Properties

Label 85.2.a.c
Level $85$
Weight $2$
Character orbit 85.a
Self dual yes
Analytic conductor $0.679$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,2,Mod(1,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)

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: yes
Analytic conductor: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 2.73205 1.00000 1.00000 −4.73205 −2.73205 1.73205 4.46410 −1.73205
1.2 1.73205 −0.732051 1.00000 1.00000 −1.26795 0.732051 −1.73205 −2.46410 1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.2.a.c 2
3.b odd 2 1 765.2.a.g 2
4.b odd 2 1 1360.2.a.k 2
5.b even 2 1 425.2.a.e 2
5.c odd 4 2 425.2.b.d 4
7.b odd 2 1 4165.2.a.t 2
8.b even 2 1 5440.2.a.bb 2
8.d odd 2 1 5440.2.a.bl 2
15.d odd 2 1 3825.2.a.v 2
17.b even 2 1 1445.2.a.g 2
17.c even 4 2 1445.2.d.e 4
20.d odd 2 1 6800.2.a.bg 2
85.c even 2 1 7225.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.a.c 2 1.a even 1 1 trivial
425.2.a.e 2 5.b even 2 1
425.2.b.d 4 5.c odd 4 2
765.2.a.g 2 3.b odd 2 1
1360.2.a.k 2 4.b odd 2 1
1445.2.a.g 2 17.b even 2 1
1445.2.d.e 4 17.c even 4 2
3825.2.a.v 2 15.d odd 2 1
4165.2.a.t 2 7.b odd 2 1
5440.2.a.bb 2 8.b even 2 1
5440.2.a.bl 2 8.d odd 2 1
6800.2.a.bg 2 20.d odd 2 1
7225.2.a.l 2 85.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(85))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 66 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 242 \) Copy content Toggle raw display
$83$ \( T^{2} - 24T + 132 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 72 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
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