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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [120,8,Mod(49,120)] 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("120.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 120.f (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: \(37.4862030581\)
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: \( x^{12} + 7098 x^{10} + 18491457 x^{8} + 22308779968 x^{6} + 13142377385352 x^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{6}\cdot 5^{7} \)
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.

\(f(q)\) \(=\) \( q + 27 \beta_1 q^{3} + (\beta_{2} + 6 \beta_1 + 1) q^{5} + (\beta_{3} + \beta_{2} - 104 \beta_1) q^{7} - 729 q^{9} + (\beta_{5} + \beta_{4} + \beta_{2} + 276) q^{11} + (\beta_{11} - \beta_{9} + \cdots + 892 \beta_1) q^{13}+ \cdots + ( - 729 \beta_{5} - 729 \beta_{4} + \cdots - 201204) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} - 8748 q^{9} + 3312 q^{11} - 1944 q^{15} + 36144 q^{19} + 33696 q^{21} - 46740 q^{25} - 366936 q^{29} + 549984 q^{31} - 602304 q^{35} - 289008 q^{39} + 1228392 q^{41} - 8748 q^{45} + 468660 q^{49}+ \cdots - 2414448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 7098 x^{10} + 18491457 x^{8} + 22308779968 x^{6} + 13142377385352 x^{4} + \cdots + 30\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2197330571 \nu^{11} + 13719397049148 \nu^{9} + \cdots + 70\!\cdots\!00 \nu ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!43 \nu^{11} + \cdots + 23\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{11} + \cdots - 23\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!95 \nu^{11} + \cdots - 81\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!41 \nu^{11} + \cdots - 62\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!47 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!25 \nu^{11} + \cdots - 37\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57\!\cdots\!13 \nu^{11} + \cdots - 34\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!77 \nu^{11} + \cdots + 14\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\!\cdots\!45 \nu^{11} + \cdots - 88\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 57\!\cdots\!55 \nu^{11} + \cdots - 51\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 3\beta_{8} - 3\beta_{6} - 21\beta_{5} + \beta_{3} + 35\beta_{2} + 160\beta_1 ) / 480 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{9} - 11 \beta_{8} - 20 \beta_{7} - 114 \beta_{6} + 57 \beta_{5} - 40 \beta_{4} + \cdots - 354900 ) / 300 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1303 \beta_{11} - 200 \beta_{10} + 1220 \beta_{9} - 2405 \beta_{8} + 2405 \beta_{6} + \cdots + 1788880 \beta_1 ) / 240 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 35277 \beta_{9} + 11056 \beta_{8} + 55695 \beta_{7} + 406394 \beta_{6} + 56973 \beta_{5} + \cdots + 669934500 ) / 300 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1652409 \beta_{11} + 511730 \beta_{10} - 1830620 \beta_{9} + 2657087 \beta_{8} + \cdots - 4177808360 \beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 140313967 \beta_{9} + 17209644 \beta_{8} - 156286345 \beta_{7} - 1147990444 \beta_{6} + \cdots - 1538893986900 ) / 300 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2165955437 \beta_{11} - 1024148085 \beta_{10} + 2340552102 \beta_{9} - 3261583821 \beta_{8} + \cdots + 7223153692420 \beta_1 ) / 60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 458579067617 \beta_{9} - 138537185524 \beta_{8} + 444832680395 \beta_{7} + 3106323654774 \beta_{6} + \cdots + 38\!\cdots\!00 ) / 300 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5778079683322 \beta_{11} + 3549426882125 \beta_{10} - 5863794642422 \beta_{9} + \cdots - 22\!\cdots\!80 \beta_1 ) / 60 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14\!\cdots\!07 \beta_{9} + 556298704775624 \beta_{8} + \cdots - 98\!\cdots\!00 ) / 300 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15\!\cdots\!42 \beta_{11} + \cdots + 68\!\cdots\!20 \beta_1 ) / 60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 45.1745i
28.7481i
− 28.2498i
12.6959i
− 22.5743i
52.5545i
45.1745i
− 28.7481i
28.2498i
− 12.6959i
22.5743i
− 52.5545i
0 − 27.0000i 0 −258.944 − 105.228i 0 − 274.886i 0 −729.000 0
49.2 0 − 27.0000i 0 −188.657 + 206.236i 0 1611.15i 0 −729.000 0
49.3 0 − 27.0000i 0 −98.1363 − 261.714i 0 − 940.981i 0 −729.000 0
49.4 0 − 27.0000i 0 130.600 + 247.120i 0 − 168.547i 0 −729.000 0
49.5 0 − 27.0000i 0 162.191 − 227.638i 0 920.713i 0 −729.000 0
49.6 0 − 27.0000i 0 258.946 + 105.223i 0 − 523.448i 0 −729.000 0
49.7 0 27.0000i 0 −258.944 + 105.228i 0 274.886i 0 −729.000 0
49.8 0 27.0000i 0 −188.657 − 206.236i 0 − 1611.15i 0 −729.000 0
49.9 0 27.0000i 0 −98.1363 + 261.714i 0 940.981i 0 −729.000 0
49.10 0 27.0000i 0 130.600 − 247.120i 0 168.547i 0 −729.000 0
49.11 0 27.0000i 0 162.191 + 227.638i 0 − 920.713i 0 −729.000 0
49.12 0 27.0000i 0 258.946 − 105.223i 0 523.448i 0 −729.000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.8.f.b ✓ 12
3.b odd 2 1 360.8.f.d 12
4.b odd 2 1 240.8.f.g 12
5.b even 2 1 inner 120.8.f.b ✓ 12
5.c odd 4 1 600.8.a.v 6
5.c odd 4 1 600.8.a.w 6
15.d odd 2 1 360.8.f.d 12
20.d odd 2 1 240.8.f.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.8.f.b ✓ 12 1.a even 1 1 trivial
120.8.f.b ✓ 12 5.b even 2 1 inner
240.8.f.g 12 4.b odd 2 1
240.8.f.g 12 20.d odd 2 1
360.8.f.d 12 3.b odd 2 1
360.8.f.d 12 15.d odd 2 1
600.8.a.v 6 5.c odd 4 1
600.8.a.w 6 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 4706928 T_{7}^{10} + 6916380633696 T_{7}^{8} + \cdots + 11\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 29\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
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