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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [535,2,Mod(1,535)] 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("535.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(535, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 535 = 5 \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 535.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.27199650814\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 18 x^{13} + 85 x^{12} + 103 x^{11} - 685 x^{10} - 108 x^{9} + 2595 x^{8} - 846 x^{7} + \cdots + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{6} + \beta_{9} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{12} + 2) q^{9}+ \cdots + (4 \beta_{14} - \beta_{13} - 2 \beta_{12} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 4 q^{2} + 22 q^{4} + 15 q^{5} + 4 q^{6} - 3 q^{7} + 9 q^{8} + 31 q^{9} + 4 q^{10} + 11 q^{11} - 16 q^{12} - 4 q^{13} + 32 q^{16} + 13 q^{17} - 6 q^{18} + 5 q^{19} + 22 q^{20} + 12 q^{21} - 13 q^{22}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 18 x^{13} + 85 x^{12} + 103 x^{11} - 685 x^{10} - 108 x^{9} + 2595 x^{8} - 846 x^{7} + \cdots + 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29653 \nu^{14} - 153204 \nu^{13} - 324074 \nu^{12} + 2964401 \nu^{11} - 1080009 \nu^{10} + \cdots - 2309336 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32419 \nu^{14} + 11440 \nu^{13} - 1072930 \nu^{12} - 142901 \nu^{11} + 13579225 \nu^{10} + \cdots - 26375624 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15518 \nu^{14} - 35441 \nu^{13} - 349251 \nu^{12} + 705375 \nu^{11} + 3124804 \nu^{10} + \cdots + 535919 ) / 856351 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 124531 \nu^{14} - 828916 \nu^{13} - 1602182 \nu^{12} + 17702427 \nu^{11} - 153159 \nu^{10} + \cdots + 30649100 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 139933 \nu^{14} - 149896 \nu^{13} - 3505794 \nu^{12} + 3275501 \nu^{11} + 34811519 \nu^{10} + \cdots - 28617760 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71053 \nu^{14} + 185508 \nu^{13} + 1438380 \nu^{12} - 3776335 \nu^{11} - 10968261 \nu^{10} + \cdots + 765232 ) / 1712702 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 91364 \nu^{14} + 396952 \nu^{13} + 1516221 \nu^{12} - 8324583 \nu^{11} - 6696679 \nu^{10} + \cdots - 13024898 ) / 1712702 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 198447 \nu^{14} - 893542 \nu^{13} - 3486266 \nu^{12} + 19575191 \nu^{11} + 17408807 \nu^{10} + \cdots + 16691312 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 198517 \nu^{14} + 660272 \nu^{13} + 3744228 \nu^{12} - 14281667 \nu^{11} - 23206445 \nu^{10} + \cdots - 23973320 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 392261 \nu^{14} + 1444900 \nu^{13} + 7344226 \nu^{12} - 30548177 \nu^{11} - 46045883 \nu^{10} + \cdots - 32246576 ) / 3425404 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 455253 \nu^{14} - 1701562 \nu^{13} - 8461512 \nu^{12} + 35975803 \nu^{11} + 52532931 \nu^{10} + \cdots + 26355836 ) / 3425404 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 10\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{13} + 2\beta_{11} - \beta_{9} + 13\beta_{6} - 10\beta_{5} - 8\beta_{4} + 3\beta_{3} + 55\beta_{2} - \beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{14} - \beta_{13} - 13 \beta_{12} + \beta_{11} + 16 \beta_{10} - 12 \beta_{8} + \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{14} + 29 \beta_{13} - \beta_{12} + 26 \beta_{11} + \beta_{10} - 10 \beta_{9} + 2 \beta_{8} + \cdots + 502 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{14} - 14 \beta_{13} - 123 \beta_{12} + 16 \beta_{11} + 173 \beta_{10} + \beta_{9} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 40 \beta_{14} + 298 \beta_{13} - 19 \beta_{12} + 244 \beta_{11} + 27 \beta_{10} - 73 \beta_{9} + \cdots + 3141 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 88 \beta_{14} - 122 \beta_{13} - 1038 \beta_{12} + 174 \beta_{11} + 1596 \beta_{10} + 20 \beta_{9} + \cdots - 256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 518 \beta_{14} + 2677 \beta_{13} - 249 \beta_{12} + 2028 \beta_{11} + 413 \beta_{10} - 485 \beta_{9} + \cdots + 20352 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 432 \beta_{14} - 801 \beta_{13} - 8319 \beta_{12} + 1612 \beta_{11} + 13555 \beta_{10} + 245 \beta_{9} + \cdots - 1896 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5547 \beta_{14} + 22483 \beta_{13} - 2793 \beta_{12} + 15895 \beta_{11} + 4900 \beta_{10} + \cdots + 135529 \) 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
−2.65525
−2.26066
−2.21112
−1.66280
−1.10132
−0.249807
−0.159430
0.294826
0.929835
1.37219
1.75312
2.17355
2.34534
2.69560
2.73594
−2.65525 1.05116 5.05036 1.00000 −2.79109 −2.21519 −8.09946 −1.89506 −2.65525
1.2 −2.26066 −3.37419 3.11058 1.00000 7.62790 3.05004 −2.51063 8.38518 −2.26066
1.3 −2.21112 −2.06361 2.88907 1.00000 4.56290 −3.86956 −1.96583 1.25850 −2.21112
1.4 −1.66280 2.48222 0.764913 1.00000 −4.12744 4.00330 2.05371 3.16141 −1.66280
1.5 −1.10132 1.43544 −0.787100 1.00000 −1.58087 −1.10040 3.06948 −0.939517 −1.10132
1.6 −0.249807 −2.83718 −1.93760 1.00000 0.708747 −4.50012 0.983637 5.04962 −0.249807
1.7 −0.159430 3.42127 −1.97458 1.00000 −0.545452 −3.14072 0.633666 8.70507 −0.159430
1.8 0.294826 −1.05609 −1.91308 1.00000 −0.311361 5.08213 −1.15368 −1.88468 0.294826
1.9 0.929835 −2.06331 −1.13541 1.00000 −1.91854 −0.411004 −2.91541 1.25724 0.929835
1.10 1.37219 2.85855 −0.117106 1.00000 3.92246 2.38589 −2.90506 5.17131 1.37219
1.11 1.75312 1.25725 1.07343 1.00000 2.20411 2.36067 −1.62439 −1.41933 1.75312
1.12 2.17355 −0.299223 2.72430 1.00000 −0.650374 2.06931 1.57430 −2.91047 2.17355
1.13 2.34534 2.21934 3.50063 1.00000 5.20511 −4.22682 3.51949 1.92546 2.34534
1.14 2.69560 0.292496 5.26625 1.00000 0.788451 0.751252 8.80451 −2.91445 2.69560
1.15 2.73594 −3.32411 5.48534 1.00000 −9.09455 −3.23876 9.53568 8.04970 2.73594
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(107\) \( +1 \)

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 535.2.a.d ✓ 15
3.b odd 2 1 4815.2.a.v 15
4.b odd 2 1 8560.2.a.bk 15
5.b even 2 1 2675.2.a.l 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
535.2.a.d ✓ 15 1.a even 1 1 trivial
2675.2.a.l 15 5.b even 2 1
4815.2.a.v 15 3.b odd 2 1
8560.2.a.bk 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 4 T_{2}^{14} - 18 T_{2}^{13} + 85 T_{2}^{12} + 103 T_{2}^{11} - 685 T_{2}^{10} - 108 T_{2}^{9} + \cdots + 24 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(535))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 4 T^{14} + \cdots + 24 \) Copy content Toggle raw display
$3$ \( T^{15} - 38 T^{13} + \cdots + 1280 \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 3 T^{14} + \cdots - 407552 \) Copy content Toggle raw display
$11$ \( T^{15} - 11 T^{14} + \cdots - 1525884 \) Copy content Toggle raw display
$13$ \( T^{15} + 4 T^{14} + \cdots - 24704 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 215084160 \) Copy content Toggle raw display
$19$ \( T^{15} - 5 T^{14} + \cdots + 26498980 \) Copy content Toggle raw display
$23$ \( T^{15} - 8 T^{14} + \cdots - 135168 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 8477470632 \) Copy content Toggle raw display
$31$ \( T^{15} - 6 T^{14} + \cdots + 32768 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 3905607808 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 1465913730 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 501325696 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 16063905792 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 12481769088 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 1013608611840 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 16405315786 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 311984045440 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 41297859035136 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 4781058304 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 10613237888 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 13225721856 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 798087510 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 869915648704 \) Copy content Toggle raw display
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