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Properties

Label 4730.2.a.bc
Level $4730$
Weight $2$
Character orbit 4730.a
Self dual yes
Analytic conductor $37.769$
Analytic rank $0$
Dimension $11$
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] = [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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 24x^{9} - x^{8} + 200x^{7} + 14x^{6} - 653x^{5} - 26x^{4} + 620x^{3} - 177x^{2} - 90x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 24x^{9} - x^{8} + 200x^{7} + 14x^{6} - 653x^{5} - 26x^{4} + 620x^{3} - 177x^{2} - 90x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52 \nu^{10} - 35 \nu^{9} - 1205 \nu^{8} + 947 \nu^{7} + 10132 \nu^{6} - 8658 \nu^{5} - 38070 \nu^{4} + \cdots - 9602 ) / 674 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 75 \nu^{10} + 44 \nu^{9} + 1900 \nu^{8} - 815 \nu^{7} - 17128 \nu^{6} + 4568 \nu^{5} + 63891 \nu^{4} + \cdots + 15884 ) / 674 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 91 \nu^{10} - 23 \nu^{9} + 2193 \nu^{8} + 786 \nu^{7} - 18068 \nu^{6} - 8270 \nu^{5} + 55333 \nu^{4} + \cdots + 3492 ) / 674 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 91 \nu^{10} - 23 \nu^{9} + 2193 \nu^{8} + 786 \nu^{7} - 18068 \nu^{6} - 8270 \nu^{5} + 55333 \nu^{4} + \cdots + 2818 ) / 674 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 210 \nu^{10} + 79 \nu^{9} - 4983 \nu^{8} - 2099 \nu^{7} + 40814 \nu^{6} + 18618 \nu^{5} - 128884 \nu^{4} + \cdots - 17650 ) / 674 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 374 \nu^{10} + 176 \nu^{9} - 8913 \nu^{8} - 4608 \nu^{7} + 73028 \nu^{6} + 40177 \nu^{5} + \cdots - 26780 ) / 337 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 843 \nu^{10} - 476 \nu^{9} + 20008 \nu^{8} + 12003 \nu^{7} - 162674 \nu^{6} - 101438 \nu^{5} + \cdots + 49236 ) / 674 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 487 \nu^{10} - 249 \nu^{9} + 11551 \nu^{8} + 6458 \nu^{7} - 93905 \nu^{6} - 55905 \nu^{5} + \cdots + 30083 ) / 337 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 8\beta_{2} + \beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} - 2\beta_{8} + 3\beta_{7} + 11\beta_{6} - 11\beta_{5} - \beta_{4} - \beta_{3} + 65\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{10} + \beta_{9} + 17 \beta_{8} + 18 \beta_{7} - 12 \beta_{6} - \beta_{5} + 16 \beta_{4} + \cdots + 229 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{10} - 19 \beta_{9} - 33 \beta_{8} + 47 \beta_{7} + 105 \beta_{6} - 108 \beta_{5} - 16 \beta_{4} + \cdots + 128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 188 \beta_{10} + 21 \beta_{9} + 207 \beta_{8} + 233 \beta_{7} - 116 \beta_{6} - 22 \beta_{5} + \cdots + 1872 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 48 \beta_{10} - 256 \beta_{9} - 396 \beta_{8} + 556 \beta_{7} + 968 \beta_{6} - 1015 \beta_{5} + \cdots + 1177 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1967 \beta_{10} + 299 \beta_{9} + 2218 \beta_{8} + 2642 \beta_{7} - 1056 \beta_{6} - 318 \beta_{5} + \cdots + 15643 \) 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
−3.00504
−2.74176
−2.26805
−1.44320
−0.419974
0.429703
0.510396
0.604446
2.60939
2.70832
3.01577
−1.00000 −3.00504 1.00000 −1.00000 3.00504 −1.41479 −1.00000 6.03028 1.00000
1.2 −1.00000 −2.74176 1.00000 −1.00000 2.74176 3.45531 −1.00000 4.51722 1.00000
1.3 −1.00000 −2.26805 1.00000 −1.00000 2.26805 −4.08168 −1.00000 2.14403 1.00000
1.4 −1.00000 −1.44320 1.00000 −1.00000 1.44320 −0.640387 −1.00000 −0.917168 1.00000
1.5 −1.00000 −0.419974 1.00000 −1.00000 0.419974 3.08102 −1.00000 −2.82362 1.00000
1.6 −1.00000 0.429703 1.00000 −1.00000 −0.429703 2.14075 −1.00000 −2.81536 1.00000
1.7 −1.00000 0.510396 1.00000 −1.00000 −0.510396 −1.71358 −1.00000 −2.73950 1.00000
1.8 −1.00000 0.604446 1.00000 −1.00000 −0.604446 −3.36004 −1.00000 −2.63465 1.00000
1.9 −1.00000 2.60939 1.00000 −1.00000 −2.60939 2.64901 −1.00000 3.80893 1.00000
1.10 −1.00000 2.70832 1.00000 −1.00000 −2.70832 4.04483 −1.00000 4.33498 1.00000
1.11 −1.00000 3.01577 1.00000 −1.00000 −3.01577 −4.16045 −1.00000 6.09484 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.bc ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bc ✓ 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{11} - 24T_{3}^{9} - T_{3}^{8} + 200T_{3}^{7} + 14T_{3}^{6} - 653T_{3}^{5} - 26T_{3}^{4} + 620T_{3}^{3} - 177T_{3}^{2} - 90T_{3} + 32 \) Copy content Toggle raw display
\( T_{7}^{11} - 50 T_{7}^{9} + 7 T_{7}^{8} + 928 T_{7}^{7} - 190 T_{7}^{6} - 7819 T_{7}^{5} + 1066 T_{7}^{4} + \cdots - 21632 \) Copy content Toggle raw display
\( T_{13}^{11} + T_{13}^{10} - 102 T_{13}^{9} - 98 T_{13}^{8} + 3468 T_{13}^{7} + 3936 T_{13}^{6} + \cdots - 161792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} - 24 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} - 50 T^{9} + \cdots - 21632 \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + T^{10} + \cdots - 161792 \) Copy content Toggle raw display
$17$ \( T^{11} + 15 T^{10} + \cdots - 497152 \) Copy content Toggle raw display
$19$ \( T^{11} - 14 T^{10} + \cdots - 852948 \) Copy content Toggle raw display
$23$ \( T^{11} + 2 T^{10} + \cdots - 29730304 \) Copy content Toggle raw display
$29$ \( T^{11} - 6 T^{10} + \cdots + 127136 \) Copy content Toggle raw display
$31$ \( T^{11} - 13 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( T^{11} - 16 T^{10} + \cdots + 29842432 \) Copy content Toggle raw display
$41$ \( T^{11} + 7 T^{10} + \cdots + 88832 \) Copy content Toggle raw display
$43$ \( (T + 1)^{11} \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 182585888 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 119622112 \) Copy content Toggle raw display
$59$ \( T^{11} - 7 T^{10} + \cdots + 3444736 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 1210970272 \) Copy content Toggle raw display
$67$ \( T^{11} - 9 T^{10} + \cdots - 4046848 \) Copy content Toggle raw display
$71$ \( T^{11} - 13 T^{10} + \cdots + 3249536 \) Copy content Toggle raw display
$73$ \( T^{11} - 20 T^{10} + \cdots + 49266304 \) Copy content Toggle raw display
$79$ \( T^{11} - 13 T^{10} + \cdots + 57392212 \) Copy content Toggle raw display
$83$ \( T^{11} + 6 T^{10} + \cdots + 42752 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 9949351936 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 25203889664 \) Copy content Toggle raw display
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