Properties

Label 245.4.a.m
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{3} + 7) q^{4} - 5 q^{5} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 3) q^{6}+ \cdots + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{3} + 7) q^{4} - 5 q^{5} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 3) q^{6}+ \cdots + ( - 15 \beta_{4} + \beta_{3} + \cdots + 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9} - 5 q^{10} + 47 q^{11} - 98 q^{12} + q^{13} + 40 q^{15} + 171 q^{16} - 2 q^{17} - 51 q^{18} - 21 q^{19} - 175 q^{20} + 523 q^{22} + 201 q^{23} + 848 q^{24} + 125 q^{25} - 47 q^{26} - 518 q^{27} + 190 q^{29} - 80 q^{30} + 388 q^{31} - 95 q^{32} - 262 q^{33} + 130 q^{34} + 1229 q^{36} - 145 q^{37} + 835 q^{38} + 14 q^{39} - 165 q^{40} - 281 q^{41} + 568 q^{43} + 1091 q^{44} - 405 q^{45} + 337 q^{46} - 473 q^{47} + 70 q^{48} + 25 q^{50} + 732 q^{51} - 379 q^{52} + 351 q^{53} - 774 q^{54} - 235 q^{55} + 954 q^{57} + 1818 q^{58} + 708 q^{59} + 490 q^{60} + 1944 q^{61} + 448 q^{62} - 125 q^{64} - 5 q^{65} + 1482 q^{66} + 1118 q^{67} - 3118 q^{68} + 374 q^{69} + 864 q^{71} - 2219 q^{72} - 1652 q^{73} - 3285 q^{74} - 200 q^{75} + 691 q^{76} - 5574 q^{78} + 218 q^{79} - 855 q^{80} - 455 q^{81} + 1027 q^{82} - 1502 q^{83} + 10 q^{85} - 4264 q^{86} + 390 q^{87} + 2131 q^{88} + 2322 q^{89} + 255 q^{90} - 2957 q^{92} - 2288 q^{93} - 2677 q^{94} + 105 q^{95} + 4592 q^{96} + 598 q^{97} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 21\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 29\nu^{2} + 21\nu + 112 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} + 21\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 29\beta_{3} + 4\beta_{2} + 327 \) 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
−5.02529
−2.79706
−0.227497
3.84623
5.20362
−5.02529 −8.34382 17.2535 −5.00000 41.9301 0 −46.5017 42.6193 25.1265
1.2 −2.79706 6.21383 −0.176445 −5.00000 −17.3805 0 22.8700 11.6117 13.9853
1.3 −0.227497 −1.80858 −7.94824 −5.00000 0.411448 0 3.62818 −23.7290 1.13749
1.4 3.84623 −8.96795 6.79345 −5.00000 −34.4927 0 −4.64067 53.4241 −19.2311
1.5 5.20362 4.90652 19.0777 −5.00000 25.5317 0 57.6442 −2.92609 −26.0181
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( +1 \)

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 245.4.a.m 5
3.b odd 2 1 2205.4.a.bu 5
5.b even 2 1 1225.4.a.bg 5
7.b odd 2 1 245.4.a.n 5
7.c even 3 2 35.4.e.c 10
7.d odd 6 2 245.4.e.o 10
21.c even 2 1 2205.4.a.bt 5
21.h odd 6 2 315.4.j.g 10
28.g odd 6 2 560.4.q.n 10
35.c odd 2 1 1225.4.a.bf 5
35.j even 6 2 175.4.e.d 10
35.l odd 12 4 175.4.k.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.c 10 7.c even 3 2
175.4.e.d 10 35.j even 6 2
175.4.k.d 20 35.l odd 12 4
245.4.a.m 5 1.a even 1 1 trivial
245.4.a.n 5 7.b odd 2 1
245.4.e.o 10 7.d odd 6 2
315.4.j.g 10 21.h odd 6 2
560.4.q.n 10 28.g odd 6 2
1225.4.a.bf 5 35.c odd 2 1
1225.4.a.bg 5 5.b even 2 1
2205.4.a.bt 5 21.c even 2 1
2205.4.a.bu 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{5} - T_{2}^{4} - 37T_{2}^{3} + 21T_{2}^{2} + 288T_{2} + 64 \) Copy content Toggle raw display
\( T_{3}^{5} + 8T_{3}^{4} - 76T_{3}^{3} - 462T_{3}^{2} + 1731T_{3} + 4126 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{5} + 8 T^{4} + \cdots + 4126 \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 47 T^{4} + \cdots - 1022160 \) Copy content Toggle raw display
$13$ \( T^{5} - T^{4} + \cdots + 307317696 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} + \cdots - 10657408 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 1626396544 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 11116980717 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 13190815450 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 2519500032 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 111213849600 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 77000100765 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 72928266842 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 376942584320 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 15657780928 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 9224535040 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 5292345093084 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 17139028321380 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 6439908260352 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 16172766031616 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 32568043234176 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 17832616128012 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 292828216813434 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 863391264288 \) Copy content Toggle raw display
show more
show less