↑

Properties

Label 170.3.j.c
Level $170$
Weight $3$
Character orbit 170.j
Analytic conductor $4.632$
Analytic rank $0$
Dimension $18$
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] = [170,3,Mod(47,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 170.j (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: \(4.63216449413\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 112 x^{16} + 5012 x^{14} + 115076 x^{12} + 1458640 x^{10} + 10354784 x^{8} + 40404568 x^{6} + \cdots + 160000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + 1) q^{2} - \beta_{3} q^{3} + 2 \beta_{4} q^{4} + (\beta_{10} + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{6} + (\beta_{7} - 2) q^{7} + (2 \beta_{4} - 2) q^{8} + ( - \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 1) q^{2} - \beta_{3} q^{3} + 2 \beta_{4} q^{4} + (\beta_{10} + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{6} + (\beta_{7} - 2) q^{7} + (2 \beta_{4} - 2) q^{8} + ( - \beta_{2} + 3) q^{9} + (\beta_{10} - \beta_{9} + \beta_{4} + 1) q^{10} - \beta_{16} q^{11} - 2 \beta_1 q^{12} + (\beta_{12} + \beta_{11} - 2 \beta_{9} + \cdots - 2) q^{13}+ \cdots + ( - 3 \beta_{16} + 8 \beta_{14} + \cdots + 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{2} - 8 q^{3} + 18 q^{5} - 8 q^{6} - 28 q^{7} - 36 q^{8} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{2} - 8 q^{3} + 18 q^{5} - 8 q^{6} - 28 q^{7} - 36 q^{8} + 62 q^{9} + 26 q^{10} - 8 q^{11} - 22 q^{13} - 28 q^{14} + 28 q^{15} - 72 q^{16} - 58 q^{17} + 62 q^{18} + 48 q^{19} + 16 q^{20} - 64 q^{21} + 16 q^{24} - 62 q^{25} + 28 q^{27} + 74 q^{29} + 28 q^{30} - 16 q^{31} - 72 q^{32} + 84 q^{33} - 70 q^{34} - 28 q^{35} + 48 q^{38} - 60 q^{39} - 20 q^{40} + 18 q^{41} - 64 q^{42} - 48 q^{43} + 16 q^{44} + 270 q^{45} - 40 q^{46} - 44 q^{47} + 32 q^{48} + 214 q^{49} - 46 q^{50} + 52 q^{51} + 44 q^{52} + 58 q^{53} + 28 q^{54} + 52 q^{55} + 56 q^{56} - 136 q^{57} - 80 q^{59} - 254 q^{61} - 32 q^{62} - 312 q^{63} + 194 q^{65} + 336 q^{67} - 24 q^{68} - 12 q^{70} - 8 q^{71} - 124 q^{72} - 772 q^{73} + 144 q^{74} - 268 q^{75} - 76 q^{77} - 444 q^{79} - 72 q^{80} + 450 q^{81} + 244 q^{83} + 126 q^{85} - 96 q^{86} + 92 q^{87} + 32 q^{88} + 446 q^{90} + 288 q^{91} - 80 q^{92} - 88 q^{94} + 92 q^{95} + 32 q^{96} + 214 q^{98} + 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 112 x^{16} + 5012 x^{14} + 115076 x^{12} + 1458640 x^{10} + 10354784 x^{8} + 40404568 x^{6} + \cdots + 160000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92960557 \nu^{16} - 9939994174 \nu^{14} - 413321934679 \nu^{12} - 8431280470492 \nu^{10} + \cdots + 6979090501600 ) / 197508629938540 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8723863127 \nu^{17} + 995664781624 \nu^{15} + 45712000827324 \nu^{13} + \cdots + 78\!\cdots\!00 \nu ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30819617231 \nu^{17} - 3483994053572 \nu^{15} - 157802099215922 \nu^{13} + \cdots - 23\!\cdots\!00 \nu ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!91 \nu^{17} + \cdots - 88\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!38 \nu^{16} + \cdots - 69\!\cdots\!80 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\!\cdots\!38 \nu^{17} + \cdots - 50\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 83\!\cdots\!35 \nu^{17} + \cdots - 68\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!83 \nu^{17} + \cdots - 13\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!76 \nu^{17} + \cdots - 32\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53\!\cdots\!41 \nu^{17} + \cdots + 68\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 63\!\cdots\!27 \nu^{17} + \cdots - 22\!\cdots\!00 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!21 \nu^{17} + \cdots + 16\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 86\!\cdots\!21 \nu^{17} + \cdots + 75\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 86\!\cdots\!21 \nu^{17} + \cdots + 75\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 21\!\cdots\!61 \nu^{17} + \cdots - 21\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} + \cdots + 267 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6 \beta_{17} + 39 \beta_{16} - 39 \beta_{15} - 44 \beta_{14} + 29 \beta_{13} + 27 \beta_{12} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{16} + 30 \beta_{15} - 75 \beta_{14} - 98 \beta_{13} - 23 \beta_{12} - 30 \beta_{11} + \cdots - 7013 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 242 \beta_{17} - 1286 \beta_{16} + 1286 \beta_{15} + 1526 \beta_{14} - 667 \beta_{13} - 656 \beta_{12} + \cdots + 1526 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 678 \beta_{16} - 678 \beta_{15} + 3087 \beta_{14} + 3611 \beta_{13} + 378 \beta_{12} + 885 \beta_{11} + \cdots + 195067 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6980 \beta_{17} + 40390 \beta_{16} - 40390 \beta_{15} - 49158 \beta_{14} + 13426 \beta_{13} + \cdots - 49158 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13008 \beta_{16} + 13008 \beta_{15} - 106706 \beta_{14} - 119814 \beta_{13} - 2858 \beta_{12} + \cdots - 5567166 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 175544 \beta_{17} - 1241922 \beta_{16} + 1241922 \beta_{15} + 1533408 \beta_{14} - 227882 \beta_{13} + \cdots + 1533408 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 193914 \beta_{16} - 193914 \beta_{15} + 3431984 \beta_{14} + 3789882 \beta_{13} - 121234 \beta_{12} + \cdots + 161096108 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4066972 \beta_{17} + 37750946 \beta_{16} - 37750946 \beta_{15} - 47029584 \beta_{14} + \cdots - 47029584 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 835400 \beta_{16} + 835400 \beta_{15} - 106573434 \beta_{14} - 116987712 \beta_{13} + 8413682 \beta_{12} + \cdots - 4699957958 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 87696380 \beta_{17} - 1139510556 \beta_{16} + 1139510556 \beta_{15} + 1428131424 \beta_{14} + \cdots + 1428131424 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 99923028 \beta_{16} + 99923028 \beta_{15} + 3246814198 \beta_{14} + 3562548430 \beta_{13} + \cdots + 137832701390 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1721327360 \beta_{17} + 34239033208 \beta_{16} - 34239033208 \beta_{15} - 43096850704 \beta_{14} + \cdots - 43096850704 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(-\beta_{4}\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
− 5.47122i
− 4.68886i
− 2.16902i
− 1.66297i
− 1.66096i
− 0.0504809i
2.54818i
3.72749i
5.42785i
5.47122i
4.68886i
2.16902i
1.66297i
1.66096i
0.0504809i
− 2.54818i
− 3.72749i
− 5.42785i
1.00000 − 1.00000i −5.47122 − 2.00000i 4.29349 − 2.56240i −5.47122 + 5.47122i −7.32642 −2.00000 − 2.00000i 20.9342 1.73109 − 6.85590i
47.2 1.00000 − 1.00000i −4.68886 − 2.00000i −1.00386 + 4.89819i −4.68886 + 4.68886i 9.04638 −2.00000 − 2.00000i 12.9854 3.89433 + 5.90205i
47.3 1.00000 − 1.00000i −2.16902 − 2.00000i −4.97231 − 0.525481i −2.16902 + 2.16902i 3.57697 −2.00000 − 2.00000i −4.29533 −5.49779 + 4.44683i
47.4 1.00000 − 1.00000i −1.66297 − 2.00000i −0.268565 + 4.99278i −1.66297 + 1.66297i −11.8517 −2.00000 − 2.00000i −6.23452 4.72422 + 5.26135i
47.5 1.00000 − 1.00000i −1.66096 − 2.00000i 4.89718 − 1.00880i −1.66096 + 1.66096i 3.64307 −2.00000 − 2.00000i −6.24121 3.88838 − 5.90597i
47.6 1.00000 − 1.00000i −0.0504809 − 2.00000i −2.05932 − 4.55623i −0.0504809 + 0.0504809i −8.66451 −2.00000 − 2.00000i −8.99745 −6.61555 − 2.49691i
47.7 1.00000 − 1.00000i 2.54818 − 2.00000i 3.97426 + 3.03401i 2.54818 − 2.54818i 7.34240 −2.00000 − 2.00000i −2.50679 7.00828 − 0.940252i
47.8 1.00000 − 1.00000i 3.72749 − 2.00000i 1.77115 − 4.67579i 3.72749 − 3.72749i 1.00262 −2.00000 − 2.00000i 4.89416 −2.90464 − 6.44694i
47.9 1.00000 − 1.00000i 5.42785 − 2.00000i 2.36797 + 4.40371i 5.42785 − 5.42785i −10.7688 −2.00000 − 2.00000i 20.4615 6.77169 + 2.03574i
123.1 1.00000 + 1.00000i −5.47122 2.00000i 4.29349 + 2.56240i −5.47122 − 5.47122i −7.32642 −2.00000 + 2.00000i 20.9342 1.73109 + 6.85590i
123.2 1.00000 + 1.00000i −4.68886 2.00000i −1.00386 − 4.89819i −4.68886 − 4.68886i 9.04638 −2.00000 + 2.00000i 12.9854 3.89433 − 5.90205i
123.3 1.00000 + 1.00000i −2.16902 2.00000i −4.97231 + 0.525481i −2.16902 − 2.16902i 3.57697 −2.00000 + 2.00000i −4.29533 −5.49779 − 4.44683i
123.4 1.00000 + 1.00000i −1.66297 2.00000i −0.268565 − 4.99278i −1.66297 − 1.66297i −11.8517 −2.00000 + 2.00000i −6.23452 4.72422 − 5.26135i
123.5 1.00000 + 1.00000i −1.66096 2.00000i 4.89718 + 1.00880i −1.66096 − 1.66096i 3.64307 −2.00000 + 2.00000i −6.24121 3.88838 + 5.90597i
123.6 1.00000 + 1.00000i −0.0504809 2.00000i −2.05932 + 4.55623i −0.0504809 − 0.0504809i −8.66451 −2.00000 + 2.00000i −8.99745 −6.61555 + 2.49691i
123.7 1.00000 + 1.00000i 2.54818 2.00000i 3.97426 − 3.03401i 2.54818 + 2.54818i 7.34240 −2.00000 + 2.00000i −2.50679 7.00828 + 0.940252i
123.8 1.00000 + 1.00000i 3.72749 2.00000i 1.77115 + 4.67579i 3.72749 + 3.72749i 1.00262 −2.00000 + 2.00000i 4.89416 −2.90464 + 6.44694i
123.9 1.00000 + 1.00000i 5.42785 2.00000i 2.36797 − 4.40371i 5.42785 + 5.42785i −10.7688 −2.00000 + 2.00000i 20.4615 6.77169 − 2.03574i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.3.j.c yes 18
5.c odd 4 1 170.3.e.c ✓ 18
17.c even 4 1 170.3.e.c ✓ 18
85.i odd 4 1 inner 170.3.j.c yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.3.e.c ✓ 18 5.c odd 4 1
170.3.e.c ✓ 18 17.c even 4 1
170.3.j.c yes 18 1.a even 1 1 trivial
170.3.j.c yes 18 85.i odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + 4T_{3}^{8} - 48T_{3}^{7} - 194T_{3}^{6} + 578T_{3}^{5} + 2576T_{3}^{4} - 672T_{3}^{3} - 9672T_{3}^{2} - 8410T_{3} - 400 \) acting on \(S_{3}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{9} \) Copy content Toggle raw display
$3$ \( (T^{9} + 4 T^{8} + \cdots - 400)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{9} + 14 T^{8} + \cdots - 7031000)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 45\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( (T^{9} - 24 T^{8} + \cdots - 35583039488)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 684032606698752)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 45\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 57\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
show more
show less