Properties

Label 180.8.a.f
Level $180$
Weight $8$
Character orbit 180.a
Self dual yes
Analytic conductor $56.229$
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] = [180,8,Mod(1,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(56.2293045871\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29569}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 7392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(\beta = 6\sqrt{29569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 125 q^{5} + ( - \beta - 280) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 125 q^{5} + ( - \beta - 280) q^{7} + (5 \beta + 1860) q^{11} + (2 \beta + 3680) q^{13} + (10 \beta + 2970) q^{17} + (10 \beta + 15272) q^{19} + ( - 20 \beta - 13440) q^{23} + 15625 q^{25} + ( - 140 \beta - 78330) q^{29} + ( - 140 \beta - 89164) q^{31} + (125 \beta + 35000) q^{35} + (210 \beta + 25580) q^{37} + ( - 60 \beta - 217320) q^{41} + ( - 52 \beta - 227920) q^{43} + (920 \beta - 56760) q^{47} + (560 \beta + 319341) q^{49} + ( - 520 \beta - 990690) q^{53} + ( - 625 \beta - 232500) q^{55} + ( - 575 \beta - 1548900) q^{59} + ( - 2460 \beta - 237166) q^{61} + ( - 250 \beta - 460000) q^{65} + (3186 \beta + 1363520) q^{67} + ( - 570 \beta - 3332040) q^{71} + (1140 \beta + 1640270) q^{73} + ( - 3260 \beta - 5843220) q^{77} + (300 \beta - 1805356) q^{79} + (3790 \beta - 4293120) q^{83} + ( - 1250 \beta - 371250) q^{85} + (6180 \beta - 5403540) q^{89} + ( - 4240 \beta - 3159368) q^{91} + ( - 1250 \beta - 1909000) q^{95} + ( - 524 \beta - 1847830) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 250 q^{5} - 560 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 250 q^{5} - 560 q^{7} + 3720 q^{11} + 7360 q^{13} + 5940 q^{17} + 30544 q^{19} - 26880 q^{23} + 31250 q^{25} - 156660 q^{29} - 178328 q^{31} + 70000 q^{35} + 51160 q^{37} - 434640 q^{41} - 455840 q^{43} - 113520 q^{47} + 638682 q^{49} - 1981380 q^{53} - 465000 q^{55} - 3097800 q^{59} - 474332 q^{61} - 920000 q^{65} + 2727040 q^{67} - 6664080 q^{71} + 3280540 q^{73} - 11686440 q^{77} - 3610712 q^{79} - 8586240 q^{83} - 742500 q^{85} - 10807080 q^{89} - 6318736 q^{91} - 3818000 q^{95} - 3695660 q^{97}+O(q^{100}) \) 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
86.4782
−85.4782
0 0 0 −125.000 0 −1311.74 0 0 0
1.2 0 0 0 −125.000 0 751.738 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 180.8.a.f 2
3.b odd 2 1 180.8.a.h yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.8.a.f 2 1.a even 1 1 trivial
180.8.a.h yes 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(180))\):

\( T_{7}^{2} + 560T_{7} - 986084 \) Copy content Toggle raw display
\( T_{11}^{2} - 3720T_{11} - 23152500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 560T - 986084 \) Copy content Toggle raw display
$11$ \( T^{2} - 3720 T - 23152500 \) Copy content Toggle raw display
$13$ \( T^{2} - 7360 T + 9284464 \) Copy content Toggle raw display
$17$ \( T^{2} - 5940 T - 97627500 \) Copy content Toggle raw display
$19$ \( T^{2} - 30544 T + 126785584 \) Copy content Toggle raw display
$23$ \( T^{2} + 26880 T - 245160000 \) Copy content Toggle raw display
$29$ \( T^{2} + 156660 T - 14728297500 \) Copy content Toggle raw display
$31$ \( T^{2} + 178328 T - 12913667504 \) Copy content Toggle raw display
$37$ \( T^{2} - 51160 T - 46289408000 \) Copy content Toggle raw display
$41$ \( T^{2} + 434640 T + 43395840000 \) Copy content Toggle raw display
$43$ \( T^{2} + 455840 T + 49069161664 \) Copy content Toggle raw display
$47$ \( T^{2} + 113520 T - 897757560000 \) Copy content Toggle raw display
$53$ \( T^{2} + 1981380 T + 693630202500 \) Copy content Toggle raw display
$59$ \( T^{2} + 3097800 T + 2047146187500 \) Copy content Toggle raw display
$61$ \( T^{2} + 474332 T - 6385583662844 \) Copy content Toggle raw display
$67$ \( T^{2} - 2727040 T - 8945960242064 \) Copy content Toggle raw display
$71$ \( T^{2} + 6664080 T + 10756639710000 \) Copy content Toggle raw display
$73$ \( T^{2} - 3280540 T + 1307082266500 \) Copy content Toggle raw display
$79$ \( T^{2} + 3610712 T + 3163506726736 \) Copy content Toggle raw display
$83$ \( T^{2} + 8586240 T + 3140524710000 \) Copy content Toggle raw display
$89$ \( T^{2} + 10807080 T - 11456954190000 \) Copy content Toggle raw display
$97$ \( T^{2} + 3695660 T + 3122193950116 \) Copy content Toggle raw display
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