Properties

Label 320.10.a.x
Level $320$
Weight $10$
Character orbit 320.a
Self dual yes
Analytic conductor $164.811$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,10,Mod(1,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 320.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: \(164.811467572\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{7}, \sqrt{418})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 850x^{2} + 168921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 5\cdot 13 \)
Twist minimal: no (minimal twist has level 160)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 625 q^{5} + ( - 9 \beta_{2} + 8 \beta_1) q^{7} + (7 \beta_{3} + 3941) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 625 q^{5} + ( - 9 \beta_{2} + 8 \beta_1) q^{7} + (7 \beta_{3} + 3941) q^{9} + ( - 49 \beta_{2} - 51 \beta_1) q^{11} + (61 \beta_{3} + 33186) q^{13} - 625 \beta_1 q^{15} + (139 \beta_{3} - 195710) q^{17} + ( - 841 \beta_{2} - 777 \beta_1) q^{19} + (542 \beta_{3} + 134344) q^{21} + (169 \beta_{2} + 3992 \beta_1) q^{23} + 390625 q^{25} + ( - 2184 \beta_{2} - 8966 \beta_1) q^{27} + ( - 2914 \beta_{3} + 2246362) q^{29} + ( - 6810 \beta_{2} + 27652 \beta_1) q^{31} + (2289 \beta_{3} - 1502352) q^{33} + (5625 \beta_{2} - 5000 \beta_1) q^{35} + (8282 \beta_{3} - 6584854) q^{37} + ( - 19032 \beta_{2} + 92234 \beta_1) q^{39} + ( - 7347 \beta_{3} - 10455294) q^{41} + ( - 14006 \beta_{2} - 97305 \beta_1) q^{43} + ( - 4375 \beta_{3} - 2463125) q^{45} + (36969 \beta_{2} + 198234 \beta_1) q^{47} + ( - 6923 \beta_{3} + 3814657) q^{49} + ( - 43368 \beta_{2} - 61158 \beta_1) q^{51} + ( - 36065 \beta_{3} - 23317414) q^{53} + (30625 \beta_{2} + 31875 \beta_1) q^{55} + (39975 \beta_{3} - 23462400) q^{57} + (119145 \beta_{2} + 242221 \beta_1) q^{59} + (57228 \beta_{3} + 27545506) q^{61} + (8043 \beta_{2} + 501536 \beta_1) q^{63} + ( - 38125 \beta_{3} - 20741250) q^{65} + ( - 15690 \beta_{2} + 1143669 \beta_1) q^{67} + (18818 \beta_{3} + 95333176) q^{69} + (30672 \beta_{2} + 6446 \beta_1) q^{71} + ( - 1777 \beta_{3} - 140896710) q^{73} + 390625 \beta_1 q^{75} + ( - 88941 \beta_{3} + 227768688) q^{77} + ( - 531208 \beta_{2} + 72628 \beta_1) q^{79} + ( - 82607 \beta_{3} - 302644735) q^{81} + (220012 \beta_{2} + 2927085 \beta_1) q^{83} + ( - 86875 \beta_{3} + 122318750) q^{85} + (909168 \beta_{2} - 574390 \beta_1) q^{87} + (215690 \beta_{3} - 5088310) q^{89} + (80502 \beta_{2} + 4361272 \beta_1) q^{91} + (561304 \beta_{3} + 611900528) q^{93} + (525625 \beta_{2} + 485625 \beta_1) q^{95} + (65405 \beta_{3} - 121248382) q^{97} + (250299 \beta_{2} + 1717233 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2500 q^{5} + 15764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2500 q^{5} + 15764 q^{9} + 132744 q^{13} - 782840 q^{17} + 537376 q^{21} + 1562500 q^{25} + 8985448 q^{29} - 6009408 q^{33} - 26339416 q^{37} - 41821176 q^{41} - 9852500 q^{45} + 15258628 q^{49} - 93269656 q^{53} - 93849600 q^{57} + 110182024 q^{61} - 82965000 q^{65} + 381332704 q^{69} - 563586840 q^{73} + 911074752 q^{77} - 1210578940 q^{81} + 489275000 q^{85} - 20353240 q^{89} + 2447602112 q^{93} - 484993528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 850x^{2} + 168921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{3} - 3677\nu ) / 137 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61\nu^{3} - 40753\nu ) / 411 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 16\nu^{2} - 6800 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -27\beta_{2} + 61\beta_1 ) / 1040 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6800 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11031\beta_{2} + 40753\beta_1 ) / 1040 \) 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
−23.0908
17.7993
−17.7993
23.0908
0 −189.052 0 −625.000 0 −5673.17 0 16057.7 0
1.2 0 −107.272 0 −625.000 0 7493.44 0 −8175.72 0
1.3 0 107.272 0 −625.000 0 −7493.44 0 −8175.72 0
1.4 0 189.052 0 −625.000 0 5673.17 0 16057.7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.10.a.x 4
4.b odd 2 1 inner 320.10.a.x 4
8.b even 2 1 160.10.a.d 4
8.d odd 2 1 160.10.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.10.a.d 4 8.b even 2 1
160.10.a.d 4 8.d odd 2 1
320.10.a.x 4 1.a even 1 1 trivial
320.10.a.x 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 47248T_{3}^{2} + 411278400 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 47248 T^{2} + \cdots + 411278400 \) Copy content Toggle raw display
$5$ \( (T + 625)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 88336528 T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{4} - 2764259392 T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} - 66372 T - 10047638908)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 391420 T - 19587639804)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 804604403200 T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} - 800031279248 T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4492724 T - 20395982253660)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 81400225018432 T^{2} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + 13169708 T - 162155268204060)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20910588 T - 52418231901180)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 691305167698192 T^{2} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 46634828 T - 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} - 55091012 T - 90\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + 281793420 T + 19\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10176620 T - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 242496764 T + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
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