_1) q^{41} + ( - 13 \beta_{3} + 5 \beta_1) q^{43} + (11 \beta_{3} - 13 \beta_1) q^{46} + (34 \beta_{2} - 4) q^{47} + (32 \beta_{2} - 85) q^{49} + (12 \beta_{3} - 18 \beta_1) q^{50} + ( - 34 \beta_{3} + 11 \beta_1) q^{52} + ( - 12 \beta_{2} - 35) q^{53} + ( - 43 \beta_{2} + 39) q^{56} + (26 \beta_{2} - 47) q^{58} + (19 \beta_{2} - 71) q^{59} + (\beta_{3} - 27 \beta_1) q^{61} - 10 \beta_{3} q^{62} + (31 \beta_{2} - 14) q^{64} + ( - 58 \beta_{3} - 7 \beta_1) q^{65} + ( - 19 \beta_{2} - 49) q^{67} + (57 \beta_{3} - 18 \beta_1) q^{68} + (15 \beta_{2} - 59) q^{70} + (31 \beta_{2} - 1) q^{71} + ( - 85 \beta_{3} + 67 \beta_1) q^{73} + 7 \beta_{3} q^{74} + ( - 28 \beta_{3} + 8 \beta_1) q^{76} + (32 \beta_{3} - 4 \beta_1) q^{79} + (13 \beta_{2} + 70) q^{80} + ( - 40 \beta_{2} + 51) q^{82} + (56 \beta_{3} - 44 \beta_1) q^{83} + (99 \beta_{3} + 6 \beta_1) q^{85} + (5 \beta_{2} + 3) q^{86} + ( - 23 \beta_{2} - 84) q^{89} + ( - 13 \beta_{2} + 165) q^{91} + (31 \beta_{2} + 51) q^{92} + (34 \beta_{3} - 72 \beta_1) q^{94} + ( - 52 \beta_{3} + 8 \beta_1) q^{95} + ( - 29 \beta_{2} + 28) q^{97} + (32 \beta_{3} - 149 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 8 q^{5} + 44 q^{14} - 4 q^{16} + 52 q^{20} + 36 q^{23} + 24 q^{25} - 36 q^{26} - 80 q^{31} + 48 q^{34} + 56 q^{37} - 16 q^{47} - 340 q^{49} - 140 q^{53} + 156 q^{56} - 188 q^{58} - 284 q^{59} - 56 q^{64} - 196 q^{67} - 236 q^{70} - 4 q^{71} + 280 q^{80} + 204 q^{82} + 12 q^{86} - 336 q^{89} + 660 q^{91} + 204 q^{92} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 0.267949 −3.19615 0 13.7632i 8.24504i 0 6.17449i
604.2 0.517638i 0 3.73205 7.19615 0 8.86422i 4.00240i 0 3.72500i
604.3 0.517638i 0 3.73205 7.19615 0 8.86422i 4.00240i 0 3.72500i
604.4 1.93185i 0 0.267949 −3.19615 0 13.7632i 8.24504i 0 6.17449i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.g 4
3.b odd 2 1 363.3.c.c 4
11.b odd 2 1 inner 1089.3.c.g 4
33.d even 2 1 363.3.c.c 4
33.f even 10 4 363.3.g.b 16
33.h odd 10 4 363.3.g.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.c.c 4 3.b odd 2 1
363.3.c.c 4 33.d even 2 1
363.3.g.b 16 33.f even 10 4
363.3.g.b 16 33.h odd 10 4
1089.3.c.g 4 1.a even 1 1 trivial
1089.3.c.g 4 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T - 23)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 268 T^{2} + 14884 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 444 T^{2} + 47961 \) Copy content Toggle raw display
$17$ \( T^{4} + 1116 T^{2} + 281961 \) Copy content Toggle raw display
$19$ \( T^{4} + 192T^{2} + 2304 \) Copy content Toggle raw display
$23$ \( (T^{2} - 18 T - 282)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2284 T^{2} + 32761 \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 28 T + 49)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5124 T^{2} + \cdots + 4835601 \) Copy content Toggle raw display
$43$ \( T^{4} + 516T^{2} + 4356 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 3452)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 70 T + 793)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 142 T + 3958)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 2812 T^{2} + 386884 \) Copy content Toggle raw display
$67$ \( (T^{2} + 98 T + 1318)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 2882)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 24076 T^{2} + \cdots + 122456356 \) Copy content Toggle raw display
$79$ \( T^{4} + 3648 T^{2} + 278784 \) Copy content Toggle raw display
$83$ \( T^{4} + 10432 T^{2} + \cdots + 22886656 \) Copy content Toggle raw display
$89$ \( (T^{2} + 168 T + 5469)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 56 T - 1739)^{2} \) Copy content Toggle raw display
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