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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,3,Mod(23,110)] 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("110.23"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 110.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(2.99728290796\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 685x^{4} + 1232x^{3} + 968x^{2} - 968x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + (\beta_{6} - \beta_{4} + \beta_{3} + 1) q^{3} - 2 \beta_{3} q^{4} + (\beta_{6} + \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{6} + \beta_{4} + \beta_1 - 2) q^{6} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots - 2) q^{7}+ \cdots + ( - 3 \beta_{7} + 4 \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 6 q^{3} + 12 q^{5} - 12 q^{6} - 12 q^{7} + 16 q^{8} - 6 q^{10} + 12 q^{12} - 4 q^{13} - 60 q^{15} - 32 q^{16} - 72 q^{17} - 48 q^{18} - 12 q^{20} + 168 q^{21} + 50 q^{23} + 6 q^{25} + 8 q^{26}+ \cdots + 272 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 685x^{4} + 1232x^{3} + 968x^{2} - 968x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -43\nu^{7} + 108\nu^{6} + 222\nu^{5} - 7052\nu^{4} - 18037\nu^{3} - 19228\nu^{2} + 17798\nu - 267168 ) / 78100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 239 \nu^{7} + 204 \nu^{6} + 301 \nu^{5} - 24641 \nu^{4} - 189126 \nu^{3} - 471589 \nu^{2} + \cdots + 125356 ) / 273350 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1191 \nu^{7} - 1076 \nu^{6} + 2916 \nu^{5} + 108704 \nu^{4} + 951919 \nu^{3} + 2682616 \nu^{2} + \cdots + 1009536 ) / 546700 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{7} + 6\nu^{6} - 116\nu^{5} + 2126\nu^{4} + 17371\nu^{3} + 44704\nu^{2} + 37906\nu - 11616 ) / 7700 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1739 \nu^{7} - 2634 \nu^{6} + 5354 \nu^{5} + 159526 \nu^{4} + 1306201 \nu^{3} + 3039544 \nu^{2} + \cdots + 778184 ) / 546700 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1001 \nu^{7} + 3736 \nu^{6} - 7901 \nu^{5} - 89614 \nu^{4} - 498934 \nu^{3} - 405076 \nu^{2} + \cdots + 864974 ) / 273350 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{5} - 2\beta_{4} - 14\beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{7} + 25\beta_{6} - 11\beta_{5} - 25\beta_{4} - 34\beta_{3} - 3\beta_{2} - 4\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 35\beta_{7} + 105\beta_{6} - 35\beta_{5} - 70\beta_{4} - 60\beta_{2} - 105\beta _1 - 389 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 165\beta_{6} + 185\beta_{5} + 165\beta_{4} + 1650\beta_{3} - 185\beta_{2} - 894\beta _1 - 1650 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1429\beta_{7} - 3059\beta_{6} + 3507\beta_{5} + 4488\beta_{4} + 15936\beta_{3} - 3059\beta _1 - 1429 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13132 \beta_{7} - 34985 \beta_{6} + 21459 \beta_{5} + 34985 \beta_{4} + 69066 \beta_{3} + \cdots + 55934 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
4.49083 − 4.49083i
0.348467 − 0.348467i
−1.50678 + 1.50678i
−2.33252 + 2.33252i
4.49083 + 4.49083i
0.348467 + 0.348467i
−1.50678 − 1.50678i
−2.33252 − 2.33252i
−1.00000 + 1.00000i −3.49083 − 3.49083i − 2.00000i 3.44736 − 3.62156i 6.98166 −9.40357 + 9.40357i 2.00000 + 2.00000i 15.3718i 0.174205 + 7.06892i
23.2 −1.00000 + 1.00000i 0.651533 + 0.651533i − 2.00000i 1.94226 − 4.60735i −1.30307 2.50420 − 2.50420i 2.00000 + 2.00000i − 8.15101i 2.66509 + 6.54960i
23.3 −1.00000 + 1.00000i 2.50678 + 2.50678i − 2.00000i −3.91719 + 3.10735i −5.01356 −5.50420 + 5.50420i 2.00000 + 2.00000i 3.56789i 0.809845 − 7.02454i
23.4 −1.00000 + 1.00000i 3.33252 + 3.33252i − 2.00000i 4.52758 + 2.12156i −6.66503 6.40357 − 6.40357i 2.00000 + 2.00000i 13.2113i −6.64914 + 2.40602i
67.1 −1.00000 − 1.00000i −3.49083 + 3.49083i 2.00000i 3.44736 + 3.62156i 6.98166 −9.40357 − 9.40357i 2.00000 − 2.00000i − 15.3718i 0.174205 − 7.06892i
67.2 −1.00000 − 1.00000i 0.651533 − 0.651533i 2.00000i 1.94226 + 4.60735i −1.30307 2.50420 + 2.50420i 2.00000 − 2.00000i 8.15101i 2.66509 − 6.54960i
67.3 −1.00000 − 1.00000i 2.50678 − 2.50678i 2.00000i −3.91719 − 3.10735i −5.01356 −5.50420 − 5.50420i 2.00000 − 2.00000i − 3.56789i 0.809845 + 7.02454i
67.4 −1.00000 − 1.00000i 3.33252 − 3.33252i 2.00000i 4.52758 − 2.12156i −6.66503 6.40357 + 6.40357i 2.00000 − 2.00000i − 13.2113i −6.64914 − 2.40602i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 23.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.3.e.a ✓ 8
3.b odd 2 1 990.3.j.a 8
5.b even 2 1 550.3.e.c 8
5.c odd 4 1 inner 110.3.e.a ✓ 8
5.c odd 4 1 550.3.e.c 8
15.e even 4 1 990.3.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.3.e.a ✓ 8 1.a even 1 1 trivial
110.3.e.a ✓ 8 5.c odd 4 1 inner
550.3.e.c 8 5.b even 2 1
550.3.e.c 8 5.c odd 4 1
990.3.j.a 8 3.b odd 2 1
990.3.j.a 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 6T_{3}^{7} + 18T_{3}^{6} - 22T_{3}^{5} + 593T_{3}^{4} - 3564T_{3}^{3} + 10952T_{3}^{2} - 11248T_{3} + 5776 \) acting on \(S_{3}^{\mathrm{new}}(110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 5776 \) Copy content Toggle raw display
$5$ \( T^{8} - 12 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{7} + \cdots + 11022400 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 49561600 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 15956742400 \) Copy content Toggle raw display
$19$ \( T^{8} + 344 T^{6} + \cdots + 10240000 \) Copy content Toggle raw display
$23$ \( T^{8} - 50 T^{7} + \cdots + 126736 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 123341440000 \) Copy content Toggle raw display
$31$ \( (T^{4} + 116 T^{3} + \cdots - 900400)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 7253164648900 \) Copy content Toggle raw display
$41$ \( (T^{4} - 36 T^{3} + \cdots + 1469248)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 7478037160000 \) Copy content Toggle raw display
$47$ \( T^{8} - 48 T^{7} + \cdots + 1600 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 5715637747600 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 18666893070400 \) Copy content Toggle raw display
$61$ \( (T^{4} + 28 T^{3} + \cdots + 880000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 743175029776 \) Copy content Toggle raw display
$71$ \( (T^{4} - 126 T^{3} + \cdots + 27115280)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 11524123878400 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 585489001734400 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 5957699905600 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
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