↑

Properties

Label 4802.2.a.o
Level $4802$
Weight $2$
Character orbit 4802.a
Self dual yes
Analytic conductor $38.344$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 9 x^{10} + 61 x^{9} + 36 x^{8} - 273 x^{7} - 125 x^{6} + 511 x^{5} + 302 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{11} - \beta_{6} + 1) q^{5} + ( - \beta_1 + 1) q^{6} + q^{8} + (\beta_{11} + \beta_{8} - \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{11} - \beta_{6} + 1) q^{5} + ( - \beta_1 + 1) q^{6} + q^{8} + (\beta_{11} + \beta_{8} - \beta_{6} + \cdots + 2) q^{9}+ \cdots + (\beta_{10} - \beta_{8} + \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 7 q^{3} + 12 q^{4} + 14 q^{5} + 7 q^{6} + 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 7 q^{3} + 12 q^{4} + 14 q^{5} + 7 q^{6} + 12 q^{8} + 9 q^{9} + 14 q^{10} + 4 q^{11} + 7 q^{12} + 28 q^{13} + 8 q^{15} + 12 q^{16} + 14 q^{17} + 9 q^{18} + 14 q^{19} + 14 q^{20} + 4 q^{22} + q^{23} + 7 q^{24} + 8 q^{25} + 28 q^{26} + 7 q^{27} + 10 q^{29} + 8 q^{30} + 28 q^{31} + 12 q^{32} - 21 q^{33} + 14 q^{34} + 9 q^{36} + 8 q^{37} + 14 q^{38} + 14 q^{39} + 14 q^{40} + 21 q^{41} + 10 q^{43} + 4 q^{44} + 56 q^{45} + q^{46} + 28 q^{47} + 7 q^{48} + 8 q^{50} - 46 q^{51} + 28 q^{52} + 4 q^{53} + 7 q^{54} - 7 q^{55} - 20 q^{57} + 10 q^{58} + 14 q^{59} + 8 q^{60} + 49 q^{61} + 28 q^{62} + 12 q^{64} + 7 q^{65} - 21 q^{66} - 24 q^{67} + 14 q^{68} + 49 q^{69} + 10 q^{71} + 9 q^{72} - 7 q^{73} + 8 q^{74} + 35 q^{75} + 14 q^{76} + 14 q^{78} - 6 q^{79} + 14 q^{80} - 32 q^{81} + 21 q^{82} + 14 q^{83} + 29 q^{85} + 10 q^{86} + 21 q^{87} + 4 q^{88} + 56 q^{89} + 56 q^{90} + q^{92} + 3 q^{93} + 28 q^{94} - 9 q^{95} + 7 q^{96} + 49 q^{97} - 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 9 x^{10} + 61 x^{9} + 36 x^{8} - 273 x^{7} - 125 x^{6} + 511 x^{5} + 302 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4515 \nu^{11} + 37055 \nu^{10} + 57726 \nu^{9} - 768203 \nu^{8} - 213302 \nu^{7} + 5433429 \nu^{6} + \cdots - 419970 ) / 436813 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1213 \nu^{11} - 5676 \nu^{10} - 7114 \nu^{9} + 49551 \nu^{8} - 5178 \nu^{7} - 84694 \nu^{6} + \cdots - 24254 ) / 33601 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19039 \nu^{11} + 106430 \nu^{10} + 231521 \nu^{9} - 1701202 \nu^{8} - 1323598 \nu^{7} + \cdots + 816557 ) / 436813 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32713 \nu^{11} + 93850 \nu^{10} + 630995 \nu^{9} - 1519205 \nu^{8} - 4478153 \nu^{7} + \cdots + 355872 ) / 436813 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44056 \nu^{11} + 285625 \nu^{10} + 70153 \nu^{9} - 3136856 \nu^{8} + 1835372 \nu^{7} + \cdots + 762149 ) / 436813 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55291 \nu^{11} - 225455 \nu^{10} - 609976 \nu^{9} + 2498662 \nu^{8} + 2931473 \nu^{7} + \cdots + 130516 ) / 436813 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4322 \nu^{11} - 28922 \nu^{10} - 8284 \nu^{9} + 336276 \nu^{8} - 158089 \nu^{7} - 1515977 \nu^{6} + \cdots - 33344 ) / 33601 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57286 \nu^{11} + 353571 \nu^{10} - 4499 \nu^{9} - 3193646 \nu^{8} + 3313145 \nu^{7} + \cdots - 427827 ) / 436813 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7312 \nu^{11} + 30614 \nu^{10} + 72634 \nu^{9} - 313681 \nu^{8} - 336071 \nu^{7} + 1005384 \nu^{6} + \cdots - 4322 ) / 33601 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 96972 \nu^{11} + 628969 \nu^{10} + 38806 \nu^{9} - 6451405 \nu^{8} + 5281441 \nu^{7} + \cdots - 1052886 ) / 436813 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{9} + 3\beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{11} - \beta_{10} + 4 \beta_{9} + 15 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} + 6 \beta_{5} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28 \beta_{11} - 4 \beta_{10} + 20 \beta_{9} + 48 \beta_{8} + 23 \beta_{7} - 10 \beta_{6} + 31 \beta_{5} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 107 \beta_{11} - 21 \beta_{10} + 73 \beta_{9} + 182 \beta_{8} + 66 \beta_{7} - 36 \beta_{6} + 109 \beta_{5} + \cdots + 351 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 339 \beta_{11} - 84 \beta_{10} + 288 \beta_{9} + 601 \beta_{8} + 266 \beta_{7} - 102 \beta_{6} + \cdots + 1091 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1197 \beta_{11} - 344 \beta_{10} + 1032 \beta_{9} + 2102 \beta_{8} + 871 \beta_{7} - 353 \beta_{6} + \cdots + 3880 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3976 \beta_{11} - 1328 \beta_{10} + 3795 \beta_{9} + 7030 \beta_{8} + 3175 \beta_{7} - 1165 \beta_{6} + \cdots + 12789 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13718 \beta_{11} - 5074 \beta_{10} + 13507 \beta_{9} + 23996 \beta_{8} + 10860 \beta_{7} - 4078 \beta_{6} + \cdots + 44131 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 46445 \beta_{11} - 19037 \beta_{10} + 48452 \beta_{9} + 80803 \beta_{8} + 38416 \beta_{7} + \cdots + 148751 \) 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
3.50745
3.14730
2.13159
2.11690
1.07968
0.0326067
−0.169636
−0.685084
−1.04135
−1.28105
−1.84758
−1.99083
1.00000 −2.50745 1.00000 2.98184 −2.50745 0 1.00000 3.28731 2.98184
1.2 1.00000 −2.14730 1.00000 2.10332 −2.14730 0 1.00000 1.61089 2.10332
1.3 1.00000 −1.13159 1.00000 1.03330 −1.13159 0 1.00000 −1.71950 1.03330
1.4 1.00000 −1.11690 1.00000 1.58423 −1.11690 0 1.00000 −1.75253 1.58423
1.5 1.00000 −0.0796818 1.00000 −0.446858 −0.0796818 0 1.00000 −2.99365 −0.446858
1.6 1.00000 0.967393 1.00000 −3.71581 0.967393 0 1.00000 −2.06415 −3.71581
1.7 1.00000 1.16964 1.00000 0.0830521 1.16964 0 1.00000 −1.63195 0.0830521
1.8 1.00000 1.68508 1.00000 2.27205 1.68508 0 1.00000 −0.160491 2.27205
1.9 1.00000 2.04135 1.00000 −1.30703 2.04135 0 1.00000 1.16710 −1.30703
1.10 1.00000 2.28105 1.00000 3.60853 2.28105 0 1.00000 2.20320 3.60853
1.11 1.00000 2.84758 1.00000 2.47638 2.84758 0 1.00000 5.10872 2.47638
1.12 1.00000 2.99083 1.00000 3.32700 2.99083 0 1.00000 5.94505 3.32700
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.o 12
7.b odd 2 1 4802.2.a.l 12
49.e even 7 2 686.2.e.c 24
49.f odd 14 2 686.2.e.d 24
49.g even 21 2 98.2.g.b ✓ 24
49.g even 21 2 686.2.g.f 24
49.h odd 42 2 686.2.g.d 24
49.h odd 42 2 686.2.g.e 24
147.n odd 42 2 882.2.z.b 24
196.o odd 42 2 784.2.bg.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.b ✓ 24 49.g even 21 2
686.2.e.c 24 49.e even 7 2
686.2.e.d 24 49.f odd 14 2
686.2.g.d 24 49.h odd 42 2
686.2.g.e 24 49.h odd 42 2
686.2.g.f 24 49.g even 21 2
784.2.bg.b 24 196.o odd 42 2
882.2.z.b 24 147.n odd 42 2
4802.2.a.l 12 7.b odd 2 1
4802.2.a.o 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 7 T_{3}^{11} + 2 T_{3}^{10} + 84 T_{3}^{9} - 150 T_{3}^{8} - 273 T_{3}^{7} + 820 T_{3}^{6} + \cdots - 41 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4802))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 7 T^{11} + \cdots - 41 \) Copy content Toggle raw display
$5$ \( T^{12} - 14 T^{11} + \cdots - 125 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots - 6047 \) Copy content Toggle raw display
$13$ \( T^{12} - 28 T^{11} + \cdots - 8183 \) Copy content Toggle raw display
$17$ \( T^{12} - 14 T^{11} + \cdots + 32677 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} + \cdots + 2128939 \) Copy content Toggle raw display
$23$ \( T^{12} - T^{11} + \cdots - 848987 \) Copy content Toggle raw display
$29$ \( T^{12} - 10 T^{11} + \cdots + 3514813 \) Copy content Toggle raw display
$31$ \( T^{12} - 28 T^{11} + \cdots - 17289131 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} + \cdots + 1106827 \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots - 61933403 \) Copy content Toggle raw display
$43$ \( T^{12} - 10 T^{11} + \cdots - 6369719 \) Copy content Toggle raw display
$47$ \( T^{12} - 28 T^{11} + \cdots + 43800961 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 157857127 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 208533527 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1011018289 \) Copy content Toggle raw display
$67$ \( T^{12} + 24 T^{11} + \cdots + 756799 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 109808707 \) Copy content Toggle raw display
$73$ \( T^{12} + 7 T^{11} + \cdots - 5889281 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12283494637 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 124445251 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1586799229 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 51111469241 \) Copy content Toggle raw display
show more
show less