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Properties

Label 1183.2.c.d
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
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] = [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: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

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
0.707107 − 0.707107i
−0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
− 1.41421i −1.41421 0 − 4.41421i 2.00000i 1.00000i − 2.82843i −1.00000 −6.24264
337.2 − 1.41421i 1.41421 0 1.58579i − 2.00000i − 1.00000i − 2.82843i −1.00000 2.24264
337.3 1.41421i −1.41421 0 4.41421i − 2.00000i − 1.00000i 2.82843i −1.00000 −6.24264
337.4 1.41421i 1.41421 0 − 1.58579i 2.00000i 1.00000i 2.82843i −1.00000 2.24264
\(n\): 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.d 4
13.b even 2 1 inner 1183.2.c.d 4
13.d odd 4 1 91.2.a.c ✓ 2
13.d odd 4 1 1183.2.a.d 2
39.f even 4 1 819.2.a.h 2
52.f even 4 1 1456.2.a.q 2
65.g odd 4 1 2275.2.a.j 2
91.i even 4 1 637.2.a.g 2
91.i even 4 1 8281.2.a.v 2
91.z odd 12 2 637.2.e.f 4
91.bb even 12 2 637.2.e.g 4
104.j odd 4 1 5824.2.a.bl 2
104.m even 4 1 5824.2.a.bk 2
273.o odd 4 1 5733.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c ✓ 2 13.d odd 4 1
637.2.a.g 2 91.i even 4 1
637.2.e.f 4 91.z odd 12 2
637.2.e.g 4 91.bb even 12 2
819.2.a.h 2 39.f even 4 1
1183.2.a.d 2 13.d odd 4 1
1183.2.c.d 4 1.a even 1 1 trivial
1183.2.c.d 4 13.b even 2 1 inner
1456.2.a.q 2 52.f even 4 1
2275.2.a.j 2 65.g odd 4 1
5733.2.a.s 2 273.o odd 4 1
5824.2.a.bk 2 104.m even 4 1
5824.2.a.bl 2 104.j odd 4 1
8281.2.a.v 2 91.i even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 22T^{2} + 49 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 54T^{2} + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$37$ \( T^{4} + 44T^{2} + 196 \) Copy content Toggle raw display
$41$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$43$ \( (T - 5)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 22T^{2} + 49 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T - 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} + 172T^{2} + 196 \) Copy content Toggle raw display
$73$ \( T^{4} + 86T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T - 23)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 198T^{2} + 3969 \) Copy content Toggle raw display
$89$ \( T^{4} + 22T^{2} + 49 \) Copy content Toggle raw display
$97$ \( T^{4} + 326 T^{2} + 25921 \) Copy content Toggle raw display
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