Properties

Label 1170.2.o.b
Level $1170$
Weight $2$
Character orbit 1170.o
Analytic conductor $9.342$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(53,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.53");
 
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.o (of order \(4\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 107x^{8} + 240x^{6} + 151x^{4} + 30x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{4} + (\beta_{8} + \beta_{7} - \beta_{3} - 1) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{4} + (\beta_{8} + \beta_{7} - \beta_{3} - 1) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + \beta_{3} q^{8} + (\beta_{11} + \beta_{9} + \beta_{8} + 1) q^{10} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} - \beta_{2} - \beta_1) q^{11} - \beta_1 q^{13} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{2} - \beta_1 - 1) q^{14} - q^{16} + ( - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{17} + ( - \beta_{10} + \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{19} + (\beta_{11} - \beta_{10} - \beta_{4} + \beta_1) q^{20} + ( - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{22} + (\beta_{10} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{23} + (\beta_{10} - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{25}+ \cdots + (\beta_{11} + \beta_{8} + 4 \beta_{6} - \beta_{3} + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} - 4 q^{7} + 8 q^{10} - 8 q^{14} - 12 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} + 20 q^{23} + 8 q^{25} - 4 q^{28} + 32 q^{29} + 16 q^{31} - 28 q^{35} + 32 q^{38} + 4 q^{40} + 8 q^{43} - 16 q^{47} - 16 q^{50} + 44 q^{53} - 4 q^{55} + 4 q^{58} + 16 q^{59} + 16 q^{61} + 4 q^{62} - 8 q^{65} + 28 q^{67} + 8 q^{68} - 16 q^{70} + 24 q^{73} - 32 q^{74} + 16 q^{76} + 40 q^{77} + 4 q^{80} + 8 q^{82} + 16 q^{83} - 20 q^{85} - 4 q^{88} + 8 q^{91} + 20 q^{92} - 76 q^{95} - 8 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 107x^{8} + 240x^{6} + 151x^{4} + 30x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{11} - 9 \nu^{10} + 85 \nu^{9} - 159 \nu^{8} + 450 \nu^{7} - 910 \nu^{6} + 740 \nu^{5} - 1860 \nu^{4} - 115 \nu^{3} - 769 \nu^{2} - 145 \nu - 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{11} + 7 \nu^{10} + 53 \nu^{9} + 127 \nu^{8} + 300 \nu^{7} + 760 \nu^{6} + 570 \nu^{5} + 1690 \nu^{4} + 53 \nu^{3} + 937 \nu^{2} - 171 \nu + 71 ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{11} + 9 \nu^{10} + 85 \nu^{9} + 159 \nu^{8} + 450 \nu^{7} + 910 \nu^{6} + 740 \nu^{5} + 1860 \nu^{4} - 115 \nu^{3} + 769 \nu^{2} - 145 \nu + 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{11} + 6 \nu^{10} + 37 \nu^{9} + 106 \nu^{8} + 235 \nu^{7} + 610 \nu^{6} + 625 \nu^{5} + 1280 \nu^{4} + 647 \nu^{3} + 616 \nu^{2} + 176 \nu + 38 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6 \nu^{11} + 7 \nu^{10} - 106 \nu^{9} + 122 \nu^{8} - 605 \nu^{7} + 680 \nu^{6} - 1215 \nu^{5} + 1300 \nu^{4} - 411 \nu^{3} + 327 \nu^{2} + 57 \nu - 24 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{11} - 143\nu^{9} - 840\nu^{7} - 1840\nu^{5} - 1058\nu^{3} - 139\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{10} - 106\nu^{8} - 605\nu^{6} - 1215\nu^{4} - 421\nu^{2} - 3 ) / 10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6 \nu^{11} + 7 \nu^{10} + 106 \nu^{9} + 122 \nu^{8} + 605 \nu^{7} + 680 \nu^{6} + 1215 \nu^{5} + 1300 \nu^{4} + 411 \nu^{3} + 327 \nu^{2} - 57 \nu - 24 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 6 \nu^{10} - 122 \nu^{9} - 106 \nu^{8} - 680 \nu^{7} - 605 \nu^{6} - 1300 \nu^{5} - 1215 \nu^{4} - 327 \nu^{3} - 411 \nu^{2} + 44 \nu + 17 ) / 20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23 \nu^{11} - 6 \nu^{10} + 408 \nu^{9} - 106 \nu^{8} + 2355 \nu^{7} - 610 \nu^{6} + 4915 \nu^{5} - 1280 \nu^{4} + 2258 \nu^{3} - 616 \nu^{2} + 259 \nu - 38 ) / 20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23 \nu^{11} + 6 \nu^{10} + 408 \nu^{9} + 106 \nu^{8} + 2355 \nu^{7} + 610 \nu^{6} + 4915 \nu^{5} + 1280 \nu^{4} + 2258 \nu^{3} + 616 \nu^{2} + 259 \nu + 38 ) / 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} - 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 14 \beta_{6} - 3 \beta_{5} + 12 \beta_{4} + 2 \beta_{2} + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} - 8\beta_{9} + 10\beta_{7} + 8\beta_{5} - 4\beta_{3} + 8\beta_{2} - 4\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31 \beta_{11} - 51 \beta_{10} - 30 \beta_{9} - 35 \beta_{8} - 98 \beta_{6} + 5 \beta_{5} - 82 \beta_{4} - 2 \beta_{3} - 30 \beta_{2} - 32 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{11} - 15\beta_{10} + 65\beta_{9} - 89\beta_{7} - 65\beta_{5} + 13\beta_{3} - 65\beta_{2} + 52\beta _1 - 46 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 217 \beta_{11} + 395 \beta_{10} + 312 \beta_{9} + 265 \beta_{8} + 730 \beta_{6} + 47 \beta_{5} + 612 \beta_{4} + 30 \beta_{3} + 312 \beta_{2} + 342 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 162 \beta_{11} + 162 \beta_{10} - 542 \beta_{9} - 6 \beta_{8} + 766 \beta_{7} + 536 \beta_{5} - 18 \beta_{3} + 542 \beta_{2} - 524 \beta _1 + 259 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1649 \beta_{11} - 3181 \beta_{10} - 2892 \beta_{9} - 2097 \beta_{8} - 5714 \beta_{6} - 795 \beta_{5} - 4830 \beta_{4} - 336 \beta_{3} - 2892 \beta_{2} - 3228 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1552 \beta_{11} - 1552 \beta_{10} + 4571 \beta_{9} + 106 \beta_{8} - 6515 \beta_{7} - 4465 \beta_{5} - 253 \beta_{3} - 4571 \beta_{2} + 4824 \beta _1 - 1620 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13177 \beta_{11} + 26207 \beta_{10} + 25570 \beta_{9} + 17065 \beta_{8} + 46206 \beta_{6} + 8505 \beta_{5} + 39384 \beta_{4} + 3316 \beta_{3} + 25570 \beta_{2} + 28886 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.203482i
0.699479i
2.91021i
2.01185i
2.15459i
0.556948i
0.203482i
0.699479i
2.91021i
2.01185i
2.15459i
0.556948i
−0.707107 0.707107i 0 1.00000i −2.23593 0.0251942i 0 1.37981 1.37981i 0.707107 0.707107i 0 1.56322 + 1.59885i
53.2 −0.707107 0.707107i 0 1.00000i −1.28075 + 1.83294i 0 −0.213858 + 0.213858i 0.707107 0.707107i 0 2.20171 0.390462i
53.3 −0.707107 0.707107i 0 1.00000i 1.80957 + 1.31357i 0 −0.751738 + 0.751738i 0.707107 0.707107i 0 −0.350723 2.20839i
53.4 0.707107 + 0.707107i 0 1.00000i −2.16333 0.565689i 0 2.58593 2.58593i −0.707107 + 0.707107i 0 −1.12970 1.92971i
53.5 0.707107 + 0.707107i 0 1.00000i 0.362277 2.20653i 0 −2.88580 + 2.88580i −0.707107 + 0.707107i 0 1.81642 1.30408i
53.6 0.707107 + 0.707107i 0 1.00000i 1.50816 + 1.65089i 0 −2.11434 + 2.11434i −0.707107 + 0.707107i 0 −0.100929 + 2.23379i
287.1 −0.707107 + 0.707107i 0 1.00000i −2.23593 + 0.0251942i 0 1.37981 + 1.37981i 0.707107 + 0.707107i 0 1.56322 1.59885i
287.2 −0.707107 + 0.707107i 0 1.00000i −1.28075 1.83294i 0 −0.213858 0.213858i 0.707107 + 0.707107i 0 2.20171 + 0.390462i
287.3 −0.707107 + 0.707107i 0 1.00000i 1.80957 1.31357i 0 −0.751738 0.751738i 0.707107 + 0.707107i 0 −0.350723 + 2.20839i
287.4 0.707107 0.707107i 0 1.00000i −2.16333 + 0.565689i 0 2.58593 + 2.58593i −0.707107 0.707107i 0 −1.12970 + 1.92971i
287.5 0.707107 0.707107i 0 1.00000i 0.362277 + 2.20653i 0 −2.88580 2.88580i −0.707107 0.707107i 0 1.81642 + 1.30408i
287.6 0.707107 0.707107i 0 1.00000i 1.50816 1.65089i 0 −2.11434 2.11434i −0.707107 0.707107i 0 −0.100929 2.23379i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.o.b 12
3.b odd 2 1 1170.2.o.d yes 12
5.c odd 4 1 1170.2.o.d yes 12
15.e even 4 1 inner 1170.2.o.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.o.b 12 1.a even 1 1 trivial
1170.2.o.b 12 15.e even 4 1 inner
1170.2.o.d yes 12 3.b odd 2 1
1170.2.o.d yes 12 5.c odd 4 1

Hecke kernels

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

\( T_{7}^{12} + 4 T_{7}^{11} + 8 T_{7}^{10} - 8 T_{7}^{9} + 212 T_{7}^{8} + 768 T_{7}^{7} + 1408 T_{7}^{6} - 816 T_{7}^{5} + 4144 T_{7}^{4} + 10752 T_{7}^{3} + 12800 T_{7}^{2} + 4480 T_{7} + 784 \) Copy content Toggle raw display
\( T_{17}^{12} + 8 T_{17}^{11} + 32 T_{17}^{10} + 16 T_{17}^{9} + 368 T_{17}^{8} + 2960 T_{17}^{7} + 12032 T_{17}^{6} + 4832 T_{17}^{5} + 5216 T_{17}^{4} + 28736 T_{17}^{3} + 93312 T_{17}^{2} + 53568 T_{17} + 15376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + 4 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + 8 T^{10} - 8 T^{9} + \cdots + 784 \) Copy content Toggle raw display
$11$ \( T^{12} + 92 T^{10} + 3036 T^{8} + \cdots + 1032256 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + 32 T^{10} + \cdots + 15376 \) Copy content Toggle raw display
$19$ \( T^{12} + 192 T^{10} + 14096 T^{8} + \cdots + 4048144 \) Copy content Toggle raw display
$23$ \( T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 80656 \) Copy content Toggle raw display
$29$ \( (T^{6} - 16 T^{5} - 50 T^{4} + \cdots + 178300)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} - 86 T^{4} + 888 T^{3} + \cdots + 4616)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 208 T^{9} + \cdots + 2102772736 \) Copy content Toggle raw display
$41$ \( T^{12} + 184 T^{10} + 7152 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + \cdots + 29688668416 \) Copy content Toggle raw display
$47$ \( T^{12} + 16 T^{11} + 128 T^{10} + \cdots + 8111104 \) Copy content Toggle raw display
$53$ \( T^{12} - 44 T^{11} + \cdots + 6775265344 \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} - 102 T^{4} + 64 T^{3} + \cdots - 392)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} - 128 T^{4} + 144 T^{3} + \cdots + 2848)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 28 T^{11} + \cdots + 19822387264 \) Copy content Toggle raw display
$71$ \( T^{12} + 656 T^{10} + \cdots + 117336391936 \) Copy content Toggle raw display
$73$ \( T^{12} - 24 T^{11} + \cdots + 205406224 \) Copy content Toggle raw display
$79$ \( T^{12} + 480 T^{10} + \cdots + 20214016 \) Copy content Toggle raw display
$83$ \( T^{12} - 16 T^{11} + \cdots + 8209809664 \) Copy content Toggle raw display
$89$ \( (T^{6} - 240 T^{4} + 96 T^{3} + \cdots - 69376)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 8 T^{11} + \cdots + 2120856291856 \) Copy content Toggle raw display
show more
show less