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Properties

Label 1170.2.j.n
Level $1170$
Weight $2$
Character orbit 1170.j
Analytic conductor $9.342$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - 6 x^{12} - 12 x^{11} + 3 x^{10} + 63 x^{9} - 90 x^{8} + 189 x^{7} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{2} + ( - \beta_{7} - \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + \beta_{15} q^{7} + q^{8} + \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{2} + ( - \beta_{7} - \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + \beta_{15} q^{7} + q^{8} + \beta_{9} q^{9} + q^{10} + (\beta_{13} + \beta_{7} + \beta_{5} + \cdots - 1) q^{11}+ \cdots + (\beta_{15} + \beta_{13} - 2 \beta_{12} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{2} - 2 q^{3} - 8 q^{4} - 8 q^{5} + q^{6} - q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{2} - 2 q^{3} - 8 q^{4} - 8 q^{5} + q^{6} - q^{7} + 16 q^{8} + 2 q^{9} + 16 q^{10} - 5 q^{11} + q^{12} - 8 q^{13} - q^{14} + q^{15} - 8 q^{16} + 10 q^{17} - q^{18} + 2 q^{19} - 8 q^{20} - 12 q^{21} - 5 q^{22} - 3 q^{23} - 2 q^{24} - 8 q^{25} + 16 q^{26} - 2 q^{27} + 2 q^{28} - 7 q^{29} - 2 q^{30} + 4 q^{31} - 8 q^{32} + 19 q^{33} - 5 q^{34} + 2 q^{35} - q^{36} + 16 q^{37} - q^{38} + q^{39} - 8 q^{40} - 22 q^{41} - 3 q^{42} - 15 q^{43} + 10 q^{44} - q^{45} + 6 q^{46} - 17 q^{47} + q^{48} - 21 q^{49} - 8 q^{50} + 39 q^{51} - 8 q^{52} - 4 q^{53} - 8 q^{54} + 10 q^{55} - q^{56} - 29 q^{57} - 7 q^{58} + 5 q^{59} + q^{60} - 7 q^{61} - 8 q^{62} - 21 q^{63} + 16 q^{64} - 8 q^{65} - 26 q^{66} - 8 q^{67} - 5 q^{68} + 35 q^{69} - q^{70} + 64 q^{71} + 2 q^{72} + 42 q^{73} - 8 q^{74} + q^{75} - q^{76} - 30 q^{77} - 2 q^{78} + 16 q^{80} + 14 q^{81} + 44 q^{82} - 15 q^{83} + 15 q^{84} - 5 q^{85} - 15 q^{86} + 42 q^{87} - 5 q^{88} + 30 q^{89} + 2 q^{90} + 2 q^{91} - 3 q^{92} - 26 q^{93} - 17 q^{94} - q^{95} + q^{96} - 11 q^{97} + 42 q^{98} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + x^{14} - 6 x^{12} - 12 x^{11} + 3 x^{10} + 63 x^{9} - 90 x^{8} + 189 x^{7} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 6 \nu^{11} - 30 \nu^{10} - 33 \nu^{9} + \cdots + 729 \nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + 14 \nu^{14} - 5 \nu^{13} - 48 \nu^{12} - 24 \nu^{11} + 6 \nu^{10} - 150 \nu^{9} + \cdots + 34992 ) / 4374 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47 \nu^{15} + 8 \nu^{14} + 109 \nu^{13} + 33 \nu^{12} - 33 \nu^{11} + 501 \nu^{10} + 354 \nu^{9} + \cdots - 6561 ) / 8748 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{15} - 152 \nu^{14} + 35 \nu^{13} + 327 \nu^{12} + 33 \nu^{11} - 231 \nu^{10} + 1536 \nu^{9} + \cdots - 343359 ) / 8748 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{15} + 52 \nu^{14} + 11 \nu^{13} - 105 \nu^{12} - 21 \nu^{11} + 165 \nu^{10} + \cdots + 102789 ) / 2916 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{15} + 146 \nu^{14} - 29 \nu^{13} - 327 \nu^{12} - 69 \nu^{11} + 159 \nu^{10} + \cdots + 330237 ) / 4374 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13 \nu^{15} + 8 \nu^{14} - 35 \nu^{13} - 33 \nu^{12} + 3 \nu^{11} - 147 \nu^{10} - 186 \nu^{9} + \cdots + 28431 ) / 972 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25 \nu^{15} - 370 \nu^{14} + 55 \nu^{13} + 807 \nu^{12} + 237 \nu^{11} - 471 \nu^{10} + 3594 \nu^{9} + \cdots - 802629 ) / 8748 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23 \nu^{15} + 212 \nu^{14} - 5 \nu^{13} - 450 \nu^{12} - 114 \nu^{11} + 330 \nu^{10} + \cdots + 441774 ) / 4374 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76 \nu^{15} - 101 \nu^{14} + 200 \nu^{13} + 345 \nu^{12} + 15 \nu^{11} + 732 \nu^{10} + \cdots - 275562 ) / 4374 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 89 \nu^{15} + \nu^{14} - 208 \nu^{13} - 108 \nu^{12} - 12 \nu^{11} - 879 \nu^{10} - 705 \nu^{9} + \cdots + 69984 ) / 4374 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 51 \nu^{15} + 44 \nu^{14} + 97 \nu^{13} - 31 \nu^{12} - 21 \nu^{11} + 555 \nu^{10} + 30 \nu^{9} + \cdots + 56133 ) / 2916 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 199 \nu^{15} - 40 \nu^{14} - 437 \nu^{13} - 147 \nu^{12} + 3 \nu^{11} - 2019 \nu^{10} - 1284 \nu^{9} + \cdots + 67797 ) / 8748 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{13} - \beta_{10} - \beta_{8} - \beta_{4} - \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 2 \beta_{12} - 2 \beta_{11} - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} + 4 \beta_{14} - \beta_{13} - 2 \beta_{11} + \beta_{10} + \beta_{9} + 5 \beta_{8} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{15} - 4 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6 \beta_{15} + 5 \beta_{14} - 9 \beta_{13} + 2 \beta_{12} - \beta_{11} - 9 \beta_{10} - 2 \beta_{9} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7 \beta_{15} + 23 \beta_{14} - 5 \beta_{13} - 12 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20 \beta_{15} + 33 \beta_{14} + 46 \beta_{13} - 20 \beta_{12} - 27 \beta_{11} + 38 \beta_{10} + \cdots + 61 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 54 \beta_{15} + 19 \beta_{14} - 12 \beta_{13} - 44 \beta_{12} - 47 \beta_{11} - 6 \beta_{10} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 92 \beta_{15} + 31 \beta_{14} - 58 \beta_{13} + 138 \beta_{12} + 79 \beta_{11} - 8 \beta_{10} + 19 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 34 \beta_{15} - 141 \beta_{14} - 136 \beta_{13} - 52 \beta_{12} + 135 \beta_{11} - 92 \beta_{10} + \cdots + 245 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 144 \beta_{15} + 431 \beta_{14} + 30 \beta_{13} - 208 \beta_{12} + 281 \beta_{11} + 24 \beta_{10} + \cdots + 1189 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 151 \beta_{15} + 239 \beta_{14} + 247 \beta_{13} - 516 \beta_{12} + 83 \beta_{11} + 575 \beta_{10} + \cdots - 1585 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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
0.627123 + 1.61453i
0.331669 + 1.70000i
1.71843 + 0.216776i
−1.23129 + 1.21817i
1.56978 − 0.731968i
−1.72810 − 0.116875i
0.319249 − 1.70237i
−1.10687 − 1.33223i
0.627123 − 1.61453i
0.331669 − 1.70000i
1.71843 − 0.216776i
−1.23129 − 1.21817i
1.56978 + 0.731968i
−1.72810 + 0.116875i
0.319249 + 1.70237i
−1.10687 + 1.33223i
−0.500000 − 0.866025i −1.71179 − 0.264162i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.627123 + 1.61453i −1.11167 − 1.92547i 1.00000 2.86044 + 0.904379i 1.00000
391.2 −0.500000 − 0.866025i −1.63808 − 0.562765i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.331669 + 1.70000i 2.09525 + 3.62908i 1.00000 2.36659 + 1.84371i 1.00000
391.3 −0.500000 − 0.866025i −1.04695 + 1.37982i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.71843 + 0.216776i 1.27600 + 2.21009i 1.00000 −0.807794 − 2.88920i 1.00000
391.4 −0.500000 − 0.866025i −0.439322 − 1.67541i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.23129 + 1.21817i −0.477192 − 0.826521i 1.00000 −2.61399 + 1.47209i 1.00000
391.5 −0.500000 − 0.866025i −0.150989 + 1.72546i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.56978 − 0.731968i −1.86938 − 3.23786i 1.00000 −2.95440 − 0.521050i 1.00000
391.6 −0.500000 − 0.866025i 0.965268 − 1.43814i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.72810 − 0.116875i −0.0780455 − 0.135179i 1.00000 −1.13651 − 2.77639i 1.00000
391.7 −0.500000 − 0.866025i 1.31468 + 1.12766i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.319249 − 1.70237i 1.85878 + 3.21950i 1.00000 0.456744 + 2.96503i 1.00000
391.8 −0.500000 − 0.866025i 1.70718 − 0.292459i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.10687 − 1.33223i −2.19374 − 3.79967i 1.00000 2.82894 − 0.998562i 1.00000
781.1 −0.500000 + 0.866025i −1.71179 + 0.264162i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.627123 − 1.61453i −1.11167 + 1.92547i 1.00000 2.86044 − 0.904379i 1.00000
781.2 −0.500000 + 0.866025i −1.63808 + 0.562765i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.331669 − 1.70000i 2.09525 − 3.62908i 1.00000 2.36659 − 1.84371i 1.00000
781.3 −0.500000 + 0.866025i −1.04695 − 1.37982i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.71843 − 0.216776i 1.27600 − 2.21009i 1.00000 −0.807794 + 2.88920i 1.00000
781.4 −0.500000 + 0.866025i −0.439322 + 1.67541i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.23129 − 1.21817i −0.477192 + 0.826521i 1.00000 −2.61399 − 1.47209i 1.00000
781.5 −0.500000 + 0.866025i −0.150989 − 1.72546i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.56978 + 0.731968i −1.86938 + 3.23786i 1.00000 −2.95440 + 0.521050i 1.00000
781.6 −0.500000 + 0.866025i 0.965268 + 1.43814i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.72810 + 0.116875i −0.0780455 + 0.135179i 1.00000 −1.13651 + 2.77639i 1.00000
781.7 −0.500000 + 0.866025i 1.31468 − 1.12766i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.319249 + 1.70237i 1.85878 − 3.21950i 1.00000 0.456744 − 2.96503i 1.00000
781.8 −0.500000 + 0.866025i 1.70718 + 0.292459i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.10687 + 1.33223i −2.19374 + 3.79967i 1.00000 2.82894 + 0.998562i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
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.n ✓ 16
3.b odd 2 1 3510.2.j.n 16
9.c even 3 1 inner 1170.2.j.n ✓ 16
9.d odd 6 1 3510.2.j.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.j.n ✓ 16 1.a even 1 1 trivial
1170.2.j.n ✓ 16 9.c even 3 1 inner
3510.2.j.n 16 3.b odd 2 1
3510.2.j.n 16 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{16} + T_{7}^{15} + 39 T_{7}^{14} + 34 T_{7}^{13} + 1039 T_{7}^{12} + 909 T_{7}^{11} + \cdots + 46656 \) Copy content Toggle raw display
\( T_{11}^{16} + 5 T_{11}^{15} + 66 T_{11}^{14} + 305 T_{11}^{13} + 2884 T_{11}^{12} + 11787 T_{11}^{11} + \cdots + 236196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} + \cdots + 46656 \) Copy content Toggle raw display
$11$ \( T^{16} + 5 T^{15} + \cdots + 236196 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} - 5 T^{7} + \cdots - 17928)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - T^{7} - 71 T^{6} + \cdots - 998)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 3 T^{15} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( T^{16} + 7 T^{15} + \cdots + 46656 \) Copy content Toggle raw display
$31$ \( T^{16} - 4 T^{15} + \cdots + 258064 \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 148752)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 478354823424 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 44686577664 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 2873594548224 \) Copy content Toggle raw display
$53$ \( (T^{8} + 2 T^{7} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60350800896 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 302895665594404 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 7479482256 \) Copy content Toggle raw display
$71$ \( (T^{8} - 32 T^{7} + \cdots - 27184896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 21 T^{7} + \cdots + 2364256)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 2674130944 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 3764034572544 \) Copy content Toggle raw display
$89$ \( (T^{8} - 15 T^{7} + \cdots + 172296)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 126801157402404 \) Copy content Toggle raw display
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