Properties

Label 85.2.a.b
Level $85$
Weight $2$
Character orbit 85.a
Self dual yes
Analytic conductor $0.679$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - \beta - 2) q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} - \beta q^{6} + (\beta - 2) q^{7} + (\beta - 3) q^{8} + (4 \beta + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - \beta - 2) q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} - \beta q^{6} + (\beta - 2) q^{7} + (\beta - 3) q^{8} + (4 \beta + 3) q^{9} + ( - \beta + 1) q^{10} + (\beta - 4) q^{11} + (3 \beta + 2) q^{12} - 2 \beta q^{13} + ( - 3 \beta + 4) q^{14} + (\beta + 2) q^{15} + 3 q^{16} - q^{17} + ( - \beta + 5) q^{18} - 2 \beta q^{19} + (2 \beta - 1) q^{20} + 2 q^{21} + ( - 5 \beta + 6) q^{22} + ( - \beta - 2) q^{23} + (\beta + 4) q^{24} + q^{25} + (2 \beta - 4) q^{26} + ( - 8 \beta - 8) q^{27} + (5 \beta - 6) q^{28} + ( - 2 \beta - 2) q^{29} + \beta q^{30} + 3 \beta q^{31} + (\beta + 3) q^{32} + (2 \beta + 6) q^{33} + ( - \beta + 1) q^{34} + ( - \beta + 2) q^{35} + ( - 2 \beta - 13) q^{36} + (6 \beta - 2) q^{37} + (2 \beta - 4) q^{38} + (4 \beta + 4) q^{39} + ( - \beta + 3) q^{40} + ( - 6 \beta + 2) q^{41} + (2 \beta - 2) q^{42} + (4 \beta + 2) q^{43} + (9 \beta - 8) q^{44} + ( - 4 \beta - 3) q^{45} - \beta q^{46} + ( - 2 \beta - 2) q^{47} + ( - 3 \beta - 6) q^{48} + ( - 4 \beta - 1) q^{49} + (\beta - 1) q^{50} + (\beta + 2) q^{51} + ( - 2 \beta + 8) q^{52} + ( - 4 \beta + 6) q^{53} - 8 q^{54} + ( - \beta + 4) q^{55} + ( - 5 \beta + 8) q^{56} + (4 \beta + 4) q^{57} - 2 q^{58} + (2 \beta - 12) q^{59} + ( - 3 \beta - 2) q^{60} + (4 \beta + 2) q^{61} + ( - 3 \beta + 6) q^{62} + ( - 5 \beta + 2) q^{63} + (2 \beta - 7) q^{64} + 2 \beta q^{65} + (4 \beta - 2) q^{66} + (2 \beta - 6) q^{67} + (2 \beta - 1) q^{68} + (4 \beta + 6) q^{69} + (3 \beta - 4) q^{70} - 3 \beta q^{71} + ( - 9 \beta - 1) q^{72} + ( - 2 \beta - 2) q^{73} + ( - 8 \beta + 14) q^{74} + ( - \beta - 2) q^{75} + ( - 2 \beta + 8) q^{76} + ( - 6 \beta + 10) q^{77} + 4 q^{78} + (\beta + 4) q^{79} - 3 q^{80} + (12 \beta + 23) q^{81} + (8 \beta - 14) q^{82} + (8 \beta - 2) q^{83} + ( - 4 \beta + 2) q^{84} + q^{85} + ( - 2 \beta + 6) q^{86} + (6 \beta + 8) q^{87} + ( - 7 \beta + 14) q^{88} + (4 \beta - 8) q^{89} + (\beta - 5) q^{90} + (4 \beta - 4) q^{91} + (3 \beta + 2) q^{92} + ( - 6 \beta - 6) q^{93} - 2 q^{94} + 2 \beta q^{95} + ( - 5 \beta - 8) q^{96} + (4 \beta - 2) q^{97} + (3 \beta - 7) q^{98} + ( - 13 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23} + 8 q^{24} + 2 q^{25} - 8 q^{26} - 16 q^{27} - 12 q^{28} - 4 q^{29} + 6 q^{32} + 12 q^{33} + 2 q^{34} + 4 q^{35} - 26 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{39} + 6 q^{40} + 4 q^{41} - 4 q^{42} + 4 q^{43} - 16 q^{44} - 6 q^{45} - 4 q^{47} - 12 q^{48} - 2 q^{49} - 2 q^{50} + 4 q^{51} + 16 q^{52} + 12 q^{53} - 16 q^{54} + 8 q^{55} + 16 q^{56} + 8 q^{57} - 4 q^{58} - 24 q^{59} - 4 q^{60} + 4 q^{61} + 12 q^{62} + 4 q^{63} - 14 q^{64} - 4 q^{66} - 12 q^{67} - 2 q^{68} + 12 q^{69} - 8 q^{70} - 2 q^{72} - 4 q^{73} + 28 q^{74} - 4 q^{75} + 16 q^{76} + 20 q^{77} + 8 q^{78} + 8 q^{79} - 6 q^{80} + 46 q^{81} - 28 q^{82} - 4 q^{83} + 4 q^{84} + 2 q^{85} + 12 q^{86} + 16 q^{87} + 28 q^{88} - 16 q^{89} - 10 q^{90} - 8 q^{91} + 4 q^{92} - 12 q^{93} - 4 q^{94} - 16 q^{96} - 4 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−2.41421 −0.585786 3.82843 −1.00000 1.41421 −3.41421 −4.41421 −2.65685 2.41421
1.2 0.414214 −3.41421 −1.82843 −1.00000 −1.41421 −0.585786 −1.58579 8.65685 −0.414214
\(n\): 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.b 2
3.b odd 2 1 765.2.a.i 2
4.b odd 2 1 1360.2.a.o 2
5.b even 2 1 425.2.a.f 2
5.c odd 4 2 425.2.b.e 4
7.b odd 2 1 4165.2.a.q 2
8.b even 2 1 5440.2.a.bm 2
8.d odd 2 1 5440.2.a.ba 2
15.d odd 2 1 3825.2.a.p 2
17.b even 2 1 1445.2.a.f 2
17.c even 4 2 1445.2.d.f 4
20.d odd 2 1 6800.2.a.ba 2
85.c even 2 1 7225.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.a.b 2 1.a even 1 1 trivial
425.2.a.f 2 5.b even 2 1
425.2.b.e 4 5.c odd 4 2
765.2.a.i 2 3.b odd 2 1
1360.2.a.o 2 4.b odd 2 1
1445.2.a.f 2 17.b even 2 1
1445.2.d.f 4 17.c even 4 2
3825.2.a.p 2 15.d odd 2 1
4165.2.a.q 2 7.b odd 2 1
5440.2.a.ba 2 8.d odd 2 1
5440.2.a.bm 2 8.b even 2 1
6800.2.a.ba 2 20.d odd 2 1
7225.2.a.o 2 85.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(85))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 24T + 136 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$71$ \( T^{2} - 18 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
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