Properties

Label 1183.2.c.g
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.11667456256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 22\nu^{2} + 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 10\nu^{4} - 14\nu^{2} + 21 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - 38\nu^{4} - 122\nu^{2} - 33 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 38\nu^{5} - 122\nu^{3} - 41\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - 66\nu^{5} - 230\nu^{3} - 127\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} - 13\nu^{5} - 43\nu^{3} - 14\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} - 7\beta_{3} - 10\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{7} + 7\beta_{6} + 15\beta_{5} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{4} + 48\beta_{3} + 86\beta_{2} - 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86\beta_{7} - 48\beta_{6} - 152\beta_{5} - 337\beta_1 \) Copy content Toggle raw display

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
2.74108i
2.22001i
0.710287i
0.231361i
0.231361i
0.710287i
2.22001i
2.74108i
2.74108i 1.36482 −5.51353 0.741082i 3.74108i 1.00000i 9.63087i −1.13727 2.03137
337.2 2.22001i 0.549551 −2.92843 4.22001i 1.22001i 1.00000i 2.06113i −2.69799 9.36845
337.3 0.710287i 2.40788 1.49549 1.28971i 1.71029i 1.00000i 2.48280i 2.79790 −0.916066
337.4 0.231361i −3.32225 1.94647 2.23136i 0.768639i 1.00000i 0.913059i 8.03736 0.516249
337.5 0.231361i −3.32225 1.94647 2.23136i 0.768639i 1.00000i 0.913059i 8.03736 0.516249
337.6 0.710287i 2.40788 1.49549 1.28971i 1.71029i 1.00000i 2.48280i 2.79790 −0.916066
337.7 2.22001i 0.549551 −2.92843 4.22001i 1.22001i 1.00000i 2.06113i −2.69799 9.36845
337.8 2.74108i 1.36482 −5.51353 0.741082i 3.74108i 1.00000i 9.63087i −1.13727 2.03137
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
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.g 8
13.b even 2 1 inner 1183.2.c.g 8
13.d odd 4 1 1183.2.a.k 4
13.d odd 4 1 1183.2.a.l 4
13.f odd 12 2 91.2.f.c 8
39.k even 12 2 819.2.o.h 8
52.l even 12 2 1456.2.s.q 8
91.i even 4 1 8281.2.a.bp 4
91.i even 4 1 8281.2.a.bt 4
91.w even 12 2 637.2.g.j 8
91.x odd 12 2 637.2.h.h 8
91.ba even 12 2 637.2.h.i 8
91.bc even 12 2 637.2.f.i 8
91.bd odd 12 2 637.2.g.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 13.f odd 12 2
637.2.f.i 8 91.bc even 12 2
637.2.g.j 8 91.w even 12 2
637.2.g.k 8 91.bd odd 12 2
637.2.h.h 8 91.x odd 12 2
637.2.h.i 8 91.ba even 12 2
819.2.o.h 8 39.k even 12 2
1183.2.a.k 4 13.d odd 4 1
1183.2.a.l 4 13.d odd 4 1
1183.2.c.g 8 1.a even 1 1 trivial
1183.2.c.g 8 13.b even 2 1 inner
1456.2.s.q 8 52.l even 12 2
8281.2.a.bp 4 91.i even 4 1
8281.2.a.bt 4 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 13T_{2}^{6} + 44T_{2}^{4} + 21T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 13 T^{6} + 44 T^{4} + 21 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} - 9 T^{2} + 16 T - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 25 T^{6} + 140 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 19 T^{6} + 101 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} - 12 T^{2} + 60 T - 53)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 111 T^{6} + 4025 T^{4} + \cdots + 250000 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} - 38 T^{2} + 56 T + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - T^{3} - 22 T^{2} - 21 T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 42 T^{6} + 485 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$37$ \( T^{8} + 66 T^{6} + 1137 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 186 T^{6} + 11129 T^{4} + \cdots + 318096 \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{3} - 3 T^{2} - 8 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 106 T^{6} + 2753 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} - 140 T^{2} + 890 T - 1389)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 218 T^{6} + 13317 T^{4} + \cdots + 498436 \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} - 15 T^{2} + 176 T - 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 458 T^{6} + \cdots + 121220100 \) Copy content Toggle raw display
$71$ \( T^{8} + 228 T^{6} + 16128 T^{4} + \cdots + 4129024 \) Copy content Toggle raw display
$73$ \( T^{8} + 302 T^{6} + 19113 T^{4} + \cdots + 3139984 \) Copy content Toggle raw display
$79$ \( (T^{4} + 26 T^{3} + 134 T^{2} - 1024 T - 7680)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 194 T^{6} + 8557 T^{4} + \cdots + 181476 \) Copy content Toggle raw display
$89$ \( T^{8} + 143 T^{6} + 5193 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{8} + 619 T^{6} + \cdots + 246238864 \) Copy content Toggle raw display
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