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Properties

Label 170.2.k.b
Level $170$
Weight $2$
Character orbit 170.k
Analytic conductor $1.357$
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] = [170,2,Mod(111,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.k (of order \(8\), degree \(4\), 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: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 286x^{12} + 1412x^{10} + 3709x^{8} + 5264x^{6} + 3780x^{4} + 1072x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + (\beta_{15} - \beta_{6}) q^{3} - \beta_{3} q^{4} + \beta_{5} q^{5} + (\beta_{8} + \beta_{6}) q^{6} + (\beta_{15} - \beta_{12} + \cdots + \beta_{2}) q^{7}+ \cdots + ( - \beta_{13} - \beta_{7} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + (\beta_{15} - \beta_{6}) q^{3} - \beta_{3} q^{4} + \beta_{5} q^{5} + (\beta_{8} + \beta_{6}) q^{6} + (\beta_{15} - \beta_{12} + \cdots + \beta_{2}) q^{7}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{11} - 8 q^{14} + 8 q^{15} - 16 q^{16} + 8 q^{18} - 8 q^{22} + 8 q^{23} - 24 q^{27} - 8 q^{28} + 8 q^{29} + 32 q^{31} + 16 q^{33} + 16 q^{34} + 16 q^{35} - 8 q^{37} - 32 q^{39} - 32 q^{41} + 32 q^{42} - 16 q^{43} + 8 q^{44} - 16 q^{45} - 24 q^{46} - 8 q^{49} - 16 q^{50} - 8 q^{51} - 8 q^{52} - 40 q^{53} - 16 q^{57} - 8 q^{58} + 16 q^{59} - 8 q^{60} - 24 q^{61} + 32 q^{62} + 56 q^{63} - 8 q^{65} - 8 q^{66} + 16 q^{67} - 16 q^{69} + 8 q^{70} + 8 q^{71} + 16 q^{73} - 8 q^{74} + 24 q^{77} + 32 q^{78} + 40 q^{79} + 16 q^{82} + 32 q^{83} + 16 q^{84} + 16 q^{85} - 32 q^{87} + 8 q^{88} + 24 q^{91} + 24 q^{92} - 32 q^{93} + 40 q^{94} + 16 q^{95} + 24 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 286x^{12} + 1412x^{10} + 3709x^{8} + 5264x^{6} + 3780x^{4} + 1072x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{14} + 72\nu^{12} + 553\nu^{10} + 1599\nu^{8} + 1178\nu^{6} - 1415\nu^{4} - 1496\nu^{2} - 150 ) / 136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{15} + 202\nu^{13} + 2146\nu^{11} + 10990\nu^{9} + 29161\nu^{7} + 39204\nu^{5} + 23630\nu^{3} + 4512\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 606 \nu^{15} + 2681 \nu^{14} - 15156 \nu^{13} + 70430 \nu^{12} - 124847 \nu^{11} + 645475 \nu^{10} + \cdots + 27314 ) / 52496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 606 \nu^{15} + 2681 \nu^{14} + 15156 \nu^{13} + 70430 \nu^{12} + 124847 \nu^{11} + 645475 \nu^{10} + \cdots + 27314 ) / 52496 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 553 \nu^{15} + 2940 \nu^{14} - 14870 \nu^{13} + 78142 \nu^{12} - 141535 \nu^{11} + 730079 \nu^{10} + \cdots - 40002 ) / 52496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 553 \nu^{15} - 2940 \nu^{14} - 14870 \nu^{13} - 78142 \nu^{12} - 141535 \nu^{11} - 730079 \nu^{10} + \cdots + 40002 ) / 52496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2681 \nu^{15} + 1812 \nu^{14} + 70430 \nu^{13} + 48469 \nu^{12} + 645475 \nu^{11} + 457279 \nu^{10} + \cdots + 2424 ) / 52496 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2448 \nu^{15} + 2536 \nu^{14} - 65025 \nu^{13} + 68038 \nu^{12} - 608090 \nu^{11} + 645754 \nu^{10} + \cdots + 46328 ) / 52496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2448 \nu^{15} - 2536 \nu^{14} - 65025 \nu^{13} - 68038 \nu^{12} - 608090 \nu^{11} - 645754 \nu^{10} + \cdots - 46328 ) / 52496 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3546 \nu^{15} - 2067 \nu^{14} - 93298 \nu^{13} - 53807 \nu^{12} - 854926 \nu^{11} - 484394 \nu^{10} + \cdots - 25102 ) / 52496 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2536 \nu^{15} - 3519 \nu^{14} + 68038 \nu^{13} - 92038 \nu^{12} + 645754 \nu^{11} - 836808 \nu^{10} + \cdots - 62288 ) / 52496 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2536 \nu^{15} - 3519 \nu^{14} - 68038 \nu^{13} - 92038 \nu^{12} - 645754 \nu^{11} - 836808 \nu^{10} + \cdots - 9792 ) / 52496 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3546 \nu^{15} + 3295 \nu^{14} + 93298 \nu^{13} + 87053 \nu^{12} + 854926 \nu^{11} + 806556 \nu^{10} + \cdots + 29526 ) / 52496 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2681 \nu^{15} + 7692 \nu^{14} - 70430 \nu^{13} + 204753 \nu^{12} - 645475 \nu^{11} + 1917437 \nu^{10} + \cdots - 77580 ) / 52496 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{7} - \beta_{6} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} - 2 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{15} + 19 \beta_{14} - 25 \beta_{13} + 25 \beta_{12} - 19 \beta_{11} + 24 \beta_{10} + 24 \beta_{9} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 72 \beta_{15} + 32 \beta_{14} - 90 \beta_{13} - 90 \beta_{12} + 32 \beta_{11} + 75 \beta_{10} + \cdots - 331 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 109 \beta_{15} - 280 \beta_{14} + 286 \beta_{13} - 286 \beta_{12} + 280 \beta_{11} - 284 \beta_{10} + \cdots - 286 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1024 \beta_{15} - 434 \beta_{14} + 991 \beta_{13} + 991 \beta_{12} - 434 \beta_{11} - 961 \beta_{10} + \cdots + 3772 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1528 \beta_{15} + 3749 \beta_{14} - 3331 \beta_{13} + 3331 \beta_{12} - 3749 \beta_{11} + 3440 \beta_{10} + \cdots + 3331 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13444 \beta_{15} + 5608 \beta_{14} - 11534 \beta_{13} - 11534 \beta_{12} + 5608 \beta_{11} + \cdots - 44789 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 19831 \beta_{15} - 48088 \beta_{14} + 39704 \beta_{13} - 39704 \beta_{12} + 48088 \beta_{11} + \cdots - 39704 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 170660 \beta_{15} - 70838 \beta_{14} + 138147 \beta_{13} + 138147 \beta_{12} - 70838 \beta_{11} + \cdots + 542050 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 250026 \beta_{15} + 604361 \beta_{14} - 480555 \beta_{13} + 480555 \beta_{12} - 604361 \beta_{11} + \cdots + 480555 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2133296 \beta_{15} + 884184 \beta_{14} - 1678544 \beta_{13} - 1678544 \beta_{12} + 884184 \beta_{11} + \cdots - 6620449 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3113903 \beta_{15} - 7520250 \beta_{14} + 5866830 \beta_{13} - 5866830 \beta_{12} + 7520250 \beta_{11} + \cdots - 5866830 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(71\) \(137\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
− 3.51034i
0.0614939i
− 2.47571i
1.09612i
3.51034i
− 0.0614939i
2.47571i
− 1.09612i
− 1.33738i
1.46868i
0.923170i
− 1.88289i
1.33738i
− 1.46868i
− 0.923170i
1.88289i
0.707107 − 0.707107i −2.31925 − 0.960664i − 1.00000i −0.382683 + 0.923880i −2.31925 + 0.960664i −1.57873 − 3.81140i −0.707107 − 0.707107i 2.33472 + 2.33472i 0.382683 + 0.923880i
111.2 0.707107 − 0.707107i −0.980692 − 0.406216i − 1.00000i 0.382683 − 0.923880i −0.980692 + 0.406216i −0.758398 − 1.83094i −0.707107 − 0.707107i −1.32457 − 1.32457i −0.382683 − 0.923880i
111.3 0.707107 − 0.707107i 1.36338 + 0.564729i − 1.00000i 0.382683 − 0.923880i 1.36338 − 0.564729i 1.35785 + 3.27815i −0.707107 − 0.707107i −0.581445 − 0.581445i −0.382683 − 0.923880i
111.4 0.707107 − 0.707107i 1.93656 + 0.802151i − 1.00000i −0.382683 + 0.923880i 1.93656 − 0.802151i −0.434936 − 1.05003i −0.707107 − 0.707107i 0.985516 + 0.985516i 0.382683 + 0.923880i
121.1 0.707107 + 0.707107i −2.31925 + 0.960664i 1.00000i −0.382683 − 0.923880i −2.31925 − 0.960664i −1.57873 + 3.81140i −0.707107 + 0.707107i 2.33472 − 2.33472i 0.382683 − 0.923880i
121.2 0.707107 + 0.707107i −0.980692 + 0.406216i 1.00000i 0.382683 + 0.923880i −0.980692 − 0.406216i −0.758398 + 1.83094i −0.707107 + 0.707107i −1.32457 + 1.32457i −0.382683 + 0.923880i
121.3 0.707107 + 0.707107i 1.36338 − 0.564729i 1.00000i 0.382683 + 0.923880i 1.36338 + 0.564729i 1.35785 − 3.27815i −0.707107 + 0.707107i −0.581445 + 0.581445i −0.382683 + 0.923880i
121.4 0.707107 + 0.707107i 1.93656 − 0.802151i 1.00000i −0.382683 − 0.923880i 1.93656 + 0.802151i −0.434936 + 1.05003i −0.707107 + 0.707107i 0.985516 − 0.985516i 0.382683 − 0.923880i
151.1 −0.707107 − 0.707107i −0.894478 − 2.15946i 1.00000i −0.923880 + 0.382683i −0.894478 + 2.15946i −3.32333 − 1.37657i 0.707107 − 0.707107i −1.74186 + 1.74186i 0.923880 + 0.382683i
151.2 −0.707107 − 0.707107i −0.179356 − 0.433004i 1.00000i 0.923880 − 0.382683i −0.179356 + 0.433004i 1.29538 + 0.536563i 0.707107 − 0.707107i 1.96600 − 1.96600i −0.923880 − 0.382683i
151.3 −0.707107 − 0.707107i −0.0294014 − 0.0709814i 1.00000i −0.923880 + 0.382683i −0.0294014 + 0.0709814i 3.48924 + 1.44529i 0.707107 − 0.707107i 2.11715 − 2.11715i 0.923880 + 0.382683i
151.4 −0.707107 − 0.707107i 1.10324 + 2.66345i 1.00000i 0.923880 − 0.382683i 1.10324 − 2.66345i −0.0470744 − 0.0194989i 0.707107 − 0.707107i −3.75550 + 3.75550i −0.923880 − 0.382683i
161.1 −0.707107 + 0.707107i −0.894478 + 2.15946i − 1.00000i −0.923880 − 0.382683i −0.894478 − 2.15946i −3.32333 + 1.37657i 0.707107 + 0.707107i −1.74186 − 1.74186i 0.923880 − 0.382683i
161.2 −0.707107 + 0.707107i −0.179356 + 0.433004i − 1.00000i 0.923880 + 0.382683i −0.179356 − 0.433004i 1.29538 − 0.536563i 0.707107 + 0.707107i 1.96600 + 1.96600i −0.923880 + 0.382683i
161.3 −0.707107 + 0.707107i −0.0294014 + 0.0709814i − 1.00000i −0.923880 − 0.382683i −0.0294014 − 0.0709814i 3.48924 − 1.44529i 0.707107 + 0.707107i 2.11715 + 2.11715i 0.923880 − 0.382683i
161.4 −0.707107 + 0.707107i 1.10324 − 2.66345i − 1.00000i 0.923880 + 0.382683i 1.10324 + 2.66345i −0.0470744 + 0.0194989i 0.707107 + 0.707107i −3.75550 − 3.75550i −0.923880 + 0.382683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.k.b ✓ 16
5.b even 2 1 850.2.l.e 16
5.c odd 4 1 850.2.o.g 16
5.c odd 4 1 850.2.o.j 16
17.d even 8 1 inner 170.2.k.b ✓ 16
17.e odd 16 1 2890.2.a.bi 8
17.e odd 16 1 2890.2.a.bj 8
17.e odd 16 2 2890.2.b.r 16
85.k odd 8 1 850.2.o.g 16
85.m even 8 1 850.2.l.e 16
85.n odd 8 1 850.2.o.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.k.b ✓ 16 1.a even 1 1 trivial
170.2.k.b ✓ 16 17.d even 8 1 inner
850.2.l.e 16 5.b even 2 1
850.2.l.e 16 85.m even 8 1
850.2.o.g 16 5.c odd 4 1
850.2.o.g 16 85.k odd 8 1
850.2.o.j 16 5.c odd 4 1
850.2.o.j 16 85.n odd 8 1
2890.2.a.bi 8 17.e odd 16 1
2890.2.a.bj 8 17.e odd 16 1
2890.2.b.r 16 17.e odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{13} - 56 T_{3}^{11} - 60 T_{3}^{10} + 24 T_{3}^{9} + 1461 T_{3}^{8} - 304 T_{3}^{7} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{15} + \cdots + 4624 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 163430656 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + 88 T^{13} + \cdots + 94322944 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2041593856 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10022412544 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 508231936 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1126451086336 \) Copy content Toggle raw display
$41$ \( T^{16} + 32 T^{15} + \cdots + 15272464 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 28853778496 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 7520231228416 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 161541659689216 \) Copy content Toggle raw display
$59$ \( T^{16} - 16 T^{15} + \cdots + 9024016 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 19613442304 \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots + 1132064)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 944701696 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 12431448982276 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 8131710976 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2100572224 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16962833011216 \) Copy content Toggle raw display
$97$ \( T^{16} - 24 T^{15} + \cdots + 1473796 \) Copy content Toggle raw display
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