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Properties

Label 1170.2.j.j
Level $1170$
Weight $2$
Character orbit 1170.j
Analytic conductor $9.342$
Analytic rank $0$
Dimension $10$
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(391,1170)]
 
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([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.391");
 
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.j (of order \(3\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8320271788800.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - 9x^{6} + 27x^{5} - 27x^{4} + 27x^{2} - 81x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{2} q^{3} + (\beta_{6} - 1) q^{4} + (\beta_{6} - 1) q^{5} - \beta_{8} q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{3}) q^{7} - q^{8} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{2} q^{3} + (\beta_{6} - 1) q^{4} + (\beta_{6} - 1) q^{5} - \beta_{8} q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{3}) q^{7} - q^{8} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{9} - q^{10} + ( - \beta_{9} + \beta_{7} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{2} + q^{3} - 5 q^{4} - 5 q^{5} + 2 q^{6} - 5 q^{7} - 10 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{2} + q^{3} - 5 q^{4} - 5 q^{5} + 2 q^{6} - 5 q^{7} - 10 q^{8} - q^{9} - 10 q^{10} - 2 q^{11} + q^{12} + 5 q^{13} + 5 q^{14} + q^{15} - 5 q^{16} + 16 q^{17} + q^{18} - 8 q^{19} - 5 q^{20} + 10 q^{21} + 2 q^{22} - q^{23} - q^{24} - 5 q^{25} + 10 q^{26} - 2 q^{27} + 10 q^{28} - q^{29} - q^{30} + 4 q^{31} + 5 q^{32} - 20 q^{33} + 8 q^{34} + 10 q^{35} + 2 q^{36} - 12 q^{37} - 4 q^{38} - q^{39} + 5 q^{40} - 11 q^{41} - q^{42} + 10 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{46} - 27 q^{47} - 2 q^{48} + 6 q^{49} + 5 q^{50} - 44 q^{51} + 5 q^{52} + 24 q^{53} - q^{54} + 4 q^{55} + 5 q^{56} - 2 q^{57} + q^{58} - 18 q^{59} - 2 q^{60} + 13 q^{61} + 8 q^{62} - 19 q^{63} + 10 q^{64} + 5 q^{65} - 4 q^{66} - 5 q^{67} - 8 q^{68} - q^{69} + 5 q^{70} + 12 q^{71} + q^{72} + 24 q^{73} - 6 q^{74} - 2 q^{75} + 4 q^{76} - 8 q^{77} + q^{78} - 2 q^{79} + 10 q^{80} + 35 q^{81} - 22 q^{82} - 9 q^{83} - 11 q^{84} - 8 q^{85} - 10 q^{86} + 8 q^{87} + 2 q^{88} + 2 q^{89} + q^{90} - 10 q^{91} - q^{92} + 52 q^{93} + 27 q^{94} + 4 q^{95} - q^{96} - 2 q^{97} + 12 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + x^{8} - 9x^{6} + 27x^{5} - 27x^{4} + 27x^{2} - 81x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - \nu^{8} + \nu^{7} - 9\nu^{5} + 27\nu^{4} - 27\nu^{3} + 27\nu - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - \nu^{8} + \nu^{7} + 9\nu^{6} - 18\nu^{5} + 9\nu^{4} - 27\nu^{2} + 108\nu - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - 4\nu^{8} + 4\nu^{7} + 6\nu^{6} - 18\nu^{5} + 36\nu^{4} - 81\nu^{3} + 54\nu^{2} + 108\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 4\nu^{8} + 4\nu^{7} + 6\nu^{6} - 18\nu^{5} + 36\nu^{4} - 81\nu^{3} - 27\nu^{2} + 189\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{9} - \nu^{8} + 10\nu^{7} - 12\nu^{6} - 54\nu^{3} + 162\nu^{2} - 54\nu - 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 4\nu^{8} + 4\nu^{7} + 6\nu^{6} - 18\nu^{5} + 36\nu^{4} - 54\nu^{3} + 135\nu - 162 ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + 3\nu^{8} - 10\nu^{6} + 21\nu^{5} - 27\nu^{4} + 27\nu^{3} + 54\nu^{2} - 189\nu + 135 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{9} - 7\nu^{8} + 25\nu^{7} + 6\nu^{6} - 54\nu^{5} + 63\nu^{4} - 162\nu^{3} + 216\nu^{2} + 513\nu - 1377 ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - 3\beta_{6} - 2\beta_{5} - 2\beta_{3} + 3\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 4\beta_{8} + 3\beta_{7} - 6\beta_{6} + \beta_{4} - 2\beta_{3} - 3\beta_{2} + 5\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} + 6\beta_{8} + 2\beta_{7} - 12\beta_{6} - \beta_{5} + 7\beta_{4} + 3\beta_{3} - 6\beta_{2} - 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{9} + 11\beta_{8} + 8\beta_{7} - 6\beta_{6} + 8\beta_{5} - 12\beta_{4} + 14\beta_{3} - 3\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8 \beta_{9} + 14 \beta_{8} - 12 \beta_{7} - 21 \beta_{6} + 18 \beta_{5} + 8 \beta_{4} + 20 \beta_{3} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{9} + 12 \beta_{8} + 7 \beta_{7} + 12 \beta_{6} + 10 \beta_{5} + 2 \beta_{4} + 42 \beta_{3} + \cdots - 114 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
−1.72991 + 0.0861500i
−0.315546 + 1.70307i
−0.268761 − 1.71107i
1.15639 + 1.28949i
1.65783 − 0.501603i
−1.72991 − 0.0861500i
−0.315546 − 1.70307i
−0.268761 + 1.71107i
1.15639 − 1.28949i
1.65783 + 0.501603i
0.500000 + 0.866025i −1.72991 − 0.0861500i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.790345 − 1.54122i −1.16800 − 2.02303i −1.00000 2.98516 + 0.298063i −1.00000
391.2 0.500000 + 0.866025i −0.315546 − 1.70307i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.31712 − 1.12480i 0.454390 + 0.787026i −1.00000 −2.80086 + 1.07479i −1.00000
391.3 0.500000 + 0.866025i −0.268761 + 1.71107i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.61621 + 0.622782i −1.65934 − 2.87405i −1.00000 −2.85554 − 0.919738i −1.00000
391.4 0.500000 + 0.866025i 1.15639 − 1.28949i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.69492 + 0.356716i −1.27139 − 2.20211i −1.00000 −0.325547 − 2.98228i −1.00000
391.5 0.500000 + 0.866025i 1.65783 + 0.501603i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.394513 + 1.68652i 1.14433 + 1.98204i −1.00000 2.49679 + 1.66314i −1.00000
781.1 0.500000 − 0.866025i −1.72991 + 0.0861500i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.790345 + 1.54122i −1.16800 + 2.02303i −1.00000 2.98516 − 0.298063i −1.00000
781.2 0.500000 − 0.866025i −0.315546 + 1.70307i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.31712 + 1.12480i 0.454390 − 0.787026i −1.00000 −2.80086 − 1.07479i −1.00000
781.3 0.500000 − 0.866025i −0.268761 − 1.71107i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.61621 − 0.622782i −1.65934 + 2.87405i −1.00000 −2.85554 + 0.919738i −1.00000
781.4 0.500000 − 0.866025i 1.15639 + 1.28949i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.69492 − 0.356716i −1.27139 + 2.20211i −1.00000 −0.325547 + 2.98228i −1.00000
781.5 0.500000 − 0.866025i 1.65783 − 0.501603i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.394513 − 1.68652i 1.14433 − 1.98204i −1.00000 2.49679 − 1.66314i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.j.j ✓ 10
3.b odd 2 1 3510.2.j.i 10
9.c even 3 1 inner 1170.2.j.j ✓ 10
9.d odd 6 1 3510.2.j.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.j.j ✓ 10 1.a even 1 1 trivial
1170.2.j.j ✓ 10 9.c even 3 1 inner
3510.2.j.i 10 3.b odd 2 1
3510.2.j.i 10 9.d odd 6 1

Hecke kernels

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

\( T_{7}^{10} + 5 T_{7}^{9} + 27 T_{7}^{8} + 58 T_{7}^{7} + 191 T_{7}^{6} + 279 T_{7}^{5} + 917 T_{7}^{4} + \cdots + 1681 \) Copy content Toggle raw display
\( T_{11}^{10} + 2 T_{11}^{9} + 30 T_{11}^{8} + 76 T_{11}^{7} + 710 T_{11}^{6} + 1530 T_{11}^{5} + \cdots + 58564 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + \cdots + 1681 \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots + 58564 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} - 8 T^{4} + \cdots - 166)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 4 T^{4} - 14 T^{3} + \cdots - 30)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{10} + T^{9} + \cdots + 383161 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + \cdots + 3006756 \) Copy content Toggle raw display
$37$ \( (T^{5} + 6 T^{4} + \cdots + 1662)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 11 T^{9} + \cdots + 1771561 \) Copy content Toggle raw display
$43$ \( T^{10} - 10 T^{9} + \cdots + 1149184 \) Copy content Toggle raw display
$47$ \( T^{10} + 27 T^{9} + \cdots + 29648025 \) Copy content Toggle raw display
$53$ \( (T^{5} - 12 T^{4} + \cdots - 768)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 18 T^{9} + \cdots + 43877376 \) Copy content Toggle raw display
$61$ \( T^{10} - 13 T^{9} + \cdots + 483025 \) Copy content Toggle raw display
$67$ \( T^{10} + 5 T^{9} + \cdots + 9455625 \) Copy content Toggle raw display
$71$ \( (T^{5} - 6 T^{4} + \cdots - 13344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 12 T^{4} + \cdots - 2568)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + \cdots + 306916 \) Copy content Toggle raw display
$83$ \( T^{10} + 9 T^{9} + \cdots + 8982009 \) Copy content Toggle raw display
$89$ \( (T^{5} - T^{4} - 146 T^{3} + \cdots + 2221)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 2 T^{9} + \cdots + 14760964 \) Copy content Toggle raw display
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