↑

Properties

Label 1183.2.c.c
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not 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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - 3 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - 3 \beta_{2} - 1) q^{9} + (3 \beta_{2} - 2) q^{10} + ( - 3 \beta_{3} - 3 \beta_1) q^{11} + ( - 6 \beta_{2} + 3) q^{12} + ( - \beta_{2} + 1) q^{14} + (2 \beta_{3} - 3 \beta_1) q^{15} + ( - 3 \beta_{2} + 5) q^{16} + ( - 4 \beta_{2} - 5) q^{17} + ( - 2 \beta_{3} + 5 \beta_1) q^{18} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + (3 \beta_{3} - 6 \beta_1) q^{20} + ( - \beta_{3} + \beta_1) q^{21} - 3 \beta_{2} q^{22} + ( - 4 \beta_{2} - 2) q^{23} + ( - 5 \beta_{3} + 9 \beta_1) q^{24} + (3 \beta_{2} + 3) q^{25} + (2 \beta_{2} + 1) q^{27} + 3 \beta_1 q^{28} + ( - 5 \beta_{2} - 1) q^{29} + ( - 8 \beta_{2} + 5) q^{30} + ( - 5 \beta_{3} - 6 \beta_1) q^{31} + ( - 6 \beta_{3} + 3 \beta_1) q^{32} + 3 \beta_1 q^{33} + (\beta_{3} + 3 \beta_1) q^{34} + ( - \beta_{2} + 1) q^{35} + (6 \beta_{2} - 9) q^{36} + 4 \beta_{3} q^{37} - 3 \beta_{2} q^{38} + ( - 9 \beta_{2} + 5) q^{40} + ( - 4 \beta_{3} - 2 \beta_1) q^{41} + (3 \beta_{2} - 2) q^{42} + (9 \beta_{2} + 2) q^{43} - 9 \beta_{3} q^{44} + ( - 2 \beta_{3} + 5 \beta_1) q^{45} + ( - 2 \beta_{3} + 6 \beta_1) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + (11 \beta_{2} - 8) q^{48} - q^{49} - 3 \beta_1 q^{50} + (3 \beta_{2} + 1) q^{51} + (2 \beta_{2} + 7) q^{53} + (\beta_{3} - 3 \beta_1) q^{54} - 3 \beta_{2} q^{55} + (4 \beta_{2} - 1) q^{56} + 3 \beta_1 q^{57} + ( - 4 \beta_{3} + 9 \beta_1) q^{58} + (\beta_{3} + 2 \beta_1) q^{59} + ( - 9 \beta_{3} + 15 \beta_1) q^{60} - 6 q^{61} + ( - 7 \beta_{2} + 1) q^{62} + ( - \beta_{3} - 3 \beta_1) q^{63} + (6 \beta_{2} + 1) q^{64} + (6 \beta_{2} - 3) q^{66} + (3 \beta_{3} - 6 \beta_1) q^{67} + ( - 3 \beta_{2} - 12) q^{68} + (6 \beta_{2} - 2) q^{69} + ( - 2 \beta_{3} + 3 \beta_1) q^{70} + ( - 2 \beta_{3} - 10 \beta_1) q^{71} + (11 \beta_{3} - 11 \beta_1) q^{72} - 2 \beta_{3} q^{73} + ( - 4 \beta_{2} + 4) q^{74} - 3 \beta_{2} q^{75} - 9 \beta_{3} q^{76} + (3 \beta_{2} + 3) q^{77} + 4 q^{79} + ( - 8 \beta_{3} + 11 \beta_1) q^{80} + (6 \beta_{2} + 4) q^{81} - 2 q^{82} + (3 \beta_{3} + 6 \beta_1) q^{83} + (3 \beta_{3} - 6 \beta_1) q^{84} + (\beta_{3} + 3 \beta_1) q^{85} + (7 \beta_{3} - 16 \beta_1) q^{86} + (9 \beta_{2} - 4) q^{87} + (3 \beta_{2} - 9) q^{88} + (13 \beta_{3} + 5 \beta_1) q^{89} + (12 \beta_{2} - 7) q^{90} + (6 \beta_{2} - 12) q^{92} + ( - \beta_{3} + 7 \beta_1) q^{93} + (3 \beta_{2} - 1) q^{94} - 3 \beta_{2} q^{95} + (9 \beta_{3} - 12 \beta_1) q^{96} + ( - 14 \beta_{3} + 3 \beta_1) q^{97} + (\beta_{3} - \beta_1) q^{98} + (12 \beta_{3} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 6 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 6 q^{4} + 2 q^{9} - 14 q^{10} + 24 q^{12} + 6 q^{14} + 26 q^{16} - 12 q^{17} + 6 q^{22} + 6 q^{25} + 6 q^{29} + 36 q^{30} + 6 q^{35} - 48 q^{36} + 6 q^{38} + 38 q^{40} - 14 q^{42} - 10 q^{43} - 54 q^{48} - 4 q^{49} - 2 q^{51} + 24 q^{53} + 6 q^{55} - 12 q^{56} - 24 q^{61} + 18 q^{62} - 8 q^{64} - 24 q^{66} - 42 q^{68} - 20 q^{69} + 24 q^{74} + 6 q^{75} + 6 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{82} - 34 q^{87} - 42 q^{88} - 52 q^{90} - 60 q^{92} - 10 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 1.61803i
0.618034i
− 0.618034i
1.61803i
− 2.61803i −2.61803 −4.85410 − 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
337.2 − 0.381966i −0.381966 1.85410 − 0.381966i 0.145898i 1.00000i − 1.47214i −2.85410 −0.145898
337.3 0.381966i −0.381966 1.85410 0.381966i − 0.145898i − 1.00000i 1.47214i −2.85410 −0.145898
337.4 2.61803i −2.61803 −4.85410 2.61803i − 6.85410i − 1.00000i − 7.47214i 3.85410 −6.85410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.c 4
13.b even 2 1 inner 1183.2.c.c 4
13.d odd 4 1 1183.2.a.c 2
13.d odd 4 1 1183.2.a.g 2
13.f odd 12 2 91.2.f.a ✓ 4
39.k even 12 2 819.2.o.c 4
52.l even 12 2 1456.2.s.h 4
91.i even 4 1 8281.2.a.n 2
91.i even 4 1 8281.2.a.bb 2
91.w even 12 2 637.2.g.c 4
91.x odd 12 2 637.2.h.g 4
91.ba even 12 2 637.2.h.f 4
91.bc even 12 2 637.2.f.c 4
91.bd odd 12 2 637.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a ✓ 4 13.f odd 12 2
637.2.f.c 4 91.bc even 12 2
637.2.g.b 4 91.bd odd 12 2
637.2.g.c 4 91.w even 12 2
637.2.h.f 4 91.ba even 12 2
637.2.h.g 4 91.x odd 12 2
819.2.o.c 4 39.k even 12 2
1183.2.a.c 2 13.d odd 4 1
1183.2.a.g 2 13.d odd 4 1
1183.2.c.c 4 1.a even 1 1 trivial
1183.2.c.c 4 13.b even 2 1 inner
1456.2.s.h 4 52.l even 12 2
8281.2.a.n 2 91.i even 4 1
8281.2.a.bb 2 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T - 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 98T^{2} + 1681 \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 95)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 31)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 162T^{2} + 81 \) Copy content Toggle raw display
$71$ \( T^{4} + 268 T^{2} + 13456 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 283T^{2} + 6241 \) Copy content Toggle raw display
$97$ \( T^{4} + 503 T^{2} + 52441 \) Copy content Toggle raw display
show more
show less