Properties

Label 85.2.j.a
Level $85$
Weight $2$
Character orbit 85.j
Analytic conductor $0.679$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,2,Mod(4,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, 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("85.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.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: \(0.678728417181\)
Analytic rank: \(1\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(-1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.00000i
1.00000i
−1.00000 −1.00000 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 + 1.00000i −3.00000 + 3.00000i 3.00000 1.00000i 2.00000 1.00000i
64.1 −1.00000 −1.00000 + 1.00000i −1.00000 −2.00000 1.00000i 1.00000 1.00000i −3.00000 3.00000i 3.00000 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 85.2.j.a 2
3.b odd 2 1 765.2.t.b 2
5.b even 2 1 85.2.j.b yes 2
5.c odd 4 1 425.2.e.a 2
5.c odd 4 1 425.2.e.b 2
15.d odd 2 1 765.2.t.a 2
17.c even 4 1 85.2.j.b yes 2
17.d even 8 2 1445.2.b.a 4
51.f odd 4 1 765.2.t.a 2
85.f odd 4 1 425.2.e.b 2
85.i odd 4 1 425.2.e.a 2
85.j even 4 1 inner 85.2.j.a 2
85.k odd 8 2 7225.2.a.p 2
85.m even 8 2 1445.2.b.a 4
85.n odd 8 2 7225.2.a.i 2
255.i odd 4 1 765.2.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.a 2 1.a even 1 1 trivial
85.2.j.a 2 85.j even 4 1 inner
85.2.j.b yes 2 5.b even 2 1
85.2.j.b yes 2 17.c even 4 1
425.2.e.a 2 5.c odd 4 1
425.2.e.a 2 85.i odd 4 1
425.2.e.b 2 5.c odd 4 1
425.2.e.b 2 85.f odd 4 1
765.2.t.a 2 15.d odd 2 1
765.2.t.a 2 51.f odd 4 1
765.2.t.b 2 3.b odd 2 1
765.2.t.b 2 255.i odd 4 1
1445.2.b.a 4 17.d even 8 2
1445.2.b.a 4 85.m even 8 2
7225.2.a.i 2 85.n odd 8 2
7225.2.a.p 2 85.k odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(85, [\chi])\). 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
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$13$ \( T^{2} \) 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
$29$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$43$ \( (T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} + 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) 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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