Properties

Label 170.2.g.d
Level $170$
Weight $2$
Character orbit 170.g
Analytic conductor $1.357$
Analytic rank $0$
Dimension $2$
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(89,170)]
 
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([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.g (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: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (i - 1) q^{3} + q^{4} + (i + 2) q^{5} + (i - 1) q^{6} + ( - i - 1) q^{7} + q^{8} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (i - 1) q^{3} + q^{4} + (i + 2) q^{5} + (i - 1) q^{6} + ( - i - 1) q^{7} + q^{8} + i q^{9} + (i + 2) q^{10} + ( - i + 1) q^{11} + (i - 1) q^{12} + 4 i q^{13} + ( - i - 1) q^{14} + (i - 3) q^{15} + q^{16} + ( - 4 i - 1) q^{17} + i q^{18} - 6 i q^{19} + (i + 2) q^{20} + 2 q^{21} + ( - i + 1) q^{22} + ( - i - 1) q^{23} + (i - 1) q^{24} + (4 i + 3) q^{25} + 4 i q^{26} + ( - 4 i - 4) q^{27} + ( - i - 1) q^{28} + ( - 5 i - 5) q^{29} + (i - 3) q^{30} + ( - 3 i - 3) q^{31} + q^{32} + 2 i q^{33} + ( - 4 i - 1) q^{34} + ( - 3 i - 1) q^{35} + i q^{36} + ( - i + 1) q^{37} - 6 i q^{38} + ( - 4 i - 4) q^{39} + (i + 2) q^{40} + (7 i - 7) q^{41} + 2 q^{42} + 4 q^{43} + ( - i + 1) q^{44} + (2 i - 1) q^{45} + ( - i - 1) q^{46} + 6 i q^{47} + (i - 1) q^{48} - 5 i q^{49} + (4 i + 3) q^{50} + (3 i + 5) q^{51} + 4 i q^{52} + 10 q^{53} + ( - 4 i - 4) q^{54} + ( - i + 3) q^{55} + ( - i - 1) q^{56} + (6 i + 6) q^{57} + ( - 5 i - 5) q^{58} + 10 i q^{59} + (i - 3) q^{60} + ( - 3 i + 3) q^{61} + ( - 3 i - 3) q^{62} + ( - i + 1) q^{63} + q^{64} + (8 i - 4) q^{65} + 2 i q^{66} - 2 i q^{67} + ( - 4 i - 1) q^{68} + 2 q^{69} + ( - 3 i - 1) q^{70} + (5 i + 5) q^{71} + i q^{72} + (i - 1) q^{73} + ( - i + 1) q^{74} + ( - i - 7) q^{75} - 6 i q^{76} - 2 q^{77} + ( - 4 i - 4) q^{78} + (i - 1) q^{79} + (i + 2) q^{80} + 5 q^{81} + (7 i - 7) q^{82} - 12 q^{83} + 2 q^{84} + ( - 9 i + 2) q^{85} + 4 q^{86} + 10 q^{87} + ( - i + 1) q^{88} + 6 q^{89} + (2 i - 1) q^{90} + ( - 4 i + 4) q^{91} + ( - i - 1) q^{92} + 6 q^{93} + 6 i q^{94} + ( - 12 i + 6) q^{95} + (i - 1) q^{96} + ( - 3 i + 3) q^{97} - 5 i q^{98} + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 4 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{17} + 4 q^{20} + 4 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} + 6 q^{25} - 8 q^{27} - 2 q^{28} - 10 q^{29} - 6 q^{30} - 6 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{35} + 2 q^{37} - 8 q^{39} + 4 q^{40} - 14 q^{41} + 4 q^{42} + 8 q^{43} + 2 q^{44} - 2 q^{45} - 2 q^{46} - 2 q^{48} + 6 q^{50} + 10 q^{51} + 20 q^{53} - 8 q^{54} + 6 q^{55} - 2 q^{56} + 12 q^{57} - 10 q^{58} - 6 q^{60} + 6 q^{61} - 6 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{65} - 2 q^{68} + 4 q^{69} - 2 q^{70} + 10 q^{71} - 2 q^{73} + 2 q^{74} - 14 q^{75} - 4 q^{77} - 8 q^{78} - 2 q^{79} + 4 q^{80} + 10 q^{81} - 14 q^{82} - 24 q^{83} + 4 q^{84} + 4 q^{85} + 8 q^{86} + 20 q^{87} + 2 q^{88} + 12 q^{89} - 2 q^{90} + 8 q^{91} - 2 q^{92} + 12 q^{93} + 12 q^{95} - 2 q^{96} + 6 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(71\) \(137\)
\(\chi(n)\) \(i\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
1.00000i
1.00000i
1.00000 −1.00000 1.00000i 1.00000 2.00000 1.00000i −1.00000 1.00000i −1.00000 + 1.00000i 1.00000 1.00000i 2.00000 1.00000i
149.1 1.00000 −1.00000 + 1.00000i 1.00000 2.00000 + 1.00000i −1.00000 + 1.00000i −1.00000 1.00000i 1.00000 1.00000i 2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.g.d yes 2
3.b odd 2 1 1530.2.n.a 2
5.b even 2 1 170.2.g.b 2
5.c odd 4 1 850.2.h.a 2
5.c odd 4 1 850.2.h.e 2
15.d odd 2 1 1530.2.n.g 2
17.c even 4 1 170.2.g.b 2
51.f odd 4 1 1530.2.n.g 2
85.f odd 4 1 850.2.h.e 2
85.i odd 4 1 850.2.h.a 2
85.j even 4 1 inner 170.2.g.d yes 2
255.i odd 4 1 1530.2.n.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.g.b 2 5.b even 2 1
170.2.g.b 2 17.c even 4 1
170.2.g.d yes 2 1.a even 1 1 trivial
170.2.g.d yes 2 85.j even 4 1 inner
850.2.h.a 2 5.c odd 4 1
850.2.h.a 2 85.i odd 4 1
850.2.h.e 2 5.c odd 4 1
850.2.h.e 2 85.f odd 4 1
1530.2.n.a 2 3.b odd 2 1
1530.2.n.a 2 255.i odd 4 1
1530.2.n.g 2 15.d odd 2 1
1530.2.n.g 2 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
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