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Properties

Label 1170.2.bj.a
Level $1170$
Weight $2$
Character orbit 1170.bj
Analytic conductor $9.342$
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] = [1170,2,Mod(199,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bj (of order \(6\), degree \(2\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.50027374224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} - \beta_{3} q^{4} + \beta_{6} q^{5} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} - \beta_{3} q^{4} + \beta_{6} q^{5} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{7} + q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{7} - \beta_{4} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 3 q^{5} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 3 q^{5} - 5 q^{7} + 8 q^{8} - 3 q^{10} + 3 q^{11} - 4 q^{13} + 10 q^{14} - 4 q^{16} - 15 q^{17} + 9 q^{19} - 3 q^{22} - 6 q^{23} + 5 q^{25} - q^{26} - 5 q^{28} + 3 q^{29} - 4 q^{32} - 15 q^{35} - 20 q^{37} + 3 q^{40} - 21 q^{41} - 18 q^{43} + 6 q^{46} + 6 q^{47} - 15 q^{49} - 4 q^{50} + 5 q^{52} + 31 q^{55} - 5 q^{56} + 3 q^{58} - 30 q^{59} - 5 q^{61} - 6 q^{62} + 8 q^{64} - 21 q^{65} + 16 q^{67} + 15 q^{68} - 18 q^{70} - 26 q^{73} - 20 q^{74} - 9 q^{76} + 4 q^{79} - 3 q^{80} + 21 q^{82} - 48 q^{83} - 29 q^{85} + 3 q^{88} + 39 q^{89} + 19 q^{91} - 3 q^{94} - 15 q^{95} + 4 q^{97} - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 10\nu^{2} + 2\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 116\nu^{3} + 172\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 16\nu^{5} + 68\nu^{3} + 8\nu^{2} + 60\nu + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 68\nu^{3} + 8\nu^{2} - 60\nu + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 8\nu^{6} + 20\nu^{5} + 128\nu^{4} + 132\nu^{3} + 544\nu^{2} + 268\nu + 544 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 8\nu^{6} + 20\nu^{5} - 128\nu^{4} + 132\nu^{3} - 544\nu^{2} + 268\nu - 544 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{5} - 10\beta_{4} + 4\beta_{2} - 2\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{7} - 24\beta_{6} + 2\beta_{5} - 2\beta_{4} + 64\beta_{3} + 44\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} + 4\beta_{6} + 92\beta_{5} + 92\beta_{4} - 64\beta_{2} + 32\beta _1 - 272 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 248\beta_{7} + 248\beta_{6} - 40\beta_{5} + 40\beta_{4} - 752\beta_{3} - 356\beta _1 + 336 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 1.83766i
1.17644i
3.07108i
− 2.40987i
1.83766i
− 1.17644i
− 3.07108i
2.40987i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −2.21022 − 0.339024i 0 −2.39871 − 4.15469i 1.00000 0 1.39871 − 1.74459i
199.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.235468 + 2.22364i 0 1.04346 + 1.80732i 1.00000 0 −2.04346 − 0.907896i
199.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.36822 − 1.76861i 0 −1.84755 − 3.20005i 1.00000 0 0.847550 + 2.06922i
199.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.10653 + 0.750022i 0 0.702803 + 1.21729i 1.00000 0 −1.70280 + 1.44930i
829.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.21022 + 0.339024i 0 −2.39871 + 4.15469i 1.00000 0 1.39871 + 1.74459i
829.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.235468 − 2.22364i 0 1.04346 − 1.80732i 1.00000 0 −2.04346 + 0.907896i
829.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.36822 + 1.76861i 0 −1.84755 + 3.20005i 1.00000 0 0.847550 − 2.06922i
829.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.10653 − 0.750022i 0 0.702803 − 1.21729i 1.00000 0 −1.70280 − 1.44930i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bj.a 8
3.b odd 2 1 130.2.m.b yes 8
5.b even 2 1 1170.2.bj.b 8
12.b even 2 1 1040.2.df.a 8
13.e even 6 1 1170.2.bj.b 8
15.d odd 2 1 130.2.m.a ✓ 8
15.e even 4 2 650.2.m.e 16
39.h odd 6 1 130.2.m.a ✓ 8
39.h odd 6 1 1690.2.c.f 8
39.i odd 6 1 1690.2.c.e 8
39.k even 12 2 1690.2.b.e 16
60.h even 2 1 1040.2.df.c 8
65.l even 6 1 inner 1170.2.bj.a 8
156.r even 6 1 1040.2.df.c 8
195.x odd 6 1 1690.2.c.f 8
195.y odd 6 1 130.2.m.b yes 8
195.y odd 6 1 1690.2.c.e 8
195.bc odd 12 2 8450.2.a.cs 8
195.bf even 12 2 650.2.m.e 16
195.bh even 12 2 1690.2.b.e 16
195.bn odd 12 2 8450.2.a.cr 8
780.cb even 6 1 1040.2.df.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a ✓ 8 15.d odd 2 1
130.2.m.a ✓ 8 39.h odd 6 1
130.2.m.b yes 8 3.b odd 2 1
130.2.m.b yes 8 195.y odd 6 1
650.2.m.e 16 15.e even 4 2
650.2.m.e 16 195.bf even 12 2
1040.2.df.a 8 12.b even 2 1
1040.2.df.a 8 780.cb even 6 1
1040.2.df.c 8 60.h even 2 1
1040.2.df.c 8 156.r even 6 1
1170.2.bj.a 8 1.a even 1 1 trivial
1170.2.bj.a 8 65.l even 6 1 inner
1170.2.bj.b 8 5.b even 2 1
1170.2.bj.b 8 13.e even 6 1
1690.2.b.e 16 39.k even 12 2
1690.2.b.e 16 195.bh even 12 2
1690.2.c.e 8 39.i odd 6 1
1690.2.c.e 8 195.y odd 6 1
1690.2.c.f 8 39.h odd 6 1
1690.2.c.f 8 195.x odd 6 1
8450.2.a.cr 8 195.bn odd 12 2
8450.2.a.cs 8 195.bc odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{8} + 5T_{7}^{7} + 34T_{7}^{6} + 29T_{7}^{5} + 214T_{7}^{4} - 187T_{7}^{3} + 1837T_{7}^{2} - 1924T_{7} + 2704 \) Copy content Toggle raw display
\( T_{17}^{8} + 15T_{17}^{7} + 83T_{17}^{6} + 120T_{17}^{5} - 144T_{17}^{4} - 336T_{17}^{3} + 572T_{17}^{2} + 84T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{7} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 15 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 9216 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{8} + 60 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$41$ \( T^{8} + 21 T^{7} + \cdots + 1106704 \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 2359296 \) Copy content Toggle raw display
$47$ \( (T^{4} - 3 T^{3} - 117 T^{2} + \cdots - 96)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 356 T^{6} + \cdots + 17783089 \) Copy content Toggle raw display
$59$ \( T^{8} + 30 T^{7} + \cdots + 369664 \) Copy content Toggle raw display
$61$ \( T^{8} + 5 T^{7} + \cdots + 28751044 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 160 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$73$ \( (T^{4} + 13 T^{3} + \cdots - 1406)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots + 1384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots - 9744)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 39 T^{7} + \cdots + 59474944 \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 327184 \) Copy content Toggle raw display
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