Properties

Label 4235.2.a.z
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.270017.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 7x^{2} + 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + 1) q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{6} - q^{7} + (\beta_{3} - \beta_{2}) q^{8} + (\beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + 1) q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{6} - q^{7} + (\beta_{3} - \beta_{2}) q^{8} + (\beta_{2} + \beta_1 - 1) q^{9} - \beta_{2} q^{10} + (2 \beta_{3} - \beta_{2} - 1) q^{12} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{13} + \beta_{2} q^{14} + \beta_1 q^{15} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{16} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 1) q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{18} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{4} + 1) q^{20} - \beta_1 q^{21} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{4} + \beta_{2} + 1) q^{24} + q^{25} + ( - \beta_{4} + 2 \beta_{2} - 4) q^{26} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{27} + ( - \beta_{4} - 1) q^{28} + ( - \beta_{4} + \beta_{3} - 4) q^{29} + ( - \beta_{3} - \beta_1 + 1) q^{30} + (\beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{31} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{32} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 6) q^{34} - q^{35} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{36} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{37} + (2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 5) q^{38} + (\beta_{4} + 2 \beta_{3} - \beta_1 - 3) q^{39} + (\beta_{3} - \beta_{2}) q^{40} + ( - \beta_{4} - 3 \beta_{2}) q^{41} + (\beta_{3} + \beta_1 - 1) q^{42} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1 - 1) q^{45} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{46} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{47} + ( - \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{48} + q^{49} - \beta_{2} q^{50} + ( - 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{51} + ( - 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{52} + (2 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{53} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 5) q^{54} + ( - \beta_{3} + \beta_{2}) q^{56} + (2 \beta_{4} + \beta_{2} - 2 \beta_1) q^{57} + ( - 2 \beta_{3} + 6 \beta_{2} - \beta_1) q^{58} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{59} + (2 \beta_{3} - \beta_{2} - 1) q^{60} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 5) q^{61} + ( - \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 2) q^{62} + ( - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 9) q^{64} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{65} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{67} + ( - \beta_{4} - \beta_{3} + 6 \beta_{2} - 5 \beta_1 - 4) q^{68} + ( - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{69} + \beta_{2} q^{70} + ( - 2 \beta_{4} + 4 \beta_{2} - 5 \beta_1 - 2) q^{71} + (2 \beta_{2} + \beta_1 - 2) q^{72} + (\beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1) q^{73} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{74} + \beta_1 q^{75} + ( - \beta_{4} - 5 \beta_{2} + 4 \beta_1 - 2) q^{76} + (\beta_{2} - \beta_1 - 1) q^{78} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{79} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{80} + (\beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{81} + (3 \beta_{4} - \beta_{3} + 2 \beta_{2} + 9) q^{82} + (2 \beta_{4} + 7 \beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{83} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{84} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 1) q^{85} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + 4 \beta_1) q^{86} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{87} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1) q^{89} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{90} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{91} + (4 \beta_{3} + \beta_{2} - 3 \beta_1 - 6) q^{92} + ( - 3 \beta_{4} - 3 \beta_{2} + \beta_1) q^{93} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 8) q^{94} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{95} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{96} + ( - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 4) q^{97} - \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - 5 q^{7} - q^{9} - 2 q^{10} - 3 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 6 q^{17} - 11 q^{18} - 7 q^{19} + 4 q^{20} - 2 q^{21} + 3 q^{23} + 6 q^{24} + 5 q^{25} - 15 q^{26} + 5 q^{27} - 4 q^{28} - 17 q^{29} + q^{30} + 12 q^{31} + 3 q^{32} - 12 q^{34} - 5 q^{35} - 3 q^{36} - 2 q^{37} + 21 q^{38} - 14 q^{39} - 5 q^{41} - q^{42} - 4 q^{43} - q^{45} - 2 q^{46} - 12 q^{48} + 5 q^{49} - 2 q^{50} - 14 q^{51} - 2 q^{52} + 8 q^{53} - 22 q^{54} - 4 q^{57} + 6 q^{58} - 16 q^{59} - 3 q^{60} - 24 q^{61} - 9 q^{62} + q^{63} - 38 q^{64} - 8 q^{65} - 9 q^{67} - 19 q^{68} + 22 q^{69} + 2 q^{70} - 10 q^{71} - 4 q^{72} - 11 q^{73} + q^{74} + 2 q^{75} - 11 q^{76} - 5 q^{78} - 9 q^{79} + 2 q^{80} - 19 q^{81} + 44 q^{82} - 10 q^{83} + 3 q^{84} - 6 q^{85} + 15 q^{86} + 2 q^{87} - 9 q^{89} - 11 q^{90} + 8 q^{91} - 26 q^{92} - q^{93} - 34 q^{94} - 7 q^{95} + 18 q^{96} + 11 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 7x^{2} + 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \) Copy content Toggle raw display

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
−1.64458
2.51766
−1.11334
1.89725
0.343016
−2.34922 −1.64458 3.51886 1.00000 3.86349 −1.00000 −3.56813 −0.295356 −2.34922
1.2 −1.82093 2.51766 1.31579 1.00000 −4.58448 −1.00000 1.24589 3.33859 −1.82093
1.3 −0.352882 −1.11334 −1.87547 1.00000 0.392879 −1.00000 1.36758 −1.76046 −0.352882
1.4 0.297682 1.89725 −1.91139 1.00000 0.564779 −1.00000 −1.16435 0.599571 0.297682
1.5 2.22536 0.343016 2.95221 1.00000 0.763334 −1.00000 2.11901 −2.88234 2.22536
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 4235.2.a.z 5
11.b odd 2 1 4235.2.a.bf yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.z 5 1.a even 1 1 trivial
4235.2.a.bf yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 10T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{5} - 2T_{3}^{4} - 5T_{3}^{3} + 7T_{3}^{2} + 7T_{3} - 3 \) Copy content Toggle raw display
\( T_{13}^{5} + 8T_{13}^{4} + 4T_{13}^{3} - 45T_{13}^{2} + 40T_{13} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 5 T^{3} - 10 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} - 5 T^{3} + 7 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} + 4 T^{3} - 45 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$17$ \( T^{5} + 6 T^{4} - 62 T^{3} + \cdots + 3465 \) Copy content Toggle raw display
$19$ \( T^{5} + 7 T^{4} - 19 T^{3} - 124 T^{2} + \cdots + 287 \) Copy content Toggle raw display
$23$ \( T^{5} - 3 T^{4} - 48 T^{3} - 101 T^{2} + \cdots + 31 \) Copy content Toggle raw display
$29$ \( T^{5} + 17 T^{4} + 92 T^{3} + 161 T^{2} + \cdots - 89 \) Copy content Toggle raw display
$31$ \( T^{5} - 12 T^{4} - 19 T^{3} + \cdots - 1823 \) Copy content Toggle raw display
$37$ \( T^{5} + 2 T^{4} - 73 T^{3} + 157 T^{2} + \cdots - 411 \) Copy content Toggle raw display
$41$ \( T^{5} + 5 T^{4} - 70 T^{3} + \cdots - 1805 \) Copy content Toggle raw display
$43$ \( T^{5} + 4 T^{4} - 77 T^{3} + 163 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$47$ \( T^{5} - 91 T^{3} + 40 T^{2} + \cdots - 2573 \) Copy content Toggle raw display
$53$ \( T^{5} - 8 T^{4} - 89 T^{3} + \cdots + 5889 \) Copy content Toggle raw display
$59$ \( T^{5} + 16 T^{4} + 14 T^{3} + \cdots + 1833 \) Copy content Toggle raw display
$61$ \( T^{5} + 24 T^{4} + 135 T^{3} + \cdots + 2475 \) Copy content Toggle raw display
$67$ \( T^{5} + 9 T^{4} - 55 T^{3} - 546 T^{2} + \cdots + 67 \) Copy content Toggle raw display
$71$ \( T^{5} + 10 T^{4} - 255 T^{3} + \cdots + 102033 \) Copy content Toggle raw display
$73$ \( T^{5} + 11 T^{4} - 47 T^{3} + \cdots - 1671 \) Copy content Toggle raw display
$79$ \( T^{5} + 9 T^{4} - 147 T^{3} + \cdots + 151 \) Copy content Toggle raw display
$83$ \( T^{5} + 10 T^{4} - 352 T^{3} + \cdots - 7719 \) Copy content Toggle raw display
$89$ \( T^{5} + 9 T^{4} - 69 T^{3} + \cdots - 3181 \) Copy content Toggle raw display
$97$ \( T^{5} - 11 T^{4} - 227 T^{3} + \cdots - 47833 \) Copy content Toggle raw display
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