Properties

Label 9075.2.a.dl
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.9444552.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 7x^{2} + 9x - 6 \) 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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 8\nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 8\beta_{2} + 16 \) 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
2.65987
1.39056
0.583819
−1.09369
−2.54056
−2.65987 1.00000 5.07489 0 −2.65987 −0.454950 −8.17880 1.00000 0
1.2 −1.39056 1.00000 −0.0663358 0 −1.39056 4.73026 2.87337 1.00000 0
1.3 −0.583819 1.00000 −1.65916 0 −0.583819 −4.38942 2.13628 1.00000 0
1.4 1.09369 1.00000 −0.803848 0 1.09369 1.13843 −3.06653 1.00000 0
1.5 2.54056 1.00000 4.45445 0 2.54056 2.97568 6.23568 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(-1\)

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 9075.2.a.dl yes 5
5.b even 2 1 9075.2.a.dm yes 5
11.b odd 2 1 9075.2.a.dn yes 5
55.d odd 2 1 9075.2.a.dk 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.dk 5 55.d odd 2 1
9075.2.a.dl yes 5 1.a even 1 1 trivial
9075.2.a.dm yes 5 5.b even 2 1
9075.2.a.dn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{5} + T_{2}^{4} - 8T_{2}^{3} - 7T_{2}^{2} + 9T_{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{5} - 4T_{7}^{4} - 18T_{7}^{3} + 77T_{7}^{2} - 32T_{7} - 32 \) Copy content Toggle raw display
\( T_{13}^{5} + 4T_{13}^{4} - 37T_{13}^{3} - 51T_{13}^{2} + 464T_{13} - 509 \) Copy content Toggle raw display
\( T_{17}^{5} + 8T_{17}^{4} - 7T_{17}^{3} - 130T_{17}^{2} - 108T_{17} + 216 \) Copy content Toggle raw display
\( T_{19}^{5} - 6T_{19}^{4} - 64T_{19}^{3} + 378T_{19}^{2} - 41T_{19} - 1244 \) Copy content Toggle raw display
\( T_{23}^{5} + 9T_{23}^{4} - 13T_{23}^{3} - 236T_{23}^{2} - 312T_{23} - 96 \) Copy content Toggle raw display
\( T_{37}^{5} - T_{37}^{4} - 101T_{37}^{3} + 221T_{37}^{2} + 112T_{37} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 8 T^{3} - 7 T^{2} + 9 T + 6 \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} - 18 T^{3} + 77 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 37 T^{3} - 51 T^{2} + \cdots - 509 \) Copy content Toggle raw display
$17$ \( T^{5} + 8 T^{4} - 7 T^{3} - 130 T^{2} + \cdots + 216 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} - 64 T^{3} + \cdots - 1244 \) Copy content Toggle raw display
$23$ \( T^{5} + 9 T^{4} - 13 T^{3} - 236 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} - 75 T^{3} + 92 T^{2} + \cdots - 768 \) Copy content Toggle raw display
$31$ \( T^{5} + 3 T^{4} - 128 T^{3} + \cdots - 9040 \) Copy content Toggle raw display
$37$ \( T^{5} - T^{4} - 101 T^{3} + 221 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$41$ \( T^{5} - T^{4} - 157 T^{3} + \cdots - 16032 \) Copy content Toggle raw display
$43$ \( T^{5} + 7 T^{4} - 57 T^{3} + \cdots + 1756 \) Copy content Toggle raw display
$47$ \( T^{5} + 14 T^{4} - 56 T^{3} + \cdots - 1536 \) Copy content Toggle raw display
$53$ \( T^{5} - 3 T^{4} - 97 T^{3} + \cdots - 8112 \) Copy content Toggle raw display
$59$ \( T^{5} - 18 T^{4} - 67 T^{3} + \cdots - 480 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} - 135 T^{3} + \cdots - 1352 \) Copy content Toggle raw display
$67$ \( T^{5} - 12 T^{4} - 63 T^{3} + \cdots - 17104 \) Copy content Toggle raw display
$71$ \( T^{5} - 8 T^{4} - 227 T^{3} + \cdots - 15600 \) Copy content Toggle raw display
$73$ \( T^{5} + 16 T^{4} - 159 T^{3} + \cdots - 47507 \) Copy content Toggle raw display
$79$ \( T^{5} + 11 T^{4} - 3 T^{3} - 303 T^{2} + \cdots - 844 \) Copy content Toggle raw display
$83$ \( T^{5} - 39 T^{4} + 423 T^{3} + \cdots + 89328 \) Copy content Toggle raw display
$89$ \( T^{5} + T^{4} - 157 T^{3} + \cdots + 16032 \) Copy content Toggle raw display
$97$ \( T^{5} - 11 T^{4} - 94 T^{3} + \cdots - 2000 \) Copy content Toggle raw display
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