Properties

Label 5.18.a.b
Level $5$
Weight $18$
Character orbit 5.a
Self dual yes
Analytic conductor $9.161$
Analytic rank $0$
Dimension $3$
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] = [5,18,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, 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("5.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(9.16110436723\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 50686x + 2014936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 39) q^{2} + (\beta_{2} + 8 \beta_1 + 5317) q^{3} + (6 \beta_{2} - 194 \beta_1 + 5572) q^{4} + 390625 q^{5} + ( - 124 \beta_{2} - 13124 \beta_1 - 810688) q^{6} + ( - 261 \beta_{2} - 25832 \beta_1 + 704579) q^{7} + (708 \beta_{2} + 41148 \beta_1 + 21696960) q^{8} + (3812 \beta_{2} + 385312 \beta_1 + 108091313) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 39) q^{2} + (\beta_{2} + 8 \beta_1 + 5317) q^{3} + (6 \beta_{2} - 194 \beta_1 + 5572) q^{4} + 390625 q^{5} + ( - 124 \beta_{2} - 13124 \beta_1 - 810688) q^{6} + ( - 261 \beta_{2} - 25832 \beta_1 + 704579) q^{7} + (708 \beta_{2} + 41148 \beta_1 + 21696960) q^{8} + (3812 \beta_{2} + 385312 \beta_1 + 108091313) q^{9} + ( - 390625 \beta_1 + 15234375) q^{10} + ( - 17430 \beta_{2} + 891920 \beta_1 + 596885622) q^{11} + ( - 42904 \beta_{2} - 1150280 \beta_1 + 1037023904) q^{12} + (46020 \beta_{2} - 4427488 \beta_1 + 1803407762) q^{13} + (174828 \beta_{2} - 2347272 \beta_1 + 3501550404) q^{14} + (390625 \beta_{2} + 3125000 \beta_1 + 2076953125) q^{15} + ( - 1087128 \beta_{2} + 3703672 \beta_1 - 5399636384) q^{16} + ( - 165948 \beta_{2} + 76541216 \beta_1 - 9108532266) q^{17} + ( - 2601584 \beta_{2} - 82855117 \beta_1 - 47609051573) q^{18} + (6888684 \beta_{2} + 99735584 \beta_1 - 9954572800) q^{19} + (2343750 \beta_{2} - 75781250 \beta_1 + 2176562500) q^{20} + ( - 2772888 \beta_{2} - 402852288 \beta_1 - 74965174008) q^{21} + ( - 4026840 \beta_{2} - 300948812 \beta_1 - 98337289092) q^{22} + ( - 27034611 \beta_{2} + 785797416 \beta_1 + 5688969957) q^{23} + (26415312 \beta_{2} + 893024112 \beta_1 + 299431636800) q^{24} + 152587890625 q^{25} + (23067408 \beta_{2} - 2906011342 \beta_1 + 671484540402) q^{26} + (31049098 \beta_{2} + 5905579088 \beta_1 + 1045863330610) q^{27} + (35006496 \beta_{2} - 2061194576 \beta_1 + 372382772048) q^{28} + ( - 175959384 \beta_{2} - 1805173184 \beta_1 - 843302681850) q^{29} + ( - 48437500 \beta_{2} - 5126562500 \beta_1 - 316675000000) q^{30} + ( - 20697990 \beta_{2} - 13226283440 \beta_1 - 178153880038) q^{31} + ( - 32399280 \beta_{2} + 10415601904 \beta_1 - 3623317258176) q^{32} + (791229452 \beta_{2} + 12657852256 \beta_1 + 581427928124) q^{33} + ( - 446635248 \beta_{2} + 22473752302 \beta_1 - 10708155093426) q^{34} + ( - 101953125 \beta_{2} - 10090625000 \beta_1 + 275226171875) q^{35} + (195204622 \beta_{2} + 7799424422 \beta_1 - 4992591100364) q^{36} + ( - 388153272 \beta_{2} - 3949553088 \beta_1 + 10586395166894) q^{37} + ( - 1121953488 \beta_{2} + \cdots - 13431274105860) q^{38}+ \cdots + (2673007356274 \beta_{2} + \cdots + 97\!\cdots\!86) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 118 q^{2} + 15944 q^{3} + 16916 q^{4} + 1171875 q^{5} - 2419064 q^{6} + 2139308 q^{7} + 65050440 q^{8} + 323892439 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 118 q^{2} + 15944 q^{3} + 16916 q^{4} + 1171875 q^{5} - 2419064 q^{6} + 2139308 q^{7} + 65050440 q^{8} + 323892439 q^{9} + 46093750 q^{10} + 1789747516 q^{11} + 3112179088 q^{12} + 5414696794 q^{13} + 10507173312 q^{14} + 6228125000 q^{15} - 16203699952 q^{16} - 27402303962 q^{17} - 142746901186 q^{18} - 29956565300 q^{19} + 6607812500 q^{20} - 224495442624 q^{21} - 294714945304 q^{22} + 16254077844 q^{23} + 897428301600 q^{24} + 457763671875 q^{25} + 2017382699956 q^{26} + 3131715461840 q^{27} + 1119244517216 q^{28} - 2528278831750 q^{29} - 944946875000 q^{30} - 521256054664 q^{31} - 10880399775712 q^{32} + 1732417161568 q^{33} - 32147385667828 q^{34} + 835667187500 q^{35} - 14985377520892 q^{36} + 31762746900498 q^{37} - 40258035935240 q^{38} + 43003853320688 q^{39} + 25410328125000 q^{40} + 86833482954446 q^{41} + 153974403759936 q^{42} + 89258046385744 q^{43} - 124942202946448 q^{44} + 126520483984375 q^{45} - 323339762673024 q^{46} + 348182738140228 q^{47} - 729510516165056 q^{48} - 387320833396229 q^{49} + 18005371093750 q^{50} - 12409602773744 q^{51} + 552556858385688 q^{52} + 44014499212594 q^{53} - 22\!\cdots\!00 q^{54}+ \cdots + 29\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 50686x + 2014936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} + 114\nu - 67619 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} - 57\beta _1 + 67562 ) / 2 \) 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
202.308
41.0886
−242.397
−364.616 20979.7 1872.98 390625. −7.64953e6 −1.29668e7 4.71081e7 3.11007e8 −1.42428e8
1.2 −42.1772 −13886.4 −129293. 390625. 585688. 3.78919e6 1.09815e7 6.36910e7 −1.64755e7
1.3 524.793 8850.68 144336. 390625. 4.64478e6 1.13170e7 6.96092e6 −5.08056e7 2.04997e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 5.18.a.b 3
3.b odd 2 1 45.18.a.c 3
4.b odd 2 1 80.18.a.g 3
5.b even 2 1 25.18.a.c 3
5.c odd 4 2 25.18.b.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.18.a.b 3 1.a even 1 1 trivial
25.18.a.c 3 5.b even 2 1
25.18.b.c 6 5.c odd 4 2
45.18.a.c 3 3.b odd 2 1
80.18.a.g 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 118T_{2}^{2} - 198104T_{2} - 8070528 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 118 T^{2} - 198104 T - 8070528 \) Copy content Toggle raw display
$3$ \( T^{3} - 15944 T^{2} + \cdots + 2578483943424 \) Copy content Toggle raw display
$5$ \( (T - 390625)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2139308 T^{2} + \cdots + 55\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{3} - 1789747516 T^{2} + \cdots - 25\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{3} - 5414696794 T^{2} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{3} + 27402303962 T^{2} + \cdots + 37\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{3} + 29956565300 T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 16254077844 T^{2} + \cdots - 46\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{3} + 2528278831750 T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + 521256054664 T^{2} + \cdots - 29\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{3} - 31762746900498 T^{2} + \cdots - 72\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{3} - 86833482954446 T^{2} + \cdots + 65\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{3} - 89258046385744 T^{2} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{3} - 348182738140228 T^{2} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{3} - 44014499212594 T^{2} + \cdots + 57\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{3} - 374173462156688 T^{2} + \cdots - 20\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 48\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 54\!\cdots\!32 \) Copy content Toggle raw display
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