↑

Properties

Label 4802.2.a.h
Level $4802$
Weight $2$
Character orbit 4802.a
Self dual yes
Analytic conductor $38.344$
Analytic rank $0$
Dimension $9$
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] = [4802,2,Mod(1,4802)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4802, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4802.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4802 = 2 \cdot 7^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4802.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: \(38.3441630506\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 11x^{7} + 48x^{6} + 31x^{5} - 191x^{4} + 20x^{3} + 258x^{2} - 126x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + \beta_{8} q^{5} + ( - \beta_1 + 1) q^{6} + q^{8} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + \beta_{8} q^{5} + ( - \beta_1 + 1) q^{6} + q^{8} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{9}+ \cdots + ( - 3 \beta_{8} - 3 \beta_{6} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 5 q^{3} + 9 q^{4} + 4 q^{5} + 5 q^{6} + 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 5 q^{3} + 9 q^{4} + 4 q^{5} + 5 q^{6} + 9 q^{8} + 12 q^{9} + 4 q^{10} - 4 q^{11} + 5 q^{12} + 7 q^{13} - 8 q^{15} + 9 q^{16} + 19 q^{17} + 12 q^{18} + 18 q^{19} + 4 q^{20} - 4 q^{22} - q^{23} + 5 q^{24} + 13 q^{25} + 7 q^{26} + 29 q^{27} - 10 q^{29} - 8 q^{30} - 2 q^{31} + 9 q^{32} + 60 q^{33} + 19 q^{34} + 12 q^{36} - 15 q^{37} + 18 q^{38} - 21 q^{39} + 4 q^{40} + 21 q^{41} - 17 q^{43} - 4 q^{44} - 11 q^{45} - q^{46} - 10 q^{47} + 5 q^{48} + 13 q^{50} + 46 q^{51} + 7 q^{52} - 4 q^{53} + 29 q^{54} + 34 q^{55} + 13 q^{57} - 10 q^{58} + 19 q^{59} - 8 q^{60} + 4 q^{61} - 2 q^{62} + 9 q^{64} - 7 q^{65} + 60 q^{66} + 17 q^{67} + 19 q^{68} - 13 q^{69} - 10 q^{71} + 12 q^{72} + 75 q^{73} - 15 q^{74} - 20 q^{75} + 18 q^{76} - 21 q^{78} - q^{79} + 4 q^{80} + 53 q^{81} + 21 q^{82} + 28 q^{83} - 29 q^{85} - 17 q^{86} - 18 q^{87} - 4 q^{88} + 4 q^{89} - 11 q^{90} - q^{92} - 10 q^{93} - 10 q^{94} + 9 q^{95} + 5 q^{96} + 28 q^{97} + 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 11x^{7} + 48x^{6} + 31x^{5} - 191x^{4} + 20x^{3} + 258x^{2} - 126x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 8\nu^{7} - 32\nu^{6} - 102\nu^{5} + 94\nu^{4} + 352\nu^{3} + 149\nu^{2} - 294\nu - 261 ) / 117 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{8} + 7\nu^{7} + 50\nu^{6} - 21\nu^{5} - 181\nu^{4} - 43\nu^{3} + 184\nu^{2} + 45\nu - 9 ) / 39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{8} - 28\nu^{7} - 239\nu^{6} + 201\nu^{5} + 1036\nu^{4} - 374\nu^{3} - 1399\nu^{2} + 132\nu + 270 ) / 117 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{8} + 14\nu^{7} + 178\nu^{6} - 276\nu^{5} - 1103\nu^{4} + 1474\nu^{3} + 2162\nu^{2} - 2289\nu - 369 ) / 117 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\nu^{8} - 76\nu^{7} + 70\nu^{6} + 579\nu^{5} - 1049\nu^{4} - 848\nu^{3} + 2387\nu^{2} - 912\nu - 153 ) / 117 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{8} + 8\nu^{7} + 22\nu^{6} - 87\nu^{5} - 89\nu^{4} + 292\nu^{3} + 158\nu^{2} - 291\nu - 81 ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -14\nu^{8} + 44\nu^{7} + 175\nu^{6} - 444\nu^{5} - 731\nu^{4} + 1429\nu^{3} + 1034\nu^{2} - 1461\nu - 90 ) / 39 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - 2\beta_{5} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} + 3\beta_{6} - 6\beta_{5} + \beta_{4} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + 12\beta_{7} + 16\beta_{6} - 33\beta_{5} + 2\beta_{4} - 2\beta_{3} + 17\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{8} + 37\beta_{7} + 58\beta_{6} - 119\beta_{5} + 12\beta_{4} - 6\beta_{3} + 6\beta_{2} + 78\beta _1 + 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 78 \beta_{8} + 162 \beta_{7} + 243 \beta_{6} - 503 \beta_{5} + 35 \beta_{4} - 38 \beta_{3} + 18 \beta_{2} + \cdots + 320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 321 \beta_{8} + 581 \beta_{7} + 919 \beta_{6} - 1900 \beta_{5} + 157 \beta_{4} - 131 \beta_{3} + \cdots + 1041 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1240 \beta_{8} + 2329 \beta_{7} + 3631 \beta_{6} - 7522 \beta_{5} + 548 \beta_{4} - 592 \beta_{3} + \cdots + 4323 \) 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
3.85659
2.09547
1.94252
1.26572
1.04037
−0.162177
−1.76319
−2.09502
−2.18028
1.00000 −2.85659 1.00000 2.93247 −2.85659 0 1.00000 5.16013 2.93247
1.2 1.00000 −1.09547 1.00000 −2.01007 −1.09547 0 1.00000 −1.79994 −2.01007
1.3 1.00000 −0.942519 1.00000 −2.16697 −0.942519 0 1.00000 −2.11166 −2.16697
1.4 1.00000 −0.265716 1.00000 3.91700 −0.265716 0 1.00000 −2.92939 3.91700
1.5 1.00000 −0.0403672 1.00000 −0.473617 −0.0403672 0 1.00000 −2.99837 −0.473617
1.6 1.00000 1.16218 1.00000 4.29712 1.16218 0 1.00000 −1.64935 4.29712
1.7 1.00000 2.76319 1.00000 −0.304996 2.76319 0 1.00000 4.63524 −0.304996
1.8 1.00000 3.09502 1.00000 0.343086 3.09502 0 1.00000 6.57918 0.343086
1.9 1.00000 3.18028 1.00000 −2.53403 3.18028 0 1.00000 7.11416 −2.53403
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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 4802.2.a.h 9
7.b odd 2 1 4802.2.a.e 9
49.e even 7 2 98.2.e.a ✓ 18
49.f odd 14 2 686.2.e.a 18
49.g even 21 4 686.2.g.i 36
49.h odd 42 4 686.2.g.j 36
147.l odd 14 2 882.2.u.j 18
196.k odd 14 2 784.2.u.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.e.a ✓ 18 49.e even 7 2
686.2.e.a 18 49.f odd 14 2
686.2.g.i 36 49.g even 21 4
686.2.g.j 36 49.h odd 42 4
784.2.u.c 18 196.k odd 14 2
882.2.u.j 18 147.l odd 14 2
4802.2.a.e 9 7.b odd 2 1
4802.2.a.h 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 5T_{3}^{8} - 7T_{3}^{7} + 57T_{3}^{6} - 10T_{3}^{5} - 145T_{3}^{4} + T_{3}^{3} + 105T_{3}^{2} + 29T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4802))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 5 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} - 4 T^{8} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 4 T^{8} + \cdots - 78273 \) Copy content Toggle raw display
$13$ \( T^{9} - 7 T^{8} + \cdots + 4459 \) Copy content Toggle raw display
$17$ \( T^{9} - 19 T^{8} + \cdots + 20952 \) Copy content Toggle raw display
$19$ \( T^{9} - 18 T^{8} + \cdots + 500387 \) Copy content Toggle raw display
$23$ \( T^{9} + T^{8} + \cdots + 1107 \) Copy content Toggle raw display
$29$ \( T^{9} + 10 T^{8} + \cdots + 1161 \) Copy content Toggle raw display
$31$ \( T^{9} + 2 T^{8} + \cdots - 13 \) Copy content Toggle raw display
$37$ \( T^{9} + 15 T^{8} + \cdots - 1305191 \) Copy content Toggle raw display
$41$ \( T^{9} - 21 T^{8} + \cdots + 25728381 \) Copy content Toggle raw display
$43$ \( T^{9} + 17 T^{8} + \cdots - 252281 \) Copy content Toggle raw display
$47$ \( T^{9} + 10 T^{8} + \cdots + 3051 \) Copy content Toggle raw display
$53$ \( T^{9} + 4 T^{8} + \cdots - 311499 \) Copy content Toggle raw display
$59$ \( T^{9} - 19 T^{8} + \cdots + 289521 \) Copy content Toggle raw display
$61$ \( T^{9} - 4 T^{8} + \cdots - 2107393 \) Copy content Toggle raw display
$67$ \( T^{9} - 17 T^{8} + \cdots - 149884139 \) Copy content Toggle raw display
$71$ \( T^{9} + 10 T^{8} + \cdots - 3753 \) Copy content Toggle raw display
$73$ \( T^{9} - 75 T^{8} + \cdots + 18746897 \) Copy content Toggle raw display
$79$ \( T^{9} + T^{8} + \cdots + 5529329 \) Copy content Toggle raw display
$83$ \( T^{9} - 28 T^{8} + \cdots + 50887872 \) Copy content Toggle raw display
$89$ \( T^{9} - 4 T^{8} + \cdots - 3181977 \) Copy content Toggle raw display
$97$ \( T^{9} - 28 T^{8} + \cdots + 8869 \) Copy content Toggle raw display
show more
show less