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Properties

Label 1089.3.c.e
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: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + (4 \beta_{2} + 1) q^{4} - 4 q^{5} + (2 \beta_{3} + 4 \beta_1) q^{7} + (3 \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + (4 \beta_{2} + 1) q^{4} - 4 q^{5} + (2 \beta_{3} + 4 \beta_1) q^{7} + (3 \beta_{3} + \beta_1) q^{8} + (4 \beta_{3} - 4 \beta_1) q^{10} + (2 \beta_{3} - 4 \beta_1) q^{13} + ( - 2 \beta_{2} - 6) q^{14} + (8 \beta_{2} + 5) q^{16} + 13 \beta_{3} q^{17} + ( - 5 \beta_{3} + 2 \beta_1) q^{19} + ( - 16 \beta_{2} - 4) q^{20} + ( - 2 \beta_{2} + 4) q^{23} - 9 q^{25} + ( - 10 \beta_{2} + 10) q^{26} + (10 \beta_{3} + 12 \beta_1) q^{28} + (8 \beta_{3} + 10 \beta_1) q^{29} + ( - 18 \beta_{2} + 4) q^{31} + (23 \beta_{3} + \beta_1) q^{32} + ( - 39 \beta_{2} + 13) q^{34} + ( - 8 \beta_{3} - 16 \beta_1) q^{35} + ( - 36 \beta_{2} - 18) q^{37} + (17 \beta_{2} - 9) q^{38} + ( - 12 \beta_{3} - 4 \beta_1) q^{40} + ( - 16 \beta_{3} - 27 \beta_1) q^{41} + (18 \beta_{3} + \beta_1) q^{43} + ( - 8 \beta_{3} + 6 \beta_1) q^{46} + ( - 2 \beta_{2} - 26) q^{47} + ( - 44 \beta_{2} - 23) q^{49} + (9 \beta_{3} - 9 \beta_1) q^{50} + ( - 22 \beta_{3} + 4 \beta_1) q^{52} + ( - 30 \beta_{2} + 30) q^{53} + ( - 26 \beta_{2} - 38) q^{56} + ( - 14 \beta_{2} - 12) q^{58} + (15 \beta_{2} - 7) q^{59} + ( - 16 \beta_{3} - 2 \beta_1) q^{61} + ( - 40 \beta_{3} + 22 \beta_1) q^{62} + ( - 36 \beta_{2} + 41) q^{64} + ( - 8 \beta_{3} + 16 \beta_1) q^{65} + (17 \beta_{2} - 49) q^{67} + ( - 39 \beta_{3} + 52 \beta_1) q^{68} + (8 \beta_{2} + 24) q^{70} + ( - 62 \beta_{2} - 38) q^{71} + (60 \beta_{3} + 17 \beta_1) q^{73} + ( - 54 \beta_{3} + 18 \beta_1) q^{74} + (23 \beta_{3} - 18 \beta_1) q^{76} + ( - 26 \beta_{3} + 14 \beta_1) q^{79} + ( - 32 \beta_{2} - 20) q^{80} + (21 \beta_{2} + 38) q^{82} + ( - 2 \beta_{3} + 45 \beta_1) q^{83} - 52 \beta_{3} q^{85} + ( - 53 \beta_{2} + 16) q^{86} + ( - 83 \beta_{2} - 72) q^{89} + (20 \beta_{2} + 40) q^{91} + (22 \beta_{2} - 4) q^{92} + (22 \beta_{3} - 24 \beta_1) q^{94} + (20 \beta_{3} - 8 \beta_1) q^{95} + ( - 67 \beta_{2} - 36) q^{97} + ( - 65 \beta_{3} + 21 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 16 q^{5} - 20 q^{14} + 4 q^{16} + 16 q^{20} + 20 q^{23} - 36 q^{25} + 60 q^{26} + 52 q^{31} + 130 q^{34} - 70 q^{38} - 100 q^{47} - 4 q^{49} + 180 q^{53} - 100 q^{56} - 20 q^{58} - 58 q^{59} + 236 q^{64} - 230 q^{67} + 80 q^{70} - 28 q^{71} - 16 q^{80} + 110 q^{82} + 170 q^{86} - 122 q^{89} + 120 q^{91} - 60 q^{92} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{10}^{3} + \zeta_{10}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{10}^{3} + \zeta_{10}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{10}^{3} - \zeta_{10}^{2} + 2\zeta_{10} - 1 \) Copy content Toggle raw display

Display \(\zeta_{10}^j\) in terms of \(\beta_i\)

\(\zeta_{10}\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{10}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{10}^{3}\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{10}^j\)

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
−0.309017 + 0.951057i
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 − 0.951057i
− 3.07768i 0 −5.47214 −4.00000 0 − 0.898056i 4.53077i 0 12.3107i
604.2 − 0.726543i 0 3.47214 −4.00000 0 − 9.95959i − 5.42882i 0 2.90617i
604.3 0.726543i 0 3.47214 −4.00000 0 9.95959i 5.42882i 0 − 2.90617i
604.4 3.07768i 0 −5.47214 −4.00000 0 0.898056i − 4.53077i 0 − 12.3107i
\(n\): 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.e 4
3.b odd 2 1 121.3.b.b 4
11.b odd 2 1 inner 1089.3.c.e 4
11.c even 5 1 99.3.k.a 4
11.d odd 10 1 99.3.k.a 4
33.d even 2 1 121.3.b.b 4
33.f even 10 1 11.3.d.a ✓ 4
33.f even 10 1 121.3.d.a 4
33.f even 10 1 121.3.d.c 4
33.f even 10 1 121.3.d.d 4
33.h odd 10 1 11.3.d.a ✓ 4
33.h odd 10 1 121.3.d.a 4
33.h odd 10 1 121.3.d.c 4
33.h odd 10 1 121.3.d.d 4
132.n odd 10 1 176.3.n.a 4
132.o even 10 1 176.3.n.a 4
165.o odd 10 1 275.3.x.e 4
165.r even 10 1 275.3.x.e 4
165.u odd 20 2 275.3.q.d 8
165.v even 20 2 275.3.q.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a ✓ 4 33.f even 10 1
11.3.d.a ✓ 4 33.h odd 10 1
99.3.k.a 4 11.c even 5 1
99.3.k.a 4 11.d odd 10 1
121.3.b.b 4 3.b odd 2 1
121.3.b.b 4 33.d even 2 1
121.3.d.a 4 33.f even 10 1
121.3.d.a 4 33.h odd 10 1
121.3.d.c 4 33.f even 10 1
121.3.d.c 4 33.h odd 10 1
121.3.d.d 4 33.f even 10 1
121.3.d.d 4 33.h odd 10 1
176.3.n.a 4 132.n odd 10 1
176.3.n.a 4 132.o even 10 1
275.3.q.d 8 165.u odd 20 2
275.3.q.d 8 165.v even 20 2
275.3.x.e 4 165.o odd 10 1
275.3.x.e 4 165.r even 10 1
1089.3.c.e 4 1.a even 1 1 trivial
1089.3.c.e 4 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 10T_{2}^{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 10T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 4)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 100T^{2} + 80 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 100T^{2} + 2000 \) Copy content Toggle raw display
$17$ \( T^{4} + 845 T^{2} + 142805 \) Copy content Toggle raw display
$19$ \( T^{4} + 145T^{2} + 605 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 820T^{2} + 9680 \) Copy content Toggle raw display
$31$ \( (T^{2} - 26 T - 236)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 1620)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4925 T^{2} + 8405 \) Copy content Toggle raw display
$43$ \( T^{4} + 1625 T^{2} + 581405 \) Copy content Toggle raw display
$47$ \( (T^{2} + 50 T + 620)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 90 T + 900)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 29 T - 71)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 1300 T^{2} + 403280 \) Copy content Toggle raw display
$67$ \( (T^{2} + 115 T + 2945)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 14 T - 4756)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 19445 T^{2} + 93787805 \) Copy content Toggle raw display
$79$ \( T^{4} + 4360 T^{2} + 67280 \) Copy content Toggle raw display
$83$ \( T^{4} + 10145 T^{2} + 22281605 \) Copy content Toggle raw display
$89$ \( (T^{2} + 61 T - 7681)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 5 T - 5605)^{2} \) Copy content Toggle raw display
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