Properties

Label 320.8.a.u
Level $320$
Weight $8$
Character orbit 320.a
Self dual yes
Analytic conductor $99.963$
Analytic rank $0$
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] = [320,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(99.9632081549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 5)
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 \(\beta = 16\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 10) q^{3} + 125 q^{5} + (7 \beta + 50) q^{7} + (20 \beta + 2777) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 10) q^{3} + 125 q^{5} + (7 \beta + 50) q^{7} + (20 \beta + 2777) q^{9} + ( - 50 \beta + 2272) q^{11} + ( - 76 \beta - 1770) q^{13} + (125 \beta + 1250) q^{15} + (148 \beta - 13670) q^{17} + (40 \beta + 19380) q^{19} + (120 \beta + 34548) q^{21} + ( - 51 \beta + 62070) q^{23} + 15625 q^{25} + (790 \beta + 103180) q^{27} + (2440 \beta + 36130) q^{29} + ( - 350 \beta - 153412) q^{31} + (1772 \beta - 220480) q^{33} + (875 \beta + 6250) q^{35} + (3192 \beta + 61510) q^{37} + ( - 2530 \beta - 387364) q^{39} + (7100 \beta + 132182) q^{41} + ( - 5399 \beta + 211650) q^{43} + (2500 \beta + 347125) q^{45} + (5687 \beta + 52730) q^{47} + (700 \beta - 582707) q^{49} + ( - 12190 \beta + 583172) q^{51} + ( - 6676 \beta + 1195790) q^{53} + ( - 6250 \beta + 284000) q^{55} + (19780 \beta + 388360) q^{57} + (28420 \beta - 560060) q^{59} + (20000 \beta - 1128522) q^{61} + (20439 \beta + 819810) q^{63} + ( - 9500 \beta - 221250) q^{65} + (9923 \beta + 2258230) q^{67} + (61560 \beta + 372636) q^{69} + ( - 8750 \beta - 310892) q^{71} + (28276 \beta + 2284530) q^{73} + (15625 \beta + 156250) q^{75} + (13404 \beta - 1588800) q^{77} + ( - 59060 \beta - 2166520) q^{79} + (67340 \beta - 1198939) q^{81} + ( - 61299 \beta - 4896510) q^{83} + (18500 \beta - 1708750) q^{85} + (60530 \beta + 12229460) q^{87} + ( - 39720 \beta + 3012810) q^{89} + ( - 16190 \beta - 2676148) q^{91} + ( - 156912 \beta - 3236520) q^{93} + (5000 \beta + 2422500) q^{95} + ( - 70212 \beta + 2304770) q^{97} + ( - 93410 \beta + 1445344) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{3} + 250 q^{5} + 100 q^{7} + 5554 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{3} + 250 q^{5} + 100 q^{7} + 5554 q^{9} + 4544 q^{11} - 3540 q^{13} + 2500 q^{15} - 27340 q^{17} + 38760 q^{19} + 69096 q^{21} + 124140 q^{23} + 31250 q^{25} + 206360 q^{27} + 72260 q^{29} - 306824 q^{31} - 440960 q^{33} + 12500 q^{35} + 123020 q^{37} - 774728 q^{39} + 264364 q^{41} + 423300 q^{43} + 694250 q^{45} + 105460 q^{47} - 1165414 q^{49} + 1166344 q^{51} + 2391580 q^{53} + 568000 q^{55} + 776720 q^{57} - 1120120 q^{59} - 2257044 q^{61} + 1639620 q^{63} - 442500 q^{65} + 4516460 q^{67} + 745272 q^{69} - 621784 q^{71} + 4569060 q^{73} + 312500 q^{75} - 3177600 q^{77} - 4333040 q^{79} - 2397878 q^{81} - 9793020 q^{83} - 3417500 q^{85} + 24458920 q^{87} + 6025620 q^{89} - 5352296 q^{91} - 6473040 q^{93} + 4845000 q^{95} + 4609540 q^{97} + 2890688 q^{99}+O(q^{100}) \) 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
−4.35890
4.35890
0 −59.7424 0 125.000 0 −438.197 0 1382.15 0
1.2 0 79.7424 0 125.000 0 538.197 0 4171.85 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 320.8.a.u 2
4.b odd 2 1 320.8.a.l 2
8.b even 2 1 80.8.a.g 2
8.d odd 2 1 5.8.a.b 2
24.f even 2 1 45.8.a.h 2
40.e odd 2 1 25.8.a.b 2
40.f even 2 1 400.8.a.bb 2
40.i odd 4 2 400.8.c.m 4
40.k even 4 2 25.8.b.c 4
56.e even 2 1 245.8.a.c 2
88.g even 2 1 605.8.a.d 2
120.m even 2 1 225.8.a.w 2
120.q odd 4 2 225.8.b.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.b 2 8.d odd 2 1
25.8.a.b 2 40.e odd 2 1
25.8.b.c 4 40.k even 4 2
45.8.a.h 2 24.f even 2 1
80.8.a.g 2 8.b even 2 1
225.8.a.w 2 120.m even 2 1
225.8.b.m 4 120.q odd 4 2
245.8.a.c 2 56.e even 2 1
320.8.a.l 2 4.b odd 2 1
320.8.a.u 2 1.a even 1 1 trivial
400.8.a.bb 2 40.f even 2 1
400.8.c.m 4 40.i odd 4 2
605.8.a.d 2 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20T_{3} - 4764 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 20T - 4764 \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 100T - 235836 \) Copy content Toggle raw display
$11$ \( T^{2} - 4544 T - 6998016 \) Copy content Toggle raw display
$13$ \( T^{2} + 3540 T - 24961564 \) Copy content Toggle raw display
$17$ \( T^{2} + 27340 T + 80327844 \) Copy content Toggle raw display
$19$ \( T^{2} - 38760 T + 367802000 \) Copy content Toggle raw display
$23$ \( T^{2} - 124140 T + 3840033636 \) Copy content Toggle raw display
$29$ \( T^{2} - 72260 T - 27652933500 \) Copy content Toggle raw display
$31$ \( T^{2} + 306824 T + 22939401744 \) Copy content Toggle raw display
$37$ \( T^{2} - 123020 T - 45775154396 \) Copy content Toggle raw display
$41$ \( T^{2} - 264364 T - 227722158876 \) Copy content Toggle raw display
$43$ \( T^{2} - 423300 T - 96985991164 \) Copy content Toggle raw display
$47$ \( T^{2} - 105460 T - 154530884316 \) Copy content Toggle raw display
$53$ \( T^{2} - 2391580 T + 1213130224836 \) Copy content Toggle raw display
$59$ \( T^{2} + 1120120 T - 3614968086000 \) Copy content Toggle raw display
$61$ \( T^{2} + 2257044 T - 672038095516 \) Copy content Toggle raw display
$67$ \( T^{2} - 4516460 T + 4620664454244 \) Copy content Toggle raw display
$71$ \( T^{2} + 621784 T - 275746164336 \) Copy content Toggle raw display
$73$ \( T^{2} - 4569060 T + 1330152816836 \) Copy content Toggle raw display
$79$ \( T^{2} + 4333040 T - 12272229720000 \) Copy content Toggle raw display
$83$ \( T^{2} + 9793020 T + 5699002341636 \) Copy content Toggle raw display
$89$ \( T^{2} - 6025620 T + 1403196358500 \) Copy content Toggle raw display
$97$ \( T^{2} - 4609540 T - 18666217374716 \) Copy content Toggle raw display
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