Properties

Label 85.2.c.a
Level $85$
Weight $2$
Character orbit 85.c
Analytic conductor $0.679$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,2,Mod(84,85)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.84");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.c (of order \(2\), degree \(1\), 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: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + (\beta_{4} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{6} + 2 \beta_1) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + (\beta_{4} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{6} + 2 \beta_1) q^{8} - \beta_{3} q^{9} + ( - \beta_{7} - \beta_{4} - \beta_{2}) q^{10} - \beta_{7} q^{11} + (\beta_{5} + 5 \beta_{4} - \beta_{2}) q^{12} + (\beta_{6} - \beta_1) q^{13} + \beta_{7} q^{14} + (\beta_{3} + \beta_1 - 2) q^{15} + ( - 2 \beta_{3} + 3) q^{16} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + ( - \beta_{7} - 3 \beta_{5} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 16 q^{15} + 24 q^{16} + 16 q^{19} + 8 q^{21} - 24 q^{26} + 32 q^{30} + 32 q^{34} - 24 q^{35} - 56 q^{36} - 16 q^{49} - 24 q^{50} - 40 q^{51} + 8 q^{55} + 48 q^{59} + 88 q^{60} - 48 q^{64} + 88 q^{66} + 8 q^{69} - 8 q^{70} - 32 q^{76} - 16 q^{81} - 72 q^{84} + 8 q^{85} + 24 q^{86} - 24 q^{89} - 104 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 9\nu^{4} + \nu^{2} - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 9\nu^{5} - 37\nu^{3} + 9\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 19\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{5} - \nu^{3} + 3\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 3\nu^{5} + \nu^{3} + 39\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} + 9\nu^{4} - 5\nu^{2} - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 9\nu^{5} - 43\nu^{3} + 99\nu ) / 108 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + 6\beta_{4} - 8\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{6} + 10\beta _1 + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} - 4\beta_{5} - 18\beta_{4} - 7\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19\beta_{6} + 3\beta_{3} - 19\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17\beta_{7} - 17\beta_{5} + 60\beta_{4} - 26\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
84.1
−0.420861 + 1.68014i
0.420861 1.68014i
−1.68014 0.420861i
1.68014 + 0.420861i
−1.68014 + 0.420861i
1.68014 0.420861i
−0.420861 1.68014i
0.420861 + 1.68014i
2.57794i −2.37608 −4.64575 1.95522 1.08495i 6.12538i −1.53436 6.82058i 2.64575 −2.79694 5.04042i
84.2 2.57794i 2.37608 −4.64575 −1.95522 + 1.08495i 6.12538i 1.53436 6.82058i 2.64575 2.79694 + 5.04042i
84.3 1.16372i −0.595188 0.645751 −1.08495 1.95522i 0.692633i 2.76510 3.07892i −2.64575 −2.27533 + 1.26258i
84.4 1.16372i 0.595188 0.645751 1.08495 + 1.95522i 0.692633i −2.76510 3.07892i −2.64575 2.27533 1.26258i
84.5 1.16372i −0.595188 0.645751 −1.08495 + 1.95522i 0.692633i 2.76510 3.07892i −2.64575 −2.27533 1.26258i
84.6 1.16372i 0.595188 0.645751 1.08495 1.95522i 0.692633i −2.76510 3.07892i −2.64575 2.27533 + 1.26258i
84.7 2.57794i −2.37608 −4.64575 1.95522 + 1.08495i 6.12538i −1.53436 6.82058i 2.64575 −2.79694 + 5.04042i
84.8 2.57794i 2.37608 −4.64575 −1.95522 1.08495i 6.12538i 1.53436 6.82058i 2.64575 2.79694 5.04042i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 84.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.2.c.a 8
3.b odd 2 1 765.2.d.c 8
4.b odd 2 1 1360.2.o.f 8
5.b even 2 1 inner 85.2.c.a 8
5.c odd 4 2 425.2.d.d 8
15.d odd 2 1 765.2.d.c 8
17.b even 2 1 inner 85.2.c.a 8
17.c even 4 2 1445.2.b.b 8
20.d odd 2 1 1360.2.o.f 8
51.c odd 2 1 765.2.d.c 8
68.d odd 2 1 1360.2.o.f 8
85.c even 2 1 inner 85.2.c.a 8
85.f odd 4 2 7225.2.a.bl 8
85.g odd 4 2 425.2.d.d 8
85.i odd 4 2 7225.2.a.bl 8
85.j even 4 2 1445.2.b.b 8
255.h odd 2 1 765.2.d.c 8
340.d odd 2 1 1360.2.o.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.c.a 8 1.a even 1 1 trivial
85.2.c.a 8 5.b even 2 1 inner
85.2.c.a 8 17.b even 2 1 inner
85.2.c.a 8 85.c even 2 1 inner
425.2.d.d 8 5.c odd 4 2
425.2.d.d 8 85.g odd 4 2
765.2.d.c 8 3.b odd 2 1
765.2.d.c 8 15.d odd 2 1
765.2.d.c 8 51.c odd 2 1
765.2.d.c 8 255.h odd 2 1
1360.2.o.f 8 4.b odd 2 1
1360.2.o.f 8 20.d odd 2 1
1360.2.o.f 8 68.d odd 2 1
1360.2.o.f 8 340.d odd 2 1
1445.2.b.b 8 17.c even 4 2
1445.2.b.b 8 85.j even 4 2
7225.2.a.bl 8 85.f odd 4 2
7225.2.a.bl 8 85.i odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 22T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 26 T^{2} + 162)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 36 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T - 2)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 38 T^{2} + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 54 T^{2} + 162)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 116 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 108 T^{2} + 648)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 266 T^{2} + 882)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 54 T^{2} + 162)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 44 T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 54)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 208 T^{2} + 10368)^{2} \) Copy content Toggle raw display
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