Properties

Label 20.10.a.b
Level $20$
Weight $10$
Character orbit 20.a
Self dual yes
Analytic conductor $10.301$
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] = [20,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(\beta = 16\sqrt{79}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 130) q^{3} - 625 q^{5} + (69 \beta - 190) q^{7} + ( - 260 \beta + 17441) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 130) q^{3} - 625 q^{5} + (69 \beta - 190) q^{7} + ( - 260 \beta + 17441) q^{9} + (30 \beta + 51360) q^{11} + (564 \beta + 89570) q^{13} + ( - 625 \beta + 81250) q^{15} + (348 \beta + 158010) q^{17} + (5160 \beta + 68636) q^{19} + ( - 9160 \beta + 1420156) q^{21} + ( - 6393 \beta - 332730) q^{23} + 390625 q^{25} + (31558 \beta - 4966780) q^{27} + ( - 5880 \beta - 3446874) q^{29} + ( - 26250 \beta + 145916) q^{31} + (47460 \beta - 6070080) q^{33} + ( - 43125 \beta + 118750) q^{35} + ( - 59976 \beta + 5630690) q^{37} + (16250 \beta - 237764) q^{39} + (72180 \beta + 14886726) q^{41} + ( - 99615 \beta - 5854090) q^{43} + (162500 \beta - 10900625) q^{45} + (42573 \beta + 31246650) q^{47} + ( - 26220 \beta + 55968957) q^{49} + (112770 \beta - 13503348) q^{51} + ( - 212532 \beta + 4708890) q^{53} + ( - 18750 \beta - 32100000) q^{55} + ( - 602164 \beta + 95433160) q^{57} + ( - 373980 \beta - 46465428) q^{59} + (721440 \beta + 97836962) q^{61} + (1252829 \beta - 366132350) q^{63} + ( - 352500 \beta - 55981250) q^{65} + (203139 \beta - 109883710) q^{67} + (498360 \beta - 86037132) q^{69} + ( - 1397610 \beta + 155603508) q^{71} + ( - 2047716 \beta - 49612030) q^{73} + (390625 \beta - 50781250) q^{75} + (3538140 \beta + 32105280) q^{77} + (70020 \beta + 271130888) q^{79} + ( - 3951740 \beta + 940619189) q^{81} + (192957 \beta - 628457850) q^{83} + ( - 217500 \beta - 98756250) q^{85} + ( - 2682474 \beta + 329176500) q^{87} + (5421960 \beta - 231145926) q^{89} + (6073170 \beta + 770018884) q^{91} + (3558416 \beta - 549849080) q^{93} + ( - 3225000 \beta - 42897500) q^{95} + ( - 2902956 \beta + 835858370) q^{97} + ( - 12830370 \beta + 738022560) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 260 q^{3} - 1250 q^{5} - 380 q^{7} + 34882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 260 q^{3} - 1250 q^{5} - 380 q^{7} + 34882 q^{9} + 102720 q^{11} + 179140 q^{13} + 162500 q^{15} + 316020 q^{17} + 137272 q^{19} + 2840312 q^{21} - 665460 q^{23} + 781250 q^{25} - 9933560 q^{27} - 6893748 q^{29} + 291832 q^{31} - 12140160 q^{33} + 237500 q^{35} + 11261380 q^{37} - 475528 q^{39} + 29773452 q^{41} - 11708180 q^{43} - 21801250 q^{45} + 62493300 q^{47} + 111937914 q^{49} - 27006696 q^{51} + 9417780 q^{53} - 64200000 q^{55} + 190866320 q^{57} - 92930856 q^{59} + 195673924 q^{61} - 732264700 q^{63} - 111962500 q^{65} - 219767420 q^{67} - 172074264 q^{69} + 311207016 q^{71} - 99224060 q^{73} - 101562500 q^{75} + 64210560 q^{77} + 542261776 q^{79} + 1881238378 q^{81} - 1256915700 q^{83} - 197512500 q^{85} + 658353000 q^{87} - 462291852 q^{89} + 1540037768 q^{91} - 1099698160 q^{93} - 85795000 q^{95} + 1671716740 q^{97} + 1476045120 q^{99}+O(q^{100}) \) 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
−8.88819
8.88819
0 −272.211 0 −625.000 0 −10002.6 0 54415.9 0
1.2 0 12.2111 0 −625.000 0 9622.57 0 −19533.9 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 20.10.a.b 2
3.b odd 2 1 180.10.a.e 2
4.b odd 2 1 80.10.a.j 2
5.b even 2 1 100.10.a.c 2
5.c odd 4 2 100.10.c.c 4
8.b even 2 1 320.10.a.t 2
8.d odd 2 1 320.10.a.l 2
20.d odd 2 1 400.10.a.l 2
20.e even 4 2 400.10.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.b 2 1.a even 1 1 trivial
80.10.a.j 2 4.b odd 2 1
100.10.a.c 2 5.b even 2 1
100.10.c.c 4 5.c odd 4 2
180.10.a.e 2 3.b odd 2 1
320.10.a.l 2 8.d odd 2 1
320.10.a.t 2 8.b even 2 1
400.10.a.l 2 20.d odd 2 1
400.10.c.l 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 260T_{3} - 3324 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(20))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 260T - 3324 \) Copy content Toggle raw display
$5$ \( (T + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 380 T - 96250364 \) Copy content Toggle raw display
$11$ \( T^{2} - 102720 T + 2619648000 \) Copy content Toggle raw display
$13$ \( T^{2} - 179140 T + 1589611396 \) Copy content Toggle raw display
$17$ \( T^{2} - 316020 T + 22517952804 \) Copy content Toggle raw display
$19$ \( T^{2} - 137272 T - 533765233904 \) Copy content Toggle raw display
$23$ \( T^{2} + 665460 T - 715854707676 \) Copy content Toggle raw display
$29$ \( T^{2} + 6893748 T + 11181707706276 \) Copy content Toggle raw display
$31$ \( T^{2} - 291832 T - 13914308520944 \) Copy content Toggle raw display
$37$ \( T^{2} - 11261380 T - 41043496652924 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 116248533661476 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 166415379974300 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 939697938528804 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 891341422077276 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 669513681826416 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 954028889496956 \) Copy content Toggle raw display
$67$ \( T^{2} + 219767420 T + 11\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} - 311207016 T - 15\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + 99224060 T - 82\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{2} - 542261776 T + 73\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} + 1256915700 T + 39\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + 462291852 T - 54\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{2} - 1671716740 T + 52\!\cdots\!36 \) Copy content Toggle raw display
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