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Properties

Label 1170.2.j.i
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: 10.0.8528759163648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + ( - \beta_{4} - \beta_1) q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + \beta_{4} q^{6} + (\beta_{8} + \beta_{2} - 1) q^{7} + q^{8} + (\beta_{8} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + ( - \beta_{4} - \beta_1) q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + \beta_{4} q^{6} + (\beta_{8} + \beta_{2} - 1) q^{7} + q^{8} + (\beta_{8} - \beta_{6}) q^{9} - q^{10} + ( - \beta_{8} - 2 \beta_{2} + 2) q^{11} + \beta_1 q^{12} - \beta_{2} q^{13} + ( - \beta_{8} - \beta_{5} + \cdots + \beta_1) q^{14}+ \cdots + ( - 2 \beta_{9} + 4 \beta_{8} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - q^{3} - 5 q^{4} + 5 q^{5} - q^{6} - 4 q^{7} + 10 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} - q^{3} - 5 q^{4} + 5 q^{5} - q^{6} - 4 q^{7} + 10 q^{8} - q^{9} - 10 q^{10} + 9 q^{11} + 2 q^{12} - 5 q^{13} - 4 q^{14} - 2 q^{15} - 5 q^{16} - 18 q^{17} + 2 q^{18} - 10 q^{19} + 5 q^{20} - 14 q^{21} + 9 q^{22} + 12 q^{23} - q^{24} - 5 q^{25} + 10 q^{26} + 2 q^{27} + 8 q^{28} + 6 q^{29} + q^{30} - 4 q^{31} - 5 q^{32} + 15 q^{33} + 9 q^{34} - 8 q^{35} - q^{36} + 20 q^{37} + 5 q^{38} + 2 q^{39} + 5 q^{40} + 9 q^{41} - 2 q^{42} - q^{43} - 18 q^{44} + q^{45} - 24 q^{46} + 6 q^{47} - q^{48} + 3 q^{49} - 5 q^{50} - 15 q^{51} - 5 q^{52} - 12 q^{53} - q^{54} + 18 q^{55} - 4 q^{56} + 25 q^{57} + 6 q^{58} + 21 q^{59} + q^{60} + 2 q^{61} + 8 q^{62} - 20 q^{63} + 10 q^{64} + 5 q^{65} - 7 q^{67} + 9 q^{68} - 36 q^{69} + 4 q^{70} - 12 q^{71} - q^{72} - 58 q^{73} - 10 q^{74} - q^{75} + 5 q^{76} + 36 q^{77} - q^{78} + 8 q^{79} - 10 q^{80} + 35 q^{81} - 18 q^{82} + 6 q^{83} + 16 q^{84} - 9 q^{85} - q^{86} - 48 q^{87} + 9 q^{88} - 36 q^{89} + q^{90} + 8 q^{91} + 12 q^{92} + 22 q^{93} + 6 q^{94} - 5 q^{95} + 2 q^{96} + 11 q^{97} - 6 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{8} + 4\nu^{7} + 12\nu^{6} + 45\nu^{5} - 63\nu^{4} + 27\nu - 567 ) / 486 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 10\nu^{8} - 14\nu^{7} - 6\nu^{6} + 18\nu^{5} + 72\nu^{4} - 162\nu^{3} + 81\nu^{2} + 27\nu + 162 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} - 12\nu^{6} + 9\nu^{5} + 9\nu^{4} + 135\nu + 81 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} + 13\nu^{8} - 2\nu^{7} - 42\nu^{6} - 63\nu^{5} + 99\nu^{4} - 162\nu^{3} - 324\nu^{2} + 351\nu + 891 ) / 486 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} + 4\nu^{8} + 16\nu^{7} - 24\nu^{6} - 36\nu^{5} + 126\nu^{4} + 81\nu^{3} - 162\nu^{2} - 135\nu + 648 ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{9} + \nu^{8} + 4\nu^{7} + 30\nu^{6} - 225\nu^{5} + 207\nu^{4} + 324\nu^{3} + 162\nu^{2} - 945\nu + 1863 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -19\nu^{9} - \nu^{8} - 4\nu^{7} + 6\nu^{6} - 153\nu^{5} + 333\nu^{4} + 162\nu^{3} + 162\nu^{2} - 513\nu + 2025 ) / 486 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{9} + 5\nu^{8} + 2\nu^{7} - 81\nu^{5} + 135\nu^{4} + 108\nu^{2} - 351\nu + 891 ) / 162 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 2\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - 3\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + 2\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} - 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} + 8\beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{5} - 11\beta_{4} + \beta_{3} + 6\beta_{2} + 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{9} - 4 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} - 8 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{9} - 25 \beta_{8} + 26 \beta_{7} + 10 \beta_{6} - \beta_{5} + 10 \beta_{4} + 19 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13 \beta_{9} - 19 \beta_{8} + 8 \beta_{7} + 16 \beta_{6} - 25 \beta_{5} - 29 \beta_{4} + 19 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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
0.756905 − 1.55791i
1.72806 − 0.117480i
−1.41743 − 0.995434i
1.06839 + 1.36328i
−1.13593 + 1.30754i
0.756905 + 1.55791i
1.72806 + 0.117480i
−1.41743 + 0.995434i
1.06839 − 1.36328i
−1.13593 − 1.30754i
−0.500000 − 0.866025i −1.72765 + 0.123458i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.970741 + 1.43446i 0.153174 + 0.265305i 1.00000 2.96952 − 0.426584i −1.00000
391.2 −0.500000 − 0.866025i −0.965772 − 1.43781i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.762291 + 1.55529i 1.17123 + 2.02864i 1.00000 −1.13457 + 2.77718i −1.00000
391.3 −0.500000 − 0.866025i −0.153356 + 1.72525i −0.500000 + 0.866025i 0.500000 − 0.866025i 1.57079 − 0.729814i −0.230793 − 0.399745i 1.00000 −2.95296 − 0.529154i −1.00000
391.4 −0.500000 − 0.866025i 0.646443 − 1.60689i −0.500000 + 0.866025i 0.500000 − 0.866025i −1.71483 + 0.243611i −2.48656 − 4.30684i 1.00000 −2.16422 − 2.07753i −1.00000
391.5 −0.500000 − 0.866025i 1.70033 + 0.329969i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.564403 − 1.63751i −0.607060 − 1.05146i 1.00000 2.78224 + 1.12211i −1.00000
781.1 −0.500000 + 0.866025i −1.72765 − 0.123458i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.970741 − 1.43446i 0.153174 − 0.265305i 1.00000 2.96952 + 0.426584i −1.00000
781.2 −0.500000 + 0.866025i −0.965772 + 1.43781i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.762291 − 1.55529i 1.17123 − 2.02864i 1.00000 −1.13457 − 2.77718i −1.00000
781.3 −0.500000 + 0.866025i −0.153356 − 1.72525i −0.500000 − 0.866025i 0.500000 + 0.866025i 1.57079 + 0.729814i −0.230793 + 0.399745i 1.00000 −2.95296 + 0.529154i −1.00000
781.4 −0.500000 + 0.866025i 0.646443 + 1.60689i −0.500000 − 0.866025i 0.500000 + 0.866025i −1.71483 − 0.243611i −2.48656 + 4.30684i 1.00000 −2.16422 + 2.07753i −1.00000
781.5 −0.500000 + 0.866025i 1.70033 − 0.329969i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.564403 + 1.63751i −0.607060 + 1.05146i 1.00000 2.78224 − 1.12211i −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.i ✓ 10
3.b odd 2 1 3510.2.j.j 10
9.c even 3 1 inner 1170.2.j.i ✓ 10
9.d odd 6 1 3510.2.j.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.j.i ✓ 10 1.a even 1 1 trivial
1170.2.j.i ✓ 10 9.c even 3 1 inner
3510.2.j.j 10 3.b odd 2 1
3510.2.j.j 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{10} + 4T_{7}^{9} + 24T_{7}^{8} + 129T_{7}^{6} + 138T_{7}^{5} + 240T_{7}^{4} + 48T_{7}^{3} + 33T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{10} - 9 T_{11}^{9} + 63 T_{11}^{8} - 174 T_{11}^{7} + 414 T_{11}^{6} - 522 T_{11}^{5} + \cdots + 324 \) 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
$7$ \( T^{10} + 4 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{10} - 9 T^{9} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} + 9 T^{4} + 18 T^{3} + \cdots - 18)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 5 T^{4} + \cdots + 334)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 12 T^{9} + \cdots + 2178576 \) Copy content Toggle raw display
$29$ \( T^{10} - 6 T^{9} + \cdots + 13749264 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} + \cdots + 633616 \) Copy content Toggle raw display
$37$ \( (T^{5} - 10 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 9 T^{9} + \cdots + 57017601 \) Copy content Toggle raw display
$43$ \( T^{10} + T^{9} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{10} - 6 T^{9} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( (T^{5} + 6 T^{4} + \cdots - 8928)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 21 T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$61$ \( T^{10} - 2 T^{9} + \cdots + 1094116 \) Copy content Toggle raw display
$67$ \( T^{10} + 7 T^{9} + \cdots + 332929 \) Copy content Toggle raw display
$71$ \( (T^{5} + 6 T^{4} + \cdots + 51552)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 29 T^{4} + \cdots - 126968)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 8 T^{9} + \cdots + 64192144 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 2351474064 \) Copy content Toggle raw display
$89$ \( (T^{5} + 18 T^{4} + \cdots + 196506)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 11 T^{9} + \cdots + 2390116 \) Copy content Toggle raw display
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