Properties

Label 49.8.a.e
Level $49$
Weight $8$
Character orbit 49.a
Self dual yes
Analytic conductor $15.307$
Analytic rank $1$
Dimension $4$
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] = [49,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(15.3068662487\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 102x^{2} + 240x + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 8) q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 86) q^{4} + (\beta_{3} - 3 \beta_{2} - 7 \beta_1 - 65) q^{5} + (3 \beta_{3} - 7 \beta_{2} + \cdots - 253) q^{6}+ \cdots + (3 \beta_{3} - 31 \beta_{2} + \cdots + 494) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 8) q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 86) q^{4} + (\beta_{3} - 3 \beta_{2} - 7 \beta_1 - 65) q^{5} + (3 \beta_{3} - 7 \beta_{2} + \cdots - 253) q^{6}+ \cdots + (5385 \beta_{3} - 86599 \beta_{2} + \cdots + 3848636) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} - 28 q^{3} + 348 q^{4} - 252 q^{5} - 1022 q^{6} - 984 q^{8} + 2008 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{2} - 28 q^{3} + 348 q^{4} - 252 q^{5} - 1022 q^{6} - 984 q^{8} + 2008 q^{9} - 4774 q^{10} - 3972 q^{11} + 5404 q^{12} + 1176 q^{13} - 16556 q^{15} + 57264 q^{16} - 56364 q^{17} - 35908 q^{18} - 41748 q^{19} - 162372 q^{20} - 152954 q^{22} + 131748 q^{23} - 389592 q^{24} - 68016 q^{25} - 113652 q^{26} - 189028 q^{27} + 34056 q^{29} + 605378 q^{30} - 401212 q^{31} - 453408 q^{32} - 24052 q^{33} - 82110 q^{34} - 360088 q^{36} - 5396 q^{37} - 31794 q^{38} + 364840 q^{39} + 443688 q^{40} - 410424 q^{41} + 46544 q^{43} - 1465836 q^{44} + 1209656 q^{45} + 1379202 q^{46} - 1470084 q^{47} + 2018800 q^{48} + 1395528 q^{50} - 1724796 q^{51} + 5269768 q^{52} - 642372 q^{53} + 4567318 q^{54} + 3063340 q^{55} + 3359068 q^{57} - 1972220 q^{58} - 752220 q^{59} + 2123324 q^{60} - 1325772 q^{61} + 3016314 q^{62} - 865856 q^{64} - 4868808 q^{65} + 6013294 q^{66} - 290916 q^{67} - 2453556 q^{68} - 4915260 q^{69} + 3377760 q^{71} + 4557744 q^{72} - 6706588 q^{73} + 3616878 q^{74} - 7223888 q^{75} - 2923004 q^{76} - 16288132 q^{78} + 3946244 q^{79} - 10695888 q^{80} - 11731292 q^{81} + 17563252 q^{82} - 9542064 q^{83} + 7006068 q^{85} + 23237832 q^{86} - 17836840 q^{87} - 8043336 q^{88} - 16165212 q^{89} - 3415748 q^{90} - 19442628 q^{92} + 12552852 q^{93} - 5720442 q^{94} - 3268452 q^{95} + 8448608 q^{96} - 1533112 q^{97} + 15213976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 102x^{2} + 240x + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 72\nu + 165 ) / 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 26\nu^{2} - 288\nu - 840 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 31\nu^{2} + 492\nu - 1815 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 5\beta _1 + 10 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 4\beta _1 + 100 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 36\beta_{3} - 29\beta_{2} + 362\beta _1 - 1250 ) / 14 \) 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
−10.3192
5.14290
8.23552
−2.05919
−20.8241 45.5009 305.642 −193.403 −947.513 0 −3699.23 −116.672 4027.44
1.2 −7.38077 −74.2434 −73.5243 290.819 547.974 0 1487.40 3325.09 −2146.47
1.3 3.18552 40.2041 −117.852 0.588975 128.071 0 −783.169 −570.629 1.87619
1.4 19.0193 −39.4615 233.735 −350.005 −750.531 0 2011.00 −629.788 −6656.85
\(n\): e.g. 2-40 or 990-1000
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 49.8.a.e 4
3.b odd 2 1 441.8.a.t 4
7.b odd 2 1 49.8.a.f 4
7.c even 3 2 49.8.c.g 8
7.d odd 6 2 7.8.c.a 8
21.c even 2 1 441.8.a.s 4
21.g even 6 2 63.8.e.b 8
28.f even 6 2 112.8.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.8.c.a 8 7.d odd 6 2
49.8.a.e 4 1.a even 1 1 trivial
49.8.a.f 4 7.b odd 2 1
49.8.c.g 8 7.c even 3 2
63.8.e.b 8 21.g even 6 2
112.8.i.c 8 28.f even 6 2
441.8.a.s 4 21.c even 2 1
441.8.a.t 4 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{4} + 6T_{2}^{3} - 412T_{2}^{2} - 1704T_{2} + 9312 \) Copy content Toggle raw display
\( T_{3}^{4} + 28T_{3}^{3} - 4986T_{3}^{2} - 43092T_{3} + 5359473 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6 T^{3} + \cdots + 9312 \) Copy content Toggle raw display
$3$ \( T^{4} + 28 T^{3} + \cdots + 5359473 \) Copy content Toggle raw display
$5$ \( T^{4} + 252 T^{3} + \cdots + 11594625 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 7298108160225 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 11\!\cdots\!27 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 52\!\cdots\!47 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 77\!\cdots\!63 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 43\!\cdots\!31 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 38\!\cdots\!93 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 33\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 43\!\cdots\!13 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 56\!\cdots\!11 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 89\!\cdots\!13 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 17\!\cdots\!23 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 62\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 24\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
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