Properties

Label 85.4.a.c
Level $85$
Weight $4$
Character orbit 85.a
Self dual yes
Analytic conductor $5.015$
Analytic rank $0$
Dimension $1$
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] = [85,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + 10 q^{3} + q^{4} + 5 q^{5} + 30 q^{6} - 22 q^{7} - 21 q^{8} + 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + 10 q^{3} + q^{4} + 5 q^{5} + 30 q^{6} - 22 q^{7} - 21 q^{8} + 73 q^{9} + 15 q^{10} - 30 q^{11} + 10 q^{12} - 46 q^{13} - 66 q^{14} + 50 q^{15} - 71 q^{16} + 17 q^{17} + 219 q^{18} + 104 q^{19} + 5 q^{20} - 220 q^{21} - 90 q^{22} + 42 q^{23} - 210 q^{24} + 25 q^{25} - 138 q^{26} + 460 q^{27} - 22 q^{28} - 66 q^{29} + 150 q^{30} + 194 q^{31} - 45 q^{32} - 300 q^{33} + 51 q^{34} - 110 q^{35} + 73 q^{36} + 206 q^{37} + 312 q^{38} - 460 q^{39} - 105 q^{40} - 126 q^{41} - 660 q^{42} - 388 q^{43} - 30 q^{44} + 365 q^{45} + 126 q^{46} - 540 q^{47} - 710 q^{48} + 141 q^{49} + 75 q^{50} + 170 q^{51} - 46 q^{52} + 78 q^{53} + 1380 q^{54} - 150 q^{55} + 462 q^{56} + 1040 q^{57} - 198 q^{58} + 432 q^{59} + 50 q^{60} - 610 q^{61} + 582 q^{62} - 1606 q^{63} + 433 q^{64} - 230 q^{65} - 900 q^{66} + 848 q^{67} + 17 q^{68} + 420 q^{69} - 330 q^{70} - 174 q^{71} - 1533 q^{72} + 362 q^{73} + 618 q^{74} + 250 q^{75} + 104 q^{76} + 660 q^{77} - 1380 q^{78} + 398 q^{79} - 355 q^{80} + 2629 q^{81} - 378 q^{82} + 828 q^{83} - 220 q^{84} + 85 q^{85} - 1164 q^{86} - 660 q^{87} + 630 q^{88} + 630 q^{89} + 1095 q^{90} + 1012 q^{91} + 42 q^{92} + 1940 q^{93} - 1620 q^{94} + 520 q^{95} - 450 q^{96} - 1486 q^{97} + 423 q^{98} - 2190 q^{99}+O(q^{100}) \) 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
0
3.00000 10.0000 1.00000 5.00000 30.0000 −22.0000 −21.0000 73.0000 15.0000
\(n\): 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 85.4.a.c 1
3.b odd 2 1 765.4.a.a 1
4.b odd 2 1 1360.4.a.a 1
5.b even 2 1 425.4.a.a 1
5.c odd 4 2 425.4.b.d 2
17.b even 2 1 1445.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.a.c 1 1.a even 1 1 trivial
425.4.a.a 1 5.b even 2 1
425.4.b.d 2 5.c odd 4 2
765.4.a.a 1 3.b odd 2 1
1360.4.a.a 1 4.b odd 2 1
1445.4.a.f 1 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(85))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{3} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T - 10 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 22 \) Copy content Toggle raw display
$11$ \( T + 30 \) Copy content Toggle raw display
$13$ \( T + 46 \) Copy content Toggle raw display
$17$ \( T - 17 \) Copy content Toggle raw display
$19$ \( T - 104 \) Copy content Toggle raw display
$23$ \( T - 42 \) Copy content Toggle raw display
$29$ \( T + 66 \) Copy content Toggle raw display
$31$ \( T - 194 \) Copy content Toggle raw display
$37$ \( T - 206 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T + 388 \) Copy content Toggle raw display
$47$ \( T + 540 \) Copy content Toggle raw display
$53$ \( T - 78 \) Copy content Toggle raw display
$59$ \( T - 432 \) Copy content Toggle raw display
$61$ \( T + 610 \) Copy content Toggle raw display
$67$ \( T - 848 \) Copy content Toggle raw display
$71$ \( T + 174 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 398 \) Copy content Toggle raw display
$83$ \( T - 828 \) Copy content Toggle raw display
$89$ \( T - 630 \) Copy content Toggle raw display
$97$ \( T + 1486 \) Copy content Toggle raw display
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