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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [242,4,Mod(3,242)] 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("242.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,2,-8,12,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2784622214\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{2} + (\beta_{7} - \beta_{2}) q^{3} + ( - 4 \beta_{6} - 4 \beta_{4} + \cdots - 4) q^{4} + ( - 6 \beta_{6} - 7 \beta_1) q^{5} + ( - 2 \beta_{6} + 2 \beta_1) q^{6} + (3 \beta_{6} + 3 \beta_{4} + 11 \beta_{3} + \cdots + 3) q^{7}+ \cdots + ( - 132 \beta_{5} + 58) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 2 q^{3} - 8 q^{4} + 12 q^{5} + 4 q^{6} + 6 q^{7} - 16 q^{8} + 46 q^{9} - 96 q^{10} - 32 q^{12} + 114 q^{13} + 12 q^{14} + 30 q^{15} - 32 q^{16} - 72 q^{17} + 92 q^{18} + 150 q^{19} + 48 q^{20}+ \cdots + 464 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(-1 - \beta_{2} - \beta_{4} - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.40126 + 1.01807i
−1.40126 − 1.01807i
0.535233 + 1.64728i
−0.535233 − 1.64728i
0.535233 − 1.64728i
−0.535233 + 1.64728i
1.40126 − 1.01807i
−1.40126 + 1.01807i
−1.61803 + 1.17557i −0.844250 − 2.59833i 1.23607 − 3.80423i −4.95471 − 3.59981i 4.42055 + 3.21172i −6.81462 + 20.9732i 2.47214 + 7.60845i 15.8049 − 11.4829i 12.2487
3.2 −1.61803 + 1.17557i 0.226216 + 0.696222i 1.23607 − 3.80423i 14.6629 + 10.6532i −1.18448 − 0.860577i 4.96051 − 15.2669i 2.47214 + 7.60845i 21.4099 − 15.5552i −36.2487
9.1 0.618034 − 1.90211i −0.592242 + 0.430289i −3.23607 − 2.35114i −5.60073 − 17.2373i 0.452432 + 1.39244i −12.9868 − 9.43546i −6.47214 + 4.70228i −8.17786 + 25.1689i −36.2487
9.2 0.618034 − 1.90211i 2.21028 − 1.60586i −3.23607 − 2.35114i 1.89253 + 5.82461i −1.68850 − 5.19667i 17.8409 + 12.9622i −6.47214 + 4.70228i −6.03692 + 18.5797i 12.2487
27.1 0.618034 + 1.90211i −0.592242 − 0.430289i −3.23607 + 2.35114i −5.60073 + 17.2373i 0.452432 − 1.39244i −12.9868 + 9.43546i −6.47214 − 4.70228i −8.17786 − 25.1689i −36.2487
27.2 0.618034 + 1.90211i 2.21028 + 1.60586i −3.23607 + 2.35114i 1.89253 − 5.82461i −1.68850 + 5.19667i 17.8409 − 12.9622i −6.47214 − 4.70228i −6.03692 − 18.5797i 12.2487
81.1 −1.61803 − 1.17557i −0.844250 + 2.59833i 1.23607 + 3.80423i −4.95471 + 3.59981i 4.42055 − 3.21172i −6.81462 − 20.9732i 2.47214 − 7.60845i 15.8049 + 11.4829i 12.2487
81.2 −1.61803 − 1.17557i 0.226216 − 0.696222i 1.23607 + 3.80423i 14.6629 − 10.6532i −1.18448 + 0.860577i 4.96051 + 15.2669i 2.47214 − 7.60845i 21.4099 + 15.5552i −36.2487
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.c.p 8
11.b odd 2 1 242.4.c.t 8
11.c even 5 1 242.4.a.l yes 2
11.c even 5 3 inner 242.4.c.p 8
11.d odd 10 1 242.4.a.i ✓ 2
11.d odd 10 3 242.4.c.t 8
33.f even 10 1 2178.4.a.bn 2
33.h odd 10 1 2178.4.a.bd 2
44.g even 10 1 1936.4.a.r 2
44.h odd 10 1 1936.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.4.a.i ✓ 2 11.d odd 10 1
242.4.a.l yes 2 11.c even 5 1
242.4.c.p 8 1.a even 1 1 trivial
242.4.c.p 8 11.c even 5 3 inner
242.4.c.t 8 11.b odd 2 1
242.4.c.t 8 11.d odd 10 3
1936.4.a.r 2 44.g even 10 1
1936.4.a.s 2 44.h odd 10 1
2178.4.a.bd 2 33.h odd 10 1
2178.4.a.bn 2 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 6T_{3}^{6} - 16T_{3}^{5} + 44T_{3}^{4} + 32T_{3}^{3} + 24T_{3}^{2} + 16T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} - 12 T_{5}^{7} + 255 T_{5}^{6} - 4392 T_{5}^{5} + 81009 T_{5}^{4} + 487512 T_{5}^{3} + \cdots + 151807041 \) Copy content Toggle raw display
\( T_{7}^{8} - 6 T_{7}^{7} + 390 T_{7}^{6} - 4464 T_{7}^{5} + 164844 T_{7}^{4} + 1580256 T_{7}^{3} + \cdots + 15704099856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} - 12 T^{7} + \cdots + 151807041 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15704099856 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 97335607906161 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1229457398481 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1008024032016 \) Copy content Toggle raw display
$23$ \( (T^{2} + 222 T + 10446)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 53\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 30\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 75\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{2} + 432 T + 19008)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{2} - 374 T - 309794)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1578 T + 616173)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
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