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Properties

Label 4335.2.a.bl
Level $4335$
Weight $2$
Character orbit 4335.a
Self dual yes
Analytic conductor $34.615$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4335,2,Mod(1,4335)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4335.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4335, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4335 = 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4335.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4,-16,20,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.6151492762\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 88 x^{13} + 93 x^{12} - 728 x^{11} + 50 x^{10} + 2788 x^{9} - 1694 x^{8} + \cdots + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 255)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_1 q^{6} + \beta_{13} q^{7} + (\beta_{14} - \beta_{12} + \cdots + \beta_1) q^{8} + q^{9} - \beta_1 q^{10} + ( - \beta_{7} + \beta_1 - 2) q^{11}+ \cdots + ( - \beta_{7} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 16 q^{3} + 20 q^{4} - 16 q^{5} - 4 q^{6} + 16 q^{9} - 4 q^{10} - 24 q^{11} - 20 q^{12} + 24 q^{13} - 4 q^{14} + 16 q^{15} + 28 q^{16} + 4 q^{18} + 16 q^{19} - 20 q^{20} + 20 q^{22} - 16 q^{23}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 18 x^{14} + 88 x^{13} + 93 x^{12} - 728 x^{11} + 50 x^{10} + 2788 x^{9} - 1694 x^{8} + \cdots + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 208 \nu^{15} + 685 \nu^{14} + 4098 \nu^{13} - 15045 \nu^{12} - 27112 \nu^{11} + 123701 \nu^{10} + \cdots - 3825 ) / 1649 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 147 \nu^{15} + 354 \nu^{14} + 3259 \nu^{13} - 7768 \nu^{12} - 27723 \nu^{11} + 64972 \nu^{10} + \cdots + 3536 ) / 1649 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2638 \nu^{15} + 6909 \nu^{14} + 56833 \nu^{13} - 153038 \nu^{12} - 452814 \nu^{11} + 1281918 \nu^{10} + \cdots + 23902 ) / 1649 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5359 \nu^{15} - 13327 \nu^{14} - 116552 \nu^{13} + 295008 \nu^{12} + 943021 \nu^{11} - 2468847 \nu^{10} + \cdots - 61523 ) / 1649 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6369 \nu^{15} - 16268 \nu^{14} - 137531 \nu^{13} + 359694 \nu^{12} + 1099913 \nu^{11} - 3004874 \nu^{10} + \cdots - 72267 ) / 1649 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7165 \nu^{15} + 17981 \nu^{14} + 155150 \nu^{13} - 397469 \nu^{12} - 1246162 \nu^{11} + \cdots + 64447 ) / 1649 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8109 \nu^{15} + 20090 \nu^{14} + 176584 \nu^{13} - 444634 \nu^{12} - 1432505 \nu^{11} + \cdots + 91103 ) / 1649 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9413 \nu^{15} + 23448 \nu^{14} + 204857 \nu^{13} - 519424 \nu^{12} - 1659684 \nu^{11} + \cdots + 102323 ) / 1649 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13317 \nu^{15} + 33089 \nu^{14} + 289884 \nu^{13} - 732855 \nu^{12} - 2349438 \nu^{11} + \cdots + 147356 ) / 1649 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20179 \nu^{15} + 50178 \nu^{14} + 439041 \nu^{13} - 1110957 \nu^{12} - 3555788 \nu^{11} + \cdots + 226389 ) / 1649 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 24092 \nu^{15} + 59518 \nu^{14} + 525386 \nu^{13} - 1318616 \nu^{12} - 4271133 \nu^{11} + \cdots + 274499 ) / 1649 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25538 \nu^{15} + 63505 \nu^{14} + 555593 \nu^{13} - 1405965 \nu^{12} - 4498809 \nu^{11} + \cdots + 287912 ) / 1649 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28669 \nu^{15} - 71235 \nu^{14} - 624052 \nu^{13} + 1577411 \nu^{12} + 5058060 \nu^{11} + \cdots - 325176 ) / 1649 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{6} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 8\beta_{14} - 10\beta_{12} + \beta_{11} - \beta_{7} + 9\beta_{6} - \beta_{5} + \beta_{2} + 30\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} + 9 \beta_{14} + \beta_{13} - 22 \beta_{11} + 10 \beta_{10} - 11 \beta_{9} + \beta_{8} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{15} + 56 \beta_{14} + \beta_{13} - 80 \beta_{12} + 10 \beta_{11} + \beta_{10} - 12 \beta_{7} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{15} + 67 \beta_{14} + 13 \beta_{13} - 3 \beta_{12} - 184 \beta_{11} + 81 \beta_{10} + \cdots + 621 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 108 \beta_{15} + 384 \beta_{14} + 14 \beta_{13} - 595 \beta_{12} + 70 \beta_{11} + 22 \beta_{10} + \cdots - 53 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 144 \beta_{15} + 480 \beta_{14} + 121 \beta_{13} - 62 \beta_{12} - 1404 \beta_{11} + 617 \beta_{10} + \cdots + 4092 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 882 \beta_{15} + 2636 \beta_{14} + 137 \beta_{13} - 4298 \beta_{12} + 400 \beta_{11} + 300 \beta_{10} + \cdots - 337 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1319 \beta_{15} + 3422 \beta_{14} + 996 \beta_{13} - 823 \beta_{12} - 10314 \beta_{11} + 4592 \beta_{10} + \cdots + 27421 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6907 \beta_{15} + 18197 \beta_{14} + 1175 \beta_{13} - 30677 \beta_{12} + 1795 \beta_{11} + \cdots - 2061 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11375 \beta_{15} + 24490 \beta_{14} + 7752 \beta_{13} - 8966 \beta_{12} - 74516 \beta_{11} + \cdots + 185737 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 52970 \beta_{15} + 126391 \beta_{14} + 9484 \beta_{13} - 218012 \beta_{12} + 3974 \beta_{11} + \cdots - 11658 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.63553
−2.56608
−1.95093
−1.93806
−0.665122
−0.554529
−0.346680
−0.0626665
0.974431
1.23981
1.51302
1.51693
1.77172
2.45839
2.53103
2.71427
−2.63553 −1.00000 4.94602 −1.00000 2.63553 −0.0813704 −7.76431 1.00000 2.63553
1.2 −2.56608 −1.00000 4.58477 −1.00000 2.56608 3.02102 −6.63273 1.00000 2.56608
1.3 −1.95093 −1.00000 1.80613 −1.00000 1.95093 −2.93737 0.378219 1.00000 1.95093
1.4 −1.93806 −1.00000 1.75608 −1.00000 1.93806 −0.117550 0.472741 1.00000 1.93806
1.5 −0.665122 −1.00000 −1.55761 −1.00000 0.665122 −1.47304 2.36625 1.00000 0.665122
1.6 −0.554529 −1.00000 −1.69250 −1.00000 0.554529 5.24705 2.04760 1.00000 0.554529
1.7 −0.346680 −1.00000 −1.87981 −1.00000 0.346680 −0.931239 1.34505 1.00000 0.346680
1.8 −0.0626665 −1.00000 −1.99607 −1.00000 0.0626665 −0.772900 0.250420 1.00000 0.0626665
1.9 0.974431 −1.00000 −1.05048 −1.00000 −0.974431 −3.93632 −2.97249 1.00000 −0.974431
1.10 1.23981 −1.00000 −0.462871 −1.00000 −1.23981 2.61134 −3.05349 1.00000 −1.23981
1.11 1.51302 −1.00000 0.289222 −1.00000 −1.51302 4.74088 −2.58844 1.00000 −1.51302
1.12 1.51693 −1.00000 0.301074 −1.00000 −1.51693 −2.11666 −2.57715 1.00000 −1.51693
1.13 1.77172 −1.00000 1.13900 −1.00000 −1.77172 −4.12564 −1.52546 1.00000 −1.77172
1.14 2.45839 −1.00000 4.04366 −1.00000 −2.45839 −2.40591 5.02411 1.00000 −2.45839
1.15 2.53103 −1.00000 4.40611 −1.00000 −2.53103 −0.552938 6.08995 1.00000 −2.53103
1.16 2.71427 −1.00000 5.36729 −1.00000 −2.71427 3.83066 9.13974 1.00000 −2.71427
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(17\) \( -1 \)

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 4335.2.a.bl 16
17.b even 2 1 4335.2.a.bm 16
17.e odd 16 2 255.2.w.b ✓ 32
51.i even 16 2 765.2.be.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.w.b ✓ 32 17.e odd 16 2
765.2.be.c 32 51.i even 16 2
4335.2.a.bl 16 1.a even 1 1 trivial
4335.2.a.bm 16 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4335))\):

\( T_{2}^{16} - 4 T_{2}^{15} - 18 T_{2}^{14} + 88 T_{2}^{13} + 93 T_{2}^{12} - 728 T_{2}^{11} + 50 T_{2}^{10} + \cdots + 17 \) Copy content Toggle raw display
\( T_{7}^{16} - 68 T_{7}^{14} - 56 T_{7}^{13} + 1744 T_{7}^{12} + 2792 T_{7}^{11} - 19888 T_{7}^{10} + \cdots - 1024 \) Copy content Toggle raw display
\( T_{11}^{16} + 24 T_{11}^{15} + 168 T_{11}^{14} - 312 T_{11}^{13} - 8768 T_{11}^{12} - 25120 T_{11}^{11} + \cdots - 10863104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 17 \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 68 T^{14} + \cdots - 1024 \) Copy content Toggle raw display
$11$ \( T^{16} + 24 T^{15} + \cdots - 10863104 \) Copy content Toggle raw display
$13$ \( T^{16} - 24 T^{15} + \cdots + 10402816 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 200225264 \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{15} + \cdots - 4553728 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 117755420416 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 2296218352 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 5149212416 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 2884509952 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 104405106688 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 52688718716 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4607667152516 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 30652112896 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 7924993914128 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 1630642767872 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 1123436068864 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 998765010688 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 36914471301376 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 5171690970176 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 57520946176 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 180471144779776 \) Copy content Toggle raw display
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