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Properties

Label 1170.2.bj.f
Level $1170$
Weight $2$
Character orbit 1170.bj
Analytic conductor $9.342$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 12 x^{13} + 13 x^{12} + 24 x^{11} - 26 x^{10} - 12 x^{9} - 686 x^{8} - 60 x^{7} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{7} - 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{15} + \beta_{5}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{7} - 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{15} + \beta_{5}) q^{7} - q^{8} + \beta_{6} q^{10} + ( - 2 \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{11}+ \cdots + (\beta_{14} - 2 \beta_{13} + \cdots - \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 8 q^{4} - 16 q^{8} - 8 q^{16} + 12 q^{17} - 18 q^{19} - 6 q^{23} + 4 q^{25} + 8 q^{32} - 24 q^{35} - 6 q^{46} + 48 q^{47} - 6 q^{49} + 2 q^{50} + 14 q^{55} - 16 q^{61} - 6 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{68} + 18 q^{76} + 20 q^{79} + 24 q^{83} - 18 q^{85} - 46 q^{91} + 24 q^{94} - 30 q^{95} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 12 x^{13} + 13 x^{12} + 24 x^{11} - 26 x^{10} - 12 x^{9} - 686 x^{8} - 60 x^{7} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2081 \nu^{15} + 50865 \nu^{14} - 95831 \nu^{13} + 76962 \nu^{12} - 276808 \nu^{11} + \cdots + 2504531250 ) / 395625000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1417 \nu^{15} + 48843 \nu^{14} - 129703 \nu^{13} + 67164 \nu^{12} - 93690 \nu^{11} + \cdots + 3888937500 ) / 237375000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9211 \nu^{15} + 64880 \nu^{14} - 225364 \nu^{13} + 220723 \nu^{12} - 932142 \nu^{11} + \cdots + 9388046875 ) / 1186875000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59 \nu^{15} + 360 \nu^{14} - 1084 \nu^{13} + 1443 \nu^{12} - 3612 \nu^{11} + 6564 \nu^{10} + \cdots + 33984375 ) / 5625000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1429 \nu^{15} + 10173 \nu^{14} - 20179 \nu^{13} + 27546 \nu^{12} - 68528 \nu^{11} + \cdots + 500906250 ) / 79125000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10173 \nu^{15} - 18750 \nu^{14} + 10398 \nu^{13} - 49951 \nu^{12} + 121224 \nu^{11} + \cdots + 162578125 ) / 395625000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34109 \nu^{15} + 163000 \nu^{14} - 232084 \nu^{13} + 464183 \nu^{12} - 1503642 \nu^{11} + \cdots + 4953359375 ) / 1186875000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34784 \nu^{15} + 109625 \nu^{14} - 475991 \nu^{13} + 372217 \nu^{12} - 77208 \nu^{11} + \cdots + 15832421875 ) / 1186875000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24599 \nu^{15} - 90120 \nu^{14} + 283901 \nu^{13} - 118932 \nu^{12} + 107028 \nu^{11} + \cdots - 10070156250 ) / 593437500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 750 \nu^{15} + 9 \nu^{14} + 2885 \nu^{13} - 441 \nu^{12} + 1402 \nu^{11} - 39078 \nu^{10} + \cdots - 158953125 ) / 15825000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24831 \nu^{15} - 47340 \nu^{14} + 68806 \nu^{13} - 91437 \nu^{12} + 389858 \nu^{11} + \cdots + 780703125 ) / 395625000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27363 \nu^{15} + 47340 \nu^{14} - 8038 \nu^{13} + 121821 \nu^{12} - 359474 \nu^{11} + \cdots - 1967578125 ) / 395625000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 86963 \nu^{15} + 42295 \nu^{14} - 441437 \nu^{13} + 138614 \nu^{12} + 116454 \nu^{11} + \cdots + 19724375000 ) / 1186875000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 88754 \nu^{15} - 199665 \nu^{14} + 37729 \nu^{13} - 348213 \nu^{12} + 764382 \nu^{11} + \cdots + 5906015625 ) / 1186875000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{15} - 3\beta_{14} - 4\beta_{10} + \beta_{9} + \beta_{8} + 7\beta_{7} - 6\beta_{6} + 7\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{15} - 6 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} + 6 \beta_{11} - 6 \beta_{10} - 6 \beta_{9} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{15} - 20 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 24 \beta_{11} - 4 \beta_{10} + 12 \beta_{9} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 30 \beta_{15} - 36 \beta_{14} - 8 \beta_{13} + 16 \beta_{12} + 12 \beta_{10} + 6 \beta_{9} + \cdots + 19 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40 \beta_{15} - 19 \beta_{14} + 24 \beta_{13} - 12 \beta_{12} + 30 \beta_{11} - 46 \beta_{10} + \cdots + 245 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 269 \beta_{15} - 41 \beta_{14} - 21 \beta_{13} - 21 \beta_{12} + 415 \beta_{11} - 67 \beta_{10} + \cdots + 927 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 604 \beta_{15} + 511 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} + 520 \beta_{10} - 239 \beta_{9} + \cdots + 612 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 768 \beta_{15} + 1440 \beta_{14} - 712 \beta_{13} + 356 \beta_{12} + 1488 \beta_{11} - 60 \beta_{10} + \cdots - 876 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2152 \beta_{15} + 668 \beta_{14} - 672 \beta_{13} - 672 \beta_{12} + 348 \beta_{11} - 2744 \beta_{10} + \cdots + 4393 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7332 \beta_{15} + 984 \beta_{14} - 3028 \beta_{13} + 6056 \beta_{12} + 408 \beta_{10} + \cdots - 2195 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 176 \beta_{15} - 1503 \beta_{14} - 8280 \beta_{13} + 4140 \beta_{12} - 10776 \beta_{11} - 6884 \beta_{10} + \cdots - 23341 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15047 \beta_{15} - 16667 \beta_{14} + 1327 \beta_{13} + 1327 \beta_{12} - 9683 \beta_{11} - 23113 \beta_{10} + \cdots + 15891 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1 + \beta_{7}\)

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
0.226937 − 2.22452i
1.81302 − 1.30879i
−1.31070 − 1.81164i
2.22428 + 0.229280i
−2.19357 − 0.433866i
1.47253 + 1.68276i
−1.22571 + 1.87020i
−1.00678 + 1.99659i
1.81302 + 1.30879i
0.226937 + 2.22452i
2.22428 − 0.229280i
−1.31070 + 1.81164i
1.47253 − 1.68276i
−2.19357 + 0.433866i
−1.00678 − 1.99659i
−1.22571 − 1.87020i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.03996 − 0.915728i 0 −0.138610 − 0.240080i −1.00000 0 −1.81302 + 1.30879i
199.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.03996 + 0.915728i 0 0.138610 + 0.240080i −1.00000 0 −0.226937 + 2.22452i
199.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.913578 − 2.04092i 0 −2.50711 − 4.34244i −1.00000 0 −2.22428 − 0.229280i
199.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.913578 + 2.04092i 0 2.50711 + 4.34244i −1.00000 0 1.31070 + 1.81164i
199.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.721047 − 2.11662i 0 0.889112 + 1.53999i −1.00000 0 −1.47253 − 1.68276i
199.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.721047 + 2.11662i 0 −0.889112 − 1.53999i −1.00000 0 2.19357 + 0.433866i
199.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.23249 − 0.126396i 0 −0.809123 − 1.40144i −1.00000 0 1.00678 − 1.99659i
199.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.23249 + 0.126396i 0 0.809123 + 1.40144i −1.00000 0 1.22571 − 1.87020i
829.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.03996 − 0.915728i 0 0.138610 − 0.240080i −1.00000 0 −0.226937 − 2.22452i
829.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.03996 + 0.915728i 0 −0.138610 + 0.240080i −1.00000 0 −1.81302 − 1.30879i
829.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.913578 − 2.04092i 0 2.50711 − 4.34244i −1.00000 0 1.31070 − 1.81164i
829.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.913578 + 2.04092i 0 −2.50711 + 4.34244i −1.00000 0 −2.22428 + 0.229280i
829.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.721047 − 2.11662i 0 −0.889112 + 1.53999i −1.00000 0 2.19357 − 0.433866i
829.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.721047 + 2.11662i 0 0.889112 − 1.53999i −1.00000 0 −1.47253 + 1.68276i
829.7 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.23249 − 0.126396i 0 0.809123 − 1.40144i −1.00000 0 1.22571 + 1.87020i
829.8 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.23249 + 0.126396i 0 −0.809123 + 1.40144i −1.00000 0 1.00678 + 1.99659i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
39.h odd 6 1 inner
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.f yes 16
3.b odd 2 1 1170.2.bj.e ✓ 16
5.b even 2 1 1170.2.bj.e ✓ 16
13.e even 6 1 1170.2.bj.e ✓ 16
15.d odd 2 1 inner 1170.2.bj.f yes 16
39.h odd 6 1 inner 1170.2.bj.f yes 16
65.l even 6 1 inner 1170.2.bj.f yes 16
195.y odd 6 1 1170.2.bj.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.bj.e ✓ 16 3.b odd 2 1
1170.2.bj.e ✓ 16 5.b even 2 1
1170.2.bj.e ✓ 16 13.e even 6 1
1170.2.bj.e ✓ 16 195.y odd 6 1
1170.2.bj.f yes 16 1.a even 1 1 trivial
1170.2.bj.f yes 16 15.d odd 2 1 inner
1170.2.bj.f yes 16 39.h odd 6 1 inner
1170.2.bj.f yes 16 65.l even 6 1 inner

Hecke kernels

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

\( T_{7}^{16} + 31T_{7}^{14} + 805T_{7}^{12} + 4396T_{7}^{10} + 17500T_{7}^{8} + 33328T_{7}^{6} + 45904T_{7}^{4} + 3520T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{8} - 6T_{17}^{7} - 25T_{17}^{6} + 222T_{17}^{5} + 753T_{17}^{4} - 4884T_{17}^{3} - 7216T_{17}^{2} + 46464T_{17} + 123904 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{6} - 6 T^{5} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 31 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 454371856 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 6 T^{7} + \cdots + 123904)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots + 166464)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 3 T^{7} - 61 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 381671897616 \) Copy content Toggle raw display
$31$ \( (T^{8} + 171 T^{6} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 172 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 794123370496 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1358954496 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + \cdots - 42)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 245 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + 8 T^{7} + 61 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 726672516714496 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5473632256 \) Copy content Toggle raw display
$73$ \( (T^{8} - 190 T^{6} + \cdots + 73984)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 5 T^{3} + \cdots + 208)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6 T^{3} + \cdots + 2352)^{4} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 22419997720576 \) Copy content Toggle raw display
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