Properties

Label 10.20.a.d
Level $10$
Weight $20$
Character orbit 10.a
Self dual yes
Analytic conductor $22.882$
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] = [10,20,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(22.8816696556\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2925852 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
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 \(\beta = 10\sqrt{11703409}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 512 q^{2} + ( - \beta + 16862) q^{3} + 262144 q^{4} + 1953125 q^{5} + ( - 512 \beta + 8633344) q^{6} + (417 \beta + 41530646) q^{7} + 134217728 q^{8} + ( - 33724 \beta + 292406477) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 512 q^{2} + ( - \beta + 16862) q^{3} + 262144 q^{4} + 1953125 q^{5} + ( - 512 \beta + 8633344) q^{6} + (417 \beta + 41530646) q^{7} + 134217728 q^{8} + ( - 33724 \beta + 292406477) q^{9} + 1000000000 q^{10} + ( - 208506 \beta + 1774740072) q^{11} + ( - 262144 \beta + 4420272128) q^{12} + (1579188 \beta + 15125112782) q^{13} + (213504 \beta + 21263690752) q^{14} + ( - 1953125 \beta + 32933593750) q^{15} + 68719476736 q^{16} + (19214148 \beta + 164827889706) q^{17} + ( - 17266688 \beta + 149712116224) q^{18} + ( - 14948364 \beta + 191538404900) q^{19} + 512000000000 q^{20} + ( - 34499192 \beta + 212257597552) q^{21} + ( - 106755072 \beta + 908666916864) q^{22} + (203872359 \beta + 1041428048442) q^{23} + ( - 134217728 \beta + 2263179329536) q^{24} + 3814697265625 q^{25} + (808544256 \beta + 7744057744384) q^{26} + (301200902 \beta + 24801081670220) q^{27} + (109314048 \beta + 10887009665024) q^{28} + ( - 1729970808 \beta + 83293842301110) q^{29} + ( - 1000000000 \beta + 16862000000000) q^{30} + ( - 38480322 \beta + 209075559519332) q^{31} + 35184372088832 q^{32} + ( - 5290568244 \beta + 273948766789464) q^{33} + (9837643776 \beta + 84391879529472) q^{34} + (814453125 \beta + 81114542968750) q^{35} + ( - 8840544256 \beta + 76652603506688) q^{36} + (10386828648 \beta - 646109818847314) q^{37} + ( - 7653562368 \beta + 98067663308800) q^{38} + (11503155274 \beta - 15\!\cdots\!16) q^{39}+ \cdots + ( - 120819839081490 \beta + 87\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1024 q^{2} + 33724 q^{3} + 524288 q^{4} + 3906250 q^{5} + 17266688 q^{6} + 83061292 q^{7} + 268435456 q^{8} + 584812954 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1024 q^{2} + 33724 q^{3} + 524288 q^{4} + 3906250 q^{5} + 17266688 q^{6} + 83061292 q^{7} + 268435456 q^{8} + 584812954 q^{9} + 2000000000 q^{10} + 3549480144 q^{11} + 8840544256 q^{12} + 30250225564 q^{13} + 42527381504 q^{14} + 65867187500 q^{15} + 137438953472 q^{16} + 329655779412 q^{17} + 299424232448 q^{18} + 383076809800 q^{19} + 1024000000000 q^{20} + 424515195104 q^{21} + 1817333833728 q^{22} + 2082856096884 q^{23} + 4526358659072 q^{24} + 7629394531250 q^{25} + 15488115488768 q^{26} + 49602163340440 q^{27} + 21774019330048 q^{28} + 166587684602220 q^{29} + 33724000000000 q^{30} + 418151119038664 q^{31} + 70368744177664 q^{32} + 547897533578928 q^{33} + 168783759058944 q^{34} + 162229085937500 q^{35} + 153305207013376 q^{36} - 12\!\cdots\!28 q^{37}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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
1711.01
−1710.01
512.000 −17348.2 262144. 1.95312e6 −8.88230e6 5.57963e7 1.34218e8 −8.61300e8 1.00000e9
1.2 512.000 51072.2 262144. 1.95312e6 2.61490e7 2.72650e7 1.34218e8 1.44611e9 1.00000e9
\(n\): 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 10.20.a.d 2
3.b odd 2 1 90.20.a.g 2
4.b odd 2 1 80.20.a.d 2
5.b even 2 1 50.20.a.f 2
5.c odd 4 2 50.20.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.20.a.d 2 1.a even 1 1 trivial
50.20.a.f 2 5.b even 2 1
50.20.b.f 4 5.c odd 4 2
80.20.a.d 2 4.b odd 2 1
90.20.a.g 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 33724T_{3} - 886013856 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 512)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 33724 T - 886013856 \) Copy content Toggle raw display
$5$ \( (T - 1953125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 47\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 40\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 47\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 39\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 21\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 50\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 52\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
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