Properties

Label 80.10.a.f
Level $80$
Weight $10$
Character orbit 80.a
Self dual yes
Analytic conductor $41.203$
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] = [80,10,Mod(1,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, 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("80.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(41.2028668931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{1009}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 130) q^{3} + 625 q^{5} + ( - 107 \beta - 850) q^{7} + (260 \beta + 1253) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 130) q^{3} + 625 q^{5} + ( - 107 \beta - 850) q^{7} + (260 \beta + 1253) q^{9} + (950 \beta - 11992) q^{11} + (676 \beta + 57510) q^{13} + ( - 625 \beta - 81250) q^{15} + ( - 6428 \beta + 206410) q^{17} + ( - 1420 \beta + 148260) q^{19} + (14760 \beta + 542352) q^{21} + (9699 \beta + 524610) q^{23} + 390625 q^{25} + ( - 15370 \beta + 1346540) q^{27} + (53480 \beta - 1833490) q^{29} + (77350 \beta - 806572) q^{31} + ( - 111508 \beta - 2275240) q^{33} + ( - 66875 \beta - 531250) q^{35} + ( - 102648 \beta - 10560970) q^{37} + ( - 145390 \beta - 10204636) q^{39} + (77900 \beta - 13478638) q^{41} + (12899 \beta - 26444850) q^{43} + (162500 \beta + 783125) q^{45} + ( - 261667 \beta - 29206090) q^{47} + (181900 \beta + 6577057) q^{49} + (629230 \beta - 889892) q^{51} + ( - 568724 \beta - 19517570) q^{53} + (593750 \beta - 7495000) q^{55} + (36340 \beta - 13542680) q^{57} + ( - 772360 \beta + 27497780) q^{59} + (346000 \beta - 137289858) q^{61} + ( - 355071 \beta - 113346570) q^{63} + (422500 \beta + 35943750) q^{65} + (1208353 \beta + 159290) q^{67} + ( - 1785480 \beta - 107344464) q^{69} + ( - 3139250 \beta + 3565468) q^{71} + (4415476 \beta + 60429090) q^{73} + ( - 390625 \beta - 50781250) q^{75} + (475644 \beta - 400066200) q^{77} + (9387820 \beta - 3438760) q^{79} + ( - 4466020 \beta - 137679679) q^{81} + ( - 1374201 \beta - 701174370) q^{83} + ( - 4017500 \beta + 129006250) q^{85} + ( - 5118910 \beta + 22508420) q^{87} + (6690840 \beta + 415044330) q^{89} + ( - 6728170 \beta - 340815452) q^{91} + ( - 9248928 \beta - 207330240) q^{93} + ( - 887500 \beta + 92662500) q^{95} + ( - 1311108 \beta + 319197290) q^{97} + ( - 1927570 \beta + 981866024) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9} - 23984 q^{11} + 115020 q^{13} - 162500 q^{15} + 412820 q^{17} + 296520 q^{19} + 1084704 q^{21} + 1049220 q^{23} + 781250 q^{25} + 2693080 q^{27} - 3666980 q^{29} - 1613144 q^{31} - 4550480 q^{33} - 1062500 q^{35} - 21121940 q^{37} - 20409272 q^{39} - 26957276 q^{41} - 52889700 q^{43} + 1566250 q^{45} - 58412180 q^{47} + 13154114 q^{49} - 1779784 q^{51} - 39035140 q^{53} - 14990000 q^{55} - 27085360 q^{57} + 54995560 q^{59} - 274579716 q^{61} - 226693140 q^{63} + 71887500 q^{65} + 318580 q^{67} - 214688928 q^{69} + 7130936 q^{71} + 120858180 q^{73} - 101562500 q^{75} - 800132400 q^{77} - 6877520 q^{79} - 275359358 q^{81} - 1402348740 q^{83} + 258012500 q^{85} + 45016840 q^{87} + 830088660 q^{89} - 681630904 q^{91} - 414660480 q^{93} + 185325000 q^{95} + 638394580 q^{97} + 1963732048 q^{99}+O(q^{100}) \) 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
16.3824
−15.3824
0 −193.530 0 625.000 0 −7647.66 0 17770.7 0
1.2 0 −66.4705 0 625.000 0 5947.66 0 −15264.7 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 80.10.a.f 2
4.b odd 2 1 5.10.a.b 2
5.b even 2 1 400.10.a.t 2
5.c odd 4 2 400.10.c.p 4
8.b even 2 1 320.10.a.s 2
8.d odd 2 1 320.10.a.k 2
12.b even 2 1 45.10.a.f 2
20.d odd 2 1 25.10.a.b 2
20.e even 4 2 25.10.b.b 4
28.d even 2 1 245.10.a.d 2
60.h even 2 1 225.10.a.h 2
60.l odd 4 2 225.10.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.b 2 4.b odd 2 1
25.10.a.b 2 20.d odd 2 1
25.10.b.b 4 20.e even 4 2
45.10.a.f 2 12.b even 2 1
80.10.a.f 2 1.a even 1 1 trivial
225.10.a.h 2 60.h even 2 1
225.10.b.h 4 60.l odd 4 2
245.10.a.d 2 28.d even 2 1
320.10.a.k 2 8.d odd 2 1
320.10.a.s 2 8.b even 2 1
400.10.a.t 2 5.b even 2 1
400.10.c.p 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 260T_{3} + 12864 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(80))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 260T + 12864 \) Copy content Toggle raw display
$5$ \( (T - 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1700 T - 45485664 \) Copy content Toggle raw display
$11$ \( T^{2} + 23984 T - 3498681936 \) Copy content Toggle raw display
$13$ \( T^{2} - 115020 T + 1463044964 \) Copy content Toggle raw display
$17$ \( T^{2} - 412820 T - 124159138524 \) Copy content Toggle raw display
$19$ \( T^{2} - 296520 T + 13842837200 \) Copy content Toggle raw display
$23$ \( T^{2} - 1049220 T - 104453293536 \) Copy content Toggle raw display
$29$ \( T^{2} + 3666980 T - 8181719994300 \) Copy content Toggle raw display
$31$ \( T^{2} + 1613144 T - 23496920418816 \) Copy content Toggle raw display
$37$ \( T^{2} + 21121940 T + 69008321696356 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 157181579575044 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 698658564887264 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 576652311252096 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 924496505573436 \) Copy content Toggle raw display
$59$ \( T^{2} - 54995560 T - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 274579716 T + 18\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} - 318580 T - 58\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} - 7130936 T - 39\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} - 120858180 T - 75\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} + 6877520 T - 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 1402348740 T + 48\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} - 830088660 T - 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} - 638394580 T + 94\!\cdots\!96 \) Copy content Toggle raw display
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