Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(79,112)] 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("112.79"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \( x^{12} - 2 x^{11} + 12 x^{10} - 56 x^{9} - 1704 x^{8} + 10928 x^{7} + 58248 x^{6} + 93416 x^{5} + \cdots + 297907600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 7^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{9} + \beta_{4} + 3 \beta_1 - 3) q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{9} + \beta_{6} + \beta_{4} + \cdots + 47) q^{9} + ( - \beta_{11} + 2 \beta_{8} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - 13 \beta_{11} + \cdots + 437 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 284 q^{9} - 520 q^{13} + 870 q^{17} - 1078 q^{21} - 1552 q^{25} + 1848 q^{29} - 1358 q^{33} + 126 q^{37} + 8472 q^{41} - 5756 q^{45} + 10892 q^{49} - 4722 q^{53} - 21532 q^{57} - 14290 q^{61}+ \cdots + 95384 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 12 x^{10} - 56 x^{9} - 1704 x^{8} + 10928 x^{7} + 58248 x^{6} + 93416 x^{5} + \cdots + 297907600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!34 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!02 \nu^{11} + \cdots + 43\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!72 \nu^{11} + \cdots - 18\!\cdots\!00 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 55572658053848 \nu^{11} - 188585642032788 \nu^{10} + \cdots - 31\!\cdots\!80 ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!08 \nu^{11} + \cdots - 30\!\cdots\!80 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 123034702103143 \nu^{11} + 405811605828090 \nu^{10} + \cdots - 20\!\cdots\!92 ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!89 \nu^{11} + \cdots + 38\!\cdots\!00 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!26 \nu^{11} + \cdots - 34\!\cdots\!80 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!60 \nu^{11} + \cdots - 42\!\cdots\!00 ) / 68\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22\!\cdots\!32 \nu^{11} + \cdots - 23\!\cdots\!80 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!12 \nu^{11} + \cdots - 92\!\cdots\!88 ) / 69\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{11} - 6 \beta_{10} + 7 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} + \beta_{5} - 58 \beta_{3} + \cdots + 63 \beta_1 ) / 196 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 31 \beta_{11} + 13 \beta_{10} + 56 \beta_{9} - 80 \beta_{8} + 9 \beta_{7} - 7 \beta_{6} + \cdots - 651 ) / 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 95 \beta_{11} - 202 \beta_{10} - 18 \beta_{8} + 89 \beta_{7} + 91 \beta_{6} + 1786 \beta_{5} + \cdots + 819 ) / 98 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{11} - 663 \beta_{10} + 3115 \beta_{9} + 195 \beta_{8} - 221 \beta_{7} + 429 \beta_{5} + \cdots + 120302 \beta_1 ) / 98 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 571 \beta_{11} - 2465 \beta_{10} + 16716 \beta_{9} - 752 \beta_{8} + 947 \beta_{7} - 9709 \beta_{6} + \cdots - 631281 ) / 98 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6929 \beta_{11} - 35828 \beta_{10} - 32955 \beta_{8} - 4056 \beta_{7} - 20111 \beta_{6} + \cdots - 2303679 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 71343 \beta_{11} - 151314 \beta_{10} - 131355 \beta_{9} + 193124 \beta_{8} - 50438 \beta_{7} + \cdots - 5112975 \beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 78241 \beta_{11} + 120417 \beta_{10} + 336774 \beta_{9} + 355140 \beta_{8} - 99329 \beta_{7} + \cdots - 13165069 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2465044 \beta_{11} + 9143300 \beta_{10} + 6319818 \beta_{8} - 358438 \beta_{7} - 12436060 \beta_{6} + \cdots - 1046128356 ) / 49 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3753602 \beta_{11} - 31276026 \beta_{10} - 249174506 \beta_{9} + 2918138 \beta_{8} + \cdots - 9507653064 \beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 153950918 \beta_{11} + 401292902 \beta_{10} - 1254934996 \beta_{9} + 863145656 \beta_{8} + \cdots + 47805550842 ) / 49 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
6.93383 + 2.92080i
2.16778 − 2.21076i
1.20930 − 5.50834i
−5.37501 − 1.70688i
−2.99846 + 0.771968i
−0.937433 + 7.46527i
6.93383 − 2.92080i
2.16778 + 2.21076i
1.20930 + 5.50834i
−5.37501 + 1.70688i
−2.99846 − 0.771968i
−0.937433 − 7.46527i
0 −14.6688 + 8.46905i 0 11.1163 − 19.2540i 0 48.8257 + 4.12972i 0 102.950 − 178.314i 0
79.2 0 −8.47381 + 4.89236i 0 −22.2519 + 38.5414i 0 −48.9299 − 2.61935i 0 7.37036 − 12.7658i 0
79.3 0 −1.33044 + 0.768130i 0 6.63562 − 11.4932i 0 −13.5984 − 47.0753i 0 −39.3200 + 68.1042i 0
79.4 0 1.33044 − 0.768130i 0 6.63562 − 11.4932i 0 13.5984 + 47.0753i 0 −39.3200 + 68.1042i 0
79.5 0 8.47381 − 4.89236i 0 −22.2519 + 38.5414i 0 48.9299 + 2.61935i 0 7.37036 − 12.7658i 0
79.6 0 14.6688 − 8.46905i 0 11.1163 − 19.2540i 0 −48.8257 − 4.12972i 0 102.950 − 178.314i 0
95.1 0 −14.6688 − 8.46905i 0 11.1163 + 19.2540i 0 48.8257 − 4.12972i 0 102.950 + 178.314i 0
95.2 0 −8.47381 − 4.89236i 0 −22.2519 − 38.5414i 0 −48.9299 + 2.61935i 0 7.37036 + 12.7658i 0
95.3 0 −1.33044 − 0.768130i 0 6.63562 + 11.4932i 0 −13.5984 + 47.0753i 0 −39.3200 − 68.1042i 0
95.4 0 1.33044 + 0.768130i 0 6.63562 + 11.4932i 0 13.5984 − 47.0753i 0 −39.3200 − 68.1042i 0
95.5 0 8.47381 + 4.89236i 0 −22.2519 − 38.5414i 0 48.9299 − 2.61935i 0 7.37036 + 12.7658i 0
95.6 0 14.6688 + 8.46905i 0 11.1163 + 19.2540i 0 −48.8257 + 4.12972i 0 102.950 + 178.314i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.r.c ✓ 12
4.b odd 2 1 inner 112.5.r.c ✓ 12
7.c even 3 1 inner 112.5.r.c ✓ 12
7.c even 3 1 784.5.d.h 6
7.d odd 6 1 784.5.d.g 6
28.f even 6 1 784.5.d.g 6
28.g odd 6 1 inner 112.5.r.c ✓ 12
28.g odd 6 1 784.5.d.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.r.c ✓ 12 1.a even 1 1 trivial
112.5.r.c ✓ 12 4.b odd 2 1 inner
112.5.r.c ✓ 12 7.c even 3 1 inner
112.5.r.c ✓ 12 28.g odd 6 1 inner
784.5.d.g 6 7.d odd 6 1
784.5.d.g 6 28.f even 6 1
784.5.d.h 6 7.c even 3 1
784.5.d.h 6 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 385T_{3}^{10} + 119854T_{3}^{8} - 10793181T_{3}^{6} + 779955246T_{3}^{4} - 1839206817T_{3}^{2} + 4202539929 \) acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 4202539929 \) Copy content Toggle raw display
$5$ \( (T^{6} + 9 T^{5} + \cdots + 172423161)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( (T^{3} + 130 T^{2} + \cdots - 1635208)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 10\!\cdots\!89)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!89 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 87\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( (T^{3} - 462 T^{2} + \cdots - 63592200)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!81)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 2118 T^{2} + \cdots + 2256215256)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 30\!\cdots\!61)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 66\!\cdots\!49)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 36\!\cdots\!41)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 98\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 39\!\cdots\!41)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 23846 T^{2} + \cdots + 482665937048)^{4} \) Copy content Toggle raw display
show more
show less