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Properties

Label 836.2.j.b
Level $836$
Weight $2$
Character orbit 836.j
Analytic conductor $6.675$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [836,2,Mod(229,836)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(836, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("836.229"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 836 = 2^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 836.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] 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: no
Analytic conductor: \(6.67549360898\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
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: \( x^{20} + 3 x^{18} - x^{17} + 54 x^{16} - 67 x^{15} + 423 x^{14} - 418 x^{13} + 1762 x^{12} - 726 x^{11} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{6} + \beta_1) q^{3} + ( - \beta_{17} + \beta_{3}) q^{5} + (\beta_{9} - \beta_{8} + \beta_{3} + \cdots + 1) q^{7} + (\beta_{19} - \beta_{17} + \cdots - \beta_{2}) q^{9} + ( - \beta_{19} + \beta_{17} - \beta_{16} + \cdots + 1) q^{11}+ \cdots + (\beta_{19} + \beta_{18} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{5} + 15 q^{7} + 9 q^{9} + 2 q^{11} + 4 q^{13} + 15 q^{15} + 7 q^{17} - 5 q^{19} - 12 q^{21} - 16 q^{23} - 5 q^{25} - 3 q^{27} + 16 q^{29} + q^{31} - 27 q^{33} - 13 q^{35} + 14 q^{37} - 29 q^{39}+ \cdots - 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 3 x^{18} - x^{17} + 54 x^{16} - 67 x^{15} + 423 x^{14} - 418 x^{13} + 1762 x^{12} - 726 x^{11} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!06 \nu^{19} + \cdots + 77\!\cdots\!07 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!87 \nu^{19} + \cdots + 37\!\cdots\!96 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!78 \nu^{19} + \cdots + 13\!\cdots\!10 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!45 \nu^{19} + \cdots + 27\!\cdots\!14 ) / 11\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!51 \nu^{19} + \cdots + 10\!\cdots\!33 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40\!\cdots\!10 \nu^{19} + \cdots + 39\!\cdots\!22 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 78\!\cdots\!56 \nu^{19} + \cdots + 26\!\cdots\!86 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!61 \nu^{19} + \cdots + 25\!\cdots\!31 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!37 \nu^{19} + \cdots + 75\!\cdots\!63 ) / 10\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 90\!\cdots\!39 \nu^{19} + \cdots - 15\!\cdots\!04 ) / 10\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 92\!\cdots\!53 \nu^{19} + \cdots - 10\!\cdots\!87 ) / 10\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!50 \nu^{19} + \cdots - 18\!\cdots\!96 ) / 10\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{19} + \cdots - 24\!\cdots\!51 ) / 11\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 42\!\cdots\!23 \nu^{19} + \cdots - 74\!\cdots\!75 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!53 \nu^{19} + \cdots + 27\!\cdots\!10 ) / 10\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 51\!\cdots\!68 \nu^{19} + \cdots - 37\!\cdots\!99 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 55\!\cdots\!14 \nu^{19} + \cdots + 20\!\cdots\!70 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 19\!\cdots\!29 \nu^{19} + \cdots - 67\!\cdots\!75 ) / 11\!\cdots\!97 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{7} + 4\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} + \beta_{16} - \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{9} + 6 \beta_{8} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{19} + 7 \beta_{17} - 8 \beta_{16} + 7 \beta_{13} + 18 \beta_{12} + 2 \beta_{10} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{19} + 7 \beta_{18} - 10 \beta_{17} + 10 \beta_{16} - 10 \beta_{13} - 9 \beta_{12} + 10 \beta_{11} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{18} + 12 \beta_{17} + 2 \beta_{15} - 2 \beta_{14} - 64 \beta_{11} + 64 \beta_{10} + \cdots - 102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 64 \beta_{19} + 30 \beta_{18} + 19 \beta_{17} - 94 \beta_{16} - 2 \beta_{15} + \beta_{14} + \cdots + 226 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 332 \beta_{19} - 88 \beta_{18} - 332 \beta_{17} + 542 \beta_{16} + 70 \beta_{15} - 43 \beta_{14} + \cdots - 692 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 715 \beta_{19} + 1035 \beta_{17} - 1370 \beta_{16} - 25 \beta_{15} + 1327 \beta_{13} + \cdots - 856 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1570 \beta_{19} + 781 \beta_{18} - 2746 \beta_{17} + 2746 \beta_{16} + 277 \beta_{14} - 2479 \beta_{13} + \cdots + 5666 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3378 \beta_{18} + 3378 \beta_{17} + 365 \beta_{15} - 365 \beta_{14} - 13198 \beta_{11} + 13198 \beta_{10} + \cdots - 23049 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14835 \beta_{19} + 11091 \beta_{18} + 7761 \beta_{17} - 25926 \beta_{16} - 4365 \beta_{15} + \cdots + 76721 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 70491 \beta_{19} - 22096 \beta_{18} - 70491 \beta_{17} + 125239 \beta_{16} + 7024 \beta_{15} + \cdots - 136379 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 229358 \beta_{19} + 294307 \beta_{17} - 398006 \beta_{16} - 21313 \beta_{15} + 361621 \beta_{13} + \cdots - 155210 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 421747 \beta_{19} + 199219 \beta_{18} - 731484 \beta_{17} + 731484 \beta_{16} + 28564 \beta_{14} + \cdots + 1289617 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 966656 \beta_{18} + 966656 \beta_{17} + 189171 \beta_{15} - 189171 \beta_{14} - 3679973 \beta_{11} + \cdots - 6544115 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3929324 \beta_{19} + 2912653 \beta_{18} + 2097616 \beta_{17} - 6841977 \beta_{16} - 750563 \beta_{15} + \cdots + 19609956 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 19575196 \beta_{19} - 5577335 \beta_{18} - 19575196 \beta_{17} + 34156598 \beta_{16} + 2779575 \beta_{15} + \cdots - 37680303 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 59084060 \beta_{19} + 76068928 \beta_{17} - 103344055 \beta_{16} - 4526062 \beta_{15} + \cdots - 43480629 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/836\mathbb{Z}\right)^\times\).

\(n\) \(419\) \(705\) \(761\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
1.84440 + 1.34004i
1.74846 + 1.27033i
−0.239002 − 0.173645i
−0.883547 − 0.641935i
−2.47032 − 1.79479i
1.84440 − 1.34004i
1.74846 − 1.27033i
−0.239002 + 0.173645i
−0.883547 + 0.641935i
−2.47032 + 1.79479i
0.617374 + 1.90008i
0.444729 + 1.36874i
−0.0872835 − 0.268631i
−0.347546 − 1.06964i
−0.627273 − 1.93055i
0.617374 − 1.90008i
0.444729 − 1.36874i
−0.0872835 + 0.268631i
−0.347546 + 1.06964i
−0.627273 + 1.93055i
0 −1.84440 + 1.34004i 0 −0.493490 − 1.51881i 0 0.169114 + 0.122868i 0 0.679071 − 2.08996i 0
229.2 0 −1.74846 + 1.27033i 0 0.106146 + 0.326685i 0 0.228407 + 0.165947i 0 0.516328 − 1.58909i 0
229.3 0 0.239002 − 0.173645i 0 1.26047 + 3.87933i 0 1.45673 + 1.05838i 0 −0.900082 + 2.77017i 0
229.4 0 0.883547 − 0.641935i 0 −0.280823 − 0.864285i 0 1.85508 + 1.34779i 0 −0.558475 + 1.71881i 0
229.5 0 2.47032 − 1.79479i 0 0.643763 + 1.98130i 0 2.83576 + 2.06030i 0 1.95414 − 6.01423i 0
533.1 0 −1.84440 − 1.34004i 0 −0.493490 + 1.51881i 0 0.169114 − 0.122868i 0 0.679071 + 2.08996i 0
533.2 0 −1.74846 − 1.27033i 0 0.106146 − 0.326685i 0 0.228407 − 0.165947i 0 0.516328 + 1.58909i 0
533.3 0 0.239002 + 0.173645i 0 1.26047 − 3.87933i 0 1.45673 − 1.05838i 0 −0.900082 − 2.77017i 0
533.4 0 0.883547 + 0.641935i 0 −0.280823 + 0.864285i 0 1.85508 − 1.34779i 0 −0.558475 − 1.71881i 0
533.5 0 2.47032 + 1.79479i 0 0.643763 − 1.98130i 0 2.83576 − 2.06030i 0 1.95414 + 6.01423i 0
609.1 0 −0.617374 + 1.90008i 0 −2.28987 + 1.66369i 0 1.18991 + 3.66218i 0 −0.802106 − 0.582764i 0
609.2 0 −0.444729 + 1.36874i 0 1.18149 − 0.858404i 0 0.910570 + 2.80245i 0 0.751396 + 0.545921i 0
609.3 0 0.0872835 − 0.268631i 0 −0.103494 + 0.0751925i 0 0.0497553 + 0.153131i 0 2.36251 + 1.71646i 0
609.4 0 0.347546 − 1.06964i 0 −1.86775 + 1.35700i 0 −0.371359 − 1.14292i 0 1.40372 + 1.01986i 0
609.5 0 0.627273 − 1.93055i 0 −0.156441 + 0.113661i 0 −0.823967 − 2.53591i 0 −0.906495 − 0.658608i 0
685.1 0 −0.617374 − 1.90008i 0 −2.28987 − 1.66369i 0 1.18991 − 3.66218i 0 −0.802106 + 0.582764i 0
685.2 0 −0.444729 − 1.36874i 0 1.18149 + 0.858404i 0 0.910570 − 2.80245i 0 0.751396 − 0.545921i 0
685.3 0 0.0872835 + 0.268631i 0 −0.103494 − 0.0751925i 0 0.0497553 − 0.153131i 0 2.36251 − 1.71646i 0
685.4 0 0.347546 + 1.06964i 0 −1.86775 − 1.35700i 0 −0.371359 + 1.14292i 0 1.40372 − 1.01986i 0
685.5 0 0.627273 + 1.93055i 0 −0.156441 − 0.113661i 0 −0.823967 + 2.53591i 0 −0.906495 + 0.658608i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 836.2.j.b ✓ 20
11.c even 5 1 inner 836.2.j.b ✓ 20
11.c even 5 1 9196.2.a.s 10
11.d odd 10 1 9196.2.a.t 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.j.b ✓ 20 1.a even 1 1 trivial
836.2.j.b ✓ 20 11.c even 5 1 inner
9196.2.a.s 10 11.c even 5 1
9196.2.a.t 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 3 T_{3}^{18} + T_{3}^{17} + 54 T_{3}^{16} + 67 T_{3}^{15} + 423 T_{3}^{14} + 418 T_{3}^{13} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(836, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 3 T^{18} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} - 15 T^{19} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 15021198721 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2025090001 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$23$ \( (T^{10} + 8 T^{9} + \cdots - 133389)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 87176002302481 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 158634025 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{10} - 14 T^{9} + \cdots - 1045049)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 54429793299025 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 986274025 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 339996389098081 \) Copy content Toggle raw display
$61$ \( T^{20} - 27 T^{19} + \cdots + 44235801 \) Copy content Toggle raw display
$67$ \( (T^{10} + 17 T^{9} + \cdots - 4909999)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 66902409025 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 24244047025 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 2389156641 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 5546020290001 \) Copy content Toggle raw display
$89$ \( (T^{10} + 35 T^{9} + \cdots + 19428559)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 29\!\cdots\!41 \) Copy content Toggle raw display
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