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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,2,Mod(136,137)] 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("137.136"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 137.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.09395050769\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 23x^{8} + 188x^{6} + 670x^{4} + 989x^{2} + 436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_1 q^{3} + ( - \beta_{9} + \beta_{6}) q^{4} - \beta_{3} q^{5} + (\beta_{7} + \beta_1) q^{6} + ( - \beta_{9} + \beta_{6} + \beta_{2} - 1) q^{7} + (\beta_{9} - \beta_{6} - \beta_{2}) q^{8}+ \cdots + (2 \beta_{9} - 2 \beta_{6} - \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 8 q^{4} + 2 q^{7} - 12 q^{8} - 16 q^{9} + 6 q^{11} - 8 q^{14} - 8 q^{15} - 4 q^{16} + 8 q^{17} + 30 q^{18} + 4 q^{19} - 22 q^{22} - 26 q^{25} + 30 q^{28} + 34 q^{30} + 2 q^{32} - 10 q^{34}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 23x^{8} + 188x^{6} + 670x^{4} + 989x^{2} + 436 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{8} + 153\nu^{6} + 1339\nu^{4} + 4452\nu^{2} + 4030 ) / 343 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{9} + 153\nu^{7} + 1339\nu^{5} + 4452\nu^{3} + 4030\nu ) / 343 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{8} + 313\nu^{6} + 1439\nu^{4} + 2093\nu^{2} + 1557 ) / 343 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{9} - 52\nu^{7} - 253\nu^{5} - 266\nu^{3} + 337\nu ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -27\nu^{8} - 517\nu^{6} - 3110\nu^{4} - 6657\nu^{2} - 3729 ) / 343 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -27\nu^{9} - 517\nu^{7} - 3110\nu^{5} - 6657\nu^{3} - 4072\nu ) / 343 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\nu^{9} + 466\nu^{7} + 2778\nu^{5} + 6545\nu^{3} + 5930\nu ) / 343 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -46\nu^{8} - 830\nu^{6} - 4549\nu^{4} - 8407\nu^{2} - 3914 ) / 343 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{3} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{9} + 15\beta_{6} - 9\beta_{4} + 4\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{8} + 15\beta_{7} - 12\beta_{5} + 13\beta_{3} + 56\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 126\beta_{9} - 175\beta_{6} + 75\beta_{4} - 59\beta_{2} - 112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -51\beta_{8} - 175\beta_{7} + 126\beta_{5} - 134\beta_{3} - 488\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1277\beta_{9} + 1857\beta_{6} - 646\beta_{4} + 669\beta_{2} + 689 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 631\beta_{8} + 1857\beta_{7} - 1277\beta_{5} + 1315\beta_{3} + 4469\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/137\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
3.11962i
− 3.11962i
1.52597i
− 1.52597i
2.34888i
− 2.34888i
0.863526i
− 0.863526i
2.16251i
− 2.16251i
−2.45281 − 3.11962i 4.01626 − 2.89884i 7.65182i 3.94549 −4.94549 −6.73203 7.11030i
136.2 −2.45281 3.11962i 4.01626 2.89884i − 7.65182i 3.94549 −4.94549 −6.73203 − 7.11030i
136.3 −1.87455 − 1.52597i 1.51392 3.70063i 2.86050i −1.91117 0.911174 0.671410 − 6.93700i
136.4 −1.87455 1.52597i 1.51392 − 3.70063i − 2.86050i −1.91117 0.911174 0.671410 6.93700i
136.5 −0.409390 − 2.34888i −1.83240 − 0.618824i 0.961608i −2.56895 1.56895 −2.51725 0.253340i
136.6 −0.409390 2.34888i −1.83240 0.618824i − 0.961608i −2.56895 1.56895 −2.51725 − 0.253340i
136.7 0.840478 − 0.863526i −1.29360 − 3.50746i − 0.725774i 1.76819 −2.76819 2.25432 − 2.94794i
136.8 0.840478 0.863526i −1.29360 3.50746i 0.725774i 1.76819 −2.76819 2.25432 2.94794i
136.9 1.89626 − 2.16251i 1.59582 1.79355i − 4.10069i −0.233563 −0.766437 −1.67646 3.40104i
136.10 1.89626 2.16251i 1.59582 − 1.79355i 4.10069i −0.233563 −0.766437 −1.67646 − 3.40104i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 136.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.2.b.a ✓ 10
3.b odd 2 1 1233.2.d.b 10
4.b odd 2 1 2192.2.c.d 10
137.b even 2 1 inner 137.2.b.a ✓ 10
411.c odd 2 1 1233.2.d.b 10
548.d odd 2 1 2192.2.c.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.2.b.a ✓ 10 1.a even 1 1 trivial
137.2.b.a ✓ 10 137.b even 2 1 inner
1233.2.d.b 10 3.b odd 2 1
1233.2.d.b 10 411.c odd 2 1
2192.2.c.d 10 4.b odd 2 1
2192.2.c.d 10 548.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(137, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 2 T^{4} - 5 T^{3} + \cdots + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} + 23 T^{8} + \cdots + 436 \) Copy content Toggle raw display
$5$ \( T^{10} + 38 T^{8} + \cdots + 1744 \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} - 14 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 3 T^{4} - 14 T^{3} + \cdots - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 74 T^{8} + \cdots + 15696 \) Copy content Toggle raw display
$17$ \( (T^{5} - 4 T^{4} - 24 T^{3} + \cdots - 18)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 2 T^{4} + \cdots - 788)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 139 T^{8} + \cdots + 436 \) Copy content Toggle raw display
$29$ \( T^{10} + 109 T^{8} + \cdots + 1744 \) Copy content Toggle raw display
$31$ \( T^{10} + 121 T^{8} + \cdots + 474804 \) Copy content Toggle raw display
$37$ \( (T^{5} - 8 T^{4} + \cdots - 1774)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 177 T^{8} + \cdots + 6976 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 247215924 \) Copy content Toggle raw display
$47$ \( T^{10} + 199 T^{8} + \cdots + 52756 \) Copy content Toggle raw display
$53$ \( T^{10} + 234 T^{8} + \cdots + 48638416 \) Copy content Toggle raw display
$59$ \( (T^{5} + 16 T^{4} + \cdots - 6780)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 19 T^{4} + \cdots + 31966)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 342 T^{8} + \cdots + 82502100 \) Copy content Toggle raw display
$71$ \( T^{10} + 156 T^{8} + \cdots + 111616 \) Copy content Toggle raw display
$73$ \( (T^{5} - 17 T^{4} + \cdots + 158)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 3745571796 \) Copy content Toggle raw display
$83$ \( T^{10} + 525 T^{8} + \cdots + 5766100 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1495438144 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1028653056 \) Copy content Toggle raw display
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