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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,11,Mod(161,270)] 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("270.161"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 270.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(171.546458222\)
Analytic rank: \(0\)
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: \( x^{12} + 280622 x^{10} + 28843684599 x^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{18}\cdot 5^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - 512 q^{4} - \beta_{6} q^{5} + ( - \beta_{2} - 2707) q^{7} + 512 \beta_{7} q^{8} + \beta_1 q^{10} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{6}) q^{11} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 108389) q^{13}+ \cdots + ( - 43456 \beta_{11} + \cdots - 614656 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6144 q^{4} - 32484 q^{7} + 1300668 q^{13} + 3145728 q^{16} - 6019356 q^{19} + 9967104 q^{22} - 23437500 q^{25} + 16631808 q^{28} + 134473488 q^{31} - 63786240 q^{34} + 136809060 q^{37} - 352685280 q^{43}+ \cdots - 16140182268 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 280622 x^{10} + 28843684599 x^{8} + \cdots + 18\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 282275193364540 \nu^{10} + \cdots - 19\!\cdots\!00 ) / 71\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 42\!\cdots\!30 \nu^{10} + \cdots - 26\!\cdots\!30 ) / 17\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37\!\cdots\!77 \nu^{10} + \cdots + 23\!\cdots\!10 ) / 31\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 85\!\cdots\!00 \nu^{10} + \cdots + 50\!\cdots\!60 ) / 51\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!77 \nu^{10} + \cdots - 23\!\cdots\!90 ) / 81\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!25 \nu^{11} + \cdots + 10\!\cdots\!50 \nu ) / 50\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!44 \nu^{11} + \cdots + 10\!\cdots\!80 \nu ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!39 \nu^{11} + \cdots - 18\!\cdots\!30 \nu ) / 92\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!21 \nu^{11} + \cdots - 12\!\cdots\!70 \nu ) / 55\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!61 \nu^{11} + \cdots - 12\!\cdots\!70 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!84 \nu^{11} + \cdots - 62\!\cdots\!30 \nu ) / 11\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -750\beta_{11} + 1125\beta_{9} + 875\beta_{8} - 375\beta_{7} + 92\beta_{6} ) / 540000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8170\beta_{5} - 11255\beta_{4} - 860\beta_{3} + 65900\beta_{2} + 45711\beta _1 - 5051196000 ) / 108000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5112360 \beta_{11} + 1587720 \beta_{10} - 9984900 \beta_{9} - 7030300 \beta_{8} + \cdots + 276004184 \beta_{6} ) / 54000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1021681970 \beta_{5} + 1464794155 \beta_{4} - 1118490740 \beta_{3} - 13879587100 \beta_{2} + \cdots + 379104078348000 ) / 108000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 184594903170 \beta_{11} - 110460222480 \beta_{10} + 440523038595 \beta_{9} + \cdots - 14437735740556 \beta_{6} ) / 21600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 28214890632160 \beta_{5} - 44604828053540 \beta_{4} + 57680298811720 \beta_{3} + \cdots - 91\!\cdots\!00 ) / 27000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 94\!\cdots\!70 \beta_{11} + \cdots + 83\!\cdots\!56 \beta_{6} ) / 108000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12\!\cdots\!10 \beta_{5} + \cdots + 39\!\cdots\!00 ) / 108000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 51\!\cdots\!60 \beta_{11} + \cdots - 47\!\cdots\!28 \beta_{6} ) / 54000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 28\!\cdots\!30 \beta_{5} + \cdots - 90\!\cdots\!00 ) / 21600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11\!\cdots\!90 \beta_{11} + \cdots + 11\!\cdots\!12 \beta_{6} ) / 108000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
174.048i
172.959i
− 344.771i
− 157.866i
257.420i
− 101.790i
157.866i
− 257.420i
101.790i
− 174.048i
− 172.959i
344.771i
− 22.6274i 0 −512.000 − 1397.54i 0 −26013.6 11585.2i 0 −31622.8
161.2 − 22.6274i 0 −512.000 − 1397.54i 0 3841.48 11585.2i 0 −31622.8
161.3 − 22.6274i 0 −512.000 − 1397.54i 0 14051.1 11585.2i 0 −31622.8
161.4 − 22.6274i 0 −512.000 1397.54i 0 −16036.8 11585.2i 0 31622.8
161.5 − 22.6274i 0 −512.000 1397.54i 0 −9729.78 11585.2i 0 31622.8
161.6 − 22.6274i 0 −512.000 1397.54i 0 17645.6 11585.2i 0 31622.8
161.7 22.6274i 0 −512.000 − 1397.54i 0 −16036.8 − 11585.2i 0 31622.8
161.8 22.6274i 0 −512.000 − 1397.54i 0 −9729.78 − 11585.2i 0 31622.8
161.9 22.6274i 0 −512.000 − 1397.54i 0 17645.6 − 11585.2i 0 31622.8
161.10 22.6274i 0 −512.000 1397.54i 0 −26013.6 − 11585.2i 0 −31622.8
161.11 22.6274i 0 −512.000 1397.54i 0 3841.48 − 11585.2i 0 −31622.8
161.12 22.6274i 0 −512.000 1397.54i 0 14051.1 − 11585.2i 0 −31622.8
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 161.12
Significant digits:
Format:

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 270.11.d.a ✓ 12
3.b odd 2 1 inner 270.11.d.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.11.d.a ✓ 12 1.a even 1 1 trivial
270.11.d.a ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 16242 T_{7}^{5} - 644153439 T_{7}^{4} - 7115945184880 T_{7}^{3} + \cdots - 38\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 512)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1953125)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 12\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 92\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!21)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
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