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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1026,2,Mod(341,1026)] 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("1026.341"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1026, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1026 = 2 \cdot 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1026.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.19265124738\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
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: \( x^{20} - x^{19} + x^{18} + 3 x^{17} - 15 x^{16} + 33 x^{15} - 42 x^{14} + 72 x^{12} - 243 x^{11} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: no (minimal twist has level 342)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} + \beta_1 q^{5} + \beta_{12} q^{7} + q^{8} + ( - \beta_{4} - \beta_1) q^{10} + ( - \beta_{11} + \beta_{6}) q^{11} + \beta_{2} q^{13} + (\beta_{14} - \beta_{12}) q^{14}+ \cdots + ( - \beta_{19} + \beta_{18} + \beta_{15} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{2} - 10 q^{4} + 2 q^{7} + 20 q^{8} + 3 q^{11} + 2 q^{14} - 10 q^{16} + q^{19} - 3 q^{22} + 12 q^{23} + 10 q^{25} - 4 q^{28} - 18 q^{31} - 10 q^{32} + 15 q^{34} + 10 q^{38} - 3 q^{41} - 5 q^{43}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + x^{18} + 3 x^{17} - 15 x^{16} + 33 x^{15} - 42 x^{14} + 72 x^{12} - 243 x^{11} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 554 \nu^{19} + 3854 \nu^{18} - 2549 \nu^{17} - 3177 \nu^{16} + 4638 \nu^{15} - 47046 \nu^{14} + \cdots - 56313063 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1237 \nu^{19} - 11653 \nu^{18} + 18628 \nu^{17} - 28368 \nu^{16} + 35121 \nu^{15} + \cdots + 102764943 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2099 \nu^{19} + 5633 \nu^{18} - 14912 \nu^{17} + 19935 \nu^{16} - 3129 \nu^{15} + \cdots - 115066818 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2426 \nu^{19} - 9479 \nu^{18} + 12386 \nu^{17} - 12537 \nu^{16} - 16923 \nu^{15} + \cdots + 83062260 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 830 \nu^{19} + 2987 \nu^{18} - 6677 \nu^{17} + 2196 \nu^{16} + 11370 \nu^{15} - 54804 \nu^{14} + \cdots - 55013985 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 836 \nu^{19} + 2984 \nu^{18} - 5594 \nu^{17} + 2709 \nu^{16} + 6762 \nu^{15} - 55353 \nu^{14} + \cdots - 45211851 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1178 \nu^{19} - 4271 \nu^{18} + 8744 \nu^{17} - 11538 \nu^{16} + 15 \nu^{15} + 56028 \nu^{14} + \cdots + 41314617 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1178 \nu^{19} + 4271 \nu^{18} - 8744 \nu^{17} + 11538 \nu^{16} - 15 \nu^{15} - 56028 \nu^{14} + \cdots - 41314617 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1226 \nu^{19} - 4976 \nu^{18} + 11744 \nu^{17} - 7623 \nu^{16} - 300 \nu^{15} + 58233 \nu^{14} + \cdots + 77688801 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4388 \nu^{19} - 10685 \nu^{18} + 21287 \nu^{17} - 31572 \nu^{16} - 6015 \nu^{15} + \cdots + 134966331 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4658 \nu^{19} - 12737 \nu^{18} + 20666 \nu^{17} - 24039 \nu^{16} - 34851 \nu^{15} + \cdots + 95639697 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1603 \nu^{19} + 4180 \nu^{18} - 8698 \nu^{17} + 10818 \nu^{16} - 1281 \nu^{15} + \cdots - 54876204 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1655 \nu^{19} - 2210 \nu^{18} + 6116 \nu^{17} - 4572 \nu^{16} - 5439 \nu^{15} + 39531 \nu^{14} + \cdots + 65583756 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5062 \nu^{19} - 10834 \nu^{18} + 24775 \nu^{17} - 28449 \nu^{16} - 45339 \nu^{15} + \cdots + 196121412 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1700 \nu^{19} + 4010 \nu^{18} - 10751 \nu^{17} + 8055 \nu^{16} + 14619 \nu^{15} + \cdots - 63812286 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 620 \nu^{19} + 1634 \nu^{18} - 3029 \nu^{17} + 3924 \nu^{16} + 2793 \nu^{15} - 29028 \nu^{14} + \cdots - 21828447 ) / 1594323 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 6575 \nu^{19} + 12872 \nu^{18} - 23474 \nu^{17} + 25011 \nu^{16} + 38820 \nu^{15} + \cdots - 91919610 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2401 \nu^{19} + 3052 \nu^{18} - 6085 \nu^{17} + 3960 \nu^{16} + 22272 \nu^{15} + \cdots - 23462136 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 8651 \nu^{19} + 26414 \nu^{18} - 50435 \nu^{17} + 66186 \nu^{16} + 23466 \nu^{15} + \cdots - 230212368 ) / 14348907 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} + \beta_{18} - 2 \beta_{17} - \beta_{16} - \beta_{13} - \beta_{11} + \beta_{10} + \cdots - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{19} + 2 \beta_{18} - \beta_{17} + \beta_{16} - 3 \beta_{15} + \beta_{13} + \beta_{11} + \cdots + 3 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{19} + 4 \beta_{18} - 5 \beta_{17} + 5 \beta_{16} + 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + \cdots + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{19} + 2 \beta_{18} - \beta_{17} + 7 \beta_{16} + 6 \beta_{14} + \beta_{13} - 3 \beta_{12} + \cdots - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{19} + 4 \beta_{18} - 2 \beta_{17} - 13 \beta_{16} - 9 \beta_{15} - 6 \beta_{14} + 2 \beta_{13} + \cdots + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5 \beta_{19} - \beta_{18} + 23 \beta_{17} + \beta_{16} - 18 \beta_{15} - 21 \beta_{14} + 31 \beta_{13} + \cdots - 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28 \beta_{19} - 2 \beta_{18} - 44 \beta_{17} + 11 \beta_{16} + 9 \beta_{15} + 3 \beta_{14} + 8 \beta_{13} + \cdots + 66 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20 \beta_{19} - 13 \beta_{18} + 20 \beta_{17} - 50 \beta_{16} - 36 \beta_{15} + 60 \beta_{14} + \cdots - 111 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 41 \beta_{19} + 28 \beta_{18} - 14 \beta_{17} - 100 \beta_{16} - 18 \beta_{15} - 42 \beta_{14} + \cdots - 249 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 82 \beta_{19} + 110 \beta_{18} + 53 \beta_{17} + 178 \beta_{16} - 117 \beta_{15} - 138 \beta_{14} + \cdots - 444 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 380 \beta_{19} + 58 \beta_{18} - 110 \beta_{17} + 5 \beta_{16} + 117 \beta_{15} + 156 \beta_{14} + \cdots + 894 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 490 \beta_{19} - 424 \beta_{18} + 1076 \beta_{17} - 341 \beta_{16} + 234 \beta_{15} + 150 \beta_{14} + \cdots + 411 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 154 \beta_{19} - 38 \beta_{18} - 35 \beta_{17} - 601 \beta_{16} + 225 \beta_{15} - 1482 \beta_{14} + \cdots + 1146 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 551 \beta_{19} + 977 \beta_{18} - 232 \beta_{17} + 94 \beta_{16} - 522 \beta_{15} - 1344 \beta_{14} + \cdots - 543 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 1669 \beta_{19} + 172 \beta_{18} - 3785 \beta_{17} + 593 \beta_{16} + 2277 \beta_{15} + 1605 \beta_{14} + \cdots + 4260 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 179 \beta_{19} + 1559 \beta_{18} + 530 \beta_{17} + 3130 \beta_{16} - 2979 \beta_{15} + 1023 \beta_{14} + \cdots + 987 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 5555 \beta_{19} + 9112 \beta_{18} - 6311 \beta_{17} + 6098 \beta_{16} + 117 \beta_{15} - 2814 \beta_{14} + \cdots - 16008 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 580 \beta_{19} + 2024 \beta_{18} - 1120 \beta_{17} + 13330 \beta_{16} + 4608 \beta_{15} + \cdots - 28371 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1026\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
1.70846 + 0.284919i
−0.424393 − 1.67925i
−0.0754025 + 1.73041i
−1.73094 − 0.0619519i
1.06721 − 1.36421i
1.57590 + 0.718716i
−1.72527 + 0.153167i
0.0940817 − 1.72949i
−1.00512 + 1.41058i
1.01548 + 1.40314i
1.70846 − 0.284919i
−0.424393 + 1.67925i
−0.0754025 − 1.73041i
−1.73094 + 0.0619519i
1.06721 + 1.36421i
1.57590 − 0.718716i
−1.72527 − 0.153167i
0.0940817 + 1.72949i
−1.00512 − 1.41058i
1.01548 − 1.40314i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −3.41314 − 1.97058i 0 1.07232 + 1.85732i 1.00000 0 3.94116i
341.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.76481 − 1.59626i 0 −0.965307 − 1.67196i 1.00000 0 3.19253i
341.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.58264 − 0.913740i 0 0.101997 + 0.176663i 1.00000 0 1.82748i
341.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.18812 − 0.685961i 0 2.26558 + 3.92410i 1.00000 0 1.37192i
341.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.446642 − 0.257869i 0 −0.534840 − 0.926369i 1.00000 0 0.515738i
341.6 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.379645 + 0.219188i 0 −2.44148 − 4.22876i 1.00000 0 − 0.438376i
341.7 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.43477 + 0.828367i 0 −1.29562 − 2.24408i 1.00000 0 − 1.65673i
341.8 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.13944 + 1.23520i 0 2.17969 + 3.77534i 1.00000 0 − 2.47041i
341.9 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.66819 + 1.54048i 0 −0.142695 − 0.247156i 1.00000 0 − 3.08096i
341.10 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.77331 + 1.60117i 0 0.760344 + 1.31695i 1.00000 0 − 3.20235i
683.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −3.41314 + 1.97058i 0 1.07232 − 1.85732i 1.00000 0 − 3.94116i
683.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −2.76481 + 1.59626i 0 −0.965307 + 1.67196i 1.00000 0 − 3.19253i
683.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.58264 + 0.913740i 0 0.101997 − 0.176663i 1.00000 0 − 1.82748i
683.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.18812 + 0.685961i 0 2.26558 − 3.92410i 1.00000 0 − 1.37192i
683.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.446642 + 0.257869i 0 −0.534840 + 0.926369i 1.00000 0 − 0.515738i
683.6 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.379645 − 0.219188i 0 −2.44148 + 4.22876i 1.00000 0 0.438376i
683.7 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.43477 − 0.828367i 0 −1.29562 + 2.24408i 1.00000 0 1.65673i
683.8 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.13944 − 1.23520i 0 2.17969 − 3.77534i 1.00000 0 2.47041i
683.9 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.66819 − 1.54048i 0 −0.142695 + 0.247156i 1.00000 0 3.08096i
683.10 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.77331 − 1.60117i 0 0.760344 − 1.31695i 1.00000 0 3.20235i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 341.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1026.2.p.a 20
3.b odd 2 1 342.2.p.b yes 20
9.c even 3 1 342.2.p.a ✓ 20
9.c even 3 1 3078.2.b.c 20
9.d odd 6 1 1026.2.p.b 20
9.d odd 6 1 3078.2.b.a 20
19.b odd 2 1 1026.2.p.b 20
57.d even 2 1 342.2.p.a ✓ 20
171.l even 6 1 inner 1026.2.p.a 20
171.l even 6 1 3078.2.b.c 20
171.o odd 6 1 342.2.p.b yes 20
171.o odd 6 1 3078.2.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.p.a ✓ 20 9.c even 3 1
342.2.p.a ✓ 20 57.d even 2 1
342.2.p.b yes 20 3.b odd 2 1
342.2.p.b yes 20 171.o odd 6 1
1026.2.p.a 20 1.a even 1 1 trivial
1026.2.p.a 20 171.l even 6 1 inner
1026.2.p.b 20 9.d odd 6 1
1026.2.p.b 20 19.b odd 2 1
3078.2.b.a 20 9.d odd 6 1
3078.2.b.a 20 171.o odd 6 1
3078.2.b.c 20 9.c even 3 1
3078.2.b.c 20 171.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{20} - 63 T_{13}^{18} + 2832 T_{13}^{16} + 3357 T_{13}^{15} - 58500 T_{13}^{14} - 111942 T_{13}^{13} + \cdots + 57972996 \) acting on \(S_{2}^{\mathrm{new}}(1026, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 30 T^{18} + \cdots + 82944 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 9604 \) Copy content Toggle raw display
$11$ \( T^{20} - 3 T^{19} + \cdots + 254016 \) Copy content Toggle raw display
$13$ \( T^{20} - 63 T^{18} + \cdots + 57972996 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 27575591481 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 256064004 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13947137604 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 152288236422144 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 529184016 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 6085773228096 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 637987982963776 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2499200544996 \) Copy content Toggle raw display
$53$ \( (T^{10} - 219 T^{8} + \cdots + 2207412)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 423797094009 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 42955545186304 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 37008140625 \) Copy content Toggle raw display
$71$ \( (T^{10} - 12 T^{9} + \cdots + 298878336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 17 T^{9} + \cdots - 584699069)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 263390662656 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{10} - 18 T^{9} + \cdots + 73284912)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
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