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Properties

Label 1335.2.a.h
Level $1335$
Weight $2$
Character orbit 1335.a
Self dual yes
Analytic conductor $10.660$
Analytic rank $0$
Dimension $9$
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] = [1335,2,Mod(1,1335)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1335, 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("1335.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1335 = 3 \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1335.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: \(10.6600286698\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} - q^{5} + (\beta_1 - 1) q^{6} + (\beta_{8} - \beta_{3} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 3) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} - q^{5} + (\beta_1 - 1) q^{6} + (\beta_{8} - \beta_{3} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 3) q^{8}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} - 9 q^{3} + 11 q^{4} - 9 q^{5} - 5 q^{6} - 3 q^{7} + 18 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{2} - 9 q^{3} + 11 q^{4} - 9 q^{5} - 5 q^{6} - 3 q^{7} + 18 q^{8} + 9 q^{9} - 5 q^{10} + 4 q^{11} - 11 q^{12} + 5 q^{13} - q^{14} + 9 q^{15} + 15 q^{16} + 21 q^{17} + 5 q^{18} - 18 q^{19} - 11 q^{20} + 3 q^{21} + 6 q^{22} + 16 q^{23} - 18 q^{24} + 9 q^{25} + 8 q^{26} - 9 q^{27} + 6 q^{28} + 3 q^{29} + 5 q^{30} - 6 q^{31} + 46 q^{32} - 4 q^{33} + 12 q^{34} + 3 q^{35} + 11 q^{36} + 11 q^{37} + 20 q^{38} - 5 q^{39} - 18 q^{40} - q^{41} + q^{42} - 3 q^{43} + 38 q^{44} - 9 q^{45} + 16 q^{46} + 27 q^{47} - 15 q^{48} + 24 q^{49} + 5 q^{50} - 21 q^{51} + 17 q^{52} + 43 q^{53} - 5 q^{54} - 4 q^{55} + 5 q^{56} + 18 q^{57} + 34 q^{58} + 3 q^{59} + 11 q^{60} - 30 q^{61} + 36 q^{62} - 3 q^{63} + 50 q^{64} - 5 q^{65} - 6 q^{66} - 12 q^{67} + 64 q^{68} - 16 q^{69} + q^{70} - 4 q^{71} + 18 q^{72} + 26 q^{73} + 2 q^{74} - 9 q^{75} - 12 q^{76} + 34 q^{77} - 8 q^{78} + q^{79} - 15 q^{80} + 9 q^{81} - 51 q^{82} + 24 q^{83} - 6 q^{84} - 21 q^{85} + 18 q^{86} - 3 q^{87} + 64 q^{88} + 9 q^{89} - 5 q^{90} - 50 q^{91} + 10 q^{92} + 6 q^{93} - 11 q^{94} + 18 q^{95} - 46 q^{96} - 4 q^{97} + 75 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 13\nu^{3} + 3\nu^{2} - 12\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 4\nu^{4} - 12\nu^{3} - 5\nu^{2} + 8\nu + 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - 5\nu^{7} + \nu^{6} + 23\nu^{5} - 16\nu^{4} - 28\nu^{3} + 12\nu^{2} + 10\nu + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 5\nu^{7} + 27\nu^{5} - 15\nu^{4} - 44\nu^{3} + 19\nu^{2} + 22\nu - 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{8} + 6\nu^{7} - 5\nu^{6} - 25\nu^{5} + 35\nu^{4} + 24\nu^{3} - 35\nu^{2} - 5\nu + 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 12\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - \beta_{6} + 9\beta_{5} + 10\beta_{4} + 3\beta_{3} + 20\beta_{2} + 44\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{7} - 3\beta_{6} + 22\beta_{5} + 26\beta_{4} + 13\beta_{3} + 63\beta_{2} + 111\beta _1 + 106 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + 14\beta_{7} - 13\beta_{6} + 72\beta_{5} + 90\beta_{4} + 39\beta_{3} + 171\beta_{2} + 346\beta _1 + 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5 \beta_{8} + 44 \beta_{7} - 38 \beta_{6} + 191 \beta_{5} + 254 \beta_{4} + 129 \beta_{3} + 504 \beta_{2} + \cdots + 787 \) 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
2.88084
2.81142
1.89144
1.09195
0.0690386
−0.286244
−1.02867
−1.70432
−1.72546
−1.88084 −1.00000 1.53757 −1.00000 1.88084 −0.422231 0.869761 1.00000 1.88084
1.2 −1.81142 −1.00000 1.28123 −1.00000 1.81142 −0.549078 1.30199 1.00000 1.81142
1.3 −0.891444 −1.00000 −1.20533 −1.00000 0.891444 3.64096 2.85737 1.00000 0.891444
1.4 −0.0919478 −1.00000 −1.99155 −1.00000 0.0919478 −4.79397 0.367014 1.00000 0.0919478
1.5 0.930961 −1.00000 −1.13331 −1.00000 −0.930961 −1.89313 −2.91699 1.00000 −0.930961
1.6 1.28624 −1.00000 −0.345575 −1.00000 −1.28624 −0.646623 −3.01698 1.00000 −1.28624
1.7 2.02867 −1.00000 2.11549 −1.00000 −2.02867 2.64395 0.234298 1.00000 −2.02867
1.8 2.70432 −1.00000 5.31333 −1.00000 −2.70432 3.91511 8.96030 1.00000 −2.70432
1.9 2.72546 −1.00000 5.42813 −1.00000 −2.72546 −4.89499 9.34325 1.00000 −2.72546
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(89\) \(-1\)

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 1335.2.a.h ✓ 9
3.b odd 2 1 4005.2.a.q 9
5.b even 2 1 6675.2.a.x 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.h ✓ 9 1.a even 1 1 trivial
4005.2.a.q 9 3.b odd 2 1
6675.2.a.x 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 5T_{2}^{8} - 2T_{2}^{7} + 39T_{2}^{6} - 25T_{2}^{5} - 91T_{2}^{4} + 83T_{2}^{3} + 56T_{2}^{2} - 50T_{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1335))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 5 T^{8} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 3 T^{8} + \cdots - 251 \) Copy content Toggle raw display
$11$ \( T^{9} - 4 T^{8} + \cdots + 8768 \) Copy content Toggle raw display
$13$ \( T^{9} - 5 T^{8} + \cdots + 14159 \) Copy content Toggle raw display
$17$ \( T^{9} - 21 T^{8} + \cdots - 347777 \) Copy content Toggle raw display
$19$ \( T^{9} + 18 T^{8} + \cdots - 425536 \) Copy content Toggle raw display
$23$ \( T^{9} - 16 T^{8} + \cdots + 10623808 \) Copy content Toggle raw display
$29$ \( T^{9} - 3 T^{8} + \cdots - 669871 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} + \cdots - 10432 \) Copy content Toggle raw display
$37$ \( T^{9} - 11 T^{8} + \cdots + 44729 \) Copy content Toggle raw display
$41$ \( T^{9} + T^{8} + \cdots + 276811 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} + \cdots + 4443017 \) Copy content Toggle raw display
$47$ \( T^{9} - 27 T^{8} + \cdots - 8178979 \) Copy content Toggle raw display
$53$ \( T^{9} - 43 T^{8} + \cdots + 1590223 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} + \cdots + 8076287 \) Copy content Toggle raw display
$61$ \( T^{9} + 30 T^{8} + \cdots - 242590144 \) Copy content Toggle raw display
$67$ \( T^{9} + 12 T^{8} + \cdots - 1216 \) Copy content Toggle raw display
$71$ \( T^{9} + 4 T^{8} + \cdots + 1069760 \) Copy content Toggle raw display
$73$ \( T^{9} - 26 T^{8} + \cdots + 58649152 \) Copy content Toggle raw display
$79$ \( T^{9} - T^{8} + \cdots + 43158037 \) Copy content Toggle raw display
$83$ \( T^{9} - 24 T^{8} + \cdots + 22784576 \) Copy content Toggle raw display
$89$ \( (T - 1)^{9} \) Copy content Toggle raw display
$97$ \( T^{9} + 4 T^{8} + \cdots - 514310656 \) Copy content Toggle raw display
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