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Properties

Label 4114.2.a.bg
Level $4114$
Weight $2$
Character orbit 4114.a
Self dual yes
Analytic conductor $32.850$
Analytic rank $1$
Dimension $8$
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] = [4114,2,Mod(1,4114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4114, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4114.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4114.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: \(32.8504553916\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 374)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} - \beta_{6} q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{4} + \beta_{2} - 1) q^{7} - q^{8} + (\beta_{7} + \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} - \beta_{6} q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{4} + \beta_{2} - 1) q^{7} - q^{8} + (\beta_{7} + \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{7} + \beta_{4} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{7} + \nu^{6} + 90\nu^{5} + 87\nu^{4} - 378\nu^{3} - 500\nu^{2} + 470\nu + 505 ) / 91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{7} - 48\nu^{6} - 134\nu^{5} + 374\nu^{4} + 308\nu^{3} - 934\nu^{2} - 174\nu + 694 ) / 91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{7} - 18\nu^{6} - 255\nu^{5} + 72\nu^{4} + 980\nu^{3} + 82\nu^{2} - 998\nu - 354 ) / 91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{7} + 79\nu^{6} + 103\nu^{5} - 589\nu^{4} + 77\nu^{3} + 1177\nu^{2} - 544\nu - 600 ) / 91 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -18\nu^{7} + 94\nu^{6} + 88\nu^{5} - 831\nu^{4} - 42\nu^{3} + 2049\nu^{2} - 46\nu - 1215 ) / 91 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -23\nu^{7} + 19\nu^{6} + 345\nu^{5} + 15\nu^{4} - 1358\nu^{3} - 491\nu^{2} + 1377\nu + 586 ) / 91 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{4} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + \beta_{5} + 2\beta_{4} + 2\beta_{3} - 4\beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{7} + \beta_{6} + 2\beta_{5} + 13\beta_{4} + 6\beta_{3} - 15\beta_{2} + 17\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 52\beta_{7} + 4\beta_{6} + 12\beta_{5} + 37\beta_{4} + 33\beta_{3} - 62\beta_{2} + 76\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 209\beta_{7} + 21\beta_{6} + 36\beta_{5} + 167\beta_{4} + 114\beta_{3} - 220\beta_{2} + 249\beta _1 + 173 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 760\beta_{7} + 78\beta_{6} + 152\beta_{5} + 562\beta_{4} + 475\beta_{3} - 864\beta_{2} + 982\beta _1 + 508 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.17063
−1.78029
−1.21054
−0.718217
1.22721
1.87555
2.04360
3.73332
−1.00000 −3.17063 1.00000 −2.18713 3.17063 3.13019 −1.00000 7.05290 2.18713
1.2 −1.00000 −2.78029 1.00000 3.41024 2.78029 −3.71831 −1.00000 4.72999 −3.41024
1.3 −1.00000 −2.21054 1.00000 −2.95515 2.21054 −3.36619 −1.00000 1.88651 2.95515
1.4 −1.00000 −1.71822 1.00000 3.65620 1.71822 0.780133 −1.00000 −0.0477314 −3.65620
1.5 −1.00000 0.227215 1.00000 −3.76434 −0.227215 −2.36764 −1.00000 −2.94837 3.76434
1.6 −1.00000 0.875550 1.00000 −0.120305 −0.875550 −1.45888 −1.00000 −2.23341 0.120305
1.7 −1.00000 1.04360 1.00000 3.29527 −1.04360 −3.68858 −1.00000 −1.91090 −3.29527
1.8 −1.00000 2.73332 1.00000 0.665210 −2.73332 −0.310718 −1.00000 4.47101 −0.665210
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)
\(17\) \(-1\)

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 4114.2.a.bg 8
11.b odd 2 1 4114.2.a.bi 8
11.d odd 10 2 374.2.g.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.g.f ✓ 16 11.d odd 10 2
4114.2.a.bg 8 1.a even 1 1 trivial
4114.2.a.bi 8 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4114))\):

\( T_{3}^{8} + 5T_{3}^{7} - 5T_{3}^{6} - 51T_{3}^{5} - 24T_{3}^{4} + 115T_{3}^{3} + 39T_{3}^{2} - 98T_{3} + 19 \) Copy content Toggle raw display
\( T_{5}^{8} - 2T_{5}^{7} - 30T_{5}^{6} + 51T_{5}^{5} + 289T_{5}^{4} - 360T_{5}^{3} - 920T_{5}^{2} + 560T_{5} + 80 \) Copy content Toggle raw display
\( T_{7}^{8} + 11T_{7}^{7} + 32T_{7}^{6} - 50T_{7}^{5} - 409T_{7}^{4} - 610T_{7}^{3} - 18T_{7}^{2} + 431T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots + 19 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 80 \) Copy content Toggle raw display
$7$ \( T^{8} + 11 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 2179 \) Copy content Toggle raw display
$17$ \( (T - 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 15 T^{7} + \cdots - 36784 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots - 30305 \) Copy content Toggle raw display
$29$ \( T^{8} + 22 T^{7} + \cdots - 880 \) Copy content Toggle raw display
$31$ \( T^{8} + 7 T^{7} + \cdots + 1290256 \) Copy content Toggle raw display
$37$ \( T^{8} - 15 T^{7} + \cdots - 180224 \) Copy content Toggle raw display
$41$ \( T^{8} + 17 T^{7} + \cdots - 668464 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 2163376 \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} + \cdots + 254720 \) Copy content Toggle raw display
$53$ \( T^{8} - 20 T^{7} + \cdots + 1316591 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 1520 \) Copy content Toggle raw display
$61$ \( T^{8} - 5 T^{7} + \cdots + 543664 \) Copy content Toggle raw display
$67$ \( T^{8} - 13 T^{7} + \cdots - 671744 \) Copy content Toggle raw display
$71$ \( T^{8} + 18 T^{7} + \cdots + 661520 \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + \cdots + 1325936 \) Copy content Toggle raw display
$79$ \( T^{8} + 21 T^{7} + \cdots + 375131 \) Copy content Toggle raw display
$83$ \( T^{8} + 38 T^{7} + \cdots + 22667920 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots - 1074256 \) Copy content Toggle raw display
$97$ \( T^{8} + 66 T^{7} + \cdots - 8458576 \) Copy content Toggle raw display
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