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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9801,2,Mod(1,9801)] 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("9801.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9801, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9801 = 3^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9801.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-1,0,11,2,0,-9,0,0,-8,0,0,-9,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2613790211\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 17 x^{10} + 16 x^{9} + 100 x^{8} - 84 x^{7} - 248 x^{6} + 152 x^{5} + 272 x^{4} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 891)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + ( - \beta_{8} + \beta_1 - 1) q^{7} + (\beta_{10} + \beta_{7} - \beta_{4} + \cdots + 1) q^{8} + (\beta_{6} + \beta_{3} - \beta_1) q^{10} + (\beta_{11} + \beta_{10} - \beta_{9} + \cdots - 1) q^{13}+ \cdots + (\beta_{11} - 4 \beta_{9} + \beta_{8} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 11 q^{4} + 2 q^{5} - 9 q^{7} - 8 q^{10} - 9 q^{13} - 21 q^{14} + 21 q^{16} + 5 q^{17} - 18 q^{19} + 11 q^{20} + 3 q^{23} + 14 q^{25} + 8 q^{26} - 17 q^{28} + 6 q^{29} + 6 q^{31} + 9 q^{32}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 17 x^{10} + 16 x^{9} + 100 x^{8} - 84 x^{7} - 248 x^{6} + 152 x^{5} + 272 x^{4} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 181 \nu^{11} - 236 \nu^{10} - 2989 \nu^{9} + 3805 \nu^{8} + 16688 \nu^{7} - 20299 \nu^{6} - 37412 \nu^{5} + \cdots - 525 ) / 67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 433 \nu^{11} + 599 \nu^{10} + 7149 \nu^{9} - 9663 \nu^{8} - 39874 \nu^{7} + 51573 \nu^{6} + \cdots + 811 ) / 67 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 525 \nu^{11} - 706 \nu^{10} - 8689 \nu^{9} + 11389 \nu^{8} + 48695 \nu^{7} - 60788 \nu^{6} + \cdots - 1612 ) / 67 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 753 \nu^{11} + 977 \nu^{10} + 12456 \nu^{9} - 15783 \nu^{8} - 69743 \nu^{7} + 84435 \nu^{6} + \cdots + 2193 ) / 67 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 931 \nu^{11} - 1235 \nu^{10} - 15421 \nu^{9} + 19941 \nu^{8} + 86549 \nu^{7} - 106603 \nu^{6} + \cdots - 2847 ) / 67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1060 \nu^{11} + 1428 \nu^{10} + 17525 \nu^{9} - 23053 \nu^{8} - 98024 \nu^{7} + 123185 \nu^{6} + \cdots + 2861 ) / 67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1520 \nu^{11} - 2030 \nu^{10} - 25158 \nu^{9} + 32755 \nu^{8} + 140990 \nu^{7} - 174888 \nu^{6} + \cdots - 4856 ) / 67 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1545 \nu^{11} + 2070 \nu^{10} + 25559 \nu^{9} - 33409 \nu^{8} - 143111 \nu^{7} + 178475 \nu^{6} + \cdots + 4317 ) / 67 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1668 \nu^{11} - 2240 \nu^{10} - 27615 \nu^{9} + 36155 \nu^{8} + 154822 \nu^{7} - 193167 \nu^{6} + \cdots - 4924 ) / 67 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 2\beta_{10} + \beta_{8} - 2\beta_{3} + 6\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{10} - \beta_{9} + \beta_{8} - 9 \beta_{7} - \beta_{6} + 3 \beta_{5} + 8 \beta_{4} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{11} - 21 \beta_{10} + \beta_{9} + 10 \beta_{8} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{11} - 70 \beta_{10} - 14 \beta_{9} + 12 \beta_{8} - 67 \beta_{7} - 12 \beta_{6} + 42 \beta_{5} + \cdots - 99 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 95 \beta_{11} - 179 \beta_{10} + 13 \beta_{9} + 79 \beta_{8} - 29 \beta_{7} - 14 \beta_{6} + \cdots + 533 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 32 \beta_{11} - 529 \beta_{10} - 140 \beta_{9} + 108 \beta_{8} - 482 \beta_{7} - 109 \beta_{6} + \cdots - 723 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 761 \beta_{11} - 1442 \beta_{10} + 125 \beta_{9} + 590 \beta_{8} - 308 \beta_{7} - 141 \beta_{6} + \cdots + 3461 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 357 \beta_{11} - 3990 \beta_{10} - 1226 \beta_{9} + 898 \beta_{8} - 3466 \beta_{7} - 902 \beta_{6} + \cdots - 4997 \) 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
2.77842
2.03299
1.78945
1.56844
1.11354
−0.0510396
−0.334588
−0.340647
−1.07045
−1.41949
−2.43734
−2.62930
−2.77842 0 5.71964 2.64920 0 −0.0798909 −10.3347 0 −7.36059
1.2 −2.03299 0 2.13305 −2.57770 0 −1.48408 −0.270482 0 5.24043
1.3 −1.78945 0 1.20212 2.48236 0 3.20696 1.42776 0 −4.44205
1.4 −1.56844 0 0.460016 −0.537100 0 2.82856 2.41538 0 0.842411
1.5 −1.11354 0 −0.760035 0.940197 0 −3.77631 3.07340 0 −1.04694
1.6 0.0510396 0 −1.99739 −0.616133 0 2.43522 −0.204025 0 −0.0314472
1.7 0.334588 0 −1.88805 −3.75811 0 −0.308496 −1.30090 0 −1.25742
1.8 0.340647 0 −1.88396 3.45969 0 −4.96180 −1.32306 0 1.17853
1.9 1.07045 0 −0.854145 3.50075 0 0.782056 −3.05521 0 3.74737
1.10 1.41949 0 0.0149385 −3.74473 0 −1.39244 −2.81777 0 −5.31559
1.11 2.43734 0 3.94061 0.441126 0 −5.06395 4.72992 0 1.07517
1.12 2.62930 0 4.91321 −0.239560 0 −1.18583 7.65971 0 −0.629874
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( +1 \)

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 9801.2.a.cg 12
3.b odd 2 1 9801.2.a.ck 12
11.b odd 2 1 9801.2.a.cl 12
11.c even 5 2 891.2.f.c ✓ 24
33.d even 2 1 9801.2.a.cf 12
33.h odd 10 2 891.2.f.d yes 24
99.m even 15 4 891.2.n.k 48
99.n odd 30 4 891.2.n.j 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.f.c ✓ 24 11.c even 5 2
891.2.f.d yes 24 33.h odd 10 2
891.2.n.j 48 99.n odd 30 4
891.2.n.k 48 99.m even 15 4
9801.2.a.cf 12 33.d even 2 1
9801.2.a.cg 12 1.a even 1 1 trivial
9801.2.a.ck 12 3.b odd 2 1
9801.2.a.cl 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):

\( T_{2}^{12} + T_{2}^{11} - 17 T_{2}^{10} - 16 T_{2}^{9} + 100 T_{2}^{8} + 84 T_{2}^{7} - 248 T_{2}^{6} + \cdots - 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 2 T_{5}^{11} - 35 T_{5}^{10} + 74 T_{5}^{9} + 403 T_{5}^{8} - 885 T_{5}^{7} - 1616 T_{5}^{6} + \cdots + 95 \) Copy content Toggle raw display
\( T_{7}^{12} + 9 T_{7}^{11} - 7 T_{7}^{10} - 225 T_{7}^{9} - 214 T_{7}^{8} + 1823 T_{7}^{7} + 2659 T_{7}^{6} + \cdots + 99 \) Copy content Toggle raw display
\( T_{17}^{12} - 5 T_{17}^{11} - 108 T_{17}^{10} + 469 T_{17}^{9} + 4489 T_{17}^{8} - 15727 T_{17}^{7} + \cdots + 4811895 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + \cdots + 95 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 99 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} + \cdots - 12419 \) Copy content Toggle raw display
$17$ \( T^{12} - 5 T^{11} + \cdots + 4811895 \) Copy content Toggle raw display
$19$ \( T^{12} + 18 T^{11} + \cdots + 2352955 \) Copy content Toggle raw display
$23$ \( T^{12} - 3 T^{11} + \cdots + 2770119 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 1287891 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots - 390955 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1129932919 \) Copy content Toggle raw display
$41$ \( T^{12} + 5 T^{11} + \cdots - 5545045 \) Copy content Toggle raw display
$43$ \( T^{12} + 15 T^{11} + \cdots - 42549345 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 165587789 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 104990155 \) Copy content Toggle raw display
$59$ \( T^{12} - 23 T^{11} + \cdots + 5675725 \) Copy content Toggle raw display
$61$ \( T^{12} + 31 T^{11} + \cdots + 96781959 \) Copy content Toggle raw display
$67$ \( T^{12} + 7 T^{11} + \cdots - 31618791 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 16142503725 \) Copy content Toggle raw display
$73$ \( T^{12} + 25 T^{11} + \cdots - 949549 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1573845281 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + \cdots - 17871579 \) Copy content Toggle raw display
$89$ \( T^{12} - 17 T^{11} + \cdots + 84645 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 7038380871 \) Copy content Toggle raw display
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