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Properties

Label 1183.2.c.i
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_{7}) q^{6}+ \cdots + (2 \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_{7}) q^{6}+ \cdots + (2 \beta_{11} + 3 \beta_{10} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} + 8 q^{9} - 24 q^{10} - 4 q^{12} - 8 q^{14} + 16 q^{16} + 8 q^{17} - 12 q^{22} + 24 q^{23} - 20 q^{25} + 12 q^{27} - 16 q^{29} - 16 q^{30} - 12 q^{35} + 20 q^{36} + 4 q^{38} + 92 q^{40} - 8 q^{42} - 4 q^{43} + 4 q^{48} - 12 q^{49} + 52 q^{51} - 44 q^{53} + 12 q^{55} + 24 q^{56} - 28 q^{61} + 8 q^{62} - 52 q^{64} - 52 q^{66} + 16 q^{68} - 8 q^{69} - 12 q^{74} - 92 q^{75} + 8 q^{77} - 56 q^{79} - 4 q^{81} - 28 q^{82} + 4 q^{87} + 28 q^{88} + 24 q^{90} + 24 q^{92} - 8 q^{94} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 3\nu^{9} - \nu^{8} + 9\nu^{7} + \nu^{6} - 23\nu^{5} + 16\nu^{4} + 26\nu^{3} - 32\nu^{2} - 28\nu + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 6 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 3 \nu^{7} + 52 \nu^{6} - 34 \nu^{5} - 88 \nu^{4} + \cdots - 160 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 4 \nu^{10} - 17 \nu^{9} - 6 \nu^{8} + 51 \nu^{7} + 6 \nu^{6} - 122 \nu^{5} + 68 \nu^{4} + \cdots + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} + 17 \nu^{9} - 4 \nu^{8} - 55 \nu^{7} + 24 \nu^{6} + 126 \nu^{5} - 128 \nu^{4} + \cdots - 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 8 \nu^{10} - 5 \nu^{9} + 30 \nu^{8} + 15 \nu^{7} - 78 \nu^{6} + 18 \nu^{5} + 124 \nu^{4} + \cdots + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} + 7 \nu^{9} - 19 \nu^{8} - 19 \nu^{7} + 49 \nu^{6} + 14 \nu^{5} - 106 \nu^{4} + \cdots - 128 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} + 9 \nu^{9} + 18 \nu^{8} - 11 \nu^{7} - 18 \nu^{6} + 6 \nu^{5} + 68 \nu^{4} - 76 \nu^{3} + \cdots + 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + \cdots - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} + 12 \nu^{9} - 3 \nu^{8} - 28 \nu^{7} + 19 \nu^{6} + 43 \nu^{5} - 52 \nu^{4} + \cdots - 16 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + \beta_{10} + 3\beta_{7} - 2\beta_{5} - 3\beta_{4} - \beta_{3} + \beta_{2} - 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} + 4\beta_{10} + 2\beta_{9} - 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - \beta_{2} - 2\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5 \beta_{11} + 5 \beta_{10} + \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + 8 \beta_{6} - \beta_{5} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5 \beta_{11} + 12 \beta_{9} + 3 \beta_{8} - \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8 \beta_{11} + 3 \beta_{10} + \beta_{9} + 11 \beta_{8} + 9 \beta_{7} - 13 \beta_{5} - 8 \beta_{4} + \cdots - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4 \beta_{11} + 13 \beta_{10} + 24 \beta_{9} + 4 \beta_{8} - 25 \beta_{7} + 8 \beta_{6} - 2 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7 \beta_{11} - 6 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} - 16 \beta_{7} - 40 \beta_{6} + 6 \beta_{5} + \cdots - 20 ) / 2 \) 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
−1.12906 − 0.851598i
−1.30089 − 0.554694i
0.759479 − 1.19298i
1.40744 − 0.138282i
−1.08105 − 0.911778i
1.34408 + 0.439820i
1.34408 − 0.439820i
−1.08105 + 0.911778i
1.40744 + 0.138282i
0.759479 + 1.19298i
−1.30089 + 0.554694i
−1.12906 + 0.851598i
− 2.70320i −0.345949 −5.30727 − 3.25812i 0.935168i − 1.00000i 8.94020i −2.88032 −8.80735
337.2 − 2.10939i 2.26165 −2.44952 − 3.60178i − 4.77070i − 1.00000i 0.948212i 2.11505 −7.59755
337.3 − 1.38595i −2.82577 0.0791355 − 0.518957i 3.91639i 1.00000i − 2.88158i 4.98500 −0.719250
337.4 − 1.27656i −1.16793 0.370384 1.81487i 1.49093i − 1.00000i − 3.02595i −1.63595 2.31680
337.5 − 0.823556i 2.66029 1.32176 3.16209i − 2.19090i 1.00000i − 2.73565i 4.07715 2.60416
337.6 − 0.120360i −0.582292 1.98551 1.68817i 0.0700846i − 1.00000i − 0.479696i −2.66094 0.203187
337.7 0.120360i −0.582292 1.98551 − 1.68817i − 0.0700846i 1.00000i 0.479696i −2.66094 0.203187
337.8 0.823556i 2.66029 1.32176 − 3.16209i 2.19090i − 1.00000i 2.73565i 4.07715 2.60416
337.9 1.27656i −1.16793 0.370384 − 1.81487i − 1.49093i 1.00000i 3.02595i −1.63595 2.31680
337.10 1.38595i −2.82577 0.0791355 0.518957i − 3.91639i − 1.00000i 2.88158i 4.98500 −0.719250
337.11 2.10939i 2.26165 −2.44952 3.60178i 4.77070i 1.00000i − 0.948212i 2.11505 −7.59755
337.12 2.70320i −0.345949 −5.30727 3.25812i − 0.935168i 1.00000i − 8.94020i −2.88032 −8.80735
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
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.i 12
13.b even 2 1 inner 1183.2.c.i 12
13.c even 3 1 91.2.q.a ✓ 12
13.d odd 4 1 1183.2.a.m 6
13.d odd 4 1 1183.2.a.p 6
13.e even 6 1 91.2.q.a ✓ 12
39.h odd 6 1 819.2.ct.a 12
39.i odd 6 1 819.2.ct.a 12
52.i odd 6 1 1456.2.cc.c 12
52.j odd 6 1 1456.2.cc.c 12
91.g even 3 1 637.2.u.h 12
91.h even 3 1 637.2.k.h 12
91.i even 4 1 8281.2.a.by 6
91.i even 4 1 8281.2.a.ch 6
91.k even 6 1 637.2.k.h 12
91.l odd 6 1 637.2.k.g 12
91.m odd 6 1 637.2.u.i 12
91.n odd 6 1 637.2.q.h 12
91.p odd 6 1 637.2.u.i 12
91.t odd 6 1 637.2.q.h 12
91.u even 6 1 637.2.u.h 12
91.v odd 6 1 637.2.k.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a ✓ 12 13.c even 3 1
91.2.q.a ✓ 12 13.e even 6 1
637.2.k.g 12 91.l odd 6 1
637.2.k.g 12 91.v odd 6 1
637.2.k.h 12 91.h even 3 1
637.2.k.h 12 91.k even 6 1
637.2.q.h 12 91.n odd 6 1
637.2.q.h 12 91.t odd 6 1
637.2.u.h 12 91.g even 3 1
637.2.u.h 12 91.u even 6 1
637.2.u.i 12 91.m odd 6 1
637.2.u.i 12 91.p odd 6 1
819.2.ct.a 12 39.h odd 6 1
819.2.ct.a 12 39.i odd 6 1
1183.2.a.m 6 13.d odd 4 1
1183.2.a.p 6 13.d odd 4 1
1183.2.c.i 12 1.a even 1 1 trivial
1183.2.c.i 12 13.b even 2 1 inner
1456.2.cc.c 12 52.i odd 6 1
1456.2.cc.c 12 52.j odd 6 1
8281.2.a.by 6 91.i even 4 1
8281.2.a.ch 6 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 206T_{2}^{6} + 208T_{2}^{4} + 72T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - 11 T^{4} - 2 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 40 T^{10} + \cdots + 3481 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 50 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} + \cdots - 491)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 58 T^{10} + \cdots + 55696 \) Copy content Toggle raw display
$23$ \( (T^{6} - 12 T^{5} + \cdots + 6208)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 8 T^{5} + \cdots + 3169)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 136 T^{10} + \cdots + 913936 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1755945216 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 884705536 \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots + 1552)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 272 T^{10} + \cdots + 9461776 \) Copy content Toggle raw display
$53$ \( (T^{6} + 22 T^{5} + \cdots - 2339)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4571923456 \) Copy content Toggle raw display
$61$ \( (T^{6} + 14 T^{5} + \cdots + 2368)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 613651984 \) Copy content Toggle raw display
$71$ \( T^{12} + 152 T^{10} + \cdots + 46895104 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1386221824 \) Copy content Toggle raw display
$79$ \( (T^{6} + 28 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 141324544 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1834580224 \) Copy content Toggle raw display
$97$ \( T^{12} + 382 T^{10} + \cdots + 53465344 \) Copy content Toggle raw display
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