Properties

Label 49.22.a.c
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
Analytic rank $0$
Dimension $5$
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] = [49,22,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(136.943898701\)
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: \( x^{5} - x^{4} - 2414834x^{3} - 673460920x^{2} + 1181661460000x + 457592123870336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 456) q^{2} + (\beta_{2} + 18 \beta_1 + 1155) q^{3} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 73 \beta_1 + 1974185) q^{4} + ( - 8 \beta_{4} + \beta_{3} + 69 \beta_{2} - 3149 \beta_1 + 12044620) q^{5} + ( - 19 \beta_{4} + 355 \beta_{3} - 1590 \beta_{2} + \cdots + 69927277) q^{6}+ \cdots + ( - 7688 \beta_{4} + 9071 \beta_{3} + 28900 \beta_{2} + \cdots + 7078964330) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 456) q^{2} + (\beta_{2} + 18 \beta_1 + 1155) q^{3} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 73 \beta_1 + 1974185) q^{4} + ( - 8 \beta_{4} + \beta_{3} + 69 \beta_{2} - 3149 \beta_1 + 12044620) q^{5} + ( - 19 \beta_{4} + 355 \beta_{3} - 1590 \beta_{2} + \cdots + 69927277) q^{6}+ \cdots + (1345662611416 \beta_{4} + \cdots - 78\!\cdots\!65) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2278 q^{2} + 5810 q^{3} + 9870772 q^{4} + 60216716 q^{5} + 349679764 q^{6} - 1125152616 q^{8} + 35387410477 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2278 q^{2} + 5810 q^{3} + 9870772 q^{4} + 60216716 q^{5} + 349679764 q^{6} - 1125152616 q^{8} + 35387410477 q^{9} - 87948705880 q^{10} - 59831489168 q^{11} + 228081081592 q^{12} + 356284703992 q^{13} + 4509317220064 q^{15} - 328945700336 q^{16} + 8470669648266 q^{17} - 87071272512214 q^{18} + 47944229742094 q^{19} + 189108884590528 q^{20} - 131880926658656 q^{22} - 254010042736656 q^{23} + 16\!\cdots\!24 q^{24}+ \cdots - 39\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 2414834x^{3} - 673460920x^{2} + 1181661460000x + 457592123870336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -133\nu^{4} + 128433\nu^{3} + 220543934\nu^{2} - 115467571056\nu - 80515775642848 ) / 145369728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 169\nu^{4} - 85125\nu^{3} - 255335126\nu^{2} + 24185419824\nu + 46617370517536 ) / 72684864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 187\nu^{4} - 63471\nu^{3} - 418100450\nu^{2} + 39526945104\nu + 170073556086112 ) / 36342432 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + 3\beta_{3} + 2\beta_{2} + 839\beta _1 + 3863401 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 319\beta_{4} + 2767\beta_{3} + 8826\beta_{2} + 2871431\beta _1 + 1620955429 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1350179\beta_{4} + 7646661\beta_{3} + 7467358\beta_{2} + 2427727189\beta _1 + 5550155798503 ) / 4 \) 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
−1037.95
−832.356
−433.779
824.937
1480.15
−2531.90 −181011. 4.31334e6 1.64561e7 4.58301e8 0 −5.61117e9 2.23046e10 −4.16651e10
1.2 −2120.71 180918. 2.40027e6 3.72326e7 −3.83675e8 0 −6.42816e8 2.22710e10 −7.89597e10
1.3 −1323.56 −42814.6 −345345. −6.10959e6 5.66676e7 0 3.23279e9 −8.62726e9 8.08640e9
1.4 1193.87 −73552.1 −671816. 5.38643e6 −8.78120e7 0 −3.30580e9 −5.05044e9 6.43072e9
1.5 2504.29 122270. 4.17432e6 7.25114e6 3.06199e8 0 5.20184e9 4.48948e9 1.81590e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

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 49.22.a.c 5
7.b odd 2 1 7.22.a.a 5
21.c even 2 1 63.22.a.d 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.22.a.a 5 7.b odd 2 1
49.22.a.c 5 1.a even 1 1 trivial
63.22.a.d 5 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{5} + 2278T_{2}^{4} - 7583624T_{2}^{3} - 17655966080T_{2}^{2} + 8182869766144T_{2} + 21247798784819200 \) Copy content Toggle raw display
\( T_{3}^{5} - 5810 T_{3}^{4} - 43827710196 T_{3}^{3} - 192762253993176 T_{3}^{2} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2278 T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{5} - 5810 T^{4} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{5} - 60216716 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 59831489168 T^{4} + \cdots + 31\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{5} - 356284703992 T^{4} + \cdots - 22\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{5} - 8470669648266 T^{4} + \cdots - 70\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{5} - 47944229742094 T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + 254010042736656 T^{4} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 25\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 41\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 40\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 29\!\cdots\!56 \) Copy content Toggle raw display
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