Properties

Label 20.8.a.b
Level $20$
Weight $8$
Character orbit 20.a
Self dual yes
Analytic conductor $6.248$
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] = [20,8,Mod(1,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("20.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(6.24770050968\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1129}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 10) q^{3} + 125 q^{5} + (9 \beta + 830) q^{7} + (20 \beta + 2429) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 10) q^{3} + 125 q^{5} + (9 \beta + 830) q^{7} + (20 \beta + 2429) q^{9} + ( - 90 \beta + 1800) q^{11} + (36 \beta + 6590) q^{13} + ( - 125 \beta - 1250) q^{15} + (468 \beta + 2730) q^{17} + (180 \beta - 20236) q^{19} + ( - 920 \beta - 48944) q^{21} + (63 \beta - 20910) q^{23} + 15625 q^{25} + ( - 442 \beta - 92740) q^{27} + (360 \beta + 59334) q^{29} + (3150 \beta - 57964) q^{31} + ( - 900 \beta + 388440) q^{33} + (1125 \beta + 103750) q^{35} + ( - 4536 \beta + 153470) q^{37} + ( - 6950 \beta - 228476) q^{39} + (4860 \beta - 176574) q^{41} + (3195 \beta + 607670) q^{43} + (2500 \beta + 303625) q^{45} + (153 \beta - 1034250) q^{47} + (14940 \beta + 231153) q^{49} + ( - 7410 \beta - 2140788) q^{51} + ( - 13428 \beta + 700230) q^{53} + ( - 11250 \beta + 225000) q^{55} + (18436 \beta - 610520) q^{57} + ( - 10440 \beta - 996252) q^{59} + ( - 36720 \beta - 839338) q^{61} + (38461 \beta + 2828950) q^{63} + (4500 \beta + 823750) q^{65} + (25569 \beta + 1831970) q^{67} + (20280 \beta - 75408) q^{69} + (18630 \beta + 897468) q^{71} + ( - 44604 \beta + 2531090) q^{73} + ( - 15625 \beta - 156250) q^{75} + ( - 58500 \beta - 2163960) q^{77} + ( - 38340 \beta + 5089112) q^{79} + (53420 \beta - 2388751) q^{81} + (1143 \beta - 3607050) q^{83} + (58500 \beta + 341250) q^{85} + ( - 62934 \beta - 2219100) q^{87} + (44280 \beta - 7665414) q^{89} + (89190 \beta + 6932884) q^{91} + (26464 \beta - 13645760) q^{93} + (22500 \beta - 2529500) q^{95} + (2124 \beta + 7012010) q^{97} + ( - 182610 \beta - 3756600) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{3} + 250 q^{5} + 1660 q^{7} + 4858 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 20 q^{3} + 250 q^{5} + 1660 q^{7} + 4858 q^{9} + 3600 q^{11} + 13180 q^{13} - 2500 q^{15} + 5460 q^{17} - 40472 q^{19} - 97888 q^{21} - 41820 q^{23} + 31250 q^{25} - 185480 q^{27} + 118668 q^{29} - 115928 q^{31} + 776880 q^{33} + 207500 q^{35} + 306940 q^{37} - 456952 q^{39} - 353148 q^{41} + 1215340 q^{43} + 607250 q^{45} - 2068500 q^{47} + 462306 q^{49} - 4281576 q^{51} + 1400460 q^{53} + 450000 q^{55} - 1221040 q^{57} - 1992504 q^{59} - 1678676 q^{61} + 5657900 q^{63} + 1647500 q^{65} + 3663940 q^{67} - 150816 q^{69} + 1794936 q^{71} + 5062180 q^{73} - 312500 q^{75} - 4327920 q^{77} + 10178224 q^{79} - 4777502 q^{81} - 7214100 q^{83} + 682500 q^{85} - 4438200 q^{87} - 15330828 q^{89} + 13865768 q^{91} - 27291520 q^{93} - 5059000 q^{95} + 14024020 q^{97} - 7513200 q^{99}+O(q^{100}) \) 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
17.3003
−16.3003
0 −77.2012 0 125.000 0 1434.81 0 3773.02 0
1.2 0 57.2012 0 125.000 0 225.189 0 1084.98 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 20.8.a.b 2
3.b odd 2 1 180.8.a.g 2
4.b odd 2 1 80.8.a.i 2
5.b even 2 1 100.8.a.c 2
5.c odd 4 2 100.8.c.b 4
8.b even 2 1 320.8.a.t 2
8.d odd 2 1 320.8.a.k 2
20.d odd 2 1 400.8.a.w 2
20.e even 4 2 400.8.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.8.a.b 2 1.a even 1 1 trivial
80.8.a.i 2 4.b odd 2 1
100.8.a.c 2 5.b even 2 1
100.8.c.b 4 5.c odd 4 2
180.8.a.g 2 3.b odd 2 1
320.8.a.k 2 8.d odd 2 1
320.8.a.t 2 8.b even 2 1
400.8.a.w 2 20.d odd 2 1
400.8.c.n 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 20T_{3} - 4416 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(20))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20T - 4416 \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1660 T + 323104 \) Copy content Toggle raw display
$11$ \( T^{2} - 3600 T - 33339600 \) Copy content Toggle raw display
$13$ \( T^{2} - 13180 T + 37575364 \) Copy content Toggle raw display
$17$ \( T^{2} - 5460 T - 981659484 \) Copy content Toggle raw display
$19$ \( T^{2} + 40472 T + 263177296 \) Copy content Toggle raw display
$23$ \( T^{2} + 41820 T + 419304096 \) Copy content Toggle raw display
$29$ \( T^{2} - 118668 T + 2935249956 \) Copy content Toggle raw display
$31$ \( T^{2} + 115928 T - 41450184704 \) Copy content Toggle raw display
$37$ \( T^{2} - 306940 T - 69364995836 \) Copy content Toggle raw display
$41$ \( T^{2} + 353148 T - 75487736124 \) Copy content Toggle raw display
$43$ \( T^{2} - 1215340 T + 323163388000 \) Copy content Toggle raw display
$47$ \( T^{2} + 2068500 T + 1069567347456 \) Copy content Toggle raw display
$53$ \( T^{2} - 1400460 T - 323963254044 \) Copy content Toggle raw display
$59$ \( T^{2} + 1992504 T + 500302949904 \) Copy content Toggle raw display
$61$ \( T^{2} + 1678676 T - 5384698256156 \) Copy content Toggle raw display
$67$ \( T^{2} - 3663940 T + 403671776224 \) Copy content Toggle raw display
$71$ \( T^{2} - 1794936 T - 761950469376 \) Copy content Toggle raw display
$73$ \( T^{2} - 5062180 T - 2578241352956 \) Copy content Toggle raw display
$79$ \( T^{2} - 10178224 T + 19260741458944 \) Copy content Toggle raw display
$83$ \( T^{2} + 7214100 T + 13004909778816 \) Copy content Toggle raw display
$89$ \( T^{2} + 15330828 T + 49903967496996 \) Copy content Toggle raw display
$97$ \( T^{2} - 14024020 T + 49147910866084 \) Copy content Toggle raw display
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