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Properties

Label 1525.2.a.o
Level $1525$
Weight $2$
Character orbit 1525.a
Self dual yes
Analytic conductor $12.177$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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] = [1525,2,Mod(1,1525)] 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("1525.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1525 = 5^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1525.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,26,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.1771863082\)
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: \( x^{16} - 29x^{14} + 339x^{12} - 2042x^{10} + 6724x^{8} - 11877x^{6} + 10407x^{4} - 3852x^{2} + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 305)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{8} - \beta_{6} - \beta_{2}) q^{6} - \beta_{9} q^{7} + (\beta_{10} - \beta_{9} + \beta_1) q^{8} + ( - \beta_{13} - \beta_{11} - \beta_{7} + \cdots + 2) q^{9}+ \cdots + ( - 3 \beta_{13} - 2 \beta_{11} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 26 q^{4} + 28 q^{9} + 30 q^{11} + 14 q^{14} + 42 q^{16} + 18 q^{19} - 12 q^{21} - 12 q^{24} + 4 q^{26} + 34 q^{29} + 10 q^{31} - 58 q^{34} + 62 q^{36} + 18 q^{39} + 54 q^{41} + 52 q^{44} - 2 q^{49}+ \cdots + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 29x^{14} + 339x^{12} - 2042x^{10} + 6724x^{8} - 11877x^{6} + 10407x^{4} - 3852x^{2} + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{15} - 298 \nu^{13} + 3099 \nu^{11} - 15049 \nu^{9} + 31397 \nu^{7} - 10380 \nu^{5} + \cdots + 15651 \nu ) / 3024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{15} + 95\nu^{13} - 831\nu^{11} + 3044\nu^{9} - 2221\nu^{7} - 13140\nu^{5} + 28092\nu^{3} - 12627\nu ) / 756 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} - 23\nu^{10} + 204\nu^{8} - 875\nu^{6} + 1822\nu^{4} - 1506\nu^{2} + 249 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{12} - 23\nu^{10} + 204\nu^{8} - 875\nu^{6} + 1858\nu^{4} - 1794\nu^{2} + 573 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} + 26\nu^{12} - 261\nu^{10} + 1259\nu^{8} - 2923\nu^{6} + 2724\nu^{4} - 339\nu^{2} - 117 ) / 72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{14} - 34\nu^{12} + 445\nu^{10} - 2843\nu^{8} + 9227\nu^{6} - 14444\nu^{4} + 9795\nu^{2} - 1947 ) / 144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{15} - 704 \nu^{13} + 7887 \nu^{11} - 44603 \nu^{9} + 133093 \nu^{7} - 198792 \nu^{5} + \cdots - 16857 \nu ) / 1512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25 \nu^{15} - 704 \nu^{13} + 7887 \nu^{11} - 44603 \nu^{9} + 133093 \nu^{7} - 198792 \nu^{5} + \cdots - 24417 \nu ) / 1512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} + 26\nu^{12} - 269\nu^{10} + 1419\nu^{8} - 4067\nu^{6} + 6212\nu^{4} - 4443\nu^{2} + 939 ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43 \nu^{15} - 1226 \nu^{13} + 14115 \nu^{11} - 83753 \nu^{9} + 269749 \nu^{7} - 450756 \nu^{5} + \cdots - 50661 \nu ) / 3024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5\nu^{14} - 134\nu^{12} + 1421\nu^{10} - 7543\nu^{8} + 20779\nu^{6} - 27580\nu^{4} + 14271\nu^{2} - 1995 ) / 144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 29 \nu^{15} + 862 \nu^{13} - 10293 \nu^{11} + 62767 \nu^{9} - 204803 \nu^{7} + 340380 \nu^{5} + \cdots + 51291 \nu ) / 1512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 73 \nu^{15} - 2054 \nu^{13} + 23193 \nu^{11} - 134051 \nu^{9} + 418591 \nu^{7} - 684300 \nu^{5} + \cdots - 130815 \nu ) / 3024 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 8\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{14} + 2\beta_{12} + 10\beta_{10} - 9\beta_{9} + \beta_{4} + \beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{13} - 2\beta_{11} + 2\beta_{8} - \beta_{7} + 13\beta_{6} - 11\beta_{5} + 60\beta_{2} + 146 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{15} + 14\beta_{14} + 26\beta_{12} + 82\beta_{10} - 71\beta_{9} + 15\beta_{4} + 17\beta_{3} + 180\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -29\beta_{13} - 29\beta_{11} + 32\beta_{8} - 13\beta_{7} + 129\beta_{6} - 94\beta_{5} + 448\beta_{2} + 966 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 94 \beta_{15} + 139 \beta_{14} + 255 \beta_{12} + 633 \beta_{10} - 548 \beta_{9} + 155 \beta_{4} + \cdots + 1162 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -294\beta_{13} - 300\beta_{11} + 354\beta_{8} - 108\beta_{7} + 1157\beta_{6} - 743\beta_{5} + 3355\beta_{2} + 6550 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 737 \beta_{15} + 1211 \beta_{14} + 2260 \beta_{12} + 4769 \beta_{10} - 4224 \beta_{9} + \cdots + 7711 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2596 \beta_{13} - 2734 \beta_{11} + 3364 \beta_{8} - 707 \beta_{7} + 9848 \beta_{6} - 5680 \beta_{5} + \cdots + 45205 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5542 \beta_{15} + 9925 \beta_{14} + 19066 \beta_{12} + 35555 \beta_{10} - 32629 \beta_{9} + \cdots + 52284 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 21427 \beta_{13} - 23449 \beta_{11} + 29512 \beta_{8} - 3710 \beta_{7} + 81207 \beta_{6} + \cdots + 316395 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 40652 \beta_{15} + 78854 \beta_{14} + 156351 \beta_{12} + 264027 \beta_{10} - 252466 \beta_{9} + \cdots + 360815 \beta_1 \) 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
−2.74917
−2.54850
−2.40258
−2.18309
−1.53788
−1.15201
−0.717325
−0.449667
0.449667
0.717325
1.15201
1.53788
2.18309
2.40258
2.54850
2.74917
−2.74917 2.70846 5.55793 0 −7.44602 −4.31281 −9.78134 4.33577 0
1.2 −2.54850 −1.62236 4.49487 0 4.13458 1.00601 −6.35819 −0.367964 0
1.3 −2.40258 −2.46863 3.77237 0 5.93108 −0.299319 −4.25824 3.09415 0
1.4 −2.18309 0.373034 2.76589 0 −0.814368 −2.28258 −1.67200 −2.86085 0
1.5 −1.53788 3.20341 0.365073 0 −4.92646 2.65077 2.51432 7.26186 0
1.6 −1.15201 −3.12867 −0.672883 0 3.60424 0.274351 3.07918 6.78855 0
1.7 −0.717325 −0.150152 −1.48544 0 0.107708 4.77862 2.50020 −2.97745 0
1.8 −0.449667 1.31375 −1.79780 0 −0.590749 0.388290 1.70774 −1.27407 0
1.9 0.449667 −1.31375 −1.79780 0 −0.590749 −0.388290 −1.70774 −1.27407 0
1.10 0.717325 0.150152 −1.48544 0 0.107708 −4.77862 −2.50020 −2.97745 0
1.11 1.15201 3.12867 −0.672883 0 3.60424 −0.274351 −3.07918 6.78855 0
1.12 1.53788 −3.20341 0.365073 0 −4.92646 −2.65077 −2.51432 7.26186 0
1.13 2.18309 −0.373034 2.76589 0 −0.814368 2.28258 1.67200 −2.86085 0
1.14 2.40258 2.46863 3.77237 0 5.93108 0.299319 4.25824 3.09415 0
1.15 2.54850 1.62236 4.49487 0 4.13458 −1.00601 6.35819 −0.367964 0
1.16 2.74917 −2.70846 5.55793 0 −7.44602 4.31281 9.78134 4.33577 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(61\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1525.2.a.o 16
5.b even 2 1 inner 1525.2.a.o 16
5.c odd 4 2 305.2.b.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
305.2.b.d ✓ 16 5.c odd 4 2
1525.2.a.o 16 1.a even 1 1 trivial
1525.2.a.o 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1525))\):

\( T_{2}^{16} - 29T_{2}^{14} + 339T_{2}^{12} - 2042T_{2}^{10} + 6724T_{2}^{8} - 11877T_{2}^{6} + 10407T_{2}^{4} - 3852T_{2}^{2} + 441 \) Copy content Toggle raw display
\( T_{7}^{16} - 55T_{7}^{14} + 1040T_{7}^{12} - 8019T_{7}^{10} + 24806T_{7}^{8} - 23034T_{7}^{6} + 5681T_{7}^{4} - 520T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 29 T^{14} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{16} - 38 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 55 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{8} - 15 T^{7} + \cdots - 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 130 T^{14} + \cdots + 5345344 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 272118016 \) Copy content Toggle raw display
$19$ \( (T^{8} - 9 T^{7} - 2 T^{6} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 238 T^{14} + \cdots + 12194064 \) Copy content Toggle raw display
$29$ \( (T^{8} - 17 T^{7} + \cdots + 26736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 5 T^{7} + \cdots + 504)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1020930304 \) Copy content Toggle raw display
$41$ \( (T^{8} - 27 T^{7} + \cdots - 96768)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 163430656 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 290907136 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2981119934464 \) Copy content Toggle raw display
$59$ \( (T^{8} - 27 T^{7} + \cdots - 191558)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 9981608464 \) Copy content Toggle raw display
$71$ \( (T^{8} - 23 T^{7} + \cdots + 16888)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 72610847296 \) Copy content Toggle raw display
$79$ \( (T^{8} - 14 T^{7} + \cdots - 142182822)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 667737391104 \) Copy content Toggle raw display
$89$ \( (T^{8} - 8 T^{7} + \cdots - 147392)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
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