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Properties

Label 9675.2.a.da
Level $9675$
Weight $2$
Character orbit 9675.a
Self dual yes
Analytic conductor $77.255$
Analytic rank $1$
Dimension $20$
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] = [9675,2,Mod(1,9675)] 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("9675.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,16,0,0,-8,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2552639556\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 28 x^{18} + 326 x^{16} - 2052 x^{14} + 7613 x^{12} - 17056 x^{10} + 22796 x^{8} - 17428 x^{6} + \cdots + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1935)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{9} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{19} + \beta_{17} + \cdots - \beta_{3}) q^{11} + (\beta_{11} - \beta_{2} - 2) q^{13} + (\beta_{18} - \beta_{17} + \cdots - \beta_1) q^{14}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} - 8 q^{7} - 32 q^{13} + 8 q^{16} - 16 q^{22} - 24 q^{28} - 12 q^{31} + 24 q^{34} - 52 q^{37} + 20 q^{43} + 8 q^{46} + 32 q^{49} - 92 q^{52} - 28 q^{58} + 12 q^{61} - 8 q^{64} - 40 q^{67}+ \cdots - 76 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 28 x^{18} + 326 x^{16} - 2052 x^{14} + 7613 x^{12} - 17056 x^{10} + 22796 x^{8} - 17428 x^{6} + \cdots + 100 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{19} + 28 \nu^{17} - 326 \nu^{15} + 2052 \nu^{13} - 7613 \nu^{11} + 17056 \nu^{9} + \cdots + 748 \nu ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{18} + 27 \nu^{16} - 299 \nu^{14} + 1757 \nu^{12} - 5928 \nu^{10} + 11616 \nu^{8} - 12748 \nu^{6} + \cdots + 108 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7 \nu^{19} + 191 \nu^{17} - 2147 \nu^{15} + 12869 \nu^{13} - 44506 \nu^{11} + 89752 \nu^{9} + \cdots + 1076 \nu ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{18} + 28 \nu^{16} - 325 \nu^{14} + 2028 \nu^{12} - 7380 \nu^{10} + 15882 \nu^{8} - 19524 \nu^{6} + \cdots + 268 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{18} + 28 \nu^{16} - 325 \nu^{14} + 2028 \nu^{12} - 7382 \nu^{10} + 15914 \nu^{8} - 19706 \nu^{6} + \cdots + 372 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{18} - 83 \nu^{16} + 949 \nu^{14} - 5809 \nu^{12} + 20624 \nu^{10} - 43020 \nu^{8} + 50952 \nu^{6} + \cdots - 708 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{18} + 83 \nu^{16} - 949 \nu^{14} + 5809 \nu^{12} - 20624 \nu^{10} + 43024 \nu^{8} - 51004 \nu^{6} + \cdots + 748 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2 \nu^{19} - 55 \nu^{17} + 625 \nu^{15} - 3803 \nu^{13} + 13431 \nu^{11} - 27906 \nu^{9} + \cdots - 420 \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13 \nu^{19} - 359 \nu^{17} + 4103 \nu^{15} - 25171 \nu^{13} + 89984 \nu^{11} - 190528 \nu^{9} + \cdots - 4544 \nu ) / 20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16 \nu^{19} - 443 \nu^{17} + 5076 \nu^{15} - 31207 \nu^{13} + 111678 \nu^{11} - 236156 \nu^{9} + \cdots - 5028 \nu ) / 20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7 \nu^{18} + 193 \nu^{16} - 2201 \nu^{14} + 13459 \nu^{12} - 47864 \nu^{10} + 100432 \nu^{8} + \cdots + 1836 ) / 8 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 7 \nu^{18} - 193 \nu^{16} + 2201 \nu^{14} - 13463 \nu^{12} + 47936 \nu^{10} - 100916 \nu^{8} + \cdots - 2068 ) / 8 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 39 \nu^{19} + 1077 \nu^{17} - 12299 \nu^{15} + 75283 \nu^{13} - 267842 \nu^{11} + 561724 \nu^{9} + \cdots + 9292 \nu ) / 40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 41 \nu^{19} + 1133 \nu^{17} - 12951 \nu^{15} + 79387 \nu^{13} - 283068 \nu^{11} + 595836 \nu^{9} + \cdots + 11468 \nu ) / 40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 3 \nu^{19} + 83 \nu^{17} - 950 \nu^{15} + 5831 \nu^{13} - 20812 \nu^{11} + 43804 \nu^{9} + \cdots + 772 \nu ) / 2 \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - \beta_{17} - \beta_{5} + 9\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{16} - \beta_{15} + \beta_{11} - 2\beta_{9} + \beta_{8} - \beta_{6} + 10\beta_{4} + 45\beta_{2} + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{18} - 12\beta_{17} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{7} - 10\beta_{5} + 66\beta_{3} + 165\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13 \beta_{16} - 13 \beta_{15} + 15 \beta_{11} + 2 \beta_{10} - 26 \beta_{9} + 13 \beta_{8} + \cdots + 507 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{19} + 117 \beta_{18} - 106 \beta_{17} + 15 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} + \cdots + 1001 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 117 \beta_{16} - 117 \beta_{15} + 149 \beta_{11} + 32 \beta_{10} - 236 \beta_{9} + 119 \beta_{8} + \cdots + 3111 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 34 \beta_{19} + 913 \beta_{18} - 828 \beta_{17} + 153 \beta_{14} + 117 \beta_{13} - 117 \beta_{12} + \cdots + 6187 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 913 \beta_{16} - 913 \beta_{15} + 1245 \beta_{11} + 334 \beta_{10} - 1858 \beta_{9} + 947 \beta_{8} + \cdots + 19421 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 368 \beta_{19} + 6639 \beta_{18} - 6058 \beta_{17} + 1317 \beta_{14} + 911 \beta_{13} + \cdots + 38727 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6639 \beta_{16} - 6641 \beta_{15} + 9495 \beta_{11} + 2894 \beta_{10} - 13606 \beta_{9} + \cdots + 122573 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3262 \beta_{19} + 46393 \beta_{18} - 42602 \beta_{17} + 10311 \beta_{14} + 6603 \beta_{13} + \cdots + 244553 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 46393 \beta_{16} - 46445 \beta_{15} + 68609 \beta_{11} + 22662 \beta_{10} - 95546 \beta_{9} + \cdots + 779169 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 25924 \beta_{19} + 316375 \beta_{18} - 291998 \beta_{17} + 76117 \beta_{14} + 45999 \beta_{13} + \cdots + 1553979 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 316375 \beta_{16} - 317181 \beta_{15} + 479123 \beta_{11} + 166942 \beta_{10} - 653566 \beta_{9} + \cdots + 4976945 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 192866 \beta_{19} + 2123573 \beta_{18} - 1966946 \beta_{17} + 540639 \beta_{14} + 312987 \beta_{13} + \cdots + 9918945 \beta_1 \) 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
−2.54466
−2.53914
−2.19006
−1.83296
−1.75805
−1.28983
−1.12283
−0.622746
−0.590749
−0.411606
0.411606
0.590749
0.622746
1.12283
1.28983
1.75805
1.83296
2.19006
2.53914
2.54466
−2.54466 0 4.47529 0 0 −2.18751 −6.29878 0 0
1.2 −2.53914 0 4.44722 0 0 1.01277 −6.21382 0 0
1.3 −2.19006 0 2.79638 0 0 −4.99426 −1.74413 0 0
1.4 −1.83296 0 1.35974 0 0 3.64733 1.17356 0 0
1.5 −1.75805 0 1.09072 0 0 −1.78552 1.59855 0 0
1.6 −1.28983 0 −0.336338 0 0 3.11206 3.01348 0 0
1.7 −1.12283 0 −0.739246 0 0 −0.206332 3.07572 0 0
1.8 −0.622746 0 −1.61219 0 0 −0.320518 2.24948 0 0
1.9 −0.590749 0 −1.65102 0 0 2.48939 2.15683 0 0
1.10 −0.411606 0 −1.83058 0 0 −4.76741 1.57669 0 0
1.11 0.411606 0 −1.83058 0 0 −4.76741 −1.57669 0 0
1.12 0.590749 0 −1.65102 0 0 2.48939 −2.15683 0 0
1.13 0.622746 0 −1.61219 0 0 −0.320518 −2.24948 0 0
1.14 1.12283 0 −0.739246 0 0 −0.206332 −3.07572 0 0
1.15 1.28983 0 −0.336338 0 0 3.11206 −3.01348 0 0
1.16 1.75805 0 1.09072 0 0 −1.78552 −1.59855 0 0
1.17 1.83296 0 1.35974 0 0 3.64733 −1.17356 0 0
1.18 2.19006 0 2.79638 0 0 −4.99426 1.74413 0 0
1.19 2.53914 0 4.44722 0 0 1.01277 6.21382 0 0
1.20 2.54466 0 4.47529 0 0 −2.18751 6.29878 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(43\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9675.2.a.da 20
3.b odd 2 1 inner 9675.2.a.da 20
5.b even 2 1 9675.2.a.db 20
5.c odd 4 2 1935.2.b.g ✓ 40
15.d odd 2 1 9675.2.a.db 20
15.e even 4 2 1935.2.b.g ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1935.2.b.g ✓ 40 5.c odd 4 2
1935.2.b.g ✓ 40 15.e even 4 2
9675.2.a.da 20 1.a even 1 1 trivial
9675.2.a.da 20 3.b odd 2 1 inner
9675.2.a.db 20 5.b even 2 1
9675.2.a.db 20 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9675))\):

\( T_{2}^{20} - 28 T_{2}^{18} + 326 T_{2}^{16} - 2052 T_{2}^{14} + 7613 T_{2}^{12} - 17056 T_{2}^{10} + \cdots + 100 \) Copy content Toggle raw display
\( T_{7}^{10} + 4 T_{7}^{9} - 35 T_{7}^{8} - 110 T_{7}^{7} + 426 T_{7}^{6} + 890 T_{7}^{5} - 1772 T_{7}^{4} + \cdots + 176 \) Copy content Toggle raw display
\( T_{11}^{20} - 128 T_{11}^{18} + 6912 T_{11}^{16} - 205848 T_{11}^{14} + 3712786 T_{11}^{12} + \cdots + 33477796 \) Copy content Toggle raw display
\( T_{13}^{10} + 16 T_{13}^{9} + 50 T_{13}^{8} - 390 T_{13}^{7} - 2584 T_{13}^{6} - 1818 T_{13}^{5} + \cdots - 20240 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 28 T^{18} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 4 T^{9} + \cdots + 176)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} - 128 T^{18} + \cdots + 33477796 \) Copy content Toggle raw display
$13$ \( (T^{10} + 16 T^{9} + \cdots - 20240)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 334158400 \) Copy content Toggle raw display
$19$ \( (T^{10} - 113 T^{8} + \cdots + 18068)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 4000309504 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 74997108736 \) Copy content Toggle raw display
$31$ \( (T^{10} + 6 T^{9} + \cdots - 230848)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 26 T^{9} + \cdots + 5825600)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 57980062007296 \) Copy content Toggle raw display
$43$ \( (T - 1)^{20} \) Copy content Toggle raw display
$47$ \( T^{20} - 326 T^{18} + \cdots + 20502784 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1317463204864 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1340145664 \) Copy content Toggle raw display
$61$ \( (T^{10} - 6 T^{9} + \cdots - 8687744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 20 T^{9} + \cdots + 4673152)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{10} + 24 T^{9} + \cdots + 4564736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 12 T^{9} + \cdots - 473022848)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 54627413153296 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + 38 T^{9} + \cdots + 77216744)^{2} \) Copy content Toggle raw display
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