↑

Properties

Label 4730.2.a.bf
Level $4730$
Weight $2$
Character orbit 4730.a
Self dual yes
Analytic conductor $37.769$
Analytic rank $0$
Dimension $13$
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] = [4730,2,Mod(1,4730)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4730.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4730.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: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 32 x^{11} - 5 x^{10} + 376 x^{9} + 100 x^{8} - 1985 x^{7} - 576 x^{6} + 4708 x^{5} + 889 x^{4} + \cdots - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_{8} + 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_{8} + 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9} - q^{10} + q^{11} - \beta_1 q^{12} + ( - \beta_{4} + 1) q^{13} + ( - \beta_{8} + 1) q^{14} + \beta_1 q^{15} + q^{16} + \beta_{12} q^{17} + (\beta_{2} + 2) q^{18} + ( - \beta_{6} + \beta_{5}) q^{19} - q^{20} + ( - \beta_{12} + \beta_{9} + \beta_{5} + 1) q^{21} + q^{22} + ( - \beta_{5} + 1) q^{23} - \beta_1 q^{24} + q^{25} + ( - \beta_{4} + 1) q^{26} + ( - \beta_{9} - \beta_{8} + \beta_{6} + \cdots - 1) q^{27}+ \cdots + (\beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 13 q^{4} - 13 q^{5} + 7 q^{7} + 13 q^{8} + 25 q^{9} - 13 q^{10} + 13 q^{11} + 11 q^{13} + 7 q^{14} + 13 q^{16} - 2 q^{17} + 25 q^{18} + 7 q^{19} - 13 q^{20} + 12 q^{21} + 13 q^{22} + 12 q^{23} + 13 q^{25} + 11 q^{26} - 15 q^{27} + 7 q^{28} + 14 q^{29} + 20 q^{31} + 13 q^{32} - 2 q^{34} - 7 q^{35} + 25 q^{36} + 17 q^{37} + 7 q^{38} - 4 q^{39} - 13 q^{40} + 9 q^{41} + 12 q^{42} + 13 q^{43} + 13 q^{44} - 25 q^{45} + 12 q^{46} - 9 q^{47} + 30 q^{49} + 13 q^{50} - 3 q^{51} + 11 q^{52} + 22 q^{53} - 15 q^{54} - 13 q^{55} + 7 q^{56} + 17 q^{57} + 14 q^{58} + 19 q^{59} + 2 q^{61} + 20 q^{62} + 12 q^{63} + 13 q^{64} - 11 q^{65} + 9 q^{67} - 2 q^{68} - 6 q^{69} - 7 q^{70} + 6 q^{71} + 25 q^{72} + 7 q^{73} + 17 q^{74} + 7 q^{76} + 7 q^{77} - 4 q^{78} + 50 q^{79} - 13 q^{80} + 85 q^{81} + 9 q^{82} + q^{83} + 12 q^{84} + 2 q^{85} + 13 q^{86} - 21 q^{87} + 13 q^{88} + 5 q^{89} - 25 q^{90} + 5 q^{91} + 12 q^{92} + 3 q^{93} - 9 q^{94} - 7 q^{95} + 20 q^{97} + 30 q^{98} + 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 32 x^{11} - 5 x^{10} + 376 x^{9} + 100 x^{8} - 1985 x^{7} - 576 x^{6} + 4708 x^{5} + 889 x^{4} + \cdots - 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1209723 \nu^{12} + 95374748 \nu^{11} - 33549200 \nu^{10} - 2743522569 \nu^{9} + \cdots + 1796287264 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5481053 \nu^{12} + 43657412 \nu^{11} + 142029872 \nu^{10} - 1180471903 \nu^{9} + \cdots - 10321684480 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1552491 \nu^{12} + 12856529 \nu^{11} + 28305288 \nu^{10} - 353658669 \nu^{9} + \cdots - 126650044 ) / 513645764 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7498623 \nu^{12} + 119840596 \nu^{11} + 152682496 \nu^{10} - 3489939941 \nu^{9} + \cdots + 9257057216 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 727017 \nu^{12} + 1618382 \nu^{11} - 22069016 \nu^{10} - 54545229 \nu^{9} + 235343334 \nu^{8} + \cdots + 617584666 ) / 128411441 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37859597 \nu^{12} + 18959624 \nu^{11} - 1105975856 \nu^{10} - 870485553 \nu^{9} + \cdots + 1147504160 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21709793 \nu^{12} - 1131240 \nu^{11} + 660508136 \nu^{10} + 179115477 \nu^{9} + \cdots - 2778200784 ) / 1027291528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57521037 \nu^{12} + 2964296 \nu^{11} + 1781628496 \nu^{10} + 152007089 \nu^{9} + \cdots + 4872284112 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 58907007 \nu^{12} - 134757996 \nu^{11} + 1894620016 \nu^{10} + 4238230907 \nu^{9} + \cdots - 18968508576 ) / 2054583056 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34294587 \nu^{12} - 28183806 \nu^{11} + 1057712496 \nu^{10} + 1016626975 \nu^{9} + \cdots - 4427223552 ) / 1027291528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - 2\beta_{11} + 2\beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2\beta_{3} + 12\beta_{2} + 2\beta _1 + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 2 \beta_{10} + 16 \beta_{9} + 15 \beta_{8} + \beta_{7} - 12 \beta_{6} + 15 \beta_{5} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{12} - 35 \beta_{11} + 34 \beta_{10} + 18 \beta_{9} - 11 \beta_{8} - \beta_{7} - 20 \beta_{6} + \cdots + 403 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{12} + 18 \beta_{11} - 30 \beta_{10} + 214 \beta_{9} + 188 \beta_{8} + 15 \beta_{7} + \cdots + 305 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 173 \beta_{12} - 479 \beta_{11} + 467 \beta_{10} + 268 \beta_{9} - 88 \beta_{8} - 4 \beta_{7} + \cdots + 4093 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 319 \beta_{12} + 234 \beta_{11} - 302 \beta_{10} + 2714 \beta_{9} + 2231 \beta_{8} + 177 \beta_{7} + \cdots + 3953 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2171 \beta_{12} - 5994 \beta_{11} + 5991 \beta_{10} + 3754 \beta_{9} - 466 \beta_{8} + 142 \beta_{7} + \cdots + 42977 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6101 \beta_{12} + 2654 \beta_{11} - 2223 \beta_{10} + 33580 \beta_{9} + 25975 \beta_{8} + \cdots + 48941 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28049 \beta_{12} - 71881 \beta_{11} + 74580 \beta_{10} + 50863 \beta_{9} + 1043 \beta_{8} + \cdots + 461337 \) 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
3.45880
3.26258
2.61361
1.30046
1.16663
0.743787
0.0802958
−0.921464
−1.13397
−2.15443
−2.24942
−2.86226
−3.30461
1.00000 −3.45880 1.00000 −1.00000 −3.45880 −3.54413 1.00000 8.96327 −1.00000
1.2 1.00000 −3.26258 1.00000 −1.00000 −3.26258 2.73152 1.00000 7.64446 −1.00000
1.3 1.00000 −2.61361 1.00000 −1.00000 −2.61361 0.881450 1.00000 3.83093 −1.00000
1.4 1.00000 −1.30046 1.00000 −1.00000 −1.30046 3.42310 1.00000 −1.30880 −1.00000
1.5 1.00000 −1.16663 1.00000 −1.00000 −1.16663 −3.85859 1.00000 −1.63898 −1.00000
1.6 1.00000 −0.743787 1.00000 −1.00000 −0.743787 2.37023 1.00000 −2.44678 −1.00000
1.7 1.00000 −0.0802958 1.00000 −1.00000 −0.0802958 2.53237 1.00000 −2.99355 −1.00000
1.8 1.00000 0.921464 1.00000 −1.00000 0.921464 −3.84192 1.00000 −2.15090 −1.00000
1.9 1.00000 1.13397 1.00000 −1.00000 1.13397 −0.848165 1.00000 −1.71412 −1.00000
1.10 1.00000 2.15443 1.00000 −1.00000 2.15443 5.20049 1.00000 1.64159 −1.00000
1.11 1.00000 2.24942 1.00000 −1.00000 2.24942 −1.43432 1.00000 2.05989 −1.00000
1.12 1.00000 2.86226 1.00000 −1.00000 2.86226 4.06491 1.00000 5.19253 −1.00000
1.13 1.00000 3.30461 1.00000 −1.00000 3.30461 −0.676945 1.00000 7.92047 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)
\(43\) \(-1\)

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 4730.2.a.bf ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bf ✓ 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{13} - 32 T_{3}^{11} + 5 T_{3}^{10} + 376 T_{3}^{9} - 100 T_{3}^{8} - 1985 T_{3}^{7} + 576 T_{3}^{6} + \cdots + 128 \) Copy content Toggle raw display
\( T_{7}^{13} - 7 T_{7}^{12} - 36 T_{7}^{11} + 321 T_{7}^{10} + 297 T_{7}^{9} - 5274 T_{7}^{8} + \cdots - 45248 \) Copy content Toggle raw display
\( T_{13}^{13} - 11 T_{13}^{12} - 94 T_{13}^{11} + 1334 T_{13}^{10} + 2238 T_{13}^{9} - 60656 T_{13}^{8} + \cdots + 33478272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{13} \) Copy content Toggle raw display
$3$ \( T^{13} - 32 T^{11} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T + 1)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} - 7 T^{12} + \cdots - 45248 \) Copy content Toggle raw display
$11$ \( (T - 1)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} - 11 T^{12} + \cdots + 33478272 \) Copy content Toggle raw display
$17$ \( T^{13} + 2 T^{12} + \cdots - 2410032 \) Copy content Toggle raw display
$19$ \( T^{13} - 7 T^{12} + \cdots - 1723840 \) Copy content Toggle raw display
$23$ \( T^{13} - 12 T^{12} + \cdots - 5988864 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 410840256 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 295763968 \) Copy content Toggle raw display
$37$ \( T^{13} - 17 T^{12} + \cdots + 13109248 \) Copy content Toggle raw display
$41$ \( T^{13} - 9 T^{12} + \cdots + 142080 \) Copy content Toggle raw display
$43$ \( (T - 1)^{13} \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 4116067392 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 307036416 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 1235101440 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 24865148480 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 160441606144 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 3704020608 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 50153784320 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 1281567189040 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 202117552128 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 21859766784 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 48689278976 \) Copy content Toggle raw display
show more
show less