Properties

Label 10.18.a.d
Level $10$
Weight $18$
Character orbit 10.a
Self dual yes
Analytic conductor $18.322$
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] = [10,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
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: \(18.3222087345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2941}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 735 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 5 \)
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 \(\beta = 240\sqrt{2941}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 256 q^{2} + ( - \beta + 8814) q^{3} + 65536 q^{4} - 390625 q^{5} + ( - 256 \beta + 2256384) q^{6} + ( - 117 \beta + 13842098) q^{7} + 16777216 q^{8} + ( - 17628 \beta + 117948033) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{2} + ( - \beta + 8814) q^{3} + 65536 q^{4} - 390625 q^{5} + ( - 256 \beta + 2256384) q^{6} + ( - 117 \beta + 13842098) q^{7} + 16777216 q^{8} + ( - 17628 \beta + 117948033) q^{9} - 100000000 q^{10} + (90882 \beta + 32257632) q^{11} + ( - 65536 \beta + 577634304) q^{12} + (74988 \beta + 1447919234) q^{13} + ( - 29952 \beta + 3543577088) q^{14} + (390625 \beta - 3442968750) q^{15} + 4294967296 q^{16} + ( - 1394748 \beta + 790106298) q^{17} + ( - 4512768 \beta + 30194696448) q^{18} + (7161912 \beta + 22106856380) q^{19} - 25600000000 q^{20} + ( - 14873336 \beta + 141824238972) q^{21} + (23265792 \beta + 8257953792) q^{22} + (23366889 \beta + 243774891414) q^{23} + ( - 16777216 \beta + 147874381824) q^{24} + 152587890625 q^{25} + (19196928 \beta + 370667323904) q^{26} + ( - 144181062 \beta + 2887563970980) q^{27} + ( - 7667712 \beta + 907155734528) q^{28} + (2589624 \beta + 1993657431750) q^{29} + (100000000 \beta - 881400000000) q^{30} + ( - 48426966 \beta - 2746130669668) q^{31} + 1099511627776 q^{32} + (768776316 \beta - 15111237442752) q^{33} + ( - 357055488 \beta + 202267212288) q^{34} + (45703125 \beta - 5407069531250) q^{35} + ( - 1155268608 \beta + 7729842290688) q^{36} + (503271432 \beta - 31357643818942) q^{37} + (1833449472 \beta + 5659355233280) q^{38} + ( - 786975002 \beta + 58872947676) q^{39} - 6553600000000 q^{40} + ( - 5025537684 \beta + 11705738638662) q^{41} + ( - 3807574016 \beta + 36307005176832) q^{42} + (2880330687 \beta - 62428461595546) q^{43} + (5956042752 \beta + 2114036170752) q^{44} + (6885937500 \beta - 46073450390625) q^{45} + (5981923584 \beta + 62406372201984) q^{46} + ( - 4923046557 \beta - 92973306061782) q^{47} + ( - 4294967296 \beta + 37855841746944) q^{48} + ( - 3239050932 \beta - 38707898443203) q^{49} + 39062500000000 q^{50} + ( - 13083415170 \beta + 243236539707372) q^{51} + (4914413568 \beta + 94890834919424) q^{52} + (25138319508 \beta + 179669890323834) q^{53} + ( - 36910351872 \beta + 739216376570880) q^{54} + ( - 35500781250 \beta - 12600637500000) q^{55} + ( - 1962934272 \beta + 232231868039168) q^{56} + (41018235988 \beta - 10\!\cdots\!80) q^{57}+ \cdots + (10150715598210 \beta - 26\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 512 q^{2} + 17628 q^{3} + 131072 q^{4} - 781250 q^{5} + 4512768 q^{6} + 27684196 q^{7} + 33554432 q^{8} + 235896066 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 512 q^{2} + 17628 q^{3} + 131072 q^{4} - 781250 q^{5} + 4512768 q^{6} + 27684196 q^{7} + 33554432 q^{8} + 235896066 q^{9} - 200000000 q^{10} + 64515264 q^{11} + 1155268608 q^{12} + 2895838468 q^{13} + 7087154176 q^{14} - 6885937500 q^{15} + 8589934592 q^{16} + 1580212596 q^{17} + 60389392896 q^{18} + 44213712760 q^{19} - 51200000000 q^{20} + 283648477944 q^{21} + 16515907584 q^{22} + 487549782828 q^{23} + 295748763648 q^{24} + 305175781250 q^{25} + 741334647808 q^{26} + 5775127941960 q^{27} + 1814311469056 q^{28} + 3987314863500 q^{29} - 1762800000000 q^{30} - 5492261339336 q^{31} + 2199023255552 q^{32} - 30222474885504 q^{33} + 404534424576 q^{34} - 10814139062500 q^{35} + 15459684581376 q^{36} - 62715287637884 q^{37} + 11318710466560 q^{38} + 117745895352 q^{39} - 13107200000000 q^{40} + 23411477277324 q^{41} + 72614010353664 q^{42} - 124856923191092 q^{43} + 4228072341504 q^{44} - 92146900781250 q^{45} + 124812744403968 q^{46} - 185946612123564 q^{47} + 75711683493888 q^{48} - 77415796886406 q^{49} + 78125000000000 q^{50} + 486473079414744 q^{51} + 189781669838848 q^{52} + 359339780647668 q^{53} + 14\!\cdots\!60 q^{54}+ \cdots - 53\!\cdots\!88 q^{99}+O(q^{100}) \) 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
27.6155
−26.6155
256.000 −4201.44 65536.0 −390625. −1.07557e6 1.23193e7 1.67772e7 −1.11488e8 −1.00000e8
1.2 256.000 21829.4 65536.0 −390625. 5.58834e6 1.53649e7 1.67772e7 3.47384e8 −1.00000e8
\(n\): 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 10.18.a.d 2
3.b odd 2 1 90.18.a.j 2
4.b odd 2 1 80.18.a.b 2
5.b even 2 1 50.18.a.c 2
5.c odd 4 2 50.18.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.d 2 1.a even 1 1 trivial
50.18.a.c 2 5.b even 2 1
50.18.b.f 4 5.c odd 4 2
80.18.a.b 2 4.b odd 2 1
90.18.a.j 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 17628T_{3} - 91715004 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 17628 T - 91715004 \) Copy content Toggle raw display
$5$ \( (T + 390625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 189284738539204 \) Copy content Toggle raw display
$11$ \( T^{2} - 64515264 T - 13\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{2} - 2895838468 T + 11\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{2} - 1580212596 T - 32\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} - 44213712760 T - 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} - 487549782828 T - 33\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{2} - 3987314863500 T + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + 5492261339336 T + 71\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} + 62715287637884 T + 94\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{2} - 23411477277324 T - 41\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + 124856923191092 T + 24\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{2} + 185946612123564 T + 45\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} - 359339780647668 T - 74\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{2} - 902179170360600 T - 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 87\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 83\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + 67588560434136 T - 91\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 46\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 90\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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